Introduction to Flight
Fifth Edition
J o h n D . A n d e r so n , Jr.
Curator fo r Aerodynamics, National A ir and Space Museum
Smithsonian Institution
P r o fe sso r E m e r itu s
University o f Maryland
Higher Education
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INTRODUCTION TO FLIGHT, FIFTH EDITION
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L ibrary of Congress Cataloging-in-Publication Data
Anderson, John David.
Introduction to flight / John D. Anderson, Jr. — 5th ed.
p. cm. — (McGraw-Hill series in aeronautical and aerospace engineering)
Includes bibliographical references and index.
ISBN 0-07-282569-3 (hard copy : alk. paper)
1. Aerodynamics. 2. Airplanes— Design and construction. 3. Spaceflight.
TL570.A68 2005
629.1—dc22
www.mhhe.com
I. Title.
II. Series.
2003026151
CIP
ABOUT TH E AUTHOR
was born in Lancaster, Pennsylvania, on October 1,
1937. He attended the University of Florida, graduating in 1959 with high hon­
ors and a Bachelor of Aeronautical Engineering Degree. From 1959 to 1962, he
was a lieutenant and task scientist at the Aerospace Research Laboratory at
Wright-Patterson Air Force Base. From 1962 to 1966, he attended the Ohio State
University under the National Science Foundation and NASA Fellowships, grad­
uating with a Ph.D. in aeronautical and astronautical engineering. In 1966, he
joined the U.S. Naval Ordnance Laboratory as Chief of the Hypersonic Group.
In 1973, he became Chairman of the Department of Aerospace Engineering at
the University of Maryland, and since 1980 has been professor of Aerospace
Engineering at Maryland. In 1982, he was designated a Distinguished Scholar/
Teacher by the University. During 1986-1987, while on sabbatical from the
University, Dr. Anderson occupied the Charles Lindbergh chair at the National
Air and Space Museum of the Smithsonian Institution. He continues with the
Museum in a part-time appointment as curator for aerodynamics. In addition to
his appointment in aerospace engineering, in 1993 he was elected to the faculty
of the Committee on the History and Philosophy of Science at Maryland, and is
an affiliate faculty member in the Department of History. In July 1999 he retired
from the University and is now Professor Emeritus.
Dr. Anderson has published nine books: Gasdynamic Lasers: An Introduction,
Academic Press (1976), A History o f Aerodynamics and Its Impact on Flying
Machines, Cambridge University Press (1997), The Airplane: A History o f Its
Technology, American Institute of Aeronautics and Astronautics (2003), and with
McGraw-Hill, Introduction to Flight, 5th edition (2004), Modern Compressible
Flow, 3d Edition (2003), Fundamentals o f Aerodynamics, 3d edition (2001),
Hypersonic and High Temperature Gas Dynamics (1989), Computational Fluid
Dynamics: The Basics with Applications (1995), and Aircraft Performance and
Design (1999). He is the author of over 120 papers on radiative gasdynamics,
entry aerothermodynamics, gas dynamic and chemical lasers, computational
fluid dynamics, applied aerodynamics, hypersonic flow, and the history of aero­
dynamics. Dr. Anderson is in W ho’s Who in America and is an Honorary Fellow
of the American Institute of Aeronautics and Astronautics (AIAA) and a Fellow
of The Royal Aeronautical Society. He is also a Fellow of the Washington Acad­
emy of Sciences and a member of Tau Beta Pi, Sigma Tau, Phi Kappa Phi, Phi
Eta Sigma, The American Society for Engineering Education (ASEE), the
Society for the History of Technology, and the History of Science Society. He has
received the Lee Atwood Award for excellence in Aerospace Engineering
Education from the AIAA and the ASEE, the Pendray Award for Aerospace
Literature from the AIAA, the von Karman Lectureship from the AIAA, and the
Gardner-Lasser History Book Award from the AIAA.
John D. Anderson, Jr.,
To Sarah-Alien, Katherine, and Elizabeth Anderson
___________ For All Their Love and Understanding
JO H N D. A N D E R SO N , JR.
CONTENTS
About the Author
v
Preface to the Fifth Edition
xv
Preface to the First Edition
xvii
2.2
C hapter 1
The First A eronautical E ngineers
2.3
1
1.1
1.2
1.3
Introduction 1
Very Early Developments 4
Sir George Cayley (1773-1857)— The
True Inventor of the Airplane 6
1.4 The Interregnum— From 1853 to 1891 13
1.5 Otto Lilienthal (1848-1896)— The Glider
Man 17
1.6 Percy Pilcher (1867-1899)— Extending
the Glider Tradition 20
1.7 Aeronautics Comes to America 21
1.8 Wilbur (1867-1912) and Orville
(1871-1948) Wright— Inventors of the
First Practical Airplane 27
1.9 The Aeronautical Triangle— Langley, the
Wrights, and Glenn Curtiss 36
1.10 The Problem of Propulsion 45
1.11 Faster and Higher 46
1.12 Summary 49
Bibliography 50
2.4
2.5
2.6
2.7
2.8
2.9
C hapter 3
The Standard A tm osphere
3.1
3.2
3.3
3.4
3.5
C hapter 2
Fundam ental T houghts
2.1
3.6
52
Fundamental Physical Quantities of a
Flowing Gas 56
2.1.1 Pressure 56
2.1.2 Density 57
2.1.3 Temperature 58
2.1.4 Flow Velocity and
Streamlines 59
The Source of All Aerodynamic
Forces 61
Equation of State for a
Perfect Gas 63
Discussion of Units 65
Specific Volume 70
Anatomy of the Airplane 76
Anatomy of a Space Vehicle 87
Historical Note: The NACA
and NASA 95
Summary 98
Bibliography 98
Problems 98
3.7
101
Definition of Altitude 103
Hydrostatic Equation 104
Relation Between Geopotential and
Geometric Altitudes 106
Definition of the Standard
Atmosphere 107
Pressure, Temperature, and Density
Altitudes 114
Historical Note: The Standard
Atmosphere 117
Summary 119
Bibliography 120
Problems 120
ix
X
Contents
C hapter
4
Basic A erodynam ics
122
4.1
Continuity Equation
126
4.2
Incompressible and Compressible
Flow 127
4.3
4.4
4.5
4.6
4.7
4.8
Momentum Equation 130
A Comment 134
Elementary Thermodynamics
Isentropic Flow 147
Energy Equation 152
Summary of Equations 155
4.26 Historical Note: Prandtl and the
Development of the Boundary Layer
Concept 239
4.27 Summary 242
Bibliography 244
Problems
245
141
4.9 Speed of Sound 156
4.10 Low-Speed Subsonic Wind Tunnels 162
4.11 Measurement of Airspeed 168
4.11.1 Incompressible Flow 171
4.11.2 Subsonic Compressible Flow 174
4.11.3 Supersonic Flow 178
4.11.4 Summary 182
4.12 Some Additional Considerations 183
4.12.1 More on Compressible Flow 183
4.12.2 More on Equivalent Airspeed 185
4.13 Supersonic Wind Tunnels and Rocket
Engines 187
4.14 Discussion of Compressibility 195
4.15 Introduction to Viscous Flow 196
4.16 Results for a Laminar Boundary
Layer 205
4.17 Results for a Turbulent Boundary
Layer 210
4.18 Compressibility Effects on Skin
Friction 213
4.19 Transition 216
4.20 Flow Separation 219
4.21 Summary of Viscous Effects on Drag 224
4.22 Historical Note: Bernoulli and Euler 225
4.23 Historical Note: The Pitot Tube 226
4.24 Historical Note: The First Wind
Tunnels 229
4.25 Historical Note: Osborne Reynolds and
His Number 235
A irfoils, W ings, and O ther
A erodynam ic Shapes 251
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
5.10
5.11
5.12
5.13
5.14
5.15
5.16
5.17
5.18
5.19
5.20
Introduction 251
Airfoil Nomenclature 253
Lift, Drag, and Moment Coefficients 257
Airfoil Data 263
Infinite Versus Finite Wings 271
Pressure Coefficient 273
Obtaining Lift Coefficient from C,, 278
Compressibility Correction for Lift
Coefficient 282
Critical Mach Number and Critical
Pressure Coefficient 283
Drag-Divergence Mach Number 294
Wave Drag (at Supersonic Speeds) 302
Summary of Airfoil Drag 310
Finite Wings 312
Calculation of Induced Drag 315
Change in the Lift Slope 321
Swept Wings 329
Flaps— A Mechanism for High Lift 342
Aerodynamics of Cylinders and
Spheres 348
How Lift Is Produced— Some Alternate
Explanations 352
Historical Note: Airfoils and Wings 362
5.20.1 The Wright Brothers
363
5.20.2 British and U.S. Airfoils
(1910 to 1920) 363
5.20.3 1920 to 1930 364
5.20.4 Early NACA Four-Digit
Airfoils 364
xi
Contents
5.21
5.22
5.23
5.24
5.20.5 Later NACA Airfoils 365
5.20.6 Modern Airfoil Work 366
5.20.7 Finite Wings 366
Historical Note: Ernst Mach
and His Number 369
Historical Note: The First Manned
Supersonic Flight 372
Historical Note: The X-15— First Manned
Hypersonic Airplane and Stepping-Stone
to the Space Shuttle 376
Summary 379
Bibliography 380
Problems 380
Chapter 6
Elem ents o f A irplane Perform ance
6.1
6.2
6.3
6.4
6.5
6.6
6.14
6.15
6.16
6.17
6.18
6.19
6.20
6.21
6.7
Altitude Effects on Power Required
and Available 414
6.8 Rate of Climb 419
6.9 Gliding Flight 428
6.10 Absolute and Service Ceilings 432
6.11 Time to Climb 435
6.12 Range and Endurance— Propeller-Driven
Airplane 436
6.12.1 Physical Considerations 437
6.23
6.24
6.25
6.26
7.2.3 Control
521
523
7.2.4 Partial Derivative
7.6
519
520
7.2.2 Dynamic Stability
438
6.12.3 Breguet Formulas (Propeller-Driven
Airplane) 440
513
Introduction 513
Definition of Stability and Control
7.2.1 Static Stability
7.3
7.4
7.5
446
Relations Between CD,o and Co.¡ 450
Takeoff Performance 458
Landing Performance 464
Turning Flight and the V-n Diagram 467
Accelerated Rate of Climb (Energy
Method) 474
Special Considerations for Supersonic
Airplanes 481
Uninhabited Aerial Vehicles (UAVs) 485
A Comment, and More on the Aspect
Ratio 494
Historical Note: Drag Reduction— The
NACA Cowling and the Fillet 494
Historical Note: Early Predictions of
Airplane Performance 499
Historical Note: Breguet and the Range
Formula 500
Historical Note: Aircraft Design—
Evolution and Revolution 501
Summary 507
Bibliography 509
Problems 510
Chapter 7
Principles o f Stability and Control
7.1
7.2
444
445
6.13.2 Quantitative Formulation
6.22
413
6.12.2 Quantitative Formulation
6.13.1 Physical Considerations
385
Introduction: The Drag Polar 385
Equations of Motion 392
Thrust Required for Level, Unaccelerated
Flight 394
Thrust Available and Maximum
Velocity 402
Power Required for Level, Unaccelerated
Flight 405
Power Available and Maximum
Velocity 410
6.6.1 Reciprocating Engine-Propeller
Combination 410
6.6.2 Jet Engine
6.13 Range and Endurance— Jet Airplane
523
Moments on the Airplane 524
Absolute Angle of Attack 525
Criteria for Longitudinal Static
Stability 527
Quantitative Discussion: Contribution of
the Wing to Mcg 532
xii
Contents
7.7
Contribution of the Tail to Mcg
7.8
Total Pitching Moment About the Center
of Gravity 539
Equations for Longitudinal Static
Stability 541
Neutral Point 543
Static Margin 544
Concept of Static Longitudinal
Control 548
Calculation of Elevator Angle to
Trim 553
Stick-Fixed Versus Stick-Free Static
Stability 555
Elevator Hinge Moment 556
Stick-Free Longitudinal Static
Stability 558
Directional Static Stability 562
7.9
7.10
7.11
7.12
7.13
7.14
7.15
7.16
7.17
536
7.18 Lateral Static Stability 563
7.19 A Comment 565
7.20 Historical Note: The Wright Brothers
Versus the European Philosophy on
Stability and Control 566
7.21 Historical Note: The Development of
Flight Controls 567
7.22 Historical Note: The “Tuck-Under”
Problem 569
7.23 Summary 570
Bibliography 571
Problems 571
C hapter 8
Space Flight (Astronautics)
8.1
8.2
8.3
8.4
573
Introduction 573
Differential Equations 580
Lagrange’s Equation 581
Orbit Equation 584
8.4.1 Force and Energy 584
8.4.2 Equation o f Motion 586
8.5
Space Vehicle Trajectories— Some Basic
Aspects 590
8.6
Kepler’s Laws
8.7
Introduction to Earth and Planetary
Entry 601
8.8
8.9
Exponential Atmosphere 604
General Equations of Motion for
Atmospheric Entry 604
Application to Ballistic Entry 608
Entry Heating 614
Lifting Entry, with Application to the
Space Shuttle 621
Historical Note: Kepler 625
8.10
8.11
8.12
8.13
597
8.14 Historical Note: Newton and the Law
of Gravitation 627
8.15 Historical Note: Lagrange 629
8.16 Historical Note: Unmanned Space
Flight 629
8.17 Historical Note: Manned Space
Flight 634
8.18 Summary 636
Bibliography 637
Problems 637
C hapter
9
Propulsion
9.1
9.2
9.3
9.4
9.5
9.6
9.7
9.8
9.9
639
Introduction 639
Propeller 642
Reciprocating Engine 650
Jet Propulsion— The Thrust
Equation 660
Turbojet Engine 663
Turbofan Engine 668
Ramjet Engine 670
Rocket Engine 674
Rocket Propellants— Some
Considerations 681
9.9.1 Liquid Propellants 681
9.9.2 Solid Propellants 684
9.9.3 A Comment 686
9.10 Rocket Equation 687
9.11 Rocket Staging 688
xiii
Contents
9.12 Electric Propulsion 692
9.12.1 Electron-Ion Thruster 693
9.12.2 Magnetoplasmadynamic
Thruster 694
9.12.3 Arc-Jet Thruster 694
9.12.4 A Comment 694
9.13 Historical Note: Early Propeller
Development 695
9.14 Historical Note: Early Development of the
Internal Combustion Engine
for Aviation 698
9.15 Historical Note: Inventors of Early Jet
Engines 700
9.16 Historical Note: Early History of Rocket
Engines 703
9.17 Summary 709
Bibliography 710
Problems 710
10.6
C hapter 11
H ypersonic Vehicles
11.1
11.2
10.1
10.2
10.3
10.4
10.5
Introduction 713
Some Physics of Solid Materials 714
10.2.1 Stress 714
10.2.2 Strain 716
10.2.3 Other Cases 717
10.2.4 Stress-Strain Diagram 718
Some Elements of an Aircraft
Structure 721
Materials 724
Fatigue 728
731
Introduction 731
Physical Aspects of Hypersonic
Flow 735
11.2.1 Thin Shock Layers 735
11.2.2 Entropy Layer 736
11.2.3 Viscous Interaction 737
11.2.4 High-Temperature Effects
11.2.5 Low-Density Flow
11.2.6 Recapitulation
11.3
11.4
C hapter 10
Flight Vehicle Structrures
and M aterials 713
Some Comments 729
Bibliography 729
Problems 730
11.5
738
739
743
Newtonian Law for Hypersonic
Flow 743
Some Comments on Hypersonic
Airplanes 749
Summary 758
Bibliography 758
Problems 758
A ppendix A Standard Atm osphere,
SI Units 760
Appendix B Standard A tm osphere, English
E ngineering Units 770
Appendix C Sym bols and Conversion
Factors 778
Appendix D Airfoil Data
Index
808
779
PREFACE TO THE FIFTH EDITION
he purpose of the present edition is the same as that of the first four: to
present the basic fundamentals of aerospace engineering at the introductory level in the clearest, simplest, and most motivating way possible.
Since the book is meant to be enjoyed as well as understood, I have made every
effort to ensure a clear and readable text. The choice of subject matter and its
organization, the order in which topics are introduced, and how these ideas are
explained have been carefully planned with the uninitiated reader in mind.
Because the book is intended as a self-contained text at the first- and second-year
levels, I avoid tedious details and massive “handbook” data. Instead, I introduce
and discuss fundamental concepts in a manner that is as straightforward and
clear-cut as possible, knowing that the book has also found favor with those who
wish to learn something about this subject outside the classroom.
The overwhelmingly favorable response to the earlier editions from stu­
dents, teachers, and practicing professionals both here and abroad is a source of
gratification. Particularly pleasing is the fact that those using the book have en­
joyed reading its treatment of the fascinating, challenging, and sometimes awe­
some discipline of aerospace engineering.
Thanks to this response, the contents of the fourth edition have been carried
over into the fifth edition with only minor modifications. A hallmark of this book
is the use of specially designed devices to enhance the readers’ understanding of
the material. In particular, carried over from the fourth edition are
1.
2.
Roadmaps placed at the beginning of each chapter to help guide the reader
through the logical flow of the material.
Design Boxes containing discussions that deal with interesting and
important applications of the fundamental material; these are literally set
apart in boxes.
In the same spirit, the fifth edition contains new material intended to further
enhance the education and interest of the reader, as follows:
1.
A second type of box appears in this edition, namely Preview Boxes,
located at the beginning of each chapter. The preview boxes are intended to
provide the reader with insight about what each chapter is about and why
the material is so important. I intend the preview boxes to be motivational,
to make the reader interested and curious enough to pay extra-close
attention to the content of the chapter. These preview boxes blatantly
“market” the material of each chapter. They are written in a particularly
informal manner to help turn the reader on to the content. In these preview
boxes, I am unabashedly admitting to providing some fun for the readers.
XV
Preface to the Fifth Edition
2.
3.
4.
5.
6.
Uninhabited aerial vehicles (UAVs) and uninhabited combat aerial vehicles
(UCAVs) are becoming an important part of aerospace engineering. A
whole section on UAVs and UCAVs has been added to Chapter 6.
Enhanced explanations of some of the more difficult concepts, aimed at
faster and better understanding of the material, have been added.
More emphasis has been placed on the better understanding of various
units and their proper use in calculations.
A number of additional worked examples have been added to further help
the reader understand how to use what he or she has been reading.
Included in this new material are about 20 new figures to help illustrate the
basic ideas.
All told, the new material represents a meaningful enhancement of Introduction
to Flight.
At the University of Maryland, this text is used for an introductory course for
sophomores in aerospace engineering. It leads directly into a second book by the
author, Fundamentals o f Aerodynamics, 3d ed. (McGraw-Hill, 2000), which is
used in a two-semester junior-senior aerodynamics course. This, in turn, feeds
into a third text, Modern Compressible Flow: With Historical Perspective, 3d ed.
(McGraw-Hill, 2003), used in a course for advanced undergraduates and firstyear graduate students. The complete triad is intended to give students a reason­
able technical and historical perspective on aerospace engineering in general and
aerodynamics in particular.
I am very grateful to Mrs. Susan Cunningham, who did such an excellent job
typing the manuscript. I am fortunate to have such dedicated and professional
help from one of the best scientific typists in the world. My gratitude also goes
out to my wife of 43 years, Sarah-Alien, who has helped to motivate and expedite
the effort that has gone into this book.
I also thank the following reviewers for their valuable feedback: B. Terry
Beck, Kansas State University; Ira M. Cohen, University of Pennsylvania; Bruce
Conway, University of Illinois, Urbana-Champaign; Fred R. DeJarnette, North
Carolina State University; Don Edberg, California State Polytechnic University,
Pomona; Ratneshwar Jha, Clarkson University; Thomas J. Mueller, University
of Notre Dame; Eugene E. Niemi, Jr., University of Massachusetts, Lowell;
Lakshmi Sankar, Georgia Institute of Technology; and Paavo Sepri, Florida
Institute of Technology.
Finally, emphasizing that the study, understanding, and practice of the pro­
fession of aerospace engineering is one of the most gratifying of human endeav­
ors and that my purpose is to instill a sense of enthusiasm, dedication, and love
of the subject, let me simply say to the reader: read, learn, and enjoy.
John D. Anderson, Jr.
PREFACE TQ THE FIRST EDITION
his book is an introduction to aerospace engineering from both the tech­
nological and historical points of view. It is written to appeal to several
.A . groups of people: (1) students of aerospace engineering in their freshman
or sophomore years in college who are looking for a comprehensive introduction
to their profession, (2) advanced high-school seniors who simply want to learn
what aerospace engineering is all about, (3) both college undergraduate and
graduate students who want to obtain a wider perspective on the glories, the in­
tellectual demands, and the technical maturity of aerospace engineering, and
(4) working engineers who simply want to obtain a firmer grasp on the funda­
mental concepts and historical traditions that underlie their profession.
As an introduction to aerospace engineering, this book is unique in at least
three ways. First, the vast majority of aerospace engineering professionals and
students have little knowledge or appreciation of the historical traditions and
background associated with the technology that they use almost every day. To fill
this vacuum, the present book attempts to marble some history of aerospace en­
gineering into the parallel technical discussions. For example, such questions as
who was Bernoulli, where did the Pitot tube originate, how did wind tunnels
evolve, who were the first true aeronautical engineers, and how did wings and
airfoils develop are answered. The present author feels strongly that such mater­
ial should be an integral part of the background of all aerospace engineers.
Second, this book incorporates both the SI and the English engineering sys­
tem of units. Modern students of aerospace engineering must be bilingual— on
one hand, they must fully understand and feel comfortable with the SI units, be­
cause most modern and all future literature will deal with the SI system; on the
other hand, they must be able to read and feel comfortable with the vast bulk of
existing literature, which is predominantly in engineering units. In this book, the
SI system is emphasized, but an honest effort is made to give the reader a feeling
for and understanding of both systems. To this end, some example problems are
worked out in the SI system and others in the English system.
Third, the author feels that technical books do not have to be dry and sterile
in their presentation. Instead, the present book is written in a rather informal
style. It attempts to talk to the reader. Indeed, it is intended to be almost a self­
teaching, self-pacing vehicle that the reader can use to obtain a fundamental un­
derstanding of aerospace engineering.
This book is a product of several years of teaching the introductory course in
aerospace engineering at the University of Maryland. Over these years, students
have constantly encouraged the author to write a book on the subject, and their
repeated encouragement could not be denied. The present book is dedicated in
part to these students.
xvii
xviii
Preface to the First Edition
Writing a book of this magnitude is a total commitment of time and effort for
a longer time than the author likes to remember. In this light, this book is dedi­
cated to my wife, Sarah-Alien, and my two daughters, Katherine and Elizabeth,
who relinquished untold amounts of time with their husband and father so that
these pages could be created. To them I say thank you, and hello again. Also, hid­
den between the lines, but ever-so-much present is Edna Brothers, who typed the
manuscript in such a dedicated fashion. In addition, the author wishes to thank
Dr. Richard Hallion and Dr. Thomas Crouch, curators of the National Air and
Space Museum of the Smithsonian Institution, for their helpful comments on the
historical sections of this manuscript, and especially Dick Hallion, for opening
the vast archives of the museum for the author’s historical research. Also, many
thanks are due to the reviewers of this manuscript, Professor J. J. Azar of the
University of Tulsa, Dr. R. F. Brodsky of Iowa State University, Dr. David
Caughey of Sibley School of Mechanical and Aerospace Engineering, and Pro­
fessor Francis J. Hale of North Carolina State University; their comments have
been most constructive, especially those of Dr. Caughey and Professor Hale.
Finally, the author wishes to thank his many colleagues in the profession for
stimulating discussions about what constitutes an introduction to aerospace
engineering. Hopefully, this book is a reasonable answer.
John D. Anderson, Jr.
CHAPTER
The First Aeronautical
Engineers
Nobody will fly for a thousand years!
Wilbur Wright, 1901, in a fit o f despair
SUCCESS FOUR FLIGHTS THURSDAY MORNING ALL AGAINST TWENTY
ONE MILE WIND STARTED FROM LEVEL WITH ENGINE POWER ALONE
AVERAGE SPEED THROUGH AIR THIRTY ONE MILES LONGEST 57 SEC­
ONDS INFORM PRESS HOME CHRISTMAS.
OREVELLE WRIGHT
A telegram, with the original misprints,
from Orville Wright to his father,
December 17,1903
1.1 INTRODUCTION
The scene: Wind-swept sand dunes of Kill Devil Hills, 4 mi south of Kitty Hawk,
North Carolina. The time: About 10:35 a m on Thursday, December 17, 1903.77»?
characters: Orville and Wilbur Wright and five local witnesses. The action:
Poised, ready to make history, is a flimsy, odd-looking machine, made from
spruce and cloth in the form of two wings, one placed above the other, a horizon­
tal elevator mounted on struts in front of the wings, and a double vertical rudder
1
chapter
1 The First Aeronautical Engineers
F igu re 1.1 T hree view s o f the Wright Flyer /, 1903.
behind the wings (see Fig. 1.1). A 12-hp engine is mounted on the top surface of
the bottom wing, slightly right of center. To the left of this engine lies a man—
Orville Wright— prone on the bottom wing, facing into the brisk and cold De­
cember wind. Behind him rotate two ungainly looking airscrews (propellers),
driven by two chain-and-pulley arrangements connected to the same engine. The
machine begins to move along a 60-ft launching rail on level ground. Wilbur
Wright runs along the right side of the machine, supporting the wing tip so that it
will not drag the sand. Near the end of the starting rail, the machine lifts into the
air; at this moment, John Daniels of the Kill Devil Life Saving Station takes a
photograph that preserves for all time the most historic moment in aviation his­
tory (see Fig. 1.2). The machine flies unevenly, rising suddenly to about 10 ft, then
ducking quickly toward the ground. This type of erratic flight continues for 12 s,
when the machine darts to the sand, 120 ft from the point where it lifted from the
starting rail. Thus ends a flight that, in Orville Wright’s own words, was “the first
1.1
Introduction
F igure 1.2 T he first heavier-than-air flight in history: the Wright Flyer I w ith O rville W right
at the controls, D ecem ber 17, 1903.
(Source: National A ir and Space Museum.)
in the history of the world in which a machine carrying a man had raised itself by
its own power into the air in full flight, had sailed forward without reduction of
speed, and had finally landed at a point as high as that from which it started.”
The machine was the Wright Flyer I, which is shown in Figs. 1.1 and 1.2 and
which is now preserved for posterity in the Air and Space Museum of the
Smithsonian Institution in Washington, District of Columbia. The flight on that
cold December 17 was momentous: It brought to a realization the dreams of cen­
turies, and it gave birth to a new way of life. It was the first genuine powered
flight of a heavier-than-air machine. With it, and with the further successes to
come over the next five years, came the Wright brothers’ clear right to be con­
sidered the premier aeronautical engineers of history.
However, contrary to some popular belief, the Wright brothers did not truly
invent the airplane; rather, they represent the fruition of a century’s worth of prior
aeronautical research and development. The time was ripe for the attainment of
powered flight at the beginning of the 20th century. The Wright brothers’ inge­
nuity, dedication, and persistence earned them the distinction of being first. The
purpose of this chapter is to look back over the years that led up to successful
powered flight and to single out an important few of those inventors and thinkers
who can rightfully claim to be the first aeronautical engineers. In this manner,
some of the traditions and heritage that underlie modern aerospace engineering
will be more appreciated when we develop the technical concepts of flight in
subsequent chapters.
It is somehow fitting that the fifth edition of this book is being prepared in
the year 2003, the centennial year of the Wright brothers’ momentous accom­
plishment shown in Fig. 1.2. At the time of writing, special celebratory events are
taking place worldwide, with many more to come throughout the year. The cen­
tennial year will reach its zenith when, on December 17, 2003, an exact replica
of the 1903 Wright Flyer, exact in every respect including the same fuel formu­
lation used to power the Wrights’ engine, is scheduled to fly at Kitty Hawk.
chapter
1 The First Aeronautical Engineers
When this happens, the curtain will momentarily close on a century of spectacu­
lar progress in aerospace engineering, but it will instantly open again on the next
century of powered flight, a new century of great promise with yet unimagined
advances in flight, advances to which the young readers of this book can look
forward to making major and exciting contributions.
1.2 VERY EARLY DEVELOPMENTS
Since the dawn of human intelligence, the idea of flying in the same realm as
birds has possessed human minds. Witness the early Greek myth of Daedalus and
his son Icarus. Imprisoned on the island of Crete in the Mediterranean Sea,
Daedalus is said to have made wings fastened with wax. With these wings, they
both escaped by flying through the air. However, Icarus, against his father’s
warnings, flew too close to the sun; the wax melted, and Icarus fell to his death
in the sea.
All early thinking of human flight centered on the imitation of birds. Various
unsung ancient and medieval people fashioned wings and met with sometimes
disastrous and always unsuccessful consequences in leaping from towers or
roofs, flapping vigorously. In time, the idea of strapping a pair of wings to arms
fell out of favor. It was replaced by the concept of wings flapped up and down by
various mechanical mechanisms, powered by some type of human arm, leg, or
body movement. These machines are called ornithopters. Recent historical re­
search has uncovered that Leonardo da Vinci himself was possessed by the idea
of human flight and that he designed vast numbers of ornithopters toward the end
of the 15th century. In his surviving manuscripts, more than 35,000 words and
500 sketches deal with flight. One of his ornithopter designs is shown in Fig. 1.3,
which is an original da Vinci sketch made sometime between 1486 and 1490. It
is not known whether da Vinci ever built or tested any of his designs. However,
human-powered flight by flapping wings was always doomed to failure. In this
sense, da Vinci’s efforts did not make important contributions to the technical
advancement of flight.
Human efforts to fly literally got off the ground on November 21, 1783,
when a balloon carrying Pilatre de Rozier and the Marquis d ’Arlandes ascended
into the air and drifted 5 mi across Paris. The balloon was inflated and buoyed up
by hot air from an open fire burning in a large wicker basket underneath. The de­
sign and construction of the balloon were due to the Montgolfier brothers, Joseph
and Etienne. In 1782, Joseph Montgolfier, gazing into his fireplace, conceived
the idea of using the “lifting pow er” of hot air rising from a flame to lift a person
from the surface of the earth. The brothers instantly set to work, experimenting
with bags made of paper and linen, in which hot air from a fire was trapped. After
several public demonstrations of flight without human passengers, including the
8-min voyage of a balloon carrying a cage containing a sheep, a rooster, and a
duck, the Montgolfiers were ready for the big step. At 1:54 pm on November 21,
1783, the first flight with human passengers rose majestically into the air and
lasted for 25 min (see Fig. 1.4). It was the first time in history that a human being
Figure 1.3 A n o rnithopter design by L eonardo da Vinci, 1486-1490.
Figure 1.4 T he first aerial voyage in history: T he M ontgolfier hotballoon lifts from the ground near Paris on N ovem ber 21, 1783.
chapter
1 The First Aeronautical Engineers
had been lifted off the ground for a sustained period of time. Very quickly after
this, the noted French physicist J. A. C. Charles (of Charles’ gas law in physics)
built and flew a hydrogen-filled balloon from the Tuileries Gardens in Paris on
December l, 1783.
So people were finally off the ground! Balloons, or “aerostatic machines,” as
they were called by the Montgolfiers, made no real technical contributions to
human heavier-than-air flight. However, they served a major purpose in trigger­
ing the public’s interest in flight through the air. They were living proof that peo­
ple could really leave the ground and sample the environs heretofore exclusively
reserved for birds. Moreover, balloons were the only means of human flight for
almost 100 years.
1.3 SIR GEORGE CAYLEY (1773-1857)— THE TRUE
INVENTOR OF THE AIRPLANE
The modern airplane has its origin in a design set forth by George Cayley in
1799. It was the first concept to include a fixed wing for generating lift, another
separate mechanism for propulsion (Cayley envisioned paddles), and a com­
bined horizontal and vertical (cruciform) tail for stability. Cayley inscribed his
idea on a silver disk (presumably for permanence), shown in Fig. 1.5. On the re­
verse side of the disk is a diagram of the lift and drag forces on an inclined plane
(the wing). The disk is now preserved in the Science Museum in London. Before
this time, thought of mechanical flight had been oriented toward the flapping
wings of ornithopters, where the flapping motion was supposed to provide both
lift and propulsion. (Da Vinci designed his ornithopter wings to flap simultane­
ously downward and backward for lift and propulsion.) However, Cayley is
responsible for breaking this unsuccessful line of thought; he separated the
Figure 1.5 T he silver disk on w hich C ayley engraved his concept for a fixed-w ing
aircraft, the first in history, in 1799. T he reverse side o f the disk show s the resultant
aerodynam ic force on a w ing resolved into lift and drag com ponents, indicating
C a y le y ’s full understanding o f the function o f a fixed w ing. T he disk is presently in
the Science M useum in L ondon.
1 .3
Sir George Cayley (1773-1857)— The True Inventor of the Airplane
concept of lift from propulsion and, in so doing, set into motion a century of
aeronautical development that culminated in the Wright brothers’ success in
1903. George Cayley is a giant in aeronautical history: He is the parent of mod­
ern aviation and is the first to introduce the basic configuration of the modern air­
plane. Let us look at him more closely.
Cayley was born at Scarborough in Yorkshire, England, on December 27,
1773. He was educated at York and Nottingham and later studied chemistry and
electricity under several noted tutors. He was a scholarly man of some rank, a
baronet who spent much of his time on the family estate, called Brompton. A por­
trait of Cayley is shown in Fig. 1.6. He was a well-preserved person, of extreme
intellect and open mind, active in many pursuits over a long life of 84 years. In
1825, he invented the caterpillar tractor, forerunner of all modern tracked vehi­
cles. In addition, he was chairman of the Whig Club of York, founded the
Yorkshire Philosophical Society (1821), cofounded the British Association for
the Advancement of Science (1831), was a member of Parliament, was a leading
authority on land drainage, and published papers dealing with optics and railroad
safety devices. Moreover, he had a social conscience: He appealed for, and
donated to, the relief of industrial distress in Yorkshire.
F igu re 1.6 A portrait o f S ir G eorge C ayley, painted by
H enry Perronet B riggs in 1841. T he portrait now hangs in
the N ational Portrait G allery in London.
chapter
1 The First Aeronautical Engineers
Figure 1.7 G eorge C a y le y ’s w hirling-arm apparatus for testing airfoils.
However, by far his major and lasting contribution to humanity was in aero­
nautics. After experimenting with model helicopters beginning in 1796, Cayley
engraved his revolutionary fixed-wing concept on the silver disk in 1799 (see
Fig. 1.5). This was followed by an intensive 10-year period of aerodynamic in­
vestigation and development. In 1804, he built a whirling-arm apparatus, shown
in Fig. 1.7, for testing airfoils; this was simply a lifting surface (airfoil) mounted
on the end of a long rod, which was rotated at some speed to generate a flow of
air over the airfoil. In modern aerospace engineering, wind tunnels now serve
this function, but in Cayley’s time the whirling arm was an important develop­
ment, which allowed the measurement of aerodynamic forces and the center of
pressure on a lifting surface. O f course, these measurements were not very ac­
curate, because after a number of revolutions of the arm, the surrounding air
would begin to rotate with the device. Nevertheless, it was a first step in aerody­
namic testing. (Cayley did not invent the whirling arm; that honor belongs to the
English military engineer Benjamin Robins in 1742.) Also in 1804, Cayley de­
signed, built, and flew the small model glider shown in Fig. 1.8; this may seem
trivial today, something that you may have done as a child, but in 1804, it repre­
sented the first modern-configuration airplane o f history, with a fixed wing, and
a horizontal and vertical tail that could be adjusted. (Cayley generally flew his
glider with the tail at a positive angle of incidence, as shown in his sketch in
Fig. 1.8.) A full-scale replica of this glider is on display at the Science Museum
in London— the model is only about 1 m long.
1 .3
Sir George Cayley (1773-1857)— The True Inventor of the Airplane
F igu re 1.8 T he first m odern-configuration airplane in history: C a y le y ’s m odel glider, 1804.
Cayley’s first outpouring of aeronautical results was documented in his
momentous triple paper of 1809-1810. Entitled “On Aerial Navigation” and
published in the November 1809, February 1810, and March 1810 issues of
Nicholson’s Journal o f Natural Philosophy, this document ranks as one of the
most important aeronautical works in history. (Note that the words natural phi­
losophy in history are synonymous with physical science.) Cayley was prompted
to write his triple paper after hearing reports that Jacob Degen had recently flown
in a mechanical machine in Vienna. In reality, Degen flew in a contraption that
was lifted by a balloon. It was of no significance, but Cayley did not know the de­
tails. In an effort to let people know of his activities, Cayley documented many
aspects of aerodynamics in his triple paper. It was the first treatise on theoretical
and applied aerodynamics in history to be published. In it, Cayley elaborates on
his principle of the separation of lift and propulsion and his use of a fixed wing to
generate lift. He states that the basic principle of a flying machine is “to make a
surface support a given weight by the application of power to the resistance of
air.” He notes that a surface inclined at some angle to the direction of motion will
generate lift and that a cambered (curved) surface will do this more efficiently
than a flat surface. He also states for the first time in history that lift is generated
by a region of low pressure on the upper surface of the wing. The modern
technical aspects of these phenomena will be developed and explained in
Chaps. 4 and 5; however, stated by Cayley in 1809-1810, these phenomena were
new and unique. His triple paper also addressed the matter of flight control and
was the first document to discuss the role of the horizontal and vertical tail planes
in airplane stability. Interestingly enough, Cayley goes off on a tangent in dis­
cussing the use of flappers for propulsion. Note that on the silver disk (see
Fig. 1.5) Cayley shows some paddles just behind the wing. From 1799 until his
death in 1857, Cayley was obsessed with such flappers for aeronautical propul­
sion. He gave little attention to the propeller (airscrew); indeed, he seemed to have
an aversion to rotating machinery of any type. However, this should not detract
from his numerous positive contributions. Also in his triple paper, Cayley tells us
of the first successful full-size glider of history, built and flown without passen­
gers by him at Brompton in 1809. However, there is no clue as to its configuration.
Curiously, the period from 1810 to 1843 was a lull in Cayley’s life in regard
to aeronautics. Presumably, he was busy with his myriad other interests and
activities. During this period, he showed interest in airships (controlled bal­
loons), as opposed to heavier-than-air machines. He made the prophetic state­
ment that “balloon aerial navigation can be done readily, and will probably, in the
chapter
1 The First Aeronautical Engineers
order of things, come into use before mechanical flight can be rendered suffi­
ciently safe and efficient for ordinary use.” He was correct; the first successful
airship, propelled by a steam engine, was built and flown by the French engineer
Henri Giffard in Paris in 1852, some 5 1 years before the first successful airplane.
Cayley’s second outpouring of aeronautical results occurred in the period
from 1848 to 1854. In 1849, he built and tested a full-size airplane. During some
of the flight tests, a l()-year-old boy was carried along and was lifted several me­
ters off the ground while gliding down a hill. Cayley’s own sketch of this ma­
chine, called the boy carrier, is shown in Fig. 1.9. Note that it is a triplane (three
wings mounted on top o f one another). Cayley was the first to suggest such mul­
tiplanes (i.e., biplanes and triplanes), mainly because he was concerned with the
possible structural failure of a single large wing (a monoplane). Stacking smaller,
more compact, wings on top of one another made more sense to him, and his con­
cept was perpetuated into the 20th century. It was not until the late 1930s that the
monoplane became the dominant airplane configuration. Also note from Fig. 1.9
that, strictly speaking, this was a “powered” airplane; that is, it was equipped
with propulsive flappers.
One of Cayley’s most important papers was published in M echanics’Maga­
zine on September 25, 1852. By this time he was 79 years old! The article was
entitled “Sir George Cayley’s Governable Parachutes.” It gave a full description
of a large human-carrying glider that incorporated almost all the features of the
modern airplane. This design is shown in Fig. 1.10, which is a facsimile of the
illustration that appeared in the original issue of M echanics’ Magazine. This
F igu re 1.9 C a y le y 's triplane from 1849— the boy carrier. N ote the vertical and horizontal
tail surfaces and the flapperlike propulsive m echanism .
1.3
Sir George Cayley (1773-1857)— The True Inventor of the Airplane
jWerijanics’ fftnga^ine,
MUSEUM,
No. 1520.]
REGISTER,
JOURNAL,
AND
S A T U R D A Y , S E P T E M B E R 25, 1852.
GAZETTE.
[Price 3 d ., Stamped i d .
E d ite d b y J . C. R o b e rt* o n , 1C6, F le e t-itre * » .
SIR G EO R G E C A Y L E Y 'S G O V E R N A B L E P A R A C H U T E S.
Fig. 2.
Flf . 1.
C H A P T E R 1 The First Aeronautical Engineers
airplane had ( l) a main wing at an angle of incidence for lift, with a dihedral for
lateral stability; (2) an adjustable cruciform tail for longitudinal and directional
stability; (3) a pilot-operated elevator and rudder; (4) a fuselage in the form of a
car, with a pilot’s seat and three-wheel undercarriage; and (5) a tubular beam and
box beam construction. These combined features were not to be seen again until
the Wright brothers' designs at the beginning of the 20th century. Incredibly, this
1852 paper by Cayley went virtually unnoticed, even though M echanics’ Maga­
zine had a large circulation. It was rediscovered by the eminent British aviation
historian Charles H. Gibbs-Smith in I960 and republished by him in the June 13,
1960, issue of The Times.
Sometime in 1853— the precise date is unknown— George Cayley built and
flew the world’s first human-carrying glider. Its configuration is not known, but
Gibbs-Smith states that it was most likely a triplane on the order of the earlier
boy carrier (see Fig. 1.9) and that the planform (top view) of the wings was prob­
ably shaped much as the glider in Fig. 1.10. According to several eyewitness
accounts, a gliding flight of several hundred yards was made across a dale at
Brompton with Cayley’s coachman aboard. The glider landed rather abruptly,
and after struggling clear of the vehicle, the shaken coachman is quoted as say­
ing: “Please, Sir George, I wish to give notice. I was hired to drive, and not to
fly.” Very recently, this flight of Cayley’s coachman was reenacted for the public
in a special British Broadcasting Corporation television show on Cayley’s life.
While visiting the Science Museum in London in August of 1975, the present au­
thor was impressed to find the television replica of Cayley’s glider (minus the
coachman) hanging in the entranceway.
George Cayley died at Brompton on December 15, 1857. During his almost
84 years of life, he laid the basis for all practical aviation. He was called the father
o f aerial navigation by William Samuel Henson in 1846. However, for reasons
that are not clear, the name of George Cayley retreated to the background soon
after his death. His works became obscure to virtually all later aviation enthusiasts
in the latter half of the 19th century. This is incredible, indeed unforgivable, con­
sidering that his published papers were available in known journals. Obviously,
many subsequent inventors did not make the effort to examine the literature be­
fore forging ahead with their own ideas. (This is certainly a problem for engineers
today, with the virtual explosion of written technical papers since World War II.)
However, Cayley’s work has been brought to light by the research of several mod­
ern historians in the 20th century. Notable among them is C. H. Gibbs-Smith,
from whose book entitled Sir George Cayley's Aeronautics (1962) much of this
material in Sec. 1.3 has been gleaned. Gibbs-Smith states that had Cayley’s work
been extended directly by other aviation pioneers and had they digested ideas
espoused in his triple paper of 1809-1810 and in his 1852 paper, successful
powered flight would have most likely occurred in the 1890s. Probably so!
As a final tribute to George Cayley, we note that the French aviation histo­
rian Charles Dollfus said the following in 1923:
T h e a e ro p la n e is a B ritish in v en tio n : it w a s c o n c e iv e d in all e sse n tia ls b y G e o rg e
C ay ley , th e g re a t E n g lish e n g in e e r w h o w o rk e d in th e first h a lf o f last c en tu ry . T h e
1.4
The Interregnum— From 1853 to 1891
n a m e o f C a y le y is little k n o w n , e v e n in h is o w n c o u n try , a n d th ere are v e ry few w h o
k n o w th e w o rk o f th is a d m ira b le m an , the g re a te s t g e n iu s o f a v ia tio n . A stu d y o f h is
p u b lic a tio n s fills o n e w ith a b so lu te a d m ira tio n b o th f o r h is in v e n tiv e n e ss , a n d fo r his
lo g ic a n d c o m m o n se n se . T h is g re a t e n g in e e r, d u rin g the S e c o n d E m p ire , d id in fa ct
n o t o n ly in v en t th e a e ro p la n e e n tire , as it n o w e x ists, b u t he re a liz e d th a t th e p ro b le m
o f a v ia tio n h a d to be d iv id e d b e tw e e n th e o re tic a l re se a rc h — C a y le y m a d e the first
a e ro d y n a m ic e x p e rim e n ts fo r a e ro n a u tic a l p u rp o se s— a n d p ra ctic a l tests, e q u a lly in
th e e a se o f th e g lid e r as o f th e p o w e re d a ero p lan e .
1.4 THE INTERREGNUM— FROM 1853 TO 1891
For the next 50 years after Cayley’s success with the coachman-carrying glider,
there were no major advances in aeronautical technology comparable to those of
the previous 50 years. Indeed, as stated in Sec. 1.3, much of Cayley’s work be­
came obscure to all but a few dedicated investigators. However, there was con­
siderable activity, with numerous people striking out (sometimes blindly) in var­
ious uncoordinated directions to conquer the air. Some of these efforts are noted
in the following paragraphs, just to establish the flavor of the period.
William Samuel Henson (1812-1888) was a contemporary of Cayley. In
April 1843, he published in England a design for a fixed-wing airplane powered
by a steam engine driving two propellers. Called the aerial steam carriage, this
design received wide publicity throughout the 19th century, owing mainly to a
series of illustrative engravings that were reproduced and sold around the world.
This was a publicity campaign of which Madison Avenue would have been
proud; one of these pictures is shown in Fig. 1.11. Note some of the qualities of
modern aircraft in Fig. 1.11: the engine inside a closed fuselage, driving two pro­
pellers; tricycle landing gear; and a single rectangular wing of relatively high
F igu re 1.11 H enson’s aerial steam carriage, 1842-1843.
(Source: National A ir and Space Museum.)
chapter
1 The First Aeronautical Engineers
aspect ratio. (We will discuss the aerodynamic characteristics of such wings in
Chap. 5.) Henson’s design was a direct product of George Cayley’s ideas and re­
search in aeronautics. The aerial steam carriage was never built, but the design,
along with its widely published pictures, served to engrave George Cayley’s
fixed-wing concept on the minds of virtually all subsequent workers. Thus, even
though Cayley’s published papers fell into obscurity after his death, his major
concepts were partly absorbed and perpetuated by following generations of in­
ventors, even though most of these inventors did not know the true source of the
ideas. In this manner, Henson’s aerial steam carriage was one of the most influ­
ential airplanes in history, even though it never flew!
John Stringfellow, a friend of Henson, made several efforts to bring
Henson’s design to fruition. Stringfellow built several small steam engines and
attempted to power some model monoplanes off the ground. He was close, but
unsuccessful. However, his most recognized work appeared in the form of a
steam-powered triplane, a model of which was shown at the 1868 aeronautical
exhibition sponsored by the Aeronautical Society at the Crystal Palace in
London. A photograph of Stringfellow’s triplane is shown in Fig. 1.12. This air­
plane was also unsuccessful, but again it was extremely influential because of
worldwide publicity. Illustrations of this triplanc appeared throughout the end of
the 19th century. Gibbs-Smith, in his book Aviation: An Historical Survey from
Its Origins to the End o f World War II (1970), states that these illustrations were
later a strong influence on Octave Chanute, and through him the Wright broth­
ers, and strengthened the concept of superimposed wings. Stringfellow’s tri­
plane was the main bridge between George Cayley’s aeronautics and the mod­
ern biplane.
During this period, the first powered airplanes actually hopped off the
ground, but for only hops. In 1857-1858, the French naval officer and engineer
Figure 1.12 S trin g fello w ’s m odel triplane exhibited at the first aeronautical exhibition in
L ondon, 1868.
1.4
The Interregnum— From 1853 to 1891
Figure 1.13 Du T em ple’s airplane: the first aircraft to m ake a pow ered but assisted
takeoff. 1874.
Felix Du Temple flew the first successful powered model airplane in history; it
was a monoplane with swept-forward wings and was powered by clockwork!
Then, in 1874, Du Temple achieved the world’s first powered takeoff by a
piloted, full-size airplane. Again, the airplane had swept-forward wings, but this
time it was powered by some type of hot-air engine (the precise type is un­
known). A sketch of Du Temple’s full-size airplane is shown in Fig. 1.13. The
machine, piloted by a young sailor, was launched down an inclined plane at
Brest, France; it left the ground for a moment but did not come close to anything
resembling sustained flight. In the same vein, the second powered airplane with
a pilot left the ground near St. Petersburg, Russia, in July 1884. Designed by
Alexander F. Mozhaiski, this machine was a steam-powered monoplane, shown
in Fig. 1.14. Mozhaiski’s design was a direct descendant from Henson’s aerial
steam carriage— it was even powered by an English steam engine! With 1. N.
Golubev as pilot, this airplane was launched down a ski ramp and flew for a few
seconds. As with Du Temple’s airplane, no sustained flight was achieved. At
various times, the Russians have credited Mozhaiski with the first powered
flight in history, but of course it did not satisfy the necessary criteria to be called
such. Du Temple and Mozhaiski achieved the first and second assisted powered
takeoffs, respectively, in history, but neither experienced sustained flight. In his
book The World’s First Aeroplane Flights (1965), C. H. Gibbs-Smith states the
following criteria used by aviation historians to judge a successful powered
flight:
In order to qualify for having made a simple powered and sustained llight, a con­
ventional aeroplane should have sustained itself freely in a horizontal or rising flight
path—without loss of airspeed—beyond a point where it could be influenced by any
momentum built up before it left the ground: otherwise its performance can only be
rated as a powered leap, i.e., it will not have made a fully self-propelled flight, but
C H A P T E R 1 The First Aeronautical Engineers
F igure 1.14 T he second airplane to m ake an assisted takeoff: M ozhaiski’s aircraft,
R ussia, 1884.
w ill o n ly h a v e fo llo w e d a b a llistic tra je c to ry m o d ifie d by th e th ru st o f its p ro p e lle r
a n d by th e a e ro d y n a m ic fo rc e s a c tin g u p o n its a ero fo ils. F u rth e rm o re , it m u st be
sh o w n th a t th e m a c h in e c a n b e k e p t in sa tis fa c to ry e q u ilib riu m . S im p le su sta in e d
flight o b v io u sly n e e d n o t in c lu d e full c o n tro lla b ility , b u t th e m a in te n a n c e o f a d e ­
q u a te e q u ilib riu m in flig h t is p a rt a n d p a rce l o f su ste n tio n .
Under these criteria, there is no doubt in the mind of any major aviation historian
that the first powered flight was made by the Wright brothers in 1903. However,
the assisted “hops” just described put two more rungs in the ladder of aeronauti­
cal development in the 19th century.
O f particular note during this period is the creation in London in 1866 of the
Aeronautical Society of Great Britain. Before this time, work on “aerial naviga­
tion” (a phrase coined by George Cayley) was looked upon with some disdain by
many scientists and engineers. It was too out of the ordinary and was not to be
taken seriously. However, the Aeronautical Society soon attracted scientists of
stature and vision, people who shouldered the task of solving the problems of
mechanical flight in a more orderly and logical fashion. In turn, aeronautics took
on a more serious and meaningful atmosphere. The society, through its regular
meetings and technical journals, provided a cohesive scientific outlet for the pre­
sentation and digestion of aeronautical engineering results. The society is still
flourishing today in the form of the highly respected Royal Aeronautical Society.
Moreover, it served as a model for the creation of both the American Rocket So­
ciety and the Institute of Aeronautical Sciences in the United States in this cen­
tury; both of these societies merged in 1964 to form the American Institute of
1.5
Otto Ulienthal (1848-1896)— The Glider Man
Aeronautics and Astronautics (AIAA), one of the most influential channels for
aerospace engineering information exchange today.
In conjunction with the Aeronautical Society of Great Britain, at its first
meeting on June 27, 1866, Francis H. Wenham read a paper entitled “Aerial
Locomotion,” one of the classics in the aeronautical engineering literature.
Wenham was a marine engineer who later was to play a prominent role in the
society and who later designed and built the first wind tunnel in history
(see Sec. 4.24). His paper, which was also published in the first annual report of
the society, was the first to point out that most of the lift of a wing was obtained
from the portion near the leading edge. He also established that a wing with high
aspect ratio was the most efficient for producing lift. (We will see why in Chap. 5.)
As noted in our previous discussion about Stringfellow, the Aeronautical
Society started out in style: When it was only two years old, in 1868, it put on the
first aeronautical exhibition in history at the Crystal Palace. It attracted an
assortment of machines and balloons and for the first time offered the general
public a firsthand overview of the efforts being made to conquer the air.
S t r i n g f c l l o w ’s t r i p l a n c ( d i s c u s s e d e a r l ie r ) w a s o f p a r t i c u l a r i n te r e s t . Z i p p i n g o v e r
the heads of the enthralled onlookers, the triplane moved through the air along an
inclined cable strung below the roof of the exhibition hall (see Fig. 1.12). How­
ever, it did not achieve sustained flight on its own. In fact, the 1868 exhibition did
nothing to advance the technical aspects of aviation, but it was a masterstroke of
good public relations.
1.5 OTTO LIL1ENTHAL (1848-1896)— THE
GLIDER MAN
With all the efforts that had been made in the past, it was still not until 1891 that
a human literally jum ped into the air and flew with wings in any type of con­
trolled fashion. This person was Otto Lilienthal, one of the giants in aeronautical
engineering (and in aviation in general). Lilienthal designed and flew the first
successful controlled gliders in history. He was a man of aeronautical stature
comparable to Cayley and the Wright brothers. Let us examine the man and his
contributions more closely.
Lilienthal was born on May 23, 1848, at Anklam, Prussia (Germany). He
obtained a good technical education at trade schools in Potsdam and Berlin, the
latter at the Berlin Technical Academy, graduating with a degree in mechanical
engineering in 1870. After a one-year stint in the army during the FrancoPrussian War, Lilienthal went to work designing machinery in his own factory.
However, from early childhood he was interested in flight and performed some
youthful experiments on ornithopters of his own design. Toward the late 1880s,
his work and interests took a more mature turn, which ultimately led to fixedwing gliders.
In 1889, Lilienthal published a book entitled Der Vogelflug als Grundlage
der Fliegekunst (Bird Flight as the Basis of Aviation). This is another of the early
chapter
1 The First Aeronautical Engineers
classics in aeronautical engineering, because not only did he study the structure
and types of birds’ wings, but also he applied the resulting aerodynamic infor­
mation to the design of mechanical flight. Lilienthal’s book contained some of
the most detailed aerodynamic data available at that time. Translated sections
were later read by the Wright brothers, who incorporated some of his data in their
first glider designs in 1900 and 1901.
By 1889, Lilienthal had also come to a philosophical conclusion that was to
have a major impact on the next two decades of aeronautical development. He
concluded that to learn practical aerodynamics, he had to get up in the air and ex­
perience it himself. In his own words,
O n e c a n g e t a p ro p e r in sig h t in to the p ra c tic e o f flying o n ly b y a ctu al fly in g e x p e ri­
m e n ts ------T h e m a n n e r in w h ic h w e h a v e to m ee t th e irre g u la ritie s o f th e w in d , w h en
so a rin g in th e air, c an o n ly b e lea rn t by b e in g in th e a ir i t s e l f . . . . T h e o n ly w ay
w h ic h lea d s us to a q u ic k d e v e lo p m e n t in h u m a n flight is a sy ste m a tic a n d e n e rg e tic
pra ctic e in actu al flying e x p e rim e n ts.
To put this philosophy into practice, Lilienthal designed a glider in 1889, and an­
other in 1890— both were unsuccessful. However, in 1891, Lilienthal’s first suc­
cessful glider flew from a natural hill at Derwitz, Germany. (Later, he was to
build an artificial hill about 50 ft high near Lichterfelde, a suburb of Berlin; this
conically shaped hill allowed glider flights to be made into the wind, no matter
what the direction.) The general configuration of his monoplane gliders is shown
in Fig. 1.15, which is a photograph showing Lilienthal as the pilot. Note the
rather birdlike planform of the wing. Lilienthal used cambered (curved) airfoil
shapes on the wing and incorporated vertical and horizontal tail planes in the
back for stability. These machines were hang gliders, the grandparents of the
Figure 1.15 A m onoplane hang g lider by L ilienthal, 1894.
1.5
Otto Lilienthal (1848-1896)— The Glider Man
sporting vehicles of today. Flight control was exercised by one’s shifting one’s
center of gravity under the glider.
Contrast Lilienthal’s flying philosophy with those of previous would-be avi­
ators before him. During most of the 19th century, powered flight was looked
upon in a brute-force manner: Build an engine strong enough to drive an air­
plane, slap it on an airframe strong enough to withstand the forces and to gener­
ate the lift, and presumably you could get into the air. What would happen after
you got into the air would be just a simple matter of steering the airplane around
the sky like a carriage or automobile on the ground— at least this was the general
feeling. Gibbs-Smith called the people taking this approach the chauffeurs. In
contrast were the airmen— Lilienthal was the first— who recognized the need to
get up in the air, fly around in gliders, and obtain the “feel” of an airplane before
an engine was used for powered flight. The chauffeurs were mainly interested in
thrust and lift, whereas the airmen were firstly concerned with flight control in
the air. The airmen’s philosophy ultimately led to successful powered flight; the
chauffeurs were singularly unsuccessful.
Lilienthal made over 2000 successful glider flights. The aerodynamic data
he obtained were published in papers circulated throughout the world. In fact, his
work was timed perfectly with the rise of photography and the printing industry.
In 1871, the dry-plate negative was invented, which by 1890 could “freeze” a
moving object without a blur. Also, the successful halftone method of printing
photographs in books and journals had been developed. As a result, photographs
of Lilienthal’s flights were widely distributed; indeed, Lilienthal was the first
human to be photographed in an airplane (see, e.g., Fig. 1.15). Such widespread
dissemination of his results inspired other pioneers in aviation. The Wright
brothers’ interest in flight did not crystallize until Wilbur first read some of
Lilienthal’s papers in about 1894.
On Sunday, August 9, 1896, Lilienthal was gliding from the Gollenberg hill
near Stollen in Germany. It was a fine sum m er’s day. However, a temporary gust
of wind brought Lilienthal’s monoplane glider to a standstill; he stalled and
crashed to the ground. Only the wing was crumpled; the rest of the glider was
undamaged. However, Lilienthal was carried away with a broken spine. He died
the next day in the Bergmann Clinic in Berlin. During the course of his life,
Lilienthal remarked several times that “sacrifices must be made.” This epitaph is
carved on his gravestone in Lichterfelde cemetery.
There is some feeling that had Lilienthal lived, he would have beaten the Wright
brothers to the punch. In 1893, he built a powered machine; however, the prime
mover was a carbonic acid gas motor that twisted six slats at each wing tip, obvi­
ously an ornithopter-type idea to mimic the natural mode of propulsion for birds. In
the spring of 1895, he built a second, but larger, powered machine of the same type.
Neither one of these airplanes was ever flown with the engine operating. It seems to
this author that this mode of propulsion was doomed to failure. If Lilienthal had
lived, would he have turned to the gasoline engine driving a propeller and thus
achieved powered flight before 1903? It is a good question for conversation.
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1 The First Aeronautical Engineers
1.6 PERCY PILCHER (1867-1899)— EXTENDING
THE GLIDER TRADITION
In June 1895, Otto Lilienthal received a relatively young and very enthusiastic
visitor in Berlin— Percy Pilcher, a Scot who lived in Glasgow and who had al­
ready built his first glider. Under Lilienthal’s guidance, Pilcher made several
glides from the artificial hill. This visit added extra fuel to Pilcher’s interest in
aviation; he returned to the British Isles and over the next four years built a series
of successful gliders. His most noted machine was the Hawk, built in 1896 (see
Fig. 1.16). Pilcher’s experiments with his hang gliders made him the most dis­
tinguished British aeronautical engineer since George Cayley. Pilcher was an air­
man, and along with Lilienthal he underscored the importance of learning the
practical nature of flight in the air before lashing an engine to the machine.
However, Pilcher’s sights were firmly set on powered flight. In 1897, he
calculated that an engine of 4 hp weighing no more than 40 lb, driving a 5-ftdiameter propeller, would be necessary to power his Hawk off the ground. Since
no such engine was available commercially, Pilcher (who was a marine engineer
by training) spent most of 1898 designing and constructing one. It was com­
pleted and bench-tested by the middle of 1899. Then, in one of those quirks of
fate that dot many aspects of history, Pilcher was killed while demonstrating his
Hawk glider at the estate of Lord Braye in Leicestershire, England. The weather
was bad, and on his first flight the glider was thoroughly water-soaked. On his
second flight, the heavily sodden tail assembly collapsed, and Pilcher crashed to
the ground. Like Lilienthal, Pilcher died one day after this disaster. Hence,
England and the world also lost the only man other than Lilienthal who might
have achieved successful powered flight before the Wright brothers.
F igu re 1.16 P ilch e r’s hang glider, the Hawk, 1896.
1.7
Aeronautics Comes to America
1.7 AERONAUTICS COMES TO AMERICA
Look at the geographic distribution of the early developments in aeronautics as
portrayed in Secs. 1.2 through 1.6. After the advent of ballooning, due to the
Montgolfiers’ success in France, progress in heavier-than-air machines was fo­
cused in England until the 1850s: Witness the contributions of Cayley, Henson,
and Stringfellow. This is entirely consistent with the fact that England also gave
birth to the industrial revolution during this time. Then the spotlight moved to
the European continent with Du Temple, Mozhaiski, Lilienthal, and others.
There were some brief flashes again in Britain, such as those due to Wenham and
the Aeronautical Society. In contrast, throughout this time virtually no important
progress was being made in the United States. The fledgling nation was busy
consolidating a new government and expanding its frontiers. There was not
much interest or time for serious aeronautical endeavors.
However, this vacuum was broken by Octave Chanute (1832-1910), a
French-born naturalized citizen who lived in Chicago. Chanute was a civil engi­
neer who became interested in mechanical flight in about 1875. For the next
35 years, he collected, absorbed, and assimilated every piece of aeronautical in­
formation he could find. This culminated in 1894 with the publishing of his book
entitled Progress in Flying Machines, a work that ranks with Lilienthal’s Der
Vogelflug as one of the great classics in aeronautics. Chanute’s book summarized
all the important progress in aviation up to that date; in this sense, he was the first
serious aviation historian. In addition, Chanute made positive suggestions as to
the future directions necessary to achieve success in powered flight. The Wright
brothers avidly read Progress in Flying Machines and subsequently sought out
Chanute in 1900. A close relationship and interchange of ideas developed be­
tween them. A friendship developed that was to last in various degrees until
Chanute’s death in 1910.
Chanute was an airman. Following this position, he began to design hang
gliders, in the manner of Lilienthal, in 1896. His major specific contribution to
aviation was the successful biplane glider shown in Fig. 1.17, which introduced
the effective Pratt truss method of structural rigging. The Wright brothers were
directly influenced by this biplane glider, and in this sense Chanute provided the
natural bridge between Stringfellow’s triplane (1868) and the first successful
powered flight (1903).
About 500 mi to the east, in Washington, District of Columbia, the United
States’ second noted pre-Wright aeronautical engineer was hard at work. Samuel
Pierpont Langley (1834-1906), secretary of the Smithsonian Institution, was
tirelessly designing and building a series of powered aircraft, which finally cul­
minated in two attempted piloted flights, both in 1903, just weeks before the
Wrights’ success on December 17.
Langley was born in Roxbury, Massachusetts, on August 22, 1834. He re­
ceived no formal education beyond high school, but his childhood interest in as­
tronomy spurred him to a lifelong program of self-education. Early in his career,
chapter
1 The First Aeronautical Engineers
F igu re 1.17 C h a n u te ’s hang glider, 1896.
(Source: National Air and Space Museum.)
he worked for 13 years as an engineer and architect. Then, after making a tour of
European observatories, Langley became an assistant at Harvard Observatory in
1865. He went on to become a mathematics professor at the U.S. Naval Acad­
emy, a physics and astronomy professor at the University of Pittsburgh, and the
director of the Allegheny Observatory at Pittsburgh. By virtue of his many sci­
entific accomplishments, Langley was appointed secretary of the Smithsonian
Institution in 1887.
In this same year, Langley, who was by now a scientist of international rep­
utation, began his studies of powered flight. Following the example of Cayley, he
built a large whirling arm, powered by a steam engine, with which he made force
tests on airfoils. He then built nearly 100 different types of rubber-band-powered
model airplanes, graduating to steam-powered models in 1892. However, it was
not until 1896 that Langley achieved any success with his powered models; on
May 6 one of his aircraft made a free flight of 3300 ft, and on November 28 an­
other flew for more than | mi. These Aerodromes (a term due to Langley) were
tandem-winged vehicles, driven by two propellers between the wings, powered
by a 1-hp steam engine of Langley’s own design. (However, Langley was influ­
enced by one of John Stringfellow’s small aerosteam engines, which was pre­
sented to the Smithsonian in 1889. After studying this historic piece of machin­
ery, Langley set out to design a better engine.)
Langley was somewhat satisfied with his success in 1896. Recognizing that
further work toward a piloted aircraft would be expensive in both time and
money, he “made the firm resolution not to undertake the construction of a large
man-carrying machine.” (Note that it was in this year that the Wright brothers be­
came interested in powered flight, another example of the flow and continuity of
ideas and developments in physical science and engineering. Indeed, Wilbur and
1.7
Aeronautics Comes to America
Orville were directly influenced and encouraged by Langley’s success with pow­
ered aircraft. After all, here was a well-respected scientist who believed in the
eventual attainment of mechanical flight and who was doing something about it.)
Consequently, there was a lull in Langley’s aeronautical work until Decem­
ber 1898. Then, motivated by the Spanish-American War, the War Department,
with the personal backing of President McKinley himself, invited Langley to
build a machine for passengers. It backed up its invitation with $50,000. Langley
accepted.
Departing from his earlier use of steam, Langley correctly decided that the
gasoline-fueled engine was the proper prime mover for aircraft. He first com­
missioned Stephan Balzer of New York to produce such an engine, but dissatis­
fied with the results, Langley eventually had his assistant, Charles Manly, re­
design the power plant. The resulting engine produced 52.4 hp and yet weighed
only 208 lb, a spectacular achievement for that time. Using a smaller, l .5-hp,
gasoline-fueled engine, Langley made a successful flight with a quarter-scale
model aircraft in June 1901, and then an even more successful flight of the model
powered by a 3.2-hp engine in August 1903.
Encouraged by this success, Langley stepped directly to the full-size air­
plane, top and side views of which are shown in Fig. 1.18. He mounted this
tandem-winged aircraft on a catapult in order to provide an assisted takeoff. In
turn, the airplane and catapult were placed on top of a houseboat on the Potomac
River (see Fig. 1.19). On October 7, 1903, with Manly at the controls, the airplane
was ready for its first attempt. The launching was given wide advance publicity,
and the press was present to watch what might be the first successful powered
flight in history. A photograph of the Aerodrome a moment after launch is shown
in Fig. 1.20. Here is the resulting report from the Washington Post the next day:
A few y a rd s fro m th e h o u se b o a t w ere th e b o a ts o f the re p o rte rs, w h o fo r th re e m o n th s
h a d b e en sta tio n e d at W id ew ater. T h e n e w sp a p e rm e n w a v e d th e ir h a n d s. M anly
lo o k e d d o w n a n d sm ile d . T h e n h is face h a rd e n e d as he b ra c e d h im s e lf fo r the flight,
w h ic h m ig h t h a v e in sto re fo r him fa m e o r d e ath . T h e p ro p e lle r w h e els, a fo o t fro m
h is h e a d , w h irre d a ro u n d h im o n e th o u sa n d tim e s to th e m in u te . A m an fo rw a rd fired
tw o sk y ro c k e ts. T h e re c a m e an a n sw e rin g “to o t, to o t,” fro m th e tu g s. A m ec h an ic
sto o p e d , cut th e c a b le h o ld in g th e c a ta p u lt; th ere w as a ro a rin g , g rin d in g n o ise — a n d
the L an g ley a irs h ip tu m b le d o v e r the e d g e o f th e h o u se b o a t and d isa p p e a re d in the
riv er, six tee n fe e t below . It sim p ly slid in to th e w a te r lik e a h a n d fu l o f m o r ta r .. . .
Manly was unhurt. Langley believed the airplane was fouled by the launching
mechanism, and he tried again on December 8, 1903. Figure 1.21, a photograph
taken moments after launch, shows the rear wings in total collapse and the Aero­
drome going through a 90' angle of attack. Again, the Aerodrome fell into the
river, and again Manly was fished out, unhurt. It is not entirely certain what hap­
pened this time; again the fouling of the catapult was blamed, but some experts
maintain that the tail boom cracked due to structural weakness. (A recent struc­
tural analysis by Dr. Howard Wolko, now retired from the National Air and
Space Museum, has proven that the large Langley Aerodrome was clearly
C H A P T E R 1 The First Aeronautical Engineers
F igu re 1.18 D raw ing o f the L angley full-size Aerodrom e.
(Source: National Air and Space Museum.)
structurally unsound.) At any rate, that was the end of Langley’s attempts. The
War Department gave up, stating that “we are still far from the ultimate goal (of
human flight).” Members of Congress and the press leveled vicious and unjusti­
fied attacks on Langley (human flight was still looked upon with much derision
by most people). In the face of this ridicule, Langley retired from the aeronauti­
cal scene. He died on February 27, 1906, a man in despair.
In contrast to Chanute and the Wright brothers, Langley was a chauffeur.
Most modern experts feel that his Aerodrome would not have been capable of
sustained, equilibrium flight, had it been successfully launched. Langley made
1.7
Aeronautics Comes to America
F ig u re 1.19 L angley’s full-size A erodrom e on the houseboat launching catapult, 1903.
(Source: National Air and Space Museum.)
v
\* V |
F ig u re 1.20 L an g ley ’s first launch o f the full-size A erodrom e, O ctober 7, 1903.
(Source: National Air and Space Museum.)
chapter
1 The First Aeronautical Engineers
F ig u re 1.21 L an g ley ’s second launch o f the full-size Aerodrome, D ecem ber 8, 1903.
(Source: National A ir and Space Museum.)
no experiments with gliders with passengers to get the feel of the air. He ignored
completely the important aspects of flight control. He attempted to launch
Manly into the air on a powered machine without M anly’s having one second of
flight experience. Nevertheless, Langley’s aeronautical work was of some im­
portance because he lent the power of his respected technical reputation to the
1.8
Wilbur (1867-1912) and Orville (1871 -1948) Wright— Inventors of the First Practical Airplane
cause of mechanical flight, and his Aerodromes were to provide encouragement
to others.
Nine days after Langley’s second failure, the Wright Flyer I rose from the
sands of Kill Devil Hills.
1.8 WILBUR (1867-1912) AND ORVILLE
(1871-1948) WRIGHT— INVENTORS
OF THE FIRST PRACTICAL AIRPLANE
The scene now shifts to the Wright brothers, the premier aeronautical engineers
of history. Only George Cayley may be considered comparable. In Sec. 1.1, it
was stated that the time was ripe for the attainment of powered flight at the
beginning of the 20th century. The ensuing sections then provided numerous his­
torical brushstrokes to emphasize this statement. Thus, the Wright brothers drew
on an existing heritage that is part of every aerospace engineer today.
Wilbur Wright was born on April 16, 1867 (two years after the Civil War),
on a small farm in Millville, Indiana. Four years later, Orville was born on
August 19, 1871, in Dayton, Ohio. The Wrights were descendants of an old
Massachusetts family, and their father was a bishop of the United Brethren
Church. The two brothers benefited greatly from the intellectual atmosphere of
their family. Their mother was three months short of a college degree. She had
considerable mechanical ability, enhanced by spending time in her father’s car­
riage shop. She later designed and built simple household appliances and made
toys for her children. In the words of Tom Crouch, the definitive biographer of
the Wright brothers: “When the boys wanted mechanical advice or assistance,
they came to their mother.” Their father. Crouch says, “was one of those men
who had difficulty driving a nail straight.” (See T. Crouch, The Bishop’s Boys,
Norton, New York, 1989.) Interestingly enough, neither Wilbur nor Orville offi­
cially received a high-school diploma; Wilbur did not bother to go to the com­
mencement services, and Orville took a special series of courses in his junior
year that did not lead to a prescribed degree, and he did not attend his senior year.
Afterward, the brothers immediately sampled the business world. In 1889, they
first published a weekly four-page newspaper on a printing press of their own de­
sign. However, Orville had talent as a prize-winning cyclist, and this prompted
the brothers to set up a bicycle sales and repair shop in Dayton in 1892. Three
years later they began to manufacture their own bicycle designs, using home­
made tools. These enterprises were profitable and helped to provide the financial
resources for their later work in aeronautics.
In 1896, Otto Lilienthal was accidently killed during a glider flight (see
Sec. 1.5). In the wake of the publicity, the Wright brothers’ interest in aviation,
which had been apparent since childhood, was given much impetus. Wilbur and
Orville had been following Lilienthal’s progress intently; recall that Lilienthal’s
gliders were shown in flight by photographs distributed around the world. In
chapter
1 The First Aeronautical Engineers
fact, an article on Lilienthal in an issue of M cClure’s Magazine in 1894 was
apparently the first to trigger W ilbur’s mature interest; but it was not until 1896
that Wilbur really became a serious thinker about human flight.
Like several pioneers before him, Wilbur took up the study of bird flight as
a guide on the path toward mechanical flight. This led him to conclude in 1899
that birds “regain their lateral balance when partly overturned by a gust of wind,
by a torsion of the tips of the wings.” Thus emerged one of the most important
developments in aviation history: the use of wing twist to control airplanes in
lateral (rolling) motion. Ailerons are used on modern airplanes for this purpose,
but the idea is the same. (The aerodynamic fundamentals associated with wing
twist or ailerons are discussed in Chaps. 5 and 7.) In 1903, Chanute, in describ­
ing the work of the Wright brothers, coined the term wing warping for this idea,
a term that was to become accepted but which was to cause some legal confu­
sion later.
Anxious to pursue and experiment with the concept of wing warping, Wilbur
wrote to the Smithsonian Institution in May 1899 for papers and books on aero­
nautics; in turn he received a brief bibliography of flying, including works by
Chanute and Langley. Most important among these was Chanute’s Progress in
Flying Machines (see Sec. 1.7). Also at this time, Orville became as enthusiastic
as his brother, and they both digested all the aeronautical literature they could
find. This led to their first aircraft, a biplane kite with a wingspan of 5 ft, in
August 1899. This machine was designed to test the concept of wing warping,
which was accomplished by means of four controlling strings from the ground.
The concept worked!
Encouraged by this success, Wilbur wrote to Chanute in 1900, informing
him of their initial, but fruitful, progress. This letter began a close friendship be­
tween the Wright brothers and Chanute, a friendship that was to benefit both par­
ties in the future. Also, following the true airman philosophy, the Wrights were
convinced they had to gain experience in the air before applying power to an air­
craft. By writing to the U.S. Weather Bureau, they found an ideal spot for glider
experiments, the area around Kitty Hawk, North Carolina, where there were
strong and constant winds. A full-size biplane glider was ready by September
1900 and was flown in October of that year at Kitty Hawk. Figure 1.22 shows a
photograph of the Wrights’ number 1 glider. It had a 17-ft wingspan and a hori­
zontal elevator in front of the wings and was usually flown on strings from the
ground; only a few brief piloted flights were made.
With some success behind them, Wilbur and Orville proceeded to build their
number 2 glider (see Fig. 1.23). Moving their base of operations to Kill Devil
Hills, 4 mi south of Kitty Hawk, they tested number 2 during July and August of
1901. These were mostly manned flights, with Wilbur lying prone on the bottom
wing, facing into the wind, as shown in Fig. 1.23. (Through 1901, Wilbur did
w h a t little fly in g w a s a c c o m p lis h e d ; O rv ille flow fo r th e fir3t tim e a y e a r la te r.)
This new glider was somewhat larger, with a 22-ft wingspan. As with all Wright
machines, it had a horizontal elevator in front of the wings. The Wrights felt that
a forward elevator would, among other functions, protect them from the type of
fatal nosedive that killed Lilienthal.
1.8
Wilbur (1867-1912) and Orville (1871-1948) Wright— Inventors of the First Practical Airplane
F ig u re 1.22 T he W right bro th ers’ num ber I glid er at K itty H aw k, N orth
C arolina, 1900.
(Source: National Air and Space Museum.)
Figure 1.23 T he W right bro th ers’ num ber 2 glider at Kill Devil Hills, 1901.
(Source: National A ir and Space Museum.)
During these July and August test flights, Octave Chanute visited the
Wrights’ camp. He was much impressed by what he saw. This led to Chanute’s
invitation to Wilbur to give a lecture in Chicago. In giving this paper on Septem­
ber 18, 1901, Wilbur laid bare their experiences, including the design of their
gliders and the concept of wing warping. Chanute described W ilbur’s presenta­
tion as “a devilish good paper which will be extensively quoted.” Chanute, as
usual, was serving his very useful function as a collector and disseminator of
aeronautical data.
However, the Wrights were not close to being satisfied with their results.
When they returned to Dayton after their 1901 tests with the number 2 glider,
both brothers began to suspect the existing data that appeared in the aeronautical
literature. To this date, they had faithfully relied upon detailed aerodynamic in­
formation generated by Lilienthal and Langley. Now they wondered about its
chapter
1 The First Aeronautical Engineers
accuracy. Wilbur wrote that “having set out with absolute faith in the existing sci­
entific data, we were driven to doubt one thing after another, until finally, after
two years of experiment, we cast it all aside, and decided to rely entirely upon our
own investigations.” And investigate they did! Between September 1901 and
August 1902, the Wrights undertook a major program of aeronautical research.
They built a wind tunnel (see Chap. 4) in their bicycle shop in Dayton and tested
more than 200 different airfoil shapes. They designed a force balance to measure
accurately the lift and drag. This period of research was a high-water mark in
early aviation development. The Wrights learned, and with them ultimately so
did the world. This sense of learning and achievement by the brothers is apparent
simply from reading through The Papers o f Wilbur and Orville Wright (l 953),
edited by Marvin W. McFarland. The aeronautical research carried out during
this period ultimately led to their number 3 glider, which was flown in 1902. It
was so successful that Orville wrote that “our tables of air pressure which we
made in our wind tunnel would enable us to calculate in advance the performance
of a machine.” Here is the first example in history of the major impact of wind
tunnel testing on the flight development of a given machine, an impact that has
been repeated for all major airplanes of the 20th century. [Very recently, it has
been shown by Anderson in A History o f Aerodynamics and Its Impact on Flying
Machines (Cambridge University Press, 1997) that Lilienthal’s data were rea­
sonable, but the Wrights misinterpreted them. Applying the data incorrectly, the
Wrights obtained incorrect results for their 1900 and 1901 gliders. However, this
is irrelevant because the Wrights went on to discover the correct results.]
The number 3 glider was a classic. It was constructed during August and
September of 1902. It first flew at Kill Devil Hills on September 20, 1902. It was
a biplane glider with a 32-ft 1-in wingspan, the largest of the Wright gliders to
date. This number 3 glider is shown in Fig. 1.24. Note that, after several modifi­
cations, the Wrights added a vertical rudder behind the wings. This rudder was
movable, and when connected to move in unison with the wing warping, it en­
abled the number 3 glider to make a smooth, banked turn. This combined use of
rudder with wing warping (or later, ailerons) was another major contribution of
the Wright brothers to flight control in particular, and aeronautics in general.
So the Wrights now had the most practical and successful glider in history.
During 1902, they made more than 1000 perfect flights. They set a distance
record of 622.5 ft and a duration record of 26 s. In the process, both Wilbur and
Orville became highly skilled and proficient pilots, something that would later be
envied worldwide.
Powered flight was now just at their fingertips, and the Wrights knew it!
Flushed with success, they returned to Dayton to face the last remaining prob­
lem: propulsion. As with Langley before them, they could find no commercial
engine that was suitable. So they designed and built their own during the winter
months of 1903. It produced 12 hp and weighed about 200 lb. Moreover, they
conducted their own research, which allowed them to design an effective pro­
peller. These accomplishments, which had eluded people for a century, gushed
forth from the Wright brothers like natural spring water.
1.8
Wilbur (1867-1912) and Orville (1871 -1948) Wright— Inventors of the First Practical Airplane
F igu re 1.24 T he W right b ro th ers’ n um ber 3 glider, 1902.
(Source: National Air and Space Museum.)
With all the major obstacles behind them, Wilbur and Orville built their
Wright Flyer I from scratch during the summer of 1903. It closely resembled the
number 3 glider but had a wingspan of 40 ft 4 in and used a double rudder behind
the wings and a double elevator in front of the wings. And of course, there was
the spectacular gasoline-fueled Wright engine, driving two pusher propellers by
means of bicycle-type chains. A three-view diagram and a photograph of the
Wright Flyer I are shown in Figs. 1.1 and 1.2, respectively.
From September 23 to 25, the machine was transported to Kill Devil Hills,
where the Wrights found their camp in some state of disrepair. Moreover, their
number 3 glider had been damaged over the winter months. They set about to
make repairs and afterward spent many weeks of practice with their number 3
glider. Finally, on December 12, everything was in readiness. However, this time
the elements interfered: Bad weather postponed the first test of the Wright Flyer I
until December 14. On that day, the Wrights called witnesses to the camp and
then flipped a coin to see who would be the first pilot. Wilbur won. The Wright
Flyer I began to move along the launching rail under its own power, picking up
flight speed. It lifted off the rail properly but suddenly went into a steep climb,
stalled, and thumped back to the ground. It was the first recorded case of pilot
error in powered flight: Wilbur admitted that he put on too much elevator and
brought the nose too high.
With minor repairs made, and with the weather again favorable, the Wright
Flyer I was again ready for flight on December 17. This time it was Orville’s turn
at the controls. The launching rail was again laid on level sand. A camera was
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1 The First Aeronautical Engineers
adjusted to take a picture of the machine as it reached the end of the rail. The en­
gine was put on full throttle, the holding rope was released, and the machine
began to move. The rest is history, as portrayed in the opening paragraphs of this
chapter.
One cannot read or write of this epoch-making event without experiencing
some of the excitement of the time. Wilbur Wright was 36 years old; Orville was
32. Between them, they had done what no one before them had accomplished.
By their persistent efforts, their detailed research, and their superb engineering,
the Wrights had made the world’s first successful heavier-than-air flight, satisfy­
ing all the necessary criteria laid down by responsible aviation historians. After
Orville’s first flight on that December 17, three more flights were made during
the morning, the last covering 852 ft and remaining in the air for 59 s. The world
of flight— and along with it the world of successful aeronautical engineering—
had been born!
It is interesting to note that even though the press was informed of these
events via Orville’s telegram to his father (see the introduction to this chapter),
virtually no notice appeared before the public; even the Dayton newspapers did
not herald the story. This is a testimonial to the widespread cynicism and disbe­
lief among the general public about flying. Recall that just nine days before,
Langley had failed dismally in full view of the public. In fact, it was not until
Amos I. Root observed the Wrights flying in 1904 and published his inspired ac­
count in a journal of which he was the editor, Gleanings in Bee Culture (Janu­
ary 1, 1905, issue), that the public had its first detailed account of the W rights’
success. However, the article had no impact.
The Wright brothers did not stop with the Wright Flyer I. In May 1904, their
second powered machine, the Wright Flyer II, was ready. This aircraft had a
smaller wing camber (airfoil curvature) and a more powerful and efficient en­
gine. In outward appearance, it was essentially like the 1903 machine. During
1904, more than 80 brief flights were made with the Wright Flyer II, all at a
90-acre field called Huffman Prairie, 8 mi east of Dayton. (Huffman Prairie still
exists today; it is on the huge Wright-Patterson Air Force Base, a massive aero­
space development center named in honor of the Wrights.) These tests included
the first circular flight— made by Wilbur on September 20. The longest flight
lasted 5 min 4 s, traversing more than 2 | mi.
More progress was made in 1905. The Wright Flyer HI was ready by June.
The wing area was slightly smaller than that of the Flyer II, the airfoil camber
was increased back to what it had been in 1903, the biplane elevator was made
larger and was placed farther in front of the wings, and the double rudder was
also larger and placed farther back behind the wings. New, improved propellers
were used. This machine, the Flyer III, was the first practical airplane in history.
It made more than 40 flights during 1905, the longest being 38 min 3 s and cov­
ering 24 mi. These flights were generally terminated only after gas was used up.
C. H. Gibbs-Smith writes about the Flyer III: “The description of this machine
as the world’s first practical powered aeroplane is justified by the sturdiness of its
structure, which withstood constant takeoffs and landings; its ability to bank,
1.8
Wilbur (1867-1912) and Orville (1871 -1948) Wright— Inventors of the First Practical Airplane
turn, and perform figures of eight; and its reliability in remaining airborne (with
no trouble) for over half an hour.”
Then the Wright brothers, who heretofore had been completely open about
their work, became secretive. They were not making any progress in convincing
the U.S. government to buy their airplane, but at the same time various people
and companies were beginning to make noises about copying the W rights’ de­
sign. A patent applied for by the Wrights in 1902 to cover their ideas of wing
warping combined with rudder action was not granted until 1906. So, between
October 16, 1905, and May 6, 1908, neither Wilbur nor Orville flew, nor did they
allow anybody to view their machines. However, their aeronautical engineering
did not stop. During this period, they built at least six new engines. They also de­
signed a new flying machine that was to become the standard Wright type A,
shown in Fig. 1.25. This airplane was similar to the Wright Flyer III, but it had a
40-hp engine and provided for two people seated upright between the wings. It
also represented the progressive improvement of a basically successful design, a
concept of airplane design carried out to present day.
The public and the Wright brothers finally had their meeting, and in a big
way, in 1908. The Wrights signed contracts with the U.S. Army in February
1908, and with a French company in March of the same year. After that, the
wraps were off. Wilbur traveled to France in May, picked up a crated type A that
had been waiting at Le Havre since July 1907, and completed the assembly in a
friend’s factory at Le Mans. With supreme confidence, he announced his first
public flight in advance— to take place on August 8, 1908. Aviation pioneers
from all over Europe, who had heard rumors about the Wrights’ successes since
1903, the press, and the general public all flocked to a small race course at
Hunaudieres, 5 mi south of Le Mans. On the appointed day, Wilbur took off,
made an impressive, circling flight for almost 2 min, and landed. It was like a rev­
olution. Aeronautics, which had been languishing in Europe since Lilienthal’s
death in 1896, was suddenly alive. The Frenchman Louis Blériot, soon to be­
come famous for being first to fly across the English Channel, exclaimed: “For us
in France and everywhere, a new era in mechanical flight has commenced— it is
marvelous.” The French press, after being skeptical for years of the Wrights’ sup­
posed accomplishments, called W ilbur’s flight “one of the most exciting specta­
cles ever presented in the history of applied science.” More deeply echoing the
despair of many would-be French aviators who were in a race with the Wrights
to be first with powered flight, Leon Delagrange said: “Well, we are beaten. We
just don’t exist.” Subsequently, Wilbur made 104 flights in France before the end
of 1908. The acclaim and honor due the Wright brothers since 1903 had finally
arrived.
Orville was experiencing similar success in the United States. On September
3, 1908, he began a series of demonstrations for the Army at Fort Myer, near
Washington, District of Columbia. Flying a type A machine, he made 10 flights,
the longest for 1 h 14 min, before September 17. On that day, Orville experienced
a propeller failure that ultimately caused the machine to crash, seriously injuring
himself and killing his passenger, Lt. Thomas E. Selfridge. This was the first crash
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1 The First Aeronautical Engineers
of a powered aircraft, but it did not deter either Orville or the Army. Orville made
a fast recovery and was back to flying in 1909; and the Army bought the airplane.
The public flights made by Wilbur in France in 1908 electrified aviators
in Europe. European airplane designers immediately adopted two of the most
important technical features of the Wright machine— lateral control and the
propeller. Prior to 1908, European flying-machine enthusiasts had no concept of
1.8
Wilbur (1867-1912) and Orville (1871 -1948) Wright— Inventors of the First Practical Airplane
the importance of lateral control (rolling of the airplane— see Sec. 7.1) and cer­
tainly no mechanical mechanism to achieve it; the Wrights achieved lateral con­
trol by their innovative concept of wing warping. By 1909, however, the French­
man Henri Farman designed a biplane named the H en ri F a rm a n III that included
flaplike ailerons at the trailing edge near the wing tips; ailerons quickly became
the favored mechanical means for lateral control, continuing to the present day.
Similarly, the European designers were quick to adopt the long, slender shape of
the Wrights' propellers, quite different from the wide paddlelike shapes then in
use with low propeller efficiencies (defined in Sec. 6.6.1) on the order of 40 to
50 percent. In 1909, the efficiency of the W rights’ propeller was measured by an
engineer in Berlin to be a stunning 76 percent. Recent wind tunnel experiments
at the NASA Langley Research Center (carried out by researchers from Old
Dominion University in 2002) indicate an even more impressive 84 percent effi­
ciency for the W rights’ propeller. These two technical features— the appreciation
for, and a mechanical means to achieve, lateral control, and the design of a highly
efficient propeller— are the two most important technical legacies left by the
Wrights to future airplanes, and European designers quickly seized upon them.
(See the book by Anderson, T he A irp la n e : A H isto ry o f Its Technology, American
Institute of Aeronautics and Astronautics, 2002, for more details.)
The accomplishments of the Wright brothers were monumental. Their zenith
occurred during the years 1908 to 1910; after that, European aeronautics quickly
caught up and went ahead in the technological race. The main reason for this was
that all the Wrights’ machines, from the first gliders, were statically unstable (see
Chap. 7). This meant that the W rights’ airplanes would not fly “by themselves” ;
rather, they had to be constantly, every instant, controlled by the pilot. In con­
trast, European inventors believed in inherently stable aircraft. After their lessons
in flight control from Wilbur in 1908, workers in France and England moved
quickly to develop controllable, but stable, airplanes. These were basically safer
and easier to fly. The concept of static stability has carried over to virtually all
airplane designs through the present century. (It is interesting to note that the new
designs for military fighters, such as the Lockheed-Martin F-22, are statically u n ­
stable, which represents a return to the Wrights’ design philosophy. However,
unlike the W right F lyers, these new aircraft are flown constantly, every moment,
by electrical means, by the new “fly-by-wire” concept.)
To round out the story of the Wright brothers, Wilbur died in an untimely
fashion of typhoid fever on May 30, 1912. In a fitting epitaph, his father said:
“This morning, at 3:15 Wilbur passed away, aged 45 years, 1 month, and 14 days.
A short life full of consequences. An unfailing intellect, imperturbable temper,
great self-reliance and as great modesty. Seeing the right clearly, pursuing it
steadily, he lived and died.”
Orville lived on until January 30, 1948. During World War I, he was com­
missioned a major in the Signal Corps Aviation Service. Although he sold all his
interest in the Wright company and “retired” in 1915, he afterward performed re­
search in his own shop. In 1920, he invented the split flap for wings, and he con­
tinued to be productive for many years.
chapter
1 The First Aeronautical Engineers
As a final footnote to this story of two great men, there occurred a dispute
between Orville and the Smithsonian Institution concerning the proper historical
claims on powered flight. As a result, Orville sent the historic W right F ly e r I, the
original, to the Science Museum in London in 1928. It resided there, through the
bombs of World War II, until 1948, when the museum sent it to the Smithsonian.
It is now part of the National Air and Space Museum and occupies a central po­
sition in the gallery.
1.9 THE AERONAUTICAL TRIANGLE— LANGLEY,
THE WRIGHTS, AND GLENN CURTISS
In 1903— a milestone year for the Wright brothers, with their first successful
powered flight— Orville and Wilbur faced serious competition from Samuel P.
Langley. As portrayed in Sec. 1.7, Langley was the secretary of the Smithsonian
Institution and was one of the most respected scientists in the United States at
that time. Beginning in 1886, Langley mounted an intensive aerodynamic
research and development program, bringing to bear the resources of the
Smithsonian and later the War Department. He carried out this program with a
dedicated zeal that matched the fervor that the Wrights themselves demonstrated
later. Langley’s efforts culminated in the full-scale Aerodrome shown in
Figs. 1.18, 1.19, and 1.20. In October 1903, this Aerodrome was ready for its first
attempted flight, in the full glare of publicity in the national press.
The Wright brothers were fully aware of Langley’s progress. During their
preparations with the Wright Flyer at Kill Devil Hills in the summer and fall of
1903, Orville and Wilbur kept in touch with Langley’s progress via the newspa­
pers. They felt this competition keenly, and the correspondence of the Wright
brothers at this time indicates an uneasiness that Langley might become the first
to successfully achieve powered flight, before they would have a chance to test
the Wright Flyer. In contrast, Langley felt no competition at all from the Wrights.
Although the aeronautical activity of the Wright brothers was generally known
throughout the small circle of aviation enthusiasts in the United States and
Europe— thanks mainly to reports on their work by Octave Chanute— this activ­
ity was not taken seriously. At the time of Langley’s first attempted flight on
October 7, 1903, there is no recorded evidence that Langley was even aware of
the Wrights’ powered machine sitting on the sand dunes of Kill Devil Hills, and
certainly no appreciation by Langley of the degree of aeronautical sophistication
achieved by the Wrights. As it turned out, as was related in Sec. 1.7, Langley’s
attempts at manned powered flight, first on October 7 and again on December 8,
resulted in total failure. A photograph of Langley’s Aerodrome, lying severely
damaged in the Potomac River on October 7, is shown in Fig. 1.26. In hindsight,
the Wrights had nothing to fear from competition with Langley.
Such was not the case in their competition with another aviation pioneer—
Glenn H. Curtiss— beginning five years later. In 1908— another milestone year
for the Wrights, with their glorious first public flights in France and the United
States— Orville and Wilbur faced a serious challenge and competition from
1.9
The Aeronautical Triangle— Langley, the Wrights, and Glenn Curtiss
F igu re 1.26 L angley’s Aerodrome resting in the Potom ac R iver after its first
unsuccessful flight on O ctober 7, 1903. C harles M anly, the pilot, w as fished out o f the
river, fortunately unhurt.
Curtiss, which was to lead to acrimony and a flurry of lawsuits that left a smudge
on the Wrights’ image and resulted in a general inhibition of the development of
early aviation in the United States. By 1910, the name of Glenn Curtiss was as
well known throughout the world as Orville and Wilbur Wright, and indeed,
Curtiss-built airplanes were more popular and easier to fly than those produced
by the Wrights. How did these circumstances arise? Who was Glenn Curtiss, and
what was his relationship with the Wrights? What impact did Curtiss have on the
early development of aviation, and how did his work compare and intermesh
with that of Langley and that of the Wrights? Indeed, the historical development
of aviation in the United States can be compared to a triangle, with the Wrights
on one apex, Langley at another, and Curtiss at the third. This “aeronautical tri­
angle” is shown in Fig. 1.27. What was the nature of this triangular relationship?
These and other questions are addressed in this section. They make a fitting con­
clusion to the overall early historical development of aeronautical engineering as
portrayed in this chapter.
Let us first look at Glenn Curtiss, the man. Curtiss was born in
Hammondsport, New York, on May 21, 1878. Hammondsport at that time was a
small town— population less than 1000— bordering on Keuka Lake, one of the
Finger Lakes in upstate New York. (Later, Curtiss was to make good use of
Keuka Lake for the development of amphibian aircraft— one of his hallmarks.)
The son of a harness maker who died when Curtiss was five years old, Curtiss
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1 The First Aeronautical Engineers
Wilbur (left) and Orville (right) Wright
Samuel P. Langley
Glenn H. Curtiss
F igu re 1.27 T he “aeronautical tria n g le,” a relationship that dom inated the early
developm ent o f aeronautics in the U nited States during the period from 1886 to 1916.
was raised by his mother and grandmother. Their modest financial support came
from a small vineyard that grew in their front yard. His formal education ceased
with the eighth grade, after which he moved to Rochester, where he went to work
for Eastman Dry Plate and Film Company (later to become Kodak), stenciling
numbers on the paper backing of film. In 1900, he returned to Hammondsport,
where he took over a bicycle repair shop (shades of the Wright brothers). At this
time, Glenn Curtiss began to show a passion that would consume him for his
lifetime— a passion for speed. He became active in bicycle racing and quickly
1.9
The Aeronautical Triangle— Langley, the Wrights, and Glenn Curtiss
earned a reputation as a winner. In 19 0 1, he incorporated an engine on his
bicycles and became an avid motorcycle racer. By 1902, his fame was spreading,
and he was receiving numerous orders for motorcycles with engines of his own
design. By 1903, Curtiss had established a motorcycle factory at Hammondsport,
and he was designing and building the best (highest horsepower-to-weight ratio)
engines available anywhere. In January 1904, at Ormond Beach, Florida, Curtiss
established a new world’s speed record for a ground vehicle— 67 mi/h over a
10-mi straightaway— a record that was to stand for seven years.
Curtiss “backed into” aviation. In the summer of 1904, he received an order
from Thomas Baldwin, a California balloonist, for a two-cylinder engine.
Baldwin was developing a powered balloon— a dirigible. The Baldwin dirigi­
bles, with the highly successful Curtiss engines, soon became famous around the
country. In 1906, Baldwin moved his manufacturing facilities to Hammondsport,
to be close to the source of his engines. A lifelong friendship and cooperation
developed between Baldwin and Curtiss and provided Curtiss with his first
experience in aviation— as a pilot of some of Baldwin’s powered balloons.
In August 1906, Baldwin traveled to the Dayton Fair in Ohio for a week of
dirigible flight demonstrations; he brought Curtiss along to personally maintain
the engines. The Wright brothers also attended the fair— specifically to watch
Thomas Baldwin perform. They even lent a hand in retrieving the dirigible when
it strayed too far afield. This was the first face-to-face encounter between Curtiss
and the Wrights. During that week, Baldwin and Curtiss visited the Wrights at
the brothers’ bicycle shop and entered into long discussions on powered flight.
Recall from Sec. 1.8 that the Wrights had discontinued flying one year earlier,
and at the time of their meeting with Curtiss, Orville and Wilbur were actively
trying to interest the United States, as well as England and France, in buying
their airplane. The Wrights had become very secretive about their airplane and
allowed no one to view it. Curtiss and Baldwin were no exceptions. However,
that week in Dayton, the Wrights were relatively free with Curtiss, giving him in­
formation and technical suggestions about powered flight. Years later, these con­
versations became the crux of the Wrights’ claim that Curtiss had stolen some of
their ideas and used them for his own gain.
This claim was probably not entirely unjustified, for by that time Curtiss
had a vested interest in powered flight; a few months earlier he had supplied
Alexander Graham Bell with a 15-hp motor to be used in propeller experiments,
looking toward eventual application to a manned, heavier-than-air, powered air­
craft. The connection between Bell and Curtiss is important. Bell, renowned as
the inventor of the telephone, had an intense interest in powered flight. He was a
close personal friend of Samuel Langley and, indeed, was present for Langley’s
successful unmanned Aerodrome flights in 1896. By the time Langley died in
1906, Bell was actively carrying out kite experiments and was testing air pro­
pellers on a catamaran at his Nova Scotia coastal home. In the summer of 1907,
Bell formed the Aerial Experiment Association, a group of five men whose offi­
cially avowed purpose was simply “to get into the air.” The Aerial Experiment
Association (AEA) consisted of Bell himself, Douglas McCurdy (son of Bell’s
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1 The First Aeronautical Engineers
personal secretary, photographer, and very close family friend), Frederick
W. Baldwin (a freshly graduated mechanical engineer from Toronto and close
friend of McCurdy), Thomas E. Selfridge (an Army lieutenant with an extensive
engineering knowledge of aeronautics), and Glenn Curtiss. The importance of
Curtiss to the AEA is attested by the stipends that Bell paid to each member of
the association— Curtiss was paid five times more than the others. Bell had asked
Curtiss to join the association because of Curtiss’s excellent engine design and
superb mechanical ability. Curtiss was soon doing much more than just design­
ing engines. The plan of the AEA was to conduct intensive research and devel­
opment on powered flight and to build five airplanes— one for each member. The
first aircraft, the Red Wing, was constructed by the AEA with Selfridge as the
chief designer. On March 12, 1908, the Red Wing was flown at Hammondsport
for the first time, with Baldwin at the controls. It covered a distance of 318 ft and
was billed as “the first public flight” in the United States.
Recall that the tremendous success of the Wright brothers from 1903 to 1905
was not known by the general public, mainly because of indifference in the press
as well as the W rights’ growing tendency to be secretive about their airplane de­
sign until they could sell an airplane to the U.S. government. However, the
W rights’ growing apprehension about the publicized activities of the AEA is re­
flected in a letter from Wilbur to the editor of the Scientific American after the
flight of the Red Wing. In this letter, Wilbur states
In 1904 a n d 1905, w e w e re fly in g e v e ry fe w d a y s in a field a lo n g s id e th e m ain w a g o n
ro a d a n d e le c tric tro lle y lin e fro m D a y to n to S p rin g field , a n d h u n d re d s o f tra v e le rs
a n d in h a b ita n ts saw th e m a c h in e in flight. A n y o n e w h o w ish e d c o u ld look. W e
m ere ly d id n o t a d v e rtis e th e flig h ts in th e n e w sp a p e rs.
On March 17, 1908, the second flight of the Red Wing resulted in a crash that
severely damaged the aircraft. Work on the Red Wing was subsequently aban­
doned in lieu of a new design of the AEA, the White Wing, with Baldwin as the
chief designer. Members of the AEA were acutely aware of the W rights’ patent
on wing warping for lateral control, and Bell was particularly sensitive to mak­
ing certain that his association did not infringe upon this patent. Therefore, in­
stead of using wing warping, the White Wing utilized triangular movable surfaces
that extended beyond the wing tips of both wings of the biplane. Beginning on
May 18, 1908, the White Wing successfully made a series of flights piloted by
various members of the AEA. One of these flights, with Glenn Curtiss at the con­
trols, was reported by Selfridge to the Associated Press as follows:
G . H . C u rtis s o f th e C u rtis s M a n u fa c tu rin g C o m p a n y m a d e a flig h t o f 3 3 9 y a rd s in
tw o ju m p s in B a ld w in ’s W h ite W in g th is a fte rn o o n at 6 :4 7 pm . In th e first ju m p he
c o v e re d 205 y a rd s th en to u c h e d , ro se im m e d ia te ly a n d flew 134 y a rd s fu rth e r w hen
th e flig h t e n d e d o n the e d g e o f a p lo u g h e d field. T h e m a c h in e w as in p e rfe c t c o n tro l
at all tim e s a n d w a s ste e re d first to th e rig h t a n d th en to the left b e fo re lan d in g . T h e
3 3 9 y a rd s w a s c o v e re d in 19 se c o n d s o r 37 m ile s p e r hour.
Two days later, with an inexperienced McCurdy at the controls, the White Wing
crashed and never flew again.
1.9
The Aeronautical Triangle— Langley, the Wrights, and Glenn Curtiss
However, by this time, the Wright brothers’ apprehension about the AEA
was growing into bitterness toward its members. Wilbur and Orville genuinely
felt that the AEA had pirated their ideas and was going to use them for commer­
cial gain. For example, on June 7, 1908, Orville wrote to Wilbur (who was in
France preparing for his spectacular first public flights that summer at Le
Mans— see Sec. 1.8): “I see by one of the papers that the Bell outfit is offering
Red Wings for sale at $5,000 each. They have some nerve.” On June 28, he re­
lated to Wilbur: “Curtiss et al. are using our patents, I understand, and are now
offering machines for sale at $5,000 each, according to the Scientific American.
They have got good cheek.”
The strained relations between the Wrights and the AEA— particularly
Curtiss— were exacerbated on July 4, 1908, when the AEA achieved its crown­
ing success. A new airplane had been constructed— the June Bug— with Glenn
Curtiss as the chief designer. In the previous year, the Scientific American had of­
fered a trophy, through the Aero Club of America, worth more than $3000 to the
first aviator making a straight flight of l km (3281 ft). On Independence Day in
1908, at Hammondsport, New York, Glenn Curtiss at the controls of his June
Bug was ready for an attempt at the trophy. A delegation of 22 members of the
Aero Club was present, and the official starter was none other than Charles
Manly, Langley’s dedicated assistant and pilot of the ill-fated Aerodrome (see
Sec. 1.7 and Fig. 1.26). Late in the day, at 7:30 p m , Curtiss took off and in l min
40 s had covered a distance of more than l mi, easily winning the Scientific
American prize. A photograph of the June Bug during this historic flight is shown
in Fig. 1.28.
F ig u re 1.28 G lenn C urtiss flying Ju n e B ug on July 4, 1908, on his w ay to the Scientific
A m erican prize for the first public flight o f g reater than 1 km.
chapter
1 The First Aeronautical Engineers
The Wright brothers could have easily won the Scientific American prize
long before Curtiss; they simply chose not to. Indeed, the publisher of the Scien­
tific American, Charles A. Munn, wrote to Orville on June 4, inviting him to
make the first attempt at the trophy, offering to delay Curtiss’s request for an at­
tempt. On June 30, the Wrights responded negatively— they were too involved
with preparations for their upcoming flight trials in France and at Fort Myer in
the United States. However, Curtiss’s success galvanized the Wrights’ opposi­
tion. Remembering their earlier conversations with Curtiss in 1906, Orville
wrote to Wilbur on July 19;
I had been thinking of writing to Curtiss. I also intended to call attention of the Sci­
entific American to the fact that the Curtiss machine was a poor copy of ours; that we
had furnished them the information as to how our older machines were constructed,
and that they had followed this construction very closely, but have failed to mention
the fact in any of their writings.
Curtiss’s publicity in July was totally eclipsed by the stunning success of
Wilbur during his public flights in France beginning August 8, 1908, and by
Orville’s Army trials at Fort Myer beginning on September 3, 1908. During the
trials at Fort Myer, the relationship between the Wrights and the AEA took an
ironic twist. One member of the evaluation board assigned by the Army to ob­
serve Orville’s flights was Lt. Thomas Selfridge. Selfridge had been officially de­
tailed to the AEA by the Army for a year and was now back at his duties of being
the Arm y’s main aeronautical expert. As part of the official evaluation, Orville
was required to take Selfridge on a flight as a passenger. During this flight, on
September 17, one propeller blade cracked and changed its shape, thus losing
thrust. This imbalanced the second propeller, which cut a control cable to the tail.
The cable subsequently wrapped around the propeller and snapped it off. The
Wright type A went out of control and crashed. Selfridge was killed, and Orville
was severely injured; he was in the hospital for 1\ months. For the rest of his life,
Orville would walk with a limp as a result of this accident. Badly shaken by
Selfridge’s death, and somewhat overtaken by the rapid growth of aviation
after the events of 1908, the Aerial Experiment Association dissolved itself on
March 31, 1909. In the written words of Alexander Graham Bell, “The A.E. A. is
now a thing of the past. It has made its mark upon the history of aviation and its
work will live."
After this, Glenn Curtiss struck out in the aviation world on his own. Form­
ing an aircraft factory at Hammondsport, Curtiss designed and built a new air­
plane, improved over the June Bug and named the Golden Flyer. In August 1909,
a massive air show was held in Reims, France, attracting huge crowds and the
crown princes of Europe. For the first time in history, the Gordon Bennett trophy
was offered for the fastest flight. Glenn Curtiss won this trophy with his Golden
Flyer, averaging a speed of 75.7 km/h (47.09 mi/h) over a 20-km course and de­
feating a number of pilots flying the W rights’ airplanes. This launched Curtiss on
a meteoric career as a daredevil pilot and a successful airplane manufacturer. His
motorcycle factory at Hammondsport was converted entirely to the manufacture
1.9
The Aeronautical Triangle— Langley, the Wrights, and Glenn Curtiss
of airplanes. His airplanes were popular with other pilots of that day because
they were statically stable and hence easier and safer to fly than the Wrights’ air­
planes, which had been intentionally designed by the Wright brothers to be stat­
ically unstable (see Chap. 7). By 19 10, aviation circles and the general public
held Curtiss and the Wrights in essentially equal esteem. At the lower right of
Fig. 1.27 is a photograph of Curtiss at this time; the propeller ornament in his cap
was a good luck charm which he took on his flights. By I9l l, a Curtiss airplane
had taken off from and landed on a ship. Also in that year, Curtiss developed the
first successful seaplanes and forged a lasting relationship with the U.S. Navy. In
June 19 11, the Aero Club of America issued its first official pilot’s license to
Curtiss in view of the fact that he had made the first public flight in the United
States, an honor which otherwise would have gone to the Wrights.
In September 1909, the Wright brothers filed suit against Curtiss for patent
infringements. They argued that their wing warping patent of 1906, liberally in­
terpreted, covered all forms of lateral control, including the ailerons used by
Curtiss. This triggered five years of intensive legal maneuvering, which dissi­
pated much of the energies of all the parties. Curtiss was not alone in this regard.
The Wrights brought suit against a number of fledgling airplane designers during
this period, both in the United States and in Europe. Such litigation consumed
W ilbur’s attention, in particular, and effectively removed him from being a pro­
ductive worker toward technical aeronautical improvements. It is generally
agreed by aviation historians that this was not the Wrights’ finest hour. Their
legal actions not only hurt their own design and manufacturing efforts, but also
effectively discouraged the early development of aeronautics by others, particu­
larly in the United States. (It is quite clear that when World War I began in 1914,
the United States— birthplace of aviation— was far behind Europe in aviation
technology.) Finally, in January 1914, the courts ruled in favor of the Wrights,
and Curtiss was forced to pay royalties to the Wright family. (By this time,
Wilbur was dead, having succumbed to typhoid fever in 1912.)
In defense of the Wright brothers, their actions against Curtiss grew from a
genuine belief on their part that Curtiss had wronged them and had consciously
stolen their ideas, which Curtiss had subsequently parlayed into massive eco­
nomic gains. This went strongly against the grain of the Wrights’ staunchly ethi­
cal upbringing. In contrast, Curtiss bent over backward to avoid infringing on the
letter of the Wrights' patent, and there is numerous evidence that Curtiss was
consistently trying to mend relations with the Wrights. It is this author’s opinion
that both sides became entangled in a complicated course of events that followed
those heady days after 1908, when aviation burst on the world scene, and that
neither Curtiss nor the Wrights should be totally faulted for their actions. These
events simply go down in history as a less-than-glorious, but nevertheless im­
portant, chapter in the early development of aviation.
An important postscript should be added here regarding the triangular rela­
tionship between Langley, the Wrights, and Curtiss, as shown in Fig. 1.27. In
Secs. 1.7 and 1.8, we have already seen the relationship between Langley and the
Wrights and the circumstances leading up to the race for the first flight in 1903.
C H A P T E R 1 The First Aeronautical Engineers
This constitutes side A in Fig. 1.27. In this section, we have seen the strong con­
nection between Curtiss and the work of Langley, via Alexander Graham Bell—
a close friend and follower of Langley and creator of the Aerial Experiment As­
sociation, which gave Curtiss a start in aviation. We have even noted that Charles
Manly, Langley’s assistant, was the official starter for Curtiss’s successful com­
petition for the Scientific American trophy. Such relationships form side B of the
triangle in Fig. 1.27. Finally, we have seen the relationship, although somewhat
acrimonious, between the Wrights and Curtiss, which forms side C in Fig. 1.27.
In 1914, an event occurred that simultaneously involved all three sides of the
triangle in Fig. 1.27. When the Langley Aerodrome failed for the second time in
1903 (see Fig. 1.26), the wreckage was simply stored away in an unused room in
the back of the Smithsonian Institution. When Langley died in 1906, he was re­
placed as secretary of the Smithsonian by Dr. Charles D. Walcott. Over the en­
suing years, Secretary Walcott felt that the Langley Aerodrome should be given
a third chance. Finally, in 1914, the Smithsonian awarded a grant of $2000 for
the repair and flight of the Langley Aerodrome to none other than Glenn Curtiss.
The Aerodrome was shipped to Curtiss’s factory in Hammondsport, where not
only was it repaired, but also 93 separate technical modifications were made,
aerodynamically, structurally, and to the engine. For help during this restoration
and modification, Curtiss hired Charles Manly. Curtiss added pontoons to the
Langley Aerodrome and on May 28, 1914, personally flew the modified aircraft
for a distance of 150 ft over Keuka Lake. Figure 1.29 shows a photograph of the
Langley Aerodrome in graceful flight over the waters of the lake. Later, the
F ig u re 1.29 T he m odified L angley Aerodrome in flight o ver K euka L ake in 1914.
1 .1 0
The Problem of Propulsion
Aerodrome was shipped back to the Sm ithsonian, where it was carefully restored
to its original configuration and in 19 18 was placed on display in the old Arts and
Industries Building. U nderneath the Aerodrome was placed a plaque reading:
“O riginal Langley flying m achine, 1903. The first m an-carrying aeroplane in the
history o f the w orld capable o f sustained free flight.” The plaque did not mention
that the Aerodrome dem onstrated its sustained-flight capability only after the 93
m odifications m ade by C urtiss in 1914. It is no surprise that O rville W right was
deeply upset by this state o f affairs, and this is the principal reason why the orig­
inal 1903 Wright Flyer was not given to the Sm ithsonian until 1948, the year o f
O rville’s death. Instead, from 1928 to 1948, the Flyer resided in the Science
M useum in London.
This section ends with tw o ironies. In 1915, O rville sold the W right A ero­
nautical C orporation to a group o f New York businesspeople. D uring the 1920s,
this corporation becam e a losing com petitor in aviation. Finally, on June 26,
1929, in a New York office, the W right A eronautical C orporation was officially
m erged with the successful C urtiss A eroplane and M otor C orporation, form ing
the Curtiss-W right C orporation. Thus, ironically, the nam es o f C urtiss and
W right finally cam e together after all those earlier turbulent years. The CurtissW right Corporation w ent on to produce num erous fam ous aircraft, perhaps the
m ost notable being the P-40 o f W orld War II fame. U nfortunately, the com pany
could not survive the lean years im m ediately after World War II, and its aircraft
developm ent and m anufacturing ceased in 1948. This leads to the second irony.
A lthough the very foundations o f pow ered flight rest on the work o f O rville and
W ilbur W right and G lenn C urtiss, there is not an airplane either produced or in
standard operation today that bears the nam e o f either W right or Curtiss.
1.10 THE PROBLEM OF PROPULSION
During the 19th century, num erous visionaries predicted that m anned heavierthan-air flight was inevitable once a suitable pow er plant could be developed to
lift the aircraft o ff the ground. It was ju st a m atter o f developing an engine hav­
ing enough horsepow er w hile at the sam e tim e not w eighing too much, that is, an
engine with a high horsepow er-to-w eight ratio. This indeed was the main stum ­
bling block to such people as Stringfellow, Du Temple, and M ozhaiski— the
steam engine sim ply did not fit the bill. Then, in 1860, the Frenchm an Jean
Joseph Etienne Lenoir built the first practical gas engine. It was a single-cylinder
engine, burning ordinary street-lighting gas for fuel. By 1865, 400 o f L enoir’s
engines were doing odd jobs around Paris. Further im provem ents in such inter­
nal com bustion engines cam e rapidly. In 1876, N. A. O tto and E. Langen of
G erm any developed the four-cycle engine (the ancestor o f all modern autom o­
bile engines), which also used gas as a fuel. This led to the sim ultaneous but sep­
arate developm ent in 1885 o f the four-cycle gasoline-burning engine by Gottlieb
D aim ler and Karl Benz, both in Germany. Both Benz and D aim ler put their
engines in m otor cars, and the autom obile industry was quickly born. A fter these
“horseless carriages” were given legal freedom o f the roads in 1896 in France
C H A P T E R 1 The First Aeronautical Engineers
and Britain, the autom obile industry expanded rapidly. Later, this industry was to
provide m uch o f the technology and m any o f the trained m echanics for the future
developm ent o f aviation.
This developm ent o f the gasoline-fueled internal com bustion engine was a
godsend to aeronautics, w hich was beginning to gain m om entum in the 1890s. In
the final analysis, it was the W right brothers’ custom -designed and -constructed
gasoline engine that was responsible for lifting their Flyer I off the sands o f Kill
D evil Hills that fateful day in D ecem ber 1903. A proper aeronautical propulsion
device had finally been found.
It is interesting to note that the brotherhood betw een the autom obile and the
aircraft industries persists to present day. For exam ple, in June 1926, Ford intro­
duced a very successful three-engine, high-w ing transport airplane— the Ford 4AT Trimotor. D uring W orld War II, virtually all the m ajor autom obile com panies
built airplane engines and airfram es. G eneral M otors m aintained an airplane en ­
gine division for m any decades— the A llison D ivision in Indianapolis, Indiana—
noted for its turboprop designs. Today, A llison is ow ned by Rolls-R oyce and
constitutes its N orth A m erican branch. M ore recently, autom obile designers are
turning to aerodynam ic stream lining and w ind tunnel testing to reduce drag,
hence increase fuel economy. Thus, the parallel developm ent o f the airplane and
the autom obile over the past 1 0 0 years has been m utually beneficial.
It can be argued that propulsion has paced every m ajor advancem ent in the
speed o f airplanes. Certainly, the advent o f the gasoline engine opened the doors
to the first successful flight. Then as the pow er o f these engines increased from
the 12-hp, W rights-designed engine o f 1903 to the 2200-hp, radial engines o f
1945, airplane speeds correspondingly increased from 28 to m ore than 500 mi/h.
Finally, je t and rocket engines today provide enough thrust to propel aircraft at
thousands o f miles per hour— m any tim es the speed o f sound. So throughout the
history o f m anned flight, propulsion has been the key that has opened the doors
to flying faster and higher.
1.11 FASTER AND HIGHER
The developm ent o f aeronautics in general, and aeronautical engineering in par­
ticular, was exponential after the W rights’ m ajor public dem onstrations in 1908
and has continued to be so to present day. It is beyond the scope o f this book to
go into all the details. However, m arbled into the engineering text in Chaps. 2
through 11 are various historical highlights o f technical im portance. It is hoped
that the follow ing parallel presentations o f the fundam entals o f aerospace engi­
neering and som e o f their historical origins will be synergistic and that, in com ­
bination with the present chapter, they will give the reader a certain appreciation
for the heritage o f this profession.
As a final note, the driving philosophy o f m any advancem ents in aeronautics
since 1903 has been to fly fa ste r and higher. This is dram atically evident from
Fig. 1.30, which gives the flight speeds for typical aircraft as a function o f
1.11
Faster and Higher
Year
Figure 1.30 Typical flight velocities over the years.
chronological time. N ote the continued push for increased speed over the years
and the particular increase in recent years m ade possible by the je t engine. Sin­
gled out in Fig. 1.30 are the w inners o f the Schneider Cup races betw een 1913
and 1931 (with a m oratorium during W orld War I). The Schneider Cup races
were started in 1913 by Jacques Schneider o f France as a stim ulus to the devel­
opm ent o f high-speed float planes. They prom pted som e early but advanced de­
velopm ent of high-speed aircraft. The w inners are shown by the dashed line in
Fig. 1.30, for com parison with standard aircraft o f the day. Indeed, the w inner o f
the last Schneider race in 1931 was the Supermarine S.6B, a forerunner o f the fa­
mous Spitfire o f W orld War II. O f course, today the m axim um speed o f flight has
been pushed to the extrem e value o f 36,000 ft/s, which is the escape velocity
47
chapter
1 The First Aeronautical Engineers
from the earth, by the A pollo lunar spacecraft.
N ote that the alm ost exponential increase in speed that occurred from 1903
to 1970 has not continued in recent years. Indeed, the m axim um speed o f m od­
em m ilitary fighters has actually been decreasing since 1970, as shown in
Fig. 1.30. This is not due to a degradation in technology, but rather is a reflection
that other airplane perform ance param eters (not speed) are dictating the design.
For exam ple, air-to-air com bat between opposing fighter airplanes capable of
high supersonic speeds quickly degenerates to flying at subsonic or near sonic
speeds because o f enhanced m aneuverability at these low er speeds. Today,
fighter airplanes are being optim ized for this low er-speed com bat arena. On the
com m ercial side, m ost transport airplanes are subsonic, even the new est (at the
tim e o f this w riting) such as the Boeing 777. There was only one type o f super­
sonic transport to provide extensive service, the A nglo-French Concorde. The
Concorde was designed with 1960s’ technology, and carried a relatively small
num ber o f passengers. H ence, it was not profitable. The Concorde was w ith­
draw n from service in 2003. A t the tim e o f this w riting, there is no com m itm ent
from any country to build a second-generation supersonic transport, although in
the U nited States, NA SA has been carrying out an extensive research program to
develop the basic technology for an econom ical high-speed supersonic transport.
Even if an econom ically viable supersonic transport could be designed, its speed
w ould be lim ited to about M ach 2.2 or less. A bove this M ach number, aerody­
nam ic heating becom es severe enough that titanium rather than alum inum would
have to be used for the aircraft skin and for som e internal structure. Titanium is
expensive and hard to m achine; it is not a preferred choice for a new supersonic
transport. For these reasons, it is unlikely that the speed curve in Fig. 1.30 will be
pushed up by a new supersonic transport.
As a com panion to speed, the m axim um altitudes o f typical m anned aircraft
are shown in Fig. 1.3 1 as a function o f chronological time. The sam e push to
higher values in the decades betw een 1903 and 1970 is evident; so far, the record
is the m oon in 1969. However, the sam e tendency to plateau after 1970, as in the
speed data, can be seen in the altitude data in Fig. 1.31.
H ence, the philosophy o f fa ste r and higher that has driven aeronautics
throughout m ost o f the 2 0 th century is now being m itigated by practical con­
straints. To this we m ust add safer, cheaper, more reliable, and more environ­
m entally clean. On the other hand, the eventual prospect o f hypersonic aircraft
(with M ach num ber greater than 5) in the 21st century is intriguing and exciting.
H ypersonic airplanes may well be a new frontier in aeronautics in the future cen­
tury. See Chap. 11 for a discussion o f hypersonic aircraft.
In this chapter, we have been able to briefly note only several im portant
events and people in the historical developm ent o f aeronautics. M oreover, there
are m any other places, people, and accom plishm ents that w e sim ply could not
m ention in the interest o f brevity. Therefore, the reader is urged to consult the
short Bibliography at the end o f this chapter for additional m odern reading on the
history o f aeronautics.
1.12
Summary
314,750 ft ■
Year
Figure 1.31 Typical flight altitudes over the years.
1.12 Summary
You are about to embark on a study o f aerospace engineering. This chapter has presented
a short historical sketch o f som e o f the heritage underlying modern aerospace engineer­
ing. The major stepping-stones to controlled, heavier-than-air, powered (light with a
human pilot are summarized as follows:
1.
2.
3.
4.
5.
Leonardo da Vinci conceives the ornithopter and leaves more than 500 sketches o f
his design, drawn from 1486 to 1490. However, this approach to flight proves to be
unsuccessful over the ensuing centuries.
The Montgolfier hot-air balloon floats over Paris on Novem ber 21, 1783. For the
first time in history, a human being is lifted and carried through the air for a
sustained period.
A red-letter date in the progress o f aeronautics is 1799. In that year, Sir George
Cayley in England engraves on a silver disk his concept o f a fuselage, a fixed wing,
and horizontal and vertical tails. He is the first person to propose separate
mechanisms for the generation o f lift and propulsion. He is the grandparent o f the
concept of the modern airplane.
The first two powered hops in history are achieved by the Frenchman Felix Du
Temple in 1874 and the Russian Alexander F. M ozhaiski in 1884. However, they
do not represent truly controlled, sustained flight.
Otto Lilienthal designs the first fully successful gliders in history. During the
period from 1891 to 1896, he achieves more than 2000 successful glider flights. If
49
chapter
6.
7.
8.
1 The First Aeronautical Engineers
he had not been killed in a glider crash in 1896, Lilienthal might have achieved
powered flight before the Wright brothers.
Samuel Pierpont Langley, secretary o f the Smithsonian Institution, achieves the
first sustained heavier-than-air, unmanned, powered flight in history with his smallscale Aerodrome in 1896. However, his attempts at manned flight are unsuccessful,
the last one failing on December 8, 1903— just nine days before the Wright
brothers’ stunning success.
Another red-letter date in the history o f aeronautics, indeed in the history o f
humanity, is December 17, 1903. On that day, at Kill D evil Hills in North Carolina,
Orville and Wilbur Wright achieve the first controlled, sustained, powered, heavierthan-air, manned flight in history. This flight is to revolutionize life during the 20th
century.
The developm ent o f aeronautics takes o ff exponentially after the Wright brothers’
public demonstrations in Europe and the United States in 1908. The ongoing work
o f Glenn Curtiss and the Wrights and the continued influence o f L angley’s early
work form an important aeronautical triangle in the developm ent o f aeronautics
before World War I.
Throughout the remainder o f this book, various historical notes w ill appear, to con­
tinue to describe the heritage o f aerospace engineering as its technology advances over
the 20th century. It is hoped that such historical notes will add a new dimension to your
developing understanding o f this technology.
Bibliography
Anderson, John D., Jr.: The Airplane: A History of Its Technology, American Institute o f
Aeronautics and Astronautics, Reston, VA, 2002.
Anderson, John D., Jr.: A History of Aerodynamics and Its Impact on Flying Machines,
Cambridge University Press, N ew York, 1997.
Angelucci, E.: Airplanes from the Dawn of Flight to the Present Day, M cGraw-Hill,
N ew York, 1973.
Combs, H.: Kill Devil Hill, Houghton Mifflin, Boston, 1979.
Crouch, T. D.: The Bishop's Boys, Norton, N ew York, 1989.
______ : A Dream of Wings, Norton, N ew York, 1981.
Gibbs-Smith, C. H.: Sir George Cayley’s Aeronautics 1796-1855, Her M ajesty’s
Stationery Office, London, 1962.
______ : The Invention of the Aeroplane (1799-1909), Faber, London, 1966.
______ : Aviation: An Historical Survey from Its Origins to the End of World War II, Her
M ajesty’s Stationery Office, London, 1970.
______ : Flight Through the Ages, Crowell, N ew York, 1974.
The follow ing are a series o f small booklets prepared for the British Science Mu­
seum by C. H. Gibbs-Smith, published by Her M ajesty’s Stationery Office, London:
The Wright Brothers, 1963
The World’s First Aeroplane Flights, 1965
Leonardo da Vinci’s Aeronautics, 1967
Bibliography
A Brief History of Flying, 1967
Sir George Cayley, 1968
Jakab, Peter L.: Visions of a Flying Machine, Smithsonian Institution Press,
Washington, 1990.
Josephy, A. M., and A. Gordon: The American Heritage History of Flight, Sim on and
Schuster, N ew York, 1962.
McFarland, Marvin W. (ed.): The Papers of Wilbur and Orville Wright, McGraw-Hill,
N ew York.
Roseberry, C. R.: Glenn Curtiss: Pioneer of Flight, Doubleday, Garden City, NY, 1972.
Taylor, J. W. R., and K. Munson: History of Aviation, Crown, N ew York, 1972.
C
H
A
P
T
E
R
Fundamental Thoughts
Engineering: “The application of scientific principles to practical ends.” From the
Latin word “ingenium,” meaning inborn talent and skill, ingenious.
The American Heritage Dictionary
o f the English Language, 1969
he language o f engineering and physical science is a logical collection and
assim ilation o f sym bols, definitions, form ulas, and concepts. To the aver­
age person in the street, this language is frequently esoteric and incom pre­
hensible. In fact, when you becom e a practicing engineer, do not expect to con­
verse with your spouse across the dinner table about your great technical
accom plishm ents o f the day. Chances are that he or she will not understand what
you are talking about. The language is intended to convey physical thoughts. It is
our way o f describing the phenom ena o f nature as observed in the world around us.
It is a language that has evolved over at least 2500 years. For exam ple, in 400 b c ,
the G reek philosopher D em ocritus introduced the word and concept o f the atom,
the sm allest bit o f m atter that could not be cut. The purpose o f this chapter is to in­
troduce some o f the everyday language used by aerospace engineers; in turn, this
language will be extended and applied throughout the rem ainder o f this book.
T hroughout this book, you will be provided with road m aps to guide you
through the thoughts and intellectual developm ent that constitute this introduc­
tion to flight. Please use these road maps frequently. They will tell you where you
are in our discussions, w here you have been, and w here you are going. For
52
chapter
2
Fundamental Thoughts
S3
PREVIEW BOX
The purpose o f this chapter is to help you get going.
For many o f us, when we have a job to do or a goal to
accomplish, the most important thing is simply to get
started— to get going— and hopefully to get going in
the right direction. This chapter deals with som e fun­
damental thoughts to help you start learning about
airplanes and space vehicles.
For example, we have to start with som e basic
definitions that are absolutely necessary for us to
speak the same “language” when we describe, dis­
cuss, analyze, and design airplanes and space vehi­
cles. When w e talk about the pressure in the airflow
around a B oeing 111 in flight, do we know what pres­
sure means? Really? If we talk about the airflow ve­
locity around the airplane, do w e really know what
w e are talking about? Definitions are important, so
this chapter pushes definitions.
Another example: When you walk down the
sidewalk in the face o f a 4 0 mile per hour gale, the
wind is pushing you around— exerting an aerody­
namic force on you. Every vehicle that m oves
through the air feels an aerodynamic force. How does
the wind reach out and grab you? How does nature
exert an aerodynamic force on a B oeing 747 cruising
at 500 miles per hour at an altitude o f 35,000 feet? In
this chapter, we w ill examine the sources o f aerody­
namic force and answer the question, how?
Dimensions and units— what dry and dull sub­
jects! Yet they are subjects o f the utmost importance
in engineering and science. You have to get them
right. In December 1999, the Mars Polar Lander was
lost during entry into the Martian atmosphere be­
cause o f a miscommunication between the contractor
in Denver and the Jet Propulsion Laboratory in
Pasadena involving feet and meters, costing the space
program a loss o f dollars and valuable scientific data.
Dimensions and units are fundamental considera­
tions and are discussed at length in this chapter.
Airplanes and space vehicles: Some readers are
enthusiasts; they recognize many o f these vehicles by
sight and even know som e o f their performance char­
acteristics. Other readers are not so sure about what
they are seeing and are not so familiar with their char­
acteristics. Just to put all readers on the same footing,
on the same page so to speak, this chapter ends with
a brief description o f the anutomy o f airplanes and
space vehicles— identifying various parts and fea­
tures o f these vehicles.
This is how we get going— looking at som e o f
the most fundamental thoughts that will be with us
for the remainder o f the book. Read on, and enjoy.
exam ple, Fig. 2.1 is an overall road map for the com plete book. Exam ining this
road map, we can obtain an overall perspective for our introduction to flight as
presented in this book. First we start out with som e necessary prelim inaries—
som e fundam ental thoughts that are used throughout the rem ainder o f the book.
This is the subject o f this chapter. Since flight vehicles spend all, or at least some
of, their tim e operating in the atm osphere, next we have to consider the proper­
ties o f the atm osphere, as discussed in Chap. 3. (A irplanes spend all their time in
the atm osphere. Space vehicles have to ascend through the atm osphere to get out
to space, and if they carry hum ans or other payloads that we wish to recover on
earth, space vehicles have to descend— at very high speeds— back through the
atm osphere.) N ow im agine a vehicle flying through the atm osphere. One o f the
first thoughts that com es to m ind is that there is a rush o f air over the vehicle. This
rush o f air generates a force— an aerodynam ic force— on the vehicle. A study of
chapter
2
Fundamental Thoughts
Figure 2.1 Road map for the book.
aerodynam ics is the subject o f Chaps. 4 and 5. The vehicle itself feels not only
this aerodynam ic force, but also the force o f gravity— its ow n weight. If the ve­
hicle is pow ered in some fashion, it will also feel the force (called thrust) from the
pow er plant. The vehicle m oves under the influence o f these forces. The study o f
the m otion o f the flight vehicle is labeled flight dynamics, w hich is further divided
into considerations o f airplane perform ance (Chap. 6 ) and stability and control
(Chap. 7). In contrast, a space vehicle m oving in space will, for all practical pur­
poses, feel only the force o f gravity (except when som e on-board propulsion de­
vice is turned on for trajectory adjustm ent). The motion o f a vehicle in space due
to gravitational force is the subject o f Chap. 8 . C onsidering again a flight vehicle
m oving through the atm osphere, there needs to be som ething to push it along—
som ething to keep it going. T his is the function o f the engine, w hich generates
thrust to keep the vehicle going. Space vehicles also need engines— to accelerate
them into orbit or deep space and for m idcourse trajectory corrections. Engines
and how they generate thrust represent the discipline o f propulsion, the subject o f
Chap. 9. A dditionally, as the flight vehicle m oves and responds to the forces
chapter
2
Fundamental Thoughts
acting on it, the physical structure o f the vehicle is under a lot o f stress and strain.
You w ant this structure to be strong enough to not fall apart under these stresses
and strains, but at the sam e tim e not to be so heavy as to render the flight vehicle
inefficient. We address som e aspects o f flight structures in Chap. 10. All these
m ajor disciplines— aerodynam ics, flight dynam ics, propulsion, and structures—
are integrated into the design o f a flight vehicle. Such design is indeed the final
objective o f most aerospace research and developm ent. Throughout this book, at
appropriate places, w e address pertinent aspects o f vehicle design. We highlight
these aspects by placing them in accented design boxes. You cannot m iss them in
your reading. Finally, looking tow ard the future, we discuss some advanced ve­
hicle concepts in Chap. 11. All the previous discussion is diagram m ed in Fig. 2.1.
This is the road map for our excursions throughout this book. From tim e to time,
as you proceed through this book, flip back to Fig. 2.1 for a rem inder o f how the
material you are reading fits into the whole scheme.
Returning to our considerations at hand, w e look at the road m ap for this
chapter in Fig. 2.2. We treat tw o avenues o f thought in this chapter. As shown in
the left colum n o f Fig. 2.2, w e exam ine som e basic ideas and definitions that are
rooted in physics. These include definitions o f the physical quantities o f a flow­
ing gas, that is, the language w e use to talk about aerodynam ics and propulsion.
We discuss the fundam ental sources o f aerodynam ic force— how aerodynam ic
Figure 2.2 Road map for Chap. 2.
chapter
2
Fundamental Thoughts
force is exerted on a vehicle. We look at som e equations that relate the physical
quantities, and we also discuss the m undane (but essential) consideration o f units
for these physical quantities. We then m ove to the right colum n in Fig. 2.2 and
discuss som e fundam ental aspects o f flight vehicles them selves, taking a look at
the anatom y o f typical airplanes and space vehicles.
2.1 FUNDAMENTAL PHYSICAL QUANTITIES
OF A FLOWING GAS
As you read through this book, you will soon begin to appreciate that the flow o f
air over the surface o f an airplane is the basic source o f the lifting or sustaining
force that allow s a heavier-than-air m achine to fly. In fact, the shape o f an air­
plane is designed to encourage the airflow over the surface to produce a lifting
force in the m ost efficient m anner possible. (You will also begin to appreciate
that the design o f an airplane is in reality a compromise betw een m any different
requirem ents, the production o f aerodynam ic lift being ju st one.) T he science
that deals w ith the flow o f air (or, for that matter, the flow o f any gas) is called
aerodynamics, and the person w ho practices this science is called an aerodynamicist. The study o f the flow o f gases is im portant in many other aerospace ap­
plications, for exam ple, the design o f rocket and jet engines, propellers, vehicles
entering planetary atm ospheres from space, wind tunnels, and rocket and projec­
tile configurations. Even the motion o f the global atm osphere and the flow o f
effluents through sm okestacks fall w ithin the realm o f aerodynam ics. The appli­
cations are alm ost lim itless.
Four fundam ental quantities in the language o f aerodynam ics are pressure,
density, tem perature, and velocity. Let us look at each one.
2.1.1 P ressu re
W hen you hold your hand outside the window o f a m oving autom obile, with
your palm perpendicular to the incom ing airstream , you can feel the air pressure
exerting a force and tending to push your hand rearw ard, in the direction o f the
airflow. The fo rce p e r unit area on your palm is defined as the pressure. The pres­
sure exists basically because air m olecules (oxygen and nitrogen m olecules) are
striking the surface o f your hand and transferring som e o f their momentum to the
surface. M ore precisely,
Pressure is the normal force per unit area exerted on a surface due to the time rate
of change of momentum of the gas molecules impacting on that surface.
It is im portant to note that even though pressure is defined as force per unit
area, for exam ple, new tons per square m eter or pounds per square foot, you do not
need a surface that is actually 1 m 2 or 1 ft 2 to talk about pressure. In fact, pressure
is usually defined at a point in the gas or a point on a surface and can vary from one
point to another. We can use the language o f differential calculus to see this more
2.1
Fundamental Physical Quantities of a Flowing Gas
clearly. Referring to Fig. 2.3, we consider a point B in a volum e o f gas. Let
d A = an increm ental area around B
d F = force on one side o f d A due to pressure
Then the pressure p at point B in the gas is defined as
p = lim
dA-* 0
(2 .1)
Equation (2.1) says that, in reality, the pressure p is the lim iting form o f the force
per unit area w here the area o f interest has shrunk to zero around point B. In this
form alism , it is easy to see that p is a point property and can have a different
value from one point to another in the gas.
Pressure is one o f the m ost fundam ental and im portant variables in aerody­
nam ics, as we shall soon see. Com m on units o f pressure are new tons per square
meter, dynes per square centim eter, pounds per square foot, and atm ospheres.
A bbreviations for these quantities are N /m 2, dyn/cm 2, lb/ft2, and atm, respec­
tively. See App. C for a list o f com m on abbreviations for physical units.
2.1.2 D ensity
The density of a substance (including a gas) is the mass of that substance per unit
volume.
D ensity will be designated by the sym bol p . For exam ple, consider air in a
room that has a volum e o f 250 m 3. If the m ass o f the air in the room is 306.25 kg
and is evenly distributed throughout the space, then p = 306.25 kg/250m 3 =
1.225 kg/m 3 and is the sam e at every point in the room.
CHAPTER 2
Fundamental Thoughts
V o lu m e o f gas
Figure 2.4 Definition of density.
A nalogous to the previous discussion o f pressure, the definition o f density
does not require an actual volum e o f 1 m 3 or 1 ft3. Rather, p is a point property
and can be defined as follow s. Referring to Fig. 2.4, we consider point B inside
a volum e o f gas. Let
d v = elem ental volum e around point B
dm = mass o f gas inside d v
Then p at point B is
(2 .2 )
Therefore, p is the m ass per unit volum e where the volum e o f interest has shrunk
to zero around point B. The value o f p can vary from point to point in the gas.
C om m on abbreviated units o f density are kg/m 3, slug/ft3, g/cm 3, and lbm/ f t 3.
(The pound mass, lb,„ will be discussed in Sec. 2.4.)
2.1.3 T em p eratu re
C onsider a gas as a collection o f m olecules and atoms. These particles are in con­
stant motion, m oving through space and occasionally colliding with one another.
Since each particle has m otion, it also has kinetic energy. If we watch the motion
o f a single particle over a long tim e during w hich it experiences num erous colli­
sions with its neighboring particles, then we can m eaningfully define the average
kinetic energy o f the particle over this long duration. If the particle is m oving
rapidly, it has a higher average kinetic energy than if it were m oving slowly. The
tem perature T o f the gas is directly proportional to the average m olecular kinetic
energy. In fact, w e can define T as follows:
Temperature is a measure o f the average kinetic energy o f the particles in the gas. If
KE is the mean molecular kinetic energy, then temperature is given by KE = \ k T ,
where k is the Boltzmann constant.
The value o f k is 1.38 x 10 23 J/K, where J is an abbreviation for joule.
2.1
Fundamental Physical Quantities of a Flowing Gas
H ence, we can qualitatively visualize a high-tem perature gas as one in which
the particles are random ly rattling about at high speeds, w hereas in a lowtem perature gas, the random m otion o f the particles is relatively slow. Tem pera­
ture is an im portant quantity in dealing w ith the aerodynam ics o f supersonic and
hypersonic flight, as we shall soon see. C om m on units o f tem perature are the
kelvin (K), degree C elsius (°C), degree R ankine (°R), and degree Fahrenheit (°F).
2 .1.4 Flow V elocity and S tream lin es
The concept o f speed is com m onplace: It represents the distance traveled by
som e object per unit time. For exam ple, we all know w hat is m eant by traveling
at a speed o f 55 mi/h down the highway. H ow ever, the concept o f the velocity o f
a flowing gas is som ew hat m ore subtle. First, velocity connotes direction as well
as speed. The autom obile is m oving at a velocity o f 55 mi/h due north in a hori­
zontal plane. To designate velocity, we m ust quote both speed and direction. For
a flowing gas, we m ust further recognize that each region o f the gas does not
necessarily have the sam e velocity; that is, the speed and direction o f the gas may
vary from point to point in the flow. H ence, flow velocity, along with p , p, and
T, is a point property.
To see this m ore clearly, consider the flow o f air over an airfoil or the flow of
com bustion gases through a rocket engine, as sketched in Fig. 2.5. To orient
yourself, lock your eyes on a specific, infinitesim ally small elem ent o f m ass in
the gas, and watch this elem ent m ove with tim e. Both the speed and direction of
this elem ent (usually called a fluid elem ent) can vary as it m oves from point to
point in the gas. Now, fix your eyes on a specific fixed point in the gas flow, say,
point B in Fig. 2.5. We can now define flow velocity as follows:
The velocity at any fixed point B in a flowing gas is the velocity o f an infinitesimally
small fluid elem ent as it sw eeps through B.
A gain, we em phasize that velocity is a point property and can vary from point to
point in the flow.
Referring again to Fig. 2.5, we note that as long as the flow is steady (as
long as it does not fluctuate with tim e), a m oving fluid elem ent is seen to trace
out a fixed path in space. This path taken by a m oving fluid elem ent is called a
R o c k e t en g in e
Figure 2.5 Flow velocity and streamlines.
F lo w o v er an a irfo il
59
60
Figure 2.6 Smoke
photograph o f the lowspeed flow over a
Lissaman 7769 airfoil
at 10° angle o f attack.
The Reynolds number
based on chord is
150.000. This is the
airfoil used on the
Gossamer Condor
human-powered
aircraft.
(The photograph was
taken in one o f the Notre
Dame University smoke
tunnels by Dr. T. J.
Mueller, Professor o f
Aerospace Engineering at
Notre Dame, and is
shown here through his
courtesy.)
Figure 2.7 An oil
streak photograph
showing the surface
streamline pattern for a
fin mounted on a flat
plate in supersonic
flow. The parabolic
curve in front of the fin
is due to the bow shock
wave and flow
separation ahead o f the
fin. Note how clearly
the streamlines can be
seen in this complex
flow pattern. Flow is
from right to left. The
Mach number is 5, and
the Reynolds number is
6.7 x 106.
( C o u rtesy o f A llen E.
Winkelmann, University o f
Maryland, and the Naval
Surface Weapons Center.)
chapter
2
Fundamental Thoughts
2.2
The Source of all Aerodynamic Forces
streamline o f the flow. D raw ing the stream lines o f the flow field is an im portant
way o f visualizing the m otion o f the gas; we will frequently sketch the stream ­
lines o f the flow about various objects. For exam ple, the stream lines o f the flow
about an airfoil are sketched in Fig. 2.5 and clearly show the direction o f motion
o f the gas. Figure 2.6 is an actual photograph o f stream lines over an airfoil
model in a low -speed subsonic w ind tunnel. T he stream lines are m ade visible by
injection o f filam ents o f sm oke upstream o f the model; these sm oke filaments
follow the stream lines in the flow. Using another flow field visualization tech­
nique, Fig. 2.7 show s a photograph o f a flow w here the surface stream lines are
made visible by coating the m odel with a m ixture o f w hite pigm ent in m ineral
oil. Clearly, the visualization o f flow stream lines is a useful aid in the study of
aerodynam ics.
2.2 THE SOURCE OF ALL AERODYNAMIC
FORCES
We have ju st discussed the four basic aerodynam ic flow quantities: p , p , 7 \ and
V, w here V is velocity, which has both m agnitude and direction; that is, velocity
is a vector quantity. A know ledge o f p , p , T , and V at each point o f a flow fully
defines the flow field. For exam ple, if we w ere concerned with the flow about a
sharp-pointed cone, as shown in Fig. 2.8, we could im agine a cartesian x y z
three-dim ensional space, w here the velocity far ahead o f the cone Voo is in the
x direction and the cone axis is also along the x direction. The specification of
y
p
p
T
V
=
=
=
=
Figure 2.8 Specifications o f a flow field.
p(x,y,z)
p(x,y,z)
T( x, y , z)
\(x,y,z)
Flow field
chapter
2
Fundamental Thoughts
the follow ing quantities then fully defines the flow field:
p = p(x,y,z)
p = p(x,y,z)
T = T( x, y, z)
V
= \(x,y,z)
(In practice, the flow field about a right circular cone is m ore conveniently de­
scribed in term s o f cylindrical coordinates, but we are concerned only with the
general ideas here.)
T heoretical and experim ental aerodynam icists labor to calculate and m ea­
sure flow fields o f m any types. W hy? W hat practical inform ation does know l­
edge o f the flow field yield with regard to airplane design or to the shape o f a
rocket engine? A substantial part o f the first five chapters o f this book endeavors
to answ er these questions. H ow ever, the roots o f the answ ers lie in the follow ing
discussion.
Probably the m ost practical consequence o f the flow o f air over an object is
that the object experiences a force, an aerodynam ic force, such as your hand feels
outside the open w indow o f a m oving car. Subsequent chapters discuss the nature
and consequences o f such aerodynam ic forces. The purpose here is to state that
the aerodynam ic force exerted by the airflow on the surface o f an airplane, m is­
sile, etc., stem s from only tw o sim ple natural sources:
1.
Pressure distribution on the surface
2.
Shear stress (friction) on the surface
We have already discussed pressure. Referring to Fig. 2.9, we see that pres­
sure exerted by the gas on the solid surface o f an object always acts normal to the
Figure 2.9 Pressure and shear-stress distributions.
2.3
Equation of State for a Perfect Gas
surface, as shown by the directions o f the arrow s. The lengths o f the arrow s de­
note the m agnitude o f the pressure at each local point on the surface. Note that
the surface pressure varies with location. The net unbalance o f the varying pres­
sure distribution over the surface creates a force, an aerodynam ic force. The sec­
ond source, shear stress acting on the surface, is due to the frictional effect o f the
flow “rubbing” against the surface as it m oves around the body. The shear stress
rw is defined as the force per unit area acting tangentially on the surface due to
friction, as shown in Fig. 2.9. It is also a point property; it varies along the sur­
face; and the net unbalance o f the surface shear stress distribution creates an
aerodynam ic force on the body. No m atter how complex the flow field, and no
m atter how complex the shape o f the body, the only way nature has o f communi­
cating an aerodynam ic force to a solid object o r surface is through the pressure
and shear stress distributions that exist on the surface. These are the basic
fundam ental sources o f all aerodynam ic forces. The pressure and shear-stress
distributions are the tw o hands o f nature that reach out and grab the body, exert­
ing a force on the body— the aerodynam ic force.
Finally, we can state that a prim ary function o f theoretical and experim ental
aerodynam ics is to predict and m easure the aerodynam ic forces on a body. In many
cases, this im plies prediction and m easurem ent o f p and xw along a given surface.
Furtherm ore, a prediction o f p and rw on the surface frequently requires know l­
edge o f the com plete flow field around the body. This helps to answ er our earlier
question as to w hat practical inform ation is yielded by know ledge o f the flow field.
2.3 EQUATION OF STATE FOR A PERFECT GAS
Air under norm al conditions o f tem perature and pressure, such as those encoun­
tered in subsonic and supersonic (light through the atm osphere, behaves very
much as a perfect gas. The definition o f a perfect gas can best be seen by return­
ing to the m olecular picture. A gas is a collection o f particles (m olecules, atoms,
electrons, etc.) in random m otion, where each particle is, on average, a long dis­
tance aw ay from its neighboring particles. Each m olecule has an intermolecular
force fie ld about it, a ram ification o f the com plex interaction o f the electrom ag­
netic properties o f the electrons and nucleus. T he interm olecular force field o f a
given particle extends a com paratively long distance through space and changes
from a strong repulsive force at close range to a weak attractive force at long
range. The interm olecular force field o f a given particle reaches out and is felt by
the neighboring particles. On one hand, if the neighboring particles are far away,
they feel only the tail o f the weak attractive force; hence the motion o f the neigh­
boring particles is only negligibly affected. On the other hand, if they are close,
their m otion can be greatly affected by the interm olecular force field. Since the
pressure and tem perature o f a gas are tangible quantities derived from the motion
o f the particles, then p and T are directly influenced by interm olecular forces,
especially when the m olecules are packed closely together (i.e., at high densi­
ties). This leads to the definition o f a perfect gas:
A perfect gas is one in which intermolecular forces are negligible.
64
chapter
2
Fundamental Thoughts
Clearly, from the previous discussion, a gas in nature in w hich the particles
are w idely separated (low densities) approaches the definition o f a perfect gas.
The air in the room about you is one such case; each particle is separated, on
average, by m ore than 10 m olecular diam eters from any other. Hence, air at stan­
dard conditions can be readily approxim ated by a perfect gas. Such is also the
case for the flow o f air about ordinary flight vehicles at subsonic and supersonic
speeds. T herefore, in this book, w e always deal with a perfect gas for our aero­
dynam ic calculations.
The relation am ong p , p , and T for a gas is called the equation o f state. For
a perfect gas, the equation o f state is
p = pR T
(2.3)
w here R is the specific gas constant, the value o f which varies from one type of
gas to another. For norm al air we have
J
ft - lb
R = 2 8 7 - ----- — = 1716(kg)(K )
(slug)(°R)
From your earlier studies in chem istry and physics, you may be m ore fam iliar
with the universal gas constant 2ft, w here 2ft = 8314 J/(kg • mole K) = 4.97 x
104 (ft lb)/(slug • m ole °R), a universal value for all gases. The specific and uni­
versal gas constants are related through R = 2ft/M where M is the m olecular
w eight (or m ore properly, the m olecular m ass) o f the gas. For air, M = 28.96 kg/
(kg • mole). N ote that kg • m ole is a single unit; it stands for a kilogram -m ole,
identifying w hat type o f m ole w e are talking about. (It does not m ean kilogram s
m ultiplied by m oles.) A kilogram -m ole contains 6.02 x 1026 m olecules—
A vogadro’s num ber for a kilogram -m ole. A kilogram -m ole is that am ount o f a
gas that has a m ass in kilogram s equal to the m olecular w eight o f the gas. For air,
since M = 28.96, one kilogram -m ole o f air has a m ass o f 28.96 kilogram s and
consists o f 6.02 x 1026 m olecules. Similarly, a slug • mole o f gas is that am ount
o f gas that has a mass in slugs equal to the m olecular w eight o f the gas. For air,
one slug-m ole has a m ass o f 28.96 slugs. The sam e litany applies to the gramm ole, with w hich you may be m ore fam iliar from chemistry. The values o f R for
air given at the beginning o f this paragraph are obtained from
_
8314 J/(kg • mole K)
J
R = 2ft/A/ =
--------- r r = 28728.96 kg/(kg • mole)
(kg)(K)
and
_
4.97 x 10 4 (ft • lb)/(slug • mole °R)
_
ft • lb
R = <31/M = ---------------- --------—— ---------------- = 1716—------- —
28.96 slug/(slug • m ole)
(slug)(°R)
It is interesting that the deviation o f an actual gas in nature from perfect gas
b e h a v io r can b e e x p r e sse d a p p ro x im a tely by the m o d ified B erth elo t eq u ation o f
state:
p
pR T
_ J
o p _ ¿7?
T
T3
2.4
Discussion of Units
where a and b are constants o f the gas. Thus, the deviations becom e sm aller as p
decreases and T increases. This m akes sense, because if p is high, the m olecules
are packed closely together, interm olecular forces becom e im portant, and the gas
behaves less as a perfect gas. On the other hand, if T is high, the m olecules move
faster. Thus, their average separation is larger, interm olecular forces becom e less
significant in com parison to the inertia forces o f each m olecule, and the gas
behaves m ore as a perfect gas.
A lso, note that w hen the air in the room around you is heated to tem peratures
above 2500 K, the oxygen m olecules begin to dissociate (tear apart) into oxygen
atom s; at tem peratures above 4000 K, the nitrogen begins to dissociate. For these
tem peratures, air becom es a chem ically reacting gas, such that its chem ical com ­
position becom es a function o f both p and T ; that is, it is no longer normal air.
As a result, R in Eq. (2.3) becom es a variable— R = R( p , T ) — sim ply because
the gas com position is changing. T he perfect gas equation o f state, Eq. (2.3), is
still valid for such a case, except that R is no longer a constant. This situation is
encountered in very high speed flight, for exam ple, the atm ospheric entry o f the
Apollo capsule, in w hich case the tem peratures in som e regions o f the flow field
reach 1 1 ,0 0 0 K.
A gain, in this book, we alw ays assum e that air is a perfect gas, obeying
Eq. (2.3), with a constant R = 287 J/(kg)(K ) or 1716 ft ■lb/(slug)(°R).
2.4 DISCUSSION OF UNITS
Physical units are vital to the language o f engineering. In the final analysis, the
end result o f m ost engineering calculations or m easurem ents is a num ber that
represents som e physical quantity, for exam ple, pressure, velocity, or force. The
num ber is given in term s o f com binations o f units, for exam ple, 105 N /m 2,
300 m/s, or 5 N, where the new ton, meter, and second are exam ples o f units. (See
App. C.)
Historically, various branches o f engineering have evolved and favored sys­
tems o f units that seem ed to m ost conveniently fit their needs. These various sets
o f “engineering” units usually differ am ong them selves and are different from
the m ctric system , preferred for years by physicists and chem ists. In the modern
world o f technology, where science and engineering interface on alm ost all
fronts, such duplicity and variety o f units have becom e an unnecessary burden.
M etric units are now the accepted norm in both science and engineering in m ost
countries outside the U nited States. M ore im portantly, in 1960 the Eleventh
General Conference on W eights and M easures defined and officially established
the Systéme International d ’Unités— the SI units— which was adopted as the
preferred system o f units by 36 participating countries, including the United
States. Since then, the U nited States has m ade progress tow ard the voluntary
im plem entation o f SI units in engineering. For exam ple, several N A SA (National
A eronautics and Space A dm inistration) laboratories have made SI units virtually
m andatory for all results reported in technical reports, although engineering units
can be shown as a duplicate set. The A IA A (A m erican Institute o f A eronautics
chapter
2
Fundamental Thoughts
and A stronautics) has m ade a policy o f encouraging SI units for all papers
reported in their technical journals. It is apparent that in a few decades, the
U nited States, along with the rest o f the world, will be using SI units alm ost
exclusively. Indeed, the aerospace and autom obile industries in the U nited States
are now making extensive use o f SI units, driven by the realities o f an interna­
tional m arket for their products.
So, here is the situation. M uch o f the past engineering literature generated in
the U nited States and Britain used engineering units, w hereas m uch o f the cur­
rent work uses SI units. Elsew here in the world, SI units have been, and continue
to be, the norm. As a result, m odem engineering students m ust do “double duty”
in regard to fam iliarization with units. They m ust be fam iliar with engineering
units in order to read, understand, and use the vast bulk o f existing literature
quoted in such units. A t the sam e tim e, they m ust be intim ately fam iliar with SI
units for present and future work. C onclusion: engineering students m ust be
bilingual with regard to units.
To prom ote fluency in both the engineering and SI units, this book incorpo­
rates both sets. It is im portant that you develop a natural feeling for both sets of
units; for exam ple, you should feel as at hom e with pressures quoted in newtons
per square m eter (pascals) as you probably already do with pounds per square
inch (psi). A m ark o f successful experienced engineers is their feel for correct
m agnitudes o f physical quantities in fam iliar units. It is im portant for you to start
gaining this feeling for units now, to m ake yourself feel com fortable w ith both
the engineering and SI units. A purpose o f this book is to help you develop this
feeling o f com fort. In the process, we will be putting a bit m ore em phasis on SI
units in deference to their extensive international use.
For all practical purposes, SI is a m etric system based on the meter, kilo­
gram , second, and kelvin as basic units o f length, m ass, time, and tem perature,
respectively. It is a coherent, or consistent, set o f units. Such consistent sets o f
units allow physical relationships to be w ritten w ithout the need for “conversion
factors” in the basic form ulas. For exam ple, in a consistent set o f units, N ew ton’s
second law can be w ritten
F = m x a
Force = m ass x acceleration
In SI units,
F = ma
1
newton = (1 kilogram )(l m eter/second2)
(2.4)
The newton is a force defined such that it accelerates a m ass o f 1 kilogram by
1 m eter per second squared.
The English engineering system o f units is another consistent set o f units.
Here the basic units o f m ass, length, time, and tem perature are the slug, foot,
2.4
Discussion of Units
second, and degree Rankine, respectively. In this system,
F = ma
1
(2.5)
pound = (1 slug)(l foot/second2)
The pound is a force defined such that it accelerates a m ass o f 1 slug by 1 foot per
second squared. N ote that in both system s, N ew ton’s second law is w ritten sim ­
ply as F = m a, w ith no conversion factor on the right-hand side.
In contrast, a nonconsistent set o f units defines force and m ass such that
N ew ton’s second law m ust be w ritten with a conversion factor, or constant, as
1
F =
—
x m xa
gc
It
t
Force
C onversion
t t
M ass
Acceleration
factor
A nonconsistent set o f units frequently used in the past by m echanical engineers
includes the pound force, pound m ass, foot, and second, w hereby
gt. = 32.2 (lbm)(ft)/(s 2 )(lb/ )
F =
1
m
X
a
( 2 .6 )
gc
t
t
t
t
lb /
yn
1
lbm
ft/s2
In this nonconsistent system , the unit o f m ass is the pound m ass lbm. C om paring
Eqs. (2.5) and (2.6), we see that 1 slug = 32.2 lb,„. A slug is a large hunk o f mass,
w hereas the pound m ass is considerably sm aller, by a factor o f 32.2. This is
illustrated in Fig. 2.10.
A nother nonconsistent set o f units that is used in international engineering
circles deals with the kilogram force, the kilogram mass, m eter and second.
1
lb m
1 slug = 3 2 .2 lb„
Figure 2.10 Comparison between the slug and pound mass.
68
chapter
2
Fundamental Thoughts
whereby
gc = 9.8 (kg)(m )/(s 2 )(kg/ )
F =
m
x
a
(2.7)
gc
t
t
kg/ 931
t
kg
t
m/s‘
In this nonconsistent system , the unit o f force is the kilogram force, k g /.
It is easy to understand why people use these nonconsistent units, the pound
m ass (lbra) and the kilogram force (kg/). It has to do with weight. By definition,
the w eight o f an object, W, is
W = mg
(2.8)
where g is the acceleration o f gravity, a variable that depends on location around
the earth, indeed throughout the universe. At standard sea level on earth, the
standard value o f g is 9.8 m /(s)2, or 32.2 ft/(s)2. Eq (2.8) is w ritten in consistent
units; it is sim ply a natural statem ent o f N ew ton’s second law, Eq. (2.4), where
the acceleration a is the acceleration o f gravity g. H ence, if you held a kilogram
o f chocolate candy in your hand at a location on earth where the acceleration o f
gravity is the standard value o f 9.8 m/sec, that “kilo” o f candy w ould weigh
W = m g = (1 kg)(9.8 m /sec) = 9.8 N
T he “kilo” box o f candy w ould weigh 9.8 N; this is the force exerted on your
hands holding the candy. In contrast, if we used the nonconsistent units em bod­
ied in Eq. (2.7) to calculate the force exerted on your hands, we obtain
ma
(1)(9.8)
F = — =
/7To\ = 1 kg /
“g T ”
(9.8)
T he “kilo” box o f candy w ould weigh 1 k g /; the force exerted on your hands is
1 k g /. W hat a com m on convenience— the force you feel on your hands is the
same number o f k g / as is the m ass in kg. Presto— the use o f the kilogram force
in engineering w ork. Similarly, im agine you are holding 1 pound o f chocolates
in your hand. In the U nited States, we go to the store and pick a “pound” box o f
candy off the shelf. We feel the pound force in our hands. From Eq. (2.8), the
m ass o f the candy is
W
1 lb
m = — = ------ -— r- = 0.031 slug
g
32.2 ft/(s )
B
But if you go into a store and ask the attendant for a “0.031 slug” box o f candy,
im agine the reply you will get. On the other hand, using Eq. (2.6) with the non­
consistent unit o f lbm, the m ass o f a 1-pound box o f candy is
F gc
(1 lb) (32.2)
, ]u
m = ----- = ------------------= 1 lb,„
a
(32.2)
2.4
Discussion of Units
O nce again, we have the ^veryday convenience o f the m ass in your hands being
the same number in lbm as is the force on your hands being 1 lb. Presto— the use
o f the lbm in engineering work. This m akes sense in com m on everyday life, but
in the technical w orld o f engineering calculations, using Eq. (2.7) with the nonconsistent unit o f k g /, or Eq. (2.6) with the nonconsistent unit o f lbm, m akes gc
appear in m any o f the equations. N ature did not plan on this; the use o f gc is a
hum an invention. In nature, N ew ton’s second law appears in its pure form,
F = m a, not F = \ / g c(ma). H ence, to use nature in its pure form, we must
always use consistent units. W hen we do this, gc will never appear in any o f our
equations, and there is never any confusion in our calculations in regard to con­
version factors— there sim ply are no conversion factors needed.
F or these reasons, we will alw ays deal w ith a consistent set o f units in this
book. We will use both the SI units from Eq. (2.4) and the English engineering
units from Eq. (2.5). As stated before, you will frequently encounter the engi­
neering units in the existing literature, w hereas you will be seeing SI units with
increasing frequency in the future literature; that is, you m ust becom e bilingual.
To sum m arize, w e will deal with the English engineering system units— lb, slug,
ft, s, °R — and the Systém e International (SI) units— N, kg, m, s, K.
Therefore, returning to the equation o f state, Eq. (2.3), where p = p R T , we
see that the units are as follows:
Knglish Kngineering System
SI
p
p
lb/ft2
slugs/ft3
N/m2
kg/m3
T
°r
K
R (for air)
1716 ft • lb/(slug)(°R)
287 J/(kg)(K)
There are tw o final points about units to note. First, the units o f a physical
quantity can frequently be expressed in m ore than one com bination sim ply by
appealing to N ew ton’s second law. From N ew ton’s law, the relation between N,
kg, m, and s is
F = ma
N = kg • m /s 2
Thus, a quantity such as R = 287 J/(kg)(K ) can also be expressed in an equiva­
lent way as
m
ITT
N •m
kg • m
J
= 287= 2 8 7 - ----- — = 2 8 7 R = 287(kg)(K )
' (kg)(K )
"■ s 2 (kg)(K )
” (s 2 )(K)
Thus, R can also be expressed in the equivalent term s o f velocity squared
divided by tem perature. In the sam e vein,
ft lb
_
ft 2
R = 1716= 1716(slug)(°R)
(s 2 )(°R)
70
chapter
2
Fundamental Thoughts
Second, in the equation o f state, Eq. (2.3), T is alw ays the absolute tem pera­
ture, w here zero degrees is the absolutely low est tem perature possible. Both K and
°R are absolute tem perature scales, where 0°R = 0 K = tem perature at w hich al­
m ost all m olecular translational motion theoretically stops. On the other hand, the
fam iliar Fahrenheit (°F) and C elsius (°C) scales are not absolute scales. Indeed,
0 °F
= 460 R
0°C = 273 K = 32°F
For exam ple,
90°F is sam e as 460 + 90 = 550°R
(2.9)
and
10°C is sam e as 2 7 3 -I- 10 = 283 K
(2.10)
Please rem em ber: T in Eq. (2.3) must be the absolute tem perature, either kelvins
or degrees Rankine.
2.5 SPECIFIC VOLUME
D ensity p is the mass p e r unit volume. The inverse o f this quantity is also fre­
quently used in aerodynam ics. It is called the specific volume v and is defined as
the volume p e r unit mass. By definition,
1
P
H ence, from the equation o f state
p = pR T = - RT
v
we also have
pv = RT
(2.11)
A bbreviated units for v are m 3/kg and ft 3/slug.
EXAM PLE 2.1
The air pressure and density at a point on the w ing o f a B oeing 747 are 1.10 x 105 N/m 2
and 1.20 kg/m3, respectively. What is the temperature at that point?
■ Solution
From Eq. (2.3), p = p R T ; hence, T = p / ( p R ) , or
T =
1.10 x 105 N /m 2
(1 .2 0 kg/m )[287 J/(kg)(K)]
319 K
2.5
71
Specific Volume
EXAM PLI 2 2.2
The high-pressure air storage tank for a supersonic wind tunnel has a volum e o f 1000 ft3.
If air is stored at a pressure o f 30 atm and a temperature o f 530 R, what is the mass o f gas
stored in the tank in slugs? In pound mass?
■ Solution
The unit o f atm for pressure is not a consistent unit. You w ill find it helpful to remember
that, in the English engineering system,
l atm = 2116 lb /ft2
Hence, p = (3 0 )(2 116) lb/ft2 = 6.348 x 104 lb/ft2 . A lso, from Eq. (2.3), p = p RT\ hence,
p = (p / R T ), or
6.348 x 104 lb/ft2
P=
= 6.98 x 10-2 slug/ft3
1716 ft • lb/(slug)(°R )](530°R)
This is the density, which is mass per unit volume. The total mass M in the tank o f volume
V is
M = p V = (6.98 x 10-2 slug/ft3) (1000 ft3) =
69.8 slugs
Recall that 1 slug = 32.2 lbm. Hence,
M = (69.8)(32.2) =
2248 lb„
EXAMPLIE 2.3
Air flowing at high speed in a wind tunnel has pressure and temperature equal to 0.3 atm
and —100UC, respectively. What is the air density? What is the specific volume?
■ Solution
You are reminded again that the unit o f atm for pressure is not a consistent unit. You will
find it helpful to memorize that, in the SI system,
1 atm = 1.01 x 105 N/m 2
Hcncc,
p = (0.3)( 1.01 x 105) = 0.303 x 105 N /m 2
Note that T = —100°C is not an absolute temperature. Hence,
T = - 1 0 0 + 273 = 173 K
From Eq. (2.3), p = p R T ; hence, p = p / ( R T ) , or
0.303 x 105 N /m 2
P=
V
|287 J/(kg)(K )](173 K)
1
1
p
0.610
= 0.610 kg/m
= 1.64 m3/kg
72
CHAPTER 2
Fundamental Thoughts
Note: We reiterate: It is worthwhile to remember that
1
atm = 2116 lb/ft2
1 atm = 1.01 x 105 N/m 2
EXAMPL
Note: In Examples 2.1-2.3, the units for each number that appears internally in the cal­
culations were explicitly written out next to each of the numbers. This was done to give
you practice in thinking about the units. In the present example, and in all the remaining
worked examples in this book, we discontinue this practice except where necessary for
clarity. We are using consistent units in our equations, so we do not have to worry about
keeping track of all the units internally in the mathematics. If you feed numbers ex­
pressed in terms of consistent units into your equations at the beginning of your calcula­
tion and you go through a lot of internal mathematical operations (addition, subtraction,
multiplication, differentiation, integration, division, etc) to get your answer, that answer
will automatically be in the proper consistent units.
Consider the Concorde supersonic transport flying at twice the speed of sound at an
altitude of 16 km. At a point on the wing, the metal surface temperature is 362 K. The im­
mediate layer of air in contact with the wing at that point has the same temperature and is
at a pressure of 1.04 x 104 N/m2. Calculate the air density at this point.
■ Solution
From Eq. (2.3),
P
p = ——,
R T'
-1
where R = 287 (kg)(K)
The given pressure and temperature are in the appropriate consistent SI units. Hence,
P=
1.04 x 104
(287X362)
0.100 kg/m-’
We know the answer must be in kilograms per cubic meter because these are the consis­
tent units for density in the SI system. We simply write the answer as 0.100 kg/m 3 with­
out needing to trace the units through the mathematical calculation.
EXAMPLE
This example deals with the conversion of units from one system to another.
An important design characteristic of an airplane is its wing loading, defined as the
weight of the airplane, W , divided by its planform wing area (the projected wing area
you see by looking directly down on the top of the wing), S. (The importance of wing
loading, W IS, on the performance of an airplane is discussed at length in Chap. 6 .)
Consider the Lockheed-Martin F-117A stealth fighter, shown in Fig. 2.11. In most mod­
ern international aeronautical publications, the wing loading is given in units of kg/ / n r .
For the F-l 17A, the wing loading is 280.8 kg//m 2. Calculate the wing loading in units
of lb/ft2.
2.5
Specific Volume
Figure 2.11 Three-view of the Lockheed-Martin FI 17A stealth fighter.
■ Solution
We want to convert from kg/ to lb and from m2 to ft2. Some useful intermediate conver­
sion factors obtained from App. C are itemized in the following.
1 ft = 0.3048 m
1 lb = 4.448 N
In addition, from Eq. (2.7), a mass of 1 kg weighs 1 kg/, and from Eq. (2.8), the same 1 kg
mass weighs 9.8 N. Thus, we have as an additional conversion factor,
1kgy = 9.8 N
I recommend the following ploy to carry out conversions of units easily and accurately.
Consider the ratio (1 ft/0.3048 m). Since 1 foot is exactly the same length as 0.3048 m,
this is a ratio of the “same things”; hence, philosophically you can visualize this ratio as
like “unity” (although the actual number obtained by dividing 1 by 0.3048 is obviously
not one). Hence, we can visualize that the ratios
(
l ft \ / 1 lb \ / l k g / \
\0 .3 0 4 8 m / ' \4 .4 4 8 N / \ 9 . 8 N /
are like “unity.” Then, to convert the wing loading given in kg//m 2 to lb/ft2, we simply
take the given wing loading in kg//m 2 and multiply it by the various factors of “unity” in
just the right fashion so that various units cancel out, and we end up with the answer in
lb/ft2. That is,
73
74
CHAPTER 2
Fundamental Thoughts
The quantitative number for W/S is, from Eq. (2.12)
W _ (280.8)(9.8)(0.3048)2
S ~
4448
“ 57'3
The units that go along with this number are obtained by canceling various units as they
appear in the numerators and denominators of Eq. (2.12), that is,
Z = 2.0.8 i t ( M i ) ( - U U
S
( ° 1ft
- ^ )) ! =
m2 \ 1 kg^/ \ 4.448N / \
57
3—
ft2
EXAM PLE 2.6
This example also deals with the conversion of units.
In common everyday life in the United States, we frequently quote velocity in units
of miles per hour. The speedometer in our car is primarily calibrated in miles per hour
(although in many new cars, the dial also shows kilometers per hour in finer print). In the
popular aeronautical literature, airplane velocities are frequently given in miles per hour.
(After their successful flight on December 17, 1903, Orville telegraphed home that the
speed of the Wright Flyer was 31 miles per hour, and miles per hour has been used for air­
plane flight speeds since that time.) Miles per hour, however, is not in consistent units;
neither miles nor hours are consistent units. In order to make proper calculations using
consistent units, we must convert miles per hour into feet per second or meters per
second.
Consider a Piper Cub, a small, light, general aviation airplane shown in Fig. 2.12a;
the Piper Cub is a design that dates before World War II, and many are still flying today.
When the airplane is flying at 60 mi/h, calculate the velocity in terms of (a) ft/s, (b) m/s.
■ Solution
We recall the commonly known conversion factors
1 mi = 5280 ft
1 h = 3600 s
Also, from App. C,
1 ft = 0.3048 m
a.
=
(60™)
( - £ - ) ( 2 *2 )
\
h / \ 3600 s / \ 1 mi /
2 .5
Specific Volume
Figure 2.12a P ip e r C u b , o n e o f th e m o st fa m o u s lig h t, g e n e ra l a v ia tio n a irc ra ft.
(Source: From the collection o f Hal Andrews and David Ostrowski.)
Figure 2.12b N o rth A m e ric a n P-51D M u s ta n g o f W o rld W ar II fam e.
76
chapter
2
Fundamental Thoughts
This answer provides a very useful conversion factor by itself. It is very simple and help­
ful to memorize that
60 mi/h —88 ft/s
For example, consider a World War II P-51 Mustang (Fig. 2.12b) flying at 400 mi/h. Its
velocity in ft/s can easily be calculated from
y = 400
b.
V
=
( 88 ft/s \
^
= 586.7 ft/s
\ 60 mi/h /
( 60^ \
( ” * * \ r °-3Q48m^
V h ) \ 3600 8 / V i mi y v
i ft )
V=
26.82 m/s
Hence,
60 mi/h - 26.82 m/s
2.6 ANATOMY OF THE AIRPLANE
In regard to fundam ental thoughts, it is appropriate to discuss som e basic nom en­
clature associated with airplanes and space vehicles— nam es for the machine
them selves. In this section we deal with airplanes; space vehicles are discussed
in Sec. 2.7.
The m ajor com ponents o f a conventional airplane are identified in Fig. 2.13.
The fuselage is the center body, where m ost o f the usable volum e o f the airplane
is found. The fuselage carries people, baggage, other payload, instrum ents, fuel,
and anything else that the airplane designer puts there. The wings are the main
Figure 2.13 Basic com ponents o f an airplane.
2.6
Anatomy of the Airplane
Figure 2.14 Control surfaces and flaps.
lift-producing com ponents o f the airplanes; the left and right wings are so iden­
tified as you w ould see them from inside the airplane, facing forward. The inter­
nal volum e o f the w ings can be used for such item s as fuel tanks and storage o f
the main landing gear (the w heels and supporting struts) after the gear is
retracted. The horizontal and vertical stabilizers are located and sized so as to
provide the necessary stability for the airplane in flight (we consider stability in
Chap. 7). Som etim es these surfaces are called the horizontal and vertical tails, or
fins. W hen the engines are m ounted from the wings, as show n in Fig. 2.13, they
are usually housed in a type o f shroud called a nacelle. As a historical note, the
French w orked hard on flying m achines in the late 19th and early 20th centuries;
as a result, som e o f our conventional airplane nom enclature today com es from
the French. Fuselage is a French word, m eaning a “spindle” shape. So is the
word nacelle, m eaning a “sm all boat.”
Flaps and control surfaces are highlighted in Fig. 2.14. These are hinged sur­
faces, usually at the trailing edge (the back edge) o f the wings and tail, that can
be rotated up or down. The function o f a flap is to increase the lift force on the
airplane; flaps are discussed in detail in Sec. 5.17. Som e aircraft are designed
with flaps at the leading edge (the front edge) o f the wings as well as at the trail­
ing edge. Leading-edge flaps are not shown in Fig. 2.14. The ailerons are control
surfaces that control the rolling m otion o f the airplane around the fuselage. For
exam ple, when the left aileron is deflected dow nw ard and the right aileron is
deflected upw ard, lift is increased on the left wing and decreased on the right
wing, causing the airplane to roll to the right. The elevators are control surfaces
that control the nose up-and-dow n pitching m otion; when the elevator is de­
flected dow nw ard, the lift on the tail is increased, pulling the tail up and the nose
of the airplane dow n. The rudder is a control surface that can turn the nose o f the
airplane to the right or left (called yaw ing). The nature and function o f these con­
trol surfaces are discussed in greater detail in Chap. 7.
In aeronautics, it is com m on to convey the shape o f an airplane by m eans of
a three-view diagram , such as that show n in Fig. 2.11 and in Fig. 2.15. Proceeding
chapter
2
Fundamental Thoughts
from the top to the bottom o f Fig. 2.15, we see a front view, top view, and side
view, respectively, o f the N orth A m erican F- 8 6 H, a fam ous je t fighter from the
Korean w ar era. A three-view diagram is particularly im portant in the design
process o f a new airplane because it conveys the precise shape and dim ensions o f
the aircraft.
2.6
Anatomy of the Airplane
79
DESIGN BOX
This is the first o f many design boxes in this book.
These design boxes highlight information pertinent
to the philosophy, process, and details o f flight vehi­
cle design, as related to the local discussion at that
point in the text. The purpose o f these design boxes is
to reflect on the design im plications o f various topics
being discussed. This is not a book on design. But the
fundamental information in this book certainly has
applications to design. The design boxes are here to
bring these applications to your attention. D esign is a
vital function, indeed usually the end product, o f en­
gineering. These design boxes can give you a better
understanding o f aerospace engineering.
This design box is associated with our discus­
sion o f the anatomy o f the airplane and three-views.
An example o f a much more detailed three-view dia­
gram is that in Fig. 2.16, which show s the Vought
F4U Corsair, the famous Navy fighter from World
War II. Figure 2.16 is an example o f what, in the air­
plane design process, is called a configuration layout.
In Fig. 2.16, we see not only the front view, side view,
top view, and bottom view o f the airplane, but also
the detailed dim ensions, the cross-sectional shape
o f the fuselage at different locations, the airfoil shape
o f the wing at different locations, landing gear de­
tails, and the location o f various lights, radio antenna,
etc. (A discussion o f the role o f the configuration lay­
out in airplane design can be found in Anderson, Air­
craft Performance and Design, McGraw-Hill, New
York. 1999.)
The internal structure o f an airplane is frequently illustrated by a cutaway
draw ing, such as that shown in Fig. 2.17. Here the fam ous Boeing B-17 bom ber
from W orld War II is shown with a portion o f its skin cut away so that the inter­
nal structure is visible. A lthough the B-17 is a late 1930s’ design, it is show n here
because o f its historical significance and because it represents a conventional air­
plane structure. A cutaw ay o f the Lockheed-M artin F-117A stealth fighter is
shown in Fig. 2.18; this is a m odern airplane, yet its internal structure is not un­
like that for the B-17 shown in Fig. 2.17. C utaw ay diagram s usually contain
m any details about the internal structure and packaging for the airplane.
Any student o f the history o f aeronautics knows that airplanes have been de­
signed with a wide variety o f shapes and configurations. It is generally true that
form follow s function, and airplane designers have configured their aircraft to
meet specific requirem ents. However, airplane design is an open-ended
problem — there is no single “right w ay” or “right configuration” to achieve the
design goals. A lso, airplane design is an exercise in com prom ise; to achieve
good airplane perform ance in one category, other aspects o f perform ance may
have to be partly sacrificed. For exam ple, an airplane designed for very high
speed may have poor landing and takeoff perform ance. A design feature that
optim izes the aerodynam ic characteristics may overly com plicate the structural
design. C onvenient placem ent o f the engines may disrupt the aerodynam ics of
the airplane. And so forth. For this reason, airplanes com e in all sizes and shapes.
An exhaustive listing o f all the different types o f airplane configurations is not
our purpose here. O ver the course o f your studies and work, you will sooner or
later encounter m ost o f these types. However, there are several general classes of
airplane configurations that we do mention here.
E N G IN E E R U G
SPECIFICATIONS
N O TE
ONLY M AJO R P A N E LIN G SHOW N
ON T H E S E TH R E E PAGES OF DRAWINGS
H AM ILTO N S T A N D AR D HYD R OM ATIC
P R O P E L L E R . N O M IN A L D IA I 3 ' 4 "
B L AD ES F L A T B L A C K . C H R O M E Y E LLO W T IP P E D . N A T U R A L M E T A L
HUB A S S E M B L Y
SCALE
B A R , m «1 «rs
»‘2V
A L T E R N A T IN G B L A C K
ft W H ITE S T R IP E S
STANDARD BU T NOT
ALWAYS E M P L O Y E D
rs%m
12* AVERAGE
J_____
3 2 * 8 “ G O O D Y E A R T IR E .
STAND AR D
Figure 2.16 Vought F4U-1D Corsair. Drawing by Paul Matt.
(Courtesy o f Aviation Heritage, Inc., Destín, F L )
20* 3 “
N OTE
D U E T O P R O N O U N C E D IN B O A R D A N m E D R A l
A N O O U T B O A R D D IH E D R A L , W IN G S D RAW N
IN P E R S P E C T IV E - T R U E M E C H A N IC A L P L A N
V IE W L A Y O U T OF O U T E R P A N E L SH O W N IN
D O T T E O L IN E
P O S IT IO N O F F O R M A T IO N
L IG H T . W HEN E M P L O Y E D
Figure 2.16 (continued)
00
N
E M P T Y SM ELL
E J E C T IO N C H U TE S
M AC H IN E GUNS 7 ” ON C E N T E R
T IX E O A T 0 * 4 4 ' PO IN TIN G IN B O A R D
O R IG IN A L A R E A U S E O FOR M E C H A N IC A L ft T E C H N IC A L
O A TA R E C O R O IN G . L A T E R E M P L O Y E D FOR A IR C R A F T
STA TU S NUMBER
T R A P D OOR V E N T . C O O L E R S
T OPSID E
C O N T R O LL E D
B O M B R AC K
C A T A P U L T HOOK
T R A P D OOR V E N T . E N G M E C O M P O N E N T S
S O C K E T F IT T IN G FOR M E C H A N IC S
S E R V IC E S T A N D
SCALE
BAR
m e te rs
6
WEIGHT E M P T Y
U S E R JL LO A D , N O R M A L
GROSS WEIGHT. N O R M A L
O V ER LO A O
WWG AREA
M AXIMUM SPEED A T 2 3 , 0 0 0 fe e t
CRUISING SPEEO
S T A LL IN G SPEED
C L IM B . IN IT IA L . S L .
SERVIC E C EILIN G
F U E L CAPACITY. N O R M A L
RANGE - A T C R U ISE. IN T E R N A L F U E L
P O W E R - P R A T T » W H ITN E Y R - 2 8 0 0 - 8
R -2 8 0 0 -8 W
AR M AM EN T -
6
6 , 9 6 2 lbs.
3 . 0 5 7 16»
1 2 .0 3 9 lb *
1 4 .0 0 9 lbs
31 4 s * f f
3 9 5 / 4 1 7 m p h.
« 0 /1 0 5 m p h
87 m Ah.
2 . 8 9 0 f e e t/n a n 3 6 . 0 0 0 fe e t
2 2 5 90IS
1.015 m ile s
2 . 0 0 0 h p. a t T A K E O F F
1 . 8 0 0 h p o t 1 5 .5 0 0 fe e t
( w a te r in je c tio n ). 2 . 1 0 0 tip.
5 0 C A L I0 R E BROW NINGS. 3 9 l r p g
2 0 0 0 lb BO M B LO A D OR E O U IV E L E N T
Figure 2.16 (concluded )
A iR F O IL SEC T IO N C E N T E R S E C T IO N N A C A 2 3 0 1 0
A T R O O T TO 2 3 0 1 5
OUTER PANEL N AC A 2 3 0 1 5 A T
R O O T T O 2 3 0 0 0 A T T IP
T A t SURFACES C V
S P E C IA L .
S Y M M E T R IC A L
S P O IL E R B O AR D . C O R R E C T E D
N O T IC A B L E T W IT C H T O R IGH T
U N D E R C E R T A IN C O N D IT IO N S
N A T IO N A L IN S IG N IA V A R IE D IN S IZ E
A N O P O S IT IO N . S T A N D A R D 5 0 " S TA R
ON 5 4 " D ISC P O S IT IO N V A R IE D A S
M UCH AS 9 ' IN B O A R D A N D 16 O U T B O A R D
R E C O G N IT IO N
L IG H T S
Figure 2.17 Cutaway drawing of the Boeing B-17.
(Source: From Bill Gunston, Classic World War II Aircraft Cutaw ays, Osprey Publishing, London, England, 1995.)
chapter
2
Fundamental Thoughts
Figure 2.18 Cutaway view of the Lockheed-Martin F-l 17A stealth fighter.
The first is the conventional configuration. This is exem plified by the aircraft
shown in Figs. 2.13 through 2.17. Here we see monoplanes (a single set o f
w ings) with a horizontal and vertical tail at the back o f the aircraft. The aircraft
may have a straight wing, as seen in Figs. 2.13, 2.14, 2.16, and 2.17, or a sw ept
wing, as seen in Fig. 2.15. W ing sweep is a design feature that reduces the aero­
dynam ic drag at speeds near to or above the speed o f sound, and that is why most
high-speed aircraft today have som e type o f sw ept wing. Swept w ings are dis­
cussed in greater detail in Sec. 5.16. It is an idea that goes back as far as 1935.
Figure 2.15 illustrates an airplane w ith a swept-back wing. A erodynam ically,
the sam e benefit can be obtained by sw eeping the w ing forward. Figure 2.19 is a
three-view o f the X -29A , a research aircraft with a sw ept-forw ard wing. Sweptforw ard wings are not a new idea. H owever, sw ept-forw ard wings have com bined
aerodynam ic and structural features that tend to cause the wing to tw ist and fail
structurally. This is why m ost sw ept-w ing airplanes have used sw ept-back wings.
W ith the new, high-strength com posite m aterials o f today, sw ept-forw ard wings
can be designed strong enough to resist this problem ; the sw ept-forw ard wing of
the X -29A is a com posite wing. T here are som e advantages aerodynam ically to a
sw ept-forw ard w ing, which are discussed in Sec. 5.16. A lso note by com paring
Figs. 2.15 and 2.19 that the juncture o f the w ing and the fuselage is farther back
on the fuselage for the airplane with a sw ept-forw ard wing than for an airplane
with a sw ept-back wing. A t the w ing-fuselage juncture, there is extra structure
(such as a wing spar that goes through the fuselage) that can interfere with the in­
ternal packaging in the fuselage. The sw ept-forw ard wing configuration, with its
m ore rearw ard fuselagc-w ing juncture, can allow the airplane designer greater
flexibility in placing the internal packaging inside the fuselage. In spite o f these
advantages, at the tim e o f w riting, no new civilian transports or m ilitary airplanes
are being designed with sw ept-forw ard wings.
2.6
Anatomy of the Airplane
Figure 2.19 T h re e -v ie w o f th e G ru m m a n X -2 9 A r e se a rc h a irc ra ft.
The X -29A show n in Fig. 2.19 illustrates another som ew hat unconventional
feature— the horizontal stabilizer is m ounted ahead o f the wing rather than at the
rear o f the airplane. This is defined as a canard configuration, and the horizontal
stabilizer in this location is called a canard surface. The 1903 Wright Flyer was
a canard design, as clearly seen in Figs. 1.1 and 1.2. However, other airplane
designers after the W rights quickly placed the horizontal stabilizer at the rear o f
the airplane. (There is som e evidence that this was done m ore to avoid patent dif­
ficulties with the W rights than for technical reasons.) T he rear horizontal tail
location is part o f the conventional aircraft configuration; it has been used on the
vast m ajority o f airplane designs since the Wright Flyer. One reason for this is
the feeling am ong som e designers that a canard surface has a destabilizing effect
on the airplane (to call the canard a horizontal “stabilizer” m ight be considered
by som e a m isnom er). However, a properly designed canard configuration can be
just as stable as a conventional configuration. This is discussed in detail in
Chap. 7. Indeed, there are som e inherent advantages o f the canard configuration,
as we outline in Chap. 7. Because o f this, a num ber o f new canard airplanes have
been designed in recent years, ranging from private, general aviation airplanes to
military, high-perform ance fighters. (The word canard com es from the French
word for duck.)
Look again at the Wright Flyer in Figs. 1.1 and 1.2. This aircraft has two
wings m ounted one above the other. The W rights called this a double-decker
configuration. H ow ever, w ithin a few years such a configuration was called a
biplane, nom enclature that persists to the present. In contrast, airplanes with just
one set o f wings are called monoplanes; Figs. 2.13 through 2.19 illustrate m ono­
planes, w hich have becom e the conventional configuration. However, this was
not true through the 1930s; until about 1935, biplanes were the conventional
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Fundamental Thoughts
Figure 2.20 Three-view of the Grumman F3F-2, the last U.S. Navy
biplane fighter.
configuration. Figure 2.20 is a three-view o f the G rum m an F3F-2 biplane de­
signed in 1935. It was the U.S. N avy’s last biplane fighter; it was in service as a
front-line fighter with the Navy until 1940. The popularity o f biplanes over
m onoplanes in the earlier years was m ainly due to the enhanced structural
strength o f tw o shorter w ings trussed together com pared to that o f a single,
longer-span wing. However, as the cantilevered wing design, introduced by the
G erm an engineer H ugo Junkers as early as 1915, gradually becam e more ac­
cepted, the main technical reason for the biplane evaporated. However, old habits
are som etim es hard to change, and the biplane rem ained in vogue far longer than
any technical reason would justify. Today, biplanes still have som e advantages as
sport aircraft for aerobatics and as agricultural spraying aircraft. So the biplane
design still lives on.
2 .7
Anatomy of a Space Vehicle
2.7 ANATOMY OF A SPACE VEHICLE
In Sec. 2.6, we discussed the conventional airplane configuration. In contrast, it
is difficult to define a “conventional” spacecraft configuration. The shape, size,
and arrangem ent o f a space vehicle are determ ined by its particular m ission, and
there are as m any (if not m ore) different spacecraft configurations as there are
m issions. In this section we discuss a few o f the better-know n space vehicles; al­
though our coverage is far from com plete, it does provide som e perspective on
the anatom y o f space vehicles.
To date, all hum an-m ade space vehicles are launched into space by rocket
boosters. A rather conventional booster is the D elta three-stage rocket, show n in
Fig. 2.21. Built by M cD onnell-D ouglas (now m erged with Boeing), the Delta
rocket is a product o f a long design and developm ent evolution that can be traced
to the T hor interm ediate-range ballistic m issile in the late 1950s. The spacecraft
to be launched into space is housed inside a fairing at the top o f the booster,
which falls aw ay after the booster is out o f the earth’s atm osphere. The rocket
booster is really three rockets m ounted on top o f one another. The technical rea­
sons for having such a m ultistage booster (as opposed to a single-stage rocket)
are discussed in Sec. 9.11. A lso, the fundam entals o f the rocket engines that
pow er these boosters are discussed in Chap. 9.
A not-so-conventional booster is the air-launched Pegasus, shown in
Fig. 2.22. The Pegasus is a three-stage rocket that is carried aloft by an airplane.
The booster is then launched from the airplane at som e altitude within the sensi­
ble atm osphere. The first stage o f the Pegasus has wings, which assist in boost­
ing the rocket to higher altitudes w ithin the sensible atm osphere.
The Delta rocket in Fig. 2.21 and the Pegasus in Fig. 2.22 are exam ples o f
expendable launch vehicles; no part o f these boosters is recovered for reuse.
There are certain econom ies to be realized by recovering part o f (if not all) the
booster and using it again. T here is great interest today in such recoverable
launch vehicles. An exam ple o f such a vehicle is the experim ental X-34, shown
in Fig. 2.23. This is basically a w inged booster that will safely fly back to earth
after it has launched its payload, to be used again for another launch.
In a sense, the Space Shuttle is partly a reusable system. The Space Shuttle
is part airplane and part space vehicle. The Space Shuttle flight system is shown
in Fig. 2.24. The shuttle orbiter is the airplanelike configuration that sits on the
side o f the rocket booster. The system is pow ered by tw o solid rocket boosters
(SRBs) that burn out and are jettisoned after the first 2 min o f flight. The SRBs
are recovered and refurbished for use again. The external tank carries liquid oxy­
gen and liquid hydrogen for the main propulsion system , w hich com prises the
rocket engines m ounted in the orbiter. The external tank is jettisoned ju st before
the system goes into orbit; the tank falls back through the atm osphere and is
destroyed. The orbiter carries on with its m ission in space. W hen the mission is
com plete, the orbiter reenters the atm osphere and glides back to earth, m aking a
horizontal landing, as a conventional unpow ered airplane would.
Let us now exam ine the anatom y o f the payload itself— the functioning
spacecraft that m ay be a satellite in orbit around earth or a deep-space vehicle on
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2
Fundamental Thoughts
Delta 3914
Delta 3920
Spacecraft
fairing
Third stage
Thiokol TE 364-4
motor
Spin table
Guidance
system (DIGS)
Support cone
M ini-skirt
TRW
TR-201 engine
Fuel tank
Center body
LOX tank
Thrust
augmentation
Thiokol Castor IV
motors
(nine locations)
Engine
com partm ent
Thrust
augm entation
Thiokol Castor IV
motors
(nine locations)
s
\
R ocketdyne-------j g g g g g
RS-27 engine
Figure 2.21 Delta 3914 and 3920 rocket booster configurations.
(Source: From M. D. Griffin and J. R. French, Space Vehicle Design, AIAA: Reslon, Virginia, 1991.)
its w ay to another planet or to the sun. As m entioned earlier, these spacecraft are
point designs for different specific m issions, and therefore it is difficult to define
a conventional configuration for spacecraft. However, let us exam ine the
anatom y o f a few o f these point designs, ju st to obtain som e idea o f their nature.
A com m unications satellite is shown in Fig. 2.25. This is the FLTSATCOM
spacecraft produced by TRW for the U.S. Navy. It is placed in a geostationary
2.7
Anatomy of a Space Vehicle
Payload
Figure 2.22 Orbital Sciences Pegasus, an air-launched rocket booster.
(Source: From C. H. Eldred et al.. “Future Space Transportation Systems and Launch, ”
in Future Aeronautical and Space System s, eds. A. K. Noor and S. L. Vennera, AlAA,
Progress in Astronautics and Aeronautics, vol. 172, 1997.)
Figure 2.23 Orbital Sciences X-34 small reusable rocket booster.
(Source: From Eldred et al.)
orbit— an orbit in the plane o f the equator w ith a period (tim e to execute one
orbit) o f 24 h. Hence, a satellite in geostationary orbit appears above the same lo­
cation on earth at all tim es— a desirable feature for a com m unications satellite.
Orbits and trajectories for space vehicles are discussed in Chap. 8 . The construc­
tion is basically alum inum . T he tw o hexagonal com partm ents (buses) m ounted
chapter
2
Fundamental Thoughts
(2) solid rocket
Figure 2.24 The Space Shuttle.
(Source: From Griffin and French.)
NORMAL ORBIT:
LAUNCH:
GEOSTATIONARY
1978
ETR
ATLAS-CENTAUR
CENTAUR STANDARD FAIRING
SPACECRAFT WEIGHT
4100 LB
IN ORBITAL OPERATION
THE SPACECRAFT IS 3 AXIS STABILIZED
WITH THE BODY FIXED ANTENNA
POINTING CONSTANTLY AT THE EARTH
AND THE SOLAR ARRAY ROTATED TO
POINT AT THE SUN
Figure 2.25 The TRW communications satellite FLTSATCOM.
(Source: From Griffin and French.)
2.7
Anatomy of a Space Vehicle
o
Rover UHF antenna
c>
Im ager for Mars
Pathfinder (IMP)
O '
Lander LGA
Lander HGA
M icrorover
Solar panel
Figure 2.26 The Mars Pathfinder on the surface o f Mars.
(Source: From M. K. Olsen el al„ "Spacecraft fo r Solar System Exploration, "in Future
Aeronautical and Space Systems, eds. A. K. Noor and S. L. Venneri, AIAA Progress in
Astronautics and Aeronautics, vol. 172, 1997.)
one above the other at the center o f the satellite contain all the engineering sub­
system s necessary for control and com m unications. The two antennas that pro­
ject outw ard from the top o f the bus are pointed at earth. The two solar array arms
(solar panels) that project from the sides o f the bus constantly rotate to remain
pointed at the sun at all times. The solar panels provide pow er to run the equip­
m ent on the spacecraft.
The M ars Pathfinder spacecraft is sketched in Figs. 2.26 and 2.27. This
spacecraft successfully landed on the surface o f M ars in 1997. The package that
entered the M artian atm osphere is shown in an exploded view in Fig. 2.27. The
aeroshell and backshell make up the aerodynam ic shape o f the entry body, with
the lander packaged in a folded position inside. The function o f this aerodynam ic
entry body is to create drag to slow the vehicle as it approaches the surface of
M ars and to protect the package inside from aerodynam ic heating during atm o­
spheric entry. The dynam ics o f a spacecraft entering a planetary atm osphere, and
entry aerodynam ic heating, are discussed in Chap. 8 . Figure 2.26 shows the
Pathfinder lander after it has been deployed on Ihe M artian surface. The rover,
solar panel, high-gain and low -gain antennas, and im ager for taking the pictures
transm itted from the surface are shown in Fig. 2.26.
chapter
2
Fundamental Thoughts
2.65 m
Cruise stage
1.0 m
Backshell
I 3 in
Folded lander
Aeroshell
Figure 2.27 Components of the Mars
Pathfinder space vehicle.
(Source: From Olsen et at.)
Som e spacecraft are designed sim ply to fly by (rather than land on) planets
in the solar system , taking pictures and transm itting detailed scientific data back
to earth. C lassic exam ples are the M ariner 6 and 7, tw o identical spacecraft
launched in 1969 to study the surface and atm osphere o f Mars. The configura­
tion o f these spacecraft is show n in Fig. 2.28. M ariner 6 flew past M ars with a
distance o f closest approach o f 3429 km on July 28, 1969, and M ariner 7
zipped by M ars with a distance o f closest approach o f 3430 km on A ugust 5,
1969, both sending back im portant inform ation on the M artian atm ospheric
com position, pressure and tem perature, and on the nature o f M ars’ heavily
cratered surface. Exam ining Fig. 2.28, we see the eight-sided m agnesium centerbody supporting four rectangular solar panels; the centerbody housed the
Control C om puter and Sequencer designed to operate M ariner independently
w ithout intervention from ground control o f earth. A ttached to the centerbody
are tw o television cam eras for w ide-angle and narrow -angle scanning o f the
M artian surface.
Voyager 2, arguably our m ost spectacular and successful deep-space probe,
is show n in Fig. 2.29. Launched on A ugust 20, 1977, this spacecraft was de­
signed to explore the outer planets o f our solar system . In April 1979, it began to
transm it im ages o f Jupiter and its m oons. Speeding on to Saturn, Voyager pro­
vided detailed im ages o f S aturn’s rings and m oons in A ugust 1981. Although
these tw o planetary encounters fulfilled Voyager's prim ary m ission, the m ission
planners at NA SA ’s Jet Propulsion Laboratory sent it on to U ranus, where clos­
est approach o f 71,000 km occurred on January 24, 1986. From the data sent
back to earth, scientists discovered 10 new m oons o f Uranus. A fter a m id-course
correction, Voyager skim m ed 4500 km over the cloud tops o f Neptune and then
headed on a course that w ould take it out o f the solar system . A fter the Neptune
2 .7
Anatomy of a Space Vehicle
Low-gain antenna
High-gain antenna
Canopus sensor
Solar panels
Temperature control
louvers
M idcourse motor nozzle
Low-gain antenna
Attitude control
gas jets
High-gain antenna
Solar panels
W ide-angle television
IK Radiom eter
UV S pectro m eter-------
IR Spectrom eter
-----Narrow-angle television
Figure 2.28 TWo views o f the Mariner 6 and 7, identical spacecraft that flew by Mars in 1969.
encounter, N A SA form ally renam ed the entire project the Voyager Interstellar
M ission, and the spacecraft’s instrum ents w ere put on low pow er to conserve en­
ergy. In N ovem ber 1998, m ost instrum ents w ere turned off, leaving only seven
essential instrum ents still operating. Today, Voyager is m ore than 10 billion km
from earth, and still going. A lthough data from the rem aining operating instru­
m ents could be obtained as late as 2 0 2 0 when pow er levels are expected to dip
c h a p te r
L O W -F ia D
MAGNETOMETERS
2
Fundamental Thoughts
H IG H -G A IN A N T E N N A
(Boom length, 13m)
PLASMA
COSM IC RAY
IM A G IN G WIDE A N G LE
IM A G IN G NARROW A N G LE
ULTRAVIOLET SPECTROMETER
INFRARED
SPECTROMETER
A N D RADIOMETER
PHOTOPOLARIMETER
LO W -EN ER G Y
CHARGED
PARTICLE
OPTICAL CALIBRATION
TARGET
PLANETARY RADIO
A S TR O N O M Y A N D
PLASMA WAVE
A N TE N N A S
RADIOISOTOPE
THERMOELECTRIC
GENERATORS
(10m length)
Figure 2.29 Voyager 2 spacecraft.
too low for receiving on earth, Jet Propulsion Laboratory engineers in early 2003
finally turned o ff the sw itches; Voyager had provided m ore than enough pioneer­
ing scientific data.
Exam ining the configuration o f Voyager 2 show n in Fig. 2.29, we see a clas­
sic spacecraft arrangem ent. Because o f the m ultiplanet flyby, the scientific in­
strum ents show n in Fig. 2.29 had to have an unobstructed view o f each planet
with the planet at any position w ith respect to the spacecraft. This led to the de­
sign o f an articulated instrum ent platform show n on the right side o f the space­
craft in Fig. 2.29. The high-gain antenna shown at the top in Fig. 2.29 was
pointed tow ards earth by m aneuvering the Voyager.
In summary, there are about as many different spacecraft configurations as
there are different m issions in space. Spacecraft fly in the near vacuum o f space
where there is virtually no aerodynam ic force, no lift or drag, exerted on the
vehicle. H ence, the spacecraft designer can make the external configuration
w hatever he or she w ants. This is not true for the airplane designer. The external
configuration o f an airplane (fuselage, w ings, etc.) dictates the aerodynam ic lift
and drag on the airplane, and the airplane designer m ust optim ize the configura­
tion for efficient flight through the atm osphere. A irplanes, therefore, share a
much more com m on anatom y than spacecraft. The anatom y o f spacecraft is all
over the map. This section on the anatom y o f spacecraft contains ju st a sam pling
o f different configurations, ju st to give you a feeling for their design.
2.8
Historical Note: The NACA and NASA
2.8 HISTORICAL NOTE: THE NACA AND NASA
NASA — four letters that have m eaning to virtually the entire world. Since its in­
ception in 1958, the N ational A eronautics and Space A dm inistration has been
front-page news, m any tim es good new s and som etim es not so good, with the
Apollo space flight program to the m oon, the Space Shuttle, the space station,
etc. Since 1958, N A SA has also been in charge o f developing new technology for
airplanes— technology thqt allow s us to fly farther, faster, safer, and cheaper. In
short, the professional w orld o f aerospace engineering is driven by research car­
ried out by NASA. Before N A SA , there was the NA CA , the National Advisory
C om m ittee for A eronautics, which carried out seminal research pow ering tech­
nical advancem ents in flight during the first h alf o f the 20th century. Before we
progress further in this book dealing with an introduction to flight, you should
have an understanding o f the historical underpinnings o f NACA and NA SA and
a better appreciation for the im pact these tw o agencies have had on aerospace
engineering. The N A C A and N A SA have been fundam ental to the technology of
flight. It is fitting, therefore, that we place this particular historical note in the
chapter dealing with “fundam ental thoughts.”
Let us pick up the thread o f aeronautical engineering history from Chap. 1.
A fter O rville and W ilbur W right’s dram atic public dem onstrations in the United
States and Europe in 1908, there w as a virtual explosion in aviation develop­
ments. In turn, this rapid progress had to be fed by new technical research in aero­
dynam ics, propulsion, structures, and flight control. It is im portant to realize that
then, as well as today, aeronautical research was som etim es expensive, always
dem anding in term s o f intellectual talent, and usually in need o f large testing fa­
cilities. Such research in m any cases either was beyond the financial resources of,
or seem ed too out o f the ordinary for, private industry. Thus, the fundam ental re­
search so necessary to fertilize and pace the developm ent o f aeronautics in the
20th century had to be established and nurtured by national governm ents. It is in­
teresting to note that G eorge Cayley him self (see Chap. 1) as long ago as 1817
called for “public subscription” to underw rite the expense o f the developm ent o f
airships. Responding about 80 years later, the British governm ent set up a school
for ballooning and m ilitary kite flying at Farnborough, England. By 1910, the
Royal A ircraft Factory was in operation at Farnborough with the noted Geoffrey
de H avilland as its first airplane designer and test pilot. This was the first major
governm ent aeronautical facility in history. This operation was soon to evolve
into the Royal A ircraft E stablishm ent (RAE), which conducted viable aeronauti­
cal research for the British governm ent for alm ost a century.
In the U nited States, aircraft developm ent as well as aeronautical research
languished after 1910. During the next decade, the United States em barrassingly
fell far behind Europe in aeronautical progress. This set the stage for the U.S.
governm ent to establish a formal m echanism for pulling itself out o f its aero­
nautical “dark ages.” On M arch 3, 1915, by an act o f C ongress, the National
A dvisory C om m ittee for A eronautics (N A CA ) was created, with an initial ap­
propriation o f $5000 per year for five years. This was at first a true com m ittee,
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2
Fundamental Thoughts
consisting o f 12 distinguished m em bers who were know ledgeable about aero­
nautics. A m ong the charter m em bers in 1915 were Professor Joseph S. Am es o f
Johns H opkins U niversity (later to becom e president o f Johns H opkins) and Pro­
fessor W illiam F. D urand o f Stanford University, both o f whom w ere to make
m ajor im pressions on aeronautical research in the first half century o f pow ered
flight. This advisory com m ittee, NACA, was originally to m eet annually in
W ashington, D istrict o f Colum bia, on “the Thursday after the third M onday o f
O ctober o f each year,” with any special m eetings to be called by the chair. Its
purpose was to advise the governm ent on aeronautical research and developm ent
and to bring som e cohesion to such activities in the U nited States.
T he com m ittee im m ediately noted that a single advisory group o f 12 m em ­
bers was not sufficient to breathe life into U.S. aeronautics. T heir insight is ap­
parent in the letter o f subm ittal for the first annual report o f NACA in 1915,
w hich contained the follow ing passage:
There are many practical problems in aeronautics now in too indefinite a form to en­
able their solution to be undertaken. The comm ittee is o f the opinion that one o f the
first and most important steps to be taken in connection with the com m ittee’s work
is the provision and equipment o f a flying field together with aeroplanes and suitable
testing gear for determining the forces acting on full-sized machines in constrained
and in free flight, and to this end the estimates submitted contemplate the develop­
ment o f such a technical and operating staff, with the proper equipment for the con­
duct o f full-sized experiments.
It is evident that there w ill ultimately be required a well-equipped laboratory
specially suited to the solving o f those problems which are sure to develop, but since
the equipment o f such a laboratory as could be laid down at this time might well
prove unsuited to the needs o f the early future, it is believed that such provision
should be the result o f gradual development.
So the first action o f this advisory com m ittee was to call for m ajor govern­
m ent facilities for aeronautical research and developm ent. The clouds o f w ar in
Europe— W orld War I had started a year earlier— m ade their recom m endations
even more im perative. In 1917, when the U nited States entered the conflict, ac­
tions follow ed the com m ittee’s w ords. We find the follow ing entry in the third
annual N A CA report: “To carry on the highly scientific and special investiga­
tions contem plated in the act establishing the com m ittee, and which have, since
the outbreak o f the war, assum ed greater im portance, and for which facilities do
not already exist, or exist in only a lim ited degree, the com m ittee has contracted
for a research laboratory to be erected on the Signal C orps Experim ental Station,
Langley Field, H am pton, Virginia.” The report goes on to describe a single,
tw o-story laboratory building with physical, chem ical, and structural testing lab­
oratories. The building contract was for $80,900; actual construction began in
1917. Two w ind tunnels and an engine test stand were contem plated “in the near
future.” The selection o f a site 4 mi north o f Ham pton, Virginia, was based on
general health conditions and the problem s o f accessibility to W ashington and
the larger industrial centers o f the east, protection from naval attack, clim atic
conditions, and cost o f the site.
2.8
Historical Note: The NACA and NASA
Thus, the Langley M em orial A eronautical R esearch Laboratory was born. It
was to rem ain the only N A CA laboratory and the only m ajor U.S. aeronautical
laboratory o f any type for the next 20 years. Nam ed after Sam uel Pierpont
Langley (see Sec. 1.7), it pioneered in wind tunnel and flight research. O f partic­
ular note is the airfoil and wing research perform ed at Langley during the 1920s
and 1930s. We return to the subject o f airfoils in Chap. 5, at w hich tim e the
reader should note that the airfoil data included in App. D were obtained at
Langley. W ith the w ork that poured out o f the Langley laboratory, the United
States took the lead in aeronautical developm ent. High on the list o f accom plish­
ments, along with the system atic testing o f airfoils, was the developm ent o f the
N A C A engine cow l (see Sec. 6.19), an aerodynam ic fairing built around radial
piston engines that dram atically reduced the aerodynam ic drag o f such engines.
In 1936, Dr. G eorge Lew is, w ho was then N A CA D irector o f A eronautical
R esearch (a position he held from 1924 to 1947), toured m ajor European labora­
tories. He noted that N A CA ’s lead in aeronautical research was quickly disap­
pearing, especially in light o f advances being m ade in Germany. As W orld War II
drew close, N A C A clearly recognized the need for two new laboratory operations:
an advanced aerodynam ics laboratory to probe into the m ysteries o f high-speed
(even supersonic) flight and a m ajor engine-testing laboratory. These needs even­
tually led to the construction o f A m es A eronautical Laboratory at M offett Field,
near M ountain View, C alifornia (authorized in 1939), and Lew is Engine Research
Laboratory at Cleveland, O hio (authorized in 1941). A long with Langley, these
two new N A C A laboratories again helped to spearhead the U nited States to the
forefront o f aeronautical research and developm ent in the 1940s and 1950s.
The dawn o f the space age occurred on O ctober 4, 1957, when Russia
launched Sputnik I, the first artificial satellite to orbit the earth. Sw allow ing its
som ew hat em barrassed technical pride, the U nited States m oved quickly to com ­
pete in the race for space. On July 29, 1958, by another act o f Congress (Public
Law 85-568), the N ational A eronautics and Space A dm inistration (N ASA) was
bom . A t this sam e m om ent, N A C A cam e to an end. Its program s, people, and
facilities w ere instantly transferred to N A SA , lock, stock, and barrel. However,
N A SA was a larger organization than ju st the old NA CA; it absorbed in addition
num erous A ir Force, Navy, and A rm y projects for space. W ithin two years o f its
birth, NA SA was authorized four new m ajor installations: an existing Army
facility at H untsville, A labam a, renam ed the G eorge C. M arshall Space Flight
Center; the G oddard Space Flight C enter at G reenbelt, M aryland; the M anned
Spacecraft C enter (now the Johnson Spacecraft Center) in Houston, Texas; and
the Launch O perations C enter (now the John F. Kennedy Space C enter) at Cape
Canaveral, Florida. These, in addition to the existing but slightly renam ed
Langley, Am es, and Lewis research centers, w ere the backbone o f NASA. Thus,
the aeronautical expertise o f N A C A now form ed the seeds for N A SA , shortly
thereafter to becom e one o f the w orld’s most im portant forces in space
technology.
This capsule sum m ary o f the roots o f N A CA and N A SA is included in this
chapter on fundam ental thoughts because it is virtually im possible for a student
chapter
2
Fundamental Thoughts
or practitioner o f aerospace engineering in the U nited States not to be influenced
o r guided by N A CA or NA SA data and results. The extended discussion on air­
foils in Chap. 5 is a case in point. Thus, N A CA and NASA are “fundam ental” to
the discipline o f aerospace engineering, and it is im portant to have som e im pres­
sion o f the historical roots and tradition o f these organizations. Hopefully, this
short historical note provides such an im pression. A m uch better im pression can
be obtained by taking a journey through the NACA and NASA technical reports
in the library, going all the w ay back to the first NACA report in 1915. In so
doing, a panoram a o f aeronautical and space research through the years will un­
fold in front o f you.
2.9 Summary
Som e o f the major ideas in this chapter are listed as follow s.
1.
2.
3.
The language o f aerodynamics involves pressure, density, temperature, and
velocity. In turn, the illustration o f the velocity field can be enhanced by drawing
streamlines for a given flow.
The source o f all aerodynamic forces on a body is the pressure distribution and the
shear-stress distribution over the surface.
A perfect gas is one in which intermolecular forces can be neglected. For a perfect
gas, the equation o f state which relates p , p , and T is
p = pR T
4.
(2.3)
where R is the specific gas constant.
To avoid confusion, errors, and a number o f unnecessary “conversion factors” in
the basic equations, always use consistent units. In this book, SI units (newton,
kilogram, meter, second) and the English engineering system (pound, slug, foot,
second) are used.
Bibliography
Anderson, John D., Jr.: Aircraft Performance and Design, W CB/M cGraw-Hill, N ew
York, 1999.
Gray, George W.: Frontiers o f Flight, Knopf, N ew York, 1948.
Griffin, Michael D., and James R. French: Space Vehicle Design, American Institute o f
Aeronautics and Astronautics, Reston, VA, 1991.
Hartman, E. R: Adventures in Research: A History o f Ames Research Center
1940-1965, N A SA SP-4302, 1970.
Mechtly, E. A.: The International System o f Units, N A SA SP-7012, 1969.
Problems
2.1
Consider the low -speed flight o f the Space Shuttle as it is nearing a landing. If the
air pressure and temperature at the nose o f the shuttle are 1.2 atm and 300 K,
respectively, what are the density and specific volume?
2.2
Consider 1 kg o f helium at 500 K. A ssum ing that the total internal energy o f
helium is due to the mean kinetic energy o f each atom summed over all the atoms,
Problems
3
4
5
6
7
calculate the internal energy of this gas. Note: The molecular weight of helium
is 4. Recall from chemistry that the molecular weight is the mass per mole of gas;
that is, 1 mol of helium contains 4 kg of mass. Also, 1 mol of any gas contains
6.02 x 1026 molecules or atoms (Avogadro’s number).
Calculate the weight of air (in pounds) contained within a room 20 ft long, 15 ft
wide, and 8 ft high. Assume standard atmospheric pressure and temperature of
2116 lb/ft2 and 59UF, respectively.
Comparing with the case of Prob. 2.3, calculate the percentage change in the total
weight of air in the room when the air temperature is reduced to —10°F (a very
cold winter day), assuming the pressure remains the same at 2116 lb/ft2.
If 1500 lb,„ of air is pumped into a previously empty 900 ft3 storage tank and the
air temperature in the tank is uniformly 70°F, what is the air pressure in the tank in
atmospheres?
In Prob. 2.5, assume the rate at which air is being pumped into the tank is
0.5 lb„,/s. Consider the instant in time at which there is 1000 lb,„ of air in the tank.
Assume the air temperature is uniformly 50°F at this instant and is increasing at
the rate of I F/min. Calculate the rate of change of pressure at this instant.
A ssum e that, at a point on the wing o f the Concorde supersonic transport, the air
temperature is —10°C and the pressure is 1.7 x 104 N /m 2. Calculate the density at
this point.
At a point in the test section of a supersonic wind tunnel, the air pressure and
temperature are 0.5 x 10* N/m 2 and 240 K, respectively. Calculate the specific
volume.
9 Consider a Hat surface in an aerodynamic flow (say a flat sidewall of a wind tunnel).
The dimensions of this surface are 3 ft in the flow direction (the x direction) and 1 ft
perpendicular to the flow direction (the y direction). Assume that the pressure
distribution (in pounds per square foot) is given by p = 2116 — 10* and is
independent of y. Assume also that the shear-stress distribution (in pounds per
square foot) is given by t w = 9 0 /(jt -I- 9 ) 1/2 and is independent of y. In the above
expressions, x is in feet, and x — 0 at the front of the surface. Calculate the
magnitude and direction of the net aerodynamic force on the surface.
10 A pitcher throws a baseball at 85 miles per hour. The flowfield over the baseball
moving through the stationary air at 85 miles per hour is the same as that over a
stationary baseball in an airstream that approaches the baseball at 85 miles per
hour. (This is the principle of wind tunnel testing, as will be discussed in Chap. 4.)
This picture of a stationary body with the flow moving over it, is what we adopt
here. Neglecting friction, the theoretical expression for the flow velocity over the
surface of a sphere (hence the baseball) is V = | V,» sin ft. Here, Vx is the
airstream velocity (the freestream velocity far ahead of the sphere). An arbitrary
point on the surface of the sphere is located by the intersection of the radius of the
sphere with the surface, and G is the angular position of the radius measured from
a line through the center in the direction of the freestream (i.e., the most forward
and rearward points on the spherical surface correspond to 8 = 0 ° and 180°,
respectively. The velocity V is the flow velocity at that arbitrary point on the
surface. Calculate the values of the minimum and maximum velocity at the
surface, and the location of the points at which these occur.
11 Consider an ordinary, helium-filled party balloon with a volume of 2.2 ft3. The
lifting force on the balloon due to the outside air is the net resultant of the pressure
8
100
CHAPTER 2
Fundamental Thoughts
distribution exerted on the exterior surface of the balloon. Using this fact,
Archimedes principle can be derived, namely that the upward force on the balloon
is equal to the weight of the air displaced by the balloon. Assuming the balloon is
at sea level, where the air density is 0.002377 slug/ft3, calculate the maximum
weight that can be lifted by the balloon. Note: The molecular weight of air is 28.8
and that of helium is 4.
2.12 In the four-stroke, reciprocating, internal combustion engine that powers most
automobiles as well as most small general aviation aircraft, combustion of the
fuel-air mixture takes place in the volume between the top of the piston and the
top of the cylinder. (Reciprocating engines are discussed in Chap. 9.) The gas
mixture is ignited when the piston is essentially at the end of the compression
stroke (called top dead center), when the gas is compressed to a relatively high
pressure and is squeezed into the smallest volume that exists between the top of
the piston and the top of the cylinder. Combustion takes place rapidly, before the
piston has much time to start down on the power stroke. Hence, the volume of the
gas during combustion stays constant; that is, the combustion process is at
constant volume. Consider the case where the gas density and temperature at the
instant combustion starts are 11.3 kg/m 1 and 625 K, respectively. At the end of the
constant volume combustion process, the gas temperature is 4000 K. Calculate the
gas pressure at the end of the constant volume combustion. Assume that the
specific gas constant for the fuel-air mixture is the same as that for pure air.
2.13 For the conditions of Prob. 2.12, calculate the force exerted on the top of the
piston by the gas at (a) the beginning of combustion and (b) the end of
combustion. The diameter of the circular piston face is 9 cm.
2.14 In a gas turbine jet engine, the pressure of the incoming air is increased by flowing
through a compressor; the air then enters a combustor that vaguely looks like a
long can (sometimes called the combustion can). Fuel is injected in the combustor,
burns with the air, and then the burned fuel-air mixture exits the combustor at a
higher temperature than the air coming into the combustor. (Gas turbine jet
engines are discussed in Chap. 9.) The pressure of the flow through the combustor
remains relatively constant; that is, the combustion process is at constant pressure.
Consider the case where the gas pressure and temperature entering the combustor
are 4 x 106 N/m 2 and 900 K, respectively, and the gas temperature existing the
combustor is 1500 K. Calculate the gas density at (a) the inlet to the combustor
and (b) the exit of the combustor. Assume the specific gas constant for the fuel-air
mixture is the same as that for pure air.
____________________ ______________________ C
H
A
P
T
E
R
The Standard Atmosphere
Sometimes gentle, sometimes capricious, sometimes awful, never the same for two
moments together; almost human in its passions, almost spiritual in its tenderness,
almost divine in its infinity.
John Rus kin, The Sky
erospace vehicles can be divided into tw o basic categories: atm ospheric
vehicles such as airplanes and helicopters, which always fly within the
sensible atm osphere, and space vehicles such as satellites, the Apollo
lunar vehicle, and deep-space probes, which operate outside the sensible atm o­
sphere. However, space vehicles do encounter the earth’s atm osphere during
their blastoffs from the earth ’s surface and again during their reentries and
recoveries after com pletion o f their m issions. If the vehicle is a planetary probe,
then it may encounter the atm ospheres o f Venus, M ars, Jupiter, etc. Therefore,
during the design and perform ance o f any aerospace vehicle, the properties o f the
atm osphere m ust be taken into account.
The earth’s atm osphere is a dynam ically changing system, constantly in a
state o f flux. The pressure and tem perature o f the atm osphere depend on altitude,
location on the globe (longitude and latitude), tim e o f day, season, and even solar
sunspot activity. To take all these variations into account when considering the
design and perform ance o f flight vehicles is im practical. Therefore, a standard
atmosphere is defined in order to relate flight tests, wind tunnel results, and
general airplane design and perform ance to a com m on reference. The standard
101
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3
The Standard Atmosphere
PREVIEW BOX
Before you jump into a strange water pond or dive
into an unfamiliar swimming pool, there are a few
things you might like to know. How cold is the
water? How clean is it? How deep is the water? These
are things that might influence your swimming per­
formance in the water, or even your decision to go
swimming at all. Similarly, before we can study the
performance of a flight vehicle speeding through the
air, we need to know something about the properties
of the air itself. Consider an airplane flying in the
atmosphere, or a space vehicle blasting through the
atmosphere on its way up to space, or a vehicle com­
ing back from space through the atmosphere. In all
these cases, the performance of the flight vehicle is
going to be dictated in part by the properties of the
atmosphere—the temperature, density, and pressure
of the atmosphere.
What are the properties of the atmosphere? We
know they change with altitude, but how do they
change? How do we find out? These are important
questions, and they are addressed in this chapter. Be­
fore you can go any further in your study of flight ve­
hicles, you need to know about the atmosphere. Here
is the story—please read on.
atm osphere gives m ean values o f pressure, tem perature, density, and other
properties as functions o f altitude; these values are obtained from experim ental
balloon and sounding-rocket m easurem ents com bined with a m athem atical
model o f the atm osphere. To a reasonable degree, the standard atm osphere re­
flects average atm ospheric conditions, but this is not its main im portance. Rather,
its m ain function is to provide tables o f com m on reference conditions that can be
used in an organized fashion by aerospace engineers everyw here. The purpose o f
this chapter is to give you som e feeling for what the standard atm osphere is all
about and how it can be used for aerospace vehicle analyses.
We m ight pose the rather glib question: Just what is the standard atm o­
sphere? A rather glib answ er is: The tables in Apps. A and B at the end o f this
book. Take a look at these two appendixes. They tabulate the tem perature, pres­
sure, and density for different altitudes. A ppendix A is in SI units, and App. B is
in English engineering units. W here do these num bers com e from ? Were they
sim ply pulled out o f thin air by som ebody in the distant past? A bsolutely not. The
num bers in these tables w ere obtained on a rational, scientific basis. O ne purpose
o f this chapter is to develop this rational basis. A nother purpose is to show you
how to use these tables.
T he road map for this chapter is given in Fig. 3.1. We first run dow n the left
side o f the road map, establishing som e definitions and an equation from basic
physics (the hydrostatic equation) that are necessary tools for constructing the
num bers in the standard atm osphere tables. Then we move to the right side o f the
road m ap and discuss how the num bers in the tables are actually obtained. We go
through the construction o f the standard atm osphere in detail. Finally, we define
s o m e term s that are d eriv ed from the n u m b ers in the ta b les— the pressu re, d en ­
sity, and tem perature altitudes— which are in alm ost everyday use in aeronautics.
Finally, we note that the details o f this chapter are focused on the determ ina­
tion o f the standard atm osphere for earth. The tables in Apps. A and B are for the
3.1
Definition of Altitude
Figure 3.1 Road map for Chap. 3.
earth’s atm osphere. However, the physical principles and techniques discussed
in this chapter are also applicable to constructing model atm ospheres for other
planets, such as Venus, M ars, and Jupiter. So the applicability o f this chapter
reaches far beyond the earth.
It should be m entioned that several different standard atm ospheres exist,
com piled by different agencies at different tim es, each using slightly different
experim ental data in the models. F or all practical purposes, the differences are
insignificant below 30 km (100,000 ft), which is the dom ain o f contem porary air­
planes. A standard atm osphere in com m on use is the 1959 A RD C model
atm osphere. (A RD C stands for the U.S. A ir Force’s previous Air Research and
D evelopm ent C om m and, which is now the A ir Force Research Laboratory.) The
atm ospheric tables used in this book are taken from the 1959 A RD C model
atm osphere.
3.1 DEFINITION OF ALTITUDE
Intuitively, we all know the m eaning o f altitude. We think o f it as the distance
above the ground. But like so m any other general term s, it m ust be more pre­
cisely defined for quantitative use in engineering. In fact, in the follow ing sec­
tions we define and use six different altitudes: absolute, geom etric, geopotential,
pressure, tem perature, and density altitudes.
First, im agine that we are at D aytona Beach, Florida, where the ground is at
sea level. If we could lly straight up in a helicopter and drop a tape m easure to
the ground, the m easurem ent on the tape would be. by definition, the geom etric
altitude h u , that is, the geom etric height above sea level.
Now, if we bored a hole through the ground to the center o f the earth and ex ­
tended our tape m easure until it hit the center, then the m easurem ent on the tape
would be, by definition, the absolute altitude hu. If r is the radius o f the earth,
then h„ = hG + r . This is illustrated in Fig. 3.2.
The absolute altitude is im portant, especially for space flight, because the
local acceleration o f gravity g varies with h„. From N ew ton’s law o f gravitation,
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chapter
3
The Standard Atmosphere
g varies inversely as the square o f the distance from the center o f the earth. By
letting go be the gravitational acceleration at sea level, the local gravitational ac­
celeration g at a given absolute altitude ha is
*■*"(£) =*°(^)
(3 j)
T he variation o f g with altitude must be taken into account when you are dealing
with m athem atical m odels o f the atm osphere, as discussed in the follow ing
sections.
3.2 HYDROSTATIC EQUATION
We will now begin to piece together a m odel that will allow us to calculate
variations o f p , p , and T as functions o f altitude. The foundation o f this model is
the hydrostatic equation, which is nothing m ore than a force balance on an ele­
m ent o f fluid at rest. C onsider the small stationary fluid elem ent o f air shown in
Fig. 3.3. We take for convenience an elem ent with rectangular faces, where the
top and bottom faces have sides o f unit length and the side faces have an infinitesim ally small height d h G- On the bottom face, the pressure p is felt, which
gives rise to an upw ard force o f p x l x 1 exerted on the fluid elem ent. The top
face is slightly higher in altitude (by the distance d h G), and because pressure
varies with altitude, the pressure on the top face will be slightly different from
that on the bottom face, differing by the infinitesim ally small value dp. Hence,
3.2
Hydrostatic Equation
P + dp
<lT
s
Cd
GO
C
* —H
C/3
C3
U
B
o
C
Figure 3.3 Force diagram for the hydrostatic equation.
on the top face, the pressure p + d p is felt. It gives rise to a dow nw ard force o f
(/; + é//?)( I ) ( l ) on the fluid elem ent. M oreover, the volum e o f the fluid elem ent
is ( l ) ( l ) d h G = d h c , and since p is the m ass per unit volum e, then the m ass o f
the fluid elem ent is sim ply p ( l ) ( l ) d h G = p d h G. If the local acceleration o f
gravity is g, then the w eight o f the fluid elem ent is gp d h G, as shown in Fig. 3.3.
The three forces shown in Fig. 3.3, pressure forces on the top and bottom , and the
weight m ust balance because the fluid elem ent is not moving. Hence,
p = p + dp + p g d h c
Thus,
dp = ~ P g d h G
(3.2)
Equation (3.2) is the hydrostatic equation and applies to any fluid o f density p,
for exam ple, w ater in the ocean as well as air in the atm osphere.
Strictly speaking, Eq. (3.2) is a differential equation; that is, it relates an infinitesim ally sm all change in pressure d p to a corresponding infinitesim ally
small change in altitude d h G, w here in the language o f differential calculus, d p
and d h G are differentials. A lso note that g is a variable in Eq. (3.2); g depends on
hG as given by Eq. (3 .1).
To be m ade useful, Eq. (3.2) should be integrated to give what we want,
namely, the variation o f pressure with altitude p = p ( h G). To sim plify the inte­
gration, we m ake the assumption that g is constant throughout the atm osphere,
equal to its value at sea level go- This is som ething o f a historical convention in
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chapter
3
The Standard Atmosphere
aeronautics. H ence, we can w rite Eq. (3.2) as
dp = -pgodh
(3.3)
However, to m ake Eqs. (3.2) and (3.3) num erically identical, the altitude h in
Eq. (3.3) m ust be slightly different from hG in Eq. (3.2), to com pensate for the
fact that g is slightly different from gQ. Suddenly, we have defined a new altitude
h, which is called the geopotential altitude and which differs from the geom etric
altitude. To better understand the concept o f geopotential altitude, consider a
given geom etric altitude, hG, where the value o f pressure is p. Let us now in­
crease the geom etric altitude by an infinitesim al am ount, d h G, such that the new
geom etric altitude is hG + d h G. At this new altitude, the pressure is p + d p ,
where the value o f d p is given by Eq. (3.2). Let us now put this same value o f d p
in Eq. (3.3). Dividing Eq. (3.3) by (3.2), we have
■-(?)(«)
Clearly, since go and g are different, then dh and d h G m ust be different; that is,
the num erical values o f dh and d h G that correspond to the same change in pres­
sure, d p , are different. As a consequence, the num erical values o f h and hG that
correspond to the sam e actual physical location in the atm osphere are different
values.
For the practical mind, geopotential altitude is a “fictitious” altitude, defined
by Eq. (3.3) for ease o f future calculations. However, many standard atm osphere
tables quote their results in term s o f geopotential altitude, and care m ust be taken
to m ake the distinction. Again, geopotential altitude can be thought o f as that ficti­
tious altitude that is physically com patible with the assum ption o f g = const = go-
3.3 RELATION BETWEEN GEOPOTENTIAL
AND GEOMETRIC ALTITUDES
We still seek the variation o f p with geom etric altitude p = p { h G). However, our
calculations using Eq. (3.3) will give, instead, p = p (h ). Therefore, we need to
relate h to hG, as follow s. Dividing Eq. (3.3) by (3.2), we obtain
l =
go dh
g dhc
or
8_
dh = — d h G
go
(3.4)
We substitute Eq. (3 .1) into (3.4):
dh =
r2
(■r + hGy
dhG
(3.5)
By convention, we set both h and h G equal to zero at sea level. Now, consider a
given point in the atm osphere. This point is at a certain geom etric altitude hG,
3 .4
Definition of the Standard Atmosphere
and associated w ith it is a certain value o f h (different from h G). Integrating
Eq. (3.5) betw een sea level and the given point, we have
rh
I
Jo
pho
dh =
I
Jo
Jl
■■
/•ha
dhG = r 2
(r + |f * o )
Jo
dh(
(r + h G)2
h
h = r 2 ( ------ — )
\ r + h a Jo
Thus,
= r 2 ( ------^ - + - >) = r 2 f
\ r + hG
r )
\
h =
r + r + hc \
( r + h G) r )
(3.6)
r + hc
where h is geopotential altitude and hG is geom etric altitude. This is the desired
relation betw een the tw o altitudes. W hen we obtain relations such as p = p(h ),
we can use Eq. (3.6) to subsequently relate p to hG.
A quick calculation using Eq. (3.6) show s that there is little difference be­
tween h and hG for low altitudes. For such a case, hG <SC r, r / ( r + h G) % 1;
hence, h
hG. Putting in num bers, r = 6.356766 x 106 m (at a altitude o f 45°),
and if hG = 1 km (about 23,000 ft), then the corresponding value o f h is, from
Eq. (3.6), h — 6.9923 km , about 0.1 o f 1 percent difference! Only at altitudes
above 65 km (213,000 ft) does the difference exceed 1 percent. (N ote that 65 km
is an altitude at which aerodynam ic heating o f NASA’s Space Shuttle becom es
im portant during reentry into the earth ’s atm osphere from space.)
3.4 DEFINITION OF THE STANDARD ATMOSPHERE
We are now in a position to obtain p, T, and p as functions o f h for the standard
atm osphere. The keystone o f the standard atm osphere is a defined variation o f
T with altitude, based on experim ental evidence. This variation is shown in
Fig. 3.4. Note that it consists o f a series o f straight lines, som e vertical (called the
constant-tem perature, or isothermal, regions) and others inclined (called the
gradient regions). Given T = T {h) as defined by Fig. 3.4, then p = p (h ) and
p — p (h ) follow from the laws o f physics, as shown in the following.
First, consider again Eq. (3.3):
d p = —p g o d h
Divide by the equation o f state, Eq. (2.3):
dp
~p =
pgo dh
g„
~pRTr = ~~RT
(3.7)
C onsider first the isotherm al (constant-tem perature) layers o f the standard
atm osphere, as given by the vertical lines in Fig. 3.4 and sketched in Fig. 3.5. The
tem perature, pressure, and density at the base o f the isotherm al layer shown in
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The Standard Atmosphere
T e m p e ra tu re , K
Figure 3.4 Temperature distribution in the standard atmosphere.
Fig. 3.5 are T\, p \, and p \, respectively. The base is located at a given geopoten­
tial altitude h\. N ow consider a given point in the isotherm al layer above the
base, w here the altitude is h. The pressure p at h can be obtained by integrating
Eq. (3.7) betw een h\ and h.
JPl
P
fd „
R T Jhi
(3.8)
3 .4
Definition of the Standard Atmosphere
Note that go, R, and T are constants that can be taken outside the integral. (This
clearly dem onstrates the sim plification obtained by assum ing that g = go =
const, and therefore dealing w ith geopotential altitude h in the analysis.) Per­
form ing the integration in Eq. (3.8), we obtain
P = ~ ^S =o (t hu ~ hui )\
In —
Pi
RT
i
or
p_ . e -[*o/(«D](*-*i)
(3.9)
Pi
From the equation o f state,
<3.
P i^ i
Pi
Thus,
p_
_
ii
II
p
Pi
e -lg0/(RTm-h,)
(3.10)
Pi
Equations (3.9) and (3.10) give the variation o f p and p versus geopotential alti­
tude for the isotherm al layers o f the standard atm osphere.
C onsidering the gradient layers, as sketched in Fig. 3.6, we find the tem per­
ature variation is linear and is geom etrically given as
T — T\ _ d T
h — h\
_
dh
w here a is a specified constant for each layer obtained from the defined tem per­
ature variation in Fig. 3.4. The value o f a is som etim es called the lapse rate for
109
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The Standard Atmosphere
T i, p i, Pi
Figure 3.6 Gradient layer.
the gradient layers.
a =
Thus,
—
~dh
dh = - d T
a
We substitute this result into Eq. (3.7):
dp> _ _ g o _ dT^
p
aR
T
(3.11)
Integrating betw een the base o f the gradient layer (shown in Fig. 3.6) and some
point at altitude h, also in the gradient layer, Eq. (3.11) yields
r p dp
pi
p
In —
Pi
Thus,
=._ Ü L dL
a R Jit ,T~
T
T
■_ i l
=
aR
.In —
7 1,
(3.12)
3 .4
Definition of the Standard Atmosphere
From the equation o f state,
P
pT
P\
P\T\
Hence, Eq. ( 3 .12) becom es
- g n /U tR )
-txo/(fl*)]-l
-([« o /(fl« )]+ l)
(3.13)
or
Recall that the variation p f T is linear with altitude and is given the specified
relation
T = T\ + a(h — h\)
(3.14)
Equation (3.14) gives T
T (h) for the gradient layers; when it is plugged into
Eq. (3.12), we obtain p = p (h )\ sim ilarly from Eq. (3.13) we obtain p = p(h ).
Now we can see how the standard atm osphere is pieccd together. Looking at
Fig. 3.4, start at sea level (h = 0), where standard sea level values o f pressure,
density, and tem p eratu re^/?.,, p.,, and 7’(, respectively— are
Ps = 1.01325 x 105 N /ni 2 = 2116.2 lb/ft 2
pf = 1.2250 kg/m 3 = 0.002377 slug/ft 3
7; = 288.16 K = 518.69'JR
These are the base values for the first gradient region. Use Eq. (3.14) to obtain
values o f T as a function o f h until T = 216.66 K, which occurs at h = 11.0km .
With these values o f T. use Eqs. (3.12) and (3.13) to obtain the corresponding
values o f p and p in the first gradient layer. Next, starting at h = 11.0 km as the
base o f the first isotherm al region (see Fig. 3.4), use Eqs. (3.9) and (3.10) to cal­
culate values o f p and p versus h, until h = 25 km, which is the base o f the next
gradient region. In this manner, with Fig. 3.4 and Eqs. (3.9), (3.10), and (3.12) to
(3.14), a table o f values for the standard atm osphere can be constructed.
Such a table is given in App. A for SI units and App. B for English engineer­
ing units. Look at these tables carefully and becom e fam iliar with them . They are
the standard atm osphere. The first colum n gives the geom etric altitude, and the
second colum n gives the corresponding geopotential altitude obtained from
Eq. (3.6). The third through fifth colum ns give the corresponding standard values
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The Standard Atmosphere
DESIGN BOX
The first step in the design process o f a new aircraft is
the determination o f a set o f specifications, or re­
quirements, for the new vehicle. These specifications
may include such performance aspects as a stipulated
maximum velocity at a given altitude or a stipulated
maximum rate o f climb at a given altitude. These per­
formance parameters depend on the aerodynamic
characteristics o f the vehicle, such as lift and drag. In
turn, the lift and drag depend on the properties o f the
atmosphere. When the specifications dictate certain
performance at a given altitude, this altitude is taken
to be the standard altitude in the tables. Therefore, in
the preliminary design o f an airplane, the designer
uses the standard atmosphere tables to define the
pressure, temperature, and density at the given alti­
tude. In this fashion, many calculations made during
the preliminary design o f an airplane contain infor­
mation from the standard altitude tables.
o f tem perature, pressure, and density, respectively, for each altitude, obtained
from the previous discussion.
We em phasize again that the standard atm osphere is a reference atm o­
sphere only and certainly does not predict the actual atm ospheric properties
that may exist at a given tim e and place. For exam ple, App. A says that at an al­
titude (geom etric) o f 3 km, p = 0.70121 x ÍO5 N /m 2, T = 268.67 K, and p =
0.90926 kg/m 3. In reality, situated where you are, if you could right now levitate
yourself to 3 km above sea level, you w ould m ost likely feel a p , T, and p dif­
ferent from the values obtained from App. A. The standard atm osphere allow s us
only to reduce test data and calculations to a convenient, agreed-upon reference,
as will be seen in subsequent sections o f this book.
Comment: Geometric and Geopotential Altitudes Revisited We now can
appreciate better the m eaning and significance o f the geom etric altitude, hG, and
the geopotential altitude, h. The variation o f the properties in the standard atm o­
sphere are calculated from Eqs. (3.9) to (3.14). These equations are derived using
the sim plifying assum ption o f a constant value o f the acceleration o f gravity
equal to its value at sea level; that is, g = constant = g0. Consequently, the
altitude that appears in these equations is, by definition, the geopotential altitude,
h. Exam ine these equations again— you see go and h appearing in these equa­
tions, not g and hG. The sim plification obtained by assum ing a constant value o f
g is the sole reason for defining the geopotential altitude. This is the only use o f
geopotential altitude we will make in this book— for the calculation o f the num ­
bers that appear in Apps. A and B. M oreover, since h and hG are related via
Eq. (3.6), w e can alw ays obtain the geom etric altitude, hG, that corresponds to a
specified value o f geopotential altitude, h. The geom etric altitude, hG, is the
actual height above sea level and therefore is m ore practical. That is why the first
colum n in Apps. A and B is h G, and the entries are in even intervals o f h G. The
second colum n gives the corresponding values o f h, and these are the values used
to generate the corresponding num bers for p, p , and T via Eqs. (3.9) to (3.14).
3 .4
Definition of the Standard Atmosphere
113
In the subsequent chapters in this book, any dealings with altitude involv­
ing the use o f the standard atm osphere tables in Apps. A and B will be couched
in term s o f the geom etric altitude, ha ■ For exam ple, if reference is m ade to a
“standard altitude” o f 5 km, it m eans a geom etric altitude o f hc = 5 km. Now
that we have seen how the standard atm osphere tables are generated, after
the present chapter w e will have no reason to deal anym ore with geopotential
altitude.
H opefully, you now have a better understanding o f the statem ent m ade at
the end o f Sec. 3.2 that geopotential altitude is sim ply a “fictitious” altitude,
defined by Eq. (3.3) for the single purpose o f sim plifying the subsequent
derivations.
EXAM PLE 3.1
Calculate the standard atmosphere values o f T, p, and p at a geopotential altitude o f 14 km.
■ Solution
Remember that T is a defined variation for the standard atmosphere. Hence, w e can im­
mediately refer to Fig. 3.4 and find that at h = 14 km,
T = 216.66 K
To obtain p and p , w e must use Eqs. (3.9) to (3.14), piecing together the different regions
from sea level up to the given altitude with which we are concerned. Beginning at sea
level, the first region (from Fig. 3.4) is a gradient region from h = 0 to h = 11.0 km. The
lapse rate is
dT
2 1 6 . 6 6 - 288.16
a = -tr- = ------------------------= —6.5 K/km
dh
1 1 .0 - 0
a = - 0 .0 0 6 5 K/m
or
Therefore, using Eqs. (3.12) and (3.13), which are for a gradient region and where the
base o f the region is sea level (hence p\ = 1.01 x 105 N /m 2 and p\ = 1.23 kg/m 3), we
find at h = 11.0 km
( T\
p « p , y
m /(« K>
~
, f , \\ -9.8/1-0.0065(287)]
// r,.
216.66
*
= ( i ° i * 1^ ( 28 ^ )
where #0 = 9.8 m /s2 in SI units. Hence, p (at h = 11.0km ) = 2.26 x l()4 N /m 2.
. -|tfo/(aft)+l]
/2 1 6 .6 6 V
= (1 .2 3 )|.................
1V 2 8 8 .1 6 /
= 0.367 kg/m3
-(9.8/|-0.0065(287)]+l)
at * = 11.0 km
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The Standard Atmosphere
The above values o f p and p now form the base values for the first isothermal region
(see Fig. 3.4). The equations for the isothermal region are Eqs. (3.9) and (3.10),
where now p\ = 2.26 x 104 N /m 2 and p\ = 0.367 kg/m3. For h = 14 km, h — h\ =
14 - 11 = 3 km = 3000 m. From Eq. (3.9),
p = p |g -l*o/(«7')](A-*i) _ (2.26 x 104)«_[98/287(216-66)K3000)
p = 1.41 x 104 N/m2
From Eq. (3.10),
Hence,
p
P_
P_
P\
P\
1.41 x 104
P = P\ —
Í7v»
p i = ° - 3 6 7 ^2.26 —x 104
0.23 kg/m3
These values check, within roundoff error, with the values given in App. A. Note: This
exam ple demonstrates how the numbers in Apps. A and B are obtained!
3.5 PRESSURE, TEMPERATURE, AND
DENSITY ALTITUDES
W ith the tables o f Apps. A and B in hand, we can now define three new
“altitudes” — pressure, tem perature, and density altitudes. This is best done
by exam ple. Im agine that you are in an airplane flying at som e real, geom etric
altitude. The value o f your actual altitude is im m aterial for this discussion.
H ow ever, at this altitude, you m easure the actual outside air pressure to be
6.16 x 104 N /m 2. From App. A, you find that the standard altitude that corre­
sponds to a pressure o f 6.16 x 10 4 N /m 2 is 4 km. Therefore, by definition, you say
that you are flying at a pressure altitude o f 4 km. Sim ultaneously, you m easure
the actual outside air tem perature to be 265.4 K. From App. A, you find that the
standard altitude that corresponds to a tem perature o f 265.4 K is 3.5 km. T here­
fore, by definition, you say that you are flying at a temperature altitude o f 3.5 km.
Thus, you are sim ultaneously flying at a pressure altitude o f 4 km and a tem ­
perature altitude o f 3.5 km w hile your actual geom etric altitude is yet a differ­
ent value. The definition o f density altitude is m ade in the same vein. These
quantities— pressure, tem perature, and density altitudes— are ju st convenient
num bers that, via App. A or B, are related to the actual p, T, and p for the actual
altitude at which you are flying.
EXAM PLE 3.2
If an airplane is flying at an altitude where the actual pressure and temperature are 4 .7 2 x
104 N /m 2 and 255.7 K, respectively, what are the pressure, temperature, and density altitudes?
3.5
Pressure, Temperature, and Density Altitudes
115
■ Solution
For the pressure altitude, look in App. A for the standard altitude value corresponding to
p = 4.72 x 104 N /m 2. This is 6000 m. Hence,
Pressure altitude = 6000 m = 6 km
For the temperature altitude, look in App. A for the standard altitude value corresponding
to T = 255.7 K. This is 5000 m. Hence,
Temperature altitude = 5000 m = 5 km
For the density altitude, w e must first calculate p from the equation o f state:
4.72 x 104
P =
RT
287(255.7)
= 0.643 kg/m
Looking in App. A and interpolating between 6.2 and 6.3 km, we find that the standard al­
titude value corresponding to p = 0.643 kg/m3 is about 6.240 m. Hence,
Density altitude = 6240 m = 6.24 km
Note that temperature altitude is not a unique value. The answer for temperature altitude
could equally well be 5.0, 38.2, or 59.5 km because o f the multivalued nature o f the
altitude-versus-temperature function. In this section, only the lowest value o f temperature
altitude is used.
EXAM PLE 3.3
The flight test data for a given airplane refer to a level-flight maximum-velocity run made
at an altitude that simultaneously corresponded to a pressure altitude o f 30,000 ft and
density altitude o f 28,500 ft Calculate the temperature o f the air at the altitude at which
the airplane was flying for the test.
■ Solution
From App. B:
For pressure altitude = 30.000 ft:
p = 629.66 lb/ft2
For density altitude = 28,500 ft:
p = 0.9408 x 10~3 slug/ft3
These are the values o f p and p that simultaneously existed at the altitude at which the
airplane was flying. Therefore, from the equation o f state,
629.66
p R f (0.94082 x I0~3) ( l 7 16)
390 R
EXAM PLE 3.4
Consider an airplane flying at som e real, geometric altitude. The outside (ambient) pres­
sure and temperature are 5.3 x 104 N /m 2 and 253 K, respectively. Calculate the pressure
and density altitudes at which this airplane is flying.
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chapter
3
The Standard Atmosphere
■ Solution
Consider the ambient pressure o f 5.3 x 104 N /m 2. In App. A, there is not a precise entry for
this number. It lies between the value p\ = 5 .3 3 1 x 1()4 N/m 2 at altitude Ac, i = 5100 m and
P 2 = 5.2621 x 104 N/m 2 at altitude/ic ,2 = 5200 m. We have at least two choices. We could
simply use the nearest entry in the table, which is for an altitude h e ,2 = 5100 m, and say that
the answer for pressure altitude is 5100 m. This is acceptable if we are making only approx­
imate calculations. However, if we need greater accuracy, w e can interpolate between en­
tries. Using linear interpolation, the value o f he corresponding to p = 5.3 x 104 N/m 2 is
he = ha,i + (ha ,2 — ha,\) ( — — —
\ P 1 “ P 2J
fto . 5 , 0 0 + (520 0 - 5 l 0 0 , ( 5| f i = J | r )
= 5 1 0 0 + 100(0.4662) = 5146.6 m
The pressure altitude at which the airplane is flying is 5146.6 m. (N ote that in this example
and in Examples 3.2 and 3.3, w e are interpreting the word altitude in the tables to be the
geometric altitude ha rather than the geopotential altitude h. This is for convenience, be­
cause h e is tabulated in round numbers, in contrast to the colum n for h. Again, at the alti­
tudes for conventional flight, the difference between he and h is not significant.)
To obtain the density altitude, calculate the density from the equation o f state.
p
5.3 x 104
,
p = - £ - = ■ ■■ ■= 0 .72992 kg/m 3
RT
(2 87)(253)
5
Once again w e note that this value o f p falls between two entries in the table. It falls be­
tween ha, 1 = 5000 m where pi = 0.73643 kg/m3 and ho ,2 = 5100 m where P2 =
0.72851 kg/m3. (Note that these subscripts denote different lines in the table from those
used in the first part o f this example. It is good never to becom e a slave to subscripts and
sym bols. Just always keep in mind the significance o f what you are doing.) We could take
the nearest entry, which is for an altitude he = 5100 m, and say that the answer for the
density altitude is 5100 m. However, for greater accuracy, let us linearly interpolate be­
tween the tw o entries.
he = he, 1 + (he.i —hc.i) ( —— —'j
\/t>i - P i )
= 5000 + (5100 - 5000)
/ 0 . 7 3 6 4 3 - 0 .7 2 9 9 2 \
--------------------------- )
\ 0.73643 — 0.72851 /
= 5 0 0 0 + 100(0.82197) = 5082.2 m
The density altitude at which the airplane is flying is 5082.2 m.
3 .6
Historical Note: The Standard Atmosphere
3.6 HISTORICAL NOTE: THE STANDARD
ATMOSPHERE
With the advent o f ballooning in 1783 (see Chap. I), people suddenly becam e in­
terested in acquiring a greater understanding o f the properties o f the atm osphere
above ground level. However, a com pelling reason for such know ledge did not
arise until the com ing o f heavier-than-air flight in the 20th century. As we shall
see in subsequent chapters, the flight perform ance o f aircraft is dependent upon
such properties as the pressure and density o f the air. Thus, a know ledge o f these
properties, or at least som e agreed-upon standard for w orldw ide reference, is ab­
solutely necessary for intelligent aeronautical engineering.
The situation in 1915 was sum m arized by C. F. M arvin, C hief o f the U.S.
W eather Bureau and chairm an o f an N A CA subcom m ittee to investigate and report
upon the existing status of atm ospheric data and knowledge. In his “Preliminary
Report on the Problem o f the A tm osphere in Relation to A eronautics,” NACA
Report No. 4, 1915, M arvin writes:
The Weather Bureau is already in possession o f an im mense amount o f data con­
cerning atmospheric conditions, including wind m ovem ents at the earth’s surface.
This information is no doubt o f distinct value to aeronautical operations, but it needs
to be collected and put in form to meet the requirements o f aviation.
The follow ing four years saw such efforts to collect and organize atm ospheric
data for use by aeronautical engineers. In 1920, the Frenchm an A. Toussaint,
director o f the A erodynam ic Laboratory at Saint-C yr-l’Ecole, France, suggested
the follow ing form ula for the tem perature decrease with height:
T = 15 - 0.0065h
where T is in degrees Celsius and h is the geopotential altitude in meters.
Toussaint’s form ula was form ally adopted by France and Italy with the Draft o f
Inter-Allied A greem ent on Law A dopted for the D ecrease o f Tem perature with
Increase o f A ltitude, issued by the M inistere de la G uerre, A eronautique
M ilitaire, Section Technique, in M arch 1920. One year later, England followed
suit. The U nited States was close behind. Since M arvin’s report in 1915, the
U.S. W eather Bureau had com piled m easurem ents o f the tem perature distribu­
tion and found T oussaint’s form ula to be a reasonable representation o f the
observed mean annual values. Therefore, at its executive com m ittee meeting of
D ecem ber 17, 1921, NACA adopted T oussaint’s form ula for airplane perfor­
mance testing, with the statement: “The subcom m ittee on aerodynam ics recom ­
mends that for the sake o f uniform practice in different countries that Tous­
saint’s form ula be adopted in determ ining the standard atm osphere up to 10 km
(33,000 ft). . .
M uch o f the technical data base that supported Toussaint’s form ula was re­
ported in NACA R eport No. 147, “Standard A tm osphere,” by W illis Ray G regg in
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The Standard Atmosphere
1922. Based on free-flight tests at M cCook Field in D ayton, O hio, and at Langley
Field in H am pton, Virginia, and on the other flights at W ashington, District o f
Colum bia, as well as artillery data from Aberdeen, M aryland, and Dahlgren,
Virginia, and sounding-balloon observations at Fort Om aha, N ebraska, and
St. Louis, M issouri, G regg was able to com pile a table o f mean annual atm o­
spheric properties. An exam ple o f his results follows:
Altitude,
m
0
1,000
2,000
5,000
10,000
Mean Annual
Temperature in
United States,
K
284.5
281.0
277.0
260.0
228.5
Temperature
from Toussaint’s
Formula,
K
288
281.5
275.0
255.5
223.0
Clearly, T oussaint’s form ula provided a sim ple and reasonable representation of
the m ean annual results in the U nited States. This was the prim ary m essage in
G reg g ’s report in 1922. However, the report neither gave extensive tables nor at­
tem pted to provide a docum ent for engineering use.
Thus, it fell to W alter S. Diehl (w ho later becam e a w ell-know n aerodynam icist and airplane designer as a captain in the Naval Bureau o f A eronautics) to
provide the first practical tables for a standard atm osphere for aeronautical use.
In 1925, in NACA Report No. TR 218, entitled (again) “Standard A tm osphere,”
D iehl presented extensive tables o f standard atm ospheric properties in both m et­
ric and English units. The tables w ere in increm ents o f 50 m up to an altitude of
10 km and then in increm ents o f 100 m up to 20 km. In English units, the tables
w ere in increm ents o f 100 ft up to 32,000 ft and then in increm ents o f 200 ft up
to a m axim um altitude o f 65,000 ft. Considering the aircraft o f that day
(see Fig. 1.31), these tables were certainly sufficient. M oreover, starting from
T oussaint’s form ula for T up to 10,769 m, then assum ing T = const = - 5 5 ° C
above 10,769 m, D iehl obtained p and p in precisely the same fashion as
described in the previous sections o f this chapter.
T he 1940s saw the beginning o f serious rocket flights, with the G erm an V-2
and the initiation o f sounding rockets. M oreover, airplanes w ere flying higher
than ever. Then, with the advent o f intercontinental ballistic m issiles in the 1950s
and space flight in the 1960s, altitudes began to be quoted in terms o f hundreds
o f m iles rather than feet. T herefore, new tables o f the standard atm osphere were
created, mainly extending the old tables to higher altitudes. Popular am ong the
various tables is the A RD C 1959 Standard A tm osphere, w hich is used in this
book and is given in Apps. A and B. For all practical purposes, the old and new
tables agree for altitudes o f greatest interest. Indeed, it is interesting to com pare
3.7
Summary
values, as shown in the follow ing:
Altitude,
m
T from
Diehl, 1925,
K
T from
ARDC, 1959,
K
288
281.5
275.0
255.5
223.0
218.0
218.0
218.0
288.16
281.66
275.16
255.69
223.26
218.03
216.66
216.66
0
1,000
2,000
5,000
10,000
10,800
11,100
20,000
So D iehl’s standard atm osphere from 1925, at least up to 20 km, is ju st as
good as the values today.
3.7 Summary
Som e o f the major ideas o f this chapter are listed as follow s.
2.
The standard atmosphere is defined in order to relate flight tests, wind tunnel
results, and general airplane design and performance to a com m on reference.
The definitions o f the standard atmospheric properties are based on a given
temperature variation With altitude, representing a mean o f experimental data. In
turn, the pressure and density variations with altitude are obtained from this
empirical temperature variation by using the laws o f physics. One o f these laws is
the hydrostatic equation:
dp = - p g d h G
(3.2)
In the isothermal regions o f the standard atmosphere, the pressure and density
variations are given by
—
=
Pi
4.
—
=
* - fo > / < * r > ] < A - * i>
(3 9 ) a n d (3
jo )
P\
In the gradient regions o f the standard atmosphere, the pressure and density
variations are given by, respectively,
-ga/(uR)
* « / * ¥
Pi
\ T\)
(3.12)
-(t(*o/(o«)]+D
(3.13)
Pi
5.
\
tJ
where T = T\ + a(h — hi) and ci is the given lapse rate.
The pressure altitude is that altitude in the standard atmosphere that corresponds to
the actual ambient pressure encountered in flight or laboratory experiments. For
119
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CHAPTER 3
The Standard Atmosphere
example, if the ambient pressure o f a flow, no matter where it is or what it is doing,
is 393.12 lb/ft2, the flow is said to correspond to a pressure altitude o f 4 0 ,0 0 0 ft
(see App. B). The same idea can be used to define density and temperature
altitudes.
Bibliography
Minzner, R. A., K. S. W. Champion, and H. L. Pond: The ARDC Model Atmosphere,
1959, Air Force Cambridge Research Center Report No. T R -59-267, U.S. Air
Force, Bedford, M A, 1959.
Problems
3.1
At 12 km in the standard atmosphere, the pressure, density, and temperature are
1.9399 x 104 N /m 2, 3.1194 x 10_l kg/m3, and 216.66 K, respectively. Using
these values, calculate the standard atmospheric values o f pressure, density, and
temperature at an altitude o f 18 km, and check with the standard altitude tables.
3.2
Consider an airplane flying at som e real altitude. The outside pressure and
temperature are 2.65 x 104 N /m 2 and 220 K, respectively. What are the pressure
and density altitudes?
3.3
During a flight test o f a new airplane, the pilot radios to the ground that she is in
level flight at a standard altitude o f 35,000 ft. What is the ambient air pressure far
ahead o f the airplane?
3.4
Consider an airplane flying at a pressure altitude o f 33,500 ft and a density altitude
o f 32,000 ft. Calculate the outside air temperature.
3.5
At what value o f the geom etric altitude is the difference h — hc, equal to 2 percent
o f the geopotential altitude, h i
3.6
U sing Toussaint’s formula, calculate the pressure at a geopotential altitude o f 5 km.
3.7
The atmosphere o f Jupiter is essentially made up o f hydrogen, H 2 . For H2, the
specific gas constant is 4157 J/(kg)(K). The acceleration o f gravity o f Jupiter is
24.9 m/s2. A ssum ing an isothermal atmosphere with a temperature o f 150 K and
assuming that Jupiter has a definable surface, calculate the altitude above that
surface where the pressure is one-half the surface pressure.
3.8
An F-15 supersonic fighter aircraft is in a rapid climb. At the instant it passes
through a standard altitude o f 25,000 ft, its time rate o f change o f altitude is
500 ft/s, which by definition is the rate o f climb, discussed in Chap. 6.
Corresponding to this rate o f climb at 2 5,000 ft is a time rate o f change o f ambient
pressure. Calculate this rate o f change o f pressure in units o f pounds per square
foot per second.
3.9
A ssum e that you are ascending in an elevator at sea level. Your eardrums are very
sensitive to minute changes in pressure. In this case, you are feeling a 1-percent
decrease in pressure per minute. Calculate the upward speed o f the elevator in
meters per minute.
3.10 Consider an airplane flying at an altitude where the pressure and temperature are
530 lb/ft2 and 390°R , respectively. Calculate the pressure and density altitudes at
which the airplane is flying.
Problems
3.11 Consider a large rectangular-shaped tank o f water open to the atmosphere, 10 ft
deep, with walls o f length 30 ft each. When the tank is filled to the top with water,
calculate the force (in tons) exerted on the side o f each wall in contact with the
water. The tank is located at sea level. (Note: The specific weight o f water is
62.4 lby/ft3, and 1 ton = 2000 lbf.) (Hint: U se the hydrostatic equation.)
3.12 A discussion o f the entry o f a space vehicle into the earth’s atmosphere after it has
completed its mission in space is given in Chap. 8. An approximate analysis o f the
vehicle motion and aerodynamic heating during atmospheric entry assumes an
approximate atmospheric m odel called the “exponential atmosphere,” where the
air density variation with altitude is assumed to be
J L = c -fo h/(RT)
Po
where po is the sea-level density and h is the altitude measured above sea level.
This equation is only an approximation for the density variation with altitude
throughout the whole atmosphere, but its sim ple form makes it very useful for
approximate analyses. U sing this equation, calculate the density at an altitude o f
45 km. Compare your result with the actual value o f density from the standard
altitude tables. In the preceding equation, assume T — 240 K (a reasonable
representation for the value o f the temperature between sea level and 45 km,
which you can see by scanning down the standard atmosphere table).
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_________________
Basic Aerodynamics
Mathematics up to the present day have been quite useless to us in regard to flying.
From the fourteenth Annual Report
of the Aeronautical Society
of Great Britain, 1879
Mathematical theories from the happy hunting grounds of pure mathematicians are
found suitable to describe the airflow produced by aircraft with such excellent
accuracy that they can be applied directly to airplane design.
Theodore von Karman, 1954
onsider an airplane flying at an altitude o f 3 km (9840 ft) at a velocity o f
112 m /s (367 ft/s or 251 mi/h). At a given point on the wing, the pressure
and airflow velocity are specific values, dictated by the laws o f nature.
One o f the objectives o f the science o f aerodynam ics is to decipher these laws
and to give us m ethods to calculate the flow properties. In turn, such inform ation
allow s us to calculate practical quantities, such as the lift and drag on the air­
plane. A nother exam ple is the flow through a rocket engine o f a given size and
shape. If this engine is sitting on the launch pad at C ape Canaveral and given
am ounts o f fuel and oxidizer are ignited in the c o m b u stio n cham b er, the flo w ve­
locity and pressure at the nozzle exit are again specific values, dictated by the
law s o f nature. The basic principles o f aerodynam ics allow us to calculate the
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123
PREVIEW BOX
At the beginning o f Chapi 2, w e imagined a vehicle
flying through the atmosphere, and one o f the first
thoughts was that there is a rush o f air over the vehi­
cle. This rush o f air generates a force— an aerody­
namic force— on the vehicle. This is an example o f
aerodynamics in action. We went on to say that aero­
dynamics was one o f the four major disciplines that
go into the design o f a flight vehicle, the others being
flight dynamics, propulsion, and structures.
What is aerodynamics? The American Heritage
Dictionary of the English Language defined aero­
dynamics as “the dynamics o f gases, especially o f
atmospheric interactions with m oving objects.’’ What
does this mean? Dynamics means motion. Gases are
a squishy substance. Is aerodynamics the dynamics
o f a squishy substance? To som e extent, yes. In con­
trast, this book is a solid object; it is easy to pick it
up and throw it across the room. In so doing, you
can easily track its velocity, acceleration, and path
through the air. This involves the dynamics o f a solid
body and is a subject you might be somewhat fam il­
iar with from a previous study o f physics. But just try
to scoop up a handful o f air and throw it across the
room. D oesn ’t make sense, does it? The air, being a
squishy substance, is just going to flow through your
fingers and go nowhere. Obviously, the dynamics o f
air (or a fluid in general) is different than the dynam­
ics o f a solid body. Aerodynamics requires a w hole
new intellectual perspective. A purpose o f this chap­
ter is to give you som e o f this new perspective.
So how do you get air to m ove? It obviously does,
because when an airplane streaks past you, the air
flows over the airplane and basically does everything
ncccssary to get out o f the way o f the airplane. From a
different perspective, imagine that you are riding in­
side the airplane, and the airplane is flying at 400 miles
per hour. If you look ahead, you see the atmospheric
air com ing towards you at 400 m iles per hour. Then it
flows up, down, and around the airplane, locally ac­
celerating and decelerating as it passes over the fuse­
lage, w ings, tail, and through the engines. The air does
more than this. It also creates a pressure distribution
and a shear stress distribution over the surface o f the
airplane that results in aerodynamic lift and drag ex­
erted on the vehicle (see again Sec. 2.2). So the air
m oves, and we repeat the question: how do you get the
air to move? Keep reading this chapter to find out.
Many engineers and scientists have spent their
professional lifetim es working on aerodynamics, so
aerodynamics must be important. Moreover, there is
a lot to aerodynamics. This chapter is long, one o f the
longest in the book, because there is a lot to aerody­
namics, and because it is important. Aerodynamics is
the dominant feature that drives the external shape o f
any flight vehicle. You can hardly take your first step
into aerospace engineering without serious consider­
ation and understanding o f aerodynamics. The pur­
pose o f this chapter is to help you take this first step
and obtain som e understanding o f aerodynamics. In
this chapter you w ill learn how to get air to move.
You will learn how to predict the pressure exerted on
the surface o f a body immersed in the flow and how
this pressure is related to the velocity o f the air. You
w ill learn about the high-speed flow o f air, with ve­
locities greater than the speed o f sound (supersonic
flow), and about shock waves that frequently occur in
supersonic flow. You will learn how to measure the
flight speed o f an airplane during flight. You will
learn why the nozzles o f rocket engines are shaped
the way they are (all due to aerodynamics). You will
learn about many applications o f aerodynamics, but
you will have to learn som e o f the fundamentals— the
concepts and equations— o f aerodynamics in the first
part o f this chapter before you can deal with applica­
tions. For all these reasons, this chapter is important;
please treat it with serious study.
A word o f caution. This chapter is going to be a
challenge to you. M ost likely the subject matter is
different than you have dealt with before. There are a
lot o f new concepts, ideas, and ways o f looking at
things. There are a lot o f new equations to help
describe all this new stuff. The material is definitely
not boring, and it can be great fun if you let it be. Ex­
pect it to be different, and go at it with enthusiasm.
Sim ply read on, and step through the door into the
world o f aerodynamics.
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exit flow velocity and pressure, which, in turn, allow us to calculate the thrust.
For reasons such as these, the study o f aerodynam ics is vital to the overall un­
derstanding o f flight. The purpose o f this chapter is to provide an introduction to
the basic laws and concepts o f aerodynam ics and to show how they are applied
to solving practical problem s.
The road m ap for this chapter is given in Fig. 4.1. Let us w alk through this
road m ap so that we can get a better idea o f w hat this chapter on aerodynam ics is
all about. First, w e can identify two basic types o f aerodynam ic flows: (1) flow
with no friction (called inviscid flow) and ( 2 ) flow with friction (called viscous
flow). T hese tw o types o f flow are represented by the tw o boxes show n near the
top o f the road map. This is an im portant distinction in aerodynam ics. Any reallife aerodynam ic flow has friction acting on the fluid elem ents m oving w ithin the
flow field. H ow ever, there are m any practical aerodynam ic problem s in which
the influence o f this internal friction is very small, and it can be neglected. Such
flows can be assumed to have no friction and hence can be analyzed as inviscid
flows. This is an idealization, but for m any problem s, a good one. By not dealing
with friction, the analysis o f the flow is usually simplified. However, for some
Figure 4.1 Road map for Chap. 4.
chapter
4
Basic Aerodynamics
flows the influence o f friction is dom inant, and it m ust be included in any analy­
sis o f such flows. T he inclusion o f friction usually m akes the analysis o f the flow
more com plicated.
This chapter deals w ith basics. We will start out with the statem ent o f three
fundam ental physical principles from physics:
1.
M ass is conserved.
2.
N ew ton’s second law (force = m ass x acceleration) holds.
3.
Energy is conserved.
W hen these fundam ental principles are applied to an aerodynam ic flow, certain
equations result, w hich, in m athem atical language, are statem ents o f these princi­
ples. We will see how this can be accom plished. We will start with the physical
principle that m ass is conserved and obtain a governing equation labeled the con­
tinuity equation. This is represented by the center box in Fig. 4.1. The continuity
equation says, in m athem atical sym bols, that m ass is conserved in an aerody­
nam ic flow. M ass is conserved, no m atter w hether the flow involves friction.
H ence, the continuity equation is equally applicable to both types of flow, and that
is why it is centered beneath the top tw o boxes in Fig. 4.1. Then we will work our
way dow n the left side o f the road map, m aking the assum ption o f an inviscid
flow. We will invoke N ew ton’s second law and obtain the m om entum equation
for an inviscid flow, called E u ler’s equation (pronounced like “oilers”). A spe­
cialized, but im portant, form o f E u ler’s equation is B ernoulli’s fam ous equation.
Then we will invoke the principle o f conservation of energy and obtain the energy
equation for a flow. However, since the science o f energy is thermodynamics, we
have to first exam ine som e o f the basic concepts of therm odynam ics.
A fter the basic equations are in hand, we will continue dow n the left side of
Fig. 4.1 with som e applications for inviscid flows, ranging from the speed of
sound to wind tunnels and rocket engines.
Finally, we will m ove to the right side o f our road map and discuss some
im portant aspects o f viscous flows. We will introduce the idea o f a viscous
boundary layer, the region o f flow im m ediately adjacent to a solid surface, where
friction is particularly dom inant. We will exam ine two types o f viscous flows
with quite different natures— laminar flow and turbulent flow— and how a lam ­
inar flow will transist to a turbulent flow. We will discuss how these flows have
an im pact on the aerodynam ic drag on a body. Finally, we will see how a viscous
aerodynam ic flow can actually lift off (separate) from the surface— the phenom ­
enon o f flow separation.
This has been a rather long discussion o f a som ew hat intricate road map.
However, the experience o f the author has been that readers being introduced to
the w orld o f basic aerodynam ics can find the subject m atter som etim es bew il­
dering. In reality, aerodynam ics is a beautifully organized intellectual subject,
and the road m ap in Fig. 4.1 is designed to prevent som e o f the possible bew il­
derm ent. As we progress through this chapter, it will be very im portant for you to
frequently return to this road map for guidance and orientation.
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Figure 4.2 Stream tube with mass conservation.
4.1 CONTINUITY EQUATION
The laws o f aerodynam ics are form ulated by applying to a flowing gas several
basic principles from physics. F or exam ple,
Physical principle: Mass can be neither created nor destroyed .1
To apply this principle to a flowing gas, consider an im aginary circle draw n per­
pendicular to the flow direction, as shown in Fig. 4.2. N ow look at all the stream ­
lines that go through the circum ference o f the circle. These stream lines form a
tube, called a stream tube. As we m ove along with the gas confined inside the
stream tube, w e see that the cross-sectional area o f the tube may change, say, in
m oving from point 1 to point 2 in Fig. 4.2. However, as long as the flow is steady
(invariant with tim e), the m ass that flows through the cross section at point 1 must
be the sam e as the m ass that flows through the cross section at point 2 , because by
the definition o f a stream line, there can be no flow across stream lines. The mass
flowing through the stream tube is confined by the stream lines o f the boundary,
m uch as the flow o f w ater through a flexible garden hose is confined by the wall
o f the hose. This is a case o f “w hat goes in one end m ust com e out the other end.”
Let A i be the cross-sectional area o f the stream tube at point 1. Let V¡ be the
flow velocity at point 1. Now, at a given instant in time, consider all the fluid
elem ents that are m om entarily in the plane o f A \ . A fter a lapse o f tim e d t , these
sam e fluid elem ents all m ove a distance V¡ d t , as show n in Fig. 4.2. In so doing,
the elem ents have sw ept out a volum e A \ V \ d t dow nstream o f point 1. The mass
o f gas dm in this volum e is equal to the density tim es the volum e; that is,
dm = p \ ( A \ V \ d t )
(4.1)
This is the m ass o f gas that has swept through area A\ during tim e interval dt.
Definition: The mass flow m through area A is the mass crossing A per unit time.
'O f course, Einstein has shown that e = m e2, and hence mass is truly not conserved in situations where
energy is released. However, for any noticeable change in m ass to occur, the energy release m ust be
trem endous, such as occurs in a nuclear reaction. We are generally not concerned with such a case in
practical aerodynamics.
4 .2
Incompressible and Compressible Flow
Figure 4.3 A stream tube.
T herefore, from Eq. (4.1), for area A \,
dm
M ass flow =s —— = m i = p { A \V\
dt
kg/s or slugs/s
Also, the mass flow through A2, bounded by the same stream lines that go
through the circum ference o f A ,, is obtained in the sam e fashion, as
m.2 = P2 A 2 V2
Since m ass can be neither created nor destroyed, we have m\ = m 2 . Hence,
P \A \V \ = P 2 A 2 V2
(4.2)
This is the continuity equation for steady fluid flow. It is a sim ple algebraic equa­
tion that relates the values o f density, velocity, and area at one section o f the
stream tube to the same quantities at any other section.
There is a caveat in the previous developm ent. In Fig. 4.2, velocity V\ is as­
sum ed to be uniform over the entire area A ,. Sim ilarly, the density p\ is assum ed
to be uniform over area A\. In the same vein, V2 and P2 are assum ed to be uni­
form over area A2. In real life, this is an approxim ation; in reality, V and p vary
across the cross-sectional area A. However, when using Eq. (4.2), we assum e
that p and V represent mean values o f density and velocity over the crosssectional area A. F or m any flow applications, this is quite reasonable. The conti­
nuity equation in the form o f Eq. (4.2) is a w orkhorse in the calculation o f flow
through all types o f ducts and tubes, such as wind tunnels and rocket engines.
T he stream tube sketched in Fig. 4.2 does not have to be bounded by a solid
wall. For exam ple, consider the stream lines o f flow over an airfoil, as sketched
in Fig. 4.3. The space betw een tw o adjacent stream lines, such as the shaded
space in Fig. 4.3, is a stream tube. Equation (4.2) applies to the stream tube in
Fig. 4.3, where p\ and V, are appropriate mean values over A 1, and p 2 and V2 are
appropriate values over A2.
4.2 INCOMPRESSIBLE AND COMPRESSIBLE FLOW
Before we proceed, it is necessary to point out that all m atter in real life is com­
pressible to som e greater or lesser extent. That is, if we take an elem ent o f m at­
ter and squeeze it hard enough with some pressure, the volum e o f the elem ent o f
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(III
«1
V2 * t>i — d v
P2 > Pi
—►
—►
m
= m /v \
m
*•—
11111
P2 = m /v 2
Figure 4.4 Illustration of compressibility.
m atter will decrease. H ow ever, its mass will stay the same. This is shown
schem atically in Fig. 4.4. As a result, the density p o f the elem ent changes as it
is squeezed. The am ount by which p changes depends on the nature o f the m ate­
rial o f the elem ent and how hard we squeeze it, that is, the m agnitude o f the pres­
sure. If the material is solid, such as steel, then the change in volum e is insignif­
icantly small and p is constant for all practical purposes. If the m aterial is a
liquid, such as water, then the change in volum e is also very small and again p is
essentially constant. (Try pushing a tight-fitting lid into a container o f liquid, and
you will find out ju st how “solid” the liquid can be.) But, if the m aterial is a gas,
the volum e can readily change and p can be a variable.
The preceding discussion allow s us to characterize tw o classes o f aerody­
nam ic flow: com pressible flow and incom pressible flow.
1.
Compressible flow — flow in which the density o f the fluid elem ents can
change from point to point. Referring to Eq. (4.2), we see if the flow is
com pressible, p\ ^ p 2. T he variability o f density in aerodynam ic flows is
particularly im portant at high speeds, such as for high-perform ance
subsonic aircraft, all supersonic vehicles, and rocket engines. Indeed, all
real-life flows, strictly speaking, are com pressible. However, there are
som e circum stances in w hich the density changes only slightly. These
circum stances lead to the second definition, as follows.
2.
Incompressible flow — flow in w hich the density o f the fluid elem ents is
alw ays constant .2 Referring to Eq. (4.2), we see if the flow is
incom pressible, p\ = p 2, hence,
A | V , = A 2 V2
(4.3)
Incom pressible flow is a myth. It can never actually occur in nature, as previ­
ously discussed. However, for those flows in w hich the actual variation o f p is
negligibly sm all, it is convenient to make the assumption that p is constant, to
sim plify our analysis. (Indeed, it is an everyday activity o f engineering and phys­
ical science to m ake idealized assum ptions about real physical system s in order
2In more advanced studies o f aerodynam ics, you will find the definition of incom pressible flow is given
by a more general statement. For our purposes in this book, we will consider incompressible flow to
be constant density flow.
4 .2
Incompressible and Compressible Flow
A i < A\
H ence,
V2 > V\
A,
Figure 4.5 Incompressible flow in a convergent duct.
to m ake such system s am enable to analysis. However, care m ust alw ays be taken
not to apply results obtained from such idealizations to those real problem s in
which the assum ptions are grossly inaccurate or inappropriate.) The assum ption
o f incom pressible flow is an excellent approxim ation for the flow o f liquids, such
as w ater or oil. M oreover, the low -speed flow o f air, where V < 100 m/s (or
V < 225 m i/h) can also be assum ed to be incom pressible to a close approxim a­
tion. A glance at Fig. 1.30 show s that such velocities w ere the dom ain o f alm ost
all airplanes from the Wright Flyer (1903) to the late 1930s. Hence, the early
developm ent o f aerodynam ics alw ays dealt with incom pressible flows, and for
this reason there exists a huge body o f incom pressible flow literature with its
attendant technology. A t the end o f this chapter, we will be able to prove why air­
flow at velocities less than 100 m /s can be safely assum ed to be incom pressible.
In solving and exam ining aerodynam ic flows, you will constantly be faced
with m aking distinctions betw een incom pressible and com pressible flows. It is
im portant to start that habit now, because there are som e striking quantitative and
qualitative differences betw een the tw o types o f flow.
As a parenthetical com m ent, for incom pressible flow, Eq. (4.3) explains why
all com m on garden hose nozzles are convergent shapes, such as shown in
Fig. 4.5. From Eq. (4.3),
If A 2 is less than A i , then the velocity increases as the w ater flows through the
nozzle, as desired. The sam e principle is used in the design o f nozzles for sub­
sonic w ind tunnels built for aerodynam ic testing, as will be discussed in
Sec. 4.10.
Consider a convergent duct with an inlet area A¡ = 5 m2. Air enters this duct with a
velocity V¡ = 10 m/s and leaves the duct exit with a velocity Vi = 30 m/s. What is the
area of the duct exit?
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■ Solution
Since the flow velocities are less than 100 in/s, we can assume incompressible flow. From
Eq. (4.3),
A,V, = a 2v2
Vi
,1 0
A2 = A\ — = (5 m2) — = 1.67m2
V2
JU
EXAM PLE
Consider a convergent duct with an inlet area A ¡ = 3 ft2 and an exit area A2 = 2.57 ft2. Air
enters this duct with a velocity V| = 700 ft/s and a density p\ = 0.002 slug/ft3, and air
leaves with an exit velocity V2 = 1070 ft/s. Calculate the density of the air pn at the exit.
■ Solution
An inlet velocity of 700 ft/s is a high-speed flow, and we assume that the flow has to be
treated as compressible. This implies that the resulting value for p2 will be different from
P \ . From Eq. (4.2),
P \ A \ V \ — P 2 A 2 V2
or
™
3(700)
P2 = Pi - 7 7 7 = 0.002;
A2V2
2.57(1070)
0.00153 slug/ft3
Note: The value of p 2 is indeed different from p¡, which clearly indicates the flow in this
example is a compressible flow. If the flow were essentially incompressible, then the cal­
culation of p 2 from Eq. (4.2) would have produced a value essentially equal to p\. But
this is not the case. Keep in mind that Eq. (4.2) is more general than Eq. (4.3). Eq. (4.2)
applies for both compressible and incompressible flows; Eq. (4.3) is valid for an incom­
pressible flow only.
Reminder: In this example, and in all the worked examples in this book, we use
consistent units in the calculations. Hence, we do not need to explicitly show all the units
carried with each term in the mathematical calculations because we know the answer will
be in the same consistent units. In this example, the calculation involves the continuity
equation; A , and A 2 are given in ft2, V\ and V2 in ft/s, and p\ in slug/ft3. When these numbers
are fed into the equation, we know the answer for P2 will be in slug/ft3. It has to be, because
we know the consistent units for density in the English engineering system are slug/ft3.
4.3 MOMENTUM EQUATION
The continuity equation, Eq. (4.2), is only part o f the story. For exam ple, it says
nothing about the pressure in the flow; yet, we know, ju st from intuition, that
pressure is an im portant flow variable. Indeed, differences in pressure from one
point to another in the flow create forces that act on the fluid elem ents and cause
them to move. H ence, there m ust be som e relation betw een pressure and veloc­
ity, and that relation is derived in this section.
4 .3
Momentum Equation
P'
.J -
dz
dx
y
V
p
X
z
Figure 4.6 Force diagram for the momentum equation.
A gain, we first state a fundam ental law o f physics, namely, N ew ton’s sec­
ond law.
Physical principle:
or
Force = m ass x acceleration
F = ma
(4.4)
To apply this principle to a flowing gas, consider an infinitesim ally small
fluid elem ent m oving along a stream line with velocity V, as shown in Fig. 4.6.
At som e given instant, the elem ent is located at point P . The elem ent is m oving
in the x direction, where the x axis is oriented parallel to the stream line at point
P. The y and z axes are m utually perpendicular to x. The fluid elem ent is very
small, infinitesim ally small. However, looking at it through a m agnifying glass,
we see the picture shown at the upper right o f Fig. 4.6. Q uestion: W hat is the
force on this elem ent? Physically, the force is a com bination o f three phenom ena:
1.
Pressure acting in a norm al direction on all six faces o f the elem ent
2.
Frictional shear acting tangentially on all six faces o f the elem ent
3.
G ravity acting on the m ass inside the elem ent
F or the tim e being, we will ignore the presence o f frictional forces; m oreover, the
gravity force is generally a sm all contribution to the total force. Therefore, we
will assum e that the only source o f a force on the fluid elem ent is pressure.
To calculate this force, let the dim ensions o f the fluid elem ent be d x , d y , and
d z, as show n in Fig. 4.6. C onsider the left and right faces, which are perpendic­
ular to the x axis. The pressure on the left face is p. The area o f the left face is
d y d z , hence the force on the left face is p ( d y d z )■ This force is in the positive x
direction. Now recall that pressure varies from point to point in the flow. Hence,
there is some change in pressure per unit length, sym bolized by the derivative
d p / d x . Thus, if we m ove aw ay from the left face by a distance d x along the jc
axis, the change in pressure is { d p / d x ) d x . Consequently, the pressure on the
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right face is p + (d p / d x ) d x . The area o f the right face is also d y d z \ hence, the
force on the right face is [p + ( d p / d x ) d x ] ( d y d z ) . This force acts in the nega­
tive x direction, as shown in Fig. 4.6. The net force in the x direction F is the
sum o f the two:
F = p dy dz — y p + — dx ) dy dz
- { p+t dx)'
dp
F = —— (dx d y d z )
dx
or
(4.5)
Equation (4.5) gives the force on the fluid elem ent due to pressure. B ecause o f
the convenience o f choosing the x axis in the flow direction, the pressures on the
faces parallel to the stream lines do not affect the motion o f the elem ent along the
stream line.
T he m ass o f the fluid elem ent is the density p m ultiplied by the volum e
d x d y dz'.
m = p (d x dy dz)
(4.6)
Also, the acceleration a o f the fluid elem ent is, by definition o f acceleration (rate
o f change o f velocity), a = d V / d t . N oting that, also by definition, V = d x / d t ,
we can write
dV
a =
d V dx
dV
lü = T x l¡ = T x V
( 4 '7 )
Equations (4.5) to (4.7) give the force, m ass, and acceleration, respectively,
that go into N ew ton’s second law, Eq. (4.4):
F = ma
- ^ ( d x d y dz) = p (d x d y d z)V ^ y dx
dx
or
dp = - p V dV
(4.8)
Equation (4.8) is E uler’s equation. Basically, it relates rate o f change o f
m om entum to the force; hence it can also be designated as the momentum equa­
tion. It is im portant to keep in m ind the assum ptions utilized in obtaining
Eq. (4.8); w e neglected friction and gravity. For flow that is frictionless, aerodynam icists som etim es use another term , inviscid flow. Equation (4.8) is the m o­
m entum equation for inviscid (frictionless) flow. M oreover, the flow field is as­
sum ed to be steady, that is, invariant with respect to time.
Please note that Eq. (4.8) relates pressure and velocity (in reality, it relates
a change in pressure d p to a change in velocity d V ) . Equation (4.8) is a
4.3
Momentum Equation
2
Figure 4.7 Two points at different locations along a
streamline.
differential equation, and hence, it describes the phenom ena in an infinitesim ally
small neighborhood around the given point P in Fig. 4.6. Now consider two
points, 1 and 2 , far rem oved from each other in the flow but on the sam e stream ­
line. To relate p\ and V\ at point 1 to p 2 and V2 at the other, far-rem oved point 2,
Eq. (4.8) m ust be integrated betw een points 1 and 2. This integration is different
depending on w hether the flow is com pressible or incom pressible. E u ler’s equa­
tion itself, Eq. (4.8), holds for both cases. For com pressible flow, p in Eq. (4.8)
is a variable; for incom pressible flow, p is a constant.
First, consider the case o f incom pressible flow. Let points 1 and 2 be located
along a given stream line, such as that show n over an airfoil in Fig. 4.7. From
Eq. (4.8),
dp + p V d V = 0
where p = constant. Integrating betw een points 1 and 2, we obtain
r pi
pV2
/
dp + p
VdV = 0
Jp,
Jv i
Vi
V,2
Pi
vp + p — = const along stream line
(4.9a)
(4.9 b)
Either form , Eq. (4.9a) or (4.9b), is called Bernoulli’s equation. Historically,
B ernoulli’s equation is one o f the m ost fundam ental equations in fluid m e­
chanics.
The follow ing im portant points should be noted:
1.
Equations (4 .9 a) and (4.9b) hold only for inviscid (frictionless),
incom pressible flow.
2.
Equations (4.9 a) and (4.9/?) relate properties betw een different points along
a stream line.
133
134
chapter
4
Basic Aerodynamics
3.
For a com pressible flow, Eq. (4.8) m ust be used, w ith p treated as a
variable. B ernoulli’s equation must not be used for com pressible flow.
4.
R em em ber that Eqs. (4.8) and (4.9a) and (4.9b) say that F = ma for a fluid
flow. They are essentially N ew ton’s second law applied to fluid dynam ics.
To return to Fig. 4.7, if all the stream lines have the same values o f p and V
far upstream (far to the left in Fig. 4.7), then the constant in B ernoulli’s equation
is the same f o r all streamlines. This w ould be the case, for exam ple, if the flow
far upstream were uniform flow, such as that encountered in flight through the
atm osphere and in the test sections o f w ell-designed wind tunnels. In such cases,
Eqs. (4.9a) and (4.9b) are not lim ited to the sam e stream line. Instead, points l
and 2 can be anyw here in the flow, even on different stream lines.
For the case o f com pressible flow also, E uler’s equation, Eq. (4.8), can be in­
tegrated betw een points l and 2 ; however, because p is a variable, we m ust in
principle have som e extra inform ation on how p varies with V before the inte­
gration can be carried out. This inform ation can be obtained; however, there is an
alternate, m ore convenient route to treating m any practical problem s in com ­
pressible flow that does not explicitly require the use o f the m om entum equation.
H ence, in this case, we will not pursue the integration o f Eq. (4.8) further.
4.4 A COMMENT
It is im portant to m ake a philosophical distinction betw een the nature o f the
equation o f state, Eq. (2.3), and the flow equations o f continuity, Eq. (4.2), and
m om entum , such as Eq. (4.9a). The equation o f state relates p , T , and p to one
another at the same point; in contrast, the flow equations relate p and V (as in
the continuity equation) and p and V (as in B ernoulli’s equation) at one point in
the flow to the sam e quantities at another point in the flow. There is a basic dif­
ference here, and one does well to keep it in m ind when setting up the solution of
aerodynam ic problem s.
EXAM PLE 4.3
Consider an airfoil (the cross section o f a wing, as shown in Fig. 4.7) in a flow o f air,
where far ahead (upstream) o f the airfoil, the pressure, velocity, and density are
2 116 lb/ft2, 100 mi/h, and 0.002377 slug/ft3, respectively. At a given point A on the air­
foil, the pressure is 2070 lb/ft2. What is the velocity at point A?
■ Solution
First, we must deal in consistent units; V\ = 100 mi/h is not in consistent units. However,
a convenient relation to remember is that 6 0 mi/h = 88 ft/s. Hence, V\ = 1 0 0 (8 8 /6 0 ) =
146.7 ft/s. This velocity is low enough that w e can assume incompressible flow. Hence,
Bernoulli's equation, Eq. (4.9), is valid
P„ l +4.
—
PV*
= PnA +
+ —
4.4
VA =
Thus,
'2 ( p , - p A)
A Comment
135
1/2
+ V,2]
2 ( 2 1 1 6 - 2 0 7 0 ) + ( i 4 6 .7 )21 ,/2
0.002377
= 245.4 ft/s
EXAMPLE : 4 .4
Consider the same convergent duct and conditions as in Example 4.1. If the air pressure
and temperature at the inlet are p\ = 1.2 x 105 N/m 2 and T\ = 330 K, respectively,
calculate the pressure at the exit.
■ Solution
First, we must obtain the density. From the equation o f state,
Pi -
1.2 x 105
Pi
/?7|
287(330)
= 1.27 kg/m J
Still assuming incompressible flow, w e find from Eq. (4.9)
„
,
PV
,2 _
pV
Pi + ~ r j- — P2 H—
22
P2 = Pi + 5 P(V,2 - Vi) = 1.2 x 105 + ( i ) (1 .2 7 X 1 0 2 - 302)
p 2 = 1.195 x 105 N /m 2
Note: In accelerating from 10 to 30 m/s, the air pressure decreases only a small
amount, less than 0.45 percent. This is a characteristic o f very low velocity airflow.
EXAMPLE : 4 .5
Consider a long dowel with a semicircular cross section, as sketched in Fig. 4.8o. The
dowel is immersed in a How o f air, with its axis perpendicular to the flow, as shown in
perspective in Fig. 4.8a. The rounded section o f the dow el is facing into the flow, as
shown in Fig. 4.8« and 4 .8 b. We call this rounded section the front face o f the dowel. The
radius o f the semicircular cross section is R = 0.5 ft. The velocity o f the flow far ahead
o f the dow el (called the free stream) is V^ — 100 ft/s. A ssum e inviscid flow; that is,
neglect the effect o f friction. The velocity o f the flow along the surface o f the rounded
front face o f the dow el is a function o f location on the surface; location is denoted by
angle 0 in Fig. 4.8 b. Hence, along the front rounded surface, V — V(6). This variation is
given by
V = 2^00 sin ft
(E4.5.1)
136
chapter
4
Basic Aerodynamics
(p ds) cos 8
(e)
Back
face
D
(h)
Figure 4.8 Diagrams for the construction o f the aerodynamic force on a dow el (Exam ple 4.5).
4 .4
A Comment
The pressure distribution exerted over the surface o f the cross section is sketched in
Fig. 4.8c. On the front face, p varies with location along the surface, where the location
is denoted by the angle 9; that is, p = p(6) on the front face. On the flat back face, the
pressure, denoted by /?«, is constant. The back face pressure is given by
P b = Poo - 0 . 1 p x
(E4.5.2)
where p x and px are the pressure and density, respectively, in the free stream, far ahead
o f the dow el. The free-stream density is given as Poo = 0.002378 slug/ft3. Calculate the
aerodynamic force exerted by the surface pressure distribution (illustrated in Fig. 4.8c) on
a l-ft segment o f the dow el, shown by the shaded section in Fig. 4.8a.
■ Solution
For this solution, w e appeal to the discussions in Secs. 2.2 and 4.3. Examine Fig. 4.8c.
Because o f the symmetry o f the semicircular cross section, the pressure distribution
over the upper surface is a mirror image o f the pressure distribution over the lower sur­
face; that is, p = p (9) for 0 < 9 < n / 2 is the same as p = p(9) for 0 > 9 > —n / 2 .
Owing to this symmetry, there is no net force on the cross section in the direction per­
pendicular to the free stream; that is, the force due to the pressure pushing down on the
upper surface is exactly canceled by the equal and opposite force due to the pressure
pushing up on the lower surface. Therefore, ow ing to this symmetry, the resultant aero­
dynamic force is parallel to the free-stream direction. This resultant aerodynamic force
is illustrated by the arrow labeled D in Fig. 4.8c.
Before feeding the numbers into our calculation, we obtain an analytical formula for
D in terms o f Voo and R, as follow s. Our calculations w ill proceed in a number o f logical
steps.
Step O ne: C alculation o f the force du e to pressure acting on the front face.
Here, we w ill integrate the pressure distribution over the surface area o f the front face. We
will set up an expression for the pressure force acting on an inlinitesimally small element
o f surface area, take the component o f this force in the horizontal flow direction (the
direction o f
in Fig. 4.8), and then integrate this expression over the surface area o f the
front face. Consider the infinitesimal arclength segment o f the surface ds and the pressure
p exerted locally on this segm ent, as drawn in Fig. 4.Hd. A magnified view o f this seg­
ment is shown in Fig. 4.8e. Recall from Fig. 4.8a that we wish to calculate the aerody­
namic force on a 1-ft length o f the dow el, as shown by the shaded region in Fig. 4.8a. As
part o f the shaded region, consider a small sliver o f area o f width ds and length equal to
1 ft on the curved face o f the dow el, as shown in Fig. 4 .8 / The surface area o f this sliver
is 1 ds. The force due to the pressure p on this area is p ( \ ) d s = p d s . This force is
shown in Fig. 4.8e, acting perpendicular to the segm ent ds. The component o f this force
in the horizontal direction is ( p d s ) c o s 9, also shown in Fig. 4.8e. From the geometric
construction shown in Fig. 4.8g, w e have
ds = Rd9
(E4.5.3)
d y = d s cos 8
(E4.5.4)
137
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c hapter
4
Basic Aerodynamics
Substituting Eq. (E4.5.3) into (E4.5.4), we have
dy = R c o s9 d 9
(E4.5.5)
We put Eq. (E4.5.5) on the shelf temporarily. It will be used later, in Step Two of this cal­
culation. However, we use Eq. (E4.5.3) immediately, as follows.
In light of Eq. (E.4.5.3), the horizontal force (p d s) cos 9 in Fig. 4.8e can be ex­
pressed as
(p d s ) c o s 9 = p R c o s 9 d 9
(E4.5.6)
Returning to Fig. 4.8c, we see that the net horizontal force exerted by the pressure distri­
bution on the rounded front face is the integral of Eq. (E4.5.6) over the front surface.
Denote this force by D F.
f ’r/2
Df =
pR cos9d9
(E4.5.7)
J-n/2
This force is shown in Figure 4.8h.
In Eq. (E4.5.7), p is obtained from Bernoulli’s equation, Eq. (4.9), written between
a point in the free stream where the pressure and velocity are /?«, and Vx , respectively,
and the point on the body surface where the pressure and velocity are p and V , respec­
tively.
Poo +
or
= P + {pV 2
P = Poo + \ p { V l - V2)
(E4.5.8)
Note: We can use Bernoulli’s equation for this solution because the free-stream velocity
of Vx = 100 ft/s is low, and we can comfortably assume that the flow is incompressible.
Also, because p is constant, the value of p in Eq. (E4.5.8) is the same as p^ in the free
stream. Substituting Eq. (E4.5.8) into Eq. (E4.5.7), we have
/
r*/2
jt/2 r
1,
[Poo + 2 p ( y i - v 2) R cos 9 dG
-n /2
I
(E4.5.9)
Recall that the variation of the surface velocity is given by Eq. (E4.5.1), repeated here:
V' = 2Voosin0
(E4.5.1)
Substituting Eq. (E4.5.1) into Eq. (E4.5.9), we have
D f = J ’ n [/>«, + -l p ( v l - 4 V l sin2 0 )] R c o s9 d 9
or
° F = J / [ P“ + \ pV°°{X “ 4sin2
* cos 0 d 9
(E4.5.10)
Let us put this expression for D F on the shelf for a moment; we will come back to it
shortly.
4 .4
A Comment
Step Two: Calculation of the force due to pressure acting on the back face.
Here, we will integrate the pressure distribution over the surface area of the back face.
Similar to Step One, we will set up an expression for the pressure force acting on an infinitesimally small element of surface area and then integrate this expression over the sur­
face area of the back face.
Returning to Fig. 4.8c, we now direct our attention to the pressure on the back face
of the cross-section p B. This pressure exerts a force Dfl on the l-ft length of dowel, as
sketched in Fig. 4.8/i. Force D B acts toward the left, opposite to the direction of D . Pres­
sure pu is constant over the back face. The rectangular area of the 1-ft length of the back
face is (1 )(2/?). Because p B is constant over this back face, we can directly write
D b = (D(,2R)P b
(E4.5.11)
However, because the resultant aerodynamic force on the cross section is given by
D f — Db, as seen in Fig. 4.8/i, and because D F is expressed in terms of an integral in
Eq. (E4.5.10), it will be convenient to couch D b in terms of an integral also, as follows.
Returning to Figure 4.8d, we consider a segment of the back surface area of height dy on
which pD is exerted. Over a 1-ft length of dowel, (perpendicular to the page in Fig. 4.8¿I),
the area of a small sliver of surface is I dy, and the force on this sliver is p H( \ ) d y . The
total force on the back face is obtained by integrating with respect to y from point a to
point b, as noted in Fig. 4.8c/.
Db =
f
p B( l ) d y
(E4.5.12)
J a
However, recall from Eq. (E4.5.5) that dy = R cos 9 dO. Hence, Eq. (E4.5.12) becomes
prr/2
DB = I
P b R cosOdO
J-n/2
(E4.5.13)
Please note that Eqs. (E4.5.13) and (E4.5.11) are both valid expressions for DB—they
just look different. To see this, carry out the integration in Eq. (E4.5.13); you will obtain
the result in Eq. (E4.5.11). Also recall that p B is given by Eq. (E4.5.2), repeated here (and
dropping the subscript oo on p since p is constant):
Pb = P o o - O . l p V l
(E4.5.2)
Hence, Eq. (E4.5.13) becomes
fn /2
Db = \
(p00- Q . l p 2V l) R c o s 9 d 9
J-n/2
(E4.5.14)
Step Three: Calculation of the resultant aerodynamic force.
Here, we will combine the results obtained in Steps One and Two. In Step One, we ob­
tained an expression for the pressure force acting on the front face. In Step Two, we
obtained an expression for the pressure force acting on the back force. Because the force
on the front face acts in one direction and the force on the back face acts in the opposite
139
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4
Basic Aerodynamics
DESIGN BOX
The results of Example 4.5 illustrate certain aspects
important to the general background of airplane
design:
1.
2.
It reinforces the important point made in Sec. 2.2,
namely, that the resultant aerodynamic force ex­
erted on any object immersed in a flowing fluid is
due only to the net integration of the pressure dis­
tribution and the shear stress distribution exerted
all over the body surface. In Example 4.5, we as­
sumed the flow to be inviscid; that is, we ne­
glected the effect of friction. So the resultant
aerodynamic force was due to just the integrated
effect of the pressure distribution over the body
surface. Indeed, this is precisely how we calcu­
lated the force on the dowel in Example 4.5—we
integrated Ihe pressure distribution over the sur­
face of the dowel. Instead of a dowel, if we had
dealt with a Boeing 747 jumbo jet, the idea would
have been the same. In airplane design, the shape
of the airplane is influenced by the desire to create
a surface pressure distribution that will minimize
drag while at the same time creating the neces­
sary amount of lift. We return to this basic idea
several times throughout the book.
Equation (E4.5.17) shows that the aerodynamic
force on the body is
(a) Directly proportional to the density of the
fluid p.
(b) Directly proportional to the square of the
freestream velocity; D oc V¿.
(c) Directly proportional to the size of the
body, as reflected by the radius R.
These results are not just specialized to the dowel in
Example 4.5; they are much more general in their ap­
plication. We will see in Chap. 5 that the aerody­
namic force on airfoils, wings, and whole airplanes is
indeed proportional to p^, V¿, and the size of the
body, where size is couched in terms of a surface
area. [In Eq. (E4.5.17), R really represents an area
equal to R( 1) for the unit length of the dowel over
which the aerodynamic force is calculated.] It is in­
teresting to note that Eq. (E4.5.17) does not contain
the free-stream pressure Poo. Indeed, Poo canceled
out in our derivation of Eq. (E4.5.17). This is not just
a characteristic of the dowel used in Example 4.5; in
general, we will see in Chap. 5 that we do not need
the explicit value of free-stream pressure to calculate
the aerodynamic force on a flight vehicle, in spite
of the fact that the aerodynamic force fundamentally
is due (in part) to the pressure distribution over the
surface. In the final result, it is always the value of the
free-stream density px that appears in the expres­
sions for aerodynamic force, not Pno.
direction, as shown in Fig. 4.8h, the net, resultant aerodynamic force is the difference
between the two.
Returning to Fig. 4.8/t, we see that the resultant aerodynamic force D is given by
D = Df - D
b
(E4.5.15)
Substituting Eqs. (E4.5.10) and (E4.5.14) into Eq. (E4.5.15), we have
D = f[
’
I"/),»
9) I R
[p,o + 5/9v ¿( 1 —
- 4 siiri
sin2 (9)J
/?(c o sd d O
rn /2
-
J—
n/2
(Poo- 0.7p V l ) R cosd d6
(E4.5.16)
4.5
Elementary Thermodynamics
Combining the two integrals in Eq. (E 4.5.16) and noting that the tw o terms involving p <*,
cancel, we have
r n/2
° = F -71/2
'r [ G P + °'7/0) V°° ~ 2pV°°S'"2 °\ RcM6dS
[r”/2
/ n/¿
cosOdO - I p V ^ R /
= l-2 pV c2
■it/2
sin2 0 cos 9 dd
J-n /2
¡«3 a 1^/2
2 A p V lR -2 p V l
= 2 A p V x2 R - 2 p V l
R[!T£r
t t /2
sG +5)= 1jo e i p v ^ R
Highlighting the preceding result, we have just derived an analytical expression for
the aerodynamic force D, per unit length o f the dow el. It is given by
D = l.O blpV ^ R
(E4.5.17)
Putting in the numbers given in the problem, where p = p ^ = 0.002378 slug/ft3,
Voo = 100 ft/s, and R = 0.5 ft, w e obtain from Eq. (E 4 .5 .17),
D = (1 .0 6 7 )(0 .0 0 2 3 7 7 )(1 0 0 )2(0.5) =
12.681b
per foot o f length o f dowel
4.5 ELEMENTARY THERMODYNAMICS
As stated earlier, when the airflow velocity exceeds 100 m/s, the flow can no
longer be treated as incom pressible. Later we will restate this criterion in terms
o f the M ach num ber, which is the ratio o f the flow velocity to the speed o f sound,
and we will show that the flow m ust be treated as com pressible when the M ach
num ber exceeds 0.3. This is the situation with the vast m ajority o f current aero­
dynam ic applications; hence, the study o f com pressible flow is o f extrem e
importance.
A high-speed flow o f gas is also a high-energy flow. The kinetic energy o f
the fluid elem ents in a high-speed flow is large and m ust be taken into account.
W hen high-speed flows are slow ed dow n, the consequent reduction in kinetic en­
ergy appears as a substantial increase in tem perature. As a result, high-speed
flows, com pressibility, and vast energy changes are all related. Thus, to study
com pressible flows, w e m ust first exam ine som e o f the fundam entals o f energy
changes in a gas and the consequent response o f pressure and tem perature to
these energy changes. Such fundam entals are the essence o f the science o f ther­
modynam ics.
H ere the assum ption is m ade that the reader is not fam iliar with therm ody­
nam ics. Therefore, the purpose o f this section is to introduce those ideas and
results o f therm odynam ics that are absolutely necessary for our further analysis
141
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chapter
4
Basic Aerodynamics
Boundary
S urroundings
Figure 4.9 System o f unit mass.
o f high-speed, com pressible flows. Caution: The material in Secs. 4.5 to 4.7 can
be intim idating; if you find it hard to understand, do not w orry— you are in good
company. T herm odynam ics is a sophisticated and extensive subject, we are ju st
introducing som e basic ideas and equations here. View these sections as an intel­
lectual challenge, and study them with an open mind.
The pillar o f therm odynam ics is a relationship called the first law, which is
an em pirical observation o f natural phenom ena. It can be developed as follows.
C onsider a fixed m ass o f gas (for convenience, say a unit m ass) contained within
a flexible boundary, as shown in Fig. 4.9. This m ass is called the system, and
everything outside the boundary is the surroundings. Now, as in Chap. 2, con­
sider the gas that m akes up the system to be com posed o f individual m olecules
m oving about w ith random motion. T he energy o f this m olecular motion,
sum m ed over all the m olecules in the system , is called the internal energy o f the
system . Let e denote the internal energy per unit m ass o f gas. The only m eans by
w hich e can be increased (or decreased) are the following:
1.
H eat is added to (or taken aw ay from ) the system . This heat com es from
the surroundings and is added to the system across the boundary. Let 8q be
an increm ental am ount o f heat added per unit mass.
2.
W ork is done on (or by) the system . This w ork can be m anifested by the
boundary o f the system being pushed in (w ork done on the system ) or
pushed out (w ork done by the system ). Let <5w be an increm ental am ount
o f work done on the system per unit mass.
A lso, let d e be the corresponding change in internal energy per unit mass.
Then sim ply on the basis o f com m on sense, confirm ed by laboratory results, we
can write
(4.10)
8q + 8w = de
Equation (4.10) is term ed the first law o f thermodynamics. It is an energy equa­
tion that states the change in internal energy is equal to the sum o f the heat added
to and the w ork done on the system . (N ote in the previous discussion that <5 and
d both represent infinitesim ally small quantities; however, d is a “perfect differ­
ential” and 8 is not.)
4 .5
Elementary Thermodynamics
A
B oundary
surface
Figure 4.10 Work being done on the system by pressure.
Equation (4.10) is very fundam ental; how ever, it is not in a practical form for
use in aerodynam ics, w hich speaks in term s o f pressures, velocities, etc. To
obtain m ore useful form s o f the first law, we m ust first derive an expression for
Sw in term s o f p and v (specific volum e), as follows. C onsider the system
sketched in Fig. 4.10. Let d A be an increm ental surface area o f the boundary. A s­
sum e that w ork A W is being done on the system by d A being pushed in a small
distance s, as also shown in Fig. 4.10. Since work is defined as force times
distance, we have
A W = (force)(distance)
(4.11)
AW = (p d A )s
N ow assum e that m any elem ental surface areas o f the type shown in Fig. 4.10 are
distributed over the total surface area A o f the boundary. A lso assum e that all the
elem ental surfaces are being sim ultaneously displaced a small distance s into the
system . Then the total work Sw done on the unit mass o f gas inside the system is
the sum (integral) o f each elem ental surface over the whole boundary; that is,
from Eq. (4.11),
(4.12)
A ssum e p is constant everyw here in the system (which, in therm odynam ic
term s, contributes to a state o f therm odynam ic equilibrium ). Then, from
Eq. (4.12),
(4.13)
The integral f A s d A has physical m eaning. G eom etrically, it is the change in
volum e o f the unit mass o f gas inside the system , created by the boundary surface
143
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chapter
4
Basic Aerodynamics
being displaced inw ard. Let d v be the change in volum e. Since the boundary is
pushing in, the volum e decreases (dv is a negative quantity) and w ork is done on
the gas (hence Sw is a positive quantity in our developm ent). Thus,
s d A = —d v
(4.14)
L
Substituting Eq. (4.14) into Eq. (4.13), w e obtain
Sw = —p d v
(4.15)
Equation (4.15) gives the relation for work done strictly in term s o f the therm o­
dynam ic variables p and v.
W hen Eq. (4.15) is substituted into Eq. (4.10), the first law becom es
Sq = d e + p d v
(4.16)
Equation (4.1 6 ) is an alternate form o f the first law o f therm odynam ics.
It is convenient to define a new quantity, called enthalpy h, as
h = e + p v = e + RT
(4.17)
w here p v = R T , assum ing a perfect gas. Then differentiating the definition,
Eq. (4.17), we find
dh = d e + p d v + v d p
(4.18)
Substituting Eq. (4.18) into (4.1 6 ), we obtain
Sq = d e + p d v = (dh — p d v — v d p ) + p d v
Sq = dh — v d p
(4.19)
Equation (4.19) is yet another alternate form o f the first law.
B efore we go further, rem em ber that a substantial part o f science and engi­
neering is sim ply the language. In this section, we are presenting som e o f the lan­
guage o f therm odynam ics essential to our future aerodynam ic applications. We
continue to develop this language.
Figures 4.9 and 4.10 illustrate system s to which heat Sq is added and on
which work Sw is done. At the sam e time, Sq and Sw may cause the pressure,
tem perature, and density o f the system to change. The way (or m eans) by which
changes o f the therm odynam ic variables (p, T, p , v) o f a system take place is
called a process. For exam ple, a constant-volume process is illustrated at the left
in Fig. 4.11. Here, the system is a gas inside a rigid boundary, such as a hollow
steel sphere, and therefore the volum e o f the system alw ays rem ains constant. If
an am ount o f heat Sq is added to this system , p and T will change. Thus, by
definition, such changes take place at constant volum e; this is a constant-volum e
process. A nother exam ple is given at the right in Fig. 4.11. H ere the system is a
gas inside a cylinder-piston arrangem ent. C onsider that heat Sq is added to the
system , and at the sam e tim e assum e the piston is m oved in ju st exactly the right
w ay to m aintain a constant pressure inside the system. W hen Sq is added to this
4 .5
6 q Heat
added
£
£
^
Rigid boundary
(such as a hollow
sphere of
constant volume)
Elementary Thermodynamics
Assume the piston
is moving in just
the right way to
keep p constant
p, T , v
Constant-pressure process
Constant-volum e process
Figure 4.11 Illustration o f constant-volume and constant-pressure processes.
system , T and v (hence p ) will change. By definition, such changes take place at
constant pressure; this is a constant-pressure process. T here are m any different
kinds o f processes treated in therm odynam ics. These are only two exam ples.
The last concept to be introduced in this section is that o f specific heat. C on­
sider a system to w hich a sm all am ount o f heat 8q is added. The addition o f 8q
will cause a sm all change in tem perature d T o f the system. By definition,
specific heat is the heat added per unit change in tem perature o f the system. Let
c denote specific heat. Thus,
c = —
However, with this definition, c is m ultivalued. T hat is, for a fixed quantity 8q,
the resulting value o f d T can be different, depending on the type o f process in
which 8q is added. In turn, the value o f c depends on the type of process. T here­
fore, in principle we can define m ore precisely a different specific heat for each
type o f process. We will be interested in only tw o types o f specific heat, one at
constant volum e and the other at constant pressure, as follows.
If the heat 8q is added at constant volume and it causes a change in tem per­
ature d T , the specific heat at constant volume c\, is defined as
constant volume
or
8q — c v d T (constant volume)
(4.20)
On the other hand, if 8q is added at constant pressure and it causes a change in
tem perature d T (w hose value is different from the preceding d T ) , the specific
heat at constant pressure cp is defined as
constant pressure
or
8q = cp d T (constant pressure)
(4.21)
145
146
chapter
4
Basic Aerodynamics
The preceding definitions o f c„ and cp, w hen com bined with the first law,
yield useful relations for internal energy e and enthalpy h as follows. First, con­
sider a constant-volum e process, w here by definition d v = 0. Thus, from the
alternate form o f the first law, Eq. (4.16),
Sq = d e + p d v = d e + 0 = d e
(4.22)
Substituting the definition o f c v, Eq. (4.20), into Eq. (4.22), we get
de = cvd T
(4.23)
By assum ing that c v is a constant, which is very reasonable for air at norm al con­
ditions, and letting e = 0 when 7 = 0, Eq. (4.23) may be integrated to
e = c vT
(4.24)
Next, consider a constant-pressure process, where by definition d p = 0. From
the alternate form o f the first law, Eq. (4.19),
Sq = dh — v d p = dh — 0 = dh
(4.25)
Substituting the definition o f cp, Eq. (4.21), into Eq. (4.25), we find
dh = cp d T
(4.26)
A gain, assum ing that cp is constant and letting h = 0 at T = 0, we see that
Eq. (4.26) yields
h = cpT
(4.27)
Equations (4.23) to (4.27) are very im portant relationships. They have been
derived from the first law, into w hich the definitions o f specific heat have been
inserted. Look at them! They relate therm odynam ic variables only (e to 7’ and h
to 7 ) ; work and heat do not appear in these equations. In fact, Eqs. (4.23) to
(4.27) are quite general. Even though we used exam ples o f constant volum e and
constant pressure to obtain them , they hold in general as long as the gas is a per­
fect gas (no interm olecular forces). H ence, for any process,
de — cvd T
dh = cp d T
e = c uT
h = cpT
This generalization o f Eqs. (4.23) to (4.27) to any process may not seem logical
and m ay be hard to accept; nevertheless, it is valid, as can be show n by good
therm odynam ic argum ents beyond the scope o f this book. For the rem ainder of
our discussions, w e will m ake frequent use o f these equations to relate internal
energy and enthalpy to tem perature.
4.6
Isentropic Flow
147
EXAM PLE 4.6
Calculate the internal energy and enthalpy, per unit mass, for air at standard sea-level
conditions in (a) SI units and ( b ) English engineering units. For air at standard conditions,
cv = 720 J/(kg)(K) = 4290 ft lb/(slug)(°R), and cp = I008 J/(kg)(K) = 6006 ft lb/
(slug)(°R).
■ Solution
At standard sea level, the air temperature is
T = 288 K = 5 l9 °R
a. From Eqs. (4.24) and (4.27), w e have
e = L t = (720) (288) = 2.07 x 105 J/kg
h — cpT = (1008) (28) = 2.90 x 105 J/kg
b. A lso from Eqs. (4.24) and (4.27),
e = cvT = (4290)(519) = 2.23 x 106 ft lb/slug
h = cpT = (6006)( 5 19) = 3.12 x 106 ft lb/slug
Note: For a perfect gas, e and h are functions o f temperature only, as emphasized in this
worked example. If you know the temperature o f the gas, you can directly calculate e and
h from Eqs. (4.24) and (4.27). You do not have to be concerned whether the gas is going
through a constant volum e process, or a constant pressure process, or whatever. Internal
energy and enthalpy are state variables, that is, properties that depend only on the local
state o f the gas as described, in this case, by the given temperature o f the gas.
4.6 ISENTROPIC FLOW
We are alm ost ready to return to our consideration o f aerodynam ics. However,
there is one m ore concept we m ust introduce, a concept that bridges both ther­
m odynam ics and com pressible aerodynam ics, namely, that o f isentropic flow.
First, consider three m ore definitions:
An adiabatic process is one in which no heat is added or taken away; Sq = 0.
A reversible process is one in which no frictional or other dissipative effects occur.
An isentropic process is one which is both adiabatic and reversible.
Thus, an isentropic process is one in which there is neither heat exchange nor
any effect due to friction. (The source o f the w ord isentropic is another defined
therm odynam ic variable, called entropy. E ntropy is constant for an isentropic
process. A discussion o f entropy is not vital to our discussion here; therefore, no
further elaboration is given.)
148
CHAPTER 4
Basic Aerodynamics
Isentropic processes are very im portant in aerodynam ics. For exam ple, con­
sider the flow o f air over the airfoil show n in Fig. 4.7. Im agine a fluid elem ent
m oving along one o f the stream lines. There is no heat being added or taken away
from this fluid elem ent; heat-exchange m echanism s such as heating by a flame,
cooling in a refrigerator, or intense radiation absorption are all ruled out by the
nature o f the physical problem we are considering. Thus, the flow o f the fluid
elem ent along the stream line is adiabatic. At the sam e time, the shearing stress
exerted on the surface o f the fluid elem ent due to friction is generally quite small
and can be neglected (except very near the surface, as will be discussed later).
Thus, the flow is also frictionless. [Recall that this sam e assum ption was used in
obtaining the m om entum equation, Eq. (4.8).] Hence, the flow o f the fluid ele­
m ent is both adiabatic and reversible (frictionless); that is, the flow is isentropic.
O ther aerodynam ic flows can also be treated as isentropic, for exam ple, the flows
through wind tunnel nozzles and rocket engines.
N ote that even though the flow is adiabatic, the tem perature need not be con­
stant. Indeed, the tem perature o f the fluid elem ent can vary from point to point in
an adiabatic, com pressible flow. This is because the volum e o f the fluid elem ent
(o f fixed m ass) changes as it m oves through regions o f different density along the
stream line; when the volum e varies, work is done [Eq. (4.15)], hence the internal
energy changes [Eq. (4.10)], and hence the tem perature changes [Eq. (4.23)].
This argum ent holds for com pressible flows, where the density is variable. On
the other hand, for incom pressible flow, where p = constant, the volum e o f the
fluid elem ent o f fixed m ass does not change as it m oves along a stream line;
hence, no w ork is done and no change in tem perature occurs. If the flow over the
airfoil in Fig. 4.7 w ere incom pressible, the entire flow field w ould be at constant
tem perature. For this reason, tem perature is not an im portant quantity for fric­
tionless incom pressible flow. M oreover, our present discussion o f isentropic
flows is relevant to compressible flows only, as explained in the following.
An isentropic process is m ore than ju st another definition. It provides us with
several im portant relationships am ong the therm odynam ic variables T, p , and p
at tw o different points (say, points 1 and 2 in Fig. 4.7) along a given stream line.
T hese relations are obtained as follows. Since the flow is isentropic (adiabatic
and reversible), 8q = 0. Thus, from Eq. ( 4 .16),
8q = d e + p d v = 0
(4.28)
—p d v = d e
Substitute Eq. (4.23) into (4.28);
—p d v = c v d T
(4.29)
In the sam e manner, using the fact that 8q = 0 in Eq. (4.19), we also obtain
8q = dh — v d p = 0
v d p = dh
(4.30)
4 .6
Isentropic Flow
Substitute Eq. (4.26) into (4.30):
(4 .3 1)
v d p = CpdT
D ivide Eq. (4.29) by (4 .3 1):
—p d v
vdp
dp_
or
cp d v
(4.32)
P
The ratio o f specific heats cp/ c v appears so frequently in com pressible flow
equations that it is given a sym bol all its ow n, usually y , cp/ c v = y . For air at
normal conditions, w hich apply to the applications treated in this book, both cp
and cv are constants, and hence y = constant = l .4 (for air). A lso cp/ c v =
y = 1.4 (for air at norm al conditions). Thus, Eq. (4.32) can be w ritten as
dp
dv
p ~
(4.33)
Y v
Referring to Fig. 4.7, we integrate Eq. (4.33) betw een points 1 and 2:
f P2d p _ _
J p iP
f V2 d v
Jvi
V
, Pi
, v2
In — = —y In —
Pi
Pi
P\
Since i>i = 1 /p i and
(4.34)
t>t
-G)
= 1 /Pi, Eq. (4.34) becom es
Pi _
P\
isentropic flow
(4.35)
\ p j
From the equation o f state, w e have p = p / { R T ) . Thus, Eq. (4.35) yields
p i _ ^ _pl RT1} y
RT2 P\
Y
(r = (iH r
r \ Y l i Y - 1)
or
H = (l
Pi
rfi /)
isentropic flow
(4.36)
149
150
CHAPTER 4
Basic Aerodynamics
C om bining Eqs. (4.35) and (4.36), we obtain
(4.37)
The relationships given in Eq. (4.37) are powerful. They provide im portant in­
form ation for p, T, and p betw een tw o different points on a stream line in an
isentropic flow. M oreover, if the stream lines all em anate from a uniform flow far
upstream (far to the left in Fig. 4.7), then Eq. (4.37) holds for any two points in
the flow, not necessarily those on the sam e stream line.
We em phasize again that the isentropic flow relations, Eq. (4.37), are rele­
vant to com pressible flows only. By contrast, the assum ption o f incom pressible
flow (rem ember, incom pressible flow is a myth, anyw ay) is not consistent with
the sam e physics that w ent into the developm ent o f Eq. (4.37). To analyze in­
com pressible flows, we need only the continuity equation [say, Eq. (4.3)] and the
m om entum equation [B ernoulli’s equation, Eqs. (4.9a) and (4.9¿)]. To analyze
com pressible flows, we need the continuity equation, Eq. (4.2), the m om entum
equation [E uler’s equation, Eq. (4.8)], and another soon-to-be-derived relation
called the energy equation. If the com pressible flow is isentropic, then Eq. (4.37)
can be used to replace either the m om entum or the energy equation. Since
Eq. (4.37) is a sim pler, m ore useful algebraic relation than E u ler’s equation,
Eq. (4.8), w hich is a differential equation, w e frequently use Eq. (4.37) in place
o f Eq. (4.8) for the analysis o f com pressible flows in this book.
As ju st m entioned, to com plete the developm ent o f the fundam ental rela­
tions for the analysis o f com pressible flow, w e m ust now consider the energy
equation.
EXAM PLE 4.7
An airplane is flying at standard sea-level conditions. The temperature at a point on the
wing is 250 K. What is the pressure at this point?
■ Solution
The air pressure and temperature, p\ and T\, far upstream of the wing correspond to stan­
dard sea level. Hence, P\ = 1.01 x 105 N/m 2 and T\ = 288.16 K. Assume the flow is
isentropic (hence, compressible). Then, the relation between points 1 and 2 is obtained
from Eq. (4.37):
H
! ) ........
p 2 = 6 .1 4 x 10 4 N /m 2
4 .6
Isentropic Flow
151
EXAM PLE 4.8
In a rocket engine, the fuel and oxidizer are burned in the combustion chamber, and then
the hot gas expands through a nozzle to high velocity at the exit of the engine. (Jump
ahead and see the sketch of a rocket-engine nozzle in Fig. 4.26.) The flow through the
rocket-engine nozzle downstream of the combustion chamber is isentropic. Consider the
case when the pressure and temperature of the burned gas in the combustion chamber are
20 atm and 3500 K, respectively. If the pressure of the gas at the exit of the nozzle is
0.5 atm, calculate the gas temperature at the exit. Note: The combustion gas is not air, so
the value for y will be different than for air; that is, y will not be equal to l .4. For the
combustion gas in this example, y = 1.15.
■ Solution
From Eq. (4.36),
n t r
where we will designate condition 1 to be the combustion chamber and condition 2 to be
the nozzle exit. Hence, p\ = 20 atm, T\ = 3500 K, and p 2 — 0.5 atm. Rearranging
Eq. (4.36), we have
2167 K
Question: Atmospheres is a nonconsistent unit for pressure. Why then did we not convert
Pi and P 2 to N/m2 before inserting into Eq. (4.36)? The answer is that p¡ and p 2 appear
as a ratio in the preceding calculation, namely P \ / P2 - As long as we use the same units
for the numerator and the denominator, the ratio is the same value, independent of what
units are used. To prove this, let us convert atmospheres to the consistent units of N/m2.
One atmosphere is by definition the pressure at standard sea level. From the listing of sealevel properties in Sec. 3.4, we see that
1 atm = 1 .0 1 x 105 N/m 2
(rounded to 3 significant figures)
Thus,
Pi = 20(1.01 x 105) = 2.02 x 106 N/m 2
p2 = 0.5(1.01 x 105) = 5.05 x 104 N/m 2
From Eq. (4.36),
/
\( y - O / y
T2 = T\ ( — )
\p |/
/ S O S x in 4 \ (1'15_1)//1'15
= 3500 ( — — — )
\ 2.02 x 10° /
which is the same answer as first obtained.
= 3500(0.025)°13 = 2167 K
152
CHAPTER 4
Basic Aerodynamics
4.7 ENERGY EQUATION
Recall that our approach to the derivation o f the fundam ental equations for fluid
flow is to state a fundam ental principle and then to proceed to cast that principle
in term s o f flow variables p, T, p , and V. A lso recall that com pressible flow,
high-speed flow, and m assive changes in energy go hand in hand. Therefore, our
last fundam ental physical principle that we m ust take into account is as follows.
Physical principle: Energy can be neither created nor destroyed. It can only change
form.
In quantitative form , this principle is nothing more than the first law o f ther­
m odynam ics, Eq. (4.10). To apply this law to fluid flow, consider again a fluid
elem ent m oving along a stream line, as show n in Fig. 4.6. L et us apply the first
law o f therm odynam ics
Sq + Sw = d e
to this fluid elem ent. Recall that an alternate form o f the first law is Eq. (4.19),
Sq = dh — v d p
A gain, we consider an adiabatic flow, where Sq = 0. H ence, from Eq. (4.19),
dh — v d p = 0
(4.38)
R ecalling E u ler’s equation, Eq. (4.8),
dp = - p V d V
w e can com bine Eqs. (4.38) and (4.8) to obtain
dh + v p V d V = 0
(4.39)
H ow ever, v = 1/ p , hence Eq. (4.39) becom es
dh + V d V = 0
(4.40)
Integrating Eq. (4.40) betw een tw o points along the stream line, we obtain
dh+ /
VdV = 0
(4.41)
h + — = const
2
4.7
Energy Equation
153
Equation (4 .4 1) is the energy equation for frictionless, adiabatic flow. It can be
written in term s o f T by using Eq. (4.27), h = cp T . H ence, Eq. (4.41) becom es
c„T, + \ V ¡ = cp T2 + \ V l
(4.42)
cpT + j V 2 = const
Equation (4.42) relates the tem perature and velocity at two different points
along a stream line. A gain, if all the stream lines em anate from a uniform flow far
upstream , then Eq. (4.42) holds for any tw o points in the flow, not necessarily on
the sam e stream line. M oreover, Eq. (4.42) is ju s t as pow erful and necessary for
the analysis o f com pressible flow as Eq. (4.37).
EXAMPI.I <: 4.9
A supersonic wind tunnel is sketched in Fig. 4.26. The air temperature and pressure in the
reservoir o f the wind tunnel are To = 1000 K and po = 10 atm, respectively. The static
temperatures at the throat and exit are T* = 833 K and Te = 300 K, respectively. The
mass flow through the nozzle is 0.5 kg/s. For air, cp = 1008 J/(kg)(K). Calculate
a. Velocity at the throat V*
b. Velocity at the exit V,
c. Area o f the throat A *
d. Area o f the exit Ae
m Solution
Since the problem deals with temperatures and velocities, the energy equation seems
useful.
a. From Eq. (4.42), written between the reservoir and the throat,
CpTo + j V q 2 = C pT* + | V *2
However, in the reservoir, Vo ** 0. Hence,
V* = y/2cp(To - T*)
= 7 2 ( 1 0 0 8 X 1 0 0 0 - 833) = 580 m/s
b. From Eq. (4.42) written between the reservoir and the exit,
CpT) = cpTe + \ V}
K = ¿2cp(T0 - Te)
= 7 2 ( 1 0 0 8 X 1 0 0 0 - 3 0 0 ) = 1188 m/s
c. The basic equation dealing with mass flow and area is the continuity equation,
Eq. (4.2). Note that the velocities are certainly large enough that the flow is compressible,
154
chapter
4
Basic Aerodynamics
hence, Eq. (4.2), rather than Eq. (4.3), is appropriate
m = p*A*V*
or
A* =
m
p* V*
In the preceding, m is given and V* is known from part a. However, p* must be obtained
before we can calculate A* as desired. To obtain p *, note that, from the equation of state,
po
lO O .O lx lO 5)
, 3
m ~KTs ~
287(1000)
= 3.52 kg/m'
Assuming the nozzle flow is isentropic, which is a very good approximation for the real
case, from Eq. (4.37), we get
(£)-( 5 P
/ 7’* \ i/(y—i)
/ o'jt \ i/(i-4—l)
P* = * ( j r )
= (3.52) ( j ^ q )
= 2.23 kg/m 3
Thus,
A* =
p*V*
(2.23)(580)
= 3 ’87 x 10-4 m 2 = 3.87cm2
d. Finding Ae is similar to the above solution for A*
m = peAeVe
where, for isentropic flow,
/
t
\ ' / ( y - *)
/ i n n \ 1/(I-4—1)
= ° - 52){ m
or
A, =
0.5
peVe
0.174(1188)
= ° - i74k^
:
= 24.2 x 10-4 m2 = 24.2 cm 2
E X A M P L E 4 .1 0
Consider an airfoil in a flow of air, where far ahead of the airfoil (the free stream), the
pressure, velocity, and density are 2116 lb/ft2, 500 mi/h, and 0.002377 slug/ft3, respec­
tively. At a given point A on the airfoil, the pressure is 1497 lb/ft2. What is the velocity at
point A? Assume isentropic flow. For air, cp = 6006 ft • lb/(slug)(°R).
■ Solution
This example is identical to Example 4.3, except here the velocity is 500 mi/h—high
enough that we have to treat the flow as compressible, in contrast to Example 4.3 where
4.8
Summary of Equations
we dealt with incompressible flow. Since the flow is isentropic, w e can use Eq. (4.37)
evaluated between the free stream and point A.
-Poo= {V¥Too)/
Z ± m ( P L ) (y- ' )/y = ( M
Too
\PooJ
T
*
= ( 0 . 7 0 7 5 ) 0 - = 0 .9058
\ 2 1 16/
The value o f T,x can be found from the equation o f state.
Hence,
P2
2116
PaoR
0 .002377(1716)
= 519°R
Ta = 0.9 0 5 8 (5 1 9 ) = 4 7 0 .1 °R
From the energy equation, Eq. (4.42), evaluated between the free stream and point A, and
noting that
= 5 0 0 (8 8 /6 0 ) = 733.3 ft/s, we have
V2
V?
CpToo + - f = c pTA + ^ VA = p C p i T n - Ta) + VI
= 72(6006) (519 —470.1) + (733.3)2 = 1061 ft/s
Note: The calculational procedure for this problem, where w e are dealing with com ­
pressible flow, is com pletely different from that for Example 4.3, where we were dealing
with incompressible flow. In Example 4.3, we could use Bernoulli’s equation, which
holds only for incompressible flow. We cannot use Bernoulli's equation to solve the pre­
sent problem, because this is a compressible flow problem and Bernoulli’s equation is not
valid for a compressible flow. If we had used Bernoulli’s equation to solve the present
problem, follow ing exactly the method in Example 4.3, we would have obtained a veloc­
ity o f 1029 ft/s at point A — an incorrect answer. Check this yourself.
4.8 SUMMARY OF EQUATIONS
We have ju st finished applying som e very basic physical principles to obtain
equations for the analysis o f flowing gases. The reader is cautioned not to be con­
fused by the m ultiplicity o f equations; they are useful, indeed necessary, tools to
exam ine and solve various aerodynam ic problem s o f interest. It is im portant for
an engineer or scientist to look at such equations and see not ju st a mathem atical
relationship, but prim arily a physical relationship. These equations talk! For ex­
am ple, Eq. (4.2) says that m ass is conserved; Eq. (4.42) says that energy is con­
served for an adiabatic, frictionless flow; etc. N ever lose sight o f the physical
im plications and lim itations o f these equations.
155
156
CHAPTER 4
Basic Aerodynamics
To help set these equations in your mind, here is a com pact sum m ary o f our
results so far:
1.
For the steady incom pressible flow o f a frictionless fluid in a stream tube o f
varying area, p and V are the m eaningful flow variables; p and T are
constants throughout the flow. To solve for p and V, use
= a 2v2
P\ + \ p v f = P 2 + \ p V i
2.
continuity
B ernoulli’s equation
For steady isentropic (adiabatic and frictionless) com pressible flow in a
stream tube o f varying area, p , p , T , and V are all variables. They are
obtained from
P \A \V \ — p 2A 2V2
E i = ( E)i
P2
y
continuity
/ j \ y/(y-D
= Í — )
isentropic relations
\P 2
cpT\ + 5 V,2 = cp T2 + 5 V¡
energy
P\ = P\RT\
P 2 = P 2 RT 2
equation o f state
Let us now apply these relations to study som e basic aerodynam ic phenom ena
and problem s.
4.9 SPEED OF SOUND
Sound w aves travel through the air at a definite speed— the speed o f sound. This
is obvious from natural observation; a lightning bolt is observed in the distance,
and thunder is heard at som e later instant. In many aerodynam ic problem s, the
speed o f sound plays a pivotal role. How do we calculate the speed o f sound?
W hat does it depend on: pressure, tem perature, density, or som e com bination
thereof? W hy is it so im portant? A nsw ers to these questions are discussed in this
section.
First let us derive a form ula to calculate the speed o f sound. C onsider a
sound w ave m oving into a stagnant gas, as show n in Fig. 4.12. T his sound wave
is created by som e source, say, a small firecracker in the corner o f a room. The
air in the room is m otionless and has density p , pressure p , and tem perature T.
If you are standing in the m iddle o f the room , the sound w ave sw eeps by you at
velocity a m/s, ft/s, etc. The sound wave itself is a thin region o f disturbance in
the air, across w hich the pressure, tem perature, and density change slightly. (The
change in pressure is w hat activates your eardrum and allow s you to hear the
sound wave.) Im agine that you now hop on the sound w ave and m ove with it. As
you are sitting on the m oving wave, look to the left in Fig. 4.12, that is, look in
4 .9
P
Speed of Sound
p + dp
a
P
p + dp
T
T + dT
S o u rce o f
so u n d wave
S o und wave m oving
to th e left w ith
velocity a in to
a sta g n an t gas
Figure 4.12 Model o f a sound wave moving into a stagnant gas.
2
Ahead of wave
P
P
Behind the wave
P + dp
a
a + da
T
P + dp
T + dT
Motionless sound wave
Figure 4.13 Model with the sound wave stationary.
the direction in w hich the w ave is moving. From your vantage point on the wave,
the sound w ave seem s to stand still, and the air in front o f the wave appears to be
com ing at you with velocity a ; that is, you see the picture shown in Fig. 4.13,
where the sound w ave is standing still and the air ahead o f the wave is m oving
tow ard the w ave w ith velocity a. Now, return to Fig. 4.12 for a m om ent. Sitting
on top o f and riding w ith the m oving wave, look to the right, that is, look behind
the wave. From your vantage point, the air appears to be m oving away from you.
This appearance is sketched in Fig. 4.13, w here the wave is standing still. Here,
the air behind the m otionless w ave is m oving to the right, away from the wave.
However, in passing through the wave, the pressure, tem perature, and density
o f the air are slightly changed by the am ounts d p , d T , and d p , respectively. From
our previous discussions, you w ould then expect the airspeed a to change
slightly, say, by an am ount da . Thus, the air behind the w ave is m oving away
from the w ave with velocity a + d a , as shown in Fig. 4.13. Figures 4.12 and 4.13
are com pletely analogous pictures; only their perspectives are different. Fig­
ure 4.12 is w hat you see by standing in the m iddle o f the room and w atching the
w ave go by; Fig. 4.13 is w hat you see by riding on top o f the wave and watching
the air go by. Both pictures are equivalent. However, Fig. 4.13 is easier to work
with, so w e will concentrate on it.
Let us apply our fundam ental equations to the gas flow shown in Fig. 4.13.
O ur objective is to obtain an equation for a, where a is the speed o f the sound
wave, the speed o f sound. Let points 1 and 2 be ahead o f and behind the wave,
respectively, as show n in Fig. 4.13. A pplying the continuity equation, Eq. (4.2),
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we find
P \A \V \ = p 2A 2V2
or
p A \ a = (p + d p ) A 2(a + d a )
(4.43)
Here, A | and A 2 are the areas o f a stream tube running through the wave. Just
looking at the picture show n in Fig. 4.13, w e see no geom etric reason why the
stream tube should change area in passing through the wave. Indeed, it does not;
the area o f the stream tube is constant; hence, A = A\ = A 2 = constant. (This is
an exam ple o f a type o f flow called one-dim ensional, or constant-area, flow.)
Thus, Eq. (4.43) becom es
p a = (p + d p )(a + da )
or
pa = pa + a dp + p da + dp da
(4.44)
The product o f tw o small quantities d p d a is very small in com parison to the
other term s in Eq. (4.44) and hence can be ignored. Thus, from Eq. (4.44),
da
a = —p —
(4.45)
Now apply the m om entum equation in the form o f E uler’s equation, Eq. (4.8):
d p = —p a da
or
pa
(4.46)
Substitute Eq. (4.46) into (4.45):
p_dp_
dp pa
or
dp
(4.47)
On a physical basis, the flow through a sound wave involves no heat addition,
and the effect o f friction is negligible. Hence, the flow through a sound wave is
isentropic. Thus, from Eq. (4.47), the speed o f sound is given by
(4.48)
Equation (4.48) is fundam ental and im portant. However, it does not give us a
straightforw ard form ula for com puting a num ber f o r a . We must proceed further.
4.9
Speed of Sound
F or isentropic flow, Eq. (4.37) gives
Pi _
Y
Pi
or
(4.49)
Equation (4.49) says that the ratio p / p Y is the sam e constant value at every point
in an isentropic flow. Thus, we can write everyw here
(4.50)
Hence,
(4 .5 1)
Substituting for c in Eq. (4 .5 1) the ratio o f Eq. (4.50), we obtain
(4.52)
Substitute Eq. (4.52) into (4.48):
(4.53)
However, for a perfect gas, p and p are related through the equation o f state;
p = p R T ; hence, p / p =c R T . Substituting this result into Eq. (4.53) yields
a = y /y R T
(4.54)
E quations (4.48), (4.53), and (4.54) are im portant results for the speed o f
sound; however, Eq. (4.54) is the m ost useful. It also dem onstrates a fundam ental
result: The speed o f sound in a perfect gas depends only on the temperature o f the
gas. This sim ple result may appear surprising at first. However, it is to be expected
on a physical basis, as follow s. The propagation o f a sound w ave through a gas
takes place via m olecular collisions. For exam ple, consider again a small fire­
cracker in the corner o f the room . W hen the firecracker is set off, som e o f its en­
ergy is transferred to the neighboring gas m olecules in the air, thus increasing
their kinetic energy. In turn, these energetic gas m olecules are m oving random ly
about, colliding with som e o f their neighboring m olecules and transferring some
o f their extra energy to these new m olecules. Thus, the energy o f a sound wave is
transm itted through the air by m olecules that collide with one another. Each m ol­
ecule is m oving at a different velocity, but if they are sum m ed over a large num ­
ber o f m olecules, a mean or average m olecular velocity can be defined. Therefore,
looking at the collection o f m olecules as a w hole, we see that the sound energy
released by the firecrackers will be transferred through the air at som ething
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4
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approxim ating this m ean m olecular velocity. Recall from Chap. 2 that tem pera­
ture is a m easure o f the m ean m olecular kinetic energy, hence o f the mean m ole­
cular velocity; then tem perature should also be a m easure o f the speed o f a sound
w ave transm itted by m olecular collisions. Equation (4.54) proves this to be a fact.
For exam ple, consider air at standard sea-level tem perature Ts = 288.16 K.
From Eq. (4.54), the speed o f sound is a = y / y R T = V I .4(287)(288.16) =
340.3 m/s. From the results o f the kinetic theory o f gases, the mean m olecular ve­
locity can be obtained as V =
(%/ tt) R T = 7 (8 /^ ) 2 8 7 (2 8 8 .1 6 ) = 458.9 m/s.
Thus, the speed o f sound is o f the same order o f m agnitude as the mean m olecular
velocity and is sm aller by about 26 percent.
Again w e em phasize that the speed o f sound is a point property o f the flow,
in the sam e vein as T is a point property as described in Chap. 2. It is also a ther­
m odynam ic property o f the gas, defined by Eqs. (4.48) to (4.54). In general, the
value o f the speed o f sound varies from point to point in the flow.
T he speed o f sound leads to another, vital definition for high-speed gas
flows, namely, the Mach number. C onsider a point B in a flow field. The flow
velocity at B is V, and the speed o f sound is a. By definition, the M ach num ber
M at point B is the flow velocity divided by the speed o f sound:
(4.55)
We will find that M is one o f the m ost pow erful quantities in aerodynam ics. We
can im m ediately use it to define three different regim es o f aerodynam ic flows:
1.
If M < 1, the flow is subsonic.
2.
If M = 1, the flow is sonic.
3.
If M > 1, the flow is supersonic.
Each o f these regim es is characterized by its own special phenom ena, as will
be discussed in subsequent sections. In addition, tw o other specialized aerody­
nam ic regim es are com m only defined, namely, transonic flow, where M gener­
ally ranges from slightly less than to slightly greater than 1 (for exam ple,
0.8 < M < 1.2), and hypersonic flow, where generally M > 5. The definitions
o f subsonic, sonic, and supersonic flows in term s o f M as given are precise; the
definitions o f transonic and hypersonic flows in term s o f M are a bit m ore im ­
precise and really refer to sets o f specific aerodynam ic phenom ena, rather than to
ju st the value o f M. This distinction will be clarified in subsequent sections.
EXAM PLE 4 .11
A jet transport is flying at a standard altitude of 30,000 ft with a velocity of 550 mi/h.
What is its Mach number?
■ Solution
From the standard atmosphere table, App. B, at 30,000 ft, Tx = 4 1 1.86 R. Hence, from
Eq. (4.54),
ax = \ f y R T = 71.4(1716X411.86) = 995 ft/s
4 .9
Speed of Sound
161
The airplane velocity is
= 550 mi/h; however, in consistent units, remembering that
88 ft/s = 60 mi/h, we find that
807 ft/s
From Eq. (4.55),
Voo = 807
floe
995
0.811
EXAM PLE 4.12
In the nozzle flow described in Example 4.9, calculate the Mach number of the flow at the
throat, M* and at the exit Me.
■ Solution
From Example 4.9, at the throat, V* = 580 m/s and T* = 833 K. Hence, from Eq. (4.54),
a * = y/yR T * = v/l.4(287)(833) = 580m/s
From Eq. (4.55),
a*
580
LLJ
Note: The flow is sonic at the throat. We will soon prove that the Mach number at the
throat is always sonic in supersonic nozzle flows (except in special, nonequilibrium,
high-temperature flows, which are beyond the scope of this book).
Also from Example 4.9, at the exit, Ve = I 188 m/s and Te = 300 K. Hence,
a , = J y R Te =
'
Ve _ 1188
ae
347
.4(287) (300) = 347 m/s
3.42
C o m m e n t Exam ples 4.11 and 4.12 illustrate tw o com m on uses o f M ach num ­
ber. The speed o f an airplane is frequently given in term s o f Mach number. In Ex­
am ple 4.11, the M ach num ber o f the je t transport is calculated; here the Mach
num ber o f the airplane is the velocity o f the airplane through the air divided by
the speed o f sound in the am bient air far ahead o f the airplane. This use o f Mach
num ber is frequently identified as the free-stream M ach number. In Exam ple 4.12,
the local M ach num ber is calculated at tw o different points in a flow field, at the
throat and at the exit o f the nozzle flow. At any given point in a flow, the local
M ach num ber is the local flow velocity at that point divided by the local value o f
the speed o f sound at that point. Here, M ach num ber is used as a local flow prop­
erty in a flow field, and its value varies from point to point throughout the flow
because both velocity and the local speed o f sound (which depends on the local
tem perature) vary throughout the flow.
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4
Basic Aerodynamics
4.10 LOW-SPEED SUBSONIC WIND TUNNELS
Throughout the rem ainder o f this book, the aerodynam ic fundam entals and tools
(equations) developed in previous sections will be applied to specific problem s
o f interest. The first will be a discussion o f low -speed subsonic wind tunnels.
W hat are w ind tunnels, any kind o f w ind tunnels? In the m ost basic sense,
they are ground-based experim ental facilities designed to produce flows o f air
(or som etim es other gases), w hich sim ulate natural flows occurring outside the
laboratory. For m ost aerospace engineering applications, wind tunnels are de­
signed to sim ulate flows encountered in the flight o f airplanes, m issiles, or space
vehicles. Since these flows have ranged from the 27 m i/h speed o f the early
Wright Flyer to the 25,000 m i/h reentry velocity o f the A pollo lunar spacecraft,
obviously m any different types o f wind tunnels, from low subsonic to hyper­
sonic, are necessary for the laboratory sim ulation o f actual flight conditions.
However, referring again to Fig. 1.30, we see that flow velocities o f 300 mi/h or
less w ere the flight regim e o f interest until about 1940. Hence, during the first
four decades o f hum an flight, airplanes were tested and developed in w ind tun­
nels designed to sim ulate low -speed subsonic flight. Such tunnels are still in use
today but are com plem ented by transonic, supersonic, and hypersonic wind
tunnels as well.
The essence o f a typical low -speed subsonic w ind tunnel is sketched in
Fig. 4.14. The airflow with pressure p\ enters the nozzle at a low velocity Vi,
w here the area is A \. The nozzle converges to a sm aller area A 2 at the test sec­
tion. Since we are dealing w ith low -speed flows, where M is generally less than
0.3, the flow will be assum ed to be incom pressible. H ence, Eq. (4.3) dictates that
the flow velocity increases as the air flows through the convergent nozzle. The
velocity in the test section is then, from Eq. (4.3),
(4.56)
A fter flowing over an aerodynam ic model (w hich may be a model o f a com plete
airplane or part o f an airplane, such as a w ing, tail, or engine nacelle), the air
Settling
cham ber
(reservoir)
Pl
Pi< A i
P3
T est section
Nozzle
Figure 4.14 Sim ple schem atic o f a subsonic wind tunnel.
D iffuser
4.1 0
Low-Speed Subsonic Wind Tunnels
passes into a diverging duct called a diffuser, w here the area increases and ve­
locity decreases to A 3 and V3, respectively. A gain, from continuity,
T he pressure at various locations in the w ind tunnel is related to the velocity,
through B ernoulli’s equation, Eq. (4.9a), for incom pressible flow.
P\ + \ p V} = P2 + \ p V } = P3 + { PV32
(4.57)
From Eq. (4.57), as V increases, p decreases; hence, p 2 < p\\ that is, the test
section pressure is sm aller than the reservoir pressure upstream o f the nozzle. In
m any subsonic wind tunnels, all or part o f the test section is open, or vented, to
the surrounding air in the laboratory. In such cases, the outside air pressure is
com m unicated directly to the flow in the test section, and p 2 = l atm. D ow n­
stream o f the test section, in the diverging area diffuser, the pressure increases as
velocity decreases. Hence, p 3 > p 2. If A 3 = A \, then from Eq. (4.56), V3 = V,;
and from Eq. (4.57), /?3 = p\. (Note: In actual wind tunnels, the aerodynam ic
drag created by the flow over the m odel in the test section causes a loss o f m o­
m entum not included in the derivation o f B ernoulli’s equation; hence, in reality,
P3 is slightly less than p\, because o f such losses.)
In practical operation o f this type o f wind tunnel, the test section velocity is
governed by the pressure difference p\ — p 2 and the area ratio o f the nozzle
A 2/A [ as follows. From Eq. (4.57),
V2 = -p (P\ ~ P 2 ) + V 2
(4.58)
From Eq. (4.56), Vi = (A 2/ A \ ) V 2. Substituting this into the right-hand side o f
Eq. (4.58), we obtain
V2 = - p (Pi
+ ( 77)
V22
(4-59)
Solving Eq. (4.59) for V2 yields
V2 = J
2{P \
P2^ ~
r
y p [ i - (A 2/ A \ ) 2]
(4.60)
The area ratio A 2/ A \ is a fixed quantity for a wind tunnel o f given design.
Hence, the “control knob” o f the wind tunnel basically controls p\ — p 2, which
allows the wind tunnel operator to control the value o f test section velocity V2
via Eq. (4.60).
In subsonic wind tunnels, a convenient method o f m easuring the pressure
difference p\ — p 2, hence o f m easuring V2 via Eq. (4.60), is by means o f a
manometer. A basic type o f m anom eter is the U tube shown in Fig. 4 . 15. Here,
the left side o f the tube is connected to a pressure p¡, the right side o f the tube is
connected to a pressure p 2, and the difference Ah in the heights o f a fluid in both
163
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4
Basic Aerodynamics
Pi
Pi
PiA
A
B
p 2A + (Ah)Aw
B
B
B
Fluid
Figure 4.15 Force diagram for a manometer.
sides o f the U tube is a m easurem ent o f the pressure difference p 2 - p \. This can
easily be dem onstrated by considering the force balance on the liquid in the tube
at the two cross sections cut by plane B-B. shown in Fig. 4.15. Plane B-B is
draw n tangent to the top o f the colum n o f fluid on the left. If A is the crosssectional area o f the tube, then p \A is the force exerted on the left colum n
o f fluid. The force on the right colum n at plane B-B is the sum o f the w eight o f
the fluid above plane B-B and the force due to the pressure p 2A. The volum e o f
the fluid in the right colum n above B-B is A Ah. The specific w eight (w eight per
unit volum e) o f the fluid is w = p ig , where p¡ is the density o f the fluid and g is
the acceleration o f gravity. H ence, the total w eight o f the colum n o f fluid above
B-B is the specific w eight tim es the volum e, that is, w A Ah. The total force on
the right-hand cross section at plane B-B is then p 2A + wA Ah. Since the fluid
is stationary in the tube, the forces on the left- and right-hand cross sections m ust
balance; that is, they are the same. Hence,
P \ A = p 2A + w A Ah
or
P\ — p 2 = w Ah
(4.61)
If the left-hand side o f the U -tube m anom eter were connected to the reservoir in
a subsonic tunnel (point 1 in Fig. 4.14) and the right-hand side w ere connected to
the test section (point 2), then Ah o f the U tube w ould directly m easure the
velocity o f the airflow in the test section via Eqs. (4.61) and (4.60).
In m odern wind tunnels, m anom eters have been replaced by pressure trans­
ducers and electrical digital displays for reading pressures and pressure differ­
ences. The basic principle o f the m anom eter, however, rem ains an integral part
o f the study o f fluid dynam ics, and that is why we discuss it here.
In a low -speed subsonic wind tunnel, one side o f a mercury manometer is connected to
the settling chamber (reservoir) and the other side is connected to the test section. The
contraction ratio o f the nozzle A^/A \ equals -¡^. The reservoir pressure and temperature
4 .1 0
Low-Speed Subsonic Wind Tunnels
165
area p\ — 1.1 atm and T\ = 300 K, respectively. When the tunnel is running, the height
difference between the two columns of mercury is 10 cm. The density of liquid mercury
is 1.36 x 104 kg/m3. Calculate the airflow velocity in the test section V2.
Solution
A h = 10 cm = 0.1 m
w (for mercury) = p/g = (1.36 x 104 kg/m3)(9.8 m/s2)
= 1.33 x 105 N/m3
From Eq. (4.61),
Pi - p 2 = w Ah = (1.33 x 105 N/m3)(0.1m) = 1.33 x 104 N/m2
To find the velocity V2, use Eq. (4.60). However, in Eq. (4.60) we need a value of den­
sity p. This can be found from the reservoir conditions by using the equation of state.
(Remember: 1 atm = 1.01 x 105N/m2.)
Pi =
P1
RTt
. 1( 1.01 x 105)
= 1.29 kg/m3
287(300)
Since we are dealing with a low-speed subsonic flow, assume p\ — p — constant. Hence,
from Eq. (4.60),
V2 =
/
2(|p, - p 2)
1 p U - M 2M 1)2]
2(1.33 x 104)
144 m/s
l - 2 9 [ l - ( ¿ ) 2]
Note: This answer corresponds to approximately a Mach number of 0.4 in the test section,
one slightly above the value of 0.3 that bounds incompressible flow. Hence, our assump­
tion of p = constant in this example is inaccurate by about 8 percent.
EXAM PLE 4.14
Referring to Fig. 4.14, consider a low-speed subsonic wind tunnel designed with a reser­
voir cross-section area A \ — 2 n r and a test-section cross-section area A2 — 0.5 m2. The
pressure in the test section is p 2 = 1 atm. Assume constant density equal to standard sealevel density. (a) Calculate the pressure required in the reservoir, p \ , necessary to achieve
a flow velocity V2 = 40 m/s in the test section, (h) Calculate the mass flow through the
wind tunnel.
■ Solution
a. From the continuity equation, Eq. (4.3),
= A2V2
or
V| “ 14 ( 3 7 ) - (4® ( Ü ) - l 0 »l/*
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chapter
4
Basic Aerodynamics
From Bernoulli’s equation, Eq. (4.9a),
V,2
V2
P2 + P~y = Pi + P y
Using consistent units,
Pi = l a tm = 1.01 x 105 N/m2
and at standard sea level,
p = 1.23 kg/m3
we have
As a check on this calculation, let us insert p\ = 1.019 x 105 N/m2 into Eq. (4.60) and
see if we obtain the required value of V2 = 40 m/s. From Eq. (4.60),
2(pi - pi)
2 (1 .0 1 9 - 1.01) x 105
— 40 m/s
This checks.
Note: The pressure difference, p2 —p \, required to produce a velocity of 40 m/s in
the test section is very small, equal to 1.019 x 105 — 1.01 x 105 = 900N/m2. In atmo­
spheres, this is 900/(1.01 x 105) = 0.0089 atm, less than a hundredth of an atmosphere
pressure difference. This is characteristic of low-speed flows, where it takes only a small
pressure difference to produce a substantial flow velocity.
b. From Eq. (4.2), the mass flow can be calculated from the product pAV evaluated at
any location in the wind tunnel. We choose the test section, where Ai = 0.5 m2, Vi =
40 m/s, and p = 1.23 kg/m3.
m = pA iV i = (1.23)(0.5)(40) = 24.6 kg/s
We could just as well have chosen the reservoir to evaluate the mass flow, where A\ =
2 m2 and V\ = 10 m/s.
m = pA , V, = (1.23)(2)(10) = 24.6kg/s
which checks with the result obtained in the test section.
For the wind tunnel in Example 4.14, (a) If the pressure difference (p\ — p2) is doubled,
calculate the flow velocity in the test section, (b) The ratio A \/A i is defined as the
4 .1 0
Low-Speed Subsonic Wind Tunnels
contraction ratio for the wind tunnel nozzle. If the contraction ratio is doubled, keeping
the same pressure difference as in Example 4 .14, calculate the flow velocity in the test
section.
■ Solution
a. From Eq. (4.60), Vi is clearly proportional to the square root of the pressure difference
Vi OC VPoo - P\
When p 2 — pi is doubled from its value in Example 4.14, where Vi = 40 m/s, then
Vi = s i i (40) =
56.6 m/s
b. The original contraction ratio from Example 4.14 is A \/A i = 2 . 0/0.5 = 4. Doubling
this value, we have A \/A i — 8. The original pressure difference is Pi —P\ = 900 N/m2.
From Eq. (4.60), we have
2(900)
2(pJ - Pi)
1.23
' [
-
t o
\
N ) ’]
Note: By doubling only the pressure difference, a 42 percent increase in velocity in the
test section occurred. In contrast, by doubling only the contraction ratio, a 3.5 percent
decrease in the velocity in the test section occurred. Once again we see an example
of the power of the pressure difference in dictating flow velocity in a low-speed flow.
Also, the decrease in the test section velocity when the contraction ratio is increased,
keeping the pressure difference the same, seems counterintuitive. Why does not the ve­
locity increase when the nozzle is “necked down” further? To resolve this apparent anom­
aly, let us calculate the velocity in the reservoir for the increased contraction ratio. From
the continuity equation, A\ V\ = AiVi. Hence,
When the contraction ratio is increased, keeping the pressure difference constant, the
reservoir velocity decreases even more than the test section velocity, resulting in a larger
velocity change across the nozzle. For the case in Example 4.14 with a contraction ratio
of 4,
V i - V | = 40 - 10 = 30 m/s
For the present case with a contraction ratio of 8,
V i - V | = 38.6 - 4.83 = 33.8 m/s
By increasing the contraction ratio while keeping the pressure difference constant, the
velocity difference across the nozzle is increased, although the actual velocities at the
inlet and exit of the nozzle are decreased.
167
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Basic Aerodynamics
4.11 MEASUREMENT OF AIRSPEED
In Sec. 4.10, we dem onstrated that the airflow velocity in the test section o f a
low -speed wind tunnel (assum ing incom pressible flow) can be obtained by
m easuring p\ — p 2. H ow ever, the previous analysis im plicitly assum es that the
flow properties are reasonably constant over any given cross section o f the flow
in the tunnel (so-called quasi-one-dim ensional flow). If the flow is not constant
over a given cross section, for exam ple, if the flow velocity in the m iddle o f the
test section is higher than that near the walls, then V2 obtained from the pre­
ceding section is only a m ean value o f the test section velocity. For this reason
and for m any other aerodynam ic applications, it is im portant to obtain a point
m easurem ent o f velocity at a given spatial location in the flow. This m easure­
m ent can be m ade by an instrum ent called a P itot-static tube, as described in
the follow ing.
First, we m ust add to our inventory o f aerodynam ic definitions. We have
been glibly talking about the pressures at points in flows, such as points 1 and 2
in Fig. 4.7. However, these pressures are of a special type, called static. Static
pressure at a given point is the pressure we w ould feel if we were m oving along
with the flow at that point. It is the ram ification o f gas m olecules m oving about
with random m otion and transferring their m om entum to or across surfaces, as
discussed in Chap. 2. If we look more closely at the m olecules in a flowing gas,
we see that they have a purely random m otion superim posed on a directed m o­
tion due to the velocity o f the flow. Static pressure is a consequence o f ju st the
purely random m otion o f the m olecules. W hen an engineer or scientist uses the
word pressure, it alw ays m eans static pressure unless otherw ise identified, and
we will continue such practice here. In all our previous discussions, the pressures
have been static pressures.
A second type o f pressure is com m only utilized in aerodynam ics, namely,
total pressure. To define and understand total pressure, consider again a fluid
elem ent m oving along a stream line, as show n in Fig. 4.6. The pressure o f the gas
in this fluid elem ent is the static pressure. However, now im agine that we grab
this fluid elem ent and slow it down to zero velocity. M oreover, im agine that we
do this isentropically. Intuitively, the therm odynam ic properties p , T , and p of
the fluid elem ent will change as w e bring the elem ent to rest; they will follow the
conservation laws w e have discussed previously in this chapter. Indeed, as the
fluid elem ent is isentropically brought to rest, p , T , and p w ould all increase
above their original values when the elem ent is m oving freely along the stream ­
line. The values o f p , T , and p o f the fluid elem ent after it has been brought to
rest are called total values, that is, total pressure po, total tem perature T0, etc.
Thus, w e are led to the follow ing precise definition:
Total pressure at a given point in a flow is the pressure that would exist if the flow
were slowed down isentropically to zero velocity.
There is a perspective to be gained here. Total pressure po is a property o f the
gas flow at a given point. It is som ething that is associated with the flow itself.
4.11
Measurement of Airspeed
The process o f isentropically bringing the fluid elem ent to rest is ju st an im agi­
nary m ental process we use to define the total pressure. It does not mean that we
actually have to do it in practice. In other w ords, if we consider again the flow
sketched in Fig. 4.7, there are tw o pressures w e can consider at points 1, 2, etc.,
associated with each point o f the flow: a static pressure p and a total pressure po,
where p 0 > p .
For the special case o f a gas that is not m oving, that is, the fluid elem ent has
no velocity in the first place, then static and total pressures are synonym ous:
Po = p . This is the case in com m on situations such as the stagnant air in the
room and gas confined in a cylinder.
T he follow ing analogy m ight help to further illustrate the difference between
the definitions o f static and total pressure. A ssum e that you are driving down the
highw ay at 60 m i/h. The w indow s o f your autom obile are closed. Inside the au­
tom obile, along with you, there is a fly buzzing around in a very random fashion.
Your speed is 60 m i/h, and in the mean, so is that o f the fly, m oving down the
highw ay at 60 m i/h. H ow ever, the fly has its random buzzing-about motion
superim posed on top o f its m ean directed speed o f 60 mi/h. To you in the auto­
m obile, all you see is the random , buzzing-about motion o f the fly. If the fly hits
your skin with this random m otion, you will feel a slight impact. This slight
im pact is analogous to the static pressure in a flowing gas, w here the static pres­
sure is due sim ply to the random m otion o f the m olecules. Now assum e that you
open the w indow o f your autom obile, and the fly buzzes out. T here is a person
standing along the side o f the road. If the fly that has ju st left your autom obile
hits the skin o f this person, the im pact will be strong (it may even really hurt) be­
cause the fly hits this person w ith a mean velocity o f 60 mi/h plus w hatever its
random velocity may be. The strength o f this im pact is analogous to the total
pressure o f a gas.
There is an aerodynam ic instrum ent that actually m easures the total pressure
at a point in the flow, namely, a P itot tube. A basic sketch o f a Pitot tube is shown
in Fig. 4.16. It consists o f a tube placed parallel to the flow and open to the flow
at one end (point A). The other end o f the tube (point B) is closed. Now im agine
that the flow is first started. Gas will pile up inside the tube. A fter a few m om ents,
there will be no m otion inside the tube because the gas has now here to go— the
gas will stagnate once steady-state conditions have been reached. In fact, the gas
will be stagnant everyw here inside the tube, including at point A. As a result, the
flow field sees the open end o f the Pitot tube (point A) as an obstruction, and a
fluid elem ent m oving along the stream line, labeled C , has no choice but to stop
when it arrives at point A. Since no heat has been exchanged, and friction is neg­
ligible, this process will be isentropic; that is, a fluid elem ent m oving along
stream line C will be isentropically brought to rest at point A by the very presence
o f the Pitot tube. Therefore, the pressure at point A is, truly speaking, the total
pressure p 0. This pressure will be transm itted throughout the Pitot tube, and if a
pressure gauge is placed at point B, it will in actuality m easure the total pressure
o f the flow. In this fashion, a Pitot tube is an instrum ent that m easures the total
pressure o f a flow.
169
170
chapter
4
Basic Aerodynamics
T o ta l pressure m easured here
Figure 4.17 Schematic of a Pitot-static measurement.
By definition, any point o f a flow w here V = 0 is called a stagnation point.
In Fig. 4.16, point A is a stagnation point.
C onsider the arrangem ent shown in Fig. 4.17. H ere we have a uniform flow
w ith velocity Vi m oving over a flat surface parallel to the flow. T here is a small
hole in the surface at point A, called a static pressure orifice. Since the surface
is parallel to the flow, only the random m otion o f the gas m olecules will be felt
by the surface itself. In other w ords, the surface pressure is indeed the static pres­
sure p . This will be the pressure at the orifice at point A. On the other hand, the
Pitot tube at point B in Fig. 4.17 will feel the total pressure po, as previously
4 .11
T o ta l p ressure
felt h ere
Measurement of Airspeed
S tatic pressure felt here
L
£
P ito t-sta tie probe
Figure 4.18 S c h e m a tic o f a P ito t-s ta tic p ro b e.
discussed. If the static pressure orifice at point A and the Pitot tube at point B are
connected across a pressure gauge, as show n in Fig. 4.17, the gauge will measure
the difference betw een total and static pressure p 0 — p .
Now we arrive at the main thrust o f this section. The pressure difference
Po — p, as m easured in Fig. 4.17, gives a m easure o f the flow velocity V\. A com ­
bination o f a total pressure m easurem ent and a static pressure m easurem ent allows
us to m easure the velocity at a given point in a flow. These two m easurem ents can
be com bined in the same instrum ent, a P itot-static probe, as illustrated in Fig. 4.18.
A Pitot-static probe m easures p 0 at the nose o f the probe and p at a point on the
probe surface dow nstream o f the nose. The pressure difference p 0 — p yields the
velocity V\, but the quantitative form ulation differs depending on w hether the flow
is low speed (incom pressible), high-speed subsonic, or supersonic.
4.11.1 In co m p ressib le F low
C onsider again the sketch shown in Fig. 4.17. A t point A , the pressure is p and
the velocity is V\. A t point B, the pressure is p 0 and the velocity is zero. A pply­
ing B ernoulli’s equation, Eq. (4.9a), at points A and B , we obtain
p
=
+
Po
S ta tic
D y n a m ic
T o ta l
p ressu re
p ressu re
p ressu re
(4.62)
In Eq. (4.62), for dynamic pressure q we have the definition
q = \p V 2
(4.63)
which is frequently em ployed in aerodynam ics; the grouping \ p V 2 is term ed the
dynamic pressure f o r flow s o f a ll types, incom pressible to hypersonic. From
Eq. (4.62),
(4.64)
Po = P + q
This relation holds f o r incompressible flow only. The total pressure equals the
sum o f the static and the dynam ic pressure. A lso from Eq. (4.62),
(4.65)
1 71
172
chapter
4
Basic Aerodynamics
P ito t probe
(6 ) O pen test sectio n
Figure 4.19 Pressure measurements in open and closed test sections
of subsonic wind tunnels.
Equation (4.65) is the desired result; it allow s the calculation o f flow velocity
from a m easurem ent o f p 0 — p , obtained from a Pi tot-static tube. A gain, we em ­
phasize that Eq. (4.65) holds only for incom pressible flow.
A Pitot tube can be used to m easure the flow velocity at various points in the
test section o f a low -speed w ind tunnel, as show n in Fig. 4.19. The total pressure
at point B is obtained by the Pitot probe, and the static pressure, also at point B,
is obtained from a static pressure orifice located at point A on the wall o f the
closed test section, assum ing that the static pressure is constant throughout the
test section. This assum ption o f constant static pressure is fairly good for sub­
sonic wind tunnel test sections and is com m only made. If the test section is open
to the room , as also sketched in Fig. 4.19, then the static pressure at all points in
the test section is p = 1 atm. In either case, the velocity at point A is calculated
from Eq. (4.65). The density p in Eq. (4.65) is a constant (incom pressible flow).
Its value can be obtained by m easurem ents o f p and T som ew here in the tunnel,
using the equation o f state to calculate p = p / ( R T ) . These m easurem ents are
usually m ade in the reservoir upstream o f the nozzle.
E ither a Pitot tube or a Pitot-static tube can be used to m easure the airspeed
o f airplanes. Such tubes can be seen extending from airplane wing tips, with the
tube oriented in the flight direction, as show n in Fig. 4.20. If a Pitot tube is used,
then the am bient static pressure in the atm osphere around the airplane is obtained
from a static pressure orifice placed strategically on the airplane surface. It is
4.1 1
Measurement of Airspeed
P ito t tu b e
L
F light
d irectio n
Figure 4.20 Sketch of wing-mounted Pitot
probe.
placed w here the surface pressure is nearly the same as the pressure o f the sur­
rounding atm osphere. Such a location is found by experience. It is generally on
the fuselage som ew here betw een the nose and the wing. The values o f p 0 ob­
tained from the wing tip Pitot probe and p obtained from the static pressure
orifice on the surface enable the calculation o f the airplane’s speed through the
air using Eq. (4.65), as long as the a irp la n e’s velocity is low enough to justify the
assumption o f incom pressible flow, that is, for velocities less than 300 ft/s. In ac­
tual practice, the m easurem ents o f p 0 and p are joined across a differential pres­
sure gauge that is calibrated in term s o f airspeed, using Eq. (4.65). This airspeed
indicator is a dial in the cockpit, with units o f velocity, say miles per hour, on the
dial. However, in determ ining the calibration, that is, in determ ining what values
o f miles per hour go along with given values o f p 0 — p , the engineer must decide
what value o f p to use in Eq. (4.65). If p is the true value, som ehow m easured in
the actual air around the airplane, then Eq. (4.65) gives the true airspeed o f the
airplane:
(4.66)
However, the m easurem ent o f atm ospheric air density directly at the airplane’s
location is difficult. Therefore, for practical reasons, the airspeed indicators on
low-speed airplanes are calibrated by using the standard sea-level value o f p s in
Eq. (4.65). This gives a value o f velocity called the equivalent airspeed:
(4.67)
The equivalent airspeed Ve differs slightly from V^e, the difference being the
factor ( p / p i ) l/2. At altitudes near sea level, this difference is small.
173
174
chapter
4
Basic Aerodynamics
EXAM PLE 4.16
The altimeter on a low-speed Cessna 150 private aircraft reads 5000 ft. By an indepen­
dent measurement, the outside air temperature is 505° R. If a Pitot tube mounted on the
wing tip measures a pressure of 1818 lb/ft2, what is the true velocity of the airplane? What
is the equivalent airspeed?
■ Solution
An altimeter measures the pressure altitude (see discussion in Chap. 3). From the stan­
dard atmosphere table, App. B, at 5000 ft, p = 1761 lb/ft2. Also, the Pitot tube measures
total pressure; hence,
p o - p = 1818 - 1761 = 57 lb/ft2
The true airspeed can be obtained from Eq. (4.66): however, we need p, which is obtained
from the equation of state. For the outside, ambient air,
P=
1761
= 2.03 x 10“3 slug/ft
1716(505)
RT
From Eq. (4.66),
VW =
¡2(p0 - p)
1
p
2(57)
V 2.03 x 10- 3
237 ft/s
Note: Since 88 ft/s = 60mi/h, Vlrue = 237(60/88) = 162mi/h.
The equivalent airspeed (that which would be read on the airspeed indicator in the
cockpit) is obtained from Eq. (4.67), where ps = 0.002377 slug/ft3 (the standard sealevel value). Hence, from Eq. (4.67),
V, =
2(po - p)
2(57)
Ps
2.377 x 10- 3
219 ft/s
Note that there exists a 7.6 percent difference between Vtnie and Ve.
4 .11.2 S u b so n ic C o m p ressib le Flow
T he results o f Sec. 4.11.1 are valid for airflows where M < 0.3, that is, where the
flow can be reasonably assum ed to be incom pressible. This is the flight regim e
of, for exam ple, sm all, piston-engine private aircraft. For higher-speed flows, but
w here the M ach num ber is still less than 1 (high-speed subsonic flows), other
equations m ust be used. This is the flight regim e o f com m ercial jet transports
such as the Boeing 747 and the M cD onnell-D ouglas D C -10 and o f m any m ilitary
aircraft. For these cases, com pressibility m ust be taken into account, as follows.
C onsider the definition o f enthalpy h = e + p v . Since h = cpT and e =
c„7\ then cp T = c vT + R T , or
cp — c v = R
(4.68)
4.11
Measurement of Airspeed
Divide Eq. (4.68) by cp:
l
R
-
Cp/Cv
l -
Cp
i . m
Y
Y
R
yR _
or
Cn ---
(4.69)
Y ~ 1
Equation (4.69) holds for a perfect gas with constant specific heats. It is a necessary therm odynam ic relatiion for use in the energy equation, as follows,
C onsider again a Pit¿)it tube in a flow, as show n in Figs. 4.16 and 4.17. A s ­
sum e the flow velocity V\ ís high enough that com pressibility m ust be taken into
account. As usual, the liew is isentropically com pressed to zero velocity at the
stagnation point on the n ose o f the probe. The values o f the stagnation, or total,
pressure and tem perature at this point are p ti and To, respectively. From the energy equation, Eq. (4.42) w ritten betw een a point in the free-stream flow where
the tem perature and veloCiity are T\ and V |, respectively, and the stagnation point,
where the velocity is zero and the tem perature is T0,
cp T\ + jV |2 = CpTo
5 = i + vT,
2cpT\
or
(4.70)
Substitute Eq. (4.69) for cp in Eq. (4.70):
^
r,
= 1+
Y - 1 V,2
V,2
= 1+
2 [y R /(y -l)]T t
2
y RT\
(4.71)
However, from Eq. (4.54
(4.54) for the speed o f sound,
a] = Y RT,
Thus, Eq. (4.71) becom es
To
'1 "
7,
, Y ~ 1 V?
^2
Uj
a?
(4.72)
Since the M ach num ber M\ = V \/a \, Eq. (4.72) becom es
— = 1 + ------ - M f
Ti
2
1
(4.73)
175
176
chapter
4
Basic Aerodynamics
Since the gas is isentropically com pressed at the nose o f the Pitot probe in
Figs. 4 .16 and 4 . 17, Eq. (4.37) holds betw een the free stream and the stagnation
point. That is, p o /p \ = (p o /P \)y = (T q/ T \ Y I(y~X). Therefore, from Eq. (4.73),
we obtain
(4.74)
(4.75)
Equations (4.73) to (4.75) are fundam ental and im portant relations for com ­
pressible, isentropic flow. They apply to many other practical problem s in addition
to the Pitot tube. Note that Eq. (4.73) holds for adiabatic flow, w hereas Eqs. (4.74)
and (4.75) contain the additional assum ption o f frictionless (hence isentropic)
flow. A lso, from a slightly different perspective, Eqs. (4.73) to (4.75) determ ine
the total tem perature, density, and pressure— T0, p0, and po— at any point in the
flow w here the static tem perature, density, and pressure are T\, p \, and p\ and
where the M ach num ber is M \. In other w ords, reflecting the earlier discussion of
the definition o f total conditions, Eqs. (4.73) to (4.75) give the values o f p 0, T0,
and At that are associated with a point in the flow where the pressure, tem perature,
density, and M ach num ber are p \ , T \ , p \ , and M \, respectively. These equations
also dem onstrate the pow erful influence o f M ach num ber in aerodynam ic flow
calculations. It is very im portant to note that the ratios T0/T i, p o /p i.a n d p o /p i are
functions o f M\ only (assum ing y is known; y = l .4 for norm al air).
Returning to our objective o f measuring airspeed, and solving Eq. (4.74) for
M i, we obtain
(4.76)
H ence, for subsonic com pressible flow, the ratio o f total to static pressure p o/P i
is a direct m easure o f M ach number. Thus, individual m easurem ents o f po and p\
in conjunction w ith Eq. (4.76) can be used to calibrate an instrum ent in the cock­
pit o f an airplane called a Mach meter, where the dial reads directly in terms o f
the flight M ach num ber o f the airplane.
To obtain the actual flight velocity, recall that M\ = V \/a \ ; hence, Eq. (4.76)
becom es
(4.77a)
E q u ation (4 .7 7 ) can b e rearranged a lg e b r a ic a lly as
(4 .1 1 b )
4.11
Measurement of Airspeed
177
Equations (4.77a) and (4.11b) give the true airspeed o f the airplane. However,
they require a know ledge o f a \, hence T\. T he static tem perature in the air sur­
rounding the airplane is difficult to m easure. H ence, all high-speed (but sub­
sonic) airspeed indicatorjs are calibrated from Eq. (4.11b), assum ing that a\ is
equal to the standard sea-level value as = 340.3 m/s = 1116 ft/s. M oreover, the
airspeed indicator is designed to sense the actual pressure difference po — p \, in
Eq. (4.11b), not the pressure ratio p o / p \ , as appears in Eq. (4.11a). Hence, the
form o f Eq. (4.77b) is used to define a calibrated airspeed as follows:
2a]
Vc1a l —
- ,— í—
- 1
\Y - \
(4.78)
where as and p s are the standard sea-level values o f the speed o f sound and sta­
tic pressure, respectively.
A gain, we em phasizp that Eqs. (4.76) to (4.78) m ust be used to m easure air­
speed when M\ > 0.3, that is, when the flow is com pressible. Equations based on
B ernoulli’s equation, such as Eqs. (4.66) and (4.67), are not valid when M\ > 0 .3 .
EXAM PLE 4.17
A high-speed subsonic M cD onnell-Douglas D C -10 airliner is flying at a pressure altitude
o f 10 km. A Pitot tube on the wing tip measures a pressure o f 4.24 x 104 N /m 2. Calculate
the Mach number at which the airplane is flying. If the ambient air temperature is 230 K,
calculate the true airspeed and the calibrated airspeed.
■ Solution
From the standard atmosphere table, App. A, at an altitude o f 10,000 m, p = 2.65 x
104 N /m 2. Hence, from Eq. (4.76),
1
Jy-V/y
(2)
M} =
Y —1
o
1 .4 - 1
r /4
/ a
m 4 \
.2 4 „
x 104
\
0:286
\ 2 .6 5 x 104 /
1
-
= 0.719
M i = 0.848
Thus,
It is given that T\ = 230 K; hence,
a , = J y R T \ = ^ 1,4 (287)(230) = 304.0 m/s
From Eq. (4.77),
1
2(304.0)2 7 4.24 \ 0'286
1
\P \J
X
y - l
TX
2af
Í
v ,2 =
V\ = 258 m/s
1 .4 - 1
true airspeed
\2 .6 5 /
'
178
CHAPTER 4
Basic Aerodynamics
Note: As a check, from the definition of Mach number,
V\ = M\d\ = 0.848(304.0) = 258 m/s
The calibrated airspeed can be obtained from Eq. (4.78).
(y - D / y
2a2
V
vc»\ =
4- 1
1
+ ‘)
^
y - 1.(
^
2(340.3)2
1 .4 - 1
/4 .2 4 x II
104 —2.65 x 104
+ 1
V
L 01 x 105
0.286
Vcal = 157 m/s
The difference between true and calibrated airspeeds is 39 percent. Note: Just out of
curiosity, let us calculate Vi the wrong way; that is, let us apply Eq. (4.66), which was ob­
tained from Bernoulli’s equation for incompressible flow. Equation (4.66) does not apply
to the high-speed case of this problem, but let us see what result we get anyway.
Pi
2.65 x 104
f) — ~ ~ = “ 'oil "
= 0 .4 kg/m3
~RT\ ~ 287(230)
From Eq. (4.66),
/ 2(p0 - P)
/2(4.24 —2.65) x 104 „„„ ,
Vjrue = , / -------------= \ ------------- — ------------— 282 m/s
y
p
V
0.4
incorrect answer
Compared with V\ = 258 m/s obtained above, an error of 9.3 percent is introduced in the
calculation of true airspeed by using the incorrect assumption of incompressible flow.
This error grows very rapidly as the Mach number approaches unity, as discussed in a
subsequent section.
4.11.3 S u p erso n ic Flow
A irspeed m easurem ents in supersonic flow, that is, for M > 1, are qualitatively
different from those for subsonic flow. In supersonic flow, a shock wave will
form ahead o f the Pitot tube, as shown in Fig. 4.21. Shock w aves are very thin
regions o f the flow (for exam ple, I0 -4 cm ), across which som e very severe
Shock wave
Figure 4.21 Pitot tube in supersonic flow.
4 .11
Measurement of Airspeed
S h o ck wave
Mx >
Pi
IfM2 < Mi
P i > P\
Ti > r ,
Vi < y,
o2 <Po
P ito t
tu b e
T o2 = T0
\
Figure 4.22 Changes across a shock wave in
front of a Pitot tube in supersonic flow.
changes in the flow properties take place. Specifically, as a fluid elem ent flows
through a shock wave,
1.
2.
The M ach num ber decreases.
The static pressure increases.
3.
The static tem perature increases.
4.
The flow velocity decreases.
5.
The total pressure po decreases.
6.
The total tem perature T0 stays the same for a perfect gas.
These changes across a shock w ave are shown in Fig. 4.22.
How and why does a shock w ave form in supersonic flow? These are vari­
ous answ ers with various degrees o f sophistication. However, the essence is as
follow s. Refer to Fig. 4.16, which show s a Pitot tube in subsonic flow. The gas
m olecules that collide with the probe set up a disturbance in the flow. This dis­
turbance is com m unicated to other regions o f the flow, away from the probe, by
m eans o f w eak pressure w aves (essentially sound waves) propagating at the local
speed o f sound. If the flow velocity V) is less than the speed o f sound, as in
Fig. 4.16, then the pressure disturbances (w hich are traveling at the speed o f
sound) will work their way upstream and eventually will be felt in all regions of
the flow. On the other hand, refer to Fig. 4.21, which show s a Pitot tube in su­
personic flow. Here V\ is greater than the speed o f sound. Thus, pressure distur­
bances that are created at the probe surface and that propagate away at the speed
o f sound cannot work their way upstream . Instead, these disturbances coalesce at
a finite distance from the probe and form a natural phenom enon called a shock
wave, as shown in Figs. 4.21 and 4.22. The flow upstream o f the shock wave (to
the left o f the shock) does not feel the pressure disturbance; that is, the presence
o f the Pitot tube is not com m unicated to the flow upstream o f the shock. The
presence o f the Pitot tube is felt only in the regions o f flow behind the shock
wave. Thus, the shock w ave is a thin boundary in a supersonic flow, across which
m ajor changes in flow properties take place and which divides the region o f
undisturbed flow upstream from the region o f disturbed flow dow nstream .
179
180
Figure 4.23 (a) Shock
waves on a swept-wing
airplane (left) and on a
straight-wing airplane
(right). Schlieren
pictures taken in a
supersonic wind tunnel
at NASA Ames
Research Center.
(b) Shock waves on a
blunt body (left) and
sharp-nosed body
(right), (c) Shock
waves on a model of
the Gemini manned
space capsule.
Parts b and c are
shadow graphs of the
flow.
CHAPTER 4
Basic Aerodynamics
(a)
(Courtesy o f NASA Ames
Research Center.)
(b)
(c)
4.11
Measurement of Airspeed
181
W henever a solid body is placed in a supersonic stream , shock waves will
occur. An exam ple is shown in Fig. 4.23, which shows photographs o f the super­
sonic flow over several aerodynam ic shapes. The shock w aves, w hich are gener­
ally not visible to the naked eye, are m ade visible in Fig. 4.23 by m eans o f a
specially designed optica} system , called a schlieren system, and a shadow graph
system. (An exam ple in w hich shock w aves are som etim es visible to the naked
eye is on the wing o f a high-speed subsonic transport such as a Boeing 707. As
we will discuss shortly, tjhere are regions o f local supersonic flow on the upper
surface o f the w ing, and these supersonic regions are generally accom panied by
weak shock waves. If the sun is alm ost directly overhead and if you look out the
window along the span o f the w ing, you can som etim es see these w aves dancing
back and forth on the wing surface.)
C onsider again the m easurem ent o f airspeed in a supersonic flow. The m ea­
surem ent is com plicated by the presence o f the shock wave in Fig. 4 .2 1 because
the flow through a shock wave is nonisentropic. W ithin the thin structure o f a
shock w ave itself, very large friction and therm al conduction effects are taking
place. H ence, neither adiabatic nor frictionless conditions hold; therefore, the
flow is not isentropic. As a result, Eq. (4.74) and hence Eqs. (4.76) and (4.77a)
do not hold across the sljiock wave. A m ajor consequence is that the total pres­
sure po is sm aller behind the shock w ave than in front o f it. In turn, the total
pressure m easured at the nose o f the Pitot probe in supersonic flow will not
be the sam e value as that associated with the free stream , that is, as associated
with M i. Consequently, a separate shock w ave theory m ust be applied to relate
the Pitot tube m easurem ent to the value o f M \. This theory is beyond the
scope o f our presentation, but the resulting form ula is given here for the sake of
com pleteness:
P(h
P\
( y + \ ) 2M f
_Ty ,(y- n
4 y M 2 —2(y —
- i 1)
) JJ
l-y + 2 y M [
y +
(4 ? 9 )
1
This equation is called the Rayleigh Pitot tube formula. It relates the Pitot tube
m easurem ent o f total pressure behind the shock wave, po2 and a m easurem ent o f
free-stream static pressure (again obtained by a static pressure orifice som ew here
on the surface o f the airplane) to the free-stream supersonic M ach num ber M i. In
this fashion, m easurem ents o f po2 and p i , along with Eq. (4.79), allow the cali­
bration o f a M ach m eter for supersonic flight.
EXAM PLE 4.18
An experimental rocket-powered aircraft is flying at a velocity o f 3000 mi/h at an altitude
where the ambient pressure and temperature are 151 lb/ft2 and 390 R, respectively. A
Pitot tube is mounted in the nose o f the aircraft. What is the pressure measured by the
Pitot tube?
182
chapter
4
Basic Aerodynamics
■ Solution
First, we ask, Is the flow supersonic or subsonic; that is, what is M\? From Eq. (4.54),
a\ = J y R T \ = ^I.4 (l7 l6 )(3 9 0 ) = 968.0 ft/s
V, = 3000 ( — ) = 4400 ft/s
\ 6o ;
V,
4400
M, = — = ------ = 4.54
a,
968.0
Hence, M\ > 1; the flow is supersonic. There is a shock wave in front of the Pitot tube;
therefore, Eq. (4.74) developed for isentropic flow does not hold. Instead, Eq. (4.79) must
be used.
Po¿
P\
r
(y
+ \)2m ]
|_4y A/ f —2(y — 1)
i y / ( y - 1)
J
(2.4)2(4.54)2
4(1.4)(4.54)2
- 2(0.4)
■[
Thus,
1 —y + 2yM \
Y+ 1
,3.5
1 - 1.4 + 2(1.4)(4.54)2
= 27
2A
p(h = 2 1 p l = 27(151) =
4077 lb/ft2
Note: Again, out of curiosity, let us calculate the wrong answer. If we had not taken into
account the shock wave in front of the Pitot tube at supersonic speeds, then Eq. (4.74)
would give
Thus,
po = 304.2/Ji = 304.2(151) = 45,931 lb/ft2
incorrect answer
Note that the incorrect answer is off by a factor of more than 10!
4.11.4 S um m ary
As a sum m ary on the m easurem ent o f airspeed, note that different results apply
to different regim es o f flight: low speed (incom pressible), high-speed subsonic,
and supersonic. T hese differences are fundam ental and serve as excellent ex­
am ples o f the application o f the different laws o f aerodynam ics developed in
4.12
Some Additional Considerations
183
previous sections. M oreover, m any o f the form ulas developed in this section
apply to other practical problem s, as discussed in Sec. 4 .12.
4.12 SOME ADDITIONAL CONSIDERATIONS
Section 4.11 contains inform ation that is considerably more general than ju st the
application to airspeed m easurem ents. The purpose o f this section is to elaborate
on som e o f the ideas and results discussed in Sec. 4.11.
4.12.1 M ore on C o m p ressib le Flow
Equations (4.73) through (4.75), relating the ratios o f T0/ T \ , p o / p \ , and Po/Pi to
the local M ach num ber M i, apply in general to any isentropic flow. We state
w ithout proof that the values o f T0, po, and p0 are constant throughout a given
isentropic flow. In conjunction w ith Eqs. (4.73) to (4.75), this fact gives us a
pow erful tool for the analysis o f an isentropic flow. For exam ple, let us again
consider the isentropic flow over an airfoil, which w as the problem solved in
Exam ple 4.10. But now we have m ore inform ation and a broader perspective
from w hich to approach tjiis problem .
EXAM PLE 4.19
Consider the isentropic flow over the airfoil sketched in Fig. 4.24. The free-stream pres­
sure, velocity, and density are 2116 lb/ft2, 500 mi/h, and 0.002377 slug/ft3, respectively.
At a given point A on the airfoil, the pressure is 1497 lb/ft2. What are the Mach number
and the velocity at point A1
m Solution
This example is the same as Example 4.10, with the additional requirement to calculate
the Mach number at point A. However, we use a different solution procedure in this
Isentropic flow
p 0 = constant = Po_
T0 = constant = 7(>„
AC
Figure 4.24 Total pressure and total temperature are
constant throughout an isentropic flow.
184
CHAPTER 4
Basic Aerodynamics
example. First, we calculate the free-stream Mach number, as follows.
Poo
°° “
2116
~ 0.002377(1716) “ 51 ' R
ax = s /y R f Z = v'(1.4)(1716)(518.8) = 1116.4 ft/s
Voo = 500 mi/h = 500
f,ys = 733.3 ft/s
V*
733.3
Moo = — = -¡-rr-7 = 0.6568
floo
1116.4
The free-stream total temperature is, from Eq. (4.73),
~ ~ —1+
MX)
= 1 + 0.2(0.6568)2 = 1.0863
2
T0x = 1.08637^ = 1.0863(518.8) = 563.6°R
The free-stream total pressure is, from Eq. (4.74),
ESs. = ( 1 + ^ -í- M
Poo
V
2
l ) y /y
= (1.0863)35 = 1.336
/
p0x = 1.336(2116) = 2827 lb/ft2
Since the total temperature and total pressure are constant throughout the isentropic flow
over the airfoil, the total temperature and total pressure at point A are the same as the freestream values.
To, = T0k = 563.6CR
Po* = Po„ = 2827 lb/ft2
We can solve for the Mach number at point A by applying Eq. (4.74) at point A:
2827
, ,,
—
= ( 1 + 0.2M2)”
or
1 + 0 .2 M 2 = (1 8 8 8 )1/3 5 = (1 .8 8 8 )0'2857 = 1.1991
ma
1 1 9 9 1 -1
= J -
0.9977
4. 12
Some Additional Considerations
Note: The Mach number at point A is essentially 1; w e have nearly sonic flow at
point A. The static temperature at point A can be obtained from Eq. (4.73).
—T = 1 -II
a
2
= 1 + 0 .2 (0 .9 9 5 5 )2 = 1.1982
a
1982
(Note: The above result for| TA = 470.4°R agrees w ell with the value o f 470.1 °R calcu­
lated in Example 4.10; the difference is due to roundoff error produced by carrying just
four significant figures and the author’s doing the calculations on a hand calculator.)
The velocity at point A\ can be obtained as follow s.
a A = s/y R T A = y / \ , 4(1716) (470.4) = 1063 ft/s
VA = aAMA = 1063(0.9955) =
1058 ft/s
(Note: This agrees well witfi the result VA = 1061 ft/s calculated in Example 4.10.)
The calculation procedure used in Example 4.19 is slightly longer than that used in
Example 4.10; however, it]1 is a more fundamental approach than that used in Exampie 4.10. Return to Example 4.10, and note that w e had to em ploy a value o f the specific
heat cr to solve the problem. However, in the present calculation, w e did not need a value
o f cp. Indeed, the explicit use o f cp is not necessary in solving isentropic compressible
flows. Instead, we used y and M to solve this example. The ratio o f specific heats y and
the Mach number M are both exam ples o f similarity parameters in aerodynamics. The
concept and power o f the similarity parameters for governing fluid flows are something
you will study in more advanced treatments than this book. Suffice it to say here that
Mach number is a powerful governing parameter for compressible flow and that the re­
sults depend on the value o f y , which is usually a fixed value for a given gas (y = 1.4 for
air, as we use here). Example 4.19 show s the power o f using M and y for solving com ­
pressible flow problems. We w ill continue to see the power o f M and y in som e o f our
subsequent discussions.
4 .1 2 .2 M ore o n E q u iv alen t A irspeed
Equivalent airspeed was introduced in Sec. 4.11.1 and expressed by Eq. (4.67)
for low -speed flight, where the flow is assum ed to be incom pressible. However,
the concept o f equivalent airspeed has a broader m eaning than ju st a value that
com es from an airspeed indicator, which uses the standard sea-level density to
determ ine its readout, as first explained in Sec. 4.11.1.
The general definition o f equivalent airspeed can be introduced by the fol­
lowing exam ple. C onsider a L ockheed-M artin F-16 fighter cruising at a velocity
of 300 m/s at an altitude o f 7 km, w here the free-stream density is 0.59 kg/m ’.
I
185
186
chapter
4
Basic Aerodynamics
The velocity o f 300 m/s is the airplane’s true airspeed. At this speed and altitude,
the dynam ic pressure is
= |(0 .5 0 )(3 0 0 )2 = 2.655 x 104 N /m 2. It is im ­
portant to reinforce that dynam ic pressure is a definition, defined by the quantity
\ p <x>V%o' This definition holds no m atter what the flight regim e is— subsonic,
supersonic, etc.— and w hether the flow is incom pressible or com pressible.
D ynam ic pressure q<x, is ju st the definition
Now im agine the F-16 flying at standard sea level, w here the free-stream density
is l . 23 kg/m 3. Question: W hat velocity w ould it have to have at standard sea
level to experience the same dynamic pressure that it had when flying at 300 m/s
at the altitude o f 7 km ? T he answ er is easy to calculate, as follows.
( ^ o o ) s e a le v e l —
( ^ 0 0 ) 7 km
Dropping the subscripts oo for convenience, we have
VSea level =
^ 7 km (
)
\PsJ
w here p is the density at 7 km and p s is the standard sea-level density. Putting in
the num bers, we have
H ence, the F-16 flying at 300 m/s at 7-km altitude w ould have to fly at a velocity
o f 207.8 m/s at standard sea level to experience the sam e dynam ic pressure. By
definition, the F-16 flying at 300 m/s at 7-km altitude has an equivalent airspeed
o f 207.8 m/s.
This leads to the m ore general definition o f equivalent airspeed, as follows.
C onsider an airplane flying at som e true airspeed at som e altitude. Its equivalent
airspeed at this condition is defined as the velocity at w hich it would have to fly
at standard sea level to experience the same dynamic pressure. The equation for
equivalent airspeed is straightforw ard, as obtained in the preceding. It is
where Ve is the equivalent airspeed, V is the true velocity at som e altitude, p is
the density at that altitude, and p s is the standard sea-level density.
In retrospect, our first discussion o f Ve in Sec. 4.11.1 is consistent with our
discussions here; however, in Sec. 4.11.1, our discussion was focused on air­
speed m easurem ents in an incom pressible flow.
The concept o f equivalent airspeed is useful in studies o f airplane perfor­
mance that involve the aerodynam ic lift and drag o f airplanes. The lift and drag
4 .1 3
Supersonic Wind Tunnels and Rocket Engines
depend on the dynam ic pressure,
as w e will see in Chap. 5. G iving the equiv­
alent airspeed o f an airplane is the sam e as stating its dynam ic pressure, as dis­
cussed previously. Hence, equivalent airspeed is som etim es used as a conve­
nience in reporting and analyzing airplane perform ance data.
4.13 SUPERSONIC WIND TUNNELS
AND ROCKET ENGINES
For m ore than a century, projectiles such as bullets and artillery shells have been
fired at supersonic velocities. However, the m ain aerodynam ic interest in supersonic flows arose after W prld W ar II with the advent o f je t aircraft and rocketpropelled guided m issiles As a result, alm ost every aerodynam ic laboratory has
an inventory o f supersonic and hypersonic w ind tunnels to sim ulate m odern
high-speed flight. In addition to their practical im portance, supersonic wind tun­
nels are an excellent exam ple o f the application o f the fundam ental laws o f aeroig h rocket engine nozzles is another exam ple o f the
sam e laws. In fact, the basjc aerodynam ics o f supersonic w ind tunnels and rocket
engines are essentially the sam e, as discussed in the following.
First, consider isentropic flow in a stream tube, as sketched in Fig. 4.2. From
the continuity equation, E q. (4.2),
p A V = const
lr p + In A + In V = In (const)
or
Differentiating, we obtain
dp
dA
dV
— H-------H-------- = 0
p
A
V
(4.80)
Recalling the m om entum equation, Eq. (4.8) (E u ler’s), we obtain
d p = —p V d V
dp
Hence,
P = ~
V dV
(4.81)
Substitute Eq. (4.81) into (4.80):
dpVdV
dA
dV
n
------------- -f- — -|- — — 0
dp
A
V
Since the flow is isentropic,
dp
dp
i
1
1
dp/dp
( d p / d p ) is en tro p ic
(4.82)
187
188
CHAPTER 4
Basic Aerodynamics
Thus, Eq. (4.82) becom es
VdV
a2
dA
dV
+ — =0
A
V
'
Rearranging, we get
V dV
dV
a2
V
dA
or
~A
( V2
U 2
1
(4.83)
= (M 2 - 1) —
V
Equation (4.83) is called the area-velocity relation, and it contains a wealth o f in­
form ation about the flow in the stream tube show n in Fig. 4.2. First, note the
m athem atical convention that an increasing velocity and an increasing area cor­
respond to positive values o f d V and d A , respectively, w hereas a decreasing ve­
locity and a decreasing area correspond to negative values o f d V and d A . This is
the norm al convention for differentials from differential calculus. With this in
m ind, Eq. (4.83) yields the follow ing physical phenom ena:
1.
If the flow is subsonic (M < 1), for the velocity to increase (d V positive),
the area m ust decrease (dA negative); that is, w hen the flow is subsonic,
the area m ust converge for the velocity to increase. This is sketched in
Fig. 4.25a. This sam e result was observed in Sec. 4.2 for incom pressible
flow. O f course, incom pressible flow is, in a sense, a singular case o f
subsonic flow, where M -*■ 0 .
2.
If the flow is supersonic (M > 1), for the velocity to increase (d V
positive), the area m ust also increase (dA positive); that is, when the flow
is supersonic, the area m ust diverge for the velocity to increase. This is
sketched in Fig. 4.25b.
3.
If the flow is sonic (M = 1), then Eq. (4.83) yields for the velocity
(a )
(b)
Figure 4.25 Results from the area-velocity relation.
(c )
Supersonic Wind Tunnels and Rocket Engines
4 .1 3
w hich at first glance says that d V / V is infinitely large. However, on a
physical basis, the velocity, and hence the change in velocity d V , at all
tim es m ust be finite. This is only com m on sense. Thus, looking at
Eq. (4.84), w e see that the only way for d V/ V to be finite is to have
d A / A = 0 , so
d\
~V
l dA
=
---------=
0 A
0
-
0
=
f in ite n u m b e r
that is, in the language o f differential calculus, d V / V is an indeterm inate
form o f 0 /0 and hen(j:e can have a finite value. In turn, if d A / A = 0, the
stream tube has a minimum area at M = 1. This m inim um area is called a
throat and is sketched in Fig. 4.25c.
Therefore, to expand a gas to supersonic speeds, starting with a stagnant gas
in a reservoir, the preceding discussion says that a duct o f a sufficiently
converging-diverging shape m ust be used. This is sketched in Fig. 4.26, where
typical shapes for supersonic wind tunnel nozzles and rocket engine nozzles are
shown. In both cases, th^ flow starts out with a very low velocity V « 0 in the
reservoir, expands to hi^h subsonic speeds in the convergent section, reaches
M ach 1 at the throat, and then goes supersonic in the divergent section dow n­
stream o f the throat. In ii supersonic wind tunnel, sm ooth, uniform flow at the
nozzle exit is usually desired, and therefore, a long, gradually converging and
diverging nozzle is em ployed, as shown at the top o f Fig. 4.26. For rocket en ­
gines, the flow quality at the exit is not quite as im portant, but the w eight of the
nozzle is a m ajor concern. For the w eight to be m inim ized, the engine’s length is
\
E xit
C o m b u stio n
ch am b er
Po
To
K*0
M < |j M = 1
S upersonic
( b ) R o c k e t engine nozzle
Figure 4.26 S u p e rs o n ic n o z z le sh a p e s.
Flow
189
190
chapter
4
Basic Aerodynamics
Figure 4.27 A typical rocket engine. Shown is a small rocket designed by MesserschmittBolkow-Blohm for European satellite launching.
m inim ized, which gives rise to a rapidly diverging, bell-like shape for the super­
sonic section, as shown at the bottom o f Fig. 4.26. A photograph o f a typical
rocket engine is show n in Fig. 4.27.
The real flow through nozzles such as those sketched in Fig. 4.26 is closely
approxim ated by isentropic flow because little or no heat is added or taken away
through the nozzle walls and a vast core o f the flow is virtually frictionless.
Therefore, Eqs. (4.73) to (4.75) apply to nozzle flows. H ere, the total pressure
and tem perature po and T0 rem ain constant throughout the flow, and Eqs. (4.73)
to (4.75) can be interpreted as relating conditions at any point in the flow to the
stagnation conditions in the reservoir. For exam ple, consider Fig. 4.26, which
illustrates the reservoir conditions p Q and 7¡, w here V
0. C onsider any cross
section dow nstream o f the reservoir. The static tem perature, density, and pres­
sure at this section are T\, p \, and p \, respectively. If the M ach num ber M\ is
know n at this point, then T\, p \ , and p\ can be found from Eqs. (4.73) to (4.75) as
T\ = 7 o[l + \ ( y — 1)A/2] 1
(4.85)
fH = po[ l + \ { y ~ l ) M 2] - ' /(v_I)
(4.86)
P \ = P v [ \ + \ { Y - \ ) M 2]~Y'{Y~X)
(4.87)
Again, Eqs. (4.85) to (4.87) dem onstrate the pow er o f the M ach num ber in m ak­
ing aerodynam ic calculations. The variation o f M ach num ber itself through the
nozzle is strictly a function o f the ratio o f the cross-sectional area to the throat
area A / A , . This relation can be developed from the aerodynam ic fundam entals
already discussed; the resulting form is
( A \2
U )
1 T
2
= ^ .7 T
/
t
y —1
( i+ V
, \ - | ° ' +1)/(,,- 1)
" 2) .
<4'88>
4.1 3
Supersonic Wind Tunnels and Rocket Engines
Flow
(a)
(*)
(c) _P
Po
(d )
To
(e)
_P
Po
Figure 4.28 Variation o f Mach number, pressure, temperature,
and density through a supersonic nozzle.
Therefore, the analysis o f isentropic flow through a nozzle is relatively
straightforw ard. The procedure is sum m arized in Fig. 4.28. C onsider that the
nozzle shape, hence A / A , , is given as shown in Fig. 4.28a. Then from Eq. (4.88),
the M ach num ber can be obtained (im plicitly). Its variation is sketched in
Fig. 4.28b. Since M is now know n through the nozzle, Eqs. (4.85) to (4.87) give
the variations o f T, p, and p, w hich are sketched in Fig. 4.28c to e. The direc­
tions o f these variations are im portant and should be noted. From Fig. 4.28, the
M ach num ber continuously increases through the nozzle, going from near zero
in the reservoir to M = l at the throat and to supersonic values dow nstream o f
the throat. In turn, p, T, dnd p begin with their stagnation values in the reservoir
191
192
chapter
4
Basic Aerodynamics
and continuously decrease to low values at the nozzle exit. Hence, a supersonic
nozzle flow is an expansion process in w hich pressure decreases through the noz­
zle. In fact, it is this pressure decrease that provides the m echanical force for
pushing the flow through the nozzle. If the nozzle show n in Fig. 4.28a is sim ply
set out by itself in a laboratory, obviously nothing will happen; the air will not
start to rush through the nozzle o f its ow n accord. Instead, to establish the flow
sketched in Fig. 4.28, we m ust provide a high-pressure source at the inlet, and/or
a low -pressure source at the exit, with the pressure ratio just the right value, as
prescribed by Eq. (4.87) and sketched in Fig. 4.28c.
EXAMPL
You are given the job of designing a supersonic wind tunnel that has a Mach 2 flow at
standard sea-level conditions in the test section. What reservoir pressure and temperature
and what area ratio Ae/A, are required to obtain these conditions?
■ Solution
The static pressure pe = l atm = 1.01 x 105 N/m2, and the static temperature T, =
288.16 K, from conditions at standard sea level. These are the desired conditions at the
exit of the nozzle (the entrance to the test section). The necessary reservoir conditions are
obtained from Eqs. (4.85) and (4.87):
T0
Y e
y - 1
=
1 +
—
,
M
<
1.4- 1
=
1 +
—
2 —
,
(
>
=
T0 = 1.87; = 1.8(288.16) = 518.7 K
Thus,
v _ i
x ^/(y-0
1 + ^ — Ai2 )
= (1.8)35 = 7 .8 2
Thus,
po = 7.82 pe = 7.82(1.01 x 105) =
7.9 x 105 N/m2
The area ratio is obtained from Eq. (4.88):
(y+D/O'-D
k2
^ y
\A,J
= _ L p _ ( 1+ L z ij,2 \
M2 my + I \
2
)_
2.4/0.4
= 2.85
Hence,
^ = 1.69
A,
4.1 3
Supersonic Wind Tunnels and Rocket Engines
193
EXAM PLE 4.21
The reservoir temperature and pressure o f a supersonic wind tunnel are 600° R and 10 atm,
respectively. The Mach number o f the flow in the test section is 3. A blunt-nosed model
like that shown at the left in Fig. 4.236 is inserted in the test section flow. Calculate the
pressure, temperature, and density at the stagnation point (at the nose o f the body).
■ Solution
The flow conditions in the test section are the same as those at the nozzle exit. Hence, in the
test section, the exit pressure is obtained from Eq. (4.87), recalling that 1 atm = 2116 lb/ft2:
p i = p o [l + i ( y = 1 0 (2 1 16)[1 + 0 .5 (0 .4 )(3 )2]-3,5
= 576 lb/ft2
The pressure at the stagnation point on the model is the total pressure behind a normal
wave because the stagnation streamline has traversed the normal portion o f the curved
bow shock wave in Fig. 4.23 b and then has been isentropically compressed to zero
velocity between the shock and the body. This is the same situation as that existing at the
mouth o f a Pitot tube in supersonic flow, as described in Sec. 4.11.3. Hence, the stagna­
tion pressure is given by Eq. (4.79):
Pth _
P stu g
P\
P s ,a g
Pe
pr
= [
J
(X + 1)2A*2
[ 4 y M j - 2( y -
2 .4 2(32)
1- y
+ 2y M ]
Y + 1
1)J
f 5 1 - 1.4 + 2 (1 .4 )(3 2) =
L 4(1 .4 )(3 2) - 2(0 .4 )
J
pm = 12.06/),, = 12.06(576) =
2.4
6947 lb/ft2
The total temperature (not the static temperature) at the nozzle exit is the same as the
reservoir temperature
To,, = T0
because the flow through the nozzle is isentropic and hence adiabatic. For an adiabatic
flow, the total temperature is constant, as demonstrated by Eq. (4.42), where at two dif­
ferent points in an adiabatic flow with different velocities if the flow is adiabatically
slowed to zero velocity at both points, w e obtain
cPT0l = CpTlh
hence, 7b, = 7b2; that is, the total temperature at the two different points is the same.
Therefore, in the present problem, the total temperature associated with the test section
flow is equal to the total temperature throughout the nozzle expansion: 7b, = 7b = 600 R.
[Note that the static temperature o f the test section flow is 214.3°R , obtained from
194
chapter
4
Basic Aerodynamics
Eq. (4.85).] Moreover, in traversing a shock wave (see Fig. 4.22), the total temperature is
unchanged; that is, the total temperature behind the shock wave on the model is also
600"R (although the static temperature behind the shock is less than 600”R). Finally,
since the flow is isentropically compressed to zero velocity at the stagnation point, the
stagnation point temperature is the total temperature, which also stays constant through
the isentropic compression. Hence, the gas temperature at the stagnation point is
r stag = T0 =
600'R
From the equation of state,
6947
P stag
A itag —
0.0067 slug/ft3
/?7SIag “ 1716(600)
EXAMPL
In the combustion chamber of a rocket engine, kerosene and oxygen are burned, resulting
in a hot, high-pressure gas mixture in the combustion chamber with the following condi­
tions and properties: 7b = 3144 K, p0 = 20 atm, R = 378 J/(kg)(K), and y = 1.26. The
pressure at the exit of the rocket nozzle is 1 atm, and the throat area of the nozzle is
0.1 m2. Assuming isentropic flow through the rocket nozzle, calculate (a) the velocity at
the exit and (b) the mass flow through the nozzle.
■ Solution
a. To obtain the velocity at the exit, let us first obtain the temperature, next the speed of
sound, and then the Mach number, leading to the velocity. We note that the combustion
chamber conditions are the “reservoir” conditions sketched in Fig. 4.26; this is why the
combustion chamber pressure and temperature have been denoted by po and To, respec­
tively. Since the flow is isentropic, from Eq. (3.46) we have
y /y -i
M£)
r- r-(£)
0 .26/ 1.26
y/y-i
or
= (3144)
(±)
= 1694 K
ae = JyR T e = V 1.26(378)(1694) = 898.2 m/s
The Mach number at the exit is given by Eq. (4.73).
- = 1+ ^
Te
2
=
*
Me
= _i_/3144
y - \\T e
)
1.26 — 1 \ 1694
Me = 2.566
X =
)
4 .1 4
Hence,
M,,ae = 2.566(898.2) =
Discussion of Compressibility
2305 m/s
b. The mass flow is given by the product pAV evaluated at any cross section of the noz­
zle. Since we are given the area of the throat, the obvious location at which to evaluate
pAV is the throat; that is,
m = p*A*V*
where p *, A*, and V* are the density, area, and velocity, respectively, at the throat. We
will make use of the fact that the Mach number at the throat is M* = l . The pressure at
the throat p* is given by Eq. (4.74);
Y l l Y - 1)
1+
Hence,
pa
P' = 1.808
0.26
- ( l 2)
1. 26 / 0.26
= (1.13)4'846 = 1.808
20( 1.01 x 105)
= 1,117 x 10ft N/m
1.808
The temperature at the throat is given by Eq. (4.73):
r*
T* =
T
----- -A T2 = 1.13
1.13
3144
= 2782.3 K
1.13
a* = y / y R T * = , / l .26(378) (2782.3) = 1151 m/s
^ 8 ( 2 7 8 2 . ^ - ' 062 kg/ml
Since M* = 1, then V* = a* = 1151 m/s. Hence,
m = /oM*V* = 1.062(0.1)(1151) =
122.2 kg/s
4.14 DISCUSSION OF COMPRESSIBILITY
We have been stating all along that flows in which M < 0.3 can be treated as es­
sentially incom pressible and, conversely, flows w here M > 0.3 should be treated
as com pressible. We are now in a position to prove this.
C onsider a gas at rest ( V = 0) w ith density
Now accelerate this gas isentropically to som e velocity V and M ach num ber M. O bviously, the therm ody­
namic properties o f the gas will change, including the density. In fact, the change
in density will be given by Eq. (4.75):
i/(y-D
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4
Basic Aerodynamics
M ach
number
Figure 4.29 Density variation with Mach number for y = l .4, showing region where
the density change is less than 5 percent.
F or y = 1.4, this variation o f p / p o is given in Fig. 4.29. Note that for M < 0.3,
the density change in the flow is less than 5 percent; that is, the density is essen­
tially constant for M < 0.3, and for all practical purposes the flow is incom ­
pressible. Therefore, we have ju st dem onstrated the validity o f the statem ent
For M < 0.3, the flow can be treated as incompressible.
4.15 INTRODUCTION TO VISCOUS FLOW
This is a good tim e to look back to our road map in Fig. 4.1. We have now com ­
pleted the left side o f this road map— inviscid flow with som e applications. Ex­
am ine again the boxes on the left side, and make certain that you feel com fortable
with the material represented by each box. T here are a large num ber o f aerody­
nam ic applications in which the neglect o f friction is quite reasonable and in which
the assum ption o f inviscid flow leads to useful and reasonably accurate results.
4 .1 5
□
F rictionless flow : no drag
Introduction to Viscous Flow
D
O
Separated
flow
Real flow: finite drag
Figure 4.30 Comparison between ideal frictionless flow and real flow with the effects of
friction.
The streamline that is right on
the surface slips over the surface.
Figure 4.31 Frictionless flow.
However, there are num erous other practical problem s in w hich the effect of
friction is dom inant, and we now turn our attention to such problem s. This con­
stitutes the right side o f our road m ap in Fig. 4 . 1— viscous flow, that is, flow with
friction. Indeed, there are som e flows in which the fundam ental behavior is gov­
erned by the presence o f friction betw een the airflow and a solid surface. A clas­
sic exam ple is sketched in Fig. 4.30, which shows the low -speed flow over a
sphere. At the left is sketched the flow field that would exist if the flow were inviscid. For such an ideal, frictionless flow, the stream lines are sym m etric, and,
amazingly, there is no aerodynam ic force on the sphere. The pressure distribution
over the forw ard surface exactly balances that over the rearw ard surface, and
hence there is no drag (no force in the flow direction). However, this purely the­
oretical result is contrary to com m on sense; in real life there is a drag force on the
sphere tending to retard the m otion o f the sphere. The failure o f the theory to pre­
dict drag was bothersom e to early 19th-century aerodynam icists and was even
given a name: d ’A lem bert’s paradox. The problem is caused by not including
friction in the theory. The real flow over a sphere is sketched on the right in
Fig. 4.30. The flow separates on the rearw ard surface o f the sphere, setting up a
com plicated flow in the wake and causing the pressure on the rearward surface to
be less than that on the forw ard surface. Hence, there is a drag force exerted on
the sphere, as shown by D in Fig. 4.30. The difference betw een the two flows in
Fig. 4.30 is sim ply friction, but w hat a difference!
C onsider the flow o f a gas over a solid surface, such as the airfoil sketched in
Fig. 4.31. A ccording to our previous considerations o f frictionless flows, we con­
sidered the flow velocity at the surface as being a finite value, such as V2 shown
in Fig. 4.31; that is, because o f the lack o f friction, the stream line right at the sur­
face slips over the surface. In fact, we stated that if the flow is incom pressible, V2
197
198
chapter
4
Basic Aerodynamics
Figure 4.32 Flow in real life, with friction. The thickness of the boundary layer is greatly
overemphasized for clarity.
can be calculated from B ernoulli’s equation:
P\ + \p V ? = P2 + {P V ?
However, in real life, the flow at the surface adheres to the surface because
o f friction betw een the gas and the solid m aterial; that is, right at the surface, the
flow velocity is zero, and there is a thin region o f retarded flow in the vicinity of
the sutface, as sketched in Fig. 4.32. This region o f viscous flow w hich has been
retarded ow ing to friction at the surface is called a boundary layer. The inner
edge o f the boundary layer is the solid surface itself, such as point a in Fig. 4.32,
w here V = 0. T he outer edge o f the boundary layer is given by point b, where
the flow velocity is essentially the value given by V2 in Fig. 4 .3 1. That is, point b
in Fig. 4.32 is essentially equivalent to point 2 in Fig. 4.31. In this fashion, the
flow properties at the outer edge o f the boundary layer in Fig. 4.32 can be calcu­
lated from a frictionless flow analysis, as pictured in Fig. 4.31. This leads to an
im portant conceptual point in theoretical aerodynam ics: A flow field can be split
into tw o regions, one region in w hich friction is im portant, namely, in the bound­
ary layer near the surface, and another region o f frictionless flow (som etim es
called potential flow ) outside the boundary layer. This concept was first intro­
duced by Ludw ig Prandtl in 1904, and it revolutionized m odern theoretical
aerodynam ics.
It can be show n experim entally and theoretically that the pressure through
the boundary layer in a direction perpendicular to the surface is constant. T hat is,
letting p„ and p¡, be the static pressures at points a and b, respectively, in
Fig. 4.32, then p a = ph. This is an im portant phenom enon. This is why a surface
pressure distribution calculated from frictionless flow (Fig. 4.31) m any tim es
gives accurate results for the real-life surface pressures; it is because the fric­
tionless calculations give the correct pressures at the outer edge o f the boundary
layer (point b), and these pressures are im pressed w ithout change through the
boundary layer right dow n to the surface (point a). T he preceding statem ents are
reasonable for slender aerodynam ic shapes such as the airfoil in Fig. 4.32; they
do not hold for regions o f separated flow over blunt bodies, as previously
sketched in Fig. 4.30. Such separated flows are discussed in Sec. 4.20.
R efer again to Fig. 4.32. The boundary layer thickness 8 grows as the flow
m oves over the body; that is, m ore and m ore o f the flow is affected by friction as
4.15
Introduction to Viscous Flow
Vb = V2
Figure 4.33 Velocity profile through a boundary layer.
the distance along the surface increases. In addition, the presence o f friction
creates a shear stress at the surface t w. This shear stress has dim ensions of
force/area and acts in a direction tangential to the surface. Both <5 and rw are im ­
portant quantities, and a large part o f boundary layer theory is devoted to their
calculation. As we will see, rw gives rise to a drag force called skin friction drag,
hence attesting to its im portance. In subsequent sections, equations for the cal­
culation o f 8 and r w will be given.
Looking m ore closely at the boundary layer, w e see that a velocity profile
through the boundary layer is sketched in Fig. 4.33. The velocity starts out at
zero at the surface and increases continuously to its value o f V2 at the outer edge.
Let us set up coordinate axes jc and y such that x is parallel to the surface and y
is norm al to the surface, as show n in Fig. 4.33. By definition, a velocity profile
gives the variation o f velocity in the boundary layer as a function o f y. In gen­
eral, the velocity profiles at different x stations are different.
The slope o f the velocity profile at the wall is o f particular im portance be­
cause it governs the wall shear stress. Let ( d V / d y )y=0 be defined as the velocity
gradient at the wall. Then the shear stress at the wall is given by
<489>
where /x is called the absolute viscosity coefficient (or sim ply the viscosity) o f the
gas. T he viscosity coefficient has dim ensions o f m ass/(length)(tim e), as can be
verified from Eq. (4.89) com bined with N ew ton’s second law. It is a physical
property o f the fluid; n is different for different gases and liquids. A lso, ¡x varies
with T. For liquids, /u decreases as T increases (we all know that oil gets “thinner”
when the tem perature is increased). But for gases, yu increases as T increases
(air gets “th ick er” when tem perature is increased). For air at standard sea-level
tem perature,
¡i = 1.7894 x 1(T5 kg/(m )(s) = 3.7373 x 1(T7 slug/(ft)(s)
The variation o f ¡x w ith tem perature for air is given in Fig. 4.34.
In this section, we are sim ply introducing the fundam ental concepts o f
boundary layer flows; such concepts are essential to the practical calculation of
aerodynam ic drag, as we will soon appreciate. In this spirit, we introduce another
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4
Basic Aerodynamics
Figure 4.34 Variation of viscosity coefficient with temperature.
L eading
edge
Figure 4.35 Growth o f the boundary layer thickness.
im portant dim ensionless “num ber,” a num ber o f im portance and im pact on aero­
dynam ics equal to those o f the M ach num ber discussed earlier— the Reynolds
number. C onsider the developm ent o f a boundary layer on a surface, such as the
flat plate sketched in Fig. 4.35. Let x be m easured from the leading edge, that is,
the front tip o f the plate. Let Voo be the flow velocity far upstream of the plate. (The
subscript oo is com m only used to denote conditions far upstream o f an aerody­
nam ic body, thtfree-strea m conditions.) The Reynolds number R ev is defined as
R er =
Poo VooX
(4.90)
Moo
N ote that Re* is dim ensionless and that it varies linearly with x. For this reason,
Re,v is som etim es called a local Reynolds number, because it is based on the local
coordinate x.
.
4 15
Introduction to Viscous Flow
201
Up to this point in our discussion on aerodynam ics, we have always consid­
ered flow stream lines to be sm ooth and regular curves in space. However, in a
viscous flow, and particularly in boundary layers, life is not quite so simple.
There are tw o basic types o f viscous flow:
1.
Laminar flow; in which the stream lines are sm ooth and regular and a fluid
elem ent m oves sm oothly along a stream line
2.
Turbulent flow, in which the stream lines break up and a fluid elem ent
m oves in a random , irregular, and tortuous fashion
If you observe sm oke rising from a lit cigarette, as sketched in Fig. 4.36, you see
first a region o f sm ooth flow— lam inar flow— and then a transition to irregular,
m ixed-up flow— turbulent flow. The differences betw een lam inar and turbulent
flow are dram atic, and they have a m ajor im pact on aerodynam ics. For exam ple,
consider the velocity profiles through a boundary layer, as sketched in Fig. 4.37.
The profiles are different, depending on w hether the flow is lam inar or turbulent.
The turbulent profile is ‘ffatter,” or fuller, than the lam inar profile. For the turbu­
lent profile, from the outer edge to a point near the surface, the velocity rem ains
reasonably close to the free-stream velocity; it then rapidly decreases to zero at
Figure 4.37 Velocity profiles for laminar and turbulent boundary
layers. Note that the turbulent boundary layer thickness is larger
than the laminar boundary layer thickness.
Figure 4.36 Smoke
pattern from a cigarette.
202
CHAPTER 4
Basic Aerodynamics
the surface. In contrast, the lam inar velocity profile gradually decreases to zero
from the outer edge to the surface. Now consider the velocity gradient at the
wall. ( d V / d y ) y=o, w hich is the reciprocal o f the slope o f the curves shown in
Fig. 4.37 evaluated at y = 0. From Fig. 4.37, it is clear that
for lam inar flow < ( — |
for turbulent flow
\ d y ) y=o
R ecalling Eq. (4.89) for zw leads us to the fundam ental and highly im portant fact
that lam inar shear stress is less than turbulent shear stress.
Tw
la m in a r
turbu len t
This obviously im plies that the skin friction exerted on an airplane wing or body
will depend on w hether the boundary layer on the surface is lam inar or turbulent,
w ith lam inar flow yielding the sm aller skin friction drag.
(a)
Figure 4.38 Comparison o f conventional and laminar flow airfoils. The pressure
distributions shown are the theoretical results obtained by NACA and are for 0° angle of
attack. The airfoil shapes are drawn to scale.
4 .1 5
Introduction to Viscous Flow
It appears to be alm pst universal in nature that system s with the m axim um
am ount o f disorder are favored. For aerodynam ics, this m eans that the vast m a­
jority o f practical viscous flows are turbulent. T he boundary layers on m ost prac­
tical airplanes, m issiles, ship hulls, etc., are turbulent, with the exception o f small
regions near the leading edge, as we will soon see. Consequently, the skin friction
on these surfaces is the higher, turbulent value. For the aerodynam icist, who is
usually striving to reduc^ drag, this is unfortunate. H owever, the skin friction on
slender shapes, such as wing cross sections (airfoils), can be reduced by design­
ing the shape in such a m anner as to encourage lam inar flow. Figure 4.38 indi­
cates how this can be achieved. H ere tw o airfoils are shown; the standard airfoil
on the left has a m axim ufn thickness near the leading edge, whereas the lam inar
flow airfoil has its m axim um thickness near the middle o f the airfoil. The pres­
sure distributions on the top surface on the airfoils are sketched above the airfoils
in Fig. 4.38. Note that fo^ the standard airfoil, the m inim um pressure occurs near
(h)
Figure 4.38 (continued)
203
204
c hapter
4
Basic Aerodynamics
the leading edge, and there is a long stretch o f increasing pressure from this point
to the trailing edge. Turbulent boundary layers are encouraged by such increasing
pressure distributions. H ence, the standard airfoil is generally bathed in long re­
gions o f turbulent flow, w ith the attendant high skin friction drag. However, note
that for the lam inar flow airfoil, the m inim um pressure occurs near the trailing
edge, and there is a long stretch o f decreasing pressure from the leading edge to
the point o f m inim um pressure. Lam inar boundary layers are encouraged by such
decreasing pressure distributions. H ence, the lam inar flow airfoil can be bathed in
long regions o f lam inar flow, thus benefiting from the reduced skin friction drag.
The North A m erican P-51 M ustang (Fig. 4.39), designed at the outset o f
W orld W ar II, was the first production aircraft to em ploy a lam inar flow airfoil.
H ow ever, lam inar flow is a sensitive phenom enon; it readily gets unstable and
tries to change to turbulent flow. For exam ple, the slightest roughness o f the air­
foil surface caused by such real-life effects as protruding rivets, im perfections in
m achining, and bug spots can cause a prem ature transition to turbulent flow in
advance o f the design condition. Therefore, m ost lam inar flow airfoils used on
production aircraft do not yield the extensive regions o f lam inar flow that are ob­
tained in controlled laboratory tests using airfoil m odels with highly polished,
sm ooth surfaces. From this point o f view, the early lam inar flow airfoils were not
successful. However, they w ere successful from an entirely different point o f
view; namely, they were found to have excellent high-speed properties, postpon­
ing to a higher flight M ach num ber the large drag rise due to shock w aves and
flow separation encountered near M ach 1. (Such high-speed effects are discussed
in Secs. 5.9 to 5.11.) As a result, the early lam inar flow airfoils w ere extensively
used on jet-propelled airplanes during the 1950s and 1960s and are still em ­
ployed today on som e m odern high-speed aircraft.
Figure 4.39 The first airplane to incorporate a laminar flow airfoil for the wing section, the
North American P-51 Mustang. Shown is a late model Mustang, the P-5 ID.
4 .1 6
Results for a Laminar Boundary Layer
G iven a lam inar or turbulent flow over a surface, how do we actually calcu­
late the skin friction drag? The answ er is given in the follow ing tw o sections.
4.16 RESULTS FOR A LAMINAR BOUNDARY LAYER
C onsider again the boundary layer flow over a flat plate, as sketched in Fig. 4.35.
A ssum e that the flow is laminar. The tw o physical quantities o f interest are the
boundary layer thickness S and shear stress z w at location ;t. Form ulas for these
quantities can be obtained from lam inar boundary layer theory, w hich is beyond
the scope o f this book. However, the results, w hich have been verified by exper­
iment, are as follow s. The lam inar boundary layer thickness is
(4.91)
where ReA = A» Voo*//-«oo> as defined in Eq. (4.90). It is rem arkable that a phe­
nom enon as com plex as the developm ent o f a boundary layer, w hich depends at
least on density, velocity, viscosity, and length o f the surface, should be de­
scribed by a form ula as sim ple as Eq. (4.91). In this vein, Eq. (4.91) dem onstrates
the pow erful influence o f the Reynolds num ber R et in aerodynam ic calculations.
N ote from Eq. (4.91) that the lam inar boundary layer thickness varies
inversely as the square root o f the Reynolds number. Also since Re* =
PooVoo*/Moo> then from Eq. (4.91) 8 <x jc 1/2; that is, the lam inar boundary layer
grow s parabolically.
The local shear stress r u, is also a function o f x, as sketched in Fig. 4.40.
R ather than deal with xw directly, aerodynam icists find it more convenient to de­
fine a local skin friction coefficient cyx as
TW
2 Poo Kx,
(4.92)
9oo
The skin friction coefficient is dim ensionless and is defined as the local shear
stress divided by the dynam ic pressure at the outer edge o f the boundary. From
rw
X
Figure 4.40 Variation of shear stress with distance
along the surface.
205
206
chapter
4
Basic Aerodynamics
lam inar boundary layer theory,
(4.93)
w here, as usual, R e, = PooVoo^/Moo- Equation (4.93) dem onstrates the conve­
nience o f defining a dim ensionless skin friction coefficient. On one hand, the di­
m ensional shear stress zw (as sketched in Fig. 4.40) depends on several quanti­
ties such as Poo* Voo, and R e*; on the other hand, from Eq. (4.93), C/x is a function
o f Re* only. This convenience, obtained from using dim ensionless coefficients
and num bers, reverberates throughout aerodynam ics. Relations betw een dim en­
sionless quantities such as those given in Eq. (4.93) can be substantiated by d i­
mensional analysis, a form al procedure to be discussed in Sec. 5.3.
C om bining Eqs. (4.92) and (4.93), we can obtain values o f tw from
0.664^00
rw = f ( x ) =
(4.94)
V R eJ
N ote from Eqs. (4.93) and (4.94) that both tyx and rw for lam inar boundary layers
vary as
that is, c¡x and xw decrease along the surface in the flow direction,
as sketched in Fig. 4.40. T he shear stress near the leading edge o f a flat plate is
greater than that near the trailing edge.
The variation o f local shear stress r U) along the surface allow s us to calculate
the total skin friction drag due to the airflow over an aerodynam ic shape. Recall
from Sec. 2.2 that the net aerodynam ic force on any body is fundam entally due
to the pressure and shear stress distributions on the surface. In m any cases, it is
this total aerodynam ic force that is o f prim ary interest. For exam ple, if you
m ount a flat plate parallel to the airstream in a wind tunnel and m easure the force
exerted on the plate, by m eans o f a balance o f som e sort, you are not m easuring
the local shear stress tw\ rather, you are m easuring the total drag due to skin fric­
tion being exerted over all the surface. This total skin friction drag can be ob­
tained as follows.
C onsider a flat plate o f length L and unit w idth oriented parallel to the flow,
as shown in perspective in Fig. 4.41. C onsider also an infinitesim ally sm all sur­
face elem ent o f the plate o f length d x and w idth unity, as shown in Fig. 4.41. The
local shear stress on this elem ent is rx, a function o f x . H ence, the force on this
elem ent due to skin friction is zw d x ( \ ) = tw d x . The total skin friction drag is
yoo
Figure 4.41 Total drag is the integral of the local shear stress over the
surface.
4 .1 6
Results for a Laminar Boundary Layer
the sum o f the forces on all the infinitesim al elem ents from the leading to the
trailing edge; that is, the total skin friction drag D ¡ is obtained by integrating z x
along the surface:
(4.95)
D , . f l Tw d x
JO
C om bining Eqs. (4.94) atjd (4.95) yields
Df
= 0.664<?c
Í
Jo
0.664q0
dx
\/R e 7
[ L—
s / p 00 Voo/flo Jo
(4.96)
\.32&qooL
D, =
vO
T O T m
Let us define a total skin friction drag coefficient C / as
Cr =
. Df
(4.97)
where S is the total area o f the plate, S = L ( l) . Thus, from Eqs. (4.96) and (4.97),
Cr
or
=
Dr
1.328</ooL
<?oo¿0)
^ ¿ ( P ooVooL / M oo) 172
1.328
Cr =
v /R e l
lam inar
(4.98)
where the Reynolds num ber is now based on the total length L , that is, Re¿ =
Poo V o o L / ¡X00*
D o not confuse Eq. (4.98) w ith Eq. (4.93); they are different quantities. The
local skin friction coefficient Cfx in Eq. (4.93) is based on the local Reynolds
num ber Re* = PooVoo*/¡¿oo and is a function o f x. However, the total skin fric­
tion coefficient C f is based on Reynolds num ber for the plate length L :
Re/. = Poo VooL f fi00 •
We em phasize that Eqs. (4.91), (4.93), and (4.98) apply to lam inar boundary
layers only; for turbulent flow, the expressions are different. Also these equations
are exact only for low -speed (incom pressible) flow. However, they have been
show n to be reasonably accurate for high-speed subsonic flows as well. For su­
personic and hypersonic flows, w here the velocity gradients within the boundary
layer are so extrem e and w here the presence o f frictional dissipation creates very
large tem peratures within the boundary layer, the form o f these equations can
still be used for engineering approxim ations, but p and p must be evaluated at
som e reference conditions germ ane to the flow inside the boundary layer. Such
m atters are beyond the scope o f this book.
207
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4
Basic Aerodynamics
EXAM PLE 4.23
Consider the flow of air over a small flat plate that is 5 cm long in the flow direction and
1 m wide. The free-stream conditions correspond to standard sea level, and the flow ve­
locity is 120 m/s. Assuming laminar flow, calculate
a. The boundary layer thickness at the downstream edge (the trailing edge)
b. The drag force on the plate
■ Solution
a. At the trailing edge of the plate, where x = 5 cm = 0.05 m, the Reynolds number is,
from Eq. (4.90),
^
_ PooV<x>x _ (1.225 kg/m3)(120 m/s)(0.05 m)
At<x
1.789 x 10-5 kg/(m)(s)
= 4.11 x 105
From Eq. (4.91),
8=
5.2x
5.2(0.05)
Re'/2
(4.11 x 105) '/2
4.06 x 10
m
Note how thin the boundary layer is—only 0.0406 cm at the trailing edge.
b. To obtain the skin friction drag, Eq. (4.98) gives, with L = 0.05 m,
C, =
1.328
Re 1/2
1.328
= 2.07 x 10-3
(4.11 x 105) '/2
The drag can be obtained from the definition of the skin friction drag coefficient,
Eq. (4.97), once
and S are known.
<7oo =
= i(1.225)(120)2 = 8820 N/m2
S = 0.05(1) = 0.05 m2
Thus, from Eq. (4.97), the drag on one surface of the plate (say the top surface) is
Top
Df = qooSCf = 8820(0.05)(2.07 x 10~3) = 0.913 N
Since both the top and bottom surfaces are exposed to the flow, the total friction drag will
be double the above result:
Total
Df = 2(0.913) =
1.826 N
EXAM PLE 4.24
For the flat plate in Example 4.23, calculate and compare the local shear stress at the
locations 1 and 5 cm from the front edge (the leading edge) of the plate, measured in the
flow direction.
Results for a Laminar Boundary Layer
4 .1 6
■ Solution
The location x = l cm is near the front edge o f the plate. The local Reynolds number at
this location, where x = l cjn = 0.01 m, is
R ev =
pJV,ooX
1.225(120)(0.01)
r'O
1.789 x 10 -5
= 8.217 x 104
From Eq. (4.93),
0 .6'
cf , =
/R Í
0.664
0.664
-s/8.217 x 104
286.65
= 0.002316
From Eq. (4.92), with qx = 8820 N /m 2 from Example 4.23,
*u> = <7olCfx = 8820(0.002316) —
20.43 N/m
At the location x = 5 cm = 0.05 m, the local Reynolds number is
R e, =
PdoVooX
1.225(120)(0.05)
1.789 x 10-5
= 4.11 x 10,5
(This is the same value as that calculated in Example 4.23.) From Eq. (4.93),
cf ,
0.664
0.664
v/Ré7
V 4 . l l x 105
= 0.001036
From Eq. (4.92),
rw = c/opCfx = 8820(0.001036)
9.135 N/m 2
By comparison, note that the local shear stress at x = 5 cm, that is, at the back end o f the
plate (the trailing edge), is less than that at x — I cm near the front edge. This confirms
the trend sketched in Fig. 4.40 that rw decreases with distance in the flow direction along
the plate.
A s a check on our calculation, we note from Eq. (4.94) that r„, varies inversely as
xl/2.Thus, once w e
have calculated zw — 20.43 N /m 2 at * = 1 cm, we can directly ob­
tain t w at x = 5 cm from the ratio
Setting condition 1 at jc = 1 cm and condition 2 at x = 5 cm, we have
- r * , , / — = 2 0 . 4 3 , / t = 9 .1 3 5 N /m 2
which verifies our original calculation o f rw at x = 5 cm.
209
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4.17 RESULTS FOR A TURBULENT
BOUNDARY LAYER
U nder the sam e flow conditions, a turbulent boundary layer w ill be thicker than
a lam inar boundary layer. This com parison is sketched in Fig. 4.42. U nlike in the
case for lam inar flows, no exact theoretical results can be presented for turbulent
boundary layers. T he study o f turbulence is a m ajor effort in fluid dynam ics
today; so far, turbulence is still an unsolved theoretical problem and is likely to
rem ain so for an indefinite time. In fact, turbulence is one o f the m ajor unsolved
problem s in theoretical physics. As a result, our know ledge o f 8 and rw for tur­
bulent boundary layers m ust rely on experim ental results. Such results yield the
follow ing approxim ate form ula for turbulent flow:
8—
031x
n-
turbulent
(4.99)
Re°
N ote from Eq. (4.99) that a turbulent boundary grows approxim ately as x 4/5. This
is in contrast to the slow er x 1/2 variation for a lam inar boundary layer. As a result,
turbulent boundary layers grow faster and are thicker than lam inar boundary layers.
The local skin friction coefficient for turbulent flow over a flat plate can be
approxim ated by
(4.100)
T he total skin friction coefficient is given approxim ately as
0.074
turbulent
« " - R t f
N ote that for turbulent flow, C¡ varies as
this is in contrast to the L ~ l/1 vari­
ation for lam inar flow. H ence C / is larger for turbulent flow, w hich precisely con­
firms our reasoning at the end o f Sec. 4.15, where we noted that rw (lam inar) < zw
(turbulent). A lso note that C¡ in Eq. (4.101) is once again a function o f Re/,. Val­
ues o f Cf for both lam inar and turbulent flows are com m only plotted in the form
shown in Fig. 4.43. N ote the m agnitude o f the num bers involved in Fig. 4.43. The
values o f Re¿ for actual flight situations may vary from 10s to 108 or higher; the
values o f C / are generally m uch less than unity, on the order o f 10“ 2 to 10-3 .
y oo
Figure 4.42 Turbulent boundary layers are thicker than laminar boundary layers.
Results for a Turbulent Boundary Layer
4. 17
211
Figure 4.43 Variation o f skin friction coefficient with Reynolds
number for low-speed flow. Comparison of laminar and turbulent flow.
E X A M P U 5 4.25
Consider the same flow over the same flat plate as in Example 4.23; however, assume that
the boundary layer is now com pletely turbulent. Calculate the boundary layer thickness
at the trailing edge and the drag force on the plate.
■ Solution
From Example 4.23, Re* = 4 .1 1 x 105. From Eq. (4.99), for turbulent flow,
8=
0.37*
0.3 7 (0 .0 5 )
Re“f
(4.11 x 105)0-2
1.39 x 10-3 m
Note: Compare this result with the laminar flow result from Example 4.23.
Slurb
1.39 x l O - 3
5
4 .0 6 x 1 0 -4
< lam
= 3.42
Note that the turbulent boundary layer at the trailing edge is 3.42 times thicker than the
laminar boundary layer— quite a sizable amount! From Eq. (4.101),
Cf =
0.074
0.074
R e°2
(4.11 x 10s )02
= 0.00558
On the top surface,
Df = qx SCf = 8820(0 .0 5 )(0 .0 0 5 5 8 ) = 2.46 N
Considering both top and bottom surfaces, w e have
Total Df = 2(2 .4 6 ) =
4.92 N
N ote that the turbulent drag is 2.7 times larger than the laminar drag.
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Basic Aerodynamics
EXAMPLE 4.26
Repeat Example 4.24, except now assume that the boundary layer is completely turbulent.
■ Solution
From Example 4.24, at x = l cm, Re* = 8.217 x 104. The local turbulent skin friction
coefficient at this location is, from Eq. (4.100),
0.0592
Cfx ~
0.0592
.............
” (8.217 x I04)'’-2 ” 000616
From Example 4.24, qx = 8820 N/m . Hence,
*w -
< 7oo C /,
=
8 8 2 0 ( 0 .0 0 6 1 6 )
=
54.33 N/m2
Note: In comparison to the laminar flow result from Example 4.24, the turbulent shear
stress is 54.33/20.43 = 2.7 times larger. By coincidence, this is the same ratio as the
total drag comparison made between turbulent and laminar boundary layer cases in
Example 4.24.
At x = 5 cm, from Example 4.24, Re, = 4.11 x 105. The local turbulent skin fric­
tion coefficient at this location is, from Eq. (4.100),
0.0592
°fx ~
Hence,
0.0592
______
Re®'2 “ (4.11 x 104)0'2 “
rw = q<x>c/x = 8820(0.00446) =
'
39.34 N/m2
Note: In comparison to the laminar flow result from Example 4.24, the turbulent shear
stress at x = 5 cm is 39.34/9.135 = 4.3 times larger.
Comparing the present results with those of Example 4.24, we see that, over a given
length of plate, the percentage drop in shear stress for the laminar case is larger than that
for the turbulent case. Specifically, the percentage drop over the 4-cm space from
x = I cm to x = 5 cm for the laminar case (Example 4.24) is
20.43 - 9.135
Decrease — ----- -——----- x 100 = 55.3%
20.43
F or the turbulent case (Exam ple 4.26),
54 33 - 39 34
Decrease = ----- —— ----- x 100 = 27.6%
54.33
4 .1 8
4.18
Compressibility Effects on Skin Friction
COMPRESSIBILITY EFFECTS
ON SKIN FRICTION
Let us exam ine again the expressions for lam inar and turbulent skin friction co­
efficients given by Eqs. (|L93) and (4.100), respectively. These equations shout
the im portant fact that c¡x is a function o f Reynolds num ber only; that is,
Lam inar
Turbulent
tv a — -r-.
Re“-
O nce again we see the pow er o f the Reynolds num ber in governing viscous flows.
However, this is not the w hole story. Equations (4.91), (4.93), and (4.98) give
expressions for <5, c¡x, and C /, respectively, for a flat-plate boundary layer in an
incompressible lam inar flow. Sim ilarly, Eqs. (4.99), (4.100), and (4.101) give
expressions for 8, c/x, and C¡, respectively, for a flat-plate boundary layer in an
incompressible turbulent flow. M ainly for the benefit o f simplicity, we did not
em phasize in Secs. 4.16 and 4.17 that these equations apply to an incom pressible
flow. However, we are no\v bringing this to your attention. Indeed, you m ight want
to go back to these equations and m ark them in the margins as “incom pressible.”
This raises the question, W hat are the effects o f com pressibility on a flatplate boundary layer? The answ er lies in the Much number, w hich, as we have
already seen in Secs. 4.11 to 4.13, is the pow erful param eter governing high­
speed, com pressible invikcid flows. Specifically, for a flat-plate boundary layer
in a com pressible flow, á, Cfx , and C / are functions o f both M ach num ber and
Reynolds number. The effect o f M ach num ber is not given by a nice, clean for­
mula; rather it m ust be evaluated from detailed num erical solutions o f the com ­
pressible boundary layer How, which is beyond the scope o f this book. It is suffi­
cient to note that for a flat-plate com pressible boundary layer, the constant 0.664
in the num erator o f Eq. (4.93) is replaced by som e other num ber that depends on
the value o f the free-stream M ach num ber; that is,
Cfx =
laminar, com pressible
(4.102)
Similarly, the constant Cj.0592 in the num erator o f Eq. (4.100) is replaced by
som e other num ber that depends on the value o f M<*,; that is,
Cfx = ------¡j-j—
R e;
turbulent, com pressible
(4.103)
These variations are plotted in Fig. 4.44. Here the ratio o f com pressible to in­
com pressible skin friction coefficients at the sam e Reynolds num ber is plotted
213
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4
Basic Aerodynamics
Figure 4.44 Approximate theoretical results for the
compressibility effect on laminar and turbulent flat plate
skin friction coefficients.
versus free-stream M ach num ber for both lam inar and turbulent flows. N ote the
follow ing trends, shown in Figure 4.44:
1.
For a constant Reynolds number, the effect o f increasing M x is to decrease Cfx.
2.
The decrease in c /t is m uch more pronounced for turbulent flow than for
lam inar flow.
EXAM PLE 4.27
A three-view of the Lockheed F -104A Starfighter is shown in Fig. 4.45. This was the first
fighter aircraft designed for sustained Mach 2 flight. The airfoil section of the wing is
very thin, with an extremely sharp leading edge. Assume the wing is an infinitely thin flat
Figure 4.45 Three-view o f the Lockheed F-104 supersonic fighter.
4 .1 8
Compressibility Effects on Skin Friction
plate. Consider the F-104 flying at Mach 2 at a standard altitude o f 35,000 ft. Assum e the
boundary layer over the wing is turbulent. Estimate the shear stress at a point 2 ft dow n­
stream o f the leading edge.
■ Solution
At 35,000 ft, from App. B, px = 7.382 x K)-4 slug/ft3 and T , = 3 9 4 .0 8 °R. To calculate
the Reynolds number, we need both
and the viscosity coefficient
velocity is obtained from the speed o f sound as follow s.
The free-stream
floo = V y RT,x = v / 1.4(1716)(394.08) = 973 ft/s
V » = floo^oo = 9 7 3(2) = 1946 ft/s
We obtain fix from Fig. 4.34, which show s the variation o f n with T. N ote that the
ambient temperature in kelvins is obtained from 394.08/1.8 = 219 K. Extrapolating the
linear curve in Fig. 4.34 to a temperature o f 219 K, w e find that £ioo = 1.35 x
10-5 kg/(m )(s). Converting to English engineering units, we note that as given in Sec. 4.15
at standard sea level, n = 1.7894 x 10-5 kg/(m )(s) = 3.7373 x 10~7 slug/(ft)(s). The
ratio o f these two values gives us the conversion factor, so at T — 2 1 9 K - 394.08' R ,
H = [1.35 x 10 5 kg/(m)(s)]
3.7373 x 1 0 -7 slug/(ft)(s)
1.7894 x I 0 - 5 kg/(m)(s)
= 2.82 x 10-7 slug/(ft)(s)
u
Hence,
D
Poo ^ 00i'
Re* = ---------
(7.382 x 10-4)(1946)(2)
= 1.02 x 107
2.82 x 10-7
From Eq. (4.100), the incompressible skin friction coefficient is
(r/,)mc
0.0592
0.0592
Re" 2
1.02 x 107
= 0.00235
From Fig. 4.44, for a turbulent boundary layer at AÍ» = 2,
Hence, the value o f Cfx at Mach 2 is
¿y, = 0.7 4 (0 .0 0 2 3 5 ) = 0.00174
The dynamic pressure is
q°o = 3P00V& =
Thus,
5
(7.382 x 1Q-4)(1 9 4 6 2) = 1398 lb/ft2
= Q'ooCf, = 1 3 9 8 (0 .0 0 1 7 4 ) =
2 .4 3 lb/ft2
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Basic Aerodynamics
4.19 TRANSITION
In Sec. 4.16 w e discussed the flow over a flat plate as if it w ere all laminar. Sim ­
ilarly, in Sec. 4.17 we assum ed all-turbulent flow. In reality, the flow always
starts out from the leading edge as laminar. Then at som e point dow nstream of
the leading edge, the lam inar boundary layer becom e unstable and small
“bursts” o f turbulent flow begin to grow in the flow. Finally, over a certain region
called the transition region, the boundary layer becom es com pletely turbulent.
F or purposes o f analysis, we usually draw the picture show n in Fig. 4.46, where
a lam inar boundary starts out from the leading edge o f a flat plate and grow s parabolically dow nstream . Then at the transition point, it becom es a turbulent
boundary layer grow ing at a faster rate, on the order o f x 4/5 dow nstream . The
value o f x w here transition is said to take place is the critical value Jtcr. In turn,
*cr allow s the definition o f a critical Reynolds number for transition as
(4.104)
Volumes o f literature have been w ritten on the phenom enon o f transition
from lam inar to turbulent flow. Obviously, because tw is different for the two
flows, know ledge o f where on the surface the transition occurs is vital to an ac­
curate prediction o f skin friction drag. The location o f the transition point (in re­
ality, a finite region) depends on many quantities, such as the Reynolds number,
M ach num ber, heat transfer to or from the surface, turbulence in the free stream,
surface roughness, pressure gradient. A com prehensive discussion o f transition is
beyond the scope o f this book. H owever, if the critical R eynolds num ber is given
to you (usually from experim ents for a given type o f flow), then the location o f
transition xct can be obtained directly from the definition, Eq. (4.104).
For exam ple, assum e that you have an airfoil o f given surface roughness
in a flow at a free-stream velocity o f 150 m/s and you wish to predict how far
from the leading edge the transition will take place. A fter searching through the
literature for low -speed flows over such surfaces, you may find that the criti­
cal Reynolds num ber determ ined from experience is approxim ately Re*cr =
5 x 105. A pplying this “experience” to your problem , using Eq. (4.104), and as­
sum ing the therm odynam ic conditions o f the airflow correspond to standard sea
Figure 4.46 Transition from laminar to turbulent flow. The boundary
layer thickness is exaggerated for clarity.
4 .1 9
Transition
217
level, you find
MooRe^
[ l .789 x 10 5 kg/(m )(s)](5 x 105)
Poo V*
( l . 225 kg/m ) ( 150 m/s)
= 0.047 m
Note that the region o f lam inar flow in this exam ple is sm all— only 4.7 cm
betw een the leading edge and the transition point. If now you double the freestream velocity to 300 m/s, the transition point is still governed by the critical
Reynolds num ber ReXcr = 5 x 105. Thus,
( .789 x I 0 " s)(5 x 105)
xcr = —-------------------------------- = 0.0235 m
c
I
1.225(300)
Hence, when the velocity jis doubled, the transition point m oves forw ard one-half
the distance to the leading edge.
In sum mary, once you know the critical Reynolds number, you can find jccr
from Eq. (4.104). However, an accurate value o f Re*cr applicable to your problem
m ust com e from som ew here— experim ent, free flight, or some sem iem pirical
theory— and this m ay be difficult to obtain. This situation provides a little insight
into why basic studies o f transition and turbulence are needed to advance our un­
derstanding o f such flows and to allow us to apply more valid reasoning to the
prediction o f transition in practical problem s.
EXAMIM.I I 4.28
The wingspan o f the Wright Flyer I biplane is 4 0 ft 4 in, and the planform area o f each
wing is 255 ft2 (see Figs. 1.1 and 1.2). Assum e the wing is rectangular (obviously not
quite the case, but not bad), as shown in Fig. 4.47. If the Flyer is m oving with a velocity
o f 30 mi/h at standard sea-level conditions, calculate the skin friction drag on the wings.
Assum e the transition Reynolds number is 6.5 x 105. The areas o f laminar and turbulent
flow are illustrated by areas A and B, respectively, in Fig. 4.47.
Figure 4.47 Planform view of surface experiencing transition from
laminar to turbulent flow.
218
chapter
4
Basic Aerodynamics
■ Solution
The general procedure is
a. Calculate Df for the combined area A + B, assuming the flow is completely
turbulent.
b. Obtain the turbulent Df for area B only, by calculating the turbulent D¡ for area A and
subtracting this from the result of part (a).
c. Calculate the laminar Df for area A.
d. Add results from parts (b) and (c) to obtain the total drag on the complete surface
A + B. First, obtain some useful numbers in consistent units: b = 40 ft 4 in = 40.33 ft.
Let S = planform area = A + B = 255 ft2. Hence, c = S /b = 255/40.33 = 6.32 ft. At
standard sea level, p = 0.002377 slug/ft3 and p.^ = 3.7373 x 10~7 slug/(ft)(s). Also
Voo = 30 mi/h = 30(88/60) = 44 ft/s. Thus,
p
e<
PccVx c
0.002377(44X6.32)
Poo
3.7373 x 10- 7
= 1.769 x 106
This is the Reynolds number at the trailing edge. To find xcr,
Poo Voo-tcr
R e <c, =
Xrr --
OO
Re^/ioo
Poo Voo
(6.5 x 105)(3.7373 x 10"7)
= 2.32 ft
0.002377(44)
We are now ready to calculate the drag. Assume that the wings of the Wright Flyer I are
thin enough that the flat-plate formulas apply.
a. To calculate turbulent drag over the complete surface S = A + B, use Eq. (4.101):
0.074
0.074
____
f ~ Re® 2 “ (1.769 x 106)0 2 " a00417
qoo = {PooVl = i (0.002377)(442) = 2.30 lb/ft2
(£>/), = qooSCf = 2.30(255)(0.00417) = 2.446 lb
b. For area A only, assuming turbulent flow,
0.074
0.074
_____ _
f ~ R eJ2
Xcr ~ (6.5 x 10’ )0 2 “ 0 00509
(Df )A = qoo ACf = 2.30(2.32 x 40.33)(0.00509) = 1.095 lb
4.20
Flow Separation
Hence, the turbulent drag on area B only is
(.Df )g = (Df )s - (Df )A = 2 .4 4 6 - 1.095 = 1.351 lb
c. Considering the drag on area A, which is in reality a laminar drag, we obtain from
Eq. (4.98)
Cf =
1.328
Re®'5
1.328
= 0.00165
(6.5 x 105)05
(Df )A = qooACf = 2.30(2.32 x 40.33)(0.00165) = 0.354 lb
d. The total drag D¡ on the surface is
Df = (laminar drag on A) + (turbulent drag on B)
= 0.334 lb + 1.351 lb = 1.705 lb
This is the drag on one surface. Each wing has a top and bottom surface, and there are two
wings. Hence, the total skin friction drag on the complete biplane wing configuration is
Df = 4(1.705) =
6.820 lb
4.20 FLOW SEPARATION
We have seen that the presence o f friction in the flow causes a shear stress at the
surface o f a body, w hich, in turn, contributes to the aerodynam ic drag o f the
body: skin friction drag. However, friction also causes another phenom enon,
called flo w separation , w hich, in turn, creates another source o f aerodynam ic
drag, called pressu re d ra g due to separation . T he real flow field about a sphere
sketched in Fig. 4.30 is dom inated by the separated flow on the rearw ard surface.
Consequently, the pressure on the rearw ard surface is less than the pressure on
the forw ard surface, and this im balance o f pressure forces causes a drag, hence
the term pressu re d ra g due to separation . In com parison, the skin friction drag
on the sphere is very small.
A nother exam ple o f where flow separation is im portant is the flow over an
airfoil. C onsider an airfoil at a low angle o f attack (low angle o f incidence) to the
flow, as sketched in Fig. 4.48. The stream lines move sm oothly over the airfoil.
The pressure distribution over the top surface is also shown in Fig. 4.48. Note
that the pressure at the leading edge is high; the leading edge is a stagnation re­
gion, and the pressure is essentially stagnation pressure. This is the highest pres­
sure anyw here on the airfoil. As the flow expands around the top surface o f the
airfoil, the surface pressure decreases dram atically, dipping to a m inim um pres­
sure, which is below the free-stream static pressure p 00. Then as the flow moves
farther dow nstream , the pressure gradually increases, reaching a value slightly
219
220
chapter
4
Basic Aerodynamics
N A SA LS( 1) - 0 4 1 7 airfo il
A ngle o f a tta c k • 0°
Figure 4.48 Pressure distribution over the top surface for attached flow over an
airfoil. Theoretical data for a modern NASA low-speed airfoil, from NASA
Conference Publication 2046, Advanced Technology Airfoil Research, vol. II,
March 1978, p. U .
(Source: After McGhee, Beasley, and Whitcomb.)
above free-stream pressure at the trailing edge. This region o f increasing pres­
sure is called a region o f adverse pressure gradient, defined as a region where
d p / d x is positive. This region is so identified in Fig. 4.48. The adverse pressure
gradient is m oderate; that is, d p / d x is sm all, and for all practical purposes
the flow rem ains attached to the airfoil surface, as sketched in Fig. 4.48. The drag
on this airfoil is therefore m ainly skin friction drag D¡ .
Now consider the sam e airfoil at a very high angle o f attack, as shown in
Fig. 4.49. First, assum e that w e had som e m agic fluid that would rem ain attached
to the surface— purely an artificial situation. If this w ere the case, then the pres­
sure distribution on the top surface would follow the dashed line in Fig. 4.49. The
pressure would drop precipitously dow nstream o f the leading edge to a value far
below the free-stream static pressure poo. Farther dow nstream the pressure
would rapidly recover to a value above p^o. However, in this recovery, the
adverse pressure gradient w ould no longer be m oderate, as was the case in
Fig. 4.48. Instead, in Fig. 4.49, the adverse pressure gradient w ould be severe;
that is, d p / d x would be large. In such cases, the real flow field tends to separate
4 .2 0
Flow Separation
NASA LS( l ) - 0 4 17 airfoil
Angle o f attack = 18.4°
Figure 4.49 Pressure distribution over the top surface for separated
How over an airfoil. Theoretical data for a modern NASA low-speed
airfoil, from NASA Conference Publication 2045, Part \, Advanced
Technology Airfoil Research, vol. I, March 1978, p. 380.
(Source: After Zumwalt and Nack.)
from the surface. Therefore, in Fig. 4.49, the real flow field is sketched with a
large region o f separated flow over the top surface o f the airfoil. In this real sep­
arated flow, the actual surface pressure distribution is given by the solid curve. In
com parison to the dashed curve, note that the actual pressure distribution does
not dip to as low a pressure m inim um and that the pressure near the trailing edge
does not recover to a value above p ^ . This has two m ajor consequences, as can
221
222
CHAPTER 4
Basic Aerodynamics
A ttached f l o w ----------------
pressure on the airfoil
Figure 4.50 Qualitative comparison of pressure distribution, lift, and drag for attached and
separated flows. Note that for separated flow, the lift decreases and the drag increases.
be seen from Fig. 4.50. Here the airfoil at a large angle o f attack (thus, with flow
separation) is shown with the real surface pressure distribution, sym bolized by
the solid arrows. Pressure always acts norm al to a surface. Hence the arrows are
all perpendicular to the local surface. The length o f the arrow denotes the m ag­
nitude o f the pressure. A solid curve is drawn through the base o f the arrows to
form an “envelope” to m ake the pressure distribution easier to visualize. H ow ­
ever, if the flow were not separated, that is, if the flow were attached, then the
pressure distribution would be that shown by the dashed arrows (and the dashed
envelope). The solid and dashed arrows in Fig. 4.50 qualitatively correspond to
the solid and dashed pressure distribution curves, respectively, in Fig. 4.49.
The solid and dashed arrows in Fig. 4.50 should be looked at carefully. They
explain the two m ajor consequences of separated flow over the airfoil. The first
consequence is a loss o f lift. The aerodynam ic lift (the vertical force shown in
Fig. 4.50) is derived from the net com ponent o f a pressure distribution in the ver­
tical direction. High lift is obtained when the pressure on the bottom surface is
large and the pressure on the top surface is small. Separation does not affect the
bottom surface pressure distribution. However, com paring the solid and dashed
arrows on the top surface ju st downstream o f the leading edge, we find the solid
arrows indicate a higher pressure when the flow is separated. This higher pres­
sure is pushing down, hence reducing the lift. This reduction o f lift is also
4 .2 0
Flow Separation
com pounded by the geom etric effect that the portion o f the top surface o f the air­
foil near the leading edge is approxim ately horizontal in Fig. 4.50. W hen the flow
is separated, causing a higher pressure on this part o f the airfoil surface, the di­
rection in which the pressure is acting is closely aligned to the vertical, and
hence, alm ost the full effpct o f the increased pressure is felt by the lift. The com ­
bined effect o f the increased pressure on the top surface near the leading edge,
and the fact that this portion o f the surface is approxim ately horizontal, leads to
the rather dram atic loss o f lift w hen the flow separates. Note in Fig. 4.50 that the
lift for separated flow (the solid vertical arrow ) is sm aller than the lift that would
exist if the flow w ere attached (the dashed vertical arrow).
N ow let us concentrate on that portion o f the top surface near the trailing
edge. On this portion o f the airfoil surface, the pressure for the separated flow is
now sm aller than the pressure that would exist if the flow were attached. M ore­
over, the top surface near the trailing edge geom etrically is inclined to the hori­
zontal, and, in fact, som ew hat faces in the horizontal direction. Recall that drag
is in the horizontal direction in Fig. 4.50. B ecause o f the inclination o f the top
surface near the trailing edge, the pressure exerted on this portion o f the surface
has a strong com ponent in the horizontal direction. This com ponent acts tow ard
the left, tending to counter the horizontal com ponent o f force due to the high
pressure acting on the nose o f the airfoil pushing tow ard the right. The net pres­
sure drag on the airfoil i|> the difference betw een the force exerted on the front
pushing tow ard the right and the force exerted on the back pushing tow ard the
left. W hen the flow is separated, the pressure at the back is low er than it would
be if the flow w ere attacHed. H ence, for the separated flow, there is less force on
the back pushing tow ard the left, and the net drag acting tow ard the right is there­
fore increased. N ote in Fig. 4.50 that the drag for separated flow (the solid hori­
zontal arrow ) is larger than the drag that would exist if the flow were attached
(the dashed horizontal arrow).
Therefore, tw o m ajor consequences o f the flow separating over an airfoil are
1.
A drastic loss o f lift (stalling)
2.
A m ajor increase in drag, caused by pressure drag due to separation
W hen the wing o f an airplane is pitched to a high angle o f attack, the wing can
stall, that is, there can be a sudden loss o f lift. O ur previous discussion gives the
physical reasons for this stalling phenom enon. Additional ram ifications o f stalling
are discussed in Chap. 5.
Before ending this discussion o f separated flow, we ask the question, W hy
does a flow separate frorp a surface? The answ er is com bined in the concept of
an adverse pressure gradient (d p / d x is positive) and the velocity profile through
the boundary layer, as shown in Fig. 4.37. If d p / d x is positive, then the fluid el­
em ents m oving along a stream line have to work their way “uphill” against an
increasing pressure. Consequently, the fluid elem ents will slow down under the
influence o f an adverse pressure gradient. For the fluid elem ents m oving outside
the boundary layer, where the velocity (and hence kinetic energy) is high, this is
not m uch o f a problem . The fluid elem ent keeps m oving dow nstream . However,
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consider a fluid elem ent deep inside the boundary layer. Looking at Fig. 4.37, we
see its velocity is small. It has been retarded by friction forces. The fluid elem ent
still encounters the same adverse pressure gradient, but its velocity is too low to
negotiate the increasing pressure. As a result, the elem ent com es to a stop som e­
w here dow nstream and then reverses its direction. Such reversed flow causes the
flow field in general to separate from the surface, as shown in Fig. 4.49. This is
physically how separated flow develops.
Reflecting once again on Fig. 4.37, we note that turbulent boundary layers
have fuller velocity profiles. At a given distance from the surface (a given value
o f y), the velocity o f a fluid elem ent in a turbulent boundary is higher than that in
a lam inar boundary layer. H ence, in turbulent boundary layers, there is m ore flow
kinetic energy nearer the surface. Hence, the flow is less inclined to separate.
This leads to a very fundam ental fact. Lam inar boundary layers separate more
easily than turbulent boundary layers. Hence, to help prevent flow field separa­
tion, we want a turbulent boundary layer.
4.21 SUMMARY OF VISCOUS EFFECTS ON DRAG
We have seen that the presence o f friction in a flow produces tw o sources o f drag:
1.
Skin friction drag D f due to shear stress at the wall
2.
Pressure drag due to flow separation D p, som etim es identified as form drag
T he total drag which is caused by viscous effects is then
D
=
Df
+
Dp
T o ta l d ra g
D ra g d u e
D ra g d u e to
d u e to v is c o u s
to sk in
separatio n
e ffe c ts
frictio n
(pressu re drag)
(4.105)
Equation (4.105) contains one o f the classic com prom ises o f aerodynam ics.
In previous sections, w e pointed out that skin friction drag is reduced by m ain­
taining a lam inar boundary layer over a surface. H owever, we also pointed out at
the end o f Sec. 4.20 that turbulent boundary layers inhibit flow separation; hence,
pressure drag due to separation is reduced by establishing a turbulent boundary
layer on the surface. Therefore, in Eq. (4.105) we have the follow ing com prom ise:
D —
Df
+
Dp
L e s s fo r lam inar,
M o re fo r lam inar,
m o re fo r turbu len t
le ss fo r turbulen t
Consequently, as discussed at the end o f Sec. 4.15, it cannot be said in general
that either lam inar or turbulent flow is preferable. Any preference depends on
the specific application. On one hand, for a blunt body such as the sphere in
Fig. 4.30, the drag is m ainly pressure drag due to separation; hence, turbulent
boundary layers reduce the drag on spheres and are therefore preferable. (We dis­
cuss this again in Chap. 5). On the other hand, for a slender body such as a sharp,
slender cone or a thin airfoil at small angles o f attack to the flow, the drag is
4 .2 2
Historical Note: Bernoulli and Euler
mainly skin friction drag; hence, lam inar boundary layers are preferable in this
case. For in-betw een cases, the ingenuity o f the designer along with practical ex­
perience helps to determ ine w hat com prom ises are best.
As a final note to this section, the total drag D given by Eq. (4.105) is called
profile drag because both skin friction and pressure drag due to separation are
ram ifications o f the shape and size o f the body, that is, the “profile” o f the body.
The profile drag D is the total drag on an aerodynam ic shape due to viscous ef­
fects. However, it is not in general the total aerodynam ic drag on the body. There
is one m ore source o f drag, induced drag, w hich is discussed in Chap. 5.
4.22 HISTORICAL NOTE: BERNOULLI AND EULER
Equation (4.9) is one o f the oldest and most pow erful equations in fluid dynam ­
ics. It is credited to Daniel Bernoulli, w ho lived during the 18th century; little did
Bernoulli know that his concept w ould find w idespread application in the aero­
nautics o f the 20th century. W ho was Bernoulli, and how did B ernoulli’s equa­
tion com e about? Let us briefly look into these questions; the answ ers will lead
us to a rather unexpected conclusion.
Daniel Bernoulli (1700-1 7 8 2 ) was born in G roningen, N etherlands, on
January 29, 1700. He was a m em ber o f a rem arkable family. His father, Johann
Bernoulli, was a noted m athem atician who m ade contributions to differential and
integral calculus and who later becam e a doctor o f m edicine. Jakob Bernoulli,
who was Joh an n ’s brother (D aniel’s uncle), was an even more accom plished
m athem atician; he m ade m ajor contributions to the calculus; he coined the term
integral. Sons o f both Jakob and Johann, including D aniel, went on to becom e
noted m athem aticians and physicists. The entire fam ily was Swiss and made its
hom e in Basel, Sw itzerland, w here they held various professorships at the U ni­
versity o f Basel. Daniel Bernoulli was born aw ay from Basel only because his fa­
ther spent 10 years as professor o f m athem atics in the N etherlands. With this
type o f pedigree, D aniel could hardly avoid m aking contributions to m athem at­
ics and science him self.
And indeed, he did make contributions. For exam ple, he had insight into the
kinetic theory o f gases; he theorized that a gas was a collection o f individual par­
ticles m oving about in an agitated fashion, and he correctly associated the in­
creased tem perature o f a gas with the increased energy o f the particles. These
ideas, originally published in 1738, were to lead a century later to a mature un­
derstanding o f the nature o f gases and heat and helped to lay the foundation for
the elegant kinetic theory o f gases.
D aniel’s thoughts on the kinetic m otion o f gases were published in his book
Hydrodynamica (1738). However, this book was to etch his name more deeply
in association with fluid m echanics than with kinetic theory. The book was
started in 1729, when Daniel was a professor o f m athem atics at Leningrad (then
St. Petersburg) in Russia. By this tim e he was already well recognized; he had
won 10 prizes offered by the Royal A cadem y o f Sciences in Paris for his solution
o f various m athem atical problem s. In his H ydrodynamica (which was written en ­
tirely in Latin), Bernoulli ranged over such topics as je t propulsion, m anom eters,
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and flow in pipes. He also attem pted to obtain a relationship betw een pressure
and velocity, but his derivation was obscure. In fact, even though B ernoulli’s
equation, Eq. (4.9), is usually ascribed to Daniel via his Hydrodynamica, the pre­
cise equation is not to be found in the book! The picture is further com plicated by
his father, Johann, w ho published a book in 1743 entitled Hydraulica. It is clear
from this latter book that the father understood B ernoulli’s theorem better than
the son; D aniel thought o f pressure strictly in term s o f the height o f a m anom eter
colum n, w hereas Johann had the more fundam ental understanding that pressure
was a force acting on the fluid. However, neither o f the Bernoullis understood
that pressure is a point property. T hat was to be left to Leonhard Euler.
Leonhard E uler (1707-1 7 83) was also a Sw iss m athem atician. He was born
in Basel, Sw itzerland, on April 15, 1707, seven years after the birth o f Daniel
Bernoulli. E uler w ent on to becom e one o f the m athem atical giants o f history, but
his contributions to fluid dynam ics are o f interest here. E uler was a close friend
o f the B ernoullis; indeed, he was a student o f Johann Bernoulli at the U niversity
o f Basel. Later, E uler follow ed Daniel to St. Petersburg, where he becam e a pro­
fessor o f m athem atics. It was here that E uler was influenced by the w ork o f the
B ernoullis in hydrodynam ics, but more influenced by Johann than by Daniel.
Euler originated the concept o f pressure acting at a point in a gas. This quickly
led to his differential equation for a fluid accelerated by gradients in pressure, the
sam e equation we have derived as Eq. (4.8). In turn, E uler integrated the differ­
ential equation to obtain, for the first time in history, B ernoulli’s equation, ju st as
we have obtained Eq. (4.9). H ence, we see that B ernoulli’s equation, Eq. (4.9), is
really a historical m isnom er. C redit for B ernoulli’s equation is legitim ately
shared by Euler.
4.23 HISTORICAL NOTE: THE PITOT TUBE
The use o f a Pitot tube to m easure airspeed is described in Sec. 4.11; indeed, the
Pitot tube today is so com m only used in aerodynam ic laboratories and on aircraft
that it is alm ost taken for granted. However, this sim ple little device has had a
rather interesting and som ew hat obscure history.
The Pitot tube is nam ed after its inventor, Henri Pitot (1695-1771). Born in
A ram on, France, in 1695, Pitot began his career as an astronom er and m athe­
matician. He was accom plished enough to be elected to the Royal A cadem y o f
Sciences, Paris, in 1724. A bout this tim e, Pitot becam e interested in hydraulics
and, in particular, in the flow o f w ater in rivers and canals. However, he was not
satisfied with the existing technique o f m easuring the flow velocity, which was to
observe the speed o f a floating object on the surface o f the water. So he devised
an instrum ent consisting o f tw o tubes. O ne was sim ply a straight tube open at one
end, which was inserted vertically into the w ater (to m easure static pressure), and
the other was a tube with one end bent at right angles, with the open end facing
directly into the flow (to m easure total pressure). In 1732, betw een two piers o f
a bridge over the Seine River in Paris, he used this instrum ent to m easure the
flow velocity o f the river. This invention and first use o f the Pitot tube was
4 .2 3
Historical Note: The Pitot Tube
announced by Pitot to the A cadem y on N ovem ber 12, 1732. In his presentation,
he also presented som e data o f m ajor im portance on the variation o f w ater flow
velocity with depth. C ontem porary theory, based on experience o f som e Italian
engineers, held that the flow velocity at a given depth was proportional to the
m ass above it; hence, th¿ velocity was thought to increase with depth. Pitot re­
ported the stunning (and correct) results, m easured w ith his instrum ent, that in
reality the flow velocity decreased as the depth increased. Hence, the Pitot tube
was introduced w ith styl$.
Interestingly enough, P itot’s invention soon fell into disfavor with the engi­
neering com m unity. A num ber o f investigators attem pted to use ju st the Pitot
tube itself, w ithout a local static pressure m easurem ent. O thers, using the device
under uncontrolled conditions, produced spurious results. Various shapes and
form s other than a sim ple tube were som etim es used for the mouth o f the instru­
ment. M oreover, there was no agreed-upon rational theory o f the Pitot tube. Note
that Pitot developed his instrum ent in 1732, six years before Daniel B ernoulli’s
Hydrodynamica and well before Euler had developed the B ernoullis’ concepts
into Eq. (4.9), as discussed in Sec. 4.22. H ence, Pitot used intuition, not theory,
to establish that the pressure difference m easured by his instrum ent was an in­
dication o f the square d f the local flow velocity. O f course, as described in
Sec. 4.11, we now clearly understand that a Pitot-static device m easures the dif­
ference betw een total and static pressures and that for incom pressible flow, this
difference is related to the velocity squared through B ernoulli’s equation; that is,
from Eq. (4.62),
P o - P = 5p V 2
However, for m ore than 150 years after P itot’s introduction o f the instrum ent,
various engineers attem pted to interpret readings in term s o f
Po - p = \ K p V 2
where K was an em pirical constant, generally much different from unity. C on­
troversy was still raging as late as 1913, when John Airey, a professor o f m e­
chanical engineering from the U niversity o f M ichigan, finally perform ed a series
o f w ell-controlled experim ents in a w ater tow tank, using Pitot probes o f six dif­
ferent shapes. These shapes are shown in Fig. 4.51, which is taken from A irey’s
paper in the April 17, 1913, issue o f the Engineering News, entitled “N otes on the
Pitot T ube.” In this paper, Aircy states that all his m easurem ents indicate that
K = 1.0 within 1 percent accuracy, independent o f the shape o f the tube. M ore­
over, he presents a rational theory based on B ernoulli’s equation. Further com ­
ments on these results are m ade in a paper entitled “O rigin and Theory of the
Pitot T ube” by A. E. Guy, the ch ief engineer o f a centrifugal pum p com pany in
Pittsburgh, in a later, June 5, 1913, issue o f the Engineering News. This paper
also helped to establish the Pitot tube on firm er technical grounds.
It is interesting to note that neither o f these papers in 1913 m entioned what
was to becom e the m ost prevalent use o f the Pitot tube, namely, the m easurem ent
o f airspeed for airplanes and wind tunnels. The first practical airspeed indicator.
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Water surface
|**— 12 in— ►
I**— 12 in — * i
1------------ — f
H— 12 in— i
p
— -i
(a )
( b)
(c)
U ----------- 24 in
228
Figure 4.5 Í S ix fo rm s o f P ito t tu b e s te s te d by J o h n A irey .
(Source: From Engineerings News, vol. 69, no. 16, p. 783, April 1913.)
a Venturi tube, was used on an aircraft by the French Captain A. Eteve in Janu­
ary 1911, more than seven years after the first pow ered flight. Later in 1911,
British engineers at the Royal A ircraft Establishm ent (RAE) at Farnborough em ­
ployed a Pitot tube on an airplane for the first time. This was to eventually evolve
into the prim ary instrum ent for flight speed m easurem ent.
T here was still controversy over Pitot tubes, as well as the need for reliable
airspeed m easurem ents, in 1915, when the brand-new National A dvisory
C om m ittee for A eronautics (N A CA ) stated in its First A nnual Report that “an
im portant problem to aviation in general is the devising o f accurate, reliable and
durable air speed m e te rs .. . . The Bureau o f Standards is now engaged in inves­
tigation o f such m eters, and attention is invited to the report o f Professor
H erschel and Dr. Buckingham o f the bureau on Pitot tubes.” The aforem entioned
report was N A CA Report No. 2, Part 1, “The Pitot Tube and other A nem om eters
for A eroplanes,” by W. H. H erschel, and Part 2, “T he Theory o f the Pitot and
Venturi T ubes,” by E. B uckingham . Part 2 is o f particular interest. In clear terms,
it gives a version o f the theory we developed in Sec. 4.11 for the Pitot tube;
m oreover, it develops for the first tim e the theory for com pressible subsonic
4.2 4
Historical Note: The First Wind Tunnels
flow— quite unusual for 1915! B uckingham shows that to obtain 0.5 percent
accuracy with the incom pressible relations,
should not exceed 148 mi/h =
66 .1 m/s. However, he goes on to state that “since the accuracy o f better than
1.0 percent can hardly be dem anded o f an airplane speedom eter, it is evident that
for all ordinary speeds o f flight, no correction for com pressibility is n e e d e d .. . . ”
This was certainly an appropriate com m ent for the “ordinary” airplanes o f that
day; indeed, it was accurate for m ost aircraft until the 1930s.
In retrospect, we see that the Pitot tube was invented alm ost 250 years ago
but that its use was controversial and obscure until the second decade o f powered
flight. Then, betw een 1911 and 1915, one o f those “explosions” in technical ad­
vancem ent occurred. Pitot tubes found a m ajor home on airplanes, and the appro­
priate theory for their correct use was finally established. Since then, Pitot tubes
have becom e com m onplace. Indeed, the Pitot tube is usually the first aerody­
namic instrum ent introduced to students o f aerospace engineering in laboratory
studies.
4.24 HISTORICAL NOTE: THE FIRST
WIND TUNNELS
A erospace engineering in general, and aerodynam ics in particular, is an em piri­
cally based discipline. D iscovery and developm ent by experim ental m eans have
been its lifeblood, extending all the way back to G eorge C ayley (see Chap. 1). In
turn, the w orkhorse for such experim ents has been predom inantly the wind tun­
nel, so m uch so that today most aerospace industrial, governm ent, and university
laboratories have a com plete spectrum o f wind tunnels ranging from low sub­
sonic to hypersonic speeds.
It is interesting to reach back briefly into history and look at the evolution of
wind tunnels. A m azingly enough, this history goes back more than 400 years, be­
cause the cardinal principle o f wind tunnel testing was stated by Leonardo da
Vinci near the beginning o f the 16th century as follows:
For since the action o f the medium upon the body is the same whether the body
m oves in a quiescent medium, or whether the particles o f the medium impinge
with the same velocity upon the quiescent body; let us consider the body as if it were
quiescent and see with what force it would be impelled by the m oving medium.
This is alm ost self-evident today, that the lift and drag o f an aerodynam ic
body are the sam e w hether it m oves through the stagnant air at 100 mi/h or
w hether the air m oves over the stationary body at 100 mi/h. This concept is the
very foundation o f wind tunnel testing.
The first actual wind tunnel in history was designed and built more than
100 years ago by Francis W enham in G reenw ich, England, in 1871. We met
W enham once before, in Sec. 1.4, where his activity in the A eronautical Society
o f G reat Britain was noted. W enham ’s tunnel was nothing more than a 10-ft-long
w ooden box with a square cross section, 18 in on a side. A steam -driven fan at the
front end blew air through the duct. There was no contour, hence no aerodynam ic
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control or enhancem ent o f flow. Plane aerodynam ic surfaces w ere placed in the
airstream at the end o f the box, where W enham m easured the lift and drag on
w eighing beam s linked to the model.
Thirteen years later, H oratio F. Phillips, also an Englishm an, built the second
know n wind tunnel in history. A gain, the flow duct was a box, but Phillips used
steam ejectors (high-speed steam nozzles) dow nstream o f the test section to suck
air through the tunnel. Phillips w ent on to conduct som e pioneering airfoil test­
ing in his tunnel, w hich will be m entioned again in Sec. 5.20.
O ther wind tunnels w ere built before the turning point in aviation in 1903.
For exam ple, the first wind tunnel in R ussia was due to N ikolai Joukow ski at the
U niversity o f M oscow in 1891 (it had a 2-in diam eter). A larger, 7 in x 10 in tun­
nel was built in A ustria in 1893 by Ludw ig M ach, son o f the fam ed scientist and
philosopher Ernst M ach, after whom the M ach num ber is nam ed. The first tunnel
in the U nited States was built at the M assachusetts Institute o f Technology in
1896 by Alfred J. Wells, w ho used the m achine to m easure the drag on a flat plate
as a check on the w hirling-arm m easurem ents o f Langley (see Sec. 1.8). A nother
tunnel in the U nited States was built by Dr. A. Heb Zahm at the C atholic U ni­
versity o f A m erica in 1901. In light o f these activities, it is obvious that at the
turn o f the 20 th century, aerodynam ic testing in w ind tunnels was poised and
ready to burst forth with the same energy that accom panied the developm ent of
the airplane itself.
It is fitting that the sam e two people responsible for getting the airplane off
the ground should also have been responsible for the first concentrated series o f
w ind tunnel tests. As noted in Sec. 1.8, the W right brothers in late 1901 concluded
that a large part o f the existing aerodynam ic data were erroneous. This led to their
construction o f a 6 -ft-long 16-in-square wind tunnel pow ered by a tw o-blade fan
connected to a gasoline engine. A replica o f the W rights’ w ind tunnel is shown in
Fig. 4.52. (Their original w ind tunnel no longer exists.) They designed and built
their ow n balance to m easure the ratios o f lift to drag. Using this apparatus,
W ilbur and O rville undertook a m ajor program o f aeronautical research betw een
Septem ber 1901 and A ugust 1902. D uring this time, they tested more than 200
different airfoil shapes m anufactured out o f steel. The results from these tests con­
stitute the first m ajor im pact o f wind tunnel testing on the developm ent o f a suc­
cessful airplane. A s w e quoted in Sec. 1.8, O rville said about their results: “O ur
tables o f air pressure which we made in our w ind tunnel w ould enable us to cal­
culate in advance the perform ance o f a m achine.” W hat a fantastic developm ent!
This was a turning point in the history o f wind tunnel testing, and it had as m uch
im pact on that discipline as the D ecem ber 17, 1903, flight had on the airplane.
The rapid grow th in aviation after 1903 was paced by the rapid grow th o f
w ind tunnels, both in num bers and in technology. For exam ple, tunnels were
built at the N ational Physical Laboratory in London in 1903; in Rom e in 1903; in
M oscow in 1905; in G ottingen, G erm any (by the fam ous Dr. Ludw ig Prandtl,
originator o f the boundary layer concept in fluid dynam ics) in 1908; in Paris in
1909 (including tw o built by G ustave Eiffel, o f tow er fame); and again at the
N ational Physical Laboratory in 1910 and 1912.
4 .2 4
Historical Note: The First Wind Tunnels
Figure 4.52 A re p lic a o f th e W rig h t b r o th e rs ' w in d tu n n e l in a w o rk ro o m b e h in d the
W rig h ts ’ b ic y c le sh o p , n o w in G re e n fie ld V illag e, D e a rb o rn , M ic h ig a n .
All these tunnels, quite naturally, were low -speed facilities, but they were pi­
oneering for their time. Then in 19 15, with the creation o f NA CA (see Sec. 2.8),
the foundation was laid for som e m ajor spurts in wind tunnel design. The first
NACA w ind tunnel becam e operational at the Langley M em orial A eronautical
Laboratory at H am pton, Virginia, in 1920. It had a 5-ft-diam eter test section that
accom m odated m odels up to 3.5 ft wide. Then in 1923, in order to sim ulate the
higher Reynolds num bers associated with flight, N A CA built the first variabledensity wind tunnel, a facility that could be pressurized to 20 atm in the flow and
therefore obtain a 20-fold increase in density, hence Re, in the test section. D ur­
ing the 1930s and 1940s, subsonic wind tunnels grew larger and larger. In 1931,
an N A C A wind tunnel with a 30 ft x 60 ft oval test section went into operation
at Langley with a 129 m i/h m axim um flow velocity. This was the first milliondollar tunnel in history. Later, in 1944, a 40 ft x 80 ft tunnel with a flow velocity
o f 265 mi/h was initiated at A m es A eronautical Laboratory at M offett Field,
California. This is still the largest wind tunnel in the world today. Figure 4.53
shows the m agnitude o f such tunnels: whole airplanes can be m ounted in the test
section!
The tunnels m entioned above w ere low -speed, essentially incom pressible
flow tunnels. They w ere the cornerstone o f aeronautical testing until the 1930s
and rem ain an im portant part o f the aerodynam ic scene today. However, air­
plane speeds w ere progressively increasing, and new wind tunnels with higher
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Figure 4.53 A subsonic wind tunnel large enough to test a full-size airplane. The NASA
Langley Research Center 30 ft x 60 ft tunnel. (Recently, Old Dominion University has
taken over operation o f this tunnel from NASA.)
4 .2 4
Historical Note: The First Wind Tunnels
Figure 4 .54 The Ames 16-flj high-speed subsonic wind tunnel, illustrating the massive size
that goes along with such a wind tunnel complex.
(Courtesy NASA Ames Research\Center.)
velocity capability w ere needed. Indeed, the first requirem ent for high-speed
subsonic tunnels was established by propellers— in the 1920s and 1930s the pro­
peller diam eters and rotational speeds w ere both increasing so as to encounter
com pressibility problem s at the tips. This problem led NACA to build a 12-indiam eter high-speed tunnel at Langley in 1927. It could produce a test section
flow o f 765 mi/h. In 1936, to keep up w ith increasing airplane speeds, Langley
built a large 8-1't high-speed wind tunnel providing 500 mi/h. This was increased
to 760 m i/h in 1945. An im portant facility was built at A m es in 1941, a 16-ft tun­
nel with an airspeed o f 680 mi/h. A photograph o f the Ames 16-ft tunnel is
shown in Fig. 4.54 ju st to give a feeling for the m assive size involved with such
a facility.
In the early 1940s, the advent o f the V-2 rocket as well as the je t engine put
supersonic flight in the m inds o f aeronautical engineers. Suddenly, the require­
ment for supersonic tunnels becam e a m ajor factor. However, supersonic flows in
the laboratory and in practice date farther back than this. The first supersonic
nozzle was developed by Laval about 1880 for use with steam turbines. This is
why the convergent-divergent nozzles are frequently called Laval nozzles. In
1905, Prandtl built a small M ach 1.5 tunnel at G ottingen to be used to study
steam turbine flows and (of all things) the m oving o f saw dust around sawmills.
The first practical supersonic wind tunnel for aerodynam ic testing was de­
veloped by Dr. A. B usem ann at Braunschw eig, Germany, in the mid-1930s.
Using the “m ethod o f characteristics” technique, w hich he had developed in
1929, Busem ann designed the first sm ooth supersonic nozzle contour that
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Figure 4.55 T h e first p ra c tic a l su p e rs o n ic w in d tu n n el, b u ilt b y A. B u se m a n n in th e m id -1 9 3 0 s.
(Courtesy o f A. Busemann .)
produced shock-free isentropic flow. He had a diffuser with a second throat
dow nstream to decelerate the How and to obtain efficient operation o f the tunnel.
A photograph o f B usem ann’s tunnel is shown in Fig. 4.55. AH supersonic tunnels
today look essentially the same.
W orking from B usem ann’s exam ple, the G erm ans built two m ajor super­
sonic tunnels at their research com plex at Peenem iinde during World War II.
These w ere used for research and developm ent o f the V-2 rocket. A fter the war,
these tunnels w ere m oved alm ost in total to the U.S. Naval O rdnance Laboratory
(later, one was m oved to the U niversity o f M aryland), where they are still in use
today. However, the first supersonic tunnel built in the U nited States was de­
signed by Theodore von K arm an and his colleagues at the C alifornia Institute o f
Technology in 1944 and was built and operated at the Army B allistics Research
Laboratory at A berdeen, M aryland, under contract with Cal Tech. Then the
1950s saw a virtual harvest o f supersonic w ind tunnels, one o f the largest being
the 16 ft x 16 ft continuously operated supersonic tunnel o f the A ir Force at the
A rnold Engineering D evelopm ent C enter (A ED C) in Tennessee.
A bout this tim e, the developm ent o f the intercontinental ballistic m issile
(IC B M ) w a s o n the h o rizo n , so o n to b e fo llo w e d by the sp a ce program o f the
1960s. Flight vehicles w ere soon to encounter velocities as high as 36,000 ft/s
in the atm osphere— hypersonic velocities. In turn, hypersonic wind tunnels
(M > 5) w ere suddenly in dem and. The first hypersonic wind tunnel was operated
4 .2 5
Historical Note: Osborne Reynolds and His Number
by N A CA at Langley in 1947. It had an 11-in-square test section capable o f
M ach 7. Three years later, another hypersonic tunnel w ent into operation at the
Naval O rdnance Laboratory. T hese tunnels are distinctly different from their su­
personic relatives in that, to obtain hypersonic speeds, the flow has to be expanded
so far that the tem perature decreases to the point o f liquefying the air. To prevent
this, all hypersonic tunnels, both old and new, have to have the reservoir gas heated
to tem peratures far above room tem perature before its expansion through the noz­
zle. H eat transfer is a problem for high-speed flight vehicles, and such heating
problem s feed right dow n to the ground-testing facilities for such vehicles.
In sum m ary, m odern wind tunnel facilities range across the w hole spectrum
o f flight velocities, from low subsonic to hypersonic speeds. These facilities are
part o f the everyday life 0 f aerospace engineering; hopefully, this brief historical
sketch has provided some insight into their tradition and developm ent.
4.25 HISTORICAL NOTE: OSBORNE REYNOLDS
AND HIS NUMBER
In Secs. 4.15 to 4.19, we observed that the Reynolds number, defined in
Eq. (4.90) as Re = PooVLx/ IM<», was the governing param eter for viscous flow.
Boundary layer thickness, skin friction drag, transition to turbulent flow, and
many other characteristics o f viscous flow depend explicitly on the Reynolds
number. Indeed, we can readily show that the Reynolds num ber itself has physi­
cal m eaning— it is proportional to the ratio o f inertia forces to viscous forces in
a fluid flow. Clearly, the Reynolds num ber is an extrem ely im portant dim ensionless param eter in fluid dynam ics. W here did the Reynolds num ber com e from?
W hen was it first introduced, and under what circum stances? The Reynolds num ­
ber is nam ed after a man— O sborne Reynolds. W ho was Reynolds? This section
answ ers these questions.
First, let us look at O sborne Reynolds, the man. He was born on O ctober 23,
1842, in Belfast, Ireland. He was raised in an intellectual fam ily atm osphere; his
father had been a fellow o f Q ueens College, C am bridge, a principal o f Belfast
C ollegiate School, headm aster o f Dedham G ram m ar School in Essex, and finally
rector at D ebach-w ith-B oulge in Suffolk. Indeed, Anglican clerics w ere a tradi­
tion in the Reynolds fam ily; in addition to his father, his grandfather and great­
grandfather had been rectors at D ebach. A gainst this background, Osborne
R eynolds’s early education was carried out by his father at Dedham. In his teens,
he already show ed an intense interest in the study of m echanics, for which he had
a natural aptitude. At the age o f 19, he served a short apprenticeship in m echan­
ical engineering before attending C am bridge U niversity a year later. Reynolds
was a highly successful student at C am bridge, graduating with the highest
honors in m athem atics. In 1867, he was elected a fellow o f Q ueens College,
C am bridge (an honor earlier bestow ed upon his father). He went on to serve one
year as a practicing civil engineer in the office o f John Law son in London.
H ow ever, in 1868, O w ens C ollege in M anchester (later to becom e the University
o f M anchester) established its chair o f engineering— the second o f its kind in any
235
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English university (the first was the chair o f civil engineering established at the
U niversity College, London, in 1865). Reynolds applied for this chair, w riting in
his application:
From my earliest recollection I have had an irresistible liking for mechanics and the
physical laws on which mechanics as a science are based. In my boyhood I had the
advantage o f the constant guidance o f my father, also a lover o f mechanics and a man
o f no mean attainment in mathematics and their application to physics.
D espite his youth and relative lack o f experience, Reynolds was appointed to
the chair at M anchester. F or the next 37 years he served as a professor at
M anchester until his retirem ent in 1905.
D uring those 37 years, Reynolds distinguished him self as one o f history’s
leading practitioners o f classical m echanics. During his first years at M anchester,
he w orked on problem s involving electricity, m agnetism , and the electrom ag­
netic properties o f solar and com etary phenom ena. A fter 1873, he focused on
fluid m echanics— the area in which he m ade his lasting contributions. For exam ­
ple, he (1) developed R eynolds’s analogy in 1874, a relation betw een heat trans­
fer and frictional shear stress in a fluid; ( 2 ) m easured the average specific heat o f
w ater betw een freezing and boiling, w hich ranks am ong the classic determ ina­
tions o f physical constants; (3) studied w ater currents and w aves in estuaries;
(4) developed turbines and pum ps; and (5) studied the propagation o f sound
w aves in fluids. However, his m ost im portant work, and the one w hich gave birth
to the concept o f the Reynolds number, was reported in 1883 in a paper entitled
“An Experim ental Investigation o f the C ircum stances which D eterm ine w hether
the M otion o f W ater in Parallel Channels Shall Be D irect or Sinuous, and o f the
Law o f Resistance in Parallel C hannels.” Published in Proceedings o f the Royal
Society, this paper was the first to dem onstrate the transition from lam inar to tur­
bulent flow and to relate this transition to a critical value o f a dim ensionless
param eter— later to becom e know n as the Reynolds number. Reynolds studied
this phenom enon in w ater flow through pipes. His experim ental apparatus is
illustrated in Fig. 4.56, taken from his original 1883 paper. (N ote that before the
day o f m odern photographic techniques, som e technical papers contained rather
elegant hand sketches o f experim ental apparatus, o f which Fig. 4.56 is an exam ­
ple.) Reynolds filled a large reservoir with water, which fed into a glass pipe
through a larger bell-m outh entrance. As the w ater flowed through the pipe,
Reynolds introduced dye into the m iddle o f the stream , at the entrance o f the bell
m outh. W hat happened to this thin filam ent o f dye as it flowed through the pipe
is illustrated in Fig. 4.57, also from R eynolds’s original paper. The flow is from
right to left. If the flow velocity w ere sm all, the thin dye filament w ould travel
dow nstream in a sm ooth, neat, orderly fashion, with a clear dem arcation betw een
the dye and the rest o f the water, as illustrated in Fig. 4.57a. H ow ever, if the flow
v e lo c ity w e re in c r e a s e d b e y o n d a c e rta in v a lu e , th e d y e fila m e n t w o u ld s u d d e n ly
becom e unstable and w ould fill the entire pipe with color, as shown in Fig. 4.57b.
Reynolds clearly pointed out that the sm ooth dye filam ent in Fig. 4.57a corre­
sponded to lam inar flow in the pipe, w hereas the agitated and totally diffused dye
4 .2 5
Historical Note: Osborne Reynolds and His Number
Figure 4.56 Osboijne Reynolds’s apparatus for his famous pipe flow
experiments. This figure is from his original paper, referenced in the text.
(«)
(*)
to
Figure 4.57 Development o f turbulent flow in pipes, as
observed and sketched by Reynolds. This figure is from his
original paper, referenced in the text.
237
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4
Basic Aerodynamics
filam ent in Fig. 4.57b was due to turbulent flow in the pipe. Furtherm ore,
Reynolds studied the details o f this turbulent flow by visually observing the pipe
flow illum inated by a m om entary electric spark, m uch as w e would use a strobe
light today. H e saw that the turbulent flow consisted o f a large num ber o f distinct
eddies, as sketched in Fig. 4.57c. The transition from lam inar to turbulent flow
occurred when the param eter defined by p V D / p exceeded a certain critical
value, w here p was the density o f the water, V was the m ean flow velocity, ¡x was
the viscosity coefficient, and D was the diam eter o f the pipe. This dim ensionless
param eter, first introduced by Reynolds, later becam e known as the Reynolds
number. Reynolds m easured the critical value o f this num ber, above w hich tur­
bulent flow occurred, as 2300. This original work o f Reynolds initiated the study
o f transition from lam inar to turbulent flow as a new field o f research in fluid
dynam ics— a field that is still today one o f the m ost im portant and insufficiently
understood areas o f aerodynam ics.
Reynolds was a scholarly man with high standards. Engineering education
was new to English universities at that time, and Reynolds had definite ideas
about its proper form. He felt that all engineering students, no m atter w hat their
specialty, should have a com m on background based on m athem atics, physics,
and, in particular, the fundam entals o f classical m echanics. At M anchester, he
organized a system atic engineering curriculum covering the basics o f civil and
m echanical engineering. Ironically, despite his intense interest in education, as a
lecturer in the classroom R eynolds left som ething to be desired. His lectures
w ere hard to follow, and his topics frequently w andered w ith little or no connec­
tion. H e was know n to com e up w ith new ideas during the course o f a lecture and
to spend the rem ainder o f the lecture w orking out these ideas on the board, seem ­
ingly oblivious to the students in the classroom . That is, he did not “spoon-feed”
his students, and m any o f the poorer students did not pass his courses. In con­
trast, the best students enjoyed his lectures and found them stim ulating. M any o f
R eynolds’s successful students w ent on to becom e distinguished engineers and
scientists, the m ost notable being Sir J. J. Thom son, later the C avendish Profes­
sor o f Physics at Cam bridge; Thom son is fam ous for first dem onstrating the ex­
istence o f the electron in 1897, for which he received the N obel Prize in 1906.
In regard to R eynolds’s interesting research approach, his student colleague,
and friend Professor A. H. G ibson had this to say in his biography o f Reynolds,
written for the British Council in 1946:
Reynolds’ approach to a problem was essentially individualistic. He never began by
reading what others thought about the matter, but first thought this out for himself.
The novelty o f his approach to som e problems made som e o f his papers difficult to
follow, especially those written during his later years. His more descriptive physical
papers, however, make fascinating reading, and when addressing a popular audi­
ence, his talks were models o f clear exposition.
At the turn o f the century, R eynolds’s health began to fail, and he subse­
quently had to retire in 1905. The last years o f his life w ere ones o f considerably
dim inished physical and m ental capabilities, a particularly sad state for such a
brilliant and successful scholar. H e died at Som erset, England, in 1912. Sir Horace
4 .2 6
Historical Note: Prandtl and the Development of the Boundary Layer Concept
Lam b, one o f history’s m ost fam ous fluid dynam icists and a long-tim e colleague
o f Reynolds, w rote after R eynolds’s death:
The character o f Reynolds was, like his writings, strongly individual. He was con­
scious o f the value o f His work, but was content to leave it to the mature judgement
o f the scientific world. For advertisement he had no taste, and undue pretensions on
the part o f others only [elicited a tolerant sm ile. To his pupils he was most generous
in the opportunities for¡ valuable work which he put in their way, and in the share o f
co-operation. Somewhat reserved in serious or personal matters and occasionally
com bative and tenacious in debate, he was in the ordinary relations o f life the most
kindly and genial o f companions. He had a keen sense o f humor and delighted in
startling paradoxes, which he would maintain, half seriously and half playfully, with
astonishing ingenuity and resource. The illness which at length com pelled his retire­
ment was felt as a grievous calamity by his pupils, his colleagues and other friends
throughout the country¡
T he purpose o f this section has been to relate the historical beginnings o f the
Reynolds num ber in fluid m echanics. From now on, w hen you use the Reynolds
number, hopefully you will view it not only as a powerful dim ensionless para­
m eter governing viscous flow, but also as a testim onial to its originator— one o f
the fam ous fluid dynam icists o f the 19th century.
4.26 HISTORICAL NOTE: PRANDTL
AND THE DEVELOPMENT OF THE
BOUNDARY LAYER CONCEPT
The m odern science o f aerodynam ics has its roots as far back as Isaac Newton,
who devoted the entire second book o f his Principia (1687) to fluid dynam ics—
especially to the form ulation o f “laws o f resistance” (drag). He noted that drag is
a function o f fluid densit^, velocity, and shape o f the body in motion. However,
Newton was unable to form ulate the correct equation for drag. Indeed, he derived
a form ula that gave the drag on an inclined object as proportional to the sine
squared o f the angle o f attack. Later, N ew ton’s sine-squared law was used to
dem onstrate the “im possibility o f heavier-than-air flight” and served to hinder
the intellectual advancem ent o f flight in the 19th century. Ironically, the physical
assum ptions used by Ne\jvton in deriving his sine-squared law approxim ately re­
flect the conditions o f hypersonic flight, and the new tonian law has been used
since 1950 in the design o f high-M ach-num ber vehicles. However, N ew ton cor­
rectly reasoned the m echanism o f shear stress in a fluid. In section 9 o f book 2 o f
Principia, N ew ton states the follow ing hypothesis: “The resistance arising from
want o f lubricity in the parts o f a fluid is . . . proportional to the velocity with
which the parts o f the fluid are separated from each other.” This is the first state­
ment in history o f the friction law for lam inar flow; it is em bodied in Eq. (4.89),
which describes a “new tonian fluid.”
Further attem pts to understand fluid dynam ic drag were m ade by the French
m athem atician Jean le Rjond d ’A lem bert, who is noted for developing the cal­
culus o f partial differences (leading to the m athem atics o f partial differential
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Basic Aerodynamics
equations). In 1768, d ’A lem bert applied the equations o f motion for an incom ­
pressible, inviscid (frictionless) flow about a tw o-dim ensional body in a m oving
fluid and found that no drag is obtained. He wrote: “I do not see then, I admit,
how one can explain the resistance o f fluids by the theory in a satisfactory m an­
ner. It seem s to m e on the contrary that this theory, dealt with and studied with
profound attention gives, at least in m ost cases, resistance absolutely zero: a sin­
gular paradox w hich I leave to geom etricians to explain.” That this theoretical
result o f zero drag is truly a paradox was clearly recognized by d ’Alem bert, who
also conducted experim ental research on drag and who was am ong the first to
discover that drag was proportional to the square o f the velocity, as derived in
Sec. 5.3 and given in Eq. (5.18).
D ’A lem bert’s paradox arose due to the neglect o f friction in the classical the­
ory. It was not until a century later that the effect o f friction was properly incor­
porated in the classical equations o f motion by the work o f M. N avier
(1785-1836) and Sir G eorge Stokes (1819-1903). The so-called N avier-Stokes
equations stand today as the classical form ulation o f fluid dynam ics. H owever, in
general they are nonlinear equations and are extrem ely difficult to solve; indeed,
only with the num erical pow er o f m odern high-speed digital com puters are
“ex act” solutions o f the N avier-Stokes equations finally being obtained for gen­
eral flow fields. A lso in the 19th century, the first experim ents on transition from
lam inar to turbulent flow w ere carried out by O sborne Reynolds (1842-1912), as
related in Sec. 4.25. In his classic paper o f 1883 entitled “An Experim ental In­
vestigation o f the C ircum stances which D eterm ine w hether the M otion o f W ater
in Parallel Channels Shall Be D irect or Sinuous, and o f the Law o f R esistance in
Parallel C hannels,” Reynolds observed a filam ent o f colored dye in a pipe flow
and noted that transition from lam inar to turbulent flow alw ays corresponded to
approxim ately the sam e value o f a dim ensionless num ber p V D / n , w here D was
the diam eter o f the pipe. This was the origin o f the R eynolds number, defined in
Sec. 4.15 and discussed at length in Sec. 4.25.
T herefore, at the beginning o f the 20th century, when the W right brothers
w ere deeply involved in the developm ent o f the first successful airplane, the de­
velopm ent o f theoretical fluid dynam ics still had not led to practical results for
aerodynam ic drag. It was this environm ent into which Ludw ig Prandtl was born
on February 4, 1875, at Freising, in Bavaria, Germany. Prandtl was a genius who
had the talent o f cutting through a m aze o f com plex physical phenom ena to ex­
tract the m ost salient points and putting them in sim ple m athem atical form. E du­
cated as a physicist, Prandtl was appointed in 1904 as professor o f applied m e­
chanics at G ottingen U niversity in Germ any, a post he occupied until his death in
1953.
In the period from 1902 to 1904, Prandtl made one o f the m ost im portant
contributions to fluid dynam ics. Thinking about the viscous flow over a body, he
reasoned that the flow velocity right at the surface was zero and that if the
Reynolds num ber was high enough, the influence o f friction was lim ited to a thin
layer (Prandtl first called it a transition layer) near the surface. Therefore, the
analysis o f the flow field could be divided into two distinct regions— one close to
4 .2 6
Historical Note: Prandtl and the Development of the Boundary Layer Concept
the surface, which included friction, and the other farther away, in which friction
could be neglected. In one o f the most im portant fluid dynam ics papers in his­
tory, entitled “U ber Flu^sigkeitsbew egung bei sehr kleiner R eibung,” Prandtl
reported his thoughts tp the Third International M athem atical Congress at
Heidelberg in 1904. In this paper, Prandtl observed:
A very satisfactory explanation o f the physical process in the boundary layer
(Grenzschicht) between a fluid and a solid body could be obtained by the hypothesis
o f an adhesion o f the fluid to the walls, that is, by the hypothesis o f a zero relative
velocity between fluid and wall. If the viscosity is very small and the fluid path along
the wall not too long, the fluid velocity ought to resume its normal value at a very
short distance from the wall. In the thin transition layer however, the sharp changes
o f velocity, even with small coefficient o f friction, produce marked results.
In the sam e paper, Prandtl’s theory is applied to the prediction o f flow
separation:
In given cases, in certain points fully determined by external conditions, the fluid
flow ought to separate from the wall. That is, there ought to be a layer o f fluid which,
having been set in rotation by the friction on the wall, insinuates itself into the free
fluid, transforming com pletely the motion o f the latter. . . .
P randtl’s boundary layer hypothesis allow s the Navier-Stokes equations to be
reduced to a sim pler form ; by 1908, Prandtl and one o f his students, H. Blasius,
had solved these sim pler boundary layer equations for lam inar flow over a flat
plate, yielding the equations for boundary layer thickness and skin friction drag
given by Eqs. (4 .9 1) and ¡(4.93). Finally, after centuries o f effort, the first rational
resistance laws describing fluid dynam ic drag due to friction had been obtained.
P randtl’s w ork was a stroke o f genius, and it revolutionized theoretical aero­
dynam ics. However, possibly due to the language barrier, it only slowly diffused
through the w orldw ide technical com m unity. Serious work on boundary layer
theory did not em erge in E ngland and the U nited States until the 1920s. By that
time, Prandtl and his students at G ottingen had applied it to various aerodynam ic
shapes and w ere including the effects o f turbulence.
Prandtl has been called the fa th er o f aerodynamics, and rightly so. His con­
tributions extend far beyond boundary layer theory; for exam ple, he pioneered
the developm ent o f wing lift and drag theory, as seen in Chap. 5. M oreover, he
was interested in m ore fields than ju st fluid dynam ics— he made several im por­
tant contributions to structural m echanics as well.
As a note on Prandtl’s personal life, he had the singleness o f purpose that
seem s to drive m any giants o f humanity. However, his alm ost com plete preoccu­
pation with his w ork led to a som ew hat naive outlook on life. Theodore von
Karm an, one o f P randtl’s m ost illustrious students, relates that Prandtl would
rather find fancy in the exam ination o f children’s toys than participate in social
gatherings. W hen Prandtl was alm ost 40, he suddenly decided that it was time to
get m arried, and he wrote to a friend for the hand o f one o f his two daughters—
Prandtl did not care which one! D uring the 1930s and early 1940s, Prandtl had
mixed em otions about the political problem s o f the day. He continued his
241
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Basic Aerodynamics
research work at G ottingen under H itler’s Nazi regim e but becam e continually
confused about the course o f events. Von Karman writes about Prandtl in his
autobiography:
I saw Prandtl once again for the last time right after the Nazi surrender. He was a sad
figure. The roof o f his house in Gottingen, he mourned, had been destroyed by an
American bomb. He couldn’t understand why this had been done to him! He was
also deeply shaken by the collapse o f Germany. He lived only a few years after that,
and though he did engage in som e research work in meteorology, he died, I believe,
a broken man, still puzzled by the ways o f mankind.
Prandtl died in G ottingen on A ugust 15, 1953. O f any fluid dynam icist or
aerodynam icist in history, Prandtl cam e closest to deserving a Nobel Prize. W hy
he never received one is an unansw ered question. However, as long as there are
flight vehicles, and as long as students study the discipline o f fluid dynam ics, the
nam e o f Ludw ig Prandtl will be enshrined for posterity.
4.27 Summary
N ow that you have finished this chapter, return to our road map in Fig. 4.1, and run your
mind over all the items shown there. Make yourself feel comfortable with these items.
Then proceed with this chapter summary, putting each equation and each concept in its
proper perspective relative to our road map.
A few o f the important concepts from this chapter are summarized as follow s:
1.
The basic equations o f aerodynamics, in the form derived here, are as follow s:
p \A \ V\ = piA^Vi
Continuity
d p = —p V d V
Momentum
cpT¡ + ± V,2 = cpT2 + \ V¡
Energy
(4.2)
(4.8)
(4.42)
These equations hold for a compressible flow. For an incom pressible flow, we have
these:
2.
Continuity
A\V\ = A2V2
Momentum
p\ + p — = P2 + p —
V?
(4.3)
V}
Equation (4.9a) is called Bernoulli’s equation.
The change in pressure, density, and temperature between two points in an
isentropic process is given by
yHy -
p1
T j
1)
(4.9a)
4 .2 7
Summary
T he sp eed o f sound is giv en by
-
(4.48)
V (\ d¥p )J iisentropic
For a perfect gas, this becom es
a = s fy R T
(4.54)
The speed o f a gas flow can be measured by a Pitot tube, which senses the total
pressure po. For incompressible flow,
V, =
2(po - P i)
(4.66)
For subsonic compressible flow,
(y-n/Y
2a\_
Yi =
Y —1
- I
'(£ )
(4.77a)
For supersonic flow, a shock wave exists in front o f the Pitot tube, and Eq. (4.79)
must be used in lieu o f Eq. (4.77a) to find the Mach number o f the flow.
The area-velocity relation for isentropic flow is
(4.83)
From this relation, we observe that (1) for a subsonic flow, the velocity increases
in a convergent duct and decreases in a divergent duct, (2) for a supersonic flow,
the velocity increases in a divergent duct and decreases in a convergent duct; and
(3) the flow is sonic only at the minimum area.
The isentropic flow o f &gas is governed by
.
Pa
y/<y—i)
(4.74)
Pi
—
7*1
=
1+
Y
- 1
M
(4.73)
(4.75)
Here, 7o, po, and po are the total temperature, pressure, and density, respectively.
For an isentropic flow, po = constant throughout the flow. Similarly, po = constant
and 7b = constant throughout the flow.
V iscous effects create a boundary layer along a solid surface in a flow. In this
boundary layer, the flow m oves slow ly and the velocity goes to zero right at the
surface. The shear stress at the wall is given by
244
chapter
8.
4 Basic Aerodynamics
The shear stress is larger for a turbulent boundary layer than for a laminar boundary
layer.
For a laminar incompressible boundary layer, on a flat plate,
5.2*
s =
and
C, =
<4'91>
1.328
,4.98)
where S is the boundary layer thickness, C / is the total skin friction drag
coefficient, and Re is the Reynolds number;
Re, =
°°
local Reynolds number
Moo
Re¿ = ?°° 00
Moo
9.
plate Reynolds number
Here, x is the running length along the plate, and L is the total length of the plate.
For a turbulent incompressible boundary layer on a flat plate,
0.37x
* = TTToT
Re?
(4.99)
0.074
C/ = ü f
(4J01)
Any real flow along a surface first starts out as laminar but then changes into a
turbulent flow. The point where this transition effectively occurs (in reality,
transition occurs over a finite length) is designated xcr. In turn, the critical Reynolds
number for transition is defined as
Rex = PooVoo*cr
(4|04)
Moo
10. Whenever a boundary layer encounters an adverse pressure gradient (a region of
increasing pressure in the flow direction), it can readily separate from the surface.
On an airfoil or wing, such flow separation decreases the lift and increases the drag.
Bibliography
Airey, J.: “Notes on the Pitot Tube,” Engineering News, vol. 69, no. 16, April 17, 1913,
pp. 782-783.
Anderson, J. D., Jr.: Fundamentals of Aerodynamics, 3d ed., McGraw-Hill, New York,
2001.
Problems
Anderson, J. D., Jr.: A History of Aerodynamics and Its Impact on Flying Machines,
Cambridge University Press, New York, 1998.
Goin, K. L.: “The History, Evolution, and Use of Wind Tunnels,” AlAA Student
Journal, February 1971, pp. 3-13.
Guy, A. E.: “Origin and Theory of the Pitot Tube,” Engineering News, vol. 69, no. 23,
June 5, 1913, pp. 1172-1175.
Kuethe, A. M.. and C. Y. Chow: Foundations of Aerodynamics, 3d ed., Wiley, New
York, 1976.
Pope, A: Aerodynamics of Supersonic Flight, Pitman, New York, 1958.
von Karman, T.: Aerodynamics, McGraw-Hill, New York, 1963.
Problems
4.1
4.2
4.3
4.4
4.5
4.6
4.7
Consider the incompressible flow of water through a divergent duct. The inlet
velocity and area are 5 ft/s and 10 ft2, respectively. If the exit area is 4 times the
inlet area, calculate the water flow velocity at the exit.
In P ro b . 4 .1 , c a lc ú la te th e p r e s s u r e d if f e r e n c e b e tw e e n th e e x it a n d th e in le t. T h e
density o f water is 62.4 l b ,„ / f t 3 .
Consider an airplane flying with a velocity of 60 m/s at a standard altitude of
3 km. At a point on the wing, the airflow velocity is 70 m/s. Calculate the pressure
at this point. Assume incompressible flow.
An instrument used to measure the airspeed on many early low-speed airplanes,
principally during 1919 to 1930, was the venturi tube. This simple device is a
convergent-divergent duct. (The front section’s cross-sectional area A decreases
in the flow direction, and the back section’s cross-sectional area increases in the
flow direction. Somewhere in between the inlet and exit of the duct, there is a
minimum area, called the throat.) Let <4i and Ai denote the inlet and throat areas,
respectively. Let p\ and /;2 be the pressures at the inlet and throat, respectively.
The venturi tube is mounted at a specific location on the airplane (generally on the
wing or near the front of the fuselage), where the inlet velocity V| is essentially
the same as the freestream velocity, that is, the velocity of the airplane through the
air. With a knowledge of the area ratio A 2 /A \ (a fixed design feature) and a
measurement of the pressure difference p\ — P2 , the airplane’s velocity can be
determined. For example, assume A2 /A \ — j and p\ — P2 = 80 lb/ft2. If the
airplane is flying at standard sea level, what is its velocity?
Consider the flow of air through a convergent-divergent duct, such as the venturi
described in Prob. 4.4. The inlet, throat, and exit areas are 3, 1.5, and 2 m2,
respectively. The inlet and exit pressures are 1.02 x 105 and 1.00 x 105 N /nr,
respectively. Calculate the flow velocity at the throat. Assume incompressible
flow with standard sea-level density.
An airplane is flying at a velocity of 130 mi/h at a standard altitude of 5000 ft. At
a point on the wing, the pressure is 1750.0 lb/ft2. Calculate the velocity at that
point, assuming incompressible flow.
Imagine that you have designed a low-speed airplane with a maximum velocity at
sea level of 90 m/s. For your airspeed instrument, you plan to use a venturi tube
with a 1.3 : 1 area ratio. Inside the cockpit is an airspeed indicator—a dial that is
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CHAPTER 4
4.8
4.9
4.10
4.11
4.12
4.13
4.14
4.15
4.16
4.17
4.18
4.19
Basic Aerodynamics
connected to a pressure gauge sensing the venturi tube pressure difference
p\ — p2 and properly calibrated in terms of velocity. What is the maximum
pressure difference you would expect the gauge to experience?
A supersonic nozzle is also a convergent-divergent duct, which is fed by a large
reservoir at the inlet to the nozzle. In the reservoir of the nozzle, the pressure and
temperature are 10 atm and 300 K, respectively. At the nozzle exit, the pressure is
1 atm. Calculate the temperature and density of the flow at the exit. Assume the
flow is isentropic and, of course, compressible.
Derive an expression for the exit velocity of a supersonic nozzle in terms of the
pressure ratio between the reservoir and exit pn/ p,, and the reservoir temperature Tn.
Consider an airplane flying at a standard altitude of 5 km with a velocity of
270 m/s. At a point on the wing of the airplane, the velocity is 330 m/s. Calculate
the pressure at this point.
The mass flow of air through a supersonic nozzle is 1.5 lb,,,/s. The exit velocity is
1500 ft/s, and the reservoir temperature and pressure are 1000°R and 7 atm, respec­
tively. Calculate the area of the nozzle exit. For air, cp = 6000 ft • lb/(slug)(°R).
A supersonic transport is flying at a velocity of 1500 mi/h at a standard altitude of
50,000 ft. The temperature at a point in the flow over the wing is 793.32°R.
Calculate the flow velocity at that point.
For the airplane in Prob. 4.12, the total cross-sectional area of the inlet to the jet
engines is 20 ft2. Assume that the flow properties of the air entering the inlet are
those of the free stream ahead of the airplane. Fuel is injected inside the engine at
a rate of 0.05 lb of fuel for every pound of air flowing through the engine (i.e., the
fuel-air ratio by mass is 0.05). Calculate the mass flow (in slugs/per second) that
comes out the exit of the engine.
Calculate the Mach number at the exit of the nozzle in Prob. 4 . 11.
A Boeing 747 is cruising at a velocity of 250 m/s at a standard altitude of 13 km.
What is its Mach number?
A high-speed missile is traveling at Mach 3 at standard sea level. What is its
velocity in miles per hour?
Calculate the flight Mach number for the supersonic transport in Prob. 4.12.
Consider a low-speed subsonic wind tunnel with a nozzle contraction ratio of
1 : 20. One side of a mercury manometer is connected to the settling chamber, and
the other side to the test section. The pressure and temperature in the test section
are 1 atm and 300 K, respectively. What is the height difference between the two
columns of mercury when the test section velocity is 80 m/s?
We wish to operate a low-speed subsonic wind tunnel so that the flow in the test
section has a velocity of 200 mi/h at standard sea-level conditions. Consider two
different types of wind tunnels: (a) a nozzle and a constant-area test section,
where the flow at the exit of the test section simply dumps out to the surrounding
atmosphere; that is, there is no diffuser, and (b) a conventional arrangement of
nozzle, test section, and diffuser, where the flow at the exit of the diffuser dumps
out to the surrounding atmosphere. For both wind tunnels (a) and (b) calculate the
pressure differences across the entire wind tunnel required to operate them so as
to have the given flow conditions in the test section. For tunnel (a), the crosssectional area of the entrance is 20 ft2, and the cross-sectional area of the test
Problems
4.20
4.21
4.22
4.23
4.24
4.25
4.26
4.27
4.28
4.29
section is 4 ft2. For tunnel (ft), a diffuser is added to (a) with a diffuser area of
18 ft2. After completing your calculations, examine and compare your answers for
tunnels (a) and (b). Which requires the smaller overall pressure difference? What
does this say about the value of a diffuser on a subsonic wind tunnel?
A Pitot tube is mounted in the test section of a low-speed subsonic wind tunnel.
The flow in the test section has a velocity, static pressure, and temperature of
150 mi/h, 1 atm, and 70°F, respectively. Calculate the pressure measured by the
Pitot tube.
The altimeter on a low-speed Piper Aztec reads 8000 ft. A Pitot tube mounted on
the wing tip measures; a pressure of 1650 lb/ft2. If the outside air temperature is
500° R, what is the true velocity of the airplane? What is the equivalent airspeed?
The altimeter on a low-speed airplane reads 2 km. The airspeed indicator reads
50 m/s. If the outside air temperature is 280 K, what is the true velocity of the
airplane?
A Pitot tube is mounted in the test section of a high-speed subsonic wind tunnel.
The pressure and temperature of the airflow are 1 atm and 270 K, respectively. If
the flow velocity is 250 m/s, what is the pressure measured by the Pitot tube?
A high-speed subsonic Boeing 777 airliner is flying at a pressure altitude of
12 km. A Pitot tube on the vertical tail measures a pressure of 2.96 x 104 N/m2.
At what Mach number is the airplane flying?
A high-speed subsonic airplane is flying at Mach 0.65. A Pitot tube on the wing tip
measures a pressure of 2339 lb/ft2. What is the altitude reading on the altimeter?
A high-performance F-16 fighter is flying at Mach 0.96 at sea level. What is the
air temperature at the ¡stagnation point at the leading edge of the wing?
An airplane is flying at a pressure altitude of 10 km with a velocity of 596 m/s.
The outside air temperature is 220 K. What is the pressure measured by a Pitot
tube mounted on the nose of the airplane?
The dynamic pressure is defined as q = 0.5p V 2. For high-speed flows, where
Mach number is used frequently, it is convenient to express q in terms of
pressure p and Mach number M rather than p and V . Derive an equation for
q - q ( p , M).
After completing its rpission in orbit around the earth, the Space Shuttle enters the
earth’s atmosphere at very high Mach number and, under the influence of
aerodynamic drag, slóws as it penetrates more deeply into the atmosphere. (These
matters are discussed in Chap. 8 .) During its atmospheric entry, assume that the
shuttle is flying at Mach number M corresponding to the altitudes h:
h, km
60
50
40
M
17
9.5
5.5
30
20
1
Calculate the corresponding values of the free-stream dynamic pressure at each one
of these flight path points. Suggestion: Use the result from Prob. 4.28. Examine and
comment on the variation of qoo as the shuttle enters the atmosphere.
4.30 Consider a Mach 2 airstream at standard sea-level conditions. Calculate the total
pressure of this flow. Compare this result with (a) the stagnation pressure that
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chapter
4.31
4.32
4.33
4.34
4.35
4.36
4.37
4.38
4.39
4
Basic Aerodynamics
would exist at the nose of a blunt body in the flow and (b) the erroneous result
given by Bernoulli’s equation, which, of course, does not apply here.
Consider the flow of air through a supersonic nozzle. The reservoir pressure and
temperature are 5 atm and 500 K, respectively. If the Mach number at the nozzle
exit is 3, calculate the exit pressure, temperature, and density.
Consider a supersonic nozzle across which the pressure ratio is p j p o = 0.2.
Calculate the ratio of exit area to throat area.
Consider the expansion of air through a convergent-divergent supersonic nozzle. The
Mach number varies from essentially zero in the reservoir to Mach 2.0 at the exit.
Plot on graph paper the variation of the ratio of dynamic pressure to total pressure
as a function of Mach number; that is, plot q/pn versus M from M = 0 to M = 2.0.
The wing of the Fairchild Republic A-10A twin-jet close-support airplane is
approximately rectangular with a wingspan (the length perpendicular to the flow
direction) of 17.5 m and a chord (the length parallel to the flow direction) of 3 m.
The airplane is flying at standard sea level with a velocity of 200 m/s. If the flow
is considered to be completely laminar, calculate the boundary layer thickness at
the trailing edge and the total skin friction drag. Assume the wing is approximated
by a flat plate. Assume incompressible flow.
In Prob. 4.34, assume the flow is completely turbulent. Calculate the boundary
layer thickness at the trailing edge and the total skin friction drag. Compare these
turbulent results with the laminar results from Prob. 4.34.
If the critical Reynolds number for transition is 106, calculate the skin friction
drag for the wing in Prob. 4.34.
Let us reflect back to the fundamental equations of fluid motion dicussed in the
early sections of this chapter. Sometimes these equations were expressed in terms
of differential equations, but for the most part, we obtained algebraic relations by
integrating the differential equations. However, it is useful to think of the
differential forms as relations that govern the change in flow field variables in an
infinitesimally small region around a point in the flow, (a) Consider a point in an
inviscid flow, where the local density is 1.1 kg/m3. As a fluid element sweeps
through this point, it is experiencing a spatial change in velocity of 2 percent per
millimeter. Calculate the corresponding spatial change in pressure per millimeter
at this point if the velocity at the point is 100 m/s. (b) Repeat the calculation for
the case when the velocity at the point is 1000 m/s. What can you conclude by
comparing your results for the low-speed flow in part (a) with the results for the
high-speed flow part (b).
The type of calculation in Prob. 4.3 is a classic one for low-speed, incompressible
flow, that is, given the free-stream pressure and velocity and the velocity at some
other point in the flow, calculate the pressure at that point. In a high-speed
compressible flow, Mach number is more fundamental than velocity. Consider an
airplane flying at Mach 0.7 at a standard altitude of 3 km. At a point on the wing,
the airflow Mach number is 1.1. Calculate the pressure at this point. Assume an
isentropic flow.
Consider an airplane flying at a standard altitude of 25,000 ft at a velocity of
800 ft/sec. To experience the same dynamic pressure at sea level, how fast must
the airplane be flying?
Problems
4.40 In Section 4.9, we defined hypersonic flow as that flow where the Mach number is
4.41
4.42
4.43
4.44
five or greater. Wind tunnels with a test section Mach number of five or greater
are called hypersonic wind tunnels. From Eq. (4.88), the exit-to-throat area ratio
for supersonic exit Mach numbers increases as the exit Mach number increases.
For hypersonic Mach numbers, the exit-to-throat ratio becomes extremely large,
so hypersonic wind tunnels are designed with long, high-expansion ratio nozzles.
In this and the following problems, let us examine some special characteristics of
hypersonic wind tunnels. Assume we wish to design a Mach 10 hypersonic wind
tunnel using air as the test medium. We want the static pressure and temperature
in the test stream to be that for a standard altitude of 55 km. Calculate: (a) the
exit-to-throat area ratio, (b) the required reservoir pressure (in atm), and (c) the
required reservoir temperature. Examine these results. What do they tell you
about the special (and sometimes severe) operating requirements for a hypersonic
wind tunnel.
Calculate the exit velocity of the hypersonic tunnel in Problem 4.40.
Let us double the exit Mach number of the tunnel in Problem 4.40 simply by
adding a longer nozzle section with the requisite expansion ratio. Keep the
reservoir properties the same as those in Problem 4.40. Then we have a Mach 20
wind tunnel, with test section pressure and temperature considerably lower than in
Problem 4.40; that is, the test section flow no longer corresponds to conditions at
a standard altitude of 55 km. Be that as it may, we have at least doubled the Mach
number of the tunnel. Calculate (a) the exit-to-throat area ratio of the Mach 20
nozzle, (b) the exit velocity. Compare these values with those for the Mach 10
tunnel in Probs. 4.40 and 4.41. What can you say about the differences? In
particular, note the exit velocities for the Mach 10 and Mach 20 tunnels. You will
see that they are not much different. What is then giving the big increase in exit
Mach number?
The results of Example 4.4 showed that the aerodynamic force on a body is
proportional to the square of the free-stream velocity. This is strictly true,
however, only when the aerodynamic force is due to the pressure exerted on the
surface and when the flow is incompressible. When the aerodynamic force is also
due to the distribution of frictional shear stress over the surface and/or the flow is
compressible, the “velocity squared” law does not strictly hold. The purpose of
this problem is to examine how the friction drag on a body varies with free-stream
velocity for an incompressible flow.
Consider a square flat plate at zero incidence angle to a low-speed
incompressible flow. The length of each side is 4 m. Assume that the transition
Reynolds number is 5 x 105 and that the free-stream properties are those at
standard sea level. Calculate the friction drag on the flat plate when the freestream velocity is (a) 20 m/s and when it is (b) 40 m/s. (c) Assuming the friction
drag, Df, varies with velocity as V^, calculate the value of the exponent n based
on the answers from (a) and (b). How close does n come to 2; that is, how close is
the friction drag to obeying the velocity squared law?
Consider the incompressible viscous flow over a flat plate. Following the
theme set in Prob. 4.43, show analytically that (a) for fully turbulent flow, skin
friction drag varies as V¿8, and (b) for fully laminar flow, skin friction drag
varies as V' 5.
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Basic Aerodynamics
4.45 Consider compressible viscous flow over the same flat plate as in Prob. 4.43.
Assume a completely turbulent boundary layer on the plate. The free-stream
properties are those at standard sea level. Calculate the friction drag on the
flat plate when (a)
= 1, and (b)
= 3 .(c) Assuming the friction drag,
D f , varies with velocity as VJJ,, calculate the value of the exponent n based on
the answers from (a) and (b). Note: This problem examines the combined effect
of compressibility and friction on the “velocity squared law,” in the same spirit
that Probs. 4.43 and 4.44 isolated just the effect of friction in an incompressible
flow.
C
H
A
P
T
E
R
5
Airfoils, Wings, and Other
Aerodynamic Shapes
There can be no doubt that the inclined plane is the true principle of aerial navigation
by mechanical means.
Sir George Cayley, 1843
5.1 INTRODUCTION
It is rem arkable that the m odern airplane as we know it today, with its fixed wing
and vertical and horizontal tail surfaces, was first conceived by G eorge Cayley
in 1799, more than 200 years ago. He inscribed his first concept on a silver
disk (presum ably for perm ancnce), shown in Fig. 1.5. It is also rem arkable that
C ayley recognized that a curved surface (as shown on the silver disk) creates
more lift than a flat surface. C ay ley ’s fixed-w ing concept was a true revolution in
the developm ent o f heavier-than-air flight m achines. Prior to his time, aviation
enthusiasts had been doing their best to im itate m echanically the natural flight
o f birds, which led to a series o f hum an-pow ered flapping-w ing designs (ornithopters), which never had any real possibility o f working. In fact, even
Leonardo da Vinci devoted a considerable effort to the design o f many types of
ornithopters in the late 15th century, o f course, to no avail. In such ornithopter
designs, the flapping o f the wings was supposed to provide sim ultaneously both
lift (to sustain the m achine in the air) and propulsion (to push it along in flight).
C ayley is responsible for directing people’s m inds away from im itating bird
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CHAPTER 5
Airfoils, Wings, and Other Aerodynamic Shapes
PREVIEW BOX
This chapter deals with lift and drag on aerody­
namic bodies, principally airfoil shapes and wings.
These are real aerospace engineering applications—
applications that extend the basic material from
Chaps. 1 to 4 well into the practical engineering
world. In this chapter, you will learn
1.
2.
3.
4.
5.
6.
7.
How to calculate lift and drag on airfoil shapes.
How to calculate lift and drag on a whole wing
of an airplane.
Why lift and drag for a wing are different
values from that for the airfoil shape that makes
up the wing.
What happens to lift and drag when an airfoil or
a wing flys near or beyond the speed of sound.
Why some airplanes have swept wings and
others have straight wings.
Why some airplanes have thin airfoils and
others have thick airfoils.
Why optimum wing shapes for supersonic
flight are different than for subsonic flight.
This is all good stuff—some of the bread and butter
of aerospace engineering. You will learn all this,
and more, in this chapter. For example, at the
Smithsonian’s National Air and Space Museum, this
author is frequently asked by visitors how a wing
produces lift—-a natural and perfectly innocent
question. Unfortunately, there is no satisfactory oneliner for an answer. Even a single paragraph does
not suffice. After a hundred years since the Wright
Flyer, different people take different points of view
about what is the most fundamental mechanism that
produces lift, some pressing their views with almost
religious fervor. There is a whole section of this
chapter (Sec. 5.19) that addresses how lift is pro­
duced, what this author considers to be the most
fundamental explanation, and how it relates to alter­
nate explanations.
With this chapter, you will begin to concentrate
on airplanes, winged space vehicles such as the
Space Shuttle, and any vehicle that flies through the
atmosphere. This chapter greatly accelerates our
introduction to flight. Hang on, and enjoy the ride.
flight and for separating the two principles o f lift and propulsion. H e proposed
and dem onstrated that lift can be obtained from a fixed, straight w ing inclined to
the airstream , w hile propulsion can be provided by som e independent m echa­
nism such as paddles or airscrews. For this concept and for his m any other
thoughts and inventions in aeronautics, Sir G eorge Cayley is called the parent of
m odern aviation. A m ore detailed discussion o f C ayley’s contributions is given
in Chap. 1. H ow ever, we em phasize that m uch o f the technology discussed in the
present chapter had its origins at the beginning o f the 19th century— technology
that cam e to fruition on D ecem ber 17, 1903, near Kitty Hawk, North Carolina.
The follow ing sections develop som e o f the term inology and basic aerody­
nam ic fundam entals o f airfoils and wings. T hese concepts form the heart o f air­
plane flight, and they represent a m ajor excursion into aeronautical engineering.
The road m ap for this chapter is shown in Fig. 5.1. There are basically three main
topics in Chap. 5, each having to do with the aerodynam ic characteristics o f a
class o f geom etric shapes— airfoils, w ings, and general body shapes. These three
topics are shown in the three boxes at the top o f our road map. We first exam ine
the aerodynam ic characteristics o f airfoils and then run dow n the various aspects
noted in the left-hand colum n in Fig. 5.1. This is a long list, but we will find that
m any thoughts on this list carry over to wings and bodies as well. We then move
5 .2
Airfoil Nomenclature
Aerodynamic shapes
Airfoils
Wings
Nomenclature
Aerodynamic coefficients
Bodies
Induced drag
- Change in lift slope
Experimental data
Swept wings
O btaining lift coefficient
from pressure coefficient
Flaps
Cylinders
- Spheres
Com pressibility corrections
Transonic speeds
a. Critical Mach num ber
b. Drag-divergence
Mach number
Supersonic speeds
a. Lift
b. Wave drag
How lift is p r o d u c e d -------- 1
Figure 5.1 Road map for Chap. 5.
to the central colum n for a discussion o f finite wings, and w e will see how the
aerodynam ics o f a w ing differs from that o f an airfoil. Both airfoils and wings
can be classified as slender bodies. In contrast, the third colum n in Fig. 5.1 deals
with a few exam ples o f blunt bodies, namely, cylinders and spheres. We define
and exam ine the distinctions betw een slender and blunt aerodynam ic shapes. Fi­
nally, we discuss how aerodynam ic lift is produced. A lthough we have alluded to
this in previous chapters, it is appropriate at the end o f the chapter dealing with
the aerodynam ics o f various shapes to have a definitive discussion on how nature
generates lift. Various physical explanations have been used in the past to explain
how lift is generated, and there have been m any spirited discussions in the liter­
ature as to which is proper, or m ore fundam ental. We attem pt to put all these
views in perspective at the end o f this chapter, as represented by the box at the
bottom o f Fig. 5.1. As you progress through this chapter, m ake certain to touch
base frequently with our road map so that you can see how the details o f our dis­
cussions fit into the grand schem e laid out in Fig. 5.1.
5.2 AIRFOIL NOMENCLATURE
C onsider the w ing o f an airplane, as sketched in Fig. 5.2. The cross-sectional
shape obtained by the intersection o f the wing with the perpendicular plane
shown in Fig. 5.2 is called an airfoil. Such an airfoil is sketched in Fig. 5.3,
which illustrates som e basic term inology. The m ajor design feature o f an airfoil
is the mean cam ber line, which is the locus o f points halfway between the upper
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chapter
5
Airfoils, Wings, and Other Aerodynamic Shapes
T hickness
and low er surfaces, as m easured perpendicular to the mean cam ber line itself.
T he m ost forw ard and rearw ard points o f the mean cam ber line are the leading
and trailing edges, respectively. The straight line connecting the leading and
trailing edges is the chord line o f the airfoil, and the precise distance from the
leading to the trailing edge m easured along the chord line is sim ply designated
the chord o f the airfoil, given by the sym bol c. The cam ber is the m axim um dis­
tance betw een the mean cam ber line and the chord line, m easured perpendicular
to the chord line. The cam ber, the shape o f the mean cam ber line, and to a lesser
extent, the thickness distribution o f the airfoil essentially control the lift and m o­
m ent characteristics o f the airfoil.
M ore definitions are illustrated in Fig. 5.4a, which shows an airfoil inclined
to a stream o f air. The free-stream velocity V», is the velocity o f the air far up­
stream o f the airfoil. The direction o f
is defined as the relative wind. The
5 .2
Airfoil Nomenclature
Figure 5.4 Sketch showing the definitions of (a) lift, drag, moments, angle o f attack, and
relative wind; (b ) normal and axial force.
angle betw een the relative wind and the chord line is the angle o f attack a o f the
airfoil. As described in Chaps. 2 and 4, there is an aerodynam ic force created by
the pressure and shear stress distributions over the w ing surface. This resultant
force is shown by the vector R in Fig. 5.4a. In turn, the aerodynam ic force R can
be resolved into tw o forces, parallel and perpendicular to the relative wind. The
drag D is alw ays defined as the com ponent o f the aerodynam ic force parallel to
the relative wind. T he lift L is alw ays defined as the com ponent o f the aerody­
nam ic force perpendicular to the relative wind.
In addition to lift and drag, the surface pressure and shear stress distributions
create a moment M w hich tends to rotate the wing. To see more clearly how this
m om ent is created, consider the surface pressure distribution over an airfoil, as
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chapter
5
Airfoils, Wings, and Other Aerodynamic Shapes
the m inimum pressure on the airfoil.
Figure 5.5 The physical origin of moments on an airfoil.
sketched in Fig. 5.5 (we will ignore the shear stress for this discussion). C onsider
ju st the pressure on the top surface o f the airfoil. This pressure gives rise to a net
force F\ in the general dow nw ard direction. M oreover, F\ acts through a given
point on the chord line, point 1, which can be found by integrating the pressure
tim es distance over the surface (analogous to finding the centroid or center o f
pressure from integral calculus). Now consider just the pressure on the bottom
surface o f the airfoil. This pressure gives rise to a net force Fj in the general up­
w ard direction, acting through point 2. The total aerodynam ic force on the airfoil
is the summation o f F\ and F2, and lift is obtained w hen F2 > F\. H owever, note
from Fig. 5.5 that F\ and F2 will create a m om ent that will tend to rotate the air­
foil. M oreover, the value o f this aerodynam ically induced m om ent depends on
the point about which we choose to take m om ents. For exam ple, if we take m o­
m ents about the leading edge, the aerodynam ic m om ent is designated A/LE. It is
m ore com m on in the case o f subsonic airfoils to take m om ents about a point on
the chord at a distance c / 4 from the leading edge, the quarter-chordpoint, as il­
lustrated in Fig. 5.4a. This m om ent about the quarter chord is designated Af(./4.
In general,
^ Mc/A. Intuition will tell you that lift, drag, and m om ents on a
w ing will change as the angle o f attack a changes. In fact, the variations o f these
aerodynam ic quantities w ith a represent som e o f the m ost im portant inform ation
that an airplane designer needs to know. We will address this m atter in the fol­
low ing sections. H ow ever, we point out that although A /Le and M c/4 are both
functions o f a , there exists a certain point on the airfoil about which m om ents es­
sentially do not vary with a . This point is defined as the aerodynam ic center, and
the m om ent about the aerodynam ic center is designated A/ac. By definition,
Mac = const
5 .3
Lift, Drag, and Moment Coefficients
independent o f the angle o f attack. The location of the aerodynam ic center for
real aerodynam ic shapes can be found from experim ent. For low -speed subsonic
airfoils, the aerodynam ic center is generally very close to the quarter-chord point.
R eturning to Fig. 5.4a, we recall that the resultant aerodynam ic force R can
be resolved into com ponents perpendicular and parallel to the relative wind,
namely, the lift and drag, respectively. An alternative to this system is to resolve
R into com ponents perpendicular and parallel to the chord line, as shown in
Fig. 5 Ah. These com ponents are called the normal force and axial force and are
denoted by N and A, respectively, in Fig. 5 Ah, shown by the heavy solid arrows.
A lso shown in Fig. 5 Ah are the lift and drag, L and D , respectively, represented
by the heavy dashed arrow s. Lift and drag are easily expressed in term s o f N and
A from the geom etry show n in Fig. 5 Ah, as follows:
L = N cos a — A sin a
(5.1)
D = N sin a 4 - A cos a
(5.2)
For airfoils and wings, the use o f N and A to describe the aerodynam ic force
dates back as early as the w ork o f O tto Lilienthal in 1889, as published in his
book Bird Flight as the Basis o f Aviation (see Sec. 1.5). Indeed, the famous
“Lilienthal tables,” which w ere used by the W right brothers to design their early
gliders (see Sec. 1.8), were tables dealing with norm al and axial forces.
The W rights preferred to think in terms o f lift and drag, and they converted
L ilienthal’s results by using Eqs. (5.1) and (5.2). Today, the use o f N and A to
describe the aerodynam ic force on airfoils and w ings is generally passé; L and D
are alm ost alw ays the system used by choice. However, N and A are still fre­
quently used to denote the aerodynam ic force on bodies o f revolution, such as
m issiles and projectiles. Therefore, it is useful to be fam iliar with both systems
of expressing the aerodynam ic force on a body.
5.3 LIFT, DRAG, AND MOMENT COEFFICIENTS
Again appealing to intuition, we note that it m akes sense that for an airplane in
flight, the actual m agnitudes o f L, D , and M depend not only on or, but also on
velocity and altitude. In fact, w e can expect that the variations o f L , D, and M
depend at least on
1.
Free-stream velocity V,*,.
2.
Free-stream density p ^ , that is, altitude.
3.
Size o f the aerodynam ic surface. For airplanes, we will use the wing area S
to indicate size.
4.
A ngle o f attack a .
5.
Shape o f the airfoil.
6.
Viscosity coefficient ¡loo (because the aerodynam ic forces are generated in
part from skin friction distributions).
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CHA PT E R 5
7.
Airfoils, Wings, and Other Aerodynamic Shapes
C om pressibility o f the airflow. In Chap. 4 w e dem onstrated that
com pressibility effects are governed by the value o f the free-stream Mach
as
num ber M <*, = Voo/a,». Since V,» is already listed, w e can designate
our index for com pressibility.
H ence, we can w rite that, for a given shape airfoil at a given angle o f attack,
L = f (Voo* Poo» S* Moo» ^oo)
(5.3)
and D and M are sim ilar functions.
In principle, for a given airfoil at a given angle o f attack, we could find the
variation o f L by perform ing m yriad wind tunnel experim ents w herein V ^, Poo.
S, fXoo, and ax are individually varied, and then we could try to m ake sense out
o f the resulting huge collection o f data. This is the hard way. Instead, we ask the
question, Are there groupings o f the quantities Voo. P o o > S, /¿oo> «oo. and ¿ such
that Eq. (5.3) can be w ritten in term s o f few er param eters? The answ er is yes. In
the process o f developing this answer, w e will gain som e insight into the beauty
o f nature as applied to aerodynam ics.
The technique w e will apply is a sim ple exam ple o f a m ore general theoret­
ical approach called dim ensional analysis. Let us assum e that Eq. (5.3) is o f the
functional form
L = Z V a00p h
00Sda*00n {0
(5.4)
w here Z , a, b, d , e, and / are dim ensionless constants. H ow ever, no m atter what
the values o f these constants may be, it is a physical fact that the dim ensions o f
the left- and right-hand sides o f Eq. (5.4) m ust m atch; that is, if L is a force (say,
in new tons), then the net result o f all the exponents and m ultiplication on the
right-hand side m ust also produce a result with the dim ensions o f a force. This
constraint will ultim ately give us inform ation on the values o f a, b, etc. If we des­
ignate the basic dim ensions o f mass, length, and tim e by m, I, and t, respectively,
then the dim ensions o f various physical quantities are as given in the following.
Physical Quantity
Dimensions
L
m l / t 2 (from Newton’s second law)
Voo
s
1,1 ,
m /l3
I2
Qoo
lit
Moo
m H.lt)
Poo
Thus, equating the dimensions o f the left- and right-hand sides o f Eq. (5.4), we
obtain
(5.5)
5 .3
Lift, Drag, and Moment Coefficients
C onsider mass m. The exponent o f m on the left-hand side is 1. Thus, the expo­
nents o f m on the right m ust add to 1. H ence,
1=b + f
(5.6)
—2 = —a — c — f
(5.7)
\ = a - 3 b + 2d + e - f
(5.8)
Similarly, for tim e t we have
and for length /,
Solving Eqs. (5.6) to (5.8) for a , b, and d in term s o f e and / yields
(5.9)
b = l - f
a = 2 - e - f
(5.10)
j - i - L
(5.11)
Substituting Eqs. (5.9) to (5.11) into (5.4) gives
L =
C ) 2- ^ p x~ f S ' - M a U n L
= Z(VX
(5.12)
Rearranging Eq. (5.12), we find
i=zp“vK c)'G ±^)/
<5' ,3,
Note that a0o/Voo = l/Afoo> w here
is the free-stream M ach number. Also
note that the dim ensions o f S are l2\ hence the dim ension o f S 1/2 is /, purely a
length. Let us choose this length to be the chord c by convention. Hence,
f¿oc/(PooVooS'/2) can be replaced in our consideration by the equivalent quantity
Moo
Poo V(x¡C
However, /¿oo/(A»VooíO = 1/R e, where Re is based on the chord length c.
Hence, Eq. (5.13) becom es
/
(5.14)
L = Z PooV l S
We now define a new quantity called the lift coefficient c¡ as
2 = z ( M 7 _ L
2
\M X )
\Re
(5.15)
Then, Eq. (5.14) becomes
L = {p o o V lS c,
(5.16)
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chapter
5
Airfoils, Wings, and Other Aerodynamic Shapes
R ecalling from Chap. 4 that the dynam ic pressure is
form Eq. (5.16) into
Dynamic
pressure
Wing
area
X
S
t
8
Lift
t
S3-
L =
t
x
= 5 P 00
we trans­
Cl
t
Lift
coefficient
Look w hat has happened! Equation (5.3), w ritten from intuition but not very
useful, has cascaded to the sim ple, direct form o f Eq. (5.17), w hich contains a
trem endous am ount o f inform ation. In fact, Eq. (5.17) is one o f the m ost im por­
tant relations in applied aerodynam ics. It says that the lift is directly proportional
to the dynam ic pressure (hence to the square o f the velocity). It is also directly
proportional to the w ing area S and to the lift coefficient c¡. In fact, Eq. (5.17) can
be turned around and used as a definition for the lift coefficient:
(5.18)
That is, the lift coefficient is always defined as the aerodynam ic lift divided by
the dynam ic pressure and som e reference area (for wings, the convenient refer­
ence area S, as we have been using).
The lift coefficient is a function o f M and Re as reflected in Eq. (5.15).
M oreover, since M<*> and Re are dim ensionless and since Z was assum ed ini­
tially as a dim ensionless constant, from Eq. (5.15) C/ is dim ensionless. This is
also consistent with Eqs. (5.17) and (5.18). Also, recall that the above derivation
w as carried out for an airfoil o f given shape and at a given angle o f attack a . If a
w ere to vary, then c¡ would also vary. H ence, for a given airfoil,
d = f ( a , Moo,Re)
(5.19)
This relation is im portant. Fix in your m ind that lift coefficient is a function of
angle o f attack, M ach num ber, and Reynolds number.
To help appreciate the value o f the relationship expressed by Eq. (5.19), let
us assum e that w e are given a particular aerodynam ic shape, and we w ish to m ea­
sure the lift and how it varies with the different param eters. So w e go to the lab­
oratory and set up a series o f w ind tunnel tests to m easure the lift on our given
shape. Reflecting on Eq. (5.3), we know the lift o f the given shape at a given ori­
entation (angle o f attack) to the flow depends on the free-stream velocity, density,
reference area, viscosity coefficient, and speed o f sound, but we do not know
precisely how L varies w ith a change in these param eters. We wish to find out
how. We begin by running a set o f w ind tunnel tests, m aking m easurem ents o f L
w here Voo is varied but S,
and
are held fixed. This gives us a stack o f
wind tunnel data from which we can obtain a correlation o f the variation o f L
with Voo. Next, we run another set o f wind tunnel tests in which p & is varied but
Voo, S, /Zoo. and «oo are held fixed. This gives us a second stack o f w ind tunnel
data from which w e can obtain a correlation o f the variation o f L with p ^ . Then
w e run a third set o f wind tunnel tests in w hich S is varied, holding everything
5.3
Lift, Drag, and Moment Coefficients
else constant. This gives us a third stack o f w ind tunnel data from w hich we can
obtain a correlation o f the variation o f L w ith S. We repeat this process tw o more
tim es, alternately holding /x ^ constant and then a constant. W hen we are fin­
ished, we end up w ith five individual stacks o f w ind tunnel data from which we
can (in principle) obtain the precise variation o f L w ith Vt*,, p 00, S, ¿i,*,, and cioo,
as represented by the functional relation in Eq. (5.3). As you can probably al­
ready appreciate, this represents a lot o f personal effort and a lot o f w ind tunnel
testing at great financial expense. However, if we use our know ledge obtained
from our dim ensional analysis, namely, Eq. (5.19), we can realize a great savings
o f effort, tim e, and expense. Instead o f m easuring L in five sets o f w ind tunnel
tests as previously described, let us m easure the variation o f lift coefficient [ob­
tained from ci = L /iqooS )]. Keying on Eq. (5.19), for a given shape at a given
angle o f attack, we run a set o f w ind tunnel tests in w hich c¡ is m easured, with
Moo varied but Re held constant. This gives us one stack o f w ind tunnel data
from which we can obtain a correlation o f the variation o f c¡ with Mx . Then we
run a second set o f wind tunnel tests, varying Re and keeping M00 constant. This
gives us a second stack o f data from w hich w e can obtain a correlation o f the
variation o f c/ with Re. And this is all we need; we now know how c¡ varies with
Moo and Re for the given shape at the given angle o f attack. With c¡, w e can ob­
tain the lift from Eq. (5.17). By dealing with the lift coefficient instead o f the lift
itself, and with
and Re instead o f Poo, V,», 5, p <*,, and a<*,, we have ended up
with only two stacks o f w ind tunnel data rather than the five we had earlier.
Clearly, by using the dim ensionless quantities c¡,
and Re, we have achieved
a great econom y o f effort and wind tunnel time.
But the moral to this story goes m uch more deeply. Dim ensional analysis
shows that c¡ is a function o f Much number and Reynolds number, as stated in
Eq. (5.19), rather than ju st individually o f poo, Voo* /¿oo* «<x>» and the size o f the
body. It is the combination o f these physical variables in the form o f M 00 and Re
that counts. The M ach num ber and the Reynolds num ber are powerful quantities
in aerodynam ics. They are called sim ilarity param eters, for reasons that are dis­
cussed at the end o f this section. We have already w itnessed in Chap. 4 the power
of Moa in governing com pressible flows. For exam ple, ju st look at Eqs. (4.73)
through (4.75) and (4.79); only the Mach number and the ratio o f specific heats
appear on the right-hand sides o f these equations.
Perform ing a sim ilar dim ensional analysis on drag and m om ents, beginning
with relations analogous to Eq. (5.3), we find that
(5.20)
where q is a dim ensionless drag coefficient and
M = qooScc,,,
(5.21)
where cm is a dim ensionless moment coefficient. Note that Eq. (5.21) differs
slightly from Eqs. (5.17) and (5.20) by the inclusion o f the chord length c. This
is because L and D have dim ensions o f a force, w hereas M has dim ensions o f a
force-length product.
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chapter
5
Airfoils, Wings, and Other Aerodynamic Shapes
The im portance o f Eqs. (5.17) to (5.21) cannot be overem phasized. They are
fundam ental to all applied aerodynam ics. They are readily obtained from dim en­
sional analysis, which essentially takes us from loosely defined functional rela­
tionships [such as Eq. (5.3)] to w ell-defined relations betw een dim ensionless
quantities [Eqs. (5.17) to (5.21)]. In summary, for an airfoil o f given shape, the
dim ensionless lift, drag, and m om ent coefficients have been defined as
L
C/
4oo S
D
Cd = -----<7oo5
M
cm —
„
qooSc
(5.22)
where
c, = /,(<*, Mx , Re)
cd = / 2(a , Moo, Re)
Cm = h((X, Moo, Re)
(5.23)
Reflecting for an instant, we find there may appear to be a conflict in our
aerodynam ic philosophy. On one hand, Chaps. 2 and 4 em phasized that lift, drag,
and m om ents on an aerodynam ic shape stem from the detailed pressure and shear
stress distributions on the surface and that m easurem ents and/or calculations o f
these distributions, especially for com plex configurations, are not trivial under­
takings. O n the other hand, the equations in Eq. (5.22) indicate that lift, drag, and
m om ents can be quickly obtained from sim ple form ulas. T he bridge betw een
these tw o outlooks is, o f course, the lift, drag, and m om ent coefficients. All the
physical com plexity o f the flow field around an aerodynam ic body is im plicitly
buried in c¡, c j, and cm. Before the sim ple equations in Eq. (5.22) can be used to
calculate lift, drag, and m om ents for an airfoil, w ing, and body, the appropriate
aerodynam ic coefficients m ust be known. From this point o f view, the sim plicity
o f Eq. (5.22) is a bit deceptive. These equations sim ply shift the forces o f aero­
dynam ic rigor from the forces and m om ents them selves to the appropriate coef­
ficients instead. So we are now led to the questions, How do we obtain values o f
c/, c j, and cm for given configurations, and how do they vary with a ,
and
Re? The answ ers are introduced in the follow ing sections.
However, before we leave our discussion o f dim ensional analysis, it is im ­
portant to elaborate on why
and Re are called sim ilarity param eters. C on­
sider that we have tw o different flows, say, a red flow and a green flow, over two
bodies that are geom etrically sim ilar but are different sizes for the red and green
flows. The red and green flows have different values o f Voo, p ^ , f ioo> and a ^ , but
they both have the same
and Re. If
is the sam e for the red and green
flows and if Re is the same for the red and green flows, then from Eq. (5.23), c¡,
c j, and cm m easured in the red flow will be the same values as the c¡, Cd, and cm
m easured in the green flow, even though the red and green flows are different
flows. In this case, the red and green flows are called dynam ically sim ilar flows;
hence,
and Re are called sim ilarity param eters. The concept o f dynam ic
flow sim ilarity is elegant, and it goes well beyond the scope o f this book. But it
is m entioned here because o f its im portance in aerodynam ics. It is the concept o f
dynam ic sim ilarity that allow s m easurem ents obtained in wind tunnel tests o f a
sm all-scale m odel o f an airplane to be applied to the real airplane in free flight.
If in the wind tunnel test (say, the red flow) the values o f
and Re are the same
5 .4
Airfoil Data
as those for the real airplane in free flight (say, the green flow), then c¡, cd, and
c,„ m easured in the wind tunnel will be precisely the sam e as those values in free
flight. T he concept o f dynam ic sim ilarity is essential to w ind tunnel testing.
In m ost wind tunnel tests o f sm all-scale m odels o f real airplanes, every effort
is m ade to sim ulate the values o f AiM and Re encountered by the real airplane in
free flight. U nfortunately, due to the realities o f wind tunnel design and opera­
tion, this is frequently not possible. In such cases, the w ind tunnel data must be
“extrapolated” to the conditions o f free flight. Such extrapolations are usually ap­
proxim ations, and they introduce a degree o f error when the wind tunnel data are
used to describe the conditions o f full-scale free flight. The problem o f not being
able to sim ultaneously sim ulate free-flight values o f
and Re in the same
wind tunnel is still pressing today in spite o f the fact that wind tunnel testing has
been going on for alm ost 150 years. A m ong other reasons, this is why there are
so m any different w ind tunnels at different laboratories around the world.
5.4 AIRFOIL DATA
A goal o f theoretical aerodynam ics is to predict values o f c¡, c¿, and cm from the
basic equations and concepts o f physical science, som e o f which were discussed
in previous chapters. However, sim plifying assum ptions are usually necessary to
m ake the m athem atics tractable. Therefore, when theoretical results are ob­
tained, they are generally not “exact.” The use o f high-speed digital com puters to
solve the governing flow equations is now bringing us m uch closer to the accu­
rate calculation o f aerodynam ic characteristics; however, there are still lim ita­
tions im posed by the num erical m ethods them selves, and the storage and speed
capacity o f current com puters are still not sufficient to solve m any com plex aero­
dynam ic flows. As a result, the practical aerodynam icist has to rely upon direct
experim ental m easurem ents o f c/, q , and cm for specific bodies o f interest.
A large bulk o f experim ental airfoil data was com piled over the years by the
National A dvisory C om m ittee for A eronautics (N A CA ), which was absorbed in
the creation o f the N ational A eronautics and Space A dm inistration (N A SA ) in
1958. Lift, drag, and m om ent coefficients w ere system atically m easured for
m any airfoil shapes in low -speed subsonic wind tunnels. These m easurem ents
were carried out on straight, constant-chord w ings that com pletely spanned the
tunnel test section from one side wall to the other. In this fashion, the flow es­
sentially “saw ” a wing with no wingtips, and the experim ental airfoil data were
thus obtained for “infinite w ings.” (The distinction betw een infinite and finite
wings will be m ade in subsequent sections.) Som e results o f these airfoil m ea­
surem ents are given in App. D. The first page o f App. D gives data for c¡ and
Cm,c/4 versus angle o f attack for the NACA 1408 airfoil. The second page gives
c,i and c„,,ac versus c¡ for the sam e airfoil. Since c¡ is know n as a function o f a
from the first page, the data from both pages can be cross-plotted to obtain the
variations o f c¿ and cmac versus a . The rem aining pages o f App. D give the same
type o f data for different standard N A C A airfoil shapes.
Let us exam ine the variation o f c¡ with a m ore closely. This variation is
sketched in Fig. 5.6. The experim ental data indicate that c¡ varies linearly with a
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chapter
5
Airfoils, Wings, and Other Aerodynamic Shapes
Figure 5.6 Sketch of a typical lift curve.
Cam bered airfoil
Sym m etric airfoil
Figure 5.7 Comparison of lift curves for cambered and symmetric airfoils.
over a large range o f angle o f attack. Thin-airfoil theory, which is the subject o f
m ore advanced books on aerodynam ics, also predicts the same type o f linear
variation. The slope o f the linear portion o f the lift curve is designated as
aQ = d c i/d a = lift slope. N ote that in Fig. 5.6, when a = 0, there is still a posi­
tive value o f ci\ that is, there is still som e lift even when the airfoil is at zero angle
o f attack to the flow. This is due to the positive cam ber o f the airfoil. All airfoils
w ith such cam ber have to be pitched to som e negative angle o f attack before zero
lift is obtained. The value o f a when lift is zero is defined as the zero-lift angle o f
attack Ul=o and is illustrated in Fig. 5.6. This effect is further dem onstrated in
Fig. 5.7, w here the lift curve for a cam bered airfoil is com pared with that for a
5 .4
Airfoil Data
Separated flow
sym m etric (no cam ber) airfoil. N ote that the lift curve for a sym m etric airfoil
goes through the origin. Refer again to Fig. 5.6, at the other extrem e: For large
values o f a , the linearity of the lift curve breaks down. As a is increased beyond
a certain value, c/ peaks at som e m axim um value c/,max and then drops precipi­
tously as a is further increased. In this situation, where the lift is rapidly de­
creasing at high a , the airfoil is stalled.
The phenom enon o f airfoil stall is o f critical im portance in airplane design.
It is caused by flow separation on the upper surface o f the airfoil. This is illus­
trated in Fig. 5.8, which again show s the variation o f c\ versus a for an airfoil. At
point 1 on the linear portion o f the lift curve, the flow field over the airfoil is at­
tached to the surface, as pictured in Fig. 5.8. H ow ever, as discussed in Chap. 4,
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c hapter
5
Airfoils, Wings, and Other Aerodynamic Shapes
the effect o f friction is to slow the airflow near the surface; in the presence o f an
adverse pressure gradient, there will be a tendency for the boundary layer to sep­
arate from the surface. As the angle o f attack is increased, the adverse pressure
gradient on the top surface o f the airfoil will becom e stronger, and at som e value
o f a — the stalling angle o f attack— the flow becom es separated from the top sur­
face. W hen separation occurs, the lift decreases drastically and the drag increases
suddenly. This is the picture associated with point 2 in Fig. 5.8. (It would be well
for the reader to review at this stage the discussion o f flow separation and its ef­
fect on pressure distribution, lift, and drag in Sec. 4.21.)
T he nature o f the flow field over the wing o f an airplane that is below, ju st
beyond, and w ay beyond the stall is show n in Fig. 5.9a, b, and c, respectively.
T hese figures are photographs o f a w ind tunnel model with a w ingspan o f 6 ft.
The entire model has been painted with a m ixture o f mineral oil and a fluorescent
powder, w hich glow s under ultraviolet light. A fter the wind tunnel is turned
on, the fluorescent oil indicates the stream line pattern on the surface o f the
m odel. In Fig. 5.9a, the angle o f attack is below the stall; the flow is fully
attached, as evidenced by the fact that the high surface shear stress has scrubbed
m ost o f the oil from the surface. In Fig. 5.9b, the angle o f attack is slightly
(a)
Figure 5.9 Surface oil flow patterns on a wind tunnel model o f a Grumman American Yankee,
taken by Dr. Allen Winkelmann in the Glenn L. Martin Wind Tunnel at the University of
Maryland. The mixture is mineral oil and a fluorescent powder, and the photographs were
taken under ultraviolet light, (a) Below the stall. The wing is at a = 4°, where the flow is
attached. (continued)
5 .4
Airfoil Dala
Figure 5.9 ( Continued) (b ) Very near the stall. The wing is at a = 11°, where the highly
three-dimensional separated flow is developing in a mushroom cell pattern, (c) Far above the
stall. The wing is at a = 24°, where the flow over almost the entire wing has separated.
267
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CHA p T ER 5
Airfoils, Wings, and Of her Aerodynamic Shapes
beyond the stall. A large, m ushroom -shaped, separated flow pattern has devel­
oped over the w ing, with attendant highly three-dim ensional, low-energy recir­
culating flow. In Fig. 5.9c, the angle o f attack is far beyond the stall. The flow
over alm ost the entire wing has separated. These photographs are striking exam ­
ples o f different types o f flow that can occur over an airplane wing at different
angles o f attack, and they graphically show the extent o f the flow field separation
that can occur.
The lift curves sketched in Figs. 5.6 to 5.8 illustrate the type o f variation ob­
served experim entally in the data o f App. D. Returning to App. D, we note that
the lift curves are all virtually linear up to the stall. Singling out a given airfoil,
say, the N A C A 2412 airfoil, also note that c/ versus a is given for three different
values o f the Reynolds num ber from 3.1 x 106 to 8.9 x 10h. The lift curves for
all three values o f Re fall on top o f one another in the linear region; that is, Re
has little influence on c¡ when the flow is attached. However, flow separation is a
viscous effect, and as discussed in Chap. 4, Re is a governing param eter for vis­
cous flow. T herefore, it is not surprising that the experim ental data for c/,max in
the stalling region are affected by Re, as can be seen by the slightly different
variations o f c/ at high a for different values o f Re. In fact, these lift curves at dif­
ferent Re values answ er part o f the question posed in Eq. (5.19): The data repre­
sent ci = /( R e ) . A gain, Re exerts little or no effect on c¡ except in the stalling
region.
On the sam e page as c¡ versus a, the variation o f c m,c/4 versus a is also given.
It has only a slight variation with a and is alm ost com pletely unaffected by Re.
A lso note that the values o f c m,c/4 are slightly negative. By convention, a positive
m om ent is in a clockw ise direction; it pitches the airfoil tow ard larger angles of
attack, as show n in Fig. 5.4. Therefore, for the NACA 2412 airfoil, with c m,t./4
negative, the m om ents are counterclockw ise, and the airfoil tends to pitch dow n­
ward. This is characteristic o f all airfoils with positive camber.
On the page follow ing c¡ and cm c/4, the variation o f q and cm,ac is given ver­
sus c/. Because c/ varies linearly with a , the reader can visualize these curves o f
c¿ and c,„iac as being plotted versus a as well; the shapes will be the same. Note
that the drag curves have a “bucket” type o f shape, with m inim um drag occur­
ring at small values o f c; (hence there are sm all angles o f attack). As a goes to
large negative or positive values, c j increases. A lso note that c,i is strongly af­
fected by Re, there being a distinct drag curve for each Re. This is to be expected
because the drag for a slender aerodynam ic shape is m ainly skin friction drag,
and from Chap. 4 we have seen that Re strongly governs skin friction. W ith
regard to cm,ac, the definition o f the aerodynam ic center is clearly evident; cm,ac
is constant with respect to a . Also, it is insensitive to Re and has a small negative
value.
R efer to the equations in Eq. (5.23); the airfoil data in App. D experim entally
provide the variation o f c/, q , and cm with a and Re. The effect o f
on the air­
foil coefficients will be discussed later. H owever, w e em phasize that the data in
App. D w ere m easured in low -speed subsonic wind tunnels. Hence, the flow was
5 .4
Airfoil Data
essentially incom pressible. Thus, c¡, c m ,c/4 > Q . and c m ,ac given in App. D are
incom pressible flow values. It is im portant to keep this in m ind during our sub­
sequent discussions.
In this section, we have discussed the properties o f an airfoil. As already
noted in Fig. 5.2, an airfoil is sim ply the shape o f a w ing section. The airfoils in
Figs. 5.3 through 5.5 and Figs. 5.7 and 5.8 are paper-thin sections— sim ple draw ­
ings on a sheet o f paper. So w hat does it mean when we talk about the lift, drag,
and m om ents on an airfoil? How can there be a lift on an airfoil that is paperthin? W hen we w rite Eq. (5.17) for the lift o f an airfoil, what really is L ? The an­
swer is given in Fig. 5.10. H ere we see a section o f a wing o f constant chord c.
The length o f the section along the span o f the wing is unity (1 ft, 1 m, etc.). The
lift on this wing section L, as show n in Fig. 5.10a, is the lift p er unit span. The
lift, drag, and m om ents on an airfoil are alw ays understood to be the lift, drag,
and m om ents p e r unit span, as sketched in Fig. 5.10. The planform area o f the
segm ent o f unit span is the projected area seen by looking at the wing from
above, namely, S — c ( l ) = c, as sketched in Fig. 5.10¿>. H ence, when we write
Eq. (5 .1 7 ) for an a irfo il, w e interpret L as the lift per unit span and S as the
L (per unit span)
c
Figure 5.10 A wing segment o f unit span.
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planform area o f a unit span; that is,
L (per unit span) = <7<x>c(l)c/
L (per unit span)
or
Cl =
(5.24)
(5.25)
<7ooC
Finally, return to our road map in Fig. 5.1. We have begun to w ork our way
dow n the left colum n under airfoils. We have already accom plished a lot. We
have becom e fam iliar with airfoil nom enclature. U sing dim ensional analysis,
we have introduced the very im portant concept o f aerodynam ic coefficients, and
we have exam ined som e experim ental data for these coefficients. M ake certain
that you feel com fortable with these concepts before you continue.
EXAM PLE 5.1
A model wing of constant chord length is placed in a low-speed subsonic wind tunnel,
spanning the test section. The wing has an NACA 2412 airfoil and a chord length of
1.3 m. The flow in the test section is at a velocity of 50 m/s at standard sea-level condi­
tions. If the wing is at a 4° angle of attack, calculate (a) c¡, c¡¡, and cm,c/4 and (b) the lift,
drag, and moments about the quarter chord, per unit span.
■ Solution
a. From App. D, for an NACA 2412 airfoil at a 4° angle of attack,
Cl = 0.63
Cm,c/4 = -0.035
To obtain q , we must first check the value of the Reynolds number:
._ AxVooC
Moo
(1.225 kg/m3)(50 m/s)(1.3 m)
= 4.45 x 10fi
1.789 x 10-5 kg/(m)(s)
For this value of Re and for c¡ = 0.63, from App. D,
cd = 0.007
b. Since the chord is 1.3 m and we want the aerodynamic forces and moments per unit
span (a unit length along the wing, perpendicular to the flow), S = c (l) = 1.3(1) = 1.3 m2.
Also,
o = |/ 0ocV¿ = i(1.225)(50)2 = 1531 N/m 2
From Eq. (5.22),
L
=
q x Sci =
1531(1.3)(0.63) =
1254 N
Since 1 N = 0.2248 lb, also
L = (1254 N)(0.2248 lb/N) = 281.9 lb
D — q o o S cd =
1531(1.3)(0.007) = 13.9 N
= 13.9(0.2248) = 3.13 lb
5 .5
271
Note: The ratio of lift to drag, which is an important aerodynamic quantity, is
L
ci
~D = Vd
1254
= 90.2
13.9
Mc/a = qooScm,c/4c = 15 3 1(1.3)(—0.035)( 1.3)
Mc/4 = —90.6 N • m
EXAMPL1 i 5.2
The same wing in the same flow as in Example 5.1 is pitched to an angle of attack such
that the lift per unit span is 700 N (157 lb).
a. What is the angle of attack?
b. To what angle of attack must the wing be pitched to obtain zero lift?
■ Solution
a. From the previous example,
^
Thus,
ci =
= 1531 N /m 2
qx S
5 = 1.3 m 2
700
= 0.352
1531(1.3)
From App. D for the NACA 2412 airfoil, the angle of attack corresponding to c¡ — 0.352 is
a=T
b. Also from App. D, for zero lift, that is, c¡ = 0,
« t - o = - 2 .2 °
5.5 INFINITE VERSUS FINITE WINGS
As stated in Sec. 5.4, the airfoil data in App. D w ere m easured in low -speed sub­
sonic wind tunnels w here the m odel wing spanned the test section from one side­
wall to the other. In this fashion, the flow sees essentially a wing with no wing
tips; that is, the w ing in principle could be stretching from plus infinity to minus
infinity in the spanw ise direction. Such an infinite wing is sketched in Fig. 5.11,
where the wing stretches to ± o o in the z direction. The flow about this wing
varies only in the jc and y directions; for this reason, the flow is called twodimensional. Thus, the airfoil data in App. D apply only to such infinite (or twodim ensional) wings. This is an im portant point to keep in mind.
On the other hand, all real airplane wings are obviously finite, as sketched in
Fig. 5.12. Here, the top view (planform view ) o f a finite wing is shown, where
the distance betw een the tw o wing tips is defined as the wingspan b. The area of
the w ing in this planform view is designated, as before, by S. This leads to an im ­
portant definition that pervades all aerodynam ic w ing considerations, namely,
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+00
Figure 5.11 Infinite (two-dimensional) wing.
R ight-w ing tip
L eft-w ing tip
Figure 5.12 Finite wing; plan view (top).
the aspect ratio AR:
b2
A spect ratio = AR = —
(5.26)
The im portance o f A R will com e to light in subsequent sections.
T he flow field about a finite wing is three-dim ensional and is therefore in­
herently different from the tw o-dim ensional flow about an infinite wing. As a re­
sult, the lift, drag, and m om ent coefficients for a finite wing with a given airfoil
5 .6
Pressure Coefficient
shape at a given a are different from the lift, drag, and m om ent coefficients for
an infinite wing with the sam e airfoil shape at the sam e a . For this reason, the
aerodynam ic coefficients for a finite wing are designated by capital letters C L,
C d >C m\ this is in contrast to those for an infinite wing, w hich we have been des­
ignating as c¡, Cd, and cm. N ote that the data in App. D are for infinite (twodim ensional) wings; that is, the data are for c/, q , and cm. In a subsequent sec­
tion, we will show how to obtain the finite-w ing aerodynam ic coefficients from
the infinite-w ing data in App. D. O ur purpose in this section is sim ply to under­
score that there is a difference.
5.6 PRESSURE COEFFICIENT
We continue with our parade o f aerodynam ic definitions. C onsider the pressure
distribution over the top surface o f an airfoil. Instead o f plotting the actual pres­
sure (say, in units o f new tons per square m eter), we define a new dim ensionless
quantity, called the pressure coefficient Cp, as
C = P ~ Po° = P ~ P°°
The pressure distribution is sketched in term s o f Cp in Fig. 5.13. This figure is
worth looking at closely because pressure distributions found in the aerodynam ic
literature are usually given in term s o f the dim ensionless pressure coefficient.
Figure 5.13 Distribution of pressure coefficient over the top and
bottom surfaces o f an NACA 0012 airfoil at 3.93° angle of attack.
Moo = 0.345, Re = 3.245 x 106. Experimental data from Ohio
State University, in NACA Conference Publication 2045, part I,
Advanced Technology Airfoil Research, vol. I, p. 1590.
(Source: After Freuler and Gregorek.)
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N ote from Fig. 5.13 that Cp at the leading edge is positive because p > /?<».
H ow ever, as the flow expands around the top surface o f the airfoil, p decreases
rapidly, and Cp goes negative in those regions where p < Poo. By convention,
plots o f Cp for airfoils are usually shown with negative values above the
abscissa, as show n in Fig. 5.13.
The pressure coefficient is an im portant quantity; for exam ple, the distribu­
tion o f C p over the airfoil surface leads directly to the value o f c¡, as will be dis­
cussed in Sec. 5.11. M oreover, considerations o f Cp lead directly to the calcula­
tion o f the effect o f M ach num ber Mx on the lift coefficient. To set the stage for
this calculation, consider C p at a given point on an airfoil surface. The airfoil is
a given shape at a fixed angle o f attack. The value o f Cp can be m easured by test­
ing the airfoil in a wind tunnel. Assum e that, at first, Voo in the tunnel test section
is low, say, M ^ < 0.3, such that the flow is essentially incom pressible. The m ea­
sured value o f Cp at the point on the airfoil will therefore be a low -speed value.
Let us designate the low -speed (incom pressible) value o f Cp by C p,0- If Voo is in­
creased but Mqo is still less than 0.3, then Cp will not change; that is, Cp is es­
sentially constant w ith velocity at low speeds. However, if we now increase
such that Mx > 0.3, then com pressibility becom es a factor, and the effect o f
com pressibility is to increase the absolute m agnitude o f Cp as M ^ increases.
This variation o f Cp with M<*, is shown in Fig. 5.14. N ote that at M^ « 0,
Cp = CPfi. As Moo increases to M 00 % 0.3, essentially Cp is constant. However,
as Mao is increased beyond 0.3, C p increases dramatically. (That is, the absolute
___________ i___________ I___________ I___________I___________ I________
0
0.2
0 .4
0.6
Figure 5.14 Plot o f the Prandtl-Glaucrt rule for CPto = 0.5.
0.8
1.0
5 .6
Pressure Coefficient
275
m agnitude increases; if C /;,0 is negative, then Cp will becom e an increasingly
m ore negative num ber as M<*, increases, w hereas if C p%0 is positive, then Cp will
becom e an increasingly m ore positive num ber as M x increases.) T he variation
o f Cp with Moo for high subsonic M ach num bers was a m ajor focus o f aerody­
nam ic research after W orld W ar II. An approxim ate theoretical analysis yields
(5.28)
Equation (5.28) is called the Prandtl-G lauert rule. It is reasonably accurate
for 0.3 < Moo < 0.7. For
> 0.7, its accuracy rapidly dim inishes; indeed,
Eq. (5.28) predicts that C p becom es infinite as M ^ goes to unity— an im possible
physical situation. (It is well to note that nature abhors infinities as w ell as dis­
continuities that are som etim es predicted by m athem atical, but approxim ate,
theories in physical science.) T here are m ore accurate, but m ore com plicated,
form ulas than Eq. (5.28) for near-sonic M ach num bers. However, Eq. (5.28) will
be sufficient for our purposes.
Form ulas such as Eq. (5.28), w hich attem pt to predict the effect o f M ^ on
Cp for subsonic speeds, are called com pressibility corrections; that is, they m od­
ify (correct) the low -speed pressure coefficient C ;),0 to take into account the ef­
fects o f com pressibility, which are so im portant at high subsonic M ach numbers.
EXAM PLE 5.3
The pressure at a point on the wing o f an airplane is 7.58 x l()4 N/m 2. The airplane is fly­
ing with a velocity o f 70 m/s at conditions associated with a standard altitude o f 2000 m.
Calculate the pressure coefficient at this point on the wing.
■ Solution
For a standard altitude o f 2000 m,
Poo = 7.95 x 104 N /m 2
Poo = 1.0066 kg/m 1
Thus, qoo =
= j(1 .0 0 6 6 )(7 0 )2 = 2466 N/m 2. From Eq. (5.27),
_ P ~ Poo _ (7 .5 8 -7 .9 5 ) x 104
''
(too
2466
EXAM PLE 5.4
Consider an airfoil mounted in a low -speed subsonic wind tunnel. The flow velocity in
the test section is 100 ft/s, and the conditions are standard sea level. If the pressure at a
point on the airfoil is 2102 lb/ft2, what is the pressure coefficient?
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Airfoils, Wings, and Other Aerodynamic Shapes
■ Solution
4oc =
= 5(0.002377 slug/ft3)( 100 ft/s)2 = 11.89 lb/ft2
From Eq. (5.27),
p -p o o
2 1 0 2 -2 1 1 6
11.89
-1 .1 8
EXAM PLE 5.5
In Example 5.4, if the flow velocity is increased such that the free-stream Mach number
is 0 .6, what is the pressure coefficient at the same point on the airfoil?
■ Solution
First, what is the Mach number of the flow in Example 5.4? At standard sea level,
T, =518.69° R
Hence,
floe = sI y RTZ = A 4(1716)(518.69) = 1 1 1 6 ft/s
Thus, in Example 5.4,
= VooA*oo = 100/1116 = 0.09—a very low value. Hence,
the flow in Example 5.4 is essentially incompressible, and the pressure coefficient is a
low-speed value; that is, CIKo = —1.18. Thus, if the flow Mach number is increased to
0.6, from the Prandtl-Glauert rule, Eq. (5.28),
Cp
C„,o
1.18
(1 - M l )'/2
(1 - 0 . 6 2)'/2
Cp = -1 .4 8
EXAM PLE 5.6
An airplane is flying at a velocity of 100 m/s at a standard altitude of 3 km. The pressure
coefficient at a point on the fuselage is —2.2. What is the pressure at this point?
■ Solution
For a standard altitude of 3 km = 3000 m, px = 7.0121 x 104 N/m2, and Poo =
0.90926 kg/m1. Thus,
<7oo = 5P 0 0 = ±(0.90926)(100)2 = 4546 N/m 2
From Eq. (5.27),
C„ =
or
P
Poo
tfoo
p = q^ C p + Poo = (4546)(—2.2) + 7.10121 x 104 =
6.01 x 104 N/m 2
N ote: This example illustrates a useful physical interpretation of pressure coefficient. The
pressure coefficient represents the local pressure in terms of the “number of dynamic
5 .6
Pressure Coefficient
277
pressure units” above or below the free-stream pressure. In this example, the local pressure
was found to be 6.01 x 104 N/m2. This value of p is equivalent to the free-stream pressure
minus 2.2 times the dynamic pressure; p is 2.2 “dynamic pressures” below the free-stream
pressure. So when you see a number for Cp, that number gives you an instant feel for the
pressure itself in terms of multiples of
above or below the free-stream pressure. In this
example, Cp is negative, so the pressure is below the free-stream pressure. If Cp = 1.5, the
pressure would be 1.5 “dynamic pressures” above the free-stream pressure.
EXAMPLI5 5.7
Consider two different points on the surface of an airplane wing flying at 80 m/s. The
pressure coefficient and flow velocity at point 1 are —1.5 and 110 m/s, respectively. The
pressure coefficient at point 2 is —0.8. Assuming incompressible flow, calculate the flow
velocity at point 2 .
■ Solution
From Eq. 5.27,
Cp, —
P i - Poo
,
(too
„
Pi - Poc
__
or
p\
„
Pqq — CjocCp
p2
Poo — Qoo^pi
Similarly,
Cp2 —
Qoo
>
_
or
„
Subtracting,
P \ - Pl = ?oo(Cp, - CP2)
From Bernoulli’s equation,
P\ + \ p V \ = P 2 + \ p v l
P \ ~ P 2 = \p { V l ~ V\ )
Since qoo =
we have
Pi ~ Pi
tfoo
- m
- m
'
Substituting the earlier expression for p\ — P2 in terms of CPl and CP2>we have
W C , , ~ Cp2)
<7oo
-m -m
Note: This expression by itself is interesting. In a low-speed incompressible flow, the dif­
ference between the pressure coefficients at two different points is equal to the difference
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5
Airfoils, Wings, and Other Aerodynamic Shapes
in the squares of the velocities, nondimensionalized by the free-stream velocity, between
the two points.
Putting in the numbers, we have
V22 = 1.19V ¿ = 1.19 (8 0 )2
V2 = 87.3 m/s
Note: The solution did not require the explicit knowledge of the density. This is because
we dealt with pressure difference in terms of the difference in pressure coefficient, which,
in turn, is related to the difference of the squares of the nondimensional velocity through
Bernoulli’s equation.
5.7 OBTAINING LIFT COEFFICIENT FROM CP
If you are given the distribution o f the pressure coefficient over the top and bot­
tom surfaces o f an airfoil, you can calculate c¡ in a straightforw ard manner. C on­
sider a segm ent o f an infinite w ing, as shown in Fig. 5.15. A ssum e the segm ent
Figure 5.15 Sketch showing how the pressure distribution can be
integrated to obtain normal force per unit span, leading to lift per unit span.
5 .7
Obtaining Lift Coefficient from CP
has unit span and chord c. The w ing is at an angle o f attack o f a . L et x be the di­
rection m easured along the chord, and let s be the distance m easured along the
surface from the leading edge, as show n in Fig. 5.15. C onsider the infinitesim ally
small sliver o f surface area o f length d s and unit length in the span direction, as
show n by the shaded area in Fig. 5.15. T he area o f this surface is U s . The
dashed line ab is perpendicular to chord c. T he solid line a c is locally perpen­
dicular to the shaded area. T he angle betw een a b and a c is 9. The aerodynam ic
force on the shaded area is p ( l ) d s , w hich acts in the direction o f ac, normal to
the surface. Its com ponent in the direction norm al to the chord is (p c o s # ) ( l) d s.
A dding a subscript u to designate the pressure on the upper surface o f the airfoil,
as well as a m inus sign to indicate the force is directed dow nw ard (we use the
convention that a positive force is directed upw ard), w e see that the contribution
to the norm al force o f the pressure on the infinitesim al strip is —p„ cos 9 d s . If all
the contributions from all the strips on the upper surface are added from the lead­
ing edge to the trailing edge, we obtain, by letting d s approach 0 , the integral
/.T E
I
p u cos 9 d s
J LE
This is the force in the normal direction due to the pressure distribution acting
on the upper surface o f the w ing, per unit span. Recall the definition o f normal
and axial forces N and A, respectively, discussed in Sec. 5.2 and sketched in
Fig. 5.4a. The above integral is that part o f N that is due to the pressure acting on
the upper surface. A sim ilar term is obtained that is due to the pressure distribu­
tion acting on the low er surface o f the airfoil. Letting pi denote the pressure on
the low er surface, we can write for the total norm al force acting on an airfoil o f
unit span
(5.29)
From the small triangle in the box in Fig. 5.15, we see the geom etric relationship
d s cos 9 = d x . Thus, in Eq. (5.29), the variable o f integration s can be replaced
by x , and at the sam e tim e the x coordinates o f the leading and trailing edges be­
com e 0 and c, respectively. Thus, Eq. (5.29) becom es
(5.30)
A dding and subtracting p 0a, we find that Eq. (5.30) becom es
(5.31)
Putting Eq. (5.31) on the shelf for a m om ent, we return to the definition o f
norm al and axial forces N and A, respectively, in Fig. 5.4a. We can define the
norm al and axial force coefficients for an airfoil, c„ and ca, respectively, in the
279
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chapter
5
Airfoils, Wings, and Other Aerodynamic Shapes
sam e m anner as the lift and drag coefficients given by Eq. (5.22); that is,
N
Cn —
Ca =
_
N
qsS
A
qooC
A
qsS
qx c
(5.32)
(5.33)
H ence, the norm al force coefficient c„ can be calculated from Eqs. (5.31) and
(5.32) as
Cn = ~1
Pi
f
P oo
,
dx
tfoo
C Jo
i1 ff
Pu
Poo
dx
(5.34)
c Jo
N ote that
Pi
~
Poo
= CPti = pressure coefficient on low er surface
qoo
Pu
-
Poo
= CP)U = pressure coefficient on upper surface
qoo
H ence, Eq. (5.34) becom es
(5.35)
Equation (5.35) gives the normal force coefficient directly in term s o f the inte­
gral o f the pressure coefficient over the surface o f the airfoil.
H ow is this related to the lift coefficient? The answ er is given by Eq. (5.1),
repeated here:
L = N cos a — A sin a
(5.1)
D ividing Eq. (5.1) by q ^ S = q ^ c , we have
L
N
A .
------= ------- cos a ---------- sin a
q ooC
or
qoo£
qooC
c¡ = c„ cos a — ca sin a
(5.36)
G iven c„ and ca, Eq. (5.36) allow s the direct calculation o f c¡. Equation (5.35) is an
expression forc„ in term s of the integral o f the pressure coefficients. [In Eq. (5.35),
we have ignored the influence o f shear stress, which contributes very little to nor­
mal force.] A sim ilar expression can be obtained for ca involving an integral o f
the pressure coefficient and an integral o f the skin friction coefficient. Such an
expression is derived in Chap. 1 o f A nderson, Fundamentals o f Aerodynamics,
3d ed., M cG raw -H ill, 2001; this is beyond the scope o f our discussion here.
C onsider the case o f small angle o f attack, say, a < 5°. Then in Eq. (5.36),
c o s a «í 1 and sin a % 0, and Eq. (5.36) yields
ci ss c„
(5.37)
5 .7
Obtaining Lift Coefficient from CP
281
Figure 5.16 Sketch of the pressure coefficient over the upper and
lower surfaces of an airfoil showing that the area between the two
curves is the lift coefficient for small angles of attack.
and com bining Eqs. (5.37) and (5.35), we have
(5.38)
M ost conventional airplanes cruise at angles o f attack o f less than 5°, so for such
cases, Eq. (5.38) is a reasonable representation o f the lift coefficient in term s of the
integral o f the pressure coefficient. This leads to a useful graphical construction
for c¡. C onsider a com bined plot o f CPu and CPl as a function o f .*/c , as sketched in
Fig. 5.16. The area betw een these curves is precisely the integral on the right-hand
side o f Eq. (5.35). H ence, this area, shown as the shaded region in Fig. 5.16, is pre­
cisely equal to the norm al force coefficient. In turn, for small angles o f attack,
from Eq. (5.38), this area is essentially the lift coefficient, as noted in Fig. 5.16.
EXAM PLE 5.X
Consider an airfoil with chord length c and the running distance x measured along the
chord. The leading edge is located at x /c = 0 and the trailing edge at x /c = 1. The pres­
sure coefficient variations over the upper and lower surfaces are given, respectively, as
Cp,u = 1 - 300 ( - ) 2
for 0 < ^ < 0.1
C„ „ = -2.2277 + 2.2777c
for 0.1 < - < 1.0
c
Co, = 1 - 0 . 9 5 c
Calculate the normal force coefficient.
forO < - < 1.0
c
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5
Airfoils, Wings, and Other Aerodynamic Shapes
■ Solution
From Eq. (5.35),
= c f0
c„
-
Cp,“)dx = J (Cp,i-C p,u) d ( ¿ \
=/’(■-°95f)"(i) - r 1 [i- 3o°(i)2]"(f)
-jf
^ - 2.2277 + 2.2777^
, 0.1
1.0
1.0
+ 2.2277c
- 1.1388
o.i
c„ = 1 -0 .4 7 5 - 0 .1 + 0 .1 + 2.2277 - 0 .2 2 2 7 7 - 1.1388 + 0.011388 =
(?y
0.1
1.40
Note that the Cp variations given analytically in this problem are only crude representa­
tions of a realistic case and should not be taken too seriously, since the purpose of this
example is simply to illustrate the use of Eq. (5.35).
5.8 COMPRESSIBILITY CORRECTION
FOR LIFT COEFFICIENT
The pressure coefficients in Eq. (5.38) can be replaced by the com pressibility
correction given in Eq. (5.28) as follows:
Cl =
l f
( C p 'l
C p ,u ) 0
dx =
1
C Jo
-f
(Cpj - C „ ' U)o d x
(5.39)
c Jo
where again the subscript 0 denotes low -speed incom pressible flow values.
However, referring to the form o f Eq. (5.38), w e see that
1
Jf o
(CPii
Cp,u)o d x — C/,o
where c/,o is the low -speed value o f the lift coefficient. Thus, Eq. (5.39) becom es
c, = - - - - - -
(5.40)
Equation (5.40) gives the com pressibility correction for the lift coefficient. It is
subject to the sam e approxim ations and accuracy restrictions as the PrandtlG lauert rule, Eq. (5.28). A lso note that the airfoil data in App. D w ere obtained at
low speeds; hence, the values o f lift coefficient obtained from App. D are ctS).
5 .9
Critical Mach Number and Critical Pressure Coefficient
283
Finally, in reference to Eq. (5.19), w e now have a reasonable answ er to how
ci varies with M ach number. F or subsonic speeds, except near M ach 1, the lift
coefficient varies inversely as (1 — M ^ ) l/'2.
EXAM PLE 5.9
Consider an NACA 4412 airfoil at an angle of attack of 4°. If the free-stream Mach num­
ber is 0.7, what is the lift coefficient?
■ Solution
From App. D, for a = 4°, C/ = 0.83. However, the data in App. D were obtained at low
speeds, hence the lift coefficient value obtained, namely, 0.83, is really c¡,Q:
ci, o = 0.83
For high Mach numbers, this must be corrected according to Eq. (5.40):
c/.o
0.83
C' = (1 - A O '/2 = (1 - 0 . 7 2)'/2
Ci = 1.16
at Mao = 0-7
5.9 CRITICAL MACH NUMBER AND CRITICAL
PRESSURE COEFFICIENT
C onsider the flow o f air over an airfoil. We know that as the gas expands around
the top surface near the leading edge, the velocity and hence the M ach num ber
will increase rapidly. Indeed, there are regions on the airfoil surface where the
local M ach num ber can be greater than M<*,. Im agine that we put a given airfoil
in a w ind tunnel w here M TO = 0.3 and that w e observe the peak local M ach num ­
ber on the top surface o f the airfoil to be 0.435. This is sketched in Fig. 5.17a.
Im agine that we now increase M<*, to 0.5; the peak local M ach num ber will cor­
respondingly increase to 0.772, as shown in Fig. 5.17/?. If we further increase
M„u to a value o f 0.61, we observe that the peak local M ach num ber is 1.0, lo­
cally sonic flow on the surface o f the airfoil. This is sketched in Fig. 5.17c. Note
that the flow over an airfoil can locally be sonic (or higher), even though the freestream M ach num ber is subsonic. By definition, that free-stream Mach num ber
at which sonic flow is first obtained som ew here on the airfoil surface is called the
critical Mach number o f the airfoil. In the preceding exam ple, the critical Mach
num ber M cr for the airfoil is 0.61. As we will see later, Ma is an im portant quan­
tity, because at som e free-stream M ach num ber above Mcr the airfoil will expe­
rience a dram atic increase in drag.
Returning to Fig. 5.17, we see that the point on the airfoil where the local M
is a peak value is also the point o f m inim um surface pressure. From the defini­
tion o f the pressure coefficient, Eq. (5.27), Cp will correspondingly have its m ost
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Critical Maeh number
for the airfoil
<r)
Figure 5.17 Illustration of critical Mach number.
negative value at this point. M oreover, according to the Prandtl-G lauert rule,
Eq. (5.28), as
is increased from 0.3 to 0.61, the value o f C,, at this point will
becom e increasingly negative. This is sketched in Fig. 5.18. The specific value of
Cp that corresponds to sonic flow is defined as the critical pressure coefficient
Cp,„. In Fig. 5.17a and 5.176, Cp at the m inim um pressure point on the airfoil is
less negative than Cp,a \ however, in Fig. 5.17c, Cp = C p,cr (by definition).
C onsider now three different airfoils ranging from thin to thick, as shown in
Fig. 5.19. Concentrate first on the thin airfoil. Because o f the thin, stream lined pro­
file, the flow over the thin airfoil is only slightly perturbed from its free-stream
5.9
Critical Mach Number and Critical Pressure Coefficient
Thick
Figure 5.19 Critical pressure coefficient and critical Mach numbers for airfoils of different
thicknesses.
values. The expansion over the top surface is mild, the velocity increases only
slightly, the pressure decreases only a relatively small amount, and hence the mag­
nitude of Cp at the minimum pressure point is small. Thus, the variation of Cp with
Moo is shown as the bottom curve in Fig. 5 .19. For the thin airfoil, CP)o is small in
magnitude, and the rate of increase of Cp as
increases is also relatively small.
In fact, because the flow expansion over the thin airfoil surface is mild, Mx can be
increased to a large subsonic value before sonic flow is encountered on the airfoil
surface. The point corresponding to sonic flow conditions on the thin airfoil is
labeled point a in Fig. 5.19. The values of Cp and
at point a are CPiCt and Mcr,
respectively, for the thin airfoil, by definition.
Now consider the airfoil of medium thickness. The flow expansion over the
leading edge for this medium airfoil will be stronger, the velocity will increase to
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larger values, the pressure will decrease to lower values, and the absolute magni­
tude of Cp is larger. Thus, the pressure coefficient curve for the medium-thickness
airfoil will lie above that for a thin airfoil, as demonstrated in Fig. 5.19. More­
over, because the flow expansion is stronger, sonic conditions will be obtained
sooner (at a lower M 00). Sonic conditions for the medium airfoil are labeled as
point b in Fig. 5.19. Note that point b is to the left of point a\that is, the critical
Mach number for the medium-thickness airfoil is less than M cr for the thin airfoil.
The same logic holds for the pressure coefficient curve for the thick airfoil, where
C p cr and M cr are given by point c. We emphasize that the thinner airfoils have
higher values of MCT. As we will see, this is desirable, and hence all airfoils on
modern, high-speed airplanes are relatively thin.
The pressure coefficient curves in Fig. 5.19 are shown as solid curves. On
these curves, only points a, b, and c are critical pressure coefficients, by defini­
tion. However, these critical points by themselves form a locus represented by
the dotted curve in Fig. 5.19; that is, the critical pressure coefficients themselves
are given by a curve of CpM = f(M 00), as labeled in Fig. 5.19. Let us proceed to
derive this function. It is an important result, and it also represents an interesting
application of our aerodynamic relationships developed in Chap. 4.
First, consider the definition of Cp from Eq. (5.27):
(5.41)
From the definition o f dynamic pressure,
However, from Eq. (4.53), a
= y Poo/Poo- Thus,
IV 2
y
qoo = ~ - f r P ° o = -zPooMoo
(5.42)
We will return to Eq. (5.42) in a moment. Now, recall Eq. (4.74) for isentropic
flow,
This relates the total pressure po at a point in the flow to the static pressure p and
local Mach number M at the same point. Also, from the same relation,
This relates the total pressure po in the free stream to the free-stream static
pressure p^, and Mach number M » . For an isentropic flow, which is a close ap­
proximation to the actual, real-life, subsonic flow over an airfoil, the total pressure
remains constant throughout. (We refer to more advanced books in aerodynamics
for a proof of this fact.) Thus, if the two previous equations are divided, pa will
5 .9
Critical Mach Number and Critical Pressure Coefficient
cancel, yielding
(5.43)
Substitute Eqs. (5.42) and (5.43) into Eq. (5.41):
{ y PooM ^1
VMl
_ | + i(y _ l)M 2
-
1
(5.44)
For a given free-stream Mach number Mx , Eq. (5.44) relates the local value of
Cp to the local M at any given point in the flow field, hence at any given point on
the airfoil surface. Let us pick that particular point on the surface where M = 1.
Then, by definition, Cr = CPXI. Putting M = 1 into Eq. (5.44), we obtain
(5.45)
Equation (5.45) gives the desired relation CP)Cr = / ( M ^ ) . When numbers are
fed into Eq. (5.45), the dotted curve in Fig. 5.19 results. Note that as M ^ in­
creases, Cp,cr decreases.
Com m entary Pause for a moment, and let us review what all this means.
In the author’s experience, the concepts of critical Mach number and critical
pressure coefficients are difficult for the first-time reader to fully understand.
So let us elaborate. Equations (5.44) and (5.45) are strictly aerodynamics; they
have nothing to do with the shape or angle of attack of a given airfoil. Indeed,
Eq. (5.44) for a compressible flow plays a role analogous to that of Bernoulli’s
equation for an incompressible flow. For an incompressible flow, Bernoulli’s
equation, Eq. (4.9), written between the free-stream point where the pressure and
velocity are /?oc and Vqo, respectively, and another arbitrary point in the flow field
where the pressure and velocity are p and V, respectively, is
P ~ Poo = |/o(V¿ - V2)
(5.46)
For the given free-stream conditions of p ^ and V^,, at any other point in the
incompressible flow where the local velocity is V, the pressure p at that point is ob­
tained from Eq. (5.46). Now focus on Eq. (5.44). Here we are dealing with a com­
pressibleflow, where Mach number rather than velocity plays the controlling role.
For the given free-stream M 00, at any other point in the compressible flow where
the local Mach number is M, the pressure coefficient at that point is obtained from
Eq. (5.44). Hence the analogy with Bernoulli’s equation. In turn, this reflects on
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Eq. (5.45). Consider a flow with a free-stream Mach number M ^. Assume that at
some local point in this flow, the local Mach number is 1. Equation (5.45) gives
the value of the pressure coefficient at this local point where we have Mach 1.
Again, we define the value of the pressure coefficient at a point where M = 1 as
the critical pressure coefficient Cpxr. Hence, when M in Eq. (5.44) is set equal to
1, the corresponding value of the pressure coefficient at that same point where
M = 1 is, by definition, the critical pressure coefficient. It is given by Eq. (5.45),
obtained by setting M — 1 in Eq. (5.44). If we graph the function given in
Eq. (5.45), that is, if we make a plot of Cpxr versus M ^, we obtain the dashed
curve in Fig. 5.19.
The fact that C,,XI decreases as
increases makes physical sense. For ex­
ample, consider a free stream at M& = 0.5. To expand this flow to Mach 1 re­
quires a relatively large pressure change p — p ^ , and therefore a relatively large
(in magnitude) pressure coefficient because, by definition, Cp = (p — Poo)/QooHowever, consider a free stream at
= 0.9. To expand this flow to Mach 1 re­
quires a much smaller pressure change; that is, p — Poo is much smaller in mag­
nitude. Hence, the pressure coefficient Cp = (p — /?oo)/<7oo will be smaller in
magnitude. As a result, Cpxr decreases with M^,, as shown by the dashed curved
in Fig. 5.19. Moreover, this dashed curve is a fixed “universal” curve— it is sim­
ply rooted in pure aerodynamics, independent of any given airfoil shape or angle
of attack.
How to Estimate the Critical Mach Number for an Airfoil Consider a given
airfoil at a given angle of attack. How can we estimate the critical Mach number
for this airfoil at the specified angle of attack? We will discuss two approaches to
the solution: a graphical solution and an analytical solution.
The graphical solution involves several steps, itemized as follows.
1.
2.
Obtain a plot of Cpxt versus M ^ from Eq. (5.45). This is illustrated by
curve A in Fig. 5.20. As discussed previously, this curve is a fixed
“universal” curve that you can use for all such problems.
For low-speed, essentially incompressible flow, obtain the value o f the
minimum pressure coefficient on the surface of the airfoil. The minimum
pressure coefficient corresponds to the point of maximum velocity on the
airfoil surface. This minimum value of Cp must be given to you, from
either experimental measurement or theory. This is Cp,0 shown as point B
in Fig. 5.20.
3.
Using Eq. (5.28), plot the variation of this minim um coefficient versus
Moo. This is illustrated by curve C in Fig. 5.20.
4.
Where curve C intersects curve A, the minimum pressure coefficient on
the surface of the airfoil is equal to the critical pressure coefficient. This
intersection point is denoted by point D in Fig. 5.20. For the conditions
associated with this point, the maximum velocity on the airfoil surface is
exactly sonic. The value o f M ^ at point D is then, by definition, the critical
Mach number.
5 .9
Critical Mach Number and Critical Pressure Coefficient
Figure 5.20 Determination of critical Mach number.
The analytical solution for MQS is obtained as follows. Equation (5.28), re­
peated here, gives the variation of C,, at a given point on the airfoil surface as a
function of M ^.
At some location on the airfoil surface, Cp%0 will be a minim um value, corre­
sponding to the point of maximum velocity on the surface. The value of the
minim um pressure coefficient will increase in absolute magnitude as Moo is
increased, owing to the compressibility effect discussed in Sec. 5.6. Hence,
Eq. (5.28) with Cp_o being the minimum value on the surface of the airfoil at es­
sentially incompressible flow conditions (M00 < 0.3) gives the value of the
m inim um pressure coefficient at a higher Mach number M ^ . However, at some
value of Moo, the flow velocity will become sonic at the point of minim um
pressure coefficient. The value of the pressure coefficient at sonic conditions is
the critical pressure coefficient, given by Eq. (5.45). When the flow becomes
sonic at the point of m inim um pressure, the pressure coefficient given by
Eq. (5.28) is precisely the value given by Eq. (5.45). Equating these two relations,
we have
Cp,o
y/l ~ M l
_
2
YMIo
- 1 ) < 1 yHy-1) |
y+1 J
')
r 2 + (y
L
The value of Moo that satisfies Eq. (5.47) is that value when the flow becomes
sonic at the point of maximum velocity (m inim um pressure). That is, the value of
Moo obtained from Eq. (5.47) is the critical Mach number for the airfoil. To
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emphasize this, we write Eq. (5.47) with M00 replaced by A/cr.
(5.48)
Equation (5.48) allows a direct analytical estimate for the critical Mach number
of a given airfoil at a given angle of attack. Note that Eq. (5.48) must be solved
implicitly for A/cr; for example, by trial and error, guessing at a value of M cr, see­
ing if it satisfies Eq. (5.48), and then trying again.
Please note that Eq. (5.48) is simply an analytical representation of point D
in Fig. 5.20, where curves A and C intersect.
EXAM PLE 5.10
Consider the N A C A 0012 airfoil, the shape o f which is shown at the top of Fig. 5.21. The
pressure coefficient distribution over the surface o f the airfoil at zero angle o f attack is
shown at the bottom o f Fig. 5.21. These arc low-spccd values measured in a wind tunnel
at Re = 3.65 x 106. From this information, estimate the critical Mach number o f the
N A C A 0012 airfoil at zero angle o f attack.
x
c
Figure 5.21 Low-speed pressure coefficient
distribution over the surface of an NACA 0012
airfoil at zero angle of attack. R e = 3.65 x 106.
(Source: After R. J. Freuler and G. M. Gregorek, "An
Evaluation of Four Single Element Airfoil Analytical
Methods ”, in Advanced Technology Airfoil Research,
NASA CP 2045, 1978, pp. 133-162.)
5 .9
Critical Mach Number and Critical Pressure Coefficient
■ Solution
First we w ill carry out a graphical solution, and then we w ill check the answer by carry­
ing out an analytical solution.
a. Graphical solution
Let us accurately plot the curve o f C p,cr versus M ^ , represented by curve <4 in
Fig. 5.20. From Eq. (5.45), repeated here,
_
p'cr
2
Y M ¡o
Í ["2 + (y — l)A /¿ "||//(y-1)
|L
Y +
1
J
for y = 1.4, we can tabulate
A*00
0.4
0.5
0.6
0.7
0.8
0.9
Cp,cr
-3.66
-2.13
- 1.29
-0.779
-0.435
-0.188
1.0
0
The curve generated by these numbers is given in Fig. 5.22, labeled curve A.
Next, let us measure the m inim um Cp on the surface o f the airfoil from Fig. 5.21;
this value is (Cp)mm — —0.43. The experimental values for pressure coefficient shown in
Fig. 5.21 arc for low-spccd, essentially incompressible flow. Hence, in Eq. (5.28),
(C/>.o)min = —0.43. As the Mach number is increased, the location o f the point o f m inim um
pressure stays essentially the same, but the value o f the m inim um pressure coefficient
Figure 5.22 Graphical solution for the critical Mach number,
from Example 5.10.
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varies according to Eq. (5.28). Hence,
,^ s _
v '-'-w /m in —
(Cp.o)min
rz--------.
.
\/! - Mlo
_
—0.43
—
t— -------- -tF-
V l ~ Mlo
Some values of (Cp)min are tabulated in the following.
M«0
(Cp)min
0
-0.43
0.2
0.4
0.6
0.8
-0.439
-0.469
-0.538
-0.717
The curve generated by these numbers is given in Fig. 5.22, labeled curve C. The inter­
section of curves A and C is at point D. The free-stream Mach number associated with
point D is the critical Mach number. From Fig. 5.22, we have
Mcr = 0.74
b. Analytical solution
Solve Eq. (5.48) for
with C,,,o = —0.43. We can do this by trial and error. As­
sume different values for Mcr, and find by iteration the value that satisfies Eq. (5.48).
Afcr
0.72
0.73
0.74
0.738
0.737
0.7371
/I
-0.43
- ^cr
..
2 |I L
YMlr
—0.6196
—0.6292
-0.6393
-0.6372
-0.6362
-0.6363
J
I"2 + (y — l)M jr 1
Y+ 1
Y /(y - 1)
)
“ 'I
-0.6996
-0.6621
-0.6260
-0.6331
-0.6367
-0.6363
To four-place accuracy, when M„ = 0.7371, both the left- and right-hand sides of
Eq. (5.48) agree, also to four-place accuracy. Hence, from the analytical solution, we have
M„ = 0.7371
Note: Compare the results from the graphical solution and the analytical solution. To the
two-place accuracy of the graphical solution, both answers agree.
Question: How accurate is the estimate of the critical Mach number ob­
tained in Example 5.10? The pressure coefficient data in Fig. 5.23a and b provide
an answer. W ind tunnel measurements of the surface pressure distributions on an
N A C A 0012 airfoil at zero angle of attack in a high-speed flow are shown
in Fig. 5.23; for Fig. 5.23a, M <*, = 0.575, and for Fig. 5.23b,
= 0.725. In
Fig. 5.23a, the value of Cp,cr = -1.465 at
= 0.575 is shown as the dashed
horizontal line. From the definition of critical pressure coefficient, any local value
of Cp above this horizontal line corresponds to locally supcrsonic flow, and any
local value below the horizontal line corresponds to locally subsonic flow. Clearly
from the measured surface pressure coefficient distribution at M = 0.575 shown
5 .9
Critical Mach Number and Critical Pressure Coefficient
x
c
X
c
Figure 5.23 Wind tunnel measurements of surface
pressure coefficient distribution for the NACA
0012 airfoil at zero angle of attack.
(Source: Experimental data of Frueler and Gregorek,
NASA CP 2045 (a) Mx = 0.575, (b)
= 0.725.)
in Fig. 5.23a, the flow is locally subsonic at every point on the surface. Hence,
Moo = 0.575 is below the critical Mach number. In Fig. 5.23b, which is for a
higher Mach number, the value of C,,.cr = —0.681 at M * = 0.725 is shown as the
dashed horizontal line. Here, the local pressure coefficient is higher than Cl>xr at
every point on the surface except at the point of minimum pressure, where (C,,),™
is essentially equal to Cp>cr. This means that for M <*, = 0.725, the flow is locally
subsonic at every point on the surface except the point of minimum pressure,
where the flow is essentially sonic. Hence, these experimental measurements
indicate that the critical Mach number of the N A C A 0012 airfoil at zero angle
of attack is approximately 0.73. Comparing this experimental result with the
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calculated value of M cr = 0,74 from Example 5.10, we see that our calculations
are amazingly accurate, to within about 1 percent.
Location of Point of Maximum Velocity (Minimum Pressure) One final
observation in this section can be made from studying the pressure coefficient
distributions, shown in Figs. 5.21 and 5.23, and the shape of the N A C A 0012
airfoil, shown at the top of Fig. 5.21. Note that the m inim um pressure (hence
maximum velocity) does not occur at the location o f maximum thickness of the
airfoil. From the airfoil shape given in Fig. 5.21, the maximum thickness is at
x/c = 0.3. From the surface pressure coefficient distributions shown in
Figs. 5.21 and 5.23, the point of minimum pressure (maximum velocity) on the
surface is at x/c = 0 . 11, considerably ahead of the point of maximum thickness.
Your intuition might at first suggest that the point of maximum velocity (m ini­
mum pressure) might be at the point of maximum thickness, but this intuition is
wrong. Nature places the maximum velocity at a point that satisfies the physics
of the whole flow field, not just what is happening in a local region of flow. The
point of maximum velocity is dictated by the complete shape of the airfoil, not
just by the shape in a local region.
5.10 DRAG-DIVERGENCE M ACH NUMBER
We now turn our attention to the airfoil drag coefficient c¿¡. Figure 5.24 sketches
the variation o f c¿ with
At low Mach numbers, less than Mct, q is virtually
constant and is equal to its low-speed value given in App. D. The flow field about
the airfoil for this condition (say, point a in Fig. 5.24) is noted in Fig. 5.25a,
divergence
Figure 5.24 Variation of drag coefficient with Mach number.
5 .1 0
M
qi
^ A/qo
Drag-Divergence Mach Number
-^ drag
divergence
(b)
Shock wave
M < 1
Mqo■’ Ai(jrag
Separated flow
divergence
(<■)
Figure 5.25 Physical mechanism of drag divergence.
(a) Flow field associated with point a in Fig. 5.19.
(b) Flow field associated with point b in Fig. 5.19.
(c) Flow field associated with point c in Fig. 5.19.
where A/ < 1 everywhere in the How. If M ^ is increased slightly above M cr, a
“bubble” of supersonic flow will occur, surrounding the minimum pressure
point, as shown in Fig. 5.25b. Correspondingly, c¡¡ will still remain reasonably
low, as indicated by point b in Fig. 5.24. However, if
is still further in­
creased, a very sudden and dramatic rise in the drag coefficient will be observed,
as noted by point c in Fig. 5.24. Here, shock waves suddenly appear in the flow,
as sketched in Fig. 5.25c. The effect of the shock wave on the surface pressure
distribution can be seen in the experimental data given in Fig. 5.26. Here, the sur­
face pressure coefficient is given for an N A C A 0012 airfoil at zero angle of at­
tack in a free stream with
= 0.808. (Figure 5.26 is a companion figure to
Figs. 5.21 and 5.23.) Comparing the result of Example 5.10 and the data shown
in Fig. 5.23b, we know that
= 0.808 is above the critical Mach number for
the N A C A 0012 airfoil at zero angle of attack. The pressure distribution in
Fig. 5.26 clearly shows that fact; the shape of the pressure distribution curve is
quite different from that in the previous figures. The dashed horizontal line in
Fig. 5.26 corresponds to the value of Cp ct at M<*, = 0.808. Note that the flow
velocity at the surface is locally supersonic in the region 0.11 < x/c < 0.45.
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x
c
Figure 5.26 Wind tunnel measurements of the
surface pressure coefficient distribution for the
N A CA 0012 airfoil at zero angle of attack for
Mao = 0.808, which is above the critical Mach
number.
(Soruce: Experimental data arefrom Freulerand
Gregorek, NASA CP 2045, and are a companion to the
data shown in Figs. 5.21 and 5.23.)
Recall from our discussion of shock waves in Sec. 4.11.3 that the pressure
increases and the velocity decreases across a shock wave. We clearly see these
phenomena in Fig. 5.26; the large and rather sudden increase in pressure at
x/c = 0.45 indicates the presence o f a shock wave at that location, and the flow
velocity drops from supersonic in front of the shock to subsonic behind the
shock. (The drop in velocity to subsonic behind the shock, rather than just a de­
crease to a smaller supersonic value, is a characteristic of shock waves that are
essentially normal to the flow, as occurs here.)
The shock waves themselves are dissipative phenomena, which result in
an increase in drag on the airfoil. But in addition, the sharp pressure increase
across the shock waves creates a strong adverse pressure gradient, causing the
flow to separate from the surface. As discussed in Sec. 4.20, such flow separation
can create substantial increases in drag. Thus, the sharp increase in cd shown in
Fig. 5.24 is a combined effect of shock waves and flow separation. The freestream Mach number at which q begins to increase rapidly is defined as the
drag-divergence Mach number and is noted in Fig. 5.24. Note that
Mct < M j raed iv e r g e n c e < 1.0
The shock pattern sketched in Fig. 5.25c is characteristic of a flight regime
called transonic. When 0.8 < M ^ < 1.2, the flow is generally designated as
5 .1 0
0
0.1
0.2
0.3
0.4
0.5
0.6
Mach number
(a)
0.7
0.8
0.9
1.0
Drag-Divergence Mach Number
0
0.1
0.2
0.3
0.4
0.5
297
0.6
0.7
0.8
0.9
Mach number M „
(6)
Figure 5.27 Variation of (a) lift coefficient and (b) drag coefficient versus Mach number with angle of attack as a
parameter for an N A CA 2315 airfoil.
(Source: Wind tunnel measurements at the NACA Langley Memorial Laboratory.)
transonic flow, and it is characterized by some very complex effects only hinted
at in Fig. 5.25c. To reinforce these comments, Fig. 5.27 shows the variation of
both ci and c¡¡ as a function of Mach number with angle of attack as a parameter.
The airfoil is a standard N A C A 2315 airfoil. Figure 5.27, which shows actual
wind tunnel data, illustrates the massive transonic-flow effects on both lift and
drag coefficients. The analysis of transonic flows has been one of the major
challenges in modern aerodynamics. Only in recent years, since about 1970,
have computer solutions for transonic flows over airfoils come into practical use;
these numerical solutions are still in a state of development and improvement.
Transonic flow has been a hard nut to crack.
1.0
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DESIGN BOX
The designers of transonic airplanes are frequently
looking for ways to get the speed closer to Mach 1
without encountering the large transonic drag rise.
These designers have two options in regard to the
choice of an airfoil that will delay drag divergence to
a higher Mach number: (I) Make the airfoil thin and
(2) adopt a specially shaped airfoil called a supercrit­
ical airfoil. These options can be used singly or in
combination.
In regard to airfoil thickness, the generic trend
sketched in Fig. 5.19 clearly shows that Mcr is in­
creased by making the airfoil thinner. An increase in
M„ usually means an increase in the drag-divergence
Mach number. Hence, everything else being equal, a
transonic airplane with a thinner airfoil can fly at a
higher Mach number before encountering drag diver­
gence. Indeed, this knowledge was incorporated in
the design of the famous Bell X-1, which was the first
airplane to fly faster than sound (see Sec. 5.22). The
X-l was designed with two sets of wings, one with a
10 percent thick airfoil for more routine flights and
another with an 8 percent thick airfoil for flights in­
tended to penetrate through Mach 1. The airfoil sec­
tions were NACA 65-110 and NACA 65-108, respec­
tively. Moreover, the horizontal tail was even thinner
in both cases, being an NACA 65-008 (8 percent
thickness) and an NACA 65-006 (6 percent thick­
ness), respectively. This was done to ensure that
when the wing cncountcred major compressibility ef­
fects, the horizontal tail and elevator would still be
free of such problems and would be functional for
stability and control. A three-view of the Bell X-l is
shown in Fig. 5.28.
The adverse compressibility effects that cause
the dramatic increase in drag and precipitous de­
crease in lift, shown in Fig. 5.27. can be delayed by
decreasing the airfoil thickness. The knowledge of
this fact dates back as early as 1918. In that year, as
World War I was coming to an end, Frank Caldwell
and Elisha Fales, two engineers at the U.S. Army’s
McCook Field in Dayton, Ohio, measured these ef­
fects in a high-speed wind tunnel capable ot produc­
ing a test stream of 465 mi/h. This knowledge was re­
inforced by subsequent high-speed wind tunnel tests
carried out by NACA in the 1920s and 1930s. (For a
detailed historical treatment of the evolution of our
understanding of compressibility effects during this
period, see Anderson, A History ofAerodynamics and
Its Impact on Flying Machines, Cambridge Univer­
sity Press, 1997. See also Anderson, “Research in Su­
personic Flight and the Breaking of the Sound Bar­
rier,” chapter 3 in From Engineering Science to Big
Science, edited by Pamela Mack, NASA SP-4219,
1998.)
Thinner airfoils are also advantageous for super­
sonic airplanes, for reasons to be discussed in
Sec. 5.11. Indeed, in airplane design, the higher the
design Mach number, usually the thinner the airfoil
section. This is dramatically shown in Fig. 5.29,
which is a plot of airfoil thickness versus design Mach
number for a variety of high-speed airplanes since
World War II. As the design Mach number of airplanes
increased, thinner airfoils became a design necessity.
The supercritical airfoil is a different approach
to the increase in drag-divergence Mach number.
Here, the shape of the airfoil is designed with a rela­
tively flat top surface, as shown in Fig. 5.30. When
the free-stream Mach number exceeds Mcr, a pocket
of supersonic flow occurs over the top surface as
usual; but because the top is relatively flat, the local
supersonic Mach number is a lower value than would
exist in the case of a conventional airfoil. As a result,
the shock wave that terminates the pocket of super­
sonic flow is weaker. In turn, the supercritical airfoil
can penetrate closer to Mach I before drag diver­
gence occurs. In essence, the increment in Mach
number (the “grace period”) between Mcr and
^dragdivergence (see Fig. 5.24) is increased by the shape
of the supercritical airfoil. One way to think about
this is that the supercritical airfoil is “more comfort­
able” than conventional airfoils in the region above
M„, and it can fly closer to Mach 1 before drag di­
vergence is encountered. Because they are more
comfortable in the flight regime above the critical
Mach number and because they can penetrate closer
to Mach I after exceeding M„, these airfoils are
called supercritical airfoils. They are designed to
cruise in the Mach number range above Mcr.
The pressure coefficient distribution over the top
surface of a supercritical airfoil flying above Mc, but
Drag-Divergence Mach Number
299
Figure 5.28
Three-view of the
Bell X-l.
UO i 0‘
»>ftN
(continued on next page)
300
chapter
5 Airfoils, Wings, and Other Aerodynamic Shapes
(continuedfrom page 299)
Figure 5.29 Variation of thickness-to-chord ratio with Mach number for a representative
sampling of different airplanes.
(Source: After Ray Whitford, Design for Air Combat. Jane’s Information Group, Surry, England. 1989.)
below Mdragdivcrgencc is sketched in Fig. 5.30. After a
sharp decrease in pressure around the leading edge,
the pressure remains relatively constant over a sub­
stantial portion o f the top surface. This is in contrast
to the pressure coefficient distribution for a conven­
tional airfoil flying above A/cr, such as shown in
Fig. 5.26. Clearly, the flow over the supercritical
airfoil is carefully tailored to achieve the desired
results.
The early aerodynamic research on supercritical
airfoils was carried out by Richard Whitcomb, an
aeronautical engineer at N A S A Langley Research
Center, during the middle 1960s. This work by
W hitcomb is described in a N A S A document entitled
“An Airfoil Shape for Efficient Flight at Supercritical
Mach Numbers” (N A S A TM X-l 109, July 1965, by
R. T. W hitcomb and L. R. Clark). W hitcomb's design
o f supercritical airfoils was pioneering; today, all
modern civilian jet transports are designed with su­
percritical wings, incorporating custom-designed su­
percritical airfoil sections that have their genes in the
original design by Richard Whitcomb.
The effectiveness of the supercritical airfoil was
clearly established by an Air Force/NASA wind tun­
nel and flight test program carried out in the early
1970s called the Transonic Aircraft Technology
(TACT) program. A standard General Dynamics
F-111 (sketched at the top o f Fig. 5.31) was modified
5.10
301
Drag-Divergence Mach Number
c
Figure 5.30 Shape of a typical supercritical airfoil and its pressure
coefficient distribution over the top surface.
0.60
0.70
0.80
0.90
l.O
Free-strcam Mach number
Figure 5.31 Increase in drag-divergence Mach number obtained by the TACT aircraft
with a supercritical wing compared to a standard F-l 11. Wind tunnel data obtained at
the NASA Langley Research Center. Wing sweep = 26°. C l held constant at 0.0465.
(Source: Reported in Symposium on Transonic Aircraft Technology (TACT), AFFDL-TR-7H100, Air Force Flight Dynamics Laboratory, August 197ft.)
(continued on next pane)
302
chapter
5
Airfoils, Wings, and Other Aerodynamic Shapes
(continuedfrom page 301)
with a supercritical wing. Wind tunnel data for the
variation of Co with
for both the standard
F-lll and the TACT aircraft (the F-lll modified
with a supercritical wing) are shown in Fig. 5.31.
The standard airfoil on the F-lll is an NACA 64210; the supercritical airfoil on the TACT aircraft
had the same 10 percent thickness. The use of the
supercritical wing increased the drag-divergence
Mach number from 0.76 to 0.88, a stunning 16 per­
cent increase, as noted in Fig. 5.31.
Designers of transonic aircraft can use supercrit­
ical airfoils to accomplish one of two objectives:
(1) For a given airfoil thickness, the supercritical air­
foil shape allows a higher cruise velocity; or (2) for a
given lower cruise velocity, the airfoil thickness can
be larger. The latter option has some design advan­
tages. The structural design of a thicker wing is more
straightforward and actually results in a more light­
weight (albeit thicker) wing. Also, a thicker wing
provides more volume for an increased fuel capacity.
Clearly, the use of a supercritical airfoil provides a
larger “design space” for transonic airplanes.
5.11 WAVE DRAG (AT SUPERSONIC SPEEDS)
To this point, we have discussed airfoil properties at subsonic speeds, that is, for
Moo < 1• When Moo is supersonic, a major new physical phenomenon is intro­
duced: shock waves. We previously alluded to shock waves in Sec. 4.11.3 in con­
junction with the Pitot tube measurement of supersonic airspeeds. W ith respect
to airfoils (as well as all other aerodynamic bodies), shock waves in supersonic
flow create a new source o f drag, called wave drag. In this section, we will sim­
ply highlight some o f the ideas involving shock waves and the consequent wave
drag; a detailed study of shock wave phenomena is left to more advanced texts in
aerodynamics.
To obtain a feel for how a shock is produced, imagine that we have a small
source of sound waves, a tiny “beeper” (something like a tuning fork). At time
t = 0, assume the beeper is at point P in Fig. 5.32. At this point, let the beeper
emit a sound wave, which will propagate in all directions at the speed of sound
a. Also let the beeper move with velocity V, where V is less than the speed of
sound. At time t, the sound wave will have moved outward by a distance at, as
shown in Fig. 5.32. At the same time t, the beeper will have moved a distance Vt,
to point Q. Since V < a, the beeper will always stay inside the sound wave. If
the beeper is constantly emitting sound waves as it moves along, these waves
will constantly move outward, ahead of the beeper. As long as V < a, the beeper
will always be inside the envelope formed by the sound waves.
On the other hand, assume the beeper is moving at supersonic speed; that is,
V > a. At time t = 0, assume the beeper is at point R in Fig. 5.33. At this point,
let the beeper emit a sound wave, which, as before, will propagate in all direc­
tions at the speed of sound a. At time t, the sound wave will have moved outward
by a distance a t , as shown in Fig. 5.33. At the same time t, the beeper will have
moved a distance V i, to point S. However, since V > a, the beeper will now be
outside the sound wave. If the beeper is constantly emitting sound waves as it
moves along, these waves will now pile up inside an envelope formed by a line
5 .11
Wave Drag (at Supersonic Speeds)
Location of sound
the sound wave
Figure 5.32 Beeper moving at less than the speed of sound.
the sound wave
Figure 5.33 The origin of Mach waves and shock waves. Beeper is moving faster
than the speed of sound.
from point S tangent to the circle formed by the first sound wave, centered at
point R. This tangent line, the line where the pressure disturbances are piling up,
is called a Mach wave. The vertex of the wave is fixed to the moving beeper at
point S. In supersonic flight, the air ahead of the beeper in Fig. 5.33 has no
warning of the approach of the beeper. Only the air behind the Mach wave has
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Airfoils, Wings, and Other Aerodynamic Shapes
felt the presence of the beeper, and this presence is communicated by pressure
(sound) waves confined inside the conical region bounded by the Mach wave. In
contrast, in subsonic flight, the air ahead of the beeper in Fig. 5.32 is forewarned
about the oncoming beeper by the sound waves. In this case, there is no piling up
of pressure waves; there is no Mach wave.
Hence, we can begin to feel that the coalescing, or piling up, of pressure
waves in supersonic flight can create sharply defined waves o f some sort. In
Fig. 5.33, the Mach wave that is formed makes an angle /i with the direction of
movement of the beeper. This angle, defined as the Mach angle, is easily
obtained from the geometry of Fig. 5.33, as follows:
at
a
Hence,
1
(5.49)
In real life, a very thin object (such as a thin needle) moving at M ^ > 1 cre­
ates a very weak disturbance in the flow, limited to a Mach wave. This is sketched
in Fig. 5.34. On the other hand, an object with some reasonable thickness, such
as the wedge shown in Fig. 5.35, moving at supersonic speeds will create a strong
disturbance, called a shock wave. The shock wave will be inclined at an oblique
angle ¡3, where /J > p, as shown in Fig. 5.35. As the flow moves across the
oblique shock wave, the pressure, temperature, and density increase, and the ve­
locity and Mach number decrease.
Consider now the pressure on the surface of the wedge, as sketched in
Fig. 5.36. Since p increases across the oblique shock wave, at the wedge surface,
p > Poo. Since the pressure acts normal to the surface and the surface itself is in­
clined to the relative wind, there will be a net drag produced on the wedge, as
seen by simple inspection of Fig. 5.36. This drag is called wave drag, because it
is inherently due to the pressure increase across the shock wave.
Mee> l
Mao> 1
Figure 5.34 Mach waves on a needlelike body.
Figure 5.35 Oblique shock waves on a wedge-type body.
5 .11
Wave Drag (at Supersonic Speeds)
D
Wave drag
net drag due to
higher pressure
behind the
shock wave
PoO
Figure 5.36 Pressure distribution on a wedge at supersonic speeds; origin of
wave drag.
Expansion
Pressure
distribution
Figure 5.37 Flow field and pressure distribution for a flat plate at angle of attack in
supersonic flow. There is a net lift and drag due to the pressure distribution set up by the
shock and expansion waves.
To minimize the strength of the shock wave, all supersonic airfoil profiles
are thin, with relatively sharp leading edges. (The leading edge of the Lockheed
F-104 supersonic fighter is almost razor-thin.) Let us approximate a thin super­
sonic airfoil by the flat plate illustrated in Fig. 5.37. The flat plate is inclined at a
small angle of attack a to the supersonic free stream. On the top surface of the
plate, the flow field is turned away from the free stream through an expansion
305
306
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5
Airfoils, Wings, and Other Aerodynamic Shapes
wave at the leading edge; an expansion wave is a fan-shaped region through
which the pressure decreases. At the trailing edge on the top side, the flow is
turned back towards the free-stream direction through an oblique shock wave.
On the bottom surface of the plate, the flow is turned into the free stream, caus­
ing an oblique shock wave with an increase in pressure. At the trailing edge, the
flow is turned back towards the free-stream direction through an expansion
wave. (Details and theory for expansion waves, as well as shock waves, are be­
yond the scope of this book— you will have to simply accept on faith the flow
field sketched in Fig. 5.37 until your study of aerodynamics becomes more
advanced.) The expansion and shock waves at the leading edge result in a sur­
face pressure distribution in which the pressure on the top surface is less than
Poo, whereas the pressure on the bottom surface is greater than poo- The net ef­
fect is an aerodynamic force normal to the plate. The components of this force
perpendicular and parallel to the relative wind are the lift and supersonic wave
drag, respectively. Approximate relations for the lift and drag coefficients are,
respectively,
« - IS E r iju
<5'“ >
4a2
and
=
iM l -
D 'f l
<5' 5 I )
A subscript w has been added to the drag coefficient to emphasize that it is the
wave drag coefficient. Equations (5.50) and (5.51) are approximate expressions,
useful for thin airfoils at small to moderate angles of attack in supersonic flow.
Note that as M ^ increases, both c, and c¡¡ decrease. This is not to say that the lift
and drag forces themselves decrease with Moo• Quite the contrary. For any flight
regime, as the flight velocity increases, L and D usually increase because the dy­
namic pressure
increases. In the supersonic regime, L and D in­
crease with velocity, even though c/ and cd,w decrease with M according to
Eqs. (5.50) and (5.51).
E X A M P L E 5.11
Consider a thin supersonic airfoil with chord length c — 5 ft in a Mach 3 free stream at a
standard altitude of 20,000 ft. The airfoil is at an angle of attack of 5°.
a. Calculate the lift and wave drag coefficients and the lift and wave drag per unit span.
b. Compare these results with the same airfoil at the same conditions, except at Mach 2.
■ Solution
a. In Eqs. (5.50) and (5.51), the angle of attack a must be in radians. Hence,
a = 5° = — rad = 0.0873 rad
57.3
5 .11
Wave Drag (at Supersonic Speeds)
Also,
- 1 = x/32 - 1 = 2.828
4a
4(0.0873)
y/Ml-l ~
C' “
Cd.w —
2.828
0.123
=
4a2
4(0.0873)2
s / M l- 1
2.828
0.0108
At 20,000 ft, p.*, = 1.2673 x 10~3 slug/ft3, and T = 447.43°R. Hence,
«oo = s/yRT 0o = v/l.4(1716)(447.43) = 1037 ft/s
Voo = A/oofloo = 3(1037) = 3111 ft/s
q°o =
= I (1.2673 x 10-3)(3 1 11)2 = 6133 lb/ft2
L (per unit span) = q^cct = 6133(5)(0.123) =
3772 lb
Dw (per unit span) = qooccd,w = 6133(5)(0.0108) =
b.
331.2 lb
M 2, - 1 = y/22 - 1 = 1.732
Cl -
Cd,ui —
Note: At Mach 2, c, and
4a
4(0.0873)
sj M l — I
1.732
4a
0.207
4(0.0873)2
0.0176
1-732
are higher than at Mach 3. This is a general result; both c¡
and Cd,w decrease with increasing Mach number, as clearly seen from Eqs. (5.50) and
(5.51). Does this mean that L and D w also decrease with increasing Mach number? Intu­
itively this does not seem correct. Let us find out.
Voo = flooMoo = 1037(2) = 2074 ft/s
qoo = \paoV¿ = j (1.2673
x 10~3)(2074)2 = 2726 lb/ft2
L (per unit span) = q^cci = 2726(5)(0.207) =
2821 lb
D w (per unit span) = q<x,ccd,w = 2726(5)(0.0176) =
2401b
Hence, there is no conflict with our intuition. As the supersonic Mach numbers increase,
L and Dw also increase although the lift and drag coefficients decrease.
307
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5
Airfoils, Wings, and Other Aerodynamic Shapes
E X A M P L E 5.12
The Lockheed F-104 supersonic fighter is shown in three-view in Fig. 4.45 and in the
photograph in Fig. 5.38. It is the first fighter aircraft designed for sustained flight at
Mach 2. Its wing planform area is 19.5 m2. Consider the F-104 in steady, level flight, and
assume its weight is 7262 kg^ Calculate its angle of attack at Mach 2 when it is flying at
(a) sea level and (b) 10 km.
■ Solution
We assume the F-104 wing in supersonic flight can be represented by a flat plate and that
the wing lift coefficient is given by Eq. (5.50). Although this equation holds for a flat-plate
airfoil section, we assume that it gives a reasonable estimate for the straight wing of the
F-104. Please keep in mind that Eq. (5.50) is only an approximation for the finite wing.
The weight is given in kgy, a nonconsistent unit. As shown in Example 2.5,
1 kg| = 9.8 N. Also, in steady, level flight, the lift equals the weight of the airplane. Hence,
L = W = 7262 (9.8) = 7.12 x 104 N
a. At sea level, poo = 1 -23 kg/m3, and T,x = 288 K. The speed of sound is given by
aoo = JyRToo = v/(1.4)(287)(288) = 340 m/s
Thus,
Voo = OooMoo = (340) (2) = 680 m/s
q0c = \pooVl = i(1.23)(680)2 = 2.84 x 105 N/m2
Figure 5.38 The first airplane to be designed for sustained flight at Mach 2, the Lockheed
F-104 Starfighter.
Wave Drag (at Supersonic Speeds)
5 .11
309
From Eq. (5.50),
4a
Cl =
ci /77>
7
0.014 r—z---
.
a = j y j M l - \= — — V (2 )2 - l = 6.06 x 10~3 rad
or
In degrees,
a = (6.06 x l0 -3)(57.3) =
0.35°
Note: This is a very small angle of attack. At Mach 2 at sea level, the dynamic pressure is
so large that only a very small lift coefficient, hence very small angle o f attack, is needed
to sustain the airplane in the air.
b. At 10 km, from App. A,
= 0 .4 13 5 1 kg/m3, and T,x = 223.26 K.
a0o = y/yRToo = v/(1.4)(287)(223.26) = 300 m/s
Voo = flooW;» = (300) (2) = 600 m/s
<7oo = jA » V £ = 5 (0.41351) (600)2 = 7.44 x l0 4 N/m2
c, -
L
7.12 x 104
qooS
(7.44 x 105)( 19.5)
= 0.049
Ci I ~
0.049 /— z--— 1I == 1 - V ( 2 ) 2 - 1 = 0 .0 2 rad
“ = 74 V.IM'M°° ~
In degrees,
a = (0.021 )(57.3) =
1. 2 °
Note: At an altitude o f 10 km, where the dynamic pressure is smaller than at sea level, the
required angle o f attack to sustain the airplane in flight is still relatively small, only
slightly above 1 degree. We learn from this example that airplanes in steady level flight at
supersonic speeds fly at very small angles o f attack.
KXAMPLK 5.13
If the pilot o f the F-104 in Example 5.12, flying in steady, level flight at M ach 2 at an al­
titude o f 10 km, suddenly pitched the airplane to an angle o f attack o f 10°, calculate the
instantaneous lift exerted on the airplane, and comment on the possible consequences.
Solution
10
a = ——— = 0 .1 7 5 rad
57.3
From Eq. (5.50),
ci =
4a
4(0.175)
V (2 )2 - 1
= 0.404
310
chapter
5 Airfoils, Wings, and Other Aerodynamic Shapes
From Example 5.12, at Mach 2 and an altitude of 10 km, qoo = 7.44 x 104 N/m2
L = qooSci = (7.44 x 104)(19.5)(0.404) =
5.86 x 105 N
Compare this value of lift with the weight of the airplane:
L
5.86
W ~ 7.12
x 105
o„
x 104 “
'
When the pilot suddenly increases the angle of attack to 10°, the lift increases to a
value 8.2 larger than the weight. The pilot will feel a sudden acceleration equal to 8.2
times the acceleration of gravity, or sometimes stated as an acceleration of 8.2 g’s. The
human body can withstand this acceleration only for a few seconds before becoming
unconscious. Moreover, the structure of the airplane will be under great stress. These
are reasons why, in supersonic flight, the angle of attack is usually maintained at low
values.
5.12 SUMMARY OF A IR F O IL DRAG
Amplifying Eq. (4.105), we can write the total drag of an airfoil as the sum of
three contributions:
D = Df + Dp + Dw
where
D =
Df =
Dp =
Dw=
total drag on airfoil
skin friction drag
pressure drag due to flow separation
wave drag (present only at transonic and supersonic speeds; zero
for subsonic speeds below the drag-divergence Mach number)
In terms of the drag coefficients, we can write
Cd = Cd,f
“I"
Cd,p
” 1”
Cd,w
where q , c¿j , cdp, and cdw are the total drag, skin friction drag, pressure drag,
and wave drag coefficients, respectively. The sum cdj + cd,p is called the profile
drag coefficient; this is the quantity that is given by the data in App. D. The pro­
file drag coefficient is relatively constant with
at subsonic speeds.
The variation of cd with
from incompressible to supersonic speeds is
sketched in Fig. 5.39. It is important to note the qualitative variation of this curve.
For A/oo ranging from zero to drag divergence, cd is relatively constant; it con­
sists entirely of profile drag. For M00 from drag divergence to slightly above I,
the value of cd skyrockets; indeed, the peak value of cd around Moo = 1 can be
an order of magnitude larger than the profile drag itself. This large increase in cd
5 .1 2
Summary of Airfoil Drag
311
DESIGN BOX
Good design of supersonic airplanes concentrates on
minimizing wave drag. It is emphasized in Fig. 5.39
that a substantial portion of the total drag at super­
sonic speeds is wave drag. In turn, the way to reduce
wave drag is to reduce the strength of the shock
waves that occur at the nose, along the leading edges
of the wing and tail, and at any other part of the air­
craft that protrudes into the locally supersonic flow.
The shock wave strength is reduced by having a sharp
nose, slender (almost needlelike) fuselage, and very
sharp wing and tail leading edges. The Lockheed F104, shown in three-view in Fig. 4.45 and in the pho­
tograph in Fig. 5.38, is an excellent example of good
supersonic airplane design. The F-104 was the first
aircraft designed for sustained speeds at Mach 2.
Examining Figs. 4.45 and 5.38, we see an aircraft
with a sharp, needlelike nose, slender fuselage, and
very thin wings and tails with sharp leading edges.
The wing airfoil section is a thin biconvex shape with
a thickness-to-chord ratio of 0.035— 3.5 percent
thickness. The leading edge is almost razor-sharp, in­
deed sharp enough to pose a hazard to ground crew
working around the airplane. Design of the F-104
began in 1953 at the famous Lockheed “Skunk
Works”; it entered service with the U.S. Air Force in
1958. Now retired from the Air Force inventory, at
the time of writing, F-I04’s are still in service with
the air forces of a few other nations around the globe.
divergence
Figure 5.39 Variation of drag coefficient with Mach number for
subsonic and supersonic speeds.
is due to wave drag associated with the presence of shock waves. For supersonic
Mach numbers, q decreases approximately as (A /¿ — l ) -1/2.
The large increase in the drag coefficient near Mach l gave rise to the term
sound barrier in the 1940s. There was a camp of professionals who at that time
felt that the sound barrier could not be pierced, that we could not lly faster than
the speed of sound. Certainly, a glance at Eq. (5.28) for the pressure coefficient
in subsonic flow, as well as Eq. (5.51) for wave drag in supersonic flow, would
hint that the drag coefficient might become infinitely large as
approaches 1
from either the subsonic or supersonic side. However, such reasoning is an
312
chapter
5
Airfoils, Wings, and Other Aerodynamic Shapes
example of a common pitfall in science and engineering, namely, the application
o f equations outside their ranges of validity. Neither Eq. (5.28) nor Eq. (5.51) is
valid in the transonic range near
= l . Moreover, remember that nature ab­
hors infinities. In real life, c¿ does not become infinitely large. To get past the
sound barrier, all that is needed (in principle) is an engine with enough thrust to
overcome the high (but finite) drag.
5.13 FINITE WINGS
We now return to the discussion initiated in Sec. 5.5. Our considerations so far
have dealt mainly with airfoils, where the aerodynamic properties are directly
applicable to infinite wings. However, all real wings are finite, and for practical
reasons, we must translate our knowledge about airfoils to the case where the
wing has wing tips. This is the purpose of Secs. 5.14 and 5.15.
Let us pose the following questions. Consider a finite wing with a specified
aspect ratio [defined by Eq. (5.26)] at an angle of attack o f 6 °. The airfoil section
of the finite wing is an N A C A 2412 section. For a = 6 °, the airfoil lift and drag
coefficients, from App. D, are
c¡ = 0.85
cj = 0.077
Question: Since the finite wing is made up of the N A C A 2412 airfoil section,
should not the wing lift and drag coefficients be the same as those for the airfoil?
That is, for the wing at a = 6 °, are the following true?
CL = 0.85
CD = 0.0077
(Recall from Sec. 5.5 that it is conventional to denote the aerodynamic coefficients
for a finite wing with capital letters, as above.) On an intuitive basis, it may sound
reasonable that C¿ and Cd for the wing might be the same as c¡ and q , respectively,
for the airfoil section that makes up the wing. But intuition is not always correct.
We will answer the preceding questions in the next few paragraphs.
The fundamental difference between flows over finite wings as opposed to
infinite wings can be seen as follows. Consider the front view of a finite wing as
sketched in Fig. 5.40a. If the wing has lift, then obviously the average pressure
over the bottom surface is greater than that over the top surface. Consequently,
there is some tendency for the air to “leak,” or flow, around the wing tips from
the high- to the low-pressure sides, as shown in Fig. 5.40a. This flow establishes
a circulatory motion that trails downstream of the wing. The trailing circular mo­
tion is called a vortex. There is a major trailing vortex from each wing tip, as
sketched in Fig. 5.40b and as shown in the photograph in Fig. 5.41.
These wing tip vortices downstream of the wing induce a small downward
component of air velocity in the neighborhood o f the wing itself. This can be
seen intuitively from Fig. 5.406; the two wing tip vortices tend to drag the sur­
rounding air around with them, and this secondary movement induces a small
Vortex
(a)
Figure 5.40 O r ig in o f w in g tip vortices o n a finite w ing .
F ig u re 5.41 W in g tip vortices m ade visible by sm oke ejected at the w ing tips o f a B oeing
727 test airplane.
(Source: NASA.)
313
314
chapter
5
Airfoils, Wings, and Other Aerodynamic Shapes
Relative wind
(free stream)
Tip vortex
'T
Local flow in
vicinity of wing
rm
n n r
The wing tip
vortex that trails
downstream causes
downwash, w
Figure 5.42 The origin of downwash.
velocity component in the downward direction at the wing. This downward com­
ponent is called downwash, given the symbol w.
An effect of downwash can be seen in Fig. 5.42. As usual, V » designates the
relative wind. However, in the immediate vicinity of the wing, V» and w add
vectorally to produce a “local” relative wind that is canted downward from the
original direction of V ». This has several consequences:
1.
The angle o f attack of the airfoil sections of the wing is effectively reduced
in comparison to the angle of attack of the wing referenced to Vx>-
2.
There is an increase in the drag. The increase is called induced drag, which
has at least three physical interpretations. First, the wing tip vortices simply
alter the flow field about the wing in such a fashion as to change the
surface pressure distributions in the direction of increased drag. An
alternate explanation is that because the local relative wind is canted
downward (see Fig. 5.42), the lift vector itself is “tilted back.” Hence it
contributes a certain component of force parallel to Voo, that is, a drag
force. A third physical explanation of the source o f induced drag is that the
wing tip vortices contain a certain amount of rotational kinetic energy. This
energy has to come from somewhere; indeed, it is supplied by the aircraft
propulsion system, where extra power has to be added to overcome the
extra increment in drag due to induced drag. A ll three of these outlooks of
the physical mechanism o f induced drag are synonymous.
We can now answer the questions posed at the beginning of this section. Re­
turning to the finite wing made up of the N A C A 2412 airfoil section, where the
wing is at a = 6 °, we now recognize that, because of the downwash, the local
airfoil sections of the wing see an angle of attack lower than 6 °. Clearly, the local
airfoil lift coefficient will then be less than 0.85. Since the lift o f the wing is an
integration of the lift from each local segment, we can state that for the finite wing
CL < 0.85
Also, the presence of induced drag for the finite wing, which is not present for an
infinite wing, adds to the already existing skin friction drag and pressure drag
due to flow separation, which is experienced by the airfoil section itself. The
value c,¡ — 0.0077 is the profile drag coefficient, which is the sum of the skin
5 .1 4
Calculation of Induced Drag
315
DESIGN BOX
For some airplane designs, the shape of the airfoil
section changes along the span of the wing. For ex­
ample, for the F-111 shown at the top of Fig. 5.31, the
airfoil section at the root of the wing is an NACA
64A210. whereas the airfoil section at the tip of the
wing is an NACA 64A209. The famous British Spit­
fire of World War 11 fame had a 13 percent thick air­
foil at the root and a 7 percent thick airfoil at the tip.
When a designer chooses to vary the airfoil shape
along the span, it is usually for one or both of the fol­
lowing reasons:
1.
To achieve a particular distribution of lift
across the span of the wing, which will improve
the aerodynamic efficiency of the wing and/or
reduce the structural weight of the wing.
2.
To delay the onset of high-speed compressibility
effects in the region near the wing tips. A thinner
airfoil in the tip region will result in the “shock
stall” pattern shown in Fig. 5.25c being delayed
in that region to a higher Mach number, hence
preserving aileron control effectiveness while
the section of the wing closer to the root may be
experiencing considerable flow separation.
In reference to our previous discussion, note that the
possible variation of the airfoil shape along the span
of a finite wing is yet another reason why the aero­
dynamic coefficients for a finite wing are different
from those for an airfoil making up part of the wing
itself.
friction and pressure drag due to flow separation. For the finite wing, the induced
drag must be added to the profile drag. Clearly, for the finite wing in this case,
CD > 0.0077
So we can rest our case. The lift coefficient for a finite wing is less than that for
its airfoil section, and the drag coefficient for a finite wing is greater than that for
its airfoil section.
In Secs. 5.14 and 5.15, we will show how the drag coefficient and the lift
coefficient, respectively, for a finite wing can be calculated. With this, we now
move to the center column of our chapter road map in Fig. 5.1. Return to Fig. 5.1
for a moment, and note all the different aspects o f airfoils that we have covered,
as represented by the left-hand column of the road map. We are now ready to use
this knowledge to examine the characteristics of finite wings, as represented by
the middle column.
5.14 CALCULATION OF INDUCED DRAG
A way of conceptualizing induced drag is shown in Fig. 5.43. Consider a finite
wing as sketched in Fig. 5.43. The dashed arrow labeled R| represents the resul­
tant aerodynamic force on the wing for the imaginary situation of no vortices
from the wing tips. The component of R| parallel to
is the drag D \
, which in
this imaginary case is due to skin friction and pressure drag due to flow separa­
tion. The solid arrow labeled R represents the actual resultant aerodynamic
force, including the effect of wing tip vortices. The presence of the vortices
changes the pressure distribution over the surface of the wing in such a fashion
that R is tilted backward relative to R | . The component of R parallel to V,*,,
316
chapter
5
Airfoils, Wings, and Other Aerodynamic Shapes
denoted by D in Fig. 5.43, is the actual total drag, which includes the effect of
the changed pressure distribution due to the wing tip vortices as well as friction
drag and pressure drag due to flow separation. Since R is tilted backward relative
to R|, then D > D\. The induced drag, D¡ is the difference between D and D\:
D¡ = D — D i . Keep in mind that induced drag is a type of pressure drag.
To calculate the magnitude of D¡, we will take the following perspective.
Consider a section of a finite wing as shown in Fig. 5.44. The angle of attack
defined between the mean chord of the wing and the direction of Voo (the rela­
tive wind) is called the geometric angle of attack a. However, in the vicinity
of the wing, the local flow is (on the average) deflected downward by angle
a¡ because of downwash. This angle a¡, defined as the induced angle of attack, is
the difference between the local flow direction and the free-stream direction.
5 .1 4
Calculation of Induced Drag
Hence, although the naked eye sees the wing at an angle of attack a, the airfoil
section itself is seeing an effective angle ofattack, which is smaller than a . Letting
a cff denote the effective angle of attack, we see from Fig. 5.44 that a eff = a — a, .
Let us now adopt the point of view that, because the local flow direction in
the vicinity of the wing is inclined downward with respect to the free stream, the
lift vector remains perpendicular to the local relative wind and is therefore tilted
back through angle a¡. This is shown in Fig. 5.44. However, still considering
drag to be parallel to the free stream, we see that the tilted-lift vector contributes
a certain component of drag. This drag is the induced drag D¡. From Fig. 5.44,
D¡ = L sin a,
Values of a, are generally small; hence, sin a, «
. Thus,
D¡ = Lot,
(5.52)
Note that in Eq. (5.52), a¡ must be in radians. Hence, D, can be calculated from
Eq. (5.52) once a, is obtained.
The calculation of a, is beyond the scope of this book. However, it can be
shown that the value of a¡ for a given section of a finite wing depends on the dis­
tribution of downwash along the span of the wing. In turn, the downwash distri­
bution is governed by the distribution of lift over the span of the wing. To see this
more clearly, consider Fig. 5.45, which shows the front view of a finite wing. The
lift per unit span may vary as a function of distance along the wing because
1.
The chord may vary in length along the wing.
2.
The wing may be twisted such that each airfoil section of the wing is at a
different geometric angle o f attack.
3.
The shape of the airfoil section may change along the span.
Shown in Fig. 5.45 is the case of an elliptical lift distribution (the lift per unit
span varies elliptically along the span), which, in turn, produces a uniform
downwash distribution. For this case, incompressible flow theory predicts that
a, —
CL
;rA R
(5.53)
where C l is the lift coefficient of the finite wing and A R = b2/S is the aspect
Front view o f wing
fT
zL
L ift per unit span as a function o f
distance along the span-this is the
lift distribution
w, the downwash distribution, which
results from the given lift distribution
Figure 5.45 Lift distribution and downwash distribution.
317
318
CHAPTER 5
Airfoils, Wings, and Other Aerodynamic Shapes
ratio, defined in Eq. (5.26). Substituting Eq. (5.53) into (5.52) yields
Di = La, = L —¡r¿
7tA R
(5.54)
However, L = q^SCL, hence, from Eq. (5.54),
C2
A = (JooS7rAR
TTr
or
D¡
C2
—
7
=
qx S
7rA R
(5-55)
Defining the induced drag coefficient as Co., = A /(,q<x>S), we can write
Eq. (5.55) as
(5.56)
This result holds for an elliptical lift distribution, as sketched in Fig. 5.45. For a
wing with the same airfoil shape across the span and with no twist, an elliptical
lift distribution is characteristic of an elliptical wing planform. (The famous
British Spitfire of World War II was one of the few aircraft in history designed
with an elliptical wing planform. Wings with straight leading and trailing edges
are more economical to manufacture.)
For all wings in general, a span efficiency factor e can be defined such that
r
....
C l2
c
(- d ,i — --- 7 7 7
7reAR
(5.57)
For elliptical planforms, e = l; for all other planforms, e < 1. Thus, C d.i and
hence induced drag is a minimum for an elliptical planform. For typical subsonic
aircraft, e ranges from 0.85 to 0.95. Equation (5.57) is an important relation. It
demonstrates that induced drag varies as the square o f the lift coefficient; at high
lift, such as near C¿,max, the induced drag can be a substantial portion of the total
drag. Equation (5.57) also demonstrates that as A R is increased, induced drag is
decreased. Hence, subsonic airplanes designed to minimize induced drag have
high-aspect ratio wings (such as the long, narrow wings of the Lockheed U-2
high-altitude reconnaissance aircraft).
It is clear from Eq. (5.57) that induced drag is intimately related to lift. In
fact, another expression for induced drag is drag due to lift. In a fundamental
sense, that power provided by the engines of the airplane to overcome in­
duced drag is the power required to sustain a heavier-than-air vehicle in the
air, the power necessary to produce lift equal to the weight of the airplane in
flight.
5 .1 4
Calculation of Induced Drag
319
Figure 5.46 Sketch of a drag polar, that is, a plot of drag
coefficient versus lift coefficient.
In light of Eq. (5.57), we can now write the total drag coefficient for a finite
wing at subsonic speeds as
C2
II
Q
Total
drag
Cd
Profile
drag
+
7rcAR
Induced
drag
(5.58)
Keep in mind that profile drag is composed of two parts: drag due to skin friction
Cdj and pressure drag due to separation cd-p\that is, cd = cdj + cd,p. Also keep
in mind that cd can be obtained from the data in App. D. The quadratic variation of
Co with C i given in Eq. (5.58), when plotted on a graph, leads to a curve as shown
in Fig. 5.46. Such a plot of CD versus CL is called a drag polar. Much of the basic
aerodynamics of an airplane is reflected in the drag polar, and such curves are es­
sential to the design of airplanes. You should become familiar with the concept of
drag polar. Note that the drag data in App. D are given in terms of drag polars for
infinite wings; that is, cd is plotted versus c¡. However, induced drag is not in­
cluded in App. D because C dj for an infinite wing (infinite aspect ratio) is zero.
E X A M P L E 5.14
Consider the Northrop F-5 fighter airplane, which has a wing area o f 170 ft2. The wing is
generating 18,000 lb o f lift. For a flight velocity o f 250 mi/h at standard sea level, calcu­
late the lift coefficient.
■ Solution
The velocity in consistent units is
366.7 ft/s
</oo =
= j(0.002377)(366.7)2 = 159.8 lb/ft2
320
chapter
5 Airfoils, Wings, and Other Aerodynamic Shapes
Hence,
CL =
L
18,000
q^S ~ 159.8(170)
0.6626
E X A M P L E 5.15
The wingspan of the Northrop F-5 is 25.25 ft. Calculate the induced drag coefficient and
the induced drag itself for the conditions of Example 5.14. Assume e = 0.8.
■ Solution
The aspect ratio is AR = b2/S = (25.25)2/170 = 3.75. Since Cl = 0.6626 from Exam­
ple 5.14, then from Eq. (5.57),
Cd.í =
C2
'~L
ttM
(0.6626)2
R
jt(0.8)(3.75)
0.0466
From Example 5.14, qx = 159.8 lb/ft2. Hence,
D¡ = q^SCpj = 159.8 (170) (0.0466) =
1266 lb
E X A M P L E 5.1 6
Consider a flying wing (such as the Northrop YB-49 of the early 1950s) with a wing area
of 206 m2, an aspect ratio of 10, a span effectiveness factor of 0.95, and an NACA 4412
airfoil. The weight of the airplane is 7.5 x 105 N. If the density altitude is 3 km and the
flight velocity is 100 m/s, calculate the total drag on the aircraft.
■ Solution
First, obtain the lift coefficient. At a density altitude of 3 km = 3000 m, p^ =
0.909 kg/m3 (from App. A).
9oo = J A » = i(0.909)(100)2 = 4545 N/m2
L = W = 7.5 x 105 N
C
L
L
q^S
7-5 X l Q 5
QQ
4545(206)
Note: This is a rather high lift coefficient, but the velocity is low— near the landing
speed— hence, the airplane is pitched to a rather high angle of attack to generate enough
lift to keep the airplane flying.
Next, obtain the induced drag coefficient:
C „ , = - £ L = -- — ----- 0.021
'
ttM R
jt(0.95)(10)
The profile drag coefficient must be estimated from the aerodynamic data in App. D.
Assume that q is given by the highest Reynolds number data shown for the NACA 4412
airfoil in App. D; furthermore, assume that it is in the drag bucket. Hence, from App. D,
cj » 0.006
5 .1 5
Change in the Lift Slope
Thus, from Eq. (5.58). the total drag coefficient is
CD = cd + CD,, = 0.006 + 0.021 = 0.027
Note that the induced drag is about 3.5 times larger than profile drag for this case, thus
underscoring the importance of induced drag.
Therefore, the total drag is
D = qx SCD = 4545(206)(0.027) =
2.53 x 104N
5.15 CHANGE IN THE LIFT SLOPE
The aerodynamic properties of a finite wing differ in two major respects from the
data of App. D, which apply to infinite wings. The first difference has already
been discussed, namely, the addition of induced drag for a finite wing. The sec­
ond difference is that the lift curve for a finite wing has a smaller slope than the
corresponding lift curve for an infinite wing with the same airfoil cross section.
This change in the lift slope can be examined as follows. Recall that because of
the presence of downwash, which is induced by the trailing wing tip vortices, the
flow in the local vicinity of the wing is canted downward with respect to the freestream relative wind. As a result, the angle of attack that the airfoil section effec­
tively sees, called the effective angle ofattack a cff, is less than the geometric angle
of attack a. This situation is sketched in Fig. 5.47. The difference between or and
a eff is the induced angle of attack a ,, first introduced in Sec 5.14, where
a¡ = a — a eff. Moreover, for an elliptical lift distribution, Eq. (5.53) gives values
for the induced angle of attack cü, = CL/(n A R ). Extending Eq. (5.53) to wings of
any general planform, we can define a new span effectiveness factor e\such that
CL
«<■ = --TE
n e \AR
<5-59)
where et and e [defined for induced drag in Eq. (5.57)] are theoretically different
but are in practice approximately the same value for a given wing. Note that
Eq. (5.59) gives a, in radians. For a, in degrees,
a¡ =
51.3Cl
ne, AR
(5.60)
321
322
CHA P T E R 5
Airfoils, Wings, and Other Aerodynamic Shapes
Figure 5.48 Distinction between the lift curve slopes for infinite and finite wings.
We emphasize that the flow over a finite wing at an angle of attack a is es­
sentially the same as the flow over an infinite wing at an angle of attack a etrKeeping this in mind, assume that we plot the lift coefficient for the finite wing
CL versus the effective angle of attack aen = a - a¡, as shown in Fig. 5.48a.
Because we are using a eff, the lift curve should correspond to that for an infinite
wing; hence, the lift curve slope in Fig. 5.48a is a0, obtained from App. D for the
given airfoil. However, in real life our naked eyes cannot see a efr; instead, what
we actually observe is a finite wing at the geometric angle o f attack a (the actual
angle between the free-stream relative wind and the mean chord line). Hence, for
a finite wing, it makes much more sense to plot CL versus a , as shown in
Fig. 5.48b, than C¿ versus a elf, as shown in Fig. 5.48a. For example, C7. versus
a would be the result most directly obtained from testing a finite wing in a wind
tunnel, because a (and not a cff) can be measured directly. Hence, the lift curve
slope for a finite wing is defined as a = d C t/d a , where a ^ a0. Noting that
a > a eir from Fig. 5.47, we see that the abscissa of Fig. 5.48/? is stretched out
5 .1 5
Change in the Lift Slope
more than the abscissa of Fig. 5.48a; hence, the lift curve of Fig. 5.48b is less in­
clined; that is, a < a(). The effect of a finite wing is to reduce the lift curve slope.
However, when the lift is zero, C/. = 0, and from Eq. (5.53), a¡ = 0. Thus, at
zero lift, a = a eff. In terms of Fig. 5.48a and 5.48b, this means that the angle of
attack for zero lift a¿=o is the same for the finite and infinite wings. Thus, for
finite wings, a L=o can be obtained directly from App. D.
Question: If we know a 0 (say, from App. D), how do we find a for a finite
wing with a given aspect ratio? The answer can be obtained by examining
Fig. 5.48. From Fig. 5.48a,
dCL
----d(a
— a,)7 = a°
Integrating, we find
C l = ao(a —«/) + const
(5.61)
Substituting Eq. (5.60) into Eq. (5.61), we obtain
/
57.3C/ \
V
7retA R J
CL = a„ ( a ----- — ) + const
(5.62)
Solving Eq. (5.62) for C l yields
1
a0ot
const
1 + 57 .3a(,/( 7re,A R )
1 + 57.3a0/(^ e 'l AR)
Differentiating Eq. (5.63) with respect to a, we get
dCL
da
ao
1 + 57.3a0/ ( ^ f i AR)
(5.64)
However, from Fig. 5.48b, by definition, dCL/da = a. Hence, from Eq. (5.64),
a =
ao
1 + 57.3ao/(jrei A R)
(5.65)
Equation (5.65) gives the desired lift slope for a finite wing o f given aspect ratio
A R when we know the corresponding slope a 0 for an infinite wing. Remember:
ao is obtained from airfoil data such as in App. D. Also note that Eq. (5.65) veri­
fies our previous qualitative statement that a < a0.
In summary, a finite wing introduces two major changes to the airfoil data in
App. D:
1.
Induced drag must be added to the finite wing:
C2
Co = cd H----j —
neAR
Total
drag
2.
Profile
drag
Induced
drag
The slope of the lift curve for a finite wing is less than that for an infinite
wing; a < a0.
323
324
CHA PT E R 5
Airfoils, Wings, and Other Aerodynamic Shapes
E X A M P L E 5.17
Consider a wing with an aspect ratio of 10 and an NACA 23012 airfoil section. Assume
Re % 5 x 106. The span efficiency factor is e = e¡ = 0.95. If the wing is at a 4° angle of
attack, calculate CL and Co­
rn
Solution
Since we are dealing with a finite wing but have airfoil data (App. D) for infinite wings
only, the first job is to obtain the slope of this lift curve for the finite wing, modifying the
data from App. D.
The infinite wing lift slope can be obtained from any two points on the linear curve.
For the NACA 23012 airfoil, for example (from App. D),
ci — 1.2
at aefr = 10°
c /= 0 .1 4
atQrCff = 0°
dc,
1.2-0.14
1.06
„
a0 = — = ——— — =
= 0.106 per degree
da
10 — 0
10
Hence,
Also from App. D,
a¿=o = —1.5°
and
c¿ Rs 0.006
The lift slope for the finite wing can now be obtained from Eq. (5.65).
«o
0.106
1 + 57.300/(^1 AR)
1 +57.3(0. 106)/[tt (0.95)( 10)]
a -----------------= ----------------------- = 0.088 per degree
K
6
At a = 4°,
CL = a(a - aL=o) = 0.088[4° - (-1.5)] = 0.088(5.5)
CL = 0.484
The total drag coefficient is given by Eq. (5.58):
C]
0.4842
CD = cd + — — = 0.006 + -------- = 0.006 + 0.0078 =
°
nek R
;r (0.95) (10)
0.0138
E X A M P L E 5.18
In Example 4.28, we calculated the skin friction drag exerted on the biplane wings of the
1903 Wright Flyer. For the flight conditions given in Example 4.28, that is,
= 30 mi/h
at sea level, calculate the induced drag exerted on the wings of the Wright Flyer, and
compare this with the friction drag calculated earlier. For its historic first flight on
5 .1 5
Change in the Lift Slope
December 17, 1903, the total weight of the Flyer including the pilot (Orville) was 750 lb.
Assume the span efficiency for the wing is e = 0.93.
■ Solution
From the data given in Example 4.28, for the Wright Flyer the wingspan is b = 40.33 ft
and the planform area of each wing is 255 ft2. Hence, the aspect ratio of each wing is
ar
= ^
= (4033)2 =
S
255
For level flight, the airplane must produce a lift to counter its weight; hence, for the flight
of the Wright Flyer, the lift was equal to its weight, namely 750 lb. Also, the Flyer is a
biplane configuration, and both wings produce lift. Let us assume that the lift is evenly di­
vided between the two wings; hence, the lift of each wing is 750/2 = 375 lb. The veloc­
ity is Voo = 30 mi/h = 44 ft/s. The dynamic pressure is
<7,» = ^Poo V i = i(0.002377)(442) = 2.3 lb/ft2
Hence, the lift coefficient of each wing is
/.
Cz. = —
375
= T-T-TT- = 0.639
qaoS 2.3(255)
From Eq. (5.57),
______
C2
(0.639)2
DJ ~ neAR ~ ;r(0.93)(6.38) “ °'° ' 9
The induced drag of each wing is
D, = qoeSCcj = 2.3(255)(0.0219) = 12.84 lb
The induced drag, accounting for both wings, is
D, =2(12.84) =
25.71b
Compare this with the friction drag of 6.82 lb calculated in Example 4.28. Clearly, the in­
duced drag is much larger than the friction drag; this is because the velocity of 30 mi/h
was relatively small, requiring a rather large lift coefficient to help generate the 750 lb of
lift, and because the induced drag coefficient varies as the square of Cl, the induced drag
is large compared to the friction drag at the relatively low flight speed.
Note: There is an aerodynamic interaction between the two wings of a biplane that
is relatively complex; a discussion of the phenomenon is beyond the scope of this book.
Because of this interaction, the induced drag of the biplane configuration is not equal to
the sum of the induced drags acting on the single wings individually in isolation, as we
have assumed in this example. Rather, the induced drag of the biplane configuration is
slightly higher than the sum based on our calculations, and there is also a loss of lift.
However, the preceding calculation is a reasonable first approximation for the biplane’s
induced drag.
325
326
chapter
5 Airfoils, Wings, and Other Aerodynamic Shapes
DESIGN BOX
It is good practice to design conventional subsonic
airplanes with high-aspect ratio wings. The reasons
are clearly seen in Eqs. (5.57) and (5.65). The in­
duced drag coefficient Cpj is inversely proportional
to AR, as seen in Eqs. (5.57) and (5.58). This is a
strong effect; if the aspect ratio is doubled, CD,f is
reduced by a factor of 2. By comparison, the impact
of the span efficiency factor e is minor, because
changes in the wing planform and airfoil design re­
sult in only a few percent change in e, and, in turn,
through Eq. (5.57), result in only a few percent
change in Co,,-. (Of course, when the designer is
looking for every ounce of performance, the wing is
designed to have a lift distribution as close to ellipti­
cal as practical, i.e., making e as close to unity as
practical.) The aspect ratio is the big design feature
that controls Co.t■The same can he said about the
lift slope. Increasing the aspect ratio increases the lift
slope, as seen from Eq. (5.65). Clearly, on an aero­
dynamic basis, the designer of a conventional sub­
sonic airplane would prefer to make the aspect ratio
as large as possible.
However, what does as large as possible mean?
W hy do not the wings o f existing airplanes look like
the long and narrow slats from a Venetian blind, which
have very large aspect ratios? The answer is driven by
structural considerations. Imagine the left and right
wings on an airplane in flight; the lift acting on each
wing acts to bend the wing upward, creating a bend­
ing moment where the wing joins the fuselage. The
wing structure and the structure through the fuselage
must be strong enough to resist this bending moment.
Now imagine the lift acting on a Venetian blind; the
Venetian blind slat will easily buckle under the load,
unless the designer adds enough material stiffness to
resist the buckling. This increase in wing stiffness can
be obtained at the cost o f increased wing structural
weight. Consequently, the design aspect ratio for a
conventional airplane is a compromise between com­
peting values in aerodynamics and structures.
The usual outcome of this compromise is sub­
sonic airplanes with aspect ratios on the order of 5 to
7. The following is a tabulation of wing aspect ratios
for various subsonic airplane designs.
Airplane
Aspect Ratio
Wright Flyer (Fig. 1.1)
Vought F4U Corsair (Fig. 2.16)
Boeing B-17 (Fig. 2.17)
Grumman X-29 (Fig. 2.19)
Grumman F3F-2 (Fig. 2.20)
Boeing 727 (Fig. 5.41)
6.4
5.35
7.58
3.91
7.85
7.1
A dramatic example of the importance of a high
aspect ratio can be seen in the Lockheed U-2 highaltitude reconnaissance airplane, shown in the threeview in Fig. 5.49. The U-2 was designed with an
unusually high aspect ratio of 14.3. This was because
of its mission. In 1954, the United States had an
urgent need for a reconnaissance vehicle that could
overfly the Soviet Union; the time was at an early
stage of the cold war, and Russia had recently tested
a hydrogen bomb. However, such a reconnaissance
vehicle would have to fly at an altitude high enough
that it could not be reached by interceptor aircraft or
ground-to-air missiles; in 1954, this meant cruising
at 70,000 ft or higher. The U-2 was designed by
Lockheed Skunk Works, a small elite design group at
Lockheed known for its innovative and advanced
thinking. The airplane was essentially a point design;
it was designed to achieve this extremely highaltitude cruise. In turn, the need for incorporating a
very high-aspect ratio wing was paramount. The rea­
son is explained in the following.
In steady, level flight, the airplane lift must equal
its weight L = W. In this case, from Eq. (5.18) writ­
ten for the whole airplane,
L= W= \
PooV^SCl
(5.66)
Consider an airplane at a constant velocity V,*,. As it
flies higher, p00 decreases and hence from Eq. (5.66)
Ci. must be increased in order to keep the lift constant,
equal to the weight; that is, as p00 decreases, the angle
of attack of the airplane increases in order to increase
Cl. There is some maximum altitude (minimum Poo)
at which Cl reaches its maximum value; if the angle
of attack is increased beyond this point, the airplane
will stall. At its high-altitude cruise condition, the IJ-2
5 .1 5
Change in the Lift Slope
327
Figure 5.49 Three-view of the Lockheed U-2 high altitude reconnaissance airplane. Aspect ratio = 14.3.
is flying at a high value of C¿ with a concurrent high
angle of attack, just on the verge of stalling. (This is
in stark contrast to the normal cruise conditions of
conventional airplanes at conventional altitudes,
where the cruise lift coefficient and angle of attack
are relatively small.) A high value of C¿ means a high
induced drag coefficient; note from Eq. (5.57) that
Coj varies directly as the square of C¿. As a result,
at the design high-altitude cruise condition of the
U-2, the induced drag is a major factor. To reduce the
cruise value of Cpj, the designers of the U-2 had to
opt for as high an aspect ratio as possible. The wing
design shown in Fig. 5.49 was the result.
It is interesting to note that at the high-altitude
operating condition of the U-2, the highest velocity
allowed by drag divergence and the lowest velocity
allowed by stalling were almost the same; only about
7 mi/h separated these two velocities, which was not
an easy situation for the pilot.
In contrast to the extreme high-altitude mission
of the U-2, the opposite extreme is high-speed flight
at altitudes on the order of hundreds of feet above the
ground. Consider a subsonic military aircraft de­
signed for low-altitude, high-speed penetration of an
enemy’s defenses, flying close enough to the ground
to avoid radar detection. The aircraft is flying at high
speed in the high-density air near sea level, so it is
flying at a very low C¡. and very small angle of at­
tack, as dictated by Eq. (5.66). Under these condi­
tions, induced drag is very small compared to profile
drag. At this design point, it is beneficial to have a
low-aspect ratio wing with a relatively small surface
area, which will reduce the profile drag. Moreover,
the low aspcct ratio provides another advantage under
these flight conditions— it makes the aircraft less
sensitive to atmospheric turbulence encountered at
low altitudes. This is achieved through the effect of
the aspect ratio on the lift slope, given by Eq. (5.65).
The lift slope is smaller for a low-aspect ratio
wing, as sketched in Fig. 5.50. Imagine the airplane
(continued on next page)
328
chapter
5
Airfoils, Wings, and Other Aerodynamic Shapes
(continuedfrom page 327)
encountering an atmospheric gust that momentarily
perturbs its angle of attack by an amount Aa, as
sketched in Fig. 5.50. The lift coefficient will be cor­
respondingly perturbed by the amount AC¿. How­
ever, because of its larger lift slope, the high-aspect
ratio wing will experience a larger perturbation
(ACl)i than the low-aspect ratio wing, which expe­
riences the smaller perturbation (ACt )|. This is
shown schematically in Fig. 5.50. The smaller
change in CL due to a gust for the low-aspect ratio
wing results in a smoother ride, which is good for
both the flight crew and the structure of the airplane.
In summary, the consideration of aspect ratio in
airplane design is not a matter of “one size fits all.”
Quite the contrary; we have just discussed two totally
different flight conditions that reflect two different
design points, one leading to the need for a highaspect ratio wing and the other to a low-aspect ratio
wing. It is clear that aspect ratio is one of the most
important considerations for an airplane designer.
The choice of what aspect ratio to use for a given air­
plane design depends on a number of factors and
compromises. We have pointed out some of these
considerations in this discussion.
Figure 5.50 Effect of aspcct ratio on the lift slope.
For a given perturbation in or. the high-aspect ratio
wing experiences a larger perturbation in C¿ than the
low-aspect ratio wing.
E X A M P L E 5.19
Consider two wings with an NACA 23012 airfoil section, (a) one with an aspect ratio of
4 and (b) the other with an aspect ratio of 10. The span efficiency factor for both wings is
e = e\= 0.95. Both wings are flying at an angle of attack of 2°. Calculate and compare
the change in lift coefficient for both wings if the angle of attack is perturbed by an
amount Aa = 0.5°; that is, referring to Fig. 5.50, calculate (AC ¿>2 and (A C a )i for
Aa =0.5°.
■ Solution
a. Let us first deal with the wing with aspect ratio 4. The lift slope and zero-lift angle of
attack for the NACA 23012 airfoil were obtained in Example 5.17 as
an = 0.106 per degree
and
a¿=o = —1.5°
5 .1 6
Swept Wings
The lift slope for the finite wing with AR = 4 is, from Eq. (5.65),
«0
1 + 513ao/(Tie \AR)
0.106
1 + 57.3(0. 106)/[jt(0.95)(4)]
= 0.07 per degree
At a = 2°, the lift coefficient is
CL = a (a - a L=0) = 0.07 [2 - (-1.5)] = 0.245
When the angle of attack is perturbed by Aa = 0.5°, the new angle of attack is 2.5°. The
lift coefficient for this angle of attack is
CL =0.7 [2.5 -(-1.5)] =0.28
Hence, referring to Fig. 5.50,
(AC¿), = 0.28 -0.245 =
0.035
b. For the wing with aspect ratio 10, the lift slope was obtained in Example 5.17 as
a = 0.088 per degree
A ta = 2°,
CL = a (a - aL=o) = 0.088 [2 - (-1.5)] = 0.308
A ta = 2.5°,
CL = a(a - a L=o) = 0.088 [2.5 - (-1.5)] = 0.352
(ACLh = 0.352 - 0.308 =
0.044
Comparing the results from parts (a) and (b), the high-aspect ratio wing experiences a
26 percent higher increase in C¿than the low-aspect ratio wing.
5.16 SWEPT W INGS
Almost all modern high-speed aircraft have swept-back wings, such as shown in
Fig. 5.51b. W hy? We are now in a position to answer this question.
We first consider subsonic flight. Consider the planview of a straight wing,
as sketched in Fig. 5.51a. Assume this wing has an airfoil cross section with a
critical Mach number M cr = 0.7. (Remember from Sec. 5.10 that for
slightly greater than M cr, there is a large increase in drag. Hence, it is desirable
to increase Mcr as much as possible in high-speed subsonic airplane design.)
Now assume that we sweep the wing back through an angle of, say, 30°, as
shown in Fig. 5.51b. The airfoil, which still has a value of A/cr = 0.7, now “sees”
essentially only the component of the flow normal to the leading edge of the
wing; that is, the aerodynamic properties of the local section of the swept wing
329
330
chapter
5 Airfoils, Wings, and Other Aerodynamic Shapes
wing = 0.7
Airfoil section
with
= 0.7
Now sweep the same wing by 30°
(a)
Mct for swept wing
0.7
cos 30°
Figure 5.51 Effect of a swept wing on critical Mach number.
are governed mainly by the flow normal to the leading edge. Hence, if M is the
free-stream Mach number, the airfoil in Fig. 5.516 is seeing effectively a smaller
Mach number, Mx cos 30°. As a result, the actual free-stream Mach number can
be increased above 0.7 before critical phenomena on the airfoil are encountered.
In fact, we could expect that the critical Mach number for the swept wing itself
would be as high as 0 .7 /c o s 30° = 0.808, as shown in Fig. 5.516. This means
that the large increase in drag (as sketched in Fig. 5.24) would be delayed to Mx
much larger than M „ for the airfoil— in terms of Fig. 5.51, something much
larger than 0.7 and maybe even as high as 0.808. Therefore, we see the main
function of a swept wing. By sweeping the wings of subsonic aircraft, drag di­
vergence is delayed to higher Mach numbers.
In real life, the flow over the swept wing sketched in Fig. 5.516 is a fairly
complex three-dimensional flow, and to say that the airfoil sees only the compo­
nent normal to the leading edge is a sweeping simplification. However, it leads to
a good rule of thumb. If Q is the sweep angle, as shown in Fig. 5.516, the actual
critical Mach number for the swept wing is bracketed by
Mcr for airfoil < actual MCI for swept wing <
M cr for airfoil
cos £2
5 .1 6
Swept Wings
Figure 5.52 By sweeping the wing, a streamline effectively sees a thinner airfoil.
There is an alternate explanation of how the critical Mach number is in­
creased by sweeping the wing. Consider the segment of a straight wing sketched
in Fig. 5.52a. The airfoil section, with a thickness-to-chord ratio of t\/c\ = 0.15,
is sketched at the left. The arrowed line AB represents a streamline flowing over
the straight wing. This streamline “sees” the airfoil section with a 15 percent
thickness. Now consider this same wing, but swept through the angle £2 = 45°, as
shown in Fig. 5.52b. The arrowed line CD represents a streamline flowing over
the swept wing. (We draw streamlines AB and CD as straight lines in the freestream direction, ignoring for simplicity any three-dimensional flow effects.)
Streamline CD now travels a longer distance over the swept wing. The airfoil sec­
tion that streamline CD effectively “sees” is sketched at the left in Fig. 5.52b. It
has the same thickness, but a longer effective chord. Hence, the effective airfoil
section that streamline CD sees is thinner than that seen in the case of the straight
wing. Indeed, for the case of a sweep angle of 45u, the effective airfoil section
seen by streamline CD has a thickness-to-chord ratio of h jc i = 0.106. Simply by
taking the straight wing in Fig. 5.52a and sweeping it through an angle of 45°, the
swept wing looks to the flow as if the effective airfoil section is almost one-third
thinner than it is when the sweep angle is 0°. From our discussion in Sec. 5.9,
making the airfoil thinner increases the critical Mach number. Hence, by sweep­
ing the wing, we can increase the critical Mach number o f the wing.
Following the usual axiom that “we cannot get something for nothing,” for
subsonic flight, increasing the wing sweep reduces the lift. Hence, although wing
sweep is beneficial in terms o f increasing the drag-divergence Mach number, it
331
332
CHA p T E R 5
Airfoils, Wings, and Other Aerodynamic Shapes
20 r-
16
12
A
8
20
_L
_L
_L
40
60
80
100
120
Wing sweepback angle, deg
Figure 5.53 Variation of lift-to-drag ratio with wing sweep. Wind tunnel measurements
at the NASA Langley Research Center.
(Source: From Loflin, NASA SP468, 1985.)
serves to decrease CL. This is demonstrated in Fig. 5.53, which gives the varia­
tion of L /D with sweep angle for a representative airplane configuration at
Moo = 0.6 flying at 30,000 ft. There is a considerable decrease in L /D as the
sweep angle increases, mainly due to the decrease in C¿.
For supersonic flight, swept wings are also advantageous, but not quite from
the same point of view as just described for subsonic flow. Consider the two
swept wings sketched in Fig. 5.54. For a given Mx > 1, there is a Mach cone
with vertex angle /¿, equal to the Mach angle [recall Eq. (5.49)]. If the leading
edge of a swept wing is outside the Mach cone, as shown in Fig. 5.54a, the com­
ponent of the Mach number normal to the leading edge is supersonic. As a result,
a fairly strong oblique shock wave will be created by the wing itself, with an at­
tendant large wave drag. On the other hand, if the leading edge of the swept wing
is inside the Mach cone, as shown in Fig. 5.546, the component of the Mach
number normal to the leading edge is subsonic. As a result, the wave drag pro­
duced by the wing is less. Therefore, the advantage of sweeping the wings for su­
personic flight is in general to obtain a decrease in wave drag; and if the wing is
swept inside the Mach cone, a considerable decrease can be obtained.
The quantitative effects of maximum thickness and wing sweep on the wave
drag coefficient are shown in Fig. 5.55a and b, respectively. For all cases, the
wing aspect ratio is 3.5, and the taper ratio (tip to root chord) is 0.2. Clearly, thin
wings with large angles of sweepback have the smallest wave drag.
5 .1 6
Swept Wings
333
Moa> l
Moa> l
( b)
(a)
Figure 5.54 Swept wings for supersonic flow, (a) Wing swept outside the Much cone.
(,b) Wing swept inside the Mach cone.
Sweep angle measured at quarter-chord line
« = 35°
n = 47°
A = 3.5; \= 0.2
.s
6
oQ‘
Í2, deg
O
u
11
60
35
c
Í
e
3
E
'5
S
0.6
0.8
_L
l.O
Mach number M
(a)
_L
l .2
l.4
0.6
_L
0.8
l.O
l.2
1.4
Mach number M
(b)
Figure 5.55 Sketch of the variation of minimum wing drag coefficient versus Mach number with (a ) wing
thickness as a parameter (Í2 = 47°) and (b) wing sweepback angle as a parameter (t/c = 4 percent).
(Source: From L. Loftin, Quest for Performance. NASA SP 468, 1985.)
334
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5
Airfoils, Wings, and Other Aerodynamic Shapes
DESIGN BOX
The designer of supcrsonic airplanes has two basic
choices of wing planform: low-aspcct ratio straight
wing, or swept wing (including a delta wing). Both
classes of wing planform result in lower wave drag
compared to a high-aspect ratio straight wing. Let us
examine these choices in greater detail.
First consider a low-aspect ratio straight wing at
supersonic speeds. From Eq. (5.51), the wave drag
coefficient for a flat plate of infinite span is
Q.u
a2
1
/M l
(5.67)
where a is the angle of attack in radians. The same
theory gives the wave drag coefficient for a flat plate
of finite aspect ratio AR as
C D.W
(5.68)
JM .
where
R s AR J
(See, e.g., Hilton, High-Speed Aerodynamics,
Longman, Green and Co., 1951.) Note that Eq. (5.68)
reduces to Eq. (5.67) for an aspect ratio going to
infinity. Equation (5.68) is graphed in Fig. 5.56, giv­
ing Co.w/ot2 as a function of the aspect ratio for the
case of
= 2. Note the rather dramatic drop in the
wave drag coefficient at very low-aspect ratios. This
curve, which is for an infinitely thin flat plate, should
be viewed as mainly qualitative when dealing with
real wings with thickness. However, it clearly shows
the advantage of low-aspect ratio wings for super­
sonic flight. This is the exact opposite to the recom­
mended practice for subsonic airplane design, as dis­
cussed earlier. However, because of the occurrence of
shock waves at supersonic speeds, supersonic wave
drag is usually much more important than induced
drag; hence, the use of low-aspect ratio wings is
good practice in supersonic airplane design. A case in
point is the Lockheed F-104 supersonic fighter,
shown in Figs. 5.38 and 4.45. Return to Fig. 4.45, and
study the wing planform for the F-104. This airplane
2.5
A R —» ©o
2.0
1.5
Cp, w
a
1.0
0.5
Aspect ratio
Figure 5.56 Variation of supersonic wave drag with aspect ratio for flat
plates.
was the first to be designed for sustained flight at
Mach 2, and the designers at I .ockheed Skunk Works
chose to go with a straight wing o f low aspect ratio.
The F-104 wing has an aspect ratio of 2.45. The air­
foil section is a very thin biconvex shape; the
thickness-to-chord ratio is only 0.0336. The leading
edge is exceptionally sharp; the leading-edge radius
o f 0.016 is so small that it poses some danger to the
ground crew working around the airplane. All these
features have one goal— to reduce the supersonic
wave drag. They are classic examples o f good super­
sonic airplane design.
We note that the supersonic lift coefficient is
also reduced when the aspect ratio is reduced. This is
illustrated in Fig. 5.57a, which gives the variation of
Ihe lift slope d C i/d a as a function o f aspect ratio for
straight, tapered wings at M = 1.53. Shown here
are some of the first experimental data obtained in the
United States on wings at supersonic speeds. These
data were obtained in the 1-ft by 3-ft supersonic tun­
nel at N A C A Ames Laboratory by Walter Vincenti in
1947, but owing to military classification were not re­
leased until 1949. In Fig. 5.57a, the dashed triangles
shown emanating from the wing leading-edge apex
represent the Mach cones at Mx = 1.53. (The Mach
cones are cones with a semivertex angle equal to the
Mach angle p.) Note that as A R is reduced, more of
the wing is contained inside the Mach cones. The ef­
fect o f decreasing A R on the lift slope at supersonic
speeds is qualitatively the same as that for subsonic
speeds. Recall from Sec. 5.15 that the lift slope is
smaller for lower-aspect ratio wings in subsonic
flight. Clearly, from Fig. 5.51a the same trend pre­
vails for supersonic flight, even though the physical
nature o f the aerodynamic flow field is completely
different.
The other option for a wing planform for super­
sonic airplanes is the swept wing. (We will consider
the delta, or triangular planform, as a subset under
swept wings.) In regard to Fig. 5.54, we have already
discussed that supersonic wave drag can be consider­
ably reduced by sweeping the wing inside the Mach
cone, that is, by having a subsonic leading edge. This
is clearly seen in the experimental data shown in
Fig. 5.57 b, taken from the pioneering supersonic wind
tunnel work o f Vincenti. In Fig. 5.57 b, the m inim um
total drag coefficient is plotted versus wing sweep
angle for Mr*> = 1-53. Keep in mind that the total drag
coefficient is due to both pressure drag (essentially
wave drag) and skin friction drag. Positive sweep an­
gles represent swept-buck wings, and negative sweep
angles represent swept-forward wings. Note the near
symmetry o f the data in regard to positive and nega­
tive sweep angles; the supersonic drag coefficient is
essentially the same for the same degree o f sweepback
as it is for the same degree of sweepforward. The im ­
portant message in Fig. 5.51b is the decrease in C 0|nin
at sweep angles greater than 49° or less than —49°.
The Mach angle for
= 1.53 is given by p —
sin-1 (l/Afoo) = sin- 1(1/1.53) = 41®. Hence, wings
with a sweep angle of 49" or larger will be inside the
Mach cone. Note the lower drag coefficient at a sweep
angle o f ±60°; for this case, the wing is comfortably
inside the Mach cone, with a subsonic leading edge.
These data also show that when the wings are swept
outside the Mach cone (supersonic leading edge), the
drag coefficient is relatively flat, independent o f the
sweep angle. So for supersonic flight, in order to real­
ize the drag reduction associated with a swept wing,
the sweep angle must be large enough that the wing is
swept inside the M ach cones.
A classic example o f this design feature is the
English Electric Lightning, a Mach 2 interceptor used
by the British Royal Air Force in the 1960s and 1970s.
As seen in Fig. 5.58, the Lightning has a highly swept
wing, with a sweep angle £2 = 60'. At Mach 2. the
Mach angle is p = s in ~ '( l/ M » ) = sin 1 | — 30°. A
swept wing, to bejust inside the Mach cone at A/oo = 2,
must have a sweep angle o f £2 = 60" or larger.
Since Mach 2 was the design point, it is no surprise
that the designers o f the Lightning chose a sweep
angle o f 60°. In addition, the wing o f the Lightning
has a relatively low aspect ratio o f 3.19. and the air­
foil section is thin, with a thickness-to-chord ratio of
5 percent— both good design practices for supersonic
airplanes.
(continued on next page)
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chapter
5
Airfoils, Wings, and Other Aerodynamic Shapes
(continuedfrom page 335)
Ai00= 1.53
---- Experiment
----Linear theory (wing alone)
0.08
- C*c !
isV
0.06
dC,
da
per 0.04
deg.
0.02
' '
i
/ 'i \
Theo ■y and 1
<
/
i
>— '—
/
exper intent -
r
coinc de
0
1
2
3
4
5
6
Aspect ratio AR
(a)
1.53
Experiment
Linear theory (wing alone)
Sweep angle at midchord A|/2 , deg.
(b)
Figure 5.57 (a) Effect of aspect ratio on the lift curve for straight wings at supersonic speeds. M'x = 1.53. After W. G.
Vincenti, “Comparison between Theory and Experiment for Wings at Supersonic Speeds”, NA CAT R 1033. (b) Effect
of wing sweep on supersonic drag. The drag coefficient quoted is for an angle of attack that gives minimum drag.
(Source: Data from Vincenti.)
Look closely at the Lightning in Fig. 5.58, and
then go back and closely examine the F-104 in
Fig. 4.45. Here we see classic examples o f the two
aspect ratio straight wing, from which designers o f
supersonic airplanes can choose.
We examined the effect of wing sweep on the
different wing planforms, swept wing and low-
subsonic lift coefficient (via the lift-to-drag ratio) in
5 .1 6
Swept Wings
337
AC =1.53
—
Experiment
Sweep angle at midchord A l/2, deg.
Figure 5.59 Effect of wing sweep on the lift slope at
supersonic speed. Data from Vincenti.
Fig. 5.53. Question: W hat is the effect o f sweep on
the supersonic lift coefficient? The answer is pro­
vided by the experimental data o f Vincenti, shown in
Fig. 5.59. In a trend similar to that for the drag coef­
ficient, we see from Fig. 5.59 that as long as the wing
is swept outside the Mach cone (supersonic leading
edge), the lift slope is relatively constant, indepen­
dent o f sweep angle. When the wing is swept inside
(continued on next page)
338
chapter
5 Airfoils, Wings, and Other Aerodynamic Shapes
(continuedfrom page 337)
the Much cone (subsonic leading edge), the lift slope
decreases with increasing sweep angle, similar to the
case for subsonic flight.
The results shown in Figs. 5.57 and 5.59 clearly
show a distinct change in the wing aerodynamic char­
acteristics when the sweep angle is large enough that
the wing is inside the Mach cone. This is because the
pressure distribution over the wing surface changes
radically when the transition is made from a super­
sonic to a subsonic leading edge. The nature of this
change is sketched in Fig. 5.60, which shows three
flat-plate wing planforms labeled A, B, and C of pro­
gressively increased sweep angle in a supersonic free
stream. The shaded region shows that portion of the
wing surface not inside the Mach cones emanating
from the wing tips and the apex (midspan location).
Wing A is a straight wing. The influence of the Mach
cones is limited to a small region at the tips; most of
the wing is feeling the type of two-dimensional su­
personic flow over a flat plate that was discussed in
Sec. 5.11 and sketched in Fig. 5.37. Hence, the pres­
sure distribution over most of the surface of wing A
is the constant-pressure distribution illustrated by the
vertical lines of constant-length shown near the right-
i
i Region o f two-dimensional flow
I'-.'....--I Lift distribution
Figure 5.60 Change in chordwise pressure
distribution as a wing at supersonic speeds is
progressively swept from outside to inside the Mach
cone, that is, as the leading edge progressively
changes from supersonic to subsonic.
hand tip of the wing. Wing B is a swept wing, but
with a supersonic leading edge. A considerable por­
tion of the wing is still outside the Mach cones; this
portion is the shaded region shown on wing B. In this
shaded region, the same constant-pressure distribu­
tion associated with a flat plate in supersonic flow
still prevails. However, wing C is a swept wing with
a subsonic leading edge; the entire wing is swept in­
side the Mach cone from the apex, and there is no
shaded region. The pressure distribution over this
wing is similar to that for subsonic flow, even though
the wing is immersed in a supersonic free stream.
Note that the vertical lines at the right on wing C
trace out the type of subsonic pressure coefficient dis­
tribution familiar to us from our earlier discussions;
for example, compare with Fig. 5.13. It is this change
in the aerodynamic behavior of the flow over a wing
swept inside the Mach cone that leads to the decrease
in wave drag and lift coefficient associated with
swept wings in supersonic flow.
There is yet another design benefit of a wing with
a subsonic leading edge; namely, the leading-edge ra­
dius can be larger, similar to that for a subsonic air­
plane. This has benefits at low speeds, especially for
landing and takeoff, for airplanes designed for super­
sonic flight. A wing with a sharp leading edge and a
thin airfoil, such as that used on the F-104 (Figs. 4.45
and 5.38), experiences early flow separation at moder­
ate angles of attack at subsonic speeds. This reduces
the value of (C/,)max and forces the airplane to have
higher landing and takeoff speeds. (For example, over
its operational history, the F-104 experienced an inor­
dinate number of accidents due to wing stall at lowspeed flight conditions.) In contrast, a wing with a
blunter, more rounded leading edge has much better
low-speed stall characteristics. Supersonic airplanes
with swept wings with subsonic leading edges can be
designed with blunter, more rounded leading edges,
and hence have better low-speed stalling behavior.
Recall from Figs. 5.57 and 5.59 that the super­
sonic drag and lift coefficients associated with sweptforward wings are essentially the same as those for
swept-back wings. Indeed, the same can be said for
high-speed subsonic flight. However, airplane design­
ers have almost always chosen sweepback rather
than sweepforward. Why? The answer has to do with
aeroelastic deformation of swept wings under load.
5 .1 6
For a swept-back wing, the location o f the effective lift
force causes the wing to twist near the tips so as to de­
crease the angle o f attack o f the outer portion o f the
wing. This tends to unload that portion of the wing
when lift is increased— a stable situation. In contrast,
for a swept-forward wing, the location o f the effective
lift force causes the wing to twist near the tips so as to
increase the angle of attack of the outer portion o f the
wing, thus causing the lift to increase, which further
increases the wing twist. This is an unstable situation
that tends to twist the swept-forward wing right off the
airplane. These aeroelastic deformation effects are
even seen in the experimental data shown in Fig. 5.59.
Note that the experimental data are not symmetric for
swept-forward and swept-back wings. The lift slope is
smaller for the swept-back wings, due to aeroelastic
deformation o f the wind tunnel models. Hence, for
structural reasons, swept-forward wings have not
been the planform o f choice. However, modern ad­
vances in composite materials now allow the design o f
very strong, lightweight wings, and this has opened
the design space for high-speed airplanes to consider
the use o f swept-forward wings. Indeed, sweptforward wings have certain design advantages. For
example, the wing root can be placed farther back on
the fuselage, allowing greater flexibility in designing
the internal packaging inside the fuselage. Also, the
details o f the three-dimensional flow over a sweptforward wing result in flow separation occurring first
near the root, preserving aileron control at the tips; in
Swept Wings
339
contrast, for a swept-back wing, flow separation tends
to occur first near the tips, hence causing a loss of
aileron control. In the 1980s, an experimental air­
plane, the Grum man X-29, was designed with sweptforward wings in order to examine more closely the
practical aspects o f swept-forward wing design. A
three-view o f the X-29 is shown in Fig. 5.61. The X-29
research program has been successful, but as yet there
has been no rush on the part o f airplane designers to go
to swept-forward wings.
Return to Fig. 5.58, and examine again the
highly swept wing o f the English Electric Lightning.
It is not much o f an intellectual leap to imagine the
empty notch between the wing trailing edge and the
fuselage filled in with wing structure, producing a
wing with a triangular planform. Such wings are
called delta wings. Since the advent of the jet engine,
there has been interest in delta wings for high-speed
airplanes, both subsonic and supersonic. One design
advantage o f the delta wing is that by filling in that
notch, as previously discussed, the chord length of
the wing root is considerably lengthened. For a fixed
t/c ratio, this means the wing thickness at the root
can be made larger, providing greater volume for
structure, fuel, etc. The list o f advantages and disad­
vantages o f a delta wing is too long to discuss here.
See the following book for a thorough and readable
discussion o f this list: Ray W hitford, Design for Air
(continued on next page)
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5
Airfoils, Wings, and Other Aerodynamic Shapes
(continuedfrom page 339)
Combat, Janes Information Group Limited, 1989.
Suffice it to say that a number of subsonic and super­
sonic delta wing aircraft have been designed and used
extensively. An example is the French DassaultBrequet Mirage 2000C, shown in Fig. 5.62. The
Mirage 2000C is a supersonic fighter with a top speed
of Mach 2.2. The leading-edge sweep angle is
£2 = 58°. Dassault is well known for its long line of
successful delta wing airplanes since the 1950s. Note
from Fig. 5.62 that the Mirage 2000C has no hori­
zontal stabilizer; this is characteristic of many delta
wing airplanes. The trailing-edge control surfaces are
called elevons, which, when deflected uniformly in
the same direction (up or down), act as elevators and
when deflected in opposite directions (one up and the
other down) act as ailerons.
In many respects, the wing is the heart of the air­
plane. Great care goes into the design of the wing.
Today, the design of wing shapes for supersonic air­
planes is sophisticated and fine-tuned. Consider, for
example, the Anglo-French Concorde supersonic
transport, shown in Fig. 5.63. The Concorde was the
only commercial supersonic transport in regular
service. Manufactured jointly by British Aircraft
Corporation in England and Aerospatiale in France,
the Concorde first flew on March 2, 1969, and went
into service with British Airways and Air France in
1976. As shown in Fig. 5.63, the wing of the Con­
corde is a highly swept ogival delta planform with
complex camber and wing droop (anhedral). The air­
foil section is thin, with a thickness-to-chord ratio of
3 percent at the root and 2.15 percent from the nacelle
outward. (A personal note: this author and his wife
flew on the Concorde during the summer of 1998—
what an exciting experience! The flight time between
New York and London was only 3 h 15 min— too
short to even show an in-flight movie. Unfortunately,
the Concorde fare was very expensive, and by most
measures, the airplane was an economic failure. For
this reason, in 2003, the Concorde was phased out of
service. It will be one of the most demanding design
challenges in the 21st century to design an economi­
cally and environmentally viable second-generation
supersonic transport. Perhaps some of the young
readers of this book will successfully rise to this
challenge.)
Figure 5.62 An example of a delta wing. The French Dassault-Breguel Mirage 20(X)C, with
an added side view (lower right) of the Mirage 2000N.
Figure 5.63 The Anglo-French Aerospatiale/BAC Concorde supersonic transport.
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5.17 FLAPS— A M ECHANISM FOR H IG H LIFT
An airplane normally encounters its lowest flight velocities at takeoff or landing,
two periods that are most critical for aircraft safety. The slowest speed at which
an airplane can fly in straight and level flight is defined as the stalling speed Vstau.
Hence, the calculation of Vstaii, as well as aerodynamic methods of making Vslai|
as small as possible, is o f vital importance.
The stalling velocity is readily obtained in terms of the maximum lift coeffi­
cient, as follows. From the definition of CL,
L = qooSCL = { P ooV ^ S C l
Thus,
(5.69)
In steady, level flight, the lift is just sufficient to support the weight W of the air­
craft, that is, L = W. Thus,
(5.70)
Examining Eq. (5.70), for an airplane of given weight and size at a given altitude,
we find the only recourse to minimize Vao is to maximize C¿. Hence, stalling
speed corresponds to the angle o f attack that produces C¿,max:
(5.71)
Flap
line
"Virtual"
'Virtual" increase in
angle of
o f attack
(d)
Figure 5.64 When a plain flap is deflected, the increase in lift is due to
an effective increase in camber and a virtual increase in angle of attack.
5 .1 7
Flaps—A Mechanism for High Lift
343
In order to decrease Vstali, C/.,max must be increased. However, for a wing
with a given airfoil shape, C¿,max is fixed by nature; that is, the lift properties of
an airfoil, including maximum lift, depend on the physics of the flow over the
airfoil. To assist nature, the lifting properties of a given airfoil can be greatly en­
hanced by the use of “artificial” high-lift devices. The most common o f these de­
vices is the flap at the trailing edge of the wing, as sketched in Fig. 5.64. When
the flap is deflected downward through the angle <5, as sketched in Fig. 5.64b, the
Figure 5.65 Illustration of the effect of flaps on the lift curve. The numbers shown are typical of a modern mediumrange jet transport.
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lift coefficient is increased, for the following reasons:
1.
The camber of the airfoil section is effectively increased, as sketched in
Fig. 5.64c. The more camber an airfoil shape has at a given angle of attack,
the higher the lift coefficient.
2.
When the flap is deflected, we can visualize a line connecting the leading
edge of the airfoil and the trailing edge o f the flap, points A and B ,
respectively, in Fig. 5.64d. Line A B constitutes a virtual chord line, rotated
clockwise relative to the actual chord line o f the airfoil, making the airfoil
section with the deflected flap see a “virtual” increase in angle o f attack.
Hence, the lift coefficient is increased.
For these reasons, when the flap is deflected downward through the flap deflec­
tion angle 5, the value of C¿,max is increased and the zero-lift angle of attack is
shifted to a more negative value, as shown in Fig. 5.65. In Fig. 5.65, the lift
curves for a wing with and without flaps are compared. Note that when the flaps
0.8
-
0.4 -
o -------------------------------------Figure 5.66 Typical values of airfoil maximum lift coefficient for
various types of high-lift devices: (1) airfoil only, (2) plain flap,
(3) split flap, (4) leading-edge slat, (5) single-slotted flap, (6) double­
slotted flap, (7) double-slotted flap in combination with a leading-edge
slat, (8) addition of boundary-layer suction at the top of the airfoil.
(Source: From Loftin, NASA SP468, 1985.)
5 .1 7
Flaps—A Mechanism for High Lift
345
are d e flected , the lift cu rv e sh ifts to the left, the v a lu e o f C /.,max in crea ses, and the
sta llin g a n g le o f attack at w h ic h C L mm is a c h ie v e d is d ecrea sed . H o w e v er , the
lift slo p e rem ain s u n ch a n g ed ; tr a ilin g -e d g e flaps d o not ch a n g e the v a lu e o f
d C i / d a . A ls o n o te that for so m e o f the a irfo ils g iv e n in A p p . D , lift cu r v e s are
sh o w n w ith the e ffe c t o f flap d e flec tio n in clu d ed .
T h e in crea se in C /,,max d u e to flaps ca n b e d ram atic. If the flap is d e sig n e d not
o n ly to rotate d o w n w a rd , but a lso to translate rearward so as to in crea se the e f ­
fe c tiv e w in g area, then C /,,max can b e in crea sed by a p p ro x im a tely a fa cto r o f 2. If
ad d ition al h ig h -lift d e v ic e s are u sed , su ch as sla ts at the lea d in g e d g e , slo ts in the
su rfa ce, or m ech a n ica l m ea n s o f b oundary layer co n tro l, then C ¿ ,max can so m e ­
tim es b e in creased b y a factor o f 3 or m ore, as sh o w n in F ig . 5 .6 6 . For an inter­
e stin g and m ore d eta iled d isc u ssio n o f va rio u s h ig h -lift d e v ic e s , the reader is
referred to the b o o k s by M cC o rm ick and S h e v e ll (se e the B ib lio g ra p h y at the en d
o f this ch ap ter), as w e ll as the a u th o r’s recen t b ook ; A n d erso n , A ircraft P erfo r­
m an ce a n d D esign, M c G r a w -H ill, B o sto n , 19 9 9 .
EXAM PLE 5.20
Consider the Lockheed F -I0 4 shown in three-view in Fig. 4.45 and in the photograph in
Fig. 5.38. With a full load o f fuel, the airplane w eighs 10,258 k g /. Its empty weight (no
fuel) is 6071 k g /. The wing area is 18.21 m2. The wing o f the F-104 is very thin, with a
thickness o f 3.4 percent, and has a razor-sharp leading edge, both designed to minimize
wave drag at supersonic speeds. A thin wing with a sharp leading edge, however, has very
poor low -speed aerodynamic performance; such w ings tend to stall at low angle o f attack,
thus limiting the maximum lift coefficient. The F-104 has both leading-edge and trailingedge flaps, but in spite o f these high-lift devices, the maximum lift coefficient at subsonic
speeds is only 1.15. Calculate the stalling speed at standard sea level when the airplane
has (a) a full fuel tank and (b ) an empty fuel tank. Compare the results.
■ Solution
a. Recall that k g/ is a nonconsistent unit o f force; w e need to convert it to newtons, re­
membering from Sec. 2.4 that 1 kg/ = 9.8 N.
W = 10,258(9.8) = 1.005 x 105 N
At standard sea level, Poo = 1.23k g/m 3. Thus, from Eq. (5.71),
2W
Vs,all =
px SCL,mM
2(1.005 x I05)
V (1 .2 3 )(1 8 .2 1 )(1 .1 5 )
88.3 m/s
In miles per hour, using the conversion factor from Example 2.6 that 60 mi/h = 26.82 m/s,
Vstaii = (88.3 m/s)
/ 60 mi/h \
V
\ Z26.82
O .B
m/s ) ~
197.6 mi/h
b.
W = 6071(9.8) = 5.949 x W4N
V's.all —
/
2VV
PoqSC l max
2(5.949 x 104)
V (1 .2 3 )(1 8 .2 1 )(1 .1 5 )
68 m/s
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VIU11| = (68) ( ¿ ^ 2 ) = I 152 mi/h
Note: The difference between parts (a) and (b ) is the weight. Since Vstaii <x W l/2 from
Eq. (5.71), a shorter calculation for part (b ), using the answer from part (a), is simply
/5 949 x 104
v -
=
( 8 ^
T
^
W
=
6 8 m
/ s
which is a check on the preceding result.
Comparing the results from parts (a) and (b ), we note the trend that the lighter the
airplane, everything else being equal, the lower the stalling speed. Because stalling speed
varies with the square root o f the weight, however, the reduction in stalling speed is pro­
portionally less than the reduction in weight. In this exam ple, a 41 percent reduction in
weight leads to a 23 percent reduction in stalling speed.
EXAM PLE 5.21
Consider the B oeing 727 trijet transport shown in the photograph in Fig. 5.41 and in the
three-view in Fig. 5.67. This airplane was designed in the 1960s to operate out o f airports
with relatively short runways, bringing jet service to smaller municipal airports. To min­
im ize the takeoff and landing distances, the 727 had to be designed with a relatively low
stalling speed. From Eq. (5.71), a low Vstan can be achieved by designing a wing with a
large planform area, S, and/or with a very high value o f C ;. max. A large wing area, how ­
ever, leads to a structurally heavier w ing and increased skin friction drag— both undesir­
able features. The B oeing engineers, instead, opted to achieve the highest possible C l ,max
by designing the most sophisticated high-lift mechanism at that time, consisting o f triple­
slotted flaps at the wing trailing edge and flaps and slots at the leading edge. With these
devices fully deployed, the B oeing 727 had a maximum lift coefficient o f 3.0. For a
Figure 5.67 Three-view o f the Boeing 727 three-engined commercial jet transport.
5 .1 7
Flaps—A Mechanism for High Lift
weight o f 160,000 lb and a wing planform area o f 1650 ft2, calculate the stalling speed o f
the B oeing 727 at standard sea level. Compare this result with that obtained for the F-104
in Example 5.20.
■ Solution
From Eq. (5.71),
2 (1 6 0 ,0 0 0)
V
165 ft/s
In miles per hour,
In Example 5.20 a for the Lockheed F-104, we found Vstan = 197.6 mi/h, a much higher
value than the Boeing 727. The airplanes in these tw o exam ples, a point-designed Mach
2 fighter and a short-field commercial jet transport, represent high and low extremes in
stalling speeds for conventional jet airplanes.
Note: Computed streamline patterns over the Boeing 727 airfoil section are shown in
Fig. 5.68, showing the high-lift devices deployed for landing configuration at an angle o f
attack o f 6°, takeoff configuration at an angle o f attack o f 10°, and with the clear config­
uration (no deploym ent o f the high-lift devices) for cruise at an angle o f attack o f 3‘ . N o­
tice how much the flow field is changed when the high-lift devices are deployed; the
streamline curvature is greatly increased, reflecting the large increase in lift coefficient.
Foreflap
Midflap
Aft flap
Figure 5.68 Streamline patterns over the Boeing 727 airfoil with
and without high-lift devices deployed, comparing the cases for
landing, takeoff, and cruise.
(Source: AIAA, with permission.)
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5.18 AERODYNAMICS OF CYLINDERS
AND SPHERES
C onsider the low -speed subsonic flow over a sphere or an infinite cylinder with
its axis norm al to the flow. If the flow were inviscid (frictionless), the theoretical
flow pattern would look qualitatively as sketched in Fig. 5.69a. The stream lines
would form a sym m etric pattern; hence, the pressure distributions over the front
and rear surfaces w ould also be sym m etric, as sketched in Fig. 5.69b. This sym ­
m etry creates a m om entous phenom enon, namely, that there is no pressure drag
on the sphere if the flow is frictionless. This can be seen by sim ple inspection of
Fig. 5.69b: The pressure distribution on the front face (—90° < 6 < 90°) creates
a force in the drag direction, but the pressure distribution on the rear face
(90° < 6 < 270°), which is identical to that on the front face, creates an equal
and opposite force. Thus, we obtain the curious theoretical result that there is no
drag on the body, quite contrary to everyday experience. This conflict between
theory and experim ent was well known at the end o f the 19th century and is
called d ’A le m b e rt’s paradox.
The actual flow over a sphere or cylinder is sketched in Fig. 4.30; as discussed
in Sec. 4.20, the presence o f friction leads to separated flows in regions o f adverse
pressure gradients. Exam ining the theoretical inviscid pressure distribution shown
in Fig. 5.69b, we find on the rear surface (90° < 0 < 270°) the pressure increases
in the flow direction; that is, an adverse pressure gradient exists. Thus, it is entirely
Figure 5.69 Ideal frictionless flow over a sphere.
(a) Flow field. (b ) Pressure coefficient distribution.
5 .1 8
Aerodynamics of Cylinders and Spheres
Figure 5.70 Real separated How over a sphere;
separation is due to friction, (a) Flow field.
Cb ) Pressure coefficient distribution.
reasonable that the real-life flow over a sphere or cylinder would be separated from
the rear surface. This is indeed the case, as first shown in Fig. 4.30 and as sketched
again in Fig. 5.70a. The real pressure distribution that corresponds to this sepa­
rated flow is shown as the solid curve in Fig. 5.70b. Note that the average pressure
is m uch higher on the front face (—90° < 0 < 90°) than on the rear face
(90" < 6 < 270°). As a result, there is a net drag force exerted on the body. Hence,
nature and experience are again reconciled, and d ’A lem bert’s paradox is rem oved
by a proper account o f the presence o f friction.
It is em phasized that the flow over a sphere or cylinder, and therefore the
drag, is dom inated by flow separation on the rear face. This leads to an interest­
ing variation o f C D with the Reynolds number. Let the Reynolds num ber be de­
fined in term s o f the sphere diam eter D: Re = p o o V ^ D /f ix . If a sphere is
m ounted in a low -speed subsonic wind tunnel and the free-stream velocity is var­
ied such that Re increases from 105 to 106, then a curious, alm ost discontinuous
drop in C o is observed at about Re = 3 x 105. This behavior is sketched in
Fig. 5 .7 1. W hat causes this precipitous decrease in drag? The answ er lies in the
different effects o f lam inar and turbulent boundary layers on flow separation. At
the end o f Sec. 4.20, we note that lam inar boundary layers separate m uch more
readily than turbulent boundary layers. In the flow over a sphere at Re < 3 x 10s ,
the boundary layer is laminar. H ence, the flow is totally separated from the rear
face, and the w ake behind the body is large, as sketched in Fig. 5.72a. In turn, the
value o f C 0 is large, as noted at the left o f Fig. 5 .7 1 for Re < 3 x 105. On the
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Re
Figure 5.71 Variation o f drag coefficient with Reynolds
number for a sphere in low-speed flow.
Separation
Laminar boundary
Fat wake
(a)
Turbulent boundary ‘
Laminar boundary
Voc
(b)
Figure 5.72 Laminar and turbulent flow over a sphere.
DESIGN BOX
The large pressure drag associated with blunt bodies
such as the cylinders and spheres discussed in this sec­
tion leads to the design concept o f streamlining. Con­
sider a body o f cylindrical cross section o f diameter d,
with the axis o f the cylinder oriented perpendicular to
the flow. There w ill be separated flow on the back face
o f the cylinder, with a relatively fat wake and with the
associated high-pressure drag; this case is sketched in
Fig. 5.73«. The bar to the right o f Fig. 5.73« denotes
the total drag o f the cylinder; the shaded portion o f the
bar represents skin friction drag, and the open portion
represents the pressure drag. Note that for the case o f a
blunt body, the drag is relatively large, and most o f this
drag is due to pressure drag. However, look at what
happens when we wrap a long, mildly tapered after­
body on the back o f the cylinder, creating the teardrop­
shaped body sketched in Fig. 5.73/?. This shape is a
streamlined body, o f the same thickness d as the cylin­
der. However, because o f the tapered afterbody, ihe
adverse pressure gradient along the back o f the
streamlined body will be much milder than that for the
back surface o f the cylinder, and hence flow separa­
tion on the streamlined body will be delayed until
much closer to the trailing edge, as sketched in
Fig. 5.73 b, with an attendant, much smaller wake. As
a result, the pressure drag o f the streamlined body will
be much smaller than that for the cylinder. Indeed, as
shown by the bar to the right o f Fig. 5.73 b, the total
drag o f the streamlined body in a low-speed flow will
be alm ost a factor or 10 smaller than that o f the cylin­
der o f same thickness. The friction drag o f the stream­
lined body will be larger due to its increased surface
area, but the pressure drag is so much less that it dom ­
inates this comparison.
This is why so much attention is placed on
streamlining in airplane design. The value o f stream­
lining was not totally recognized by airplane design­
ers until the late 1920s. Jump ahead to Figs. 6.75 and
6.76. In Fig. 6.75, a typical strut-and-wire biplane
from World War I, the French SPAD XIII, is shown.
This airplane is definitely not streamlined. In con­
trast, by the middle 1930s, streamlined airplanes
were in vogue, and the Douglas DC-3 shown in
Fig. 6.76 is a classic example. The evolution o f our
understanding o f streamlining, and how it was even­
tually applied in airplane design, is one o f the more
interesting stories in the history o f aerodynamics. For
this story, see Anderson, A History o f Aerodynamics
and Its Impact on Flying Machines, Cambridge Uni­
versity Press, New York, 1997.
Relative drag
force
(I?) Streamlined body
Code
Skin friction drag
Pressure drag
Figure 5.73 Comparison o f the drag for a blunt body and a streamlined body.
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other hand, as Re is increased above 3 x 105, transition takes place on the front
face, the boundary layer becom es turbulent, and the separation point m oves rear­
ward. (Turbulent boundary layers rem ain attached for longer distances in the face
o f adverse pressure gradients.) In this case, the wake behind the body is much
sm aller, as sketched in Fig. 5.12b. In turn, the pressure drag is less and C D de­
creases, as noted as the right o f Fig. 5.71.
Therefore, to decrease the drag on a sphere or cylinder, a turbulent boundary
layer m ust be obtained on the front surface. This can be made to occur naturally
by increasing Re until transition occurs on the front face. It can also be forced ar­
tificially at low er values o f Re by using a rough surface to encourage early tran­
sition or by w rapping w ire or other protuberances around the surface to create
turbulence. (The use o f such artificial devices is som etim es called tripping the
boundary layer.)
It is interesting to note that the dim ples on the surface o f a golf ball are
designed to prom ote turbulence and hence reduce the drag on the ball in flight.
Indeed, some recent research has shown that polygonal dim ples result in less
drag than the conventional circular dim ples on g o lf balls; but a dim ple o f any
shape leads to less pressure drag than a sm ooth surface does (table-tennis balls
have m ore drag than g olf balls).
5.19 HOW LIFT IS PRODUCED— SOME ALTERNATE
EXPLANATIONS
Return to our road map in Fig. 5.1. We have covered all the m ilestones on this
m ap except the one at the bottom labeled “How lift is produced.” This is the sub­
je c t o f this present section.
It is am azing that today, alm ost 100 years after the first flight o f the Wright
Flyer, groups o f engineers, scientist, pilots, and others can gather together and
have a spirited debate on how an airplane w ing generates lift. Various explana­
tions are put forth, and the debate centers on which explanation is the m ost fun­
dam ental. The purpose o f this section is to attem pt to put these various explana­
tions in perspective and to resolve the debate. Indeed, in our previous discussions
in this book, we have consistently put forth one explanation as the m ost funda­
m ental, and we have intentionally not burdened your thinking with any alterna­
tives. So you may be w ondering w hat the big deal is here. You already know how
lift is produced. However, because the literature is replete w ith various different,
and som etim es outright incorrect explanations as to how lift is produced, you
need to be aw are o f som e o f the alternative thinking.
First, let us consider w hat this author advocates as the m ost fundam ental ex­
planation o f lift. It is clear from our discussion in Sec. 2.2 that the two hands o f
nature that reach out and grab hold o f a body m oving through a fluid (liquid or
gas) are the pressure and shear stress distributions exerted all over the exposed
surface o f the body. The resultant aerodynam ic force on the body is the net, inte­
grated effect o f the pressure and shear stress distributions on the surface. Be­
cause the lift is the com ponent o f this resultant force perpendicular to the relative
w ind and because the pressure on the surface o f an airfoil at reasonable angles o f
5 .1 9
How Lift Is Produced—Some Alternate Explanations
attack acts mainly in the lift direction w hereas the shear stress acts mainly in the
drag direction, we are com fortable in saying that, for lift, the effect o f shear stress
is secondary and that lift is m ainly due to the im balance o f the pressure distribu­
tions over the top and bottom surfaces o f the airfoil. Specifically, the pressure on
the top surface o f the airfoil is low er than the pressure on the bottom surface, and
presto— lift! H ow ever, this raises the question as to why the pressure is low er on
the top o f the airfoil and higher on the bottom . The answ er is sim ply that the
aerodynam ic flow over the airfoil is obeying the laws o f nature, namely, m ass
continuity and N ew ton’s second law. Let us look at this m ore closely and see
how nature applies these laws so as to produce lift on an airplane wing. There are
three intellectual thoughts that follow in sequence:
1.
C onsider the flow over an airfoil as sketched in Fig. 5.74a. C onsider the
stream tubes A and B shown here. T he shaded stream tube A flows over the
top surface, and the unshaded stream tube B flows over the bottom surface.
Both stream tubes originate in the free stream ahead o f the airfoil. As stream
tube A flows tow ard the airfoil, it senses the upper portion o f the airfoil as
an obstruction, and stream tube A m ust m ove out o f the way o f this
Note: The length o f the arrows denoting pressure is proportional to
P -P rc f, where prcf is an arbitrary reference pressure slightly
less than the minimum pressure on the airfoil.
Figure 5.74 (a) Flow velocity on upper surface is on the
average higher than that on the bottom surface due to
squashing o f streamline A compared to streamline B. (b) As a
result, pressure on the top surface is lower than the pressure on
the bottom surface, hence creating lift in the upward direction.
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obstruction. In so doing, stream tube A is squashed to a sm aller crosssectional area as it flows over the nose o f the airfoil. In turn, because o f
m ass continuity ( p A V = constant), the velocity o f the flow in the stream
tube m ust increase in the region where the stream tube is being squashed.
T his higher velocity is show n by the long arrow at point a in Fig. 5.74a. As
the stream tube flows dow nstream o f point a, its cross-sectional area
gradually increases and the flow velocity decreases, such as shown by the
shorter arrow at point 6 . N ote that stream tube A is squashed the m ost in the
nose region, ahead o f the m axim um thickness o f the airfoil. H ence, the
m axim um velocity occurs ahead o f the m axim um thickness o f the airfoil.
N ow consider stream tube B, w hich flows over the bottom surface o f the
airfoil. The airfoil is designed with positive cam ber; hence, the bottom
surface o f the airfoil presents less o f an obstruction to stream tube B, and so
stream tube B is not squashed as m uch as stream tube A in flowing over the
nose o f the airfoil. As a result, the flow velocity in stream tube B rem ains
less than that in stream tube A. Therefore, w e can state the following:
Because o f the law o f mass continuity, that is, the continuity equation, the flow
velocity increases over the top surface o f the airfoil more than it does over the
bottom surface.
To see the squashing o f the stream tube in an actual flow, return to the
sm oke flow photograph in Fig. 2.6. Clearly the stream tube flowing over
the top surface o f the airfoil is being squashed in the region just
dow nstream o f the leading edge, and this is w here the m axim um flow
velocity is occurring.
2.
For an incom pressible flow, from B ernoulli’s equation p + \ p V 2 =
constant, clearly where the velocity increases, the static pressure decreases.
This trend is the sam e for com pressible flow. From E uler’s equation
d p = —p V d V , when the velocity increases (d V positive), the pressure
decreases (d p negative). We can label this general trend— namely, when the
velocity increases, the pressure decreases— the Bernoulli effect. Recall that
B ernoulli’s equation and E uler’s equation are statem ents o f N ew ton’s second
law. B ecause we have show n in item 1 that the flow velocity is higher over
the top surface than it is over the bottom surface, we can state the following:
Because o f the Bernoulli effect, the pressure over the top surface o f the airfoil is
less than the pressure over the bottom surface.
This is illustrated in Fig. 5.746, which is a schem atic o f the pressure
distribution over the top and bottom surfaces. Note that the m inim um
pressure occurs at point a.
3.
Finally, it follow s that
Owing to the lower pressure over the top surface and the higher pressure over
the bottom surface, the airfoil feels a lift force in the upward direction.
This lift force is shown schem atically in Fig. 5.746.
5 .1 9
How Lift Is Produced— Some Alternate Explanations
The sequence o f preceding item s 1 through 3 are the fundam ental laws of
nature that result in lift being produced on an airplane wing. You cannot get m ore
fundam ental than this— m ass conservation and N ew ton’s second law.
We also note that the above explanation shows why m ost o f the lift o f the
wing is produced by the first 20 or 30 percent o f the wing ju st dow nstream of
the leading edge. This is seen in Fig. 5.746 w here the largest difference in pres­
sure betw een the top and bottom surfaces is on the front part of the airfoil. That
most o f the lift is generated by the forw ard portion o f the airfoil is also seen in
Figs. 5.16, 4.48, and 4.49, w hich dem onstrate that the m inim um pressure on the
top surface occurs over the forw ard portion o f the airfoil ju st dow nstream o f the
leading edge. In a sense, the m ain function o f the back portion o f the airfoil is to
sim ply form a stream lined shape to avoid flow separation.
We dispel here a com m on m isconception about why the flow velocity in­
creases over the top surface o f the airfoil. It is som etim es found in the literature
that a fluid elem ent that com es into the stagnation region splits into two ele­
ments, one o f w hich flows over the top surface and the other over the bottom sur­
face. It is then assum ed that these tw o elem ents m ust m eet up at the trailing edge,
and because the running distance over the top surface o f the airfoil is longer than
that over the bottom surface, the elem ent over the top surface m ust m ove faster.
This is sim ply not true. Experim ental results and com putational fluid dynam ic
calculations clearly show that a fluid elem ent m oving over the top surface o f an
airfoil leaves the trailing edge long before its com panion m oving over the bottom
surface arrives at the trailing edge. This is illustrated in Fig. 5.75. C onsider a
com bined fluid elem ent CD at tim e t, in the stagnation region at the leading edge
o f the airfoil, as sketched in Fig. 5.75. This elem ent splits into elem ent C m ov­
ing over the top surface and elem ent D m oving over the bottom surface. At a
later tim e t2, elem ent C has m oved dow nstream o f the trailing edge, and elem ent
D has not yet arrived at the trailing edge. T he tw o elem ents sim ply do not m eet
at the trailing edge, and so any explanation that depends on their m eeting is
flawed.
The preceding explanation o f the generation o f lift applies also to flat plates
as well as curved airfoil shapes. C ontrary to statem ents in some o f the popular lit­
erature, the curved shape o f an airfoil is not necessary for the production o f lift.
A thin flat plate at an angle o f attack produces lift. A schem atic o f the stream line
pattern over a flat plate at angle o f attack is shown in Fig. 5.76. The stagnation
point (labeled s.p. in Fig. 5.76) is located on the bottom surface, dow nstream o f
Time f.
Figure 5.75 The tracking o f two fluid elements in the flow over
an airfoil. Element C moves over the top, and element D over the
bottom.
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Airfoils, Wings, and Other Aerodynamic Shapes
the leading edge. The stream line through the stagnation point is called the divid­
ing stream line; the flow above the dividing stream line flows up and over the top
o f the plate, w hereas the flow below the dividing stream line flows over the bot­
tom o f the plate. The shaded stream tube shown in Fig. 5.76 is analogous to the
shaded stream tube A in Fig. 5.74. The flow in the shaded stream tube in
Fig. 5.76 m oves upstream from the stagnation point along the surface, curls
around the leading edge w here, in term s o f our previous discussions, it experi­
ences extrem e squashing, and then flows dow nstream over the top o f the plate.
As a result at the squashing, the flow velocity at the leading edge is very large,
with a correspondingly low pressure. As the stream tube flows dow nstream over
the top o f the plate, its cross-sectional area gradually increases; hence, the flow
velocity gradually decreases from its initially high value at the leading edge, and
the surface pressure gradually increases from its initially low value. The pressure
on the top surface, however, rem ains, on the average, low er than that on the bot­
tom surface; as usual, this pressure difference produces lift on the plate. The
question is naturally raised: Why, then, do we not fly around on thin flat plates as
airplane wings. The answer, besides the obvious practical requirem ent for wing
thickness to allow room for internal structure and for fuel and landing gear stor­
age, is that the flat plate also produces drag— and a lot o f it. The flow over the top
surface tends to separate at the leading edge at fairly sm all angles o f attack, caus­
ing m assive pressure drag. Consequently, although the flat plate at angle o f attack
produces lift, the lift-to-drag ratio is m uch low er than conventional thick airfoils
with their stream lined shapes.
There are several alternate explanations o f the generation o f lift that are in
reality not the fu n d a m en ta l explanation, but rather are m ore o f an effect o f lift
being produced, not the cause. Let us exam ine these alternate explanations.
The follow ing alternate explanation is som etim es given: The wing deflects
the airflow such that the m ean velocity vector behind the w ing is canted slightly
5 .1 9
How Lift Is Produced—Some Alternate Explanations
L = tim e rate o f change o f
m om entum o f airflow in
the downward direction
AV
Figure 5.77 Relationship o f lift to the time rate o f change o f momentum o f the airflow.
dow nw ard, as sketched in Fig. 5.77. H ence, the wing im parts a dow nw ard com ­
ponent o f m om entum to the air; that is, the w ing exerts a force on the air, push­
ing the flow dow nw ard. From N ew ton’s third law, the equal and opposite reac­
tion produces a lift. H ow ever, this explanation really involves the effect o f lift,
and not the cause. In reality, the air pressure on the surface is pushing on the sur­
face, hence creating lift in the upw ard direction. As a result o f the equal-andopposite principle, the airfoil, surface pushes on the air, im parting a dow nw ard
force on the airflow, which deflects the velocity dow nw ard. Hence, the net rate of
change o f dow nw ard m om entum created in the airflow because o f the presence
o f the wing can be thought o f as an effect due to the surface pressure distribution;
the pressure distribution by itself is the fundam ental cause o f lift.
A third argum ent, called the circulation theory o f lift, is som etim es given for
the source o f lift. H ow ever, this turns out to be not so m uch an explanation o f lift
per se, but rather m ore o f a m athem atical form ulation for the calculation o f lift
for an airfoil o f given shape. M oreover, it is mainly applicable to incom pressible
flow. The circulation theory o f lift is elegant and well developed; it is also be­
yond the scope o f this book. H ow ever, som e o f its flavor is given as follows.
C onsider the flow over a given airfoil, as show n in Fig. 5.78. Im agine a
closed curve C draw n around the airfoil. At a point on this curve, the flow veloc­
ity is V , and the angle betw een V and a tangent to the curve is 0. Let d s be an in­
crem ental distance along C . A quantity called the circulation f is defined as
(5.72)
that is, T is the line integral o f the com ponent o f flow velocity along the closed
curve C . A fter a value o f T is obtained, the lift p e r unit span can be calculated
from
(5.73)
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C H A p T E R 5 Airfoils, Wings, and Other Aerodynamic Shapes
Uniform flow
Pure circulation
Incompressible flow over an airfoil
Figure 5.79 Addition of two elementary flows to synthesize a more complex flow. If one or more o f the
elementary flows have circulation, then the synthesized flow also has the same circulation. The lift is directly
proportional to the circulation.
Equation (5.73) is the K utta-Joukow sky theorem; it is a pivotal relation in the cir­
culation theory o f lift. T he object o f the theory is to (som ehow ) calculate T for a
given Voo and airfoil shape. Then Eq. (5.73) yields the lift. A m ajor thrust o f ideal
incom pressible flow theory, m any tim es called potential flo w theory, is to calcu­
late T. Such m atters are discussed in m ore-advanced aerodynam ic texts (see,
e.g., A nderson, Fundam entals o f Aerodynam ics, 3d ed., M cG raw -H ill, 2001).
The circulation theory o f lift is com patible with the true physical nature of
the flow over an airfoil, as any successful m athem atical theory m ust be. In the
sim plest sense, we can visualize the true flow over an airfoil, show n at the right
o f Fig. 5.79, as the superposition o f a uniform flow and a circulatory flow, shown
at the left o f Fig. 5.79. The circulatory flow is clockw ise, which when added to
5 .1 9
How Lift Is Produced—Some Alternate Explanations
359
the uniform flow, yields a higher velocity above the airfoil and a low er velocity
below the airfoil. From B ernoulli’s equation, this im plies a low er pressure on the
top surface o f the airfoil and a higher pressure on the bottom surface, hence gen­
erating lift in the upw ard direction. The strength o f the circulatory contribution,
defined by Eq. (5.72), is ju st the precise value such that when it is added to the
uniform flow contribution, the actual flow over the airfoil leaves the trailing edge
smoothly, as sketched at the right o f Fig. 5.79. This is called the Kutta condition
and is one o f the m ajor facets o f the circulation theory o f lift.
A gain, keep in m ind that the actual m echanism that nature has o f com m uni­
cating a lift to the airfoil is the pressure distribution over the surface o f the air­
foil, as sketched in Fig. 5.74b. In turn, this pressure distribution ultim ately
causes a tim e rate o f change o f m om entum o f the airflow, as shown in Fig. 5.77,
a principle which can be used as an alternate w ay o f visualizing the generation of
lift. Finally, even the circulation theory o f lift stem s from the pressure dis­
tribution over the surface o f the airfoil, because the derivation o f the KuttaJoukow sky theorem , Eq. (5.73), involves the surface pressure distribution. Again,
for more details, consult A nderson, Fundam entals o f Aerodynam ics, 3rd ed.,
M cG raw -H ill, 2 0 0 1.
EXAM PLE 5.22
Can an airfoil produce lift when it is flying upside down? This question is frequently
asked by the general public. The answer is yes, but not as effectively. Consider the NACA
4412 airfoil shown right-side up and upside down in Fig. 5.80a and b, respectively. Both
are shown at the same angle o f attack relative to the free stream. For an angle o f attack o f
6°, obtain the lift coefficient for each case shown in (a) and (b ).
Figure 5.80 Illustration o f (c/) an airfoil flying rightside up and (b) flying upside down. Both are at the
same angle o f attack.
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Airfoils, Wings, and Other Aerodynamic Shapes
■ Solution
а. From App. D, for the NACA 4412 airfoil at a = 6°,
ci = 1.02
б. Take Fig. 5.806, and turn it upside down. What you see is the N A C A 4412 airfoil rightside up, but at a negative angle o f attack. Therefore, the lift coefficient for the upside down
airfoil at the positive angle o f attack shown in Fig. 5.80b is given by the data in App. D for
negative angles o f attack. For a = —6°, App. D show s c¡ = —0.22; the negative c¡ con­
notes a downward lift on the ordinary right-side up airfoil when pitched to a negative angle
o f attack o f —6°. In the upside-down orientation shown in Fig. 5.80 b, this lift is directed
upward. Hence, for the N AC A 4412 airfoil flying upside down at an angle o f attack o f 6°,
c, = 0.22
Note: The airfoil flying upside down as shown in Fig. 5.806 produces lift, but not as much
as the same airfoil flying right-side up at the same angle o f attack. In order for the upsidedown airfoil in Fig. 5.806 to produce the same lift as the right-side up airfoil in
Fig. 5.80a, it must be pitched to a higher angle o f attack.
Aerobatic airplanes spend a lot o f time flying upside down. For this reason, the de­
signers o f such airplanes frequently choose a symmetric airfoil for the wing scction. A lso,
the horizontal and vertical tails on airplanes o f all types usually have symmetric airfoil
shapes. An aerobatic airplane, flown by the famous aerobatic pilot and three-time U.S.
National Champion Patty Wagstaff, is shown in Fig. 5.81, which show s the w ing with a
symmetric airfoil section.
(«)
Figure 5.81 Patty Wagstaff’s aerobatic airplane, the Extra 260, on display at the National
Air and Space Museum, (a) Full view o f the airplane. (continued)
5 .1 9
How Lift Is Produced— Some Alternate Explanations
Figure 5.81 (continued) (b) Left wing, showing the squared-off wing tip. (c) Detail of the
left wing tip, showing the symmetric airfoil section.
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5.20 HISTORICAL NOTE: AIRFOILS AND WINGS
We know that G eorge C ayley introduced the concept o f a fixed-wing aircraft in
1799; this has been discussed at length in Secs. 1.3 and 5.1. M oreover, Cayley
appreciated the fact that lift is produced by a region o f low pressure on the top
surface o f the w ing and high pressure on the bottom surface and that a cam bered
shape produces m ore lift than a flat surface. Indeed, Fig. 1.5 shows that Cayley
was thinking o f a curved surface for a w ing, although the curvature was due
to the wind billow ing against a loosely fitting fabric surface. However, neither
C ayley nor any o f his im m ediate follow ers perform ed work even closely resem ­
bling airfoil research or developm ent.
It was not until 1884 that the first serious airfoil developm ents w ere made. In
this year, H oratio F. Phillips, an Englishm an, was granted a patent for a series o f
double-surface, cam bered airfoils. Figure 5.82 shows Phillip’s patent draw ings
for his airfoil section. Phillips was an im portant figure in late 19th-century aero­
nautical engineering; we m et him before, in Sec. 4.24, in conjunction with his
ejector-driven w ind tunnel. In fact, the airfoil shapes in Fig. 5.82 were the result
o f num erous wind tunnel experim ents in w hich Phillips exam ined curved wings
o f “every conceivable form and com bination o f form s.” Phillips w idely pub­
lished his results, w hich had a m ajor im pact on the aeronautics com m unity. C on­
tinuing with his work, Phillips patented m ore airfoil shapes in 1891. Then, m ov­
ing into airplane design in 1893, he built and tested a large m ultiplane model,
consisting o f a large num ber o f wings, each with a 19-ft span and a chord o f only
1j in, which looked like a Venetian blind! The airplane was pow ered by a steam
Figure 5.82 Double-surface airfoil sections by Phillips. The six
upper shapes were patented by Phillips in 1884; the lower
airfoil was patented in 1891.
5 .2 0
Historical Note: Airfoils and Wings
engine with a 6.5-ft propeller. The vehicle ran on a circular track and actually
lifted a few feet off the ground, mom entarily. A fter this dem onstration, Phillips
gave up until 1907, when he m ade the first tentative hop flight in England in a
similar, but gasoline-pow ered, m achine, staying airborne for about 500 ft. This
was his last contribution to aeronautics. H ow ever, his pioneering work during the
1880s and 1890s clearly designates Phillips as the grandparent o f the modern
airfoil.
A fter Phillips, the w ork on airfoils shifted to a search for the m ost efficient
shapes. Indeed, w ork is still being done today on this very problem , although
m uch progress has been made. This progress covers several historical periods, as
described in the follow ing. Secs. 5.20.1 to 5.20.6.
5.20.1 T he W right B rothers
As noted in Secs. 1.8 and 4.24, W ilbur and O rville W right, after their early ex­
perience with gliders, concluded in 1901 that m any o f the existing “air pressure”
data on airfoil sections w ere inadequate and frequently incorrect. To rectify these
deficiencies, they constructed their own w ind tunnel (see Fig. 4.51), in which
they tested several hundred different airfoil shapes betw een Septem ber 1901 and
August 1902. From their experim ental results, the W right brothers chose an air­
foil w ith a 1 /2 0 m axim um cam ber-to-chord ratio for their successful Wright
Flyer 1 in 1903. T hese wind tunnel tests by the W right brothers constituted a
m ajor advance in airfoil technology at the turn o f the century.
5.20 .2 B ritish and U .S. A irfo ils (1 9 1 0 to 1920)
In the early days o f pow ered flight, airfoil design was basically custom ized and
personalized; very little concerted effort was m ade to find a standardized, effi­
cient section. However, som e early work was perform ed by the British govern­
ment at the N ational Physical Laboratory (N PL), leading to a series o f Royal
A ircraft Factory (RAF) airfoils used on W orld War I airplanes. Figure 5.83 illus­
trates the shape o f the RAF 6 airfoil. In the U nited States, m ost aircraft until 1915
used either an RA F section or a shape designed by the Frenchm an Alexandre
G ustave Eiffel. This tenuous status o f airfoils led the N ACA , in its First Annual
Figure 5.83 Typical airfoils in 1917.
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Report in 1915, to em phasize the need for “the evolution o f more efficient wing
sections o f practical form, em bodying suitable dim ensions for an econom ical
structure, with m oderate travel o f the center o f pressure and still affording a large
angle o f attack com bined w ith efficient action.” To this day, more than 85 years
later, N A SA is still pursuing such work.
T he first N A CA w ork on airfoils was reported in NA CA Report No. 18,
“A erofoils and Aerofoil Structural C om binations,” by Lt. Col. Edgar S. Gorrell
and M ajor H. S. M artin, prepared at the M assachusetts Institute o f Technology
(M IT) in 1917. G orrell and M artin sum m arized the contem porary airfoil status as
follows:
Mathematical theory has not, as yet, been applied to the discontinuous motion past a
cambered surface. For this reason, we are able to design aerofoils only by consider­
ation o f those forms which have been successful, by applying general rules learned
by experience, and by then testing the aerofoils in a reliable wind tunnel.
In N A C A Report No. 18, Gorrell and M artin disclosed a series o f tests on the
largest single group o f airfoils to that date, except for the work done at N PL and
by Eiffel. They introduced the USA airfoil series and reported wind tunnel data
for the USA 1 through USA 6 sections. Figure 5.83 illustrates the shape o f the
USA 6 airfoil. The airfoil m odels w ere m ade o f brass and were finite w ings with
a span o f 18 in and a chord o f 3 in; that is, A R = 6 . Lift and drag coefficients
w ere m easured at a velocity o f 30 mi/h in the M IT w ind tunnel. T hese airfoils
represented the first system atic series originated and studied by NACA.
5.2 0 .3 1920 to 1930
Based on their wind tunnel observations in 1917, Gorrell and M artin stated that
slight variations in airfoil design m ake large differences in the aerodynam ic per­
form ance. This is the underlying problem o f airfoil research, and it led in the 1920s
to a proliferation o f airfoil shapes. In fact, as late as 1929, F. A. Louden, in his
NA CA Report No. 331, entitled “Collection o f W ind Tunnel Data on Com m only
Used W ing Sections,” stated that “the w ing sections m ost com m only used in this
country are the Clark Y, Clark Y-15, G ottingen G -387, G -398, G -436, N.A.C.A.
M -12, Navy N-9, N-10, N-22, R.A.F.-15, Sloane, U .S.A .-27, U .S.A .-35A ,
U .S.A .-35B .” However, help was on its way. As noted in Sec. 4.24, the NACA
built a variable-density wind tunnel at Langley A eronautical Laboratory in 1923,
a wind tunnel that was to becom e a w orkhorse in future airfoil research, as
em phasized in Sec. 5.20.4.
5 .2 0 .4 E arly N A C A F ou r-D igit A irfoils
In a classical w ork in 1933, order and logic were finally brought to airfoil design
in the U nited States. This was reported in NA CA Report No. 460, “The C harac­
teristics o f 78 Related A irfoil Sections from Tests in the Variable-Density W ind
T unnel,” by Eastm an N. Jacobs, Kenneth E. Ward, and Robert M. Pinkerton.
5 .2 0
Historical Note: Airfoils and Wings
T heir philosophy on airfoil design was as follows:
Airfoil profiles may be considered as made up o f certain profile-thickness forms dis­
posed about certain mean lines. The major shape variables then becom e two, the
thickness form and the m ean-line form. The thickness form is o f particular impor­
tance from a structural standpoint. On the other hand, the form o f the mean line de­
termines almost independently som e o f the most important aerodynamic properties
o f the airfoil section, e.g., the angle o f zero lift and the pitching-moment character­
istics. The related airfoil profiles for this investigation were derived by changing sys­
tematically these shape variables.
They then proceeded to define and study for the first tim e in history the fa­
m ous N A C A four-digit airfoil series, som e o f w hich are given in App. D o f this
book. F or exam ple, N A C A 2412 is defined as a shape that has a m axim um cam ­
ber o f 2 percent o f the chord (the first digit); the m axim um cam ber occurs at a po­
sition o f 0.4 chord from the leading edge (the second digit), and the m axim um
thickness is 12 percent (the last tw o digits). Jacobs and his colleagues tested
these airfoils in the N A C A variable-density tunnel using a 5-in by 30-in finite
wing (again, an aspect ratio o f 6 ). In N A CA Report No. 460, they gave curves of
C l , C o , and L / D for the finite wing. M oreover, using the same form ulas devel­
oped in Sec. 5.15, they corrected their data to give results for the infinite wing
case also. A fter this work was published, the standard NACA four-digit airfoils
w ere w idely used. Indeed, even today the N A C A 2412 is used on several light
aircraft.
5.20.5 L ater N A C A A irfoils
In the late 1930s, NACA developed a new cam ber line fam ily to increase m axi­
m um lift, with the 230 cam ber line being the m ost popular. Com bining with the
standard NACA thickness distribution, this gave rise to the NACA five-digit air­
foil series, such as the 23012, som e o f which are still flying today, for exam ple,
on the C essna Citation and the Beech King Air. This w ork was follow ed by fam ­
ilies o f high-speed airfoils and lam inar flow airfoils in the 1940s.
To reinforce its airfoil developm ent, in 1939 NACA constructed a new lowturbulence tw o-dim ensional wind tunnel at Langley exclusively for airfoil test­
ing. This tunnel has a rectangular test section 3 ft w ide and 7^ ft high and can be
pressurized up to 10 atm for h igh-R eynolds num ber testing. M ost importantly,
this tunnel allows airfoil m odels to span the test section com pletely, thus directly
providing infinite wing data. This is in contrast to the earlier tests previously de­
scribed, which used a finite w ing o f A R = 6 and then corrected the data to cor­
respond to infinite wing conditions. Such corrections are alw ays com prom ised
by tip effects. (For exam ple, w hat is the precise span efficiency factor for a given
w ing?) W ith the new tw o-dim ensional tunnel, vast num bers o f tests were per­
form ed in the early 1940s on both old and new airfoil shapes over a Reynolds
num ber range from 3 to 9 m illion and at M ach num bers less than 0.17 (incom ­
pressible flow). The airfoil m odels generally had a 2-ft chord and com pletely
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Airfoils, Wings, and Other Aerodynamic Shapes
spanned the 3-ft width o f the test section. It is interesting to note that the lift and
drag are not obtained on a force balance. Rather, the lift is calculated by inte­
grating the m easured pressure distribution on the top and bottom walls o f the
w ind tunnel, and the drag is calculated from Pitot pressure m easurem ents made
in the w ake dow nstream o f the trailing edge. H ow ever, the pitching m om ents are
m easured directly on a balance. A vast am ount o f airfoil data obtained in this
fashion from the tw o-dim ensional tunnel at Langley w ere com piled and pub­
lished in a book entitled Theory o f Wing Sections Including a Sum m ary o f Airfoil
Data, by A bbott and von D oenhoff, in 1949 (see Bibliography at end o f this
chapter). It is im portant to note that all the airfoil data in App. D are obtained
from this reference; that is, all the data in App. D are direct m easurem ents for an
infinite wing at essentially incom pressible flow conditions.
5 .2 0 .6 M odern A irfoil W ork
Priorities for supersonic and hypersonic aerodynam ics put a stop to the NACA
airfoil developm ent in 1950. O ver the next 15 years, specialized equipm ent for
airfoil testing was dism antled. Virtually no system atic airfoil research was done
in the United States during this period.
H ow ever, in 1965, Richard T. W hitcom b made a breakthrough with the
N A SA supercritical airfoil. This revolutionary developm ent, which allow ed the
design o f wings with high critical Mach num bers (see Sec. 5.10), served to reac­
tivate the interest in airfoils within NASA. Since that tim e, a healthy program in
m odern airfoil developm ent has been reestablished. The low -turbulence, pres­
surized, tw o-dim ensional wind tunnel at Langley is back in operation. M oreover,
a new dim ension has been added to airfoil research: the high-speed digital com ­
puter. In fact, com puter program s for calculating the flow field around airfoils at
subsonic speeds are so reliable that they shoulder som e o f the routine testing
duties heretofore exclusively carried by w ind tunnels. The sam e cannot yet be
said about transonic cases, but current research is focusing on this problem . An
interesting survey o f m odern airfoil activity w ithin N A SA is given by Pierpont in
A stronautics and Aeronautics (see Bibliography).
O f special note is the m odern low -speed airfoil series, designated LS( 1), de­
veloped by NA SA for use by general aviation on light airplanes. The shape o f a
typical LS(1) airfoil is contrasted with a “conventional” airfoil in Fig. 5.84. Its lift­
ing characteristics, illustrated in Fig. 5.85, are clearly superior and should allow
sm aller wing areas, hence less drag, for airplanes o f the type shown in Fig. 5.84.
In sum m ary, airfoil developm ent over the past 100 years has m oved from an
ad hoc individual process to a very system atic and logical engineering process. It
is alive and well today, with the prom ise o f m ajor advancem ents in the future
using both wind tunnels and com puters.
5.20.7 F inite W ings
Som e historical com m ents on the finite wing are in order. Francis Wenham
(see Chap. 1), in his classic paper entitled A erial Locom otion, given to the
5 .2 0
Historical Note: Airfoils and Wings
Figure 5.84 Shape comparison between the modern LS( 1)-0417 and a
conventional airfoil. The higher lift obtained with the LS(1)-0417 allows a
smaller wing area and hence lower drag.
(Source: NASA.)
R eynolds num ber Re X 10 6
Low speed, M = 0.2
Figure 5.85 Comparison o f maximum lift coefficients
between the LS(1)-0417 and conventional airfoils.
(Source: NASA.)
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A eronautical Society o f G reat Britain on June 27, 1866, theorized (correctly) that
m ost o f the lift o f a wing occurs from the portion near the leading edge; hence, a
long, narrow w ing w ould be m ost efficient. In this fashion, he was the first per­
son in history to appreciate the value o f high-aspect ratio wings for subsonic
flight. M oreover, he suggested stacking a num ber o f long, thin wings above one
another to generate the required lift; hence, he becam e an advocate o f the m ulti­
plane concept. In turn, he built two full-size gliders in 1858, both with five wings
each, and successfully dem onstrated the validity o f his ideas.
However, the true aerodynam ic theory and understanding o f finite wings did
not com e until 1907, when the Englishm an Frederick W. Lanchester published
his book Aerodynam ics. In it, he outlined the circulation theory o f lift, developed
independently about the sam e tim e by Kutta in G erm any and by Joukow sky in
Russia. M ore im portantly, L anchester discussed for the first time the effect o f
wing tip vortices on finite-wing aerodynam ics. Unfortunately, L anchester was
not a clear writer; his ideas w ere extrem ely difficult to understand, and they did
not find application in this aeronautical community.
In 1908, Lanchester visited G ottingen, Germ any, and fully discussed his
wing theory with Ludw ig Prandtl and his student, Theodore von Karman. Prandtl
spoke no English, L anchester spoke no G erm an, and in light o f L anchester’s un­
clear ways o f explaining his ideas, there appeared to be little chance o f under­
standing betw een the two parties. H ow ever, in 1914, Prandtl set forth a simple,
clear, and correct theory for calculating the effect o f tip vortices on the aerody­
nam ic characteristics o f finite wings. It is virtually im possible to assess how
m uch Prandtl was influenced by Lanchester, but to Prandtl must go the credit of
first establishing a practical finite wing theory, a theory w hich is fundam ental to
our discussion o f finite w ings in Secs. 5.13 to 5.15. Indeed, P randtl’s first pub­
lished w ords on the subject w ere these:
The lift generated by the airplane is, on account o f the principle o f action and reac­
tion, necessarily connected with a descending current in all its details. It appears that
the descending current is formed by a pair o f vortices, the vortex filaments o f which
start from the airplane wingtips. The distance o f the two vortices is equal to the span
o f the airplane, their strength is equal to the circulation o f the current around the air­
plane, and the current in the vicinity o f the airplane is fully given by the superposi­
tion o f the uniform current with that o f a vortex consisting o f three rectilinear
sections.
P randtl’s pioneering w ork on finite w ing theory, along with his ingenious
concept o f the boundary layer, has earned him the title parent o f aerodynamics.
In the four years follow ing 1914, he w ent on to show that an elliptical lift distri­
bution results in the m inim um induced drag. Indeed, the first coining o f the terms
induced drag and profile drag was made in 1918 by M ax M unk in a note entitled
“Contribution to the A erodynam ics o f the Lifting Organs o f the A irplane.” M unk
w as a colleague o f Prandtl’s, and the note was one o f several classified G erm an
w artim e reports on airplane aerodynam ics.
5.21
Historical Note: Ernst Mach and His Number
F or m ore details on the history o f airfoils and wings, see A nderson, A H is­
tory o f A erodynam ics and Its Im pact on Flying M achines, C am bridge University
Press, New York, 1997.
5.21 HISTORICAL NOTE: ERNST MACH
AND HIS NUMBER
Airplanes that fly at M ach 2 are com m onplace today. Indeed, high-perform ance
m ilitary aircraft such as the Lockheed S R -7 1 B lackbird can exceed M ach 3. As a
result, the term M ach num ber has becom e part o f our general language— the
average person in the street understands that M ach 2 m eans tw ice the speed o f
sound. On a m ore technical basis, the dim ensional analysis described in Sec. 5.3
dem onstrated that aerodynam ic lift, drag, and m om ents depend on tw o im portant
dim ensionless products— the Reynolds num ber and the M ach number. Indeed, in
a more general treatm ent of fluid dynam ics, the Reynolds num ber and M ach num ­
ber can be shown as the m ajor governing param eters for any realistic flow field;
they are am ong a series o f governing dim ensionless param eters called sim ilarity
param eters. We already exam ined the historical source o f the Reynolds num ber
in Sec. 4.25; let us do the sam e for the M ach num ber in this present section.
T he M ach num ber is nam ed after Ernst M ach, a fam ous 19th-century physi­
cist and philosopher. M ach was an illustrious figure with widely varying inter­
ests. He was the first person in history to observe supersonic flow and to under­
stand its basic nature. Let us take a quick look at this man and his contributions
to supersonic aerodynam ics.
Ernst M ach was born at Turas, M oravia, in Austria, on February 18, 1838.
M ach’s father and m other w ere both extrem ely private and introspective intel­
lectuals. His father was a student o f philosophy and classical literature; his
m other was a poet and m usician. T he fam ily was voluntarily isolated on a farm,
where M ach’s father pursued an interest o f raising silkw orm s— indeed pioneer­
ing the beginning o f silkw orm culture in Europe. At an early age, Mach was not
a particularly successful student. Later, M ach described him self as a “weak piti­
ful child who developed very slowly.” Through extensive tutoring by his father
at hom e, Mach learned Latin, G reek, history, algebra, and geometry. A fter m ar­
ginal perform ances in grade school and high school (due not to any lack of in­
tellectual ability, but rather to a lack o f interest in the material usually taught by
rote), M ach entered the U niversity o f Vienna. There he blossom ed, spurred by
interest in m athem atics, physics, philosophy, and history. In 1860, he received
a Ph.D. in physics, writing a thesis entitled "On Electrical Discharge and In­
duction.” By 1864, he was a professor o f physics at the University o f Graz.
(The variety and depth o f his intellectual interests at this time are attested by
the fact that he turned dow n the position o f a chair in surgery at the University
o f Salzburg to go to G raz.) In 1867, M ach becam e a professor o f experim ental
physics at the U niversity o f Prague— a position he would occupy for the next
28 years.
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In today’s m odern technological world, where engineers and scientists are
virtually forced, out o f necessity, to peak their know ledge in very narrow areas o f
extrem e specialization, it is interesting to reflect on the personality o f M ach, who
was the suprem e generalist. H ere is only a partial list o f M ach’s contributions,
as dem onstrated in his writings: physical optics, history o f science, m echanics,
philosophy, origins o f relativity theory, supersonic flow, physiology, therm ody­
nam ics, sugar cycle in grapes, physics o f music, and classical literature. He even
w rote on w orld affairs. (O ne o f M ach’s papers com m ented on the “absurdity
com m itted by the statesm an w ho regards the individual as existing solely for the
sake o f the state” ; for this, M ach was severely criticized at that tim e by Lenin.)
We can only sit back with aw e and envy for M ach, who— in the w ords o f U.S.
philosopher W illiam Jam es— knew “everything about everything.”
M ach’s contributions to supersonic aerodynam ics w ere highlighted in a
paper entitled “Photographische Fixierung der durch Projektile in der Luft eingeleiten Vorgange,” given to the A cadem y o f Sciences in Vienna in 1887. Here,
for the first tim e in history, M ach show ed a photograph o f a shock wave in front
o f a bullet m oving at supersonic speeds. This historic photograph, taken from
M ach’s original paper, is show n in Fig. 5.86. A lso visible are w eaker waves at the
rear o f the projectile and the structure o f the turbulent w ake dow nstream o f the
base region. The tw o vertical lines are trip wires designed to tim e the photo­
graphic light source (or spark) with the passing o f the projectile. M ach was a
precise and careful experim enter; the quality o f the picture shown in Fig. 5.86,
Figure 5.86 Photograph o f a bullet in supersonic flight, published by Ernst Mach in 1887.
5 .21
Historical Note: Ernst Mach and His Number
along with the very fact that he was able to m ake the shock w aves visible in the
first place (he used an innovative technique called the shadow gram ), attests to his
exceptional experim ental abilities. Note that M ach was able to carry out such
experim ents involving split-second tim ing w ithout the benefit o f electronics—
indeed, the vacuum tube had not yet been invented.
M ach was the first to understand the basic characteristics o f supersonic flow.
He was the first to point out the im portance o f the flow velocity V relative to the
speed o f sound a and to note the discontinuous and m arked changes in a flow
field as the ratio V/a changes from less than 1 to greater than 1. He did not, how ­
ever, call this ratio the M ach number. The term M ach num ber was not coined
until 1929, when the w ell-know n Swiss engineer Jacob Ackeret introduced this
term inology, in honor o f M ach, at a lecture at the Eidgenossiche Technische
H ochschule in Zurich. Hence, the term M ach num ber is o f fairly recent use, not
introduced into the English literature until 1932.
M ach was an active thinker, lecturer, and w riter up to the tim e o f his death
on February 19, 1916, near M unich, one day after his 78th birthday. His
contributions to hum an thought were many, and his general philosophy on
epistem ology— a study o f know ledge itself— is still discussed in college classes
in philosophy today. A eronautical engineers know him as the originator o f su­
personic aerodynam ics; the rest o f the w orld know s him as a man w ho originated
the follow ing philosophy, as paraphrased by Richard von M ises, him self a wellknow n m athem atician and aerodynam icist o f the early 20 th century:
Mach does not start out to analyze statements, systems of sentences, or theories, but
rather the world of phenomena itself. His elements are not the simplest sentences,
and hence the building stones of theories, but rather—at least according to his way
of speaking—the simplest facts, phenomena, and events of which the world in which
we live and which we know is composed. The world open to our observation and ex­
perience consists of “colors, sounds, warmths, pressure, spaces, times, etc.” and their
components in greater and smaller complexes. All we make statements or assertions
about, or formulate questions and answers to, are the relations in which these ele­
ments stand to each other. That is Mach’s point of view.1
We end this section with a photograph o f M ach, taken about 1910, shown in
Fig. 5.87. It is a picture o f a thoughtful, sensitive man; no w onder that his phi­
losophy o f life em phasized observation through the senses, as discussed by von
M ises. In honor o f his m emory, an entire research institute, the Ernst M ach Insti­
tute in Germ any, is nam ed. This institute deals with research in experim ental gas
dynam ics, ballistics, high-speed photography, and cinem atography. Indeed, for a
m uch m ore extensive review o f the technical accom plishm ents o f M ach, see the
recent paper authored by a m em ber o f the Ernst M ach Institute, H. Reichenbach,
entitled “C ontributions o f E rnst M ach to Fluid M echanics,” in A nnual Reviews
o f F luid M echanics, vol. 15, 1983, pp. 1-28 (published by A nnual Review s, Inc.,
Palo Alto, C alifornia).
'From Richard von Mises, Positivism, A Study in Human Understanding, Braziller, New York, 1956.
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Figure 5.87 Ernst Mach (1838-1916).
5.22 HISTORICAL NOTE: THE FIRST MANNED
SUPERSONIC FLIGHT
O n O ctober 14, 1947, a hum an being flew faster than the speed o f sound for
the first tim e in history. Im agine the m agnitude o f this accom plishm ent—-just
60 years after Ernst M ach observed shock w aves on supersonic projectiles (see
Sec. 5.21) and a scant 44 years after the W right brothers achieved their first suc­
cessful pow ered flight (see Secs. 1.1 and 1.8 ). It is alm ost certain that M ach was
not thinking at all about heavier-than-air m anned flight o f any kind, w hich in his
day was still considered to be virtually im possible and the essence o f foolish
dream s. It is also alm ost certain that the W right brothers had not the rem otest
idea that their fledgling 30 m i/h flight on D ecem ber 17, 1903, would ultim ately
lead to a m anned supersonic flight in O rville’s lifetim e (although W ilbur died in
1912, O rville lived an active life until his death in 1948). C om pared to the total
spectrum o f m anned flight reaching all the way back to the ideas o f Leonardo da
Vinci in the 15th century (see Sec. 1.2), this rapid advancem ent into the realm o f
supersonic flight is truly phenom enal. How did this advancem ent occur? W hat
are the circum stances surrounding the first supersonic flight? W hy w as it so im ­
portant? The purpose o f this section is to address these questions.
5.2 2
Historical Note: The First Manned Supersonic Flight
Supersonic flight did not happen by chance; rather, it was an inevitable result
o f the progressive advancem ent o f aeronautical technology over the years. On
one hand, we have the evolution o f high-speed aerodynam ic theory, starting with
the pioneering w ork o f M ach, as described in Sec. 5.21. This was follow ed by the
developm ent o f supersonic nozzles by tw o European engineers, Carl G. P. de
Laval in Sw eden and A. B. Stodola in Sw itzerland. In 1887, de Laval used a
convergent-divergent supersonic nozzle to produce a high-velocity flow o f steam
to drive a turbine. In 1903, Stodola was the first person in history to definitely
prove (by m eans o f a series o f laboratory experim ents) that such convergentdivergent nozzles did indeed produce supersonic flow. From 1905 to 1908,
Prandtl in G erm any took pictures o f M ach w aves inside supersonic nozzles and
developed the first rational theory for oblique shock w aves and expansion waves.
A fter W orld War I, Prandtl studied com pressibility effects in high-speed sub­
sonic flow. This w ork, in conjunction w ith independent studies by the English
aerodynam icist H erm an G lauert, led to the publishing o f the Prandtl-G lauert rule
in the late 1920s (see Sec. 5.6 for a discussion o f the Prandtl-G lauert rule and its
use as a com pressibility correction). These m ilestones, am ong others, established
a core o f aerodynam ic theory for high-speed flow— a core that was well estab­
lished at least 20 years before the first supersonic flight. (For m ore historical
details concerning the evolution o f aerodynam ic theory pertaining to supersonic
flight, see A nderson, M o d em Com pressible Flow: With H istorical Perspective,
3d ed„ M cG raw -H ill, 2003.)
On the other hand, we also have the evolution o f hardw are necessary for su­
personic flight. The developm ent o f high-speed wind tunnels, starting with the
small 12-in-diam eter high-speed subsonic tunnel at N A C A Langley M emorial
A eronautical L aboratory in 1927 and continuing with the first practical super­
sonic w ind tunnels developed by A dolf B usem ann in G erm any in the early
1930s, is described in Sec. 4.24. The exciting developm ents leading to the first
successful rocket engines in the late 1930s are discussed in Sec. 9.16. The con­
current invention and developm ent o f the je t engine, which would ultim ately pro­
vide the thrust necessary for everyday supersonic flight, are related in Sec. 9.15.
Hence, on the basis o f the theory and hardw are which existed at that time, the
advent o f m anned supersonic flight in 1947 was a natural progression in the
advancem ent o f aeronautics.
However, in 1947, there was one m issing link in both the theory and the
hardw are— the transonic regim e, near Mach 1. The governing equations for tran­
sonic flow are highly nonlinear and hence are difficult to solve. No practical
solution o f these equations existed in 1947. This theoretical gap was com ­
pounded by a sim ilar gap in wind tunnels. The sensitivity o f a flow near Mach 1
m akes the design o f a proper transonic tunnel difficult. In 1947, no reliable tran­
sonic wind tunnel data w ere available. This gap of know ledge was o f great con­
cern to the aeronautical engineers w ho designed the first supersonic airplane, and
it was the single m ost im portant reason for the excitem ent, apprehension, uncer­
tainty, and outright bravery that surrounded the first supersonic flight.
The unansw ered questions about transonic flow did nothing to dispel the
m yth o f the “sound barrier” that arose in the 1930s and 1940s. As discussed in
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Sec. 5.12, the very rapid increase in drag coefficient beyond the drag-divergence
M ach num ber led som e people to believe that hum ans would never fly faster than
the speed o f sound. G rist was lent to their argum ents when, on Septem ber 27,
1946, G eoffrey deH avilland, son o f the fam ous British airplane designer, took
the D.H. 108 Sw allow up for an attack on the w orld’s speed record. The Sw allow
was an experim ental jet-propelled aircraft, with sw ept wings and no horizontal
tail. A ttem pting to exceed 615 mi/h on its first high-speed, low -altitude run, the
Sw allow encountered m ajor com pressibility problem s and broke up in the air.
D eH avilland was killed instantly. The sound barrier had taken its toll. It was
against this background that the first supersonic flight was attem pted in 1947.
During the late 1930s, and all through W orld War II, som e visionaries clearly
saw the need for an experim ental airplane designed to probe the m ysteries o f su­
personic flight. Finally, in 1944, their efforts prevailed; the Arm y A ir Force, in
conjunction w ith N A CA , aw arded a contract to Bell A ircraft C orporation for the
design, construction, and prelim inary testing o f a m anned supersonic airplane.
D esignated the XS-1 (Experim ental S o n ic-1), this design had a fuselage shaped
like a 50-caliber bullet, m ated to a pair o f very thin (thickness-to-chord ratio o f
0.08), lo w -asp ect ratio, straight wings, as show n in Fig. 5.88. The aircraft was
pow ered by a four-cham ber liquid-propellant rocket engine m ounted in the tail.
This engine, m ade by R eaction M otors and designated the X L R 11, produced
6000 lb o f thrust by burning a m ixture o f liquid oxygen and diluted alcohol.
T he Bell X S -1 was designed to be carried aloft by a parent airplane, such as
the giant Boeing B-29, and then launched at altitude; in this fashion, the extra
w eight o f fuel that w ould have been necessary for takeoff and clim b to altitude
was saved, allow ing the designers to concentrate on one perform ance aspect—
speed. T hree X S -Is w ere ultim ately built, the first one being com pleted ju st after
Figure 5.88 T he B e ll X S -1 , the first su p erson ic airplane, 1947.
(Source: National A ir and Space Museum.)
5 .2 2
Historical Note: The First Manned Supersonic Flight
Christm as 1945. T here follow ed a year and a h alf o f gliding and then pow ered
tests, wherein the XS-1 was cautiously nudged tow ard the speed o f sound.
M uroc Dry Lake is a large expanse o f flat, hard lake bed in the M ojave
D esert in California. Site o f a U.S. A rm y high-speed flight test center during
World W ar II, M uroc was later to becom e know n as Edw ards A ir Force Base,
now the site o f the A ir Force Test Pilots School and the hom e o f all experim ental
high-speed flight testing for both the A ir Force and N ASA . On Tuesday, O ctober
14, 1947, the Bell X S -1, nestled under the fuselage o f a B-29, was w aiting on the
flight line at M uroc. A fter intensive preparations by a sw arm of technicians, the
B-29 with its cargo took off at 10 a m . On board the X S -1 was C aptain Charles E.
(Chuck) Yeager. T hat m orning, Yeager was in excruciating pain from tw o broken
ribs fractured during a horseback riding accident earlier that weekend; however,
he told virtually no one. At 10:26 a m at an altitude o f 20,000 ft, the Bell XS-1,
with Yeager as its pilot, was dropped from the B-29. W hat happened next is one
o f the m ajor m ilestones in aviation history. Let us see how Yeager him self re­
called events, as stated in his w ritten flight report:
Date:
Pilot:
Time:
14 October 1947
Capt. Charles E. Yeager
14 Minutes
9th Powered Flight
1. After normal pilot entry and the subsequent climb, the XS-1 was dropped from
the B-29 at 20,000' and at 250 MPH IAS. This was slower than desired.
2. Immediately after drop, all four cylinders were turned on in rapid sequence,
their operation stabilizing at the chamber and line pressures reported in the last
flight. The ensuing climb was made at .85-88 Mach, and, as usual it was necessary
to change the stabilizer setting to 2 degrees nose down from its pre-drop setting of
1 degree nose down. Two cylinders were turned off between 35,000' and 40,000', but
speed had increased to .92 Mach as the airplane was leveled off at 42,000'. Inciden­
tally, during the slight push-over at this altitude, the lox line pressure dropped per­
haps 40 psi and the resultant rich mixture caused the chamber pressures to decrease
slightly. The effect was only momentary, occurring at .6 G ’s, and all pressures re­
turned to normal at 1 G.
3. In anticipation of the decrease in elevator effectiveness at all speeds above .93
Mach, longitudinal control by means of the stabilizer was tried during the climb at
.83, .88, and .92 Mach. The stabilizer was moved in increments of
degree and
proved to be very effective; also, no change in effectiveness was noticed at the dif­
ferent speeds.
4. At 42,000' in approximately level flight, a third cylinder was turned on. Accel­
eration was rapid and speed increased to .98 Mach. The needle of the machmeter
fluctuated at this reading momentarily, then passed off the scale. Assuming that the
off-scale reading remained linear, it is estimated that 1.05 Mach was attained at this
time. Approximately 30% of fuel and lox remained when this speed was reached and
the motor was turned off.
5. While the usual lift buffet and instability characteristics were encountered in the
.8 8 -9 0 Mach range and elevator effectiveness was very greatly decreased at .94
Mach, stability about all three axes was good as speed increased and elevator effec­
tiveness was regained above .97 Mach. As speed decreased after turning off the
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motor, the various phenomena occurred in reverse sequence at the usual speeds, and
in addition, a slight longitudinal porpoising was noticed from .98-96 Mach which
was controllable by elevators alone. Incidentally, the stabilizer setting was not
changed from its 2 degrees nose down position after trial at .92 Mach.
6. After jettisoning the remaining fuel and lox at 1 G stall was performed at
45,000'. The flight was concluded by the subsequent glide and a normal landing on
the lakebed.
CHARLES E. YEAGER
Capt. Air Corps
In reality, the Bell SX-1 had reached M 00 = 1.06, as determ ined from official
N A C A tracking data. The duration o f its supersonic flight was 20.5 s, alm ost
tw ice as long as the W right brothers’ entire first flight ju st 44 years earlier. On
that day, C huck Yeager becam e the first person to fly faster than the speed o f
sound. It is a fitting testim onial to the aeronautical engineers at that tim e that the
flight was sm ooth and w ithout unexpected consequences. An aircraft had finally
been properly designed to probe the “sound barrier,” w hich it penetrated with rel­
ative ease. Less than a m onth later, Yeager reached M ach 1.35 in the sam e air­
plane. The sound barrier had not only been penetrated, it had been virtually de­
stroyed as the m yth it really was.
As a final note, the w hole story o f the hum an and engineering challenges that
revolved about the quest for and eventual achievem ent o f supersonic flight is fas­
cinating, and it is a living testim onial to the glory o f aeronautical engineering.
The story is brilliantly spelled out by Dr. Richard H allion, earlier a curator at the
A ir and Space M useum o f the Sm ithsonian Institution and now chief historian o f
the U.S. A ir Force, in his book Supersonic F light (see Bibliography at the end o f
this chapter). The reader should m ake every effort to study H allion’s story o f the
events leading to and follow ing Y eager’s flight in 1947.
5.23 HISTORICAL NOTE: THE X-15— FIRST
MANNED HYPERSONIC AIRPLANE AND
STEPPING-STONE TO THE SPACE SHUTTLE
F aster and higher— for all practical purposes, this has been the driving potential
behind the developm ent o f aviation since the W rights’ first successful flight in
1903. (See Sec. 1.11 and Figs. 1.30 and 1.31.) This credo was never m ore true
than during the 15 years follow ing Chuck Y eager’s first supersonic flight in the
Bell X S-1, described in Sec. 5.22. O nce the sound barrier was broken, it was left
far behind in the dust. The next goal becam e m anned hypersonic flight-—M ach 5
and beyond.
To accom plish this goal, N A CA initiated a series o f prelim inary studies in
the early 1950s for an aircraft to fly beyond M ach 5, the definition o f the hyper­
sonic flight regim e. This definition is essentially a rule o f thum b; unlike the se­
vere and radical flow field changes that take place w hen an aircraft flies through
M ach 1, nothing dram atic happens when M ach 5 is exceeded. Rather, the hyper­
sonic regim e is sim ply a very h igh-M ach num ber regim e, w here shock waves
are particularly strong and the gas tem peratures behind these shock w aves are
5.2 3
Historical Note: The X-15— First Manned Hypersonic Airplane
high. For exam ple, consider Eq. (4.73), which gives the total tem perature To, that
is, the tem perature o f a gas w hich was initially at a M ach num ber M\ and which
has been adiabatically slow ed to zero velocity. This is essentially the tem perature
at the stagnation point on a body. If M \ = 7, Eq. (4.73) shows that (for y = 1.4)
Tq/T \ = 10.8. If the flight altitude is, say, 100,000 ft where 7j = 419 R, then
T0 = 4525°R = 4065°F— far above the m elting point o f stainless steel. T here­
fore, as flight velocities increase far above the speed o f sound, they gradually ap­
proach a therm al barrier, that is, those velocities beyond which the skin tem per­
atures becom e too high and structural failure can occur. A s in the case o f the
sound barrier, the therm al barrier is only a figure o f speech— it is not an inherent
lim itation on flight speed. W ith proper design to overcom e the high rates o f aero­
dynam ic heating, vehicles today have flown at M ach num bers as high as 36 (e.g.,
the A pollo lunar return capsule). (For m ore details on high-speed reentry aero­
dynam ic heating, see Sec. 8.11.)
N evertheless, in the early 1950s, m anned hypersonic flight was a goal to be
achieved— an untried and questionable regim e characterized by high tem pera­
tures and strong shock waves. T he basic N A CA studies fed into an industryw ide
design com petition for a hypersonic airplane. In 1955, N orth A m erican A ircraft
Corporation was aw arded a jo in t N A C A -A ir F orce-N avy contract to design and
construct three prototypes o f a m anned hypersonic research airplane capable of
M ach 7 and a m axim um altitude o f 264,000 ft. This airplane was designated the
X -15 and is show n in Fig. 5.89. The first tw o aircraft w ere pow ered by Reaction
Figure 5 .8 9 T he N orth A m erican X -1 5 , the first m anned h yp erson ic airplane.
(Source: North American/Rockwell Corporation.)
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Airfoils, Wings, and Other Aerodynamic Shapes
M otors L R 1 1 rocket engines, with 8000 lb o f thrust (essentially the same as the
engine used for the Bell XS-1). Along with the third prototype, the tw o aircraft
w ere later reengined with a m ore pow erful rocket motor, the Reaction M otors
X LR 99, capable o f 57,000 lb o f thrust. The basic internal structure o f the airplane
was m ade from titanium and stainless steel, but the airplane skin was Inconel
X— a nickel-alloy steel capable o f withstanding tem peratures up to 1200°F. (A l­
though the theoretical stagnation tem perature at M ach 7 is 4065°F, as discussed
previously, the actual skin tem perature is cooler because o f heat sink and heat
dissipation effects.) The w ings had a low aspect ratio o f 2.5 and a thickness-tochord ratio o f 0.05— both intended to reduce supersonic w ave drag.
The first X-15 was rolled out o f the North A m erican factory at Los A ngeles
on O ctober 15, 1958. Then Vice President Richard M. Nixon was the guest of
honor at the rollout cerem onies. The X-15 had becom e a political, as well as a
technical, accom plishm ent, because the U nited States was attem pting to heal its
w ounded pride after the R ussians had launched the first successful unm anned
satellite, Sputnik 1, ju st a year earlier (see Sec. 8.15). The next day, the X-15 was
transported by truck to the nearby Edw ards A ir Force Base (the site at Muroc
which saw the first supersonic flights o f the Bell XS-1).
As in the case o f the X S -1, the X -15 was designed to be carried aloft by a
parent airplane, this tim e a B oeing B-52 je t bomber. The first free flight, w ithout
power, was m ade by Scott Crossfield on June 8 , 1959. This was soon follow ed by
the first pow ered flight on Septem ber 17, 1959, w hen the X-15 reached M ach 2.1
in a shallow clim b to 52,341 ft. Pow ered with the sm aller L R 1 1 rocket engines,
the X -15 set a speed record o f M ach 3.31 on A ugust 4, 1960, and an altitude
record o f 136,500 ft ju st eight days later. However, these records w ere ju st tran­
sitory. A fter N ovem ber 1960, the X-15 received the m ore pow erful X LR99 en­
gine. Indeed, the first flight with this rocket was made on N ovem ber 15, 1960; on
this flight, w ith pow er adjusted to its low est level and with the air brakes fully
extended, the X-15 still hit 2000 mi/h. Finally, on June 23, 1961, hypersonic
flight was fully achieved when U.S. Air Force test pilot M ajor Robert W hite flew
the X-15 at M ach 5.3 and in so doing accom plished the first “m ile-per-second”
flight in an airplane, reaching a m axim um velocity o f 3603 mi/h. This began an
illustrious series o f hypersonic flight tests, peaked by a flight at M ach 6.72 on
O ctober 3, 1967, with A ir Force M ajor Pete K night at the controls.
Experim ental aircraft are ju st that— vehicles designed for specific experi­
m ental purposes, w hich, after they are achieved, lead to the end o f the program .
This happened to the X-15 when on O ctober 24, 1968, the last flight was carried
out— the 199th o f the entire program . A 200th flight was planned, partly for rea­
sons o f nostalgia; how ever, technical problem s delayed this planned flight until
D ecem ber 20, w hen the X-15 was ready to go, attached to its B-52 parent plane
as usual. H ow ever, o f all things, a highly unusual snow squall suddenly hit
Edw ards, and the flight was canceled. The X-15 never flew again. In 1969, the
first X-15 was given to the N ational A ir and Space M useum o f the Sm ithsonian,
w here it now hangs with distinction in the M ilestones o f Flight Gallery, along
with the Bell XS-1.
5 .2 4
Summary
T he X-15 opened the w orld o f m anned hypersonic flight. The next hyper­
sonic airplane was the space shuttle. The vast bulk o f aerodynam ic and flight dy­
nam ic data generated during the X - 15 program carried over to the space shuttle
design. T he pilots’ experience w ith low -speed flights in a high-speed aircraft
with low lift-to-drag ratio set the stage for flight preparations with the space shut­
tle. In these respects, the X-15 was clearly the m ajor stepping-stone to the space
shuttle o f the 1980s.
5.24 Summary
Some of the aspects of this chapter are highlighted in the following.
1.
For an airfoil, the lift, drag, and moment coefficients are defined as
L
Cl —
~
QooS
D
Crf —
—
QooS
M
Cm —
Q ooSc
where L , D, and M are the lift, drag, and moments per unit span, respectively, and
S = c(l).
For a finite wing, the lift, drag, and moment coefficients are defined as
L
Cl = — r
QooS
2.
„
D
Co = — -
QooS
„
Cm =
M
QooSC
where L, D, and M are the lift, drag, and moments, respectively, for the complete
wing and S is the wing planform area.
For a given shape, these coefficients are a function of angle of attack, Mach
number, and Reynolds number.
The pressure coefficient is defined as
C„ = P ~ P~
' jA x V 2,
3.
(5.27)
The Prandtl-Glauert rule is a compressibility correction for subsonic flow:
where Cp,o and Cp are the incompressible and compressible pressure coefficients, re­
spectively. The same rule holds for the lift and moment coefficients, that is,
c¡ =
. C;'°
(5.40)
The critical Mach number is that free-stream Mach number at which sonic flow is
first achieved at some point on a body. The drag-divergence Mach number is that
free-stream Mach number at which the drag coefficient begins to rapidly increase
due to the occurrence of transonic shock waves. For a given body, the dragdivergence Mach number is slightly higher than the critical Mach number.
The Mach angle is defined as
p. = arcsin —
M
(5.49)
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6.
5
Airfoils, Wings, and Other Aerodynamic Shapes
The total drag coefficient for a finite wing is equal to
C2
Co = cj + —
ite AR
7.
(5. 58)
where c¡¡ is the profile drag coefficient and C ¿ /(jtM R ) is the induced drag
coefficient.
The lift slope for a finite wing a is given by
a = ------------ —-----------1 +
(5.65)
5 7 .3 ao /(7 rei A R )
where ao is the lift slope for the corresponding infinite wing.
Bibliography
Abbott, I. H., and A. E. von Doenhoff: Theory o f Wing Sections, McGraw-Hill, New
York, 1949 (also Dover, New York, 1959).
Anderson, John D„ Jr.: Fundamentals o f Aerodynamics, 3d ed„ McGraw-Hill, New
York, 2001.
______ : A History o f Aerodynamics and Its Impact on Flying Machines, Cambridge
University Press, New York, 1997.
Dommasch, D. O., S. S. Sherbey, and T. F. Connolly: Airplane Aerodynamics, 4th ed.,
Pitman, New York, 1968.
Hallion, R.: Supersonic Flight (The Story o f the Bell X -l and Douglas D-558),
Macmillan, New York, 1972.
McCormick, B. W.: Aerodynamics, Aeronautics and Flight Mechanics, Wiley, New
York, 1979.
Pierpont, P. K.: “Bringing Wings of Change,” Astronautics and Aeronautics, vol. 13,
No. 10, October 1975, pp. 20-27.
Shapiro, A. H.: Shape and Flow: The Fluid Dynamics o f Drag, Anchor, Garden City,
NY, 1961.
Shevell, R. S.: Fundamentals o f Flight, Prentice-Hall, Englewood Cliffs, NJ, 1983.
von Karman, T. (with Lee Edson): The Wind and Beyond, Little, Brown, Boston, 1967
(an autobiography).
Problems
5.1
5.2
5.3
By the method of dimensional analysis, derive the expression M = q^sccm for
the aerodynamic moment on an airfoil, where c is the chord and cm is the moment
coefficient.
Consider an infinite wing with an NACA 1412 airfoil section and a chord length
of 3 ft. The wing is at an angle of attack of 5° in an airflow velocity of 100 ft/s at
standard sea-level conditions. Calculate the lift, drag, and moment about the
quarter chord per unit span.
Consider a rectangular wing mounted in a low-speed subsonic wing tunnel. The
wing model completely spans the test section so that the flow “sees” essentially
an infinite wing. If the wing has an NACA 23012 airfoil section and a chord of
Problems
5.4
5.5
5.6
5.7
5.8
5.9
5.10
5.11
5.12
5.13
5.14
5.15
5.16
5.17
0.3 m, calculate the lift, drag, and moment about the quarter chord per unit span
when the airflow pressure, temperature, and velocity are l atm, 303 K, and 42 m/s,
respectively. The angle of attack is 8°.
The wing model in Prob. 5.3 is pitched to a new angle of attack, where the lift on
the entire wing is measured as 200 N by the wind tunnel force balance. If the
wingspan is 2 m, what is the angle of attack?
Consider a rectangular wing with an NACA 0009 airfoil section spanning the test
section of a wind tunnel. The test section airflow conditions are standard sea level
with a velocity of 120 mi/h. The wing is at an angle of attack of 4°, and the wind
tunnel force balance measures a lift of 29.5 lb. What is the area of the wing?
The ratio of lift to drag L /D for a wing or airfoil is an important aerodynamic
parameter. Indeed, it is a direct measure of the aerodynamic efficiency of the
wing. If a wing is pitched through a range of angle of attack, L /D first increases,
then goes through a maximum, and then decreases. Consider an infinite wing with
an NACA 2412 airfoil. Estimate the maximum value of L /D . Assume the
Reynolds number is 9 x 106.
Consider an airfoil in a free stream with a velocity of 50 m/s at standard sea-level
conditions. At a point on the airfoil, the pressure is 9.5 x 104 N/m2. What is the
pressure coefficient at this point?
Consider a low-speed airplane flying at a velocity of 55 m/s. If the velocity at a
point on the fuselage is 62 m/s, what is the pressure coefficient at this point?
Consider a wing mounted in the test section of a subsonic wind tunnel. The
velocity of the airflow is 160 ft/s. If the velocity at a point on the wing is 195 ft/s,
what is the pressure coefficient at this point?
Consider the same wing in the same wind tunnel as in Prob. 5.9. If the test section
air temperature is 510°R and if the flow velocity is increased to 700 ft/s, what is
the pressure coefficient at the same point?
Consider a wing in a high-speed wind tunnel. At a point on the wing, the velocity
is 850 ft/s. If the test section flow is at a velocity of 780 ft/s, with a pressure and
temperature of 1 atm and 505°R, respectively, calculate the pressure coefficient at
the point.
If the test section flow velocity in Prob. 5.11 is reduced to 100 ft/s, what will the
pressure coefficient become at the same point on the wing?
Consider an NACA 1412 airfoil at an angle of attack of 4°. If the free-stream
Mach number is 0.8, calculate the lift coefficient.
An NACA 4415 airfoil is mounted in a high-speed subsonic wind tunnel. The lift
coefficient is measured as 0.85. If the test section Mach number is 0.7, at what
angle of attack is the airfoil?
Consider an airfoil at a given angle of attack, say ori. At low speeds, the minimum
pressure coefficient on the top surface of the airfoil is —0.90. What is the critical
Mach number of the airfoil?
Consider the airfoil in Prob. 5.15 at a smaller angle of attack, say, « 2- At low
speeds, the minimum pressure coefficient is —0.65 at this lower angle of attack.
What is the critical Mach number of the airfoil?
Consider a uniform flow with a Mach number of 2. What angle does a Mach wave
make with respect to the flow direction?
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Airfoils, Wings, and Other Aerodynamic Shapes
5.18 Consider a supersonic missile flying at Mach 2.5 at an altitude of 10 km. Assume
the angle of the shock wave from the nose is approximated by the Mach angle
(i.e., a very weak shock). How far behind the nose of the vehicle will the shock
wave impinge upon the ground? (Ignore the fact that the speed of sound, hence the
Mach angle, changes with altitude.)
5.19 The wing area of the Lockheed F-104 straight-wing supersonic fighter is
approximately 210 ft2. If the airplane weighs 16,000 lb and is flying in level flight
at Mach 2.2 at a standard altitude of 36,000 ft, estimate the wave drag on the
wings.
5.20 Consider a flat plate at an angle of attack of 2 in a Mach 2.2 airflow. (Mach 2.2 is
the cruising Mach number, e.g., of the Concorde supersonic transport.) The length
of the plate in the flow direction is 202 ft, which is the length of the Concorde.
Assume that the free-stream conditions correspond to a standard altitude of
50,000 ft. The total drag on this plate is the sum of wave drag and skin friction
drag. Assume that a turbulent boundary layer exists over the entire plate. The
results given in Chap. 4 for skin friction coefficients hold for incompressible
flow only; there is a compressibility effect on C¡ such that its value decreases with
increasing Mach number. Specifically, at Mach 2.2 assume that the C/ given in
Chap. 4 is reduced by 20 percent.
(a) Given all the preceding information, calculate the total drag coefficient for
the plate.
(b) If the angle of attack is increased to 5°, assuming C¡ stays the same, calculate
the total drag coefficient.
(c) For these cases, what can you conclude about the relative influence of wave
drag and skin friction drag?
5.21 The Cessna Cardinal, a single-engine light plane, has a wing with an area of
16.2 m2 and an aspect ratio of 7.31. Assume the span efficiency factor is 0.62. If
the airplane is flying at standard sea-level conditions with a velocity of 251 km/h,
what is the induced drag when the total weight is 9800 N?
5.22 For the Cessna Cardinal in Prob. 5.21, calculate the induced drag when the
velocity is 85.5 km/h (stalling speed at sea level with flaps down).
5.23 Consider a finite wing with an area and aspect ratio of 21.5 m2 and 5, respectively
(this is comparable to the wing on a Gates Learjet, a twin-jet executive transport).
Assume the wing has an NACA 65-210 airfoil, a span efficiency factor of 0.9, and
a profile drag coefficient of 0.004. If the wing is at a 6° angle of attack, calculate
C l and Cp5.24 During the 1920s and early 1930s, the NACA obtained wind tunnel data on
different airfoils by testing finite wings with an aspect ratio of 6 . These data were
then “corrected” to obtain infinite wing airfoil characteristics. Consider such a
finite wing with an area and aspect ratio of 1.5 ft2 and 6, respectively, mounted in
a wind tunnel where the test section flow velocity is 100 ft/s at standard sea-level
conditions. When the wing is pitched to a = —2°, no lift is measured. When the
wing is pitched to a = 10°, a lift of 17.9 lb is measured. Calculate the lift slope
for the airfoil (the infinite wing) if the span effectiveness factor is 0.95.
5.25 A finite wing of area 1.5 ft2 and aspect ratio of 6 is tested in a subsonic wind
tunnel at a velocity of 130 ft/s at standard sea-level conditions. At an angle of
attack of —1°, the measured lift and drag are 0 and 0.181 lb, respectively. At
Problems
5.26
5.27
5.28
5.29
5.30
5.31
5.32
an angle of attack of 2°, the lift and drag are measured as 5.0 and 0.23 lb,
respectively. Calculate the span efficiency factor and the infinite-wing lift slope.
Consider a light, single-engine airplane such as the Piper Super Cub. If the
maximum gross weight of the airplane is 7780 N, the wing area is 16.6 m2, and
the maximum lift coefficient is 2.1 with flaps down, calculate the stalling speed at
sea level.
The airfoil on the Lockheed F-104 straight-wing supersonic fighter is a thin
symmetric airfoil with a thickness ratio of 3.5 percent. Consider this airfoil in a
flow at an angle of attack of 5°. The incompressible lift coefficient for the airfoil is
given approximately by a = 2not, where a is the angle of attack in radians.
Estimate the airfoil lift coefficient for (a) M = 0.2, (b) M = 0.7, (c) M = 2.0.
The whirling arm test device used in 1804 by Sir George Cayley is shown in
Figure 1.7. Cayley was the first person to make measurements of the lift on
inclined surfaces. In his 1804 notebook, he noted that, on a flat surface moving
through the air at 21 .8 ft/s at 3° angle of attack, a lift force of 1 ounce was
measured. The flat surface was a 1 ft by 1 ft square. Calculate the lift coefficient
for this condition. Compare this measured value with that predicted by the
expression for lift coefficient for a flat plate airfoil in incompressible flow given
by ci = 2n a , where a is in radians. What are the reasons for the differences in the
two results? (See Anderson, A History o f Aerodynamics and Its Impact on Flying
Machines, Cambridge University Press, 1997, pp. 68-71 for a detailed discussion
of this matter.)
Consider a finite wing at an angle of attack of 6°. The normal and axial force
coefficients are 0.8 and 0.06, respectively. Calculate the corresponding lift and
drag coefficients. What comparison can you make between the lift and normal
force coefficients?
Consider a finite wing with an aspect of ratio of 7; the airfoil section of the wing is
a symmetric airfoil with an infinite wing lift slope of 0.11 per degree. The lift-todrag ratio for this wing is 29 when the lift coefficient is equal to 0.35. If the angle
of attack remains the same and the aspect ratio is simply increased to 10 by adding
extensions to the span of the wing, what is the new value of the lift-to-drag ratio?
Assume the span efficiency factors e = e¡ = 0.9 for both cases.
Consider a flat plate oriented at a 90" angle of attack in a low-speed
incompressible flow. Assume the pressure exerted over the front of the plate
(facing into the flow) is a constant value over the front surface, equal to the
stagnation pressure. Assume the pressure exerted over the back of the plate is also
a constant value, but equal to the free-stream static pressure. (In reality, these
assumptions are only approximations to the real flow over the plate. The pressure
over the front face is not exactly constant nor exactly equal to the stagnation
pressure, and the pressure over the back of the plate is not constant nor exactly
equal to the free-stream pressure. The preceding approximate model of the flow,
however, is useful for our purpose here.) Note that the drag is essentially all
pressure drag; due to the 90° orientation of the plate, skin friction drag is not a
player. For this model of the flow, prove the drag coefficient for the flat plate is
CD = 1.
In some aerodynamic literature, the drag of an airplane is couched in terms of
the “drag area” instead of the drag coefficient. By definition, the drag area, / , is
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chapter
5
Airfoils, Wings, and Other Aerodynamic Shapes
the area of a flat plate at 90° to the flow that has a drag force equal to the drag
of the airplane. As part of this definition, the drag coefficient of the plate is
assumed to be equal to 1, as shown in Prob. 5.31. If Cp is the drag coefficient
of the airplane based on wing planform area 5, prove that / = CpS.
5.33 One of the most beautifully streamlined airplanes ever designed is the North
American P-51 Mustang shown in Fig. 4.39. The Mustang has one of the lowest
minimum drag coefficients of any airplane in history, namely, Cp = 0.0163. The
wing planform area of the Mustang is 233 ft2. Using the result from Prob. 5.32,
show that the drag area for the Mustang is 3.8 ft2. That is, drag on the whole P-51
airplane is the same as the drag on a flat plate perpendicular to the flow of an area
of only 3.8 ft2.
C
H
A
P
T
E
R
Elements of Airplane
Performance
First Europe, and then the globe, will be linked by flight, and nations so knit together
that they will grow to be next-door neighbors. This conquest of the air will prove,
ultimately, to be m an’s greatest and most glorious triumph. What railways have done
for nations, airways will do for the world.
Claude Grahame-White
British aviator, 1914
6.1 INTRODUCTION: THE DRAG POLAR
H enson’s aerial steam carriage o f the m id-19th century (see Fig. 1.11) was
pictured by contem porary artists as flying to all corners o f the world. O f course,
questions about how it w ould fly to such distant locations w ere not considered by
the designers. A s with most early aeronautical engineers o f that time, their main
concern was sim ply to lift or otherw ise propel the airplane from the ground; what
happened once the vehicle was airborne was view ed with secondary im portance.
However, w ith the success o f the W right brothers in 1903, and with the subse­
quent rapid developm ent o f aviation during the pre-W orld W ar I era, the airborne
perform ance o f the airplane suddenly becam e o f prim ary im portance. Som e ob­
vious questions were, and still are, asked about a given design. W hat is the
m axim um speed o f the airplane? How fast can it clim b to a given altitude? How
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6
Elements of Airplane Performance
PREVIEW BOX
You are a passenger, or perhaps the pilot, in an air­
plane standing at the beginning of the runway, ready
to take off. The engine throttle is pushed wide open,
and you accelerate down the runway. How do you
know if you will be able to lift off the ground and get
into the air before you use up all the runway length?
In this chapter, you will learn how to answer this
question.
Now you are in the air, but there are thunder­
storms off in the distance, and you will need to climb
over them as quickly as possible. How do you know
if your airplane can do this? How long will it take for
you to climb to a safe altitude? In this chapter, you
will learn how to answer these questions.
Once you are comfortably at altitude and you are
winging your way to your destination, how do you
know if you can get there without running out of
fuel? Alternatively, how can you estimate how far
you can fly on a tank of fuel? Or perhaps you arc sim­
ply up for a joy ride, and you want to stay up for as
long as possible. How can you estimate how long you
can stay in the air on a tank of fuel? In this chapter,
you will learn how to answer these questions.
Maybe you are a speed freak. You push the throt­
tle wide open, getting maximum power from your
engine (or engines). You accelerate like mad, at least
for a while, until the airplane reaches the fastest ve­
locity at which it can fly. How do you estimate this
“fastest” velocity? In this chapter, you will learn how
to answer this question.
Suddenly you are the “Red Baron” in your "hot”
fighter airplane, locked in mortal air combat with an
adversary. In order to defeat your adversary in a dog­
fight, you want to be able to make turns with a small
radius (turn “inside” your adversary) and be able to
make a turn faster. How do you know your airplane
can do this? In this chapter, you will learn how to an­
swer this question.
Unfortunately, your engine goes out; you are at
some altitude, and you lose all your power. You have
to glide back to your base. Can your airplane make
it, or will you have to land short of your destination?
In this chapter, you will learn how to answer this
question.
Fortunately, your engine does not go out, and
you are now ready to complete your flight and land.
You approach the runway. Is the runway long enough
for you to land safely and come to a stop? Or are you
going to zip past the end of the runway into the
woods beyond, holding on for dear life? In this chap­
ter, you will learn how to answer this question.
This chapter is full of such important questions,
and equally important answers. They all have to do
with the performance of the airplane. In this chap­
ter, at last, we deal with the whole airplane, not just
an airfoil or a wing. Finally, in the middle of this
book on the introduction to flight, we are actually
going to take flight. Buckle up, and read on. Let’s
go for a ride.
far can it fly on a given tank o f fuel? How long can it stay in the air? A nsw ers to
these and sim ilar questions constitute the study o f airplane perform ance, which
is the subject o f this chapter.
In previous chapters, the physical phenom ena producing lift, drag, and m o­
m ents o f an airplane were introduced. We em phasized that the aerodynam ic forces
and m om ents exerted on a body m oving through a fluid stem from two sources:
1.
The pressure distribution
2.
T he shear stress distribution
both acting over the body surface. The physical laws governing such phenom ena
w ere exam ined, with various applications to aerodynam ic flows.
6 .1
Introduction: The Drag Polar
In this chapter, we begin a new phase o f study. The airplane is considered a
rigid body on which is exerted four natural forces: lift, drag, propulsive thrust,
and w eight. C oncern is focused on the m ovem ent o f the airplane as it responds to
these forces. Such considerations form the core o f flig h t dynamics, an im portant
discipline o f aerospace engineering. Studies o f airplane perform ance (this
chapter) and stability and control (Chap. 7) both fall under the heading o f flight
dynam ics.
In these studies, we will no longer be concerned with aerodynam ic details;
rather, we will generally assum e that the aerodynam icists have done their work
and provided us with the pertinent aerodynam ic data for a given airplane. These
data are usually packaged in the form o f a drag p o la r for the com plete airplane,
given as
C2
neA R
C d = C d i(,-I-------- (6 . l a )
Equation ( 6 . 1a ) is an extension o f Eq. (5.58) to include the whole airplane. Here
C o is the drag coefficient for the com plete airplane; C L is the total lift coefficient,
including the sm all contributions from the horizontal tail and fuselage; and C p,e
is defined as the parasite drag coefficient, w hich contains not only the profile
drag o f the wing [c¡¡ in Eq. (5.58)] but also the friction and pressure drag o f the
tail surfaces, fuselage, engine nacelles, landing gear, and any other com ponent o f
the airplane that is exposed to the airflow. At transonic and supersonic speeds,
Co.e also contains w ave drag. Because o f changes in the flow field around the
airplane— especially changes in the am ount o f separated flow over parts o f the
airplane— as the angle o f attack is varied, C o,e will change with angle o f attack;
that is, Co.e is itself a function o f lift coefficient. A reasonable approxim ation for
this function is
Cd,« = Co, o + r C ]
where r is an em pirically determ ined constant. H ence, Eq. ( 6 . la ) can be w ritten as
C 0 = C D,0 + ( r + — L _ ) c 2
(6.1 6 )
In Eqs. (6.1a) and (6.16), e is the fam iliar span efficiency factor, w hich takes
into account the nonelliptical lift distribution on wings o f general shape (see
Sec. 5.14). Let us now redefine e such that it also includes the effect o f the vari­
ation o f parasite drag with lift; that is, let us w rite Eq. (6.16) in the form
C2
C d = C o ,o 4- —
neA R
( 6. 1c)
where C D,a is the parasite drag coefficient at zero lift and the term C 2L /( n e A R )
includes both induced drag and the contribution to parasite drag due to lift. In
Eq. ( 6 . lc ), our redefined e, w hich now includes the effect o f r from Eq. (6.16), is
called the O sw ald efficiency fa c to r (nam ed after W. Bailey O swald, who first
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CHAPTER 6
Elements of Airplane Performance
established this term inology in NACA Report No. 408 in 1932). In this chapter,
the basic aerodynam ic properties o f the airplane are described by Eq. ( 6 . lc), and
we consider both Co,o and e as known aerodynam ic quantities, obtained from the
aerodynam icist. We will continue to designate C l / ( n e A R ) by C D |, w here C D,¡
now has the expanded interpretation as the coefficient o f drag due to lift, includ­
ing both the contributions due to induced drag and the increm ent in parasite drag
due to angle o f attack different from a L = 0 . We designate C D,0 sim ply as the
zero-lift drag coefficient, which is obvious from Eq. (6.1c) when C L = 0; how ­
ever, we recognize C o ,o m ore precisely as the parasite drag coefficient at zero
lift, that is, the value o f the drag coefficient when a = a L~0.
The graph o f Eq. (6.1c), show n in Fig. 6.1, is also called the drag polar. With
the approxim ations m ade in Eq. (6.1c), the drag polar is a parabola with its axis
on the zero-lift axis, and its vertex is Co,o- In Fig. 6.1a, C D is plotted versus C¿;
in Fig. 6.1b, C i is plotted versus C o. The tw o representations are identical;
Fig. 6.1b is sim ply a m irror im age o f Fig. 6.1a rotated on its side. Both repre­
sentations are found in the literature. In Fig. 6 .1, negative values o f C¿ pertain to
negative lift, w hich occurs w hen the angle o f attack o f the airplane is less than
Figure 6.1 Schematic o f the drag polar.
6.1
Introduction: The Drag Polar
®l =o- This situation is not encountered frequently in the analysis o f airplane
perform ance; hence, only that portion o f the drag polar associated with positive
C L is usually shown.
An illustration o f the drag polar for a specific airplane is shown in Fig. 6.2,
w hich gives the actual data for the L ockheed C -14 1A, shown in three-view at the
top o f the figure. U pon close exam ination, the drag polar for an actual airplane
exhibits a subtle difference from our approxim ation given in Eq. ( 6 .lc ) as
graphed in Fig. 6 . 1. Note that the zero-lift drag coefficient in Fig. 6.2 is not the
m inim um drag coefficient; that is, the axis o f the parabolic drag polar is not the
zero-lift axis, but rather is displaced slightly above the zero-lift axis. In Fig. 6.2,
the m inim um drag coefficient is C'o.min = 0.015, and it occurs for a value o f the
lift coefficient C Lmjn dru# = 0 . 16. T he zero-lift drag coefficient is Cp.o = 0 .0 17, at
C L = 0. And C D,o is not the m inim um drag coefficient because a L=o for m ost
airplane designs is a small but finite negative value; that is, the airplane is pitched
A
I
r r f i r r
mis id
1.2
C,
Cl. 0 . 6
0
0
0.02 0.04 0.06 0.08 0.10 0.12
Cp
0
4
8
12
16
20
24
L
D
Figure 6.2 Low-speed drag polar and variation o f liftto-drag ratio for the Lockheed C-141A. The airplane is
shown in a three-view above the drag polar.
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chapter
6
Elements of Airplane Performance
Figure 6.3 Drag polar where the zero-lift
drag coefficient is not the same as the
minimum drag coefficient.
slightly dow nw ard at this orientation, and the pressure drag due to flow separa­
tion (form drag) is slightly higher than if the airplane is at an angle o f attack
slightly larger, nearer a zero angle o f attack. The m inim um drag coefficient
occurs when the airplane is more aligned with the relative wind, that is, when a
is slightly larger than a l= 0. For this situation, the drag polar can be expressed as
.-t
<-D =
¿-i
I
(C
— C ^min dra8
*-D,m ¡
n -I-------------— ------
jreA R
ri\
(,O.Z;
The corresponding graph o f the drag polar is shown in Fig. 6.3.
Now that we have m ade the distinction betw een the two generic drag polars
sketched in Figs. 6 .1 and 6.3, for our considerations o f airplane perform ance in
this chapter we will adopt Eq. ( 6 .1 c) and Fig. 6 .1 as the representation o f the drag
polar. It sim plifies our analysis and presentation, w ithout loss o f generality.
Q uantitatively, there is only a small difference betw een the two representations.
How ever, for an industry standard detailed perform ance analysis o f a particular
airplane, you w ant to have as accurate a drag polar as you can obtain for the air­
plane, and you would be dealing with the more accurate representation shown in
Fig. 6.3 and given by Eq. (6.2).
Return for a m om ent to our overall road m ap in Fig. 2.1. W ith this chapter,
we m ove to a new main discipline, namely, flight m echanics, as item ized at the
left o f Fig. 2.1. In particular, in this chapter we deal w ith airplane perform ance,
a subheading under flight m echanics, as shown at the center o f Fig. 2.1. The road
map for this chapter is shown in Fig. 6.4. A study o f airplane perform ance is
frequently based on N ew ton’s second law, which dictates the motion o f the
airplane through the atm osphere. We will first obtain these equations o f motion.
The rem ainder o f the chapter is based on tw o form s o f these equations: (1) the
6.1
Introduction: The Drag Polar
Figure 6.4 Road map for Chap. 6.
form associated w ith the assum ption o f unaccelerated flight, leading to a study
o f static perform ance item ized on the left side o f Fig. 6.4; and (2) the form asso­
ciated with acceleration o f the airplane, leading to a study o f dynam ic p erfor­
m ance item ized on the right side o f Fig. 6.4. (The difference betw een static per­
form ance and dynam ic perform ance is analogous to taking a course in statics and
another course in dynam ics.) U nder static perform ance we will exam ine such im ­
portant aspects as how to calculate the m axim um velocity o f the airplane, how
fast it can clim b (rate o f clim b), how high it can fly (m axim um altitude), how far
it can fly (range), and how long it can stay in the air (endurance). U nder dynam ic
perform ance we will exam ine takeoff and landing characteristics, turning flight,
and accelerated rate o f clim b. W hen we arrive at the bottom o f this road map, we
will have toured through som e o f the basic aspects that dictate the design o f an
airplane and will have covered som e o f the m ost im portant territory in aerospace
engineering. So, let us hit the road.
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Elements of Airplane Performance
6.2 EQUATIONS OF MOTION
To study the perform ance o f an airplane, the fundam ental equations that govern
its translational motion through air m ust first be established. C onsider an airplane
in flight, as sketched in Fig. 6.5. The flight path (direction o f motion o f the
airplane) is inclined at an angle 9 with respect to the horizontal. In term s o f the de­
finitions in Chap. 5, the flight path direction and the relative wind are along
the sam e line. The mean chord line is at a geom etric angle o f attack a with respect
to the flight path direction. There are four physical forces acting on the airplane:
1.
Lift L, w hich is perpendicular to the flight path direction
2.
Drag D , w hich is parallel to the flight path direction
3.
W eight W , w hich acts vertically tow ard the center o f the earth (and hence
is inclined at angle 6 with respect to the lift direction)
4.
Thrust T , w hich in general is inclined at the angle a T with respect to the
flight path direction
T he force diagram shown in Fig. 6.5 is im portant. Study it carefully until you feel
com fortable with it.
T he flight path shown in Fig. 6.5 is draw n as a straight line. This is the
picture we see by focusing locally on the airplane itself. However, if w e stand
back and take a w ider view o f the space in which the airplane is traveling, the
flight path is generally curved. T his is obviously true if the airplane is m aneu­
vering; even if the airplane is flying “straight and level” with respect to the
ground, it is still executing a curved flight path with a radius o f curvature equal
to the absolute altitude ha (as defined in Sec. 3.1).
W hen an object m oves along a curved path, the m otion is called curvilinear,
as opposed to m otion along a straight line, w hich is rectilinear. N ew ton’s second
6 .2
Equations of Motion
law, w hich is a physical statem ent that force = m ass x acceleration, holds in
either case. C onsider a curvilinear path. At a given point on the path, set up two
m utually perpendicular axes, one along the direction o f the flight path and the
other normal to the flight path. A pplying N ew ton’s law along the flight path gives
/
dV
F|| = m a = m ~Jj~
(6.3)
where £ F | is the sum m ation o f all forces parallel to the flight path, a = d V / d t is
the acceleration along the flight path, and V is the instantaneous value o f the air­
plane’s flight velocity. (Velocity V is alw ays along the flight path direction, by de­
finition.) Now, applying N ew ton’s law perpendicular to the flight path, we have
y
V2
F± — m —
rc
(6.4)
w here J2
*s the sum m ation o f all forces perpendicular to the flight path and
V 2/ r c is the acceleration norm al to a curved path with radius o f curvature rc.
This norm al acceleration V 2/ r c should be fam iliar from the basic physics. The
right-hand side o f Eq. (6.4) is nothing other than the centrifugal force.
E xam ining Fig. 6.5, we see that the forces parallel to the flight path (positive
to the right, negative to the left) are
y ; Fu = T c o s (Xt - D — W sin #
(6.5)
and the forces perpendicular to the flight path (positive upward and negative
dow nw ard) are
Fj. = L + T s in a T — W co st?
( 6 .6 )
Com bining Eq. (6.3) with (6.5), and Eq. (6.4) with ( 6 .6 ) yields
T cos «7 — D
dV
W sin 0 — m —
dt
V2
L + T s \ n a T — W cos 6 = m —
rc
(6.7)
68
( . )
Equations (6.7) and ( 6 .8 ) are the equations o f m otion for an airplane in transla­
tional flight. (N ote that an airplane can also rotate about its axes; this will be
discussed in Chap. 7. A lso note that we are not considering the possible sidewise
m otion o f the airplane perpendicular to the page o f Fig. 6.5.)
Equations (6.7) and (6 .8 ) describe the general tw o-dim ensional translational
m otion o f an airplane in accelerated flight. However, in the first part o f this
chapter, we are interested in a specialized application o f these equations, namely,
the case where the acceleration is zero. The perform ance o f an airplane for such
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6
Elements of Airplane Performance
unaccelerated flight conditions is called static perform ance. This may, at first
thought, seem unduly restrictive; however, static perform ance analyses lead to
reasonable calculations o f m axim um velocity, m axim um rate o f clim b, m axi­
mum range, etc.— param eters o f vital interest in airplane design and operation.
W ith this in m ind, consider level, unaccelerated flight. Referring to Fig. 6.5,
level flight m eans that the flight path is along the horizontal; that is, 0 = 0.
U naccelerated flight means that the right-hand sides o f Eqs. (6.7) and ( 6 .8 ) are
zero. Therefore, these equations reduce to
T cos otr = D
L -I- T sin a T = W
(6.9)
( 6 . 10 )
Furtherm ore, for m ost conventional airplanes, a y is small enough that cos « r % 1
and sin « 7- 0. Thus, from Eqs. (6.9) and (6.10),
T = D
( 6 . 11)
L = W
(6. 12)
Equations (6.11) and (6.12) are the equations o f motion for level, unaccelerated
flight. They can also be obtained directly from Fig. 6.5, by inspection. In level,
unaccelerated flight, the aerodynam ic drag is balanced by the thrust o f the
engine, and the aerodynam ic lift is balanced by the w eight o f the airplane—
alm ost trivial, but very useful, results.
Let us now apply these results to the static perform ance analysis o f an air­
plane. The follow ing sections constitute the building blocks for such an analysis,
which ultim ately yields answ ers to such questions as how fast, how far, how long,
and how high a given airplane can fly. A lso, the discussion in these sections relies
heavily on a graphical approach to the calculation o f airplane perform ance. In
m odern aerospace engineering, such calculations are m ade directly on high-speed
digital com puters. However, the graphical illustrations in the follow ing sections
are essential to the program m ing and understanding o f such com puter solutions;
m oreover, they help to clarify and explain the concepts being presented.
6.3 THRUST REQUIRED FOR LEVEL,
UNACCELERATED FLIGHT
C onsider an airplane in steady, level flight at a given altitude and a given veloc­
ity. For flight at this velocity, the airplane’s pow er plant (e.g., turbojet engine or
reciprocating engine-p ro p eller com bination) m ust produce a net thrust which is
equal to the drag. T he thrust required to obtain a certain steady velocity is easily
calculated as follow s. From Eqs. (6.11) and (5.20),
T = D = qooSCp
(6.13)
6.3
Thrust Required for Level, Unaccelerated Flight
and from Eqs. (6.12) and (5.17),
L = W = qooS C L
(6.14)
D ividing Eq. (6.13) by (6.14) yields
L = £r
w
CL
(6.15)
Thus, from Eq. (6.15), the thrust required for an airplane to fly at a given veloc­
ity in level, unaccelerated flight is
(6.16)
(N ote that a subscript R has been added to thrust to em phasize that it is thrust
required.)
T hrust required TR for a given airplane at a given altitude varies with velocity
Voo- The thrust-required curve is a plot o f this variation and has the general shape
illustrated in Fig. 6 .6 . To calculate a point on this curve, proceed as follows:
1.
C hoose a value o f V ^ .
2.
F or this Vx , calculate the lift coefficient from Eq. (6.14):
w
C L = ------ —
¡P o o V * S
Voo,
ft/s
Figure 6.6 Thrust-required curve. The results on this and subsequent
figures correspond to answers for some o f the sample problems in
Chap. 6.
(6-17)
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CHAPTER 6
Elements of Airplane Performance
N ote that p <*, is known from the given altitude and S is know n from the
given airplane. The C L calculated from Eq. (6.17) is that value necessary
for the lift to balance the known w eight W o f the airplane.
3.
Calculate C o from the known drag polar for the airplane
C o — Co, o +
cl
7TCAR
where C¿ is the value obtained from Eq. (6.17).
4.
5.
Form the ratio C l / C oC alculate thrust required from Eq. (6.16).
The value o f Tg obtained from Step Five is that thrust required to fly at the
specific velocity chosen in Step One. In turn, the curve in Fig. 6.6 is the locus o f
all such points taken for all velocities in the flight range o f the airplane. The
reader should study Exam ple 6 .1 at the end o f this section in order to becom e
fam iliar with the preceding steps.
N ote from Eq. (6.16) that Tg varies inversely as L / D . Hence, m inim um
thrust required will be obtained when the airplane is flying at a velocity where
L / D is m axim um . This condition is shown in Fig. 6 .6 .
The lift-to-drag ratio L / D is a m easure o f the aerodynam ic efficiency o f an
airplane; it only m akes sense that m axim um aerodynam ic efficiency should lead
to m inim um thrust required. Consequently, the lift-to-drag ratio is an im portant
aerodynam ic consideration in airplane design. A lso note that L / D is a function
o f angle o f attack, as sketched in Fig. 6.7. For m ost conventional subsonic air­
planes, L / D reaches a m axim um at som e specific value o f a , usually on the
order o f 2° to 5°. Hence, when an airplane is flying at the velocity for m inim um
Tg, as shown in Fig. 6 .6 , it is sim ultaneously flying at the angle o f attack for m ax­
imum L / D , as shown in Fig. 6.7.
As a corollary to this discussion, note that different points on the thrustrequired curve correspond to different angles o f attack. This is em phasized in
Fig. 6 .8 , which show s that as we m ove from right to left on the thrust-required
Figure 6.7 Lift-to-drag ratio versus angle o f attack.
6 .3
Thrust Required for Level, Unaccelerated Flight
a
Increasing velocity
Figure 6.8 Thrust-required curve with associated angle-of-attack variation.
curve, the airplane angle o f attack increases. This also helps to explain physically
why T r goes through a m inim um . Recall that L = W = q ^ S C i . At high veloc­
ities (point a in Fig. 6 .8 ), m ost o f the required lift is obtained from high dynam ic
pressure q ^ ; hence, C L and therefore a are small. Also, under the sam e condi­
tions, drag (£) = qooSCD) is relatively large because qx is large. As we m ove to
the left on the thrust-required curve, q00 decreases; hence, C L and therefore a
m ust increase to support the given airplane weight. Because qx decreases, D and
hence TR initially decrease. H ow ever, recall that drag due to lift is a com ponent
o f total drag and that C D,,- varies as C 2L . At low velocities, such as at point b in
Fig. 6 .8 , goo is low, and therefore C L is large. At these conditions, C d ,¡ increases
rapidly, m ore rapidly than q x decreases, and D and hence TR increase. This is
why, starting at point a, TR first decreases as Voo decreases, then goes through a
m inim um and starts to increase, as shown at point b.
Recall from Eq. (6.1c) that the total drag of the airplane is the sum o f the
zero-lift drag and drag due to lift. The corresponding drag coefficients are C o ,o
and C D i = C l/( jr e A R ) , respectively. At the condition for m inim um TR, there
exists an interesting relation betw een C¡j ,0 and C D., , as follows. From Eq. ( 6 .11),
T r — D — qooSCo — qooS(C[)'0 + Co,¡)
(6.18)
Cl
Tr = qooSCo.O
+ qooS
qooSCo.o +
qooS—
ttt
;r<?AR
Zero-Iift T/t
Lift-induced 7#
397
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chapter
6
Elements of Airplane Performance
Figure 6.9 Comparison of lift-induced and zero-lift thrust required.
Note that as identified in Eq. (6.18), the thrust required can be considered the
sum o f zero-lift thrust required (thrust required to balance zero-lift drag) and liftinduced thrust required (thrust required to balance drag due to lift). Exam ining
Fig. 6.9, we find lift-induced TR decreases but zero-lift TR increases as the velocity
is increased. (W hy?)
Recall that C¿ = W / i q ^ S ) . From Eq. (6.18),
A lso,
W2
T r = qooSCn.o H------- ----- —
^oo^TreAR
(6.19)
dTR
dT R dVx
— =■ =
( 6 -20 )
d tfo Q
d V qq cI cJ qq
From calculus, we find that the point o f m inim um TR in Fig. 6.6 corresponds to
d T R/ d V oo = 0. H ence, from Eq. (6.20), m inim um TR also corresponds to d T R/
d q ^ = 0. D ifferentiating Eq. (6.19) with respect to qoo and setting the derivative
equal to zero, we have
dT*
or
M/ 2
----- — O C o.o----- r—---- — — U
dq™
q^SneA R
Thus,
H ow ever,
W2
C n a ~ ——---------'
q^neA R
W2
(
= (
W V
_ j
=Cl.
(6.21)
6 .3
Thrust Required for Level, Unaccelerated Flight
399
Hence, Eq. (6.21) becom es
( 6 . 22 )
Equation (6.22) yields the interesting aerodynam ic result that at m inim um thrust
required, zero-lift drag equals drag due to lift. H ence, the curves for zero-lift and
lift-induced TR intersect at the velocity for m inim um TR (i.e., for m axim um
L / D ) , as shown in Fig. 6.9. We will return to this result in Sec. 6.13.
EXAM PLE 6.1
For all the exam ples given in this chapter, two types o f airplanes will be considered:
a. A light, single-engine, propeller-driven, private airplane, approximately modeled after
the C essna Skylane shown in Fig. 6.10. For convenience, w e will designate our hypo­
thetical airplane as the CP-1, having the follow ing characteristics:
Wingspan = 35.8 ft
Wing area = 174 ft2
Normal gross weight = 2950 lb
Fuel capacity: 65 gal o f aviation gasoline
Power plant: one-piston engine o f 230 hp at sea level
Specific fuel consumption = 0.45 lb/(hp)(h)
Figure 6.10 The hypothetical CP-1 studied in Chap. 6 sample problems is modeled after the
Cessna Skylane shown here.
(Source: Cessna Aircraft Corporation.)
400
chapter
6
Elements of Airplane Performance
Parasite drag coefficient Cp,o = 0.025
Oswald efficiency factor e = 0.8
Propeller efficiency = 0.8
b. A jet-powered executive aircraft, approximately modeled after the Cessna Citation 3,
shown in Fig. 6.11. For convenience, we will designate our hypothetical jet as the CJ-1,
having the following characteristics:
Wingspan = 53.3 ft
Wing area = 318 ft2
Normal gross weight = 19,815 lb
Fuel capacity: 1119 gal of kerosene
Power plant: two turbofan engines of 3650-lb thrust each at sea level
Specific fuel consumption = 0.6 lb of fuel/(lb thrust)(h)
Parasite drag coefficient C p,o = 0.02
Oswald efficiency factor e = 0.81
By the end of this chapter, all the examples taken together will represent a basic
performance analysis of these two aircraft.
In this example, only the thrust required is considered. Calculate the TR curves at sea
level for both the CP-1 and the C J-1.
Figure 6.11 The hypothetical CJ-1 studied in Chap. 6 sample problems is modeled after the
Cessna Citation 3 shown here.
(Source: Cessna Aircraft Corp.)
6 .3
Thrust Required for Level, Unacceierated Flight
■ Solution
a. For the CP-1 assume V», = 200 ft/s = 136.4 mi/h. From Eq. (6.17),
w — = -¡---------------------2950 --------= 0.357
CL = ------i(0.002377)(200)2(174)
b2
(35.8 )2
AR = — = --------- = 7.37
S
174
The aspect ratio is
F
Thus, from Eq. (6.1c),
C?
(0.357 )2
Co = Cp.o -I------ — = 0.025 H---- = 0.0319
jreAR
7t(0.8 )(7 .3 7 )
L
CL
0.357
— = — = ---------= 11.2
D
CD
0.0319
Hence,
Finally, from Eq. (6.16),
Tr =
W
L /D
2950
263 lb
11.2
To obtain the thrust-required curve, the preceding calculation is repeated for many differ­
ent values of V ^. Some sample results are tabulated in the following table.
V o o ,f» /S
Cl
Co
L /D
Tr , lb
100
1.43
0.634
0.228
0.159
0.116
0.135
0.047
0.028
0.026
0.026
10.6
279
217
359
491
652
150
250
300
350
13.6
8.21
6.01
4.53
The preceding tabulation is given so that the reader can try such calculations and compare
the results. Such tabulations are given throughout this chapter. They are taken from a
computer calculation in which 100 different velocities are used to generate the data. The
7# curve obtained from these calculations is given in Fig. 6.6 .
b. FortheCJ-1 assume V,*, = 500 ft/s = 341 mi/h. From Eq. (6.17),
W
19,815
C l = ------- t — = -¡------------- 1--------;------- = 0.210
^(0.002377)(500)2(318)
b2
(53.3 )2
AR = — = --------- = 8.93
S
318
The aspect ratio is
F
Thus, from Eq. (6.1c),
C2
neA R
nM .
(0 .2 1)2
n ( 0.81)(8.93)
C d — C D o “ I- ------- -— — 0 . 0 2 - f - --------------------------- = 0 . 0 2 2
0,0
402
chapter
6
Elements of Airplane Performance
o
x
V„, ft/s
Figure 6.12 Thrust-required curve for the CJ-1.
Finally, from Eq. (6.16),
W
19,815
Tr = ~z—n r = _
= 2075 lb
L /D
9.55
A tabulation for a few different velocities follows
Voo, ft/s
CL
CD
300
600
700
850
1000
0.583
0.146
0.107
0.073
0.052
0.035
0.021
0.021
0.020
0.020
L/D
16.7
6.96
5.23
3.59
2.61
T r > lb
1188
2848
3797
5525
7605
The thrust-required curve is given in Fig. 6.12.
6.4 THRUST AVAILABLE AND MAXIMUM
VELOCITY
Thrust required TR, described in Sec. 6.3, is dictated by the aerodynam ics and
w eight o f the airplane itself; it is an airfram e-associated phenom enon. In
contrast, the thrust available TA is strictly associated with the engine o f the
6 .4
Thrust Available and Maximum Velocity
Reciprocating
engine-propeller
com bination
(a)
Ta
Turbojet
engine
(b)
Figure 6.13 Thrust-available curves for (a) piston engine-propeller combination
and (b) a turbojet engine.
airplane; it is the propulsive thrust provided by an engine-propeller com bination,
a turbojet, a rocket, etc. Propulsion is the subject of Chap. 9. Suffice it to say here
that reciprocating piston engines with propellers exhibit a variation o f thrust with
velocity, as sketched in Fig. 6 .13a. Thrust at zero velocity (static thrust) is a m ax­
imum and decreases with forw ard velocity. At near-sonic flight speeds, the tips
o f the propeller blades encounter the sam e com pressibility problem s discussed in
Chap. 5, and the thrust available rapidly deteriorates. On the other hand, the
thrust o f a turbojet engine is relatively constant with velocity, as sketched in
Fig. 6 .13b. These tw o pow er plants are quite com m on in aviation today; recipro­
cating engine-p ro p eller com binations pow er the average light, general aviation
aircraft, w hereas the je t engine is used by alm ost all large com m ercial transports
and m ilitary com bat aircraft. For these reasons, the perform ance analyses o f this
chapter consider only these tw o propulsive m echanism s.
C onsider a je t airplane flying in level, unaccelerated flight at a given altitude
and with velocity Vi, as shown in Fig. 6.12. Point 1 on the thrust-required curve
gives the value o f TR for the airplane to fly at velocity V¡. T he pilot has adjusted
the throttle such that the je t engine provides thrust available ju st equal to the thrust
required at this point; TA = TR. This partial-throttle TA is illustrated by the dashed
curve in Fig. 6.12. If the pilot now pushes the throttle forw ard and increases the
403
404
CHAPTER 6
Elements of Airplane Performance
engine thrust to a higher value o f TA, the airplane will accelerate to a higher ve­
locity. If the throttle is increased to full position, m axim um TA will be produced
by the je t engine. In this case, the speed o f the airplane will further increase until
the thrust required equals the m axim um TA (point 2 in Fig. 6.12). It is now im ­
possible for the airplane to fly any faster than the velocity at point 2 ; otherw ise,
the thrust required w ould exceed the m axim um thrust available from the pow er
plant. H ence, the intersection o f the TR curve (dependent on the airfram e) an d the
m axim um TA curve (dependent on the engine) defines the m axim um velocity Vinax
o f the airplane a t the given altitude, as show n in Fig. 6.12. C alculating the m axi­
m um velocity is an im portant aspect o f the airplane design process.
Conventional je t engines are rated in term s o f thrust (usually in pounds).
H ence, the thrust curves in Fig. 6.12 are useful for the perform ance analysis
o f a jet-pow ered aircraft. H owever, piston engines are rated in term s o f pow er
(usually horsepow er). H ence, the concepts o f TA and TR are inconvenient for
propeller-driven aircraft. In this case, pow er required and pow er available are the
m ore relevant quantities. M oreover, considerations o f pow er lead to results such
as rate o f clim b and m axim um altitude for both je t and propeller-driven
airplanes. T herefore, for the rem ainder o f this chapter, em phasis is placed on
pow er rather than thrust, as introduced in Sec. 6.5.
E X A M P L E 6.2
Calculate the maximum velocity of the CJ-1 at sea level (see Example 6.1).
■ Solution
The information given in Example 6.1 states that the power plant for the CJ-1 consists of
two turbofan engines of 3650-lb thrust each at sea level. Hence,
Ta = 2(3650) = 7300 lb
Examining the results of Example 6.1, we see that TR = TA = 7300 lb occurs when
Vac = 975 ft/s (see Fig. 6.12). Hence,
Vmax = 975 ft/s = 665 mi/h
It is interesting to note that since the sea-level speed of sound is 1117 ft/s, the maximum
sea-level Mach number is
AJ
M max
Knax
975
= ------ = 777= = 0.87
a
1117
In the present examples, Cp.o >s assumed constant; hence, the drag polar does not include
drag-divergence effects, as discussed in Chap. 5. Because the drag-divergence Mach
number for this type of airplane is normally on the order of 0.82 to 0.85, the preceding
calculation indicates that Mmax is greater than drag divergence, and our assumption of
constant Cp.o becomes inaccurate at this high a Mach number.
6 .5
Power Required for Level, Unaccelerated Flight
6.5 POWER REQUIRED FOR LEVEL,
UNACCELERATED FLIGHT
Power is a precisely defined m echanical term; it is energy per unit time. The
pow er associated w ith a m oving object can be illustrated by a block m oving at
constant velocity V under the influence o f the constant force F, as shown in
Fig. 6.14. The block m oves from left to right through distance d in a time in­
terval h — t\. (We assum e that an opposing equal force not shown in Fig. 6.14,
say due to friction, keeps the block from accelerating.) Work is another pre­
cisely defined m echanical term; it is force m ultiplied by the distance through
which the force m oves. M oreover, w ork is energy, having the sam e units as en­
ergy. Hence,
energy
force x distance
,
distance
Pow er = --------- = ------------------------ = force x ------------time
tim e
time
A pplied to the m oving block in Fig. 6.14, this becom es
Power = F
= FV
(6.23)
where d / f a — t\) is the velocity V o f the object. Thus, Eq. (6.23) dem onstrates
that the pow er associated with a force exerted on a m oving object is force x
velocity, an im portant result.
Consider an airplane in level, unaccelerated flight at a given altitude and
with velocity Voo. The thrust required is TK. From Eq. (6.23), the po we r required
PR is therefore
P r = Tr Voí
(6.24)
T he effect o f the airplane aerodynam ics (C¿ and Co ) on P R is readily ob­
tained by com bining Eqs. (6.16) and (6.24):
Pr
= Tr V00 = — — V00
L l/ C d
(6.25)
From Eq. (6.12),
L = W = qooS C L = \ PooV l S C L
F
V
Time = t\
V
F
Time = ti
Figure 6.14 Force, velocity, and power o f a m oving body.
405
406
chapter
6
Elements of Airplane Performance
2W
H ence,
(6.26)
PooS C l
Substituting Eq. (6.26) into (6.25), we obtain
W
Pr =
2W
Cl /C d V PooSC l
(6.27)
In contrast to thrust required, which varies inversely as C l / C d [see Eq. (6.16)],
pow er required varies inversely as C ] 12/ C D.
The pow er-required curve is defined as a plot o f PR versus Vo,, as sketched
in Fig. 6.15; note that it qualitatively resem bles the thrust-required curve o f
Fig. 6 .6 . As the airplane velocity increases, PR first decreases, then goes through
a m inim um , and finally increases. At the velocity for m inim um pow er required,
y oo, ft/s
Figure 6.15 Power-required curve for the CP-1 at sea level.
6.5
Power Required for Level, Unaccelerated Flight
the airplane is flying at the angle o f attack w hich corresponds to a m axim um
c l /2/ c D.
In Sec. 6.3, w e dem onstrated that m inim um TR aerodynam ically corresponds
to equal zero-lift and lift-induced drag. An analogous but different relation holds
at m inim um PR. From Eqs. ( 6 .11) and (6.24),
P r — Pr Vqo — D V0o = q ^ S
Pr = QoqS C d ,o^oo
21ero-lift pow er
required
tfooSVoo
( Cdo + í S
r)
C2
——
n e AR
(6.28)
Lift-induced power
required
Therefore, as in the earlier case o f TR, the pow er required can be split into the
respective contributions needed to overcom e zero-lift drag and drag due to lift.
These contributions are sketched in Fig. 6.16. A lso as before, the aerodynam ic
conditions associated with m inim um PR can be obtained from Eq. (6.28) by set­
ting d P R/dV,oo = 0. To do this, first obtain Eq. (6.28) explicitly in term s o f V,»,
recalling t h a t ^ =
and C L =
V ¿ 5 ).
P r = ^ o X S C o . o + \ p o o V ¿ s í VV/
2
00
'
2
00
neAR
(6.29)
1
PR = 2^° °V
VV2/(5/0ooVoo5)
,
0+
L
Figure 6.16 Comparison o f lift-induced, zero-lift, and net power
required.
407
408
chapter
6
Elements of Airplane Performance
F or m inim um pow er required, d P R/ d Voo = 0. D ifferentiating Eq. (6.29) yields
7 v Z - 2 ' ,“ V» Í C o » ---------- ^ 7 a r —
^ 2/ ( ! ^ 5 2 V ¿) 1
Cm
^ a r -
= ^ P o o V ¿ 5 ^ C d ,o - ^ c o,;^ = o
for m inim um PR
H ence, the aerodynam ic conditions that holds at m inim um pow er required is
(6.30)
T he fact that zero-lift drag is one-third the drag due to lift at m inim um PR is
reinforced by exam ination o f Fig. 6.16. A lso note that point 1 in Fig. 6.16 corre­
sponds to C D,o = C o,;, that is, m inim um TR\ hence, Vi» for m inim um PR is less
than that for m inim um TR.
T he point on the pow er-required curve that corresponds to m inim um TR is
easily obtained by draw ing a line through the origin and tangent to the PR curve, as
show n in Fig. 6.17. The point o f tangency corresponds to m inim um TR (hence
m axim um L / D ) . To prove this, consider any line through the origin and intersect­
ing the PR curve, such as the dashed line in Fig. 6.17. The slope o f this line is
P r / V oo- As we m ove to the right along the PR curve, the slope o f an intersecting
Figure 6.17 The tangent to the power-required curve locates the point
o f minimum thrust required (and hence the point o f maximum L/D).
6.5
Power Required for Level, Unaccelerated Flight
409
line will first decrease, then reach a m inim um (at the tangent point), and again
increase. This is clearly seen sim ply by inspection o f the geom etry o f Fig. 6.17.
Thus, the point o f tangency corresponds to a m inim um slope, hence a m inim um
value o f PR/ Vqo. In turn, from calculus this corresponds to
d ( P R/Voo) = d ( T RV0q/Voq) = d T R
d V oo
d V o»
d V oo
The above result yields d T R/dVoo = 0 at the tangent point, w hich is precisely the
m athem atical criterion for m inim um TR. C orrespondingly, L /D is m axim um at
the tangent point.
E X A M P L E 6.3
Calculate the power-required curves for (a) the CP-1 at sea level and (b) the CJ-1 at an
altitude o f 22,000 ft.
■ Solution
a. For the C P -1, the values o f TR at sea level have already been tabulated and graphed in
Example 6.1. Hence, from Eq. (6.24),
P r = Tr Vx
w e obtain the follow ing tabulation:
V, ft/s
7 V lb
P„, f t • lb/s
100
150
250
300
350
279
217
359
491
652
27,860
32,580
89,860
147,200
228,100
The power-required curve is given in Fig. 6.15.
b. For the C J-1 at 22,000 ft, Poa = 0.001184 slug/ft3. The calculation o f TR is done with
the same method as given in Example 6.1, and P r is obtained from Eq. (6.24). Som e
results are tabulated in the follow ing.
Voo, ft/s
Cl
CD
300
500
600
800
1000
1.17
0.421
0.292
0.165
0.105
0.081
0.028
0.024
0.021
0.020
L/D
Tr , lb
PR, ft lb/s
14.6
15.2
12.3
7.76
5.14
1358
1308
1610
2553
3857
0.041
0.065
0.097
0 .204
0.386
The reader should attempt to reproduce these results.
The power-required curve is given in Fig. 6.18.
x
x
x
x
x
107
107
I07
107
107
410
c hapter
6
Elements of Airplane Performance
V°o, ft/s
Figure 6.18 Power-required curve for the CJ-1 at 22,000 ft.
6.6 POWER AVAILABLE AND MAXIMUM VELOCITY
N ote again that PR is a characteristic o f the aerodynam ic design and w eight o f
the aircraft itself. In contrast, the p o we r available PA is a characteristic o f the
pow er plant. A detailed discussion o f propulsion is deferred until Chap. 9; how ­
ever, the follow ing com m ents are m ade to expedite our perform ance analyses.
6 .6 .1 R ecip ro catin g E n g in e -P ro p e lle r C o m b in atio n
A piston engine generates pow er by burning fuel in confined cylinders and using
this energy to m ove pistons, which, in turn, deliver pow er to the rotating crankshaft,
as schem atically shown in Fig. 6.19. The pow er delivered to the propeller by the
crankshaft is defined as the shaft brake p o we r P (the w ord brake stems from a
m ethod o f laboratory testing that m easures the pow er o f an engine by loading it
with a calibrated brake m echanism ). However, not all P is available to drive the air­
plane; som e o f it is dissipated by inefficiencies o f the propeller itself (to be dis­
cussed in Chap. 9). H ence, the pow er available to propel the airplane PA is given by
P a = rjP
(6.31)
w here rj is the propeller efficiency, i] < \. Propeller efficiency is an im portant
quantity and is a direct product o f the aerodynam ics o f the propeller. It is always
less than unity. F or our discussions here, both r¡ and P are assum ed to be known
quantities for a given airplane.
6 .6
Power Available and Maximum Velocity
Propeller
Pa = nP
Figure 6.19 Relation between shaft brake power and power
available.
( 6)
(a)
Figure 6.20 Power available for (a) piston engine-propeller combination and (b ) the jet engine.
A rem ark about units is necessary. In the engineering system , pow er is in
foot-pounds per second (ft • lb/s); in SI, pow er is in watts |w hich are equivalent
to N ew ton-m eters per second (N • m /s)]. However, the historical evolution o f
engineering has left us with a horrendously inconsistent (but very convenient)
unit o f pow er that is w idely used, namely, horsepower. All reciprocating engines
are rated in term s o f horsepow er (hp), and it is im portant to note that
1 hp = 550 ft • lb/s = 746 W
Therefore, it is com m on to use shaft brake horsepower bhp in place o f P, and
horsepow er available h p A in place o f PA. Equation (6.31) still holds in the form
hP/i = (f?) (bhp)
(6.32)
However, be cautious. As alw ays in dealing with fundam ental physical relations,
units m ust be consistent; therefore, a good habit is to im m ediately convert horse­
pow er to foot-pounds per second or to w atts before starting an analysis. This
approach is used here.
The pow er-available curve for a typical piston engine-propeller com bination
is sketched in Fig. 6.20a.
411
412
c h apter
6
Elements of Airplane Performance
Figure 6.21 Poweravailable and powerrequired curves, and
the determination of
maximum velocity.
(a) Propeller-driven
airplane, (b) Jetpropelled airplane.
K - , ft/s
(a)
o
X
V„,
f t/ s
(b)
Power Available and Maximum Velocity
6 .6
413
6.6.2 Jet Engine
The je t engine (see Chap. 9) derives its thrust by com bustion heating an incom ­
ing stream o f air and then exhausting this hot air at high velocities through a noz­
zle. T he pow er available from a je t engine is obtained from Eq. (6.23) as
Pa
= Ta V00
(6.33)
Recall from Fig. 6 . 13b that TA for a je t engine is reasonably constant with veloc­
ity. Thus, the pow er-available curve varies essentially linearly with V^, as
sketched in Fig. 6.20b.
For both the propeller- and jet-pow ered aircraft, the m axim um flight veloc­
ity is determ ined by the high-speed intersection o f the m axim um PA and the PR
curves. This is illustrated in Fig. 6.21. B ecause o f their utility in determ ining
other perform ance characteristics o f an airplane, these pow er curves are essential
to any perform ance analysis.
E X A M P L E 6.4
Calculate the maximum velocity for (a) the C P -1 at sea level and ( b ) the C J-1 at 22,000 ft.
■ Solution
a. For the C P-1, the information in Example 6.1 gave the horsepower rating o f the power
plant at sea level as 230 hp. Hence,
hP/t = (/?)(bhp) = 0.8 0 (2 3 0 ) = 184 hp
The results o f Example 6.3 for power required are replotted in Fig. 6.21a in terms o f
horsepower. The horsepower available is also shown, and Vmax is determined by the in­
tersection o f the curves as
Vmax = 265 ft/s = 1 8 1 mi/h
b. For the CJ-1, again from the information given in Example 6.1, the sea-level static
thrust for each engine is 3650 lb. There are two engines; hence, TA = 2(3650) = 7300 lb.
From Eq. (6.33), PA = TAV^, and in terms o f horsepower, where TA is in pounds and
in feet per second,
7'a V'00
hPa =
550
Let hp^ 0 be the horsepower at sea level. A s we w ill see in Chap. 9, the thrust o f a jet
engine is, to a first approximation, proportional to the air density. If w e make this
approximation here, the thrust at altitude becom es
T A .all
=
— Ta
o
A>
hence-
h PM,al. =
~
A)
h P /t,0
414
chapter
6
Elements of Airplane Performance
For the CJ-l at 22,000 ft, where p = 0.00 l l 84 slug/ft3,
(0.001184/0.002377) (7300) V,»
, _ „
------------------ ——------------------= 6.61Vc 00
The results of Example 6.3 for power required are replotted in Fig. 6.21b in terms of
horsepower. The horsepower available, obtained from the preceding equation, is also
shown, and Vmax is determined by the intersection of the curves as
Vmax = 965 ft/s = 658 mi/h
6.7 ALTITUDE EFFECTS ON POWER REQUIRED
AND AVAILABLE
W ith regard to P r , curves at altitude could be generated by repeating the
calculations o f the previous sections, with p ^ appropriate to the given altitude.
However, once the sea-level P R curve is calculated by m eans o f this process, the
curves at altitude can be m ore quickly obtained by sim ple ratios, as follows. Let
the subscript 0 designate sea-level conditions. From Eqs. (6.26) and (6.27),
(6.34)
(6.35)
w here Vo, P r ,o, and po are velocity, power, and density, respectively, at sea level.
At altitude, w here the density is p, these relations are
(6.36)
(6.37)
Now, strictly for the purposes o f calculation, let C L rem ain fixed betw een sea
level and altitude. Hence, because C D = Co.o + C \ / { n e A R ) , also C D rem ains
fixed. Dividing Eq. (6.36) by (6.34), and Eq. (6.37) by (6.35), we obtain
6 .7
and
PR,
Altitude Effects on Power Required and Available
= PR,o ( ^ j
(6.39)
G eom etrically, these equations allow us to plot a point on the PR curve at altitude
from a given point on the sea-level curve. F or exam ple, consider point 1 on the
sea-level PR curve sketched in Fig. 6.22. By m ultiplying both the velocity and
the pow er at point 1 by ( p o /p ) l/2, a new point is obtained— point 2 in Fig. 6.22.
Point 2 is guaranteed to fall on the curve at altitude because o f our previous
analysis. In this fashion, the com plete PR curve at altitude can be readily
obtained from the sea-level curve. The results are qualitatively given in Fig. 6.23,
Figure 6.22 Correspondence o f points on sea-level and altitude power-required curves.
Figure 6.23 Effect o f altitude on power required.
415
416
CHAPTER 6
Elements of Airplane Performance
where the altitude curves tend to experience an upw ard and rightw ard translation
as well as a slight clockw ise rotation.
W ith regard to PA, the low er air density at altitude invariably causes a
reduction in pow er for both the reciprocating and je t engines. In this book we
assum e PA and TA to be proportional to am bient density, as in Exam ple 6.4. Rea­
sons for this will be m ade clear in Chap. 9. For the reciprocating engine, the loss
in pow er can be delayed by using a supercharger. N evertheless, the im pact on
airplane perform ance due to altitude effects is illustrated in Fig. 6.24a and b for
the propeller- and jet-pow ered airplanes, respectively. Both PK and m axim um
PA are show n; the solid curves correspond to sea level and the dashed curves to
altitude. From these curves, note that Vmax varies with altitude. A lso note that at
high enough altitude, the low -speed limit, w hich is usually dictated by Vstaii, may
instead be determ ined by m axim um PA. This effect is em phasized in Fig. 6.25,
w here m axim um PA has been reduced to the extent that, at velocities ju st above
stalling, PR exceeds PA. For this case, we m ake the interesting conclusion that
stalling speed cannot be reached in level, steady flight.
(a)
Figure 6.24 Effect o f altitude on maximum velocity, (a) Propeller-driven
airplane. (continued)
6 .7
Altitude Effects on Power Required and Available
(b)
Figure 6.24 ( Continued) (b ) Jet-propelled airplane.
Figure 6.25 Situation when minimum velocity at altitude is
greater than stalling velocity.
417
418
chapter
Elements of Airplane Performance
6
To this point in our discussion, only the horizontal velocity perform ance—
both m axim um and m inim um speeds in steady, level flight— has been em pha­
sized. We have seen that m axim um velocity o f an airplane is determ ined by the
high-speed intersection o f the PA and PR curves and that the m inim um velocity
is determ ined either by stalling or by the low -speed intersection o f the pow er
curves. T hese velocity considerations are an im portant part o f airplane perfor­
m ance; indeed, for som e airplanes, such as m any m ilitary fighter planes, squeez­
ing the m axim um velocity out o f the aircraft is the pivotal design feature.
H owever, this is ju st the beginning o f the perform ance story; we exam ine other
im portant characteristics in the rem aining sections o f this chapter.
EXAM PLE 6.5
Using the method of this section, from the CJ-l power-required curve at 22,000 ft in
Example 6.4, obtain the CJ-l power-required curve at sea level. Compare the maximum
velocities at both altitudes.
■ Solution
From Eqs. (6.38) and (6.39), corresponding points on the power-required curves for sea
level and altitude are, respectively,
Í P \ I/2
hp«.o — hPft.aii I ~ )
and
We are given Va|t and hps alt for 22,000 ft from the CJ-l curve in Example 6.4. Using the
above formulas, we can generate Vo and hpR 0 as in the following table, noting that
/ ^ y / 2 = / o o o u 8 4 y /2
\po)
\ 0.002377,/
Given Points
V * ,ft/s
hPw.all
200
889
741
1190
3713
7012
300
500
800
1000
1/2
(*
0.706
Generated Points
V* ft/s
hPR.o
141
628
523
840
2621
4950
212
353
565
706
These results, along with the hpA curves for sea level and 22,000 ft, are plotted in
Fig. 6.26. Looking closely at Fig. 6.26, note that point l on the hpR curve at 22,000 ft is
used to generate point 2 on the hpR curve at sea level. This illustrates the idea of this
section. Also, note that Vmax at sea level is 975 ft/s = 665 mi/h. This is slightly larger than
Vmax at 22,000 ft, which is 965 ft/s = 658 mi/h.
6 .8
Rate of Climb
K«,, ft/s
Figure 6.26 Altitude effects on Vmax for the CJ-1.
6.8 RATE OF CLIMB
Visualize a Boeing 111 transport (see Fig. 6.27) pow ering itself to takeoff speed
on an airport runway. It gently lifts o ff at about 180 mi/h, the nose rotates
upward, and the airplane rapidly clim bs out o f sight. In a m atter o f m inutes, it is
cruising at 30,000 ft. This picture prom pts the follow ing questions: How fast can
the airplane clim b? How long does it take to reach a certain altitude? The next
two sections provide som e answ ers.
C onsider an airplane in steady, unaccelerated, clim bing flight, as shown in
Fig. 6.28. The velocity along the flight path is
and the flight path itself is
419
420
chapter
D ESIGN
6
Elements of Airplane Performance
BOX
Maximum velocity at a given altitude is frequently a
part of the set of specifications for a new airplane
design. To design an airplane for a given Vmax, what
characteristics of the airplane would you, the airplane
designer, be concerned with? That is, what design as­
pects of the airplane dictate the maximum velocity?
The answer to this question reveals several critical de­
sign parameters that are important not only for Vmax
but also for other performance aspects of the airplane.
Let us answer this question by obtaining an equation
for Vmax and examining the parameters in the equation.
Combining Eqs. (6.1c) and (6.13), we have
Examine Eq. (6.44) carefully. Note that (7/1)m»x
does not appear alone, but appears only in the ratio
(Ta / W)max. Also note that the wing planform area S
does not appear alone, but only in the ratio W/S.
Hence, Vmax does not depend on thrust alone, or
weight alone, or wing area alone, but rather only on
certain ratios of these quantities, namely,
maximum thrust-to-weight ratio
W
: wing loading
i3
(6.40)
T = qocSCi) = qaoS ^Co.o +
From Eq. (6.14), we obtain for steady, level flight
W
CL
(6.41)
(/oqS
Inserting Eq. (6.41) into (6.40) yields
T = c/ooS yCp o +
W2
q ^ & n e AR,
W 2
— QooSC d.O+
(6.42)
qooSneAR
We have just identified two of the most important air­
plane design parameters, namely, thrust-to-weight
ratio and wing loading. In addition, from Eq. (6.44),
we see that Vmax depends on px (altitude), the zero-lift
drag coefficient Co o, and the product pAR. Later in
Sec. 6.15, we show that the product jtM R is equal to
4CDfi( L / D ) m
2 M, where (L / D ) mm is the maximum
value of the lift-to-drag ratio for the airplane. Hence,
(L /D )max is also an important design parameter.
From Eq. (6.44), we conclude that V^ax can be
increased by
1.
Increasing the maximum thrust-to-weight ratio
C
Multiply Eq. (6.42) by qx and rearrange.
(V2
Qoo^Cd,q —q<x>T +
Site AR
2.
3.
= 0
(6.43)
Eq. (6.43) is a quadratic equation in terms of q^.
Solving Eq. (6.43) for qoo by use of the quadratic for­
mula, recalling that q 00 =
and setting T in
Eq. (6.43) equal to the maximum thrust available
(fulI-throttle thrust) (TA)mm, we obtain for the maxi­
mum velocity (the details are left to you as a home­
work problem)
1/2
7Za\
V W /m a x
+ <w\ h U Y
\ SJ~ \
S
J
\
m ax
\ W >max
-
4Cp,o
ttM
R
PooCn.o
(6.44)
Ta / W
w
Increasing the wing loading W/S
Decreasing the zero-lift drag coefficient Cd o
These trends are almost intuitively obvious, even
without looking at Eq. (6.44), except possibly for the
benefit of increasing the wing loading. To help under­
stand the advantage of a high wing loading in this
case, imagine that W/S is increased by decreasing S.
If the planform area is made smaller, the total skin
friction drag on the wing is reduced (less surface for
the shear stress to act on), and hence V,nax is in­
creased.
The results discussed here are important to other
aspects of airplane performance. The design parame­
ters T / W and W/S have a strong cffcct on other per­
formance quantities in addition to Vmax, as we will
see in subsequent sections.
6 .8
Rate of Climb
Figure 6.28 Airplane in climbing flight.
inclined to the horizontal at angle 6. As alw ays, lift and drag are perpendicular
and parallel to V^, and the w eight is perpendicular to the horizontal. Thrust T is
assum ed to be aligned with the flight path. Here, the physical difference from our
previous discussion o f level flight is that T is not only working to overcom e the
drag, but for clim bing flight it is also supporting a com ponent o f weight.
421
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6
Elements of Airplane Performance
Sum m ing forces parallel to the flight path, we get
T = D + W sind
(6.45)
and perpendicular to the flight path, we have
L = W cos 9
(6.46)
N ote from Eq. (6.46) that the lift is now sm aller than the weight. Equations (6.45)
and (6.46) represent the equations o f m otion for steady, clim bing flight and are
analogous to Eqs. (6.11) and ( 6 .12) obtained earlier for steady, horizontal flight.
M ultiply Eq. (6.45) by V»:
T V oo = DVoo + WVoo s in 9
T V q q — D V p,
w
= V„osin 0
(6.47)
Exam ine Eq. (6.47) closely. The right-hand side, V» sin 9, is the airplane’s verti­
cal velocity, as illustrated in Fig. 6.28. This vertical velocity is called the rate o f
climb R/C:
R/C = Voo sin 9
(6.48)
On the left-hand side o f Eq. (6.47), T
is the pow er available, from Eq. (6.33),
and is represented by the PA curves in Fig. 6.20. The second term on the left-hand
side o f Eq. (6.47) is D Voo, w hich for level flight is the pow er required, as repre­
sented by the PR curve in Fig. 6.15. For clim bing flight, however, D V 00 is no
longer precisely the pow er required, because pow er m ust be applied to overcom e
a com ponent o f w eight as w ell as drag. N evertheless, for small angles o f climb,
say 9 < 2 0 ° , it is reasonable to neglect this fact and to assum e that the D Voo term
in Eq. (6.47) is given by the level-flight PR curve in Fig. 6.15. W ith this,
7 Voo — D Voo = excess power
(6.49)
w here the excess pow er is the difference betw een pow er available and pow er
required, as show n in Fig. 6.29a and 6.29b, for propeller-driven and jet-pow ered
aircraft, respectively. Com bining Eqs. (6.47) to (6.49), we obtain
R/C =
excess pow er
W
(6.50)
w here the excess pow er is clearly illustrated in Fig. 6.29.
Again, we em phasize that the PR curves in Fig. 6.29a and 6.29b are taken, for
convenience, as those already calculated for level flight. Hence, in conjunction
with these curves, Eq. (6.50) is an approximation to the rate o f clim b, good only
for sm all 9. To be more specific, a plot o f D V » versus Voo for clim bing flight
[which is exactly called for in Eq. (6.47)] is different from a plot o f D V00 versus
6 .8
Rate of Climb
( 6)
(a)
Figure 6.29 Illustration of excess power, (a) Propeller-driven airplane, (b) Jet-propelled
airplane.
Voo for level flight [which is the curve assum ed in Fig. 6.29 and used in Eq. (6.50)]
sim ply because D is smaller f o r climbing than f o r level flight at the same V*,.
To see this m ore clearly, consider an airplane w ith W = 5000 lb, S = 100 ft2,
C d ,o = 0.015, e — 0.6, and AR = 6 . If the velocity is Voo = 500 ft/s at sea level
and if the airplane is in level flight, then
= L / ^ q ^ S ) = W /ij/OooV^S) =
0.168. In turn,
C2
C D = C D o + — 7— = 0.015 + 0.0025 - 0.0175
7TéA R
Now consider the sam e airplane in a 30° clim b at sea level, with the sam e velocity
Voo = 500 ft/s. H ere the lift is sm aller than the weight, L = W cos 9, and therefore
C i = W cos 30‘7 (^ /0 oo V ¿S ) = 0.145. In turn, C D = Cp.o + C J2 ( j i e A R ) =
0.015 + 0.0019 = 0.0169. This should be com pared with the higher value of
0.0175 obtained earlier for level flight. As seen in this exam ple, for steady clim b­
ing flight, L (hence C L) is smaller, and thus induced drag is smaller. C onse­
quently, total drag for clim bing flight is sm aller than that for level flight at the
same velocity.
Return again to Fig. 6.29, which corresponds to a given altitude. Note that
the excess pow er is different at different values o f Voo. Indeed, for both the
propeller- and jet-pow ered aircraft there is som e Voo at which the excess power
is m axim um . At this point, from Eq. (6.50), R/C will be m aximum:
m axim um excess pow er
max R/C = ------------------------- - ------W
(6.51)
This situation is sketched in Fig. 6.30c/, w here the pow er available is that at full
throttle, that is, m axim um PA. The m axim um excess power, shown in Fig. 6.30a,
via Eq. (6.51) yields the m axim um rate o f clim b that can be generated by the
airplane at the given altitude. A convenient graphical m ethod o f determ ining
m axim um R/C is to plot R/C versus Voo, as shown in Fig. 6.30b. A horizontal
tangent defines the point o f m axim um R/C. A nother useful construction is the
hodograph diagram , w hich is a plot o f the airplane’s vertical velocity Vv versus
423
424
chapter
6
Elements of Airplane Performance
(a)
(b)
Figure 6.30 Determination o f maximum rate o f climb
for a given altitude.
its horizontal velocity Vh. Such a hodograph is sketched in Fig. 6.31. R em em ber
that R/C is defined as the vertical velocity, R/C = V,,; hence a horizontal tangent
to the hodograph defines the point o f m axim um R/C (point 1 in Fig. 6.31). Also,
any line through the origin and intersecting the hodograph (say, at point 2 ) has
the slope Vv/ V h4, hence, from the geom etry o f the velocity com ponents, such a
line m akes the clim b angle 9 with respect to the horizontal axis, as show n in
Fig. 6.31. M oreover, the length o f the line is equal to V^,. A s this line is rotated
counterclockw ise, R/C first increases, then goes through its m axim um , and
finally decreases. Finally, the line becom es tangent to the hodograph at point 3.
This tangent line gives the m axim um clim b angle for w hich the airplane can
m aintain steady flight, show n as 9mm in Fig. 6.31. It is interesting that m axim um
R/C does not occur at the m axim um clim b angle.
The large excess pow er and high thrust available in m odern aircraft allow
clim bing flight at virtually any angle. For large clim b angles, the previous analy­
sis is not valid. Instead, to deal with large 9 , the original equations o f motion
[Eqs. (6.45) and (6.46)] m ust be solved algebraically, leading to an exact solution
valid for any value o f 9. T he details o f this approach can be found in the books
6 .8
Rate of Climb
425
M axim um R/C
K
= R/C
^ 6 , max
^ m a x R/C
Figure 6.31 Hodograph for climb performance at a given altitude.
by D om m asch et al. and by Perkins and H age (see Bibliography at the end o f this
chapter).
R eturning briefly to Fig. 6.29a and b for the propeller-driven and jetpow ered aircraft, respectively, w e can see an im portant difference in the lowspeed rate-of-clim b perform ance betw een the two types. Due to the poweravailable characteristics o f a piston engine-p ro p eller com bination, large excess
pow ers are available at low values o f Voo, ju st above the stall. For an airplane on
its landing approach, this gives a com fortable margin o f safety in case o f a sud­
den w ave-off (particularly im portant for landings on aircraft carriers). In con­
trast, the excess pow er available to je t aircraft at low
is small, with a corre­
spondingly reduced rate-of-clim b capability.
Figures 6.30b and 6.31 give R/C at a given altitude. In Sec. 6.10, we will ask
how R/C varies with altitude. In pursuit o f an answer, w e will also find the
answ er to another question, namely, how high the airplane can fly.
EXAM PLE 6.6
Calculate the rate o f clim b versus velocity at sea level for (a) the CP-1 and (ft) the CJ-1.
■ Solution
a. For the C P-1, from Eq. (6.50)
_ excess power _
PA - PR
W
W
With power in foot-pounds per second and W in pounds, for the CP-1, this equation
becom es
R/C =
— ~ Pr
2950
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6
Elements of Airplane Performance
From Example 6.3, at Voo = 150 ft/s, Pr = 0.326 x 105 ft • lb/s. From Example 6.4,
PA = 550(hp4) = 550(184) = 1.012 x 105 ft • lb/s. Hence,
(1 .0 1 2 -0 .3 2 6 ) x 105
R/C = ----------- — — 1--------- = 23.3 ft/s
2950
In terms of feet per minute,
1398 ft/min
R/C = 23.3(60) =
at Voc = 150 ft/s
This calculation can be repeated at different velocities, with the following results:
Voo, ft/s
R/C, ft/min
100
1492
1472
1189
729
32.6
130
180
220
260
These results are plotted in Fig. 6.32.
b. For the CJ-1, from Eq. (6.50),
r /C = Pa ~ P r = 55° ( hP ¿ ~ hP«)
W
19,815
Vo., ft/s
Figure 6.32 Sea-level rate o f clim b for the C P -1.
6 .8
Rate of Climb
From the results and curves of Example 6.5, at Vx = 500 ft/s, hp^ = 1884, and
hp^ = 6636.
Hence,
/ 6636 — 1884 \
R/C = 550 (“ l 9^
R/C = 132(60) =
or
)
= 132ft/S
7914 ft/rnin
at V» = 500 ft/s
Again, a short tabulation for other velocities is given in the following for the reader to check.
Voo, ft/s
R/C, ft/min
200
3546
7031
8088
5792
1230
400
600
800
950
These results are plotted in Fig. 6.33.
10,000 -
8,000
q
6,000
1
c
u
0£
4 ,0 0 0
2,000
Voo, ft/s
Figure 6.33 Sea-level rate o f clim b for the C J- 1.
427
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6
chapter
Elements of Airplane Performance
DESIGN BOX
What airplane design parameters dictate maximum
rate of climb? The answer is not explicitly clear from
our graphical analysis carried out in this section.
However, the answer can be obtained explicitly by
deriving an equation for maximum rate of climb and
identifying the design parameters that appear in the
equation. The derivation is lengthy, and we are inter­
ested only in the final result here. For a detailed de­
rivation, see Anderson, Aircraft Performance and
Design, McGraw-Hill, New York, 1999. Denoting
maximum rate of climb by (R/C)max, and for com­
pactness identifying the symbol Z as
1+.1 +
( L / D ) * ( T / W ) 2max
where ( L / D ) mm is the maximum value of the iift-todrag ratio for the given airplane, we can show that for
a jet-propelled airplane
(R/C)n
~ ( W/ S) Z'
IpooCp.o.
6
1/2 f( z
. y /2
\
_________ i______
2(7/W )?naxa / D ) L xZ j
(6.52)
and for a propeller-driven airplane
(R/C)„
■ (* ).
-0.8776
W/S
1
,3/2
A*C»,o (L/DY„
/m u x
(6.53)
where /? is the propeller efficiency defined by
Eq. (6.31) and P is the shaft brake power from the
engine (or engines, for a multiengine airplane).
Examining Eq. (6.52), we see once again that W,
5, and T do not appear alone, but rather in ratios.
From Eq. (6.52), the design parameters that dictate
(R/C),nax for a jet-propelled airplane arc
■
■
■
■
Wing loading W/S
Maximum thrust-to-weight ratio (T/ W)max
Zero-lift drag coefficient Co,o
Maximum lift-to-drag ratio ( L / D ) mm
These are the same design parameters that dictate
Vmax from Eq. (6.44). We also note, looking ahead to
Sec. 6.14, that ( L / D ),nax is determined by Co.o, e,
and AR, namely, that (L /D )J,ax = ;reAR/(4Co,o),
as we will see. So identifying (L /D )max as a design
parameter is the same as identifying a certain combi­
nation of e, AR, and Co.o as a design parameter. We
will have more to say about the importance of
(L / D ) mm in airplane design in subsequent sections.
Recall that for a propeller-driven airplane, the rating of the engine-propeller combination in terms of
power is more germane than that in terms of thrust.
Hence, Eq. (6.53) gives maximum rate of climb for a
propeller-driven airplane in terms of the power-toweight ratio r]P/W. | Recall from Eq. (6.31) that >]P
is the power available PA for a propeller-driven air­
plane.] Therefore, for a propeller-driven airplane, an
important design parameter that dictates (R/C)max is
the power-to-weight ratio.
6.9 GLIDING FLIGHT
C onsider an airplane in a pow er-off glide, as sketched in Fig. 6.34. The forces
acting on this aircraft are lift, drag, and w eight; the thrust is zero because the
pow er is off. T he glide flight path m akes an angle 6 below the horizontal. For an
equilibrium unaccelerated glide, the sum o f the forces must be zero. Sum m ing
forces along the flight path, we have
D = W sin#
(6.54)
and perpendicular to the flight path
L = W cos 6
(6.55)
6 .9
Gliding Flight
429
Figure 6.34 Airplane in power-off gliding flight.
The equilibrium glide angle can be calculated by dividing Eq. (6.54) by (6.55),
yielding
sin #
c o s#
or
ta n # —
(6.56)
L/D
Clearly, the glide angle is strictly a function o f the lift-to-drag ratio; the higher the
L/D, the shallow er the glide angle. From this, the sm allest equilibrium glide angle
occurs at ( ¿ / D ) max, which corresponds to the m axim um range for the glide.
EXAMPLI l 6.7
The maximum lift-to-drag ratio for the CP-1 is 13.6. Calculate the minimum glide angle
and the maximum range measured along the ground covered by the CP-1 in a power-off
glide that starts at an altitude of 10,000 ft.
■ Solution
The minimum glide angle is obtained from Eq. (6.56) as
tan 6/fnin —
0
1
(L/I))mm
— 4 2°
C'mm —
—
1
13.6
430
c hapter
6
Elements of Airplane Performance
----------------------------------------------R ----------------------------------------------Figure 6.35 Range covered in an equilibrium glide.
The distance covered along the ground is R , as shown in Fig. 6.35. If h is the altitude at
the start of the glide, then
h
, L
R = ------= h —
tan#
D
Hence,
Rm n = h ( ^ \
\
=10,000(13.6)
/ max
Rmm = 136,000 ft = 25.6 mi
EXAM PLE 6.8
Repeat Example 6.7 for the CJ-I, for which the value of ( L / D )max is 16.9.
■ Solution
1
1
ta" min ~ (L /D )max “ 16.9
=
=10,000(16.9)
\
/ m ax
Rmm = 169,000 ft = 32 mi
Note the obvious fact that the CJ-1, with its higher value of (L /D )max, is capable of a
larger glide range than the CP-1.
6 .9
Gliding Flight
431
KXAMPL1 <: 6 .9
For the CP-1, calculate the equilibrium glide velocities at altitudes of 10,000 and 2000 ft,
each corresponding to the minimum glide angle.
Solution
L = { pooV^SC l
Combining this with Eq. (6.55) gives
W cos d = '*PooV*SCL
2 cos 0 W
or
PooC l S
where W/S is the by now familiar wing loading. From this equation, we see that the
higher the wing loading, the higher the glide velocity. This makes sense: A heavier
airplane with a smaller wing area is going to glide to the earth’s surface at a greater
velocity. Note, however, that the glide angle, and hence range, depends not on the weight
of the airplane and not on its wing loading but exclusively on the value of (L /D )max,
which is an aerodynamic property of the airframe design. A higher wing loading simply
means that the airplane will have a faster glide and will reach the earth’s surface sooner.
From Example 6.1, we have for the CP-1,
W
2950
= 16.95 lb/ft2
~S ~ ~VT4
Also from the tabulation in Example 6.1, we see that (L / D ) mm = 13.6 corresponds to a
lift coefficient C¿ = 0.634. (Note that both L / D and C¿ are functions of the angle of
attack of the airplane; these are aerodynamic data associated with the airframe and are not
influenced by the flight conditions. Hence, C¿ = 0.634 at maximum L / D, no matter
whether the airplane is in level flight, is climbing, or is in a glide.) Therefore, at 10,000 ft,
where p<*, = 0.0017556 slug/ft3, we have
Voo =
(2cos4.2°)(16.95)
0.0017556(0.634)
Voo = 174.3 ft/s
at/i = 10,000 ft
At 2000 ft, Poo = 0.0022409 slug/ft3. Hence,
Voc =
(2 cos4.2°)(l6.95)
0.(X)22409(0.634)
= 154.3 ft/s
at h = 2000 ft
Note that the equilibrium glide velocity decreases as altitude decreases.
432
chapter
6
Elements of Airplane Performance
6.10 ABSOLUTE AND SERVICE CEILINGS
The effects o f altitude on PA and PR w ere discussed in Sec. 6.7 and illustrated in
Fig. 6.24a and b. For the sake o f discussion, consider a propeller-driven airplane:
The results o f this section will be qualitatively the sam e for a jet. As altitude
increases, the m axim um excess pow er decreases, as shown in Fig. 6.36. In turn,
m axim um R/C decreases. This is illustrated by Fig. 6.37, w hich is a plot o f
m axim um R/C versus altitude, but with R/C as the abscissa.
Figure 6.36 Variation of excess power with altitude.
Figure 6.37 Determination o f absolute and service ceilin gs for the C P -1.
6 .1 0
Absolute and Service Ceilings
433
V.»
Figure 6.38 Power-required and power-available curves
at the absolute ceiling.
T here is som e altitude high enough at which the PA curve becom es tangent
to the PR curve (point l in Fig. 6.38). T he velocity at this point is the only value
at which steady, level flight is possible; moreover, there is zero excess power,
hence zero m axim um rate o f clim b, at this point. The altitude at w hich m axi­
m um R/C = 0 is defined as the absolute ceiling o f the airplane. A m ore useful
quantity is the service ceiling, defined as that altitude w here m axim um R/C =
100 ft/m in. The service ceiling represents the practical upper limit o f steady,
level flight.
T he absolute and service ceilings can be determ ined as follows:
1.
U sing the technique o f Sec. 6 .8, calculate values o f m axim um R/C for a
num ber o f different altitudes.
2.
Plot m axim um rate o f clim b versus altitude, as shown in Fig. 6.37.
3.
Extrapolate the curve to 100 ft/m in and 0 ft/m in to find the service and
absolute ceilings, respectively, as also shown in Fig. 6.37.
EXAM PLE 6.10
Calculate the absolute and service ceilings for (a) the CP-1 and (b) the CJ-1.
■ Solution
a. For the CP-1, as stated in Example 6.1, all the results presented in all the examples of
this chapter are taken from a computer program that deals with 100 different velocities,
each at different altitudes, beginning at sea level and increasing in 2000-ft increments. In
modern engineering, using the computer to take the drudgery out of extensive and
repeated calculations is an everyday practice. For example, note from Example 6.6 that
the maximum rate of climb at sea level for the CP-1 is 1500 ft/min. In essence, this result
is the product of all the work performed in Examples 6.1 to 6.6. Now, to obtain the
absolute and service ceilings, these calculations must be repeated at several different
altitudes in order to find where R/C = 0 and 100 ft/min, respectively. Some results are
tabulated and plotted in the table that follows; the reader should take the time to check a
few of the numbers.
434
CHAPTER 6
Elements of Airplane Performance
Altitude, ft
0
4,000
8,000
12,000
16,000
20,000
24,000
26,000
Maximum R/C, ft/min
1500
1234
987
755
537
331
135
40
These results are plotted in Fig. 6.37. From these numbers, we find
Absolute ceiling (R/C = 0) is
27,000 ft
Service ceiling (R/C = 100 ft/min) is
25,000 ft
■o
Maximum R/C, ft/min X 10-3
Figure 6.39 Determination of absolute and service ceilings for the CJ-1.
6 .11
Time to Climb
b. For the C J-l, utilizing the results from Examples 6 . 1 to 6.6 and making similar calcu­
lations at various altitudes, w e tabulate the follow ing results:
Altitude, ft
Maximum R/C, ft/min
0
6,000
12,000
18,000
24,000
30,000
36,000
8118
6699
5448
4344
3369
2502
1718
These results are plotted in Fig. 6.39.
From these results, w e find
Absolute ceiling (R/C = 0) is
4 9 ,0 0 0 ft
Service ceiling (R/C = 100 ft/min) is
4 8 ,0 0 0 ft
6.11 TIME TO CLIMB
To carry out its defensive role adequately, a fighter airplane m ust be able to clim b
from sea level to the altitude o f advancing enem y aircraft in as short a tim e as pos­
sible. In another case, a com m ercial aircraft must be able to rapidly clim b to high
altitudes to m inim ize the discom fort and risks o f inclem ent weather and to m ini­
mize air traffic problem s. As a result, the tim e for an airplane to clim b to a given al­
titude can becom e an important design consideration. The calculation o f the time to
clim b follows directly from our previous discussions, as described in the following.
The rate o f clim b was defined in Sec. 6.8 as the vertical velocity o f the air­
plane. Velocity is sim ply the tim e rate o f change o f distance, the distance here
being the altitude h. Hence, R/C = d h / d t . Therefore,
d, = m
(6-57)
In Eq. (6.57), d t is the small increm ent in tim e required to clim b a sm all incre­
ment dh in altitude. Therefore, from calculus, the time to clim b from one altitude
h\ to another h2 is obtained by integrating Eq.(6.57):
pf h2 dh_
t
~ L
R/C
Normally, time to clim b is considered from sea level, where h\ = 0. Hence, the
tim e to clim b to any given altitude h2 is
t=
Chl dh
^
(6-58)
To calculate t graphically, first plot (R/C) 1 versus h, as show n in Fig. 6.40.
The area under the curve from h = 0 to h = h2 is the tim e to clim b to altitude h2.
435
chapter
6
Elements of Airplane Performance
(R/C)- 1 , minimum, (1/ft) X 103
436
Altitude h, ft X 10~3
Figure 6.40 Determination of time to climb for the CP-1
Altitude h, ft X 10 3
Figure 6.41 Determination o f time to climb for the CJ-1.
EXAMPLE 6.11
Calculate and compare the time required for (a) the CP-1 and (b) the CJ-1 to climb to
20,000 ft.
■ Solution
a. For the CP-1, from Eq.(6.58), the time to climb is equal to the shaded area under the
curve shown in Fig. 6.40. The resulting area gives time to climb as
27.0 min.
b. For the CJ-1, Eq.(6.58) is plotted in Fig. 6.41. The resulting area gives time to climb
3.5 min.
Note that the C J-1 climbs to 20,000 ft in one-eighth of the time required by the CP-1;
this is to be expected for a high-performance executive jet transport in comparison to its
propeller-driven piston engine counterpart.
6.12 RANGE AND ENDURANCE— PROPELLERDRIVEN AIRPLANE
W hen C harles Lindbergh m ade his spectacular solo flight across the Atlantic
O cean on M ay 2 0 -2 1 , 1927, he could not have cared less about m axim um veloc­
ity, rate o f clim b, or tim e to clim b. U pperm ost in his m ind w as the m axim um dis­
tance he could fly on the fuel supply carried by the Spirit o f St. Louis. Therefore,
6 .1 2
Range and Endurance— Propeller-Driven Airplane
range was the all-pervasive consideration during the design and construction o f
L indbergh’s airplane. Indeed, throughout all 20th-century aviation, range has
been an im portant design feature, especially for transcontinental and transoceanic
transports and for strategic bom bers for the military.
Range is technically defined as the total distance (m easured with respect to
the ground) traversed by the airplane on a tank o f fuel. A related quantity is en­
durance, which is defined as the total tim e that an airplane stays in the air on a
tank o f fuel. In different applications, it may be desirable to m axim ize one or the
other o f these characteristics. T he param eters that m axim ize range are different
from those that m axim ize endurance; they are also different for propeller- and
jet-pow ered aircraft. The purpose o f this section is to discuss these variations for
the case o f a propeller-driven airplane; je t airplanes are considered in Sec. 6.13.
6.12.1 P hy sical C o n sid eratio n s
O ne o f the critical factors influencing range and endurance is the specific fuel
consumption, a characteristic o f the engine. F or a reciprocating engine, specific
fuel consum ption (com m only abbreviated SFC) is defined as the weight o f fuel
consumed p e r unit p owe r p e r unit time. As m entioned earlier, reciprocating en­
gines are rated in term s o f horsepow er, and the com m on units (although incon­
sistent) o f specific fuel consum ption are
lb o f fuel
SFC = _______
(bhp)(h)
where bhp signifies shaft brake horsepow er, discussed in Sec. 6 .6 .
First, consider endurance. On a qualitative basis, to stay in the air for the
longest tim e, com m on sense says that we m ust use the minimum num ber of
pounds o f fuel per hour. On a dim ensional basis, this quantity is proportional to
the horsepow er required by the airplane and to the SFC:
lb o f fuel
----- ------ a (SFC )(hp«)
h
Therefore, m inim um pounds o f fuel per hour are obtained with m inim um h p fi.
Since m inim um pounds o f fuel per hour give m axim um endurance, we quickly
conclude that
Maximum endurance for a propeller-driven airplane occurs when the airplane is fly­
ing at minimum power required.
This condition is sketched in Fig. 6.42. Furtherm ore, in Sec. 6.5, we have already
proved that m inim um pow er required corresponds to a m axim um value o f
C l /2/ C D [see Eq. (6.27)]. Thus,
Maximum endurance for a propeller-driven airplane occurs when the airplane is flying
at a velocity such that c ] 1 / C o is maximum.
Now, consider range. To cover the longest distance (say, in m iles), com m on
sense says that we m ust use the m inim um num ber o f pounds o f fuel per mile. On
437
438
chapter
6
Elements of Airplane Performance
a dim ensional basis, we can state the proportionality
lb o f fuel
(SFC)(hptf)
------- :— o c ---------------mi
Voo
(C heck the units yourself, assum ing
is in m iles per hour.) As a result, m ini­
m um pounds o f fuel per m ile are obtained with a m inim um hpff/ Voo. This m ini­
m um value o f h p s /Voo precisely corresponds to the tangent point in Fig. 6.17,
w hich also corresponds to m axim um L / D , as proved in Sec. 6.5. Thus,
Maximum range for a propeller-driven airplane occurs when the airplane is flying at
a velocity such that C ¿/C o is maximum.
This condition is also sketched in Fig. 6.42.
6 .12.2 Q u an titativ e F orm u lation
The im portant conclusions displayed in Sec. 6.12.1 w ere obtained from purely
physical reasoning. We will develop quantitative form ulas that substantiate these
conclusions and that allow the direct calculation o f range and endurance for
given conditions.
In this developm ent, the specific fuel consum ption is couched in units that
are consistent, that is,
lb o f fuel
(ft • lb/s)(s)
N o f fuel
°r
(J/s)(s)
For convenience and clarification, c will designate the specific fuel consum ption
w ith consistent units.
m ax im u m
en d u ran ce
m axim um
range
Figure 6.42 Points of maximum range and endurance on the
power-required curve for a propeller-driven airplane.
6 .1 2
Range and Endurance— Propeller-Driven Airplane
C onsider the product c P d t , w here P is engine pow er and d t is a small in­
crem ent o f time. The units o f this product are (in the English engineering system )
lb o f fuel ft • lb v
,
c P d t = ------------------------ (s) = b o f fue
(ft • lb/s)(s) s
Therefore, c P d t represents the differential change in w eight o f the fuel due to
consum ption over the short tim e period d t . T he total w eight o f the airplane W is
the sum o f the fixed structural and payload w eights, along with the changing fuel
weight. Hence, any change in W is assum ed to be due to the change in fuel
weight. Recall that W denotes the w eight o f the airplane at any instant. Also, let
Wo = gross w eight o f the airplane (w eight w ith full fuel and payload), W¡ =
weight o f the fuel load, and W: = w eight o f the airplane without fuel. W ith these
considerations, we have
Wi
and
or
=
Wo -
Wf
dWf = d W = - c P d t
dW
d t = -------cP
(6.59)
The m inus sign in Eq. (6.59) is necessary because d t is physically positive
(tim e cannot m ove backw ard, except in science fiction novels), while at the same
time W is decreasing (hence, d W is negative). Integrating Eq. (6.59) between
tim e t = 0, w here W = W0 (fuel tanks full), and tim e t = E, w here W = Wi
(fuel tanks em pty), w e find
/■ % ,= - r * *
Jo
Jwo
(6.60)
In Eq. (6.60), E is the endurance in seconds.
To obtain an analogous expression for range, multiply Eq. (6.59) by V,»:
( 6 .6 l)
Voadt =
cP
In Eq. (6.61), V ^ d t is the increm ental distance d s covered in time d t .
V° ° d W
d/s = ------------cP
(6.62)
The total distance covered throughout the flight is equal to the integral of
Eq. (6.62) from s = 0, where W = W0 (full fuel tank), to s = R, w here W = W¡
439
440
c hapter
6
Elements of Airplane Performance
(em pty fuel tank):
f R
/ dS = ~
JO
f W,
JW U
Vo0 d W
cP
or
(6.63)
In Eq.(6.63), R is the range in consistent units, such as feet or meters.
Equations (6.60) and (6.63) can be evaluated graphically, as shown in
Fig. 6.43a and b for range and endurance, respectively. Range can be calculated
accurately by plotting Vx,/ ( c P ) versus W and taking the area under the curve
from Wi to Wo, as show n in Fig. 6.43a. A nalogously, the endurance can be cal­
culated accurately by plotting ( c /3)-1 versus W and taking the area under the
curve from W, to Wo, as shown in Fig. 6.43b.
Equations (6.60) and (6.63) are accurate form ulations for endurance and
range. In principle, they can include the entire flight— takeoff, clim b, cruise, and
landing— as long as the instantaneous values o f W, Vx,, c, and P are know n at
each point along the flight path. However, Eqs. (6.60) and (6.63), although accu­
rate, are also long and tedious to evaluate by the m ethod ju st discussed. T here­
fore, sim pler but approxim ate analytic expressions for R and E are useful. Such
form ulas are developed in Sec. 6.12.3.
6 .12.3 B reguet F o rm u las (P ro p eller-D riv en A irplane)
For level, unaccelerated flight, we dem onstrated in Sec. 6.5 that PR = D V ^ .
Also, to m aintain steady conditions, the pilot has adjusted the throttle such that
pow er available from the engine-propeller com bination is ju st equal to the pow er
Range and Endurance— Propeller-Driven Airplane
6 .1 2
required: PA = PR = D V00. In Eq. (6.59), P is the brake pow er output o f the
engine itself. R ecall from Eq. (6.31) that PA = t] P, w here rj is the propeller
efficiency. Thus,
p = P a = DVoo
>1
(6 64)
V
Substitute Eq. (6.64) into (6.63):
R _ f w°
V oodw
J w ,
_
f w° VcovdW _
C' P
J w i
c D V o o
r Wo r¡dW
J
Wl
(6.65)
c D
M ultiplying Eq. (6.65) by W / W and noting that for steady, level flight W = L,
we obtain
rw,o
w
,w„
—----- d W =
J Wl c D W
L dw
I
------------J w¡ c D W
R = /
( 6 .66)
U nlike Eq. (6.63), w hich is exact, Eq. ( 6 .66) now contains the direct assum ption
o f level, unaccelerated flight. However, for practical use, it will be further
sim plified by assum ing that rj, L / D = C L/ C D, and c are constant throughout the
flight. This is a reasonable approxim ation for cruising flight conditions. Thus,
Eq. ( 6 .66) becom es
Wo d W
c Co J w!
nr
(6.67)
Equation (6.67) is a classic form ula in aeronautical engineering; it is called the
Breguet range formula, and it gives a quick, practical estim ate for range, which
is generally accurate to within 10 to 20 percent. Also, keep in mind that, as with
all proper physical derivations, Eq. (6.67) deals with consistent units. Hence, R
is in feet or m eters when c is in consum ption o f fuel in lb/(ft ■lb/s)(s) or
N /(J/s)(s), respectively, as discussed in Sec. 6 .12.2. If c is given in term s o f brake
horsepow er and if R is desired in m iles, the proper conversions to consistent
units should be m ade before using Eq. (6.67).
Look at Eq. (6.67). It says all the things that com m on sense would expect;
namely, to m axim ize range for a reciprocating-engine, propeller-driven airplane,
we w ant the follow ing:
1.
The largest possible propeller efficiency r¡.
2.
3.
The low est possible specific fuel consum ption c.
The highest ratio o f Wo/ W\ , which is obtained with the largest fuel
w eight W/r.
441
442
chapter
4.
6
Elements of Airplane Performance
M ost im portantly, flight at m axim um L / D . This confirm s our argum ent
in Sec. 6.12.1 that for maximum range, we m ust fly at m axim um L / D .
Indeed, the B reguet range form ula shows that range is directly proportional
to L / D . This clearly explains why high values o f L / D (high aerodynam ic
efficiency) have alw ays been o f im portance in the design o f airplanes. This
im portance was underscored in the 1970s due to the increasing aw areness
o f the need to conserve energy (hence fuel).
A sim ilar form ula can be obtained for endurance. Ifw e recall that P = D V 00/r]
and that W = L, Eq. (6.60) becom es
E _
r Wod \V _
r Wo TJ d W
Jw cP Jw,
,
c D \ oo
_
í w" rj
/w,
L
c DVoo
dW
W
Since L = W = { p ^ V ^ S C l , then K* = J 2 W / ( p ^ S C L) . Thus,
=
PooS C l d W
f W° ! l £ l
l W]1 C
fl T
Í.
W 3/2
Jw
c tCo
A ssum ing that C l , C o , r¡, c, and a » (constant altitude) are all constant, this
equation becom es
or
n C V2
E = - - ^ - ( 2 p o o S)'/2( w ; l/2 - W ~ 1'2)
( 6 .68 )
C C/j
Equation ( 6 .68) is the Breguet endurance formula, where E is in seconds (con­
sistent units).
L ook at Eq. ( 6 .68). It says that to m axim ize endurance for a reciprocatingengine, propeller-driven airplane, we want
1.
The highest propeller efficiency rj.
2.
The low est specific fuel consum ption c.
3.
The highest fuel w eight Wf, where W0 = Wi + Wf .
4.
Flight at m axim um C l3/2/ C d ■This confirm s our argum ent in Sec. 6.12.1
that for maximum endurance, we m ust fly at m axim um C¿/ 2/ C o -
5.
Flight at sea level, because E a p'J2, and p x is largest at sea level.
It is interesting to note that, subject to our approxim ations, endurance depends on
altitude, w hereas range [see Eq. (6.67)] is independent o f altitude.
R em em ber that the discussion in this section pertains only to a com bination
o f piston engine and propeller. For a jet-pow ered airplane, the picture changes,
as discussed in Sec. 6.13.
6 .1 2
Range and Endurance— Propeller-Driven Airplane
443
EXAM PLE 6.12
Estimate the maximum range and maximum endurance for the CP-1.
■ Solution
The Breguet range formula is given by Eq. (6.67) for a propeller-driven airplane. This
equation is
„
n CL
Wo
R = ------ In —
c CD Wi
with the specific fuel consumption c in consistent units, say (lb fuel)/(ft • lb/s)(s) or sim­
ply per foot. However, in Example 6.1, the SFC is given as 0.45 lb of fuel/(hp)(h). This
can be changed to consistent units, as
lb
l hp
1h
c = 0.45 = 2.27 x 10~7 ft"
(hp)(h) 550 ft • lb/s 3600 s
In Example 6.1, the variation of C l / C d = L / D was calculated versus velocity. The vari­
ation of c ] /2/ Cn can be obtained in the same fashion. The results are plotted in Fig. 6.44.
K , ft/s
Figure 6.44 Aerodynam ic ratios for the CP-1 at sea level.
chapter
6
Elements of Airplane Performance
From these curves,
max
= 13.62
max
= 12-81
These are results pertaining to the aerodynamics of the airplane; even though the preced­
ing plots were calculated at sea level (from Example 6.1), the maximum values of C l / C d
3/2
•
i* i •
and C¿ / C d are independent of altitude, velocity, etc. They depend only on the aerody­
namic design of the aircraft.
The gross weight of the CP-1 is W« = 2950 lb. The fuel capacity given in Exam­
ple 6.1 is 65 gal of aviation gasoline, which weighs 5.64 lb/gal. Hence, the weight of the
fuel W,, = 65(5.64) = 367 lb. Thus, the empty weight W\ = 2950 —367 = 2583 lb.
Returning to Eq. (6.67), we have
__
n Cl
Wo
cCD
W\
_
0.8
„„
2.27 x 1 0 ~7
'
2950\
\ n 2583/
/
R = 6.38 x 106 ft
Since 1 mi = 5280 ft,
6.38 x 106
R = -------------5280
1207 mi
The endurance is given by Eq. (6.68):
n r 3/2
E = - - * r - ( 2 PocS)l/2( w ; i/2 - W0“ 1/2)
C CD
Because of the explicit appearance of poo ¡n the endurance equation, maximum en­
durance will occur at sea level,
= 0.002377 slug/ft3. Hence,
E - 5^
7(>2.*l)f2(0.002J77,«174)]'«
E = 5.19 x 104 s
Since 3600 s = 1 h,
5.19 x 104
3600
14.4 h
6.13 RANGE AND ENDURANCE— JET AIRPLANE
For a je t airplane, the specific fuel consum ption is defined as the wight o f fuel
consumed p er unit thrust p e r unit time. Note that thrust is used here, in contradis­
tinction to power, as in the previous case for a reciprocating engine-propeller
6 .1 3
Range and Endurance— Jet Airplane
com bination. T he fuel consum ption o f a je t engine physically depends on the
thrust produced by the engine, w hereas the fuel consum ption o f a reciprocating
engine physically depends on the brake pow er produced. It is this sim ple differ­
ence that leads to different range and endurance form ulas for a je t airplane. In the
literature, thrust-specific fuel consumption (TSFC) for je t engines is com m only
given as
lb o f fuel
TSFC = ---------------------(lb o f thrust)(h)
(N ote the inconsistent unit o f time).
6.13.1 P h y sical C o n sid eratio n s
The m axim um endurance o f a je t airplane occurs for m inim um pounds o f fuel
per hour, the sam e as for propeller-driven aircraft. However, for a jet,
lb o f fuel
----- ------ = (T S F C )(rA)
n
where TA is the thrust available produced by the je t engine. Recall that in steady,
level, unaccelerated flight, the pilot has adjusted the throttle such that thrust
available TA ju st equals the thrust required TR: TA = TR. Therefore, m inim um
pounds o f fuel per hour correspond to m inim um thrust required. H ence, we con­
clude that
Maximum endurance for a jet airplane occurs when the airplane is flying at mini­
mum thrust required.
This condition is sketched in Fig. 6.45. Furtherm ore, in Sec. 6.3, m inim um thrust
required was show n to correspond to m axim um L / D . Thus,
Maximum endurance for a jet airplane occurs when the airplane is flying at a veloc­
ity such that C l / C d is maximum.
Now, consider range. As before, m axim um range occurs for a minimum
num ber o f pounds o f fuel per mile. For a jet, on a dim ensional basis,
lb o f fuel _ (T S F C )(rA)
mi
Voo
Recalling that for steady, level flight, TA = TR, we note that m inim um pounds o f
fuel per m ile correspond to a m inim um TR/ Voq. In turn, TR/ V x is the slope o f a
line through the origin and intersecting the thrust-required curve; its minimum
value occurs when the line becom es tangent to the thrust-required curve, as
sketched in Fig. 6.45. The aerodynam ic condition holding at this tangent point is
obtained as follows. Recall that for steady, level flight, TR = D. Then,
chapter
6
Elements of Airplane Performance
maximum
endurance
maximum
range
Figure 6.45 Points of maximum range and endurance on the thrustrequired curve.
Since Voo = J 2 W / (pooSC l ) , w e have
T« _ 1
i/ —
Voo
2
c
\
2W
c'/~<
V P oo S C L
r
D
u
„
1
C l/ 2/ C D
H ence, m inim um TR/Vo0 corresponds to m axim um C |/ 2/C d . In turn, we con­
clude that
Maximum range for a jet airplane occurs when the airplane is flying at a velocity
such that C ;/“/ C d is maximum.
6.1 3 .2 Q u an titativ e F o rm u lation
Let c, be the thrust-specific fuel consum ption in consistent units, for exam ple,
lb o f fuel
(lb o f thrust)(s)
or
N o f fuel
(N o f thrust)(s)
Let d W be the elem ental change in w eight o f the airplane due to fuel consum p­
tion over a tim e increm ent dt . Then,
6 .1 3
Range and Endurance— Jet Airplane
Integrating Eq. (6.69) betw een / = 0, where W = W(), a n d r = E, where W = Wlt
we obtain
E = - f W
Jw„
<Wa c, T a
(6.70)
Recalling that TA = TR = D and W = L, we have
f•W
Woa l L d W
E = J w. -c,nD~ nWr
' w,
(6 -7 »
With the assum ption o f constant c, and C L/ C o = L / D , Eq. (6 .7 1) becom es
„
I CL
W0
E = --------In —
c, C D
Wi
(6.72)
Note from Eq. (6.72) that for m axim um endurance for a je t airplane, we want
1.
M inim um thrust-specific fuel consum ption c,.
2.
M axim um fuel w eight W j .
3.
Flight at m axim um L / D . This confirm s our argum ent in Sec. 6.13.1 that
for m axim um endurance for a jet, we must fly such that L / D is maxim um .
N ote that, subject to our assum ptions, E for a je t does not depend on
that is,
E is independent o f altitude.
N ow consider range. R eturning to Eq. (6.69) and m ultiplying by Vno. we get
d s = Voodt =
(6.73)
C,Ta
where d s is the increm ent in distance traversed by the je t over the time increm ent
dt. Integrating Eq. (6.73) from s = 0, where W = W(), to s = R, where W = W],
we have
R=
fR ,
ds = JO
f w' VgodW
/
JWo
(6.74)
Ct , A
However, again noting that for steady, level flight, the engine throttle has been ad­
justed such that Ta = TR and recalling from Eq. (6.16) that TR = W / ( C l / C d ),
we rewrite Eq. (6.74) as
f w" Voo C L d W
R = Jw, c t C o VV
(6 -75)
Since Voo = s/ I W / Í P oqS C l ), Eq. (6.75) becom es
R =
r*
Jw,
2
PooS
C¿ / C D dW
c,
W V2
(6.76)
447
448
chapter
6
Elements of Airplane Performance
A gain, assum ing constant cr, C ¿, C o, and Poo (constant altitude), w e rew rite
Eq. (6.76) as
I 2
C [ /2 1 f w° d W
~ y p ¿ s ~ c ¿ c i J Wl W ñ
1/2
R = 2.
Poo S C» C d
W
2- <
'/ 2\
2)
(6.77)
N ote from Eq. (6.77) that to obtain m axim um range for a je t airplane, we want
the following:
1.
M inim um thrust-specific fuel consum ption c,.
2.
M axim um fuel w eight W
3.
Flight at m axim um c )/2/ C d ■This confirm s our argum ent in Sec. 6.13.1
that for m axim um range, a je t m ust fly at a velocity such that C ] ' 2¡ C o is
m axim um .
4.
Flight at high altitudes, that is, low p ^ . O f course, Eq. (6.77) says that
R becom es infinite as Poo decreases to zero, that is, as w e approach outer
space. This is physically ridiculous, however, because an airplane requires
the atm osphere to generate lift and thrust. Long before outer space is
reached, the assum ptions behind Eq. (6.77) break down. M oreover, at
extrem ely high altitudes, ordinary turbojet perform ance deteriorates and
c, begins to increase. All w e can conclude from Eq. (6.77) is that range
for a je t is poorest at sea level and increases with altitude, up to a point.
Typical cruising altitudes for subsonic com m ercial je t transports are
from 30,000 to 40,000 ft; for supersonic transports they are from 50,000
to 60,000 ft.
EXAM PLE 6.1
Estimate the maximum range and endurance for the CJ-I.
■ Solution
1/2
From the calculations of Example 6.1, the variation of C l / C o and CL / C o can be plot­
ted versus velocity, as given in Fig. 6.46. From these curves, for the C J-1,
(€ > =
23.4
max |
| = 16.9
\ C D)
6 .1 3
Range and Endurance— Jet Airplane
V„ , ft/s
Figure 6.46 Aerodynamic ratios for the CJ-l at sea level.
In Example 6 .1, the specific fuel consumption is given as TSFC = 0.6 (lb fuel)/
(lb thrust)(h). In consistent units,
C,
lb
l h
= 0 . 6 — — — — = l . 667 x 10-4 s_l
(lb)(h) 3600 s
A lso, the gross weight is 19,815 lb. The fuel capacity is 1119 gal o f kerosene, where 1 gal
o f kerosene w eighs 6.67 lb. Thus, Wy = 1119(6.67) = 7463 lb. Hence, the empty weight
is IV, = W(, - Wf = 19,815 - 7463 = 12,352 lb.
The range o f a jet depends on altitude, as shown by Eq. (6.77). Assume the cruising
altitude is 22,000 ft, where p00 = 0.00184 slug/ft3. From Eq. (6.77), using information
from Example 6.1, w e obtain
=
2,
Co
* 1
1
8
R = 2. / 2
r l/2
f '/2 _ U/'/2\
CD
0.001184(318) ( l . 6 6 7 x l0 -< ) (2 3 4 K ' 9' 8 ' 5l" ~ l2 ' 352',2)
R = 19.2 x 106 ft
449
450
chapter
6
Elements of Airplane Performance
In miles,
R=
The endurance can be found from Eq. (6.72):
it = ------1
W<
E
Ini —
->
c, CD Wt
1.667 x 10~4
(16.9)
V
12,352 /
E = 4.79 x 104 s
E=
or in hours
6.14 RELATIONS BETWEEN C D 0 AND C D>i
In the previous sections, we have observed that various aspects o f the perfor­
m ance o f different ty[5es o f airplanes depend on the aerodynam ic ratios
C{/2/ C D, C /./C o , or C / 2/ C i j>. M oreover, in Sec. 6.3, w e proved that at m ini­
mum Tr , drag due to lift equals zero-lift drag, that is, Co,o = Co,;. Analogously,
for m inim um PR, we proved in Sec. 6.5 that C o o = j C D<i. 1° this section,
such results are obtained strictly from aerodynam ic considerations. The rela­
tions betw een C D,o and Co,, depend purely on the conditions for m axim um
C [ /2/ C d , C L/ C D, o r c l /2/ C o \ their derivations do not have to be associated
w ith m inim um TR or PR, as they w ere in Secs. 6.3 and 6.5.
F or exam ple, consider m axim um L / D . Recalling that C o = C p ,o +
C l / ( n e A R ), w e can w rite
Cl
CD
(6.78)
C d .o + C 2L/ ( ne MC )
For m axim um C l / C o, differentiate Eq. (6.78) with respect to C¿ and set the re­
sult equal to 0 :
d ( C L/ C D) = C d ,q + C j / i n e A R ) - C J 2 C ¿ / ( ttM R ) ]
[C D.o + C £ / ( t t M R )]2
d °L
Thus,
^C]i
2 C 2,/
;r<?AR
^eA R
CDo H-------------------- — = 0
'
C2
or
C D,o =
(6.79)
H ence, Eq. (6.79), which is identical to Eq. (6.22), sim ply stem s from the fact
that L / D is m axim um . T he fact that it also corresponds to m inim um TR is only
because TR happens to be m inim um when L / D is m axim um .
6 .1 4
Relations Between CD0 and CDI
3/2,
N ow consider m axim um C¿/ 2/C o - By setting d(C'l'¿) / d C i = 0, a derivation sim ilar to that above yields
for
Co, o —
(C)
(6.80)
A gain, Eq. (6.80), w hich is identical to Eq. (6.30), sim ply stem s from the fact
that C ^ " / C o is m axim um . The fact that it also corresponds to m inim um PR is
only because PR happens to be m inim um when c [ /2/ C o is maxim um .
Sim ilarly, w hen c [ /2/ C D is m axim um , setting d ( C [ /2) /dCL = 0 yields
C d .o = 3 Co,,
i c l /2\
for
( 6 .8 1)
\C o )
You should not take Eqs. (6.80) and (6 .8 1) for granted; derive them yourself.
We stated in Exam ple 6 . 12 that the maximum values o f C |/ 2/C o> C L/ C o ,
and C ^ 2/ C o are independent o f altitude, velocity, etc.; rather, they depend only
on the aerodynam ic design o f the aircraft. The results o f this section allow us to
prove this statem ent, as follows.
First, consider again the case o f m axim um C l / C o- From Eq. (6.79),
C d .o = Co,¡ =
Thus,
C 2l
An
neAR
(6.82)
(6.83)
C L = y / n e A R C D,o
Substituting Eqs. (6.82) and (6.83) into Eq. (6.78), we obtain
Cl
Cl
7reAR
¡reA R
Co
2C¿/(7Té'AR)
2CL
l^TTeARCo.o
(6.84)
Hence, the value o f the m axim um C l / C o is obtained from Eq. (6.84) as
(r) max ■
( C p , o 7 T f A R ) l/2
(6.85)
2C d,0
Note from Eq. (6.85) that ( CL/ C D)max depends only on e, AR, and C D,o. which
are aerodynam ic design param eters o f the airplane. In particular, ( C i./C o )max
docs not depend on altitude. However, note from Figs. 6.44 and 6.46 that m axi­
mum C L/ C o occurs at a certain velocity, and the velocity at which ( C ¿ /C o ) max
is obtained does change with altitude.
In the sam e vein, it is easily shown that
( 6 . 86 )
and
Prove this yourself.
(6.87)
451
452
chapter
6
Elements of Airplane Performance
E X A M P L E 6 .1 4
From the equations given in this section, directly calculate ( C l / C o ) max and
-3/2
(C£ / Co)n,,x for the CP-1.
■ Solution
From Eq. (6.85),
(!L -
(C 0 .07rM R )1/2
2 C d ,o
[0.025tt (0.8)(7.37 )]'/2
2(0.025)
13.6
From Eq. (6.87),
(3Co.o7re,AR)3/4
/c f>
I Co
'max
4Co,0
|(3)(0.025) tt(0.8)(7.37)]3/4
4(0.025)
12.8
Return to Example 6.12, where the values of (Ci,/CD)max and (C¿/2/C d)max were ob­
tained graphically, that is, by plotting C l / C d and C\[2/ C o and finding their peak values.
Note that the results obtained from Eqs. (6.85) and (6.87) agree with the graphical values
obtained in Example 6.12 (as they should); however, the use of Eqs. (6.85) and (6.87) is
much easier and quicker than plotting a series of numbers and finding the maximum.
E X A M P L E 6 .1 5
I/2
From the equations given in this section, directly calculate (C¿ /C o ) max and
(Ct / C D)max for the C M .
■ Solution
From Eq. (6.77),
/en
V ) max
(1 C d ,o7tM R ) I/4 _ [^(0.02)7T( 0 .8 1)(8.93)]
1/4
.
23.4
\ Co, o
From Eq. (6.76),
(7reARCD,0)1/2
(\Cr0)/max "
2 Co,o
[tt( 0.8 1)(8.93)(0.02)] 1/2
2 ( 0 .02 )
16.9
These values agree with the graphically obtained máximums in Example 6.13.
E X A M P L E 6 .1 6
Using the result from this section and Eqs. (6.44), (6.52), and (6.53), analytically calculate
a. V,nax for the CP-1 at sea level.
b. (R/C)max for the CP-1 at sea level.
c. Vinax for the CJ-1 at sea level.
d. (R/C)max for the CJ-1 at sea level.
Compare with the graphical solutions obtained in Examples 6.2, 6.4, and 6.6.
6.1 4
Relations Between CD0 and CDI
■ Solution
a. The maximum velocity is given by Eq. (6.44), repeated here:
1/2
\ W j max\ s )
y
_ 4c ^
\ s j n w j mu
neAR
PooC d.o
For the CP-1, from the data given in Example 6.1,
W
2950
= 16.95 lb/ft
174
r¡P = 0.8(230)(550) = 1.012 x 10;
ft lb
From Eq. (6.85) and the result from Example 6.14,
4Q>,o
1
1
ireAR “ ( ¿ /D ) 2a* _ 13-62
Also,
c
ir>_3
'
X
PocCD,0 = 0.002377(0.025) = 5.9425 x 10~5 slug/ft3
Power available and thrust available are related by
Ta Voo = Pa = r)P
For maximum TA and PA,
= Vmux. Hence,
r,P
(7 /t)m a x =
T)P
i 1 5 ')
I » / .
1.012 x 1 0 s
1
W
/Ta\
or
1
34.305
(E6.1)
Inserting the preceding data into Eq. (6.44), we have
x 10 3
(TA U-).!dX(16.95,
Vmax —
1,
(E6.2)
5.9425 x 10- 5
1/2
Vmax = 5 5 8 .9 7
(I) +J(w)
\
/ m ax
Y \ ”
/ max
453
454
chapter
6
Elements of Airplane Performance
Equations (E6.1) and (E6.2) must be solved for Vmm by trial and error. Assume Vmax,
calculate (TA/ W ) mm from Eq. (E6.1), insert this into Eq. (E6.2), calculate Vmax from
Eq. (E6.2), and see if this matches the originally assumed Vmax. If not, assume another
value of Vmax, and try again. A few iterations are tabulated in the following.
(assumed)
Vmax
265
270
269
(n \
V W / max
[from Eq. (E6.1)]
Vmax (ft/*)
[from Eq. (E6.2)]
0.1295
0.12706
0.1275
271.6
268.5
269.1
From this, we have calculated for the CP-1 at sea level,
Vmax — 269 ft/s
This is to be compared with Vmax = 265 ft/s as obtained from the graphical solution in
Example 6.4, which is limited by “graphical accuracy.” The analytical solution of Vmax =
269 ft/s obtained here is inherently more accurate.
b. The maximum rate of climb for a propeller-driven airplane is given by Eq. (6.53),
repeated here:
(R/C)„
= (£ )-0-8776
w/s
1
PooCd.o (L / D ) 3/2
max
We have already obtained the following data:
1.012 x 105
\ w ; max
2950
PoqCo,o = 5.9425 x 10
= 34.305 ft/s
slug/ft
W
— = 16.95 lb/ft2
S
( - )
\ D ) max
= 1 3 .6
Hence, Eq. (6.53) becomes
(R/C)max = 34.305 - 0.8776
or
16.95
5.9425
(R/C)max = 34.305 - 9.345 = 24.96 ft/s
1
x 10- 5 (13.6)3/2
6 .1 4
Thus,
(R/C)max = 24.96(60)
Relations Between CD0 and CD,
1497.6 ft/min
This is to be compared with (R/C)max = 1500 ft/min as read from the peak of the graph in
Fig. 6.22 from Example 6.6.
c . From the data given about the CJ-l in Example 6.1, we have
= ^
/M
U Lmax
= 03 68 4
19.815
W
19,815
= 62.31 lb/ft
~S ~ 318
PooCD,o = 0.002377(0.02) = 4.754 x 10“ 5 slug/ft3
Also, from Example 6.15, we have
1
4C0.0
1
= 3.501 x 10-3
ttM R ~ ( L / Ü ) f ~ ~ (16.9)2
Substituting these data in Eq. (6.44), we obtain
1/2
(SLGMtm SL-
4 CDl,0
■Re AR
PooCn.o
0.3684(62.31) -I- 62.31 v/(0.3684)2 —3.501 x 10“ 3
4.754 x 1()~5
Vmax=
1/2
979.5 ft/s
This is to be compared with Vmux = 975 ft/s obtained by graphical means in Example 6.2.
d. The maximum rate of climb for a jet airplane is given by Eq. (6.52), repeated here:
(R/C)max =
where
r ( w/ s ) z i 1/2 ( T \ 3/2
L PooC /; () _
[l
v w j max L
Z = 1 + 11 +
Z
6
3
(L / D ) l n ( T / W ) i e
Putting in the data for the C J-1, w e have
Z - 1 + . / 1 +
( I 6 .9 ) 2(0 .3 6 8 4 )2
1
2 ( T / W ) l m( L / D ) l mZ \
= 2 .0 3 8
455
456
c hapter
6
Elements of Airplane Performance
DESIGN BOX
The ratio of lift to drag is a direct measure of the
aerodynamic efficiency of a given airplane. For ex­
ample, if for a given airplane ( L / D )max = 15, this
means that the airplane can lift 15 lb of weight at a
cost of only 1 lb of drag—quite a leverage. Indeed,
for atmospheric flight, the wing of an airplane (usu­
ally its strongest lifting component by far) can be
loosely likened to a lever that allows us to lift far
more weight than we have to expend in thrust from
the engine (to counterbalance the drag). The evolu­
tion of the airplane in the 20th century has been char­
acterized by a steady increase in (L / D )mm; this evo­
lution is discussed at length in Sec. 6.24. Some values
of ( L / D)max for typical airplanes are tabulated here.
Airplane
Wright Flyer (1903)
French SPAD XIII (World War I)
Douglas DC-3 (1930s)
Boeing 747 (contemporary)
(L/D)mm
5.7
7.4
14.7
20
The importance of ( L / D )max as a parameter in
airplane design cannot be overstated—it is one of the
driving aspects that dictate the configuration of the
airplane. Airplane designers usually try to squeeze as
much ( L / D ) mm into a new airplane as they can, sub­
ject to compromises with other aspects of the design.
We have already seen that ( L/ D) max plays a role in
dictating Kn,.,*. (R/C)mux, and especially range and
endurance. Indeed, historically the quest for greater
range has been the primary factor that has driven
up the design value of ( L/ D) mM. (See Anderson,
A History of Aerodynamics and Its Impact on Fly­
ing Machines, Cambridge University Press, New
York, 1997.)
Strictly speaking, we have seen in Secs. 6.12 and
6.13 that the value of ( L / D ) mM = (CL/ C D)nm dic­
tates maximum range for a propeller-driven airplane
and maximum endurance for a jet airplane, whereas
(C¿ / C o ) max dictates maximum endurance for a
propeller-driven airplane and (C¡/2/ Cp)mia dictates
maximum range for a jet airplane. However, the geo­
metric and aerodynamic features of an airplane that
maximize C l / C o will also maximize C '/ / C o and
C]12/C[ j, as seen in Eqs. (6.85) through (6.87). To
obtain maximum values of these aerodynamic ratios,
Eqs. (6.85) through (6.87) clearly indicate that the
airplane designer should, as much as possible,
1.
2.
3.
Reduce the zero-lift drag coefficient Co,oIncrease the Oswald efficiency factor e.
Increase the aspect ratio AR.
Of course, this last point—increasing the aspect
ratio— makes sense only for subsonic flight. We have
discussed previously that for transonic and super­
sonic airplanes, wave drag is dominant, and wave
drag can be somewhat reduced by using low-aspect
ratio wings. For high-speed airplanes designed for
cruise at supersonic speeds, the design wing aspect
ratio is driven by considerations other than those for
maximum range in subsonic flight. The low-aspect
ratio, Mach 2, Lockheed F-104 shown in Fig. 4.45 is
a case in point.
The value of (L / D ) mm is fixed by the aerody­
namics and geometry of the given airplane configura­
tion via Co,o, e and AR. Hence, ( L / D ) mm does not
change with altitude. However, the velocity at which
the airplane must fly to achieve ( L / D ) mm does vary
with altitude. To explain why this is so, first recall
that L / D is a function of the airplane’s angle of at­
tack. For example, the variation of L / D versus a for
the special-purpose F-111 TACT aircraft (illustrated
in Fig. 5.31) is shown in Fig. 6.47. Note that
( L / D ) mm occurs at an angle of attack of 6°; at this
angle of attack, C l = 0.44. If the airplane is flying at
sea level, in order to fly at ( L / D ) mm, it must be fly­
ing at a = 6° with CL = 0.44. For the given weight,
this condition fixes the velocity at which the airplane
must fly via the relation W = q^SCL, or
=
(<" 88)
6 .1 4
-j
o
Angle of attack a (deg)
Figure 6.47 Flight data for lift-to-drag ratio versus
angle of attack for the F-111 TACT airplane shown
in Fig. 5.31.
= 0.7. Wing sweep angle = 26 .
(Source: Data from Baldwin el al„ Symposium on Transonic
Aircraft Technology (TACT), Air Force Flight Dynamics
Laboratory Technical Report AFFDL-TR-78-100, WrightPatterson A ir Force Base, Ohio, 1978.')
To fly at (L / D ) mm at higher altitude, the airplane
must still fly at a = 6° with C l = 0.44. However,
since Poo has decreased,
must be larger, as given
by Eq. (6.88). That is, Vao must be increased to just
the right value so that the lift remains equal to the
weight for the fixed C l at a = 6°. As a result, the ve­
locity required to fly at (L / D ) mM increases with an
increase in altitude.
Relations Between CD0 and CDi
457
Although the value of (L / D ) nm is very impor­
tant in airplane design, flight at ( L/ D) mm is not
always the holy grail of aeronautical engineering
that it may seem. As usual, the airplane designer is
faced with a compromise, this time involved with
Vmax relative to the velocity for (L /D )max. The veloc­
ity for (L /D )max can be substantially smaller than
the maximum velocity. For example, from Fig. 6.46,
the velocity at sea level for (Z-/D)max for the CJ-1 is
about 300 ft/s, whereas from Fig. 6.26, Vmax =
975 ft/s—a considerable difference. For the CP-1 at
sea level, from Fig. 6.44 the velocity for (L / D ) mm is
about 150 ft/s, whereas from Fig. 6.21 a, Vmax =
265 ft/s at sea level. If the pilot of the CP-1 chooses
to fly very efficiently by flying at Vx = 150 ft/s so
that L / D is at its maximum value, then the flight will
take almost 75 percent longer to go from point A to B
compared to flying at \/rn.,x. Since time is valuable
(indeed, the reason why most passengers fly is to save
time), the design cruise speed for a given airplane
may not correspond to ( L/ D) max. The airplane de­
signer must be ready to accept a higher-speed cruise
with an L / D that is less than the value of (L /D )max.
However, this does not diminish the importance of
(L /D )max as a design parameter. For example, an air­
plane with a high value of ( L / D ) mM will still have
comparatively high values of L / D while flying at ve­
locities other than that for ( L/ D) nm. Also, Bernard
Carson, a professor of aerospace engineering at the
U.S. Naval Academy, has suggested a rational
compromise that combines the concept of long range
obtained by flying at the slower velocity for
(L/D)„m and the shorter flight times obtained by
flying at higher speeds. His analysis leads to an opti­
mum compromise for flight velocity called the
Carson speed, which can be shown to be a factor of
1.32 higher than the velocity for (L / D ) mm. The de­
tails can be found in Anderson, Aircraft Performance
and Design, McGraw-Hill, New York, 1999.
458
c hapter
6
Elements of Airplane Performance
and
(R/C),
' (62.31X2.038) 1 1/2
~ [3(4.754 x 10-5).
,,, r
2.038
(0.3684) ^ 1 ------ —
6
3
2(0.3684)2(16.9)2(2.038)
= 135.28 ft/s
or
(R/C)max = 135.28(60) =
8117 ft/min
This is to be compared with (R/C)max = 8100 ft/min read from the peak of the graph in
Fig. 6.33 from Example 6.6.
6.15 TAKEOFF PERFORMANCE
Up to this point in our discussion o f airplane perform ance, we have assum ed that
all accelerations are zero; that is, we have dealt with aspects o f static perfor­
m ance as defined in Sec. 6.2. For the rem ainder o f this chapter, we relax this re­
striction and consider several aspects o f airplane perform ance that involve finite
acceleration, such as takeoff and landing runs, turning flight, and accelerated rate
o f clim b. W ith this, we m ove to the right-hand colum n on our chapter road map,
shown in Fig. 6.4. We now take up the study o f dynam ic perform ance.
To begin, we ask the question, W hat is the running length along the ground
required by an airplane, starting from zero velocity, to gain flight speed and lift
from the ground? This length is defined as the ground roll, or liftoff distance, .s' l o To address this question, let us first consider the accelerated rectilinear m o­
tion o f a body o f m ass m experiencing a constant force F, as sketched in
Fig. 6.48. From N ew ton’s second law,
F = ma = m ——
dt
or
F
d V = — dt
m
(6.89)
A ssum e that the body starts from rest (V = 0) at location 5 = 0 at tim e t = 0 and
is accelerated to velocity V over distance s at tim e t. Integrating Eq. (6.89) be­
tw een these tw o points and rem em bering that both F and m are constant, we have
or
m
(6.90)
6 .1 5
Body at time t
Body at time t = 0
I----I
I
Takeoff Performance
F = constant
F = constant
V=0
I_______
s
s=0
Figure 6.48 Sketch of a body moving under the influence o f a constant force F, starting
from rest (V = 0) at j = 0 and accelerating to velocity V at distance s.
Solving for t, we get
Vm
(6 .9 1)
C onsidering an instant when the velocity is V, the increm ental distance d s cov­
ered during an increm ental tim e d t is d s = V d t . From Eq. (6.90), we have
ds = V dt = —t dt
m
(6.92)
Integrating Eq. (6.92) gives
t dt
Jo
or
m Jo
_ F t2
m 2
(6.93)
Substituting Eq. (6 .9 1) into (6.93), we obtain
V 2m
~2F
(6.94)
Equation (6.94) gives the distance required for a body o f m ass m to accelerate to
velocity V under the action o f a constant force F.
Now consider the force diagram for an airplane during its ground roll, as il­
lustrated in Fig. 6.49. In addition to the fam iliar forces o f lift, drag, thrust, and
weight, the airplane experiences a resistance force R due to rolling friction be­
tween the tires and the ground. This resistance force is given by
R = fir ( W - L)
(6.95)
where W — L is the net normal force exerted between the tires and the ground
and ¡i, is the coefficient o f rolling friction. Sum m ing forces parallel to the
ground and em ploying N ew ton’s second law, we have
dV
F = T - D - R = T - D - ixr( W - L) = m —
(6.96)
dt
459
460
chapter
6
Elements of Airplane Performance
Figure 6.49 Forces acting on an airplane during takeoff and landing.
L et us exam ine Eq. (6.96) m ore closely. It gives the local instantaneous ac­
celeration o f the airplane d V / d t as a function o f T, D, W, and L. For takeoff,
over m ost o f the ground roll, T is reasonably constant (this is particularly true for
a jet-pow ered airplane). A lso, W is constant. However, both L and D vary with
velocity, since
L = \ Poov l SCL
and
D = - PooV^S
{c°°+<’S r)
(6.97)
(6'98)
The quantity 0 in Eq. (6.98) requires som e explanation. W hen an airplane is fly­
ing close to the ground, the strength o f the wing tip vortices is som ew hat dim in­
ished because o f interaction with the ground. Since these tip vortices induce
dow nw ash at the wing (see Sec. 5.13), w hich, in turn, generates induced drag
(see Sec. 5.14), the dow nw ash and hence induced drag are reduced w hen the air­
plane is flying close to the ground. This phenom enon is called ground effect and
is the cause o f the tendency for an airplane to flare, or “float,” above the ground
near the instant o f landing. T he reduced drag in the presence o f ground effect is
accounted for by 0 in Eq. (6.98), w here </> < 1. An approxim ate expression for </>,
based on aerodynam ic theory, is given by M cCorm ick (see Bibliography at the
end o f this chapter) as
(\6h/b)2
<P = 1 + ( \ 6 h / b ) 2
(6.99)
w here h is the height o f the w ing above the ground and b is the wingspan.
In light o f the preceding, to accurately calculate the variation o f velocity
with tim e during the ground roll, and ultim ately the distance required for liftoff,
Eq. (6.96) m ust be integrated numerically, taking into account the proper veloc­
ity variations o f L and D from Eqs. (6.97) and (6.98), respectively, as well as any
6 .1 5
Takeoff Performance
Figure 6.50 Schematic of a typical variation of forces acting on an
airplane during takeoff.
velocity effect on T. A typical variation o f these forces with distance along the
ground during takeoff is sketched in Fig. 6.50. Note from Eq. (6.94) that s is pro­
portional to V 2, and hence the horizontal axis in Fig. 6.50 could ju st as well be
V2. Since both D and L are proportional to the dynam ic pressure
= jPoo
they appear as linear variations in Fig. 6.50. Also, Fig. 6.50 is draw n for a jetpropelled airplane; hence, T is relatively constant.
A sim ple but approxim ate expression for the liftoff distance $lo can be ob­
tained as follow s. A ssum e that T is constant. Also, assum e an average value for
the sum o f drag and resistance forces, [D + /¿r (W - ¿ ) ] av such that this average
value, taken as a constant force, produces the proper liftoff distance ,vLO- Then we
consider an effective constant force acting on the airplane during its takeoff
ground roll as
Fefr = T — [D + / i r (VV — ¿ ) ] av = const
( 6 . 100)
T hese assum ptions are fairly reasonable, as seen from Fig. 6.50. Note that the sum
o f D + fir{ W — L) versus distance (or V 2) is reasonably constant, as shown by
the dashed line in Fig. 6.50. H ence, the accelerating force T — [D + n , { W — L )],
which is illustrated by the difference betw een the thrust curve and the dashed line
in Fig. 6.50, is also reasonably constant. N ow return to Eq. (6.94). Considering F
given by Eq. ( 6 . 100), V — Vlo (the liftoff velocity), and m = W/ g , where g is the
461
462
chapter
6
Elements of Airplane Performance
acceleration o f gravity, Eq. (6.94) yields
(v¿>)(W 7 g)
SLO — T?—---- PT—------— -----TTZ “
2 [ T - \ D + n r ( W - L )]av}
( 6 . 10 1 )
To ensure a m argin o f safety during takeoff, the liftoff velocity is typically
20 percent higher than the stalling velocity. H ence, from Eq. (5.71), we have
/
2W
Vio = l-2Vsta„ = 1.2 / ----— -----
V Poo ''- 'L , max
(6.102)
Substituting Eq. (6.102) into (6.101), we obtain
(6.103)
To m ake a calculation using Eq. (6.103), Shevell (see Bibliography at the end o f
this chapter) suggests that the average force in Eq. (6.103) be set equal to its in­
stantaneous value at a velocity equal to 0.7 Vlo ; that is,
[D
+ Hr( W — L )]av = [D
+
flr( W
— Z ,)]0.7Vlo
Also, experience has shown that the coefficient o f rolling friction /i r in Eq. (6.103)
varies from 0.02 for a relatively sm ooth, paved surface to 0.10 for a grass field.
A further sim plification can be obtained by assum ing that thrust is much
larger than either D or R during takeoff. Refer to the case shown in Fig. 6.50; this
sim plification is not unreasonable. H ence, ignoring D and R com pared to T,
Eq. (6.103) becom es sim ply
1.44 W 2
*lo = ------ ^ -------(6.104)
8P oo^^
L,max*
Equation (6.104) illustrates som e im portant physical trends:
1.
L iftoff distance is very sensitive to the w eight of the airplane, varying directly
as W 2. If the w eight is doubled, the ground roll o f the airplane is quadrupled.
2.
L iftoff distance is dependent on the am bient density p,*,. If we assum e that
thrust is directly proportional to p,*,, as stated in Sec. 6.7, that is, T oc. p ^ ,
then Eq. (6.104) dem onstrates that
1
slo
oc
—
Plo
This is why on hot sum m er days, when the air density is less than that on
cooler days, a given airplane requires a longer ground roll to get off the
ground. Also, longer liftoff distances are required at airports that are located
at higher altitudes (such as at Denver, C olorado, a m ile above sea level).
3.
The liftoff distance can be decreased by increasing the wing area, increasing
C l ,max* and increasing the thrust, all o f w hich sim ply m ake com m on sense.
The total takeoff distance, as defined in the Federal Aviation Requirem ents
(FAR), is the sum o f the ground roll distance sLo and the distance (m easured
along the ground) to clear a 35-ft height (for jet-pow ered civilian transports) or a
6 .1 5
Takeoff Performance
463
50-ft height (for all other airplanes). A discussion o f these requirem ents, as well
as m ore details regarding the total takeoff distance, can be found in Anderson,
Aircraft Performance and Design, M cG raw -H ill, New York, 1999. A lso see the
books by Shevell and by M cC orm ick listed in the Bibliography at the end o f this
chapter for m ore inform ation on this topic.
EXAM PLE 6.17
Estimate the liftoff distance for the CJ-1 at sea level. Assume a paved runway; hence,
ixr = 0.02. Also, during the ground roll, the angle of attack of the airplane is restricted by
the requirement that the tail not drag the ground; therefore, assume that CL>mm during
ground roll is limited to 1.0. Also, when the airplane is on the ground, the wings are 6 ft
above the ground.
■ Solution
Use Eq. (6.103). To evaluate the average force in Eq. (6.103), first obtain the ground
effect factor from Eq. (6.99), where h/b = 6/53.3 = 0.113.
0=
(I6W
1 + (\6h/b)2
= 0 ,7 6 4
Also, from Eq. (6.102),
VLO = ‘'2V/s,a" = * / Ax.SC,..™ = 1>2/ 0 . 0 0 2 3 7 7 ( 3 1 8 ^ = 230 ^
Hence, 0.7Vlo = 160.3 ft/s. The average force in Eq. (6.103) should be evaluated at a
velocity of 160.3 ft/s. To do this, from Eq. (6.97) we get
L = { pooV^SC l = ^(0.002377)( 160.3)2(3 18)(1.0) = 9712 lb
Equation (6.98) yields
o = j ^
( c
0, + ^
)
= 4 (0.002377)( 160.3)2(3 18) i().02 + 0.764-------— — - 1 = 520.7 lb
2
L
7T(0.81)(8.93) J
Finally, from Eq. (6.103),
■'LO =
1.44 VV2
gPocSCL,imx(T ~ [ D + HAW - L)]av}
1.44(19,815)2
32.2(0.002377)(318)(1.0)(7300- 1520.7 + (0.02)( 19,815 -9 7 1 2 )])
3532 ft
Note that \D + fxr( W — ¿ )]av — 722.8 lb, which is about 10 percent of the thrust. Hence,
the assumption leading to Eq. (6.104) is fairly reasonable, that is, that I) and R can some­
times be ignored compared with T.
464
ch apter
6
Elements of Airplane Performance
6.16 LANDING PERFORMANCE
C onsider an airplane during landing. A fter the airplane has touched the ground,
the force diagram during the ground roll is exactly the sam e as that given in
Fig. 6.49, and the instantaneous acceleration (negative in this case) is given by
Eq. (6.96). H ow ever, we assum e that in order to m inim ize the distance required
to com e to a com plete stop, the pilot has decreased the thrust to zero at touch­
dow n, and therefore, the equation o f motion for the landing ground roll is ob­
tained from Eq. (6.96) with T = 0.
dV
- D - /j.r( W - L) = m —
dt
(6.105)
A typical variation o f the forces on the airplane during landing is sketched in
Fig. 6.51. D esignate the ground roll distance betw een touchdow n at velocity VT
and a com plete stop by
An accurate calculation o f s L can be obtained by
num erically integrating Eq. (6.105) along with Eqs. (6.97) and (6.98).
However, let us develop an approxim ate expression for
that parallels
the philosophy used in Sec. 6.15. A ssum e an average constant value for
D + [ir( W - L) that effectively yields the correct ground roll distance at landing
Figure 6.51 Schematic o f a typical variation o f forces acting on an airplane during landing.
6 .1 6
Landing Performance
s L. O nce again, we can assum e that [D + /xr(W - L )]av is equal to its instanta­
neous value evaluated at 0.7 VT.
F = - [ D + i i r{W - L ) ]„ = —[D + fir(W - L )]o.7Vr
(6.106)
[Note from Fig. 6 .5 1 that the net decelerating force D + n r(W — L) can vary
considerably with distance, as shown by the dashed line. Hence, our assum ption
here for landing is m ore tenuous than for takeoff.] R eturning to Eq. (6.92), we in­
tegrate betw een the touchdow n point, w here s = s L and t = 0, and the point
w here the airplane’s m otion stops, w here s = 0 and tim e equals t.
f
JSL
f
ds = tdt
m Jo
Ft2
or
s L = ----- —
m 2
(6 .107)
Note that, from Eq. (6.106), F is a negative value; hence, s L in Eq. (6.107) is
positive.
Com bining Eqs. (6 .9 1) and ( 6 .107), w e obtain
V 2m
sL = - - ? = 2F
(6.108)
Equation (6.108) gives the distance required to decelerate from an initial veloc­
ity V to zero velocity under the action o f a constant force F. In Eq. (6.108), F is
given by Eq. (6.106), and V is Vf. Thus, Eq. (6.108) becom es
V 2 ( W /g )
s L = ------------ T
---------2[D + fir (W — L)]o.7vr
(6.109)
To m aintain a factor o f safety,
VT
= 1 - 3 V , tou =
/
2W
1 . 3 / -------— --------
(6 .1 1 0 )
V PooJt-L.max.
Substituting Eq. (6.110) into (6.109), we obtain
_____________1.69 W 2____________
g P o o S C L, max [D + i i r( W - L ) lo .7 VT
During the landing ground roll, the pilot is applying brakes; hence, in
Eq. (6.111), the coefficient o f rolling friction is that during braking, w hich is ap­
proxim ately [ir = 0.4 for a paved surface.
M odern je t transports utilize thrust reversal during the landing ground roll.
Thrust reversal is created by ducting air from the jet engines and blow ing it in the
upstream direction, opposite to the usual dow nstream direction when normal
thrust is produced. As a result, with thrust reversal, the thrust vector in Fig. 6.49
is reversed and points in the drag direction, thus aiding the deceleration and
465
466
chapter
6
Elements of Airplane Performance
shortening the ground roll. D esignating the reversed thrust by TK, we see that
Eq. (6.105) becom es
dV
- T r - D - fir(W - L) = m —
(6.112)
at
A ssum ing that TH is constant, Eq. (6.111) becom es
1.69W 2
gPocSCL,maATR + [D + i i r{W — £)]o.7vr }
A nother ploy to shorten the ground roll is to decrease the lift to near zero,
hence im pose the full w eight o f the airplane betw een the tires and the ground and
increase the resistance force due to friction. The lift on an airplane wing can be
destroyed by spoilers, which are sim ply long, narrow surfaces along the span o f
the w ing, deflected directly into the flow, thus causing m assive flow separation
and a striking decrease in lift.
The total landing distance, as defined in FAR, is the sum o f the ground roll
distance and the distance (m easured along the ground) to achieve touchdow n in
a glide from a 50-ft height. Such details are beyond the scope o f this book; see
the books by Shevell and by M cCorm ick (listed in the Bibliography at the end of
this chapter) and by A nderson, Aircraft Performance and Design, M cG raw -H ill,
N ew York, 1999, for m ore inform ation.
EXAM PLE 6 .IS
Estimate the landing ground roll distance at sea level for the CJ-1. No thrust reversal is
used; however, spoilers are employed such that L = 0. The spoilers increase the zero-lift,
drag coefficient by 10 percent. The fuel tanks are essentially empty, so neglect the weight
of any fuel carried by the airplane. The maximum lift coefficient, with flaps fully em­
ployed at touchdown, is 2.5.
■ Solution
The empty weight of the CJ-1 is 12,352 lb. Hence,
* = , . 3 ^ . = 1.3
= ,.3 /
2(12’352)
= 148.6 ft/s
T
*' PocSC l, ^
V 0 002377(318)(2-5)
Thus, 0.7VV = 104 ft/s. Also, CD,0 = 0.02 + 0.1(0.02) = 0.022. From Eq. (6.98), with
C l = 0 (remember, spoilers are employed, destroying the lift),
D = { pooV^SC d.o = | (0.002377)(104)2(318)(0.022) = 89.9 lb
From Eq. (6.111), with L = 0,
Sl =
1.69W2
S P o q S C l .max
(D + p.r W)ojvT
1.69(12,352)2
3 2 .2 (0 .0 0 2 3 7 7 )(3 18 )(2 .5 )[8 9 .9 + 0.4(12,352)1
842 ft
6 .1 7
Turning Right and the V-n Diagram
6.17 TURNING FLIGHT AND THE V-n DIAGRAM
Up to this point in our discussion o f airplane perform ance, we have considered
rectilinear m otion. O ur static perform ance analyses dealt with zero acceleration
leading to constant velocity along straight-line paths. O ur discussion o f takeoff
and landing perform ance involved rectilinear acceleration, also leading to m o­
tion along a straight-line path. Let us now consider som e cases involving radial
acceleration, which leads to curved flight paths; that is, let us consider the turn­
ing flight o f an airplane. In particular, we exam ine three specialized cases; a level
turn, a pull-up, and a pull-dow n. A study o f the generalized motion o f an airplane
along a three-dim ensional flight path is beyond the scope o f this book.
A level turn is illustrated in Fig. 6.52. Here, the wings o f the airplane are
banked through angle 0 ; hence, the lift vector is inclined at angle (¡> to the verti­
cal. The bank angle 0 and the lift L are such that the com ponent o f the lift in the
vertical direction exactly equals the weight:
L cos <p = W
and, therefore, the airplane m aintains a constant altitude, m oving in the same
horizontal plane. However, the resultant o f L and W leads to a resultant force
Fr , w hich acts in the horizontal plane. This resultant force is perpendicular to
the flight path, causing the airplane to turn in a circular path with a radius of
curvature equal to R. We w ish to study this turn radius R as well as the turn
rate d O / d t .
From the force diagram in Fig. 6.52, the m agnitude o f the resultant force is
Fr = v /L 2 - W 2
(6.114)
We introduce a new term, the loadfactor-n, defined as
w
(6.115)
The load factor is usually quoted in term s o f “g ’s ” ; for exam ple, an airplane with
lift equal to 5 tim es the w eight is said to be experiencing a load factor o f 5 # ’s.
H ence, Eq. (6 .114) can be w ritten as
Fr = W ^ n 2 - 1
(6.116)
The airplane is m oving in a circular path at velocity Voo. Therefore, the radial ac­
celeration is given by V ^ /R . From N ew ton’s second law,
V2
W V2
Fr = m - f = ------f
R
g R
(6.117)
Com bining Eqs. ( 6 .116) and (6.117) and solving for R, we have
(6.118)
467
468
CH APTER 6
Elements of Airplane Performance
Figure 6.52 An airplane in a level turn.
The angular velocity, denoted by co = d 9 / d t , is called the turn rate and is given
by Voo/R. Thus, from Eq. (6.118), we have
g y/n 2 - 1
a> = -----—------
(6.119)
For the m aneuvering perform ance o f an airplane, both m ilitary and civil, it is fre­
quently advantageous to have the sm allest possible R and the largest possible a>.
6 .1 7
Turning Flight and the V-n Diagram
Equations ( 6 .118) and ( 6 .119) show that to obtain both a small turn radius and a
large turn rate, we w ant
1.
The highest possible load factor (i.e., the highest possible L / W ) .
2.
The low est possible velocity.
C onsider another case o f turning flight, in w hich an airplane initially in
straight, level flight (w here L = W) suddenly experiences an increase in lift.
Since L > W , the airplane will begin to turn upward, as sketched in Fig. 6.53.
F or this pull-up m aneuver, the flight path becom es curved in the vertical plane,
with a turn rate co = d 8 / d t . From the force diagram in Fig. 6.53, the resultant
force Fr is vertical and is given by
Fr = L - W
= W ( n - 1)
(6.120)
From N ew ton’s second law,
( 6 . 121)
C om bining Eqs. (6.120) and (6.121) and solving fo r/? give
R =
g(n - 1)
w
Figure 6.53 The pull-up maneuver.
(6. 122)
469
C H APTER 6
Elements of Airplane Performance
and since co = V ^ /R ,
g(n - 1)
CO =
(6.123)
A related case is the pull-dow n m aneuver, illustrated in Fig. 6.54. H ere an
airplane in initially level flight suddenly rolls to an inverted position, such that
both L and W are pointing dow nw ard. The airplane will begin to turn dow nw ard
in a circular flight path with a turn radius R and turn rate co = d 6 / d t . By an
analysis sim ilar to those preceding, the follow ing results are easily obtained:
V2
R = ------2 ° _
g (« + D
« (« + !)
w = — --------
(6.124)
(6.125)
MX)
Prove this to yourself.
C onsiderations o f turn radius and turn rate are particularly im portant to
m ilitary fighter aircraft; everything else being equal, those airplanes with the
sm allest R and largest u> will have definite advantages in air com bat. Highperform ance fighter aircraft are designed to operate at high load factors, typically
from 3 to 10. W hen n is large, then n + 1
n and n — 1 «s n ; for such cases,
Eqs. (6 .1 18), (6.119), and ( 6 .122) to (6.125) reduce to
Vl
6 .1 7
Turning Flight and the V-n Diagram
Let us w ork with these equations further. Since
L = [ PooV l S C L
2L
1'“ = ¿ ? C Z
then
<6' l28)
Substituting Eqs. ( 6 .128) and (6 .115) into Eqs. (6.126) and (6.127), we obtain
2L
2
W
R = ----------------------- = ---------------PooS C Lg ( L / W )
PooC Lg
s
and
co =
(6.129)
gn
y/2 L / ( PooS C l )
gn
\/[2 n / (PooC l )](W / S)
2 ( W /S )
(6.130)
N ote that in Eqs. (6.129) and (6.130), the factor W / S appears. As we have dis­
cussed in previous sections, this factor occurs frequently in airplane perform ance
analyses and is labeled
W
i
i*
—
= wing loading
S
Equations (6.129) and (6.130) clearly show that airplanes with low er w ing load­
ings will have sm aller turn radii and larger turn rates, everything else being
equal. However, the design wing loading o f an airplane is usually determ ined by
factors other than m aneuvering, such as payload, range, and m axim um velocity.
As a result, wing loadings for light, general aviation aircraft are relatively low,
but those for high-perform ance m ilitary aircraft are relatively large. W ing load­
ings for some typical airplanes are listed here.
Airplane
Wright Flyer (1903)
Beechcraft Bonanza
McDonnell Douglas F-15
General Dynamics F-16
W/S, lb/ft2
1.2
18.8
66
74
From this table, we conclude that a small, light aircraft such as the Beechcraft
B onanza can outm aneuver a larger, heavier aircraft such as the F-16 because o f
sm aller turn radius and larger turn rate. However, this is really com paring apples
and oranges. Instead, let us exam ine Eqs. (6.129) and (6.130) for a given airplane
with a given wing loading and ask the question, For this specific airplane, under
what conditions will R be m inim um and u> m axim um ? From these equations,
clearly R will be m inim um and co will be m axim um when both C¿ and n are
471
472
chapter
6
Elements of Airplane Performance
m axim um . T hat is,
2
/?min =
W
---------^ ----------=■
(6 .1 3 1 )
P o o 8 ^ L,max
/ P
" max" g \
ooC l
,m a x ^ m a x
2(W/S)
....
(
}
A lso note from Eqs. (6.131) and (6.132) that best perform ance will occur at sea
level, where
is m axim um .
T here are som e practical constraints on the preceding considerations. First,
at low speeds, nmM is a function o f C/,,max itself, because
L
n~ w =
hence,
1>SCl
w
nmM = z A x > V ¿ % 7 jr
L
W/ b
(6.133)
A t higher speeds, n max is lim ited by the structural design o f the airplane. These
considerations are best understood by exam ining Fig. 6.55, which is a diagram
show ing load factor versus velocity for a given airplane— the V-n diagram. Here,
curve A B is given by Eq. (6.133). C onsider an airplane flying at velocity V\, where
V\ is shown in Fig. 6.55. A ssum e that the airplane is at an angle o f attack such that
C l < C ¿,max. This flight condition is represented by point 1 in Fig. 6.55. Now as­
sum e that the angle o f attack is increased to that for obtaining Ci.,max, keeping the
velocity constant at V\. T he lift increases to its m axim um value for the given V\,
and hence the load factor n = L / W reaches its m axim um value n max for the given
V\. This value o f nmm is given by Eq. (6.133), and the corresponding flight condi­
tion is given by point 2 in Fig. 6.55. If the angle o f attack is increased further, the
wing stalls and the load factor drops. Therefore, point 3 in Fig. 6.55 is unobtain­
able in flight. Point 3 is in the stall region o f the V-n diagram . Consequently,
point 2 represents the highest possible load factor that can be obtained at the given
velocity Vi. Now, as V¡ is increased, say, to a value o f V4, then the m axim um pos­
sible load factor n max also increases, as given by point 4 in Fig. 6.55 and as calcu­
lated from Eq. (6.133). H ow ever, « max cannot be allow ed to increase indefinitely.
Beyond a certain value o f load factor, defined as the positive limit load factor and
shown as the horizontal line BC in Fig. 6.55, structural dam age may occur to the
aircraft. The velocity corresponding to point B is designated as V*. At velocities
higher than V*, say, V5, the airplane must fly at values o f C¿ less than C ¿,max so that
the positive limit load factor is not exceeded. If flight at C i.,max is obtained at ve­
locity V5, corresponding to point 5 in Fig. 6.55, then structural dam age will occur.
The right-hand side o f the V-n diagram , line CD, is a high-speed limit. A t veloci­
ties greater than this, the dynam ic pressure becom es so large that again structural
dam age may occur to the airplane. (This m axim um velocity limit is, by design,
6 .1 7
Turning Flight and the V-n Diagram
Figure 6.55 T h e V-n d ia g ra m f o r a ty p ic a l j e t tra in e r a irc ra ft.
(Source: U.S. Air Force Academy.)
much larger than the level-flight Vmax calculated in Secs. 6.4 to 6.6. In fact, the
structural design o f m ost airplanes is such that the m axim um velocity allow ed by
the V-n diagram is sufficiently greater than the m axim um diving velocity for the
airplane.) Finally, the bottom part o f the V-n diagram , given by curves AE and ED
in Fig. 6.55, corresponds to negative absolute angles o f attack, that is, negative
load factors. Curve AE defines the stall limit. (At absolute angles o f attack less
than zero, the lift is negative and acts in the dow nw ard direction. If the wing is
pitched dow nw ard to a large enough negative angle o f attack, the flow will sepa­
rate from the bottom surface o f the wing and the dow nw ard-acting lift will de­
crease in m agnitude; that is, the wing stalls.) Line ED gives the negative lim it load
factor, beyond w hich structural dam age will occur.
As a final note concerning the V-n diagram , consider point B in Fig. 6.55. This
point is called the maneuver point. At this point, both C L and n are sim ultaneously
at their highest possible values that can be obtained anyw here throughout the al­
low able flight envelope o f the aircraft. Consequently, from Eqs. (6 .13 1) and
(6 .132), this point corresponds sim ultaneously to the sm allest possible turn radius
and the largest possible turn rate for the airplane. The velocity corresponding to
point B is called the corner velocity and is designated by V* in Fig. 6.55. The cor­
ner velocity can be obtained by solving Eq. ( 6 .133) for velocity, yielding
V* =
2 n.
P o o C - L ,max
W
S
(6.134)
473
474
chapter
6
Elements of Airplane Performance
In Eq. (6.134), the value o f nmM corresponds to that at point B in Fig. 6.55. The
co m er velocity is an interesting dividing line. At flight velocities less than V*, it
is not possible to structurally dam age the airplane ow ing to the generation o f too
m uch lift. In contrast, at velocities greater than V*, lift can be obtained that can
structurally dam age the aircraft (e.g., point 5 in Fig. 6.55), and the pilot must
m ake certain to avoid such a case.
6.18 ACCELERATED RATE OF CLIMB
(ENERGY METHOD)'
M odern high-perform ance airplanes, such as the supersonic General Dynam ics
F-16 shown in Fig. 6.56, are capable o f highly accelerated rates o f climb. T here­
fore, the perform ance analysis o f such airplanes requires m ethods that go beyond
the static rate-of-clim b considerations given in Secs. 6.8 to 6.11. The purpose o f
this section is to introduce one such method, namely, a m ethod dealing with the
energy o f an airplane. This is in contrast to our previous discussions that have
dealt explicitly w ith forces on the airplane.
C onsider an airplane o f m ass m in flight at som e altitude h and with some
velocity V. D ue to its altitude, the airplane has potential energy PE equal to mgh.
Due to its velocity, the airplane has kinetic energy KE equal to
V 2. The total
energy o f the airplane is the sum o f these energies;
Total aircraft energy = PE + KE = mgh +
V2
(6.135)
The energy per unit w eight o f the airplane is obtained by dividing Eq. (6.135) by
W = mg. This yields the specific energy, denoted by He, as
_ P E + KE
mgh + ~ m V 2
He = -------------— --------------------W
mg
or
V2
He = h + —
2g
(6.136)
The specific energy He has units o f height and is therefore also called the energy
height o f the aircraft. Thus, let us becom e accustom ed to quoting the energy o f
an airplane in term s o f the energy height He, which is sim ply the sum o f the po­
tential and kinetic energies o f the airplane per unit weight. C ontours o f constant
He are illustrated in Fig. 6.57, w hich is an altitude-M ach num ber map. Here, the
ordinate and abscissa are altitude h and M ach num ber M, respectively, and the
dashed curves are lines o f constant energy height.
To obtain a feeling for the significance o f Fig. 6.57, consider two airplanes,
one flying at an altitude o f 30,000 ft at M ach 0.81 (point A in Fig. 6.57) and the
'This section is based in part on material presented by the faculty o f the departm ent o f aeronautics at the
U.S. Air Force Academy at its annual aerodynam ics workshop, held each July at Colorado Springs. This
author has had the distinct privilege to participate in this workshop since its inception in 1979. Special
thanks for this material go to Col. Jam es D. Lang, M ajor Thom as Parrot, and Col. Daniel Daley.
6.1 8
Accelerated Rate of Climb (Energy Method)
Figure 6.56 General Dynamics F-16 in 90° vertical accelerated climb.
(Source: U.S. Air Force.)
other Hying at an altitude o f 10,000 ft at M ach 1.3 (point B). Both airplanes have
the sam e energy height, 40,000 ft (check this yourself by calculation). However,
airplane A has m ore potential energy and less kinetic energy (per unit weight)
than airplane B. If both airplanes m aintain their sam e states o f total energy, then
both are capable o f “zoom ing” to an altitude o f 40,000 ft at zero velocity
(point C) sim ply by trading all their kinetic energy for potential energy. C onsider
475
chapter
6
Elements of Ai rplane Performance
Mach n u m b er M
Figure 6.57 Altitude-Mach number map showing curves of constant energy
height. These are universal curves that represent the variation o f kinetic and
potential energies per unit weight. They do not depend on the specific design
factors o f a given airplane.
another airplane, flying at an altitude o f 50.000 ft at M ach 1.85, denoted by
point D in Fig. 6.57. This airplane will have an energy height o f 100,000 ft and
is indeed capable o f zoom ing to an actual altitude o f 100,000 ft by trading all its
kinetic energy for potential energy. A irplane D is in a m uch higher energy state
( He = 100,000 ft) than airplanes A and B (which have He = 40,000 ft). T here­
fore, airplane D has a m uch greater capability for speed and altitude perform ance
than airplanes A and B. In air com bat, everything else being equal, it is advanta­
geous to be in a higher energy state (have a larger He) than your adversary.
How does an airplane change its energy state; for exam ple, in Fig. 6.57, how
could airplanes A and B increase their energy heights to equal that o f D ? To
answ er this question, return to the force diagram in Fig. 6.5 and the resulting
equation o f m otion along the flight path, given by Eq. (6.7). A ssum ing that a t is
sm all, Eq. (6.7) becom es
T - D - W s i n 0 = m t^ ~
at
R e c a llin g that m = W / g , w e can rearrange Eq. (6 .1 3 7 ) as
(6.137)
Accelerated Rate of Climb (Energy Method)
6.18
M ultiplying by V/ W, we obtain
T V -D V
.
V dV
--------------- = V s in # H----------—
W
g dt
,
(6.138)
Exam ining Eq. (6.138) and recalling som e o f the definitions from Sec. 6.8, we
observe that V sin # = R / C = d h / d t and that
TV —DV
excess pow er
W
W
w here the excess pow er per unit w eight is defined as the specific excess pow er
and is denoted by Ps. H ence, Eq. (6.138) can be w ritten as
„
dh
V dV
Ps — —¡----1-------- T~
dt
g dt
(6.139)
Equation (6.139) states that an airplane w ith excess pow er can use this excess for
rate o f clim b ( d h / d t ) or to accelerate along its flight path (d V / d t ) or for a com bi­
nation o f both. For exam ple, consider an airplane in level flight at a velocity o f
800 ft/s. A ssum e that when the pilot pushes the throttle all the way forw ard, an
excess pow er is generated in the am ount Ps = 300 ft/s. Equation (6.139) shows
that the pilot can choose to use all this excess pow er to obtain a m axim um unac­
celerated rate o f c lim b o f3 0 0 ft/s (i/V y i/r = 0 , hence Ps = d h / d t = R/C). In this
case, the velocity along the flight path stays constant at 800 ft/s. A lternatively, the
pilot may choose to m aintain level flight ( d h / d t = 0) and to use all this excess
pow er to accelerate at the rate o f d V / d t = g P s/ V = 3 2 .2 (3 0 0 )/8 0 0 = 12.1 ft/s2.
On the other hand, some com bination could be achieved, such as a rate of
clim b d h / d t = 100 ft/s along with an acceleration along the flight path of
d V / d t = 3 2 .2 (2 0 0 )/8 0 0 = 8.1 ft/s2. [Note that Eqs. (6.138) and (6.139) are gen­
eralizations of Eq. (6.50). In Sec. 6.8, we assum ed that d V / d t = 0, w hich resulted
in Eq. (6.50) for a steady clim b. In the present section, w e are treating the more
general case o f clim b with a finite acceleration.] Now return to Eq. (6.136) for the
energy height. D ifferentiating with respect to time, we have
dH e
dh
V dV
—- = — + - ~ r
dt
dt
g dt
(6.140)
The right-hand sides o f Eqs. (6.139) and (6.140) are identical; hence, we see that
dHe
Ps =
dt
(6.141)
That is, the time rate o f change o f energy height is equal to the specific excess
power. This is the answ er to the question at the beginning o f this paragraph. An
airplane can increase its energy state sim ply by the application o f excess power.
In Fig. 6.57, airplanes A and B can reach the energy state o f airplane D if they
have enough excess pow er to do so.
477
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chapter
6
Elements of Airplane Performance
This im m ediately leads to the next question, namely, H ow can w e ascertain
w hether a given airplane has enough P, to reach a certain energy height? To ad­
dress this question, recall the definition o f excess pow er as illustrated in
Fig. 6.29, that is, the difference betw een pow er available and pow er required.
For a given altitude, say, h, the excess pow er (hence Px) can be plotted versus
velocity (or M ach num ber). For a subsonic airplane below the drag-divergence
M ach number, the resulting curve will resem ble the sketch shown in Fig. 6.58a.
At a given altitude h i , Ps will be an inverted, U -shaped curve. (This is essentially
Figure 6.58 Construction o f the specific excess-power contours
in the altitude- Mach number map for a subsonic airplane below
the drag-divergence Mach number. These contours are
constructed for a fixed load factor; if the load factor is changed,
the Ps contours will shift.
6.18
Accelerated Rate of Climb (Energy Method)
the sam e type o f plot as shown in Figs. 6.32 and 6.33.) For progressively higher
altitudes, such as h2 and 6 3, Ps becom es smaller, as also shown in Fig. 6.58a.
Hence, Fig. 6.58a is sim ply a plot o f Px versus M ach num ber with altitude as a
param eter. These results can be cross-plotted on an altitude-M ach num ber map
using Ps as a param eter, as illustrated in Fig. 6.586. For exam ple, consider all the
points on Figure 6.58a where Ps = 0; these correspond to points along a hori­
zontal axis through Ps = 0, such as points a, 6, c, d, e, and / i n Fig. 6.58a. Now
replot these points on the altitude-M ach num ber map in Fig. 6.586. Here, points
a, 6, c, d , e, and / f o r m a bell-shaped curve, along which Px = 0. This curve is
called the Ps contour for Ps = 0. Similarly, all points with Ps = 200 ft/s are on
the horizontal line AB in Fig. 6.58a, and these points can be cross-plotted to gen­
erate the Ps = 200 ft/s contour in Fig. 6.586. In this fashion, an entire series o f
Ps contours can be generated in the altitude-M ach num ber map. For a supersonic
airplane, the Ps versus M ach num ber curves at different altitudes will appear as
sketched in Fig. 6.59a. The “d en t” in the U -shaped curves around M ach l is
due to the large drag increase in the transonic flight regim e (see Sec. 5.10). In
turn, these curves can be cross-plotted on the altitude-M ach num ber map, pro­
ducing the Ps contours as illustrated in Fig. 6.596. Due to the double-hum ped
shape o f the Ps curves in Fig. 6.59a, the Ps contours in Fig. 6.596 have different
shapes in the subsonic and supersonic regions. The shape o f the Ps contours
shown in Fig. 6.596 is characteristic o f m ost supersonic aircraft. Now we are
close to the answ er to our question at the beginning o f this paragraph. Let us
overlay the Ps contours, say, from Fig. 6.596, and the energy states illustrated in
Fig. 6.57— all on an altitude-M ach num ber map. We obtain a diagram , as illus­
trated in Fig. 6.60. In this figure, note that the Ps contours alw ays correspond to
a given airplane at a given load factor, w hereas the He lines are universal funda­
mental physical curves that have nothing to do with any given airplane. The use­
fulness o f Fig. 6.60 is that it clearly establishes w hat energy states are obtainable
by a given airplane. The regim e o f sustained flight for the airplane lies inside the
envelope form ed by the Ps = 0 contour. H ence, all values o f H,, inside this
envelope are obtainable by the airplane. A com parison o f figures like Fig. 6.60
for different airplanes will clearly show in w hat regions o f altitude and Mach
num ber an airplane has m aneuverability advantages over another.
Figure 6.60 is also useful for representing the proper flight path to achieve
m inim um tim e to clim b. For exam ple, consider tw o energy heights He ¡ and He¿,
where He2 > He \. The tim e to m ove betw een these energy states can be ob­
tained from Eq. (6 .141), w ritten as
Integrating betw een He \ and He¿ , we have
(6.142)
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(a)
M ach n u m b e r M
(b)
Figure 6.59 Specific excess-power contours for a supersonic
airplane.
From Eq. (6.142), the tim e to clim b will be a m inim um when Ps is a m axim um .
L ooking at Fig. 6.60, for each H,, curve, we see there is a point where Px is a
m axim um . Indeed, at this point, the Ps curve is tangent to the He curve. Such
points are illustrated by points A to / in Fig. 6.60. The heavy curve through these
points illustrates the variation o f altitude and M ach num ber along the flight path
for m inim um time to clim b. The segm ent o f the flight path betw een D and D'
represents a constant-energy dive to accelerate through the drag-divergence re­
gion near M ach 1.
As a final note, analyses o f m odern high-perform ance airplanes m ake exten­
sive use o f energy concepts such as those previously described. Indeed, m ilitary
6 .1 9
Special Considerations for Supersonic Airplanes
Mach num ber M
Figure 6.60 Overlay o f Ps contours and specific energy states on an altitude-Mach number
map. The Ps values shown here approximately correspond to a Lockheed F-104G supersonic
fighter. Load factor n = l. W = 18,000 lb. Airplane is at maximum thrust. The path given by
points A through / is the flight path for minimum time to climb.
pilots fly with Ps diagram s in the cockpit. O ur purpose here has been to simply
introduce som e o f the definitions and basic ideas in involving these concepts. A
m ore extensive treatm ent is beyond the scope o f this book.
6.19 SPECIAL CONSIDERATIONS FOR
SUPERSONIC AIRPLANES
The physical characteristics o f subsonic llow and supersonic How are totally
different— a contrast as striking as that betw een day and night. We have already
addressed som e o f these differences in Chaps. 4 and 5. However, these differ­
ences do not affect the airplane perform ance techniques discussed in this chap­
ter. T hese techniques are general, and they apply to both subsonic and supersonic
airplanes. The only way by which our perform ance analysis knows that the air­
plane is subsonic or supersonic is through the drag polar and the engine charac­
teristics. Recall from our discussion in Sec. 5.3 that C/ and C[> are functions o f
free-stream M ach num ber; hence, the drag polar is a function o f
A given
drag polar pertains to a specified M ach num ber; for exam ple, the drag polar for
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Figure 6.61 Generic comparison of a
subsonic drag polar with a supersonic
drag polar for the same airplane.
the Lockheed C - 141A shown in Fig. 6.2 pertains to low -speed flow
< 0.3.
A generic com parison betw een the drag polars for a given subsonic M ach num ­
ber and a given supersonic M ach num ber for the sam e airplane is sketched in
Fig. 6.61. For a given C L, C D is m uch larger at supersonic speeds than at sub­
sonic speeds because o f the presence o f supersonic w ave drag. Therefore, the su­
personic drag polar is displaced to the right o f the subsonic drag polar and is a
more tightly shaped parabola, as sketched in Fig. 6.61.
C onsider an arbitrary point on the drag polar, such as point 1 show n in
Fig. 6.61. A straight line 0 - 1 draw n from the origin to point 1 will have a slope
equal to C ¿ ,i/C o ,i; that is, the slope is equal to the lift-to-drag ratio associated
with flight at point 1. As w e m ove point 1 up the drag polar, the slope o f line 0 - 1
will increase, associated with increased values o f L / D . Let point A be the point
where the straight line becom es tangent. H ence, the slope o f the straight line OA
is the m axim um possible slope. This slope is equal to ( ¿ / D ) max, and point A cor­
responds to flight at m axim um lift-to-drag ratio. This dem onstrates the graphical
construction from which ( ¿ / D ) max can be obtained from the drag polar. Sim ply
draw a straight line from the origin tangent to the drag polar— the slope o f this
line is equal to ( L / D ) max.
W ith this in m ind, let us com pare the two drag polars in Fig. 6.61. Line OA
is draw n tangent to the subsonic drag polar, and its slope gives ( L / D ) max at
the given subsonic M ach number. Line OB is drawn tangent to the supersonic
drag polar, and its slope gives ( ¿ / D ) max at the given supersonic M ach number.
Clearly, the slope o f OB is sm aller than the slope o f OA. The values o f ( L / D ) max
at supersonic speeds are smaller than at subsonic speeds. This is dram atically
show n in Fig. 6.62. As an airplane accelerates through M ach 1, there is a consid­
erable drop in its (L / D ) max.
Perhaps the m ost severe effect on airplane perform ance associated with the
decrease in ( L /£ ) ) max at supersonic speeds is that on range. From Eq. (6.77), we
saw that range for a je t airplane is proportional to C ]/ 2/ C D■ If ( L / D )max is
6.19
Special Considerations for Supersonic Airplanes
20
(h i
io
M uch n u m b e r
Figure 6.62 V ariatio n o f ( L / D ) max w ith M a c h n u m b e r f o r se v e ra l g e n e ric a irp la n e
c o n fig u ra tio n s .
(Source: From M. R. Nichols, A. L. Keith, and W. E. Foss, "The Second-Generation Supersonic
Transport, "in Vehicle Technology for Civil Aviation: The Seventies and Beyond. NASA
SP-292, pp. 409-428.)
sm aller for a given supersonic M ach number, then so will be the value o f
(C [/2/ C o ) max . This is the prim ary reason why the range o f a given airplane cruis­
ing at supersonic speed is sm aller than that at subsonic speed, everything else
being equal.
Let us return to Eq. (6.75), repeated here:
R = r'W
^oV o o C i d w
(6.75)
Jwi ct Co W
This is the equation from which Eq. (6.77) was derived. A ssum ing flight at con­
stant Voo, c,, and C L/ C D, Eq. (6.75) becom es
R = - - I n
c, D
Wo
Wi
( 6 .143)
You will frequently see Eq. ( 6 .143) in the literature as the equation for range for
a je t airplane. Note that Eq. ( 6 .143) show s that m axim um range is obtained not
with m axim um L / D but rather with the m axim um value o f the product,
Voo( L / D ) . This product is m axim um when C |/2/C/> is m axim um , as shown
through the derivation o f Eq. (6.77). N evertheless, Eq. (6.143) is a useful ex­
pression for the range for a je t airplane.
Equations (6.77) and ( 6 .143) both indicate the obvious ways to com pen­
sate for the loss o f ( L / D ) max, hence C'L/2/ C D, on the range for a supersonic
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DESIGN BOX
Based on the preceding discussion, the designer of a
supersonic cruise airplane, such as a civil supersonic
transport, must live with the realities embodied in
Eq. (6.143). For example, during the 1990s, an ex­
tended study of a second-generation supersonic
transport, labeled the high-speed civil transport
(HSCT), was carried out by industry in the United
States, supported by the high-speed research (HSR)
program carried out by NASA. (By comparison, the
Anglo-French Concorde designed in the 1960s,
shown in Fig. 5.63, is a first-generation supersonic
transport.) The baseline design specifications for the
HSCT called for cruise at Mach 2.4 with a range of
5000 mi, carrying 300 passengers. This is an extreme
design challenge, on the cutting edge of modern aero­
nautical technology. From Eq. (6.143), a few percent
shortfall in L / D could prevent the achievement of
the specified range. This underscores the importance
of supersonic aerodynamic research aimed at im­
proving supersonic L /D . The engine must produce
the lowest possible thrust specific fuel consumption,
while at the same time producing an environmentally
acceptable low value of atmospheric pollutants in the
jet exhaust, in order to protect the atmospheric ozone
layer. Moreover, the engine noise must be an accept­
ably low value during takeoff and landing—that is a
major challenge for jet engines designed for super­
sonic flight, for which the exhaust jet velocities are
large, hence very noisy. Therefore, the design of en­
gines for the HSCT is a massive challenge in itself.
There are major structural and materials challenges
as well. The design goal of the HSCT is a structural
weight fraction (weight of the structure divided by
the gross takeoff weight) of 0.2, which is consider­
ably smaller than the more typical value of 0.25 and
higher for conventional subsonic transports. With the
smaller structural weight fraction, the HSCT can
carry more fuel and/or more passengers to meet its
other design specifications. And if this were not
enough, the size of the baseline HSCT is so large,
with a length longer than a football field, that there is
a problem with elastic bending of the fuselage (in the
longitudinal direction); as a result, stability and con­
trol are severely compromised. Indeed, this problem
is compounded by the interaction of the aerodynamic
force, the propulsive thrust, and the real-time control
inputs. Called the APSE (aeropropulsiveservoelastic)
effect, this is a problem that affects the HSCT in
flight and on the ground. (For more details on the
HSCT design challenges, see U.S. Supersonic Com­
mercial Aircraft: Assessing NASA’s High Speed
Research Program, National Research Council Re­
port, National Academy Press, Washington, DC,
1997.) We note that the sonic boom is not considered
to be a problem for the HSCT because of the up-front
decision that it would fly subsonieally over land, the
same restriction imposed on the Concorde SST.
In short, the design of an environmentally
acceptable, economically viable supersonic transport
is a major aeronautical technological problem that
has yet to be solved. It will be one of the most chal­
lenging aeronautical endeavors in the early 21st cen­
tury. and perhaps many of the young readers of this
book will have a hand in meeting this challenge.
airplane, namely,
1.
D ecrease the thrust specific fuel consum ption c,.
2.
Increase the fuel w eight Wf , thereby increasing the ratio Wo/ W\ in
Eq. (6.143) and increasing the difference W()1/2 — W¡/2 in Eq. (6.77).
Increasing the fuel w eight is usually not a desirable design solution, because the
a d d ition al fu el u su a lly m ea n s a sm a lle r u se fu l p a y lo a d fo r the airplane. A lso , for
turbojet and low -bypass ratio turbofans (see Chap. 9), the thrust specific fuel
consum ption increases with an increase in M ach num ber for supersonic speeds,
further com pounding the degradation o f range.
6.20
Uninhabited Aerial Vehicles (UAVs)
6.20 UNINHABITED AERIAL VEHICLES (UAVs)
A fter the W right brothers w orked so hard to put hum ans in the air in flying m a­
chines, a hundred years later som e aerospace engineers are w orking hard to take
hum ans out o f the flying m achine. Uninhabited aerial vehicles (UAVs) are air­
planes that have no hum ans on board, but rather are flown rem otely by pilots on
the ground or in other airplanes. Such vehicles cam e on the scene in the 1950s
with the introduction o f the rem otely controlled Ryan Firebee for reconnais­
sance, which was used extensively in Vietnam. In the early days o f their use,
these types o f aircraft w ere labeled as remotely piloted vehicles (RPVs). Israel is
the first nation to have used RPV s in a com bat situation, arguing that for recon­
naissance m issions a loss o f a relatively inexpensive RPV was better than the
loss o f a pilot and a m ultim illion-dollar airplane. In the later part o f the 20th cen­
tury, RPVs m atured and were redesignated UAVs, which at the tim e stood for
“unm anned” aerial vehicles. The term “unm anned” is, however, a m isnom er be­
cause such aircraft are m anned rem otely by a hum an pilot, albeit that pilot is not
physically in the aircraft. This led to the recent use o f the term uninhabited aerial
vehicle, a m ore proper description o f the case.
A t the tim e o f w riting, UAVs, and their spinoff, uninhabited combat aerial
vehicles (UCAVs), are becom ing a more im portant part o f aerospace engineering.
In the U nited States alone, at least five dozen UAV design program s are under­
way, w ith m any m ore throughout Europe, the M iddle East, and Asia. It is already
a m ultibillion-dollar business, and grow ing rapidly. In term s o f airplane design,
UAVs offer a w idely expanded design space, in part because the pilot, passen­
gers, and related life support, and safety and com fort equipm ent are no longer
needed, thereby saving w eight and com plexity. M oreover, the physical con­
straints im posed by the limits o f the hum an body, such as losing consciousness
when exposed to accelerations around and above 9 g 's even for a few seconds,
are rem oved. U ninhabited aerial vehicles present new and exciting design chal­
lenges to aerospace engineers; such vehicles offer the chance for greatly im ­
proved perform ance and m any new and unique applications. Because o f their
grow ing im portance, we devote this section to UAVs as part o f our overall intro­
duction to flight.
Let us take a look at a few exam ples o f existing UAVs. To date, the primary
m ission for UAVs has been reconnaissance. O ne o f the best-know n UAVs is the
G eneral A tom ics Predator, shown in the three-view in Fig. 6.63. This aircraft has
been used in cam paigns in Bosnia, A fghanistan, and Iraq. The Predator has a
w ingspan o f 14.85 m (48.7 ft), a high-aspect ratio of 19.3, and a m axim um takeoff
w eight o f 1020 kgy (2250 lb). It is pow ered by a 105-hp Rotax four-cylinder reci­
procating engine driving a tw o-blade, variable-pitch pusher propeller. Because it
is a reconnaissance vehicle, the Predator is designed to stay in the air for a long
time; its m axim um endurance is greater than 40 hours. (If a human pilot were on
board, such a long endurance would not be practical.) The high-aspect ratio is one
o f the design features allow ing such a long endurance. Endurance at low altitude
is the prim ary perform ance characteristic o f this airplane; its m axim um speed is a
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slow 204 km /h (127 m i/h), its loiter speed is betw een 111 and 130 km /h (69 and
81 m i/h), and its service ceiling is a low 7.925 km (26,000 ft).
In contrast to the low -altitude Predator, the N orthrop G rum m an Global
H aw k, shown in Fig. 6.64, is a high-altitude surveillance UAV. As seen in
Fig. 6.64, the G lobal H aw k has an exceptionally high-aspect ratio o f 25, provid­
ing the sam e beneficial aerodynam ic characteristics as that for the high-aspect
ratio wing used for the Lockheed high-altitude U-2 described in detail in the
Design Box in Sec. 5.15. The G lobal Hawk is m uch larger than the Predator, with
a 35.42 m (116.2 ft) w ingspan, and w eighing 11,612 kg (25,600 lb) at takeoff. Its
service ceiling is 19.8 km (65,000 ft), and it is designed for a loiter speed of
635 km/h (395 m i/h) at a loiter altitude o f 15.2 to 19.8 km (50,000 ft to 65,000 ft).
Its m axim um endurance is 42 hours. In contrast to the piston-engine Predator, the
Global H aw k is pow ered by a R olls-R oyce Allison AE 300 7H turbofan engine,
producing 7600 lb o f thrust at standard sea level.
There are stealth UAVs. An exam ple is the Lockheed M artin DarkStar, shown
in Fig. 6.65. This is an experim ental vehicle, and the program was term inated in
1999 after tw o prototypes w ere produced. The D arkStar nevertheless represents
the design o f a low -observable, high-altitude endurance UAV. Its size is m idway
betw een the Predator and the G lobal Hawk. The w ingspan is 21.03 m (69 ft), with
6.20
Figure 6.65 Three-view o f the DarkStar stealth UAV.
Uninhabited Aerial Vehicles (UAVs)
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an aspect ratio o f 14.8. Its takeoff w eight is 3901 kg (8600 lb). It was designed for
a loiter altitude o f 13.7 to 19.8 km (45,000 ft to 65,000 ft), with a cruising speed
o f 463 km/h (288 m i/h) at 13.7 km (45,000 ft). M axim um endurance was approx­
im ately 12 hours, low er than that for the Predator and the Global H aw k, possibly
reflecting poorer aerodynam ic characteristics that usually plague any airplane de­
signed prim arily for stealth.
Let us glance into the future. Design studies reflecting new ideas for advanced
UAVs are shown in Fig. 6.66. These are part o f the SensorC raft study by the U.S.
A ir Force R esearch Laboratory at W right-Patterson A ir Force Base in Dayton,
O hio. The SensorC raft is designed as a long-endurance, high-altitude UAV for
perform ing com m and, control, detection, tracking, relay, and targeting functions
for long durations at extended ranges. The goal is to increase the endurance 50 per­
cent above that for the Global Hawk. Exam ining Fig. 6.66, we see three basic con­
figurations considered under the SensorC raft study: a som ew hat conventional
w ing-body-tail (upper left), a Hying wing (low er left), and a joined wing (upper
and low er right). The configurations are driven by a host o f antenna-size and fieldof-view requirem ents, w hile at the same time having extrem ely high levels o f
aerodynam ic efficiency— requirem ents that can be som ew hat conflicting.
Uninhabited Combat Aerial Vehicles (UCAVs) The UAVs discussed in the
preceding section do not carry arm am ent, m issiles, or bom bs; they are noncom bat
Figure 6.66 Some advanced UAV designs.
(Source: U.S. A ir Force.)
6 .20
Uninhabited Aerial Vehicles (UAVs)
vehicles for reconnaissance, com m and, and control, etc. In contrast, specialized
uninhabited aerial vehicles are being designed for direct air-to-air and air-toground com bat. T hese vehicles are called uninhabited combat aerial vehicles
(UCAVs), and they form a distinct and different class o f vehicles. By taking the
pilot out o f a fighter or bomber, UVACs can be optim ized for com bat perfor­
m ance with greatly increased accelerations and m aneuverability at g-forces (load
factors) much higher than a hum an can tolerate. The design space for UCAVs is
greatly expanded com pared to airplanes occupied by hum ans, and com bat tactics
can be much m ore aggressive than those intended to protect the lives o f the
occupants.
An exam ple o f a UCAV is the Boeing X-45, shown in Fig. 6.67. This is an
experim ental vehicle intended to pave the w ay to future operational UCAVs. As
/
/
y
/
w
Figure 6.67 The X-45 stealth UCAV.
(Source; U.S. A ir Force.)
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Elements of Airplane Performance
seen in Fig. 6.67, the X-45 is a stealth configuration; a low -radar cross section
will be absolutely necessary for operational UCAVs. T he w ingspan o f the X-45
is 33.75 ft, and its gross w eight is 15,000 lb. Pow ered by one H oneywell F-124
turbofan engine, the X-45 can achieve M ach 0.95. The X-45, and the design
space it represents, is a paradigm shift for m ilitary aircraft. It represents a wave
o f the future.
C o m m e n t Exam ine again Figs. 6.63 through 6.67. W hat you see are configu­
rations that are unconventional com pared to ordinary airplanes but that are
“conventional” for the current generation o f UAVs and UCAVs. These are ju st
the beginning. Twenty years from now you will look back at the configurations
in Figs. 6.63 through 6.67 and view them as the “W right Flyers” o f uninhabited
aerial vehicles.
EXAM PLE 6.19
Consider the C P-1 airplane of our previous examples. Let us examine the change in per­
formance of this airplane if the pilot, passengers, seats, and instrument panel are removed
and if we convert the CP-1 to a UAV. This is purely an academic exercise. In reality, a
UAV is point-designed from the beginning to optimize its performance; it is not simply a
stripped-down CP-1 that we are considering in this example. Nevertheless, there is some
value to examining the change in performance of the CP-1 when humans and related
equipment are taken out of the airplane, but the rest of the airplane is kept the same. In
this case, calculate (a) Vmm at sea level, (b) maximum rate of climb at sea level, (c) max­
imum range, and (d) maximum endurance at sea level. The weights of the removed peo­
ple and equipment include the following: four people (including the pilot) at 180 lb each,
720 lb total; four seats at 30 lb each, 120 lb total; and the instrumentation panel at 40 lb.
The total weight decrease is 880 lb.
■ Solution
From our previous examples dealing with the CP-1, we note that the fuel empty weight is
2583 lb, and the weight of the fuel is 367 lb. For the “UAV version” of the CP-1, the fuel
empty weight is
Wi = 2583 - 8 8 0 = 1703 lb
The gross weight is
Wo = W{ + Wf = 1703 + 367 = 2070 lb
Also,
AR = 7.37, CD,0 = 0.025, e = 0.8, and S = 174 ft2
a. We could find Vmax by constructing the power-required curve and finding the intersec­
tion of this curve with the power-available curve, as discussed in Secs. 6.5 and 6.6. In­
stead, let us take the following analytical approach. Repeating Eq. (6.42),
/
T = io o S (C „ ,o + q2J
W2
\
2neA R )
6 .2 0
Uninhabited Aerial Vehicles (UAVs)
Multiplying by V», and noting that T V» — PA, we have
W2
T Voo — PA — Qoo VooS
9ÍS 27rcAR’)
IV2
Pa = 5A »V ¿SC d,0 + f-----7j~z—
(E6. 3)
jPcxj^ooSTrfAR
From Example 6.4 for the CP-1,
PA = »?(bhp) = (0.80)(230) = 184 hp
PA = (184) (550) = 1.012 x 105 ft • lb/s
or
Also,
¿PooSCd.o = 5 (0.002377)(174)(0.025) = 5.17 x 10“3
W2
(2080)2
PooS^-e-AR
\ (0.002377) (174)^ (0.8)(7.37)
1.119 x 10°
Hence, Eq. (E6.3) becomes
1.012 x 105 = 5.17 x I0 -3V¿ + 1-119 X -K)
(E6.4)
'0 0
Solving Eq. (E6.4) for Voo,
Voo = 266 ft/s
Since PA in Eq. (E6.3) is the maximum power available, then Voo = Vm
Thus,
Vm;ix = 266 ft/s
Compare this result with that for the CP-1 obtained in Example 6.4, where Vmax = 265 ft/s.
There is virtually no change! Simply reducing the weight and keeping everything else the
same did not materially influence Vmax. In particular, the wing area was kept the same, re­
sulting in a lower wing loading than for the CP-1. The new wing loading is
W
2070
,
UAV: — = ------ = 11.9 lb/ft2
S
174
com pared to
chapter
6
Elements of Airplane Performance
Maximum velocity depends on W/S; in the Design Box in Sec. 6.7, we see that V^x
increases as W/S increases. Even though the power-to-weight ratio was increased for our
UAV, which would increase Vmax, the reduced wing loading negated the increased powerto-weight ratio. If we would reduce the wing area of our sample UAV to keep W/S the
same as for the C P-1. then V,™, would increase noticeably. This illustrates the importance
of point designing a UAV from the beginning to take advantage of the new design space.
b. From Eq. (6.53), repeated here,
(R/C)max = ( — )
- 0.8776 I -tV/'lS ------
V W 7 max
Y
Px Cd.o (L/D)mm
From Eq. (6.85),
/C A
\ C 0 /m ax
_ /L \
= y/Cn.pjrgAR
\ D /m ax
2 C » .0
v/(0.025)^(0.8)(7.37)
2(0.025)
= 13.6
- 1 1 .9 lb/ft2
fr ,P \
(0.8) (230) (550)
„0ftlkl
[ W ) mu=
207Ó
= 4 8 '9ft/S
Hence, Eq. (6.53) yields
( R /C ) „ ,a x
= 48.9 - 0.8776
11.9
I
(0.002377)(0.025) (13.6)3/2
= 4 8 .9 - 7 .8 = 41.1 ft/s
or
(R/C)max = (41.1) (60) =
2466 ft/min
Compare this result with that for the CP-1 from Example 6.6. The value of (R/C)max for
the CP-1 at sea level is 1494 ft/min. By taking the humans and associated equipment out
of the CP-1, the maximum rate of climb is increased by 65 percent, a dramatic increase!
c. The maximum range is obtained from Eq. (6.67), repeated here:
6 .2 0
Uninhabited Aerial Vehicles (UAVs)
493
where r\ = 0.8, c = 2.27 x 10 7 ft 1 (from Example 6.12), (C /./C D)max = 13.6, and
Wo/Wi = 2070/1703 = 1 .2 1 6 . Eq. (6.67) yields
R=
0.8
— (13.6) ln( 1.216) = 9.37 x 106 ft
2.27 x 10
A
=
9.37 x 106
5280
1775 miles
Compare this with the maximum range of the CP-1 obtained in Example 6.12, where
R = 1207 miles. By taking the humans and associated equipment out of the airplane, the
maximum range is increased by 47 percent!
d. The maximum endurance at sea level is obtained from Eq. (6.68), repeated here:
/-I
j
-.3/2
E = - ^ '_
éL-_( 2 PocS)'/2( W - '/2
C CD
K ' /2)
From Eq. (6.87),
(3C/),o7reAR)3/4
\
C i> ) max
4 Cp, o
[3(0.025) tt(0.8)(7.37)|3/4
4(0.025)
E=
=
12.8
(0.8)(12.8)
v/2(0.002377)( 174) [(17 0 3 ) '1/2 - (2070)"1/2]
2.27 x 10-
E = 9.24 x 104 s =
9.24 x 104
3600
25.7 h
Compare this with the maximum endurance of the CP-1 obtained in Example 6.12, where
E = 14.4 hours. By taking the humans and associated equipment out of the airplane, the
maximum endurance is increased by 78 percent!
Note: This example demonstrates the substantial increases in maximum rate-ofclimb, range, and endurance that can be obtained simply by taking the humans and
associated equipment out of an existing airplane. Imagine the even larger increases in
performance that can be obtained by point-designing the UAV from the beginning, rather
than just modifying an existing airplane. This is the message of the present example.
EXAM PLE 6.20
Consider two military airplanes, one a conventional piloted airplane limited to a maxi­
mum load factor of 9, and the other a UCAV designed for a maximum load factor of 25.
At the same flight velocity, compare the turn radius and the turn rate for these two aircraft.
494
chapter
6
Elements of Airplane Performance
■ Solution
Repeating Eq. (6.118), the turn radius, R, is
n
__
00
gs/n2 - 1
Letting R\ denote the turn radius for the UCAV and Ri denote the turn radius for the
conventional airplane, we have from Eq. (6.118) for the same Vx,
Repeating Eq. (6.119) for turn rate, u>,
10 =
gy/n2 - 1
Letting coi and a>2 denote the turn rates for the UCAV and conventional airplane,
respectively, we have from Eq. (6.119) for the same Voo,
=
2.8
Note: The UCAV can turn in a circle almost one-third the radius of the conven­
tional airplane and do it at almost 3 times the turn rate—a spectacular increase in
maneuverability.
6.21 A COMMENT, AND MORE
ON THE ASPECT RATIO
We end the technical portion o f this chapter by noting that detailed com puter pro­
gram s now exist w ithin N A SA and the aerospace industry for the accurate esti­
mation o f airplane perform ance. These program s are usually geared to specific
types o f airplanes, for exam ple, general aviation aircraft (light single- or twinengine private airplanes), m ilitary fighter aircraft, and com m ercial transports.
Such considerations are beyond the scope o f this book. H owever, the principles
developed in this chapter are stepping-stones to m ore-advanced studies o f air­
plane perform ance; the Bibliography at the end o f this chapter provides some
suggestions for such studies.
6.22 HISTORICAL NOTE: DRAG REDUCTION— THE
NACA COWLING AND THE FILLET
The radial piston engine cam e into w ide use in aviation during and after World
W ar I. As described in Chap. 9, a radial engine has its pistons arranged in a cir­
cular fashion about the crankshaft, and the cylinders them selves are cooled by
airflow over the outer finned surfaces. Until 1927, these cylinders were generally
directly exposed to the main airstream o f the airplane, as sketched in Fig. 6.68.
As a result, the drag on the engine-fuselage com bination was inordinately high.
6 .22
Historical Note: Drag Reduction—The NACA Cowling and the Fillet
Figure 6.68 E n g in e m o u n te d w ith no c o w lin g .
Figure 6.69 E n g in e m o u n te d w ith full c o w lin g .
D ynam ic p ressure, lb / f t2
Figure 6.70 R e d u c tio n in d ra g d u e to a lu ll co w lin g .
The problem was severe enough that a group o f aircraft m anufacturers met at
Langley Field on May 24, 1927, to urge N A CA to undertake an investigation o f
m eans to reduce this drag. Subsequently, under the direction of Fred E. Weick, an
extensive series o f tests was conducted in the Langley 20-ft propeller research
tunnel using a W right W hirlw ind J-5 radial engine m ounted to a conventional
fuselage. In these tests, various types o f aerodynam ic surfaces, called cowlings,
were used to cover, partly or com pletely, the engine cylinders, directly guiding
part o f the airflow over these cylinders for cooling purposes but at the same time
not interfering with the smooth prim ary aerodynam ic flow over the fuselage. The
best cow ling, illustrated in Fig. 6.69, com pletely covered the engine. T he results
w ere dram atic! C om pared with the unscow led fuselage, a full cow ling reduced
the drag by a stunning 60 percent! This is illustrated in Fig. 6.70, taken directly
from W eick’s report, entitled “D rag and Cooling with Various Form s o f Cow ling
495
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chapter
6
Elements of Airplane Performance
DESIGN BOX
In Chaps. 5 and 6 we have underscored the impor­
tance of the wing aspect ratio in airplane design. In
particular, for subsonic flight we have noted that by
increasing the aspect ratio, we can obtain a lower
indueed-drag coefficient, and hence a higher maxi­
mum L / D ratio. Now that we are at the end of our
discussions of airplane aerodynamics and perfor­
mance, it is worthwhile to expand this consideration
by asking the question: For an airplane in steady,
level flight, what design parameter dictates the in­
duced drag itself (as contrasted with the induced drag
coefficient)? Is it the aspect ratio, as intuition might
indicate, or is it another design parameter? The an­
swer is developed in the following discussion, which
will help to expand our understanding of induced
drag and will provide an enhanced physical under­
standing of the definition of aspect ratio.
From Eq. (6. lc), the coefficient of drag due to lift
(which for subsonic flight at normal angles of attack is
mainly due to the induced-drag coefficient) is given by
2
/ II, \ 2
,6j461
Substituting Eq. (6.146) into (6.145), we have
\q o o S ) ireAR
(6.147)
Since AR = b2/ S , Eq. (6.147) can be written as
D¡ = — qooS
ne
(6.148)
Note that the wing area cancels out of Eq. (6.148),
and wc arc left with
(6.149)
cl
C dj =
neAR
(6.144)
In turn, the drag due to lift is
A — <?00SCtJj — C¡oqS
For steady, level flight, L = W. Hence,
Cj
TTt'AR
(6.145)
This is a rather revealing result! The drag due
to lift in steady, level flight—the force itself—
does not depend explicitly on the aspect ratio, but
rather on another design parameter, W/b, called the
for a W hirlw ind Radial A ir-C ooled E ngine,” N A CA Technical Report No. 313,
published in 1928. A fter this w ork, virtually all radial engine-equipped airplanes
since 1928 have been designed with a full N A CA cow ling. The developm ent o f
this cow ling was one o f the m ost im portant aerodynam ic advancem ents o f the
1920s; it led the way to a m ajor increase in aircraft speed and efficiency.
A few years later, a second m ajor advancem ent was m ade, but by a com ­
pletely different group and on a com pletely different part o f the airplane. In the
early 1930s, the C alifornia Institute o f Technology at Pasadena, California,
established a program in aeronautics under the direction o f Theodore von
Karm an. Von K arm an, a student o f Ludw ig Prandtl, becam e probably the leading
aerodynam icist o f the 1920-1960 tim e period. At Caltech, von K arm an estab­
lished an aeronautical laboratory o f high quality, which included a large subsonic
w ind tunnel funded by a grant from the G uggenheim Foundation. T he first m ajor
experim ental program in this tunnel was a com m ercial project for D ouglas
6.22
497
Historical Note: Drag Reduction—The NACA Cowling and the Fillet
span loading.
(6.150)
The drag due to lift varies with the square of the span
loading.
From Eq. (6.149), we see that the drag due to
lift, for a given weight airplane, can be reduced sim­
ply by increasing the wingspan. In so doing, the wing
tip vortices (the physical source of induced drag) are
simply moved farther away, hence lessening their ef­
fect on the rest of the wing and, in turn, reducing the
induced drag. This makes good intuitive sense.
In light of this, the span loading W/b takes its
place as yet another design parameter that the air­
plane designers can adjust during the conceptual
design process for a new airplane. Of course, the span
loading and the aspect ratio are related via
-(Í)
b_
AR
(6.151)
where W/S is the familiar wing loading.
Let us return to the concept of aspect ratio,
which now takes on an enhanced significance, as
follows. First, note that the zero-lift drag, which we
denoted by D„, is given by q'ooSC/j.o and hence is
proportional to the wing area, whereas the drag due to
lift for steady, level flight is proportional to the square
of the span loading via Eq. (6.149). The ratio of these
two drags is
£i_
D„
i
<
■
looSCpfi
neqx
(6.152)
In Eq. (6.152), the ratio ( W/b)2/S can be cast as
(W /b)2
( W /S)2
b2/S
(W /S)2
AR
(6.153)
S u b s titu tin g E q . ( 6 . 15 3 ) in to ( 6 . 15 2 ) , w e h a v e
El
D„
1
(W /S)2
rt eq^C p,0 AR
(6.154)
From Eq. (6.154), we can make the following state­
ment: For specified values of the design parameters
W/S and Co.o, increasing the design aspect ratio will
decrease the drag due to lift relative to the zero-lift
drag. So the aspect ratio predominately controls the
ratio of lift-induced drag to the zero-lift drag,
whereas the span loading controls the actual value of
the lift-induced drag.
A ircraft Company. D ouglas was designing the DC-1, the forerunner o f a series o f
highly successful transports (including the fam ous DC-3, which revolutionized
com m ercial aviation in the 1930s). The DC-1 was plagued by unusual buffeting
in the region w here the wing join ed the fuselage. The sharp corner at the juncture
caused severe flow field separation, which resulted in high drag as well as shed
vortices that buffeted the tail. T he Caltech solution, which was new and pioneer­
ing, was to fair the trailing edge o f the wing sm oothly into the fuselage. These
fairings, called fillets, w ere em pirically designed and were m odeled in clay on
the DC-1 wind tunnel models. The best shape was found by trial and error. The
addition o f a fillet (see Fig. 6.71) solved the buffeting problem by sm oothing out
the separated flow and hence also reduced the interference drag. Since that time,
fillets have becom e a standard airplane design feature. M oreover, the fillet is an
excellent exam ple o f how university laboratory research in the 1930s contributed
directly to the advancem ent o f practical airplane design.
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CHAPTER 6
Elements of Airplane Performance
A
Figure 6.71 Illustration o f the w ing fillet.
6.23
Historical Note: Early Predictions of Airplane Performance
6.23 HISTORICAL NOTE: EARLY PREDICTIONS
OF AIRPLANE PERFORMANCE
The airplane o f today is a m odern w ork o f art and engineering. In turn, the pre­
diction o f airplane perform ance as described in this chapter is som etim es viewed
as a relatively m odern discipline. However, contrary to intuition, som e o f the
basic concepts have roots deep in history; indeed, som e o f the very techniques
detailed in previous sections w ere being used in practice only a few years after
the W right brothers’ successful first flight in 1903. This section traces a few his­
toric paths for som e o f the basic ideas o f airplane perform ance, as follows:
1.
Som e understanding o f the p o w e r required PR for an airplane was held by
G eorge Cayley. He understood that the rate o f energy lost by an airplane in
a steady glide under gravitational attraction m ust be essentially the pow er
that m ust be supplied by an engine to m aintain steady, level flight. In 1853,
Cayley wrote
The whole apparatus when loaded by a weight equal to that of the man intended
ultimately to try the experiment, and with the horizontal rudder [the elevator]
described on the essay before sent, adjusted so as to regulate the oblique descent
from some elevated point, to its proper pitch, it may be expected to skim down,
with no force but its own gravitation, in an angle of about 11 degrees with the
horizon; or possibly, if well executed, as to direct resistance something less, at
a speed of about 36 feet per second, if loaded 1 pound to each square foot of sur­
face. This having by repeated experiments, in perfectly calm weather, been as­
certained, for both the safety of the man, and the datum required, let the wings
be plied with the man’s utmost strength; and let the angle measured by the
greater extent of horizontal range of flight be noted; when this point, by re­
peated experiments, has been accurately found, we shall have ascertained a
sound practical basis for calculating what engine power is necessary under the
same circumstances as to weight and surface to produce horizontal flight. . ..
2.
The drag polar, a concept introduced in Secs. 5.14 and 6.1, sketched in
Figs. 5.35 and 6.1 and em bodied in Eq. (6.1a), represents sim ply a plot o f
C D versus C L, illustrating that Co varies as the square o f C ¿. A knowledge
o f the drag polar is essential to the calculation o f airplane perform ance. It is
interesting that the first drag polars were drawn and published by Otto
Lilienthal (see Sec. 1.5) in 1889, although he did not call them such. The
term polar for these diagram s was first introduced by G ustave Eiffel in
1909. Eiffel, the designer o f the Eiffel Tower in Paris, built two wind
tunnels and carried out extensive aerodynam ic testing from 1909 to the
tim e o f his death in 1923.
3.
Som e understanding o f the requirem ents for rate o f climb existed as far
back as 1913, when in an address by G ranville E. Bradshaw before the
Scottish A eronautical Society in G lasgow in D ecem ber, the follow ing
com m ent was made: “A m ong the essential features o f all successful
aeroplanes [is that] it shall clim b very quickly. This depends alm ost
entirely on the w eight efficiency o f the engine. T he rate o f clim b varies
499
500
chapter
6
Elements of Airplane Performance
directly as the pow er developed and indirectly as the w eight to be lifted.”
This is essentially a partial statem ent o f Eq. (6.50).
4.
N o general understanding o f the prediction o f airplane performance
existed before the 20th century. The excellent sum m ary o f aeronautics
w ritten by O ctave C hanute in 1894, Progress in Flying Machines, does not
contain any calculational technique even rem otely resem bling the
procedures set forth in this chapter. At best, it was understood by that time
that lift and drag varied as the first pow er o f the area and as the second
pow er o f velocity, but this does not constitute a perform ance calculation.
H ow ever, this picture radically changed in 1911. In that year, the
Frenchm an D uchéne received the M onthyon Prize from the Paris Academ y
o f Sciences for his book entitled The Mechanics o f the Airplane: A Study o f
the Principles o f Flight. C aptain D uchéne was a French engineering officer,
born in Paris on D ecem ber 27, 1869, educated at the fam ous Ecole
Polytechnique, and later assigned to the fortress at Toul, one o f the centers
o f “aerostation” in France. It was in this capacity that C aptain D uchéne
w rote his book during 1910-1911. In this book, the basic elem ents o f
airplane perform ance, as discussed in this chapter, are put forth for the first
tim e. D uchéne gives curves o f pow er required and pow er available, as we
illustrated in Fig. 6.21a; he discusses airplane m axim um velocity; he also
gives the sam e relation as Eq. (6.50) for rate o f clim b. Thus, som e o f our
current concepts for the calculation o f airplane perform ance date back as
far as 1910-1911— four years before the beginning o f World War I, and
only seven years after the W right brothers’ first flight in 1903. Later, in
1917, D uchéne’s book was translated to English by John Ledeboer and
T. O ’B. H ubbard (see Bibliography at the end o f this chapter). Finally,
during 1918-1920, three additional books on airplane perform ance w ere
w ritten (again, see Bibliography), the m ost fam ous being the authoritative
Applied Aerodynamics by Leonard Bairstow. By this tim e, the foundations
discussed in this chapter had been well set.
6.24 HISTORICAL NOTE: BREG UETAND
THE RANGE FORMULA
Louis-C harles B reguet was a fam ous French aviator, airplane designer, and
industrialist. Born in Paris on January 2, 1880, he was educated in electrical
engineering at the Lycée C ondorcet, the Lycée Carnot, and the École Superieure
d ’Electricité. A fter graduation, he joined the electrical engineering firm o f his
father, M aison Breguet. H ow ever, in 1909 B reguet built his first airplane and
then plunged his life com pletely into aviation. D uring W orld War I, his airplanes
w ere m ass-produced for the French air force. In 1919, he founded a com m ercial
airline com pany that later grew into A ir France. His airplanes set several longrange records during the 1920s and 1930s. Indeed, B reguet was active in his own
aircraft com pany until his death on M ay 4, 1955, in Paris. His nam e is associated
with a substantial part o f French aviation history.
6 .2 5
Historical Note: Aircraft Design— Evolution and Revolution
The form ula for range o f a propeller-driven airplane given by Eq. (6.67) has
also becom e associated with B reguet’s nam e; indeed, it is com m only called the
B reguet range equation. However, the reason for this association is historically
obscure. In fact, the historical research o f the present author can find no sub­
stance to B reguet’s association with Eq. (6.67) until a presentation by B reguet to
the Royal A eronautical Society in London in 1922. On one hand, we find
absolutely no reference to airplane range or endurance in any o f the airplane per­
form ance literature before 1919, least o f all a reference to Breguet. T he authori­
tative books by C ow ley and Levy (1918), Judge (1919), and Bairstow (1920)
(see Bibliography at the end o f this chapter) am azingly enough do not discuss
this subject. On the other hand, in 1919, N A CA Report No. 69, entitled “A Study
o f A irplane Ranges and Useful L oads,” by J. G. Coffin, gives a com plete deriva­
tion o f the form ulas for range, Eq. (6.67), and endurance, Eq. (6.68). However,
Coffin, w ho was director o f research for C urtiss Engineering Corporation at that
time, gives absolutely no references to anybody. Coffin’s work appears to be
original and clearly seem s to be the first presentation o f the range and endurance
form ulas in the literature. However, to confuse m atters, we find a few years later,
in N A C A R eport No. 173, entitled “R eliable Form ulae for Estim ating A irplane
Perform ance and the Effects o f C hanges in W eight, W ing A rea or Pow er,” by
W alter S. D iehl (we have met Diehl before, in Sec. 3.6), the follow ing statement:
“T he com m on form ula for range, usually credited to Breguet, is easily derived.”
D iehl’s report then goes on to use Eq. (6.67), with no further reference to
Breguet. This report was published in 1923, four years after C offin’s work.
Consequently, to say the least, the proprietorship o f Eq. (6.67) is not clear. It
appears to this author that, in the U nited States at least, there is plenty o f docu­
m entation to justify calling Eq. (6.67) the Coffin-Breguet range equation. H ow ­
ever, it has com e dow n to us through the ages sim ply as B reguet’s equation,
apparently w ithout docum ented substance.
6.25 HISTORICAL NOTE: AIRCRAFT DESIGN—
EVOLUTION AND REVOLUTION
Sit back for a m om ent and think about the evolution o f the airplane, beginning
with Sir G eorge C ayley’s 1804 hand-launched glider. Indeed, Fig. 1.8 (C ayley’s
own sketch o f this aircraft) show s the first airplane with a m odern configuration.
Now ju m p ahead a century in the design o f the airplane to Fig. 1.2, the W right
brothers’ historical photograph o f their first successful flight in 1903; this is the
true beginning o f the practical airplane. Finally, jum p another 80 years to
Fig. 6 .1 1, which show s a m odern je t aircraft. Put these three aircraft side by side
in your mind: C ay ley ’s glider, the Wright Flyer, and the Cessna Citation 3. W hat
a testim onial to the evolution o f airplane design! Each machine is totally differ­
ent, each being the product o f three different w orlds o f scientific and engineering
understanding and practice. One m ust m arvel at the rapid technical progress,
especially in the 20th century, that brings us to the present status o f airplane
design represented by the m odern, fast, high-flying je t aircraft shown in
501
502
CH APTER 6
Elements of Airplane Performance
Fig. 6.11. W hat w ere the m ajor technical m ilestones in this progress? W hat were
the evolutionary (and som etim es revolutionary) developm ents that sw ept us
from C ayley’s sem inal concepts to the m odern airplane? The eye-opening and
exciting answ ers to these questions w ould require a separate book to relate, but
in this section we highlight a few aspects o f the technical progression o f airplane
design, using som e o f the technology w e have covered in this chapter on airplane
perform ance.
To provide a technical focus for our discussion, we chose two aerodynam ic
param eters as figures o f m erit to com pare and evaluate different airplane designs.
The first is the zero-lift drag coefficient Co,o, an im portant characteristic o f any
airplane because it has a strong effect on the m axim um flight speed. Recall that
at Vmax for an airplane since the angle o f attack (and hence the induced drag) is
sm all, the total drag given by the drag polar in Eq. (6. lc ) is dom inated by C 0i0 at
high speeds. Everything else being equal, the low er the C d .o, the faster the air­
plane. The other aerodynam ic figure o f m erit highlighted here is the lift-to-drag
ratio and especially its m axim um value ( L / D ) max. As we have already seen,
L / D is a m easure o f the aerodynam ic efficiency o f an airplane, and it affects
such flight characteristics as endurance and range. We will use both C D,0 and
( L / D ) max to illustrate the historical progress in airplane design.
We start with the airplanes o f Cayley early in the 19th century because they
w ere the first designs to exem plify the fixed-wing heavier-than-air aircraft we
know today. Return again to Fig. 1.8, show ing the first airplane with a m odern
configuration, w ith a fixed w ing for lift, a tail for stability, and a fuselage con­
necting the two. The m echanism o f propulsion (in this case a hand launch) is sep­
arate from the m echanism o f lift. The am ount o f technical know ledge Cayley
was able to incorporate in his design is best reflected in his fam ous “triple p ap er”
o f 1809-1810 (see Sec. 1.3). The technical concepts o f C D,o and L / D did not
exist in C ayley’s day, but he reflects a basic intuition about these quantities in his
triple paper. For exam ple, C ayley used a m ethod called new tonian theory (which
will be derived in Chap. 11) to estim ate the aerodynam ic force on an inclined
plane (the wing). This theory takes into account only the pressure acting on the
surface; surface shear stress (and hence friction drag) was not fully appreciated
in C ayley’s tim e; furtherm ore there were no m ethods for its prediction. The new ­
tonian theory predicts a net force perpendicular to the inclined plane and there­
fore contains a com ponent o f drag. C ayley m akes reference to this “retarding
force” due to the com ponent o f the aerodynam ic pressure force acting along the
flow direction. In m odern term s, we call this com ponent o f drag the drag due to
lift. C ayley goes on to say (in discussing the flight o f birds), “In addition to the
retarding force thus received is the direct resistance, w hich the bulk o f the bird
opposes to the current. This is a m atter to be entered into separately from the
principle now under consideration.” Here, C ayley is discussing w hat we would
today call the zero-lift drag (sum o f pressure drag due to separation and skin fric­
tion drag) due prim arily to the body o f the bird. A lthough C ayley was on the right
track conceptually, he had no m ethod o f calculating the zero-lift drag, and m ea­
surem ents (m ade with a w hirling arm such as sketched in Fig. 1.7) w ere wholly
6.25
Historical Note: Aircraft Design— Evolution and Revolution
unreliable. Therefore, we have no value o f Cp.o for C ayley’s 1804 glider in
Fig. 1.8.
A lthough C ayley did not identify and use the concept o f L / D directly, in his
triple paper he refers to his glider sailing “m ajestically” from the top o f a hill, de­
scending at an angle o f about 18° w ith the horizon. U sing the results o f Sec. 6.9
dealing w ith a pow er-off glide, we can today quickly calculate that the L / D ratio
for the glider was 3.08, not a very im pressive value. Typical values o f L / D for
m odern airplanes are 15 to 20, and for m odern gliders, greater than 40. Cayley
did not have an efficient airplane, nor did he know about aspect-ratio effects.
Today w e know that lo w -asp ect ratio wings such as used by C ayley (aspect ratio
about 1) are very inefficient because they produce large am ounts o f induced drag.
The technical evolution o f airplane design after Cayley was gradual and evo­
lutionary during the rem ainder o f the 19th century. The change that occurred
with the Wright Flyer (Figs. 1.1 and 1.2) w as revolutionary (1) because the
W rights ultim ately relied on virtually no previous data, doing everything them ­
selves (see Sec. 1.8) and (2) because it was the first successful flying machine.
The aerodynam ic quality o f the Wright Flyer is discussed by Culick and Jex, who
report m odern calculations and m easurem ents o f the drag polar for the Wright
Flyer (Fig. 6.72). The experim ental data w ere obtained on a model o f the Wright
Flyer m ounted in a wind tunnel at the C alifornia Institute o f Technology. The
theoretical data are supplied by a m odern vortex-lattice com puter program for
Theory
Figure 6.72 Drag polar and lift curve for the 1903 Wright Flyer. Experimental data are from
modern experiments using models o f the Wright Flyer in modern wind tunnels. The vortexlattice theory is a modern computer calculation. The values o f C/_, C ¡¡, and á correspond to
equilibrium trimmed-flight conditions (see Chap. 7), highlighted by the horizontal bar across
the figure.
(Source: From Culick and Jex.)
503
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chapter
Elements of Airplane Performance
6
calculating low -speed incom pressible inviscid flow. (Since these m ethods do not
include the effects o f friction, they cannot be used to predict flow separation.)
T he data in Fig. 6.72 show that C D,o is about 0.10 and the m axim um lift coeffi­
cient nearly 1.1. M oreover, draw ing a straight line from the origin tangent to the
drag polar curve, we see that the value o f (L / D ) mm is about 5.7. By present stan­
dards, the Wright Flyer was not an aerodynam ic m asterpiece, but in 1903, it was
the only successful flying m achine in existence. M oreover, com pared with
C ayley’s airplanes, the Wright Flyer was a revolutionary advancem ent in design.
A fter the Wright Flyer, advances in airplane design grew alm ost exponen­
tially in the last h alf o f the 20th century. U sing our two figures o f m erit, C/>i0 and
(L / D ) mm, we can identify three general periods o f progress in airplane design
during the 20th century, as show n in Figs. 6.73 and 6.74. Values o f C D,0
(Fig. 6.73) and (L / D ) mM (Fig. 6.74) for representative airplanes are show n ver­
sus tim e in years. These data are obtained from Loftin, an authoritative publica­
tion that the interested reader is encouraged to exam ine; it contains detailed case
studies o f the technical designs o f many fam ous aircraft. The data for C D,0 in
Fig. 6.73 suggest that airplane design has gone through three m ajor evolutionary
periods, distinguished from one another by a dram atic change. For exam ple, the
period o f strut-and-w ire biplanes (such as the SPAD XIII, shown in Fig. 6.75) ex­
tends from the Wright Flyer to the m iddle or end o f the 1920s. Here, values o f
0.06 r
Period of
strut-and-wire biplanes
0.05
-H
0.04
4 •
Period o f mature propeller-driven
monoplanes with NACA cowling
7 •
0.03
------------------ H
• 6
13
8•
9 .
,
15
* 12
10 « r í -----------
11
0.01
Period of modern
je t airplanes
14
______ I_________I_______
1910
1920
1930
1940
Year
1950
1960
1990
Figure 6.73 Use o f zero-lift drag coefficient to illustrate three general periods
o f 20th-century airplane design. The numbered data points correspond to the
following aircraft: (1) SPAD XIII, (2) Fokker D-VII, (3) Curtiss JN-4H Jenny,
(4) Ryan NYP (Spirit o f St. Louis), (5) Lockheed Vega, (6) Douglas DC-3,
(7) Boeing B-17, (8) Boeing B-29, (9) North American P-51, (10) Lockheed
P-80, (11) North American F-86, (12) Lockheed F-104, (13) McDonnell
F-4E, (14) Boeing B-52, (15) General Dynamics F-l 1ID.
6 .2 5
Historical Note: Aircraft Design— Evolution and Revolution
22 —
■ 14
20 g
18 -
Period o f m odern
je t airplanes
[j
8B
16 -
h----------------------- ■
.
14 -
|--------------- 7 " --------- 1
12 10 8 -
9 .
■ 5
■ 4
■ 3
_______^ £ ________ ^
6 4 -
Period o f m atu re
propeller-driven
m o noplanes w ith
NACA cowling
M 12
Period o f
strut-and-w ire
biplanes
2 0
1910
1920
1930
1940
1950
1960
Y ear
Figure 6.74 Use of lift-to-drag ratio to illustrate three general periods of 20th-century
airplane design.
Figure 6 .7 5 T h e F re n c h S P A D X III, a n e x a m p le o f th e stru t-a n d -w ire b ip la n e p erio d .
C a p ta in E d d ie R ic k e n b a c k e r is s h o w n a t th e fro n t o f th e airp la n e .
(Source: U.S. Air Force.)
1990
505
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chapter
6
Elements of Airplane Performance
Figure 6.76 The Douglas DC-3, an example of the period of mature propeller-driven
monoplanes with the NACA cowling and wing fillets.
(Source: Douglas Aircraft Company.)
C o .o are typically on the order o f 0.04, a high value due to the large form drag
(pressure drag due to flow separation) associated with the bracing struts and
w ires betw een the tw o w ings o f a biplane. In the late 1920s, a revolution in de­
sign cam e with the adoption o f the m onoplane configuration coupled with the
N A C A cow l (see Sec. 6.22). The resulting second period o f design evolution
(exem plified by the DC-3 shown in Fig. 6.76) is characterized by C d .o values on
the order o f 0.027. In the m id-1940s, the m ajor design revolution was the advent
o f the jet-propelled airplane. This period, which we are still in today (reflected in
the fam ous F-86 o f the K orean w ar era, show n in Fig. 6.77) is represented by
C d .o values on the order o f 0.015.
The use o f (L / D ) max as an aerodynam ic figure o f m erit has been discussed
in previous sections. As seen in Fig. 6.74, where ( L / D ) max is plotted versus
years, the data points for the sam e airplanes as in Fig. 6.73 group them selves in
the sam e three design periods as deduced from Fig. 6.73. N ote that com pared
with the value o f 5.7 for the Wright Flyer, the average value o f (L / D ) max for
W orld War I airplanes was about 8— not a great im provem ent. A fter the intro­
duction o f the m onoplane w ith the N A CA cow ling, typical ( L / D )max values av­
eraged substantially higher, on the order o f 12 or som etim es considerably
greater. [The B oeing B-29 bom ber o f W orld W ar II fam e had an ( L / D ) max value
o f nearly 17, the highest for this period. This was in part due to the exceptionally
large wing aspect ratio o f 11.5 in a period w hen wing aspect ratios w ere averag­
ing on the order o f 6 to 8.] Today, ( L / D )max values for m odem aircraft range
over the w hole scale, from 12 or 13 for high-perform ance m ilitary je t fighters to
nearly 20 and above for large je t bom bers and civilian transports such as the
Boeing 747.
6 .2 5
Historical Note: Aircraft Design— Evolution and Revolution
Figure 6.77 The North American F-86, one o f the most successful modern jet airplanes from
the early 1950s.
(Source: North American/Rockwell.)
This section has given you the chance to think about the progress in aircraft
design in term s o f som e o f the aerodynam ic perform ance param eters discussed
in this chapter.
6.26 Summary
A few o f the important aspects o f this chapter are listed here:
1.
For a com plete airplane, the drag polar is given as
C2
C p = C p ,o H-------¡jTT
7TÍÍAR
2.
where Cp,o is the zero-lift drag coefficient and the term C \ / { n e A R ) includes
both induced drag and the contribution o f parasite drag due to lift.
Thrust required for level, unaccelerated flight is
W
Tr = ------L /D
3.
(6.1c)
(6.16)
Thrust required is a minimum when L / D is maximum.
Power required for level, unaccelerated flight is
PR =
2 W 'C l
--------- £
w Poo S C I
Pow er required is a m inim um w hen c \ [ 2/C p is a m axim um .
(6.27)
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chapter
4.
6
Elements of Airplane Performance
The rate of climb R/C = d h /d t is given by
dh _ TV - DV
V dV
dt
g dt
W
(6 .1 3 9 )
where (TV — D V ) / W = f t , the specific excess power. For an unaccelerated
climb, d V /d t = 0; hence,
„
dh
TV - DV
~ d i=
5.
W
(6'50)
In a power-off glide, the glide angle is given by
tan* = 1 7 5
6.
7.
<6-56>
The absolute ceiling is defined as that altitude where maximum R/C = 0. The
service ceiling is that altitude where maximum R/C = 100 ft/min.
For a propeller-driven airplane, range R and endurance E are given by
c CD
(6.67)
Wi
r 3/2
and
E = ^ - ^ - ( 2 PooS)l/2{ w ; l/2 - W - i/2)
(6.68)
C CD
8.
Maximum range occurs at maximum C l / C o- Maximum endurance occurs at sea
level with maximum C\[2/ C d For a jet-propelled airplane, range and endurance are given by
1/2
K=
,
and
9.
‘6-77>
~
1 CL
Wo
E = — — In —
c, CD Wi
(6.72)
At maximum CL /C o , Co,o = \ C D i . For this case,
(£:l'2\
jl\
\ C D / max
(3CD,0neAR)}/4
4Co,o
(6.87)
At maximum C l / C d , Cp.o = C DJ. For this case,
(C d,o^ é-AR)i/2
(6.85)
2 C /),o
At maximum C^¿2/Cd> Cd, o = 3C Di, . For this case,
/C¡_/2\
\ CD / max
= (jC p ,07rMR)
\ C D,o
1/4
(6 .8 6 )
Bibliography
10. T ak eoff ground roll is giv en by
1.441V2
ÍLO
8PooScL,mdx{T — [D + M W - L ) U )
(6.103)
11. The landing ground roll is
1.69W 2
( 6 . 111)
8PnaSC¡. ,max [D + i x A W - L ) lav
12. The load factor is defined as
L
n
(6.115)
W
13. In level turning flight, the turn radius is
(6.118)
g y /n 1 - 1
and the turn rate is
gVn2- 1
(6.119)
14. The V-n diagram is illustrated in Fig. 6.55. It is a diagram showing load factor
versus velocity for a given airplane, along with the constraints on both n and V due
to structural limitations. The V-n diagram illustrates som e particularly important
aspects o f overall airplane performance.
15. The energy height (specific energy) o f an airplane is given by
(6.136)
This, in combination with the specific excess power
TV - DV
leads to the analysis o f accelerated-climb performance using energy considerations
only.
Bibliography
Anderson, J. D., Jr.: Aircraft Perform ance an d D esign, McGraw-Hill, N ew York, 1999.
Anderson, J. D„ Jr.: The A irplane: A H istory o f Its Technology, American Institute O f
Aeronautics and Astronautics, Reston, VA, 2002.
Bairstow, L.: A pplied Aerodynam ics, Longmans, London, 1920.
Cowley, W. L., and H. Levy: A eronautics in Theory and Experiment, E. Arnold,
London, 1918.
Culick, F. E. C., and H. R. Jex: “Aerodynamics, Stability and Control o f the 1903
Wright Flyer,” pp. 19-43 in Howard Wolko (ed.), The Wright Flyer; An
Engineering Perspective, Smithsonian Press, Washington, 1987.
509
510
chapter
6
Elements of Airplane Performance
Dommasch, D. O., S. S. Sherbey, and T. F. Connolly, Airplane Aerodynamics, 3d ed.,
Pitman, New York, 1961.
Duchéne, Captain: The Mechanics of the Airplane: A Study of the Principles of Flight
(transí, by J. H. Ledeboer and T. O ’B. Hubbard), Longmans, London, 1917.
Hale, F. J.: Introduction to Aircraft Performance, Selection and Design, Wiley, New
York, 1984.
Judge, A. W.: Handbook of Modern Aeronautics, Appleton, London, 1919.
Loftin, Lawrence: Quest for Performance: The Evolution of Modern Aircraft, NASA
SP-468, 1985.
McCormick, B. W.: Aerodynamics, Aeronautics and Flight Mechanics, Wiley, New
York, 1979.
Perkins, C. D., and R. E. Hage: Airplane Performance, Stability and Control, Wiley,
New York, 1949.
Shevell, R. S.: Fundamentals of Flight, Prentice-Hall, Englewood Cliffs, NJ, 1983.
Problems
6.1
6.2
6.3
6.4
Consider an airplane patterned after the twin-engine Beechcraft Queen Air
executive transport. The airplane weight is 38,220 N, wing area is 27.3 m2, aspect
ratio is 7.5, Oswald efficiency factor is 0.9, and zero-lift drag coefficient
CD,o = 0.03. Calculate the thrust required to fly at a velocity of 350 km/h at
(a) standard sea level and (b) an altitude of 4.5 km.
An airplane weighing 5000 lb is flying at standard sea level with a velocity of
200 mi/h. At this velocity, the L /D ratio is a maximum. The wing area and aspect
ratio are 200 ft2 and 8.5, respectively. The Oswald efficiency factor is 0.93.
Calculate the total drag on the airplane.
Consider an airplane patterned after the Fairchild Republic A -10, a twin-jet attack
aircraft. The airplane has the following characteristics: wing area = 47 m2, aspect
ratio = 6.5, Oswald efficiency factor = 0.87, weight = 103,047 N, and zero-lift
drag coefficient = 0.032. The airplane is equipped with two jet engines with
40,298 N of static thrust each at sea level.
a. Calculate and plot the power-required curve at sea level.
b. Calculate the maximum velocity at sea level.
c. Calculate and plot the power-required curve at 5-km altitude.
d. Calculate the maximum velocity at 5-km altitude. (Assume the engine thrust
varies directly with free-stream density.)
Consider an airplane patterned after the Beechcraft Bonanza V-tailed, single­
engine light private airplane. The characteristics of the airplane are as follows:
aspect ratio = 6.2, wing area = 181 ft2, Oswald efficiency factor = 0.91,
weight = 3000 lb, and zero-lift drag coefficient = 0.027. The airplane is powered
by a single piston engine of 345 hp maximum at sea level. Assume the power of
the engine is proportional to free-stream density. The two-blade propeller has an
efficiency of 0.83.
a. Calculate the power required at sea level.
b. Calculate the maximum velocity at sea level.
Problems
c.
Calculate the power required at 12,000-ft altitude.
d.
Calculate the maximum velocity at 12,000-ft altitude.
6.5
From the information generated in Prob. 6.3, calculate the maximum rate o f climb
for the twin-jet aircraft at sea level and at an altitude o f 5 km.
6.6
From the information generated in Prob. 6.4, calculate the maximum rate o f climb
for the single-engine light plane at sea level and at 12,000-ft altitude.
6.7
From the rate-of-climb information for the twin-jet aircraft in Prob. 6.5, estimate
the absolute ceiling o f the airplane. (Note: Assum e maximum R/C varies linearly
with altitude— not a precise assumption, but not bad either.)
6.8
From the rate-of-climb information for the single-engine light plane in Prob. 6.6,
estimate the absolute ceiling o f the airplane. (Again, make the linear assumption
described in Prob. 6.7.)
6.9
The maximum lift-to-drag ratio o f the World War I Sopwith Camel was 7.7. If the
aircraft is in flight at 5000 ft when the engine fails, how far can it glide in terms o f
distance measured along the ground?
6.10 For the Sopwith Camel in Prob. 6.9, calculate the equilibrium glide velocity at
3000 ft, corresponding to the minimum glide angle. The aspect ratio o f the
airplane is 4.11, the Oswald efficiency factor is 0.7, the weight is 1400 lb, and the
wing area is 231 ft2.
6.11 Consider an airplane with a zero-lift drag coefficient o f 0.025, an aspect ratio o f
6.72, and an Oswald efficiency factor o f 0.9. Calculate the value o f ( ¿ / D ) max.
6.12 Consider the single-engine light plane described in Prob. 6.4. If the specific fuel
consumption is 0.42 lb o f fuel per horsepower per hour, the fuel capacity is
44 gal, and the maximum gross weight is 3400 lb, calculate the range and
endurance at standard sea level.
6.13 Consider the twin-jet airplane described in Prob. 6.3. The thrust specific fuel
consumption is 1.0 N o f fuel per newton o f thrust per hour, the fuel capacity is
1900 gal, and the maximum gross weight is 136,960 N. Calculate the range and
endurance at a standard altitude o f 8 km.
6.14 Derive Eqs. (6.80) and (6.81).
6.15 Derive Eqs. (6.86) and (6.87).
6.16 Estimate the sea-level liftoff distance for the airplane in Prob. 6.3. Assum e a
paved runway. A lso, during the ground roll, the angle o f attack is restricted by the
requirement that the tail not drag the ground. Hence, assume C ¿ ,max during the
ground roll is limited to 0.8. A lso, when the airplane is on the ground, the wings
are 5 ft above the ground.
6.17 Estimate the sea-level liftoff distance for the airplane in Prob. 6.4. A ssum e a
paved runway, and Cl,mm = 1.1 during the ground roll. When the airplane is on
the ground, the w ings are 4 ft above the ground.
6.18 Estimate the sea-level landing ground roll distance for the airplane in Prob. 6.3.
A ssum e the airplane is landing at full gross weight. The maximum lift
coefficient with flaps fully em ployed at touchdown is 2.8. After touchdown,
assume zero lift.
6.19 Estimate the sea-level landing ground roll distance for the airplane in Prob. 6.4.
A ssum e the airplane is landing with a weight o f 2900 lb. The maximum lift
coefficient with flaps at touchdown is 1.8. After touchdown, assume zero lift.
51 1
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chapter
6
Elements of Airplane Performance
6.20 For the airplane in Prob. 6.3, the sea-level corner velocity is 250 mi/h, and the
6.21
6.22
6.23
6.24
6.25
6.26
6.27
maximum lift coefficient with no flap deflection is 1.2. Calculate the minimum
turn radius and maximum turn rate at sea level.
The airplane in Prob. 6.3 is flying at 15,000 ft with a velocity of 375 mi/h.
Calculate its specific energy at this condition.
Derive Eq. (6.44).
From the data shown in Fig. 6.2, estimate the value of the Oswald efficiency
factor for the Lockheed C-141A. The wing aspect ratio of the C -141A is 7.9.
Since the end of World War II, various claims have appeared in the popular
aviation literature of instances where powerful propeller-driven fighter airplanes
from that period have broken the speed of sound in a vertical, power-on dive. The
purpose of this problem is to show that such an event is technically not possible.
Consider, for example, the Grumman F6F-3 Hellcat, a typical fighter from World
War II. For this airplane, the zero-lift drag coefficient (at low speeds) is 0.0211,
the wing planform area is 334 ft2, and the gross weight is 12,441 lb. It is powered
by a Pratt and Whitney R-2800 reciprocating engine that, with supercharging to
an altitude of 17,500 ft, produces 1500 horsepower. Consider this airplane in a
full-power vertical dive at (a) 30,000 ft and then (b) 20,000 ft. Prove at these two
altitudes that the airplane can not reach Mach 1.
Note: The aerodynamic characteristics of this airplane at Mach 1 have not
been measured. So you will have to make some reasonable assumptions. For
example, what is the zero-lift drag coefficient at Mach 1? As an estimate, we can
obtain from NACA TR 916 a zero-lift drag coefficient for the North American P51 Mustang, which, when extrapolated to Mach 1, shows an increase of 7.5 over
its low-speed v