Fluid Mechanics and Its Applications
Roelof Vos
Saeed Farokhi
Introduction
to Transonic
Aerodynamics
Fluid Mechanics and Its Applications
Volume 110
Series editor
André Thess, German Aerospace Center, Institute of Engineering
Thermodynamics, Stuttgart, Germany
Founding Editor
René Moreau, Ecole Nationale Supérieure d’Hydraulique de Grenoble,
Saint Martin d’Hères Cedex, France
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It is a widely held view that it is the interdisciplinary subjects that will receive
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advancement. Fluids have the ability to transport matter and its properties as well
as to transmit force, therefore fluid mechanics is a subject that is particularly open
to cross fertilization with other sciences and disciplines of engineering. The
subject of fluid mechanics will be highly relevant in domains such as chemical,
metallurgical, biological and ecological engineering. This series is particularly
open to such new multidisciplinary domains.
The median level of presentation is the first year graduate student. Some texts
are monographs defining the current state of a field; others are accessible to final
year undergraduates; but essentially the emphasis is on readability and clarity.
More information about this series at http://www.springer.com/series/5980
Roelof Vos Saeed Farokhi
•
Introduction to Transonic
Aerodynamics
123
Roelof Vos
Faculty of Aerospace Engineering
Delft University of Technology
Delft
The Netherlands
Saeed Farokhi
Department of Aerospace Engineering
The University of Kansas
Lawrence, KS
USA
ISSN 0926-5112
ISSN 2215-0056 (electronic)
Fluid Mechanics and Its Applications
ISBN 978-94-017-9746-7
ISBN 978-94-017-9747-4 (eBook)
DOI 10.1007/978-94-017-9747-4
Library of Congress Control Number: 2015930202
Springer Dordrecht Heidelberg New York London
© Springer Science+Business Media Dordrecht 2015
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This book is dedicated to our wives,
Carola and Mariam.
Preface
Within aerodynamics, the transonic Mach range is a much-studied topic among
industrial and academic institutions. Many of today’s airliners cruise at Mach
numbers where both subsonic and supersonic flow exist. Terms such as drag
divergence, buffet, and transonic dip are all associated with transonic flow conditions. Transonic aerodynamics has, to a large degree, dominated the exterior design
of high-subsonic aircraft for the past 60 years. However, textbooks on compressible
aerodynamics often focus on the supersonic Mach range and less on the phenomena
in the transonic Mach regime. Therefore, a comprehensive course on transonic
aerodynamics is seldom part of an aerospace engineering curriculum.
This discrepancy between the importance of transonic aerodynamics in practice
and the lack of a comprehensive course on the subject is often explained by the
complexity of the subject matter. Indeed, the governing equations that include all
the relevant aerodynamic phenomena cannot be easily reduced to something
equivalent to a thin airfoil theory or a lifting-line theory. Also, many correction
factors such as the well-known Prandt-Glauert compressibility correction do not
hold in the transonic regime. In order to predict the performance of a wing or wing
section, one therefore often relies on computer programs that solve a numerical
implementation of the governing equations of motion. But teaching students how to
operate a piece of software is not the same as teaching students the physics of how
and why certain phenomena occur when a body is subjected to transonic flow
conditions. Therefore, this textbook was written to teach students about the nature
of transonic flow and how it can be captured in mathematical equations.
This textbook serves as an introduction to the subject of transonic aerodynamics.
In eight chapters we present a quantitative and qualitative assessment of subsonic,
supersonic, and transonic flow about bodies in two and three dimensions. We have
included relevant analytical analysis methods that allow students to practice with
the subject matter. The book contains numerous examples and every chapter closes
with a list of problems. Some subjects are treated more from a numerical perspective (e.g., shock and expansion theory), while others are discussed more from a
qualitative point of view (e.g., shock-boundary-layer interaction). Where possible,
numerical examples and methods have been included to enhance the understanding
vii
viii
Preface
of each subject. The book contains 60 examples and more than 200 practice
problems.
This textbook is intended primarily for senior undergraduate students or graduate
students with prior knowledge in aerodynamics. Although we repeat the fundamental equations and flow characteristics in the beginning of the book, we do
assume that the student has had a course on subsonic aerodynamics and is familiar
with its fundamentals. Even though knowledge of transonic aerodynamics is
important for many internal flow applications (e.g., turbo machinery, engine
intakes, exhausts, etc.) the present textbook primarily focuses on external aerodynamics with limited applications to internal flows. Examples are targeted mainly
toward wings and bodies exposed to a transonic flow field. Many of the examples
reference real aircraft or wing components. Therefore, a strong connection is
present between the content of this textbook and the subject of transonic aircraft
design. To understand why a modern high-subsonic aircraft is designed the way it
is, requires one to understand the subject matter of this textbook.
Acknowledgments
Writing this book has been a privilege. Naturally, we would not have been able to
do so without the support of our respective universities. We therefore would like to
thank the University of Kansas and Delft University of Technology for providing us
with the time and resources to write this book. We would also like to thank the
publisher, who, based on a 135-page summary, trusted that we would extend it to its
current form. We would also like to acknowledge Dr. Luca Guadani and Dr. Ali
Elham who proofread various chapters and performed some of the numerical calculations to support the examples in this book. A special thank you to emeritus
professor Egbert Torenbeek for providing us with meticulous feedback on the
content of various chapters. Finally, we would like to thank the following persons
on both sides of the Atlantic, who over the course of multiple years, aided in the
preparation of this document: Mr. Thomas Statsny, Ms. Lisanne van Veen,
Mr. Maarten Broekhuizen, Mr. Kevin Haagen, Ms. Maaike Weerdesteyn, and
Mr. Amool Raina.
Delft, The Netherlands, 2014
Lawrence, KS, USA
Roelof Vos
Saeed Farokhi
Contents
1
Introduction and Historic Perspective . . . . . . . . . . . . . . . . . . . . .
1.1 From Subsonic to Supersonic Flight . . . . . . . . . . . . . . . . . . .
1.2 Definition of the Transonic Flow Domain . . . . . . . . . . . . . . .
1.3 Transonic Wind Tunnel Experiments . . . . . . . . . . . . . . . . . . .
1.4 Transonic Aerodynamics of Wings and Bodies . . . . . . . . . . . .
1.5 Transonic Flow Calculations . . . . . . . . . . . . . . . . . . . . . . . .
1.6 Outline of Present Textbook. . . . . . . . . . . . . . . . . . . . . . . . .
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1
4
5
8
14
16
18
2
Review of Fundamental Equations. . . . . . . . . . . . . . . . . . . . . . . .
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Review of Partial Differential Equations . . . . . . . . . . . . . . . .
2.2.1
One-Dimensional Wave Equation and Solution
by D’Alembert . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2
One-Dimensional Heat Equation and Solution
by Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.3
Conservation Form of PDEs. . . . . . . . . . . . . . . . . . .
2.2.4
Classification of Partial Differential Equations . . . . . .
2.3 Review of Vector Algebra . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1
Vectors, Vector Fields, and Scalar Fields . . . . . . . . . .
2.3.2
Gradient of a Scalar Field . . . . . . . . . . . . . . . . . . . .
2.3.3
Divergence of a Vector Field . . . . . . . . . . . . . . . . . .
2.3.4
Curl of a Vector Field . . . . . . . . . . . . . . . . . . . . . . .
2.3.5
Relation Between Volume, Surface,
and Line Integrals . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Review of Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1
Perfect Gas Relations . . . . . . . . . . . . . . . . . . . . . . .
2.4.2
First Law of Thermodynamics . . . . . . . . . . . . . . . . .
2.4.3
Second Law of Thermodynamics . . . . . . . . . . . . . . .
2.4.4
Isentropic Relations. . . . . . . . . . . . . . . . . . . . . . . . .
21
21
22
22
29
33
35
38
38
40
41
42
42
43
44
46
48
50
ix
x
Contents
2.5
Equations of Fluid Motion . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.1
Conservation of Mass . . . . . . . . . . . . . . . . . . . . . . .
2.5.2
Conservation of Linear Momentum . . . . . . . . . . . . . .
2.5.3
Conservation of Energy . . . . . . . . . . . . . . . . . . . . . .
2.5.4
Conservation Form of the Navier-Stokes Equations . . .
2.6 Reynolds-Averaged Navier-Stokes Equations . . . . . . . . . . . . .
2.6.1
Incompressible Reynolds-Averaged Equations
of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6.2
Compressible Reynolds-Averaged Equations
of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6.3
Turbulence Modeling: The k-Epsilon Model. . . . . . . .
2.7 Equations of Motion for Inviscid Flows . . . . . . . . . . . . . . . . .
2.7.1
Euler Equations . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7.2
Potential Flow Equation. . . . . . . . . . . . . . . . . . . . . .
2.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
4
51
51
53
56
60
62
62
65
67
69
69
72
75
79
Transonic Similarity Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Linearized Compressibility Corrections . . . . . . . . . . . . . . . . .
3.2.1
2-D Subsonic Flow . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.2
Other Subsonic Compressibility Corrections . . . . . . . .
3.2.3
2-D Supersonic Flow. . . . . . . . . . . . . . . . . . . . . . . .
3.2.4
The Principle of Superposition . . . . . . . . . . . . . . . . .
3.2.5
Slender Bodies of Revolution in Subsonic
and Supersonic Flow—Linear Theory . . . . . . . . . . . .
3.3 Transonic Small Disturbance Theory . . . . . . . . . . . . . . . . . . .
3.4 Transonic Similarity Parameters . . . . . . . . . . . . . . . . . . . . . .
3.4.1
Other Transonic Similarity Parameters . . . . . . . . . . . .
3.5 3-D Planar and Axisymmetric Slender Bodies . . . . . . . . . . . .
3.6 Hodograph Transformation. . . . . . . . . . . . . . . . . . . . . . . . . .
3.7 Empirical Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.8 Approximate Location of Detached Shocks . . . . . . . . . . . . . .
3.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
81
81
82
85
89
99
103
106
114
117
120
123
126
127
130
134
144
Shock-Expansion Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Lift and Wave Drag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Bi-Convex Airfoil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Axisymmetric and Slender Bodies. . . . . . . . . . . . . . . . . . . . .
4.5 Examples and Applications . . . . . . . . . . . . . . . . . . . . . . . . .
145
145
169
173
183
188
Contents
xi
4.5.1
The Shape and Geometric Parameters of an
(Axisymmetric) Ogive Nose . . . . . . . . . . . . . . . . . . .
4.5.2
Extension to Transonic Speeds . . . . . . . . . . . . . . . . .
4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
6
Method of Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 2-D Irrotational Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Design of a 2-D Supersonic Minimum-Length
Nozzle (MLN) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4 Wave-Field Method Versus Lattice-Point Approach. . . . . . . . .
5.5 Axisymmetric Irrotational Flows . . . . . . . . . . . . . . . . . . . . . .
5.6 Examples and Applications . . . . . . . . . . . . . . . . . . . . . . . . .
5.6.1
Design of a Family of Supersonic Nozzles . . . . . . . . .
5.6.2
Deflecting Jet . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6.3
Non-uniform Inlet Condition: Example
of Source Flow. . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6.4
Streamlines and Ducts . . . . . . . . . . . . . . . . . . . . . . .
5.6.5
Curved Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Aerodynamics of Non-lifting Bodies . . . . . . . . . . . . . . . . . . . . . . .
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Pressure Distribution Over Nonlifting Bodies . . . . . . . . . . . . .
6.2.1
Subsonic Inviscid Flow . . . . . . . . . . . . . . . . . . . . . .
6.2.2
Effect of Subcritical Compressibility . . . . . . . . . . . . .
6.3 Wave Drag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3.1
Calculation of Pressure Drag
by Use of Momentum Theory . . . . . . . . . . . . . . . . .
6.3.2
Supersonic Wave Drag of a Slender
Axis-Symmetric Body . . . . . . . . . . . . . . . . . . . . . . .
6.3.3
Optimum Shape of a Slender Body of Revolution. . . .
6.3.4
Examples of Transonic Area Ruling . . . . . . . . . . . . .
6.4 Fundamentals of Boundary-Layer Flow . . . . . . . . . . . . . . . . .
6.4.1
Laminar Boundary Layer . . . . . . . . . . . . . . . . . . . . .
6.4.2
Turbulent Boundary Layer . . . . . . . . . . . . . . . . . . . .
6.4.3
Shock-Wave Boundary-Layer Interaction
in Transonic Flow. . . . . . . . . . . . . . . . . . . . . . . . . .
6.5 Boundary-Layer Computations . . . . . . . . . . . . . . . . . . . . . . .
6.5.1
Boundary Layer Equations for Steady
Incompressible Flow . . . . . . . . . . . . . . . . . . . . . . . .
6.5.2
Laminar Boundary Layer . . . . . . . . . . . . . . . . . . . . .
188
192
194
209
211
211
212
222
232
251
256
256
256
258
258
263
265
273
275
275
279
279
284
287
291
294
296
300
302
303
313
319
324
325
327
xii
Contents
6.5.3
Turbulent Boundary Layer . . . . . . . . . . . . . . . . . . . .
6.5.4
Boundary Layer Transition. . . . . . . . . . . . . . . . . . . .
6.6 Interference Drag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.6.1
Interference Drag in Subsonic Conditions . . . . . . . . .
6.6.2
Interference Drag in Transonic Conditions . . . . . . . . .
6.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
331
343
349
350
352
356
364
7
Airfoil Aerodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2 Pressure Distribution About Airfoils . . . . . . . . . . . . . . . . . . .
7.3 Laminar-Flow Airfoils . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.4 Supercritical Airfoils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.4.1
Shock-Free Supercritical Airfoil . . . . . . . . . . . . . . . .
7.4.2
Supercritical Airfoils with Shocks . . . . . . . . . . . . . . .
7.4.3
Sonic Rooftop Airfoils . . . . . . . . . . . . . . . . . . . . . .
7.4.4
Effect of Trailing Edge Geometry . . . . . . . . . . . . . . .
7.5 Low-Speed Stall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.5.1
Qualification of Low-Speed Stall . . . . . . . . . . . . . . .
7.5.2
Reynolds Effect on Maximum Lift Coefficient . . . . . .
7.5.3
Mach Effect on Maximum Lift Coefficient. . . . . . . . .
7.5.4
High-Lift Devices . . . . . . . . . . . . . . . . . . . . . . . . . .
7.6 High-Speed Stall. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.6.1
Flow Separation at the Shock Foot . . . . . . . . . . . . . .
7.6.2
Transonic Buffet . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
367
367
368
372
375
379
383
390
390
393
394
397
399
403
405
406
411
419
425
8
Aerodynamics of Swept Wings . . . . . . . . . . . . . . . . . . . . . . . . . .
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2 The Advantages of Wing Sweep. . . . . . . . . . . . . . . . . . . . . .
8.2.1
Theory of Wing Sweep . . . . . . . . . . . . . . . . . . . . . .
8.2.2
Effect on Drag Divergence Mach Number . . . . . . . . .
8.2.3
Effect on Wave Drag Coefficient . . . . . . . . . . . . . . .
8.3 Inviscid Flow over Swept Wings . . . . . . . . . . . . . . . . . . . . .
8.3.1
Flow over Infinite Swept Wings . . . . . . . . . . . . . . . .
8.3.2
Flow over Finite Swept Wings . . . . . . . . . . . . . . . . .
8.4 Viscous Flow over Swept Wings . . . . . . . . . . . . . . . . . . . . .
8.4.1
Three-Dimensional Boundary Layer
over Swept Wings . . . . . . . . . . . . . . . . . . . . . . . . .
8.4.2
Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.4.3
Separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
427
427
428
430
433
441
449
450
453
461
461
464
470
Contents
xiii
8.5
474
476
481
490
491
Aeroelasticity of Swept Wings . . . . . . . . . . . . . . . . . . . . . . .
8.5.1
Static Deformation . . . . . . . . . . . . . . . . . . . . . . . . .
8.5.2
Reduced Control-Surface Effectiveness . . . . . . . . . . .
8.5.3
Structural Divergence . . . . . . . . . . . . . . . . . . . . . . .
8.5.4
Flutter at High Subsonic Mach Numbers . . . . . . . . . .
8.5.5
Prediction and Detection of Aeroelastic
Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
496
499
509
Appendix A: Isentropic Flow Table . . . . . . . . . . . . . . . . . . . . . . . . . .
513
Appendix B: Normal Shock Table . . . . . . . . . . . . . . . . . . . . . . . . . . .
521
Appendix C: Prandtl-Meyer Function. . . . . . . . . . . . . . . . . . . . . . . . .
527
Partial Answers to Selected Problems . . . . . . . . . . . . . . . . . . . . . . . . .
529
Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
545
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
551
Chapter 1
Introduction and Historic Perspective
Abstract Airliners and jet fighters are often designed for Mach numbers close to
the speed of sound. At those speeds the flow domain is termed transonic and contains both subsonic and supersonic flow. Typical aerodynamic characteristics in this
domain are shock waves, drag divergence, and transonic buffet. To suppress these
adverse effects or postpone them to higher Mach numbers, multiple design measures
can be taken. A supercritical airfoil can be designed that postpones the onset of
shock waves. In addition, wing sweep can be introduced or the airplane can be area
ruled to further reduce the adverse transonic effects. This chapter presents some of
these practical measures and their effect. It also discusses the definition of transonic
aerodynamics and why it is beneficial to fly at Mach numbers where transonic flow
exits. Furthermore, it demonstrates the challenges associated with simulating transonic flow and transonic windtunnel experiments. Finally, an outline of the present
textbook is given.
1.1 From Subsonic to Supersonic Flight
On October 14th, 1947 a new era of flight was entered when Chuck Yeager officially
became the first person to surpass the speed of sound in fully controlled flight. The
rocket-powered Bell X-1 (Fig. 1.1) flew at a Mach number of 1.06, thereby proving
that the sound barrier was in reality not as impenetrable as sceptics had assumed.
This event was the culmination of research efforts over the preceding decades that
attempted to unveil the secrets of flow near Mach one, i.e., transonic flow. Von
Kármán first coined the word transonic to mark this peculiar regime in flight where
a mixture of subsonic and supersonic flow co-exists.
The very first appreciation of transonic flow and its effects on the drag coefficient
dates back to 1910 when the drag coefficient was shown to increase for Mach numbers
close to unity [2]. At that time, transonic and supersonic flow was encountered only by
ballistic projectiles such as ammunition. It is therefore no surprise that the fuselage of
the first supersonic aircraft was shaped like a machine-gun bullet. The first production
aircraft with an air-breathing engine that achieved supersonic speed in a dive was
the XF-86-Sabre, although it was never intended to fly that fast. Only the second
© Springer Science+Business Media Dordrecht 2015
R. Vos and S. Farokhi, Introduction to Transonic Aerodynamics,
Fluid Mechanics and Its Applications 110, DOI 10.1007/978-94-017-9747-4_1
1
2
1 Introduction and Historic Perspective
Fig. 1.1 Bell X-1, the first aircraft to officially fly supersonic (Photo USAF)
generation of jet-powered fighter aircraft were designed for (low) supersonic speeds
and featured typical design characteristics (such as an area-ruled fuselage) to reduce
their peak drag coefficient as the airplane surpassed Mach 1.
It took more than 25 years after the first supersonic flight for the first transport
aircraft to fly supersonic. The Russian Tupolev 144 and the Anglo-French Concorde
were the first aircraft that cruised beyond Mach 2 and were thought to initiate a new
era in aviation: supersonic flight [3]. As is generally known now, there never was
a true supersonic era. Concorde flew between 1976 and 2003, but when it retired
there was no other supersonic aircraft to replace it. Environmental issues (supersonic
boom) as well as fuel cost had limited the feasibility of the aircraft. Instead, highsubsonic jet transport aircraft have dominated the civil aviation market for the past
60 years and their number increased rapidly over the 1970s and 80s. Being relatively
fuel efficient, fast and quiet (no supersonic boom), they have proven to be the best
solution for past and present-day airlines.
Jet transports fly in the lower transonic regime, also known as the high-subsonic
regime. It is therefore of interest to look more closely to the flow phenomena that
are taking place at these speeds and how this effects the design and operation of the
aircraft. To be able to justify a particular aircraft design we need to have a strong
knowledge about the flow phenomena and how we can influence them to meet the
requirements set by the costumers while satisfying the constraints of the aviation
authorities. One of the general goals of aerodynamic design is to maximize the
range parameter (Mach number times lift-to-drag ratio, M × L/D). According to
the Bréguet range equation, the maximum cruise range (Rmax ) for a given thrust-toweight ratio is found when the product of Mach number (M) and lift-to-drag ratio
L/D) is maximized [17]:
Rmax =
M
L
D
×
max
a
ln
SFC
Wbegin cruise
Wend cruise
(1.1)
where SFC is the specific fuel consumption, a the ambient speed of sound, and W
the gross weight of the aircraft. Assume that we can find an airplane shape that
1.1 From Subsonic to Supersonic Flight
3
25
7-4
B74
7-3
74
10
0
A3
-11
MD
R
7E
ER
76
00
B-
A3
00
-6
IM
P
01
1-5
00
L-1
-30
-10
DC
-8
30
-30
1970
B-
1960
DC
-
10
DC
L-1 -10-1
0
01
1-1
00
B74
15
B70
7
7-1
-30
00
00
20
00
Long-Range Jet Aircraft
DC
Range Parameter, M x (L/D)max
supports a lift-to-drag ratio of 10 at a cruise Mach number of M = 0.3. Maximizing
the range parameter can then easily be done by increasing the Mach number. So
flying at high-subsonic speeds (e.g., M = 0.8) has a large impact on the range
factor as long as the aerodynamic efficiency does not deteriorate with Mach number.
We now find ourselves at the heart of the transonic challenge. Cruise speeds of
high-subsonic aircraft are often limited by the drag penalty that is encountered at
Mach numbers approaching unity. Aerodynamic designers of high-subsonic aircraft
are therefore faced with the challenge to increase the Mach number as much as
possible, while keeping the aerodynamic efficiency high. Or, vice versa, maximize
the aerodynamic efficiency at a predefined Mach number. Over the past fifty years,
the range parameter has steadily increased due to improved aerodynamic design of
high-subsonic transport aircraft (see Fig. 1.2). This improvement is attributed to a
better understanding of transonic aerodynamics and the ever improving aerodynamic
prediction techniques.
The wave drag that is associated with high subsonic speeds originates from shock
waves that terminate supersonic flow areas on the wing. The onset of wave drag
causes a slow increase in drag (known as drag creep). The local shock wave on top
of the wing creates a very strong adverse pressure gradient over the boundary layer.
When the shock wave is strong enough, this results in boundary-layer separation
at the shock foot, creating an exponential increase in drag. The Mach number at
which this occurs is called the drag divergence Mach number, Mdd . The definition of
the drag-divergence Mach number is arbitrary. Boeing defines the drag-divergence
Mach number as the Mach number at which the drag coefficient reaches a value of
20 counts above the value in the mid-subsonic region [18]. Airbus and Douglas used
to define it as the Mach number at which dC D /dM = 0.10 [18]. Figure 1.3 shows
the typical drag divergence graphs for four different aircraft. We can see that the
definition of the drag-divergence Mach number has an impact on its magnitude for
a given aircraft. Transonic airfoils that are used on modern transport aircraft (like
the Boeing 727) are designed such that local supersonic flow creates a plateau in
the pressure distribution over part the airfoil’s upper surface. In general, it is desired
1980
1990
2000
Date of Entry into Service
Fig. 1.2 Aerodynamic efficiency (M × L/D) variation with date of entry into service (after Ref. [5])
4
1 Introduction and Historic Perspective
Zero Lift Drag Rise, ΔCD0 (cts)
40
Mdd Douglas definition
Mdd Boeing definition
C-130H
F-106
C-5 A
1 count (ct) = 0.0001
30
727
20
10
0
0.4
0.5
0.6
0.7
0.8
Mach Number, M (-)
0.9
1.0
Fig. 1.3 Typical compressibility drag behavior and definition of the drag-divergence Mach number
(modified from Ref. [16])
to have a weak shock terminating this supersonic region such that the wave drag is
minimized. It was already shown in the 1960s that by careful design of the airfoil’s
upper surface it is possible to decelerate supersonic flow from as high as M = 1.4
isentropically without having a shock, thereby minimizing the wave drag [14].
1.2 Definition of the Transonic Flow Domain
We define transonic flow by a domain which can be subdivided into three subdomains: a subsonic domain, a sonic domain and a supersonic domain [13]. The
sonic domain is usually condensed to a line (two-dimensional) or a surface (threedimensional) which can contain a shock wave. Transonic phenomena occur for both
subsonic and supersonic freestream flows. Figure 1.4 schematically shows the subsonic, transonic, and supersonic flow domains around a body (in this case a twodimensional airfoil). Let us briefly observe the flow domains around the body and
how they change with increasing Mach number.
Initially, the flow is subsonic everywhere around the body. Even though the flow
is accelerated over the body, nowhere in the entire flow domain does the Mach
number increase beyond Mach = 1. When the Mach number increases beyond the
critical Mach number, Mcrit , a supersonic domain starts to form. Due to the shape
of this body, this supersonic domain is likely to form over the part of the body that
possesses the highest curvatures, which is close to the leading edge of the body.
In the present case, a pocket of supersonic flow is formed on the upper and lower
side of the body. We know from aerodynamics that the flow behind the body should
be close to the free stream velocity (which is subsonic). Therefore, the flow in the
supersonic pocket needs to be decelerated. In the present example this deceleration
occurs quite abruptly by means of a normal shock wave that is almost perpendicular to
the airfoil surface. This shock wave decelerates the supersonic flow back to subsonic
1.2 Definition of the Transonic Flow Domain
subsonic
5
upper transonic
lower transonic
shockwave
sonic line
0
Mcrit
supersonic
sonic line
supersonic
domain
bow shock
subsonic
domain
1.0
subsonic
domain
Free-stream
Mach number
transonic flows
Fig. 1.4 The basic classes of external transonic flows (after Ref. [13])
conditions. When the free-stream Mach number is even further increased, the shock
waves on either side of the airfoil start to move aft as more of the body is immersed
in supersonic flow. As the freestream Mach number exceeds unity, a detached bow
shock forms in front of the body. Only part of the flow behind the bow shock is
decelerated to subsonic conditions. The flow behind the oblique branches of this
bow shock remains supersonic downstream. When the freestream Mach number is
increased even more, the subsonic domain becomes smaller and can be deemed to be
‘sensibly’ empty. The Mach number at which this occurs marks the boundary between
transonic and supersonic flow. Even though there is no physical discontinuity when
the flow transitions between supersonic and transonic, the upper freestream Mach
number for the flow to be termed transonic is generally set at 1.3 [6].
1.3 Transonic Wind Tunnel Experiments
One of the main challenges that was associated with the flight of the Bell X-1 was
the fact that there was no valid wind tunnel data available for speeds between Mach
0.95 and Mach 1.1. The reason for this lack of transonic wind tunnel data stems from
choking of the test section at a freestream Mach number of 0.95. Until the mid 1940s
it was therefore impossible to generate accurate wind tunnel results [1]. The choking
phenomenon had hindered transonic wind tunnel tests at high Mach numbers during
the 1930s and 40s and needed to be resolved if accurate data was to be obtained. On
the other hand, open wind tunnels proved to be unsuitable for transonic flow due to
its high susceptibility to atmospheric disturbances.
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1 Introduction and Historic Perspective
A break-through came in the late 1940s (a few years after the Bell X-1 had made
its maiden flight). Slotted test sections were introduced, where thin, rectangular slots
were made such that the displaced airflow could move out of the tunnel into a socalled plenum chamber that surrounded the test section. By doing this, the blockage
problem was essentially solved and the flows could be accelerated through Mach 1
without any problems. These slotted tunnels were essentially the first transonic wind
tunnels and became operational in the early 1950s. Figure 1.5 shows the 16-foot
transonic wind tunnel that is operated by NASA Langley. Later it was found that the
improvement is even greater when the walls are not slotted but perforated [21].
Apart from wind-tunnel blockage, transonic wind tunnel tests faced another problem: reflected shock waves from the walls. This was particularly an issue for transonic
flows because the shock on the airfoil was close-to normal to the surface of the body.
Therefore, the reflected shock wave interfered with the flow over the body, rather
than with the flow downstream of the body as would be the case for oblique shock
waves [15]. The requirement on the wind tunnel walls was therefore to effectively
cancel the reflected shock waves by producing streamlines downstream of the shock
that corresponded to the flow downstream of the shock. In this respect, porous walls
proved to be superior to slotted walls in which the higher pressure in the viscous
slots extended upstream of the shocks thereby distorting the flow.
In addition to the problem of shock-wave reflection, the growth of the boundary
layer on the walls of the test section proved to be a source of inaccuracy. The thickening of the boundary layer was caused by small wind-tunnel-wall irregularities. It was
found that a slightly converging wind tunnel wall resulted in a much thinner boundary layer compared to a parallel or slightly diverging wall. The wall inclination was
therefore a parameter that could be changed to reduce the boundary-layer thickness
growth [7]. This problem was targeted in the 1970s. One transonic wind tunnel
incorporated theoretical correction models that used input from a series of test runs.
Fig. 1.5 The present-day slotted-wall test section of the NASA Langley 16-foot transonic tunnel [4]
1.3 Transonic Wind Tunnel Experiments
7
It was then calculated how the walls needed to be oriented such that the wall interference was minimized and the most accurate results were obtained [22].
Finally, it was found that considerable turbulence formed in the wind tunnel that
modified the boundary layer on the test article in an unpredictable way. Even though
the walls were positioned such that their influence was minimized, this phenomenon
had a considerable influence on the accuracy of the aerodynamic measurements. It
was found that the source of the turbulence came from acoustic disturbances from the
slots or perforations. A practical approach yielded a solution to this problem. Simple
splitter plates in each of the holes diminished the acoustic waves for the perforated
walls. Analogously, a thin wire mesh that covered the slots had the same effect in
case of slotted walls.
Wind tunnel tests are conducted on scale models of aerodynamic bodies that
resemble the true model very accurately. All the conditions in the tunnel are set such
as to match realistic conditions as well as possible. However, the Reynolds number
effect is one characteristic that is hard to simulate when scaling down to wind tunnel
sizes. Since the Reynolds number is dominant in the determination of friction drag,
boundary layer thickness, transition point, and shock position, it is evident that testing
at lower Reynolds number can have severe impact on the predicted aerodynamic
performance of the test object.
For that reason, the idea arose in the early 1970s to modify the test gas such as to
match the Reynolds number encountered by the wind tunnel model to the Reynolds
number of the full-scale body. By increasing the density of the test gas the Reynolds
number increased linearly. To increase the density, three different measures were
taken: the air in the tunnel was replaced with a heavier gas (e.g., carbondioxide), the
gas was cooled by injecting liquid nitrogen into the flow, and the air was pressurized.
Combining these measures resulted in a Reynolds number which could be twenty
times as high as for conventional wind tunnels [8].
Combine all these efforts with the measures that were described in the previous
paragraphs and it becomes clear that transonic cryogenic wind tunnels are among the
most complex aerodynamic facilities to test aerodynamic bodies. Currently, there are
only three large scale facilities in the world: one in the United States (National Transonic Facility, NTF) and two in Cologne, Germany (European Transonic Windtunnel,
ETW, and Kryo Kanal Köln). The ETW can cool the flow to 110 K in combination
with pressures up to 450 kPa and a Mach number of 1.3 [8]. Advantages of this
tunnel are that compressibility effects, friction effects and deformation effects can
be studied independently by varying Mach number, dynamic pressure and Reynolds
number independently of one another. A photograph taken in the test section of the
ETW is presented in Fig. 1.6.
8
1 Introduction and Historic Perspective
Fig. 1.6 ETW wind tunnel with aircraft model (Photo ETW; printed with permission)
1.4 Transonic Aerodynamics of Wings and Bodies
In parallel to the development of the transonic wind tunnel, a series of aircraft design
modifications was investigated to reduce the drag that occurred at high-subsonic
conditions. This included modifications to airfoil shapes, the planform layout of
wings, and the shape of nonlifting bodies such as the fuselage. Both in Europe and
in the United States, great efforts were made to postpone the onset of the sharp drag
rise and to reduce the drag peak that was encountered at high-subsonic conditions.
To appreciate wing aerodynamics in transonic flow, we examine the flow patterns
on airfoils in this regime. In Fig. 1.7 the four stages of transonic flow about an airfoil
are shown. In part (a) the lower subsonic case shows the appearance of a supersonic
bubble that is terminated in a shock on the suction surface of a generic airfoil. The
pressure surface remains entirely subsonic. In part (b) the upper subsonic case shows
the growth of the supersonic bubble on the entire suction surface as well as the
appearance of supersonic flow and a shock on the pressure surface. In part (c) the
freestream is sonic and except for a small subsonic zone near the stagnation point of
the airfoil, the upper and lower surfaces are entirely supersonic with a distinct trailing
edge shock system as shown. The case of slightly supersonic freestream is shown
in part (d). The formation of the bow shock ahead of the airfoil creates a subsonic
region, followed by a supersonic zone on the upper and lower surfaces of the airfoil.
Again the terminal shocks are at the trailing edge, as shown. Note that Fig. 1.7 shows
the Mach distributions (instead of pressure distributions) on the airfoil.
To postpone the onset of a strong shock wave, so-called “supercritical” airfoils
were developed. Supercritical airfoils were developed by Pearcey [14] at the National
1.4 Transonic Aerodynamics of Wings and Bodies
9
(b)
(a)
M >1
M∞ < 1
M∞< 1
(c)
M >1
M∞= 1
(d)
M >1
M∞ >1
M >1
M >1
M >1
Upper
Surface
M
M
M
M
1
1
1
1
Lower
Surface
0
M ∞< 1
0
M ∞< 1
0
M∞ =1
0
M∞ > 1
Fig. 1.7 Four flow patterns and corresponding Mach distributions representative of transonic flow
over an airfoil (after Ref. [12]). a M∞ < 1, b M∞ < 1, c M∞ = 1, d M∞ > 1
Physical Laboratory (UK) and Whitcomb [23] at NASA in the early 1960s. The
pressure distributions on a “conventional” (NACA 6-series) airfoil and a supercritical
airfoil are shown in Fig. 1.8. The strong shock that abruptly transitions the supersonic
flow to subsonic on the suction surface of the conventional airfoil (in transonic Mach
number) is replaced by a relatively weak shock that appears further aft on the airfoil.
The corresponding pressure distributions show the difference between a smooth
transition and an abrupt transition to subsonic flow on the two airfoils. We also note
that the supercritical airfoil has a concave curvature near the trailing edge on its
pressure surface. This curvature reversal on the aft pressure surface causes a higher
loading on the airfoil and thus a larger contribution to lift.
We have superimposed a critical pressure coefficient line on the data of Ref.
[11] in Fig. 1.9 to see (or separate) the supersonic zone on the airfoil. The pressure
distribution on a supercritical airfoil at Mach 0.8 with a lift coefficient of 0.54 and a
chord Reynolds number of three million is shown. Noteworthy features are significant
aft loading on the airfoil, as well as a relatively smooth compression on the suction
surface at the end of supersonic bubble. A footprint of flow separation, i.e., a flat C p
profile downstream of the shock, on the suction surface is visible in the data. We also
note that the pressure surface never reaches sonic flow. This pressure distribution is
typical for airfoils that were developed for high subsonic Mach numbers.
Another measure to postpone the drag divergence is the application of swept
wings. The first incarnations of swept-back wings date back to the early 1900s,
when John William Dunne (an Irish aeronautical engineer) designed and built various
tailless biplanes with aft-swept wings. In Fig. 1.10 the Dunne D.8 is shown taking off
from the airfield at the Farnborough air show. Dunne applied wing sweep to shift the
neutral point (i.e., the aerodynamic center of the aircraft) further aft, such that it would
lay behind the airplane’s center of gravity. This ensured a positive static stability
margin, which is an important prerequisite for having acceptable flying qualities for
manually controlled aircraft. Many other tailless aircraft relied on wing sweep to
10
1 Introduction and Historic Perspective
(a)
Relatively
strong
shock
(c)
M>1
M>1
Relatively
weak
shock
M>1
(d)
(b)
Cp
Cp
(-)
Cp, crit
(-)
Cp, crit
(+)
(+)
Supercritical airfoil (13% thick)
M∞= 0.79
NACA 642-A215 airfoil
M∞= 0.69
Fig. 1.8 Pressure distribution on a conventional (NACA 6-series) airfoil and a supercritical airfoil
(after Ref. [23]). a Supersonic domain over NACA 6-series airfoil. b Pressure distribution over
NACA 6-series airfoil. c Supersonic domain over supercritical airfoil. d Pressure distribution over
supercritical airfoil
Weak shock
Separation (footprint)
-1.0
Supersonic
-0.8
Reattachment
-0.6
C p, crit = - 0.435
-0.4
-0.2
Subsonic
Cp
0
0.2
0.4
0.6
0.8
0.0
0.2
0.4
x
c
0.6
0.8
1.0
Fig. 1.9 Pressure distribution on a supercritical airfoil at Mach 0.80, cl = 0.54, Rec = 3.0 × 106
(Flagged symbols are for pressure surface, data after Ref. [11])
1.4 Transonic Aerodynamics of Wings and Bodies
11
Fig. 1.10 Dunne D.8 at Farnborough Airshow, 1914 (Photo unkown)
ensure a statically stable airplane. Notable examples include aircraft designed by
Lippisch, various aircraft designed by the Horton brothers, and the Northrop X-35.
Up to the second world war, swept wings were therefore not yet associated with high
speeds.
That all changed with the introduction of the jet engine in the early 1940s. Early
jet aircraft such as the Messerschmidt 262 featured a small swept-back wing, probably due to stability reasons. Both the Me 262 (maximum operating Mach number,
MMO = 0.9) and the Me 163 (rocket-powered, MMO = 0.83) experienced adverse
compressibility effects when they flew close to the speed of sound. The first aircraft
that was truly designed with the high-speed philosophy of a swept-wing in mind is
probably the Focke-Wulff 183, designed in 1943 by Hans Multhopp. The war had
ended before this aircraft was ever fielded. However, the first generation of jet fighters, such as the North American Sabre and the MiG 15 featured many similarities to
the original design of the Focke-Wulff 183. Already prior to the war, Adolf Büsemann
had demonstrated the favorable effect of wing sweep on the drag coefficient of wings
in supersonic flow. Following these investigations, Albert Betz showed in 1939 that
the onset of compressibility problems (such as strong shock waves) could be delayed
to higher Mach numbers when sweep was introduced. These findings only became
relevant for wing design when the jet engine was introduced and aircraft started to
approach the speed of sound.
The first prototype aircraft that were designed and built with swept wings were all
German fighter planes. None of them were finished before the war ended. One sweptwing aircraft that did fly prior to the end of the war was a forward-swept jet bomber
aircraft (Junkers, Ju-287), designed by Hans Wocke. He acknowledged the advantages of wing sweep in high-subsonic flow but also knew that the stall characteristics
of aft-swept aircraft were unfavorable. His aircraft demonstrated acceptable stall
characteristics but was captured by the Soviets before high-speed testing could com-
12
1 Introduction and Historic Perspective
mence. After the war, the concept of forward-swept wings was almost completely
abandoned due to the high weight penalty that was introduced due to adverse aeroelastic effects. The aft-swept wing, on the other hand, was embraced by both military
and civil airplane manufacturers in Europe and the United States. It was Boeing that
first introduced aft-swept wings on a large bomber aircraft. This aircraft, the B-47
Stratojet (Fig. 1.11), also featured podded jet engines that were suspended below
the shoulder-mounted wing. This configuration with swept-back wings and engines
mounted in nacelles below the wing is still the dominant aircraft configuration for
high-subsonic airliners.
With the flight of the X-1 in 1947, the supersonic domain had opened up for
military aircraft. However, the jet engines at the time were not yet powerful enough
to overcome the drag rise around Mach one. Therefore, first generation of jet fighters
flew at high subsonic speeds. New specifications from the US Airforce in 1950
dictated an aircraft that could fly with a top speed of Mach 1.2. It was Convair that was
awarded the contract to design and build a prototype aircraft. The resulting aircraft
featured a delta wing and no horizontal tail plane. Test flights showed disappointing
results: the airplane could only achieve M = 0.98, far below the requirement. A major
redesign was initiated to reshape the fuselage such that the peak drag coefficient
around Mach one reduced. This reshaping, commonly referred to as area ruling,
proved to be very effective. The new prototype was able to fly beyond the speed of
sound and met the top-speed requirement of Mach 1.2. This airplane (designated
F-102A) became quite successful with 1,000 copies built. In Fig. 1.12, a comparison
between the first prototype and the second prototype of the aircraft is shown.
The area-rule was initially proposed by Richard Whitcomb as a ‘rule of thumb’
as he called it. This rule of thumb essentially stated that the wave drag of an arbitrary
body at sonic conditions is solely dependent on the cross-sectional area distribution
of that body in the direction of the free stream. He first proposed this rule in 1952
Fig. 1.11 Boeing B-47 Stratofortress (Photo USAF)
1.4 Transonic Aerodynamics of Wings and Bodies
13
Fig. 1.12 Area ruling applied to Convair F-102 (Photos NASA). a YF-102 prior to area ruling. b
YF-102A with area-ruled fuselage
based on experimental results that he had obtained in the NACA wind tunnels and the
theory of stream pipes that had been introduced the year before by Adolf Büsemann.
However, already during the second world war the effect of cross-sectional area distribution on transonic drag was acknowledged by Otto Frenzl. Working at Junkers, he
introduced so-called “displacement bodies” that were strategically positioned such
as to reduce the wave drag a high-subsonic conditions. The idea of a favorable crosssectional area distribution was to be implemented in 1945 by Waldemar Voigt on the
Messerschmidt P112, which was never completed. In 1946, Dietrich Küchemann
introduced the “pinched fuselage,” a reduction of fuselage diameter at the location
of the wing. This modification was later dubbed the Küchemann Coke-bottle design
and is one of the modifications found in the YF102A. A mathematical description of
the ideal cross-sectional area distribution was presented in two separate articles by
Wolfgang Haack (1941) [9] and William Sears (1947) [19]. For a given combination
of length and volume, the Sears-Haack body possesses the cross-sectional area distribution with the least amount of wave drag at Mach one. For speeds beyond the speed
of sound, a different area distribution should be found according to the supersonic
area rule. This rule was first introduced by Wallace Hayes in his PhD dissertation of
1947 [10].
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1 Introduction and Historic Perspective
1.5 Transonic Flow Calculations
The governing equations of fluid flow are the well-known Navier-Stokes equations.
With present-day numerical techniques, it is possible to solve the Navier-Stokes
equations numerically. However, this technique (termed direct numerical simulation
or DNS) requires a very fine numerical grid and therefore is computationally very
expensive. Therefore, to evaluate the global flow field around arbitrary bodies in
transonic flow, we would rather employ equations that are less expensive to solve
numerically. Examples of a reduced set of equations that are often used are the Euler
equations that hold for inviscid fluid. When neglecting rotation in the flow, also the
full potential equation can be employed. Both neglect the viscous and heat-transfer
characteristics of the fluid. However, to predict the aerodynamic properties of a body
in transonic flow, it is important to at least include viscous effects in the thin layer,
i.e., the boundary layer, close to the body.
In capturing aerodynamics of wings in transonic speeds by computational methods, the shock-boundary layer interaction plays a big role in accurate flow simulation.
Therefore, inviscid flow solutions, in general, are not expected to be accurate when
boundary layers separate. To test this hypothesis, computational results on a finite
(swept) wing at Mach 0.86 are shown in Fig. 1.13. The pressure distributions in five
semi-spanwise stations are examined. Two computational predictions are also shown
at those stations. The inviscid flow solution is based on the full potential equation
and the results of a coupled inviscid-boundary-layer solution are also depicted. Let
us examine the semispanwise position near the wing tip, i.e., η = 0.89, first. Data
indicates the appearance of a moderately strong shock on the suction surface, around
x/c = 0.25. The full-potential solver over predicts the peak suction and the shock
location by a wide margin. However, the coupled solution does a reasonable job
of predicting the peak suction, shock position and subsequent pressure distribution.
We further note that the pressure surface is also better predicted using the inviscidboundary layer coupled scheme. The same trend is also observed at other spanwise
stations. The discrepancy between the inviscid and coupled solution strategy gets
smaller as we approach the wing root, e.g., see η = 0.26.
Another set of data compared with inviscid and the coupled-boundary layer computational approach is shown in Fig. 1.14 for a winglet on a wing-body-winglet configuration. Pressure distributions at four sections at the base and along the winglet
span are shown. In this case the inviscid solver is based on Euler momentum equations, which means that rotational flows are properly simulated. The coupling of
Euler and a boundary layer solver at least promises to include the viscous effects,
albeit in a sequential and iterative way. We note that the agreement with the measurement is best at the base of the winglet, then it becomes weaker and the predictions
deviate more, as we approach the winglet tip. In particular, Euler solutions always
place a shock further downstream than the data indicate and often the peak suction is
over-predicted. We also note that Euler-coupled boundary-layer method does capture
the viscous dominated events on the winglet and thus better predicts the pressure distribution. The shortcoming of the coupled inviscid-boundary layer approach stems
1.5 Transonic Flow Calculations
15
-1.
-1.
-.8
-.8
-.6
-.6
Cp
-.4
Cp
-.2
0
-.4
-.2
.2
.2
.4 .6
x/c
.4
η = 0.26
.8
1.
0
M∞ = 0.86
inviscid
coupled with b.l.
.2
.6
x/c
.4
η = 0.89
-1.
-1.
-1.
-.8
-.8
-.8
-.6
-.6
-.6
-.4
-.4
Cp -.4
Cp -.2
Cp -.2
-.2
0
0
.4
.4
.8
1.
.8
1.
.6
.6
.2
.2
.2
.4 .6
x/c
η = 0.42
.8
1.
0
.2
.2
.4 .6
x/c
.4
η = 0.60
.8
.2
.2
.4
.4
.6
x/c
η = 0.73
.6
.6
.6
1.
Fig. 1.13 Impact of viscous coupling with inviscid flow solution on a finite wing in transonic flow
(after Ref. [20])
-0.8
Cp -0.4
Euler
Euler + Coupled
Boundary Layer
Test
Cp
-0.8
-0.4
0.0
0.0
0.4
0.4
M∞ = 0.84
α = 2.08
-0.8
-0.8
Cp -0.4
Cp -0.4
0.0
0.0
0.4
0.4
Fig. 1.14 Impact of viscous coupling on pressure distributions on a winglet in transonic flow (after
Ref. [20])
16
1 Introduction and Historic Perspective
from the shortcomings of the boundary layer simulation of complex viscous flows.
Therefore, such problems demand the use of fully viscous solvers that are based
on the (Reynolds-Averaged) Navier-Stokes equations rather than inviscid solutions
coupled with a boundary layer solver.
1.6 Outline of Present Textbook
In the present text, we introduce the reader to quantitative and qualitative aspects of
transonic aerodynamics. The focus lies on external aerodynamics, rather than flows
through ducts, nozzles or turbomachinery. A brief outline of the each of the chapters
follows below.
In Chap. 2, we (re-)familiarize the reader with concepts from mathematics, thermodynamics, and aerodynamics that are fundamental to the methods presented in
the remainder of the book. This includes a short review of vector algebra and partialdifferential equations to provide the mathematical insight into the governing equations of fluid flow. The Navier Stokes equations are presented and it is shown how
the Reynolds-Averaged Navier-Stokes (RANS) equations, the Euler equations, and
the full-potential equation are derived. Using examples, it is shown how well each
of these models can predict the outcome of aerodynamic experiments in transonic
conditions.
Chapter 3 presents the concept of similarity parameters between “affinely” related
bodies. The theory behind transonic similarity parameters is useful in the prediction of
lift, drag, and moment coefficients of affinely related bodies. In addition, this chapter
presents various compressibility corrections for subsonic and supersonic flow based
on the linearized potential-flow equation. The application of the linearized potentialflow equation to supersonic flow about two-dimensional and axisymmetric bodies is
demonstrated through a number of examples.
As transonic flow is characterized by the formation of expansion waves, normal
shocks, bow shocks, and/or oblique shock waves, Chap. 4 presents the theoretical
basis for the characterization of these waves. It is shown how the flow characteristics
change over a (normal) shock wave and how this differs from a fan of expansion
waves. It is shown that shock-expansion theory is a powerful tool in analyzing aerodynamic forces and moments on pointed bodies in steady supersonic flow where the
oblique shock(s) at the leading edge are attached. This method produces exact results
within the confines of inviscid flow theory. Various examples demonstrate how this
method is used and what its limitations are.
In supersonic flow with large perturbations, a unique set of lines (called characteristics) can be defined where the hyperbolic partial differential equation that describe
the flow properties reduce to an ordinary differential equation. On these lines, we
can therefore integrate and analytically solve these equations. In Chap. 5 this method
of characteristics is presented. Two approaches to the method-of-characteristics are
demonstrated: the wave-field method and the lattice-point method. Examples of both
1.6 Outline of Present Textbook
17
methods include the supersonic flow through nozzles and the supersonic exhaust
plume of an under-expanded jet.
Chapter 6 presents the sources of drag that are encountered by a nonlifting body
that is moving at transonic Mach numbers. A qualitative characterization of laminar
and turbulent boundary layers is presented along with the concept of transition. In
addition, calculation methods are presented to estimate the friction drag of bodies
subjected to a pressure distribution. Furthermore, a method for the calculation of the
pressure drag over axisymmetric bodies is presented based on the linear potential
flow equation. It is shown how the pressure drag is a function of the cross-sectional
area distribution of the body. The concept of area ruling is presented and examples
are shown of practical implementations of this knowledge.
In Chap. 7 we present the transonic aerodynamics about lift-producing airfoils. It
is shown how various design parameters influence the velocity distribution and the
shock formation over an airfoil. The development of supercritical airfoils is explained
from a historic perspective. The concept of shock-boundary layer interaction, related
shock stall, and shock oscillation are also explained. Furthermore, it is demonstrated
that transonic flow limitations can play a major role in the maximum lift coefficient
of (multi-element) airfoils at Mach numbers as low as 0.25. Examples demonstrate
how low-speed and high-speed stall limit the flight envelope of aircraft that travel at
transonic Mach numbers.
In the final chapter (Chap. 8) we discuss the aerodynamics of swept wings in
transonic flow. It is shown through simple-sweep theory why a swept wing allows for
a higher drag-divergence Mach number and what its effect is on the drag coefficient
in the transonic regime. Apart from these advantages the chapter also presents the
challenges that aerodynamic designers face to reduce the adverse effects of wing
sweep: tip stall, early formation of tip shocks, form drag at the wing root, an early
onset of boundary-layer transition, aerodynamic twist, and aileron reversal. Apart
from aft-swept wings, also the merits and drawbacks of forward-swept wings are
presented.
Each of the chapters includes a set of examples and a set of problems to practice
with the theoretical concepts that are presented. We hope that through the addition
of these examples and problems, the reader can familiarize himself or herself with
important aspects that are related to transonic flow. Answers to selected problems
are given in the back of this book (p. 527 and onwards). A comprehensive list of
symbols and nomenclature is also added (starting at p. 543) such that the reader can
check the meaning and units (all in SI) of the variables that are used throughout the
chapters. Each chapter also lists a variety of references for further reading. Naturally,
if the reader is interested to learn more about particular topics that are presented in
this book we encourage him/her to consult the original sources. They often contain
much more detail about the particular topic at hand and provide a valuable link to
more material on the subject.
18
1 Introduction and Historic Perspective
Problems
1.1 Consider an aircraft with a maximum take-off weight (MTOW) of 560 metric
tons. Assume that at the beginning of the cruise phase this aircraft has burned some
fuel and weighs 96 % of its maximum take-off weight. At the end of the cruise phase
assume that the aircraft weighs only 75 % of its MTOW.
(a) For a cruise altitude of 35,000 ft (10.67 km) calculate the speed of sound, a, using
a temperature lapse rate of −6.5 K/km, a sea-level temperature of 288 K, and a
gas constant of 287 J/kg/K.
(b) For a cruise Mach number of 0.85 and a specific fuel consumption of SFC =
0.000186 [N/N/s] calculate the range, R, when L/D = 16, L/D = 19, and
L/D = 22. Use the Bréguet range formula (1.1).
1.2 The maximum range of an airplane is achieved when V × L/D is maximized.
In the low subsonic realm the drag coefficient can be approximated as the sum of
the zero-lift drag coefficient, C D0 and the induced drag coefficient, C Di , with C Di =
kC L2 , k being a constant. For a business jet in clean configuration C D0 = 0.021,
k = 0.038 and the wing loading is W/S = 3 kN/m2 . Using a spreadsheet program
solve the following problems:
(a) Graph the relation between C L (vertical axis) and C D (horizontal axis).
(b) Now, calculate the lift-to-drag ratio (C L /C D ) for this business jet for a lift
coefficient ranging from 0 to 1.7. Graph the relation between the lift to drag ratio
(vertical axis) and lift coefficient (horizontal axis).
(c) From your graph, estimate the maximum L/D and the lift coefficient this
occurs at.
(d) Express the velocity as a function of the wing loading, density, and lift coefficient.
(e) Now, calculate the product of velocity and lift-to-drag ratio (V × C L /C D ) for
a lift coefficient ranging from 0 to 1.7. Graph this relation by putting the lift
coefficient on the horizontal axis. Assume a value of 0.3 kg/m3 for the density.
(f) From your graph, estimate the maximum V × C L /C D and the lift coefficient
this occurs at.
References
1. Anderson, J.: Modern Compressible Flow With Historic Perspective, 3rd edn. McGraw Hill,
New York (2003)
2. Bensberg, W., Cranz, C.: Uber eine photographische Mehode Zur Messung von
Geschwindigkeiten und Geschwindigskeitverlusten bei Infanteriegeschossen. Artillerische
Monatshefte 41, 333–346 (1910)
3. Burgess, E.H.: Concorde inaugurates the supersonic era. In: Proceedings of the 9th Annual
Meeting and Technical Display. AIAA-1973-16, Washington, DC (1973)
4. Cornelliussen, S.T.: The transonic wind tunnel and the NACA technical culture, Chap. 4. From
Engineering Science to Big Science, pp. 91–133. NASA, Washington (1998)
References
19
5. Denning, R., Armstrong, J.A., Allen, J.E.: The broad delta airliner. Aeronaut. J. 107(1075),
547–558 (2003)
6. Ferrari, C., Tricomi, F.G.: Transonic Aerodynamics. Academic Press, New York (1968)
7. Goethert, B.H.: Transonic Wind Tunnel Testing. No. 49 in AGARDograph. Pergamon Press,
Oxford (1961)
8. Green, J., Quest, J.: A short history of the European Transonic Wind tunnel ETW. Prog. Aerosp.
Sci. 47, 319–368 (2011)
9. Haack, W.: Geschossformen kleinsten Wellenwiderstandes. Bericht 139 der LilienthalGesellschaft, pp. 14–28 (1941)
10. Hayes, W.D.: Linearized Supersonic Flow. Ph.D. thesis, California Institute of Technology
(1947)
11. Hurley, F.X., Spaid, F.W., Roos, F.W., Stivers, L., Bandettini, A.: Detailed Transonic Flowfield
Measurements about a Supercritical Airfoil Section, NASA TMX-3244 (1975)
12. Kuechemann, D.: The Aerodynamic Design of Aircraft. Pergamon Press, Oxford (1978)
13. Moulden, T.H.: Fundamentals of Transonic Flow. Wiley, New York (1984)
14. Pearcey, H.H.: The aerodynamic design of section shapes for swept wings. Adv. Aeronaut. Sci.
3, 277–322 (1963)
15. Ramm, H.J.: Fluid Dynamics for The Study of Transonic Flow. Oxford University Press,
New York (1990)
16. Roskam, J.: Airplane Design, Part 6: Preliminary Calculation of Aerodynamic. Thrust and
Power Characteristics. DARcorp, Lawrence (2006)
17. Ruijgrok, G.J.J.: Elements of Airplane Performance. Delft University Press, Delft (1989)
18. Scholz, D., Ciornei, S.: Mach number, relative thickness, sweep and lift coefficient of the
wing—an empirical investigation of parameters and equations. DGLR Jahrbuch 2005 (2005)
19. Sears, W.R.: Projectiles of minimum wave drag. Q. Appl. Math. 4(4), 361–366 (1947)
20. Tinoco, E.N.: CFD applications to complex configurations: a survey. In: Henne, P. (ed.) Applied
Computational Aerodynamics, Chap. 15. AIAA, Washington (1990)
21. Tokaty, G.A.: Fluid Dynamics. W & J Mackay & Co Ltd., Chatham (1971)
22. Vidal, R.J.: Wall Interference Effects in Transonic Flows. Calspan Corp., Buffalo (1976)
23. Whitcomb, R.T., Clark, L.R.: An Airfoil Shape for Efficient Flight at Supercritical Mach
Numbers, NASA TM X-1109. Langley (1965)
Chapter 2
Review of Fundamental Equations
Abstract In this chapter we (re-)familiarize the reader with concepts from
mathematics, thermodynamics, and aerodynamics that are fundamental to the methods presented in the remainder of the book. This includes a short review of vector
algebra and partial-differential equations to provide the mathematical insight into
the governing equations of fluid flow. Examples include the one-dimensional wave
equation and the one-dimensional heat equation. A basic review of thermodynamics
is given including the equation of state, the first law and the second law of thermodynamics. Also the isentropic relations between pressure, density and temperature
are derived. In the aerodynamics section, the Navier-Stokes equations are derived
in integral and derivative form. To simulate transonic flow, often simplifications of
the Navier-Stokes equations are used. Therefore, it is shown how the ReynoldsAveraged Navier-Stokes (RANS) equations can be obtained and how the k-epsilon
turbulence model can be used to close the RANS equations. For inviscid flow the
Euler equations, and the full-potential equation are derived. Using examples from
the literature, it is shown how well each of these models can predict the outcome of
aerodynamic experiments in transonic conditions. This chapter contains 11 examples
and concludes with 29 practice problems.
2.1 Introduction
Transonic flow conditions are encountered by the majority of today’s jet aircraft.
Almost all major airlines use high-subsonic jet transports on their medium to long
haul flights. In addition, business jets, fighter aircraft, and military UAVs such as the
Northrop X-47 are also confronted with transonic effects. To comprehend the physics
of transonic flow, this chapter presents a review of physical and mathematical topics
that form a basis for the subsequent chapters of the book. For further reading on these
topics, the reader is referred to the reference list at the end of this chapter, that lists a
number of text books that adequately explain the subject matter in more detail. After
this short introductory section, we will present basic reviews on two mathematical
topics: partial differential equations and vector algebra. Since mathematics is often
referred to as “the language of physics,” a good understanding is mandatory for the
© Springer Science+Business Media Dordrecht 2015
R. Vos and S. Farokhi, Introduction to Transonic Aerodynamics,
Fluid Mechanics and Its Applications 110, DOI 10.1007/978-94-017-9747-4_2
21
22
2 Review of Fundamental Equations
subsequent sections where we review the laws of thermodynamics and equations of
motion in fluid flow.
2.2 Review of Partial Differential Equations
As the reader might remember from a course on elementary aerodynamics, the equations of motion of a particle in a fluid flow can be represented as a set of partial
differential equations (PDEs) called the Navier-Stokes equations. Before we discuss
these relatively complex equations we consider two more elementary equations: the
one-dimensional heat and wave equations. These equations serve as examples of partial differential equations and how they can be solved. At the same time they are also
part of a unique set of PDEs that actually have a closed-form solution. Unfortunately
this is not true for most PDEs that describe physical processes such as the motion of
a fluid. To solve these PDEs we need different solution techniques which often rely
on a (numerical) approximation of the problem. However, to test such a numerical
scheme we can always use the one-dimensional heat or wave equation because we
know their exact solution. They can therefore serve as a test case for our numerical
schemes to gain confidence in the accuracy of the approximation method.
In the subsequent two subsections, we do not attempt to review the broad subject
of PDEs and their solution methods, but merely present a basic review of the topic
within the context of the theory of aerodynamics and computational fluid dynamics.
2.2.1 One-Dimensional Wave Equation and Solution
by D’Alembert
In this section we derive the governing equation that describes the transverse vibration
of an elastic string, such as a guitar string. When solving this governing equation we
should end up with an expression that describes each individual point on the string in
both space (x, u) and time (t). First, consider a string of length L that we perturb and
at time t = 0 we release the string such that it starts to vibrate. We want to evaluate
its vertical displacement, u(x, t) as shown in Fig. 2.1. To simplify this problem we
assume that the mass per unit length (ρ) of the string is constant along the string and
that it does not have any bending stiffness. In addition, we neglect the gravitational
force and assume that the deflections remain small enough to justify the assumption
that each point on the string only moves in the vertical direction.
To come up with the equation of motion for this string, we consider the force
balance on an infinitesimal part within the string itself (P–Q in Fig. 2.1). We have
α and β denoting the deviation angles from the horizontal axis, and T1 and T2 being
the components of the internal forces at points P and Q, respectively. The horizontal
components of the forces at points P and Q must be balanced:
T1 cos α = T2 cos β = T = constant
(2.1)
2.2 Review of Partial Differential Equations
Fig. 2.1 Arbitrary
transverse deflection, u of an
elastic string at time t
23
u
Q
β
P
T2
α
T1
x x+Δx
0
L
Directing our attention to the vertical direction we know that the following force and
acceleration balance should hold over the part Δx:
− T1 sin α + T2 sin β = ρΔx
∂2u
∂t 2
(2.2)
Using (2.1) we can rewrite (2.2) as follows:
tan β − tan α =
ρΔx ∂ 2 u
T ∂t 2
(2.3)
Since tan α = ∂u/∂x|x and tan β = ∂u/∂x|x+Δx we can now write a partial
differential equation where both derivatives with respect to x and t appear:
1
Δx
∂u
∂x
−
x+Δx
∂u
∂x
=
x
ρ ∂2u
T ∂t 2
(2.4)
If we subsequently take the limit as Δx → 0 we obtain the well known onedimensional wave equation:
2
∂2u
T
2∂ u
=
c
with c2 ≡
2
2
∂t
∂x
ρ
(2.5)
The wave equation is a second order (=the highest power of one of the differential
terms), linear partial differential equation. Not only does this equation describe the
transverse motion of a vibrating string it also describes the wave motion of a plane
wave (e.g. acoustic wave) [16]. In that case we replace the excitation of
√ the string by
the sound pressure, p (x, t) and c becomes the speed of sound, c = γ RT , where
γ is the ratio of specific heats, R is the gas constant, and T is the static temperature
in the calorically perfect gas.1
1 The speed of sound will be derived in Sect. 2.5.
24
2 Review of Fundamental Equations
We can now proceed with a solution procedure of the wave equation. There are
several approaches to derive a closed form solution of the problem, one of which
is by using D’Alembert’s approach. This approach relies on a transformation to the
characteristic coordinates ξ and η according to:
ξ = x + ct,
η = x − ct
(2.6)
Therefore, u now becomes a function of ξ and η. The partial derivatives should
therefore be rewritten with respect to these new independent variables. If we apply
the chain rule to either side of the equation we transform (2.5) to:
u ξη =
∂2u
=0
∂η∂ξ
(2.7)
Note that we shift to the notation of denoting a partial derivative as a subscript to
the dependent variable. In the subscript notation the subscripts are written in the
order in which we differentiate, whereas in the “∂” notation the order is opposite.
As the reader can readily observe, the substitution of the new independent variables
have transformed the problem such that it can be (easily) solved by two successive
integration steps:
∂u
= h(ξ),
∂ξ
where h(ξ) is an arbitrary function of ξ. A second step of integration yields the
following:
u = h(ξ)dξ + ψ(η),
where ψ(η) is an arbitrary function of η. Since the integral of an arbitrary function
in ξ is another arbitrary function in ξ, say φ(ξ), the solution is of the form u(ξ, η) =
φ(ξ) + ψ(η). Substituting the original values for ξ and η in the previous equation
results in the following closed form solution:
u(x, t) = φ(x + ct) + ψ(x − ct)
(2.8)
If we assume that two arbitrary functions, f (x) and g(x) describe the initial
position and velocity, respectively, we can express the functions φ(x) and ψ(x) in
terms of the these initial conditions. We have:
u(x, 0) = f (x) = φ(x) + ψ(x)
u t (x, 0) = g(x) = φξ ξt + ψη ηt
(2.9)
(2.10)
Using (2.6), we can evaluate that ξt = c and ηt = −c. Since at t = 0 we have
ξ = η = x, (2.10) reduces to:
u t (x, 0) = g(x) = cφx (x) − cψx (x)
(2.11)
2.2 Review of Partial Differential Equations
25
If we integrate both sides of (2.11) between x0 and x and divide both sides by c
we obtain:
φ(x) − ψ(x) =
1x
g(s)ds with k(x0 ) = φ(x0 ) − ψ(x0 ).
c x0
(2.12)
Now, to get φ(x) we simply add (2.12) to (2.9) and divide both sides by two. Similarly,
to obtain ψ(x) we subtract (2.12) from (2.9) and divide both sides by two. The
following results are found:
1
1
1 x
g(s)ds + k(x0 )
f (x) +
x
2
2c 0
2
1
1 x
1
ψ(x) = f (x) −
g(s)ds − k(x0 )
2
2c x0
2
φ(x) =
(2.13)
(2.14)
Substituting φ(x + ct) and ψ(x − ct) in the above expressions and inserting everything back into (2.8) results in the following solution to this initial value problem:
u(x, t) =
1
1 x+ct
g(s)ds.
[ f (x + ct) + f (x − ct)] +
2
2c x−ct
(2.15)
For a brief moment, let us return to interpretation of this result with respect to
the original problem of the vibrating string and assume that the initial velocity,
u t (x, 0) = g(x) = 0. To come up with a valid solution that satisfies the boundary
conditions (u(0) = u(L) = 0), the following should hold:
1
[ f (ct) + f (−ct)] = 0
2
1
u(L , t) = [ f (L + ct) + f (L − ct)] = 0
2
u(0, t) =
(2.16)
(2.17)
From (2.16) we learn that the initial shape function f (x) is odd. In other words it
is anti-symmetric with respect to the u-axis. Combining this with (2.17) we obtain
that f (L + ct) = f (−L + ct), which shows that the initial function should have a
period of 2L.
Example 2.1 Calculate the solution to the homogeneous wave equation (2.5) for
u(x, 0) = f (x) = sin(2πx), u t (x, 0) = g(x) = 0, and c = 1.
Solution:
Using D’Alembert’s solution we can directly substitute f (x) = sin(πx) and g(s) =
0 in to (2.15). We obtain:
u(x, t) = sin 2π(x + t) + sin 2π(x − t)
A graphical representation of this string is sketched over the interval [0, 1] between
time t = 0 and t = 0.25 with time increments Δt = 0.05 (Fig. 2.2).
26
2 Review of Fundamental Equations
Fig. 2.2 Solution to the
wave equation corresponding
to the initial conditions of
Example 2.1
1
t=0
t=0.05
t=0.1
u (x,t)
0.5
t=0.15
t=0.2
t=0.25
0
−0.5
−1
0
0.2
0.4
0.6
0.8
1
x
To investigate the effect of a disturbance function on the excitation of the vibrating
string we consider the inhomogeneous wave equation:
u tt − c2 u xx = f (x, t),
(2.18)
where f (x, t) is the disturbance function. We assume that the string is initially
undisturbed, i.e.:
(2.19)
u(x, 0) = u t (x, 0) = 0
Using the characteristic coordinates as in (2.6), we can rewrite u(x, t) as follows:
u(x, t) = u
ξ+η ξ−η
,
2
2c
= v(ξ, η)
(2.20)
Similarly, we can transform the forcing function: f (x, t) = G(ξ, η). Note that when
t = 0 we have ξ = η and ξ = x. Therefore, we can write the first initial condition:
u(x, 0) = u(ξ, 0) = v(ξ, ξ) = 0
(2.21)
To transform the second initial condition we write the first partial derivative of v with
respect to ξ in terms of u x and u t by employing the chain rule (see Problem 2.5):
vξ =
1
1
u x + ut
2
2c
(2.22)
Substituting the equalities that hold when t = 0 (same as above) we have:
vξ (ξ, ξ) =
1
1
u x (ξ, 0) + u t (ξ, 0)
2
2c
(2.23)
2.2 Review of Partial Differential Equations
27
Now, we know that the second part of the right-hand side of the equation is zero
due to the initial condition u t (x, 0) = 0. And, since u(x, 0) = 0 we know that
1/2u(x, 0)|x=ξ must also be zero. Therefore, we now have the second initial condition
in transformed coordinates:
vξ (ξ, ξ) = 0
(2.24)
Finally, we write the PDE in characteristic coordinates according to:
vξη = −
1
G(ξ, η)
4c2
(2.25)
We are interested in the excitation at a particular time and location, u(x0 , t0 ), which
corresponds to v(ξ0 , η0 ). Therefore, we need to integrate the left-hand side of (2.25)
twice with appropriate boundary conditions:
ξ0 ξ
η0
η0
vξη (ξ, η)dηdξ
Solving the inner integral by using the fundamental theorem of calculus (FTC)2
results in:
ξ
vξη (ξ, η)dη = vξ (ξ, ξ) − vξ (ξ, η0 )
η0
We note that the first term on the right-hand side must equal zero, due to the initial
condition (2.24). The second integration step thus results in:
−
ξ0
η0
vξ (ξ, η0 )dξ = v(η0 , η0 ) − v(ξ0 , η0 )
Again, due to the initial condition (2.21) for ξ = η0 , the first term on the right-hand
side equals zero, which leaves us with the negative of what we were initially looking
for: −v(ξ0 , η0 ). Now, by performing the same integration on the right-hand side
of (2.25) and bringing the minus sign to the other side of the equality sign (which
cancels) we have a solution in characteristic coordinates:
v(ξ0 , η0 ) =
1 ξ0 ξ
G(ξ, η)dηdξ
4c2 η0 η0
(2.26)
To change back to the original coordinate system, the determinant (for explanation
of the determinant see Sect. 2.3) of the Jacobian, det(J ), needs to be calculated, since
dηdξ = det(J )dxdt. It is left to the reader (Problem 2.6) to show that this equals 2c.To
2 The fundamental theorem of calculus specifies the relationship between the two central operations
of calculus: differentiation and integration [11].
28
2 Review of Fundamental Equations
(b)
(a)
t
η
(ξ 0 ,ξ0)
ξ0
η0
(x 0 ,t 0 )
(ξ 0 ,η0)
(η0,η0)
η0
ξ0
ξ
x0 - ct0
x0+ct0
x
Fig. 2.3 Transformation of the area of integration between the characteristic and the original coordinate system. a Region of integration in the (ξ, η)-plane. b Region of integration in the (x, t)-plane
transfer the limits of the integrals it is convenient to look at the region over which we
are integrating G(ξ, η). This region is displayed in Fig. 2.3a. If we look at the vertices
of this triangle, and what they represent in the physical (x, t)-plane, we note that:
(ξ0 , η0 )
(ξ0 , ξ0 )
(η0 , η0 )
→
→
→
(x0 , t0 )
(x0 + ct0 , 0)
(x0 − ct0 , 0)
The region of integration in the (x, t)-plane that corresponds to these vertices is
shown in Fig. 2.3b. We therefore have:
u(x0 , t0 ) =
1 t0 x0 +ct0
f (x, t)dxdt
2c 0 x0 −ct0
(2.27)
The region of integration in the physical domain can be used to see whether a
point in the domain is influenced by a particular disturbance. When the area under
the graph of the disturbance function and the area of the integration region (partially)
overlap, then the point in the domain is influenced by the disturbance. However,
if there is no overlap at all, the disturbance function does not affect that particular
point in the domain. The characteristic lines (ξ = x + ct and η = x − ct) therefore
mark the boundaries of the so-called domain of dependence. The solution, therefore,
only depends on a disturbance that happens within this domain of dependence. Any
disturbance outside this region does not have any effect on the excitation of this point.
In Example 2.2 this is demonstrated.
2.2 Review of Partial Differential Equations
29
Fig. 2.4 Region of
integration (striped) for
Example 2.2
t
f(x,t)
1
(x,t)
0 x-t
-1
1 x+t
x
Example 2.2 Consider the following problem:
u tt − c u x x = f (x, t) =
2
1 for 0 < t < 1, − 1 < x < 1
0 elsewhere
u(x, 0) = 0
u t (x, 0) = 0
Calculate u(x, t) explicitly for (x, t) satisfying: x > t, x < 1, and x + t > 1.
Solution:
To solve this problem we use (2.27) with c = 1. Due to the simple
forcing function,
f (x, t), we can directly see that the solution u(x, t) = 1/2 Δ(x,t) f (x, t)dxdt,
where Δ(x, t) represents the region of integration. This region is schematically shown
in Fig. 2.4. Using simple geometry we calculate the area of this region Δ(x, t) =
t2
x2
1
2 − 2 − xt − x − t − 2 . The solution then becomes:
u(x, t) =
1 2
(t − x 2 − 2xt + 2x + 2t − 1)
4
2.2.2 One-Dimensional Heat Equation and Solution
by Fourier Series
In this section we look at the other famous PDE, the heat equation. In the onedimensional case we consider a bar with length L (Fig. 2.5). We are interested in
the temperature, T , and its distribution over the bar with time, depending on the
boundary and initial conditions. We assume that the bar is of constant cross section
0
x=L
Fig. 2.5 Model problem for the one-dimensional heat equation: a bar of length L
30
2 Review of Fundamental Equations
and homogeneous material and that it is perfectly insulated, such that heat flows only
along the x-direction.
The derivation of the heat equation starts with Fourier’s law of heat conduction
that states that the heat flux, q, is negatively proportional to the temperature gradient,
∂T /∂x, according to (in one-dimensional space):
q = −k∂T /∂x,
(2.28)
where k is the thermal conductivity of the material. The negative sign indicates
the direction of heat flow is from hot to cold, i.e. in the opposite direction to the
temperature gradient. In the absence of work done, a change in internal energy per
unit volume in the material, ΔE, is proportional to the change in temperature, ΔT .
That is,
ΔE = cρΔT
(2.29)
where c is the specific heat capacity and ρ is the mass density of the material. Choosing
zero energy at the absolute zero temperature, this can be rewritten as
E = cρT.
(2.30)
The increase in internal energy in a small spatial region of the material
x − Δx ≤ ξ ≤ x + Δx
(2.31)
t − Δt ≤ τ ≤ t + Δt
(2.32)
over the time period
is given by
cρ
x+Δx
x−Δx
[T (ξ, t + Δt) − T (ξ, t − Δt)] dξ = cρ
t+Δt x+Δx ∂T
t−Δt
x−Δx
∂τ
dξdτ , (2.33)
where the FTC was used. Additionally, with no work done and the absence of any
heat sources or sinks, the change in internal energy in the interval [x −Δx, x +Δx]
is accounted for entirely by the flux of heat across the boundaries. By Fourier’s law,
this is
t+Δt x+Δx ∂ 2 T
t+Δt ∂T
∂T
(x + Δx, τ ) −
(x − Δx, τ ) dτ = k
k
dξdτ
t−Δt
t−Δt x−Δx ∂ξ 2
∂x
∂x
(2.34)
again by the fundamental theorem of calculus. In higher dimensions, the divergence
theorem (Sect. 2.3.3) is used instead. By conservation of energy (Sect. 2.5.3),
t+Δt x+Δx
t−Δt
x−Δx
[cρTτ − kTξξ ] dξdτ = 0.
(2.35)
2.2 Review of Partial Differential Equations
31
This is true for any rectangle [t − Δt, t + Δt] × [x − Δx, x + Δx]. Consequently,
the integrand must vanish identically:
cρTt − kTx x = 0.
Which can be rewritten as:
Tt =
or:
k
Tx x ,
cρ
∂T
k
=
∂t
cρ
∂2 T
∂x 2
(2.36)
(2.37)
(2.38)
which is the heat equation. The coefficient k/(cρ) is called thermal diffusivity and
is denoted by α for convenience.
To solve the heat equation (2.38), we use the ‘separation-of-variables’ technique [18] where we substitute
T (x, t) = F(x)G(t)
(2.39)
Denoting ∂/∂t with (. ˙. .) and ∂/∂x with (. . .) , we can write (2.38) as:
F Ġ
=
= −β 2 .
αG
F
(2.40)
Here we introduce a constant, β, that is conveniently supplied with a minus sign to
aid in the solution procedure. We can see that each fraction in (2.40) can be written
as a separate ordinary differential equation (ODE) according to:
F + β 2 F = 0
(2.41)
Ġ + αβ G = 0
(2.42)
2
Equation (2.41) is second order in space and Eq. (2.42) is first order in time.
To find the eigenfunctions of the heat equation we need a set of boundary equations
that give us some details about the heat transfer conditions of the rod at the ends.
In the present case we choose to keep the ends of the rods at zero temperature:
T (0, t) = T (L , t) = 0. A general solution to (2.41) is:
F(x) = A cos(βx) + B sin(βx)
(2.43)
Using the specified boundary conditions, we know that A = 0, and B sin β L = 0.
To avoid the trivial solution (B = 0) we know that sin β L = 0 and thus, βn = nπ
L for
n = 1, 2, . . .. We have found the following solution to the boundary-value problem:
Fn (x) = Bn (x) sin
nπx L
for n = 1, 2, . . .
(2.44)
32
2 Review of Fundamental Equations
√
Let us now return to (2.42) and substitute the eigenvalue λn = βn α:
Ġ + λ2n G = 0
(2.45)
G n (t) = Cn e−λn t for n = 1, 2, . . . ,
(2.46)
This has the general solution
2
where Cn are arbitrary constants. This results in the following final solution of this
problem:
Tn (x, t) = Fn (x)G n (t) = Dn sin
nπx L
e−λn t for n = 1, 2, . . .
2
(2.47)
where Dn are arbitrary constants. These are the eigenfunctions of the problem and
λn = cnπ
L are the eigenvalues.
In order to solve the heat equation we need an initial condition, f (x). This initial
condition is prescribing the temperature distribution along the bar at time t = 0
must comply with the boundary conditions. We know that the sum of each of the
solutions in (2.47) is also a solution to (2.38) and we consider the following series
of eigenfunctions:
∞
nπx 2
e−λn t
Dn sin
(2.48)
T (x, t) =
L
n=1
Since at t = 0 we need to satisfy the initial condition the following relation must be
satisfied:
∞
nπx =
ˆ f (x)
(2.49)
T (x, 0) =
Dn sin
L
n=1
Multiplying both sides by sin nπx
L and integrating both sides over the interval [0, L]
results in the following coefficients of the Fourier sine series:
Dn =
nπx 2 L
dx
f (x) sin
L 0
L
(2.50)
Example 2.3 Solve the heat equation, Tt = Tx x on the interval 0 < x < 1 with
boundary conditions T (0, t) = Tx (1, t) = 0 and initial condition T (x, 0) = 1.
Solution:
Using the same methodology as above we separate the temperature function in the
product of F(x) and G(t). We have
F(x) = A cos(βx) + B sin(βx)
2.2 Review of Partial Differential Equations
33
Again, we state that due to the first boundary condition A = 0. The second boundary
condition implies: Bβ cos(βx) = 0. This equation is satisfied when
1
for n = 0, 1, 2, . . .
β=π n+
2
This results in the following general solution:
1
Fn (x, t) = Bn sin πx n +
2
Substituting this in (2.39), with G n (t) being the same as in (2.46) and λn =
π n + 21 , we now have the following solution:
∞
T (x, t) =
n=0
2
1
−π 2 n+ 21 t
e
Dn sin πx n +
2
(2.51)
1
Setting T (x, 0) = 1 and knowing that 0 sin π n + 21 x sin π m + 21 x dx = 0
for n = m we multiply both sides of (2.51) with sin π m + 21 x and integrate
between 0 and 1:
1
1
1
2
dx = Dm → Dm =
sin πx m +
0
2
2
π m + 21
This gives the following final solution (switching back from m to n for convenience):
∞
T (x, t) =
n=0
2
π n + 21
2
1
−π 2 n+ 21 t
e
sin πx n +
2
(2.52)
In Fig. 2.6, we see a graphical interpretation of the solution. Note how the initial
condition is satisfied by showing almost a complete horizontal line at t = 0. The
boundary condition at x = 0 is also satisfied, while we can see that the slope of the
line at x = 1 is zero, indicating that we have also satisfied this boundary condition
correctly.
2.2.3 Conservation Form of PDEs
The equations of motion of fluid flow that are presented in this chapter (and throughout the text) are in conservation form (also referred to as conservative form). This
implies that the partial differential equations (PDEs) that describe the physics of the
flow have coefficients that are either constant or, if variable, their derivatives do not
appear in the equation. We use the following example to illustrate this form.
34
2 Review of Fundamental Equations
Fig. 2.6 Graphical
representation of (2.52) for
the first 200 components of
the Fourier series expansion
1
t=0
T (x,t)
0.8
0.6
0.2
0.4
0.4
0.6
0.8
0.2
1.0
0
0
0.2
0.4
0.6
0.8
1
x
Example 2.4 To appreciate the difference between the conservative and nonconservative form of a partial differential equation we consider the one-dimensional heat
equation (2.38) without the assumption that ρ, c, or k are constant with x. In other
words, the thermal diffusivity, α is a function of x: α = α(x). First this equation is
formulated in conservative form:
∂
∂T
∂T
=
α
(2.53)
∂t
∂x
∂x
The non-conservative form of the same equation reads:
∂T
∂2 T
∂α ∂T
=α 2 +
∂t
∂x
∂x ∂x
(2.54)
Note how in (2.53) the coefficient, α, can vary with position (x) but its derivative does
not appear in the equation. Therefore, (2.53) is in conservation form. In (2.54) the
derivative on the RHS of (2.53) has been expanded and now contains the derivative
term ∂α/∂x, which is a non-conservative term in the equation. Therefore we deem
(2.54) to be in non-conservative form.
The same logic that is used for the one-dimensional heat equation can be expanded
to the equations of motion that describe fluid flows (Sect. 2.5). These governing
equations must hold at any distinct point in the flow. To formulate these equations
we take a Eulerian approach where we consider a fixed control volume through which
the fluid passes. The conservation form for each PDE at such a point also allows us
to formulate finite-difference representations that provide a good approximation to
the PDEs. These, in turn, can be used in an iterative numerical code to obtain the
global properties of a flow field. The conservative formulation is therefore often used
to solve partial differential equations by numerical methods.
2.2 Review of Partial Differential Equations
35
2.2.4 Classification of Partial Differential Equations
A second order partial differential equation (PDE) can be classified based on the
coefficient of its second-order derivative terms. For an arbitrary function, φ, a secondorder differential equation in two dimensions reads:
aφx x + bφx y + cφ yy = H
(2.55)
where a, b, and c are coefficients and H = H (φ, φx , φ y , x, y). Along the so-called
characteristic curves of the PDE the following should hold:
a(dy)2 − b(dxdy) + c(dx)2 = 0
(2.56)
Solving for dy/dx results in:
√
−b ± b2 − 4ac
dy
=
dx
2a
(2.57)
Depending on the term under the radical in (2.57), the family of characteristic curves
displays different behavior. When b2 −4ac > 0 there are two distinct families of real
characteristic curves. In this case the PDE is termed hyperbolic. When b2 − 4ac = 0
a single family of real characteristic curves is found and the PDE is called parabolic.
Finally, when b2 − 4ac < 0 the RHS of (2.57) is complex and no real characteristic
curves exist. The PDE is termed elliptic in this case.
Example 2.5 Consider the wave equation (2.5) (using χ for wave speed to avoid
confusion in notation):
u tt − χ2 u x x = 0
Determine whether this equation is hyperbolic, parabolic, or elliptic, and determine,
if possible, its characteristic curves.
Solution:
The coefficients of the wave equation are: a = 1, b = 0, and c = −χ2 . We calculate
the discriminant to be 4χ2 and since χ is real we know that 4χ2 > 0 and that the
wave equation is therefore hyperbolic. Substituting the coefficients in (2.57) gives
the two characteristic curves: dt/dx|1,2 = ± χ1 .
Every PDE that is represented in an orthogonal coordinate system can be transformed to its canonical form. A coordinate transformation takes place and every
point in the original coordinate system is mapped onto the arbitrary coordinate system. In a two-dimensional space, this would imply mapping the x and y coordinates
onto the ξ–η plane: (x, y) → (ξ, η), which we also used for the wave equation. How
do we find these characteristic coordinate transformation? Well, that is quite simple:
we employ (2.57) to find the slope of the characteristic lines and integrating this
results in an expression for the characteristic curves.
36
2 Review of Fundamental Equations
Example 2.6 Find the characteristic curves for the wave equation of Example 2.5.
Solution:
For (2.5) we have dt/dx|1,2 = ± χ1 . Integrating this gives:
1
x +η
χ
1
t2 = − x + ξ,
χ
t1 =
where, η and ξ are integration constants. Rearranging these equations results in the
relations of (2.6).
In this arbitrary (orthogonal) coordinate system, the hyperbolic, parabolic, and
elliptic PDEs assume a particular canonical (natural) form. Let us examine these
canonical forms for each class of second order PDEs. We assume that the solution
u(x, t) is written in the transformed coordinates as φ(ξ, η).
Hyperbolic PDE
Two characteristic forms exist:
φξξ − φηη = h 1 (φξ , φη , φ, ξ, η)
φξη = h 2 (φξ , φη , φ, ξ, η)
Similar to the wave equation, for a two-dimensional problem, two characteristic
lines (or characteristics) can be drawn that intersect at the point of interest (x0 , t0 )
(see Fig. 2.7a). The region between the characteristic lines that is present in a flow
that is described by hyperbolic PDEs is called the region of influence [1]. A
disturbance in the flow only propagates in the region of influence. Conversely,
any point in the flow is influenced by the initial data that falls between the two
characteristics. As we saw before, this region is termed the domain of dependence.
Any disturbance that occurs in the domain of dependence influences the state of
the point where the characteristic lines cross. Hyperbolic PDEs describe the steady
inviscid supersonic flow, as well as the unsteady inviscid flow (Euler equation).
(a)
(b)
t
region of
influence
t
region of
influence
characteristics
(x0 ,t0 )
domain of
dependence
(c)
t
region of =
influence
characteristic
(x0 ,t0 )
domain of
dependence
(x0 ,t0 )
domain of
dependence
x
x
x
Fig. 2.7 Domain of dependence and region of influence for the three different classes of second
order PDEs. a Hyperbolic. b Parabolic. c Elliptic
2.2 Review of Partial Differential Equations
Parabolic PDE
either:
37
The characteristic equation in canonical form can be written as
φξξ = h 3 (φξ , φη , φ, ξ, η) or
φηη = h 4 (φξ , φη , φ, ξ, η)
Similar to hyperbolic PDEs, the parabolic PDEs have a region of influence. However, this region is now unbounded by characteristics and therefore spans the entire
space (can also be time) beyond the point where the initial disturbance takes place
(see Fig. 2.7b). In addition, the entire region before this point influences this point
in the flow. Parabolic PDEs describe steady boundary layer equations along with
“parabolized” viscous flows.
Elliptic PDE The canonical form of an elliptic PDE can be written in its characteristic coordinates as follows:
φξξ + φηη = h 5 (φξ , φη , φ, ξ, η)
Any disturbance in a flow field that is described by elliptic PDEs is felt throughout
the entire flow. In other words, the entire flow domain forms the region of influence.
Conversely, every point in the flow domain influences any other point in the flow
domain. Therefore, the entire flow can also be viewed as the domain of dependence
(see Fig. 2.7c). Elliptic PDEs describe the steady subsonic, inviscid flow along
with an incompressible inviscid flow field.
Consider the two-dimensional potential equation for steady, irrotational, inviscid
flow (we will derive this equation in Sect. 2.7.2):
Φ y2
Φx Φ y
Φx2
1 − 2 Φx x + 1 − 2 Φ yy − 2 2 = 0
c0
c0
c0
(2.58)
The velocity potential function is denoted with Φ(x, y), with the velocities, Φx (x, y)
and Φ y (x, y), as well as the speed of sound, c0 . The three coefficients of interest can
be identified as follows:
a =1−
Φx2
c02
b=−
2Φx Φ y
c02
c =1−
Φ y2
c02
(2.59)
The discriminant for this second order PDE becomes:
b2 − 4ac =
Φx2 + Φ y2 − c02
c02
= M 2 − 1,
(2.60)
where M is the Mach number of the flow. For subsonic flows M < 1 and the
potential equation is elliptic. When the flow is supersonic M > 1 and the equation
38
2 Review of Fundamental Equations
becomes hyperbolic. At sonic condition the equation is parabolic. In transonic flow
the potential equation therefore changes type between the different flow domains.
It is important to distinguish the class of the PDE because different solution
techniques should be used to solve them. The “problem” in transonic flows is that
the most convenient equations in both supersonic and subsonic flow are not of the
same type. Any type of numerical scheme that attempts to approximate one of these
equations is therefore doomed to fail. Steady supersonic and steady subsonic flows
can therefore not be modeled with the same numerical approach. However, unsteady
flows can be described by a hyperbolic scheme whether the flow is subsonic or
supersonic. This is the reason that in transonic aerodynamics the unsteady equations
of motion form the basis for the solution of the flow field. The solution is found by
marching in time until the solution converges to a steady-state value.
2.3 Review of Vector Algebra
As demonstrated in Sect. 2.5 the equations of motion of a fluid flow can be conveniently represented in vector form. The vector notation is often used to represent
the governing flow equations and their (numerical) approximations. Vector notation
allows us to represent a set of equations in a single line, which make it easier to
perform mathematical operations in order to derive certain numerical approximation
schemes. The following sections give a review of basic vector algebra including a
review of the most common operations and their notation used in this text.
2.3.1 Vectors, Vector Fields, and Scalar Fields
Some properties of a fluid flow are represented as scalar values (such as p, T , and
ρ). Scalars have a magnitude but no direction. Velocity, however, is represented by
a vector, because it has both a magnitude and a direction. In this text a vector is
denoted with a bold font. For example, the velocity vector is denoted with V , while
its magnitude,
V = |V |. In two dimensions, the velocity vector is denoted with
u
V =
, where u and v are the magnitude of the velocity vector components in
v
the two orthogonal directions.
Addition and subtraction of vectors (and also matrices) is the same as for scalars.
Each entry of the first vector (matrix) is added or subtracted to the corresponding
entry in the second vector (matrix). This does require the two vectors (matrices) to
be of identical dimension. We distinguish different forms of vector multiplication.
For two vectors, a and b the inner product (also called dot product or scalar product)
is defined as:
2.3 Review of Vector Algebra
39
⎞
b1
⎜ ⎟
a · b = (a1 · · · an ) ⎝ ... ⎠ =
⎛
bn
n
ai bi
(2.61)
i=1
Here (a1 · · · an ) = a T , with T denoting the transposition of vector a. When two
vectors are orthogonal to each other, the inner product of these vectors is zero.
The outer product refers to the tensor product of two vectors. Given a vector
a = (a1 , . . . , am ) with m elements and a vector b = (b1 , . . . , bn ) with n elements,
their outer product a ⊗ b is defined as the m × n tensor A obtained by multiplying
each element of a by each element of b. Thus, the outer product defines every entry
of A according to Amn = am bn . Notation of the outer product often neglects the “⊗”
symbol. For example, the outer product of the vector V with itself can be denoted as
V V , as we will see in Sect. 2.5.2.
The third form of multiplication is the so called the cross product (also called
vector product). The cross product of two vectors is a vector:
a × b = ab sin θn = v
(2.62)
where θ is the measure of the smaller angle between a and b (0◦ ≤ θ ≤ 180◦ ), a and
b are the vector magnitudes of vectors a and b, and n is a unit vector perpendicular
to the plane containing a and b in the direction given by the right-hand rule. If the
vectors a and b are parallel (i.e., the angle θ between them is either 0◦ or 180◦ ), by
the above formula, the cross product of a and b is the zero vector 0. In the present
text the cross product in the three-dimensional Euclidian space is considered and
the vectors a and b therefore each have three entries. This allows us to calculate the
entries for v using the third-order determinant:
i j k
v = a1 a2 a3 = (a2 b3 − a3 b2 )i + (a3 b1 − a1 b3 )j + (a1 b2 − a2 b1 )k,
b1 b2 b3 (2.63)
where i, j, and k are the unit vectors in the respective directions x, y, and z.
Example 2.7 For the vectors a = (1, 2, 3) and b = (4, 5, 6), calculate the inner
product, the outer product and the vector product of a with b and b with a.
Solution:
The inner product is calculated according to (2.61):
a · b = 1 · 4 + 2 · 5 + 3 · 6 = 32
40
2 Review of Fundamental Equations
The outer product can be calculated according to:
⎡
⎤
4 5 6
(1, 2, 3)T ⊗ (4, 5, 6) = ⎣ 8 10 12 ⎦
12 15 18
To calculate the vector product we use (2.63):
a × b = (2 · 6 − 3 · 5, 3 · 4 − 1 · 6, 1 · 5 − 2 · 1) = (−3, 6, −3)
b × a = −(a × b) = (3, −6, 3)
In vector calculus we have two kinds of functions: scalar functions and vector functions. The output of a scalar function at a particular point P is a scalar:
f = f (P)
Vector function, g, is based on the vectorial function evaluation at a particular point P:
g = g(P) = (g1 (P), g2 (P), g3 (P))
The output of the vector function is a vector. The domain where a vector function is
defined is a region of space (can also be a surface or a curve in space). Within this
region (or on that surface or line) we say that this vector function defines a vector
field. In Cartesian coordinates a vector field can be denoted by
g(x, y, z) = (g1 (x, y, z), g2 (x, y, z), g3 (x, y, z)),
while a scalar function in Cartesian coordinates can be written as f (x, y, z).
2.3.2 Gradient of a Scalar Field
The following sections present various operations that can be performed on vector
and scalar fields. One of these operations is the gradient of a scalar field. The gradient
of a scalar field is a vector field which points in the direction of the greatest rate of
increase of the scalar field, and whose magnitude is the greatest rate of change.
The gradient (grad) of a scalar field is defined as (in three-dimensional Cartesian
coordinates):
∂f ∂f ∂f
,
,
(2.64)
grad f = ∇ f =
∂x ∂ y ∂z
The ∇ (read: nabla or del) operator is defined as the following differential vector
operator:
2.3 Review of Vector Algebra
41
∇≡
∂
∂
∂
i+
j+
k
∂x
∂y
∂z
(2.65)
We can define a scalar function whose gradient forms the three components of the
velocity field for irrotational, inviscid flow. This so-called potential-flow equation
reduces the system of equations to a single equation, thereby simplifying the problem
of finding the velocity field considerably. For example, the velocity distribution in a
fluid flow can be represented by the vector field V. At any point, P, throughout the
fluid this vector field can be related to the gradient of a scalar field, Φ, at this point:
V(P) = ∇Φ(P). Since this is true throughout the entire physical domain we define
the velocity potential function as:
V ≡ ∇Φ
(2.66)
Example 2.8 Consider a fluid flow where the velocity vector field is given by V =
(2x, yz 2 , zy 2 ). Determine the velocity potential function that describes this flow, if
it exists.
Solution:
To find Φ(x, y, z) find the primitive function of each of the components of the vector
field:
Φ = 2xdx = x 2 + C1 (y, z)
y2 z2
Φ = yz 2 dy =
+ C2 (x, z)
2
z2 y2
+ C3 (x, y)
Φ = zy 2 dz =
2
It is possible to combine the above equations into a single expression for the following
potential flow function: Φ = x 2 + 21 y 2 z 2 + C4 , where C4 is a constant. When we
employ (2.66) to Φ, we find V . This confirms that Φ exists and is indeed the potential
function of V .
2.3.3 Divergence of a Vector Field
To assess the magnitude of ‘change’ of a vector field we use the divergence (div) of
a vector field defined as:
div g = ∇ · g =
where g = (g1 , g2 , g3 ).
∂g2
∂g3
∂g1
+
+
∂x
∂y
∂z
(2.67)
42
2 Review of Fundamental Equations
Example 2.9 For the vector field of Example 2.8 calculate the divergence.
Solution:
Employing (2.67) we obtain the following:
div V = ∇ · V = 2 + z 2 + y 2
2.3.4 Curl of a Vector Field
To study the rotation of a vector field we employ the curl. The curl is defined as:
i
j
k curl g = ∇ × g = ∂/∂x ∂/∂ y ∂/∂z g1
g2
g3 =
∂g3
∂g2
−
∂y
∂z
i+
∂g1
∂g3
−
∂z
∂x
j+
∂g2
∂g1
−
∂x
∂y
(2.68)
k
In a velocity field, the curl is called the vorticity vector and is related to the rotation
of the flow. Rotation of the flow is introduced due to the viscosity of the gas and leads
to turbulent conditions. We will see later that the rotation of a flow field is directly
tied to the formation of curved shock waves in supersonic flow via Crocco’s theorem.
Example 2.10 Determine the curl of the vector field of Example 2.8.
Solution:
We can directly apply (2.68):
curl V = ∇ × V = (2zy − 2zy)i + (0 − 0)j + (0 − 0)k = 0
In Example 2.10 we see that the specified vector field is irrotational. This agrees with
the fact that we can define a potential equation. This confirms that we can define a
potential function only when the vector field is irrotational.
2.3.5 Relation Between Volume, Surface, and Line Integrals
In this section we briefly review the relation between volume, surface, and line
integrals. First, we consider an enclosed volume, V , with an outer surface S. The
divergence theorem (usually attributed to Gauss) states that the flux of a vector field
2.3 Review of Vector Algebra
43
through the surface equals the divergence of that vector field inside the enclosed
volume. For a velocity vector field, V , this theorem can be written as follows3 :
V · dS =
∇ · V dV
(2.69)
V
S
The gradient theorem states that the gradient of a scalar field, p, integrated over
the control volume, equals the scalar field integrated over the surface vector of the
control volume:
pdS =
∇ pdV
(2.70)
V
S
where dS is the element of the surface vector in the normal direction to the surface
pointing outward. Finally, we can relate the curl of a vector field over a control surface,
S, to the closed line integral of the vector field in a counter-clockwise direction over
its boundary, C. This is the Kelvin-Stokes theorem (often referred to as Stokes’
theorem):
V · dC =
(∇ × V ) · dS
(2.71)
C
S
2.4 Review of Thermodynamics
Thermodynamics is the science that studies the energy conversion between mechanical work and heat. In fluid flow, thermodynamics plays an important role in transferring energy from one state to the other. A simple example is a stagnation point at
the leading edge of a wing, where the flow is brought to rest and its kinetic energy is
transferred to heat. Similar effects arise in the boundary layer over the wing where
the friction between the air and the structure also converts kinetic energy into heat.
As we will see in Sect. 2.5 the first law of thermodynamics forms the basis of the
energy equation in fluid flow. This section presents the thermodynamic parameters
that we find in a fluid flow and their interrelation via the state law, the first law of
thermodynamics, and the second law of thermodynamics. It is emphasized that this
section is merely a review of the thermodynamic relations that we will use in subsequent sections and chapters. For a more complete and in-depth discussion of the
topic the reader may consult Sonntag and Van Wylen [21]. For more information on
the application of thermodynamics to aircraft propulsion, the reader is referred to
Farokhi [9].
3 This theorem applies to any vector field, not only V but also mass flow or force fields.
44
2 Review of Fundamental Equations
2.4.1 Perfect Gas Relations
In fluid flow, the thermodynamic properties of fluid are characterized by scalar fields.
The variables of interest for the present discussion are pressure ( p in Pa or psi), density
(ρ in kg/m3 or lb/ft3 ), temperature (T in K or ◦ F), and the specific internal energy
(e in J/kg or BTU/lb). According to the state principle of thermodynamics, the local
thermodynamic state is determined by any two independent state variables. If e and
ρ are chosen as those variables, both pressure, p, and temperature, T , are dependent
on these two variables. The subsequent paragraphs present the equations that relate
the state variables to each other via a set of laws.
For a calorically perfect gas (intermolecular forces are assumed negligible and the
volume of individual molecules is infinitesimally small) the equation of state reads:
p = ρRT,
(2.72)
where R is the gas constant for a specific gas. In terms of specific volume, v = 1/ρ,
(2.72) can also be expressed as pv = RT . A new thermodynamic state variable is
defined as the sum of internal energy, e, and the product of pressure and specific
volume and is called specific enthalpy:
h = e + pv = e + RT
(2.73)
Even though specific enthalpy is introduced here without any physical motivation,
we will see later that it is indeed a fundamental state variable for aero-thermodynamic
analysis. Both specific energy and specific enthalpy (which also has unit J/kg) are
both related to the temperature for a perfect gas:
e = e(T )
(2.74)
h = h(T )
(2.75)
In transonic flow we assume that the gas is perfect and we can consequently find
de = cv dT
(2.76)
dh = c p dT
(2.77)
where cv is the specific heat at constant volume and c p the specific heat at constant
pressure. These coefficients can be assumed to be constant for air up to temperatures
of 1,000 K, which is the basis for the calorically perfect gas assumption. Based on
this assumption we can now define the following relations between the specific heat
coefficients, cv and c p , the specific internal energy and enthalpy, e and h, the gas
constant, R, and the ratio of specific heats, γ:
2.4 Review of Thermodynamics
45
e = cv T, h = c p T, γ =
cp
R
γR
, cp =
,
, cv =
cv
γ−1
γ−1
R = c p − cv
(2.78)
Using the above relations, the temperature and pressure can now be defined in terms
of the independent state variables e and ρ:
p = (γ − 1)ρe, T =
(γ − 1)e
R
(2.79)
For dry air at temperatures that are normally encountered during flight, the gas
constant equals R = 287.04 J/kg/K.
The viscosity (μ) and thermal conductivity (k) are properties of a gas that are
both dependent on the temperature. The viscosity of the fluid is responsible for the
momentum transport on molecular level. The dynamic viscosity of an ideal gas is:
1
ρλc
3
μ=
(2.80)
Here λ is the mean free path between molecules and c is the average velocity of the
molecules. Examining this equation we can see that the viscosity increases with the
average velocity and therefore with temperature. Sutherland’s equation states that the
viscosity of air is dependent on its temperature according to the following relation:
μ
=
μ0
T
T0
3/2
T0 + S
T +S
(2.81)
where μ0 and T0 are a reference viscosity and temperature, respectively, and S is the
Sutherland temperature which is S = 110 K for air. For ISA (international standard
atmosphere) conditions these are μ0 = 1.7894 × 10−5 kg/m/s and T0 = 288.16 K.
The thermal heat conductivity was already introduced in Fourier’s law (2.28) and is a
property of the gas. For air k = 0.024 W/m/K at a temperature of 273 K. The Prandtl
number relates the viscosity and the thermal conductivity of a fluid according to:
Pr =
cp
μ
k
(2.82)
Since k, μ, and c p are dependent on the temperature, the Prandtl number is dependent
on temperature as well. However, for air over a substantial temperature range (up to
600 K) we can assume the Prandtl number remains constant at 0.71 [3]. In Fig. 2.8
the viscosity and conductivity of air are presented as a function of temperature, along
with the assumed values for the constants that have been used. We see that there is
an appreciable change in conductivity and viscosity with temperature, which can be
important if we model high-speed fluid flows.
46
2 Review of Fundamental Equations
T = 288K, µ = 17.8mg/m/ s
Pr = 0.71 cp = 1003.5J/kg/ K
0
Thermal conductivity, k (mW/m/K)
0
Viscosity, µ (mg/m/s)
30
25
20
15
10
200
300
400
500
Temperature, T (K)
600
45
40
35
30
25
20
15
200
300
400
500
Temperature, T (K)
600
Fig. 2.8 Graphical representation of (2.81) and (2.82) for dry air
2.4.2 First Law of Thermodynamics
The first law of thermodynamics states that energy can only be transformed from one
form into another. In other words, the summation of all energies in a closed system
remains constant. The system has hypothetical boundaries to the surroundings that
we can define based on the problem under consideration. The change in specific
internal energy of the system, de, can be related to the increment in specific work
done by the system, δw, and the specific heat added to the system, δ Q:
de + δw = δ Q
(2.83)
As we noted in Sect. 2.4.1, e is a state variable. That means that de is an exact
differential which only depends on the difference between the initial (e1 ) and final
state (e2 ) of the process:
2
de = e2 − e1
(2.84)
1
Contrary to that, the heat and work depend on the path between the two states.
Therefore, their changes are represented with a δ in (2.83). To demonstrate this
process dependency Fig. 2.9 shows how the total amount of work changes with the
path taken from point 1 to point 2. Note that this figure shows expansion processes
where pV n = constant. This is a polytropic process, with p and V representing
the pressure and volume of the gas, respectively. The exponent, n, depends on the
process. For example if n = 0 we have an isobaric process, when n = 1 we have an
isothermal process (using the perfect gas law). Of course, we can define an infinite
number of ways to get from state 1 to state 2, each described by a different value of n.
If work is done by a system under adiabatic conditions (meaning no heat addition
or extraction, hence δ Q = 0) the internal energy (read: temperature) of the system
2.4 Review of Thermodynamics
47
Fig. 2.9 Work depends on
the path (process) between
state 1 and state 2
p
1
A
B
Area = reversible work
for process A
2
changes according to:
de = −δw
(2.85)
For a process that is reversible (no dissipative phenomena occur), we can calculate
the specific work (w) according to:
w=
2
1
pdv =
p2 v 2 − p1 v 1
1−n
p1 v1 ln vv21
for n = 1 (non-isothermal condition)
for n = 1 (isothermal condition)
(2.86)
Substituting the perfect gas law for pv in (2.86), we get the following expression for
non-isothermal processes:
R(T2 − T1 )
(2.87)
w=−
n−1
Since we assume that the change in energy equals the work done on the system and
using (2.76), we can write:
Δe = cv (T2 − T1 ) = −w
(2.88)
Combining (2.87) and (2.88) leads, after simplification, to the following statement:
n=
cp
=γ
cv
(2.89)
which demonstrates that for an adiabatic and reversible process the following is true:
pvγ = constant
(2.90)
Let us recap the result of this process. We have made two assumptions in the
previous derivation: (1) no heat addition or extraction; (2) no dissipative phenomena
occur. Finally, we implicitly assumed that the total amount of mass within the system
48
2 Review of Fundamental Equations
remained the same, which is inherent in the definition of a thermodynamic system.
Under these assumptions, the process is adiabatic and reversible. As the reader might
remember from a course on thermodynamics, this process is called isentropic. In an
aerodynamic context this means that the dissipative phenomena such as thermal
conductivity, viscosity and mass diffusion are assumed to be absent from the flow. In
addition, no heat is added or extracted from the flow by means of heating or cooling
the flow, respectively. Flows that exhibit the formation of shock waves are therefore
per definition anisentropic as this process is irreversible and dominated by viscous
properties in the flow.
2.4.3 Second Law of Thermodynamics
The first law of thermodynamics only states that energy is conserved during a thermodynamic process. However, it does not tell us anything about the direction in which
the energy flow is taking place. This is governed by the second law of thermodynamics that states that over time, differences in temperature, pressure, and chemical
potential tend to even out in a physical system that is isolated from the outside world.
To quantify this ‘evening-out’ process, a property called entropy, S (denoted by s for
specific entropy), is introduced. Entropy is related to the change in heat in a reversible
system:
δ Q rev
,
(2.91)
ds =
T
where T is the temperature of the system. Entropy is a state property meaning that
the change in entropy of a system going from one state to another is the same for all
processes. Note that for an irreversible process where dissipative phenomena occur,
we could always assign an effective δ Q rev that relates the initial state and the end state
to each other. However, it is more revealing to say that these dissipative processes
produce their own entropy, dsirrev , i.e.:
ds = dsrev + dsirrev =
δ Q rev
+ dsirrev
T
(2.92)
This statement says that ds has two parts, one is reversible and the other is irreversible
due to dissipation phenomena. The dissipative phenomena within the system always
increase the entropy of the system:
dsirrev ≥ 0
(2.93)
If dsirrev = 0 we have a truly reversible process. If ds exceeds the reversible limit,
the process is irreversible and entropy is produced. Combining (2.92) and (2.93) we
express the second law of thermodynamics as:
2.4 Review of Thermodynamics
49
ds ≥
δQ
T
ds ≥ 0 for adiabatic processes
(2.94)
(2.95)
When a process is isentropic (literally meaning “equal disorder”) the entropy of the
system and its surroundings remains constant from state 1 to state 2.
Combining the first and second law of thermodynamics, (2.83) and (2.91), respectively, produces Gibbs’ equation:
T ds = de + pdv
(2.96)
If we use the chain rule we can find the change in enthalpy (2.73) to be dh =
de + pdv + vd p. Combining this with (2.96) we can now write two alternative forms
for the first law of thermodynamics, one in terms of specific energy and one in terms
of specific enthalpy:
T ds = de + pdv
T ds = dh − vd p
(2.97)
(2.98)
By inserting the relations between temperature and specific energy (2.76) and specific
enthalpy (2.77), respectively, and by utilizing the state law (2.72) we can find two
expressions for ds: one in terms of specific volume and one in terms of specific
pressure (see Problem 2.22). Integrating both relations between the state properties
at state 1 and state 2 results in the following expressions for the change in entropy
of a calorically perfect gas:
T2
v2
+ R ln
T1
v1
T2
p2
s2 − s1 = c p ln
− R ln
T1
p1
s2 − s1 = cv ln
(2.99)
(2.100)
These forms of Gibbs’ equation are useful in aero-thermodynamic calculations. For
an isentropic flow, i.e. s1 = s2 , we can establish the isentropic relations between
temperature, density, and pressure (see Sect. 2.4.4).
Example 2.11 We consider a one-dimensional flow field of air with a discontinuity
in state properties due to the presence of a normal shock wave. In front of the
shock, the static pressure is 50 kPa, while right behind the shock it is 75.6 kPa. The
static temperature over the shock increases with a factor of 1.128. Calculate the
corresponding change in entropy over the shock and the corresponding density ratio
between state 2 and 1.
50
2 Review of Fundamental Equations
Solution:
We calculate the pressure ratio to be p2 / p1 = 1.512 and enter this in (2.100) to find
s2 − s1 = 0.746 J/kg/K. Rewriting (2.99) we get the following:
⎤
⎡
(s1 − s2 ) + cv ln TT21
ρ2
v1
⎦
=
= exp ⎣
ρ1
v2
R
Substitution of the appropriate values and constants gives ρ2 /ρ1 = 1.34.
The example above is actually a demonstration of how the state properties of the
flow change when it goes through a normal shock wave. Pressure, density, temperature, and entropy all increase depending on the Mach number of the flow in front of
the shock wave. In the above example this Mach number is 1.2 and the entropy change
is still relatively small. Notice, though, that even at this relatively low Mach number
the shock wave creates a relatively large pressure, temperature, and density rise.
2.4.4 Isentropic Relations
Based on the relations (2.99) and (2.100) we can investigate the change in state
properties for an isentropic (adiabatic and reversible) process. In this case ds = 0
and we can rewrite (2.99) and (2.100) to show the relation between the pressure,
density and temperature ratios in an isentropic process. We set the LHS of (2.99)
and (2.100) to zero and perform some manipulations to find the following:
v2
=
v1
p2
=
p1
T2
T1
T2
T1
−cv /R
(2.101)
c p /R
(2.102)
By using the relations of (2.78), we can make the following substitutions:
cp
cv
1
γ
=
and
=
R
γ−1
R
γ−1
With these substitutions and subsequent manipulations (see Problem 2.23) we can
now state the isentropic relations:
p2
=
p1
ρ2
ρ1
γ
=
T2
T1
γ
γ−1
(2.103)
2.4 Review of Thermodynamics
51
Now that we have the isentropic relations we ask ourselves: why are they so
important? We know that in reality a gas flow is viscous and thermally conducting
and therefore dissipative phenomena always occur. However, we also know that these
phenomena are only dominant in small regions in the flow: the boundary layer and
inside the shock waves. Outside of these regions the dissipative phenomena are so
small that they can often be neglected when we want to calculate the local state
properties. However, within these regions the isentropic relations do not hold and
we cannot relate the state properties in this simple way. This becomes evident in the
subsequent section and chapters of this text.
2.5 Equations of Fluid Motion
The following sections give an overview of the fundamental equations of motion in
(transonic) fluid dynamics. For a more comprehensive treatment of this subject matter we refer to introductory texts on aerodynamics such as the work by Anderson [3].
Most of what is presented in this chapter follows the text of Tannehill et al. [19].
Before we start discussing the equations of motion we introduce two nondimensional numbers that are often used to characterize a fluid flow: the Reynolds
number and the Mach number. The Reynolds number is a measure for the ratio of
inertial forces to viscous forces in a moving fluid. Its definition is as follows:
Rel =
ρV l
μ
(2.104)
where l is a characteristic length, for example the chord length of an airfoil or the
diameter of a pipe. A second non-dimensional number to characterize a fluid flow is
its Mach number. The Mach number is defined as the ratio of the local flow velocity
to the local speed of sound:
V
M=
(2.105)
a
The local speed of sound, a, is a function of the state properties of the fluid. For a
perfect gas the speed of sound is solely dependent on the temperature of the gas:
a=
∂p
∂ρ
=
γ
p = γ RT
ρ
(2.106)
2.5.1 Conservation of Mass
We consider a fluid flow through a fixed control volume V . We consider the balance
between the mass flowing in and out of the control volume and the time rate of change
52
2 Review of Fundamental Equations
of the mass inside the control volume. We use the empirical law that mass can neither
be destroyed nor created during this process. Mass can only exit the control volume
through its enclosed surface, S. The resulting mass flowing out of the control volume
can be written as follows:
ρV · dS
(2.107)
S
where V is the velocity vector. Note that the vector dS has a magnitude of dS and
a direction n, which is perpendicular to the surface of dS. We can express the time
rate of increase in mass inside V using the following volume integral:
∂ ρdV
∂t
(2.108)
V
The change in mass within the control volume should be in balance with the net mass
flux through the control surface. This conservation principle results in the following
continuity equation in integral form:
S
ρV · dS +
∂ ρdV = 0
∂t
(2.109)
V
Equation (2.109) is the conservation of mass in integral form. Applying the divergence theorem to the surface integral combines both terms within one volume integral
(see Problem 2.24). From this integral the conservation form of the continuity equation can be found. The conservation of mass law in differential form is:
∂ρ
+ ∇ · (ρV ) = 0
∂t
(2.110)
Let us briefly analyze this equation. It is a first order partial differential equation.
At a particular point in space it describes the change in density with time and the
divergence of the product of density and velocity. We can simplify this equation
even further by introducing the substantial derivative operator, D/Dt. The substantial derivative combines the local derivative (∂/∂t) and the so-called convective
derivative (V · ∇), to represent the “total” derivative:
∂
D
=
+V ·∇
Dt
∂t
(2.111)
In the case of the density in two Cartesian dimensions it reads:
Dρ
∂ρ
∂ρ
∂ρ
∂ρ
=
+ (V · ∇)ρ =
+u
+v
Dt
∂t
∂t
∂x
∂y
(2.112)
2.5 Equations of Fluid Motion
53
This equation physically states that the change in density is due to temporal and
spatial variations. Using the definition of the substantial derivative (2.111) we can
rewrite the continuity equation (2.110) in substantial-derivative form:
Dρ
+ ρ(∇ · V ) = 0
Dt
(2.113)
When homogeneous, incompressible flow is considered, the first term on the LHS
can be dropped and only have ∇ · V = 0. However, generally speaking this is only a
valid approximation at Mach numbers below M = 0.3 where density variations are
less than 5 %. In transonic flows with typical Mach numbers in the range of 0.8–1.3
we cannot use the incompressible flow assumption.
2.5.2 Conservation of Linear Momentum
The momentum theorem in fluid mechanics is the counterpart of Newton’s second
law of motion in solid mechanics which states that the time rate of change of linear
momentum of a body of mass, m, must be equal to the net forces that act on that body:
F=
d
(mV )
dt
(2.114)
Let us again consider a fluid passing through a finite control volume. The forces
on the fluid in this control volume can be divided into forces that are acting on the
fluid (such as gravity) and forces that are acting on the control surface, S (pressure and shear forces). In Fig. 2.10 it is schematically shown how the body force,
pressure and stress vectors act on a control volume in Cartesian coordinates. Note
that τij is the stress on the surface of the control volume with normal vector i in
y
y
y
τ22
p
τ21
τ23
τ32
p
f
z
τ13
τ11
x
τ33
p
z
τ31
x
x
τ12
z
Fig. 2.10 Schematic control volume with body force (left), surface pressure (center) and shear
stresses (right)
54
2 Review of Fundamental Equations
the direction j. The stress components for which i = j are shear stress components, while the stress components normal to the surface (i = j) are associated with
the thermodynamic pressure. The thermodynamic pressure can be perceived as the
force exerted on the control volume walls as fluid molecules coincide with it during
their random movement. The latter becomes important only for such effects where
fluid compressibility is essential. Examples would include shock waves and sound
propagation. For incompressible flows the thermodynamic pressure is zero.
If we denote the body forces per unit volume by ρ f we can represent the total
body force by:
ρ f dV
(2.115)
body force =
V
The most common body force per unit volume is the gravitational force for which
ρ f = ρg. The surface force due to pressure always acts perpendicular to the surface
of the control volume:
pdS
(2.116)
surface force due to pressure = −
S
Note that the pressure force is negative, since the pressure acts in the opposite direction to the surface normal vector, n. Finally, the surface force due to friction can be
written as follows:
τ ij · dS
(2.117)
viscous surface force =
S
where τ ij is the viscous shear stress tensor. In three dimensions this tensor is expressed
as follows:
⎤
⎡
τ11 τ12 τ13
τ ij = ⎣ τ21 τ22 τ23 ⎦
(2.118)
τ31 τ32 τ33
A tensor can be perceived as a multi-dimensional vector. The inner product of a
tensor with a vector therefore results in a vector, rather than a scalar as we noted in
Sect. 2.3. In three dimensions, this tensor consists of three vectors, which form its
columns. The inner product should be applied to each of these vectors. The entries
of the resulting vector correspond to the scalar result of each of these operations. We
can combine each of the force components and form the LHS of (2.114):
F=
V
ρ f dV −
S
pdS +
τ ij · dS
(2.119)
S
Now, let us turn our attention to the RHS of (2.114) and look at the time rate of
change of linear momentum. Similar to the components in the continuity equation
we distinguish two contributions to the time rate of change of momentum: first due
to the momentum change with time of the fluid inside the control volume and second
2.5 Equations of Fluid Motion
55
due to momentum entering and leaving the control volume. These components can
be defined as follows:
d
∂ (mV ) =
ρV dV +
(ρV · dS)V
dt
∂t
V
(2.120)
S
We can now combine (2.119) and (2.120) to obtain a first version of the fluid-flow
momentum balance in integral form:
∂ ρV dV + (ρV · dS)V =
ρ f dV −
pdS +
τ ij · dS (2.121)
∂t
V
V
S
S
S
We can expand the shear stress terms in this equation in terms of state variables
and fluid constants, but let us first evaluate the differential form of the momentum
equation.
By applying the gradient and divergence theorems (Sects. 2.3.2 and 2.3.3, respectively) appropriately to each of the individual terms in (2.121) we can rewrite this
equation only in terms of volume integrals, which allows us to evaluate only the
integrand. This results in the following differential form of the momentum equation:
∂
(ρV ) + ∇ · (ρV V ) = ρ f − ∇ p + ∇ · τ ij
∂t
(2.122)
Note that V V is the outer product of V with itself, leading to the second-order
velocity tensor. For Newtonian fluids (stress-rate of strain relation is linear) the
relation between the shear stress tensor and velocity the components can be written
as follows:
∂u j
∂u k
∂u i
+ δij μ
+
i, j, k = 1, 2, 3
(2.123)
τ ij = μ
∂x j
∂xi
∂xk
where μ is the dynamic viscosity and μ is second (or bulk) coefficient of viscosity
of the fluid. Furthermore, u i is the velocity component of V in the direction of i and
δij is the Kronecker delta.4 Following Stokes’ hypothesis:
2
μ = − μ
3
(2.124)
Therefore, the viscous stress tensor, τ ij , can be written as:
τ ij = μ
∂u j
∂u i
+
∂x j
∂xi
4 The Kronecker delta is defined as follows: δ
ij =
2 ∂u k
− δij
3 ∂xk
1 for i = j
0 for i = j
.
(2.125)
56
2 Review of Fundamental Equations
By expanding the LHS of (2.122) using the chain rule and subsequent substitution
of the continuity equation (see Problem 2.25), we can write the momentum equation
in substantial-derivative notation according to:
ρ
DV
= ρ f − ∇ p + ∇ · τ ij
Dt
(2.126)
Subsequent substitution of (2.123) yields the following set of momentum equations:
ρ
∂u j
∂u i
2
DV
∂u k
∂
μ
− δij μ
+
= ρ f − ∇p +
Dt
∂x j
∂x j
∂xi
3
∂xk
(2.127)
We emphasize that (2.127) is a set of three second-order partial-differential equations (if three-dimensions are considered). These equations are often referred to as
the Navier-Stokes equation named after Claude-Louis Navier and George Gabriel
Stokes.
2.5.3 Conservation of Energy
The first law of thermodynamics (2.83) forms the basis for the energy balance
between a fluid going through a fixed control volume and its surroundings:
de + δw = δ Q
(2.83)
Note that (2.83) is the balance between the rate of change in internal (or specific)
energy with the sum of the rate of specific heat added to the fluid and the rate of
specific work done by the fluid. In order to express the first law in energy per unit
volume we need to multiply (2.83) by the density:
ρde + ρδw = ρδ Q
(2.128)
Let us expand each of those terms in terms of state variables of the fluid. We start
with the LHS of (2.128), which is the change of internal energy. Remember, that this
law is written for a gas in stationary condition. Since we are considering a moving
fluid, the kinetic energy per unit mass is to be added. If we denote the total energy
per unit volume with E t we have:
V2
Et = ρ e +
2
(2.129)
Since the subject of this text is aerodynamics, we have intentionally omitted a potential energy term in (2.129). The time rate of change of total energy in the control
volume can now be written as:
2.5 Equations of Fluid Motion
57
∂ E t dV
∂t
V
(2.130)
Since fluid is entering and leaving the control volume, the kinetic and internal energy
changes. Each particle that leaves the control volume has a volume-specific energy
density of E t . The total rate of volume-specific total energy leaving the control
volume therefore equals the volumetric flux over the boundary dS times E t : E t V ·dS.
Integrating over the entire control surface yields the following:
E t V · dS
(2.131)
time rate of energy transfer across control surface =
time rate of change of total energy inside control volume =
S
Now, let us turn our attention to the heat term in (2.128). First, we consider the
time rate of change of heat inside the control volume. Similar to the energy flux, we
define two components: the time rate of change of heat inside the control volume
and the heat flow going in and out of the volume via the control surface. We switch
in nomenclature from specific heat (δ Q, unit: J/kg) to heat per unit volume (ρδ Q,
unit: J/m3 ). The time rate of change of heat inside the control surface is given by:
time rate of change of heat inside control volume =
∂ ρQdV
∂t
(2.132)
V
By using Fourier’s law of heat conduction (see also (2.28) in Sect. 2.2.2), the heat
flow (q, unit: J/s/m2 ) is linearly related to the temperature gradient:
q = −k∇T
(2.133)
where k is the coefficient of thermal conductivity and T is the temperature of the gas.
The heat flow into the control volume is in opposite direction to the control surface
vector dS. Therefore, the rate of heat addition is negatively related to the heat flux
over the control surface dS:
q · dS
(2.134)
heat flow across control surface = −
S
Having examined the energy and heat flux in the control volume, the last part of
(2.128) is the rate of work being done on the fluid in the control volume. We consider
three different contributions that are related to the three different forces we defined
in Sect. 2.5.2. First we have the time rate of change in work due to the body forces.
We know that work can be defined as the force times a displacement. Hence, the
time rate of change of work can be expressed as the force multiplied by the velocity.
Therefore, we have the following:
time rate of change of work due to body forces =
V
ρ f · V dV
(2.135)
58
2 Review of Fundamental Equations
Secondly, we have the pressure on the control surface that does work on the fluid
inside the control surface. We take the inner product of the RHS of (2.116) with V
to obtain the work flux due to this component:
time rate of work done due to pressure on the control surface = −
pdS · V
S
(2.136)
Finally, we look at the rate of work done due to the shear stress on the control surface.
Similar to the pressure-induced work flux, the work flux due to the shear stress can
be found by taking the inner product of the RHS of (2.117) with V :
time rate of work done due to shear stress on the control surface =
(τ ij · dS) · V
S
(2.137)
Remember that τ ij is the stress tensor as defined in (2.118).
We have now defined the time derivative of each of the components of (2.128) and
we can formulate the first version of the energy balance between the control volume
and its surroundings:
∂ ∂ E t dV +
E t V · dS =
ρQdV −
q · dS +
ρ f · V dV
∂t
∂t
S
S
V
V
V
−
pdS · V +
(τ ij · dS) · V
(2.138)
S
S
We can cast (2.138) in differential form by applying the gradient and divergence
theorems to the appropriate terms and changing everything to volumetric integral
formulation. The integrand of that equation yields the following energy balance:
∂ Et
∂(ρQ)
+ ∇ · Et V =
− ∇ · q + ρ f · V − ∇( p · V ) + ∇ · (τ ij · V ) (2.139)
∂t
∂t
The first term on the LHS of (2.139) represents the rate of increase of E t in the control
volume, while the second term represents the total energy lost due to convection
through the control volume. The first term on the RHS is the amount of heat produced
by external factors. The second term is the rate of heat lost by means of conduction.
The third term represents the work done by body forces while the last two terms
represent the work done by normal and shear stresses on the surface, respectively.
By subsequently employing the continuity and momentum equation (see Problem
2.26) we can write the energy equation in a convenient substantial-derivative form:
ρ
∂(ρQ)
De
+ p(∇ · V ) =
− ∇ · q + ∇ · (τ ij · V ) − (∇ · τ ij ) · V
Dt
∂t
(2.140)
The sum of last two terms in this equation is termed the dissipation function, Φ, and
represent the rate at which mechanical energy is expended due to viscosity when
2.5 Equations of Fluid Motion
59
the fluid is deformed. The LHS of (2.140) can be rewritten in terms of enthalpy
by employing (2.73) and the continuity equation (see Problem 2.27). The following
energy balance results:
ρ
D p ∂(ρQ)
Dh
=
+
−∇·q+Φ
Dt
Dt
∂t
(2.141)
where the dissipation function is:
Φ = ∇ · (τ ij · V ) − (∇ · τ ij ) · V
(2.142)
From a mathematical point of view we classify (2.141) as a second order partial
differential equation. Note that the second order terms appear only in the dissipation
function (see Problem 2.27).
If we start from (2.139) and assume isentropic conditions (inviscid: μ = 0 and
adiabatic: d(ρQ)/dt = 0 and k = 0), we can write the energy equation as follows:
ρ
D(e + V 2 /2)
= −∇( p · V )
Dt
(2.143)
Here, we have assumed the body forces, f , to be negligible. Using the continuity
equation (2.113), we can rewrite this equation as follows:
ρ
∂p
D(h + V 2 /2)
=−
Dt
∂t
(2.144)
In steady conditions, we can therefore derive that:
h+
V2
= H = constant
2
(2.145)
where H is the definition of the total enthalpy. If we use the relation between enthalpy
and static temperature (2.77), we can write:
cpT +
V2
= c p Tt = constant
2
(2.146)
where Tt is the total temperature. For a flow where all the streamlines emanate from
the same uniform freestream, the temperature and velocity are therefore uniquely
correlated through (2.146). We can manipulate (2.146) and include the definition of
the speed of sound (2.106) to obtain:
γ−1 2
Tt
=1+
M
T
2
(2.147)
60
2 Review of Fundamental Equations
Equation (2.147) states that there exists a unique relationship between the Mach
number and the static temperature in the flow. For a given total temperature, Tt , in
the freestream, Eq. (2.147) shows that the static temperature decreases with Mach
number. If we combine (2.147) with the isentropic relations (2.103) we can express
the temperature, density and pressure as a function of the Mach number. We have
tabulated this relation for Mach numbers ranging from 0 through 10 in Appendix A.
2.5.4 Conservation Form of the Navier-Stokes Equations
We now make two assumptions about the nature of the flow we consider. The first
assumption is that the body forces (such as gravity) are so small compared to the
surface forces that it is unnecessary to account for them during calculations. In
addition, we assume that there is no change in heat per unit volume due to volumetric
heating. Accordingly, we declare that:
f =0
d(ρQ)
=0
dt
(2.148)
Let us rewrite the three equations that represent the conservation of mass (2.110),
momentum (2.122) and energy (2.139), respectively, and apply the assumptions of
(2.148):
∂ρ
+ ∇ · (ρV ) = 0
(2.110)
∂t
∂
(ρV ) + ∇ · (ρV V ) + ∇ p − ∇ · τ ij = 0
∂t
∂ Et
+ ∇ · E t V + ∇ · q + ∇ p · V − ∇ · (τ ij · V ) = 0
∂t
(2.149)
(2.150)
This shortened form permits us to explicitly rewrite all five equations in their conservation form:
∂ρ (∂ρu) ∂(ρv) ∂(ρw)
+
+
+
=0
(2.151a)
∂t
∂x
∂y
∂z
τx y
∂(ρu) ∂(ρu 2 ) ∂(ρuv) ∂(ρuw) ∂ p
τx x
τx z
+
+
+
+
−
+
+
= 0 (2.151b)
∂t
∂x
∂y
∂z
∂x
∂x
∂y
∂z
τ yy
τ yz
τx y
∂(ρv) ∂(ρuv) ∂(ρv2 ) ∂(ρvw) ∂ p
+
+
+
+
−
+
+
= 0 (2.151c)
∂t
∂x
∂y
∂z
∂y
∂x
∂y
∂z
τ yz
∂(ρw) ∂(ρuw) ∂(ρvw) ∂(ρw2 ) ∂ p
τx z
τzz
+
+
+
+
−
+
+
= 0 (2.151d)
∂t
∂x
∂y
∂z
∂z
∂x
∂y
∂z
2.5 Equations of Fluid Motion
61
∂(u E t ) ∂(vE t ) ∂(wE t ) ∂(up) ∂(v p) ∂(w p)
∂ Et
+
+
+
+
+
+
∂t
∂x
∂y
∂z
∂x
∂y
∂z
∂q y
∂
∂qz
∂qx
−
+
(uτx x + vτx y + wτx z )
−
+
∂x
∂y
∂z
∂x
∂
∂
+ (uτx y + vτ yy + wτ yz ) +
(uτx z + vτ yz + wτzz ) = 0
∂y
∂z
(2.151e)
The Navier-Stokes equations consists of a time-dependent continuity equation for
conservation of mass, three time-dependent conservation of momentum equations
and a time-dependent conservation of energy equation. There are four independent
variables in the problem: x, y, z and time t. There are six dependent variables; the
pressure p, density ρ, and temperature T (which is contained in the energy equation
through the total energy E t ) and three components of the velocity vector (u, v, w). All
of the dependent variables are functions of all four independent variables. The differential equations are therefore a set of partial differential equations. We can reduce
the number of dependent variables to five by adding two algebraic equations to close
this system, namely (2.72) and (2.76) that together relate pressure, p, to the state
variables ρ and e (see Problem 2.18). We are then left with five partial differential
equations and five unknowns: u, v, w, ρ, and e.
It can be convenient to combine these equations in vector form such that they
become more compact. By doing so, we can formulate this set of equations in the
following vector form (see Problem 2.28):
∂U ∂E ∂F ∂G
+
+
+
=0
∂t
∂x
∂y
∂z
(2.152)
where:
⎡
⎤
ρ
⎢ ρu ⎥
⎢ ⎥
U = ⎢ ρv ⎥
⎣ρw⎦
Et
⎡
⎡
⎤
ρu
ρu 2 + p − τx x
⎥
⎥
ρuv − τx y
⎥
⎦
ρuw − τx z
(E t + p)u − uτx x − vτx y − wτx z + qx
⎤
⎤
⎡
ρv
ρw
ρuv + p − τx y
ρuw − τx z
⎥
⎥
⎢
⎢
⎥
⎥
⎢
⎢
ρvw − τ yz
F=⎢
ρv2 + p − τ yy
⎥ G=⎢
⎥
⎦
⎦
⎣
⎣
2
ρvw − τ yz
ρw + p − τ yz
(E t + p)v − uτx y − vτ yy − wτ yz + q y
(E t + p)w − uτx z − vτ yz − wτzz + qz
⎢
⎢
E=⎢
⎣
(2.153)
The first and last row in these vectors represent the continuity and energy equation,
respectively. The three middle rows represent the three components of the momentum
equation. The form above is often used because it is easier to code in numerical form.
It still represents the full equations of motion. Note that this set in the subsequent
text is referred to as the Navier-Stokes (NS) equations.
62
2 Review of Fundamental Equations
2.6 Reynolds-Averaged Navier-Stokes Equations
The Navier-Stokes equations can be solved numerically by using direct numerical
simulation (DNS). This means that the whole range of spatial and temporal scales
of the turbulence must be resolved. All the spatial scales of the turbulence must be
resolved in the computational mesh, from the smallest dissipative scales, up to the
integral scale L, associated with the motions containing most of the kinetic energy.
This requires to have a high spatial density of the volumetric mesh in combination
with a small time step. It can be shown that the number of floating-point operations
required to complete this simulation is proportional to the number of mesh points
and the number of time steps. The number of operations grows with the third power
of the Reynolds number: Re3L [4]. For practical engineering problems, the number
of required operations would exceed the maximum number of the most powerful
computers that are currently available. To increase the minimum spatial and temporal
scales and hence reduce the number of floating point operations, the Reynoldsaveraged Navier-Stokes equations (RANS) can be used. In this section we show how
the Reynolds-averaged Navier-Stokes equations can be obtained. Because RANS
equations relate the turbulence in the flow to the mean flow properties , we introduce
new dependent variables. Therefore, additional equations are necessary to “close”
the RANS equations. Many turbulence models have been developed to close the
RANS equations. As an example, we present the so-called k-epsilon model, which
is often used in engineering practice.
2.6.1 Incompressible Reynolds-Averaged Equations of Motion
Conventional Reynolds-averaging is done by splitting the flow parameters into a
time-averaged part and a fluctuating part. This is called Reynolds decomposition [15]:
u = ū + u v = v̄ + v
w = w̄ + w
p = p̄ + p
h = h̄ + h
T = T̄ + T
ρ = ρ̄ + ρ
H = H̄ + H
(2.154)
(2.155)
Here, the total enthalpy is defined by H = h + u i u i /2 and the barred parameters
represent averages. The fluctuation terms that appear in the equations above become
zero when they are time-averaged. For example:
u =
1 Δt u dt = 0
Δt 0
(2.156)
The equations of motion of Sect. 2.5 are modified by substitution of the flow parameters that we have defined in (2.155). Subsequently, each of the equations is time
averaged. The result yields a set of equations that have averaged fluctuating terms.
Each of the terms in the equations of motion that have averaged fluctuating terms
2.6 Reynolds-Averaged Navier-Stokes Equations
63
are subsequently set to zero. This yields a more compact version of these equations,
including the fluctuating terms. We briefly discuss each of the governing equations
below.
When substituting the relations of (2.154) in the continuity equation (2.110) the
following averaged equation can be obtained:
∂ ρ̄
∂ ρ̄ū + ρ u j = 0
+
∂t
∂x j
(2.157)
For incompressible flow, the fluctuating and unsteady density terms can be dropped
from this equation. Furthermore, ρ̄ = ρ = constant. What we are left with is a much
reduced continuity equation:
∂ ū j
=0
(2.158)
∂x j
Starting from the Navier-Stokes equation (2.127) and neglecting body forces,
the Reynolds-averaged momentum equation can be obtained by substitution of the
relations (2.154) and applying subsequent time averaging:
∂ ∂ ρ̄ū i + ρ u j +
ρ̄ū i ū j + ū i ρ u j
∂t
∂x j
∂ p̄
∂
=−
τ̄ij − ū j ρ u i − ρ̄ j u i u j − ρ u i u j
+
∂xi
∂x j
where
τ̄ij = μ
∂ ū j
∂ ū i
+
∂x j
∂xi
2 ∂ ū k
− δij
3 ∂xk
(2.159)
(2.160)
For incompressible flow, this can be simplified to:
∂
∂ p̄
∂ ∂
τ̄ij − ρu i u j
ρū i ū j = −
+
(ρū i ) +
∂t
∂x j
∂x j
∂x j
(2.161)
As can be seen from (2.161) the Reynolds-Averaged Navier-Stokes (RANS) equation
exhibits additional terms with respect to the original NS equations. These fluctuating
terms need to be related to the average flow parameters in order to close the RANS
equations. Since the fluctuating terms are in the same bracket as the averaged stress
terms they are often termed Reynolds stresses. Closure of the Reynolds-averaged
equations via turbulence models is the topic of Sect. 2.6.3. To interpret the equations
properly, consider Eq. (2.161) in substantial-derivative form:
64
2 Review of Fundamental Equations
ρ
D ū i
Dt
=−
Particle acceleration
of mean motion
∂ p̄
∂xi
Mean pressure
gradients
+
∂ τ̄ij lam
∂x j
Laminar-like
stress gradient
for the mean motion
+
∂ τ̄ij turb
∂x j
(2.162)
Apparent stress gradients
due to transport of momentum by
turbulent fluctuations
This shows how the Reynolds stresses appear as an addition to the mean flow parameters that are very much comparable to the original NS equations. The Reynoldsaveraged stress is related to the fluctuating velocities according to:
τ̄ij turb = −ρu i u j
(2.163)
Since u i u j = u j u i , the Reynolds stress tensor, τ̄ij is symmetric. Considering three
dimensions, it consists of six new variables. Unfortunately, we do not have six additional equations. We therefore require additional equations to close this system.
The Reynolds-averaged energy equation can be found by substitution of the relations (2.154) in (2.140):
∂
∂ ρ̄ H̄ ρ H +
∂t
∂x j
∂ T̄
ρ̄ū j H̄ + ρ̄u j H + ρ u j H̄ + ρ u j H + ū j ρ H − k
∂x j
∂ ū j
∂ p̄
2
∂
∂ ū k
∂ ū i
=
+ μū i
+
ū i − μδij
+
∂t
∂t
3
∂xk
∂xi
∂x j
!
∂u j
∂
ū
∂u
∂ 2
k
i
μδij u i
−
+ μ u i
+ u i
∂t 3
∂xk
∂xi
∂x j
(2.164)
For incompressible flows the energy equation can be written as follows:
∂ρ H̄
∂
∂ T̄
ρu j H̄ + ρu j H − k
+
∂t
∂x j
∂x j
!
∂u j
∂u
∂ ū j
∂
∂ p̄
∂ ū i
+ μ u i
+
=
+
+ u i i
μū i
∂t
∂x j
∂xi
∂x j
∂xi
∂x j
(2.165)
Since the last bracketed term within the square brackets on the RHS of (2.165) is often
small compared to the other terms within the square brackets, it is usually neglected.
What remains is an energy equation comparable to the original energy equation
(2.140) except for the fluctuating term ρu j H . Since this term shares its brackets
with heat flux terms it is often termed the Reynolds heat flux. A similar analysis can
be done for the energy equation where the Reynolds heat flux component is:
− (∇ · q)turb =
∂ −ρ̄c p T u j − c p ρ T u j − ū j c p ρ T ∂x j
(2.166)
2.6 Reynolds-Averaged Navier-Stokes Equations
65
The Reynolds-averaged equations form a set of five partial differential equations with
more than five unknowns. As was briefly explained, additional fluctuating terms need
to be related to mean flow properties via appropriate turbulence models. We discuss
a typical turbulence model in Sect. 2.6.3.
2.6.2 Compressible Reynolds-Averaged Equations of Motion
The Reynolds-averaged equations in the previous section assume a constant density
of the fluid. In Mach numbers beyond 0.3 it can be shown that compressibility effects
become important in determining the correct flow parameters [3]. This section shows
how time-averaging along with mass-weighting of the equations of motion results in
an equivalent set of equations that describe an equivalent compressible flow.
In a Reynolds averaged compressible flow, it is convenient to apply a massweighted averaging in addition to the time averaging.Mass-averaging the flow parameters can be done by multiplying the parameters by the density, averaging this
product and dividing by the average density:
ũ =
ρu
ρ̄
ṽ =
ρv
ρ̄
w̃ =
ρw
ρ̄
h̃ =
ρh
ρ̄
T̃ =
ρT
ρ̄
H̃ =
ρH
ρ̄
(2.167)
(2.168)
The following fluctuating quantities are defined:
u = ũ + u v = ṽ + v
w = w̃ + w
ρ = ρ̄ + ρ
p = p̄ + p h = h̃ + h T = T̃ + T H = H̃ + H where the primed parameters stand for time dependent deviations from the average
Δt 1
which are zero when integrated over time (i.e. u = Δt
0 u dt = 0). The averages
of the doubly primed fluctuating quantities are not zero. Instead the time average
of the doubly primed fluctuations multiplied by the density equals zero (i.e. u =
Δt 1
Δt 0 ρu dt = 0).
We substitute the averaged flow parameters (2.168) in the equations of motion
of Sect. 2.5. Substituting the averaged flow parameters into the continuity equation
(Eq. 2.110) and subsequently time averaging the equation yields:
∂
∂ ρ̄
+
ρ̄ũ j = 0
∂t
∂x j
j = 1, 2, 3
(2.169)
To develop the Reynolds momentum equation in mass-weighed variables, the
flow parameters defined in (2.168) are substituted in Eq. (2.127). After averaging
and elimination of terms that are zero, the complete Reynolds momentum equation
in mass-weighted variables becomes:
66
2 Review of Fundamental Equations
∂
∂
∂ p̄
∂ ρ̄ũ j +
τ̄ij − ρu i u j
i, j = 1, 2, 3 (2.170)
ρ̄ũ i ũ j = −
+
∂t
∂x j
∂xi
∂x j
where, neglecting viscosity fluctuations, τ̄ij is as follows:
τ̄ij = μ
∂ ũ j
∂ ũ i
+
∂xi
∂x j
!
∂u j
∂u i
2 ∂u k
2 ∂ ũ k
+
− δij
− δij
+μ
3 ∂xk
∂xi
∂x j
3 ∂xk
(2.171)
A brief look at the expression reveals a more complex expression than the original,
non-averaged expression of Eq. (2.125). In practice, however, the viscous terms with
doubly primed fluctuations are likely candidates for being neglected based on their
magnitude compared to the mass-averaged variables. By substituting the Reynoldsaveraged continuity equation (2.169) in the Reynolds-averaged momentum equation
(2.170) and employing the substantial derivative yields the following [19]:
ρ̄
D ũ i
Dt
=−
Particle acceleration
of mean motion
∂ p̄
∂xi
Mean pressure
gradients
+
∂ τ̄ij lam
+
∂x j
∂ τ̄ij turb
(2.172)
∂x j
Apparent stress gradients
due to transport of momentum by
turbulent fluctuations and
deformations attributed to fluctuations
Laminar-like
stress gradient
for the mean motion
Note that the Reynolds-averaged momentum equation (above) has the same form
as the original Navier-Stokes equation, Eq. (2.127), with the addition of a turbulent
stress term. Explicitly, the laminar and turbulent stress terms are given by:
τ̄ij lam = μ
τ̄ij turb = −ρu i u j + μ
∂ ũ j
∂ ũ i
+
∂xi
∂x j
∂u j
∂u i
+
∂xi
∂x j
2 ∂ ũ k
− δij
3 ∂xk
!
2 ∂u k
− δij
3 ∂xk
(2.173)
(2.174)
To arrive at the Reynolds form of the energy equation, the mass-weighted variables
(2.168) are substituted in (2.140). Subsequent elimination of terms that go to zero
yields the following Reynolds energy equation in mass-weighed variables:
∂ p̄
∂ ∂ T̄
ρ̄ũ j H̃ + ρu j H j − k
=
+
ũ i τ̃ij + u i τij
∂x j
∂t
∂x j
(2.175)
A similar analysis can be performed on the energy equation. Apart from the laminar and turbulent stress terms a laminar and turbulent heat flux term can be
defined. The new apparent turbulent stresses and flux terms that appear in both the
Reynolds-averaged momentum and energy equation should be treated as new variables. Therefore, additional equations are required that make assumptions regarding
the apparent turbulent quantities and the mean flow variables.
∂
∂ ρ̄ H̃ +
∂t
∂x j
2.6 Reynolds-Averaged Navier-Stokes Equations
67
2.6.3 Turbulence Modeling: The k-Epsilon Model
The fluctuation terms that are introduced in the Reynolds-averaged equations of
motion need to be related to the average flow values. Several models have been
introduced that couple those two parameters. Ranging from simple algebraic models
to more rigorous models, each of these approaches attempts to describe the equivalent
behavior of an essentially random flow. The only way to verify whether a turbulence
model is effective, is by experimental verification. This section presents one of the
many turbulence models that are used in practice, the k-epsilon model.
The Reynolds stress in incompressible flow amounts to −ρu i u j , see (2.163). For
compressible flow this is −ρu i u j when the molecular viscosity (the second term of
(2.174)) is neglected. The Boussinesq assumption relates the Reynolds stress to the
mean-flow parameters according to:
2
− ρu i u j = 2μT Sij − δij
3
∂ ū k
μT
+ ρk̄
∂xk
(2.176)
where μT is the turbulent viscosity, k̄ is the kinetic energy of turbulence: k̄ = u i u i /2,
and the rate of the mean strain tensor Sij given by:
Sij =
1
2
∂ ū j
∂ ū i
+
∂x j
∂xi
(2.177)
To predict the value of the turbulent kinetic energy, k̄, a transport equation is
developed from the Reynolds-averaged Navier-Stokes equations. Using Boussinesq’s
assumption for eddy viscosity (2.176) this transport equation reads (in substantialderivative form):
D k̄
∂
ρ
=
Dt
∂x j
∂ k̄
∂u i
μT
k̄ 3/2
2
μ+
+ 2μT Sij − ρk̄δij
(2.178)
− CDρ
Prk ∂x j
3
∂x j
l
A derivation of this equation is beyond the scope of this text but can be found in
texts on turbulence modeling (i.e. [22] or [6]). In (2.178) the Prandtl number for
turbulent kinetic energy (Prk ) appears as a closure constant (Prk = 1.0). C D is the
dissipation coefficient and has been experimentally shown to be C D = 0.164. The
term on the LHS of this equation represents the particle rate of increase of k̄. The
terms on the RHS of Eq. (2.178) represent the diffusion rate, the generation rate
and the dissipation rate of k̄, respectively. Here l is a characteristic length, referred
to as the mixing length. The mixing length can be interpreted as follows: a fluid
parcel will conserve its properties for a characteristic length, l, before mixingwith the
68
2 Review of Fundamental Equations
surrounding fluid. Prandtl [14] described that the mixing length “may be considered
as the diameter of the masses of fluid moving as a whole in each individual case; or
again, as the distance traversed by a mass of this type before it becomes blended in
with neighboring masses…”
The dissipation rate of k̄ which is embedded in the last term in (2.178) is often
3/2
represented with ε: ε = C D k̄ l . Similar to the analysis of k̄, a transport equation
developed from the Reynolds-averaged Navier-Stokes equations can be established
for ε:
∂ε
∂u i
μT
∂
ε
2
ε2
Dε
μ+
+ Cε1
=
2μT Sij − ρk̄δij
− Cε2 ρ
ρ
Dt
∂x j
Prε ∂x j
3
∂x j
k̄
k̄
(2.179)
The term on the LHS of Eq. (2.179) represents the particle rate of increase in dissipation, while the terms on the RHS represent the diffusion, generation and dissipation
rates of ε. In terms of k̄ and ε the terms in Eqs. (2.178) and (2.179) are as follows:
l = CD
k̄ 3/2
ε
μT = C μ ρ
k̄ 2
ε
4/3
Cμ = C D
(2.180)
By appropriate substitution, the k̄-ε model becomes a set of two coupled partial
differential equations with two unknowns: k̄ and ε.
Complementary to closure of the Reynolds-averaged momentum equations, the
k̄-ε closure for the heat flux terms in the energy equation assumes the following
apparent turbulent conductivity:
ρu j H = ρc p x j T = −k T
∂T
∂x j
(2.181)
Analogous to the laminar heat conduction coefficient, k, the turbulent heat conductive
heat coefficient is defined according to:
kT =
c p μT
PrT
(2.182)
where PrT is the turbulent Prandtl number that is most commonly takes on the value
of PrT = 0.9. The combination of (2.181) and (2.182) form the closure for the
energy equation. That is, if the last bracketed term in (2.175) is neglected because it
is small compared to the other terms within the square brackets. Table 2.1 displays
the constants that are to be used in the k̄-ε model.
Table 2.1 Model constants
for k̄-ε model [19]
Cμ
0.09
Cε1
1.44
Cε2
1.92
Prk
1.0
Prε
1.3
PrT
0.9
2.6 Reynolds-Averaged Navier-Stokes Equations
69
The k-epsilon method that has been presented is tailored towards closure of the
incompressible Reynolds-averaged equations of motion. Similar models have been
developed for the closure of the Reynolds-averaged, mass-weighted equations of
motions that were developed in Sect. 2.6.2. The closure of the compressible Reynolds
equations is beyond the scope of this text. Even though in transonic aerodynamics the
flow is compressible, the incompressible turbulence models have been proven to give
good predictions up to Mach 5 [17]. The k̄-ε method for compressible flow is treated
extensively by Launder and Spalding [12] and Mohammadi and Prionneau [13].
2.7 Equations of Motion for Inviscid Flows
The Reynolds-averaged Navier-Stokes equations can be further simplified if heat
transfer and viscous effects are neglected. This is often done to reduce the number
of independent variables. If cast into a numerical scheme to solve these equations,
the equations of motion for inviscid flow can be solved must faster. This can be
advantageous if the computational domain is large (i.e. the problem is solved on a
grid with a large number of nodes) and/or when the analysis has to be frequently
repeated (e.g. as part of a design optimization loop). In this section we discuss two
reduced forms of the equations of motion: the Euler equations (Sect. 2.7.1) and the
potential flow equation (Sect. 2.7.2).
2.7.1 Euler Equations
In this section a reduced model is presented of the complete Navier-Stoke equations.
By neglecting the heat-transfer terms as well as the viscous terms, these equations
describe the flow of an inviscid, non-conducting fluid. The resulting set of equations
is referred to as the Euler equations, although strictly speaking, the name of Euler
should only be attached to the inviscid momentum equation. In addition to the aforementioned assumptions, it is also assumed that there is no external heat transfer, such
that the ∂(ρQ)/∂t term in the energy equation can be dropped.
The continuity equation is neither dependent on the viscosity nor on the heattransfer coefficient. The continuity equation (2.113) remains therefore unchanged in
the inviscid case:
Dρ
+ ρ(∇ · V ) = 0
(2.113)
Dt
When the viscous terms in the momentum equation (2.127) are neglected, the following equations result:
DV
ρ
= −∇ p
(2.183)
Dt
70
2 Review of Fundamental Equations
Neglecting the viscous and heat-transfer terms in the energy equation (2.140) results
in the following:
De
ρ
+ p(∇ · V ) = 0
(2.184)
Dt
Alternatively, this equation can be written in terms of enthalpy, h, by modifying
(2.141):
Dp
Dh
=
(2.185)
ρ
Dt
Dt
Equations (2.113), (2.183), and (2.184) are generally known as the Euler equations.
The compressible Euler equations can be written in conservation form according
to (2.152). However, the vector representation is now simpler than for the full NS
equations:
⎡
⎤
ρ
⎢ ρu ⎥
⎢ ⎥
⎥
U=⎢
⎢ ρv ⎥
⎣ρw⎦
Et
⎡
⎡
⎤
ρu
⎢ ρu 2 + p ⎥
⎢
⎥
⎥
E=⎢
⎢ ρuv ⎥
⎣ ρuw ⎦
(E t + p)u
⎤
ρv
⎢ ρuv + p ⎥
⎢ 2
⎥
⎥
F=⎢
⎢ ρv + p ⎥
⎣ ρvw ⎦
(E t + p)v
⎡
⎤
ρw
⎢ ρuw ⎥
⎢
⎥
⎥
G=⎢
⎢ ρvw
⎥
⎣ ρw2 + p ⎦
(E t + p)w
(2.186)
It is of interest to take a closer look at the momentum equations (2.183). Because
of the cleaner representation of the momentum equations we can say something
about the application of the Euler equations in transonic flow. From kinematics, the
acceleration of a fluid particle, DV /Dt, can be expressed as:
DV
∂V
=
+∇
Dt
∂t
V2
2
−V ×ζ
(2.187)
where ζ = ∇ × V is the vorticity of the flow. Equation (2.187) is referred to as
Lagrange’s acceleration formula. It can be substituted in (2.183) to give an alternate
form of the Euler equations:
∂V
+∇
∂t
V2
2
1
− V × ζ = − ∇p
ρ
(2.188)
The vorticity can be related to the specific entropy, s, according to Crocco’s equation:
∂V
− V × ζ = T ∇s − ∇
∂t
V2
2
(2.189)
This equation can be derived from the first and second laws of thermodynamics,
(2.83) and (2.94), respectively. Note that the LHS of (2.189) is also present in the
Lagrange form of the momentum equation (2.188). A physical interpretation of
2.7 Equations of Motion for Inviscid Flows
71
Crocco’s theorem shows that whenever an enthalpy or entropy gradient is present
in the flow, it must be rotational. This is particularly important for transonic flow,
since a shockwave often terminates a supersonic portion. From the second law of
thermodynamics it is known that there is an entropy discontinuity across the shockwave, which means the flow cannot be assumed irrotational [2]. This is an important
finding because it shows that the Euler equations are the simplest formulation of the
equations of motion which still capture the effect of the shock wave on the flow.
In the 1980s the airfoil analysis code ISES was developed by MIT (currently
further developed into the multi-element airfoil code MSES). This program solves
the two-dimensional Euler equations interacting with the boundary layer equations.
As is shown in Fig. 2.11 and from [8], the prediction of the pressure distribution
2 ] and the two-dimensional drag polar is quite accurate.
[C p = ( p − p∞ )/ 21 ρ∞ V∞
The sharp pressure gradient on the upper surface at approximately 60 % of the chord
length, indicates the presence of a shock wave. If we focus on the airfoil below the
pressure distribution, we see that the streamline that separates the inviscid flow from
the viscous boundary layer has been drawn. Up to the shock wave this line virtually
coincides with the airfoil contour, indicating a very thin boundary layer. However,
due to the sharp adverse pressure gradient, the boundary layer thickness grows and
the streamline is displaced upwards. The flow outside the boundary layer therefore
‘sees’ a different body than the airfoil. This added thickness (known as displacement
thickness) is important to accurately determine the experimentally obtained pressure
distribution. Note that in this figure the pressure distributions are shown for the same
lift coefficient (cl = 0.743), while the angles of attack are slightly different. In
predicting the pressure distribution over an airfoil it has been shown that this often
gives a better correlation to experimental results. Producing these results can be done
in under a minute of computation time, making this a suitable tool for airfoil design.
To stress the importance of including a boundary layer in the calculations we also
show results for flow about a transonic airfoil without the inclusion of the boundary
layer. In Fig. 2.12 we see that the pressure distribution has been predicted using
-2.0
Pressure coefficient, Cp (~)
Fig. 2.11 Comparison
between predicted and
measured pressure
distribution about the
RAE2822 airfoil (from
[7, 8], respectively)
-1.5
-1.0
-0.5
-0.0
0.5
1.0
RAE 2822
M =0.750
Re =6.200·10 6
α =2.734
CL =0.7431
CD =0.02284
CM =-0.0941
L/D =32.54
Experiment
3.19
0.743
0.0242
-0.106
72
2 Review of Fundamental Equations
-2.0
α=3.0°
Cl=0.60
-1.2
Cl=1.27
M=1.0
-0.8
-0.4
0.0
M∞=0.68
Expected
Inviscid
-1.2
M=1.0
-0.8
-0.4
0.0
0.4
0.4
0
Cl=0.60
-1.6
-1.6
Pressure coefficient, Cp (~)
Pressure coefficient, Cp (~)
-2.0
.2
.4
.6
.8
1.0
Chordwise position, x/c (~)
0
.2
.4
.6
.8
1.0
Chordwise position, x/c (~)
Fig. 2.12 Comparison between predicted and measured pressure distribution at constant angle of
attack (left) and constant lift coefficient (right) (after Ref. [5])
the same angle of attack as in the experiment (left) and at the same lift coefficient
as in the experiment (right). None of these predictions gives an accurate result.
This shows that we must be extremely careful when assessing the results from a
purely inviscid solver. Given the short computation time of viscous-inviscid solvers
such as MSES, it is therefore advised to include the presence of the boundary layer
when predicting the pressure distribution over a body. In particular at low-Reynolds
numbers and transonic Mach numbers this is an important prerequisite for obtaining
reliable results.
2.7.2 Potential Flow Equation
Even though the previous analysis showed that the vorticity in transonic flow cannot
be neglected, there is a body of literature that is based on steady, inviscid, irrotational
flow with no body forces [10]. It can be shown that for weak normal shocks (M1 ≈ 1)
the entropy change is relatively small. Based on this finding the approximation can be
made that transonic flow is irrotational and that that therefore the curl of the velocity
field (see Sect. 2.3.4) is zero:
∇×V =0
(2.190)
In potential flow we pose that there exists a scalar potential function, Φ, which’
partial derivatives equal the velocity components:
∇Φ = V
(2.191)
2.7 Equations of Motion for Inviscid Flows
73
In Cartesian coordinates, we align the x-axis with the freestream velocity, V ∞ , and
represent the local velocity components, u, v and w by
∂Φ
= Φx
∂x
∂Φ
= Φy
v=
∂y
∂Φ
w=
= Φz
∂z
u=
(2.192a)
(2.192b)
(2.192c)
If we assume the flow to be steady, the continuity equation (2.113) reduces to:
∇ · (ρV ) = 0
(2.193)
In Cartesian coordinates this results in the following expression:
∂
∂
∂
(ρΦx ) +
(ρΦ y ) +
(ρΦz ) = 0
∂x
∂y
∂z
(2.194)
Note in this equation that the partials ρx , ρ y , and ρz appear. We will find an expression
for each of these density derivatives by considering the potential form of the steady,
inviscid, irrotational momentum equation.
If we assume that the flow is steady (i.e. ∂/∂t = 0) the Lagrange form of the
momentum equation (2.188) can be reduced to:
∇
V2
2
1
= − ∇p
ρ
(2.195)
In differential form we can rewrite (2.195) as follows:
V2
d p = −ρd
2
= −ρd
Φx2 + Φ y2 + Φz2
2
!
(2.196)
In isentropic flow the speed of sound (a) is given by (2.106). We can substitute the
expression for the speed of sound (2.106) in (2.196) and obtain:
Φx2 + Φ y2 + Φz2
ρ
dρ = − 2 d
a
2
!
(2.197)
If we return to ∇ρ and consider each of the partials ρx , ρ y , and ρz we obtain:
ρ ∂
ρx = − 2
a ∂x
Φx2 + Φ y2 + Φz2
2
!
(2.198a)
74
2 Review of Fundamental Equations
ρ ∂
ρy = − 2
a ∂y
Φx2 + Φ y2 + Φz2
ρ ∂
ρz = − 2
a ∂z
Φx2 + Φ y2 + Φz2
2
!
(2.198b)
!
(2.198c)
2
If we substitute (2.198a–2.198c) in (2.194) we obtain the full potential equation for
steady, inviscid, irrotational and isentropic flow:
Φ y2
Φz2
Φx2
1 − 2 Φx x + 1 − 2 Φ yy + 1 − 2 Φzz
a
a
a
−2
Φx Φ y Φx y
Φ y Φz Φ yz
Φ x Φz Φ x z
−2
−2
=0
a2
a2
a2
(2.199)
1.5
1.5
1.0
1.0
Cp*
0.5
0.0
-0.5
2y/b = 0.206000
-1.0
0.2
0.4
0.6
Cp*
0.5
0.0
-0.5
2y/b = 0.521000
-1.0
Matrics-V
Flight test
-1.5
0.0
Pressure coefficient, -Cp (~)
Pressure coefficient, -Cp (~)
In the late 1980s and early 1990s the analysis code Matrics-V was developed by
the National Aerospace Laboratory of the Netherlands to estimate the aerodynamic
forces on wing-fuselage combinations (three-dimensional) in high-subsonic conditions. This code relies on a viscous-inviscid formulation, where the full potential
equation (2.199) is numerically solved in the inviscid domain outside the boundary
layer. In the boundary layer the boundary layer equations are numerically solved.
Subsequently, an interaction algorithm is used to match the flow properties at the
0.8
1.0
Chordwise position, x/c (~)
-1.5
0.0
Matrics-V
Flight test
0.2
0.4
0.6
0.8
1.0
Chordwise position, x/c (~)
Fig. 2.13 Comparison of predicted and measured pressure distributions about two sections of the
Fokker 100 wing (from [20]). Note that this is a two-dimensional representation of results that were
obtained from three-dimensional calculations (Matrics-V)
2.7 Equations of Motion for Inviscid Flows
75
edge of the boundary layer to the inviscid flow outside the boundary layer [20]. This
analysis code was specifically developed to investigate the flow about wing-body
combinations. The objective was to develop a code that could produce results in less
than week of computation time. Due to the increase in computational power, this has
now (2015) reduced to less than five minutes on a personal computer, including the
generation of the mesh. In Fig. 2.13 a comparison between the predicted and measured pressure distribution at two spanwise wing stations of the Fokker 100 can be
seen. We see that the prediction of the pressure distribution is in excellent agreement
with the measured data at the same angle of attack. The accurate prediction in combination with the short computation time make this type of code a viable candidate
in the early stages of preliminary wing design.
2.8 Summary
In the preceding sections we have briefly reviewed the fundamental tools that the
reader needs to be familiar with in order to properly understand the material that
is presented in the subsequent chapters. We have done a very basic review of partial differential equations, their solution methods, and their classification. We have
also re-acquainted ourselves with the mathematical operations in vector algebra. We
have explained the fundamental laws of thermodynamics: the state law that relates
the thermodynamic state variables to each other, the first law that balances work,
heat and internal energy, and the second law that tells us the direction in which a
thermodynamic process takes place. Finally, we have applied all this knowledge in
defining the equations of motion of fluid flow. When expanded we have shown that
the equations of motion are a coupled system of five partial differential equations
[(2.110), (2.127) (3 components), and (2.139)] with five unknowns: ρ, e, and the three
velocity components, u, v, w. These equations are often referred to as the NavierStokes equations and they can be cast in different forms: integral form, substantialderivative form, and conservative-derivative form. Because solving these equations
requires vast computational resources we have presented derivatives of these equations. In the Reynolds-averaged Navier-Stokes equations the flow is decomposed into
a mean flow with fluctuating terms. A turbulence model can then be used to relate
those fluctuating terms to the mean flow parameters. A further simplification of the
Navier-Stokes equations is obtained through the assumption of having an inviscid
and non-conducting flow. The resulting equations are often referred to as the Euler
equations and they can provide an accurate prediction of the flow about a body, in
particular when they are used in parallel with the boundary-layer equations. If also
the assumption is made that the flow is irrotational, the Euler equations reduce to
the full-potential equation. Solving the three-dimensional full-potential equation in
combination with the boundary-layer equations can still produce accurate predictions
of transonic flow about bodies, as long as the shock strength is relatively weak.
76
2 Review of Fundamental Equations
Problems
Review of Partial Differential Equations
2.1 Show that (2.7) results from (2.5) by using the chain rule and substituting ξ =
x + ct and η = x − ct.
π(x−ct)
+
sin
is a solution to the
2.2 Demonstrate that u(x, t) = sin π(x+ct)
L
L
following problem:
u tt = c2 u x x
πx u(x, 0) = 2 sin
L
u(0, t) = u(L , t) = 0
2.3 Consider a string of steel wire, measuring 1 m in length and weighing 0.5 N. It
is stretched by a force of 100 N. What is the corresponding speed c of the transverse
waves?
2.4 Consider the wave equation (2.5) with the following boundary conditions:
u(0, t) = u x (L , t) = 0 .
(a) Use separation of variables technique to calculate the eigenvalues, eigenfunctions
and general solution.
(b) Now, assume L = π and c = 1. With initial conditions u(x, 0) = 0 and
u t (x, 0) = 1, calculate the solution for u(x, t).
(c) With initial conditions u(x, 0) = sin(x/2) and u t (x, 0) = 2 sin(x/2) −
3 sin(5x/2) calculate the solution for u(x, t).
2.5 Demonstrate that (2.22) can be derived from u t = 0 by using the chain rule and
the transformation (2.6).
2.6 Show that the Jacobian for the change in characteristic coordinates (2.6)
equals 2c.
2.7 For the same problem as in Example 2.2, calculate the solution for:
(a) x < t, t > 0, x + t < 0, and check u tt − u x x = 1.
(b) x > 1, t < x < t + 1, and check u tt − u x x = 0.
2.8 Show that (2.43) is indeed a solution of (2.41).
2.9 Demonstrate that from (2.49) we can obtain (2.50) by using the method outlined
in the text.
2.10 Using the separation-of-variables technique find a solution to the following
problem:
u t = ku x x
(0 x L)
2.8 Summary
77
u(0, t) = 0
u x (L , t) = 0
u(x, 0) = f (x)
(t > 0)
(t > 0)
(0 < x < L)
2.11 For Problem 2.10, provide calculate Dn in closed form for L = π and
f (x) = x.
2.12 Consider the homogenous heat equation, c2 u x x − u t = 0. Determine whether
this equation is hyperbolic, parabolic, or elliptic.
2.13 Consider the potential equation on p. 74 (2.199). When M > 1, determine the
characteristic coordinates and write the equation in its canonical form.
Review of Vector Algebra
2.14 Let a = (−2, 5, − 9) and b = (4, 2, 7). Calculate the following:
(a) a + b
(b) a · b
(c) a × b
(d) b × a
(e) a ⊗ b
2.15 For the following velocity vector fields, find the potential function if it exists.
Check that (2.66) holds in each case. φ(x, y, z): (a) (x, 2y, 3z), (b) (yz, x z, x y),
(c) (ye x , e x , 1), (d) (2y, 5x, 0).
2.16 For the vector fields of Problem 2.15, calculate the divergence.
2.17 For the vector fields of Problem 2.15, calculate the curl. Compare your results
to those of Problem 2.15. What do you notice?
Review of Thermodynamics
2.18 With c p as given in Fig. 2.8 and R as given in the text, calculate cv , γ, e, and
h for air at an altitude of 10 km under ISA conditions.
2.19 For the values given in Problem 2.18 also calculate the viscosity of the air, μ
and the thermal conductivity, k.
2.20 Calculate the change in specific internal energy, Δe, for air that is being compressed isentropically. Assume that the initial temperature is 288 K and the final
temperature, due to compression, 340 K.
2.21 Show that (2.97) and (2.98) can be derived by combining the first and second
law of thermodynamics, (2.83) and (2.94), respectively.
78
2 Review of Fundamental Equations
2.22 Derive the following relations by starting from (2.97) and (2.98), respectively.
dv
dT
+R
T
v
dT
dp
ds = c p
−R
T
p
ds = cv
(2.200)
(2.201)
2.23 Demonstrate that (2.103) can be derived by combining (2.101) and (2.102) and
applying the substitutions as described in the text.
Equations of Fluid Motion
2.24 Use the divergence theorem (2.69) to prove that (2.109) and (2.110) are mathematically identical.
2.25 Consider the LHS of (2.122). Show that by using (2.110) this can be simplified
to ρ DV
Dt .
2.26 Consider the energy equation (2.139).
(a) By employing the continuity equation (2.110) show that the following is true:
ρ
∂ Et
D(E t /ρ)
=
+ ∇ · Et V
Dt
∂t
(b) Demonstrate that also the following relation is true:
ρ
De
DV
D(E t /ρ)
=ρ
+ρ
V
Dt
Dt
Dt
(c) Use (2.122) to show that the second term in the equation above can be written as:
ρ
DV
V = ρ f · V − ∇ p · V + (∇ · τ ij ) · V
Dt
(d) Using the three equations above, demonstrate that (2.140) is identical to (2.139).
2.27 Consider the energy equation in substantial-derivative form (2.140).
(a) Show, by employing the continuity equation, that the following identity is true:
ρ
Dh
Dp
De
+ p(∇ · V ) = ρ
−
Dt
Dt
Dt
(b) Explicitly write out the dissipation function, Φ, in terms of the velocity components in three-dimensional Cartesian coordinates. Use the assumption that
μ = 23 μ.
2.28 Write out (2.110), (2.149), and (2.150) in three-dimensional Cartesian coordinates and demonstrate that you can write this as (2.152).
2.8 Summary
79
2.29 Consider the system of Eq. (2.152).
(a) Demonstrate that pressure, p, can be expressed as a function of the the state
variables ρ and e and the ratio of specific heats, γ by using (2.72) and (2.76).
(b) Show that the heat flow, q, can expressed in the state variable e. Assume that k
is constant.
(c) Repeat the previous exercise, but now assume that k is variable and related to
temperature according to (2.81) and (2.82). Is this a conservative differential
equation?
References
1. Anderson, J.: Computational Fluid Dynamics: The Basics and Applications, 1st edn. McGraw
Hill, New York (1995)
2. Anderson, J.: Modern Compressible Flow with Historic Perspective, 3rd edn. McGraw Hill,
New York (2003)
3. Anderson, J.: Fundamentals of Aerodynamics, 5th edn. McGraw Hill, New York (2010)
4. Anon.: Direct numerical simulation (DNS) wikipedia. http://en.wikipedia.org/wiki/Direct_
numerical_simulation (2012)
5. Blackwell, J.A.: Scale effects on supercritical airfoils. In: Proceedings of ICAS, pp. 370–283
(1978)
6. Cebeci, T.: Analysis of Turbulent Flows. Elsevier, Amsterdam (2004)
7. Cook, P.H., McDonald, M.A., Firmin, M.C.P.: Aerofoil RAE 2822—pressure distributions,
and boundary layer and wake measurements. In: AGARD-138 (1979)
8. Drela, M., Giles, M.B.: Viscous-inviscid analysis of transonic and low Reynolds number airfoils. AIAA J. 25(10), 1347–1355 (1987). doi:10.2514/3.9789
9. Farokhi, S.: Aircraft Propulsion, 2nd edn. Wiley, Chichester, UK (2014)
10. Ferrari, C., Tricomi, F.G.: Transonic Aerodynamics. Academic Press, New York (1968)
11. Keener, J.P.: Principles of Applied Mathematics. Westview Press, Boulder (1999)
12. Launder, B.E., Spalding, D.B.: Lectures in Mathematical Models of Turbulence. Academic
Press, London (1972)
13. Mohammadi, B., Pironneau, O.: Analysis of the K-epsilon Turbulence Model. Wiley, Chichester, UK (1994)
14. Prandtl, L.: Über die ausgebildete turbulenz. In: Proceedings of the Second International
Congress of Applied Mechanics, pp. 62–75. Orell Füssli Verlag, Zurich (1927)
15. Reynolds, O.: On the dynamical theory of incompressible viscous fluids and the determination
of the criterion. Philos. Trans. R. Soc. Lond. 186, 123–164 (1895)
16. Ruijgrok, G.J.J.: Elements of Airplane Acoustics. Eburon, Delft (2003)
17. Schlichting, H., Gestern, K.: Boundary Layer Theory, 8th edn. Springer, Berlin (1999)
18. Strauss, W.A.: Partial Differential Equations: An Introduction. Wiley, Chichester (1992)
19. Tannehill, J.C., Anderson, D.A., Pletcher, R.H.: Computational Fluid Mechanics and Heat
Transfer, 2nd edn. Taylor & Francis, Philadelphia (1997)
20. Van Muijden, J., Broekhuizen, A.J., Van der Wees, A.J., Van der Vooren, J.: Flow analysis
and drag prediction for transonic transport wing/body configurations using a viscous-inviscid
interaction type method. In: Proceedings of the 19th ICAS Congress. Anaheim, California
(1994)
21. Van Wylen, G., Sonntag, R.: Fundamentals of Classical Thermodynamics. Wiley, Chichester,
UK (1973)
22. Wilcox, D.: Turbulence Modeling for CFD, 2nd edn. DCW Industries, Canada (1998)
Chapter 3
Transonic Similarity Laws
Abstract This chapter starts the foundation with the small disturbance theory
applied in the subsonic and supersonic flows. Both of these regimes are linear and
thus fail at sonic speed, which is at the heart of transonic flow. Nonlinear small
disturbance theory for the transonic regime is subsequently derived and discussed.
With limited analytical options for the transonic nonlinear differential equation, special attention is directed to the development of similarity laws. These laws connect
non-dimensional parameters such as pressure coefficient, lift and drag coefficients on
families of affinely related bodies of different thickness or slenderness ratio and angle
of attack. Brief discussion of hodograph transformation where switching between
the dependent and independent variables causes the governing equation to be linear is presented. Finally, semi-empirical models on lift curve slope in the transonic
regime and the approximate location of detached shocks in front of blunt bodies are
developed and compared to experimental results. This chapter contains 12 examples
and concludes with 39 practice problems.
3.1 Introduction
On a phenomenological basis, transonic flow is often characterized by the appearance of shocks, boundary layer separations, loss of lift and rise in drag. On the
mathematical basis, transonic flow is characterized by a dominant nonlinearity in its
governing equations. Even in the case of small disturbances that are caused by planar and slender bodies at small angle of attack, the transonic perturbation potential
equation is nonlinear. This, i.e., the nonlinearity in the governing differential equations, limits the options of solution techniques. The question then arises whether it is
possible to transform the transonic differential equation such that it holds the same
non-dimensional solution on bodies that are “similar” in the range of transonic Mach
numbers? The transformed bodies are then called “affinely” related to each other. We
shall further discuss affinely-related bodies in this chapter. The same solution of the
transformed differential equation can only be guaranteed when the coefficients are
held constant, from one transonic flow to another transonic flow. It is therefore the
coefficients of transformed differential equations that reveal the nature of transonic
© Springer Science+Business Media Dordrecht 2015
R. Vos and S. Farokhi, Introduction to Transonic Aerodynamics,
Fluid Mechanics and Its Applications 110, DOI 10.1007/978-94-017-9747-4_3
81
82
3 Transonic Similarity Laws
similarity parameters, as we shall see. The utility of such concepts is in the power of
generalization inherent in the similar solutions in transonic regime. This allows us to
extrapolate the aerodynamic behavior of affinely related bodies at different transonic
Mach numbers and different thickness, or slenderness ratios and angles of attack. It is
interesting to note that another nonlinear aerodynamic regime in the context of small
disturbance theory, namely hypersonic flow, also lends itself to the development of
similarity parameters on affinely-related bodies. In this chapter we examine similarity solutions from subsonic to supersonic flows including the transonic regime. The
monumental work of Shapiro [9] is a key reference to this classical topic.
3.2 Linearized Compressibility Corrections
To study the aerodynamic characteristics of thin planar and slender bodies, in small
angles of attack (and/or sideslip), the theory of small disturbances is developed. We
propose that for a thin and slender body, with small angle of attack, or sideslip, the
x-component of velocity is but a small perturbation around the freestream speed,
V∞ , and the other two velocity components, Vy and Vz are both small as compared
to freestream velocity. This proposition is the hallmark of small perturbation theory.
It suggests that a thin or slender body can only perturb the flow a small amount, as
compared to freestream. Expressed mathematically, we imply
u = V∞ + û û/V∞ = O(ε) where ε 1
v = v̂ v̂/V∞ = O(ε)
w = ŵ ŵ/V∞ = O(ε)
(3.1)
where the û, v̂ and ŵ are perturbation velocity components in x, y and z directions
respectively and they are small compared to V∞ . Obviously, this approximation
excludes the neighborhood of stagnation point where the change in velocity is large.
Therefore, excluding the neighborhood of stagnation point, we can now introduce a
perturbation potential function that describes the perturbation velocity components
in the flow according to:
û = φx
v̂ = φ y
(3.2)
ŵ = φz
Therefore, the full velocity potential, Φ and the perturbation velocity potential, φ are
related by the following set of equations:
Φx = V∞ + φx where φx /V∞ = O(ε)
Φ y = φ y where φ y /V∞ = O(ε)
Φz = φz where φz /V∞ = O(ε)
(3.3)
3.2 Linearized Compressibility Corrections
83
Now, for steady, inviscid, irrotational and isentropic flow, we have the full potential
equation, namely
Φ y2
Φz2
Φx2
1 − 2 Φx x + 1 − 2 Φ yy + 1 − 2 Φzz
a
a
a
−2
Φx Φ y Φx y
Φ y Φz Φ yz
Φ x Φz Φ x z
−2
−2
=0
2
2
a
a
a2
(2.199)
This is a second order, nonlinear partial differential equation that is best solved numerically. Also, note that the local speed of sound, a, is related to velocity components
following the energy equation in adiabatic flows according to:
a2
V2
a2
V2
+
= ∞ + ∞
γ−1
2
γ−1
2
(3.4)
Multiply both sides by (γ − 1) and isolate a 2 to get:
2
−
a 2 = a∞
γ−1 2
2
V − V∞
2
(3.5)
Now, substitute for V 2 :
2
2
V 2 = V∞ + û + v̂2 + ŵ2 = V∞
+ 2V∞ û + û 2 + v̂2 + ŵ2
(3.6)
We then obtain:
2
−
a 2 = a∞
2
a 2 = a∞
−
γ−1
2V∞ û + û 2 + v̂2 + ŵ2
2
(3.7)
û
γ−1 2
û 2
v̂2
ŵ2
(γ − 1) 2
2
2
V − V∞
V∞ 2
= a∞
−
+ 2 + 2 + 2
2
2
V∞
V∞
V∞
V∞
(3.8)
We note that the last two terms in the parenthesis on the right-hand-side of (3.8)
are second-order terms, i.e., O(ε2 ), and may be neglected; therefore the local speed
of sound may be approximated, to first order, by
a2
2
≈ 1 − (γ − 1)M∞
2
a∞
φx
V∞
+ O(ε2 )
(3.9)
84
3 Transonic Similarity Laws
The full potential equation (2.199), in the limit of small perturbations, where we
neglect the squares and products of small parameters, i.e., the second order terms,
as O(ε2 ), and for freestream Mach numbers that exclude transonic and hypersonic
flows, may be written as
2
(1 − M∞
)φx x + φ yy + φzz = 0
(3.10)
This is the classical linear differential equation that governs the dynamics of gas
in steady subsonic and supersonic flows around bodies that are thin (planar), or
slender (3-D) and are at small angles with respect to the flow. The impact of small
perturbations in velocity is also felt at the pressure level. We introduce the pressure
coefficient, C p , which is a non-dimensional coefficient for the pressure differential
at a particular location in the flow:
p − p∞
2
=
Cp ≡
2
q∞
γ M∞
T
p
2
−1 =
2
p∞
γ M∞
T∞
γ
γ−1
−1
(3.11)
And since from the energy equation, we have
T+
V2
V2
= T∞ + ∞
2c p
2c p
(3.12)
The temperature ratio is related to perturbation velocity component according to:
φx
V∞
T
2
= 1 − (γ − 1)M∞
T∞
+ O(ε2 )
(3.13)
γ
Applying binomial expansion to (T /T∞ ) γ−1 where the temperature ratio is represented by (3.13) and keeping only the linear term, i.e., O(ε) term, we get
T
T∞
γ
γ−1
2
≈ 1 − (γ − 1)M∞
φx
V∞
γ
γ−1
2
≈ 1 − γ M∞
φx
+ O(ε2 ) (3.14)
V∞
Therefore an approximate expression for the pressure coefficient, which is valid in
the linear theory, is now produced
C p ≈ −2
φx
V∞
(3.15)
The solid surface boundary condition demands that the local flow slope and the local
body slope to be matched at all points along the body surface. This is also known
as the flow tangency condition to the solid surface. An alternative description of
solid surface boundary condition calls for the normal component of the flow to the
solid surface to vanish at all points on the solid surface. For a 2-D body in the x–y
3.2 Linearized Compressibility Corrections
Fig. 3.1 Two-dimensional
body in x–y plane
y
85
V
v̂
V∞+ uˆ
dy/dx = df/dx = tan θ
θ
V∞
y = f(x)
x
plane, the body may be mathematically represented by y = f (x) with a local body
slope of dy/dx or d f /dx. This is schematically shown in Fig. 3.1. A representative
flow velocity vector is also shown in Fig. 3.1. The corresponding flow slope may be
expressed in terms of a general angle, θ, which is also valid on the body, i.e.,
tan θ =
φy
v̂
v̂
≈
=
V∞ + û
V∞
V∞
(3.16)
Therefore the flow tangency condition on the body surface demands:
φy
df
≈
dx
V∞
(3.17)
3.2.1 2-D Subsonic Flow
In the case of subsonic flow, i.e., M∞ < 0.8 that also excludes transonic regime,
Eq. (3.10) is an elliptic partial differential equation:
βφx x + φ yy = 0 where β ≡
2
1 − M∞
(3.18)
A transformation of the function φ(x, y), is proposed that converts Eq. (3.18) in
the x–y plane (known as the physical plane) to the Laplace’s equation in φ̄(ξ, η),
which governs the incompressible flow, in the transformed ξ–η plane. Consider the
following transformation:
ξ=x
η = βy
(3.19)
φ̄ = βφ
Then, by using the chain rule, we calculate the terms of Eq. (3.18) in the new variables,
for example
86
3 Transonic Similarity Laws
∂φ
∂φ ∂ φ̄ ∂ξ
1
=
= φ̄ξ
∂x
∂ξ
∂x
β
∂ φ̄
∂
∂2φ
∂ 1
=
φx x =
=
φ̄
ξ
2
∂x
∂x β
∂ξ
φx =
(3.20)
1
φ̄ξ
β
∂ξ
1
= φ̄ξξ
∂x
β
∂φ ∂ φ̄ ∂η
1
∂φ
=
= φ̄η β = φη
∂y
β
∂ φ̄ ∂η ∂ y
∂η
∂ ∂2φ
∂ = φ̄ηη β
φ̄η =
φ̄η
φ yy =
=
2
∂y
∂y
∂η
∂y
φy =
(3.21)
(3.22)
(3.23)
By substituting Eqs. (3.21) and (3.23) in (3.18), we get the transformed governing
equation in (ξ, η) plane, i.e.,
β2
1
φ̄ξξ + β φ̄ηη = 0
β
→
φ̄ξξ + φ̄ηη = 0
(3.24)
We note that the transformed potential equation, as shown in (3.24), is Laplace’s
equation, which satisfies the irrotational flow of an incompressible fluid (in ξ–η
plane). The transformed body is η = g(ξ) with the flow tangency condition requiring
φ̄η
dg
≈
dξ
V∞
(3.25)
But since φ̄η = φ y following (3.22), we note that the transformed and the physical
bodies have the same slope,
df
dg
=
(3.26)
dξ
dy
Thus we conclude that the proposed transformation as described by (3.19), which
is known as the Prandtl-Glauert transformation, does not change the shape of the
body. This is an important result, as it is most useful. The pressure coefficient in the
transformed plane is
φ̄ξ
C p M =0 ≈ −2
(3.27)
∞
V∞
Note that we have added a subscript “M∞ = 0” to C p in order to signify the Machzero or incompressible condition that prevails in the transformed plane. Replacing
φξ by βφx from (3.20) in (3.27), we relate the compressible and incompressible
pressure coefficients, namely
Cp M
∞ =0
≈ −2
φ̄ξ
βφx
φx
= −2
= β −2
V∞
V∞
V∞
= β Cp M
∞
(3.28)
3.2 Linearized Compressibility Corrections
Or alternatively,
87
C p M =0
C p M =0
∞
∞
=
Cp M =
∞
2
β
1 − M∞
(3.29)
Equation (3.29) is the celebrated Prandtl-Glauert compressibility correction, which
states that subsonic compressible C p is amplified by (1/β) factor, as compared to
the corresponding points on the same body in the incompressible (M∞ = 0) flow.
Since the (inviscid) two-dimensional lift coefficient (on an airfoil of chord length c)
is related to the integral of pressure coefficient by
cl =
1
C p dx
c
(3.30)
We conclude that the lift coefficient in subsonic compressible flow also is magnified
by the same factor, (1/β) from the incompressible lift coefficient that appeared in
C p , namely
(cl ) M∞ =0
(cl ) M∞ =0
(3.31)
=
(cl ) M∞ =
2
β
1 − M∞
Similarly, the lift curve slope, in the subsonic compressible plane is related to the
incompressible lift slope, according to
dcl
dα
=
dcl
dα
M∞ =0
β
M∞
=
dcl
dα
M∞ =0
2
1 − M∞
(3.32)
This equation also indicates that the lift curve slope is magnified in the subsonic
compressible flow as compared to the incompressible, by the factor 1/β. To complete
the discussion of two-dimensional lift in the compressible domain, we address the
issue of circulation, Γ, around an airfoil. Since the local strength of the bound vortex
sheet, representing an airfoil, is equal to the tangential velocity jump across the sheet,
which is amplified by 1/β factor in the compressible plane, the vortex sheet strength
is increased by the same factor. Therefore the circulation, which is the integral of the
vortex sheet strength along the mean camber line, is amplified by the same factor.
Another view that leads to the same conclusion is based on static pressure jump and
its proportionality to vortex sheet strength. Therefore, we may conclude that
(Γ ) M∞ =
(Γ ) M∞ =0
(Γ ) M∞ =0
=
2
β
1 − M∞
(3.33)
Finally, we note that the Kutta-Joukowski theorem relating lift to circulation will
remain valid in the compressible subsonic flow as well, i.e.,
L M = ρ∞ V∞ (Γ ) M∞
∞
(3.34)
88
3 Transonic Similarity Laws
Physical Subsonic Compressible Plane
Transformed Incompressible Plane
φξξ + φηη = 0
β φxx + φ yy = 0
2
y
C p ≈ −2
φ y ≈ V∞
η
φx
φ
V∞
C p,0 ≈ −2 ξ
V∞
dys
dx
φη ≈ V∞
V∞
dη s
dξ
V∞
ηs
ys
x
ξ
Fig. 3.2 An airfoil in the physical and the transformed planes following Prandtl-Glauert transformation
Figure 3.2 shows the same body in two planes, one subsonic compressible and the
other incompressible, following the Prandtl-Glauert transformation that preserves
the shape of the transformed body. Finally, the pitching moment coefficient on an
airfoil, say about the leading edge, is related to the integral of the moment of the
pressure coefficient according to:
1 cm, LE = − 2 xC p dx
c
(3.35)
We conclude that the subsonic compressible pitching moment, is also amplified by
the same factor, 1/β namely
cm, LE M =0
cm, LE M =0
∞
∞
=
cm, LE M =
∞
2
β
1 − M∞
(3.36)
The hallmark of all of these compressibility correction results that are derived from
the linear theory, is the singularity of the solutions at M∞ = 1. The failure to predict
a finite pressure, lift, drag, pitching moment coefficients in the vicinity of sonic flow
is due to the linearized nature of differential equation, which does not represent the
physics of transonic flow, i.e., dominated by the nonlinearity. We will examine the
suitable transonic small perturbation equation after we review a few examples from
the linear theory.
Example 3.1 The minimum (or peak suction) pressure coefficient on an airfoil at 2◦
angle of attack, in incompressible flow, is C p,min = −0.95. Calculate the pressure
coefficient at the same point on the airfoil, when freestream Mach number is 0.5.
Solution:
Applying the Prandtl-Glauert compressibility correction to this airfoil, we get:
3.2 Linearized Compressibility Corrections
89
(C p, min ) M∞ =0
C p, min M =
=
∞
2
1 − M∞
−0.95
1 − (0.5)2
≈ −1.097
Example 3.2 The lift curve slope, Clα , of an airfoil in incompressible flow is 6 rad−1 .
Calculate the lift curve slope for the same airfoil at Mach 0.6.
Solution:
The lift curve slope is also amplified by the 1/β factor according to Prandtl-Glauert
compressibility correction formula, namely (3.29), therefore
dcl
dα
=
M∞
dcl
dα
M∞ =0
=
2
1 − M∞
6 [rad−1 ]
1 − (0.6)2
≈ 7.5 [rad−1 ]
Example 3.3 The lift coefficient of an airfoil at 10◦ angle-of-attack is cl = 1.05 in
low speed. Calculate the lift coefficient of the same airfoil, at the same angle-ofattack and at Mach 0.5. Also, calculate the percent amplification of the circulation
that occurs between compressible (M∞ = 0.5) and incompressible (M∞ = 0) limits.
Solution:
According to Prandtl-Glauert compressibility correction, the lift coefficient in subsonic compressible flow is amplified by 1/β factor, therefore,
(cl ) M∞ =
(cl ) M∞ =0
2
1 − M∞
1.05
=
1 − (0.5)2
≈ 1.212
The circulation around an airfoil is also increased by the same 1/β factor, as KuttaJoukowski theorem on airfoil lift is maintained, according to (3.33) and (3.34):
(Γ ) M∞
=
(Γ ) M∞ =0
1
2
1 − M∞
=
1
2
1 − M∞
≈ 1.154
Therefore, the circulation in the compressible domain (at M∞ = 0.5) is ∼15.4 %
higher than the circulation in the incompressible limit.
3.2.2 Other Subsonic Compressibility Corrections
There are several other compressibility corrections in the subsonic flow. One such
correction is due to Laitone and another one is due to Karman-Tsien. The first, i.e.,
Laitone [4], extends the Prandtl-Glauert correction by replacing the freestream Mach
number with the local Mach number, i.e.,
90
3 Transonic Similarity Laws
C p M =0
∞
Cp M =
∞
1 − M L2
(3.37)
The local Mach number on the airfoil in the compressible domain is noted by M L in
(3.37). The effect of this improvement is to create higher suction on airfoils where
Prandtl-Glauert consistently under-predicted the experimental data. Laitone related
M L to the freestream Mach number via an isentropic relation and expressed the
revised Prandtl-Glauert compressibility correction for subsonic flow as:
Cp M =
∞
C p M =0
∞
(3.38)
γ−1 2
2
2
2
1 − M∞ + M∞ 1 + 2 M∞ /2 1 − M∞ C p M =0
∞
The Karman-Tsien compressibility correction is based on the nonlinear hodograph
solution that uses a linear approximation to pressure-density relation, instead of the
isentropic p/ργ relation. This new gas model is known as “tangent gas”, and allows
for the compressible and incompressible pressure coefficients to be related as follows:
Cp M =
∞
C p M =0
∞
2 +
1 − M∞
√
1+
(3.39)
(C p ) M∞ =0
2
M∞
2
2
1−M∞
In the limit of small pressure coefficient, (C p ) M∞ =0 , both of these models
approach the Prandtl-Glauert compressibility correction. The additions of extra terms
in the denominator in Laitone and Karman-Tsien, as compared to Prandtl-Glauert,
help create a closer match with the experimental data, as evidenced in Fig. 3.3. We
note that the Prandtl-Glauert model underpredicts and Laitone overpredicts the experimental data, whereas Karman-Tsien (i.e., based on non-linear theory) represents the
best match with the local pressure data. Also, we note that both Laitone and KarmanFig. 3.3 Three
compressibility correction
models compared to the
NACA 4412 airfoil data at
small angle of attack
(α = 1.88◦ ), data from [10]
-1.8
-1.6
-1.4
Laitone
Cp
Karman-Tsien
-1.2
Prandtl-Glauert
-1.0
Experiment
-0.8
0.2
0.4
M∞
0.6
0.8
3.2 Linearized Compressibility Corrections
91
Tsien corrections apply to the local pressure coefficient only and do not extend to the
force coefficients, as in the Prandtl-Glauert correction. However, based on point wise
correction of local pressure coefficient in the subsonic compressible domain, we may
proceed to integrate the pressure jumps and their moment to calculate lift and moment
coefficients on the airfoil using Laitone or Karman-Tsien compressibility corrections.
In subsonic compressible flow over an airfoil, we define a new parameter, called
the critical Mach number, Mcrit . The flight Mach number that corresponds to the
first appearance of a sonic flow on the airfoil is called the critical Mach number. The
first appearance of the sonic point occurs at peak suction, i.e., the point of minimum
pressure, C p, min , or the point of maximum velocity, Vmax on the airfoil, as expected.
Since the pressure coefficient at the sonic point on the airfoil is defined as
C pcrit =
2
2
γ Mcrit
p∗
p∞
−1
(3.40)
And we may relate the static pressure ratio, p ∗ / p∞ , to the Mach numbers, Mcrit and
Mach 1 (for sonic point) using an isentropic flow connection (2.147) between the
flight and the sonic point on the airfoil, i.e.,
p∗
=
p∞
2
1 + γ−1
2 Mcrit
γ
γ−1
(3.41)
γ+1
2
We can express the critical pressure coefficient as a function of the critical Mach
number as:
5
Critical Pressure Coefficient, -Cp, crit (~)
Fig. 3.4 Critical pressure
coefficient and the
corresponding flight Mach
number, Mcrit
4.5
Sonic Boundary
4
3.5
3
Supersonic
Flow
2.5
2
1.5
Subsonic
Flow
1
0.5
0
0
0.1
0.2 0.3
0.4
0.5
0.6
0.7 0.8
0.9
Critical Flight Mach number, Mcrit (~)
1
92
3 Transonic Similarity Laws
⎤
⎡
γ
γ−1 2
γ−1
2 ⎣ 1 + 2 Mcrit
C pcrit =
− 1⎦
γ+1
γ Mcrit
(3.42)
2
The graph of (3.42) for −C pcrit , as is customary, is shown in Fig. 3.4. We note that
C pcrit is the sonic boundary with the region to the right of the curve denoting supersonic and the region to the left of C pcrit curve identifying the subsonic regime. We use
the knowledge of critical Mach number on an airfoil at certain angle of attack, in
low speed, to mark the onset of transonic flow phenomenon, or the onset of shock
appearance. We can graphically construct the critical Mach number for an airfoil
where the knowledge of peak suction C pmin exists in the incompressible limit.
Example 3.4 The airfoil of Example 3.1 had a C pmin = −0.95 at the incompressible
limit. First graph the C pmin for the same airfoil, in the subsonic compressible flow,
using the Prandtl-Glauert compressibility correction and second find the intersection
of this curve and the critical pressure coefficient described by (3.42).
Solution:
We first generate
a spreadsheet of the Prandtl-Glauert compressibility correction
applied to C p M=0 = −0.95 for a range of Mach numbers (below 1) and then
graph the two functions, representing C p , as shown in Fig. 3.5.
Critical Pressure Coefficient, -Cp, crit (~)
Example 3.5 Develop a family of compressibility correction curves using KarmanTsien approach for the range of incompressible C p s between 1, corresponding to
incompressible stagnation point and −3 corresponding to strong suction at angle of
attack. Also, graph critical pressure coefficient representing the sonic point on the
airfoil.
Incompressible
C p ,min = - 0.95
5
4.5
-C p,crit
4
3.5
Prandtl - Glauert
Compressibility
Correction
3
2.5
2
1.5
1
0.5
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Critical Mach number, Mcrit (~)
Fig. 3.5 Graphical construction of critical Mach number, Mcrit
0.9
1
3.2 Linearized Compressibility Corrections
93
Solution:
We apply Karman-Tsien compressibility correction to the incompressible pressure
coefficients, described in the problem, and graph them using a spreadsheet program.
The result is shown in Fig. 3.6. The first appearance of sonic flow on the airfoil is
marked by the critical pressure coefficient, C pcrit curve. The Mach numbers beyond
critical, which is also called supercritical Mach numbers, will no longer be reached
isentropically.
Some classical experimental data, shown in Fig. 3.7 reveal the lift coefficient, k L
as a function of angle of attack (here called angle of incidence) for two airfoils.
Since k L in these graphs represents lift coefficient based on double dynamic pressure
2 ), it represents 1 of how we conventionally define lift coefficient, c , i.e., based
(ρ∞ V∞
l
2
2 ). The U in these graphs represents V and a represents
on dynamic pressure (ρ∞ V∞
∞
1
a∞ , therefore a freestream Mach umber range 0.25 < M∞ < 1.70 is investigated.
To relate the lift curve slopes between different subsonic Mach numbers, we may use
Pressure Coefficient, -Cp (~)
5
Suction
4.5
-C p,crit
4
3.5
3
2.5
2
1.5
1
0.5
0
-0.5
Compression
Incompressible
Stagnation Point
(-Cp )
-1
-1.5
-2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Freestream Mach Number, M∞ (~)
Fig. 3.6 Family of Karman-Tsien compressibility correction curves and the critical pressure coefficient
94
3 Transonic Similarity Laws
Circular Arc Airfoil
R.A.F. 31a Airfoil
0.6
0.6
kL
kL
0.4
0.2
0.4
U=0.5a1
U=0.25a1
U=0.7a1
U=0.7a1
U=0.5a1
U=0.25a1
0.2
U=0.8a1
U=1.7a1
U=1.7a1
0.0-5
0
5
10
Angle of Incidence (deg)
0.0
-5
0
5
10
Angle of Incidence (deg)
-0.2
-0.2
Fig. 3.7 Experimental data on airfoil lift in subsonic compressible flow on two airfoils (Definition
of lift coefficient k L is based on double dynamic pressure, therefore it is 21 cl ), after [11]
dcl
dα
dcl
dα
M1 =
M2
1 − M22
(3.43)
1 − M12
The lift curve slopes as a function of Mach number follow a near perfect match
with the Prandtl-Glauert prediction for the circular arc airfoil (i.e., between 0.25 and
0.50). Although, the lift curve slope increase to Mach 0.7 is less accurate, but still
it is in the right direction, i.e., slope increases with compressible subsonic Mach
number. The accuracy is however reduced in the RAF-31a airfoil. In both cases,
the main departure in the subsonic flow occurs when we cross the critical Mach
number boundary and enter the transonic regime on the airfoil. As evidenced by
data, RAF-31a has entered the transonic zone at freestream Mach number of ∼0.5
and the circular arc airfoil enjoys a higher critical Mach number, in excess of 0.7.
The example of subsonic compressible flow over a wavy wall is both instructive
in its own right, as well as serving as a building block for wind tunnel testing in
transonic flow. Here, we use an example to study the flow over a wavy wall.
Example 3.6 Consider a subsonic compressible flow over a wavy wall of infinite
extent, in the ± x-direction, with amplitude, h and the wavelength, . The wall may
be described by a cosine function according to:
yw = h cos
2πx
(3.44)
3.2 Linearized Compressibility Corrections
95
Calculate the perturbation potential function, φ(x, y), as well as pressure distribution
on the wall. Also, determine the similarity parameter based on h/.
Solution:
In the limit of small h/, we may use
β 2 φx x + φ yy = 0
(3.45)
As the governing equation for φ, subject to the flow tangency boundary condition on
the wall and total decay of the perturbations at infinity. The separation of variables
suggests that
φ(x, y) = F(x)G(y)
(3.46)
Therefore,
φx x = F (x)G(y)
φ yy = F(x)G (y)
(3.47a)
(3.47b)
Upon substitution in the governing equation and dividing by F G, we get:
F
1 G
+ 2
=0
F
β G
(3.48)
We set the pure function of x equal to a constant and the pure function of y equal to
the negative of that constant to satisfy Eq. (2.50), namely
F
= −k 2
F
1 G
= k2
β2 G
(3.49a)
(3.49b)
where k 2 is as yet an arbitrary constant. The solution to (3.49a) is
F(x) = A cos(kx) + B sin(kx)
(3.50)
The periodic function in x matches the periodic wall and boundary condition in x,
i.e., the wavy wall. The solution to (3.49b) is
G(y) = Ce−βky + De+βky
(3.51)
The coefficients A, B, C and D are as yet arbitrary and need to be determined from
our boundary conditions. The exponentially decaying part of the solution in lateral
direction matches the boundary condition at infinity for subsonic flows and is thus
physical. However, the exponentially growing term is unphysical and we thus need
to set D = 0. The wall boundary condition demands
96
3 Transonic Similarity Laws
φy
h
dyw
= −2π sin(2πx/)
≈
V∞
dx
(3.52)
The perturbation potential is the product of F and G, therefore,
φ(x, y) = [A cos(kx) + B sin(kx)]e−βky
(3.53)
The y-derivative of φ evaluated at y = 0, approximates the y-velocity component at
the wall and we may substitute it in (3.52) to get
−
h
βk
[A cos(kx) + B sin(kx) = −2π sin(2πx/)
V∞
We may satisfy the above equation, by inspection, which sets A = 0, k = 2π/ and
βk
h
B = 2π
V∞
Which gives the only unknown coefficient, B, as
B=
hV∞
β
Finally, the perturbation potential, φ(x, y) is derived to be
φ(x, y) =
hV∞
2
1 − M∞
sin
2πx
2π
e− √
2 y
1−M∞
(3.54)
The linearized pressure coefficient is −2φx /V∞ , therefore, evaluated at y ≈ 0, we get
C pw ≈ −
4π (h/)
2
1 − M∞
cos
2πx
(3.55)
Comparing the pressure coefficient at the wall to the cosine function that described
the wall, we conclude that the pressure is 180◦ out of phase (or −Cp is in-phase)
with the wall and is symmetric with respect to its peaks and valleys; therefore, the
net axial force, i.e., the (inviscid) drag, on the wall is zero. This is D’Alembert
paradox. We also note that the impact of Mach number on the pressure coefficient is
inversely proportional to β, which was predicted by Prandtl-Glauert compressibility
correction. We finally note the attenuation of pressure disturbance away from the wall
(i.e., the exponential decay function) depends on M∞ through the −β term, which
approaches zero as M∞ approaches 1 (see Eq. (3.54)). This is another hallmark of
transonic flow, namely the disturbances that are caused by the body propagate away
from the body without significant decay or attenuation.
3.2 Linearized Compressibility Corrections
Fig. 3.8 Definition sketch of
a wavy wall parallel to a
straight wall used as a model
for subsonic compressible
wind tunnel interference
studies
97
yw = H
M∞
H
y
yw = h cos ⎛⎜ 2π x⎞⎟
⎝
⎠
x
h
The general expression for pressure coefficient, invoking the wave number
k = 2π/, is
2hk
Cp ≈ −
cos(kx)e−kβ y
(3.56)
β
Therefore, the similarity parameter, f , for the wavy wall involving the pressure
coefficient, freestream Mach number and the non-dimensional wall parameter is:
2
C p 1 − M∞
= f (kx, kβ y)
hk
(3.57)
To study the compressible interference effects in a windtunnel, e.g., over an airfoil,
we may place a wall at distance H above a wavy wall, as shown in Fig. 3.8 to simulate
the windtunnel environment. We propose to solve this problem using linear theory
in Example 3.7.
Example 3.7 Calculate the perturbation potential for a subsonic compressible flow
over a wavy wall parallel to a straight wall separated by a distance H , as shown
in Fig. 3.8. Also explore the interference effect of the wind tunnel on the wavy
wall pressure coefficient, C pw , as a function of freestream Mach number and nondimensional length scales of the model and the test section, e.g., h/ and h/H .
Solution:
For the adaptation of linearized flow, we require the wall amplitude parameter, h/l 1. Consequently, our governing equation, the linearized pressure coefficient and the
wall boundary conditions are:
β 2 φx x + φ yy = 0
φx
C p ≈ −2
V∞
dys
φ y ≈ V∞
dx
98
3 Transonic Similarity Laws
We again approach the problem through separation of variables, as before, and note
that the presence of the upper wall imposes a new boundary condition and thus
remove the necessity to set the coefficient of the exponentially growing term equal
to zero. The new solution, again in terms of the wave number k, is:
φ(x, y) =
h
V∞
sin(kx)e−βky 1 + e2βk(y−H )
−2βk
H
β 1−e
(3.58)
The pressure coefficient follows the linear theory and at the wall, we approximate
the wall location by y = 0, or the x-axis, to get:
2hk 1 + e−2βk H
C pw ≈ − cos(kx)
β 1 − e−2βk H
(3.59)
Comparing (3.59) that corresponds to the windtunnel environment to the wavy wall
problem in unbounded space, i.e.,
C pw ≈ −
2hk
cos(kx)
β
We note that the wall pressure in the windtunnel is amplified as compared to
unbounded space (labelled as flight) by the following factor,
C pw W T
1 + e−2βk H
=
1 − e−2βk H
C pw f light
(3.60)
And the dimensionless length scale is k H . The wind tunnel interference is defined
as the change in C pw (or local speed on the model) caused by the presence of the
windtunnel wall, expressed as a percentage, namely
C pw W T − C pw f light
1 + e−2βk H
2e−2βk H
=
=
−
1
=
1 − e−2βk H
1 − e−2βk H
C pw f light
C pw f light
C pw
=
2
e+2βk H − 1
(3.61)
As expected, we note that the windtunnel interference effect vanishes as H tends
to infinity. But, we also note that the interference effect becomes infinitely large as
β → 0, or M∞ → 1. This conclusion, albeit reached from linearized theory, points
to the challenge of large interference effects in testing a model in a windtunnel at
transonic speeds.
3.2 Linearized Compressibility Corrections
99
3.2.3 2-D Supersonic Flow
The 2-D linearized supersonic flow is governed by the same small perturbation equation we derived earlier (see 3.10), here cast in a suitable form representing supersonic
freestream Mach number, as
λ2 φx x − φ yy = 0 where λ =
2 −1
M∞
(3.62)
This is the classical wave equation where general solutions are:
φ(x, y) = F(x − λy)
(3.63)
where F is any function of its argument, i.e., (x − λy). Also, any function of (x + λy)
is a solution to the wave equation, namely
φ(x, y) = G(x + λy)
(3.64)
Therefore following the principle of superposition for linear differential equations,
any linear combination of the above solutions is also a solution to the wave equation.
The straight lines described by (x −λy) and (x +λy) in the x–y plane have the slopes
of +1/λ and −1/λ respectively. Examining the Mach waves in the x–y plane, as
shown in Fig. 3.9, reveals that the straight lines (x − λy) and (x + λy) are indeed
two families of Mach waves.
The two wave families are known as the Right-Running and Left-Running Waves
(RRW and LRW). If we stand on a particular wave and look downstream, the wave
either appears on our right, or it appears on our left. The wave that appears on the
right is called the RRW and the one that appears on the left is called the LRW. By
this description, the Mach waves with positive slope (above the x-axis) are LRW
and the Mach waves with a negative slope, i.e., below the x-axis are RRW. A thin
bi-convex airfoil in a supersonic flow is shown in Fig. 3.10 with the wave pattern
on its upper and lower surfaces. The isentropic flow turning is accomplished by the
Mach waves, which in the limit of linearized theory are the two families of Mach
waves at freestream Mach angle, μ∞ (see Fig. 3.10).
Fig. 3.9 Two families of
Mach waves in the x–y plane
y
Mach waves
M∞
µ∞
M∞
1
dy/dx = +1/λ
1
M∞2 − 1
λ
x
M∞
Mach waves
dy/dx = -1/λ
100
3 Transonic Similarity Laws
y
y
M∞ µ∞
M∞ µ∞
x
x
µ∞
µ∞
Fig. 3.10 Wave patterns generated by a thin bi-convex airfoil in supersonic flow as viewed by
linear theory
The pressure coefficient, C p is related to the axial velocity perturbation, φx , which
may be written, in terms of the function F(x − λy), as
C p ≈ −2
φx
F
= −2
V∞
V∞
(3.65)
Here, F is the derivative of F with respect to its argument (x − λy). Now, we relate
F to φ y via
(3.66)
φ y = −λF Therefore, the pressure coefficient is
C p = −2
F
2
=
V∞
λ
φy
V∞
(3.67)
The solid surface boundary condition demanded that the flow and the body (locally)
possess the same slope (see 3.17), namely
φy
df
≈
dx
V∞
(3.68)
In which the body is represented with the function f. The body surface slope is the
tangent of the body surface angle, which for thin bodies may be approximated by
the local body angle, θ, in radians, i.e.,
2θ
(3.69)
λ
Note that a similar result would be produced had we chosen the function G(x +λy)
instead of F(x − λy). The plus and minus signs in (3.69) reflect the possibilities of
compression or expansion on a surface with the local angle θ in radians, in supersonic
flow, respectively. The determination of whether a surface is undergoing compression
or expansion by the flow is obvious as we note whether the flow is locally turning
‘into itself’ or is it locally turning ‘out of itself’ respectively. Here we note that at a
Cp ≈ ±
3.2 Linearized Compressibility Corrections
101
point on the body, as the (supersonic) Mach number increases the pressure coefficient
is reduced, by the 1/λ rule in the linear theory. We also note the singularity in C p as
freestream speed approaches sonic.
Now, let us apply the theory of linearized supersonic flow to a few examples.
Example 3.8 A flat plate is at an angle of attack, α, in a supersonic flow, as shown.
Apply the linear theory to this problem to derive an expression for the wave drag and
lift coefficient.
y
M∞ >1
Flat Plate
α
x
Solution:
From the linear theory, the upper surface experiences a suction pressure and the lower
surface experiences a compression, both proportional to local wall angle, i.e., α,
C pu ≈ −
2α
2α
and C pl ≈ +
λ
λ
(3.70)
Therefore, the flat plate experiences a normal force with the coefficient cn according to
4α
(3.71)
Cn ≈
λ
Then, we may resolve the normal force in the lift and wave drag components through
cosine and sine of the plate angle of attack respectively and in the limit of small angles,
the cosine is replaced by 1 and the sine is replaced by α (in radians) to get the final
result, i.e.,
4α
(α in radians)
(3.72)
cl ≈
λ
cd, w ≈
4α2
(α in radians)
λ
(3.73)
The flat plate, according to linear theory, creates a lift-to-drag ratio in supersonic flow
that is inversely proportional to angle of attack, and is singular at α = 0, namely
L
1
≈
Dw
α
(3.74)
102
3 Transonic Similarity Laws
Example 3.9 Apply the supersonic linear theory to the wavy wall problem (of
Example 3.6). Examine the behavior of the solution φ(x, y) and the wall pressure
coefficient, C pw (x). Also, calculate the wave drag coefficient for one wavelength
section of the wall.
Solution:
The equation for the wall is
yw = h cos(kx) where k ≡
2π
The flow on the upper surface of the wavy wall is described by
φ(x, y) = F(x − λy)
(3.75)
We apply the wall boundary condition
φy
dyw
≈
dx
V∞
(3.76)
At y = 0, as before, to get
− hk sin(kx) = −
λ F (x)
V∞
(3.77)
We may integrate (3.77) to get the function F(x), i.e.,
F(x) = −
hV∞
cos(kx)
λ
(3.78)
We can now generate the general solution for F(x − λy) as
φ(x, y) = −
hV∞
cos[k(x − λy)]
λ
(3.79)
We note that the perturbation potential solution (3.79) is periodic in x, bounded in y
and satisfies the boundary condition at the wall (here approximated at y = 0). The
pressure coefficient is
C pw ≈ −2
φx (y = 0)
hk
= −2 sin(kx)
V∞
λ
(3.80)
Note that the pressure coefficient on the wall varies as the sine function whereas the
wall is described by the cosine function. Therefore there is a 90◦ phase shift between
the wall and its pressure distribution, as shown below.
3.2 Linearized Compressibility Corrections
103
(λ /2hk) C p
y w /h
1
0.5
00
kx
1
2
3
4
5
6
7
-0.5
-1
The wall wave drag coefficient (for one wavelength, ) is
1
Cd,w =
C pw
0
dyw
dx
hk
1 2π
−2 sin(kx) [−hk sin(kx)] d(kx)
dx =
2π 0
λ
(3.81)
Or
Cd,w =
(hk)2
λ
(3.82)
Note that hk in the numerator of (3.82) is proportional to h/, which measures the
wavy wall thickness ratio. Therefore, wall wave drag coefficient scales as the square
of the wall thickness parameter, hk.
3.2.4 The Principle of Superposition
One of the fruits of linear theory is the principle of superposition. Therefore within
this framework, the aerodynamic forces, i.e., lift and drag, due to a thin symmetrical
body at angle of attack are divided into a separate symmetrical thickness problem and
a separate angle-of-attack problem on a flat plate (i.e., a body with zero thickness).
Since a symmetrical thickness problem at zero angle of attack makes no contribution
to lift, the lift of a thin symmetrical profile is modeled as the lift on a flat plate,
namely
4α
(3.72)
cl ≈
λ
The wave drag is the sum of the wave drag of a symmetrical profile at zero angle of
attack and the wave drag of a flat plate at an angle of attack.
cd, w = cd, th + cd, AoA = cd, th +
4α2
λ
(3.83)
In fact, we may even add the friction drag coefficient, c f , to (3.83) to produce the
overall drag coefficient of a thin symmetrical profile in supersonic flow at angle of
attack, namely,
104
3 Transonic Similarity Laws
cd = c f + cd, th +
4α2
λ
(3.84)
Example 3.10 Use the principle of superposition in the linear theory to develop
the drag√polar of a 10 % thick, symmetrical double-diamond airfoil, as shown, at
M∞ = 2. Vary the angle of attack between −2◦ and 12◦ and assume the friction
drag coefficient for the profile is 0.005. Also by drawing a tangent from the origin of
c L − c D coordinate system to the drag polar curve, identify the angle of attack that
corresponds to the maximum L /D . Graph L /D as a function of angle of attack
and verify this angle.
θ nose
2
1
α
3
4
t/c =10%
M ∞= 2
Solution:
The lift coefficient is calculated based on a flat plate, i.e., (3.72). The drag coefficient
is the sum of three drag coefficients, as in (3.84). The thickness contribution to drag
is calculated from the pressure coefficients on the four panels of the symmetrical
airfoil, namely
cd, th =
1
C p1 + C p3 − C p2 − C p4
2
t
c
= C p1 − C p2
t
c
(3.85)
Here we used the symmetry of the profile and lumped the contribution of the lower
half of the airfoil with its upper half. The pressure coefficients C p1 and C p2 are
related to the angles in radians that those panels make with respect to x-axis, i.e.,
± 21 vertex angle, respectively. The half-vertex angle, φnose , is
φnose = tan−1 (0.1) ≈ 0.09967 (rad) or 5.7135 (deg)
Therefore,
2(0.09976)
≈ 0.19934
√
2−1
−2(0.09976)
≈ −0.19934
C p2 =
√
2−1
t
≈ 0.039868
cd, th = C p1 − C p2
c
A simple spreadsheet calculation produces the following table (Table 3.1). and
the associated graphs.
C p1 =
3.2 Linearized Compressibility Corrections
105
Table 3.1 Lift and drag characteristics of a diamond airfoil in supersonic flow [linear theory]
α (deg)
α (rad)
cd
cl
L/D
−2
0
2
4
6
8
10
12
−0.034917
0.000000
0.034917
0.069813
0.104726
0.139636
0.174533
0.209440
−0.1396
0.0000
0.1396
0.2793
0.4189
0.5585
0.6981
0.8378
0.04974
0.04487
0.04974
0.06436
0.08873
0.12285
0.16671
0.22033
−2.8070
0.0000
2.8070
4.3387
4.7207
4.5463
4.1876
3.8023
The tangent to the drag polar from the origin finds the maximum L/D, on the drag
polar curve, which from our graph is α ∼ 6◦ .
Lift-Drag Polar
10% Thick, Double-Wedge Airfoil
1
0.8
Lift Coefficient, cl
(L/D) max
0.6
0.4
0.2
0
0.00
0.05
0.10
0.20
0.15
0.25
-0.2
Drag Coefficient, cd
The lift-drag ratio is also plotted versus the angle of attack in the following figure,
which confirms the maximum L/D for a 10 % thick symmetrical double wedge
occurs at α ∼ 6◦ .
Lift-to-Drag Ratio
10% Thick, Double-Wedge Airfoil
L/D
6
-2
-1
4
2
0
0
-2
-4
-6
Angle of Attack, degrees
1
2
3
4
5
6
7
8
9
10
11
12
106
3 Transonic Similarity Laws
3.2.5 Slender Bodies of Revolution in Subsonic
and Supersonic Flow—Linear Theory
Assume the axis of a body of revolution lies along the x-axis. The y- and z-coordinates
then describe the cross section of the body. To simulate the flowfield, sources and
sinks may be distributed along the x-axis to create the body of interest, following
the incompressible flow theory. The velocity field is repeated
V = V∞ i + ∇φ
(3.86)
where φx /V∞ , φ y /V∞ and φz /V∞ 1 are based on our linearizing assumptions.
The governing perturbation equation for incompressible flow is known as the Laplace
equation, i.e.,
(3.87)
∇ 2 φ = φx x + φ yy + φzz = 0
where the elementary solutions are tabulated and superposed to create the solution
of interest. For example, a 3-D source (or a sink) flow is an elementary solution to
the Laplace equation with the following form (where source is placed at the origin):
φ(x, y, z) =
−Q
4π
x 2 + y2 + z2
=
−Q
4πr
(3.88)
where Q is the source strength, i.e., the volume flow rate from the 3-D (incompressible) source. The flowfield associated with a source at the origin is thus purely radial
with
Q
∂φ
=
(3.89)
Vr =
∂r
4πr 2
The denominator of (3.89) is the surface area of a sphere of radius r and the product
of the area and the normal velocity component (Vr ) then constitutes the volume flow
rate, from the source as expected
Figure 3.11 shows a slender body of revolution with a pointed-nose in uniform
flow. We define slenderness and fineness ratios for the body shown in Fig. 3.11.
y
x
V∞
z
Fig. 3.11 A slender body of revolution (with a pointed-nose) in uniform flow
d
3.2 Linearized Compressibility Corrections
107
These are:
d
slenderness ratio
F=
fineness ratio
d
τ=
(3.90)
(3.91)
The governing equation (3.10) for a linearized subsonic flow is:
2
)φx x + φ yy + φzz = β 2 φx x + φ yy + φzz = 0
(1 − M∞
(3.92)
We may divide the governing equation by β 2 (and absorb the β in the y- and zderivatives) to get:
(3.93)
φx x + φ(β y)(β y) + φ(β z)(β z) = 0
A solution to this equation, analogous to the incompressible solution, is known as
the subsonic source solution, namely,
φ(x, y, z) =
−Q
x 2 + β2 y2 + β2 z2
4π
=
−Q
4π x 2 + β 2 r 2
(3.94)
where r 2 = (y 2 + z 2 ). The geometry of a compressible subsonic source is shown in
Fig. 3.12.
The constant potential surfaces are φ = const, which require the denominator of
(3.94) to be constant
(3.95)
x 2 + β 2 r 2 = const.
Thus the constant potential surfaces are evidently ellipsoids of revolution centered
on the source.
In the supersonic linearized case, we have the following governing small perturbation equation, i.e.,
2
− 1)φx x − φ yy − φzz = λ2 φx x − φ yy − φzz = 0
(M∞
3 -D Source
at origin
Circle,
y 2 +z 2 = r 2
y
r
x
M∞
z
Fig. 3.12 Definition sketch for a source in a compressible subsonic flow
(3.96)
108
3 Transonic Similarity Laws
√
We may divide (3.96) by λ2 and introduce the unit imaginary number, i ≡ −1, to
cast the supersonic small perturbation equation in a similar form as the incompressible
case, namely,
(3.97)
φx x + φ(iλ y)(iλ y) + φ(iλ z)(iλ z) = 0
By analogy to the incompressible source flow solution, we get the so-called supersonic source flow potential, which satisfies our governing equation, namely
φ(x, y, z) =
−Q
4π
x 2 − λ2 y 2 − λ2 z 2
=
−Q
= φ(x, r )
√
4π x 2 − λ2 r 2
(3.98)
where r = y 2 + x 2
The constant potential surfaces for supersonic source are described by:
x 2 − λ2 r 2 = const.
(3.99)
There are two asymptotes x = ±λr where the hyperboloids of revolution tend to as
the solution of Eq. (3.99). The slopes of dr/dx = ±1/λ describe the Mach cones
emanating from the point source both in the upstream and downstream directions, as
shown in Fig. 3.13. However, due to the existence of the zones of action and silence
associated with supersonic flow, the solution bounded by the Mach cone upstream of
the source, i.e., dr/dx = −1/λ, is not acceptable and only the solution downstream
of the point source is acceptable and is considered physical, i.e., within the zone of
action of the source with dr/dx = +1/λ.
To simulate a flow about a body of revolution we need to distribute sources/sinks
along the axis of the body. The potential at point (x, r ) due to a source/sink distribution of strength f (ξ) spread over a dξ extent along the x-axis is:
dφ(x, r ) =
f (ξ)dξ
Non -Physical Solution
Source at
Origin y
Zone of
silence
M∞ >1
(3.100)
4π (x − ξ)2 − λ2 r 2
Mach Cone
dr/dx =1/λ
r
µ∞
r
Zone of
x action
z
A hyperboloid of revolution
describes a constant potential
surface , ϕ =const. for a
supersonic source
Fig. 3.13 A Mach cone downstream of a source in a supersonic flow and a constant potential
surface, φ(x, r )
3.2 Linearized Compressibility Corrections
Fig. 3.14 Definition sketch
for a source distribution on
the x-axis and the potential at
(x, r )
109
Source/sink Distribution
Body of Revolution
(x, r)
r
r
x
dξ
x-ξ
ξ
x
Note that a minus sign is now absorbed in the general description of the source/sink
distribution, f (ξ). The parameters in (3.100) are shown in Fig. 3.14.
The overall contribution of the source/sink distribution along the x-axis is obtained
by integrating (3.100) from the leading edge (i.e., x = 0) to the vertex of the Mach
cone on the x-axis that intercepts the point (x, r ), as shown in Fig. 3.15.
Hence the potential at (x, r ) is
φ(x, r ) =
x−λr
f (ξ)dξ
0
4π (x − ξ)2 − λ2 r 2
(3.101)
The streamwise perturbation of the flow due to the source distribution along the
x-axis is û(x, r ) = φx (x, r ), or
û(x, r ) =
x−λr
f (ξ)dξ
0
4π (x − ξ)2 − λ2 r 2
(3.102)
The lateral perturbation is the r -derivative of the potential, namely
v̂(x, r ) =
x−λr f (ξ)
∂φ
=−
0
∂r
4π
x −ξ
r
dξ
(3.103)
(x − ξ)2 − λ2 r 2
The pressure coefficient C p in linearized aerodynamics is related to the streamwise
perturbation, as we derived earlier,
Nose Mach Cone
Mach Cone
Intercepting
(x, r)
(x, r)
r
µ∞
dξ
r
x
µ∞
ξ
(x-λ r)
λr
Fig. 3.15 The segment (x − λr ) contributes to the potential at (x, r ) in a supersonic flow
110
3 Transonic Similarity Laws
C p ≈ −2
û
φx
= −2
V∞
V∞
(3.104)
Therefore, the pressure coefficient is
Cp = −
1 x−λr
2πV∞ 0
f (ξ)dξ
(x − ξ)2 − λ2 r 2
(3.105)
The flow tangency condition on the body (with body radius R(x)) demands the body
slope and flow inclination to match, namely,
dR
v̂
v̂
1 x−λR
=
≈
=−
dx
V∞ + û
V∞
4πV∞ 0
x −ξ
R
f (ξ)dξ
(x − ξ)2 − λ2 r 2
(3.106)
In the limit of slender body, λR is much smaller than x − ξ, and also x, hence the
integrand and the limit of the integral simplify to:
x
dR
1
f (x)
≈−
f (ξ)dξ = −
dx
4π RV∞ 0
4π RV∞
(3.107)
Hence the strength of the source/sink distribution f (x) is related to the slope of
the body, d R/dx, the body radius, R and the freestream velocity, V∞ . We may cast
(3.107) in terms of the body cross sectional area variation, dS/dx, as
f (x) ≈ −4πV∞ R
dS
dR
= −2V∞
dx
dx
(3.108)
Note that the negative sign that we absorbed in the definition of f (x) is now recovered
in (3.107) or Eq. (3.108). Based on the slender body approximation, we may express
the perturbation potential, the streamwise and the lateral velocity perturbations as
well as the pressure coefficient in the flowfield of a slender body in supersonic flow as:
φ(x, r ) = −
û(x, r ) = −
dξ
V∞ x−λr dS
2π 0
dξ (x − ξ)2 − λ2 r 2
V x−λr d2 S
dξ
∞
2π
dξ 2
0
V∞ x−λr d2 S
2π 0
dξ 2
1 x−λr d2 S
v̂(x, r ) = −
C p (x, r ) =
π
(3.109)
0
dξ 2
(3.110)
(x − ξ)2 − λ2 r 2
x −ξ
r
dξ
(x − ξ)2 − λ2 r 2
dξ
(x − ξ)2 − λ2 r 2
(3.111)
(3.112)
Now, let us apply these results to a slender cone in supersonic flow. The cone of
semi-vertex angle δ, has a body radius R(x) according to:
3.2 Linearized Compressibility Corrections
111
R(x) = x tan δ ∼
= xδ
→
R(x)
≈δ
x
(3.113)
Therefore the cross sectional area variation S(x) along the axis of the cone is calculated from:
S(x) ≈ πx 2 δ 2
(3.114)
This is a quadratic in x; hence the second derivative of cross sectional area of the
cone is constant, namely
d2 S ∼
(3.115)
= 2πδ 2
dξ 2
Now, substitute the second derivative of area in the pressure coefficient integral
(3.112) to get:
C p, cone ≈ 2δ 2
x−λR
dξ
= −2δ 2 ln
(x − ξ)2 − λ2 R 2
0
x+
√
λR
x 2 − λ2 R 2
(3.116)
Applying a slender cone approximation of λR x, we get the slender cone pressure
coefficient in supersonic speeds:
λR
C p, cone ≈ −2δ ln
2x
2
= 2δ ln
2
2
2 −1
δ M∞
→
C p, cone
2
≈ 2 ln
2
δ
λδ
(3.117)
Here we conclude that slender cones in supersonic flow obey the following similarity
rule:
C p, cone
= f (λδ)
(3.118)
δ2
This is essentially the Göthert rule for axisymmetric bodies, where δ is replaced by
the characteristic thickness ratio of the body, τ. The similarity parameter in (3.118) is
λδ. Therefore, slender cones with different semi-vertex angles, δ1 and δ2 in different
supersonic flows, with M∞1 and M∞2 , will have their cone surface pressures related
by
δ2
C p, cone1
= 12
(3.119)
C p, cone2
δ2
If the similarity parameter is held constant in the two flows, namely
2 −1=δ
2
δ1 M∞1
2 M∞2 − 1
(3.120)
112
3 Transonic Similarity Laws
We may also apply the linear supersonic slender body theory to a parabolic body of
revolution, as a second example. The geometry of the parabolic body of revolution,
in the meridian plane, is shown in Fig. 3.16.
The equation for body radius, R, as a function of body axis coordinate, x, is
R(x) = −
2 2
(x − 1/4)
F
(3.121a)
dR
4x
=−
dx
F
(3.121b)
d2 R
4
= − = const.
dx 2
F
(3.121c)
Here F is defined as the ratio of body length to maximum diameter, or the fineness
ratio, i.e., the inverse of slenderness ratio. Note that the body length is normalized to
one, which serves as a reference length scale. The cross sectional area of the body is
that of a circle or S = π R 2 , which differentiates into
dR
dS
= 2π R
dx
dx
2
dR
d S
= 2π
dx 2
dx
(3.122)
2
+R
d2 R
dx 2
(3.123)
Now, if we substitute the body area distribution in (3.112) for the pressure coefficient and integrate the equation we get a closed form solution for the C p distribution.
Note that the new limits of the integral should begin at x = −0.5 instead of zero that
we had in our derivation, which requires x to be replaced by x + 0.5 to represent a
shift to the left on the x-axis, namely
C p (x, r ) =
1 x+0.5−λr d2 S
π −0.5
dξ 2
dξ
(3.124)
(x + 0.5 − ξ)2 − λ2 r 2
Again, upon substitution for d2 S/dξ 2 in (3.124) from (3.123), we can evaluate the
integral to get the surface pressure coefficient on a parabolic body of revolution in
supersonic flow, namely
r
Parabola
M∞
x
-0.5
+0.5
=1
Fig. 3.16 A parabolic body of revolution in the meridian plane
3.2 Linearized Compressibility Corrections
C ps =
113
4 2
2 2
−1 x + 0.5
12x
cosh
−
1
+
6λ
R
F2
λR
4
+ 2 6 [(x + 0.5) − 4x] (x + 0.5)2 − λ2 R 2
F
(3.125)
A graph of (3.125) for a parabolic body of revolution for a fineness ratio of 10 and
freestream Mach number of M∞ = 1.4 is shown in Fig. 3.17. Also the pressure
distribution on a 2-D parabolic body is also graphed in Fig. 3.17 for comparative
purposes.
The two-dimensional supersonic linear theory predicts:
Cp =
d R/dx
2θ
≈2
λ
λ
(3.126)
With d R/dx = −4x/F, we get:
8
x
Cp ∼
=−
Fλ
(3.127)
This equation indicates a linear distribution of C p with x. Also for F = 10, λ = 1,
we have
Cp ∼
(3.128)
= −0.8x
At the leading edge, we are at x = −0.5, therefore the pressure coefficient at the
nose is 0.4, as graphed in Fig. 3.17. This profile is also symmetrical with respect to
the mid chord. The three dimensional case provides for aft body compression due
to a shrinking streamtube area in 3-D and a weaker nose pressure rise due to the
well-known 3-D relieving effect. The incompressible solution over a symmetrical
body of revolution creates a symmetrical pressure distribution about the mid chord.
Fineness Ratio = 10
0.4
Tw
o
me
ns
0.2
ion
al,
Cp 0
-0.2
-0.4
Three Dimensional
M ∞ = 1.4
Di
M
∞
=1
.4
Three Dimensional
M∞ = 0
-0.5
0
0.5
Fig. 3.17 Pressure distribution on two and three-dimensional bodies with parabolic contours
114
3 Transonic Similarity Laws
Also the bodies sharp nose and the trailing edge in incompressible flow provide for
singularity at the sharp corners, as indicated in the graph of Fig. 3.17 for M∞ = 0.
3.3 Transonic Small Disturbance Theory
Transonic flow is characterized by a mixed subsonic and supersonic flow on a body
in flight. On the lower end, it begins with the first appearance of sonic flow on the
body, which we called critical Mach number. On the upper end, transonic flow regime
ends with the last appearance of sonic flow on the body. The rationale for this range
of Mach numbers is that for any Mach number below critical, the entire body is
submerged in subsonic flow and for any Mach number above the last sonic point on
the body, the entire body is submerged in supersonic flow. Therefore the existence
of mixed subsonic-supersonic flow is the definition and the hallmark of transonic
aerodynamics.
As we discussed in Sect. 3.2, thin or slender bodies at small angle of attack perturb
the flow only by a small amount. This hypothesis allowed for the linearization of
the full potential equation in the limits of subsonic and supersonic flow. However,
when M∞ ≈ 1, i.e., in transonic flow, the term that we kept in the potential equation
(Eq. 3.10), namely
2
)φx x
(3.129)
(1 − M∞
Becomes comparable in magnitude to the term that we neglected, namely
2
M∞
φx
(γ + 1)
φx x
V∞
(3.130)
Note that the term (in 3.129) can be made arbitrarily small when the freestream Mach
number is made arbitrarily close to 1. Consequently, the small perturbation equation
for steady, three-dimensional irrotational flows assumes the following form in the
transonic limit,
φx
2
2
(γ + 1)
φx x
)φx x + φ yy + φzz = M∞
(3.131)
(1 − M∞
V∞
The non-linear term on the right-hand side of (3.131) speaks to the nature of transonic
flow. Combining the left-hand side term with the right (in φx x ), we get the transonic
small disturbance equation:
2
2 φx
(1 − M∞ ) − (γ + 1)M∞
φx x + φ yy + φzz = 0
V∞
(3.132)
Figure 3.18 shows the pressure coefficient approaching the sonic flow from the
subsonic side and from the supersonic side, where linear theory was deemed valid.
3.3 Transonic Small Disturbance Theory
Linearized subsonic
flow regime
-Cp
1
Cp ∼
β
115
∞
∞
Nonlinear Transonic flow regime
Linearized supersonic flow regime
1
Cp ∼
λ
- Cp, 0
0
0
1
2
3
M∞
Fig. 3.18 Variation of pressure coefficient with Mach number according to linear theory
Transonic flow is sandwiched between the two regimes and is characterized by an
inherent nonlinearity (and a finite pressure coefficient).
Another view of transonic flow emerges when we examine the linear small perturbation equation (following Von Kármán) in the limit of sonic flow, namely,
2
)φx x + φ yy + φzz = 0
(1 − M∞
→
φ yy + φzz = 0 as M∞ → 1
(3.133)
Equation (3.133) in the limit of sonic flow shows a lack of dependence on the streamwise (i.e., x-) direction. This indicates that the disturbance caused by a body in sonic
flow propagates laterally and stays confined in the streamwise direction. This picture of near sonic flow helps us understand an important character of transonic flow.
Figure 3.19 shows a cartoon of a body in sonic flow where the disturbance caused
by the body laterally extends to large distances whereas the axial extent is limited to
the body length.
The transonic small disturbance equation (3.132) does not lend itself to analytical
solution; therefore, we seek to establish similarity laws that would hold on bodies that
are affinely-related, but at different transonic Mach numbers and different thickness
or slenderness ratios (and different angles of attack). First, let us define affinelyrelated bodies. Figure 3.20a shows a body of maximum thickness-to-chord ratio, τ1
Stream
surfaces
y
M∞ =1
x
Fig. 3.19 A slender body in sonic flow showing extensive lateral and limited axial extent of disturbance
116
3 Transonic Similarity Laws
(a)
y
(b)
τ1 = t1 /
ys(x)
y
t1
ys(x)
x
t2
τ 2 = t2 /
x
Fig. 3.20 Affinely-related profiles with different thickness/slenderness ratios. a Definition sketch
for a body with thickness ration τ1 . b An affinely-related body to (a) but with thickness ratio τ2
and in Fig. 3.20b we note that the same body is stretched in the y-direction and it
has attained a thickness-to-chord ratio of τ2 .
We define the local body slope as the product of a dimensionless function h(x/)
and the body thickness (or slenderness) ratio, τ, as
dys
= h(x/)τ
dx
(3.134)
The function h(x/) is called the (dimensionless) thickness distribution function.
Therefore, affinely-related profiles have the same dimensionless thickness distribution function but they have different thickness ratios, τ1 , τ2 , etc. This implies that
the same profile is stretched in y-direction, or z-direction or both. The implication
is that the local slopes of affinely-related bodies follow:
(dys /dx)1
(dys /dx)2
=
τ1
τ2
(3.135)
In addition, the angle of attack, α, in affine transformations is adjusted by the thickness (or slenderness) ratio according to:
α1
α2
=
τ1
τ2
(3.136)
Although the example profile shown in Fig. 3.20 is symmetrical, i.e., it has zero
camber, affine transformation of cambered profiles will change camber in proportion
to thickness ratio as well. If we define a profiles camber in the x–y plane as yc /
(recall that camber in two-dimensional profiles is customarily defined as z c /c, in x–
z plane, where c is the chord length), then affine transformation adjusts or changes
camber according to (using our notation in x–y plane):
(yc /)1
(yc /)2
=
τ1
τ2
(3.137)
We are now ready to non-dimensionalize the transonic small disturbance equation
and the corresponding boundary conditions. The transformed coordinates for a thin
(or slender) body of length, , thickness (or slenderness) ratio τ in transonic flow are
3.3 Transonic Small Disturbance Theory
117
chosen according to the following rule:
x
(3.138)
y=
yτ 1/3
(3.139)
z=
zτ 1/3
(3.140)
x=
In addition, the dimensionless perturbation potential, φ is defined as
φ=
φ
V∞ τ 2/3
(3.141)
Note that the dimensionless lateral coordinates, y and z are scaled down by the factor,
τ 1/3 , as compared to the streamwise coordinate, x, with the intention of placing them
all on the same order of magnitude. The choice of dimensionless perturbation potential φ yields the same governing equation and boundary conditions for affinely-related
bodies, as we shall see. For a rigorous proof of these choices of transformed coordinates, the reader may consult Ref. [9]. The transonic small disturbance equation is
now expressed in terms of the transformed coordinates and perturbation potential as:
2 )
(1 − M∞
2
−
(γ
+
1)M
φ
∞ x φx x + φ y y + φzz = 0
τ 2/3
(3.142)
3.4 Transonic Similarity Parameters
The first group in the bracket in (3.142) is identified as a transonic similarity parameter, K , i.e.,
2 )
(1 − M∞
(3.143)
K ≡
τ 2/3
Note that based on this definition, K can be both positive and negative. For example,
it becomes positive when M∞ < 1 and negative when M∞ > 1. The second group
in the bracket may be simplified to
2
φx ≈ (γ + 1)φx
(γ + 1)M∞
(3.144)
This approximation is certainly valid in the transonic limit, namely where M∞ ≈ 1.
As a result of these choices, the transformed transonic small disturbance equation is
now expressed as
118
3 Transonic Similarity Laws
K − (γ + 1)φx φx x + φ y y + φzz = 0
(3.145)
The boundary condition on the solid body is the flow tangency condition, which
matches the body to flow slope, namely
v̂
dys
≈
dx
V∞
(3.146)
Repeating (3.134) for body slope, we have
dys
= τ h(x)
dx
(3.147)
Now the right-hand side of (3.146) may be expressed in terms of the transformed
coordinates as
(φφ )(φ y )(∂ y/∂ y)
V∞ τ 2/3 (φ y )(τ 1/3 /)
φy
v̂
=
=
=
= τ (φ y )
V∞
V∞
V∞
V∞
(3.148)
Therefore substituting Eqs. (3.147) and (3.148) in (3.146), we get
τ h(x) = τ φ y
→
φ y = h(x)
(3.149)
Since the dimensionless thickness distribution function, h(x) is unchanged in affine
transformations, the boundary condition φ y produces the same result at corresponding points x, y and z on affinely-related bodies. In addition, we note that the boundary
condition on thin/slender bodies is often satisfied on the x-axis, which again points
to the adequacy of (3.149) on thin or slender bodies that are affinely-related. Finally,
one last point about the flow boundary condition has to address the boundary condition at infinity. In subsonic flows we demand the perturbations to decay to zero at
large distances from the body in motion. In supersonic flows there are waves, thus we
demand the perturbations to remain finite at large distances from the body. In transonic flows, since we encounter both types of flows (i.e., subsonic and supersonic),
we demand that the perturbations to remain finite or possibly decay to zero at large
distances from the body. The nature of affine transformations on similar bodies at
transonic Mach numbers is such that the condition at infinity does not change with
the thickness/slenderness ratio of a body. Therefore, the solution to (3.145) satisfies
the far-field boundary condition as caused by affinely-related bodies in transonic
flow.
The pressure coefficient from the small disturbance theory is written as
C p ≈ −2
φx
V∞
(3.150)
3.4 Transonic Similarity Parameters
119
Let us express the pressure coefficient in terms of the non-dimensional variables
according to the transformations listed in Eqs. (3.138)–(3.141).
C p ≈ −2
(φφ )(φx )(∂x/∂x)
φx
V∞ τ 2/3 (φx )(1/)
= −2
= −2
= −2τ 2/3 φx
V∞
V∞
V∞
(3.151)
This produces the transonic similarity rule for the pressure coefficient, C p , namely
Cp
≈ −2φx = f (K , x, y, z)
τ 2/3
Cp
= f (K , x, y, z)
τ 2/3
→
(3.152)
The implication of this similarity rule is that
C p1
2/3
τ1
=
C p2
2/3
τ2
(3.153)
On two bodies that are affinely-related in transonic flow, i.e., at the same x, y and z
locations, when the two transonic flows have K 1 = K 2 and the angle of attacks on
the two bodies follow (3.136), namely
α2
α1
=
τ1
τ2
Now, let us put the transonic similarity rule, (3.153), to use in the following example.
Example 3.11 Consider an airfoil with 10 % thickness-to-chord ratio in M∞ = 0.9
(transonic) flow at 4◦ angle of attack. The peak suction pressure coefficient is
C p, min = −0.55. What does transonic similarity rule say about an affinely-similar
airfoil in Mach 0.95? Also, calculate C p, crit corresponding to M∞ = 0.9 and
M∞ = 0.95.
Solution:
First, let us calculate the transonic similarity parameter, K , namely
K1 ≡
2 )
(1 − M∞1
2/3
τ1
≈ 0.88190
Now, let us consider an affinely-related airfoil with thickness τ2 in Mach 0.95 flow
at an angle of attack, α2 . In order to have the same transonic similarity parameter,
K 2 = K 1 , the freestream Mach number has to be:
K2 ≡
2 )
(1 − M∞2
2/3
τ2
≈ 0.88190
120
3 Transonic Similarity Laws
Since M∞2 is known to be 0.95, we calculate the thickness-to-chord ratio of the
affinely-related airfoil to be
τ2 ≈ 0.03676
The angle of attack follows:
α2 = α1
τ2
τ1
= 4◦
0.03676
0.10
≈ 1.47◦
The similarity rule for the pressure coefficient says
C p2 = C p1
τ2
τ1
2/3
= −0.55
0.03676
0.10
2/3
≈ −0.2822
Therefore the pressure coefficient at the peak suction point in Mach 0.95 flow is
−0.2822. The following graph shows the schematic drawing of two transonically
similar flows (not to scale).
y
K=0.88190
τ = 0.10
o
α= 4
y
Cp, min = - 0.550
M∞=0.90
K=0.88190
τ ≈ 0.03676
α ≈ 1.47 o Cp, min ≈ -0.2822
M∞= 0.95
x
x
The critical pressure coefficient follows (3.42) and gives:
(C p, crit ) M∞ =0.90 ≈ −0.18786 and (C p, crit ) M∞ =0.95 = −0.08821
3.4.1 Other Transonic Similarity Parameters
The complete transonic small disturbance equation is re-written here for reference.
2 )
(1 − M∞
2
− (γ + 1)M∞ φx φx x + φ y y + φzz = 0
τ 2/3
There are different forms of transonic similarity parameter that are possible, which
are all identical at M∞ = 1, but differ at Mach numbers slightly below and above
2 in the
the sonic point. For example, as we noted earlier, the product of (γ + 1)M∞
3.4 Transonic Similarity Parameters
121
bracket in (3.142) may be replaced with (γ + 1), in the limit of sonic flow and that
led to the definition of the transonic similarity parameter K in (3.143). Here, we first
introduce the transonic similarity parameter that Von Kármán proposed, namely
K ≡
2
2
1 − M∞
γ+1 2
2 M∞ τ
2/3
(3.154)
Alternatively, if we apply M∞ = 1 to the denominator of Von Kármán transonic
similarity parameter, we get
2
1 − M∞
K ≡ (3.155)
2/3
2 γ+1
τ
2
2 ) in the numerator of the Von Kármán parameter and write
What if we take (1 − M∞
it as the product of (1 − M∞ ) and (1 + M∞ ) and then replace (1 + M∞ ) by 2?
This is certainly a valid approximation in transonic flow; therefore another similarity
parameter emerges, namely
1 − M∞
(3.156)
K ≡
2/3
γ+1
τ
2
It may be shown that the pressure coefficient then takes on the form
C p [(γ + 1)/2]1/3
= f 1 (K , x, y, z)
τ 2/3
(3.157)
Although the derivation of the expression (3.157) for pressure coefficient is beyond
the scope of this book, the reader may consult Ref. [9] for detailed derivation. We also
note that (3.157) is more general than the basic expression (3.152), as it accounts for
the effect of different gases through ratio of specific heats, γ. The two expressions for
similarity pressure coefficients share the same functional dependency on slenderness
ratio, τ, namely τ −2/3 .
We are now ready to examine the similarity principles in force and moment coefficients on two dimensional thin/slender bodies in transonic flow. The lift coefficient
is written as
1
C p dx = C p dx
(3.158)
cl =
We divide both sides of (3.158) by τ 2/3 to get
cl
τ
=
2/3
C
p
dx = f 1 (K )
τ 2/3
(3.159)
122
3 Transonic Similarity Laws
Now, we multiply both sides by [(γ + 1)/2]1/3 to get
C [(γ + 1)/2]1/3
cl [(γ + 1)/2]1/3
p
=
dx = f 2 (K )
2/3
τ
τ 2/3
cl [(γ + 1)/2]1/3
→
= f 2 (K )
τ 2/3
(3.160)
The drag coefficient may be written as
cd =
1
Cp
dys
dx
dx =
C p τ h(x)dx
(3.161)
We divide both sides by τ 5/3 and multiply by [(γ + 1)/2]1/3 to get
C [(γ + 1/2]1/3
cd [(γ + 1)/2]1/3
p
=
h(x)dx = f 3 (K )
5/3
τ 2/3
τ
cd [(γ + 1)/2]1/3
→
= f 3 (K )
τ 5/3
(3.162)
From the transonic similarity expression for 2-D wave drag coefficient, we note
that drag coefficient on two-dimensional thin bodies in transonic flow is proportional
to τ 5/3 . The 2-D wave drag then rises as τ 5/3 , which indicates the advantage of thin
profiles in transonic flow. The 2-D drag/lift ratio is proportional to τ as well, which
again points to the advantage of thin bodies in transonic flight.
The pitching moment coefficient about the leading edge is
1 cm, LE = − 2 C p xdx = − C p xdx
(3.163)
We divide both sides by 2/3 and multiply by [(γ + 1)/2]1/3 to get
C [(γ + 1)/2]1/3
cm, LE [(γ + 1)/2]1/3
p
=
−
xdx = f 4 (K )
τ 2/3
τ 2/3
cm, LE [(γ + 1)/2]1/3
→
= f 4 (K )
τ 2/3
(3.164)
Example 3.12 The lift coefficient of a 10 %-thick airfoil at the angle of attack of 3◦
in Mach M∞ = 1. air stream (γ = 1.4) is cl = 0.12. What does transonic similarity
theory predict as the lift coefficient of an affinely similar airfoil (also called the same
airfoil family in airfoil theory) that is 15 % thick and is in transonic flight through
air (with γ = 1.4)?
3.4 Transonic Similarity Parameters
123
Solution:
First, we establish the transonic similarity parameter, K, according to (3.156), namely
K =
1 − M∞1
2/3
τ1
=
1 − 1.175
≈ −0.8123
0.12/3
The minus sign in K indicates a supersonic freestream Mach number. Now, we
propose to calculate a new Mach number that combined with 15 % thick airfoil
would preserve K , i.e.,
2/3
M∞2 = 1 − K τ2
= 1 + 0.81230.152/3 ≈ 1.229
The lift coefficients follow (3.160), namely
cl2
=
cl1
τ2
τ1
2/3
→
cl2 = 0.12(0.15)2/3 ≈ 0.1572
Now, we need to examine the angle of attack on the second airfoil, as it is adjusted
by affine transformation, i.e.,
α1
α2
=
τ2
τ1
→
α2 = α1
τ2
τ1
= 3◦ (1.5) = 4.5◦
Hence, the transonic similarity theory, predicts for K = −0.8123:
.
τ
0.1
0.15
M∞
1.175
1.229
α
3◦
4.5◦
cl
0.12
0.1572
3.5 3-D Planar and Axisymmetric Slender Bodies
We are now ready to examine the transonic similarity principle on affinely-related
bodies in three-dimensions, namely 3-D planar (e.g., wings), axisymmetric slender
bodies (e.g., projectiles and fuselage) and wing-body combinations. We anticipate the
three-dimensional relieving effect, which is a fundamental aerodynamic principle, to
be in full force when we study three-dimensional transonic flows on slender bodies.
The detailed mathematical derivations and proofs (of the similarity parameters outline
below) are beyond the scope of this book and are thus left out of this section. The
reader may consult classical texts, e.g., Ref. [9], for such details.
For axisymmetric slender bodies the similarity parameter is K :
K =
2
1 − M∞
(γ + 1)τ 2
(3.165)
124
3 Transonic Similarity Laws
The corresponding pressure coefficient on an axisymmetric body is
Cp
= f 5 (K , x, y, z)
τ2
(3.166)
The wave drag coefficient for a slender axisymmetric body is thus
CD =
1 1 1 dy
dy
C p (2π y) dx ∼ 2 2
C p y dx ∼ 2 2
C p τ x[τ h(x)]dx
A 0
dx
τ 0
dx
τ 0
(3.167)
After cancelling the τ 2 from numerator and denominator of (3.167) and taking the
2 in the integrand to change x to x, we get
CD ∼
1
0
C p h(x)xdx
→
1 C
CD
p
∼
h(x)xdx = f (K )
0 τ2
τ2
CD
= f 6 (K )
τ2
(3.168)
(3.169)
Here we note that for M∞ = 1, K = 0 and thus C D = const. τ 2 and for an
axisymmetric body of a given length, , and diameter, d,
C D ∼ d 2 and Dw ∼ d 4
(3.170)
Therefore, the wave drag of a projectile rises as the fourth power of its diameter in
transonic regime, which strongly supports the notion (or necessity) of slender bodies
in transonic flight.
For a planar body, e.g., a wing, the transonic similarity parameter is:
1 − M∞
K = 2/3
γ+1
τ
2
(3.171)
The pressure coefficient is the same as the 2-D flows, but affine transformations
change the wing aspect ratio, A, in addition to the angle of attack and thus the
following three parameters are kept constant on two wings that are affinely-related,
namely:
1 − M∞2
1 − M∞1
=
(3.172a)
2/3
[(γ1 + 1)τ1 ]
[(γ2 + 1)τ2 ]2/3
α1 /τ1 = α2 /τ2
(3.172b)
A1 [(γ1 + 1)τ1 ]1/3 = A2 [(γ2 + 1)τ2 ]1/3
(3.172c)
Thus the pressure coefficient for a wing of thickness ratio, τ, aspect ratio, A, span
of b, ratio of specific heats, γ and angle of attack, α is:
3.5 3-D Planar and Axisymmetric Slender Bodies
125
4
Predicted theoretical
critical Mach Number
5/3
(t/c)
(CD p)min
3
AR(t/c)1/3= 3.0
2.5
2.0
1.5
2
1.0
.5
1
.25
0
-2.0
-1.6
-1.2
-.8
-.4
M2- 1
0
.4
.8
1.2
2/3
(t/c)
Fig. 3.21 Curves of the generalized drag coefficient for symmetrical wings for γ = 1.4 (after
Ref. [6])
C p |α/τ ; A(γ+1)1/3 τ 1/3 =
τ 2/3
f 7 (K , x, y, z, z/b)
(γ + 1)1/3
(3.173)
The wing lift coefficient is then:
C L |α/τ ; A(γ+1)1/3 τ 1/3 =
τ 2/3
f 8 (K )
(γ + 1)1/3
(3.174)
τ 2/3
f 10 (K )
(γ + 1)1/3
(3.175)
τ 5/3
f 9 (K )
(γ + 1)1/3
(3.176)
The wing pitching moment coefficient is
C M |α/τ ; A(γ+1)1/3 τ 1/3 =
The wing wave drag coefficient is:
C D |α/τ ; A(γ+1)1/3 τ 1/3 =
The transonic zero-lift wave drag coefficient for unswept wings with a symmetrical
NACA 63 profile is presented in Fig. 3.21, as presented by McDevitt [6]. We recognize
2 )/τ 2/3 in the graph as well as the
the similarity parameters A(t/c)1/3 and (1 − M∞
5/3
drag coefficient divided by τ . For a given thickness to chord ratio, aspect ratio, and
freestream Mach number, this graph allows us to calculate the wave drag coefficient.
A similar graph can be found in the USAF Datcom [3] for general airfoils with
rounded leading edges. We will show in Chap. 8 how this method can be extended
to calculate the wave drag of swept wings at transonic speeds.
126
3 Transonic Similarity Laws
3.6 Hodograph Transformation
Small disturbance theory in transonic flow yields a nonlinear partial differential
equation in perturbation velocity potential, φ. Reverting to perturbation velocity
components û and v̂ and limiting the analysis to two-dimensional flowfields, we get
the governing equation to be:
2
)
(1 − M∞
2
∂ v̂
(γ + 1)M∞
∂ û
∂ û
+
=
û
∂x
∂y
V∞
∂x
(3.177)
The condition of irrotationality demands:
∂ û
∂ v̂
−
=0
∂y
∂x
(3.178)
The system of two equations and two unknowns in û and v̂, as described by
Eqs. (3.177) and (3.178), is still nonlinear. The independent variables are x and
y, which describe the coordinates in the physical plane. However, if we switch the
dependent variables with the independent variables, the resulting set of equations
become linear! This is the essence of Hodograph transformation in transonic small
disturbance theory. From the physical domain we have
û = û(x, y)
v̂ = v̂(x, y)
(3.179)
The differentials of these functions yield:
∂ û
∂ û
dx +
dy = û x dx + û y dy
∂x
∂y
∂ v̂
∂ v̂
dv̂ =
dx +
dy = v̂x dx + v̂ y dy
∂x
∂y
dû =
(3.180)
Therefore, by isolating dx and dy from (3.180), we get
v̂ y dû − û y dv̂
û x v̂ y − û y v̂x
−v̂x dû + û x dv̂
dy =
û x v̂ y − û y v̂x
dx =
where the denominator is the determinant of the coefficient matrix
û x û y Δ≡
v̂x v̂ y (3.181)
(3.182)
3.6 Hodograph Transformation
127
Alternatively, we may express the two independent as dependent variables according to:
x = x(û, v̂)
y = y(û, v̂)
(3.183)
And the corresponding differentials are:
dx = xû dû + xv̂ dv̂
dy = yv̂ dû + yv̂ dv̂
(3.184)
A comparison between the two sets of Eqs. (3.182) and (3.184) shows that
v̂ y
−û y
and xv̂ =
Δ
Δ
v̂x
−v̂x
and yv̂ =
yû =
Δ
Δ
xû =
(3.185)
Now, if we substitute these in (3.177) and (3.178), we get the new governing equation
set as:
2
∂x
(γ + 1)M∞
∂y
2 ∂y
(1 − M∞
+
=
(3.186)
)
û
∂ v̂
∂ û
V∞
∂ v̂
∂y
∂x
−
=0
∂ v̂
∂ û
(3.187)
The new set of governing equations, known as transonic hodograph equations are
now linear in x and y, as independent variables where the new dependent variables
are û and v̂. This clever mathematical exercise has two impediments in real life
applications. First, the physical boundary conditions are always expressed in terms
of x and y as independent variables and not the û and v̂. Second, the solution can only
exist if the determinant of the coefficient matrix Δ = 0. The determinant may only
be zero in supersonic flows and the loci of all the points with Δ = 0 is called the limit
line. Except for simple boundaries and geometries (as in a wedge), the application
of hodograph transformation to transonic problems is not practical. In addition, with
the advances in Computational Fluid Dynamics (CFD), the nonlinearity of transonic
flow poses no problem to achieving high-fidelity simulation or solution. For more
details on hodograph transformation such classical texts as Shapiro [9] and Liepmann
and Roshko [5] may be consulted.
3.7 Empirical Rules
There are many empirical rules that were developed prior to the advent of supercritical
airfoil sections and area rule in transonic flow in the mid 1950s. Most such rules
involve the old profiles and thus do not translate into a useful practice today in
128
3 Transonic Similarity Laws
modern aerodynamics. So, in this section, we concentrate on semi-empirical rules
that are post-supercritical era and are thus useful to an aerodynamicist or aircraft
designer.
An example of such semi-empirical rules is shown in Fig. 3.22 from Ref. [8] for
2-D airfoils and wings. The lift curve slope, C L α for swept and straight wings as well
as wings with low and high aspect ratio is shown in the transonic domain. Although,
the values are not exact, they may be used as guidelines and in preliminary design
studies. To extend the semi-empirical results to wing-body-nacelle configurations
and compare it to aerodynamic data, we examine Fig. 3.23. Here, the lift curve slope
as well as the percent root-chord location of the aerodynamic center is shown for
a supersonic transport (namely, Boeing B2707-300) over a wide range of the flight
Mach numbers, i.e., 0.3–2.7. The computational results are compared to experimental
data of a windtunnel model. We note that lift curve slope is reasonably well predicted
whereas the location of aerodynamic center is in general farther aft that the data
indicates. We further note that the values of the lift curve slope in the transonic
range, in Fig. 3.23, match with the semi-empirical correlation of Ref. [8] in Fig. 3.22.
Now, we wish to address the issue of the location of the sonic point on a blunt
body in supersonic and upper transonic flow. The blunt body in supersonic flow will
create a (bow) shock that is detached with the subsequent zones of subsonic and
supersonic flow that are created downstream of the shock. The boundary separating
the supersonic region from the subsonic is called the sonic line, i.e., the loci of all
the points in the flow where local Mach number is 1. Examples of blunt bodies with
a sharp corner and with continuous curvature are shown in Fig. 3.24. The straight
sonic lines shown in these examples are mere simplifications to the real sonic lines,
which are indeed curved.
There is experimental evidence that points to the sharp corner as the location of
the sonic boundary on a blunt body as shown in Fig. 3.24a. On the blunt body with a
Fig. 3.22 Semi-empirical
model for lift curve slope in
transonic Mach numbers
(from Ref. [8])
Subsonic 2-D
Theoretical
CL =
α
Supersonic 2-D
Theoretical
4
CL =
M 2 -1
α
1-M 2
α
Lift-curve slope, CL (rad-1)
10
Typical Unswept
High Aspect Ratio Wings
9
8
Thin Airfoil
Thick Airfoil
Typical Swept Wing
7
6
High Aspect Ratio
Low Aspect Ratio
5
4
3
2
1
0
0
.5
1.0
1.5
2.0
Mach number, M (~)
2.5
3.0
3.7 Empirical Rules
129
4.
α
B2707-300 three-views:
Lift-curve slope, CL (rad-1)
Lift Curve Slope
3.
Experiment
2.
Computation
Linear Potential
1.
0.
0.
0.5
1.0
1.5
2.0
2.5
aerodynamic center,
xac/cr (%)
Aerodynamic Center
65
60
Computation
Linear Potential
55
Experiment
50
0.
0.5
1.0
1.5
2.0
2.5
Mach number, M (~)
Fig. 3.23 Lift curve slope and aerodynamic center for the Boeing B2707-300 supersonic transport
aircraft (after Ref. [12])
(a)
(b)
Detached shock
M>1
Sonic line
Detached shock
M<1
M<1
M∞ > 1
M>1
Sonic line
M∞ > 1
Fig. 3.24 Two examples of a blunt body in supersonic and upper transonic flow. a Blunt body with
a sharp corner (Discontinuous curvature). b Blunt body with continuous curvature
continuous curvature however, as in Fig. 3.24b, there is no simple rule that identifies
the sonic boundary on the body. However, there are two empirically supported ideas
on the location of the sonic boundary on two-dimensional and axisymmetric bodies.
These are (see [7] or [9]):
• The location on the 2-D body where θb = θmax is approximately the sonic boundary
on the body. Here, θb is the body angle and θmax is the maximum turning angle,
or flow deflection angle, corresponding to an attached plane oblique shock at the
freestream Mach number, M∞ .
• The location on the axisymmetric body where θb = θmax is approximately the
sonic boundary on the body. Here, θmax is the maximum turning angle, or flow
deflection angle, corresponding to an attached conical shock at the freestream
Mach number, M∞ .
130
3 Transonic Similarity Laws
(a)
(b)
90o
Approximate
Location of Sonic
Boundary
σ
θ b = θ max
M∞> 1
µ
M∞ > 1
θ max
θ
Fig. 3.25 Definition sketch of the approximate location of the sonic point on a blunt body [σ is
the shock wave angle, θ is the flow turning angle and θmax is the maximum turning angle]. a Shock
wave angle-deflection diagram boundary. b The approximation location of the sonic boundary
These empirically-supported ideas are very useful in establishing the location of
detached shock waves ahead of blunt bodies as well as estimating the nose pressure
drag, up to the sonic point, on blunt bodies in supersonic and upper transonic flow.
Figure 3.25 is the definition sketch relating the body angle and the maximum turning
angle across an attached oblique/conical shock at the freestream Mach number, M∞ .
Now, we are ready to apply these empirical ideas (i.e., the location of the sonic
point on a blunt body and the straight sonic line) to establishing an approximate
location of detached shock waves as well as the pressure drag estimation of blunt
bodies, up to the sonic point.
3.8 Approximate Location of Detached Shocks
The bow shocks that are created in supersonic flow ahead of blunt bodies vary in
strength from a normal shock to a Mach wave. The normal shock represents the
strongest point on the wave and it is located on the stagnation streamline. The Mach
wave represents the weakest point, which is asymptotically arrived at a large distance
from the body. Therefore, we may assume the basic shape of a detached shock is
approximately a hyperbola, with the Mach wave (of angle μ) running as its asymptote
(see Fig. 3.26).
y
Asymptote
tan−1 β
y = (tan μ ) x; Equation of
asymptote
y =(tan μ ) x 2 −x 02 ; Equation of
hyperbola
or
M∞ > 1
β y = x 2 − x02
µ
x0
where:
μ ≡ sin−1 (1/ M ∞ )
β ≡ M∞2 − 1
and
x0 is the position of wave vertex
x
Fig. 3.26 Hyperbola as the approximate shape of a detached shock wave in a supersonic blunt-body
flow
3.8 Approximate Location of Detached Shocks
131
noindentFollowing the technique of Ref. [7] for supersonic detached shocks, we
use the assumptions:
• The shape of the detached shock is a hyperbola with the Mach wave as its asymptote
• The shape of the blunt body from the stagnation point to the sonic boundary does
not affect the location of the sonic point on the body
• The sonic point on the body has the same slope as θmax corresponding to the
maximum flow turning angle for the plane oblique shocks (or conical shocks) at
the freestream Mach number, M∞
• The sonic line is straight
• The average stagnation pressure downstream of the detached shock is represented
by the streamline that passes through the mass centroid of the streamtube bounded
by the stagnation and the sonic streamlines. This assumption is less critical in
transonic flow where total pressure change across a shock is of higher order effect
(see for example Ref. [4])
• The sonic line angle, η, is the average of θs and θmax where θs is the flow angle
downstream of the shock at its sonic boundary.
Figure 3.27 serves as the definition sketch for the estimation of detached shock
location following Ref. [7]. The intersection of the sonic line and the detached shock
is denoted by S whereas the coordinates of the sonic point on the body are x S B and
y S B , respectively. The angle of the sonic line with respect to the vertical direction
is η. Since the flow angle corresponding to the sonic point on the shock, θs , is close
to the maximum turning angle, θmax , the angle of the sonic line, is deemed to be
the average of the two angles, θs and θmax . The position of the mass centroid in a
streamtube of height ys , in 2-D plane flow, is defined as:
y
Streamline passing
through the mass
centroid in 3-D
σ
Sonic
streamtube
S
yS
M∞ > 1
θs
η≈
1
(θ s + θ max )
2
L= xSB –x0
θmax
η
SB
yc= 2y S /3
M=1
ySB
yc= yS /2
0
Streamline passing through
the mass centroid in 2-D
xSB
x0
x
L
Fig. 3.27 Definition sketch used in the calculation of the location of detached shock waves
132
3 Transonic Similarity Laws
yc ≡
1 ys
1 ρ∞ V∞ ys2 /2
ys
y dṁ =
y(ρ∞ V∞ ) dy =
=
ṁ
ṁ 0
ρ∞ V∞ ys
2
(3.188)
The mass centroid in a streamtube of radius ys in axisymmetric flow is given by:
yc ≡
1 ys
1 2πρ∞ V∞ ys2 /3
2ys
y dṁ =
y(2πρ∞ V∞ )y dy =
=
2
ṁ
ṁ 0
π ys ρ∞ V∞
3
(3.189)
These positions for the mass centroid are labeled in Fig. 3.27.
By applying the continuity equation to the sonic streamtube upstream of the
detached shock and the sonic flow through the sonic line downstream of the detached
shock, we establish the sonic flow area S − S B (normalizing with respect to the
reference length scale of the problem, namely, y S B in plane 2-D flows and reference
area scale, y S2 B in axisymmetric flows) and thus the length, L, where the vertex of
the detached shock is located upstream of the sonic boundary x S B . Step-by-step
calculation procedure based on the theoretical development of Ref. [7] is outlined in
Shapiro [9], and is not repeated here, for brevity. Once the position of the detached
shock is established, we may perform a momentum balance, in the streamwise, i.e.,
the x-direction, between the sonic streamtube upstream of the shock and the sonic
flow through the sonic line downstream of the shock. This results in an estimation
of nose pressure drag, upstream of the sonic line on the blunt body. Alternatively,
the momentum deficit in the x-direction between upstream and downstream of the
bow shock is equal to the pressure drag of the blunt nose, up to the sonic boundary.
Such calculations were performed by Ferri [2] and Moeckel [7], among others. The
momentum balance method as described here may also be applied to supersonic
inlets that operate in sub-critical mode, i.e., with external spillage (see for example,
Ref. [1]). The sub-critical mode corresponds to the operation of a supersonic inlet
where the normal (or terminal) shock is pushed outside the inlet and thus forms a
bow shock ahead of the inlet cowl lip. The momentum balance method yields inlet
cowl pressure drag in sub-critical operation, or alternatively, the spillage drag of a
supersonic inlet.
Now, we examine some results of the theoretical prediction of the detached shock
position in blunt-body supersonic and upper transonic flows and the experimental
data. Figure 3.28 (from Refs. [7, 9]) shows the detached shock position in supersonic
flow (i.e., Mach 1.7) in front of a cone, sphere and projectile.
First, we note that the shock position is well captured by the theory. Second,
we note that the location of the sonic line is reasonably well predicted, however,
the experimental data point to a curved sonic line, as compared to the straight line
assumption in the theory of Ref. [7]. Finally, we note that the location of the sonic
boundary on the three blunt bodies is nearly independent of the body shape upstream
of the sonic point. Note that the normalizing length scale is yS B , as discussed earlier.
3.8 Approximate Location of Detached Shocks
Fig. 3.28 Comparison
between the predicted
detached shock positions and
experimental data for
supersonic blunt-body
problems (from Ref. [7, 9])
133
3.6
M∞ = 1.7
Shock
3.2
2.8
Shock
2.4
y
M=1
}
2.0
Ladenburg
(Experimental)
Moeckel
(Theoretical)
ySB
1.6
Projectile
1.2
Sonic
Line
Sphere
0.8
Cone ( , )
Sphere ( , )
0.4
Projectile ( )
0
0
0.4
0.8
1.2
1.6
2.0
2.4
x-x0
ySB
Shock detachment distance upstream of plane and axisymmetric bodies over a
range of transonic and supersonic Mach numbers is shown in Fig. 3.29 [9].
In Fig. 3.29a we note that the detachment distance of bow shocks is well predicted
by the theory and the axially symmetric bodies, by virtue of 3-D relieving effect,
encounter detached shocks that are closer to their vertex (at the same Mach number as
a 2-D object), as expected. Also, the transonic range is fully explored, as the theory
and data are compared starting at M∞ ≈ 1.05. In Fig. 3.29b, the detached shock
standoff distance is shown for a sphere. We note that the theoretical prediction is still
reasonable, albeit under-predicting the detachment distance. The shock gets closer to
the body as freestream Mach number increases, as expected. In part, Fig. 3.29c shows
the comparison between the detached shock position ahead of cones (of varying
vertex angle) and at different Mach numbers. The prediction is reasonably well, but
still under-predicts the detachment distance, as in part (b) for spheres.
134
3 Transonic Similarity Laws
(b) 1.5
M∞
L’
d
(in.)
1/4
1/2
1
d
d
L’
1.0
(a) 18
Theoretical
16
Experimental
14
}
12
} Two-Dimensional
0.6
1.0
M∞
Axi-Symmetric
1.5
(c) 1.2
d
(in.)
1/4
9/16
3/4
9/16
3/4
9/16
1/4
1/2
1.1
L
ySB
10
Theoretical
1.0
8
L
6
Two-Dimentional
(Theoretical)
4
0.9
Theoretical
0.8
0.7
Axially Symmetric
(Theoretical)
2
0
1.0
d
0.6
1.9
L
d
Θ
(deg)
35
35
35
45
45
60
90
90
Θ
0.5
1.4
1.8
2.2 2.6
M∞
3.0
3.4
3.8
0.4
1.0
1.5
M∞
1.9
Fig. 3.29 Detached shock position ahead of a axisymmetric and planar objects, b spheres and c
cones (after [9])
3.9 Summary
Transonic flow is more a physical description of the flow than a mere Mach number
range near sonic. From the lower end, i.e., the subsonic side, transonic flow is entered
at the critical Mach number. On the upper end, i.e., the supersonic side, transonic
flow persists until the profile is submerged entirely in supersonic flow. It is indeed
the mixture of subsonic and supersonic flow with the appearance of shocks on the
body that characterizes and dominates transonic aerodynamics. We learned that this
flow regime is inherently nonlinear and its analysis does not lend itself to full linearization of the perturbation potential equation. Therefore, the solution techniques
are numerical in nature and the classical theoretical developments lead to a group of
transonic similarity parameters and similarity laws. These laws allow for extrapolation of experimental data at different transonic flow conditions on affinely-related
bodies.
Problems
3.1 The peak suction pressure on an airfoil at 5◦ angle of attack is C p, min = −1.25
in low-speed flow. Calculate the pressure coefficient at the same point on the airfoil, when the freestream Mach number is 0.7 using Prandtl-Glauert compressibility
correction.
3.9 Summary
135
3.2 The lift curve slope, clα , of an airfoil in incompressible flow is 1.95π rad−1 .
Calculate the lift curve slope for the same airfoil at Mach 0.75, using Prandtl-Glauert
correction.
3.3 The lift coefficient of an airfoil at 5◦ angle-of-attack is cl = 0.5 in low-speed,
i.e., incompressible flow. Calculate the lift coefficient of the same airfoil, at the same
angle-of-attack and at Mach 0.75 using Prandtl-Glauert compressibility correction.
Also, calculate the percent change in circulation caused by the compressibility effects
at M∞ = 0.75.
3.4 Calculate the critical pressure coefficient, C p, crit for a range of Mach numbers
starting at 0 and extending to Mach 1. Graph the critical pressure coefficient versus
Mach number.
3.5 An airfoil has a C p, min = −1.01 in low speed. Extend this result to the compressible domain, using three compressibility corrections:
(a) Prandtl-Glauert compressibility correction
(b) Laitone compressibility correction, and
(c) Karman-Tsien compressibility correction.
Also, use the graphical technique to determine the critical Mach number for this
profile and flow condition.
3.6 In a low-speed WT test, i.e., M∞ = 0, we measured C p, min on an airfoil.
M∞
Cp,min
γ =1.4
(a) At what free-stream Mach number, M∞ , will the minimum pressure coefficient
double?
(b) Calculate C p, crit , if the Mcrit = 0.72
(c) Calculate the lift curve slope at Mcrit if the low-speed lift curve slope is a0 =
2π rad−1 .
3.7 A thin airfoil is in a Mach 0.35 flow and develops peak suction, C p, min of −2.4
and a lift coefficient, cl of 0.78.
Calculate:
Cl.q∞ .c
M∞
Cp, min
136
3 Transonic Similarity Laws
(a) the lift coefficient at M∞ = 0.65
(b) the C p, min at M∞ = 0.1
(c) the ratio of lift curve slopes at Mach 0.65 and Mach zero, i.e., the incompressible
limit
3.8 Consider a subsonic compressible flow over a wavy wall of infinite extent. The
wall amplitude is h and its wavelength is . The wavy wall may be described by:
yw = h cos
2πx
Calculate:
(a) the streamwise velocity at (2, h)
(b) the velocity vector at (, 2h)
(c) the minimum and maximum pressure coefficient, C p, min and C p, max on the wall
3.9 A two-dimensional airfoil is in subsonic flight. The flight Mach number is M∞ =
0.5 and C p, min = −3.5.
Cp, min = - 3.5
M∞=0.5
Calculate:
(a) the Mach number, Mmax , on the airfoil, at C p, min
(b) the critical C p, crit
(c) stagnation point C p , C p, stag
(d) the incompressible value of (C p0 , min )
3.10 A thin airfoil has a minimum pressure coefficient of C p, min = −0.5 at low
speed, i.e., incompressible limit.
y
M∞
γ =1.4
Cp, min
x
Calculate:
(a) C p, min when M∞ = 0.5
(b) C p, stag when M∞ = 0.5
(c) M∞ when C p, min = −1.2
(d) the velocity vector, V/V∞ on the airfoil where C p = −0.60 and the airfoil slope
is θ = −15◦
3.9 Summary
137
3.11 An airfoil experiences a minimum pressure coefficient of C p = −1.0 when
flying at M∞ = 0.7, as shown.
Cp, min
Calculate:
(a) Mach number at the point of C p, min
(b) C p, max at M∞ = 0.7
(c) C p, max at M∞ = 0
(d) C p, min at M∞ = 0
3.12 The critical pressure coefficient on an airfoil at an angle of attack, α, is C p, crit =
1.42 in a flow where V∞ = 200 m/s and T∞ = 22 ◦ C.
Critical
point
V∞
T∞
Calculate:
(a) the critical Mach number
(b) the incompressible C p, min
(c) speed of sound at the critical point
(d) the ratio pcrit / p∞
Assume γ = 1.4 and R = 287 J/kg · K.
3.13 Consider a subsonic flow over a wavy wall as shown. The perturbation velocity
potential is known:
φ(x, y) = hV∞ e−π y sin(2πx)
y
v
u
M ∞= 3
2
a∞ =300 m/s
y = h cos(2πx)
h <<1
x
0
1/4
1/2
3/4
1
138
3 Transonic Similarity Laws
(a) show that it satisfies the governing differential equation
(b) calculate u (m/s) at (0, h)
(c) calculate C p at (1/2, 0) assume h = 0.05.
3.14 For a flat plate in a supersonic flow at an angle of attack, α, apply the linear
theory to calculate the angle of attack that corresponds to L /D = 2.
3.15 For a wavy wall in supersonic flow, the ratio of wall height-to-wave length,
h/ = 0.05 (see Fig. 3.8 for a definition sketch of wavy wall).
Calculate:
(a) the wave drag coefficient (for one wavelength) at Mach 2.0
(b) the wall height-to-wavelength ratio, h/ that gives the same wave drag coefficient
at Mach 3.
3.16 Using the principle of superposition in the linear theory, express the drag of
a 5 % thick, symmetrical double-diamond airfoil as the sum of its friction drag,
thickness drag and wave drag due to angle of attack (on a flat plate),
√ similar to
Example 3.10. To develop the drag polar for this airfoil, assume M∞ = 5 and vary
the angle of attack between −2◦ and 12◦ . For a friction drag coefficient of 0.005,
calculate and plot the drag polar for this airfoil.
√
3.17 Consider a slender cone of semi-vertex angle δ = 10◦ in Mach M∞ = 2
flow. Using linear theory, estimate the pressure coefficient on the cone, C p, cone .
√
3.18 Consider a slender cone of semi-vertex angle δ = 5◦ in Mach M∞ = 5
flow. First show that this cone has the same similarity parameter, λδ, as the cone
described in Problem 3.17. Next, use the linear theory to calculate the ratio of pressure
coefficients on the two cones (i.e., the 5◦ and 10◦ cones).
3.19 Consider an airfoil with 12 % thickness-to-chord ratio in a transonic flow with
M∞ = 0.85 at 2◦ angle of attack. The peak suction pressure for this airfoil is
C p, min = −0.35. Apply the transonic similarity rule to an affinely-similar airfoil
in Mach 0.90 to calculate the thickness ratio and the corresponding peak suction
pressure [Hint: keep similarity parameter, K , constant between the affinely-related
airfoils]. Also, calculate critical pressure coefficient, C p, crit corresponding to M∞ =
0.85 and M∞ = 0.90.
3.20 An airfoil is in a freestream Mach number, M∞ = 0.72. A sonic bubble appears
on its suction surface that terminates in a normal shock. The pressure coefficient
immediately upstream of the shock is C ps = −1.22.
3.9 Summary
139
Calculate:
(a) the pressure coefficient at the stagnation point
(b) the shock Mach number, Ms
(c) the pressure coefficient downstream of the shock
3.21 The pressure coefficient, C p , is measured at two locations on the surface of a
bi-convex airfoil, as shown.
y
CpA =0.2
CpB =0
M ∞=2.0
x
Calculate:
(a) p A / p B
(b) Ratio of freestream dynamic-to-static pressure, q∞ / p∞
3.22 A 12 % thick airfoil experiences C p, min, 1 = −0.75 at M∞1 = 0.85 at 2◦
angle of attack (i.e., α1 = 2◦ ). Using transonic similarity rule, we wish to relate
M∞2 , α2 and C p, min, 2 on an affinely-related airfoil that is 6 % thick, to airfoil 1.
Therefore, calculate M∞2 , α2 and corresponding C p, min, 2 for the 6 % thick airfoil,
as shown.
Cp, min, 1= - 0.75
M ∞1=0.85
α 1=2o
t/c =12%
Cp, min, 2
t/c=6%
α2
M∞2
3.23 A 2-D body of 5 % thickness ratio, i.e., τ1 = 0.05 is in transonic flow with the
flow Mach number of M1 = 0.90. The pressure coefficient at point A on this body
is C p1= 0.12. Now consider an affinely-related body that has a thickness ratio τ and
is in a Mach 0.95 flow, i.e., M2 = 0.95.
Calculate:
(a) the transonic similarity parameter, K
(b) the thickness ratio of the second (affinely-related) body, τ2
(c) the pressure coefficient, C p2 , on the second body that corresponds to point A on
the first body
M1
A
M2
A
140
3 Transonic Similarity Laws
3.24 Use the transonic similarity rule on wave drag coefficients (on two-dimensional bodies) and compare the wave drag coefficients of two affinely-related airfoils with 5 and 10 % thickness ratios, respectively. Assume the transonic similarity
parameter, K , remains constant between the two airfoils.
3.25 The lift coefficient of a 5 %-thick airfoil at the angle of attack of 2◦ in Mach
M∞ = 1.12 air stream (γ = 1.4) is cl = 0.10. Apply transonic similarity theory to
predict the lift coefficient of an affinely similar airfoil that is 10 % thick in transonic
flight through air (with γ = 1.4)?
3.26 A thin symmetrical half-diamond airfoil is in supersonic flow, as shown. Using
linearized supersonic flow theory; we are allowed to breakdown the wave drag problem of the airfoil into a separate thickness problem and a separate angle-of-attack
problem (on a flat plate). The lift is then purely due to the angle of attack on a flat
plate. Based on this approach to linearized supersonic aerodynamics, calculate:
(a) Airfoil wave drag coefficient due to thickness, cd, th
(b) Airfoil wave drag coefficient due to angle-of-attack, cd, Ao A
(c) Airfoil lift coefficient, cl
t/c =0.10
θ nose
α = 6o
1
2
3
c
M∞ = 5
3.27 An airfoil in Mach 0.80 flow at 2◦ AoA experiences the pressure distribution,
as shown below (not to scale). Calculate:
(a) Maximum Mach number on the airfoil, Mmax
(b) Pressure coefficient at the stagnation point, C p, stag
(c) Critical pressure coefficient, C p, crit
(d) Shock static pressure ratio on the suction surface
(e) does the shock separate the BL? And if yes what is the extent of the separated
bubble? [Hint: look for the footprint of separation]
(-)
Cp, min = -1.2
Cp
Cp, crit
1.0
x/c
(+)
0.35
Cp, stag
M >1
0.70
T.E.
3.9 Summary
141
3.28 For a family of slender axisymmetric bodies (that are affinely-related) in Mach
1 flow, use transonic similarity rule to compare their wave drag coefficients as a
function of their base diameters while keeping their length constant.
3.29 Consider a finite wing of aspect ratio A R1 = 8 and thickness ratio τ1 = 0.10
is in transonic flight at Mach 0.9 (with γ = 1.4) at an angle of attack of 6◦ . First,
calculate the transonic similarity parameter, K , for this wing. Next, consider an
affinely-transformed wing of 5 % thickness, (i.e., τ2 = 0.05), with γ = 1.4 and the
same similarity parameter, K 2 = K 1 . Using transonic similarity laws, calculate:
(a) The flight Mach number of the second wing, M∞2
(b) the corresponding angle of attack on the second wing, α2
(c) the aspect ratio of the second wing, A R2
(d) the ratio of wing lift coefficients, C L2 /C L1
(e) the ratio of the wing wave drag coefficients, C D2 /C D1
(f) the ratio of wing pitching moment coefficients, C M2 /C M1
3.30 A detached shock is formed ahead of a cylinder in a Mach 1.2 flow. Estimate
the location of the sonic boundary on the cylinder. Assume γ = 1.4.
3.31 A detached shock is formed ahead of a sphere in a Mach 1.2 flow. Estimate the
location of the sonic boundary on the sphere. Assume γ = 1.4.
3.32 Calculate the coordinate, y S /y S B , of the sonic point, S, on a (3-D) hyperbolic
detached shock ahead of a sphere. The sonic point on the shock, S, is where the
flow emerges from the shock at sonic speed. The freestream Mach number is 2.0
and γ = 1.4 [see Fig. 3.27 for definition sketch]. Also, calculate the flow angle θ S
downstream of the sonic point on the shock.
3.33 A detached shock is formed ahead of a 2-D blunt object in a Mach 1.6 flow.
Assuming the shape of the detached shock is described by a hyperbola; calculate
the shock angle, σc on the streamline passing through the mass centroid of the sonic
stream tube. Assume γ = 1.4.
Streamline passing y
through the mass
centroid in 2 -D flow
yS
σ
S
SB
θs M=1
θmax
yS /2
ySB
x0
xSB
x
3.34 Consider a transonic flow where the freestream Mach number is:
M∞ = 1 + ε where ε 1
142
3 Transonic Similarity Laws
Assuming a detached shock is formed, show that the Mach number downstream
of the normal shock is:
M2 ≈ 1 − ε
M∞ = 1 + ε
M2 ≈ 1 - ε
3.35 To demonstrate that weak shocks that appear in transonic flow are nearly isentropic, we choose the freestream Mach number to a normal shock to be:
M∞ = 1 + ε where ε 1
Use the normal shock relations to show that non-dimensional entropy rise across
the normal shock, Δs/R is of order of ε3 , i.e.,
Δs
∼ O(ε3 )
R
3.36 Concept of rotationality of the flow downstream of the curved shocks is derived
from Croccos theorem (for steady, inviscid flows), i.e.,
−V × (∇ × V) = ∇h t − T ∇s
where h t is the flow stagnation enthalpy. However, stagnation enthalpy remains
constant in gas dynamic flows involving shocks (of any shape) therefore Croccos
theorem reduces to:
V × (∇ × V) = T ∇s
Use the result of Problem 3.34 to show that the vorticity field downstream of curved
shocks in transonic flow is negligibly weak and thus the flow may be modeled as
irrotational.
3.37 A 2-D wedge of nose angle δ is in a supersonic flow. Apply the linearized
theory to the wedge and show that the expression for pressure coefficient may be put
in the transonic similarity form:
γ+1 2/3
1/3
δ
[(γ + 1) /2] C p
2
=
2
2 −1
δ 2/3
M∞
3.9 Summary
143
3.38 Show that from linear theory in supersonic flow, a thin symmetrical bi-convex
airfoil creates wave drag according to:
Cd, w =
16
2 −1
3 M∞
t
c
2
where t/c is the airfoil thickness ratio.
3.39 Prove that for a given thickness-to-chord ratio, t/c, and the location of the
maximum thickness, a symmetrical diamond airfoil creates the lowest wave drag in
supersonic flow, i.e.,
4
(Cd, w )diamond = (Cd, w )min =
2 −1
M∞
t
= const.
for
c
t
c
2
The above expression represents the minimum wave drag. You may use linear theory
in your proof.
Hint: Wave drag coefficient for a general thin airfoil shape using linear theory is
derived to be (see for example Liepmann-Roshko, or Shapiro):
Cd, w =
4
2 −1
M∞
A + B + α02
where
A≡
1c
c 0
B≡
1c 2
α (x) dx
c 0 c
dh
dx
2
dx
α0 is the geometric angle of attack, h(x) is the symmetrical thickness distribution
and αc (x) represents the angle of attack due to camber, i.e., the slope of the mean
camber line. Therefore the integrals A and B are the chordwise averages of the
square of parameters (dh/dx) and αc (x). For a symmetrical airfoil, the contribution
of camber to wave drag is identically zero, otherwise positive. Therefore, you have
to prove that A is a minimum for diamond airfoil of a given thickness-to-chord ratio
and the location of maximum thickness.
144
3 Transonic Similarity Laws
References
1. Farokhi, S.: Aircraft Propulsion, 2nd edn. Wiley, Chichester, UK (2014)
2. Ferri, A.: Elements of Aerodynamics of Supersonic Flows. Macmillan Company, New York
(1949)
3. Hoak, D.E., Anderson, R., Goss, C.R.: USAF Stability and Control Datcom, Air Force Wright
Aeronautical Laboratories. Wright Patterson Air Force Base, Ohio (1978)
4. Laitone, E., Pardee, O.: Location of Detached Shock Wave in Front of a Body Moving at
Supersonic Speeds. NACA RM A7B10 (1947)
5. Liepmann, H.W., Roshko, A.: Elements of Gas Dynamics. Wiley, New York (1957)
6. McDevitt, J.B.: A correlation by means of transonic similarity rules of experimentally determined characteristics of a series of symmetrical and cambered wings of rectangular plan form.
NACA-TR-1253 (1955)
7. Moeckel, W.E.: Approximate Method for Predicting Form and Location of Detached Shock
Waves Ahead of Plane or Axially Symmetric Bodies. NACA TN 1921 (1949)
8. Raymer, D.P.: Aircraft Design: A Conceptual Approach. AIAA, Washington (1989)
9. Shapiro, A.H.: The Dynamics and Thermodynamics of Compressible Fluid Flow. Ronald Press,
New York (1953)
10. Stack, J., Lindsey, W.F., Littell, R.E.: The Compressibility Burble and the Effect of Compressibility on Pressures and Forces Acting on an Airfoil. NACA TR 646 (1939)
11. Stanton, T.E.: A high speed wind channel for tests on aerofoils, Technical Report for the
Aeronautical Research Committee for the year 1927–1928, T&M 1130 (1928)
12. Tinoco, E.N.: CFD applications to complex configurations: a survey. In: Henne, P. (ed.) Applied
Computational Aerodynamics, Chap. 15. AIAA, Washington (1990)
Chapter 4
Shock-Expansion Theory
Abstract This chapter develops the foundation of inviscid nonlinear shockexpansion theory on pointed bodies in supersonic flows. Fundamental equations
of shock and expansion waves are derived and discussed. To extend the analysis
to three-dimensional bodies, conical shock theory is derived. Examples from diamond and bi-convex airfoils are used to highlight the shock-expansion techniques in
developing lift and wave drag on two-dimensional pointed bodies. Drag polar and
the aerodynamic performance parameter, (L/D)max are discussed. Axisymmetric
and slender bodies are the basis of 3-D shock-expansion application. Approximate
methods based on the method of local cones is introduced to estimate the effect
of 3-D expansion, using conical shocks associated with smaller nose angle. The
problem of asymmetrical flows is analyzed through the method of local cones. The
shock-expansion theory is extended to the upper transonic regime. Here, the principle
of stationary local Mach numbers near sonic freestream flow is demonstrated and
applied to wave drag of axisymmetric bodies at sonic speed. This chapter contains 9
examples and 42 practice problems at the end of the chapter.
4.1 Introduction
Supersonic flow over bodies involves the creation of waves of compression and
expansion to achieve flow turning. Blunt bodies, for example, create a detached, or
bow shock that forms ahead of the blunt body and the flow downstream of the shock is
a mixed subsonic-supersonic flow. Such mixed flows are not amenable to analytical
solutions. However, there is a class of problems in inviscid supersonic flow over
pointed bodies that lend itself to exact analytical solution (known as shock-expansion
theory). This class of inviscid supersonic flow problems requires the oblique shock
(either plane or conical) to be attached to the body in flight at the leading edge/nose.
This necessitates first the body to be pointed and second the turning imposed by the
body at the leading edge not to exceed the maximum turning angle corresponding
to a plane oblique (or conical) shock at the designated flight Mach number. Given
these stipulations, shock-expansion theory may be applied to any inviscid flow above
sonic, which includes the upper transonic regime.
© Springer Science+Business Media Dordrecht 2015
R. Vos and S. Farokhi, Introduction to Transonic Aerodynamics,
Fluid Mechanics and Its Applications 110, DOI 10.1007/978-94-017-9747-4_4
145
146
4 Shock-Expansion Theory
Our presentation of shock-expansion theory in this chapter is based on inviscid
fluid assumption. As the fluid viscosity is ignored, the no-slip boundary condition is
lost and thus the boundary layers are not formed. Consequently, the effect of boundary
layer separation on aerodynamic forces and moments is clearly beyond the scope of
inviscid flow theory, such as shock-expansion theory. In transonic flows over airfoils,
wings and bodies the local Mach number exceeds the sonic value, therefore shocks
can be formed. These shocks interact with the low momentum, low energy zone next
to the body, i.e., the boundary layer, and may cause boundary layer separation. This
subject is further elaborated in Chap. 6.
Before we get started with the applications of shock-expansion theory, we briefly
review the plane oblique and conical shocks as well as the Prandtl-Meyer expansion
wave theory. Waves of infinitesimal strength in supersonic flow are called Mach
waves, which are capable of turning the flow only infinitesimally. The angle of the
Mach wave with respect to the flow is called Mach angle and is given the symbol, μ.
Since the wave strength is established by the normal component of the flow upstream
of the wave, the normal component of a supersonic flow to a Mach wave is therefore
sonic. This geometric property of Mach wave is shown in a right triangle in Fig. 4.1.
From the definition sketch of Fig. 4.1, we deduce:
sin μ =
1
M
tan μ = √
(4.1a)
1
(4.1b)
M2 − 1
1
1
= tan−1 √
μ = sin−1
M
M2 − 1
(4.1c)
For example in a Mach-2 flow, the Mach angle is sin−1 21 , which is 30◦ . In sonic
flow at Mach 1, the Mach angle is 90◦ , i.e., normal to the flow and for Mach 1.1, the
Mach angle is reduced to 65.38◦ with respect to the local flow direction. However a
single Mach wave due to its infinitesimal strength is unable to provide finite turning
of the supersonic flow that is needed to make room for a body (of finite thickness) in
flight. Consequently, finite compression and expansion waves are created to turn the
supersonic flow. The coalescence of compression Mach waves creates shocks and
Fig. 4.1 Definition sketch of
the Mach wave and the Mach
angle, μ, in supersonic flow
Mach wave
M>1
µ
M=1
M ² –1
4.1 Introduction
(a)
147
(b)
Oblique
Shock
Compression
Mach waves
Streamline
Expansion
Mach waves
2
µ2
1
2
1
µ2
M1 > 1
M2 > 1
M2 < M 1
M2 > 1
M2 > M1
M1 > 1 µ
1
µ1
Fig. 4.2 Waves on concave and convex surfaces in supersonic flow. a Waves generated on a concave
surface. b Waves generated on a convex surface
finite expansion waves are created through many expansion Mach waves, known as
the Prandtl-Meyer expansion waves. Figure 4.2a shows the coalescence of compression Mach waves on a concave surface that culminates in the shock formation and
Fig. 4.2b shows the spreading of the expansion Mach waves on a convex surface that
create finite turning in the flow at supersonic speeds.
We note that the compression Mach waves on a concave surface coalesce, as
shown in Fig. 4.2a, and form an oblique shock wave whereas the expansion Mach
waves on a convex surface (Fig. 4.2b) spread/diverge and thus never form a surface of
discontinuity of rarefaction. The coalescence of compression Mach waves that form
the shock thus creates a region of very steep pressure and temperature gradients in the
gas where normal viscous stress and thermal conductivity of the gas play an important
role in establishing the equilibrium thickness for the shock wave. It is therefore the
dominance of viscous effects within the shock that creates an entropy increase and
thus non-isentropic flow conditions across the shock. We also know from kinetic
theory that the thickness of the shock is proportional to the mean-free path of the gas
at reference sonic temperature condition (e.g., see classical textbooks of Shapiro [6],
Liepmann-Roshko [5], among others). The steep gradients in pressure, temperature,
density, normal component of velocity, entropy and stagnation pressure across the
shock are depicted graphically in Fig. 4.3.
The structure of a normal shock in standard sea level pressure and temperature conditions with upstream Mach number of 1.1 is shown in Fig. 4.4 (from [6]). Subscripts
x and y denote the flow conditions upstream and downstream of the shock, respectively. The streamwise coordinate x shows the region where the steep gradients exist.
Therefore, noting that the x-axis has a 10−5 inch scale, we appreciate the thickness,
or rather thinness of the shock in standard air. To put these numbers in perspective,
we recall that the mean-free path in air at standard conditions is ∼3.7 × 10−5 cm
(or ∼1.46 × 10−5 in.), which makes the thickness of the shock between 3–5 times
the mean-free path in air at standard conditions. The static pressure and temperature
across the normal shock are shown in ratios and the normal velocity component in
units of ft/s. The density ratio is not shown in the graph, but it rises as well as the
gas is compressed across the shock. It is thus the product of density and temperature
148
4 Shock-Expansion Theory
(a)
u1
(b)
(c)
1
T2
2
2
2
1
1
u2
p1
T1
(d)
(e)
2
2
1
p2
(f)
s2
pt
1
2
1
1
1
s1
2
pt
2
Fig. 4.3 Schematic drawing of flow gradients across a shock, with shock thickness, δ (not-to-scale).
a Normal component of velocity. b Static pressure. c Static temperature. d Density. e Entropy.
f Total pressure
Fig. 4.4 Variation of flow properties within a normal shock (from [6]). Note that x indicates the
conditions ahead of the shock wave, while y indicates the conditions behind the shock wave
4.1 Introduction
149
Fig. 4.5 Normal shock
thickness at standard
atmospheric pressure and
temperature, as a function of
shock strength parameter,
Mx − 1, in transonic flow
(from [6])
ratios following the perfect gas law that produces the static pressure ratio across the
shock, as shown in Fig. 4.4. To demonstrate the variation of normal shock thickness
with Mach number at standard pressure and temperature, we examine Fig. 4.5 (from
[6]). The normal shock thickness on a log-log scale indicates an exponential drop in
shock thickness as shock strength or Mach number increases. As a result of shock
thinning with increasing Mach number (at a given altitude), the gradients increase
and thus total pressure losses mount.
The normal shock relations are derived through the application of the great conservation laws of nature, namely conservation of mass, linear momentum and the
energy across the shock. The conservation of mass applied to a normal shock with
stations 1 and 2 identifying up- and downstream of the normal shock, gives birth to
the first constant of motion, related to normal shock flows, namely
ρ1 u 1 = ρ2 u 2 = const.
(4.2)
The application of the Newton’s second law of motion to a control volume shown
in Fig. 4.6, gives birth to the second constant of motion for a flow through a normal
shock, namely
(4.3)
p1 + ρ1 u 21 = p2 + ρ2 u 22 = const.
Since the flow across a shock is adiabatic, i.e., there is no heat exchange between
an external source and the fluid in the control volume, and since the fluid within the
control volume undergoes no chemical reaction, the law of conservation of energy
for non-chemically reacting, steady flows across a shock produces the third constant
of motion, namely
150
4 Shock-Expansion Theory
Fig. 4.6 Definition sketch of
the control volume used in
the analysis of normal
shocks in steady flow
Normal shock
p2
T2
p1
T1
1
2
1
u1
s1
pt
1
ht
Control
Volume
u2
s2
pt
2
ht
2
2
1
h t1 = h t2 = const.
(4.4)
Here, h t is the stagnation or total enthalpy in the flow, as measured by a stationary
observer with respect to the shock. In hypersonic flow where molecular collisions
are highly energetic, the law of conservation of energy is no longer represented
by Eq. (4.4) as the collisions may produce dissociation and even ionization in air.
There are a few other constants of motion that emerge directly from the conservation
of energy, namely stagnation speed of sound, at , which is based on the stagnation
temperature of the gas, remains constant as well for calorically-prefect gas where
cp = constant, i.e.,
(4.5)
at1 = at2 = at = const.
Since the sonic state may be reached adiabatically in a gas, the total enthalpy of the
gas at sonic state that is reached adiabatically and the constant stagnation enthalpy
of the flow are equal, namely,
h∗ +
a ∗2
a ∗2
a ∗2
a ∗2
= cpT ∗ +
=
+
=
2
2
γ−1
2
γ+1
γ−1
a ∗2
= h t = const. (4.6)
2
This yields the fifth constant of motion, namely
a1∗ = a2∗ = a ∗ = const.
(4.7)
By introducing the concept of maximum velocity that is achieved through expansion
of a gas to zero static enthalpy, or inversely, by converting the entire stagnation
enthalpy of the gas to kinetic energy, we get the sixth constant of motion, namely
Vmax ≡
2h t = const.
(4.8)
Any of the constant speeds, at , a ∗ or Vmax can serve as our reference speed in gas
dynamic calculations dealing with normal (or oblique) shocks. For example a ∗ has
served as the reference speed in shock polar and hodograph solutions in gas dynamics
and Vmax is used in Taylor-Maccoll formulation of conical shocks, as we will briefly
review. Also a ∗ is used in the definition of a characteristic Mach number, M ∗ using
the following definition:
4.1 Introduction
151
M∗ ≡
u
a∗
(4.9)
The characteristic Mach number, is related to the Mach number through the law of
conservation of energy equation, namely,
(γ + 1) M 2
2 + (γ − 1) M 2
(4.10a)
2
(γ + 1) /M ∗2 − (γ − 1)
(4.10b)
M ∗2 =
Or inversely,
M2 =
Note that subsonic flows with M < 1, have M ∗ < 1, sonic flow with M = 1, has
M ∗ = 1 and supersonic flow with M > 1 has M ∗ > 1. The added advantage of M ∗
over M is that M ∗ is bounded whereas M may tend to infinity. This is seen in the
following limit:
γ+1
∗
as
M →∞
(4.10c)
M →
γ−1
In summary, primary constants of motion for a normal shock are:
ρu = const.
p + ρu 2 = const.
h t = const.
at = const.
a ∗ = const.
Vmax = const.
The simultaneous solution of the conservation laws yields the celebrated Prandtl’s
relation relating the product of gas speeds across a normal shock to the square of the
(constant) speed of sound at sonic state, a ∗ , following
u 1 u 2 = a ∗2
(4.11)
The immediate result of the Prandtl’s relation is the inverse relationship between the
upstream and downstream characteristic Mach numbers, namely
M1∗ =
1
1
or M2∗ = ∗
∗
M2
M1
(4.12)
The emergence of subsonic flow downstream of a normal shock in supersonic flow
is directly based on the above inverse relation between M2∗ and M1∗ . In terms of the
152
4 Shock-Expansion Theory
upstream Mach number, M, we may express the “jump” conditions across a normal
shock as follows:
(γ + 1) M12
ρ2
u1
=
=
ρ1
u2
2 + (γ − 1) M12
p2
2γ 2
M1 − 1
=1+
p1
γ+1
T2
2γ 2
M1 − 1
= 1+
T1
γ+1
(4.13)
(4.14)
2 + (γ − 1) M12
(4.15)
(γ + 1) M12
γ
2γ 2
s2 − s1
M1 − 1
=
ln 1 +
R
γ−1
γ+1
2γ 2
− ln 1 +
M1 − 1
γ+1
Δs
pt2
2γ 2
M1 − 1
= e− R = 1 +
pt1
γ+1
2 + (γ − 1) M12
(γ + 1) M12
(4.16)
⎡
⎢
⎢
⎣
2 + (γ − 1)
2+(γ−1)M12
2γ M12 −(γ−1)
2 + (γ − 1) M12
⎤ γ
γ−1
⎥
⎥
⎦
(4.17)
In addition, downstream Mach number is related to upstream Mach number through
the use of Prandtl’s relation, namely
M22 =
1 + [(γ − 1) /2] M12
γ M12 − (γ − 1) /2
(4.18)
The graphical depiction of the flow properties across a normal shock is very instructive and the levels and scales of these parameters may be used as an instructional
tool. For example in Fig. 4.7 we have plotted the static pressure, temperature and
density ratios across a normal shock for a calorically-perfect gas with the ratio of
specific heats, γ = 1.4. Note that the upstream Mach number range is intentionally
limited to the supersonic range of up to Mach 5.
We note that the level of pressure jump is higher than both density and temperature
ratios, since in fact it is the product of the two jumps (in density and temperature)
that create the pressure jump. In the transonic range, we note that the behavior of
these jumps in static pressure, temperature and density is nearly linear. Limiting the
Mach number range to 1.3 and re-plotting the ratios of pressure, temperature and
density in Fig. 4.8 we can show this linear behavior.
The Mach number downstream of a normal shock (for γ = 1.4) is shown in
Fig. 4.9. We note that the initial drop in downstream Mach number is rapid and it
gradually plateaus with higher Mach numbers. In fact, there is a limiting downstream
4.1 Introduction
153
31.00
26.00
p2/p1
21.00
16.00
2
/ 1
T2/T1
11.00
6.00
1.00
1
1.5
2
2.5
3
3.5
4
4.5
5
1.2
1.25
Upstream Mach Number, M1
Fig. 4.7 Normal shock parameters for a calorically-perfect gas, γ = 1.4
Fig. 4.8 The linear behavior
of normal shock (jump)
properties in transonic range
1.80
p2/p1
1.70
1.60
1.50
2
1.40
/ 1
T2/T1
1.30
1.20
1.10
1.00
1
1.05
1.1
1.15
1.3
Upstream Mach Number, M1
Mach number that can be deduced from expression (4.18) in the limit of upstream
Mach number tending to infinity, namely
M2 →
γ−1
as M1 → ∞
2γ
(4.19)
The behavior of Mach number downstream of a normal shock in transonic range
is also linear, as expected (see Fig. 4.10). The normal shock relations up to a Mach
number of 10 are tabulated in Appendix B.
Considering a moving source (i.e., a small disturbance) in a compressible fluid,
we learned that the source creates spherical acoustic waves that propagate at sonic
speed in the medium. The case of a source moving at sonic speed, a, and the source
moving faster than the sonic speed, V > a, i.e., moving supersonically, are shown
154
1.000
Downstream Mach Number, M2 (~)
Fig. 4.9 Normal shock
parameters for a
calorically-perfect gas,
γ = 1.4
4 Shock-Expansion Theory
0.900
0.800
M2
0.700
0.600
0.500
0.400
1
5
1.05
1.3
1.000
Downstream
Mach Number, M2(~)
Fig. 4.10 Downstream
Mach number in normal
shock flows in transonic
range (as compared to a
straight line)
2
3
4
Upstream Mach Number, M1 (~)
0.950
0.900
0.850
0.800
0.750
1
1.1
1.15
1.2
1.25
Upstream Mach Number, M1 (~)
in Fig. 4.11. The labels for the zones of action and silence signify the space where
disturbance is felt and the space corresponding to “forbidden” signal propagation, or
zone of silence. The Mach cone downstream of the moving source thus represents
the zone of action and the vast zone upstream of the Mach cone represents the zone
of silence.
The motion of finite disturbances at supersonic speeds, as in bodies with finite
thickness moving faster than the speed of sound, involves the creation of oblique
shocks. The analysis of plane oblique shocks is based on the application of conservation principles in the normal and tangential directions to the shock. The schematic
drawing of an oblique shock in a supersonic flow, shown in Fig. 4.12, identifies the
wave angle β, the flow turning angle, θ, and the normal and tangential components
of flow with respect to the oblique shock wave.
4.1 Introduction
155
(a)
(b)
Mach Wave
µ
a. t
Mach Wave
V=a
V>a
V. t
Zone of
Silence
Zone of Action
Zone of Silence
Zone of Action
Fig. 4.11 Delineation of zones of action and silence for a moving source (of small disturbance) in
a compressible medium. a Source moving at the speed of sound. b Source moving at supersonic
speed, V > a
(a)
O.S.
Mn1 > 1
Mt1
Mn1
O.S.
w1
M2
M1
(b)
u1
V2
V1
w2
Mt2
Mn2
1
2
u2
1
Mn2 < 1
2
w2 = w1
Fig. 4.12 An oblique shock with its normal and tangential flow components. a The components
of flow Mach number across an oblique shock. b The structure of flow velocities across an oblique
shock
The conservation of tangential momentum along the plane oblique or conical
shock (i.e., a straight shock) reveals that the tangential velocity remains constant
across the (straight) shock, namely
w2 = w1
(4.20)
However, as the static temperature and thus the speed of sound are both raised across
the shock, the tangential Mach numbers up- and downstream of the oblique shock
are unequal. Examining part (a) and part (b) of Fig. 4.12 shows the constancy of w,
but a drop in Mt across the oblique shock, while maintaining the same shock wave
angle, β and the deflection angle, θ in the two graphs. Therefore oblique shocks
enjoy an additional constant of motion in comparison to normal shocks and that is
the tangential velocity. In summary the primary constants of motion for an oblique
shock are as follows:
156
4 Shock-Expansion Theory
w = const.
ρu = const.
p + ρu 2 = const.
h t = const.
at = const.
a ∗ = const.
Vmax = const.
Since the normal component of conservation equations are the same for the
oblique and normal shock, all the jump conditions across an oblique shock are thus
solely determined from the normal component of upstream Mach number, namely,
M1 sin β. We will thus replace the upstream Mach number, M1 , in normal shock equations (4.13)–(4.17) by M1 sin β to establish the jump conditions across an oblique
shock of angle β. These are summarized as follows:
(γ + 1) M12 sin2 β
ρ2
u1
=
=
ρ1
u2
2 + (γ − 1) M12 sin2 β
2γ 2 2
p2
M1 sin β − 1
=1+
p1
γ+1
T2
2γ 2 2
M1 sin β − 1
= 1+
T1
γ+1
(4.21)
(4.22)
2 + (γ − 1) M12 sin2 β
(γ + 1) M12 sin2 β
γ
2γ 2 2
s2 − s 1
M1 sin β − 1
=
ln 1 +
R
γ−1
γ+1
2γ 2 2
M1 sin β − 1
− ln 1 +
γ+1
Δs
2γ 2 2
pt2
M1 sin β − 1
= e− R = 1 +
pt1
γ+1
⎤ γ
⎡
γ−1
2+(γ−1)M12 sin2 β
2 + (γ − 1) 2γ M 2 sin2 β−(γ−1)
⎥
⎢
1
⎥
⎢
⎦
⎣
2
2
2 + (γ − 1) M1 sin β
(4.23)
2 + (γ − 1) M12 sin2 β
(γ + 1) M12 sin2 β
(4.24)
(4.25)
From the velocity triangles up- and downstream of the oblique shock, we get the
following two equations:
4.1 Introduction
157
tan β =
u1
w1
(4.26a)
tan (β − θ) =
u2
w2
(4.26b)
The ratio of these two equations eliminates the constant tangential velocity and
replaces the normal velocity ratio by inverse density ratio, to produce:
(γ + 1) M12 sin2 β
tan β
=
tan (β − θ)
2 + (γ − 1) M12 sin2 β
(4.27)
The range of the wave angle β lies between the Mach angle, μ and 90◦ that
correspond to a normal shock. The flow turning angle, θ, thus starts from zero when
the oblique shock is an infinitesimal-strength Mach wave to a maximum turning
angle, θmax , which is shown in Fig. 4.13 (from [2]).
Example 4.1 The flow upstream of a normal shock is at Mach M1 = 2.0, in air with
γ = 1.4. The stagnation speed of sound is at = 300 m/s, calculate:
(a) Downstream Mach number, M2
(b) The ratio of speeds of sound, a2 /a1 across the normal shock
(c) The change in gas speed, Δu, across the shock in m/s
Solution:
1+[(γ−1)/2]M 2
We may use M22 = γ M 2 −(γ−1)/21 to calculate the downstream Mach number, M2 .
1
We get: M2 = 0.5774 The ratio of speeds
of sound
√ is equal to square root of the
√
static temperature ratio, i.e., a2 /a1 = T2 /T1 = 1.6875 ≈ 1.30
From stagnation speed of sound, we get
the speed of sound
upstream and downstream of the shock using a 2 = at2 / 1 + (γ − 1) M 2 /2 , from which we get
a1 ≈ 224 m/s and a2 ≈ 290 m/s, therefore Δu = u 1 −u 2 = M1 a1 −M2 a2 ≈ 280 m/s
There are two reminders in using the oblique shock charts shown in Fig. 4.13.
Namely, for a given turning angle, θ, in a supersonic flow, there are two wave angles
that satisfy the conservation equations. Therefore, β is a double-valued function of
θ (instead of a single-valued function). The smaller wave angle, known as the weak
solution, is the angle to be used in oblique shock calculations. The larger wave angle,
known as the strong solution, does not naturally occur for attached shocks, but may be
created/induced in a supersonic wind tunnel or in ducts, by raising the back pressure.
The reason for the dashed lines in the strong solution portion of the oblique shock
graphs (e.g., Fig. 4.13) is to alert the user to select the weak wave solution (shown as
solid curves). The second reminder concerns θmax . The flow turning angle beyond
θmax may not be supported by an attached oblique shock; rather a detached (bow)
shock is needed to support the flow. These are graphically summarized in Fig. 4.14.
In part (a) of Fig. 4.14, we show the possibility of two oblique wave angles, one
corresponding to a strong solution and one corresponding to a weak solution. The
158
4 Shock-Expansion Theory
90
8
M1=
2010
8
6 5
4.5
4.0
3.8
3.6
3.4
3.2
3.0
80
70
2.8
2.8
3.0
3.2
3.4
3.6
3.8
4.0
4.5
5
6
8
10
20
Weak Shock
50
40
8
Shock−wave angle,
β (deg)
60
30
20
shock wave
M2
M
10
1
streamline
0
0
4
8
12
16
20
Deflection angle, θ (deg)
Fig. 4.13 Oblique shock charts for air are from Ref. [2]
24
β
28
θ
32
4.1 Introduction
159
90
80
70
2.2
Shock−wave angle,
β (deg)
60
Weak Shock
50
40
30
shock wave
M2
M
1
streamline
20
β
θ
10
0
26
30
Fig. 4.13 (continued)
34
38
42
46
Deflection angle, θ (deg)
50
54
58
160
4 Shock-Expansion Theory
strong solution is not used in the calculations; therefore it is shown as dashed line. In
part (b) of Fig. 4.14 we are showing an attached oblique shock (the weak solution) to
a 2-D wedge or ramp where the nose angle is less than the maximum turning angle
at M1 . We also show an abrupt turning of the supersonic flow across the plane 2-D
oblique shock in Fig. 4.14b. In part (c) of Fig. 4.14 we note the relative position of
the oblique shock and the Mach wave at M1 . It shows that the wave angle for the
oblique shock, β, is greater than the Mach angle, μ, as expected. Finally, in part (d) of
Fig. 4.14, we show a detached shock formation upstream of a blunt body with a nose
angle greater than the maximum turning angle, θmax , allowed for an attached oblique
shock at freestream Mach number, M1 . We may think of bow shock formation as
the wave that is needed to create higher entropy rise across the shock in response to
higher aerodynamic drag of a blunt body.
The conical shocks, which are created on cones in supersonic flow at zero angle
of attack, create a new class of flows known as conical flowfields. Since the flowfield downstream of an attached conical shock lacks length scale, the flow variables
become a pure function of the (cone) angle θ in spherical coordinate system, as shown
in Fig. 4.15. The flow variables, e.g., p, T , ρ, V , M, are thus constant on constant
θ-cones in between the conical shock and the cone. These surfaces are indeed compression Mach cones that provide isentropic compression and continuous turning of
the flow downstream of a conical shock to accommodate the changing flow area in
the downstream direction. The changing flow area downstream of a conical shock
(a)
Oblique Shock
(strong solution)
(b)
Oblique Shock
(weak solution;
acceptable)
Oblique Shock
(weak)
strong
M1 > 1
weak
M1 > 1
<
<
max
Oblique
Shock
(c)
>
(d)
max
Detached or bow shock
(Where varies between 90° and )
Mach
wave
M1 > 1
M1 > 1
<
max
90°
nose
>
max
Fig. 4.14 Oblique and detached shock formation on pointed bodies of different nose angle in
supersonic flow. a Two theoretically possible oblique shock angles. b Attached oblique shock on
a 2-D compression ramp. c The relative positions of oblique shock and Mach wave in supersonic
flow. d The formation of a bow shock ahead of a body in supersonic flow with θnose > θmax
4.1 Introduction
is the conical shock angle
is the cone angle
is a coordinate in spherical
)
coordinate system ( ,
161
Compression Mach cones
Conical Shock
Conical Shock
r
shock
=
=
M1 > 1
cone
Streamline upstream and
within the conical flowfield
Cone at
zero angleof-attack
Compression Mach
Cones
Fig. 4.15 Formation of the conical shock on a cone at zero-angle-of-attack in supersonic flow
Mach
Cone
Conical
Shock
>
M1 > 1
Zone of Silence
Zone of Action
Fig. 4.16 Relative position of the conical shock and the Mach cone
is a new feature compared to the constant flow area downstream of a plane oblique
shock (see Fig. 4.14b). It is exactly this feature that leads to 3-D relieving effect, as
we shall see. A graphical depiction of the continuous bending of a streamline is also
shown in Fig. 4.15.
Here again, we emphasize that the angle of the conical shock is greater than the
Mach angle corresponding to M1 , as expected. To emphasize this point and also
show the zones of action and silence on a cone, we graph Fig. 4.16.
162
4 Shock-Expansion Theory
The theoretical formulation of conical flowfields downstream of conical shocks is
due to Taylor and Maccoll [7]. The flowfield is best suited for spherical coordinates
(r , θ, Φ). The velocity components are Vr and Vθ only as the VΦ component identically vanishes for axisymmetric flows corresponding to a cone at zero angle-of-attack.
Utilizing Crocco’s theorem, we deduce that the flowfield downstream of the straight
conical shock is irrotational and thus the two velocity components Vr and Vθ are
related to each other according to the irrotationality condition:
∇×V =0
→
Vθ =
dVr
dθ
(4.28)
The continuity equation for a steady, axisymmetric flow in spherical coordinates
reduces to:
∂ρ
∂Vθ
+ Vθ
=0
(4.29)
2ρVr + ρVθ cot θ + ρ
∂θ
∂θ
The density is further eliminated from the above equation by the application of Euler
momentum equation and the local speed of sound in the flow downstream of the
conical shock. The result is a second-order non-linear ordinary differential equation
in Vr that is known as the Taylor-Maccoll equation:
d2 Vr
dVr 2
dVr
2Vr +
cot θ +
dθ
dθ
dθ2
dVr d2 Vr
dVr
=0
+
Vr
dθ
dθ
dθ2
γ−1
2
− Vr2 −
Vmax
2
−
dVr
dθ
(4.30)
Note that the only independent variable is θ and the only dependent variable is Vr . The
Vmax term is the reference speed that we introduced earlier in this chapter, as a constant
of motion, and is used in conical flowfields, following Taylor-Maccoll’s theoretical
formulations. The Eq. (4.30) is still dimensional and may be non-dimensionalized
3 . The resulting non-dimensional equaby dividing both sides of the equation by Vmax
tion is:
⎡
2 ⎤
2
dV
γ−1⎣
r
⎦ 2Vr + dVr cot θ + d Vr
1 − Vr 2 −
2
dθ
dθ
dθ2
dVr d2 Vr
dVr
dVr
+
−
=0
(4.31a)
Vr
dθ
dθ
dθ
dθ2
where
Vr ≡
Vr
Vmax
(4.31b)
4.1 Introduction
163
From energy equation, V is related to the local Mach number according to:
V =
−1/2
2
+
1
(γ − 1) M 2
(4.31c)
There are many good references on the subject of conical flowfields and conical
shocks and the reader may refer to them to fill in the blanks in the above derivation.
An excellent contemporary reference is by Anderson [1] that is recommended for
further reading. The Taylor-Maccoll equation (4.31a) is solved numerically using the
standard numerical approach of Runge-Kutta fourth order technique starting from the
shock at σ with the known Mach number and flow angle immediately downstream of
the shock. These are used to get Vr and Vθ that serve as the initial conditions for the
Runge-Kutta solution of the Taylor-Maccoll equation. Starting from the immediate
downstream of the conical shock, we march in the negative theta direction by taking
small incremental steps of Δθ. Per step in the numerical procedure, we calculate
the new Vr and Vθ until we capture the cone at Vθ = 0. In this approach, we are
finding the cone that supports the conical shock at σ. The numerical solution of
Taylor-Maccoll equation has been used to develop conical shock charts shown in
Fig. 4.17. In part Fig: 3.17a, only the weak solution is plotted and we clearly note
the maximum cone angles that support attached conical shocks. In part 4.17b, the
ratio of cone surface static pressure to freestream stagnation pressure is plotted.
Part 4.17c is a graph of the surface Mach number on the cone for different cones
in supersonic flow. In part 4.17d, we have the ratio of stagnation pressures across
a conical shock for different cone angles and freestream Mach number. Part 4.17e
highlights the three-dimensional relieving effect by comparing a cone to a wedge and
their corresponding wave angles. The wave angle for the wedge is greater than the
wave angle for the cone and thus the surface of the wedge experiences higher (static)
pressure than the (static) pressure on the corresponding cone. This is the essence
of three-dimensional relieving effect. The cone wave drag coefficient, referenced to
base area, is also shown in part 4.17f.
Example 4.2 Consider a 20◦ (2-D) ramp in a Mach 2 flow of air with γ = 1.4. Also,
there is a cone of semi-vertex angle of 20◦ in the same supersonic flow condition.
Calculate and compare:
(a) the wave angles on the 2-D ramp and the cone
(b) the surface pressure on the ramp and the cone.
Solution:
From the plane oblique shock charts (Fig. 4.13) we read the weak shock wave angle
corresponding to Mach 2 freestream and a turning angle of twenty degrees as β ≈
53.4◦ . From the conical shock charts (Fig. 4.17a) we read the corresponding conical
shock angle of θs ≈ 38◦ . Comparison of these two angles reveals the weaker 3-D
shock as compared to the 2-D shock.
164
4 Shock-Expansion Theory
(b) 0.7
(a)
pS
0.6
Conical Shock Charts
= 1.4
, deg.
or
0.3
40°
30°
20°
10°
0°
0.4
in
50
m
pS
p1
M1 min or
0.2
20°
10°
0.1
2
2.0
3.0
0
4.0
1.0
2.0
0, 5°
0°
10°
20°
30°
40°
2.0
0.8
M1 min or
0S
0.2
max
2.0
0.6
0.4
50°
1.0
3.0
p01
(f)
CD , (Referred to Base
Area of Cone)
ne
Shock wave attached
to nose of cone
but not to wedge
ge
Co
, deg.
max
or
32
ed
W
24
Shock wave attached
to nose of both
cone and wedge
16
8
0
1.0
1.5
2.0
2.5
ma
x
45°
50°
3.0
4.0
M1
(e) 48 Shock wave detached
40
or
2.0
M1
from both cone
and wedge
in
2
M1
1.0
4.0
M
1m
10°
15°
20°
25°
30°
35°
40°
2
0
1.0
4.0
3.0
(d) 1.0
p0S/p01
MS
3.0
50°
40°
M1
MS
M1
30°
0°
M1
(c) 4.0
2
p1
0.5
M1
90
80
70
60
50
40
30
20 M1
10
0
1.0
M1
3.0
M1, min or M1
3.5
4.0
1.2
1.1
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1.0
40°
Shock Wave Detached
30°
2
M1
20°
15°
10°
5°
2.0
3.0
4.0
M1
Fig. 4.17 Conical shock charts for γ = 1.4 from Shapiro [6]. a Shock angle versus approach Mach
number, with cone angle as parameter. b Ratio of surface pressure to free stream stagnation pressure
versus free stream Mach number. c Surface Mach number versus free stream Mach number, with
cone angle as parameter. d Ratio of surface stagnation pressure to free stream stagnation pressure,
with cone angle as parameter. e Regions of shock attachment and detachment for cone and wedge.
f Pressure drag coefficient based on projected frontal area
4.1 Introduction
165
The surface pressure on the ramp is calculated based on the normal component of
the flow to the oblique shock, i.e., M1 sin β = 2 sin(53.4◦ ) ≈ 1.6, which produces a
shock (static) pressure ratio of ∼2.82 downstream of the 2-D shock. Using Fig. 4.17b,
we read the ratio of the cone surface static pressure to the freestream stagnation
pressure of ∼0.25. The stagnation and static pressures are related by (the isentropic
relation):
γ
γ − 1 2 γ−1
pt
= 1+
M
= 7.824 for M = 2
p
2
Therefore the ratio of cone surface pressure to freestream static pressure is: pc / p1 ∼
0.25(7.824) = 1.956 The counterpart of this ratio for the 2-D ramp was calculated to
be 2.82. Here again we reveal the 3-D relieving effect as evidenced by this example,
i.e., lower cone surface pressure.
Example 4.3 Compare the maximum turning angles of Mach-2 flow of air in 2-D
and 3-D through plane oblique and conical shocks, respectively.
Solution:
Referring to the plane oblique shock charts, we note that θmax ≈ 23◦ in 2-D. The
implication of this angle is that in two-dimensional flow a plane oblique shock in
Mach 2 can at most cause ∼23◦ turning.
For the case of 3-D flows, we refer to conical shock charts. From Fig. 4.17e, we
note that maximum turning at Mach 2 is nearly 40◦ . The implication is that conical
shocks can stay attached to cones of up to 40◦ semi-vertex angle in Mach 2 flow,
whereas in 2-D the ramp angle above ∼23◦ causes the shock to be detached. This
too is due to 3-D relieving effect.
We now direct our attention to flow expansion. The counterpart of compression
waves is the expansion waves, which cause flow acceleration and turning in supersonic flow. The direction of flow turning (with respect to local upstream flow) determines whether an expansion or a compression wave is needed to cause the necessary
flow deflection. One description that says “turning the flow into itself” is caused by
compression waves and “turning the flow out of itself” is accomplished by expansion
waves is useful when we consider right- and left-running waves. We identify rightand left-running waves in supersonic flow through the eyes of an observer who stands
on the wave and is facing downstream. The right-running wave (RRW) appears on
the right side of the observer and the left-running wave (LRW) appears on the left
side of the observer. The expansion waves that are centered at a corner are referred to
as expansion fans and are known as Prandtl-Meyer expansion waves. Let us examine some compression and expansion corners in supersonic flow and the waves that
cause the flow deflection in Fig. 4.18. The oblique shocks are marked as O.S. and
the Prandtl-Meyer expansion fans are abbreviated to P-M E.F. in Fig. 4.18. We also
have defined the flow angle, θ, with respect to horizontal, or x-axis and have assigned
positive values to θ when the flow direction is above horizontal and have assigned
a negative value to θ when the flow points in the downward (i.e., −y) direction. We
can also discern the “flow turning into itself” and “flow turning out of itself” from
compression and expansion corners that are schematically shown in Fig. 4.18.
166
4 Shock-Expansion Theory
(a)
(b)
LRW: Left-Running Waves
LRW
O.S.
> 1
2
y
X
1
2
M1 > 1
y
M2
1
M2
M1 > 1
X
2
RRW
O.S.
< 1
2
(c)
(d)
LRW
P-M E.F.
< 1
2
y
X
RRW: Right-Running Waves
M1 > 1
1
2
2
1
1
M2
2
M2
2
1
2
y
2
X
M1 > 1
RRW
P-M E.F.
> 1
2
Fig. 4.18 Definition sketch of compression and expansion corners and the corresponding waves that
are generated. a Compression corner, oblique shock, LRW. b Compression corner, oblique shock,
RRW. c Expansion corner, Prandtl-Meyer expansion fan, LRW. d Expansion corner, Prandtl-Meyer
expansion fan, RRW
Mach Wave
w
V
w
µ)
V
µ)
d
V + dV
µ-d )
Fig. 4.19 An exaggerated view of infinitesimal turning across an expansion Mach wave
4.1 Introduction
167
Expansion waves, as noted earlier are Mach waves, which cause an infinitesimal
turning of the supersonic flow. The analysis of an expansion Mach wave can be
carried out using the definition sketch in Fig. 4.19. Note that we had to exaggerate
the infinitesimal turning in order to “see” the geometrical relations involving flow
angles and velocities across the Mach wave in Fig. 4.19.
By applying the law of sines to the triangle with sides V and V + dV , we may
relate the infinitesimal turning angle, dθ, and incremental speed change, dV , in the
following equation:
V
V + dV
=
(4.32)
sin π2 + μ
sin π2 − μ − dθ
The Eq. (4.32) may be simplified by expanding the sine terms to get:
cos μ
cos μ
1
dV
∼
=
=
=
V
cos (μ + dθ)
cos μ − (dθ) sin μ
1 − (dθ) tan μ
≈ 1 + (dθ) tan μ
1+
(4.33)
By taking isentropic flow conditions across the Mach wave; we may relate incremental
turning angle, dθ, to the incremental Mach number change, dM, according to:
√
dθ =
M 2 − 1 dM
1 + γ−1 M 2 M
(4.34)
2
Equation (4.34) may be integrated from zero turning, corresponding to Mach one
(sonic flow), to a general turning angle ν (M) corresponding to supersonic Mach
number, M. The angle ν (M) is called the Prandtl-Meyer angle and has a closed
form solution, i.e.,
ν (M) =
γ+1
tan−1
γ−1
γ−1 2
M − 1 − tan−1 M 2 − 1
γ+1
(4.35)
In addition to a table for the Prandtl-Meyer function and Mach angle, we graph these
two functions in terms of Mach number in Fig. 4.20. In Appendix C the relation
between M, μ and ν are tabulated up to a Mach number of 18.
Fig. 4.20 Prandtl-Meyer
and Mach angles, in degrees,
in supersonic flow with
γ = 1.4
140
120
100
80
60
40
20
0
max
(M)
1
(M)
10
Mach Number, M
100
= 130.5°
168
4 Shock-Expansion Theory
The Prandtl-Meyer function asymptotically approaches a limiting angle, which
can be established by taking the limit of Eq. (4.35) as freestream Mach number
approaches infinity. The implication of the maximum turning angle in ν (M) is that
the flow will expand to infinite Mach number and thus pure vacuum is achieved.
Neither of these two conditions is physically realizable; thus νmax represents an ideal
limit in high-speed gas dynamics.
νmax = ν (M2 → ∞) =
π
γ+1
−1
[rad]
γ−1
2
(4.36)
νmax → 130.5◦ for γ = 1.4
Finally, since the expansion Mach waves never coalesce to create a rarefaction
shock, the flow across such waves is isentropic. Consequently, the stagnation pressure remains constant across Prandtl-Meyer expansion waves. The stepwise calculation involving expanding supersonic flows is dictated by the wall geometry/turning,
which establishes the Mach number and the pressure. Now we solve a supersonic
flow expansion problem using Prandtl-Meyer function, ν (M).
Example 4.4 A supersonic flow approaches an expansion corner, as shown. Assuming the medium is air with γ = 1.4, calculate:
(a) The Mach number downstream of the sharp corner
(b) The wave envelope, i.e., the angle of the head and tail waves
(c) The static pressure on the expansion ramp if p1 = 1 atm.
1
M1 = 2.0
2
w
= 22°
Expansion
ramp
Head Mach
wave
1
M1 = 2.0
Tail Mach
wave
30°
2
20.6°
M2 = 2.836
4.1 Introduction
169
Solution:
The P-M function in region 1 is ν1 ≈ 26.5◦ . The P-M function on the expansion ramp is thus: ν2 = ν1 + Δθ ≈ 26.5◦ + 22◦ = 46.5◦ Therefore the corresponding Mach number is M2 = 2.836. The leading Mach wave makes an
angle of μ1 = sin−1 (1/2) = 30◦ and the tail Mach wave makes an angle of
μ2 = sin−1 (1/2.836) ≈ 20.6◦ Finally, the ratio of static pressures on the two ramps
is related to their respective Mach numbers following the isentropic flow condition
up- and downstream of the corner, namely
p2
=
p1
1 + (γ − 1) M12 /2
γ
γ−1
1 + (γ − 1) M22 /2
Upon substitution for the up- and downstream Mach numbers (along with γ = 1.4),
we get: p2 / p1 ∼ 0.2729. Therefore, flow expansion around a 22◦ corner in Mach
2 upstream flow caused a flow acceleration to Mach 2.836 on the downstream ramp
and the static pressure was reduced to ∼27.3 % of upstream value, i.e., p2 = 0.2729
atm., stagnation pressure, pt , remained constant.
4.2 Lift and Wave Drag
By direct application of the shock waves and Prandtl-Meyer expansion theory to
pointed bodies, we are able to calculate local static pressure on a body in supersonic
flow. Once the static pressure distribution is obtained; the integration of pressure
around the body then establishes the components of resultant force in the lift and drag
directions. Since the inviscid drag in supersonic flow is entirely due to pressure, it is
called pressure or wave drag. Unlike its subsonic counterpart, this drag is inherently
a creature of the supersonic flow where a two-dimensional body in inviscid fluid still
produces drag. As the reader recalls D’Alembert paradox where two-dimensional
bodies in inviscid fluid and in low speed (i.e., incompressible) flow, even in purely
subsonic flow, experienced zero drag!
First, we consider a two-dimensional body in Cartesian coordinates, as shown in
Fig. 4.21. The x-axis is chosen to be parallel to (or coincide with) the chord and thus
the freestream is inclined with respect to the x-axis by the angle of attack, α. This
approach readily produces the normal and axial force components (per unit span),
N and A respectively, which are then related to the lift and drag using angle of
attack, α, according to:
L = N cos α − A sin α
D = N sin α + A cos α
(4.37)
170
4 Shock-Expansion Theory
y
A’
D’
N’
R’
L’
x
M >1
x=c
Fig. 4.21 A pointed 2-D body in supersonic flight with the resultant force (per unit span), R
resolved in normal-axial (N and A ) directions and lift-drag (L and D ) directions
In Fig. 4.21, R is the resultant aerodynamic force, which is then resolved in
different orthogonal directions, as shown. The normal force is simply the integral of
pressure difference across the airfoil along the chord, namely:
N =
c
0
( pl − pu ) dx
(4.38)
And the axial force (per unit span) is related to the integral of pressure times the
body slopes (along the chord) according to:
A =
c
0
pu
dyu
dx
− pl
dyl
dx
dx
(4.39)
In terms of normal and axial force coefficients, cn and ca , we may express Eqs. (4.38)
and (4.39) in terms of the non-dimensional pressure coefficients on the upper and
lower surfaces, namely
1c
C pl − C pu dx
c 0
dyu
dyl
1c
C pu
− C pl
dx
ca =
c 0
dx
dx
cn =
(4.40)
(4.41)
Now, we are ready to apply these principles to a typical supersonic profile, namely,
a diamond airfoil.
Example 4.5 A symmetrical diamond airfoil is shown at 5◦ angle of attack (not to
scale). Use shock-expansion theory to calculate:
(a) pressure coefficient on all four panels
(b) normal force coefficient, cn
(c) axial force coefficient, ca
(d) lift and wave drag coefficients, cl and cd
(e) lift-to-drag ratio, L /D
4.2 Lift and Wave Drag
171
y
3
1
5°
M = 2.4
15°
15°
15°
15°
2
x
4
Solution:
Freestream has to turn an additional 10◦ up to reach panel #1 (i.e., from 5◦ to 15◦ ).
This is the case of flow turning into itself. Therefore, the turning has to occur via an
oblique shock wave. From θ − β − M chart for a plane oblique shock, we get:
M∞ = 2.4 and θ = 10◦
−−→
OS
β ≈ 33◦
M∞ sin β ≈ 1.306
−−→
NS
Mn 1 ≈ 0.786 and p1 / p∞ ≈ 1.805
We use the pressure coefficient definition
Cp ≡
2
2
γ M∞
p
−1
p∞
to get:
C p1 ≈ 0.1996
Also, the Mach number on panel 1 is:
M1 =
0.786
Mn 1
≈
≈ 2.01
sin (β − θ)
sin (33◦ − 10◦ )
The freestream has to turn a total of 20◦ (from +5◦ to −15◦ ), in order to reach ramp
number 2. Reading the plane oblique shock chart, we estimate a wave angle (i.e., the
weak solution):
M∞ = 2.4 and θ = 20◦
OS
−−→
β ≈ 44.2◦
Again the normal component of flow establishes the shock jumps, namely
M∞ sin β ≈ 1.67
NS
−−→
Mn 2 ≈ 0.6458 and p2 / p∞ ≈ 3.126
The Mach number on panel 2 is:
M2 =
0.6458
Mn 2
≈
≈ 1.71
sin (β − θ)
sin (44.2◦ − 20◦ )
172
4 Shock-Expansion Theory
We use the pressure coefficient definition
Cp =
2
2
γ M∞
p
−1
p∞
to get:
C p2 ≈ 0.5273
To march to ramps 3 and 4, we recognize that the net turning angle is twice the
15◦ (i.e., 30◦ ) and since the flow is turning out of itself, we expect an expansion fan
to cause the turn. We start with ν1 which corresponds to M1 and from Prandtl-Meyer
table, we get: ν1 ≈ 26.38◦ and thus
P−M
ν3 = ν1 + 30◦ = 56.38◦
−−−→
M3 ≈ 3.35
The isentropic conditions between panels 1 and 3 can be used to establish static
pressure, p3 ,
γ
1 + (γ − 1) M12 /2 γ−1
p3
=
≈ 0.1272
p1
1 + (γ − 1) M32 /2
We use chain rule to write
p3 / p∞ = ( p3 / p1 ) ( p1 / p∞ ) = 0.1272 · 1.805 = 0.2296
Therefore, the pressure coefficient follows as
Cp =
2
2
γ M∞
p
−1
p∞
or
C p3 ≈ −0.1911
Now, marching onto panel 4 from panel 2, we note that the net turning angle is
again 30◦ . The P-M angle for panel 2 is ν2 ≈ 18◦ , which gives the P-M angle for
panel 4 to be ν4 ≈ 48◦ . From P-M tables, we get M4 ≈ 2.9. The isentropic conditions
between panels 2 and 4 can be used to establish static pressure, p4
p4
=
p2
1 + (γ − 1) M22 /2
1 + (γ − 1) M42 /2
γ
γ−1
≈ 0.15861
We use chain rule to write
p4 / p∞ = ( p4 / p2 ) ( p2 / p∞ ) = 0.15861 · 3.126 = 0.4958
4.2 Lift and Wave Drag
173
Therefore, the pressure coefficient follows as
Cp =
2
2
γ M∞
p
−1
p∞
or
C p4 ≈ −0.1251
The normal force coefficient is:
cn = 0.5 C p2 + C p4 − C p1 − C p3
Therefore, we get the normal force coefficient as cn ≈ 0.1968. To calculate the axial
force, we first establish the airfoil thickness-to-chord ratio as t/c = tan 15◦ . This
gives t/c ≈ 0.2678 (a rather thick profile for supersonic flow!). Therefore, the axial
force coefficient may be written as:
ca =
t
1
C p1 + C p2 − C p3 − C p4
2
c
Upon substitution, we get ca ≈ 0.1397.
We use Eq. (4.37) to get lift and drag coefficients from the normal and axial force
coefficients and the angle of attack, α, namely cl ≈ 0.1839 and cd ≈ 0.1563. Finally,
lift-to-drag ratio is L /D = 0.1839/0.1563 ∼ 1.18.
4.3 Bi-Convex Airfoil
We are now ready to apply the shock-expansion theory to bi-convex airfoils,
which represent a broad class of supersonic airfoil shapes, e.g., circular-arc airfoils.
The general characteristics of these supersonic airfoils are sharp leading and trailing
edges, small thickness and small camber. These are indeed the qualities that lead to a
reduced wave drag. The definition sketch in Fig. 4.22 shows a general asymmetrical
bi-convex airfoil. The upper and lower surfaces are identified by their own functions,
Fig. 4.22 Definition sketch of a general (asymmetrical) bi-convex airfoil in supersonic flow at
angle of attack
174
4 Shock-Expansion Theory
yu (x) and yl (x) respectively. The nose angles on the upper and lower surfaces, θnu
and θnl , are also identified. The chordwise position of the maximum thickness, xtmax ,
and the maximum thickness itself, tmax , are shown. The upstream supersonic flow is
at angle α (angle of attack) with respect to chord.
First, we need to identify the waves that are needed to turn the flow on the upper
and lower surfaces at the leading edge (nose). Depending on the relative values of
the upper surface nose angle and the angle of attack, we may have an oblique shock
or a Prandtl-Meyer expansion fan at the leading edge on the upper surface. This is
easily established by the following rules at the leading edge (for the case of positive
angle of attack, i.e., α > 0):
Case 1: α < θnu The wave is an oblique shock, which turns the supersonic flow
by (θnu − α) angle (note that the flow turns into itself ).
Case 2: α = θnu A Mach wave appears at the leading edge on the upper surface
with zero net turning (since α = θnu ) imposed across the Mach wave.
Case 3: α > θnu A Prandtl-Meyer expansion fan will be formed at the leading
edge on the upper surface, with a net turning of (α − θnu ). Note that the flow turns
out of itself.
(a)
(b)
(c)
Fig. 4.23 The wave pattern on a bi-convex airfoil at positive angle of attack, α. a Case 1: with
α < θnu (an oblique shock will form at the leading edge on the upper surface). b Case 2: with
α = θnu (upstream flow is aligned with the upper surface at the leading edge). c Case 3: with
α > θnu (Prandtl-Meyer expansion fan is formed at the leading edge on the upper surface)
4.3 Bi-Convex Airfoil
175
The leading edge of the lower surface will encounter an oblique shock, which
turns the flow by (θnl + α) angle. The identification of the leading-edge waves will
allow us to march across the waves and onto the upper and lower surfaces of the
bi-convex airfoil. The waves that are formed on the airfoil beyond the leading edge
are all expansion Mach waves that cause the flow to accelerate and the static pressure
to drop. In the vicinity of the airfoil, we will have the wave patterns corresponding
to Case-1, Case-2 and Case-3, shown in Fig. 4.23 where solid lines show oblique
shocks and the dashed lines are expansion Mach waves.
The waves at the trailing edge (on the upper and lower surfaces) are formed to
equalize the static pressure and flow direction downstream of the airfoil. In supersonic
airfoil theory, we define the slipstream as a (free) vortex sheet that is formed at the
trailing edge of the airfoil that separates the upper and lower flows with flow directions
on two sides coinciding with the slipstream and the static pressure being continuous
across the slipstream. Although the flows on two sides of the slipstream are parallel,
they are not equal in magnitude, i.e., V4 = V5 . It is indeed the jump in tangential
velocity across the slipstream that is called the vortex sheet strength. Also, since the
airfoil is in the zone of silence of the waves at the trailing edge, the flow on the airfoil
is not affected by the waves at the trailing edge. However, these waves impact the
formation of slipstream (i.e., vortex sheet), which may impact aircraft components
downstream of the wing (e.g., engine inlet). We show the slipstream downstream
of a supersonic airfoil at high angle of attack and its two boundary conditions in a
definition sketch in Fig. 4.24. We have (arbitrarily) chosen to show an oblique shock
wave as well as a Prandtl-Meyer expansion fan at the trailing edge of the airfoil in
Fig. 4.24. However, it is possible to have two oblique shock waves at the trailing
edge, if the flow environment downstream of the airfoil (i.e., static pressure and
flow direction) demands it. Indeed, the nature of these waves at the trailing edge is
established by the two boundary conditions on the slipstream, namely the equality
of static pressure ( p4 = p5 ), which is known as the pressure equilibrium condition,
and the flow angles (θ4 = θ5 ).
Once, we established the nature of the waves at the leading edge, we may proceed
to calculate the pressure and Mach number immediately downstream of the leading
Boundary Conditions
P-M E.F.
O.S.
M4
2
M1 > 1
4
1
p4 = p5
= 5=
4
4
p4
Slipstream
(Vortex sheet)
M4
M5
3
O.S.
P-M E.F.
5
M5
p5
5
Fig. 4.24 The slipstream downstream of a supersonic airfoil at angle of attack with possible trailing
edge waves
176
4 Shock-Expansion Theory
edge on the upper and lower surfaces of the airfoil. The flow downstream of the
leading edge on a bi-convex airfoil undergoes a continuous acceleration on both
upper and lower surfaces (as shown in Figs. 4.23 and 4.24). Thus, the flow expands
isentropically on both sides of the airfoil downstream of the leading edge where
oblique shocks may appear. The difference between the upper and lower surfaces is
therefore in the initial conditions that are set at the nose by the type and the strength of
the waves that are formed at the leading edge. Otherwise, the solution methodology
is identical on the upper and lower surfaces of the bi-convex airfoil.
Solution Methodology
We start at the leading edge, calculating the nose angle on the upper surface, according to:
(4.42)
θnu = tan−1 (dyu /dx)
x=0
Depending on the angle of attack, α, we may have one of the three cases that we
described earlier as determining the wave at the leading edge on the upper surface.
We then march across the wave at the leading edge and establish:
1. Mnu , Mach number of the flow immediately downstream of the nose on the upper
surface
2. pnu , static pressure immediately downstream of the nose on the upper surface
We then propose to march downstream on the upper surface by creating a table of
values for the surface pressure (and Mach number). For this purpose, we may divide
the airfoil into (n + 1)-chordwise calculation stations, from leading edge to trailing
edge, similar to the graph shown in Fig. 4.25. Here, we used the chord length, c,
as the normalizing length scale in the problem and graphed y/c versus x/c, as the
non-dimensional coordinates. Counting the leading and trailing edges as calculation
stations, we have (arbitrarily) chosen 11 positions along the airfoil upper surface
in Fig. 4.25, which divides the airfoil into ten segments/panels along the chord, as
shown.
Next, we calculate the surface (inclination) angle at every calculation station,
namely at n = 2 to 11 [note that we have already calculated the leading edge angle,
i.e., n = 1 in Eq. (4.42)]. For the ith station, we use:
y/c
Panel number 7
n=1
n=2
n=3
n=4
n=5
n=6
n=7
n=8
n=9
n=10 n=11
0.3
0.4
0.5
0.6
0.7
0.8
0.9
3
0.0
0.1
0.2
1.0
x/c
Fig. 4.25 The upper surface of a general bi-convex airfoil divided into 11 equally-spaced calculation
stations and 10 panels
4.3 Bi-Convex Airfoil
177
θu(i) = tan−1 (dyu /dx)
(4.43)
x=xi
The Mach number immediately downstream of the nose that we calculated earlier
is now used to determine the Prandtl-Meyer angle at the nose, i.e., νn or ν1 . Since
the flow is continuously expanding on the bi-convex airfoil, the Prandtl-Meyer angle
continually increases on the upper surface, from one station to the next, by the net
flow turning angle between those stations, namely
νu(i+1) = νu(i) + θu(i+1) − θu(i) (4.44)
The absolute value in Eq. (4.44) shows that the Prandtl-Meyer angle increases as we
march towards the trailing edge on the bi-convex airfoil. Now, we can determine the
local Mach numbers, Mu(i) based on the Prandtl-Meyer angles, νu(i) . The isentropic
flow condition on the upper (and lower) surface can be used to calculate the static
pressure according to isentropic rule:
γ
⎤ γ−1
γ−1
2
M
1
+
u(i)
2
pu(i+1)
⎦
=⎣
γ−1
2
pu(i)
1 + 2 Mu(i+1)
⎡
(4.45)
We have now created the pressure distribution on the upper surface of the airfoil at
discrete points along the airfoil, from the leading edge to the trailing edge. By defining
the effective panel areas for each calculated pressure as the half-way points between
the adjacent calculation stations (or panels), we can proceed with the calculation
of the incremental force, per effective panel area. A definition sketch is shown in
Fig. 4.26 as an aid to understand the effective panel area and the calculation of the
incremental force and pitching moment based on pu(i) . The increment of normal
components of force (per unit span) on the upper and lower surfaces due to the static
pressures pu(i) and pl(i) are:
ΔNu(i) = − pu(i) xi+1/2 − xi−1/2
ΔNl(i) = + pl(i) xi+1/2 − xi−1/2
Fig. 4.26 Definition sketch
of the effective panel for the
ith calculation station
(4.46a)
(4.46b)
Effective Panel
for pu(i)
pu(i)
y/c
i
i-1
yu/c
i+1
x/c
i-1/2
i+1/2
178
4 Shock-Expansion Theory
The increments of axial component of force (per unit span) on the upper and lower
surface due to static pressures pu(i) and pl(i) are:
ΔAu(i) = + pu(i) yu(i+1/2) − yu(i−1/2)
ΔAl(i) = − pl(i) yl(i+1/2) − yl(i−1/2)
(4.47a)
(4.47b)
The increments of pitching moment about the leading edge (per unit span) on the
upper and lower surfaces are:
ΔMu(i) = pu(i) xi+1/2 − xi−1/2 xi + pu(i) yu(i+1/2) − yu(i−1/2) yu(i) (4.48a)
ΔMl(i) = − pl(i) xi+1/2 − xi−1/2 xi − pl(i) yl(i+1/2) − yl(i−1/2) yl(i) (4.48b)
Finally, we add all the increments to get the normal and axial force components
(per unit span) on the upper surface and proceed similarly on the lower surface.
Equation (4.37) will then be used to express lift and wave drag (per unit span) in terms
of the normal and axial force and the angle of attack, α. We will now demonstrate
the shock-expansion theory applied to an asymmetric, parabolic profile bi-convex
airfoil with negative camber in the following example.
Example 4.6 Consider a 5 % thick bi-convex airfoil of parabolic arc upper and lower
profiles with negative camber, as shown in the schematic drawing. The upper surface
contributes 2 % to thickness at 50 % chord and is thus defined by:
yu (x) = 0.08x (1 − x)
The lower surface contributes 3 % thickness to the airfoil at mid chord and is thus
defined by:
yl (x) = −0.12x (1 − x)
By virtue of negative camber, a stronger oblique shock appears at the leading edge
on the lower surface than the oblique shock on the upper surface. Therefore, we
expect this airfoil to produce lift (albeit small) even at zero angle of attack. Use
shock-expansion theory to first calculate the pressure coefficient on the upper and
lower surfaces, then calculate the lift, wave drag and pitching moment (about leading
edge) coefficients for this airfoil at the freestream Mach number of M∞ = 1.4 and
at zero angle of attack. Graph the airfoil and its upper and lower surface pressure
distributions along the chord.
y
0.02c
nu
M = 1.4
x
0.03c
= 1.4
c
nl
Not-to-Scale
4.3 Bi-Convex Airfoil
179
Solution:
First, we calculate the leading edge angles for the upper and lower surfaces.
dyu = (0.08 − 0.16x)|x=0 = 0.08
dx x=0
Therefore, the nose angle on the upper surface is θnu ≈ 4.57◦ . Similarly, we calculate
the leading edge angle for the lower surface,
dyl = (−0.12 + 0.24x)|x=0 = −0.12
dx x=0
→
θnl ≈ −6.84◦
In both cases, the supersonic flow turns into itself and thus requires an oblique shock
to form at the leading edge on both upper and lower surfaces. Using oblique shock
charts (Fig. 4.13), we get:
M∞ = 1.4 and θnu ≈ 4.6◦
OS
βu ≈ 52.2◦
−→
NS
M∞ sin β ≈ 1.11 −→ Mnunose ≈ 0.907 and pn,u / p∞ ≈ 1.261
Munose = Mnunose / sin (β − θ) ≈ 1.228
The Prandtl-Meyer angle immediately downstream of the leading edge on the upper
surface corresponds to Munose = 1.228. We can use Eq. (4.35) to get:
νunose ≈ 4.25◦
u
Now, from the equation of the slope of the upper surface, dy
dx = 0.08 − 0.16x,
we calculate the surface angle at increments of 10 % chord. This procedure is easily implemented in a spreadsheet program (e.g., Excel). The results are shown in
Table 4.1.
The lower surface is then treated in the same way, i.e., we first establish the oblique
shock wave angle at the leading edge for the lower surface, i.e.,
M∞ = 1.4 and θnl ≈ −6.84◦
OS
−−→
βl ≈ 56.2◦
NS
M∞ sin β ≈ 1.16 −−→ Mnlnose ≈ 0.866 and pl / p∞ ≈ 1.4124
Mlnose = Mnlnose / sin (β − θ) ≈ 1.141
The Prandtl-Meyer angle immediately downstream of the leading edge on the lower
surface corresponds to Mlnose = 1.141. We can use Eq. (4.35) to get:
θlnose ≈ 2.185◦
1.4
1.4
1.4
1.4
1.4
1.4
1.4
1.4
1.4
1.4
1.4
1.4
1.4
1.4
1.4
1.4
1.4
1.4
1.4
1.4
1.4
0.02
0.02
0.02
0.02
0.02
0.02
0.02
0.02
0.02
0.02
0.02
0.02
0.02
0.02
0.02
0.02
0.02
0.02
0.02
0.02
0.02
0.03
0.03
0.03
0.03
0.03
0.03
0.03
0.03
0.03
0.03
0.03
0.03
0.03
0.03
0.03
0.03
0.03
0.03
0.03
0.03
0.03
0
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1
0
0.004
0.007
0.010
0.013
0.015
0.017
0.018
0.019
0.020
0.020
0.020
0.019
0.018
0.017
0.015
0.013
0.010
0.007
0.004
0
0
−0.006
−0.011
−0.015
−0.019
−0.023
−0.025
−0.027
−0.029
−0.030
−0.030
−0.030
−0.029
−0.027
−0.025
−0.023
−0.019
−0.015
−0.011
−0.006
0
0
−0.001
−0.002
−0.003
−0.003
−0.004
−0.004
−0.005
−0.005
−0.005
−0.005
−0.005
−0.005
−0.005
−0.004
−0.004
−0.003
−0.003
−0.002
−0.001
0
4.57
4.12
3.66
3.21
2.75
2.29
1.83
1.37
0.92
0.46
0.00
−0.46
−0.92
−1.37
−1.83
−2.29
−2.75
−3.21
−3.66
−4.12
−4.57
−6.84
−6.16
−5.48
−4.80
−4.12
−3.43
−2.75
−2.06
−1.37
−0.69
0.00
0.69
1.37
2.06
2.75
3.43
4.12
4.80
5.48
6.16
6.84
Table 4.1 Geometric and flow calculations on a bi-convex airfoil using Shock-expansion theory
M∞
τu
τl
x
yu
yl
ycamber
θ u (◦ )
θ l (◦ )
4.2525
4.7082
5.1645
5.6212
6.0783
6.5358
6.9936
7.4516
7.9098
8.3681
8.8264
9.2848
9.7431
10.2013
10.6593
11.1170
11.5745
12.0316
12.4884
12.9446
13.4003
ν u (◦ )
2.1848
2.8635
3.5440
4.2260
4.9094
5.5939
6.2795
6.9658
7.6527
8.3401
9.0276
9.7151
10.4024
11.0893
11.7757
12.4612
13.1458
13.8291
14.5112
15.1916
15.8703
ν l (◦ )
Mu
1.227
1.245
1.262
1.279
1.296
1.313
1.329
1.346
1.362
1.378
1.394
1.41
1.426
1.441
1.457
1.473
1.488
1.504
1.519
1.535
1.55
Ml
1.141
1.171
1.199
1.226
1.253
1.278
1.303
1.328
1.353
1.377
1.401
1.425
1.448
1.472
1.495
1.518
1.542
1.565
1.588
1.611
1.634
180
4 Shock-Expansion Theory
1
0.9773
0.9554
0.9338
0.9126
0.8917
0.8723
0.8521
0.8334
0.8150
0.7969
0.7792
0.7617
0.7456
0.7288
0.7122
0.6970
0.6811
0.6664
0.6510
0.6369
1
0.9626
0.9283
0.8960
0.8645
0.8359
0.8081
0.7809
0.7544
0.7296
0.7055
0.6819
0.6600
0.6377
0.6169
0.5966
0.5761
0.5570
0.5384
0.5204
0.5029
1.2610
1.2324
1.2048
1.1775
1.1507
1.1244
1.1000
1.0745
1.0509
1.0277
1.0049
0.9825
0.9605
0.9403
0.9190
0.8981
0.8789
0.8588
0.8403
0.8209
0.8031
1.4124
1.3596
1.3111
1.2655
1.2210
1.1807
1.1413
1.1030
1.0655
1.0305
0.9964
0.9632
0.9321
0.9006
0.8712
0.8427
0.8137
0.7867
0.7604
0.7350
0.7102
0.190
0.169
0.149
0.129
0.110
0.091
0.073
0.054
0.037
0.020
0.004
−0.013
−0.029
−0.044
−0.059
−0.074
−0.088
−0.103
−0.116
−0.131
−0.143
0.301
0.262
0.227
0.194
0.161
0.132
0.103
0.075
0.048
0.022
−0.003
−0.027
−0.049
−0.072
−0.094
−0.115
−0.136
−0.155
−0.175
−0.193
−0.211
0.0010
0.0005
0.0002
0.0001
0.0000
0.0000
0.0002
0.0004
0.0007
0.0005
0.0078
0.0051
0.0030
0.0011
−0.0006
−0.0021
−0.0035
−0.0048
−0.0058
−0.0034
dca,u
0.0007
0.0055
Table 4.2 Geometric and flow calculations on a bi-convex airfoil using Shock-expansion theory
pu / pnose
pl / pnose
pu / p∞
pl / p∞
C p, u / p∞
C p, u / p∞
dcn
dca,l
0.0012
0.0017
0.0010
0.0005
0.0001
0.0000
0.0001
0.0005
0.0012
0.0022
0.0017
−0.0072
−0.0105
−0.0071
−0.0041
−0.0017
0.0002
0.0015
0.0022
0.0022
0.0015
0
dcm,u
0.0106
0.0157
0.0108
0.0066
0.0030
0.0001
−0.0019
−0.0031
−0.0032
−0.0023
0
dcm,l
4.3 Bi-Convex Airfoil
181
182
4 Shock-Expansion Theory
y/c
l
Now, from the equation of the slope of the lower surface, dy
dx = −0.12 + 0.24x,
we calculate the surface angle at increments of 10 % chord. This procedure is easily
implemented in a spreadsheet program (e.g., Excel). Table 4.1 shows the spreadsheet
results pertaining to the geometry, Prandtl-Meyer angle and Mach numbers on the
upper and lower surfaces of the airfoil. Tau is the thickness ratio for the upper and
lower surfaces, 0.02 and 0.03 respectively.
From local Mach number calculations, we determine the static pressure and thus
static pressure coefficient on the airfoil using isentropic relation (Eq. 4.45). The increments of force and pitching moment are also calculated using Eqs. (4.46)–(4.48).
These calculations are performed using a spreadsheet program (see Table 4.2).
0.03
0.01
-0.01
-0.03
-0.05
(a)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
x/c
(b) -0.3
-0.2
Cp
-0.1
0
Cp, upper
0.1
0.2
Cp, lower
0.3
0
0.2
0.6
0.4
0.8
1
x/c
Fig. 4.27 Application of shock-expansion theory to a 5 % thick bi-convex airfoil with negative
camber. a The 5 % thick parabolic bi-convex airfoil with negative camber (2 % upper thickness and
3 % lower thickness). b Pressure distribution on the upper and lower surfaces
4.3 Bi-Convex Airfoil
183
The normal force coefficient is the sum of dcn column (in Table 4.2): cn ≈
0.00233. The axial force coefficient is the sum of dca, u and dca, l columns (in
Table 4.2): ca ≈ 0.01453. The pitching moment coefficient about the leading edge
is the sum of dcm, u and dcm, l (in Table 4.2): cm, LE ≈ 0.0132.
Since the airfoil is at zero angle of attack, the normal and axial force directions
coincide with the lift and drag directions respectively. The shape of the airfoil, including the mean camber line is shown in Fig. 4.27a and the pressure distribution is shown
in Fig. 4.27b, from the Excel files.
4.4 Axisymmetric and Slender Bodies
Flow around axisymmetric bodies in supersonic flow involves conical shocks and
expansion Mach cones. In the absence of angle of attack, a pointed axisymmetric
body that supports a conical shock may be analyzed using the conical shock charts
in Fig. 4.17. However when the body is in an angle of attack, the axis of the conical
shock will no longer coincide with the body axis. Figure 4.28 shows an example of a
cone at an angle of attack and zero yaw. The axis of the cone is along x-direction, as
shown. The cone cross section is therefore in the y-z plane. We define the meridian
angle, ϕ in the y-z plane with Φ = 0 at the top and 180◦ at the bottom of the cone,
as shown. Therefore the windward side of the cone in positive angle of attack is at
Φ = 180◦ and the leeward side of the cone is at Φ = 0◦ , as indicated. The meridian
angle becomes important in asymmetrical flows as it describes the rays along the
cone surface where flow attains unique properties.
We expect the windward side of the cone to experience higher static pressures and
the leeward side of the cone to experience lower static pressures. The same arguments
apply to a cone in sideslip or yaw. Therefore, in general we expect the static pressure
gradient that is set up (in the meridian plane) around a cone with angle of attack and
yaw to create a spiraling flow around the cone, as shown in Fig. 4.29.
Fig. 4.28 Cone at an angle of attack (and zero yaw) in a supersonic flow showing an asymmetrical
disposition of a conical shock attached to the cone at the vertex. a An axisymmetric cone at an angle
of attack. b Meridian angle, Φ at the base of the cone
184
4 Shock-Expansion Theory
z
Spiraling streamlines
on a cone with angle
of attack and yaw
Conical Shock
z
Leeward side
-
-
-
-
-
-
=0º
x
+
M >1
+
M >1
+
+
+
Windward side
=270º
=
90º
+
y
=180º
Cone
Fig. 4.29 Spiraling streamlines on a cone in supersonic flow at angle of attack and yaw
(a)
(b)
Curved shock
(Attached)
Conical shock corresponding
to a cone of semi-vertex
angle, in M
z
Local cone
is the body angle
at point A
B
A
z
M >1
0
x
M >1
0
Fig. 4.30 A (pointed) slender body of revolution in supersonic flow at zero angle of attack. a Slender
body of revolution at zero angle of attack. b Local cone (tangent) to point A has semi-vertex angle, δ
An approximate method of calculation for slender bodies of revolution is based
on the method of local cones. Consider a pointed body of revolution at zero angle
of attack as shown in Fig. 4.30. The semi-vertex angle, δ0 is assumed to be less than
the maximum turning angle, δmax , corresponding to a cone at the given flight Mach
number, M∞ . Therefore we guarantee that the conical shock at the vertex is attached
to the axisymmetric body. Any other point on the body, e.g., point A, makes an angle
δ with respect to the axis, or freestream direction. Since δ < δ0 , we expect the static
pressure at point A to be less than the static pressure at the vertex. The method of
local cones suggests that the reduced pressure at point A is due to a smaller angle
cone that is tangent to the body at point A had it been placed in freestream Mach
number of M∞ . This method then predicts a pressure coefficient of zero at point B
on the shoulder of the axisymmetric body where the angle is zero with respect to the
body axis. In the aft portion of the body, i.e., past point B, the method of local cones
does not apply. The aft region is known as the shadow zone.
4.4 Axisymmetric and Slender Bodies
185
Example 4.7 Consider a slender body of revolution at zero angle of attack in supersonic flow with M∞ = 2.0. The semi-vertex angle is δ0 = 30◦ . Calculate the pressure
coefficient at the vertex and estimate the pressure coefficient at point A, where the
body angle is δ = 10◦ using the method of local cones.
A
M = 2.0
= 10º
0
0
= 30º
x
Solution:
From conical shock charts (Fig. 4.17), for Mach 2.0 and a cone of 30◦ semi-vertex
angle, we get the conical shock angle at the vertex to be σ0 ≈ 48◦ . The normal
component of flow to the shock is therefore:
M∞ sin σ0 ≈ 2 sin 48◦ = 1.486
Therefore the static pressure ratio at the nose is calculated from
pN
2γ 2
M∞ sin2 σ0 − 1
=1+
p∞
γ+1
To be p N / p∞ = 2.4106. Assuming the pressure at point A corresponds to the cone
pressure of 10◦ semi-vertex angle in Mach 2 flow, we use Fig. 4.17b (the conical shock
charts) to get the ratio of the cone surface static pressure to the freestream stagnation
pressure (from Fig. 4.17b) to be: pc / pt∞ ≈ 0.17. For Mach 2, pt∞ / p∞ = 7.824
and therefore
pA
p A pt∞
=
≈ (0.17) (7.824) ≈ 1.330
p∞
pt∞ p∞
We note the static pressure at point A is significantly less than the pressure at the
nose, as expected. The static pressure at point A is ∼55 % of the static pressure at
the nose.
The same approximation applies to a slender axisymmetric body in angle of attack,
α. For a general axisymmetric body at an angle of attack, we define an equivalent
cone angle, δc that is tangent to point A and the freestream direction is the axis of the
equivalent cone, as shown in Fig. 4.31. The body angle at A is δ, the nose angle is δ0
and the equivalent cone angle is δc . The meridian angle is also shown in Fig. 4.31,
which affects the equivalent cone angle according to (4.49).
186
4 Shock-Expansion Theory
Equivalent cone (for point A on the
body) that creates a semi-vertex
angle, with respect to M vector
Equivalent conical shock (for point
A on the body)
Axis of the equivalent cone that
is tangent to point A
z
= 0º
270º
c
90º
0
M >1
x
A
180º
Fig. 4.31 Definition sketch used in the method of local cones for a slender body in angle of attack
The relationship between the angles δc , δ, ϕ and α (in radian) is:
δc = δ − α cos Φ +
α2
cot δ · sin2 Φ
2
(4.49)
Example 4.8 Consider a slender axisymmetric body, similar to the body shown in
Fig. 4.31, with the semi-vertex angle, δ0 = 20◦ , in a supersonic flow with M∞ = 2
and at 10 degrees angle of attack, i.e., α = 10◦ . Apply the method of local cones to
point A on the body where the local body angle is 10◦ ; and A is on the windward
side of the body with Φ = 180◦ . Estimate the pressure coefficient at A using the
method of local cones.
Solution:
The nose angle with respect to the flow direction on the windward side is the sum of
20◦ for the nose and 10◦ for the angle of attack (this may also be deduced from
Eq. (4.49)). Therefore, the conical shock angle corresponding to M∞ = 2 and
equivalent cone angle of 30◦ is (the same as Example 4.7): σ0 ≈ 48◦ on the windward side. The rest of the calculations immediately downstream of the nose on the
windward side follow the Example 4.7, namely
M∞ sin σ0 ≈ 2 sin 48◦ = 1.486
and the static pressure ratio at the nose (on the windward side) is calculated to be:
2γ 2
p N M∞ sin2 σ0 − 1 ≈ 2.411
=
1
+
p∞ ϕ=180◦
γ+1
To get the equivalent cone angle for point A, we substitute for the body angle at A,
namely δ = 10◦ and the meridian angle, Φ = 180◦ as well as the angle of attack,
α = 10◦ in Eq. (4.49). We get:
4.4 Axisymmetric and Slender Bodies
δc = δ − α cos Φ +
187
α2
cot δ · sin2 Φ
2
→
δc = 20◦
Therefore the corresponding (equivalent) conical shock produces the ratio of the cone
surface pressure to freestream stagnation pressure of pc / pt∞ ≈ 0.25 at M∞ = 2
(from Fig. 4.17b). From isentropic table, we get pt∞ / p∞ = 7.824 for Mach 2, and
therefore we estimate p A / p∞ to be:
pA
p A pt∞
=
≈ (0.25) (7.824) ≈ 1.956
p∞
pt∞ p∞
In Example 4.8, we assumed that the body surface pressure was the same as
the equivalent cone surface pressure. However, within the framework of method of
local cones; there are two hypotheses that are advanced on the flow conditions, i.e.,
pressure and velocity, at point A as compared to an equivalent cone that is tangent
to A, namely:
Hypothesis 1: pressure at A is identical as on the equivalent cone with δc , good
for high velocities
Hypothesis 2: velocity at A is identical as on the equivalent cone, which is better
for low velocities
2.0
= 6º
No. 1
1.0
No. 1
1.6
5
= 0º
R6.5
1.2
No. 2
1.0
CD
= 6º
5
= 0º
R6.5
= 6º
0.4
No. 3
No. 3
= 0º
0
1
2
3.5
1.5
No. 2
0.8
1.0
2.5
3
4
5
M
Fig. 4.32 The effect of angle of attack and body shape on nose wave drag (after Ref. [3])
188
4 Shock-Expansion Theory
We will examine these hypotheses more closely in the next section (Examples and
Applications). In addition, the base pressure is assumed to be equal to the freestream
pressure, i.e., pb = p∞ , which is a reasonably accurate assumption for most applications of truncated bodies in supersonic flow. Once the pressure distribution on the
slender axisymmetric body is obtained [using (4.49) for an equivalent cone] and
further assuming that the base pressure is equal to the ambient pressure, we may
approximate the aerodynamic forces and moments on the body, namely, lift, wave
drag, side force and three moment coefficients for the pointed slender body in supersonic flow. The effect of angle of attack and body shape (i.e., nose bluntness) on
the nose wave drag coefficient is shown through experimental data in Fig. 4.32 (from
Krasnov 1970). Body number 1 is a flat-head cylinder, whereas the bodies designated
as number 2 and 3 represent a streamlined nose of 1.5 and 2.5 (nose) fineness ratios,
respectively. Note that the lower-end range of freestream Mach numbers dips into the
transonic zone. Here, the effect of angle of attack is seen as an upward shift in C D .
4.5 Examples and Applications
4.5.1 The Shape and Geometric Parameters
of an (Axisymmetric) Ogive Nose
Consider a pointed axisymmetric body of nose length, l N , and a base diameter of
d. Tangent ogive describes such a pointed body where the body contour, in the
meridian plane, is a circular arc with zero slope at the base. Figure 4.33 shows a
definition sketch of a tangent ogive in the meridian plane.
We define the fineness ratio for the nose as the ratio of nose length-to base diameter,
following Chap. 3 notation:
lN
(4.50)
FN ≡
d
The non-dimensional coordinates describing the body are:
r
d/2
x
x≡
lN
R
R≡
d
r≡
(4.51a)
(4.51b)
(4.51c)
where R is the radius of curvature of the body (i.e., a constant for a tangent ogive).
In terms of the non-dimensional coordinates, the equation for the (circular arc) ogive
is described by:
⎧
⎫
1/2 ⎬
⎨
FN2
r = 1 − 2R 1 − 1 − 2 (1 − x)2
(4.52)
⎩
⎭
R
4.5 Examples and Applications
189
The nose fineness ratio may be related to the non-dimensional radius of curvature
via:
(4.53)
FN = R − 1/4
The local body slope is obtained by differentiating r (x) with respect to x, i.e.,
F2
1 dr
FN
dr
=
=
tan δ =
(1 − x) 1 − N2 (1 − x)2
dx
2FN dx
R
R
−1/2
(4.54)
The nose angle δ0 is related to the nose fineness ratio via:
sin δ0 =
FN
R
=
FN
2
FN + 1/4
(4.55)
Based on the method of local cones described in Sect. 4.4, we propose to examine
the two hypotheses as they apply to a tangent ogive, namely:
1. the pressure on the body is equal to the pressure on the equivalent cone, of the
same local angle δ, and the stagnation pressure corresponding to δ0 cone (at the
nose)
2. the velocity on the body is equal to the velocity on an equivalent cone, of the same
local angle, δ and the stagnation pressure corresponding to a cone of δ, semi-vertex
angle (i.e., for higher total pressure recovery than the first hypothesis)
(x)
r
dr
dx
r(x)
0
d
x
lN
R
0
Fig. 4.33 An axisymmetric tangent ogive in the (r, x) plane
x
190
4 Shock-Expansion Theory
The first hypothesis assumes the same pressure results in a pressure coefficient, C pp
and the second method assumes the same surface velocity produces C pv . These
pressure coefficients are related to each other via [3]:
C pv = C pp ν +
2 (ν − 1)
2
γ M∞
(4.56)
where,
ν0
(4.57)
ν
pt |δ0
ν0 ≡
; Stagnation pressure ratio on a cone with the nose angle δ0 (4.58)
pt∞
pt |δ
; Stagnation pressure ratio on a cone with the nose angle δ
(4.59)
ν≡
pt∞
ν≡
The second hypothesis, i.e., C pv proves to be a more accurate estimate of the local
pressure coefficient than the C pp . The C pv leads to a negative pressure coefficient
on the aft sections of the ogive, i.e., near the shoulder and the shadow region that is
borne by both the Method of Characteristics as well as the experimental data. The
graphs of C pp and C pv on tangent ogives at Mach numbers 2 and 5 are shown in
Figs. 4.34 and 4.35 respectively (from Ref. [3]). Note that the pressure coefficient
C pv near the shoulder region of the tangent ogive where δ ≈ 0, dips into the negative
territory for tan δ0 > 0.35, or semi-apex angle of >19.3◦ .
Fig. 4.34 Pressure
coefficient on a tangent ogive
based on competing
hypotheses [3]
0.8
M =2
0.65
A
0.7
Cpp
0.60
0
0.55
0.6
0.50
0.5
0.45
0.4
ta
n
Cp
0.40
0.3
Cpv
0.35
0.30
0.2
0.25
0.20
0.15
0.1
0.10
0
-0.1
0.05
0
0
0.2
0.4
tan
0.6
0
0.8
4.5 Examples and Applications
191
Fig. 4.35 Pressure
coefficient on a tangent ogive
based on competing
hypotheses [3]
0.8
M =5
0.7
Cpp
A
0
0.65
0.6
0.60
Cp
0.5
0.55
ta
n
0.50
0.4
0.45
0.3
0.40
Cpv
0.35
0.30
0.25
0.2
0.1
0.15
0
0
-0.1
0
0.2
0.4
tan
0.6
0.8
0
There is a semi-empirical correlation for the wave drag coefficient of a tangent
ogive in supersonic flow that is given in Ref. [3], which is:
C D = C pc 1 −
196FN2 − 16
14 (M∞ + 18) FN2
(4.60)
This is valid for: 1.5 ≤ M∞ ≤ 3.5 and 10◦ ≤ δ0 ≤ 45◦ .
In Eq. (4.60), C D is the pressure or wave drag coefficient of the tangent ogive
(nose) and C pc is the pressure coefficient on a cone of the (semi-) apex angle δ0 .
Example 4.9 A tangent ogive has a nose fineness ratio of FN = 2.825. Calculate:
(a) the non-dimensional radius of curvature, R for the nose
(b) the semi-vertex angle, δ0 in degrees
(c) the pressure coefficients C pp and C pv at M∞ = 2 where the body angle is 11.2◦
(d) the wave drag coefficient of this tangent ogive (nose) at M∞ = 2
Solution:
From Eq. (4.53), we establish the non-dimensional radius of curvature of the nose
to be:
R = FN2 + 1/4 = 8.23
From Eq. (4.55) we calculate the semi-vertex angle at the nose:
δ0 = sin−1 FN /R ≈ 20◦
192
4 Shock-Expansion Theory
For the body angle of 11.2◦ and the semi-apex angle of 20◦ we calculate the tangents
to be: tan δ ≈ 0.20 and tan δ0 ≈ 0.365. We use these values in Fig. 4.34 to get the
two pressure coefficients: C pp ≈ 0.13 and C pv ≈ 0.12 Finally, the pressure drag
coefficient of the tangent ogive is estimated from the semi-empirical correlation
(4.60). But first, we need the pressure coefficient on the cone of the semi-apex angle
20◦ , in Mach-2 flow (from Fig. 4.17f, noting that the cone pressure coefficient and
the wave drag coefficients are the same since the base pressure is assumed to be equal
to p∞ ), C pc ≈ 0.325
C D = C pc 1 −
196FN2 − 16
14 (M∞ + 18) FN2
≈ 0.0998 ≈ 0.1
4.5.2 Extension to Transonic Speeds
A slender body in a slightly supersonic flow, i.e., for M∞ = 1+ε creates a normal
shock that stands ahead of the body. The flow downstream of the normal shock is
subsonic with M ≈ 1 − ε. This is the result of the linear behavior of normal shocks
in slightly supersonic flows, as we demonstrated earlier in this chapter. Therefore
the local Mach number on a slender body, M, does not change appreciably when the
freestream flow condition is near sonic. This argument, first forwarded by Liepmann
and Bryson [4] can be expressed mathematically as:
∂ M =0
∂ M∞ M∞ =1
(4.61)
This is the principle of stationary local Mach numbers near sonic freestream flow
conditions on slender bodies. This condition can be used as an extrapolation tool to
(a) 1.0
(b)
0.30
0.6
(CDp)M = 1
(CDp )M = 1
0.8
0.4
c
M =1
0.2
0
0
0.20
M =1
0.10
0
0
30
60
c
(deg)
90
0
30
60
0
(deg)
90
Fig. 4.36 Wave drag coefficient, C D p , at sonic flow (after Ref. [3]). a Cone. b Ogive
4.5 Examples and Applications
193
extend the results of force and moment coefficients from subsonic and supersonic
sides through the transonic region. We will apply this principle to a slender cone and
an ogive in transonic flow. First, the experimental results for the wave drag coefficient
for cones and ogives are shown in Fig. 4.36a, b respectively at sonic freestream Mach
number [3]. Note that the cone semi-apex angle of 90◦ is the case of a flat-head
cylinder and the ogive with the semi-vertex angle of 90◦ is the case of hemisphere.
Applying the principle of stationary Mach numbers to slender axisymmetric bodies,
e.g., cone or ogive, where transonic similarity applies, produces:
2
2 +1
−1
2 M∞
γ M∞
CD =
(C D ) M∞ =1 +
2
2
(γ + 1) M∞
(γ + 1) M∞
(4.62)
This expression is valid for axisymmetric slender bodies that include both cones
and ogives. Therefore the experimental data on C D -cone at M∞ = 1, as shown in
Fig. 4.36a substituted in Eq. (4.62) allows for the extension of wave drag in transonic
range, i.e., from subsonic to supersonic. In particular the slope of the drag coefficient
near sonic flow is determined. The proof of the match between the theory and the
experimental results on a cone in the transonic range is shown in Fig. 4.37a [3]. The
rising dashed lines through sonic Mach number are produced from the theoretical
model, i.e., Eq. (4.62). Similarly, the wave drag coefficient on an ogive in the vicinity
of sonic flow may be explored using Eq. (4.62) in conjunction with the ogive wave
drag coefficient for sonic flow, i.e., Fig. 4.36b. Comparison with experimental results
shown in Fig. 4.37b is in good agreement with the theoretical model.
dmid
(a)
M
c
(b)
M
lN
1.2
c
N
= 45º
= lN /dmid
0.12
N
3
35º
0.8
4
CD
CD
p
p
0.08
25º
0.4
20º
5
6
0.04
10º
0
0.8
1.2
1.6
M
2.0
0
0.8
1.2
1.6
2.0
M
Fig. 4.37 Experimental data on wave drag coefficient in transonic flow (after Ref. [3]). Note that
λ N is the nose fineness ratio, FN in our notation. a Cone. b Ogive
194
4 Shock-Expansion Theory
4.6 Summary
Shock-expansion theory is a powerful tool in analyzing aerodynamic forces and
moments on pointed bodies in steady supersonic flow where the oblique shock(s) at
the leading edge are attached. This method produces exact results within the confines
of inviscid flow theory. The method breaks down however when the shocks become
detached where the flow at the leading edge exceeds the maximum turning angle
supported by straight oblique shocks. The two-dimensional flow problems are easily
analyzed using plane oblique shocks and expansion waves. The three-dimensional
problems are analyzed using conical shocks. The 3-D expansion through conical
Mach waves is best accomplished using the method of characteristics, i.e., the subject
of Chap. 5. Approximate method of local cones is introduced to estimate the effect
of 3-D expansion, using conical shocks associated with a smaller nose angle. The
problem of asymmetrical flows is also analyzed through the method of local cones.
The extension of aerodynamic forces through sonic velocity is accomplished through
the principle of stationary local Mach numbers. The wave drag coefficient on cones
and ogives in transonic flow are in good agreement with the theoretical predictions.
Problems
4.1 A two-dimensional projectile with a sharp nose is exposed to a Mach 3 flow, as
shown. Assuming the base pressure is p∞ , calculate:
(a) p1 / p∞
(b) p2 / p∞
(c) wave drag coefficient referenced to chord, Cd ≡ D /q∞ · c.
p1
M = 3.0
p
5°
10°
p
p2
c
4.2 Calculate the stagnation pressure measured by a Pitot tube on an inclined ramp
in a supersonic flow, as shown.
4.6 Summary
195
10°
M = 2.0
Pitot
p = 100 kPa
4.3 A symmetrical half-diamond airfoil has a leading-edge angle of 5◦ . This airfoil
is set at 5◦ angle of attack, as shown. The airfoil is placed in a windtunnel with test
section (T.S.) Mach number MT.S. = 2.0, pt,T.S. = 100 kPa and Tt,T.S. = 25 ◦ C.
Assuming γ = 1.4 and c p = 1.004 kJ/kg. K, use shock-expansion theory to calculate:
(a) Static pressure, p1 (in kPa)
(b) Static pressure, p2 (in kPa)
(c) Static pressure, p3 (in kPa).
p3
5º
MT.S. = 2.0
5º
y
p2
p1
x
4.4 A supersonic flow expands around a sharp corner, as shown. Calculate the following parameters:
(a) Downstream Mach number, M2
(b) Flow area ratio, A2 /A1
(c) The angle of Prandtl-Meyer fan envelope, δ.
1
Streamline
A1
M1 = 2.0
2
= 30º A
2
196
4 Shock-Expansion Theory
4.5 A symmetrical half-diamond airfoil has a nose angle θnose = 30◦ and is exposed
to a supersonic flow. A Pitot tube is installed on each of the three surfaces, as shown.
Calculate the Pitot tube readings on the airfoil surfaces at zero angle of attack.
Pitot tubes
2
M1 = 3.0
pt1 = 100 kPa
3
30º
30º
4
Pitot tube
4.6 Calculate the wave drag coefficient of a 2-D sharp-nosed projectile, as shown,
assuming the base static pressure is equal to ambient, i.e., pbase = p∞ . The 2-D
wave drag coefficient here is defined based on b, i.e., Cd ≡ D /q∞ · b. Assume the
angle of attack is zero.
20º
M = 2.0
b
4.7 A Prandtl-Meyer centered expansion wave is visualized with the wave angles
as shown. Calculate:
(a) the flow turning angle, i.e., wall angle, θw
(b) velocity ratio across the expansion waves V2 /V1 .
V1
30º
16.6º
w
V2
4.8 A half-diamond airfoil with a nose angle of 15◦ and a thickness-to-chord ratio
of 10 % is in a supersonic flow, as shown. Calculate:
(a) the chordwise location of the maximum thickness point (in % c)
(b) the trailing-edge angle
(c) the non-dimensional pressure on the three surfaces, p1 / p∞ , p2 / p∞ , p3 / p∞ .
4.6 Summary
197
1
2
tmax
M = 2.4
3
c
15º
4.9 Calculate the 2-D lift and drag coefficients of a two-dimensional wave rider, as
shown. Assume the base pressure is equal to the ambient, p4 = p1 . Shock wave
angle is β1 = 45◦ . The upper surface is aligned with the flight direction (surface 2).
Surface 3 is the lower surface and surface 4 is the base.
c
4
2
p4 = p1
M1 = 3.0
1
45º
3
Oblique
shock surface
4.10 A half-diamond unsymmetrical airfoil is shown at 5◦ angle-of-attack. Calculate:
(a) thickness-to-chord ratio, t/c
(b) p1 / p∞ , p2 / p∞ and p3 / p∞
(c) normal force coefficient, Cn
(d) axial force coefficient, Ca .
c
5º
1
M = 2.0
= 1.4
10º
2
t
20º
3
4.11 A symmetrical diamond airfoil is shown. Free-stream Mach number is M1 =
5.0. The angle of attack is 5◦ . Use shock-expansion theory to calculate:
(a) thickness-to-chord ratio, t/c
(b) pressure coefficients C p2 , C p3 , C p4 , C p5
(c) axial force coefficient, Ca .
198
4 Shock-Expansion Theory
t
2
M1 = 5.0
= 1.4
4
10º
10º
10º
10º
3
5
c
4.12 Consider a 10 % thick bi-convex airfoil of parabolic arc upper and lower profiles
with negative camber. The upper surface contributes 3 % to thickness at 50 % chord
and the lower surface contributes 7 % thickness to the airfoil at mid chord. First,
write the equations yu (x) and yl (x) for the upper and lower profiles. Next, use
shock-expansion theory to calculate:
(a) the pressure coefficient at the leading edge, and increments of 10 % chord until
trailing edge
(b) the lift coefficient
(c) the (wave) drag coefficient
(d) the pitching moment coefficient about the leading edge.
y
yu(x)
0.03c
M =2
x
0.07c
= 1.4
c
yl(x)
4.13 A hexagonal airfoil is a suitable shape for supersonic flow. Assuming the airfoil
shown is 10 % thick and it is at zero angle of attack in Mach 2.4 flow, use shockexpansion theory to calculate:
(a) pressure coefficients on all six panels
(b) the airfoil wave drag coefficient.
y/c
x/c
0.05
1.0
0.25
0.75
4.14 Identify the waves and draw the wave pattern around the two-dimensional
symmetrical body at points A, B and C, as shown. For expansion waves, calculate
the wave angle of the head and tail waves and with the oblique shocks calculate the
wave angle. Note that the body is symmetrical and is set at zero angle-of-attack.
4.6 Summary
199
M1 = 3.0
B
C
30º
A
10º
4.15 A supersonic flow approaches an expansion corner, as shown. The static pressure ratio p2 / p1 is 0.25. Calculate:
(a) Mach number on the expansion ramp, M2
(b) Wall angle, θw (in degrees).
1
p1
2
M1 = 2.5
p2
w
4.16 Two compression ramps in supersonic flow create two plane oblique shocks,
as shown. Calculate:
(a) shock wave angle, β1 (in degrees)
(b) ramp angle, θ1 (in degrees)
(c) shock wave angle, β2 (in degrees)
(d) estimate the second ramp angle, θ2 (in degrees).
1
2
M1n = 1.2
1
M1 = 2.4
2
M2n = 1.2
2
1
OS1
3
M2
OS2
200
4 Shock-Expansion Theory
4.17 A sharp hexagonal airfoil is placed in a supersonic flow. The Mach numbers
on its upper three surfaces are as indicated. Calculate:
(a) The semi-nose angle, θLE
(b) Static pressure ratio, p2 / p1
(c) Static pressure ratio, p3 / p1
(d) The semi trailing-edge angle, θTE .
1
2
M
M2 = 3
3
M1 = 2
M3 = 4
T.E.
L.E.
4.18 Consider a symmetrical, bi-convex parabolic profile airfoil that is of 10 %
thickness, as shown. The airfoil is in Mach 1.8 flow at zero angle of attack. Using
shock-expansion theory, calculate:
(a) Pressure distribution on the airfoil at x/c =0, 0.2, 0.4, 0.6, 0.8 and 1.0
(b) Estimate the wave drag coefficient by numerical integration of the pressures in
part (a).
y
t/c = 0.10
M = 1.8
= 1.4
x
x=0
x=1
4.19 Apply shock-expansion theory to the diamond airfoil in supersonic flow (as
shown), to calculate:
(a) upper and lower thickness-to-chord ratios, t1 /c and t2 /c
(b) the pressure coefficients, C p1 , C p2 , C p3 , C p4
(c) the normal force coefficient, cn
(d) the axial force coefficient, ca
(e) lift-to-drag ratio, L /D
(f) the pitching moment coefficient about the leading edge, cm,L E .
3
1
12º
2º
M = 2.0
t1
14º
t2
12º
14º
4
2
c
4.6 Summary
201
4.20 Consider a (two-dimensional) compression ramp in supersonic flow in air
(γ = 1.4) followed by a sharp corner that turns the flow back to horizontal, as shown.
The expansion waves that are centered at the shoulder intersect the attached oblique
shock at the nose and cause wave reflections (not shown). If we ignore the reflected
waves at the shock, we can use the shock-expansion theory to calculate the ratio of
static pressure downstream of the shoulder, p3 , to upstream of the shock, p1 .
1
3
2
Write a computer program or use the exact shock-expansion formulas in a spreadsheet, to calculate and graph p3 / p1 for a range of Mach numbers, M1 = 1.5, 3, 5, 10
and a range of ramp angles, θ, from the corresponding wave angles, μ, to near maximum turning angles. Do we recover the flow static pressure p1 when we get to
region 3?
4.21 An oblique shock is described by its upstream flow components (u and w are
normal and parallel to the shock, respectively) as shown. Calculate:
(a) Upstream speed of sound, a1
(b) Characteristic Mach number, M2∗ .
O.S.
1
u1 = 360 m/s
w1 = 410 m/s
*
= 280 m/s
= 1.4
R = 287 J/kg.K
w2
V1
u1
u2
V2
w1
2
4.22 A flat plate is in Mach 1.2 flow at 2◦ angle of attack. Use shock-expansion
theory to predict its lift-to-wave drag ratio. The ratio of specific heats is γ = 1.4.
4.23 A bi-convex airfoil has a nose angle, θnose = 10◦ . For an angle of attack of 5◦
and freestream Mach number, M∞ = 2.4, calculate:
202
4 Shock-Expansion Theory
(a) the upper surface static pressure at the nose, pnose / p∞
(b) the static pressure at the mid-chord point, pA / p∞ .
10º
A
5º
4.24 Consider a symmetrical bi-convex airfoil with a half-nose angle of 10◦ . It is
in a supersonic flow with M∞ = 2.8 and has an angle-of-attack of 2◦ . Use shockexpansion theory to calculate the static pressure difference between the lower and
upper surfaces at 50 % chord (see figure), Δp/ p∞ .
y
= 2º
x
M = 2.8
= 1.4
p
4.25 Consider a slender body of revolution at zero angle of attack in supersonic
flow with M∞ = 3.0. The semi-vertex angle is δ0 = 40◦ . Calculate the pressure
coefficient at the vertex and estimate the pressure coefficient at point A, where the
body angle is δ = 20◦ using the method of local cones.
= 1.4
M = 3.0
0
A
= 40º
0
= 20º
x
4.6 Summary
203
4.26 Consider a slender axisymmetric body, as shown, with the semi-vertex angle,
δ0 = 30◦ , in a supersonic flow with M∞ = 2.4 and at 5 degrees angle of attack, i.e.,
α = 5◦ . Apply the method of local cones to point A on the body where the local body
angle is 10◦ ; and A is on the meridian plane with Φ = 120◦ . Estimate the pressure
coefficient at A.
= 5º
0
= 30º
x
A
M = 2.4
= 1.4
= 10º
4.27 A tangent ogive has a nose fineness ratio of FN = 2.5. Calculate:
(a) the non-dimensional radius of curvature, R for the nose
(b) the semi-vertex angle, δ0 in degrees
(c) the pressure coefficients C pp and C pv at M∞ = 5 where the body angle is 10◦
(d) the wave drag coefficient of this tangent ogive (nose) at M∞ = 5.
4.28 Use the wave drag coefficient for a cone of semi-vertex angle of 45◦ at M∞ =
1.0, from Fig. 4.36a to create the pressure drag behavior of the cone in the transonic
range from M∞ = 0.8 to M∞ = 1.5. Graph your results and compare it to the
experimental data in Fig. 4.37a.
4.29 Consider four cones with semi-vertex angles of 0◦ , 15◦ , 30◦ and 40◦ , as shown.
First for a flow Mach number of M∞ = 3.0, calculate the cones’ wave drag coefficients and graph them on a chart, (as shown). You may use conical shock charts of
Shapiro (Fig. 4.17) for γ = 1.4.
204
4 Shock-Expansion Theory
M
=0º
M
=15º
M
=30º
M
=40º
1.0
0.8
0.6
0.4
0.2
0.0
0
10
20
30
40
(deg)
Finally, if we install a Pitot tube on the 30◦ cone in the Mach 3 flow, calculate the
stagnation pressure that the Pitot tube reads, as a fraction of flight stagnation pressure.
4.30 Use the wave drag coefficient for an ogive of fineness ratio 3.0 at M∞ = 1.0,
from Fig. 4.36b. You first need to calculate the semi-vertex angle for a tangent ogive
that has a fineness ratio of 3.0 before you can use Fig. 4.36b. Next, calculate the
pressure drag behavior of the ogive in the transonic range from M∞ = 1.0 to M∞ =
1.4. Graph your results and compare it to the experimental data in Fig. 4.37b for the
ogive with fineness ratio 3.0.
4.31 A cone of 30◦ semi-vertex angle is in supersonic flow at M∞ = 2.0 and zero
angle of attack. Use conical shock charts to find:
4.6 Summary
205
(a) the conical shock angle, σ, in degrees
(b) the Mach number on the surface of the cone, Mc
(c) the pressure drag coefficient of the cone, C D p .
Mc
M = 2.0
4.32 Consider a 30◦ (2-D) ramp in a Mach 3 flow of air with γ = 1.4. Also, there is
a cone of semi-vertex angle of 30◦ in the same supersonic flow condition. Calculate
and compare:
(a) The wave angles on the 2-D ramp and the cone
(b) The surface pressure on the ramp and the cone
(c) Explain the 3-D relieving effect as evidenced by this problem.
4.33 A two-dimensional wedge is in supersonic flight, as shown. The freestream
velocity is parallel to the upper surface. Assuming that the base pressure, p3 = p∞
and γ = 1.4, calculate:
(a) Pressure coefficients, C p1 and C p2
(b) Wave drag coefficient, cd , referenced to base area, i.e., cd = D /q∞ b (1)
(c) Lift coefficient, cl , referenced to planform area, i.e., cl = L /q∞ c (1).
c
30º
1
M = 3.0
b
3
l
2
4.34 Supersonic flow approaches a sharp expansion corner as shown. Calculate:
(a) Speed of gas upstream of the corner, V1
(b) Mach number downstream of the corner, M2
(c) Speed of sound downstream of the corner, a2 .
206
4 Shock-Expansion Theory
30º
2
1
M1 = 2.4
p1 = 30 kPa
T1 = -40 ºC
R = 287 J/kg.K
= 1.4
4.35 An oblique shock intersects a wall at a corner, as shown (not to scale). What
kind of wave(s) will form at the corner to accommodate the turning of the flow?
Calculate the Mach number in region 3.
1
2
M1 = 2.4
10º
30º
3
4.36 A flat plate is placed in a supersonic flow, as shown. Use shock-expansion
theory to calculate:
(a) lift-to-(wave) drag ratio, L /D
(b) the angle of the slipstream, Φ (in degrees).
P-M E.F.
O.S.
10º
M = 1.6
= 1.4
y
x
O.S.
P-M E.F.
4.6 Summary
207
4.37 In the method of local cones, we introduced a function for the cone semi-apex
angle, δc that was related to the angle of attack (in radians), local body angle and the
meridian angle following:
δc = δ − α cos Φ +
α2
cot δ · sin2 Φ
2
We wish to graph δc as a function of x for a tangent ogive, for α = 10◦ and the local
ogive angle δ varying between the nose angle, δ0 = 45◦ to the shoulder value, where
δ = 0 around the circumference of the ogive, i.e., with Φ varying between 0◦ and
360◦ . First make a table for δ versus x for a tangent ogive in increments of Δx = 0.1
along the axis of the ogive, i.e.,
x
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
δ (◦ ) 45
0
Then, calculate (e.g., using a spreadsheet) and graph δc versus Φ at each x. Note
that there is a singularity at the shoulder where δ = 0 [due to cot (δ) term], that
corresponds to x = 1. To avoid the singularity at the shoulder; we limit x to 0.99.
Identify the shadow region on the ogive at angle of attack, where δc is negative.
4.38 Consider a flat plate at angle of attack in supersonic flow. Use shock-expansion
theory to calculate and graph the drag polar (i.e., graph of Cl vs. Cd ) for the airfoil
in the range of α from 0 to 12◦ for constant freestream Mach numbers of M∞ = 2,
3 and 5.
4.39 A symmetrical diamond airfoil has a semi-vertex angle of 20◦ in a Mach 2 flow
in air (γ = 1.4).
(a) At what angle of attack, will the leading-edge shock detach?
(b) At what angle of attack will the flow downstream of the leading-edge shock be
sonic?.
4.40 Consider a 15◦ (2-D) ramp in a Mach-3 flow of air with γ = 1.4. Also, there is
a cone of semi-vertex angle of 15◦ in the same supersonic flow condition. Calculate
and compare:
(a) surface Mach number on the ramp and the cone
(b) the wave angles on the 2-D ramp and the cone
(c) the surface pressure coefficient on the ramp and the cone
(d) wave drag coefficient, referenced to base area for the ramp and the cone.
208
4 Shock-Expansion Theory
Compression Mach Waves
Free Boundary (constant pressure)
A
pa
2
1
60º
y
x
pa
4
3
M1 = 2.6
pa
H
pa
D
F
Centerline
G
C
E
B
Centered expansion fan
Fig. 4.38 Figure belonging to Problem 4.41
4.41 An over-expanded jet emerges from a symmetrical 2-D nozzle in a quiescent
ambient at pressure pa , as shown in Fig. 4.38. Due to symmetry, the centerline is a
streamline, thus it behaves like a solid wall. The oblique shocks that are formed (at
points A and B on the nozzle lip) to satisfy the pressure equilibrium condition on the
free boundary, i.e., p2 = pa , are reflected from the centerline (at point C) as oblique
shocks. The reflected oblique shocks impinge on the free boundary (at points D and
E) and are reflected as centered expansion waves (DF and DG) to maintain the free
boundary at constant pressure, i.e., p4 = pa .
Calculate:
(a) The nozzle pressure ratio, pt1 / pa
(b) The angle of the free boundary AD with respect to the x-axis
(c) Mach number in field number 3, M3
(d) The ratio of static pressure in field number 3 to ambient pressure, p3 / pa
(e) The angle of the free boundary DH with respect to the x-axis.
4.42 An airfoil is in transonic flow. The freestream Mach number is M∞ = 0.90
and the local Mach number at the peak suction point is Mmax = 1.24. Calculate:
(a) the critical pressure coefficient, C p, crit
(b) the pressure coefficient at peak suction point, C p, min
(c) the pressure coefficient at the stagnation point, C p, stag .
Cp, crit
Cp, stag
Cp, min
References
209
References
1. Anderson, J.: Modern Compressible Flow with Historic Perspective, 3rd edn. McGraw Hill,
New York (2003)
2. Anon.: Equations, Tables, and Charts for Compressible Flow. NACA TM 1135, Moffett Field
(1953)
3. Krasnov, N.F.: Aerodynamics of Bodies of Revolution. Elsevier Publication, New York (1970)
4. Liepmann, H.W., Bryson, A.E.: Transonic flow past wedge sections. J. Aeronaut. Sci. 17(12),
745 (1950)
5. Liepmann, H.W., Roshko, A.: Elements of Gas Dynamics. Wiley, New York (1957)
6. Shapiro, A.H.: The Dynamics and Thermodynamics of Compressible Fluid Flow. Ronald Press,
New York (1953)
7. Taylor, G.I., Maccoll, J.W.: The air pressure on a cone moving at high speed. In: Proceedings
of Royal Society, vol. 139. London (1933)
Chapter 5
Method of Characteristics
Abstract The governing equations of motion for two and three-dimensional inviscid
irrotational supersonic flows are derived. Although these equations are nonlinear,
there are unique curves in the physical space, known as characteristics that turn
the governing partial differential equations into ordinary differential equations that
may be integrated along the characteristics. Therefore, the method of characteristics (MOC) is an exact solution technique that is graphical in nature and since it is
nonlinear, it applies to upper transonic flow when the local Mach number is sonic or
supersonic. The theory of MOC is developed and applied to 2-D irrotational flows.
Within MOC, two competing techniques known as wave-field method and latticepoint approach are presented. The first example that is detailed is the minimum-length
supersonic nozzle design. The principle of wave cancellation at a solid boundary is
used in supersonic nozzle design and it is well described. The next application is
the wave pattern in supersonic exhaust plume. Here, wave reflection from a free
boundary, i.e., the free shear layer, is introduced and analyzed. The cases of underand over-expanded exhaust plumes are considered and MOC is applied to capture
the spatial evolution of the jet. MOC is extended to axisymmetric irrotational flows.
Family of supersonic nozzles, deflecting jets, non-uniform inlet condition, streamlines and ducts and curved shocks are the applications of MOC that are treated in
this chapter. This chapter contains 9 MOC application examples and 21 practice
problems at the end of the chapter.
5.1 Introduction
The governing partial differential equations in fluid mechanics are nonlinear, even in
the case of 2-D inviscid, irrotational flows. Although, linearization as an approximate
method is successfully developed for thin, slender bodies at small angle of attack
in subsonic and supersonic flows, the accuracy of solution is diminished when perturbations are no longer small. In addition, we learned that some flow regimes, e.g.
transonic, may not be linearized even for slender bodies. Interestingly, in supersonic
flow where the governing partial differential equation is hyperbolic, there are unique
curves in physical space called characteristics that turn the governing nonlinear
© Springer Science+Business Media Dordrecht 2015
R. Vos and S. Farokhi, Introduction to Transonic Aerodynamics,
Fluid Mechanics and Its Applications 110, DOI 10.1007/978-94-017-9747-4_5
211
212
5 Method of Characteristics
partial differential equations into ordinary differentials that can be integrated.
Therefore, we have exact solutions in supersonic flow along the characteristic curves,
once we can identify them in the flowfield. The integration of the governing equations
produces the compatibility conditions that are held constant along the characteristics.
The method of characteristics is inherently graphical since it relies on developing
a network of characteristic curves. Thus, mapping of a supersonic flowfield starts
with an initial data line (as an initial-value problem) and literally propagates downstream along characteristic directions in the flow. Since the analysis does not rely
on the linearization of the governing equations, the upper transonic flow regime
(where M ≥ 1) is also included in the analysis. In this chapter we present the twodimensional irrotational flows first and then extend the results to axisymmetric flows.
The examples and applications conclude the present chapter.
5.2 2-D Irrotational Flows
Two-dimensional flows in Cartesian coordinates are described by the velocity field:
V = ui + vj
(5.1)
For irrotational flows where the curl of the velocity field vanishes, i.e.,
∇×V =0
→
∂v
∂u
=
∂y
∂x
(5.2)
The mathematical implication of the irrotationality condition (i.e., Eq. (5.2)) is that
a scalar potential function, Φ exists such that:
V = ∇Φ
(5.3)
u≡
∂Φ
∂x
(5.4a)
v≡
∂Φ
∂y
(5.4b)
In two-dimensional flow, we have
The law of conservation of mass for steady, two-dimensional flows requires:
∂u
∂v
u ∂ρ
v ∂ρ
+
+
+
=0
∂x
∂y
ρ ∂x
ρ ∂y
(5.5)
The steady, compressible inviscid momentum equations expressed in 2-D Cartesian
coordinates are:
5.2 2-D Irrotational Flows
213
u
∂u
1 ∂p
∂u
+v
=−
∂x
∂y
ρ ∂x
(5.6a)
u
∂v
1 ∂p
∂v
+v
=−
∂x
∂y
ρ ∂y
(5.6b)
Since the flow is isentropic, we may relate the pressure gradient to the density gradient
and the local speed of sound according to:
∂p
∂ρ
= a2
∂x
∂x
(5.7a)
∂p
∂ρ
= a2
∂y
∂y
(5.7b)
Combining Eqs. (5.6) and (5.7) substituting them for the density terms in Eq. (5.5)
we get the following equation in velocity components u and v and the local speed of
sound:
u 2 ∂u
2uv ∂u
v2 ∂v
1− 2
− 2
+ 1− 2
=0
(5.8)
a
∂x
a ∂y
a
∂y
Replacing the velocity components in Eq. (5.8) with partial derivatives of the scalar
potential function, Φ, we get the full potential equation in steady two-dimensional,
irrotational flows:
Φy2
2Φx Φy
Φx2
1 − 2 Φxx −
Φxy + 1 − 2 Φyy = 0
a
a2
a
(5.9)
Consider the motion of a small disturbance in 2-D supersonic flow. Infinitesimal
pressure waves, called Mach waves, are created by the body that separate the zone
of action where the flow is influenced, i.e., perturbed, by the body and the zone
of silence where the flow is unaffected/unaware by the presence of the body. The
flow within the zone of action is different in magnitude and direction, albeit small,
from the flow in the zone of silence. Both of these zones however satisfy Eq. (5.5)
thus create two solutions, Φ1 and Φ2 representing the two regions. Since the flow
is continuous, the value of the potential and its first derivative (i.e., speed) match at
the boundary between the two regions, namely the Mach wave. However, according
to Cauchy’s theorem, the solution of the second order partial differential equations,
such as Eq. (5.5), is completely defined when the value of the function and its first
derivative are known on a line. Therefore, the second derivative of the potential, i.e.,
in our case, the first derivative of the speed, is no longer continuous nor is it uniquely
∂v
) is at best indeterminate and may
defined at the line. In fact the derivative ∂∂uy (or ∂x
be discontinuous across the Mach wave. Armed with this physical insight, we try to
establish the characteristics related to Eq. (5.5) by finding the pathways across which
214
5 Method of Characteristics
∂v
velocity derivative ∂∂uy (or ∂x
) is indeterminate. We form a system of three equations
and three unknowns by noting that:
du =
∂u
∂2Φ
∂2Φ
∂u
dx +
dy =
dy
dx +
2
∂x
∂y
∂x
∂x∂ y
(5.10a)
dv =
∂v
∂v
∂2Φ
∂2Φ
dx +
dy =
dx + 2 dy
∂x
∂y
∂x∂ y
∂ y
(5.10b)
Therefore, the system of three equations for the three unknowns, Φx x , Φx y and Φ yy
are written as:
u2
2uv
v2
1 − 2 Φxx − 2 Φxy + 1 − 2 Φyy
(5.11)
a
a
a
dxΦx x + dyΦx y = du
(5.12)
dxΦx y + dyΦ yy = dv
(5.13)
∂v
The solution for Φx y , which is our ∂∂uy or ∂x
, is expressed in terms of Cramer’s rule
as the ratio of two determinants:
(1 − u 2 ) 0 (1 − v2 )
a2
a2 dx
du
0 0
dv
dy Φxy = (1 − u 2 ) −2uv (1 − v2 )
a2
a2
a2 dx
dy
0 0
dx
dy (5.14)
The condition for the indeterminate Φx y is satisfied by demanding the numerator and
denominator of Eq. (5.14) to be simultaneously zero. First, by setting the determinant
in the denominator equal to zero, we get the equation for the characteristics, namely:
v2
u2
−2uv
dy
−
1
−
dx = 0
1 − 2 (dy)2 − dx
a
a2
a2
(5.15)
Dividing both sides of Eq. (5.14) by (dx)2 and rearranging, we get a quadratic equation for the slope of the characteristic curve, dy/dx:
2
u2
dy
v2
2uv dy
1− 2
+ 1− 2 =0
+ 2
a
dx
a
dx
a
(5.16)
5.2 2-D Irrotational Flows
215
The solution of the quadratic equation in dy/dx is:
dy
dx
=
uv
±
a2
uv
a2
2
2
1 − av 2
2
2
1 − ua 2
characteristic
=
2
− 1 − ua 2
uv
±
a2
2
u2
+ av 2 − 1
a2
2
1 − ua 2
=
uv
±
a2
√
M2 − 1
(5.17)
2
1 − ua 2
We note that the square root of (M 2 − 1) written at the end of Eq. (5.17) holds
the key to the nature of the characteristics, namely for subsonic flows the square
root becomes imaginary and thus we conclude that no (real) characteristics exist in
subsonic flows. In supersonic flows however, the square root yields real numbers and
thus we conclude that two (real) characteristics exist. At sonic flow, the square root
identically vanishes and thus one real characteristic exists. These in the language
of classifications of the second order partial differential equations (see Sect. 2.2.4)
are called elliptic, hyperbolic and parabolic respectively, corresponding to subsonic,
supersonic and sonic flows. Figure 5.1 shows the definition sketch for the velocity
vector in x − y plane, its components and the flow angle, θ that it makes with respect
to x-axis. Based on this, we express the velocity components in terms of the velocity
magnitude and the angle, θ, according to:
dy
dx
u = V cos θ
(5.18)
v = V sin θ
(5.19)
√
M 2 sin θ cos θ ± M 2 − 1
=
1 − M 2 cos2 θ
characteristic
(5.20)
We express Mach number in terms of the Mach angle, μ in Eq. (5.20), to get
dy
dx
=
characteristic
Fig. 5.1 Definition sketch of
the velocity field in
two-dimensional plane (in
Cartesian coordinates)
sin θ cos θ
cot μ
sin θ cos θ ± sin μ cos μ
sin2 μ
=
cos2 θ
sin2 μ − cos2 θ
1 − sin2 μ
(5.21)
y
V
v
θ
u
x
216
5 Method of Characteristics
The right-hand-side of Eq. (5.21) simplifies to the following compact form:
dy = tan (θ ∓ μ)
(5.22)
dx characteristic
Equation (5.22) reveals that the local characteristics about a (local) velocity vector
with flow angle, θ, are indeed Mach waves above and below the velocity vector. The
characteristic that is above the velocity vector is called C+ and the characteristic
that is below the velocity vector is called C− . The local flow at point A and its two
characteristics are shown in Fig. 5.2, as definition sketch.
Now, by setting the determinant in the numerator of Eq. (5.14) equal to zero, we
establish the compatibility condition along the two characteristics, namely
2
2 1 − ua 2 0 1 − av 2 (5.23)
dx
=0
du
0
0
dv
dy
Expanding the determinant, we get:
u2
v2
1 − 2 dudy + 1 − 2 dvdx
a
a
(5.24)
Using the slope of the characteristics dy/dx (from Eq. (5.17)) in Eq. (5.24) and the
substitutions (5.18) and (5.19), we simplify the compatibility equation to the following form:
2
1 − ua 2
dv
=
2
du
1 − v2
a
dy
dx
2
1 − ua 2
d (V sin θ)
=−
=
2
d (V cos θ)
1 − v2
characteristic
− auv2 ±
a
√
M2 − 1
2
1 − ua 2
(5.25)
y
C+ Characteristic with
Tangent to C+ characteristic at A
(Left-Running Mach Wave)
( dy / dx)A = tan( θA + µA )
VA (> a)
µA
A
θA
µA
Tangent to C- characteristic at A
(Right - Running Mach Wave)
C- Characteristic with
(dy / dx)A = tan( θA − µA )
x
Fig. 5.2 Local characteristics are right- and left-running Mach waves in two-dimensional supersonic flow
5.2 2-D Irrotational Flows
217
Further simplification yields:
√
d (V sin θ)
M 2 sin θ cos θ ∓ M 2 − 1
=
d (V cos θ)
1 − M 2 sin2 θ
(5.26)
After extensive manipulations of (5.26), we get:
dθ = ∓ M 2 − 1
dV
V
(5.27)
We recognize the right-hand-side of (5.27) as the differential of the Prandtl-Meyer
function dν, based on our derivations in Chap. 4. Therefore, the compatibility equations along the two characteristics reduce to:
dθ ∓ dν = 0
(5.28)
Upon integration of (5.28), we arrive at the two algebraic compatibility conditions
that hold along C+ and C− characteristics with the constants of integration, K + and
K − respectively, i.e., we get:
θ − ν = K + Constant along C+ characteristic
(5.29a)
θ + ν = K − Constant along C− characteristic
(5.29b)
Hence from an initial data line in 2-D supersonic flows, we establish both local characteristic directions as well as the constants (K + and K − ) along those characteristics.
Although characteristic directions change as we march in the flow direction, the constants that we established from the initial data line, (K + and K − )-, remain the same.
We had introduced the zone of action and zone of silence in Chap. 4 to lie within the
Mach cone downstream of a point in supersonic flow and the entire space upstream of
the Mach cone, respectively. In this chapter, the two characteristics that emanate from
a point in supersonic flow in the downstream direction are indeed consistent with the
boundaries of the zone of action. We can now extend this concept to upstream and
define a domain of dependence that impacts the flow at a point in supersonic flow that
lies within the upstream characteristics passing through the point. In this context, we
may regard the domain of dependence upstream of a point as the “past” events that
influenced the state of flow at point A and the region of influence in the downstream
direction to represent the “future” of the flow that is impacted by the “presence” at
point A. These physical arguments impact the finite differencing strategy (of information flow) in computational fluid dynamics. Figure 5.3 is a definition sketch that shows
the concepts of domain of dependence and region of influence in supersonic flow.
The construction of the characteristic network relies on three unit processes. These
unit processes help calculate flow properties at internal points, wall points and at
shock points. First, the internal point is created at the intersection of two characteristics that still lies within the flowfield. Consider points 1 and 2 in the x − y plane
218
5 Method of Characteristics
y
C+ Characteristic
Domain of Dependence
of point A
VA
A
Region of Influence
of point A
C- Characteristic
x
Fig. 5.3 Definition sketch of intersecting C+ and C− characteristics and their physical significance
in supersonic flow
Fig. 5.4 Intersection of C−1
and C+2 creates point 3,
which lies downstream of
points 1 and 2
y
C+1
M1
1
y1
C-1
3
y3
y2
M3
C-3
C+2
2
C+3
M2
C-2
x
x2
x1
x3
(see Fig. 5.4) for which we know the flow inclination/direction, θ1 and θ2 as well as
the Mach numbers, M1 and M2 . From local Mach numbers, we calculate the Mach
angles, μ1 and μ2 as well as the Prandtl-Meyer (P-M) angles, ν1 and ν2 . These angles
are sufficient to establish the characteristic directions for C+1 , C−1 , C+2 and C−2
as well as the compatibility constants K +1 , K −1 , K +2 and K −2 along those characteristics respectively. The intersection of C−1 and C+2 creates point 3, which shares
the same compatibility constant with K −1 and K +2 that form two equations with
two unknowns, i.e.,
(5.30a)
θ3 + ν3 = θ1 + ν1 = K −1
θ3 − ν3 = θ2 − ν2 = K +2
(5.30b)
The solution to Eqs. (5.30a) and (5.30b) yields the flow angle θ3 and (indirectly,
through P-M angle, ν3 ) Mach number at point 3:
5.2 2-D Irrotational Flows
219
θ3 =
K −1 + K +2
2
(5.31a)
ν3 =
K −1 − K +2
2
(5.31b)
To graphically construct point 3, from the known points 1 and 2, we graph the
C+2 characteristic at the average angle corresponding to points 2 and 3 (along a C+
characteristic), i.e., C+2 is graphed at the angle of
μ2 + μ3
θ2 + θ3
+
2
2
(5.32)
Similarly, the C−1 will be graphed at the average angles corresponding to points 1
and 3 on the C− characteristic, namely,
θ1 + θ3
μ1 + μ3
−
2
2
(5.33)
Note that we are averaging the angles of the characteristics between points 1 and
3 and 2 and 3 and not the slopes (i.e., the tangent of the angles). Also, since we
are drawing straight lines for the characteristic network between the adjacent grid
points, in general the accuracy will improve if the points are closer to each other.
Figure 5.5 shows the average angles used in connecting C− and C+ characteristics
to create point 3.
We next treat the unit process that involves the intersection of a characteristic and
an adjacent wall. Consider a point in the flowfield (e.g., point 1 in Fig. 5.6) that has
its local flow angle, θ1 and the local Mach number, M1 known. Based on these, we
can establish the angles μ1 and ν1 from the Mach number M1 and thus calculate the
K +1 and K −1 corresponding to point 1 in the flow. Now, consider an adjacent wall
to point 1, as shown in Fig. 5.6.
Fig. 5.5 Definition sketch
for the average angles on
connecting C− and C+
characteristics for an interior
point
y
1
y1
⎛ θ1 + θ3 ⎞ ⎛ μ1 + μ3 ⎞
⎜
⎟−⎜
⎟
⎝ 2 ⎠ ⎝ 2 ⎠
(C− )1−3
3
y3
(C+ ) 2−3
⎛ θ 2 + θ3 ⎞ ⎛ μ2 + μ3 ⎞
⎜
⎟+⎜
⎟
⎝ 2 ⎠ ⎝ 2 ⎠
2
y2
x
x2
x1
x3
220
Fig. 5.6 The unit process
involving the intersection of
a characteristic and an
adjacent wall
5 Method of Characteristics
y
y1
1
C +1
M1
C-1
2
y2
θ w2
x1
x2
x
The C−1 characteristic intersects the wall at point 2 with coordinates (x2 , y2 ), as
shown in Fig. 5.6. We thus conclude that K −2 and K −1 are equal, which means that
K −2 = θ2 + ν2 = θ1 + ν1 = K −1
(5.34)
The flow angle at point 2 coincides with the wall angle at point 2 (from flow tangency condition), which means that we have an estimate for θ2 , based on the C−1
characteristic line that was drawn at the angle of (θ1 − μ1 ) from point 1 towards
point 2. The estimate of θ2 , from the known wall angle θw at (x2 , y2 ), substituted in
Eq. (5.34) yields an estimate for P-M angle ν2 , according to:
ν2 = K −1 − θw2
(5.35)
The P-M angle ν2 from Eq. (5.35) then yields M2 and μ2 . Now based on the new
information at the wall, namely μ2 , we can redraw the C−1 characteristic for the
average angle of
θ1 + θ2
μ1 + μ2
−
(5.36)
2
2
The intersection of the new (C− )1−2 characteristic and the wall will result at a new
point 2, with its corresponding θw2 . Depending on the correction of the wall angle
in our calculations, i.e., θw2 − θw2 , we may have to iterate to reach the desired level
of accuracy. The factors that influence the correction angle of the wall are the wall
curvature and the proximity point 1 to the wall.
The next unit process involves the intersection of a characteristic, which is a
pressure wave, and an oblique shock. The drawing in Fig. 5.7 serves as a definition
sketch for this interaction.
At point 1 in the flowfield, we know the local Mach number, M1 and the flow
angle, θ1 . Therefore, we know K +1 , which remains constant on C+1 characteristic.
The point 2 is at the intersection of the C+1 characteristic and the oblique shock
wave, as shown in Fig. 5.7. The interaction of a characteristic, i.e., a Mach wave
or pressure wave, with an oblique shock causes the shock strength to change, and
a characteristic, i.e., a pressure wave, to be reflected from the shock wave. These
are schematically shown in Fig. 5.7 where the angle of the oblique shock, β∞ has
changed to β2 after the C+1 characteristic has interacted with the wave. Also note
that a C−2 wave is reflected from the oblique shock. Since point 2 is on the same C+
5.2 2-D Irrotational Flows
221
y
Oblique Shock (influenced by
C+1 pressure wave)
M2
β2
M∞
θ2
2
C-2
M∞
C+1
β∞
M1
θ1
1
C-1
Oblique Shock
x
Fig. 5.7 Definition sketch of a C+ characteristic (pressure) wave interaction with an oblique shock
wave
characteristic as 1, K +2 = K +1 . Therefore, we have
K +2 = θ2 − ν2 = θ1 − ν1 = K +1
(5.37)
The flow angle at 2 is the turning or deflection angle across the oblique shock at 2,
which is related to M∞ and β2 via oblique shock relations. The strategy is to guess a
flow deflection angle, θ2 and then find the corresponding wave angle β2 from oblique
shock relations establish M2 and thus ν2 . Then check Eq. (5.36) to see whether θ2 −ν2
is indeed equal to K +1 . We then need to iterate on our choice of flow angle at 2 to
converge on a consistent solution. Depending on the application, we may set the level
of acceptable accuracy at for example 0.01◦ . The classical textbooks of Liepmann
and Roshko [5] and Shapiro [7] as well as the contemporary book by Anderson on
compressible flow [1] should be consulted.
Example 5.1 We have experimentally measured the flow directions and Mach numbers at two points (1 and 2) in a supersonic flow, as shown. Calculate:
(a) The flow angle and Mach number at the intersection of the C− 1 and C+2 , i.e.,
point 3.
(b) The angle of the C− characteristic between the points 1 and 3.
(c) The angle of the C+ characteristic between the points 2 and 3.
(d) The coordinates of point 3.
y
y1 =1
1
M1
(C− )1− 3
3
y3
M3
M1
(C+ )2−3
y2 =0
2
x1 = x 2 =0
2
M2
x
x3
1
(deg )
M2
+5
1.8
2
(deg )
0
222
5 Method of Characteristics
Solution:
Based on the Mach number and the flow angle specified for points 1 and 2, we
complete the table:
Node M θ (deg) ν (deg) μ (deg) K − (deg) K + (deg)
1
2 5
26.38 30
31.38
−21.38
2
1.8 0
20.73 33.75 20.73
−20.73
Since at node 3, K −3 = K −1 and K +3 = K +2 , we get = 5.325◦ (from Eq. (5.31a))
+2
= 26.055◦ (from Eq. (5.31a)). From Prandtl-Meyer
and we get ν3 = K −1 −K
2
function ν3 , we get the corresponding Mach number to be M3 ≈ 1.988 and μ3 ≈
30.2◦ . The angle of the C− characteristic connecting nodes 1 and 3 is the average of
the C− angles, following Eq. (5.33), namely
θ1 + θ3
μ1 + μ3
−
≈ −24.94◦
2
2
and the angle of the C+ characteristic connecting nodes 2 and 3 follows Eq. (5.32), i.e.,
μ2 + μ3
θ2 + θ3
+
≈ 34.64◦
2
2
The coordinates x3 , y3 are calculated from the intersection of two straight lines, with
known angles, i.e., slopes. We find:
x3 ≈ 0.8652◦
and
y3 ≈ 0.5977◦
Note that the fineness of the characteristic net is set by the closeness of the points
along the initial data line.
5.3 Design of a 2-D Supersonic Minimum-Length Nozzle
(MLN)
The method of characteristics can be used to design the contour of a supersonic nozzle
or wind tunnel. A convergent-divergent nozzle accelerates the gas from subsonic flow
in the converging section, through sonic flow at the throat to supersonic flow in the
divergent section. The method of characteristics may be initiated at the sonic throat
and be extended to the diverging section with the help of the two characteristic
families, C+ and C− . The function of these characteristics, or Mach waves, is to
change the local flow direction by a small amount and thus accelerate or decelerate
the flow. Associating the turning angle with the strength of these waves, we may
assign a prescribed strength; say 1◦ or 2◦ , to each wave. We may then construct
a network of C+ and C− characteristics (i.e., of expansion Mach waves for nozzle
design) of known strength that cause the flow to change direction and thus accelerate
in supersonic flow. The local flow angle, θ, decreases when it encounters a LeftRunning Wave (LRW), which we called a C+ characteristic and the local flow angle
5.3 Design of a 2-D Supersonic Minimum-Length Nozzle (MLN)
(a)
(b)
LRW or
C+ Characteristics
y
y
V
V> a
θ =0
Counter +θ
clockwise
Direction
223
V>a
θ= 0
-θ
Clockwise
Direction
V
RRW or
C- Characteristics
x
x
Fig. 5.8 Basic expansion corners that create two families of characteristics that change the flow
angle to ±θ a Encountering Right-Running expansion Mach waves increasing the flow angle, θ.
b Encountering Left-Running expansion Mach waves decreasing the flow angle, θ
increases when it encounters a Right-Running Wave (RRW), which was called a
C− characteristic, (see Fig. 5.8). Therefore, when an axial flow in the nozzle throat
encounters the same number and strength of the LRW and RRW, i.e., C+ and C−
characteristics, the net flow turning angle at the nozzle exit is zero, therefore the
nozzle exit achieves a supersonic flow that is in the axial direction. With the goal of
achieving a wave-free, uniform, parallel flow at the nozzle exit with a design exit
Mach number of Me , we propose to generate the same number and strength of C+
and C− characteristics that accomplish the exit Mach number, Me . Figure 5.8 helps
visualize an increasing flow angle when RRWs are encountered and a decreasing flow
angle when the supersonic flow interacts with LRWs. Here we use the convention
that the flow angles above the x-axis (i.e., in counter-clockwise direction) are positive
and below the x-axis (i.e., in clockwise direction) are negative. This makes the x-axis
serve as the reference direction where the flow angle is zero, i.e., θ = 0.
Now if we apply the concept that is schematically shown in Fig. 5.8 to a sonic
throat with uniform flow, the head Mach wave would be normal to the flow (since
μ = 90◦ for sonic flow) followed by the subsequent expansion Mach waves. On
the upper corner, a centered expansion wave (C− characteristics) appears that turns
the flow in the positive θ direction. The opposite happens at the lower corner where
the centered expansion wave (C+ characteristics) turns the flow in the negative θ
direction. The flow that emerges from the first set of C− characteristics encounters
the second set, which belongs to the C+ characteristic family. All the positive θ
gained in the first encounter (on the upper corner) is reversed to achieve an axial flow
condition at the nozzle exit. Similarly, all the negative θ gained on the lower corner
is reversed to achieve axial flow condition at the exit. After the first turning at the
sonic corner where the wall attains the maximum angle, θw, max , we turn the wall
continually in the opposite direction to cancel the waves and achieve a wave-free
exit flow condition (see Fig. 5.9). That is the essence of wave cancelation at the wall
where the wall angle is adjusted to match the flow imposed by the incident wave.
An approximation that we often make at the throat of a choked nozzle is in
the shape of the sonic line at the throat. For example, here we have assumed the
224
5 Method of Characteristics
y
e
θ w, max
th
C+
C+
Mth =1.0
θ th =0
γ, R
Uniform flow
in the throat
section
C+
Uniform flow in the
Nozzle exit plane
Cx
Centerline
C-
C-
Me>1
θ e=0
γ, R
C-
Fig. 5.9 Centered expansion waves at the throat of a choked nozzle with uniform flow that form a
network of C+ and C− characteristics
sonic line is straight and the flow at the throat is uniform, as shown in Fig. 5.9.
The condition of uniform flow is reasonable for nozzles with slow area-variation
convergent sections. However, if the area variation in the converging section prior to
the throat is large, it will affect the throat curvature and thus the sonic line is curved,
which causes flow non-uniformity. For additional discussion on this topic the reader
may consult Shapiro [7] and Ferri [3].
We take advantage of flow symmetry with respect to the x-axis and use the MOC
to design only the upper contour (or upper half) of the nozzle. The lower contour is the
reflection of the upper contour through the mirror of the x-axis. In essence since the
x-axis is a streamline, we may treat it as a solid wall. The reflected C+ characteristics
are then treated as reflections from a solid wall. Note that the symmetry argument
allowed the number of the nodes in the characteristic network to be reduced by
half. Figure 5.9 shows a sonic throat with uniform flow and four C− characteristics
when we include the first, i.e., the μ = 90◦ , characteristic. The reflection of the
90◦ C− characteristic from the x-axis is upon itself, as expected. However, it is
the next C− characteristic with a prescribed incremental flow turning angle, θinitial
that creates a reflection as C+ characteristic, as shown in Fig. 5.9. The subsequent
C− characteristics emanating from the sharp corner cause additional turning and
reflections that create the network of C− and C+ characteristics. Returning to the
four characteristics shown in Fig. 5.9, we conclude that the characteristic that sets off
the MOC is the one with a non-zero θinitial . Let us use an expanded view of Fig. 5.9
and label the characteristic nodes, for further analysis (see Fig. 5.10). Note that the
exit Mach number, Me , is considered to be a nozzle design parameter, i.e., known.
Consequently, the Prandtl-Meyer angle at the nozzle exit, νe , is also a known design
parameter. The ratio of specific heats of the gas is γ and is treated as constant in our
application of MOC. In reality, the expansion process in the nozzle causes a reduction
in the gas temperature and thus chemical composition of the gas and γ change. The
same arguments apply to the gas constant, R, which changes with gas composition.
However, we treat both γ and R as constant, which means that we assume frozen
chemistry in the nozzle.
5.3 Design of a 2-D Supersonic Minimum-Length Nozzle (MLN)
y
*
A
2
Diverging Nozzle Outer Contour
7
Nozzle throat
4
Converging
Nozzle
C+
C+
θw, max
A
C+
Mth = 1.0
C3
θth = 0
Cνth = 0
6
γ = const.
Nozzle axis
2
R = const. C1
5
8
Diverging Nozzle Length
225
9
e
Ae
Me > 1
θe = 0
νe = f (Me)
γ = const.
R = const.
x
Fig. 5.10 The grid points in a characteristic network in a 2-D supersonic nozzle
According to Fig. 5.10, the flow angle at node 8 is zero, θ8 = 0, since it lies on
the x-axis (i.e., plane of symmetry). The C− characteristic that passes through node
8 and the sharp corner at point “A” has to maintain the same K − value, namely
θ 8 + ν8 = θ A + ν A
(5.38)
Node 8 is the last node on the x-axis; therefore its Prandtl-Meyer angle is the same
as the exit value, νe . Also, since the flow turning angle started from zero at the sonic
condition the value of the wall angle is the same as the corresponding Prandtl-Meyer
angle, as discussed in Chap. 4. Therefore, Eq. (5.38) may be re-written as
θ8 + ν8 = νe = θ A + ν A = 2θw, max
(5.39)
We have now established a geometrical relation between the maximum wall angle,
θw, max and the Prandtl-Meyer angle at the nozzle exit, i.e., a nozzle design parameter
corresponding to the exit Mach number, Me . We have shown:
θw, max =
νe
2
(5.40)
Since we started the MOC with the selection of a small θinitial angle, node 1 attains
that angle. However node 1 is on the x-axis and its flow angle should be identically
zero. This indeed is the small discrepancy that our MOC introduces in the design
of the supersonic nozzle. The characteristic that connects points A and 1 is a C−
characteristic that is also a Mach wave. We first determine the Mach number and the
corresponding Mach and Prandtl-Meyer angles at node 1. The flow angle was the
θinitial that we had chosen to begin the MOC procedure, i.e.,
θ1 = θinitial
(5.41)
226
5 Method of Characteristics
Since the flow expanded from the sonic throat condition and turned by the θinitial
angle, the Prandtl-Meyer angle attains the same value, namely
ν1 = θinitial
(5.42)
The angles θ1 and ν1 allow us to establish the constants K ± at node 1, namely
K −1 = θ1 + ν1 = 2θinitial
(5.43a)
K +1 = θ1 − ν1 = 0
(5.43b)
In addition, the Prandtl-Meyer angle ν1 corresponds to a Mach number, M1 , which
then establishes the Mach angle, μ1 . Now, we have calculated the flow parameters
corresponding to node 1. To graphically construct the first C− characteristic (connecting points A and 1), we use the average angle rule described in Eq. (5.32) for a
C− characteristic.
The second C− characteristic emanating from the sharp throat corner “A”, is
established by choosing a flow turning angle, Δθ. Therefore the node “A” has a new
flow turning and Prandtl-Meyer angles that are equal to each other and are described
by:
(5.44a)
θ A = θinitial + Δθ
νA = θA
(5.44b)
The corresponding K − value that is shared by node 2 is:
K −2 = θ2 + ν2 = 2(θinitial + Δθ)
(5.45)
Equation (5.45) is one equation with two unknowns, θ2 and ν2 . To establish a second
equation involving the same unknowns, we note that node 2 is on the C+ characteristic
that passes through node 1, therefore, it shares the same K + value with node 1, i.e.,
K +2 = θ2 − ν2 = K +1 = 0
(5.46)
Solving Eqs. (5.45) and (5.46) simultaneously, we get:
θ2 = ν2 = θinitial + Δθ
(5.47)
From Prandtl-Meyer angle ν2 we get the flow Mach number M2 and subsequently
the Mach angle, μ2 . Now, we have two straight characteristics to graph, one from A
to node 2 and one from node 1 to node 2. But this process is identical to the worked
Example 5.1.
The third C− characteristic takes the flow turning at “a” through an additional
turning angle, Δθ. Following the same arguments as node 2, we get the angles θ3
and ν3 and the corresponding K values as
5.3 Design of a 2-D Supersonic Minimum-Length Nozzle (MLN)
227
θ3 = ν3 = θinitial + 2Δθ
(5.48a)
K −3 = θ3 + ν3 = 2(θinitial + 2Δθ)
(5.48b)
K +3 = θ2 − ν3 = 0
(5.48c)
In this example, we have only used three C− characteristics to demonstrate the
principle of MOC applied to the design of a minimum-length nozzle. Therefore, the
flow angle at node 3, in our example, is the same as the wall angle, θw, max .
θ3 = θw, max = ν3 = θinitial + 2Δθ =
νe
2
K −3 = νe
(5.49a)
(5.49b)
The connection between the design exit P-M angle and the steps in our MOC is
revealed in Eq. (5.49a). Namely, for a give design νe , we divide the P-M angle by
two and select the initial turning angle and the subsequent steps in Δθ. In the simple
example of Fig. 5.10, we have taken two steps beyond θinitial .
Continuing with our calculations, we note that the flow properties at node 4 and
3 are the same, i.e.,
θ4 = θ3 = θw, max and ν4 = ν3 = θw, max
(5.50)
We graph the wall angle between the two nodes on the wall at the average flow angles
at the two nodes, which in the case of A and 4, is θw, max . Now, let us proceed to
node 5 on the x-axis. We conclude that the flow angle at 5 has to be zero and it lies
on the same C− characteristic as node 2, therefore
θ5 = 0
(5.51a)
K −5 = K −2
(5.51b)
We get the Prandtl-Meyer angle at node 5 from Eqs. (5.51a) and (5.51b) to be:
ν−5 = 2 (θinitial + Δθ) = νe − 2Δθ
(5.52)
The C+ characteristic that is reflected form the plane of symmetry has the value
K +5 = θ5 + ν5 = −νe + 2Δθ
(5.53)
We may proceed to node 6 as it lies at the intersection of two characteristics, namely
C−3 and C+5 . This results in the flow properties at 6 to be:
θ6 =
K −3 + K +5
= Δθ
2
(5.54a)
228
5 Method of Characteristics
ν6 =
K −3 − K +5
= νe = Δθ
2
(5.54b)
K −6 = νe
(5.54c)
K +6 = −νe + 2Δθ
(5.54d)
The flow properties at node 7 is identical to 6, therefore
θ7 = θ6 = Δθ
(5.55a)
ν7 = ν6 = νe − Δθ
(5.55b)
The wall that connects nodes 4 and 7 is drawn at the average of flow angles at 4 and
7, namely
θw, max + Δθ
θ4 + θ7
=
(5.56)
θw |4−7 =
2
2
Proceeding to node 8, we note that it lies on the x-axis and it is the last node, as we
had discussed earlier. Therefore,
(5.57a)
θ8 = 0
ν8 = ν e
(5.57b)
K −8 = νe
(5.57c)
K +8 = −νe
(5.57d)
The flow properties at node 9 on the wall are the same as those at node 8. Therefore
the wall angle connecting nodes 7 and 9 is the average of the two flow angles at 7
and 9, namely
Δθ
θ7 + θ9
=
(5.58)
θw |7−9 =
2
2
Equation (5.58) that states that the angle of the last wall segment is one half of our
step size, Δθ, is a general result. This means that regardless of the number of steps
in the Δθ selection in MOC, the last wall segment makes an angle of Δθ/2 with
respect to the x-axis.
Based on our graphical construction, the ratio y9 /ya in the x − y plane represents
the ratio of exit-to-throat height, which for two-dimensional nozzles, it is the ratio of
exit-to-throat area. This should theoretically be equal to the Ae /A∗ corresponding
to Me and γ. However, there will be a discrepancy that is attributed to our non-zero
θinitial angle as well as the finite number of steps, i.e., Δθ’s, which we took to reach
νe /2. A reduction in θinitial has the largest impact on reducing the discrepancy in
area ratios followed by the size of Δθ steps. Now, we have completed the design of
a 2-D, minimum-length isentropic nozzle with uniform inlet condition using MOC,
albeit with only three C− characteristics to demonstrate the principle.
5.3 Design of a 2-D Supersonic Minimum-Length Nozzle (MLN)
229
Example 5.2 Design an isentropic two-dimensional supersonic minimum-length
nozzle (MLN) for an exit Mach number of 2.20 using MOC. Assume that the flow
in the throat is choked and uniform. The gas is air with γ = 1.4 = constant. Compare the nozzle area ratio from the graphical solution to the Ae /A* corresponding to
Me = 2.2 from isentropic tables. Also, graph Mach number distribution along the
nozzle axis (i.e., centerline).
Solution:
The Prandtl-Meyer angle corresponding to Me = 2.2 is νe = 31.73◦ . The maximum
wall angle is one half of νe , therefore the maximum wall angle is
θw, max = 15.865◦
Table 5.1 Flow parameters in MOC at 27 characteristic nodes
Node K − (deg) K + (deg) θ (deg) ν (deg) M
μ (deg)
x
y
a
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
37.862
69.195
55.764
48.778
44.393
40.834
37.862
37.862
47.356
43.252
39.889
37.061
34.418
34.418
39.889
37.061
34.418
32.344
32.344
34.418
32.344
30.350
30.350
30.350
28.750
28.750
27.049
27.049
1
1.398
1.5436
1.6256
1.6877
1.7443
1.7979
2.3072
1.807
1.9521
2.0646
2.1691
2.2707
3.3647
2.1538
2.2935
2.4348
2.5741
4.2023
2.4655
2.6414
2.8173
5.0641
2.8552
3.0722
6.0637
3.3346
7.2537
1
0
0.30955
0.43882
0.52717
0.60498
0.67787
1.3713
0
0.15439
0.2711
0.37912
0.485
1.6334
0
0.11698
0.2365
0.35702
1.7885
0
0.1226
0.2496
1.9019
0
0.13055
1.9805
0
2.0117
31.730
1.730
7.730
13.730
19.730
25.730
31.730
31.730
7.730
13.730
19.730
25.730
31.730
31.730
13.730
19.730
25.730
31.730
31.730
19.730
25.730
31.730
31.730
25.730
31.730
31.730
31.730
31.730
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
−7.730
−7.730
−7.730
−7.730
−7.730
−7.730
−13.730
−13.730
−13.730
−13.730
−13.730
−19.730
−19.730
−19.730
−19.730
−25.730
−25.730
−25.730
−31.730
−31.730
15.865
0.865
3.865
6.865
9.865
12.865
15.865
15.865
0.000
3.000
6.000
9.000
12.000
12.000
0.000
3.000
6.000
9.000
9.000
0.000
3.000
6.000
6.000
0.000
3.000
3.000
0.000
0.000
15.865
0.865
3.865
6.865
9.865
12.865
15.865
15.865
7.730
10.730
13.730
16.730
19.730
19.730
13.730
16.730
19.730
22.730
22.730
19.730
22.730
25.730
25.730
25.730
28.730
28.730
31.730
31.730
1.630
1.070
1.210
1.330
1.430
1.530
1.630
1.630
1.360
1.460
1.560
1.660
1.770
1.770
1.560
1.660
1.770
1.870
1.870
1.770
1.870
1.980
1.980
1.980
2.080
2.080
2.200
2.200
230
5 Method of Characteristics
A suitable choice may be θinitial = 0.865◦ and the remaining 15◦ divided into five
steps of 3◦ each, i.e., let Δθ = 3◦ .
There is a rule on the number of characteristic nodes in a Minimum-Length-Nozzle
(with uniform inlet condition), which states that:
No. of nodes = [2 + (n + 1)]
n
2
(5.59)
where n is the number of C− characteristics in the problem, i.e., the number of θ
sub-divisions at the throat. In this example, we chose a total of six subdivisions (i.e.,
θinitial and five Δθ’s). Therefore, we expect (9) · (3) = 27 grid points. Had we chosen
7 subdivisions, we would have produced (10) · (7/2) = 35 characteristic nodes.
Tables 5.1, 5.2 and 5.3 show the flow parameters at the characteristic nodes, the
characteristic slopes and the wall slopes, respectively. In addition to the flow variables, the Cartesian coordinates of the grid and wall points are also tabulated. The
graph of the upper half of the nozzle contour is shown in Fig. 5.11. The lower contour
is symmetrical with respect to the centerline.
The nozzle area ratio based on the MOC and our graphical construction is thus
2.0117, since the 2-D nozzle has a unit depth. The isentropic area ratio is based on
continuity equation:
Table 5.2 Slopes of C− and
C+ characteristics
Nodes
(dy/dx) char
Nodes
(dy/dx) char
A to 1
A to 2
A to 3
A to 4
A to 5
A to 6
2 to 8
3 to 9
4 to 10
5 to 11
6 to 12
9 to 14
10 to 15
11 to 16
12 to 17
15 to 19
16 to 20
17 to 21
20 to 23
21 to 24
24 to 26
−2.51235
−1.27408
−0.897
−0.68756
−0.5307
−0.40374
−1.17509
−0.87118
−0.67937
−0.53173
−0.40801
−0.84059
−0.67344
−0.53676
−0.42178
−0.68017
−0.55126
−0.44178
−0.57345
−0.4671
−0.49611
1 to 2
2 to 3
3 to 4
4 to 5
5 to 6
6 to 7
8 to 9
9 to 10
10 to 11
11 to 12
12 to 13
14 to 15
15 to 16
16 to 17
17 to 18
19 to 20
20 to 21
21 to 22
23 to 24
24 to 25
26 to 27
2.126231
1.576156
1.424069
1.373924
1.360623
1.361322
1.064162
1.037232
1.033785
1.04338
1.049892
0.837755
0.845641
0.865011
0.879234
0.696655
0.721989
0.735424
0.601679
0.618436
0.51031
5.3 Design of a 2-D Supersonic Minimum-Length Nozzle (MLN)
Table 5.3 Slopes of the
nozzle wall segments
231
Nodes
(dy/dx) char
Nodes
(dy/dx) char
A to 7
7 to 13
13 to 18
0.284045
0.247946
0.185243
18 to 22
22 to 25
25 to 27
0.131585
0.078662
0.026173
A
1
=
A∗
M
2
1 + γ−1
2 M
γ+1
2(γ−1)
(5.60)
γ+1
2
The area ratio for an exit Mach number of 2.2 and a gas with γ = 1.4 is
(A/A∗ )isentropic = 2.004975. We note that the area ratio based on the MOC (2.0117)
is within 0.33 % of the theoretical value. This level of accuracy is remarkable when
we consider that we only did a 27-point calculation.
In Fig. 5.11, the wall point at the throat is at (1, 1) whereas the nozzle exit coordinate at the wall, i.e., grid point 27, is at:
x27 = 7.2537 and y27 = 2.0117
The geometric implication of these coordinates is that a minimum-length nozzle
with uniform inlet condition and exit Mach number of 2.2 has a physical length
(downstream of the throat) that is 6.2537 times its throat half-width.
The last grid point on the axis of symmetry is number 26. The x-coordinate for
point number 26 is:
x26 = 3.3346
Since xthroat is at 1, the physical distance of the last C− characteristic that intersects
the axis of symmetry is at 2.3346 (units of throat half-width) downstream of the throat.
The implication here is that the flow accomplishes its exit condition on the axis of
symmetry at only 2.3346/6.2537 or 37.3 % of the nozzle length. The flow continues
to evolve away from the axis up to the wall. Therefore the condition of uniform flow
is accomplished after the entire flow has accelerated to the exit condition. Figure 5.12
shows the evolution of Mach number along the centerline of the supersonic nozzle.
2.5
2
13
1.5
27
25
22
18
7
A
1
6
12
0.5
1
0
0
0.5
1
17
21
24
8
1.5
2
2.5
3
26
3.5
4
4.5
5
5.5
6
6.5
7
7.5
8
Fig. 5.11 The outer contour of a minimum-length 2-D isentropic nozzle with uniform inlet condition designed for exit Mach number of 2.2 using MOC (γ = 1.4)
232
5 Method of Characteristics
2
M 1.5
1
1
2
3
4
5
6
7
8
x
Fig. 5.12 The evolution of Mach number along the nozzle centerline (γ = 1.4)
Note that the waviness of the Mach distribution curve along the nozzle axis is caused
by sparse (only 6 in this example) data points along the centerline (remember that
we used 6 C− characteristics to accelerate the flow at the throat). Increasing the
number of the characteristics, or equivalently, reducing the turning angle, Δθ, of
each characteristic will smooth the Mach distribution curve along the nozzle axis.
5.4 Wave-Field Method Versus Lattice-Point Approach
In supersonic flow the turning is accomplished by two kinds of waves, the rightrunning and left-running waves that may be of expansion or of compression type.
Therefore there are two families of waves that dictate the flow direction and magnitude in supersonic flow. There are regions in the flow where only one type of
waves uniquely determines the flow variables and they are called simple-wave flow.
Figure 5.13 shows a convex wall where the flow upstream of the shoulder (point 1)
has been uniform and parallel. The subsequent turns in the wall are accomplished by
left-running Mach waves, as shown. Here we note that the Prandtl-Meyer expansion
fans at points 1, 2 and 3 are replaced by an “average” single Mach wave that is
based on the average Mach numbers on the two sides of the respective expansion
corners. The Mach angles that are shown in Fig. 5.13 are based on the average Mach
numbers in regions 1 and 2, 2 and 3, and 3 and 4 respectively. Therefore crossing of
each wave implies a net turning that is dictated at the wall. The left-running waves turn
the flow in clockwise, i.e., in negative θ, direction whereas right-running
waves turn the flow in the counter-clockwise, i.e., the positive, direction. For example,
V
LRW
1
M1
=0
V
LRW
Streamline
x
µ2
µ1
1
y
M3
2
1
2
1
LRW
2
M2
µ3
3
3
3
2
3
M4
4
4
Fig. 5.13 Simple-wave problem in supersonic expansion flow over a convex surface with LRWs
5.4 Wave-Field Method Versus Lattice-Point Approach
233
when the flow crosses from field 1 to field 2 in Fig. 5.13, the flow angle is reduced
by the turning at the wall at point 1. The Prandtl-Meyer angle, ν, also jumps across
the waves. In an expanding flow, such as the convex wall in Fig. 5.13, the PrandtlMeyer angle jumps up by the amount of wall turning angle, since the Mach number
in flow increases along a streamline that crosses the waves emanating from corners
1, 2 and 3.
For the general case where the flow upstream of corner 1 makes an angle, θ1 , with
respect to the x-axis, the flow angles in regions 2, 3 and 4 are:
θ2 = θ1 − Δθ1
(5.61a)
θ3 = θ1 − Δθ1 − Δθ2
(5.61b)
θ3 = θ1 − Δθ1 − Δθ2 − Δθ3
(5.61c)
The Prandtl-Meyer angles in regions 2, 3 and 4 are related to the P-M angle in
region 1 and the wall turning angles, following:
ν2 = ν1 + Δθ1
(5.62a)
ν3 = ν2 + Δθ2
(5.62b)
ν4 = ν3 + Δθ3
(5.62c)
From Eqs (5.62a), (5.62b), and (5.62c) we get the Mach numbers in regions 2, 3 and
4 respectively. The waves at the corners, as noted earlier, are C+ characteristics that
are drawn at the average Mach and flow angles according to definition:
dy = tan(θ̄ + μ̄)
(5.63)
dx C+
where the mean angles are defined as:
θ1 + θ2
θ¯1 =
2
(5.64a)
θ2 + θ3
θ¯2 =
2
(5.64b)
θ2 + θ3
θ¯3 =
2
(5.64c)
μ¯1 = sin−1
2
M1 + M2
(5.64d)
μ¯2 = sin−1
2
M2 + M3
(5.64e)
234
5 Method of Characteristics
μ¯3 = sin−1
2
M3 + M4
(5.64f)
Figure 5.14 shows the definition sketch of the physical Mach waves and the “mean”
Mach waves in an expansive supersonic flow over a convex surface.
It is important to note that the concept of “mean” Mach wave, as described by
Eqs. (5.64d)–(5.64f), is mainly used as a tool that facilitates the numerical-graphical
solution to supersonic flows using wave-field method. It is used to replace the head
and tail Mach waves that envelope the P-M expansion fans. However, these are all
physical waves, including the mean Mach wave that exist in nature and are fundamental to supersonic flows (see Fig. 5.14).
The counterpart of a convex wall is a concave surface, which in supersonic flow
creates compression Mach waves to turn the flow. The P-M angle across a (leftrunning) compression Mach wave is thus reduced by the turning angle of the wall. The
reduced P-M angle across compression Mach waves creates a reduction in flow Mach
number and an increase in Mach angle. Consequently, the compression Mach waves
on a concave wall converge, in contrast to the expansion Mach waves that diverge.
Figure 5.15 shows the simple wave problem in supersonic flow over a concave wall
and a representative streamline.
The flow deceleration across (the left-running) compression Mach waves as shown
in Fig. 5.15 satisfy the P-M relations:
ν2 = ν1 − (θ2 − θ1 )
(5.65a)
ν3 = ν2 − (θ3 − θ2 )
(5.65b)
ν4 = ν3 − (θ4 − θ3 )
(5.65c)
Note that the left-running compression Mach waves cause an increase in flow
angle, i.e., θ4 > θ3 > θ2 > θ1 , therefore P-M relations (5.65a)–(5.65c) result in a
____
1
y
x
Physical Mach Waves
Mean Mach wave
___
“Mean” Mach Waves
Head Mach wave
Tail Mach wave
M1
µ2 M
2
µ3
Physical Mach
Angle:
µ1
M3
µ = sin-1(1/M)
µ2
µ
4
2
µ3
θ1
3
θ2
M4
θ3
Convex wall
4
θ4
Fig. 5.14 Definition sketch for the physical and mean Mach waves in supersonic flow over expansion corners
5.4 Wave-Field Method Versus Lattice-Point Approach
235
Compression Mach waves
4
M1
4
3
y
Streamline
x
2
1
θ3
3
θ2
2
1
θ4
Concave wall
Fig. 5.15 Supersonic flow over a 2-D concave surface—A simple wave problem with compression
Mach waves
reduction in P-M angle, i.e., ν4 < ν3 < ν2 < ν1 , which in turn cause a reduction
in Mach number, M4 < M3 < M2 < M1 . To generalize the principle, we now use
right-running compression Mach waves on a concave surface a shown in Fig. 5.16.
The flow angle is reduced across the right-running compression Mach waves,
which is the opposite of the left-running waves of the same type. However, the flow
deceleration still applies, namely M4 < M3 < M2 < M1 . Therefore to generalize
the P-M relations that we used in the case of left-running compression Mach waves,
Eq. (5.65a)–(5.65c), may be cast in terms of the absolute value of flow turning angle
according to:
ν2 = ν1 − |θ2 − θ1 | Compression Mach wave
(5.66a)
ν3 = ν2 − |θ3 − θ2 | Compression Mach wave
(5.66b)
ν4 = ν3 − |θ4 − θ3 | Compression Mach wave
(5.66c)
1
Concave wall
2
3
Streamline
1
2
3
y
4
x
4
4
Right-running compression Mach waves
Fig. 5.16 Turning of supersonic flow over a 2-D concave surface by right-running compression
Mach waves
236
5 Method of Characteristics
The generalized Eq. (5.65a)–(5.65c) apply to both types of compression Mach waves,
i.e., RRW as well as LRW. By using the same principle, the generalized case of a
convex surface in supersonic flow where expansion Mach waves cause a flow turning
and acceleration follow the P-M relations:
ν2 = ν1 + |θ2 − θ1 | Expansion Mach wave
(5.67a)
ν3 = ν2 + |θ3 − θ2 | Expansion Mach wave
(5.67b)
ν4 = ν3 + |θ4 − θ3 | Expansion Mach wave
(5.67c)
Figure 5.17 is a definition sketch for an expansive flow over a convex surface with
right-running (Mach) waves.
The non-simple wave problem in supersonic flow involves the creation and interaction of both types of waves, i.e., left-running and right-running waves. We observed
an example of non-simple wave in the design of minimum-length supersonic nozzle
where the throat created C− characteristics (as in Fig. 5.10) and the C+ characteristics were created by the reflection of C− characteristics from the x-axis, i.e., the
nozzle centerline. Now, we are ready to apply the wave-field method to isentropic
supersonic flows with waves of both families.
The unit processes in wave-field method are: wave generation, wave reflection
from a solid boundary, wave reflection from a free boundary and wave cancelation.
The method is best learned through an example. Wave-field approach is applied to
design of a minimum-length 2-D nozzle as shown in Fig. 5.18. The exit Mach number
is 2.06, which requires 14◦ of flow turning to accelerate a sonic flow to Mach 2.06
flow. Subsequently, seven waves of each 2◦ strength are created at the sharp corner,
A (see Fig. 5.18). The numbers in cells are characteristic numbers. Once we learn the
rules for assigning the characteristic numbers, the process becomes easy. The first
rule is that the sum of the two numbers, 600 and 400, at the sonic cell adds up to
1,000. In subsequent cells the sum is less than 1,000. In the first cell after the sonic
throat, where the flow has crossed a 2◦ turn (Left-Running) Mach wave, the sum of
the two numbers is 998 (i.e., 598 and 400). Therefore the difference between 1,000
4
Convex wall
RRW
3
4
2
1
RRW
Streamline
3
y
x
2
RRW
1
RRW
Fig. 5.17 A convex wall in supersonic flow with expansion Mach waves (RRW)
5.4 Wave-Field Method Versus Lattice-Point Approach
237
Minimum-Length Nozzle
B
Wave Reflection
598
400
600
400
A
598
398
586
386 586
388
586
400
Exit Mach number
Me = 2.06
586
398
Wave Generation
Wave Cancellation Region
Fig. 5.18 A 2-D minimum-length nozzle (MLN) designed using wave-field method (adapted from
[3])
and the sum of the numbers in the cell constitute the P-M angle, ν, which is equal to
θ in magnitude. Now we present the rule about the difference of the two numbers.
The difference between the numbers in the cell minus 200 is the flow angle in the
cell. Note that at the sonic throat the difference between 600 and 400 is exactly 200,
which means that the flow angle at the throat is zero. In the next cell, the difference
between the numbers 598 and 400 is 198, which taking away 200 leaves a −2◦ of
flow angle in that cell. The next cell where the numbers 598, 398 are written mean
the following: the sum is 996, which means the P-M angle in that cell is 4◦ . The
difference between the characteristic numbers is 200, which means that the flow
angle is 0. Note that the cell in discussion is adjacent to the centerline, where flow
has to attain zero-angle due to flow symmetry. In addition the flow has crossed a
LRW and its reflection, i.e., a RRW; therefore the flow angles of −2◦ encountered
by a LRW is canceled by the +2◦ induced by the RRW that was reflected from the
centerline. The reason for the P-M angle to be 4◦ is due to crossing of two 2◦ Mach
waves, which add up to 4◦ . The cell that has the maximum wall angle is designated
with two numbers, 586 and 400. The sum of the numbers is 986, which means that
the P-M angle in that cell is 14◦ . The difference between the two numbers is 186,
which implies the flow angle in the cell is −14◦ . Note that the flow angle crossing
the LRW is reduced; therefore we expect the flow angle to increase as we cross
the RRW, as in the reflection of the LRW from the centerline. The last cell that we
examine is characterized by two numbers, 586 and 386. Their sum is 972, which
implies that the P-M angle is 28◦ . This angle corresponds to the exit Mach number
of 2.06. So far we have experienced the wave generation at the throat where seven
LRWs created the initial expansion. We also experienced the wave reflection from a
solid boundary, namely the centerline behaves as solid boundary. The rule was that
the refection from a solid boundary is in like-manner where expanding Mach waves
reflect as expanding Mach waves, except the reflections are of the opposite family,
namely the RRWs in this case.
238
5 Method of Characteristics
Table 5.4 Characteristic parameters that are used in wave-field approach (γ = 1.4)
N
ν (deg)
M
μ (deg)
p/ p ∗
1,000
999
998
997
996
995
994
993
992
991
990
989
988
987
986
985
984
983
982
981
980
979
978
977
976
975
974
973
972
971
970
969
968
967
966
965
964
963
962
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
1.000
1.081
1.133
1.176
1.219
1.258
1.294
1.330
1.365
1.400
1.435
1.469
1.503
1.537
1.571
1.605
1.638
1.672
1.706
1.741
1.775
1.810
1.845
1.880
1.915
1.950
1.986
2.024
2.060
2.097
2.132
2.173
2.211
2.249
2.288
2.329
2.369
2.411
2.453
90.000
67.640
61.988
58.217
55.153
52.665
50.617
48.763
47.114
45.585
44.176
42.901
41.701
40.595
39.546
38.551
37.620
36.730
35.876
35.061
34.283
33.542
32.828
32.143
31.483
30.847
30.233
29.617
29.041
28.484
27.972
27.405
26.897
26.407
25.918
25.428
24.965
24.508
24.063
1.000000
0.907248
0.851025
0.804867
0.761912
0.723304
0.688964
0.655897
0.624887
0.594836
0.566083
0.539221
0.513255
0.488759
0.465116
0.442326
0.420759
0.399939
0.379809
0.360539
0.342107
0.324543
0.307674
0.291536
0.276111
0.261340
0.247249
0.233236
0.220336
0.208035
0.196900
0.184785
0.174144
0.164089
0.154276
0.144668
0.135835
0.127330
0.119294
(continued)
5.4 Wave-Field Method Versus Lattice-Point Approach
239
Table 5.4 (continued)
N
ν (deg)
M
μ (deg)
p/ p ∗
961
960
959
958
957
956
955
954
953
952
951
950
949
948
947
946
945
944
943
942
941
940
939
938
937
936
935
934
933
932
931
930
929
928
927
926
925
924
923
2.495
2.537
2.581
2.626
2.671
2.718
2.765
2.812
2.860
2.911
2.961
3.012
3.065
3.119
3.174
3.226
3.287
3.346
3.407
3.468
3.530
3.595
3.661
3.728
3.798
3.870
3.943
4.018
4.095
4.175
4.256
4.340
4.427
4.517
4.610
4.704
4.802
4.905
5.010
23.632
23.211
22.794
22.387
21.985
21.591
21.202
20.833
20.465
20.090
19.736
19.389
19.042
18.701
18.367
18.06
17.71
17.39
17.07
16.76
16.46
16.15
15.85
15.56
15.27
14.98
14.69
14.41
14.13
13.86
13.59
13.32
13.06
12.79
12.53
12.27
12.02
11.76
11.51
0.111723
0.104549
0.097667
0.091182
0.084987
0.079144
0.073585
0.068507
0.063651
0.058907
0.054618
0.050599
0.046760
0.043166
0.039813
0.03684
0.03373
0.03097
0.02836
0.02597
0.02378
0.02172
0.0198
0.01803
0.01639
0.01485
0.01346
0.01218
0.01099
0.0099
0.00891
0.008
0.00717
0.0064
0.00571
0.00509
0.00452
0.004
0.00354
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
(continued)
240
5 Method of Characteristics
Table 5.4 (continued)
N
ν (deg)
M
μ (deg)
p/ p ∗
922
921
920
919
918
917
916
915
914
913
912
911
910
909
908
907
906
905
904
903
902
901
900
899
898
897
896
895
894
5.119
5.231
5.348
5.472
5.599
5.730
5.865
6.007
6.152
6.307
6.472
6.642
6.821
7.008
7.202
7.407
7.622
7.853
8.092
8.343
8.616
8.902
9.210
9.535
9.881
10.260
10.665
11.088
11.552
11.27
11.02
10.78
10.53
10.29
10.05
9.817
9.583
9.355
9.123
8.888
8.659
8.43
8.204
7.981
7.759
7.539
7.316
7.099
6.884
6.665
6.45
6.233
6.02
5.809
5.593
5.38
5.174
4.966
0.00312
0.00274
0.00241
0.0021
0.00183
0.00159
0.00138
0.00119
0.00103
0.00088
0.00075
0.00064
0.00054
0.00045
0.00038
0.00032
0.00027
0.00022
0.00018
0.00015
0.00012
9.6E-05
7.7E-05
6.1E-05
4.8E-05
3.8E-05
2.9E-05
2.2E-05
1.7E-05
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
The reflected waves from the centerline are still responsible for more flow acceleration in subsequent cells, as their expansive nature is preserved in a like-manner
reflection. We learn that the opposite is true for the reflection of waves from free
boundaries when we discuss under-expanded nozzle flows. The wave cancellation is
accomplished at the wall where we turn the wall to meet the flow angle upstream of
the wave, i.e., before it meets the wall. To demonstrate this point let us examine the
cell next to the maximum wall angle cell where the characteristic numbers are 586
and 398. The sum is 984 which means that the flow P-M angle is now 16◦ , which
is 2◦ more than the previous adjacent cell (with 586 and 400 numbers). The difference between the numbers is 188, which means that the flow angle in the cell is
5.4 Wave-Field Method Versus Lattice-Point Approach
Wave Generation
Wave Cancellation Region
241
Outer wall is a streamline
Intermediate
Streamlines
Intermediate
Streamline
Wave Reflection
Nozzle Length
Centerline is a streamline,
thus behaves as a solid wall
Fig. 5.19 Wave-field method is used to design a 2-D supersonic nozzle with an extended initial
expansion region that produces longer nozzle than MLN (adapted from [3])
−12◦ . Since the flow angle in the wall cell and the wall angle have to match, the
wall angle is −12◦ . The cell where the maximum wall angle is a boundary had a
flow and wall angle of −14◦ ; therefore the wall has turned in the positive θ direction, i.e., in counter-clockwise direction by 2◦ . This is the wave cancellation step in
wave-field method. We note that the wave strength does not change in reflections,
i.e., the 2◦ that we started the initial expansion persists through reflections. In this
example, we learned about the characteristic numbers, the wave generation step, the
wave reflection from a solid boundary step and the wave cancellation step at the wall.
To facilitate the numerical-graphical calculation, a suitable table that has a column
for characteristic number, N , which is the sum of the two numbers in a cell, is added
to a P-M table that includes Mach angle and p/ p ∗ as additional columns. Table 5.4
shows the characteristic parameters that are used in the wave-field approach in MOC
for a gas with γ = 1.4. These parameters and more detail may be found in NACA
Report 1135 [2].
The wave generation step in Fig. 5.18 was concentrated at a point that led to the
design of minimum-length nozzle (MLN). However the concentrated expansion at the
throat is very abrupt which may affect the flow quality at the nozzle exit. The MLN
is suitable for flight due to its reduced weight/cost, but in stationary applications,
such as supersonic wind tunnels, the weight or cost advantage is over-shadowed by
the flow quality advantage of accomplishing uniform, wave-free flow condition in
the test section. As a result, supersonic nozzles with an arbitrarily stretched wavegeneration region are designed for stationary applications. The example shown in
Fig. 5.19, which is adapted from Ferri [3], shows a spread of the wave generation near
the throat, followed by wave reflections from the centerline and wave cancellations
at the wall similar to the previous MLN case. In this example Ferri addresses the
case of a nozzle with fixed length and then works in the reverse order to establish
the waves that are necessary in its design. Our purpose was however to show that the
wave generation region may be spread to accomplish higher quality flow in the exit
plane/test section of a supersonic wind tunnel.
242
5 Method of Characteristics
Incident pressure
wave (expansion)
Reflected pressure
wave (expansion)
LRW
RRW
y
x
1
2
3
Solid Wall
Reflected pressure
Incident pressure
wave (compression) wave (compression)
LRW
y
RRW
x
2
1
3
So lid W all
Fig. 5.20 Incident and reflected pressure waves (of expansion and compression types) from a solid
wall
Example 5.3 Show that the strength of the reflected (isentropic) pressure wave from
a solid wall is the same as the strength of the incident wave, i.e., show:
(Δθ)reflected = −(Δθ)incident
(Δν)reflected = −(Δν)incident
Solution:
We consider two cases of expansion and compression waves, as shown in Fig. 5.20.
The proof of the first part, namely flow turning of the incident wave from a solid wall
is canceled by the reflected wave is imposed at the solid surface. Mathematically, we
state that:
θ1 = θ3 = θw
(5.68)
θ2 = θ1 ± Δθ
(5.69)
The flow angle in region 2, is
The ± sign in Eq. (5.69) depends on whether the incident wave was of compressive or
expansive nature as well as whether the incident wave was of the left- or right-running
type. For example, the two RRWs that are the graphed as incident waves in Fig. 5.20,
cause the flow turning to be in the positive direction for the expansion wave and
in the negative direction for the compression wave. Also note that by choosing the
straight wall to lie along the x-axis, the flow angles in regions 1 and 3 as well as the
wall angle are all zero. Combining Eqs. (5.68) and (5.69) proves the first assertion:
(Δθ)reflected = −(Δθ)incident
Since the origin of this condition is set at the solid wall (and not by the nature of
the wave, i.e., whether the wave was isentropic or non-isentropic), the incident wave
may even be non-isentropic, e.g., an oblique shock wave. The flow turning induced
by an oblique shock is reversed upon reflection at a plane solid wall to satisfy the
flow tangency condition at the solid surface.
5.4 Wave-Field Method Versus Lattice-Point Approach
243
In Prandtl-Meyer flow theory, we have shown that the change in P-M angle is
equal to the magnitude of the flow turning angle. The expansion waves will cause
flow acceleration and thus P-M angle increases by the absolute value of the turning
angle. In this context, the LRW or RRW both cause an increase in P-M angle for an
expansion wave. The compression waves will cause flow deceleration and thus P-M
angle decreases upon encountering of such waves. This may be stated mathematically
for expansion waves as:
ν2 = ν1 + |Δθ| Expansion waves
(5.70a)
ν3 = ν2 + |Δθ| Expansion waves
(5.70b)
The equivalent form for the compression waves is:
ν2 = ν1 − |Δθ| Compression waves
(5.71a)
ν3 = ν2 − |Δθ| Compression waves
(5.71b)
Therefore, the change in P-M angle across the incident expansion wave and its
reflection is:
(5.72a)
ν2 − ν1 == (Δν)incident = |Δθ|
ν3 − ν2 == (Δν)reflected = |Δθ|
(5.72b)
The change in P-M angle across the incident and reflected compression waves is:
ν2 − ν1 == (Δν)incident = − |Δθ|
(5.73a)
ν3 − ν2 == (Δν)reflected = − |Δθ|
(5.73b)
The Eqs. (5.72) and (5.73) prove the second assertion that (Δν)reflected = (Δν)incident .
The case of wave reflection from a free boundary is of interest when we encounter
under-expanded (or over-expanded) nozzles. The problem statement is that a supersonic stream at Mach, Me , and static pressure, pe , that is discharged in an atmosphere
with static pressure, pa , where pe = pa has to adjust its pressure to match the ambient pressure at the free boundary separating the jet from the ambient. We know that
the interface between the two fluids, i.e., the free shear layer, cannot sustain a static
pressure jump. The case of pe > pa is called under-expanded flow and the opposite is true for over-expanded flows. In under-expanded flows, there is an excess of
static pressure at the nozzle exit that needs to be (abruptly) relieved when it meets
the ambient fluid. The abrupt static pressure relief mechanism in supersonic flow is
through centered expansion fans, which are known as P-M expansion Mach waves.
The case of over-expanded flows where the nozzle exit pressure is lower than the
ambient static pressure, i.e., sub-atmospheric, the mechanism for abrupt rise in static
244
5 Method of Characteristics
pressure in supersonic flow is through shock waves. Figure 5.21 shows a definition
sketch of an under-expanded jet and its wave train in the exhaust plume. Representing
the centered expansion fan at the nozzle lip by only two (finite) expansion waves,
we arrive at field numbers 1–12. The jet centerline is the plane of symmetry, which
is a streamline and may thus be replaced by a solid wall. The reflection of expansion
waves from the centerline (i.e., a solid surface) is in like-manner; therefore further
expansions below atmospheric occur upon reflection. The jet in field number 3 is
perfectly expanded, which means that p3 = pa thereby maintaining the constant
pressure condition at the free boundary. To compensate for over expansion of the
flow in fields number 4 and 5, the expansion waves reflect as pressure waves from
the free boundary to create fields’ number 7 and 8. This is the principle of unlike
reflection at a free boundary. Compression waves that are generated at the boundary
reflect from the centerline (that behaves like a solid wall) as compression waves and
thus create fields with static pressure that are in excess to ambient pressure (in fields
number 10 and 11). The process of unlike wave reflections from the free boundary
and like reflections from the centerline repeats to form a periodic exhaust plume
shape with a spatial wavelength of λ, as depicted in Fig. 5.21.
The periodicity of the wave structures is noted in Fig. 5.21 by repeating cell
numbers 2, 3, 4 and 5 towards the end of the plume, where in theory the flowfield
is identical to their counterparts near the nozzle lip. The case of wave pattern in
an over-expanded nozzle is shown in Fig. 5.22. The waves that are formed at the
nozzle lip are shock waves (here are drawn as oblique shocks) and the subsequent jet
___
Streamline
pa
Constant-Pressure
Free Boundary
pa
pa
pe > p a
γ, R
y
2
5
8
11
6
4
x
3
9
5
1
2
12
10
4
Centerline
Wavelength, λ
pe = p1
pa
pa
7
3
Me
Free Shear layer
Expansion Waves
Compression Waves
Streamline
p2
p11
p 3 = pa
p5 < pa
p7 = pa
p8 < pa
p12 = pe
p9 = pa
Fig. 5.21 The wave pattern in the exhaust plume of an under-expanded jet and the pressure distribution along a representative streamline (not to scale)
5.4 Wave-Field Method Versus Lattice-Point Approach
Oblique Shock
Free Shear layer
Expansion Waves
Compression Waves
Streamline
pa
pa
___
Streamline
Constant-Pressure
Free Boundary
pa
pa
Me
pa
pa
pa
pa
pe < p a
γ, R
245
pa
pa
pa
y
Centerline
x
Wavelength, λ
Incident and Reflected
Shock waves
pa
pa
pa
pa
pe < pa
Fig. 5.22 The exhaust plume of an over-expanded nozzle with a representative Streamline and its
pressure distribution along the jet (not to scale)
contraction. It is through the process of like and unlike reflections from the centerline
and the free boundary respectively that gives the exhaust plume a periodic structure.
We recall that the strength of the shock wave at the lip depends on the severity of overexpansion. It is even possible to have a normal shock inside the nozzle of severely
over-expanded flows. Figure 5.22 shows the case of an over-expanded nozzle where
the pressure rise across the oblique shock at lip is sufficient to reach the ambient
static pressure, pa .
Example 5.4 Calculate the initial exhaust plume angle for an under-expanded jet
with Me = 2.2, pe / pa = 2.0 with the ratio of specific heats, γ = 1.4.
Solution:
Since the centered expansion waves are isentropic, stagnation pressure remains constant, i.e., pt2 = pt1 . From isentropic tables we get:
Me = M1 = 2.2
→
pte / pe = 10.693
pt2
pte
pte pe
=
=
= (10.693) · (20) ≈ 21.38
p2
pa
pe pa
From the ratio of pt2 / p2 and isentropic table we get M2 ≈ 2.64. The flow turning
angle across P-M waves is Δθ = Δν, therefore from ν1 = 31.73 and ν2 = 42.31,
we get θ2 ≈ 42.31◦ − 31.73◦ ≈ 10.58◦ .
246
5 Method of Characteristics
2
2
2
M2
1
1
Me = M 1
Example 5.5 A two-dimensional under-expanded nozzle has an exit Mach number
of Me = 1.435 and exit static pressure that is 134.7 % of the ambient pressure, i.e.,
pe = 1.347 pa . Assuming that the nozzle exit flow is uniform and the gas specific
heat ratio is γ = 1.4 = constant, use MOC to map out the (inviscid) flowfield in the
exhaust plume and establish the jet boundary. What is the wavelength of the periodic
plume structure, as compared to nozzle exit half-width?
y
Constant-Pressure
Free Boundary
G
D
7
yA=1
A
J
F
5
Me= 1.435
pe= 1.347pa
= 1.4
1
2
5
8
I
C
6
2
10
E
B
11
12
4
x
3
9
3
H
4
K
Solution:
The wave-field approach in MOC is best suited for supersonic problems involving a
free boundary. The flow parameters at the exit are:
Me = 1.435
→
νe = 10◦ , μe = 44.176◦ and
pte
≈ 3.344
pe
Since the expansion process at the nozzle lip is isentropic, the total pressure at the
nozzle exit remains constant along the jet, therefore
pt3
pte pe pa
=
= 3.344 · 1.347 · 1 = 4.504
p3
pe pa p3
→
M3 = 1.639
→
ν3 = 16◦
The flow turning angle across a centered expansion fan is equal to the change of P-M
angle, Δν across the fan, namely
θ3 − θ1 = ν3 − ν1 = 6◦
Assuming that the jet axis coincides with the x-axis, we use the flow angle at the
nozzle exit equal to zero (i.e., as reference), and thus the flow angle in field number
5.4 Wave-Field Method Versus Lattice-Point Approach
247
3 is +6◦ . In summary the flow angles are:
θ1 = 0 and θ3 = +6◦
Therefore, the initial expansion at point A makes an angle of +6◦ with respect to
the x-axis. The equation for the free boundary AD is therefore:
y AD = 1 + tan(6◦ )x ≈ 1 + 0.105x
To map out the remaining fields, we choose the number of waves at A to be
two and of equal strength. Therefore, two RRWs, i.e., the expansion waves (C−
characteristics) at A turn the flow +3◦ each. We march through the incident and
reflected waves and establish flow and P-M angles. The rule for the flow angle is
that θ increases when crossing an expansion LRW (as in waves AB and AC) and
θ decreases when crossing an expansion RRW (as in waves BC and CD). The rule
for the P-M angle is that ν increases when crossing an expansion wave, whether a
LRW or a RRW. This means that P-M angle in field number 4 jumped up through
both expansion waves AB and BC, although AB is a RRW and BC is a LRW. The
rule on the compression wave is that flow angle increases when crossing a LRW (as
in wave HI) and decreases when crossing a RRW (as in waves DH or GI). Note that
the compression and expansion waves turn the flow in opposite directions. The rule
on the P-M angle is that ν drops in crossing a compression wave, whether a LRW or
RRW. This means that the compression waves FH and HI both cause a reduction in ν,
although they belong to opposite families. Table 5.5 shows the flow and P-M angles
in different fields. We have also added a column for the Mach angle since we need it
for the slope of characteristics. Table 5.6 shows the average flow and P-M angles on
the waves that separate the fields. Since waves are at the boundaries of different fields,
Table 5.5 Flow variables in
designated fields
Field number
θ (deg)
ν (deg)
μ (deg)
1
2
3
4
5
6
7
8
9
10
11
12
0
3
6
0
3
0
0
−3
−6
0
−3
0
10
13
16
16
19
22
16
19
16
16
13
10
44.176
40.595
37.620
37.620
35.061
32.828
37.620
35.061
37.620
37.620
40.595
44.176
248
5 Method of Characteristics
Table 5.6 Characteristic waves and their parameters
Wave
Type
θ (deg)
μ (deg)
(θ + μ) (deg)
(θ − μ) (deg)
Slope of waves
AB
AC
BC
CD
CE
EF
FG
DF
FH
HI
GI
IK
IJ
KJ
−40.89
−34.61
−37.61
−31.84
−34.84
−32.44
−37.84
−34.84
−35.44
−37.84
−40.84
−40.61
−43.61
−43.89
−0.8658
−0.6900
0.8573
0.8644
−0.6961
0.7118
0.6961
−0.6961
−0.7118
0.6961
−0.8644
−0.8573
0.6900
0.8658
C−
C−
C+
C+
C−
C+
C+
C−
C−
C+
C−
C−
C+
C+
1.5
4.5
1.5
4.5
1.5
1.5
−1.5
1.5
−1.5
−1.5
−4.5
−1.5
−4.5
−1.5
42.39
39.11
39.11
36.34
36.34
33.94
36.34
36.34
33.94
36.34
36.34
39.11
39.11
42.39
43.89
43.61
40.61
40.84
37.84
35.44
34.84
37.84
32.44
34.84
31.84
37.61
34.61
40.89
we attribute an average θ and μ that correspond to the two neighboring fields to the
wave. We use these average angles in the construction of the characteristic network.
With the known slopes and the starting coordinate, e.g., A, we start our march
in the streamwise direction by writing the equations of straight lines (to represent
waves). For example, we start with the wave AB.
yAB = 1 + 0.86578x
→
xb = 1/0.86578 = 1.155
With the coordinate of point B known in the previous step, we write the equations
for the waves AC and BC.
yAC = 1 − 0.69005x
y BC = 0.857326 · (x − 1.155)
From the intersection of AC and BC, we get the coordinates of point C, namely
xC = 1.2862 and yC = 0.11246. We continue with the remaining waves and their
intersections to establish the coordinates of the nodes.
yCD = 0.11246 + 0.8644 · (x − 1.2862)
yAD = 1 + tan(6◦ )x ≈ 1 + 0.105x
From the intersection of AD and CD, we get the coordinates of point D, namely,
x D = 2.6327 and y D = 1.2764.
yCE = 0.11246 + 0.69606 · (x − 1.2862)
5.4 Wave-Field Method Versus Lattice-Point Approach
249
Therefore, the coordinates of point E on the x-axis is x E = 1.4478 and y E = 0.
yDF = 1.27644 + 0.69606 · (x − 2.63276)
yEF = 0.711829 · (x − 1.447768)
From the intersection of DF and EF, we get x F = 2.9403 and y F = 1.0624.
yDG = 1.27644
yFG = 1.0624 − 0.69606 · (x − 2.9402)
From the intersection of DG and FG, we get x G = 3.2477 and yG = 1.27644
yFH = 1.0624 − 0.71183 · (x − 2.9402)
The coordinates of point H on the x-axis is x H = 4.4327 and y H = 0
yHI = 0.69606 · (x − 4.4327)
yGI = 1.27644 − 0.8644 · (x − 3.2477)
From the intersection of HI and GI, we get x I = 4.5943 and y I = 0.1125
yIK = 0.112462 − 0.85733 · (x − 4.594326)
From yIK and the x-axis, we get the coordinates of point K , as: x K = 4.7255 and
y K = 0.
yIJ = 0.11246 + 0.69 · (x − 4.5943)
yKJ = 0.865779 · (x − 4.7255)
From the intersection of yIJ and yKJ , we get the coordinates of point J as x J = 5.8805
and y J = 1.0.
Table 5.7 is a summary of the coordinates of the characteristic nodes. To graph
the lower half of the jet, we have created a (−y) column that is the mirror image of
the upper half of the jet. Figure 5.23 shows the 2-D under-expanded exhaust plume,
the incident expansion waves, the reflected compression waves and the free (jet)
boundary. We also note that the waves and their reflections form a periodic plume
structure with a wavelength of 5.881 unit of jet half-width. Since we had neglected
fluid viscosity, the supersonic shear layer does not spread due to the actions of
fluid viscosity, turbulent stresses and wave interactions at the jet boundary. The
periodic plume structure is also subject to the dissipative actions of fluid viscosity
and turbulence.
250
5 Method of Characteristics
Table 5.7 The coordinates of
characteristic nodes including
the jet boundary points
Jet (Free) Boundary
x
y
−y
A
B
C
D
E
F
G
H
I
J
K
0.000
1.155
1.286
2.633
1.448
2.940
3.248
4.433
4.594
5.881
4.726
1.000
0.000
0.112
1.276
0.000
1.062
1.276
0.000
0.112
1.000
0.000
−1.000
0.000
−0.112
−1.276
0.000
−1.062
−1.276
0.000
−0.112
−1.000
0.000
Compression Waves
1.5
1.0
Node
+6
o
D
F
3
y
1
B
o
J
7
5
10
8
11
6
4
C
2
0.0
-6
G
A
0.5
Expansion Waves
H
I
12
K
9
E
-0.5
-1.0
-1.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
x
Wavelength of the periodic (inviscid) plume structure
Fig. 5.23 The exhaust plume of an under-expanded supersonic jet discharging into a stationary
ambient using wave-field approach in MOC [Uniform exit flow with Me = 1.435, pe = 1.347 pa
and γ = 1.4 = constant]
Wave-field approach is more suitable for plane flows where wave strength remains
constant, even after interaction with other waves. Also the wave method is advantageous when there is a free boundary to be determined, as in under-expanded nozzle
flows. But in 3-D or axisymmetric flows, wave strength varies along the wave and
the method is less useful. The lattice-point MOC is readily applied to 2-D as well as
3-D flowfields.
5.5 Axisymmetric Irrotational Flows
251
5.5 Axisymmetric Irrotational Flows
Axisymmetric bodies of revolution at zero angle of attack or yaw produce
axisymmetric flows. Therefore, axisymmetry in flow is purely imposed by the geometry of the body and its perfect alignment (of its axis) with the flow. Another view would
suggest that axisymmetry is a boundary condition-driven phenomenon. Regardless
of the view, axisymmetry offers the elimination of the third coordinate in the analysis
of 3-D flows and thus a major simplification to the fluid flow analysis.
Pointed axisymmetric bodies at zero angle of attack or yaw in supersonic flow
create shocks that may or may not be attached to the body. The case of attached
shocks where the body turning angle at the nose is less than θmax corresponding to
the flow Mach number would lend themselves to graphical-numerical procedure that
we had called method of characteristics. First let us define the axisymmetric body
and the cylindrical coordinates (x, r, ϕ) in Fig. 5.24. The velocity components in
cylindrical polar coordinates are: u, v and w in x, r and ϕ directions, respectively.
Note that for the flow to be axisymmetric, ϕ dependency is eliminated.
The continuity equation for a steady compressible fluid flow in cylindrical (polar)
coordinates is:
∂(ρu) 1 ∂(r ρv) 1 ∂(r ρw)
+
+
=0
∂x
r ∂r
r ∂ϕ
(5.80)
The last term in Eq. (5.80) is identically zero, since w = 0 as well as ∂/∂ϕ = 0.
The first two terms in the continuity equation may be expanded to:
∂u
∂v
ρ
+
∂x
∂r
y
+u
∂ρ ρv
∂ρ
+v
+
=0
∂x
∂r
r
Axisymmetric
Shock
x = constant planes
tan = dr/dx
r = y2 + z 2
r
r
M >1
z
(5.81)
nose
x
Axisymmetric
Body
Fig. 5.24 A pointed axisymmetric body at zero-angle-of-attack and yaw in supersonic flow
252
5 Method of Characteristics
From the Euler momentum equation for irrotational flow, we have
ρV dV = −d p
(5.82)
We may rewrite the inertia term in terms of velocity components as:
V2
ρd
2
2
u + v2
= ρd
2
(5.83)
The pressure term in Eq. (5.82) may be related to the square of the speed of sound
and density perturbation since the flowfield is isentropic, namely
d p = −a 2 dρ
(5.84)
Combining Eqs. (5.83) and (5.84) in (5.82), we get:
2
1
1
dρ
u + v2
= − 2d
= − 2 (udu + vdv)
ρ
a
2
a
(5.85)
In partial differential form, Eq. (5.85) may be written as two equations, namely
1
1 ∂ρ
=− 2
ρ ∂x
a
1 ∂ρ
1
=− 2
ρ ∂r
a
∂u
∂v
u
+v
∂x
∂x
∂u
∂v
u
+v
∂r
∂r
(5.86a)
(5.86b)
We may now eliminate the density form the continuity Eq. (5.81), to get:
∂v v
u
∂u
+
+ − 2
∂x
dr
r
a
u
∂u
∂v
+v
∂x
∂x
−
v
a2
∂u
∂v
u
+v
=0
∂r
∂r
(5.87)
Equation (5.87) may be rearranged to:
u 2 ∂u
uv ∂u
∂v
v
v2 ∂v
1− 2
− 2
+
=−
+ 1− 2
a
∂x
a
∂r
∂x
a
dr
r
(5.88)
Irrotational flow demands that the curl of the velocity field to vanish, namely:
e re e 1 ∂r ∂ϕ ∂x ∇ × V = ∂r ∂ϕ ∂x = 0
r v rw u (5.89)
Applying the conditions of axisymmetry, i.e., w = 0 and ∂/∂ϕ = 0, Eq. (5.89)
reduces to:
5.5 Axisymmetric Irrotational Flows
eϕ
253
∂v
∂u
−
∂x
∂r
=0
→
∂u
∂v
=
∂x
∂r
(5.90)
Substituting (5.90) in (5.88), we get:
u 2 ∂u
2uv ∂v
v2 ∂v
v
1− 2
− 2
+ 1− 2
=−
a
∂x
a ∂x
a
dr
r
(5.91)
The remaining two complimentary equations in du and dv are written similar to the
2-D case as:
du =
∂u
∂u
∂u
∂v
dx +
dr =
dx +
dr
∂x
∂r
∂x
∂x
(5.92)
∂v
∂v
dx +
dr
∂x
∂r
(5.93)
dv =
Now, we have formed the system of three Eqs. (5.91)–(5.93) and three unknowns
(∂u/∂x, ∂v/∂x and ∂v/∂r ) in cylindrical polar coordinates. We solve for ∂v/∂x
using Cramers rule:
(1 − u 2 /a 2 ) −v/r (1 − v2 /a 2 ) dx
du
0
0
dv
dr
N
∂v
=
=
2
2
2
2
2
∂x
D
(1 − u /a ) −2uv/a (1 − v /a ) dx
dr
0
0
dx
dr
(5.94)
By setting the denominator equal to zero, we get the equations for the characteristics,
namely:
(1 − u 2 /a 2 )dr 2 − dx
−2uv/a 2 dr − 1 − v2 /a 2 dx = 0
(5.95)
Note that Eq. (5.95) is identical to (5.15) for 2-D flow except coordinate “y” is
replaced with coordinate “r ”, therefore the slope of the characteristics in 3-D axisymmetric flow which is dr/dx follows (Eq. (5.22)):
dr
dx
= tan(θ ∓ μ)
(5.96)
characteristic
As expected, the characteristics are Mach waves that lie above and below the local
velocity vector. Since we are dealing with 3-D flows, the two characteristics are
indeed the projections of the Mach cone in the meridian plane. The C− characteristic
makes an angle θ − μ and the C+ characteristic makes an angle θ + μ with respect
to the x-axis in the meridian plane.
254
5 Method of Characteristics
By setting the numerator, N , in Eq. (5.94), equal to zero, we get the compatibility
conditions along C− and C+ characteristics in 3-D axisymmetric flows.
2
1 − ua 2
dv
=
2
du
1 − av 2
dr
dx
v dr −
characteristic
r du
2
1 − av 2
(5.97)
By substituting for the characteristics slope, dr/dx, in Eq. (5.97), and replacing velocity components u and v by the velocity magnitude and flow angle according to:
u = V cos θ and v = V sin θ
(5.98)
We get the compatibility equation as
dθ = ∓ M 2 − 1
1
dV
dr
±√
2
V
M − 1 ∓ cot θ r
(5.99)
Since the first term in Eq. (5.99) is the differential of the Prandtl-Meyer angle, dν, we
can combine it with dθ on the LHS to arrive at the simplified compatibility equation
along C− and C+ characteristics in axisymmetric irrotational flows as:
d(θ + ν) = √
1
dr
M 2 − 1 − cot θ r
d(θ − ν) = − √
1
dr
M 2 − 1 + cot θ r
Along C− characteristic
(5.100)
Along C+ characteristic
(5.101)
Note that the compatibility conditions in 3-D axisymmetric flows is no longer the
algebraic equations θ ± ν = constant that we derived in 2-D irrotational flows.
The compatibility equations are now differential equations that have to be solved
numerically. By taking small steps in r in an x = constant plane, we replace the
differential of r by Δr and the corresponding differential of θ ± ν, by Δ(θ ± ν)
and create finite differences connecting two points along the characteristics network.
Here we show an example of axisymmetric irrotational flow that is solved by the
MOC.
Example 5.6 Apply the MOC to a cone with 30◦ half-vertex angle that is matched
with a circular arc ogive (from [3]) at point 1 (i.e., x = 6), as shown in Fig. 5.25.
The freestream Mach number is 3.07.
Solution:
This graphical-numerical solution is worked out by Ferri [3]. The conical shock angle
at the nose is established by the Taylor-Maccoll equation [9]. It makes an angle of
5.5 Axisymmetric Irrotational Flows
255
y
45
37
36
8
7
6
5
4
2
4
3
2
1
s
0
1
3
47
48
38
30
31
17 29
16a 22
24 25
26
16
18 19
20
12
13 14
11
7
15
8 9
10
6
3
5
1
39
40
32
33
27
21
s
= 30o
2
46
4
5
6
7
8
x
9 10 11 12 13 14 15 16
Fig. 5.25 MOC applied to a sharp-nosed axisymmetric body of revolution in Mach 3.07 (Adapted
from [3])
39.42◦ with respect to the x-axis. The C+ characteristics at the nose form a simple
wave region and are Mach cones with different vertex angles η. Corresponding to
each η is a flow angle, θ that is established by Taylor-Maccoll equation. The angular
space between the cone and the conical shock is divided into 5 incremental steps. The
start of the calculation is at point 1 (on the cone-ogive) where the C+ characteristic
from the cone intersects the 6 characteristics (compression Mach waves) emanating
from the nose at points 2, 4, 7, 11 and 16.
The table of values produced by Ferri is shown in its original form in Table 5.8.
The fractional notation for angle is expressed in minutes (where there are 60 min in
1◦ ). Also note that the y-coordinate is the same as r in our notation and W is the
non-dimensional velocity, which is the ratio of local velocity V , to the maximum
velocity, Vmax (that we had defined in Chap. 3). Krasnov [4] is recommended for
further reading on aerodynamics of bodies of revolution.
Table 5.8 The MOC
calculations corresponding to
axisymmetric cone-ogive
example by Ferri [3]
Point
η◦
θ
W
μ
x
y
1
2
4
7
11
16
30◦
32◦
34◦
36◦
38◦
39◦ 25
30◦
28◦
26◦ 15
24◦ 40
23◦
21◦
0.6400
0.6414
0.6438
0.6477
0.6516
0.6560
32◦ 28
32◦ 20
32◦ 6
31◦ 44
31◦ 22
30◦ 58
6.000
6.220
6.509
6.876
7.390
7.902
3.462
3.880
4.380
4.982
5.753
6.473
256
5 Method of Characteristics
M=2.91
M=2.37
600
598 596 594
382 382
382 382
596
398 594 592 584
394 392 384
598
596 592 590 588 586
398
396 394 390 388 386
600
400
598
400
592
382
590
382 588
382 586
382
584
382
582
382
(b)
580
380
578
378
576
376
576
378
576
380
576
382
578
580
382
582
382
582 384
582 386
576 576
576 576 584
594 594
400 398 390 582 388
386 384
576
390
400 398
388
576
576 576
594
396 576
392 390
396
394
(a)
Fig. 5.26 Wave-field approach is used in the design of a Mach 2.37 and Mach 2.91 isentropic
nozzles (from [3])
5.6 Examples and Applications
5.6.1 Design of a Family of Supersonic Nozzles
Design of supersonic nozzles and wind tunnels are the best candidates for MOC.
With the assumption of uniform flow in the throat section, we may apply the wavefield MOC to design a family of supersonic nozzles. Figure 5.26 shows such a family
for two exit Mach numbers, 2.37 and 2.91. Note that the waves in Fig. 5.26 are all
of equal strength with Δθ = 2◦ . By choosing 9 steps of 2◦ each, we arrive at the
Mach 2.37 supersonic nozzle design (see Fig. 5.26). By taking an additional 3 steps
of 2◦ each at the throat (i.e., the wave generation section), the nozzle becomes a
Mach-2.91 design. Taking these additional 3 steps in designing the Mach 2.91 nozzle
design did not change the first 9 steps in the wave field method (see Fig. 5.26).
5.6.2 Deflecting Jet
We applied the wave-field method to an under-expanded 2-D symmetrical nozzle in
Example 5.5. However, the method may be applied to 2-D nozzles with asymmetrical
exit flow. An Example is shown by Ferri for a nozzle with exit plane inclined with
respect to the nozzle axis [3]. The interesting aspect of this configuration is that the
exhaust plume bends with respect to the axis, as shown in Fig. 5.27 (from [3]). The
cases of under-expanded and over-expanded nozzles are shown in part (a) and part
(b) of Fig. 5.27, respectively. Note that the central streamline bends and deviates
5.6 Examples and Applications
(a)
257
582
390
582
382
590
390
582
390
582
382
Horizontal Axis
Streamline
582
390
590
390
(b)
582
390
598
390
598
390
Streamline
Horizontal Axis
598
390
598
390
598
390
Fig. 5.27 Wave-field method applied to a nozzle with its exit cross section inclined with respect to
the jet axis (from [3]) a Expansion b Compression
from the horizontal axis as the exhaust plume evolves. We note that the asymmetry
in the exit plane of supersonic nozzle, as shown in Fig. 5.27, presents a case for thrust
vectoring, albeit of fixed and weak form.
y
+
r = const.
9
M>1
4 8
+
Virtual Source
C+
r1
=0
C+
C-
x
CC-
Initial Data Line
C+
C+
3 7
2 6
1 5
M=const.
C+
w
- w
CC-
r2
r3
r4
Fig. 5.28 Supersonic flow in a straight 2-D diverging duct with a few representative C± characteristics
258
5 Method of Characteristics
5.6.3 Non-uniform Inlet Condition: Example of Source Flow
MOC may also be applied to supersonic flows with non-uniform inlet flow condition.
The starting point is flow description on an initial data line, which then creates a
two-family network of characteristics. As an example, consider a 2-D supersonic
flow with radial streamlines, which resembles a source flow. Figure 5.28 shows the
definition sketch of a supersonic flow in a straight diverging duct with non-uniform
inlet condition with a few representative C± characteristics. The initial data line is
shown on a circular arc at r = r1 . The Mach number is constant on the initial data line
(of circular arc shape), but the flow angle changes from −θw to +θw corresponding to
the symmetrical diverging duct. The flow angle θ on the initial data line corresponds to
the polar coordinate θ shown in the definition sketch of Fig. 5.28. The unit processes
in this calculation involve interior and wall points that we have presented in Sect. 5.2.
5.6.4 Streamlines and Ducts
In this example, we calculate the shape of bending streamlines in supersonic flow
with the help of MOC. Upon interaction with waves, streamlines bend to conserve
mass flow rate. Since streamlines describe the constant-mass-flow boundaries in
steady flows, they may be replaced with walls, i.e., solid boundaries, thereby creating
interesting geometries for supersonic ducts or conduits with curvature, e.g., flow
passages between turbomachinery blades.
Consider a centered P-M expansion fan at a 2-D corner B, as shown in Fig. 5.29. A
representative streamline is shown at height, h 1 that is normal to AB, in region 1. This
streamline emerges at height h 2 that is normal to surface BC, in region 2. Since the
flow is two dimensional, the area ratio A2 /A1 is the same as h 2 / h 1 . Using isentropic
flow conditions between regions 1 and 2, continuity gives:
h2
M1
=
h1
M2
2
1 + γ−1
2 M2
2
1 + γ−1
2 M1
γ+1
2(γ−1)
C+ characteristics or (left-running) Mach waves.
(5.102)
For a given wall angle in region 2, θ2 , we calculate the flow Mach number in
region 2 by using the P-M expansion theory (presented in Chap. 4, shock-expansion
theory). The lengths of the C+ characteristics along the head and tail Mach waves
that are cut by the streamline are l1 and l2 respectively. Since these lengths are the
hypotenuse of a right triangle with respective heights h 1 and h 2 , they are related
according to:
l1 = h 1 / sin μ1
(5.103a)
l2 = h 2 / sin μ2
(5.103b)
5.6 Examples and Applications
259
Therefore in terms of characteristic lengths, we combine Eqs. (5.103a) and
(5.103b) with Eq. (5.102) and recall the relation between the Mach angle, μ, and
Mach number, to get:
l2
sin μ1 M1
=
l1
sin μ2 M2
2
1 + γ−1
2 M2
γ+1
2(γ−1)
=
2
1 + γ−1
2 M1
2
1 + γ−1
2 M2
γ+1
2(γ−1)
(5.104)
2
1 + γ−1
2 M1
We have established the orientation of line BF, i.e., the tail characteristic, as well as its
length, l2 . For intermediate C+ characteristics, e.g., BE, we choose an intermediate
Mach number between M1 and M2 , which immediately produces its corresponding Mach and P-M angles. The corresponding flow turning angle is the difference
between the two P-M angles (i.e., the difference between the intermediate PM angle
and the upstream value). The length BE is related to the length BD according to
Eq. (5.104) where M2 is replaced by the Mach number that we chose for the intermediate C+ characteristic. In the Cartesian coordinates attached to point B in Fig. 5.29,
we establish the coordinates of the bending streamline, DEF by noting that
x D = 1 cos μ1 and y D = 1 sin μ1
(5.105a)
x E = E cos(μ E − θ E ) and y E = E sin(μ E − θ E )
x F = 2 cos(μ2 − θ2 )
(5.105b)
(5.105c)
Example 5.7 A parallel uniform flow at Mach 1.6 approaches a sharp corner that
turns the flow in the clockwise direction by 21.89◦ , similar to the schematic drawing
of Fig. 5.29. Calculate the coordinates of the bending streamline with h 1 = 1 and
graph the streamline. Assume that the flow is planar 2-D with γ = 1.4.
C+ characteristics or (left-running) Mach waves
Streamline
D
M1
E
µ1
h1
1
y
1 =0
F
l1
M2
l2
(µ2 –
2)
2
x
A
B
µ2
Streamline
h2
2
C
Fig. 5.29 An expansion corner in supersonic flow and a representative streamline
260
5 Method of Characteristics
M2 = 1.8
M1 = 1.6
M=1.6
C+1
θ =0
C+2
M3 = 2.0
C+3
y
M4 = 2.2
C+4
M5 = 2.4
C+5
x
M=2.4
θ =21.89
o
Fig. 5.30 Definition sketch for the five C+ characteristics in the expansion corner
Solution:
The first C+ characteristic is the head Mach wave that makes Mach angle with respect
to local upstream flow:
μ1 = sin
−1
1
M1
= sin
−1
1
1.6
= 38.68◦
The P-M angle corresponding to the incoming flow is ν1 = 14.86◦ . The P-M angle
after 21.89◦ of wall turning is the sum of the upstream P-M angle and the turning
angle, i.e., ν = 36.75◦ . The corresponding Mach number is M = 2.4. We divide
the centered expansion fan into finite steps in Mach number, e.g., steps of 0.2. This
choice produces 5 C+ characteristics as shown in Fig. 5.30.
Since the 5 C+ characteristics are chosen based on Mach number, we immediately
associate a Mach angle and P-M angle with each characteristic. The flow angle, θ,
starts from zero (based on our choice of the coordinate system) and the steps in θ
are the difference between the local P-M angle and the incoming P-M angle. The
length ratio along different characteristics associated with an intersecting streamline
is calculated from Eq. (5.104). The coordinates for the bending streamline follow
Eq. (5.105). These are summarized in table form from Excel calculations (Tables 5.9,
5.10 and 5.11).
The coordinates of the wall are summarized in table form:
The coordinates of the streamline are also summarized in table form:
The graph of the expansion corner and the corresponding streamline is shown in
Fig. 5.31.
We may now select a second streamline with a different h 1 value than the
Example 5.7, say h 1 = 2.0 The second streamline bends similar to the first and
the two streamlines, as an example, form the boundaries of a bending duct that
accelerate a Mach 1.6 flow to the exit Mach number of 2.4.
5.6 Examples and Applications
261
y
Streamline
1.5
C+1
1
Wall
Streamline
C+2
C+3
0.5
C+4
C+5
0
-2
-1
-0.5 0
1
-1
2
3
4
x
5
Wall
-1.5
-2
Fig. 5.31 Expansion corner of Example 5.7 and the graph of a representative streamline (γ = 1.4)
-2
2.5
2
1.5
1
0.5
0
-0.5 0
-1
-1.5
-2
y
2
4
6
8
10
x
Fig. 5.32 The expansion corner of Example 5.7 and the graph of the second streamline (γ = 1.4)
Example 5.8 Calculate and graph a second streamline for the expansion corner of
Example 5.7 with h 1 = 2.0.
Solution:
We follow the same procedure that we outlined in Example 5.7 and produce the
following tables in a spreadsheet calculation. Note that the flow characteristics, M,
ν, μ and θ remain unchanged on the second streamline (Tables 5.12 and 5.13).
Figure 5.32 shows the second streamline, with h 1 = 2. We may use the two
streamlines as the boundaries of a curved duct and is shown in Fig. 5.33.
The curved duct of Example 5.8 is a supersonic nozzle with an exit flow angle
of 21.89◦ with respect to the entrance flow direction. By incorporating a second
expansion fan of opposite family, it is possible to expand the flow and accomplish
Table 5.9 Characteristics in
an expansion corner
M
ν (deg)
μ (deg)
θ (deg)
L/L − 1
x
C+1
C+2
C+3
C+4
C+5
1.6
14.86
38.68
0.00
1.00
1.25
1.8
20.73
33.75
5.86
1.29
1.83
2
26.38
30.00
11.52
1.69
2.56
2.2
31.73
27.04
16.87
2.21
3.47
2.4
36.75
24.62
21.89
2.88
4.61
262
5 Method of Characteristics
Table 5.10 Wall points
x
y
Table 5.11 Streamlines
x
y
Streamline 2
Streamline 1
-2
2.5
2
1.5
1
0.5
0
-0.5
-1
-1.5
-2
−2
0
−2.00
1.00
1.25
1.00
0
0
1.83
0.97
4
−1.606869
2.56
0.86
3.47
0.62
4.61
0.22
y
Curved Duct
2
4
10
8
6
Fig. 5.33 The shape of a curved duct that accelerates a uniform, parallel Mach 1.6 flow to a uniform
and parallel Mach 2.4 flow (γ = 1.4)
an axial exit flow. Figure 5.34 shows a schematic drawing of an S-duct supersonic
nozzle. The inlet flow is uniform and parallel, in field #1 and is in the x-direction.
Field #2 is a continuous (expansive) turning of the flow that is associated with a
centered expansion fan, which creates a non-uniform and non-parallel flowfield. The
flow in field #3 is again uniform and parallel, i.e., non-accelerating or decelerating.
Field #4 is a continuous expansion process associated with a centered expansion fan
of opposite family. Therefore, the turning of the flow is in the counter-clockwise
direction. Field #5 is the uniform parallel exit flow, which is in the axial direction.
Table 5.12 Characteristic
values and streamline
coordinates
Table 5.13 Streamline
coordinates
M
ν (deg)
μ (deg)
θ (deg)
L/L − 1
x
y
x
y
−2.00
2.00
C+1
C+2
C+3
C+4
C+5
1.6
14.86
38.68
0.00
1.00
2.50
2.00
1.8
20.73
33.75
5.86
1.29
3.66
1.94
2
26.38
30.00
11.52
1.69
5.12
1.71
2.2
31.73
27.04
16.87
2.21
6.95
1.25
2.4
36.75
24.62
21.89
2.88
9.22
0.44
5.12
1.71
6.95
1.25
9.22
0.44
2.50
2.00
3.66
1.94
5.6 Examples and Applications
263
5.6.5 Curved Shocks
A slender pointed body in supersonic flow creates an oblique shock at its leading edge,
as shown in Fig. 5.35. The convex curvature of the body on the upper surface creates
expansive Mach waves, or C+ characteristics, that interact with the oblique shock
and weaken it. We may use MOC to study the Mach wave—shock wave interaction.
According to Crocco’s theorem, the flow downstream of a curved shock is rotational
due to entropy gradient (see Eq. (2.189)). Therefore, to be concise, we have to use the
MOC suitable for rotational flows. However, in case of thin bodies at small angle of
attack, the shock curvature becomes slight and thus entropy gradient downstream of
the shock becomes negligibly small. Under these conditions, the MOC for potential
flows may be applied. Another acceptable simplification that is often used neglects
the reflected Mach waves, or C− characteristics, from the shock in establishing the
shape of the curved shock.
We may use the lattice-point method (or the wave-field approach) in calculating
the shock wave angle, βC upon interaction with the C+ characteristic that emanates
from point B on the body (see Fig. 5.35b). Since we neglected the reflected C−
characteristic from the shock at point C, the problem only involves the left-running
C+ characteristics and thus it becomes a simple-wave problem. Since the flow turning
is imposed at the wave, all streamlines turn the same amount. Therefore, the flow that
emerges from point C on the shock has to have the same angle as the wall at point
B. The reason for the nearly equal sign used in θc ≈ θ B in Fig. 5.35b is due to our
approximation of no reflected C− characteristic at point C on the shock, otherwise the
flow angles are identical in simple-wave problems. Now, with flow angle downstream
of the shock at point C known, we may use an oblique shock chart to establish the
shock wave angle βC (using the weak solution). At any point on the body, say B, we
know the body angle, θ B and using shock-expansion theory (Chap. 4), we calculate
the local Mach number, M B . There is a Mach angle μ B that corresponds to M B and
the sum of the two angles; θ B + μ B is the angle of the local C+ characteristic that
emanates from point B (see Fig. 5.35b). Subsequently, we move to other points on
the body, e.g., D, and repeat the process.
Fig. 5.34 An S-shaped
supersonic nozzle based on
two centered expansion
waves of opposite family
y
x
1
2
3
4
5
264
5 Method of Characteristics
(a)
(b)
G
MC
C
E
C
C+
A
M1 > 1
A
B
D
B
H
F
B
Reflected Ccharacteristic
(neglected)
C-
C
1
c
B
B
A
Fig. 5.35 The waves on a slender body in supersonic flow a Interaction of Mach waves and an
oblique shock (simplified view), b expanded view of the flow and shock angles
Example 5.9 A 2-D pointed body at zero angle of attack is in a Mach 2 flow, as
shown. Its leading edge angle on its upper surface is θ A = 10◦ . Point B on the body
has an angle of θ B = 8◦ . Calculate:
(a) The oblique shock angle, β A in degrees
(b) The oblique shock angle, βC in degrees
(c) Mach number at the nose (downstream of the O.S.)
(d) Mach number at point B on the body
(e) The angle of C+ characteristic at point B
C
o
c
C
M1 = 2.0
B
=8
C+
A
B
B
B
=8
o
o
A
A
= 10
x
Solution:
With M1 = 2 and the nose angle of 10◦ , we use the oblique shock charts (Fig. 4.13)
to get the wave angle, β A ≈ 39.5◦ . Since the flow angle downstream of the shock at
C is estimated to be 8◦ , i.e., the same as the body at B, we use the oblique Shock
charts to get βc ≈ 37.2◦ .
We march across the O.S. at the nose, by using the following relations:
M1n = M1 sin β A = 2 sin 39.5◦ ≈ 1.27
M2n ≈ 0.80 from NS table
5.6 Examples and Applications
M2 = M A =
265
M2n
≈ 1.63
sin(β A − θ A )
The Prandtl-Meyer angle corresponding to M A is ν A ≈ 15.7◦ and since the isentropic
flow turning from A to B is 2◦ , i.e., the difference between the nose angle and the
wall angle at B (10◦ − 8◦ ), the P-M angle at B is ν B ≈ 17.7◦ . The corresponding
Mach number is M B ≈ 1.7.
The angle that C+ characteristic makes with respect to the x-axis at point B is
θ B + μ B . The Mach angle at B is
μ B = sin−1 (1/M B ) = sin−1 (1/1.7) ≈ 36◦
Therefore the angle of C+ characteristic at point B is ≈ 44◦ .
5.7 Summary
Method of characteristics (MOC) is a numerical-graphical technique that allows
supersonic flowfields (including upper transonic flows) with large perturbations to
be analyzed. We showed that two real characteristics exist in supersonic flows and
we demonstrated that these characteristics are indeed local right-running and leftrunning Mach waves. With unique compatibility conditions that hold along the characteristics, we march in the flow direction and establish flow properties along a
characteristic network. We introduced two equivalent approaches to MOC, namely
wave-field and lattice-point methods where the former approach was more suitable
for problems with free boundaries than the latter method. Although our analysis was
based on potential flows, the rotational flows, e.g., with strongly-curved shocks, may
also be analyzed with a suitable MOC that include flow vorticity (e.g., see [7] or [3]).
For thin bodies at small angle of attack, potential flow characteristics may be used
to establish the shape of the curved shock.
In the applications section, we learned that supersonic nozzle/wind tunnel contours
may be designed using the MOC. The wave-field method captures the flowfield of
under- or over-expanded exhaust plumes. We learned to follow a streamline in supersonic flows and form possible supersonic passages, e.g., an S-shaped nozzle. These
applications may be extended to include turbomachinery blade design in supersonic
flows. Additional original contributions to MOC may be found in Prandtl-Busemann
[6] and Shapiro-Edelman [8].
Problems
5.1 A C+ characteristic intersects the wall of a supersonic nozzle, as shown. The
wall angle is θw = +15◦ . Calculate the Mach number at 2 (on the wall) and the slope
of the C+ characteristic, i.e., dy/dx, between points 1 and 2.
266
5 Method of Characteristics
o
w = +15
2
C-
C+
y
M1 =2.0
x
o
1 =+10
1
5.2 In a steady, 2-D, irrotational, isentropic supersonic nozzle flow, we have the
flow Mach number and direction at point 1, as shown. The nozzle wall is straight and
makes an angle, θw = −9.5◦ with respect to x-axis. Calculate the flow Mach number
on the nozzle wall that corresponds to the C− characteristic that passes through point
1 (i.e., point 2, M2 ). Also calculate the slope of the C− characteristic, dy/dx, between
points 1 and 2.
M1 =2.2
o
1 =- 4
1
y
C-
x
2
w
5.3 Based on the initial data line in a diverging section of a 2-D supersonic nozzle,
as shown, M1 = M3 = 1.6, θ1 = −θ3 = −15◦ , M2 = 1.8, θ2 = 0◦ . Use Method
of Characteristics (MOC) for 2-D irrotational flow to calculate:
(a) M5 and θ5
(b) M7 and θ7
8
y
3
x2 5
7
1 4
6
5.4 A 2-D C+ characteristic intersects an oblique shock, as shown. Assuming the
flow angle at point 2 is 12◦ , calculate:
(a) Oblique shock angle, β2
(b) Downstream Mach number, M2
(c) Constants K +1 and K +2
(d) Was the assumption about θ2 correct?
5.7 Summary
267
M ∞ = 3.2
β2
2
M∞ = 3.2
M2
θ 2 = 12o
C+ characteristic
y
x
1
M1 = 2.2
θ 1 = 9o
5.5 A 2-D, minimum-length supersonic nozzle is to be designed using the MOC.
The exit Mach number is M2 = 2.8.
(a) Calculate the maximum wall angle at the throat, θw, max .
(b) What is a good assumption for θinitial ?
(c) What is the slope dy/dx of the first C− characteristic?
(d) Calculate the slope of the last C+ characteristic.
w, max
C+ C+
y
C-
C+
C+
C-
Me=2.8
x
The first Ccharacteristic
The last C+
characteristic
5.6 We intend to design a symmetric minimum-length 2-D supersonic nozzle (with
uniform parallel flow at the throat) using the method of characteristics. The exit Mach
number is Me = 1.2 and the medium is air with γ = 1.4. The Prandtl-Meyer angle
at the exit is νe = 3.558◦ . We wish to use two waves (i.e., two C− characteristics),
at the throat to cause the necessary turning. The first wave is to accomplish an initial
turning of θinitial = 0.279◦ and the second wave will cause 1.5◦ of turning.
y
3
A
5
e
Me=1.2
2
1
Calculate:
(a) K −1 in degrees
(b) K +3 in degrees
4
x
Centerline
268
5 Method of Characteristics
(c) Nozzle wall angle between points A and 3 (in degrees)
(d) Nozzle wall angle between points 3 and 5 (in degrees)
(e) Compare the nozzle area ratio, Ae /Ath with the isentropic value.
5.7 In a 2-D irrotational supersonic flow, we have C+ and C− characteristics that
form a network, as shown. We have the following parameters known at the initial
data line:
Parameters
1
Nodes
2
3
M
θ (deg)
2.0
0.0
1.8
4.0
1.5
8.0
y
x
10
8
12
13
3
2
1
5 7
4
9
6
11
Centerline
Calculate:
(a) Flow angle θ and Mach number at node 5, i.e., θ5 and M5
(b) Wall angle connecting nodes 3 and 8 on the wall, θw |3−8
(c) Mach number, M11
5.8 We wish to design an isentropic minimum-length, two-dimensional, supersonic
nozzle for air (γ = 1.4) with uniform inlet flow condition at the choked throat. The
exit Mach number is to be Me = 1.8, as shown in Fig. 5.36. We use the Method of
Characteristics, MOC, to design the nozzle and choose the initial turning angle of
θinitial = 0.365◦ and five subsequent turns of 2◦ each from the sonic throat with
uniform parallel flow, as shown in the schematic drawing.
Calculate:
(a) The maximum wall angle, θw, max
(b) The flow angle, θ7
(c) The flow angle, θ13
5.9 A 2-D minimum-length nozzle is designed using MOC. There are 5 C− characteristics at the throat and each turn the flow equally by 2◦ (Fig. 5.37). Calculate:
(a) The exit Mach number, Me
(b) The slope dy/dx of the first C− characteristic, i.e., AB
(c) The flow angle in region 2, i.e., θ2
(d) The flow angle in region 3, i.e., θ3
5.7 Summary
269
22
18
13
7
th
M e=1.8
A
w,max
6
12
e
y
Centerline
1
8
14 19
23
26
x
Minimum-Length Nozzle with Uniform Parallel Inlet Condition
Fig. 5.36 Figure belonging to Problem 5.8
3
e
A
2
Me
1
y
B
x
Fig. 5.37 Figure belonging to Problem 5.9
e
y
Centerline
x
Me = 2.06
pe=1.621 p
a
pa
Inviscid constant-pressure ( jet) boundary
Fig. 5.38 Figure belonging to Problem 5.11
5.10 Calculate the initial exhaust plume angle for an under-expanded jet with Me =
2.6, pe / pa = 3.0 with the ratio of specific heats, γ = 1.4. Assuming the flow in
the nozzle was isentropic; calculate the nozzle pressure ratio, NPR that is defined as
pt0 / pa where pt0 is the total pressure at the nozzle entrance.
5.11 A symmetrical two-dimensional under-expanded nozzle has an exit Mach number of Me = 2.06 and exit static pressure that is 162.1 % of the ambient pressure,
i.e., pe = 1.621 pa (see Fig. 5.38). Assuming that the nozzle exit flow is uniform and
the gas specific heat ratio is γ = 1.4 = constant, use MOC (wave-field method) to
map out the (inviscid) flowfield in the exhaust plume and establish the jet boundary.
What is the wavelength of the periodic plume structure, as compared to nozzle exit
half-width?
270
5 Method of Characteristics
5.12 A symmetrical two-dimensional over-expanded nozzle has an exit Mach number of Me = 2.0 and exit static pressure that is 50 % of the ambient pressure, i.e.,
pe = 0.5 pa . Assuming that the nozzle exit flow is uniform and parallel with the gas
specific heat ratio of γ = 1.4 = constant, calculate:
(a) The oblique shock angle, β1
(b) The reflected oblique shock angle, β2
(c) Static pressure in region 3, p3 / pa
(d) The angle of the jet boundary AB with respect to x-axis
(e) The angle of the jet boundary BC with respect to the jet axis
Centerline
e
Me= 2.0
pe=0.5 pa
1
1
y
3
2
4
2
x
B
A
C
Inviscid constant - pressure jet boundary
5.13 The initial data line in a 2-D straight diverging duct is a circular arc with
radius of curvature R, where R = 1 m. The flow Mach number is M = 2.0 =
constant in magnitude on the initial data line. The straight, radial diverging duct has
its two sidewalls symmetrical with respect to the axis of the duct and their angles are
θw = ±15◦ , as shown. There are five points on the initial data line. Point number 1
is on the lower wall. Point number 3 is on the centerline. Point number 5 is on the
upper wall. Points 2 and 4 are half-way between 1 and 3 and 3 and 5 respectively.
Fill out the table below for the coordinates and flow parameters at the 5 initial data
points.
y
R =1m
w = +15
5
3
x
Initial Data Line
1
Parameters
o
w = -15
o
Initial Data Line
1 2 3 4 5
x (m)
y (m)
(deg)
(deg)
(deg)
5.14 A parallel uniform flow at Mach 1.4 approaches a sharp corner that turns the
flow in the clockwise direction. The flow downstream of the corner reaches Mach
number of 2.8. Calculate the wall turning angle and use the wave-field method to
5.7 Summary
271
calculate the coordinates of the bending streamline with h 1 = 1 (see Fig. 5.29 for
definition sketch) and graph the streamline. Assume that the flow is planar 2-D with
γ = 1.4.
5.15 Apply wave-field method to the incident-reflected expansion wave problem
from a solid wall, as shown. The Mach number in field #1 is 2.0 and the flow angle is
0◦ . The incident pressure wave turns the flow 2◦ in the clockwise direction. Assuming
γ = 1.4, calculate:
(a) Mach number in field #2
(b) Mach number in field #3
(c) Slopes of the incident and reflected waves
(d) Static pressure ratio, p3 / p1
Incident pressure
wave (expansion)
Reflected pressure
wave (expansion)
LRW
RRW
y
x
2
1
3
Solid Wall
5.16 A uniform parallel supersonic flow approaches a 2-D concave (compression)
corner, as shown. The corner is approximated by two straight ramps, AB and BC,
each making 2◦ angle with respect to their upstream flow. The two C+ characteristics
are compression waves of 2◦ strength each. Using the wave-field method, calculate:
(a) Mach number in field #2, M2
(b) Mach number in field number 3, M3
(c) The slope of the wave AD
(d) The slope of the wave BD.
D
M1=2.6
o
1 =0
=1.4
y
x
1
A
C+
C+
3
2
C
B
5.17 A parallel uniform flow at Mach 2.0 approaches a sharp corner that turns
the flow in the clockwise direction by 10.37◦ . First calculate the flow Mach number
downstream of the expansion corner. Second, calculate the coordinates of the bending
streamline with h 1 = 3.0 and graph the streamline. Assume that the flow is planar
2-D with γ = 1.4.
5.18 To accelerate a Mach 2.0 stream to Mach 2.4 isentropically, an expansion fan
that turns the flow by 10.37◦ will be formed, as shown schematically by lines AC and
272
5 Method of Characteristics
AE. This is the same problem as 5.16 with a single streamline that passes through
h 1 = 3.0. Now, construct a second streamline with h 1 = 4.0, which along with
the first streamline may form the walls of a supersonic nozzle. Calculate and graph
a second streamline with h 1 = 4.0. What is the area ratio, A2 /A1 , of this nozzle?
Assume flow is planar 2-D and γ = 1.4.
h
4.0 M =2.0 1
C
1
3.0
E
B
D
2
M 2=2.4
A
x
5.19 A 2-D S-shaped supersonic nozzle is designed based on two centered expansion
fans of opposite family, ABC and DEF, following the schematic drawing shown. The
flow at the entrance to the S-shaped nozzle is parallel and uniform in the x-direction
with M1 = 2.0. The nozzle is divided into three fields. The field #2 represents the
flow that has emerged from the first bend. It has attained a flow angle of θ2 = −20◦ .
The field #3 represents the flow that has emerged from the second bend and is purely
axial, i.e., θ3 = 0◦ . Calculate:
(a) The Mach number in field #2, M2
(b) The Mach number in field #3, M3
(c) The nozzle exit-to-entrance area ratio, A3 /A1
D
B
y
C
x 1
2
3
A
E
F
5.20 We apply wave-field theory to the half diamond airfoil in supersonic flow, as
shown. Assuming that the flow downstream of the shock is irrotational, we may use
the irrotational MOC at the shoulder with two C+ characteristics, AB and AC. Mach
number in region 2 is M2 = 2.0 and the flow angle is θ2 = +4◦ . The flow angles in
regions 3 and 4 are θ3 = 0◦ and θ4 = −4◦ respectively. Calculate:
(a) Mach numbers in regions 3 and 4
(b) The angles that C+ characteristics make with respect to the x-axis
(c) Flight Mach number, M1
5.7 Summary
273
C
B
y
3
C+
C+
2
A
1
4
x
5.21 Apply the lattice-point method to a curved (convex) body ABDF to estimate
the shock angles for the segments, AC, CE and EG. The body angles are known to be:
Body Angle (deg) A
B
8
12
D
6
G
F
4
E
C
1
M1=2.4
= 1.4
A
B
D
F
H
x
References
1. Anderson, J.: Modern Compressible Flow with Historic Perspective, 3rd edn. McGraw Hill,
New York (2003)
2. Anon: Equations, Tables, and Charts for Compressible Flow. NACA TM 1135, Moffett Field,
CA (1953)
3. Ferri, A.: Elements of Aerodynamics of Supersonic Flows. Macmillan Company, New York
(1949)
4. Krasnov, N.F.: Aerodynamics of Bodies of Revolution. Elsevier Publication, New York (1970)
5. Liepmann, H.W., Roshko, A.: Elements of Gas Dynamics. Wiley, New York (1957)
6. Prandtl, L., Busemann, A.: Näherungsverfahren zur Zeichnerischen Ermittlung von Ebenen
Strömungen mit Überscallgeschwindigkeit. Stodola Festschrift p. 499 (1929)
7. Shapiro, A.H.: The Dynamics and Thermodynamics of Compressible Fluid Flow. Ronald Press,
New York (1953)
8. Shapiro, A.H., Edelman, G.M.: Method of characteristics for two-dimensional supersonic flow
graphical and numerical procedure. J. Appl. Mech. 14(2), A-154 (1947)
9. Taylor, G.I., Maccoll, J.W.: The air pressure on a cone moving at high speed. Proc. R. Soc. Lond.
139, 298–311 (1933)
Chapter 6
Aerodynamics of Non-lifting Bodies
Abstract This chapter presents the sources of drag that are encountered by a
nonlifting body that is moving at transonic Mach numbers. In inviscid conditions
drag can be produced if the integrated pressure over a body has a component in the
opposite direction to the flow (pressure drag). First, the reader is introduced to the
relation between the geometry of a body and the pressure distribution over the body.
Subsequently, a method for the calculation of the pressure drag over axisymmetric
bodies is presented based on the linear potential flow equation. It is shown how the
pressure drag is a function of the cross-sectional area distribution of the body. The
concept of area ruling is presented and examples are shown of practical implementations of the area rule. If the flow is assumed viscous, other drag sources arise: friction
drag and drag due to boundary-layer separation. A qualitative characterization of laminar and turbulent boundary layers is presented along with the concepts of transition
and separation. Also, the interaction between shock and boundary layer (both weak
and strong) is further detailed and it is shown how this influences drag divergence in
transonic conditions. In addition, calculation methods are presented to estimate the
boundary-layer properties of laminar and turbulent boundary layers along with their
transition region for bodies subjected to an external pressure and velocity distribution.
This chapter contains 8 examples and concludes with 29 practice problems.
6.1 Introduction
There exist two types of drag: friction drag and pressure drag. Friction drag is mostly
caused by the boundary layer between the body and the outer flow and is a function of
the viscosity of the fluid. A nonlifting body in transonic flow can experience pressure
drag through two mechanisms. One is due to a wake that is formed when the flow
streamlines separate from the body surface (drag due to flow stagnation and separated
flow). The second cause is through the formation of shock waves (wave drag). Wave
drag is an aerodynamic phenomenon which is unique to supersonic flow. This type
of drag is associated with the energy which is radiated away from the vehicle in
the form of pressure waves. In the present chapter we focus on these three causes
for drag. The aim of the chapter is to explain to the reader how a nonlifting body
produces drag and what parameters are of influence.
© Springer Science+Business Media Dordrecht 2015
R. Vos and S. Farokhi, Introduction to Transonic Aerodynamics,
Fluid Mechanics and Its Applications 110, DOI 10.1007/978-94-017-9747-4_6
275
276
6 Aerodynamics of Non-lifting Bodies
u2
control surface, S
u1
C
B
F
p
body
τ
A
-F
body
D
2D control volume
control surface, σ
Fig. 6.1 Body force generated by pressure and shear stresses on the body and their relation to the
velocity deficit in the wake
In Fig. 6.1 a body is shown in a control volume. It is assumed that the upper and
lower boundary of the volume are formed by streamlines of the flow. The pressure
and shear forces that act on the body result in a total body force, F body . By Newton’s
third law, the body exerts an equal and opposite force on the fluid, −F body . Let us
investigate the effect of this body force on the momentum in the flow. We start from
Newton’s second law:
d
(2.114)
F = (mV )
dt
Remember that the right-hand side (RHS) of (2.114) physically represents the time
rate of change in momentum that is moving through the control volume. Its representation has been given in the LHS of (2.121):
F=
∂
ρV dV +
(ρV · dS)V
∂t
V
(6.1)
S+σ
The integral over the second control surface, σ is always zero (regardless of control
volume shape) due to the fact that the inner product of velocity and surface area
vanishes on the body surface. If we neglect the small body forces on the fluid ( f = 0),
the external force F can be broken down into two parts: the reaction force from the
body, −F body , and the force exerted across the outer boundary:
F=−
pdS − F body
(6.2)
S
If we assume steady conditions (∂/∂t = 0) we can rewrite (6.1) according to:
Fbody = −
S
(ρV · dS)V −
pdS
S
(6.3)
6.1 Introduction
277
In (6.3), the force exerted on the body is coupled to properties of the flow inside
the control volume. We would like to determine the x-component of the body force
from the parallel velocity components at the beginning of the control volume, u 1 , and
at the end of the control volume, u 2 . We write the x-component of (6.3) as follows:
Fbodyx = −
(ρV · dS)u −
S
pdSx
(6.4)
S
First we notice that when the boundaries of the control volume are chosen at an
infinite distance from the body the pressure, p, should be constant on this boundary.
For a constant pressure, the integral yields:
pdSx = 0
(6.5)
S
Since the upper and lower surface of the control volume are streamlines of the flow,
the velocity vector and the control surface are locally aligned. This means that the
inner produce V · dS = 0 along those boundaries. Therefore, we can write for Fbodyx :
Fbodyx =
B
ρ1 u 21 dy −
A
C
ρ2 u 22 dy
(6.6)
D
By employing the continuity equation in integral form (2.109) over the entrance
and exit of this control volume and subsequent substitution yields the following
expression:
C
(6.7)
Fbodyx = ρ2 u 2 (u 1 − u 2 )dy
D
This substitution is left as a homework problem (Problem 6.1).
The body force in x-direction is of course the drag force that a body experiences
whenever exposed to a fluid. What is demonstrated by (6.7) is that the drag of a body
is directly related to a momentum decrement in the wake of the flow. It shows that
whenever a drag force is present, it is being accompanied by a momentum decrement
in the wake. Vice versa, the measurement of the total momentum deficit in the wake
is a direct measure for the amount of drag that a body experiences. It can be shown
that the momentum deficit, in turn, is related to total pressure deficit in the wake
(see Problem 6.2). In Fig. 6.2 the total pressure deficit is shown at an outboard wing
section of a modern high-subsonic wing. Even though this experimental result does
not pertain to a nonlifting body, it is still a useful example. It can clearly be seen
how the shock wave and the boundary layer both contribute to the total pressure loss
(drag) in the wake. Note that in this example the contribution of the wave drag is
much less than the friction drag.
278
6 Aerodynamics of Non-lifting Bodies
Fig. 6.2 Drag is related to
the total pressure loss in the
wake (after Ref. [14])
52
η =0.723
M=0.84
CL=0.40
51
Probe height (in.)
0
1.0
Loss in total pressure
due to shock wave on
wing upper surface
50
49
Loss in total
pressure due to
boundary layer
48
47
46
0
.2
.4
.6
.8
1.0
1.2
1.4
ΔpT (psi)
If we subdivide the outer geometry of a typical high-subsonic transport aircraft
into different components, we can make a distinction between components that are
primarily designed to produce aerodynamic forces, and components that are designed
not to produce any aerodynamic forces. Examples of the latter are the fuselage
(including cockpit and tail cone), the pylons, the engine nacelles, and the vertical
tail in cruise conditions. The aerodynamic design of such components should be
such that the overall drag is minimized. In transonic conditions, the production of
wave drag on these components should be reduced to a minimum. A general design
rule for these components is that local supervelocities should be minimized. Why
supervelocities should be minimized is evident when this chapter is studied.
Even though this statement seems simple enough, in practice it is often not as
easy to do this due to conflicting requirements on the geometry. Cockpit design,
for example, must satisfy strict visibility constraints that ensure that the pilot has a
good view on the runway when the aircraft is at high pitch angles before the landing.
Window design should ensure that this constraint is satisfied, although this might lead
to sharp corners and high curvatures that lead to high supervelocities and possible
wave drag in high speed flight. We will present more examples where functional
and aerodynamic requirements are conflicting. How do we design these nonlifting
components on aircraft such that the wave drag that they produce in high-subsonic
conditions is minimized?
The present chapter presents the relation between geometry and supervelocities,
via the pressure coefficient, Cp . Subsequently, the pressure drag in supersonic conditions is treated with a focus on the sonic area rule. The foundations of boundary layer
flow and friction drag is discussed in the Sect. 6.3 of this chapter. Typical transonic
characteristics such as shock-boundary-layer interaction are introduced as well as
the effect of Mach and Reynolds number on friction drag and transition. The chapter
closes with a qualitative section on the relation between external shape of intersecting
components and their interference drag.
6.2 Pressure Distribution Over Nonlifting Bodies
279
6.2 Pressure Distribution Over Nonlifting Bodies
In this section we investigate the effect of the geometry and Mach number on the
supervelocities over a nonlifting body. In high subsonic conditions the flow over
curved surfaces can surpass the speed of sound creating expansion waves or shock
waves. The effect of shock waves is wave drag and possible separation of the boundary
layer. It is therefore instrumental to know how the geometric shape of a body affects
the increase in local Mach number.
6.2.1 Subsonic Inviscid Flow
This section partially follows the introduction to pressure fields and fluid acceleration
as presented by Shapiro [51]. In Chap. 2 we derived the Euler equations in Cartesian
coordinates. Along a streamline we can reduce the system of equations according to:
d p = −ρV dV
(6.8)
We can derive a relation between the local pressure in the fluid and the local velocity.
We can integrate both sides of (6.8) between two arbitrary points along the streamline.
If we assume that the flow is incompressible (i.e. ρ1 = ρ2 = ρ) we obtain Bernoulli’s
equation (see Problem 6.3):
1
1
p1 + ρV12 = p2 + ρV22
2
2
(6.9)
For incompressible flow we can therefore conclude that:
1
pt =
ˆ p + ρV 2 = constant along a streamline
2
(6.10)
The second term in (6.10) is the dynamic pressure, often denoted with q. Similar to the
static pressure, p, the dynamic pressure is a scalar. Bernoulli’s equation literally states
that the local velocity is inversely proportional to the square root of the local pressure.
The pressure coefficient may be written as
Cp =
ˆ
p − p∞
q∞
(3.11)
Note that the subscript ∞ implies that we are dealing with a point in the flow at
an infinite distance from the measuring point. This is referred to as the freestream
condition. Substituting (6.10) and assuming incompressible conditions yields the
following expression for the pressure coefficient:
Cp = 1 −
V
V∞
2
(6.11)
280
6 Aerodynamics of Non-lifting Bodies
This relation directly ties the pressure coefficient to the local velocity. We can
approximate this equation by assuming that the local velocity is a relatively small
deviation from the freestream velocity: V = V∞ + ΔV and ΔV /V∞ ≪ 1. In that
case, (3.11) reduces to:
ΔV
(6.12)
Cp = −2
V∞
We now have a very simple linear relationship between the supervelocity, ΔV , and
the pressure coefficient, Cp . Next, we have to correlate the pressure coefficient to the
local curvature of the streamlines.
Perpendicular to the streamline we can derive a relation between the pressure on
a fluid particle and its velocity along a curved streamline. A schematic drawing of
such a streamline is shown in Fig. 6.3. We define the direction perpendicular to the
streamline as n, and along the streamline with s. The free-body diagram (FBD) shows
the pressure forces that are acting on this fluid particle in the n-direction, in inviscid,
irrotational flow, while the kinetic diagram (KD) shows its centripetal acceleration.
If we assume the dimensions of the fluid particle to be Δs in length and Δr in width,
then Newton’s second law reduces to:
mV 2
= ( p + Δp − p)Δs
r
(6.13)
We know that Δp = (d p/dr )Δr and that m = ρΔsΔr . Combining this with (6.13)
yields the following relation:
ρV 2
dp
=
(6.14)
r
dr
This relation shows that the pressure always increases outward from the center of
curvature.
We find curved streamlines over curved bodies in subsonic, transonic and supersonic flows. These bodies can be concave or convex (see Fig. 6.4). If the body is
convex the normal vector to the body wall, n is defined in the same direction as r
in Fig. 6.3 and the pressure gradient, d p/dn is therefore positive. In other words,
the pressure is increasing with the distance from the wall. Because we know that
p+Δp
n
s
m
r
V
m
p
FBD
V
m
mV 2
r
KD
Fig. 6.3 Forces and accelerations on a fluid particle perpendicular to a curved streamline
6.2 Pressure Distribution Over Nonlifting Bodies
281
the pressure at an infinite distance normal to the body should equal the freestream
pressure, p∞ , we can deduce that the pressure at the wall, p, should be lower than
p∞ . In other words, a convex shape leads to a reduction in pressure and consequently
a negative pressure coefficient, Cp . A similar argument can be made for the concave
shape. However, in this case the direction normal to the body wall, is in opposite
direction to the radius vector, r . Therefore, the pressure gradient in this direction is
negative and the pressure continuously decreases when we move further away from
the wall. Consequently, we can deduce that the pressure at the wall should therefore
be higher than the pressure infinity: p > p∞ . The pressure coefficient at the body
for a concave shape is therefore positive.
We have now qualitatively determined that streamlines along convex curves have
decrease Cp and streamlines along concave curves increase Cp . In addition, we can
see from (6.14) that the there is an inverse relation between the radius of curvature,
r , and the pressure gradient, d p/dn. From this relation we can deduce that when
r → 0, the pressure gradient must tend to infinity, i.e. d p/dn → ∞. We also see that
when limr →∞ d p/dn → 0 and that V → V∞ . The effect of the radius of curvature
of the body, r = R, can be evaluated on a qualitative basis. We can plot the pressure
coefficient Cp as a function of the distance from the center of curvature r , using the
aforementioned relations at the respective boundaries at r = 0 and r → ∞. This is
shown in Fig. 6.5. We can observe that the position of the wall has an effect on the
local pressure, p, at r = R. Whenever the wall is positioned close to the center of
concave wall
R
p
p
8
Fig. 6.4 Relation between
body curvature and pressure
coefficient
dp/dn < 0
dp/dn > 0
R
Cp< 0
Cp
wall
+
0
-
Cp > 0
concave shape
0
R
r
convex shape
8
Fig. 6.5 Qualitative relation
between pressure and radius
of curvature
p>p
8
8
p<p
p
8
p
convex wall
282
6 Aerodynamics of Non-lifting Bodies
curvature (i.e. it has a large curvature itself) the pressure difference between pr = R
and p∞ is large and we have a high Cp . Conversely, when R is large (i.e. the wall
has little curvature) the pressure difference is much smaller and Cp tends to zero.
Hence, on a purely qualitative basis we can state that the radius of curvature of the
body is negatively related to the pressure coefficient. In other words, the larger the
curvature, the higher the absolute value of the pressure coefficient.
We can combine the relation between curvature and Cp and between Cp and
ΔV in a straightforward manner. We know that when a geometry displays a convex
curvature the pressure coefficient, Cp < 0. Consequently, employing (6.12) we note
that the supervelocity, ΔV > 0. A convex curvature, therefore, leads to local speeds
that surpass the freestream velocity. Similarly, when the geometry has a concave
curvature, the pressure coefficient is positive and ΔV < 0. Concave shapes therefore
reduce the speed of the fluid near the body. Finally, we can state that the absolute
curvature (whether convex or concave) has a magnification effect on ΔV . A body
with sharp corners, for example, yields a large speed fluctuations at these corners,
while a body with little curvature only induces mild speed variations.
Now that we have established the effect of local curvature on the pressure coefficient, we take a look at the distribution of the pressure coefficient over a body. The
value of the pressure coefficient at various locations on the body is often termed the
pressure distribution. Strictly speaking this is incorrect. However, we have used the
term “pressure distribution” in relation to the Cp distribution and we will continue to
do so in this chapter. In subsonic flow, the pressure at any point A has an effect on the
pressure at an arbitrary point B (point B is in point A’s “region of influence”). Every
change in pressure coefficient due to the presence of curvature at point A therefore
also affects the pressure coefficient at any other arbitrary point B on that body. This
makes the intuitive prediction of the pressure coefficient somewhat more difficult.
However, we can still relate the geometric features of a given body to its pressure
distribution by relating the change in Cp to the local curvature. For example, if the
flow encounters a convex shape with a small radius of curvature we expect a sharp
increase in local suction. Conversely, if the flow encounters a concave shape with a
large radius of curvature, we expect a gradual increase in local pressure.
This is shown in Fig. 6.6 where two axis-symmetric bodies of identical thicknessto-length ratio are placed in an axial flow field. The ellipsoid has a blunt leading
edge, which leads to a high local curvature. The Cp therefore increases rapidly.
Because the curvature over the ellipsoid is low over the center section of the body,
the pressure coefficient shows little variance. The absence of the blunt leading edge
on the paraboloid makes for a more gradual increase of the pressure coefficient.
However, due to the higher curvature in the center section, the pressure coefficient
reaches a higher peak.
In Fig. 6.7 we show the pressure distribution over an important part of the fuselage: the cockpit. In this case, the aircraft in question has been modified with the
addition of a large nose fairing (radome) to house a radar. The radome has a significant impact on the pressure distribution over cockpit. The pressure distribution over
the top of the fuselage shows the most erratic behavior. Initially the flow is accelerated from near stagnation pressure (Cp = 1) at the nose towards a minimum of about
6.2 Pressure Distribution Over Nonlifting Bodies
283
-0.08
Paraboloid
-0.06
II
-0.04
I
CP
-0.02
Ellipsoid
0
0.02
0.04
0
0.2
0.4
x
0.6
0.8
1.0
X= l
R
I
II
Fig. 6.6 Pressure distribution over an ellipsoid and paraboloid of identical maximum thickness
and length at incompressible axial flow (after Ref. [49])
-0.6
PRESENT METHOD
EXPERIMENTAL DATA
TOP
BOTTOM
-0.4
-0.2
0
50
100
150
200
300
350
400
CP
FUSELAGE STATION ~INS
0.2
0.4
0.6
0.8
1.0
BASIS FUSELAGE
Fig. 6.7 Pressure distributions on a C-135 fuselage with a large radome at zero angle of attack
(after Ref. [6])
284
6 Aerodynamics of Non-lifting Bodies
Cp = −0.24. Then, the flow decelerates once more to a value close to stagnation at the
kink between the radome and the cockpit window. Due to the high convex curvature
over the window, the flow accelerates rapidly beyond the kink. At the bottom curve
we also see the presence of the concave kink between fuselage and radome in the
pressure distribution. Before the kink the velocity on the uncurved part of the radome
is close to its freestream value (Cp ≈ 0). At the kink the flow decelerates (Cp > 0)
after which it accelerates once more over the convex part of the lower fuselage.
Example 6.1 Below an axis-symmetric body is shown. Assuming inviscid, incompressible flow, sketch the notional pressure distribution about this body.
ellipsoid
cylinder
ellipsoid
Solution:
We first identify some of the key characteristics of the body under consideration.
Since we consider inviscid conditions we have a stagnation point at the leading edge
and at the trailing edge where the pressure coefficient is 1. This body has a blunt nose,
so we expect a sharp increase in suction over the first few percent of the body. Then the
curvature decreases, and so does the flow velocity. Hence the slope dCp /dx reduces.
Once the flow reaches the cylinder it does not see any change in curvature. Therefore,
the flow decelerates towards freestream velocity. The Cp will therefore increase to
a value close to 0. Even further downstream a convex shape is present again which
accelerates the flow and creates a second suction peak. Finally, the flow decelerates
to stagnation over the last few percent of the body. Due to the symmetric shape of
the body we also expect a symmetric shape in pressure distribution. Applying this
line of reasoning results in the notional pressure distribution drawn in Fig. 6.8.
6.2.2 Effect of Subcritical Compressibility
In high-subsonic flow, the effect of compressibility on the pressure coefficient has
been discussed in Chap. 3. Using the Prandtl-Glauert compressibility correction, the
Fig. 6.8 Figure belonging to
Example 6.1 (after Ref. [69])
-
Cp
0
1
0
x/l
1
6.2 Pressure Distribution Over Nonlifting Bodies
285
compressible pressure coefficient, Cp M is related to the incompressible pressure
∞
coefficient according to:
Cp M = 0
∞
(6.15)
Cp M = ∞
2
1 − M∞
In other words, the pressure coefficient increases with the freestream Mach number
up to infinity at M∞ = 1. We know from practice that in reality this theoretical value
is not reached. However, it does show qualitatively that the pressure coefficient and
thus the supervelocities over a given body in high-subsonic flow are affected by the
compressibility of the air.
Let us clarify this qualitative statement through the relationship between the pressure coefficient and Mach number in steady isentropic flow. If we employ the relation
between Mach number and static temperature in isentropic conditions (2.147) and
the relation between pressure and temperature (2.103), we have:
γ
γ
γ − 1 2 γ−1
γ − 1 2 γ−1
= p∞ 1 +
M
M∞
pt = p 1 +
2
2
(6.16)
This relation shows that under the assumption of isentropic compression there is a
relation between the local Mach number, M, and the Mach number at infinity, M∞ ,
via the static pressure. We use the following definition of the pressure coefficient:
2
Cp =
2
γ M∞
p
−1
p∞
(3.11)
Combining (6.16) and (3.11) results in the following relation between pressure coefficient, local Mach number, and freestream Mach number:
⎤
⎡
γ−1 2
M
1
+
2 ⎣
∞
2
(6.17)
− 1⎦
M2 =
γ−1
γ−1 1
γ
2
1 + 2 γ M∞ Cp
For air flow about curved bodies this equation can be used up to a local Mach number
of 1.58. This is the (theoretical) maximum supersonic speed that can be achieved over
a body in subsonic flow. It will be derived in Sect. 7.5. A graphical overview relating
the local Mach number to the pressure coefficient is shown in Fig. 6.9. It can be seen
from this graph that high negative pressure coefficients (suction) in combination
with moderate to high subsonic Mach numbers lead to a local Mach number well
beyond 1.
If we define the increase in local Mach number as ΔM = M − M∞ then we can
also plot the relative increase in Mach number ΔM/M∞ . This is shown in Fig. 6.10.
If we take this graph and we compare the relative increase in Mach number for
Cp = −1 then we can see that for M∞ = 0.1 the increase is around 42 %, while
for M∞ = 0.8 the increase is around 62 %. In other words, there is a progressive
increase in local Mach number with increasing freestream Mach number.
286
6 Aerodynamics of Non-lifting Bodies
Fig. 6.9 Relation between
local Mach number and
pressure coefficient for
various values of M∞
1.6
0.8
0.7
0.6
1.2
M=
tr e
es
0.5
M
0.4
am
1
0.3
0.2
r, M
0.6
be
um
hn
ac
0.8
8
0.4
0.1
0.2
0
2
, γ=1.4
3-γ
fre
Local Mach number, M
1.4
0
-5
−10
-15
Pressure Coefficient, C p
8 0.8
0.7
free stream Mach number, M
0.8
0.6
0.4
0.5
0.3
0.2
8
Relative Local Incrase in Mach number, Δ M/M
Fig. 6.10 Relation between
relative local Mach number
and pressure coefficient in
regions with supervelocity
for various values of M∞
0.7
0.1
0.6
0.5
0.4
0.3
0.2
0.1
0
0
−0.5
−1
−1.5
−2
Pressure Coefficient, C p
We have now established that the curvature in a body is directly responsible for
the supervelocities about that body and the related pressure coefficient. In addition,
we have shown that the local Mach number is strongly affected by the pressure
coefficient and value of the freestream Mach number. This should give the reader
some indication about the relation between the geometry of the body, the freestream
Mach number and the expected pressure coefficient. In addition to the effect close to
the body, the body has a larger effect on the streamlines further away from the body
when the Mach number approaches 1. As can be seen in Fig. 6.11, at a Mach number
of 0.8, region of influence is much larger that at a Mach number that is well-below 1.
This is a second effect of compressibility that is important for transonic aircraft.
6.2 Pressure Distribution Over Nonlifting Bodies
287
Fig. 6.11 Streamlines over a
wavy wall (after Ref. [3])
M ∞ << 1
M ∞ ≈ 0.8
As was established in Chap. 3, this effect plays an important role when wind tunnel
tests are carried out to predict the aerodynamic coefficients of the airplane.
The displacement effect caused by the body on the streamlines far away from
it continues to grow with Mach number. In supersonic conditions, the linearized
potential theory shows that the streamlines even at an infinite distance from the body
are affected. In fact, they follow exactly the same path as the wall itself. In supersonic
conditions, it is the slope of the surface with respect to the freestream velocity vector
that determines the local pressure coefficient [see (3.69)].
6.3 Wave Drag
The aerodynamic shape of high-subsonic aircraft is to be designed such that the Machdependent drag penalty in its cruise configuration is minimized. One instrument to
do this is by creating a favorable cross-sectional area distribution of the airplane. In
1951 it was Adolf Büsemann who first described that a streamtube of air at near-sonic
speeds no longer varied in diameter with speed but remained constant. Conversely,
when a streamtube of air would flow over a curved surface, the velocity inside the
tube would increase but its diameter would stay the same.
Let us investigate this claim by looking at the steady inviscid flow equations for a
one-dimensional streamtube with a variable cross section A. The continuity equation
(2.110) states that the mass flow entering a section of the streamtube equals the mass
flow that exits the streamtube. In other words:
ρu A = constant
(6.18)
The change in mass flow is therefore:
d(ρu A) = 0
(6.19)
288
6 Aerodynamics of Non-lifting Bodies
We can elaborate (6.19) by using logarithmic differentiation1 :
dA
dρ du
+
+
=0
ρ
u
A
(6.20)
The momentum equation (2.183) in steady one-dimensional flow reduces to:
d p = −ρudu
(6.21)
Furthermore, we have already seen that the speed of sound can be written as:
a=
dp
dρ
(2.106)
If we combine (6.19), (6.21), and (2.106), we can show that (see Problem 6.8):
dA
du
= M2 − 1
A
u
(6.22)
We observe that when M = 1, dA/A = 0 even though a finite du exists. This
shows that Büsemann’s claim is indeed true for flows near the speed of sound. It
also corresponds to our earlier observation that the region of influence in transonic
conditions is much larger than in subsonic conditions. Streamtubes around a body
(or wavy wall, as in Fig. 6.11) are simply displaced rather than reduced in area. At
M = 1 this effect extends, in theory, to infinity.
Richard Whitcomb realized that the individual streamtubes over a wing-body
combination would interfere with each other due to dramatic increases in crosssectional area where the wing joined the fuselage. Rather than looking at the shape
of the wing and fuselage separately he postulated that one needed to look at the crosssectional area of the entire airplane configuration and keep it as smooth a curve as
possible as it increased and decreased around the fuselage, wing, and empennage.
This design rule to minimize wave drag was termed the Whitcomb transonic area
rule [63]. The area rule states that the wave drag of an airplane configuration depends
primarily on the longitudinal distribution of the total cross-sectional area. This tells
us that a simple body of revolution that has the same distribution of cross sectional
area as a complex airplane produces the same wave drag. To evaluate the wave drag
of such a complex aircraft we can therefore reduce the complexity of the problem
by evaluating its equivalent axis-symmetric body. The equivalent body can be constructed by measuring the cross-sectional area of a fuselage station perpendicular to
the longitudinal axis of the airplane (see Fig. 6.12).
To demonstrate the applicability and limitations of the area rule, Fig. 6.13 shows a
comparison study carried out in the wind tunnel by Chan [8, 9]. Four different wind
tunnel models are subjected to a Mach number ranging from 0.8 to 1.1. We see a
1 See a text book on calculus such as Ref. [2] for details on logarithmic differentiation.
6.3 Wave Drag
289
Fig. 6.12 Procedure for
obtaining the axis-symmetric
equivalent body of a complex
airplane (after Ref. [68])
swept-wing configuration with its equivalent body (1) and a delta-wing configuration
with its equivalent body (2). The equivalent bodies are axisymmetric and have the
same cross-sectional area distribution as the swept-wing configuration and deltawing configuration, respectively. The graph shows us the variation in zero-lift drag
coefficient of the four wind tunnel models over the Mach range. We see that when the
Mach number is increased beyond M = 0.9 all models show a significant increase
in drag. This drag is attributed to the formation of shock waves and is therefore
0.25
Zero-lift drag coefficient, CD
0
0.20
equivalent body 1
0.15
0.10
equivalent body 2
0.05
Mach number, M∞(~)
0.8
0.9
1.0
1.1
0
5
0
-20
-15
-10
-5
10
M∞2-1
K=
2 2
(γ +1)M∞τb
Fig. 6.13 Zero-lift drag coefficient of wing-body combinations and their equivalent axisymmetric
bodies (data from Refs. [8, 9])
290
6 Aerodynamics of Non-lifting Bodies
Fig. 6.14 Schlieren
visualization for swept-wing
configuration and equivalent
body at zero angle of attack
and M = 0.98 (from
Ref. [9]). a Swept wing
configuration. b Equivalent
body
termed wave drag. This wave drag can be quantified by the momentum deficiency
over the shock wave as we have seen in Sect. 6.4.3. If we compare the curves for
the delta-wing configuration and its equivalent body, we see that their wave drag
in the transonic domain is very similar. This is in line with Whitcomb’s area rule.
However, when we look at the comparison between the swept-wing configuration
and its equivalent body, we notice that there is a significant difference between the
wave drag of the two bodies.
6.3 Wave Drag
291
Let us investigate why there is such a strong difference between the swept-wing
configuration and its equivalent body. It is expected that the formation of shock waves
is an important reason for discrepancy. Since pressure waves, such as shock waves or
Mach waves, induce density gradients in the flow, schlieren optics is an often used tool
to visualize these waves and the flow field. Schlieren (German, meaning “streaks”)
optics captures density gradients in compressible flows. The physical principle behind
such visualization techniques as schlieren or shadowgraph is based on light beam
deflection as it propagates through a medium with density gradients. For details
of these flow visualization techniques, their distinguishing features, limitations and
applications, the reader may refer to classical gas dynamic books such as Shapiro [50],
Liepmann and Roshko [37], Ferri [20] or books on wind tunnel testing principles.
Figure 6.14 shows schlieren images of the two wind tunnel models at M = 0.98. We
see that both models show shock waves in the same longitudinal position. However,
the shock wave over the wing of the swept-wing model is much stronger than the
corresponding shock wave over the equivalent model. The shock wave over the
swept-wing model causes boundary-layer separation at the outer portion of the wing
which causes additional drag with respect to the equivalent body. This explains the
larger drag over the swept-wing configuration compared to its equivalent body.
Based on the observations above, we conclude that the area rule works well for
delta-wing aircraft of low aspect ratio but works less well for swept-back wing-bodies
with moderate aspect ratio. The area rule is therefore limited to low-aspect-ratio
thin-wing body combinations. Viscous effects, such as shock-wave-boundary-layer
interaction, prevent a correct application of the area rule to wing-body combinations
with higher aspect ratio or thicker wings.
6.3.1 Calculation of Pressure Drag by Use of Momentum
Theory
There is a large body of literature that supports the concept of the transonic area rule
in order to minimize the wave drag of a nonlifting body in transonic conditions, e.g.
Ref. [65]. We would like to know what cross-sectional area distribution would give
us the lowest wave drag. In the analysis that develops in this section we follow the
classic text of Ashley and Landahl for the computation of wave drag in inviscid flow
[4].
We start the derivation of wave drag in supersonic flow by looking at the axissymmetric body in uniform flow conditions. A two-dimensional representation of
such a body is presented in Fig. 6.1. The force acting on this body is governed by
(6.3). We introduce unit normal vector, n, such that we can bring both terms on the
RHS of (6.3) within a single integral:
(6.23)
F body = − [ρV (V · n) + pn]dS
S
292
6 Aerodynamics of Non-lifting Bodies
In this equation ndS = dS. Under the same assumptions, the continuity equation in
integral form (2.110) can be written as follows:
ρ(V · n)dS = 0
(6.24)
S
At this point we introduce a normalized disturbance velocity vector v, which is
defined as follows:
∇φ
V
v=
=
−i
(6.25)
V∞
V∞
Here V∞ is the freestream velocity in the x-direction and i is the unit vector parallel to
the x-axis. Note that this perturbation function v represents the normalized perturbation to the velocity field in x, y, and z direction, assuming that the freestream is aligned
with the x-axis. Substituting (6.25) into (6.23) we obtain the following expression:
2
pn + ρV∞
(6.26)
Fbody = −
(v + i) (v · n + i · n) dS
S
Similarly, the continuity (6.24) can be written as:
ρ (v + i) · ndS = 0
(6.27)
S
We can employ the continuity (6.27) to show that (6.26) can be simplified. If we
assume that S is an enclosed surface we can introduce the freestream static pressure,
p∞ , and reduce the momentum equation to (see Problem 6.9):
2
Fbody = −
v (v · n + i · n)dS
(6.28)
( p − p∞ ) n + ρV∞
S
Equation (6.28) is a compact equation that relates the total body force to the properties
of the freestream flow ( p∞ , V∞ ) and the local perturbation velocities (v), density (ρ)
and pressure p under the assumption of an inviscid and steady state flow.
We are interested in calculating the wave drag of a three-dimensional body in
supersonic flow. An example of such a body in its cylindrical control volume is
shown in Fig. 6.15. The surface area, S of the control volume is divided into three
parts marked S1 –S3 . Let us first extract the drag force, D, from (6.28):
2
u (v · n + i · n) dS
(6.29)
D = Fbodyx = −
( p − p∞ ) i + ρV∞
S
We write the normalized perturbation velocity in terms of the perturbation velocity
potential in cylindrical coordinates according to:
1
v=
V∞
φ̂x
φx
=
φr
φ̂r
(6.30)
6.3 Wave Drag
293
Fig. 6.15 Momentum
control surface around a
body in supersonic flow
(modified from Ref. [4])
z
S2
R1
y
θ
S3
S1
Furthermore, we note that in cylindrical coordinates:
dS1
0
dS3
ndS2 =
ndS3 =
ndS1 =
0
0
dS2
(6.31)
We can now rewrite (6.29) as follows:
2
2
D = − ( p − p∞ )dS1 −
ρV∞
ρV∞
φ̂x (φ̂x + 1)dS1 −
φ̂x φ̂r dS2
S1
−
S1
( p − p∞ )dS3 −
S3
S2
2
ρV∞
φ̂x (φ̂x + 1)dS3
(6.32)
S3
We can simplify this expression by acknowledging that in supersonic flow the pressure and velocity at S1 must remain unperturbed by the body (i.e. p = p∞ and
n φ̂x = 0). Similarly, we assume that at S3 is sufficiently far behind the body such
that the flow has become essentially two-dimensional in the y–z plane (“Trefftz
plane”). Therefore, φ̂x = 0 such that the last integral in (6.32) vanishes. In addition,
we employ Bernoulli (6.9) to relate the pressure to the velocity disturbance velocity
potential in cylindrical coordinates:
1 2
p − p∞ = − ρV∞
φ̂r2 + 2φ̂x + φ̂2x
2
(6.33)
We can use this relation to write p − p∞ only in terms of φ̂r since φ̂x = 0 at S3 .
Finally, we note that φ̂r2 = φ̂2y + φ̂2z . Combining all of the information above we can
now rewrite (6.32) according to:
2
D = −ρV∞
S2
φ̂x φ̂r dS2 +
2 ρV∞
(φ̂2y + φ̂2z )dS3
2
S3
(6.34)
294
6 Aerodynamics of Non-lifting Bodies
This equation expresses the total pressure drag experienced by the body. In supersonic
flow, the pressure drag stems from two sources: the wave drag and the vortex drag.
The second integral in (6.34) gives the vortex drag, which is identical to the induced
drag for subsonic flow. The first integral represents the wave drag:
2
Dwave = −ρV∞
φ̂x φ̂r dS2
(6.35)
S2
This is a remarkably simple expression for the wave drag in terms of the derivatives
of the perturbation velocity potential. We must note that the present result is only
valid for bodies that are pointed in the rear.
6.3.2 Supersonic Wave Drag of a Slender
Axis-Symmetric Body
It can be shown that flow around a body can be constructed by adding elementary
source solutions. Examples of this technique can be found in text books on the
fundamentals of aerodynamics such as Ref. [3]. In spherical coordinates a point
source should satisfy Laplace’s equation for incompressible conditions: ∇ 2 Φ = 0. In
spherical coordinates it can be shown that a source of strength Q located at r = 0 has
the potential function: Φ = −Q/4πr [see also (3.88)]. Since the disturbance velocity
potential can be expressed as a function of the velocity potential [Φ = V∞ (x + φ̂)]
we write for the disturbance velocity potential in spherical coordinates:
φ̂ = −
Q
4πr V∞
(6.36)
where r is the distance measured at any point to the origin of the source.
Since we know that the superposition of elementary sources can represent the
shape of any axis-symmetric body we would like to know what the wave drag would
be of a line of sources. Therefore, we introduce a so-called lineal source distribution:
a straight line of sources of strength Q = V∞ f (x). Here, f (x) represents the variable source strength along the line. Without derivation we show that the associated
perturbation velocity potential in supersonic flow is [4]:
x−βr
f (x1 )dx1
1 φ̂ = −
2π
(x − x1 )2 − β 2 r 2
(6.37)
0
√
where β = M 2 − 1 and r is the distance from any point in space to the origin of
the y–z plane. We assume that f is continuous and that f (0) = f (l) = 0. We can
substitute (6.37) in (6.35) to find a closed-form expression of the wave drag. The
derivation involves complex algebraic operations that are beyond the scope of this
6.3 Wave Drag
295
book. Reference [4] can be consulted for details on the exact derivation. We merely
present the reader with the result:
2 ρ∞ V∞
f (x1 ) f (x2 ) ln |x2 − x1 | dx1 dx2
4π
l l
Dwave = −
(6.38)
0 0
For f (l) = 0 we can rewrite (6.38) according to:
2 ρ∞ V∞
Dwave = −
f (x)dx f (x1 ) ln(x − x1 )dx1
2π
l
x
0
0
(6.39)
If we compare (6.37) with (6.39) we notice that the drag coefficient has become
independent of β provided that the body ends in a point (as in Fig. 6.15) or ends in
a cylindrical portion. Finally, the tangentiality condition at the surface of the bodies
relates the source strength to the first derivative of the cross-sectional area, S:
f (x) = S (x)
(6.40)
Example 6.2 We consider a body of revolution with the following radius distribution:
1 2
1
for 0 ≤ x ≤ 1
R(x) = − x −
4
2
(a) In one figure, plot the distribution of R, S, and S as function of x
(b) Calculate the drag coefficient of this body with respect to the unit square as
reference area.
Solution:
(a) We first calculate the cross-sectional area distribution that corresponds to this
radius distribution:
2 2
1
1
− x−
S(x) = π R 2 (x) = π
4
2
Subsequently, we determine the derivative of the cross-sectional area distribution
with respect to x:
2 1
1
1
x−
− x−
S (x) = −4π
4
2
2
The results are plotted in Fig. 6.16.
296
6 Aerodynamics of Non-lifting Bodies
Fig. 6.16 Distribution of
radius, cross-sectional area,
and cross-sectional area
derivative
R(x), S(x), dS/dx
0.6
R(x)
S(x)
dS/dx
0.4
0.2
0
−0.2
−0.4
−0.6
0
0.2
0.4
0.6
0.8
1
x/l
(b) To calculate the wave drag coefficient with respect to the frontal area (π/16) we
rewrite (6.39) accordingly:
C Dwave = −
16 f
(x)dx
f (x1 ) ln(x − x1 )dx1
π2
l
x
0
0
For f (x) we write:
1
f (x) = S (x) = π 12 x −
2
2
−1
When examining (6.39) we notice that for x = x1 the integrand of the second
integral tends to negative infinity. This means that we cannot evaluate this integral using standard quadrature methods. We therefore replace the integral by a
double series:
I
i−1
16 C Dwave ≈ − π2
f (x j ) ln(xi − x j )Δx
f (xi )Δx
i=2
j=1
where xi = x j = [0, Δx, 2Δx, . . . , 1]
We evaluate this double summation in Matlab and find for Δx = 0.001 and
I = 1,000:
C Dwave ≈ 2.63
Note that this drag coefficient is calculated with respect to the frontal area of the
body.
6.3.3 Optimum Shape of a Slender Body of Revolution
For a slender body of revolution, we are interested in the cross-sectional area distribution that yields the least amount of wave drag. We start from the the expression for
6.3 Wave Drag
297
the wave drag (6.38). We first perform a coordinate transformation by introducing
the Glauert variable, θ that relates to x according to:
x=
l
(1 + cos θ)
2
(6.41)
With this transformation the nose of the body is located at θ = π, while the base is
situated at θ = 0. We can now express f as a Fourier series (see Sect. 2.2.2 for an
introduction on Fourier series or consult Ref. [30]) in θ according to:
f (θ) = l
∞
An sin (nθ)
(6.42)
n=1
Likewise, we can express the Fourier coefficients, An , as a function of f (θ). We do
this by multiplying either side by sin mθ and integrating between 0 and π:
2 π
f (θ) sin nθdθ = An
(6.43)
πl 0
Notice that we use the orthogonality of the Fourier terms such that:
π
0
for n = m
sin nθ sin mθdθ =
0
π/2
for n = m
(6.44)
To obtain the area distribution as a function of θ we use (6.40). We integrate (6.42)
between θ and π, noting that dx = − 2l sin θdθ:
l2 An sin (nθ1 ) sin θ1 dθ1
2
θ n=1
∞
2
sin (n + 1) θ sin (n − 1) θ
sin (2θ)
l
−
+
=
An
A1 π − θ +
4
2
n+1
n−1
π ∞
S(θ) = −
n=2
(6.45)
If we integrate the area distribution between 0 and π we obtain an expression for the
volume of the body in terms of the first two Fourier coefficients:
πl 3
l S(θ) sin θdθ =
V =
2
8
π
0
1
A1 − A2
2
(6.46)
Finally, we can obtain an expression for the wave drag in terms of the Fourier coefficients by substituting (6.42) in (6.38):
2 l2 ρ∞ V∞
Dw = −
An n cos(nθ1 )
Am m cos(mθ2 )
16π π π
0 0 ∞
∞
n=1
m=1
298
6 Aerodynamics of Non-lifting Bodies
l
× ln (cos θ − cos θ1 ) sin θ1 sin θ2 dθ1 dθ2
2
∞
2 l2 πρ∞ V∞
=
n A2n
8
(6.47)
n=1
where we use the properties of trigonometric functions as in (6.44).
Let us reflect on what we have done here. We have defined three properties of
the body in terms of Fourier coefficients: the volume of the body, the cross-sectional
area distribution, and the wave drag. For a body of a specified cross-sectional area
distribution and length, we can find the wave drag of this body by employing (6.43)
and (6.47). To calculate the associated volume of the body we can employ (6.46).
6.3.3.1 Sears-Haack Body
We consider a body which is pointed at both ends and has a given volume, V . This
implies that the cross-sectional area is zero at θ = 0 and θ = π. Via (6.45) and
S(0) = S(π) = 0 we obtain that A1 = 0. For a given volume and length we employ
(6.46) to obtain a value for A2 :
A2 = −
16V
πl 3
(6.48)
If we scrutinize (6.47) we can immediately see that in order to minimize Dwave we
would have to ensure that as many Fourier coefficients as possible are equal to zero.
In practice this means that the drag is smallest when only A2 is non-zero while all
the others are zero. Substituting (6.48) into the equation for the wave drag, (6.47),
results in the equation for wave drag for this body:
Dwave =
64V 2
2
ρ∞ V∞
πl 4
(6.49)
In order to find the ideal area distribution for this body one can substitute A1 = 0,
A2 = −16V /πl 3 into (6.45) and let all remaining An = 0. We then obtain the
following equation:
1
4V
(6.50)
sin θ − sin 3θ
S(θ) =
πl
3
This is is the cross-sectional area distribution that yields the lowest wave drag. The
axis-symmetric body that it represents is termed the Sears-Haack body after the
English scientist William Sears and the German scientist Wolfgang Haack, who
independently derived (6.50). In Fig. 6.17 one can see a Sears-Haack body for a
fineness ratio of approximately 6.8.
To minimize the zero-lift wave drag on any body in transonic flow, the crosssectional area distribution should be as close as possible to the Sears-Haack body.
6.3 Wave Drag
299
Fig. 6.17 Shape of an
axisymmetric Sears-Haack
body for V = 0.01 and l = 1
0.08
R(x)
S(x)
R(x), S(x)
0.06
0.04
0.02
0
0
0.2
0.4
0.6
0.8
1
x/l
The actual shape of the aircraft is fairly independent of its wave drag according to the
theory of Büsemann and Whitcomb. It is therefore important that large volumetric
additions to the body (e.g. wings, vertical and horizontal tail planes) are positioned
such that a cross-sectional area distribution results that matches as closely as possible
to the Sears-Haack body. Small deviations from the ideal Sears-Haack area distribution do not lead to excessive amounts of wave drag. However, discontinuities in area
distribution can cause a large increase in wave drag and should therefore be avoided.
6.3.3.2 Von Kármán Ogive
The Sears-Haack body gives minimum drag for a body with a pointed nose and rear.
A second shape of interest is the optimum nose cone shape for a straight body of
revolution (e.g. a rocket, or bullet). If one applies different boundary conditions to
(6.45) one can find the area distribution for which the nose cone produces minimum
drag. For a nose cone with a given base area, S(l), at x = l we have θ = 0. Substituting
θ = 0 in (6.45) we obtain:
4S(l)
(6.51)
A1 =
πl 2
Since A1 is nonzero, the wave drag is minimized when An = 0 for n ≥ 2. Substituting
(6.51) into (6.47) one obtains the expression for the minimum wave drag of the nose
cone:
2 S(l)2
2ρ∞ V∞
(6.52)
Dw =
π
l2
If we substitute the expression for A1 into (6.45) we obtain the area distribution for
the minimum-drag nose cone:
1
S(l)
(6.53)
π − θ + sin 2θ
S=
π
2
This resulting body is called the Von Kármán ogive and gives the minimum drag for
a given base area and length. Figure 6.18 shows an example of a Von Kármán ogive.
300
6 Aerodynamics of Non-lifting Bodies
Fig. 6.18 Shape of an
axis-symmetric Von Kármán
ogive for S(l) = 0.1 and
l=1
R(x), S(x)
0.4
R(x)
S(x)
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
x/l
6.3.4 Examples of Transonic Area Ruling
Application of the transonic area rule had the same objective as the supercritical airfoil
and the swept wing concept: reduce wave drag in the transonic regime. Fighter aircraft
at the time (e.g. Convair F-102) could not penetrate the sound barrier because of the
exponential drag increase that they experienced near Mach 1. Their jet engines could
not produce enough thrust to overcome the exponential increase in drag. The crosssectional area distribution of the original YF-102a prototype (Fig. 1.12a) had such a
high wave drag penalty near M = 1 that the available thrust could not overcome the
drag. Flight tests showed that this aircraft could not exceed M = 0.98. This is shown
in Fig. 6.19a where the thrust and the drag of the prototype aircraft are plotted at an
altitude of 35,000 ft. Because the US Air Force was not satisfied with the maximum
speed of this interceptor airplane a redesign was commissioned that had to result in a
top speed of M = 1.2. This redesign included a narrower canopy, revised intakes, and
a “pinched” fuselage (Fig. 1.12b). This dramatically altered the cross-sectional area
distribution of the airplane resulting in a smaller increase in drag coefficient around
M = 1 (see Fig. 6.19a) and a top speed of M = 1.22 as can be seen in Fig. 6.19b.
(b)
min
Minimum drag
coefficient, CD (~)
.04
prototype
Thrust and drag (lb)
(a)
.03
revised
improved
nose
.02
.01
0
.6
.7
.8
.9
1.0
1.1
Mach number, M (~)
1.2
prototype
10,000
8,000
thrust available
6,000
revised
4,000
improved nose
2,000
0
.6
.7
.8
.9
1.0
1.1
1.2
Mach number, M (~)
Fig. 6.19 Effect of cross-sectional area distribution on performance of the F-102 prototype at
35,000 ft (after Ref. [17]). a Drag coefficient and area distribution. b Drag an thrust versus Mach
number
6.3 Wave Drag
301
In the late 1960s when commercial transports started to emerge, it became evident
that aircraft needed to be designed with the area rule and Sears-Haack body in mind.
This resulted in some particular features of commercial transports that are still very
obvious this day. An example of how the cross-sectional area distribution was altered
on the Boeing 747 is shown in Fig. 6.20. It can be seen that by extending the upper
deck the area distribution became more smooth, which resulted in a higher dragdivergence Mach number.
Another famous example of transonic area ruling is found in the anti-shock bodies
that were developed by Whitcomb and Küchemann. The anti-shock bodies were
positioned near the trailing edge of the wing to smoothen the cross-sectional area
distribution of the airplane. The effect of these bodies was a reduction in shock
strength over the upper surface of the wing, resulting in a lower wave drag. This was
demonstrated by experiments at NASA, which are shown in Fig. 6.21. Notice the
more aft position of the shock wave due to the application of the anti-shock bodies.
Even though these bodies increased the wetted area of the airplane (i.e. more friction
drag), their benefit in terms of wave drag reduction outweighed this penalty. The
socalled Küchemann carrots found their way onto several aircraft such as the Convair
990 (fastest high-subsonic transport airplane to date) and the Handley Page Victor.
Yet another aircraft that benefited from the research of Whitcomb and Küchemann
was the Soviet Union’s revolutionary Tu-95 bomber. The Tu-95 demonstrated the
ability to cruise at jet-like speeds of Mach 0.85 while using propeller-driven engines.
One of the reasons for this aircraft’s exceptional performance was the use of large
engine fairings aft of the inboard engines. These pods brought the external shape
of the Tu-95 closer to the Sears-Haack body to reduce drag and increase maximum
speed.
Extended
cab
(b)
Wing
W/B fairing
Body
Drag coefficient, CD (~)
Cross-section aera
(a) Extended cab
0.040
Basic cab
Extended cab
0.032
C L = 0.5
0.024 CL = 0.4
CL = 0.3
0.016
Body station
Wind tunnel data
(wing, body, fin)
0.80
0.90
1.00
Mach number, M (~)
Fig. 6.20 Example of transonic area ruling on the Boeing 747 [24]. a Extended cab resulted in a
smoother area distribution. b Drag-divergence Mach number increased substantially
302
6 Aerodynamics of Non-lifting Bodies
Fig. 6.21 Reduction of
shock strength through the
application of Küchemann
bodies (Photos NASA).
a Clean wing. b With
Küchemann bodies
6.4 Fundamentals of Boundary-Layer Flow
Friction between a body and the surrounding fluid is one of the main causes of aircraft
drag. On typical high-subsonic transports the friction drag is responsible for 40–60 %
of the total drag of the airplane [57]. Every component (lifting or nonlifting) of the
aircraft that is wetted by the flow contributes to the friction drag of the airplane.
An obvious way to reduce friction drag is therefore to reduce the total wetted area.
6.4 Fundamentals of Boundary-Layer Flow
303
Alternatively, the friction coefficient can be reduced. To understand the causes of
friction we need to take a detailed look into the physics of the boundary layer, the
thin viscous layer in between the virtually inviscid flow and the body.
When the boundary layer separates from the body, it creates a wake between
the body and the continuous streamlines. A large momentum deficiency exists in
the wake, which can contribute to a significant amount of drag. In addition, the
pressure distribution over the body is significantly altered when separation occurs.
In normal operating conditions, separation of the boundary layer is therefore to be
prevented on any part of the airplane, whether intended to generate lift or not. The
most noticeable limitation that boundary-layer separation poses is on the stall speed
of an aircraft. At stall, the lift over the wing reduces significantly while the drag
rises rapidly. Via the aviation regulations it has been established that the minimum
approach speed of the aircraft (which relates to the required landing length of the
aircraft) should be ≈23 % higher than the stall speed measured at 1-g conditions
according to FAR/CS-25 regulations. This means that in practice, an airplane flies
only at 2/3rd of its maximum lift coefficient. The latter is, in turn, governed by the
separation of the boundary layer on the wing and is therefore an important constraint
during the aerodynamic design of an aircraft.
Even though low-speed stall is arguably the most renown result of boundary-layer
separation, it plays an important role in determining the maximum cruise altitude
of high-subsonic aircraft. Boundary layers are prone to separate when a relatively
large adverse pressure gradient is present and the boundary layer has thickened considerably. Such conditions occur for example on the upper wing surface when a
shock wave is present. This instantaneous jump in local pressure can cause boundary layer separation at the foot of the shock wave. However, not only on the wing
do we find areas that require the attention of the aerodynamic designer. Also on
non-lifting bodies such as the rear of the fuselage that often has a considerable
amount of upsweep (read: curvature) to allow for airplane rotation during take-off.
Other areas where sharp adverse pressure gradients occur are also suspect: edges
of cockpit windows, intersections between wing, nacelle and pylon, and the engine
intake.
In the subsequent sections we discuss both qualitative and quantitative aspects of
the boundary layer and their effect on friction drag. We follow the classical presentation of Abernathy [1] on the fundamentals of boundary layers.
6.4.1 Laminar Boundary Layer
In steady potential flow the inertia forces of the fluid are balanced by pressure forces.
Viscosity does not play any role. If we consider the potential flow about a thin
plate attached to a cylinder, Fig. 6.22a, we see that near the body the streamlines
follow the contours of the body perfectly. Two stagnation points are present, one on
the leading edge of the plate and one on the opposite end of the cylinder. In highReynolds-number flow, however, the effect of viscosity dominates the flow around
304
(a)
6 Aerodynamics of Non-lifting Bodies
(b)
Fig. 6.22 Flow visualization of flow around a thin plate attached to a cylinder (after Ref. [1]).
a Potential flow streamlines. b Streamlines of water flowing past the object
the boundaries of the body Fig. 6.22b. We see that the streamlines do not follow the
curvature of the body anymore but separate from the surface near the top and bottom
of the cylinder.
If we consider the flow past a simple symmetric airfoil and we assume that the
Reynolds number is relatively large and the airfoil is aligned with the direction of the
freestream, then the effect of viscosity is confined to a very narrow region close to
the body. The streamlines about the the airfoil resemble very much the potential flow
streamlines (see Fig. 6.23a). Now, when we place this same airfoil at a significant
incidence angle with respect to the freestream flow direction, the viscous effects
become very pronounced and the flow field about the airfoil is completely changed
(Fig. 6.23b). The airfoil is said to be in a stalled condition. The boundary layer has
separated from the surface and a large wake is present between the streamlines and
the body. Separation is caused by a large adverse pressure gradient on the body
surface which causes separation of the boundary layer.
We first look at the development of a boundary layer in the absence of any pressure
gradient. In Fig. 6.24 the velocity profile of the flow about a flat plat is represented by
white areas. We clearly see that there is a no-slip condition at the wall: the fluid is at
(a)
(b)
Fig. 6.23 Flow visualization of flow past an airfoil (after Ref. [1]). a At small angle of attack. b At
high angle of attack
6.4 Fundamentals of Boundary-Layer Flow
305
Fig. 6.24 Flow visualization
of velocity profiles within
the boundary layer about a
flat plate (after Ref. [1])
a standstill there. With increasing distance from the wall, y, the velocity, u, increases
progressively until a point where it matches the freestream value, U . We typically
define the thickness of the boundary layer, δ, as the y coordinate where the velocity
vector reaches 99 % of the freestream value. If we look progressively downstream
we see that the thickness of the boundary layer increases, while the velocity gradient,
du/dy, decreases. The velocity gradient at the wall is directly related to the shear
stress, τw , the fluid is exerting on the wall:
τw = μ
du
dy
(6.54)
where μ is the dynamic viscosity of the fluid. This relation was originally proposed by Isaac Newton and confirmed by experiments carried out by Poiseuille
in 1849 using liquid flow in tubes. The local friction coefficient is the wall-stress
non-dimensionalized by the freestream dynamic pressure according to:
cf = 1
τw
2
2 ρ∞ V∞
(6.55)
Based on our observations of the velocity profiles in Fig. 6.24 and Eqs. (6.54) and
(6.55), we can deduce that the local friction coefficient is highest near the leading
edge and that it reduces further downstream.
To understand why the boundary layer is growing in thickness downstream we
look at the time history of vorticity within the boundary layer. We know that for a
given control surface, the vorticity vector (∇ × V ) is related to the velocity around
the contour of the control surface:
(∇ × V ) · dS = V · dS
(2.71)
S
C
The vorticity enclosed within the contour of the control surface is called the circula
tion, Γ :
Γ = V · dS
(6.56)
C
Let us try to estimate the circulation in the boundary layer. In Fig. 6.25 we show a
contour drawn about the velocity profile in the boundary layer. Due to the no-slip
306
6 Aerodynamics of Non-lifting Bodies
Fig. 6.25 Circulation per
unit length about a contour at
the upstream station
U0
condition, the velocity at the wall is zero. Furthermore, we assume that locally the flow
is aligned with the wall and that the perpendicular velocity component is therefore
zero. Evaluating the circulation per unit length about this contour therefore results in:
Γ =
V · dS = 0 + 0 − U0 + 0 = −U0
C
We can conclude that at any station, the circulation per unit length equals −U0 . The
vorticity is constant at any streamwise station along the plate: −U0 times a unit
length. The total amount of vorticity within each contour remains the same. Because
the flow upstream of the plate has no vorticity, we can conclude that the vorticity is
introduced at the leading edge of the plate due to the no-slip boundary condition.
Even though the circulation per unit length is constant at any streamwise station
on the plate, the distribution of vorticity normal to the plate does change. Due to the
viscosity of the fluid the vorticity is spread transversely as it travels downstream. In a
laminar boundary layer, this goes according to the principle of molecular diffusion.
The local boundary layer thickness represents the distance the vorticity has diffused
away from the plate. This growing process is a function of time, t, and the diffusivity of momentum. The latter quantity is more commonly known as the kinematic
viscosity:
μ
(6.57)
ν=
ρ
√
The transverse diffusion length, δ, can be shown to correlate with νt (a derivation
of this quantity is beyond the scope of this text2 ). At any distance from the leading
edge, x, the time that has passed can be calculated according to t ≈ x/U0 . Therefore,
the relative boundary layer thickness can be expressed as:
1
μ
δ
∝
=√
(6.58)
x
ρU0 x
Rex
This relationship is only valid at high Reynolds numbers where δ/x
1 and for
laminar boundary layers. We can note from (6.58) that with higher freestream velocity
the boundary layer becomes thinner at any station along the plate. This is simply
because the boundary layer has had less time to grow. Bodies exposed to high2 The interested reader is referred to Fick’s laws of diffusion [21].
6.4 Fundamentals of Boundary-Layer Flow
307
Reynolds number flows therefore have a thinner boundary layer with respect to their
length.
6.4.1.1 Boundary Layer Characteristics
To characterize the boundary layer, we frequently use the terms displacement thickness, momentum thickness, and shape factor. In the subsequent paragraphs we show
how these parameters relate to the velocity distribution in the boundary layer. These
properties are applicable to both laminar and turbulent boundary layers.
The displacement thickness, δ ∗ , is a measure of the missing mass flow due to the
presence of the boundary layer.
δ∗ =
δ
ρu
1−
dy
0
ρe u e
(6.59)
If we assume that the flow in the boundary layer is incompressible its displacement
thickness can be reduced to:
δ
u
1−
dy
(6.60)
δi∗ =
0
ue
The incompressibility assumption is often used in practice, since it has shown to give
good correlation to experimental results.
Similar to the displacement thickness we can also define a thickness that accounts
for the missing momentum in the boundary layer. This thickness is termed the
momentum thickness, θ, and can be found from:
δ ρu ρu
1−
dy
(6.61)
θ=
0 ρe u e
ρu e
Again, if we assume that in the boundary layer the flow is incompressible we have:
δ u u
θi =
1−
dy
(6.62)
0 ue
ue
Finally, we introduce a shape factor, H , that relates the displacement thickness to
the momentum thickness according to:
H=
δ∗
θ
(6.63)
The shape factor is a function of the shape of the velocity distribution within the
boundary layer, hence its name. Notional velocity profiles for both high and low
shape factor are shown in Fig. 6.26. Notice that for a small H the velocity gradient at
the wall is higher compared to a high H . The shape factor will prove its importance
in predicting, for example, transition and separation (Sect. 6.4).
308
6 Aerodynamics of Non-lifting Bodies
Fig. 6.26 Typical velocity
profiles within the boundary
layer at high and low values
of H
1.0
1
y/ δ
high H
low H
u/u e
6.4.1.2 Effect of Reynolds Number and Mach Number
Let us expand on the effect of the Reynolds number on the friction coefficient. As
is evident from (6.58), the Reynolds number is responsible for the relative thickness
of the boundary layer. Using Blasius’ equation (which is presented in Sect. 6.5.2)
we can deduce a velocity profile inside the incompressible laminar boundary layer.
We can relate the velocity gradient at the wall to the kinematic viscosity and the
freestream velocity according to:
U0
∂u
(6.64)
= 0.332 U0
∂ y y=0
νx
Using this result and (6.55) we have a very straightforward quantification of the
friction drag:
0.664
U0
0.664μ
cf =
=√
U0
(6.65)
2
νx
Rex
ρU0
This shows us that the friction coefficient indeed decreases with increasing Reynolds
number. The average friction coefficient of a complete plate can be found by integrating the local friction coefficient over the complete length of the plate:
Cf =
1 l
cf dx
l 0
(6.66)
Substituting (6.65) we find that:
1.328
Cf = √
Rel
(6.67)
This equation demonstrates once more that an increase in the Reynolds number
decreases the friction drag coefficient of a surface wetted by an airflow. If we relate
that to airplanes, we can conclude from this simple derivation that both an increase in
size as well as an increase in cruise velocity decreases the average friction coefficient
of the aircraft. However, we must note that the effect of surface roughness has not
yet been taken into account and that we solely speak of laminar boundary layers. It
should also be noted that absolute drag does increase with velocity and size.
6.4 Fundamentals of Boundary-Layer Flow
309
(a)
(b)
40
M∞ = 12
32
32
=
M∞
M=1
24
M∞ = 6
16
12
M∞ = 8
24
=
M∞
16
8
=6
M ∞ M ∞= 4
y
Normal Distance Parameter, x Re∞ (~)
40
M∞ = 4
8
0
8
M∞ = 2
M∞ = 0
0
0.2
0.4
0.6
0.8
Mach number ratio, M/M∞ (~)
1.0
0
0
10
M∞ = 2
20
30
Temperature ratio, T/T∞ (K/K)
Fig. 6.27 Mach number and temperature distribution across a laminar boundary layer on an insulated flat plate for various Mach numbers. Calculations carried out by Van Driest [59] assuming
Pr = 0.75 and S/Tw = 0.505. a Mach number distribution. b Temperature distribution
For compressible flows the density is not constant and the assumption of a constant viscosity and heat transfer coefficient through the boundary-layer thickness are
no longer valid. The friction between the flow and the wall generates a temperature
difference between the wall and the free-stream. The resulting temperature distribution in the boundary layer has an effect on the velocity distribution and therefore
on the velocity gradient at the wall. At the same time, the temperature has an effect
on the viscosity through Sutherland’s formula (2.81). Both effects contribute to a
change in friction coefficient.
For a laminar boundary layer and adiabatic wall conditions, Van Driest [59] calculated the change in velocity profile with free-stream Mach number using Crocco’s
method.3 Assuming adiabatic wall conditions and a constant heat transfer coefficient,
the temperature and Mach number profiles within the boundary layer could be determined for various free-stream Mach numbers. The Mach number and temperature
distribution is shown in Fig. 6.27.
If we examine Fig. 6.27a, we can observe that an increase in Mach number has
several effects. First of all, it increases the boundary layer thickness. Secondly, the
Mach number gradient at the wall decreases with Mach number. It can be shown that
also the velocity profile follows this trend [59]. This results in a lower shear stress
at the wall, which causes a reduction in friction coefficient with Mach number. The
sonic line is also shown in Fig. 6.27a. With increasing Mach number, the sonic line
moves away from the wall. However, if we compare the location of the sonic line to
the thickness of the boundary layer, we conclude that the position of the sonic line
only changes slightly. For example, for M∞ = 2 the sonic line lies at approximately
3 Details on Crocco’s method can be found in Ref. [59].
310
6 Aerodynamics of Non-lifting Bodies
0.40δ, where δ has been defined as the distance where u/V∞ = 0.995. For M∞ = 8
the sonic line is located at approximately 0.35δ.
The temperature distribution in Fig. 6.27b shows the effect of friction on the wall
temperature. Note that at M = 2 the wall temperature is 40 % higher than the freestream temperature. At M = 4 there is more than a factor of 3 between the wall
temperature and the free-stream temperature. We can therefore deduce that for adiabatic conditions, the temperature increase over at high supersonic Mach numbers
results in temperatures that are higher than conventional aluminum can sustain without degrading material properties. Therefore, supersonic aircraft that fly faster than
M ≈ 2.5 have to rely on more heat resistant materials, such as titanium. In the
hypersonic realm (M∞ > 5) the wall temperatures become extremely high, which
requires aerospace vehicles that operate in this regime to have an ablative heat shield
or a thermal-soak heat shield. In the transonic domain, aerodynamic heating is comparatively small and has little effect on the velocity gradient and friction coefficient.
Therefore, the incompressible boundary-layer equations can often be successfully
used to predict the boundary layer properties in the transonic flow domain.
Using the method of Van Driest, the compressible friction coefficient can be
correlated to the incompressible friction coefficient. An approximation of the relation between compressible and incompressible friction coefficient was derived by
Johnson and Rubesin [27]:
Cf
1
=
0.12
C fi
1 + 0.1305M 2
(6.68)
Both the approximation of Johnson and Rubesin as well as the calculation of Van
Driest is shown in Fig. 6.28. We see a gradual decay of the friction coefficient with
increasing Mach number. We also note that in the transonic flow domain the decrease
in friction coefficient is relatively small. Using (6.68), a reduction of 1.5 % at M = 1
can be computed. The change in turbulent friction coefficient is also presented in
Fig. 6.28 and is discussed in more detail in Sect. 6.4.2.
6.4.1.3 Flow Reversal in Laminar Boundary Layers
In the preceding paragraphs we have investigated the development of boundary layers over flat plates with negligible pressure gradients (i.e. ∂ p/∂x = 0). We now
investigate how the pressure gradient affects the velocity profile in the boundary
layer, the friction drag, the transition point, and the separation point.
We start our discussion with the presence of a favorable pressure gradient. In
practice this means that the further downstream one travels, the lower is the pressure
(i.e. ∂ p/∂x < 0). In this case, the flow outside the boundary layer accelerates
continuously while the favorable pressure gradient is present. This effect is illustrated
in Fig. 6.29 where we see the velocity profile of a fluid in a converging channel (twoto-one contraction ratio). The flow accelerates in the converging part of the channel
such that it reaches twice its velocity in the narrower part of the channel.
(~)
6.4 Fundamentals of Boundary-Layer Flow
311
Coles 1
Chapman and Kester 2
Monaghan and Cooke 1
Rubesi, Maydew and Varga 1
Brinich and Diaconis 1
Korkegi 1
Lobb, Winkler and Persh 1
Friction coefficient ratio, Cf /Cf
M=0
1
0.8
{
Pr = 0.75
Crocco S/T = 0.505
method T∞ =∞ 217.8 K
0.6
LAMINAR
(1+0.1305M 2 ) -0.12
(1+0.2M 2 )-0.467
0.4
Sommer & Short method
0.2
nic
ic
son transo
sub
0
0
1
TURBULENT
supersonic
2
3
hypersonic
4
5
6
7
8
9
10
Mach number, M (~)
Fig. 6.28 Variation of skin friction coefficient with Mach number. Data from Refs. [25, 27, 37, 39,
46, 53, 58, 59]. 1: Mean skin friction values. 2: Local skin friction values, for Reynolds number
Reθ = 8,000 based on momentum thickness
(a)
(b)
δ
h =.53
δ =.66
h
h
δ
UPSTREAM
h
δ
DOWNSTREAM
Fig. 6.29 Flow visualization of velocity profiles in a converging channel (after Ref. [1]). a Velocity
profile in a converging channel. b Comparison between upstream and downstream velocity profiles
When looking at the velocity profile in the boundary layer we see that the boundary
layer thickness reduces considerably. Most of this reduction can be explained by
the two-to-one decrease in flow area. However, using the distance to the nearby
streamline, h, as a reference dimension, we can see that also the relative thickness
δ/ h has decreased. If we consider the vorticity within the layer up to h we can deduce
that the vorticity increases linearly with the increase in freestream velocity. This new
vorticity is added to the vorticity already present in the boundary layer at the wall.
It is as if a new boundary layer is being formed at every increment in the converging
duct. The combined profile at the exit of the contraction is relatively thinner because
the vorticity has had little time to diffuse. In the narrow part of the duct we now have
a boundary layer with a larger percentage of vorticity close to the wall. This results
in a higher velocity gradient, shear stress, and friction coefficient.
312
6 Aerodynamics of Non-lifting Bodies
In a divergent channel, the opposite effect occurs. When the diffuser angle is
relatively low, the flow velocity in the channel will decelerate and an unfavorable (or
adverse) pressure distribution is present: ∂ p/∂x > 0. The pressure downstream is
higher than the pressure upstream. The result is a thickening of the boundary layer
both in the absolute and in the relative sense. In other words, ∂ hδ /∂x > 0. The reduced
velocity slope at the wall reduces the shear stress and the friction coefficient. The
velocity profile becomes less full: less vorticity is concentrated near the wall and more
is concentrated near the edge of the boundary layer. This can be seen by comparing
the velocity profiles at the first two stations of Fig. 6.30b.
If the deceleration of the flow continues to take place the velocity gradient (∂u/∂ y)
near the wall becomes smaller and smaller. Eventually the shear stress goes to zero,
followed by a reversal of the flow direction near the wall. This is visualized in the
3rd station of Fig. 6.30b. We see that the flow at the second station is still directed
to the right. While the flow at the third station is directed to the left. Somewhere in
between those two stations the flow at the wall has come to a complete stand still.
The corresponding shear stress at this point is zero. We call this point the separation
point. Beyond this point, the fluid, which was in contact with the wall in the upstream
boundary layer, has separated from the wall by a region of reversed (or recirculating)
flow. This is visualized in Fig. 6.31.
We have now established that the development of the boundary is influenced by
the pressure gradient. Given the fact that any body in a flow field generates a pressure
distribution, we can deduce that the boundary layer development is influenced by
the shape of the pressure distribution. In addition, the separation point (if there is
any) is affected by the pressure distribution. The shape of the velocity profile at any
point in the boundary layer is also dependent on the conditions of the boundary layer
downstream. For the simple case of a flat plate we know that the boundary layer
transitions from laminar to turbulent and subsequently grows in thickness the further
we move downstream. A thicker boundary layer has a lower velocity gradient at the
wall and is therefore more prone to separate whenever an adverse pressure gradient
is present. If we recall the lessons we learned from Sect. 6.2 we can state that in order
to prevent separation we need to modify the outer shape of the body such that the we
obtain relatively small pressure (adverse) gradients in regions where the boundary
layer is relatively thick.
(a)
(b)
Fig. 6.30 Flow visualization of velocity profiles in a diverging channels of different diffuser angle
(after Ref. [1]). a Diverging channel with small diffuser angle. b Diverging channel with larger
diffuser angle
6.4 Fundamentals of Boundary-Layer Flow
Fig. 6.31 Effect of adverse
pressure gradient d p/dx on
velocity profile, u(y)
313
dp
>0
dx
u
u
u
Point of
separation
∂u
>0
∂y
∂u
=0
∂y
∂u
<0
∂y
6.4.2 Turbulent Boundary Layer
As was demonstrated by Osbourne Reynolds in the late 19th century, fluid flow in a
pipe transitions from laminar to turbulent beyond a certain critical Reynolds number
(Fig. 6.32).4 The same happens in boundary layers. The steps in the transition from
laminar to turbulent are complicated and interdependent. First there is the growth of
the nearly two-dimensional Tollmien-Schlichting waves. These waves are the first
(infinitesimal) indicators of laminar-flow instabilities. Whether these instabilities
grow depends on the amplification factor of the instability at a particular position in
the boundary layer.
Experiments of boundary layer flow over a flat plate show that the unstable
Tollmien-Schlichting waves grow downstream and become three-dimensional. At
regions of high localized shear stress, several types of unstable vortices occur. These
vortices are longitudinally stretched and begin a cascading breakdown into smaller
units. In this fluctuating state, intense local changes occur at random locations and
at random times in the shear layer near the wall. Turbulence occurs in these spots
which further coalesce downstream into fully turbulent flow. This three-dimensional
transition process is shown in Fig. 6.33 after Kegelman and Mueller [28]. Note that
the turbulent spots mark wedges of turbulent flow. This results in the typical zig-zag
pattern of the transition line.
Since it is generally known that laminar boundary layers have a lower friction drag
than turbulent boundary layers, it is of interest to know where transition originates.
As we will see more explicitly in Sect. 6.5.4, amplification or damping of TollmienSchlichting waves largely depends on the local pressure gradient. A negative pressure
gradient reduces the amplification of disturbances, while a positive (adverse) pressure
gradient increases the amplification rates. An adverse pressure gradient therefore
hastens the onset of transition compared to boundary layer flow in the absence of a
pressure gradient.
If the critical Reynolds number is defined as the Reynolds number for which a
disturbance with a particular wave length causes an unstable growth of that particular disturbance, the minimum critical Reynolds number is the Reynolds number
4 In this context, the word critical is unrelated to the flow properties at sonic conditions.
314
Fig. 6.32 The classical dye
experiment in pipe flow as
described by Reynolds in
1883 (after Ref. [45]). a Low
speed: laminar flow, b high
speed: turbulent flow, c spark
photograph of condition (b)
6 Aerodynamics of Non-lifting Bodies
(a)
(b)
(c)
Fig. 6.33 Idealized
boundary layer transition
over a flat plat (after Ref.
[28])
T/S
waves
Threedimensional
vortex
breakdown
Turbulent
spots
Fully
turbulent
flow
Stable
laminar
flow
U
x
0
U
Spanwise
vorticity
Edge
contamination
Recrit
δ (x)
Laminar
Transition length
Turbulent
Re tr
for which any disturbance of arbitrary wave length causes the boundary layer to
become unstable. Based on a theoretical investigation of compressible boundary
layer stability over a flat plate, Lees [35] concludes that for subsonic conditions any
laminar boundary-layer flow is unstable at sufficiently high Reynolds numbers. The
minimum critical Reynolds number is largely determined by the product of mean
density and the mean vorticity across the boundary layer. The addition of heat to the
fluid through the solid surface reduces the minimum critical Reynolds number and
therefore hastens the onset of transition. Cooling the wall has the opposite effect.
For adiabatic conditions, Laufer and Vrebalovich [34] concluded that the Mach
number has no effect on the stability of the laminar boundary layer but is solely
dependent on the displacement-thickness Reynolds number, Reδ . In addition, the
amplification factor is also shown to be independent of the Mach number. However,
it is shown that the disturbances propagate faster with increasing Mach number. Consequently, a disturbance propagating to a certain x-position will not be amplified as
much in a higher Mach number flow because its time available to grow is less, even
6.4 Fundamentals of Boundary-Layer Flow
315
though its amplification coefficient is the same. Therefore, the adiabatic compressible boundary layer is shown to be more stable with respect to small disturbances
than the incompressible one.
Once the boundary layer has transitioned from laminar to turbulent, its velocity
profile changes significantly. Whereas the laminar boundary layer showed a smoothly
changing velocity profile, the large amount of vorticity in the turbulent boundary layer
creates a more rugged velocity profile. In Fig. 6.34 the instantaneous velocity of a
laminar and turbulent boundary layer are shown on opposite sides of a flat plate. The
ruggedness of the velocity profile in the turbulent boundary layer is evident. When we
superimpose a large number of these instantaneous velocity profiles over time, we can
distinguish a mean velocity profile for both the turbulent and the laminar boundary
layer. It can be seen from Fig. 6.34 that the deviation from the mean velocity profile
is much larger in the turbulent boundary layer than it is in the laminar boundary layer,
due to the randomness of the vortices that are created in the turbulent boundary layer.
In Fig. 6.35 the two velocity profiles are compared. When superimposing them
we see that the turbulent boundary layer shows a larger velocity gradient at the wall.
The turbulent boundary layer therefore generates more shear stress as per (6.54).The
incompressible friction coefficient at a particular station x on a flat plate can be
approximated by the “power law” [66]:
cf =
0.027
1/7
(6.69)
Rex
Note that this relation only holds for low subsonic conditions. For turbulent boundary
layers the incompressible friction coefficient therefore decays much slower with
−1/7
−1/2
while cf ∝ Rex
for laminar boundary layers. In
Reynolds number: cf ∝ Rex
Fig. 6.36 the local friction coefficient is plotted for both a turbulent and a laminar
boundary layer. In addition, a typical transition line is shown which shows the rapid
increase in local friction coefficient when the boundary layer transitions from laminar
to turbulent.
The circulation is the same in both the laminar and turbulent boundary layers.
However, the distribution of circulation is different. For the turbulent boundary layer
(a)
(b)
Fig. 6.34 Boundary layer around a flat plat: flow is turbulent on upper surface and laminar
on lower surface (after Ref. [1]). a Instantaneous velocity profiles. b Superposition of many
instantaneous velocity profiles
316
6 Aerodynamics of Non-lifting Bodies
(a)
(b)
LAMINAR
TURBULENT
Fig. 6.35 Comparison between velocity profiles in laminar and turbulent boundary layers (after Ref.
[1]). a Comparison of mean laminar (solid) and turbulent (dashed) velocity profiles. b Comparison
of velocity gradients at the wall for laminar and turbulent boundary layers
0.005
Local friction coefficient, cf (~)
Fig. 6.36 Local coefficient,
cf , for turbulent and laminar
boundary layers as a function
of the local Reynolds
number, Rex
“Exact”
Power-law theory
0.004
0.003
0.002
Typical
transition
curve
Laminar flow:
0.664
(Blasius)
Re
0.001
0
105
106
107
108
109
1010
Local Reynold number, Rex (~)
more vorticity is concentrated near the plate even though some vorticity has also
spread farther from the plate. The thickness of the turbulent boundary layer on a flat
plate at point x can be approximated according to [66]:
0.16
δ
≈
1/7
x
Rex
(6.70)
Comparing this to the laminar boundary layer we see that the laminar boundary layer
grows according to δ ∝ x 1/2 , while the turbulent boundary layer grows much faster
according to δ ∝ x 6/7 .
The distribution of momentum is also different between the two boundary layers.
The unsteady random rotary motions associated with vorticity that is aligned with the
flow transports high-momentum fluid towards the plate while low-momentum fluid
is transported away from the plate. This extra momentum close to the wall enables
the turbulent boundary layer to withstand much larger adverse pressure gradients
6.4 Fundamentals of Boundary-Layer Flow
317
Fig. 6.37 Gloster Javelin FAW.9 with three rows of vortex generators on the wing to delay the
onset of boundary layer separation. Photo Adrian Pingstone
before separating from the wall. This is the reason why vortex generators are often
seen on airplane wings. They generate additional vorticity in the direction of the
flow that transports momentum towards the wing surface and hence delays the onset
of separation. Rows of vortex generators can be seen on various modern transonic
airplanes. A historic example is the Gloster Javelin having three rows of vortex
generators on its outer wing (Fig. 6.37).
6.4.2.1 Effect of Reynolds Number and Mach Number on Friction Drag
We have already noted that the effect of the Reynolds number on the friction coefficient is less strong in turbulent boundary layers than in laminar boundary layers.
There exist many empirically derived relations that correlate the turbulent friction
coefficient, Cf , of a hydraulically smooth plate to the freestream Reynolds number.
Schlichting [48] proposed the following interpolation formula:
Cf =
0.472
(log10 Re)2.58
(6.71)
Note that the two coefficients in (6.71) are empirically determined. Von Kármán and
Schoenherr used the following equation to fit the empirical data:
0.242
= log10 Cf
Cf Re
(6.72)
Equation (6.72) needs to be solved iteratively to find a solution for Cf . Note that the
Von Kármán-Schoenherr relation relies on a single empirical coefficient. In Fig. 6.38
the two relations are shown together with various experimental results. Some of the
experiments date back to Froude in the late 19th century. There exists quite some
scatter between the various experimental data. According to Ref. [40] the experimental data from Smith and Walker (1959) is most reliable due to the accurate
measurement techniques. It can be seen from Fig. 6.38 that both theoretical lines fol-
318
6 Aerodynamics of Non-lifting Bodies
−3
Average Friction Coefficient, Cf (~)
4.0
x 10
Froude, 1872 -- 16, 28, and 50 foot varnished planks
Gebers, 1908 -- 160, 3601 460, 652 cm. planks
Froude Tank, NPL, 1915, -- 3, 8, 16 foot planks
Washington Unpublished, 1932 -- 20, 30, 40, 80 foot planks
Wieselsberger, 1925 -- 50, 100, 150, 200 cm. planks.
Smith & Walker, 1959 -- Flat plate, momentum defect method
Smith & Walker, 1959 -- Flat plate, floating element method.
3.8
3.6
3.4
3.2
3.0
2.8
2.6
Schlichting
2.4
Von Karman - Schoenherr
2.2
2.0
3
4
5
6
7
8
9
10
20
Reynolds number, Re (~)
30
x 10 6
Fig. 6.38 Variation of turbulent skin friction coefficient with Reynolds number. Data selected
from [40]
low approximately the same trend. It depends on the experimental data set to judge
which theoretical line gives a better prediction.
It is of practical interest to investigate the effect of the Mach number on the
friction coefficient of a turbulent boundary layer. We would like to know if the friction
coefficient in transonic domain is influenced by the Mach number. Figure 6.28 on
p. 309 shows experimental results for the compressible, turbulent friction coefficient
up to a Mach number of 10. In addition, two theoretical lines are plotted, which
are shortly discussed hereafter. It can be observed that the friction coefficient of the
turbulent boundary layer has a much stronger dependency on the Mach number than
the friction coefficient of a laminar boundary layer.
Rubesin et al. [46] base their prediction of the compressible, turbulent friction
coefficient on the interpolation curve of Schlichting (6.71):
Cf =
0.472
2
(log10 Re)2.58 1 + γ−1
2 M∞
0.467
(6.73)
If we compare the denominator of (6.73) to the isentropic Mach-temperature relation
(2.147), we recognize the effect of the static temperature on the friction coefficient.
In Fig. 6.28 this simple model is shown with a dashed line. It can be seen that up to
a Mach number of about M = 3.5 this method gives quite a good approximation of
the decrease in friction coefficient with Mach number.
Sommer and Short [53] detail a somewhat more elaborate method, referred here
as the T method. The prime superscript refers to conditions at which incompressible
flow relations must be evaluated in order to represent compressible flow. The skin friction drag of a smooth flat plate and turbulent boundary layer in combination with an
adiabatic boundary condition is assumed. The T method is based on the calculation
of a compressible friction coefficient, Cf , from a reference skin friction coefficient,
Cf , for a selected Mach number, M∞ , Reynolds number, Re∞ and adiabatic wall
6.4 Fundamentals of Boundary-Layer Flow
319
temperature, Tw . The adiabatic wall temperature is a function of the Mach number:
Tw
2
= 1 + 0.178M∞
T∞
Sommer and Short [53] obtained the following equation for T :
T
Tw
2
= 1 + 0.035M∞ + 0.45
−1
T∞
T∞
(6.74)
(6.75)
The Reynolds, Re , number then becomes:
Re
=
Re∞
1
μ
μ∞
T
T∞
(6.76)
where the viscosity ratio is given by the Sutherland equation (2.81). The relationship
between Cf and Re is given by the Von Kármán-Schoenherr relation (6.72):
0.242
= log10 Cf
Cf Re
(6.77)
Finally, the compressible skin friction coefficient is obtained from Cf according to:
Cf = Cf
T∞
T
(6.78)
If we compare the two theoretical lines we can conclude that the method of Sommer
and Short is most consistent with experimental results up to M = 10. However, in the
transonic regime equation (6.73) gives an approximation that is sufficiently accurate
given the scatter in experimental results. For M = 1, the compressible friction
coefficient is about 8 % less than its incompressible counterpart. If we compare that
to the laminar boundary layer (−1.5 % at M = 1), we can conclude that the turbulent
boundary layer is more sensitive to changes in Mach number and that the change in
friction coefficient due to compressibility effects is significant in the transonic regime.
6.4.3 Shock-Wave Boundary-Layer Interaction
in Transonic Flow
In a mixed subsonic and supersonic domain often normal shock waves form. On the
wing surface of many high-subsonic aircraft a weak normal shock (recompression
shock) terminating a supersonic region is present in the design condition. The
associated pressure distribution allows for a high cruise lift coefficient and a relatively high lift-to-drag ratio. On non-lifting bodies the presence of a shock should be
avoided. It produces wave drag and it can also cause separation of the boundary layer
at the foot of the shock wave. Shock waves on nonlifting bodies often occur when two
320
6 Aerodynamics of Non-lifting Bodies
components intersect (see Sect. 6.6.2). Minimizing supervelocities on these components to prevent the onset of supersonic flow and associated shock waves can be a
design goal to prevent such a drag penalty.
Figure 6.39a–d show shock-boundary layer interaction in transonic speeds (from
Ref. [36]). The first three flow visualizations show the effect of successively increasing
flow Mach number while the boundary layer is laminar. The last figure shows shock
boundary layer interaction for a turbulent boundary layer in transonic speeds. Note
that the weaker shocks that are formed in lower flow Mach numbers of 0.795 and
0.843 interact with the boundary layer, but do not cause significant separation (see
Fig. 6.39 Shock
boundary-layer interaction at
transonic speed for
knife-edge airfoil normal to
flow direction (from
Ref. [36]); Reprinted with
permission of the American
Institute of Aeronautics and
Astronautics, Inc. a Flow at
Mach 0.795,
Rec = 8.33 × 105 (laminar).
b Flow at Mach 0.843,
Rec = 8.45 × 105 (laminar).
c Flow at Mach 0.895,
Rec = 8.77 × 105 (laminar).
d Flow at Mach 0.895,
Rec = 1.75 × 106 (turbulent)
(a)
Flow
(b)
Flow
(c)
Flow
6.4 Fundamentals of Boundary-Layer Flow
321
(d)
Flow
Fig. 6.39 (continued)
Fig. 6.39a, b). This is referred to as weak interaction of the shock with the boundary
layer. For a slightly higher Mach number, namely 0.895, the shocks are strong enough
to cause boundary layer separation (see Fig. 6.39c). We call this strong interaction.
The turbulent eddies are visible at the interface of the separated shear layer and the
subsequent pressure waves that emanate from the wake. The effect of boundary layer
type, i.e., laminar versus turbulent, on shock-boundary layer interaction is qualitatively shown in Fig. 6.39c, d. Note that the flow Mach number is maintained between
the two cases (both are at 0.895) but the Reynolds number is doubled. The laminar
boundary layer shows earlier separation on the airfoil than the turbulent boundary
layer case, as expected. Consequently, the wake downstream of the airfoil in the
laminar boundary layer case is thicker than in the turbulent boundary layer case. We
also note the lambda shock formation at the trailing edge of the airfoil in Fig. 6.39d.
6.4.3.1 Weak Interaction
Consider Fig. 6.40, which gives a general sketch of the flow field within the turbulent
boundary layer and the interaction with a normal shock wave. In front of the shock
wave the outer flow is supersonic. This implies that the edge velocity at the boundary
layer edge is also larger than the speed of sound. Descending into the boundary layer
the velocity progressively decreases up to the point where u = a (i.e. M = 1). If we
connect the points in the streamwise direction where this condition holds we form
the sonic line. The location of the sonic line is determined by the boundary layer
velocity profile and the edge Mach number, Me . The velocity profile, in turn, is a
function of the pressure gradient under which the boundary layer developed before
encountering the interaction. Below the sonic line information can travel upstream.
The pressure rise that results from the presence of the shock is spread ahead of the
shock location resulting in the boundary layer thickening.
The flow field shown in Fig. 6.40 forms a complex equilibrium of flow parameters.
We see that at the interaction the boundary layer becomes thicker. This implies that
322
6 Aerodynamics of Non-lifting Bodies
normal shock wave
compression waves
M e <1
M e >1
M>1
large
eddy
region
M<1
M>1
sonic line: M=1
merging layer
viscous sublayer
Fig. 6.40 Detail of weak shock-wave boundary-layer interaction for a turbulent boundary layer
(after Ref. [41])
the streamlines move away from the wall. The resulting curvature in the streamlines
causes the formation of (isentropic) compression waves. These compression waves
start at the sonic line and allow the supersonic flow to change the flow direction
(remember that in supersonic conditions the direction of the flow can only be altered
through the formation of waves as we explained in Chap. 4). The fan of compression
waves merge into a shock wave outside of the boundary layer. Close to the boundary
layer edge the shock wave is still slightly oblique which reduces the flow velocity
behind the shock to low supersonic. Within the boundary layer this is also the case:
the flow behind the compressions waves can still be (slightly) supersonic. This gives
rise to a pocket of supersonic flow behind the shock wave that is termed the supersonic tongue. Within the supersonic tongue the flow decelerates isentropically to
subsonic speeds. The size of the supersonic tongue is case-dependent and is merely
notional in Fig. 6.40.
The profile of the velocity distribution in the boundary layer is altered in the short
interaction region. The outer velocity decreases substantially and the boundary layer
itself thickens. This results in a higher value for the shape factor (H ) and a lower
velocity gradient at the wall.
The outer flow through the shock wave experiences an almost instantaneous rise in
static pressure. In other words: dCp /dx → ∞. Based on our previous discussion on
the effect of the pressure distribution on boundary layer separation we might therefore
wrongfully conclude that the boundary layer would invariably separate from the surface. However, a closer look into the shock-wave boundary-layer interaction reveals
that the pressure increase is smeared out over a larger streamwise region. This smearing property is due to the formation of the compression waves within the boundary
layer. The effective pressure gradient within the boundary layer is therefore much
smaller and merely results in an increase in shape factor and boundary layer thickness.
Finally, we note that even in the case of a weak interaction the boundary layer
properties are adversely affected. Beyond the shock wave the boundary layer can
6.4 Fundamentals of Boundary-Layer Flow
323
tolerate a smaller adverse pressure gradient before separating. When a convexly
curved body is considered the weak shock interaction could still contribute to an
early onset of trailing edge separation.
6.4.3.2 Strong Interaction
When the shock wave strength increases, the interaction between shock wave and
boundary layer becomes stronger. The sonic line descends deeper into the boundary
layer and a different equilibrium results. Let us try to explain this equilibrium by
considering Fig. 6.41. Here we have a pocket of reversed flow (separation bubble).
A separation line is present between the reversed flow and the remaining boundary
layer flow. The streamlines outside of the separation bubble need to bend away from
the wall. In the supersonic domain (above the sonic line) this strong curvature can
only be supported by the formation of compression waves. Outside the boundary
layer these compression waves merge into an oblique shock. The bending of the
streamlines above the sonic line cause the boundary layer to rapidly grow in thickness. To align the streamlines with the wall a second oblique shock is present. The
two oblique shocks meet at the triple point above the boundary layer to form a normal
shock wave. This is the typical lambda shock pattern that is often seen on the upper
surface of transonic wings.
Flow outside the boundary layer that goes through the two oblique shocks experiences a different deceleration than flow going through the single normal shock.
A velocity difference is therefore present resulting in two velocity domains which
are separated by a shear layer (known as slip line). Further downstream the velocity
differences over the slip line are reduced to zero. The flow that goes through the two
oblique shocks experiences less total pressure loss than the flow going through the
normal shock wave. Typically the region between the triple point and the boundary
layer edge is relatively small. To reduce the loss in total pressure (and the corresponding wave drag) it is therefore beneficial to smear the separation region over a
normal shock
sonic line: M=1
slip line
oblique shocks
M e <1
M>1
M>1
M e >1
M>1
separation line
S
separation
bubble
M<1
R
Fig. 6.41 Detail of strong shock-wave boundary-layer interaction for a turbulent boundary layer
(after: [16])
324
6 Aerodynamics of Non-lifting Bodies
larger distance such that the triple point is moved upwards and the integrated total
pressure loss over the shock system decreases.
The velocity profile of the boundary layer changes substantially over this interaction. Initially, the profile shows a high pressure gradient near the wall. However, due
to the large adverse pressure gradient under the oblique shock, the velocity at the wall
quickly reduces to zero at point S. The flow separates and a bubble of recirculating
flow forms. The boundary layer shows a negative velocity gradient at the wall. The
reverse flow aids significantly in the upstream transmission of the post-shock pressure rise that causes the separation of the boundary layer in the first place. Beyond
the second oblique shock the deceleration of the subsonic flow in the boundary layer
causes the streamtube to expand. This aids in the reattachment of the boundary layer.
At the reattachment point (R) the velocity gradient at the wall is again zero. Over
the distance of the separation bubble the sonic line gradually moves away from the
wall towards the boundary layer edge. Further downstream of the bubble the velocity
profile has significantly thinned compared to the initial velocity distribution due to
the lower edge velocity and thicker boundary layer.
The onset of separation at the foot of the shock is dependent on the development of
the boundary layer in front of the shock and the surface curvature at the shock. However, for general purposes we can assume that on a flat plate separation starts at shock
Mach numbers (Me ) in the range of 1.3–1.35 [5]. In many practical cases the flow
behind the shock wave is further decelerated towards the trailing edge of the body.
Under influence of such an adverse pressure gradient the flow might separate close to
the trailing edge (trailing edge separation). Increasing the shock Mach number pushes
the shock further aft. In addition, the separation bubble grows, which also moves the
re-attachment point further aft. Finally, the reattached boundary layer is now more
likely to separate under the adverse pressure gradient. This moves the trailing-edge
separation point more forward. At some Mach number the reattachment point gets
close enough to the point of trailing edge separation such that the flow does not reattach anymore. This generates a large wake that originates at the foot of the shock.
This shock-induced separation is an important contributor to the exponential increase
in drag (drag divergence) that occurs at the drag divergence flight Mach number.
For more information on the interaction between the normal shock and the boundary layer the reader is referred to Refs. [5, 16].
6.5 Boundary-Layer Computations
Now that we have familiarized ourselves with the qualitative aspects of boundarylayer flow, we turn to the mathematical models that describe it. In this section we
present the boundary layer equations and show how they can be reduced to model
the flow in laminar and turbulent boundary layers. In addition, we also present
the e N method to predict where the boundary layer transitions from laminar to
turbulent.
6.5 Boundary-Layer Computations
325
6.5.1 Boundary Layer Equations for Steady
Incompressible Flow
For a body in a high Reynolds number flow, viscous forces are dominant in essentially three domains: the boundary layer, the wake of the body, and the shock wave
(if present). If the boundary layers stays attached to the airfoil, inviscid models that
permit the existence of strong shocks and rotational flow (e.g. Euler equations, see
Sect. 2.7.1) can be used in the region outside of the boundary layer. Prandtl’s equations can be used in the laminar boundary-layer region [33]. The governing continuity,
momentum and energy equations for a general unsteady, compressible, viscous flow
are a set of coupled nonlinear partial differential equations, which have been presented in Sect. 2.5. If we neglect the normal stress (τx x = τ yy = 0), and assume
steady, incompressible, two-dimensional flow, these equations can be simplified for
two-dimensional flow as follows:
continuity equation
∂v
∂u
+
=0
∂x
∂y
(6.79)
∂u
∂v
∂u
∂p
∂
∂u
ρu
+ ρv
=−
+
+
μ
∂x
∂y
∂x
∂y
∂x
∂y
(6.80a)
x-momentum equation
y-momentum equation
ρu
∂v
∂v
∂v
∂p
∂
∂u
μ
+ ρv
=−
+
+
∂x
∂y
∂y
∂x
∂x
∂y
(6.80b)
energy equation
∂(uτx y ) ∂(vτx y )
V2
∂2 T
∂2 T
ρV · ∇ e +
+
= k 2 + k 2 − ∇ · pV +
2
∂x
∂y
∂y
∂x
(6.81)
We consider an incompressible boundary layer over a flat plate of length c. We
make the assumption that the flow over this flat plate produces a boundary layer of
thickness δ, which is much smaller than c, i.e. δ
c. Furthermore, it is assumed
that the Reynolds number of the flow is large, i.e. 1/Re∞ = O(δ 2 ). Based on
these assumptions and an order of magnitude analysis of the components in the
equations above (see for example [3, 55]), the boundary layer equations can be
greatly simplified:
continuity equation
326
6 Aerodynamics of Non-lifting Bodies
∂v
∂u
+
=0
∂x
∂y
(6.82)
x-momentum equation
ρu
y-momentum equation
∂u
∂u
dpe
∂
+ ρv
=−
+
∂x
∂y
dx
∂y
μ
∂u
∂y
(6.83a)
∂p
=0
∂y
(6.83b)
2
∂h
∂u
∂h
∂2 T
d pe
ρu
+ ρv
=k 2 +u
=μ
∂x
∂y
∂y
dx
∂y
(6.84)
energy equation
Note that subscript “e” represents the conditions at the boundary-layer edge. Also
note that the y-momentum equation basically states that the pressure stays constant throughout the boundary layer. The pressure is therefore solely dependent on
x. Hence, from now on we replace its partial derivative operator (∂) with a total
derivative operator (d).
By defining the boundary values at a solid surface, this set of equations can be
seen as a boundary value problem (BVP). To assess this BVP, consider a coordinate
system which is bound to a solid object, with the y-axis perpendicular to its surface
and the x-axis aligned with the flow. The wall boundary values usually imply the
no-slip condition:
(6.85)
u(x, 0) = 0, v(x, 0) = v0
Here, v0 is the velocity perpendicular to the wall. In most conditions v0 = 0. However,
in case boundary-layer suction is applied v0 < 0. In addition, the temperature at the
wall is defined as follows:
(6.86)
T (x, 0) = Tw (x)
In the general case, Tw is a function of time, i.e. Tw = Tw (x, t). In the case of
adiabatic wall conditions the heat flow to the wall, qw is zero and the following
boundary condition results:
∂T
=0
(6.87)
∂ y y=0
If adiabatic wall conditions exist, the adiabatic wall temperature (Tw = Taw ) is a
part of the solution rather than a set boundary condition. The steady-state adiabatic
wall temperature can, however, be computed by using (6.74). At the outer edge
(subscript e) of the boundary layer, the following boundary conditions exist:
lim u(x, y) = u e (x),
y→∞
lim T (x, y) = Te (x)
y→∞
(6.88)
6.5 Boundary-Layer Computations
327
An estimate of the values of u e (x) and Te (x) can be obtained by calculating the
flow around an object using for example the Euler equations. The five boundary
conditions together with the coupled equations of motion [(6.82)–(6.84)] form the
basis for the analysis of laminar boundary layer flow. A significant simplification with
respect to the original equations of motion can be observed. It is stressed that the
above equations are derived for laminar boundary layers, assuming incompressible
flow and a thin boundary layer. A more complex order-of-magnitude analysis can
be carried out for boundary layers which exhibit a significant thickness [48, 55].
However, this is beyond the scope of the present text.
When we integrate the boundary-layer momentum Eq. (6.83a) from y = 0 to
y = δ and also employ (6.82) we obtain the Von Kármán momentum integral relation,
(also known as the momentum integral equation):
cf
θ du e
τw
dθ
+ (2 + H )
= 2 =
dx
u e dx
ρu e
2
(6.89)
The shock-wave-boundary-layer interactions that have been presented in
Sect. 6.4.3 cannot be captured by solving the Von Kármán equation. This method
only determines the effect of a velocity change on the development of the momentum thickness. However, this approach is still used in numerical codes for transonic
flow about bodies. To be able to successfully apply the method described above the
pressure rise over the normal shock should be smeared over a certain streamwise
distance. In this manner the shock is treated as a finite adverse pressure gradient. The
separation and subsequent reattachment of the boundary layer cannot be predicted
with this method, which essentially limits its application to local Mach numbers
below M = 1.3. To capture the interaction between the flow and the boundary layer
we need to employ more computationally expensive tools based on (for example)
the Reynolds-averaged Navier-Stokes equations (see Chap. 2).
6.5.2 Laminar Boundary Layer
The Blasius equation can be derived from the boundary layer equations. It reduces
the system of equations [(6.82)–(6.84)] to a single ordinary differential equation (see
Ref. [3] for complete derivation):
f +
1 ff =0
2
(6.90a)
subject to
f = f = 0
on η = 0
(6.90b)
f → 1 as η → ∞
(6.90c)
328
6 Aerodynamics of Non-lifting Bodies
where
η=y
f (η) =
V∞
νx
u
V∞
(6.90d)
(6.90e)
The resulting velocity gradient at the wall has been quoted in (6.64). The displacement
thickness of a typical Blasius velocity profile (incompressible conditions) can be
shown to be a function of the Reynolds number according to:
1.72x
δ∗ = √
Rex
(6.91)
The momentum thickness for such a velocity profile can be shown to be
0.664x
θ= √
Rex
(6.92)
We note that both
√the momentum thickness and the displacement thickness increase
proportional to x. At the trailing edge of the plate we substitute x = c. If we
compare the result to (6.67) we see that the momentum thickness at the trailing edge
is directly related to the friction drag over the plate:
Cf =
2θx=c
c
(6.93)
This is an important result because it is not limited to laminar boundary layers or
the Blasius velocity profile. It is a universal observation that holds for both turbulent
and laminar boundary layers: the reduced momentum in the boundary layer at the
trailing edge is a direct measure for the amount of friction drag of the profile. More
general, any loss of momentum can directly be correlated to the drag of a body (see
also Fig. 6.2). In this case, the momentum deficiency is caused by the friction of the
body.
The Blasius solution only holds for a flat plate where the external velocity distribution is constant. To predict the boundary-layer properties in the presence of a
pressure distribution, an engineering approach can be used. For example, the method
of Thwaites [56] predicts these properties regardless of the exact shape of the velocity profile inside the (laminar) boundary layer. We will derive this method and show
some of its properties.
Thwaites’ method starts from the Von Kármán integral relation. We first multiply
the momentum integral equation (6.89) by θu e /ν:
θ2 du e
u e θ dθ
τw θ
+ (2 + H )
=
μu e
ν dx
ν dx
(6.94)
6.5 Boundary-Layer Computations
329
We define the pressure gradient parameter, λ, as follows:
λ=
θ2 du e
ν dx
(6.95)
Subsequently, we correlate the shear stress parameter [(S(λ)] and the shape factor
[H (λ)] to λ using Thwaites empirical relations [66]:
τw θ
≈(λ + 0.09)0.62
μu e
(6.96)
H (λ) ≈2.0 + 4.14z − 83.5z 2 + 854z 3 − 3337z 4 + 4576z 5
(6.97)
S(λ) =
with z = (0.25 − λ). Note that when S → 0 the shear stress goes to zero. In other
words, the boundary layer separates when λ < −0.09. Substituting (6.95) and (6.96)
in (6.94) reduces it to:
dx
d
= 2 [S − (2 + H )λ] = F(λ)
(6.98)
λ
ue
dx
du e
Thwaites shows that F(λ) = 0.45 − 6λ. We can therefore rewrite (6.98) to the
following differential equation [employing (6.95)]:
d
ue
dx
2
2
θ
θ
du e
= 0.45
+6
ν
ν
dx
(6.99)
Both terms on the LHS can be grouped in a single differential term:
1 d
u 5e dx
2 θ 6
u = 0.45
ν e
(6.100)
Integrating this differential equation and assuming that the integration constant is
zero results in the following relation between boundary-layer edge velocity and
momentum thickness:
0.45ν x
u 5e dx
(6.101)
θ= 6
ue
0
Now, let us put this engineering method of Thwaites in practical perspective. We
start the calculation process with a velocity distribution, u e (x) about a surface. If we
consider the flow about a body we obtain such a velocity distribution from solving
the Euler equations or potential flow equations. Subsequently we employ (6.101)
to calculate the momentum thickness at any point on the body. In addition, we use
(6.95) to obtain a value for the correlation parameter λ. Using (6.96) and (6.97) we
can calculate the shear stress parameter and shape factor, respectively. These can be
330
6 Aerodynamics of Non-lifting Bodies
used to obtain estimates of the displacement thickness and shear stress at any point
on the body. By doing a number of iterations of inviscid and displacement-thickness
calculations, a converged value of the displacement thickness results. It can be shown
that the displacement thickness is about 1/3 of the boundary layer thickness [48].
The following example illustrates this process.
Example 6.3 Consider a plate with chord length of c = 1 m, which is exposed to
a flow with V∞ = 100 m/s at 0 m ISA conditions. Consider the following velocity
distribution at the edge of a boundary layer:
u e = V∞ 1 −
x
c
(6.102)
Calculate:
(a) The chordwise position of separation, xseparation .
(b) The friction coefficient, cf between x = 0 and xseparation .
Solution:
(a) The analytic nature of the velocity distribution allows us to compute θ from
(6.101):
θ=
x
0.45ν
5 1−
V∞
6
x
6
c
V∞ 1 − c 0
x
5
dx =
0.075
νc
V∞
1−
x
c
−6
−1
Knowing that du e /dx = −V∞ /c, we can calculate λ using (6.95):
λ = −0.075
1−
x
c
−6
−1
We know that separation occurs when λ = −0.09, we can therefore calculate
that:
x
= 0.123
c separation
(b) First, we calculate θ at discrete x positions using (6.101). Subsequently, we
employ (6.95) to calculate λ at each position. The kinematic viscosity at 0 m
ISA is ν = μ/ρ = 18.27 × 10−6 [Ns/m2 ]/1.225 [kg/m3 ] = 14.91 × 10−6 [m2 /s]
and the velocity gradient is −100 [1/s]. Then, we use (6.96) to compute the stress
parameter (S) at each position and deduce the shear stress, τw . Finally we use
the relation between shear stress and friction coefficient (6.55) to compute the
friction coefficient at each x-position. The results are tabulated in Table 6.1 with
intervals of 10 mm up to the separation point.
6.5 Boundary-Layer Computations
331
In the preceding example, we have demonstrated how Thwaites’ method can be
employed to calculate the friction coefficient distribution under a predefined velocity distribution. Thwaites’ method can only be applied to the properties of a laminar
boundary layer prior to separation. Because the momentum thickness at x = 0 is
zero, the local friction coefficient tends to infinity if (6.96) is used to compute τw .
Even though this is not physically accurate, Thwaites’ method provides a fairly good
estimate of the laminar boundary-layer development between 0 < x < xseparation .
It is very well possible that in reality the boundary layer transitions from turbulent
to laminar before xseparation . This is not predicted by Thwaites’ method. If the transition point is known, the boundary-layer properties should be calculated up to the
transition point.
6.5.3 Turbulent Boundary Layer
The turbulent boundary layer differs substantially from the laminar boundary layer.
Let us first examine which layers we can distinguish within the turbulent boundary
layer. Four layers can be identified: a viscous sublayer that touches the wall (denoted
with subscript w), a merging layer, a large-eddy region, and a viscous superlayer. To
represent the velocity distribution in each of these layers we need to introduce a new
variable: u τ , the wall friction velocity (see Problem 6.19):
uτ =
τw
ρ
(6.103)
We nondimensionalize the velocity in the boundary layer and the boundary layer
thickness by introducing two additional new variables:
u+ =
u
uτ
√
|τw |/ρw
yu τ
y =y
=
νw
ν
+
(6.104)
(6.105)
where ν is the kinematic viscosity of the fluid (ν = μ/ρ).
The viscous sublayer (also called laminar sublayer) is a thin laminar layer between
the wall and the turbulent flow in the boundary layer. In this layer the turbulence is
damped out and the boundary layer is dominated by viscous shear flow. This layer
allows for the no-slip condition of the turbulent boundary layer. The thickness of this
layer can be as small as δv = 1/500th of the boundary layer thickness, depending on
the velocity profile in the boundary layer, the viscosity of the flow, and the density
[66]. The velocity profile in this part of the boundary layer has been shown to match
closely to the following relation:
30
0.047
−0.0151
0.20
7.49
1.22
0
0.026
0.038
−0.0000 −0.0047 −0.0097
0.22
0.22
0.21
∞
14.84
9.85
∞
2.42
1.61
θ
λ
S
τw
cf × 103
mm
(∼)
(∼)
N/m2
(∼)
Table 6.1 Table belonging to Example 6.3
x
mm 0
10
20
40
0.056
−0.0209
0.19
6.00
0.98
50
0.064
−0.0271
0.18
4.92
0.80
60
0.071
−0.0338
0.17
4.06
0.66
70
0.078
−0.0410
0.15
3.35
0.55
80
0.085
−0.0488
0.14
2.73
0.45
90
100
110
120
0.092
0.099
0.106
0.114
−0.0571 −0.0662 −0.0760 −0.0866
0.12
0.10
0.07
0.03
2.17
1.63
1.08
0.42
0.35
0.27
0.18
0.07
332
6 Aerodynamics of Non-lifting Bodies
6.5 Boundary-Layer Computations
333
u+ = y+ y+ < 5
(6.106)
The velocity in the merging layer (sometimes called “overlap” layer or “blending”
layer) between the laminar sublayer and the large-eddy region can be predicted using
a formula provided by Spalding [54]:
+
+
y =u +e
−κB
(κu + )2
(κu + )3
κu +
+
e
− 1 − κu −
−
2
6
y + < 100 (6.107)
From experiments, the Von Kármán constant is found to be κ = 0.41 and B = 5.0
for a smooth wall. This equation is termed the law of the wall.
In the large-eddy region the velocity changes logarithmically with distance from
the wall. The following approximation by Coles and Hirst [12] is often used:
u+ =
1
ln y + + B with κ = 0.41 and B = 5.0 y + > 30
κ
(6.108)
The logarithmic region in the boundary layer extends to 10–20 % of the total boundary
layer thickness, depending on the value of Reδ . In the “wake” component of the
boundary layer, the local pressure distribution has a significant effect on the velocity
distribution. Therefore, Coles [11] proposed to add an additional quantity, Π , to
account for the local pressure distribution. The resulting law of the wake is written
as follows:
u+ ≈
1
2Π
ln y + + B +
f
κ
κ
y
δ
y + > 30
(6.109)
It was shown that Coles’ wake parameter, Π , could be related to the displacement thickness of the boundary layer by integrating (6.109) over the boundary layer
thickness:
1+Π
δ∗
≈
with λ =
δ
κλ
2
cf
(6.110)
The value of λ is a measure for the local skin friction coefficient and is useful in
the analyses which follow. Note that this λ is not the same as the λ we used in the
analysis of the laminar boundary layer with Thwaites method (Sect. 6.5.2). Finally,
the wake function f (y/δ) is of the form [11]:
f
y
πy
= sin2
δ
2δ
(6.111)
This implies that the wake function is zero at the wall and unity at the edge of the
boundary layer. This function allows for the S-shape we expect to see in the velocity
profile in the wake of the boundary layer whenever there is a pressure gradient present
(see Sect. 6.4). At the edge of the boundary layer, we can now relate the edge velocity
334
6 Aerodynamics of Non-lifting Bodies
50
Viscous sublayer y + < 5
45
+
Coles and Hirst: Large−eddy layer y > 30
Spalding: Merging layer y + < 100
40
Π=6
Π=4
Π=2
30
25
viscous
sublayer
merging
layer
Π=0
+
u = u/u τ
35
20
15
logarithmic
region
10
large-eddy
layer
5
0
0
10
outer
region
1
2
10
10
3
10
4
10
+
y = y u τ /ν
Fig. 6.42 Velocity distribution in a turbulent boundary layer for various values of Π , assuming
δ + = 10,000
to the velocity profile according to:
u + (y = δ) =
ue
1 Reδ
2Π
= ln
+B+
uτ
κ
λ
κ
(6.112)
The velocity distributions as presented in this section are summarized in Fig. 6.42
for various values of Π . If we connect the prediction of Spalding to that of Coles and
Hirst, we observe the typical S-shape in the velocity profile of the boundary layer flow.
6.5.3.1 Turbulent Boundary Layer Development
The effect of the pressure distribution on the development of the turbulent boundary
layer has been demonstrated qualitatively in Fig. 6.42. We see that for various values
of Π the velocity profile in the outer region can change significantly. We would like
to know how the value of Π is related to the pressure distribution outside of the
boundary layer. Therefore we return to the momentum integral equation as derived
by Von Kármán [62]:
cf
θ du e
τw
dθ
+ (2 + H )
= 2 =
dx
u e dx
ρu e
2
(6.89)
In order to use this equation we assume that an outer velocity distribution is known
and that we need to obtain values for cf , θ, and H . In order to do so we require at
least two additional equations. One is given by Coles’ wall-wake law: (6.112). Based
6.5 Boundary-Layer Computations
335
on this relation the following algebraic equation due to Felsch et al. [19] describes
the relation between H , Reθ and cf :
cf = 0.058Reθ−0.268 (0.93 − 1.95 log10 H )1.705
(6.113)
The third relation stems from a large body of experimental research where Π is
correlated to Clauser’s equilibrium parameter β which is defined as follows [10]:
β=
θ du e
δ ∗ d pe
= −λ2 H
τw dx
u e dx
(6.114)
where d pe /dx is the pressure gradient at the edge of the boundary layer and λ is as in
(6.110). Clauser’s equilibrium parameter is directly related to the pressure gradient.
Whenever β < 0 we have a favorable pressure gradient (d pe /dx < 0). Vice versa,
when β > 0 we have an adverse pressure distribution (d pe /dx > 0). The proposed
correlation between Π and β stems from Ref. [15]:
β = −0.4 + 0.76Π + 0.42Π 2
(6.115)
The additional variables (Π , and β) need to be correlated to H , cf , and θ. We need
two additional equations to do so. One is the definition of β, (6.114). The final
equation can be obtained from Coles’ wall-wake equation (6.112). The derivation of
this relation is beyond the scope of this text but can be found in Ref. [66]. We merely
present the reader with the result:
λ=
H 2 + 3.2Π + 1.5Π 2
H −1
κ(1 + Π )
(6.116)
Let us review this method and put this in a practical perspective. We would like to
calculate the properties of the boundary layer at various stations along a plate under
the presence of a pressure distribution. To that extent we intend to solve the ordinary
differential equation (6.89) with known boundary conditions. These boundary conditions could for example stem from an assumed value of the momentum thickness
or Reynolds number at x = 0. Subsequently, we use the algebraic equations (6.113)–
(6.116) to close the problem.
Example 6.4 Assume we have a pressure distribution over a curved plate of unit
length that can be represented by the following polynomial:
Cp = bx(ax − 1)
where a and b are coefficients. Let us further assume that the boundary layer over the
curved plate is made fully turbulent at x = 0 and that the initial Reynolds number
based on momentum thickness, Reθ = 100. Furthermore, the freestream velocity is
V∞ = 50 m/s and 0 m ISA conditions apply. Consider the following cases:
336
6 Aerodynamics of Non-lifting Bodies
1. a = 21 and b = 1
2. a = 1 and b = 21
3. a = 2 and b = 2
For these three cases perform the following tasks:
(a) Plot the pressure distribution, velocity distribution and distribution of the velocity
gradient for each of the aforementioned cases.
(b) Plot the distribution of the following boundary layer parameters for the three
aforementioned cases: H , β, Π , cf , θ, and δ ∗ .
(c) Briefly reflect on the results that have been plotted.
Solution:
(a) First we employ (6.11) to rewrite the equation of Cp for u e :
u e = V∞ 1 + bx(1 − ax)
Subsequently we take the derivative of u e with respect to x to find du e /dx:
b(1 − 2ax)
V∞
du e
=
√
dx
2
1 + bx(1 − ax)
We substitute each combination of a and b in the above equations and plot the
results in Fig. 6.43. A quick evaluation of these three graphs shows us that for
case 1 we have a completely favorable velocity gradient. For cases 2 and 3 we
have a velocity gradient that is favorable until x = 0.5. Beyond this point the
pressure gradient is negative, which implies that the outer flow is decelerated.
This effect is four times larger for case 3 than four case 2.
(b) To answer the second part of the question we must solve the first order differential
equation (6.89). We rewrite (6.89) according to:
F=
cf
cf
1 − 2ax
θ du e
θb
dθ
=
− (2 + H )
=
− (2 + H )
(6.117)
dx
2
u e dx
2
2 1 + bx(1 − ax)
−0.5
65
50
−0.2
60
0
e
−0.3
du /dx (1/s)
ue (m/s)
Cp (~)
−0.4
55
a=0.5, b=1
a=1, b=0.5
a=1, b=2
−0.1
0
0
0.5
1
50
0
0.5
1
−50
0
Fig. 6.43 Distribution of pressure coefficient, velocity and velocity gradient
0.5
1
6.5 Boundary-Layer Computations
337
We employ the Runge-Kutta5 method to solve for θ. The Runge-Kutta method is a
multi-step method where we first calculate intermediate constants k1 through k4 :
k1 = F(xn , θn )Δx
1
k1
k2 = F(xn + Δx, θn + )Δx
2
2
1
k2
k3 = F(xn + Δx, θn + )Δx
2
2
k4 = F(xn + Δx, θn + k3 )Δx
1
θn+1 = θn + (k1 + 2k2 + 2k3 + k4 )
6
In this method we march from position xn to position xn+1 with steps of Δx. At
each position we calculate the value of θ. To start this iteration we need a starting
value for θ. We compute this through the value of Reθ in combination with the
0 m ISA properties and the value of u e at x = 0. For all three cases u e = V∞ at
x = 0. We have:
Reθ μ
θ1 =
= 29 µm
ρV∞
In order to do determine the values of H at each position we need to solve the four
algebraic equations (6.113)–(6.116) simultaneously. If we substitute (6.114) in
(6.115) we are left with three equations that are written in the following functional format:
ueθ
G(1) = 0.058
ν
−0.268
(0.93 − 1.95 log10 H )1.705 − cf = 0
G(2) = −0.4 + 0.76Π + 0.42Π 2 +
G(3) =
2
θ du e
=0
H
cf u e dx
H 2 + 3.179Π + 1.5Π 2
−
H −1
κ(1 + Π )
2
=0
cf
Notice that we have three equations with three unknowns: H , cf , and Π . The
Matlab routine “fsolve” is used to compute the roots of this system of equations.
Having computed H and cf we can now evaluate the RHS of (6.117) and compute θn+1 . Finally, we use this new value of θ in combination with H to compute
5 For details on the Runge-Kutta method the reader is directed to an introductory text on numerical
methods for differential equations, which is also covered in Ref. [30].
338
6 Aerodynamics of Non-lifting Bodies
−3
−3
x 10
5
4
4
δ* (m)
θ (m)
x 10
5
3
2
1
0
2
1
0
0.2
0.4
0.6
0.8
0
1
0
0.2
0.4
0.6
0.8
1
0.01
3
2
cf (~)
H (~)
3
a = 0.5, b =1
a = 1, b = 0.5
a = 2, b = 2
0.005
1
0
0
0.2
0.4
0.6
x (m)
0.8
1
0
0
0.2
0.4
0.6
0.8
1
x (m)
Fig. 6.44 Distribution of boundary layer parameters according to three different pressure
distributions
δ ∗ via (6.63). We repeat this procedure at every xn until xn = 1. In Fig. 6.44 the
results are plotted for the three cases.
(c) If we look at case 1, a favorable pressure gradient is present. This favorable
pressure gradient results in an almost constant value of the shape factor. If we
compare this to the second case we see the effect of the adverse pressure gradient
on the momentum thickness and displacement thickness. We also see that the
friction coefficient becomes smaller due to a slightly higher value of the shape
factor. Remember that this corresponds to a lower velocity gradient at the wall
(see also Fig. 6.26). Finally, we have case 3 where the shape factor rises much
quicker due to the larger adverse pressure gradient. If we examine the friction
coefficient we see that it tends towards zero around x = 0.62. We know that this
is the condition of separation. In this example the flow separates from the surface.
We see an exponential increase in θ and δ ∗ and we notice that H goes to a value
close to 3.0. Beyond x = 0.62 the calculation of the roots of the functions G gives
a complex solution and is therefore neglected in the generation of these plots.
6.5.3.2 Effect of Roughness
If we consider real airplanes, we know that the surface of the airplane is covered with
various imperfections ranging from small rivet heads to larger antennas. To represent
these imperfections, for every airplane an equivalent sand grain roughness can be
defined. To predict the total friction drag of the airplane, the skin is assumed to be
covered with this uniform sand grain roughness instead of the real excrescences. The
6.5 Boundary-Layer Computations
339
parameter that determines the significance of the roughness is the ratio between the
sand grain height (k) and the boundary-layer thickness. We distinguish two cases:
1. Hydraulically smooth surface. If k is smaller than the height of the viscous sublayer, i.e. k < δv , k does not affect the boundary layer significantly. In other
words: the friction coefficient is unaffected by the presence of roughness. We
can imagine each sand grain to be a bluff body that is immersed in a very low
Reynolds-number flow. In such flow (see Fig. 6.22a) the streamlines are attached
over the complete body. The drag that results in this flow (Stokes flow) is due to
friction and therefore has the same dependency on Reynolds number as a perfectly
smooth surface.
2. Hydraulically rough surface. If k is larger than the height of the viscous sublayer, i.e. k > δv , the sand grains create small wakes behind them due to flow
‘separation.’ The pressure drag that is generated at each sand grain results in
the approximate friction drag. Because the pressure drag is less dependent on
the Reynolds number than the friction drag, the friction coefficient over a rough
surface also has much lower dependency on the Reynolds number. The value of
the friction coefficient is dominated by the relative size of the sand grains with
respect to the boundary layer thickness: k/δ.
If we assume the sand grains to have an average height of y = k, we can introduce
a nondimensional sand grain height according to (6.105):
k+ =
ku τ
ν
(6.118)
From experiments (e.g. Clauser [10]) it is known that the layer where velocity changes
logarithmically with wall distance still exists when roughness is present. However,
the intercept of the logarithmic curve moves downward with progressively larger
values of k + . This alters (6.108) as follows:
u+ =
1
ln y + + B − ΔB(k + )
κ
(6.119)
For sand grains ΔB changes logarithmically with k + according to the classical
Prandtl-Schlichting relation for k + > 60:
ΔBsand grains ≈
1
ln(1 + 0.3k + ) for k + > 60
κ
(6.120)
The result is the following expression for the nondimensional velocity profile in a
boundary layer that is disturbed by roughness:
u+ =
1
1
ln y + + B − ln(1 + 0.3k + )
κ
κ
(6.121)
340
6 Aerodynamics of Non-lifting Bodies
From the experiments it is know that for values below k + < 4 roughness has no
effect on the velocity profile in the boundary layer. Between 4 < k + < 60 there
is a transitional roughness regime. Finally, above k + > 60 the flow has become
hydraulically rough. Based on (6.121) it can be shown that as the sand grains become
larger, the effect of viscosity becomes negligible and the friction coefficient therefore
becomes independent of the Reynolds number (see Problem 6.23):
u+ =
y
1
+ 7.9
ln
κ
k
(6.122)
We can insert (6.121) into (6.83a) under the assumption of an absent pressure
gradient. Integrating the result over the thickness of the boundary layer and the length,
x, results in a relation between the friction coefficient, cf , the Reynolds number, Rex
and the grain size, k [66]:
Rex = 1.73(1 + 0.3k + )e Z Z 2 − 4Z + 6 −
0.3k +
(Z − 1)
1 + 0.3k +
(6.123)
Fig. 6.45 Local skin friction
on a flat plate from (6.123)
for constant values of x/k
Local friction coefficient, c (~)
f
where Z = κλ. This relation, even though it is implicit, is valid for uniform sandgrain roughness over the complete range of hydraulically smooth, transitional, and
fully rough walls in turbulent flat plate flow. If we write k + = ku τ ν = Rex (k/x)
, we
λ
can plot the local friction coefficient as a function of the local Reynolds number for
various values of x/k. This is shown in Fig. 6.45. The effect of the roughness is evident
from this figure. The larger the grains compared to the length x the higher the friction
coefficient. In addition, we see that the friction coefficient of a smooth plate decays
with Reynolds number (as expected). However, we also see that the decaying effect
of the Reynolds number diminishes for roughened plates above a certain Reynolds
number. The smaller the sand grains compared to the size of the distance x, the larger
the Reynolds number at which the Reynolds number ceases to decrease the friction
coefficient. This Reynolds number is called the cut-off Reynolds number. This is in
10
−1
x
=10
k
10
100
−2
1,000
104
105
106
10
−3
10
5
10
6
10
7
Reynolds number, Re x (~)
10
8
10
9
6.5 Boundary-Layer Computations
Fig. 6.46 Effect of cut-off
Reynolds number on friction
drag of three airplanes of the
same generation and
manufacturer but different
size (after Ref. [38])
341
.0030
Friction coefficient, Cf (~)
Critical
values
l/k s
.0025
JETSTAR
C-141
.0020
3.2 x 10
5
6.5 x 10
5
9.1 x 10
5
C-5
.0015
.0010
20
30
40
50
60
80
100
200
Reynolds number, Rel x 10-6
good agreement with classical experiments carried out by Nikuradse on pipe flow
with roughened walls [42].
The results plotted in Fig. 6.45 can be directly translated to the friction drag coefficient of full scale aircraft and wind tunnel models. In Fig. 6.46 the variation in friction
coefficient with Reynolds number is shown for three airplanes of the same generation
and built by the same company (Lockheed). It could be assumed that the manufacturing techniques for all three airplanes was of the same level leading to identical
values for the equivalent sand grain roughness of the skin of the airplane. What is
shown is the relative sand grain height, scaled with respect to a characteristic length,
l, of the airplane. Similar to the theoretical curves in Fig. 6.45, the largest airplane
(C-5) demonstrates to have the largest cut-off Reynolds number (≈100 million). This
advocates the statement that large aircraft benefit from their size when it comes to
the friction drag coefficient.
Example 6.5 Assume we have a flat plate with an equivalent sand grain roughness of
0.010 mm. Calculate using (6.123) the friction coefficient (cf ) at x = 2.5 m. Assume
that we have 0 m ISA conditions and a freestream velocity of 100 m/s. Also calculate
the wall stress, τw and the velocity gradient at the wall, (∂u/∂ y)w .
Solution:
We first calculate the Reynolds number at x = 2.5 m:
Rex=2.5 =
1.225 · 100 · 2.5
ρV x
=
= 17.1 × 106 .
μ
17.89 · 10−6
Secondly, we calculate the relative grain size:
k
0.010 × 10−3
=
= 2.5 × 105
x
2.5
342
6 Aerodynamics of Non-lifting Bodies
Because (6.123) cannot be solved directly, we use a “for” loop in Matlab according to
the following routine. We first establish a set of possible values for λ. Subsequently,
for each value of λ we calculate the value for k + , Z , and Rex :
for
λ = [1, 1.001, 1.002, . . . , 50]
do
Rex (k/x)
k+ =
λ
Z = κλ
+
Rex = 1.73(1 + 0.3k )e
Z
0.3k +
Z − 4Z + 6 −
(Z − 1)
1 + 0.3k +
2
end
Finally, we compare the outcome of Rex to the value that we computed above:
ΔRex = |Rex=2.5 − Rex |
We subsequently find one value of λ for which ΔRex is closest to zero. This results
in λ = 26.3. Substituting this in (6.110) results in:
cf =
2
= 0.00290
λ2
From the friction coefficient we can calculate the local shear stress according to
(6.55):
1
1
τw = cf ρU02 = 0.00290 · · 1.225 · 1002 = 17.7 N/m2
2
2
The local velocity gradient can be found by employing (6.54):
∂u
∂y
=
w
m/s
τw
= 0.994
μ
µm
Example 6.6 Considering the same roughened plate as in Example 6.5, calculate the
value of the total friction coefficient over the entire plate. Assume the plate has a
length of l = 3 m and that the boundary becomes turbulent at x = 0.10 m. Ignore
the contribution of the laminar boundary layer over the first 10 cm to the friction
coefficient.
Solution:
The friction coefficient of the plate is given by (6.66) with the upper bound x = 3 and
the lower bound x = 0.10. We evaluate this integral numerically by dividing the plate
6.5 Boundary-Layer Computations
343
into 29 sections of Δx = 0.1 m. On each section we evaluate the friction coefficient at
the beginning of the interval and at the end of the interval by employing the method
of Example 6.5. We approximate the integral over each section by employing the
trapezoid rule:
C fj =
x+Δx
x
Δx
cf (x)dx = cf (x) + cf (x + Δx)
2
The total friction drag over the plate is the sum of the incremental contributions:
1
C f j = 0.0030
l
29
Cf =
j=1
6.5.4 Boundary Layer Transition
From an operational perspective, there has been a long desire to minimize the drag
of an airplane to a minimum. In the context of boundary-layer theory, this desire has
translated to the objective of having as much laminar flow over a surface as possible. In a two-dimensional boundary layer, transition is governed by the formation
of Tollmien-Schlichting waves. As stated before, the amplification of these waves
depends on many factors. Experience has shown that a “fuller” boundary layer usually results in lower disturbance growth [47]. Let us rewrite (6.83a) for y ≈ 0 and
add the body force term that we have left out before:
μ
dp
dμ ∂T ∂u
∂u
∂2u
+
−
− fx
= ρv0
2
∂y
∂y
dx
dT ∂ y ∂ y
(6.124)
Note that we have split the partial derivative ∂/∂ y(μ∂u/∂ y) and have used ∂μ/∂ y =
(∂μ/∂T )·(∂T /∂ y). The left-hand side of (6.124) is a measure for the curvature of the
boundary layer profile near the wall. Generally speaking, a “fuller” boundary layer
corresponds to a convex velocity profile, i.e. a more negative value of ∂ 2 u/∂ y 2 . A negative pressure gradient (d p/dx < 0) therefore results in lower disturbance growth,
while an adverse pressure gradient (d p/dx > 0) increases disturbance growth. One
can also heat or cool the wall. Wall cooling in air (∂T /∂ y > 0, ∂μ/∂T > 0) stabilizes the boundary layer, while heating it (∂T /∂ y < 0, ∂μ/∂T > 0) destabilizes the
boundary layer. Alternatively, one can apply suction perpendicular to the boundary
layer (v0 < 0) in order to reduce the growth of disturbances. This has been successfully demonstrated to postpone transition in the wind tunnel and in flight [7, 44].
Finally, one can apply a body force to the flow in the direction of the velocity vector.
This can be done by locally ionizing the air using an electric field. By employing
an asymmetric electrode arrangement, a body force can be exerted on the resulting
plasma [13]. It has been shown that a periodic body force is effective in attenuating
the Tollmien-Schlichting waves [29]. Both suction and plasma control are examples
of active laminar flow control (LFC).
344
6 Aerodynamics of Non-lifting Bodies
In the early 20th century William Orr and Arnold Sommerfield independently
derived the linear homogeneous ODE describing the infinitesimal laminar flow
instability based on the steady continuity equation (2.113) and the unsteady linear
momentum equation (2.126). In contrast to our earlier assumption, Orr and Sommerfield allowed infinitesimal velocity perturbations perpendicular to the flow direction
within the boundary layer. They assumed that these perturbations changed harmonically with time and could grow exponentially in time and space depending on the
frequency of the wave, the wave propagation speed, the kinematic viscosity, and
the shape of the velocity profile. For negligible viscosity, the solution of the OrrSommerfield equation showed that only velocity profiles with an inflection point in
the velocity profile would amplify the instabilities. Additionally, the second derivative of the velocity with respect to y (distance perpendicular to the surface) needed
to be smaller than zero. This led to the mistaken assumption that a favorable pressure gradient could stabilize the boundary layer up to an infinite Reynolds number.
Prandtl demonstrated, however, that viscosity can indeed be destabilizing for certain
wave numbers at a finite Reynolds number.
A more elaborate discussion of the the stability of laminar boundary layers can
be found in the text of White [66]. A selection of the results for boundary layers over
smooth flat plates is listed below:
1. The minimum or critical Reynolds number for initial instability is Rex crit ≈
91,000.
2. The smallest unstable wavelength is λmin ≈ 6δ. Thus, unstable TollmienSchlichting waves are long compared to boundary layer thickness.
3. Compared to Rex crit ≈ 91,000, the point of final transition to turbulence is at
about Rex tr ≈ 3 × 106 , or about 30 times further downstream.
In practice, the transition process is influenced by many factors: surface roughness,
freestream disturbances, pressure gradients, vibration, sound, etc. Therefore, the
exact Reynolds number (or position, for a given fluid and speed) can vary with time
in a random way. Placing objects in the boundary layer results in a swift amplification
of the instabilities in the boundary layer and hastens the onset of transition. Such trip
strips are often used in wind tunnel experiments to fix the point of transition at a fixed
percentage of the wing chord. Other sources of instabilities stem from the attachment
line of the boundary layer on the leading edge of objects that are placed oblique to
the flow (such as the leading edge of a swept wing). In addition, instabilities arising
from three-dimensional cross flow (also found on swept wings) hasten the onset of
laminar to turbulent transition. This will be further discussed in Chap. 8 when we
treat the aerodynamics of swept wings.
In the derivation of the critical Reynolds number a Blasius6 profile was assumed
for the velocity distribution, u(y), in the boundary layer. However, it was shown that
the critical Reynolds number can be 90 times higher than this value when the velocity
6 The Blasius velocity profile satisfies the nonlinear ordinary differential equation that can be derived
from the boundary layer equations under specific assumptions, see (6.90a).
6.5 Boundary-Layer Computations
345
Fig. 6.47 Effect of pressure
gradients d p/dx on velocity
profile, u(y)
y
u (y)
dp
> 0 (adverse)
dx
Point of
inflection
dp
=0
dx
dp
< 0 (favorable)
dx
profile in the boundary layer satisfies the condition: ∂ 2 u/∂ y 2 < 0. Such velocity
profiles are very stable and postpone transition to a much higher Reynolds number.
Such a stable velocity profile can be formed when a favorable pressure gradient
is present as is shown in Fig. 6.47. The boundary layer profile that is subjected to
an adverse pressure gradient shows an inflection point, which means that ∂ 2 u/∂ y 2
changes sign in this point. In this case the critical Reynolds number (and the transition
Reynolds number) is decreased.
The critical Reynolds number can be shown to correlate to the shape factor of the
boundary layer at the point where instabilities start to occur. The shape factor can,
in turn, be calculated by the velocity distribution in the x-direction outside of the
boundary layer (as was shown through Thwaites method on pp. 305–306). In Fig. 6.48
this correlation is experimentally demonstrated through a range of experiments. Both
the critical Reynolds number as well as the transition Reynolds number are shown.
Fig. 6.48 Correlation
between shape factor and
transition Reynolds number
(after Ref. [64])
10 9
Reynolds number, Rex (~)
10 8
10 7
Re x, tr
10 6
Re x, crit
10 5
10 4
10 3
2.0
2.2
2.4
2.6
Shape factor, H = δ
θ
*
2.8
(~)
3.0
346
6 Aerodynamics of Non-lifting Bodies
The curve that is fitted through transition data points corresponds to the following
curve-fitted equation [64]:
log10 (Rex, tr ) ≈ −40.4557 + 64.8066H − 26.7538H 2 + 3.3819H 3
(6.125)
By using Thwaites method for the calculation of the local shape factor and combining that with (6.125), a swift prediction of the transition Reynolds number can be
obtained.
Example 6.7 Consider the flow problem of Example 6.3. Use the criterion of (6.125)
to calculate where transition is expected to occur.
Solution:
We need to calculate the local Reynolds number at each x-position, Rex . Subsequently, we need to evaluate whether the following inequality holds:
Rex < Rex, tr
We employ (6.125) to compute the transition Reynolds number, Rex, tr . We have
tabulated these values up to the position where separation is predicted (Table 6.2).
We can observe that the inequality holds until x = 70 mm. This means that transition
takes place between 60 and 70 mm. Note that the boundary layer calculations beyond
this point should therefore consider a turbulent boundary layer.
Another popular method to predict boundary layer transition is based on linear
stability theory (LST). The e9 method as pioneered by van Ingen [60] and Smith and
Gamberoni [52] assumes that transition occurs when the most unstable TollmienSchlichting waves are amplified by a certain factor. Here, e ≈ 2.71828 is the base
of the natural logarithm. The transition amplification factor is usually taken as e9 ≈
8,100, although a different value than 9 can be assigned to the exponential depending
on the disturbances in the freestream flow [61]. Therefore, the method is currently
referred to as the e N method, where N depends on the turbulence intensity in the
free stream. van Ingen [61] uses the following relating between the value of N and
the turbulence intensity, Tu:
N = 2.13 − 6.18 log10 (Tu) for 0.1 % < Tu < 2 %
(6.126)
Note that Tu should be inserted in this equation as a percentage. This equation is
based on a variety of experimental data and presents the critical reduced amplification
Table 6.2 Table belonging to Example 6.7
x
mm 0
10 20 30
40
50
60
70
80
90
100 110 120
Rex × 10−6
(∼) 0.00 0.07 0.13 0.20 0.26 0.32 0.38 0.44 0.49 0.55 0.60 0.66 0.71
Rex, tr × 10−6 (∼) 4.59 3.72 2.91 2.19 1.55 1.02 0.61 0.32 0.14 0.05 0.02 0.01 0.00
6.5 Boundary-Layer Computations
347
factor at the beginning of transition. For turbulence levels below Tu < 0.1 % the
experimental data shows too much scatter to warrant a simple relation as (6.126).
The turbulence intensity (in percentages) is defined as follows:
Tu = 100 ·
RMS(u )
ū
(6.127)
where RMS(u ) is the root-mean-square of the turbulent velocity perturbations and
ū is the mean velocity as defined in (2.154).
The amplification factor (or the logarithm thereof) can be correlated to integrated
amplification rates. For example, Giles and Drela [18] rely on the following computation for the logarithm of the maximum amplification factor (N ) [23]:
N=
dN
(Hk ) Reθ − Reθ0 (Hk )
dReθ
(6.128)
where Reθ is the momentum-thickness Reynolds number that can be calculated as
follows:
ρe u e θ
(6.129)
Reθ =
μe
dN
The following empirical formulas can be employed to calculate dRe
and Reθ0 (Hk ):
θ
dN
= 0.01 [2.4Hk − 3.7 + 2.5 tanh(1.5Hk − 4.65)]2 + 0.25 (6.130)
dReθ
1.415
20
3.295
log10 Reθ0 =
− 0.489 tanh
− 12.9 +
+ 0.44
Hk − 1
Hk − 1
Hk − 1
(6.131)
In the above equations the parameter Hk is the kinematic shape factor, which is
calculated using the following empirical relation assuming a constant density across
the boundary layer [67]:
H − 0.290Me2
Hk =
(6.132)
1 + 0.113Me2
with Me being the Mach number at the boundary-layer edge and H being the shape
factor from (6.63).
The method described above predicts the onset of transition. The laminar
boundary-layer variables can be calculated using (6.59), (6.61), and (6.63). If the
outcomes are combined with the properties at the boundary-layer edge (u e , pe , Te ),
the above equations can be used to calculate the value of N . If N surpasses a predefined threshold (e.g. N = 9), the boundary layer transitions from laminar to turbulent.
Example 6.8 Assume we have curved plate of length c = 1m, that supports a pressure distribution according to:
348
6 Aerodynamics of Non-lifting Bodies
Cp = 8
x
c
!
x
−1 +1
c
The flow over this plate can be assumed incompressible and adiabatic. The freestream
velocity of this flow is V∞ = 50 m/s and the plate is at mean sea-level conditions
of the international standard atmosphere (i.e. 0 m ISA). Carry out the following
assignments:
(a) Use Thwaites’ method to calculate the position of laminar separation and the
momentum thickness distribution.
(b) For Tu = 0.1 % and Tu = 2 % compute the logarithm of the amplification factor,
N , at which transition takes place.
(c) Use the method presented prior to this example to compute the distribution of
the logarithm of the amplification factor, N .
(d) For each turbulence level, compute the position of transition point and mark
these points in the plot of the reduced amplification factor.
Solution:
(a) We follow Example 6.3 to answer the first question. The most convenient way
to do this is to implement all the relevant equations in a spreadsheet program
or other computational software such as Matlab. We first convert the pressure
distribution to a velocity distribution by employing (6.11):
u e = V∞ 1 − Cp
Subsequently, we take the derivative of u e with respect to x and we employ
(6.101) to compute the distribution of θ and λ up to λ = −0.09. The resulting
separation point is (x/c)sep = 0.61.
(b) We employ (6.126) to compute that for Tu = 0.1 % we have N ≈ 8.3 and for
Tu = 2 % we have N ≈ 0.3.
(c) We need to solve (6.128). To that purpose, we implement (6.130)–(6.132) into our
spreadsheet program. Furthermore, we need the momentum-thickness Reynolds
number (6.129), the equation of state (2.72), and the equation for the speed
dN
, and Reθ0 . The resulting values are
of sound (2.106) to calculate Reθ , dRe
θ
subsequently substituted in (6.128) to compute the distribution of N with x as
shown in Fig. 6.49.
(d) The transition points are indicated in the graph with dotted lines. They are
(x/c)tr = 0.45 and (x/c)tr = 0.54, respectively.
Let us briefly reflect on the result of Example 6.8. First of all, note that the transition
point is ahead of the calculated laminar separation point. The predicted onset of
transition will therefore cause a turbulent boundary layer before the boundary layer
can separate. The separation point therefore has lost its meaning in this example.
Secondly, observe the arrows in Fig. 6.49 indicating the stabilizing region and the
destabilizing region. When N < 0 the amplification factor is smaller than 1, meaning
Logartithm of amplification
factor, N (~)
6.5 Boundary-Layer Computations
349
15
e9
10
Tu = 0.1%
5
destabilizing
Tu = 3.0%
0
stabilizing
−5
xtr
−10
−15
0
0.2
0.4
0.6
0.8
1
x (m)
Fig. 6.49 Distribution of logarithm of amplification factor: N = ln e N belonging to Example 6.8
that small disturbances in the boundary layer are damped. This corresponds to the
pressure distribution, which features a favorable pressure gradient up to x = 0.5. We
see that when the pressure gradient becomes less negative, the value of N starts to rise
quickly. Moreover, when the pressure gradient is positive (beyond x = 0.5) a very
rapid increase in N can be seen. This implies that initial disturbances in the boundary
layer are rapidly amplified under an adverse pressure gradient and therefore cause a
quick onset of transition. When we discuss natural-laminar flow airfoils in Sect. 7.3,
we will see how an airfoil can be designed in order to sustain laminar flow over a
large percentage of the chord.
6.6 Interference Drag
We have seen in the preceding sections that any body of a particular curvature causes
the flow over the body to accelerate or to decelerate. In addition we have seen that
the supervelocities directly (via friction) and indirectly (via shocks) affect the drag
of a body in transonic conditions. If we have two bodies that each have their own
drag (e.g. D1 and D2 ) and we would join these bodies together we mostly find that
the cumulative drag of these two bodies is larger than the sum of the individual components. If the difference, ΔD = D1+2 − (D1 + D2 ) > 0, we speak of interference
drag. This effect is not only present when two bodies intersect each other but also
when two bodies are in each others vicinity. At this point we would like to emphasize that the term ‘interference drag’ is merely a term that describes the effect on
drag when two components intersect. The physical cause for drag can, in turn, be
attributed to an increase in pressure drag or an increase in friction drag.
In the classic book of Hoerner [26] various types of interference drag are distinguished. In the context of the present chapter we focus on the interference drag
caused by the intersection of two bodies. In practice, at least one of the two bodies
is often a lifting body. For example, the intersection of the horizontal tail plane with
350
6 Aerodynamics of Non-lifting Bodies
the vertical tail can cause interference drag. For the explanation of the concept of
interference drag it is not important whether the interfering bodies are designed to
generate lift or not.
6.6.1 Interference Drag in Subsonic Conditions
Interference between aircraft components is not limited to high-subsonic aircraft. In
fact, there are several examples of low subsonic aircraft where interference is utilized
as a benefit. The most notable example is the single slotted flap that is often seen
on low-subsonic aircraft such as the propeller-powered Cessna 172 (most successful
mass produced light aircraft to date). The small slot that occurs between the flap and
the main wing causes a beneficial interference between these two components which
allows the aircraft to fly at a lower speed without stalling. This same effect is used
on many of the high-lift systems found on contemporary civil transport aircraft. The
interference between the individual components of the high-lift system causes the
maximum lift coefficient to increase substantially allowing high-subsonic aircraft to
take-off and land at relatively short airfields with acceptable take-off and landing
speeds.
Even though the interference on components that are generated to produce lift
can produce a net benefit to the airplane, on components which are not intended to
generate lift they often contribute to increased drag. An illustration of how two bodies
in subsonic flow interfere with each other is shown in Fig. 6.50. In this figure we see
that when the two streamlined bodies are at a relative large distance from each other,
the total drag of the bodies converges to a fixed value. By decreasing the distance
the bodies are forming a converging-diverging nozzle with increasing area ratio (the
ratio between nozzle exit and nozzle throat area). The flow that flows through the two
bodies decelerates beyond the “throat” such that the fluid can expand in the diverging
part of the duct. With increasing area ratio the associated pressure gradient with this
deceleration becomes larger and larger. At a certain part the adverse pressure gradient
becomes so large that the boundary layers separate from the respective bodies, which
results in a large increase in pressure drag.
0.25
interference
0
CD
y
ReC= 4 .105
0.20
0.15
0.10
t
c = 3t
0.05
0
0
1
2
3
4
5
lateral spacing, y/t (~)
Fig. 6.50 Interference drag between two streamlined bodies (after Ref. [26])
6
6.6 Interference Drag
351
The example above illustrates that two bodies do not need to intersect in order
for their mutual interference to cause separation of the boundary layer. This type
of interference is mainly based on the flow outside the boundary layer, which is
essentially inviscid. However, the magnified acceleration and deceleration of the flow
outside the boundary layer also has an effect on the development of the boundary
layer, as we have seen in Sect. 6.4.2. Therefore, the local friction coefficient and
shape factor of the boundary layer are directly affected by the interference of the two
bodies. The increase in adverse pressure gradient for the bodies in Fig. 6.50 increased
the shape factor of the boundary layer which made the boundary layer more prone
to separation than without the interference of the two bodies.
The effect of interference on the boundary layer is even greater when two bodies intersect. A relevant example is shown in Fig. 6.51a where a streamlined strut
intersects with a flat plate. Both the strut and the flat plat generate a boundary layer.
The boundary layer on the plate is, generally speaking, a lot thicker than the one on
the strut due to the longer distance it has traveled and therefore the time it has had
to develop. This results in a relatively small du/dy|wall . At the intersection of the
two components the boundary layers join. At the leading edge (stagnation) and over
(a)
0.15
increment
CD
0
0.10
l
t = .43
c
sectional drag
0
t
interference on two ends ( = 0)
0.05
0
20°
40°
strut angle,
(b)
60°
(deg)
Profile NACA 0015
d
d
= 0.5
c
c
Separation
U
Separation
Fig. 6.51 Interference drag due to the intersection of a curved body with a flat wall. a Interference
between strut and wall (after Ref. [26]). b Interference causing separation between cascades and
wall (after Ref. [22])
352
6 Aerodynamics of Non-lifting Bodies
the rear of the airfoil, the boundary layer is invariably retarded, which often leads to
separation (du/dy|wall < 0). This is illustrated in Fig. 6.51b where the interference
between airfoil cascades and the wall is shown. Near the wall the separated region
on the cascades expand more upstream. In addition, a separated region starts to form
near the leading edge of the cascades. The separated flow, in turn, causes a wake
which results in additional pressure drag. As can be seen in Fig. 6.51a the angle
between the strut and the wall is an important factor for the value of the interference
drag coefficient. When the angle between the strut and the wall becomes smaller
than 90◦ the interference drag also increases. This is due to increased separation in
the narrower one of the two corners.
In order to minimize interference drag between components we need to make
sure that the boundary layer remains attached at all times. The unfavorable addition
of pressure distributions should be avoided. One way to achieve this is to apply
fillets between the two surfaces. This fillet locally reduces the supervelocities at
the intersection of the two bodies. Thereby it reduces the magnitude of the adverse
pressure gradient, which shifts the point of separation further downstream or prevents
the formation of separation all together.
6.6.2 Interference Drag in Transonic Conditions
In transonic flow the characteristics of interference are magnified. If we look back at
Fig. 6.11 we can directly see that the region of influence of a body in transonic flow is
much larger than in subsonic conditions, giving rise to a stronger interference between
the individual components of the airplane. One can imagine two bodies of the aircraft
(e.g. horizontal and vertical tail) that intersect at a particular point. Both bodies generate high supervelocities, possibly even supersonic. However, at the intersection there
is less physical space for the flow to go and even higher supervelocities are generated
resulting in much stronger local shock waves than would be expected if either one
of the two bodies would be considered by itself. The stronger shock waves induce
an increase in wave drag that is also a form of interference drag [43]. The strong
pressure gradient that accompany strong shocks can also result in separation of the
boundary layer, as has been discussed in Sect. 6.4.3. The onset of strong shocks and
the accompanied wave drag can lead to a decrease in the drag-divergence Mach number, which generally limits the cruise Mach number of an airplane. Minimizing the
interference is therefore an important goal in the aerodynamic design of the aircraft.
Interference drag plays a role throughout the entire aircraft where two bodies
intersect or are positioned close to each other. Ideally, the velocity distributions on the
intersecting bodies should complement each other to minimize the total supervelocity
at the intersection. If one body locally displays a negative pressure coefficient, the
intersecting body should have a positive pressure coefficient. Often, local area ruling
is applied to achieve this. Particular geometric characteristics on aircraft often show
how designers have attempted to reduce the transonic interference drag. We shortly
present three such examples in this section.
6.6 Interference Drag
353
Fig. 6.52 Leading-edge fairing at the wing root of an A340-300 (photo A Pingstone)
-0.1
-0.3
-0.4
-0.5
Cp = 0.0
-0.2
upper surface
-0.6
Cp = 0.0
lower surface
Fig. 6.53 Theoretical isobar patterns on area ruled empennage for M = 0.80 and C L h = −0.117
The first example is the wing-body fairing which smoothes the angle between the
wing and the fuselage. Without the presence of such a fairing the boundary layer
that has developed over the fuselage side is likely to separate. An example of such a
fairing is presented in Fig. 6.52.
Another example is the junction between the horizontal and vertical tail plain in
a T-tail (see Fig. 6.53). Often, an additional fairing (acorn) is positioned to reduce
the added supervelocities that are caused by interference. Adding this body allows
the designer to locally change the area distribution of the intersection such that
interference is minimized.
The position of the nacelle with respect to the wing is a third example of how
interference-drag considerations dominate this geometric feature. The positioning
of the nacelle under the wing implies a strong interference between the pressure
distribution over the nacelle, pylon, and lower wing surface. An interesting example
in this respect is the development of the Convair 990. The specification for this
airplane included a very high cruise Mach number (Mcruise = 0.89). During early
experimentation it surfaced that supersonic flow accompanied by strong shock waves
354
6 Aerodynamics of Non-lifting Bodies
were present on the inboard sides of the pylons. The drag rise7 of the airplane
coincided with the drag rise of the nacelles.
To remedy this problem and postpone the onset of drag divergence to a higher
Mach number local area ruling was applied. A representative streamtube was chosen by including (part of) the cross-sectional area of nacelle, pylon and wing. This
streamtube is shown in Fig. 6.54a. A target area distribution was defined that had
Treated as solid boundaries
(a)
A
B
C
A
B
C
1.5 Rnacelle
Sect. A-A
(b)
1.5 Rnacelle
max
Sect. C-C
Sect. B-B
1200
post
modification
1000
Cross sectional area (in2)
max
wing leading edge
@ pylon centerline
800
nacelle exit
600
added
area
400
original area distribution
200
0
40
80
120
160
200
240
280
320
360
400
Longitudinal station (in)
Fig. 6.54 Local area ruling between nacelle, pylon and wing of Convair 990 (after Ref. [31]).
a Streamtube boundaries. b Area distribution prior and post modifications. c Drag curve prior and
post modification. d Proposed modifications to nacelle and pylon
7 “Drag rise” is a term that describes the exponential increase in drag coefficient with Mach number.
6.6 Interference Drag
355
(c)
Change in aircraft drag
coefficient, D (cts)
Initial configuration
Area modification
.0060
.0040
.0020
0
.80
.82
.84
.86
.88
.90
.92
Mach number, M (~)
(d)
A
WING
LE
B
Inboard
nacelle
A
forward pylon
fairings
A
Outboard
nacelle
A
terminal
fairing
B
Sect. A-A
Sect. B-B
aft pylon
fairings
Wing
LE
B
B
Sect. A-A
Sect. B-B
Fig. 6.54 (continued)
to reduce the local supervelocities. Modifications to the cross-sectional area distribution were made by thickening the pylon, modifying the nacelle aft body, and
modifying the terminal fairing [32]. This combined effort on all four pylon-nacelle
combinations significantly postponed the drag rise onset and resulted in a lower drag
coefficient at the design Mach number (Fig. 6.54c). The resulting geometry of the
pylons is shown in Fig. 6.54d. Notice the typical area-ruling characteristics of the
pylon: close to where the leading edge of the wing intersects the cross-sectional area
of the pylon reduces. It is as if the pylon gives the flow additional space to go around
the wing.
The examples presented above illustrate the importance of considering the aerodynamic interference between the various components on the airplane. The present discussion is not limited to nonlifting components of the airplane. On the contrary, when
integrating bodies that are designed to generate forces the supervelocities are generally larger and the formation of strong shock waves in combination with separation
356
6 Aerodynamics of Non-lifting Bodies
is even more likely to occur. The integration of winglets to the tip of the wing is an
example where the interference of two force-producing bodies is likely to produce
interference drag. Creating a fillet with a large radius can prevent the onset of strong
shocks in the corner of the two lifting surfaces. For more examples on interference
drag created by lifting and non-lifting components the reader is referred to the text
of Obert [43].
6.7 Summary
There are several parts of a high-subsonic airplane that do not contribute to the
production of lift during the cruise phase of the mission (e.g. fuselage, fin, nacelle).
These surfaces do provide a significant contribution to the total drag that is produced
by the airplane. We have investigated three important sources of drag: friction drag,
drag due to separation, and wave drag.
Wave drag that is associated with the formation of shock waves. A simple method
has been presented to calculate the wave drag of an arbitrary nonlifting body. This
method, which is based on linearized supersonic potential theory, uses an equivalent
axis-symmetric body with the same cross-sectional area distribution as the body
under investigation (transonic area rule). It has been shown how the cross-sectional
area should be distributed along the length of the body in order to minimize the wave
drag. Several examples of the application of area ruling have been presented for both
civil and military aircraft.
Friction drag is associated with shear forces that are generated in the boundary
layer between the outer flow and the airplane surface. It has been shown that the shape
of the velocity distribution in the boundary layer has an effect on the local friction
coefficient of both a turbulent and laminar boundary layer. In addition, it affects the
streamwise position at which the laminar boundary layer transitions to a turbulent
boundary layer. A turbulent boundary layer has a higher velocity gradient at the wall
and produces more friction drag than a laminar boundary layer. In addition, the turbulent boundary layer is much thicker than the laminar boundary layer which affects the
outer pressure distribution through the displacement effect. The shape of the velocity
profile in the boundary layer, in turn, is affected by the outer pressure distribution.
The shape of the body (which is responsible for the pressure distribution) is therefore
of pivotal importance for the total amount of friction drag that is generated. While
the turbulent boundary layer is less sensitive to changes in Reynolds number than the
laminar boundary layer, it is much more sensitive to changes in Mach number. The
presence of surface roughness has an effect on the transition point (shifts forward)
as well as on the local value of the friction drag (becomes larger). Additionally,
it is shown that the friction coefficient becomes smaller with increasing Reynolds
and Mach number. However, for roughened surfaces there exists a cut-off Reynolds
number above which the friction coefficient remains constant with Reynolds number.
6.7 Summary
357
When a boundary layer is highly loaded by an adverse pressure distribution it tends
to separate from the surface. In the aerodynamic design of an aircraft separation of
the boundary layer is to be prevented during normal operation. It has been shown
that the boundary layer shape factor grows under the influence of an adverse pressure
gradient. Furthermore, a simple method has been presented that shows that the onset
of separation starts at a shape factor of around 3.0. In the context of nonlifting bodies,
separation often occurs relatively far downstream due to a (weak) adverse pressure
gradient in combination with a relatively high shape factor. Conversely, in areas where
supersonic flow domains are terminated with a shock wave the shape factor can rise
quickly and separation can occur at the foot of the shock. Flow visualization is capable
of showing shock-boundary layer interaction in transonic speeds. The strength of the
shock and the state of the boundary layer (i.e., laminar versus turbulent) dominate
the behavior of boundary layer upon its interaction with shocks in transonic flow.
Intimately tied to the formation of shock waves and the separation of the boundary
layer is the concept of interference drag. It has been shown that in high-subsonic flow
the propagation of disturbances caused by the presence of a body is significantly
higher than in low-subsonic conditions. This interference has a magnification effect
on the supervelocities over the interfering bodies. This can result in separation of the
boundary layer in stagnation areas or areas with large adverse pressure gradients.
In addition, strong shocks might form that increase the wave drag and limit the
cruise Mach number of the airplane. To reduce interference drag, local area ruling
can be applied where the cross-sectional area distribution of the combined bodies is
smoothened. In practice, this often means that additional fairings between the joining
components are required to reduce the interference effect.
Problems
Introduction
6.1 Consider the flow about the body in the control volume of Fig. 6.1.
(a) Show that the
continuity equation (2.109) can be rewritten according to: intake
ρ1 u 1 dy = exhaust ρ2 u 2 dy.
(b) Show that (6.7) can be derived from (6.6) and the relation shown under (a).
6.2 Consider the flow about the body in the control volume of Fig. 6.1.
(a) If we assume that the velocity defect behind the wing (u 1 −u 2 ) is small compared
C
to the free stream velocity, V∞ , show that D ≈ ρV ∞ (V∞ − u 2 ) dy.
D
(b) Show that, under the same assumptions, the drag can also be derived from the
C pt∞ − pt2 dy.
total-pressure defect in the wake of the wing: D ≈
D
(c) Based on the items above, how would you measure the drag of a body at section
CD?
358
6 Aerodynamics of Non-lifting Bodies
Pressure Distribution Over Nonlifting Bodies
6.3 Consider Euler’s equation for conservation of momentum along a streamline,
(6.8).
(a) Starting from (6.8) derive Bernoulli’s equation.
(b) Derive (6.11) from (3.11).
(c) Derive (6.12) from (6.11) using the assumptions mentioned in the text.
6.4 Using the same approach as in Example 6.1, draw the notional pressure distributions about the following axis-symmetric bodies:
(a)
paraboloid
cylinder
paraboloid
(b)
ellipsoid
cylinder
paraboloid
(c) paraboloid
paraboloid
6.5 In the figure below you find a side-view of a British experimental aircraft for
radar surveillance (Early Warning), the British Aerospace Nimrod AEW. Sketch the
relation of the static pressure coefficient at the lower and upper crown line of the
forward fuselage in the plane of symmetry.
-1
Cp 0
x
1
6.6 (a) Derive (6.17) based on the expression for the pressure coefficient and the
isentropic relation between total pressure and static pressure.
(b) If M∞ = 0.8 and Cp = −2, calculate the local Mach number.
(c) If M∞ = 0.8 and Cp = −3, calculate the local Mach number.
(d) Calculate for what value of the Cp the theoretical local Mach number tends to
infinity if M∞ = 0.8.
6.7 Summary
359
6.7 For the free stream Mach numbers ranging from 0.1 to 0.8, graph the increase in
Mach number, ΔM, as a function of pressure coefficient. Use a range of Cp between
−2 and 0.
Wave Drag
6.8 Show that (6.22) can be derived from (6.19), (6.21), and (2.106).
6.9 Consider the perturbed momentum equation in integral form (6.26).
(a) Show that, by employing the continuity equation (6.27), this equation can be
simplified according to:
F body = −
!
2
pn + ρV∞
v (v · n + i · n) dS.
S
(b) Argue that the expression above is identical to (6.28) because we integrate over
an enclosed surface, S.
6.10 Consider (6.50) in terms of the Glauert variable θ.
(a) Rewrite (6.50) in terms of x.
(b) If R(x) is the distribution of the radius of the Sears-Haack body, express R(x)
in terms of the maximum radius, Rmax , and x.
6.11 Consider the body described in Example 6.2.
(a) Calculate the volume, V , and maximum radius, Rmax corresponding to the aforementioned body.
(b) Calculate the wave drag of a Sears-Haack body with the same length and volume
as the aforementioned body.
(c) Calculate the relative decrease in wave drag coefficient (in percentages) that can
be attained when modifying this body to the Sears-Haack body with identical
volume and length.
6.12 The aft fuselage of the Airbus A320 exhibits a tailored waist at the location
of the horizontal tail. In addition, the vertical tail is moved forward relative to the
horizontal tail. What are the explanations for their relative positioning?
horizontal tail
vertical tail
360
6 Aerodynamics of Non-lifting Bodies
6.13 The figure below reveals the large wing-fuselage fairing of the Airbus A380.
This fairing is so large that it received the nickname “the bathtub.” Associated with it
are a large wetted area and a weight penalty. Carefully explain the reason of existence
of a wing-fuselage fairing in general, despite the drawbacks mentioned.
Photo Creative Commons - Simon-sees
6.14 The Convair 990 (see below) was designed to be a jet transport with a cruise
Mach number of M = 0.89. In 1958 this was quite challenging due to the limited
computational resources. Wind tunnel tests on the initial design demonstrated that the
drag coefficient was too large to attain the desired cruise Mach number. A somewhat
unconventional solution was proposed and eventually adopted in the final design.
The solution was to position large bodies near the trailing edge of the wing on the
top surface.
(a) Carefully explain how the addition of these bodies can result in a lower drag
coefficient at the design Mach number.
(b) Name two disadvantages of these bodies.
(c) Why do modern transport aircraft lack these additional bodies?
6.7 Summary
361
Photo NASA
Fundamentals of Boundary-Layer Flow
6.15 Consider Fig. 6.24. If we assume that the freestream velocity value of this flow
of water equals 1 m/s and that the streamtube in this figure measures 1 cm.
(a) Calculate the velocity gradients at the three different positions.
(b) Calculate the local value of the wall stress at these three positions.
(c) Calculate the local value of the friction coefficient at these three positions.
6.16 Assume we have a plate with a total friction coefficient of Cf = 0.0030 at
M → 0. Plot the relation between the friction coefficient and the Mach number for
0 < M ≤ 1.4.
Boundary-Layer Computations
6.17 Consider a plate with chord length of c = 10 m, which is exposed to a flow with
distribution
V∞ = 100 m/s at 0 m ISA conditions. Consider the following
velocity
x over the plate at the edge of the boundary layer: u e = V∞ 1 + b c 1 − a xc .
For a = 1 and b = 2, perform the following tasks:
(a) Plot the pressure coefficient, Cp , over the plate.
(b) Calculate the position of separation, (x/c)separation .
(c) Plot the momentum thickness over the plate between x/c = 0 and (x/c)separation .
(d) Plot the displacement thickness over the plate between x/c = 0 and (x/c)separation .
(e) Plot the shear stress over the plate between x/c = 0 and (x/c)separation .
6.18 Repeat Problem 6.17 for the following conditions:
(a) V∞ = 100 m/s, c = 1 m, a = 1 and b = 2.
(b) V∞ = 10 m/s, c = 10 m, a = 1 and b = 2.
(c) V∞ = 100 m/s, c = 10 m, a = 2 and b = 1.
362
6 Aerodynamics of Non-lifting Bodies
6.19 Derive (6.103) by using the definition of u + and the definition of τw in (6.54).
6.20 Plot the velocity profile according to the law of the wall. On the vertical axis
show u + , ranging between 0 and 25. On the horizontal axis, show y + , ranging
between 1 and 1,000.
6.21 To investigate the effect of the Coles wake paramter on the velocity profile,
plot (6.109) for Π = 0, 1, 2.5, 5, 10, 15. On the vertical axis show u + , ranging from
0 to 100. On the horizontal axis, show y/δ between 0 and 1.0.
6.22 Show that (6.112) can be derived from (6.109) by substituting y = δ in (6.109).
6.23 Show that when k + → ∞ we can reduce (6.121) to (6.122).
6.24 Assume we have a flat plate with an equivalent sand grain roughness of
0.020 mm. Assume that we have 0 m ISA conditions and a freestream velocity of
100 m/s.
(a) Calculate using (6.123) the friction coefficient (cf ) at x = 2 m.
(b) Calculate the wall stress, τw .
(c) Calculate the velocity gradient at the wall, (∂u/∂ y)w .
6.25 Consider the plate and flow conditions of Problem 6.24 and the method of
Example 6.6. Imagine we have 5 plates with lengths: 1, 3, 10, 20, and 50 m. Assume
that on each plate the turbulent boundary layer starts at x = 0.10 m. Calculate
(a) The Reynolds number, Rel , for each plate
(b) The friction coefficient, Cf , for each plate.
(c) Make a plot of the Reynolds number versus the friction coefficient.
6.26 Consider a flat plate of 10 m long. Assume that we have 0 m ISA conditions,
a freestream velocity of 100 m/s, and that the turbulent boundary layer starts at
x = 0.10 m. We would like to evaluate the effect of various sand grain sizes: k = 0,
0.025, 0.050, 0.075, 0.10 mm.
(a) Plot the value of the local friction coefficient, cf as a function of the position x
along the length of the plate for the five roughness heights.
(b) Plot the value of the total friction coefficient, Cf , as a function of the roughness
height, k.
Interference Drag
6.27 In the figures below you see pictures of two versions of a British Navy-fighter,
the Hawker, later Armstrong-Whitworth “Sea Hawk.” The later version, powered
by an engine with 5400 lb static thrust at sea level, has a fairing at the junction of
the horizontal and vertical tailplane. The earlier version, with a 5000 lb static thrust,
lacked this fairing. Explain the (aerodynamic) cause for the presence of the fairing
on the later version.
6.7 Summary
363
Photos by Hawker Siddley (left) and A. Pingstone (right)
6.28 The picture below shows the Tupolev, Tu-154 in landing configuration. Visible
is the pointed cone at the intersection of the horizontal and vertical tailplane. Explain
the aerodynamic function of that cone using sketches of pressure distributions.
Photo A. Pingstone
6.29 The Lockheed P-38 Lightning (below) was in many ways innovative. An undesirable characteristic of this aircraft was heavy horizontal tail buffeting at high speed.
Initially it was expected that it was classical flutter of the horizontal tail itself and
could be cured using mass balances. However, that did not solve the problem at all.
Only after a fairing was added at the root of the wing’s leading edge, where it meets
the fuselage, did the buffet disappear. What can be concluded about the precise cause
of this phenomenon?
Photo USAF
364
6 Aerodynamics of Non-lifting Bodies
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10. Clauser, F.H.: The turbulent boundary layer. J. Aeronaut. Sci. 21, 91–108 (1954)
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12. Coles, D.E., Hirst, E.A.: Computation of turbulent boundary layers. In: Proceedings of the
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Chapter 7
Airfoil Aerodynamics
Abstract In this chapter we present the aerodynamics about two-dimensional
lift-producing wing sections (airfoils) at low transonic conditions. It is shown how
various design parameters influence the velocity distribution and the shock formation
over an airfoil. The development of supercritical airfoils is explained from a historic
perspective. First the advantages of natural laminar flow sections are explained for
subsonic conditions prior to the formation local supersonic flow. It is shown how
the supercritical airfoil distinguishes itself in terms pressure distribution and drag
behavior once transonic conditions are encountered. Furthermore, it is demonstrated
that transonic flow limitations can play a major role in the maximum lift coefficient
of (multi-element) airfoils at Mach numbers as low as 0.25. A theoretical limit of the
local Mach number is derived that effectively limits the amount of suction that can be
generated over the upper surface of an airfoil. Also the concept of shock-boundary
layer interaction is further expanded in this chapter. It is shown how periodic separation at the shock foot and shock oscillation can interact to produce a high-frequency
pressure fluctuation known as transonic buffet. This chapter contains 4 examples and
16 practice problems at the end of the chapter.
7.1 Introduction
Experimental research into transonic aerodynamics was initiated before the first
world war. Although initially tailored towards munition, at the end of WWI transonic
effects were showing up in the increased tip speeds of propellers. It was found that
thicker sections at the tip yielded much less thrust and a much higher drag coefficient.
Soon it was found that when these sections were made thinner, these adverse effects
were not encountered [2]. Even though the physical understanding of transonic aerodynamics was still non-existing at the time, practical solutions to the adverse effects
were already conceived. Experimental research in transonic aerodynamics always
prevailed above the theoretical research because of the many challenges associated
with modeling of transonic flow.
With the introduction of the jet engine on transport aircraft in 1949 on the de
Havilland Comet, the operating Mach numbers for passenger aircraft increased
© Springer Science+Business Media Dordrecht 2015
R. Vos and S. Farokhi, Introduction to Transonic Aerodynamics,
Fluid Mechanics and Its Applications 110, DOI 10.1007/978-94-017-9747-4_7
367
368
7 Airfoil Aerodynamics
substantially. It was soon found that the profile shapes of the popular natural-laminar
flow airfoils produced significant wave drag and were therefore not suitable for this
new type of aircraft. The application of such airfoils required relatively large sweep
angles in order to increase the Mach number at which strong shock waves started to
cause excessive wave drag. Even though the swept wing solution worked well from
a cruise-performance point of view, it posed drawbacks in the low-speed regime as
well as in weight and aeroelastic performance. Better airfoil sections were therefore
pivotal in the development of contemporary wings on high-subsonic aircraft.
We have seen in Chap. 6 that any body submersed in a flow field produces a
pressure distribution. When the integral of the pressure distribution over the entire
body is nonzero, a resultant force is produced. We showed that for nonlifting bodies this resulting force is comprised only of the drag force. Such conditions only
apply for (axis) symmetric bodies at zero angle of attack. Whenever either of these
requirements is violated a resultant force component perpendicular to the freestream
direction results. This discrete force (often referred to as the lift) has a point of application somewhere on the body. The resultant force is now a combination of the lift
force and the drag force, as shown in Fig. 6.1. When this body is two-dimensional
it reduces to an airfoil. For many high-aspect ratio wings, the flow over the center
section of the wing can be treated as two-dimensional. Airfoil shape therefore plays
an important role in the lift and drag characteristics of the wing.
This chapter explains the typical transonic flow characteristics about twodimensional airfoil sections (commonly referred to as airfoils). We use a historical approach to take the reader on a quest to discover the relevant aerodynamic
effects related to transonic flow about airfoils. We demonstrate how experimental
researchers such as Pearcey and Whitcomb in the late 1940s and early 1950s came
up with airfoils designed for supercritical conditions. They showed how sections
could be shaped to operate at high Mach numbers and at high lift coefficients without a large (wave) drag penalty. We present the key differences between laminar-flow
airfoils and supercritical airfoils. We also discover that transonic flow about airfoils
is not limited to high-subsonic freestream Mach numbers. On the contrary, the maximum lift coefficient is shown to be highly dependent on the freestream Mach number.
Especially multi-element airfoils are shown to yield lower maximum lift coefficients
due to transonic effects at Mach numbers as low as 0.2. Finally, we present the concept of transonic buffeting: a high-frequency, self-sustained oscillation of the shock
wave that severely limits the maximum lift coefficient of an airplane at its operating
Mach number and beyond.
7.2 Pressure Distribution About Airfoils
The pressure distribution about airfoils can be understood by following the same
rationale as in Sect. 6.2. Concave curvatures on a body cause a deceleration of the
flow, which are accompanied by an increase in static pressure. Convex curvatures
accelerate the flow and result in a decrease in static pressure (suction). The magnitude
7.2 Pressure Distribution About Airfoils
mean camber line
6%c
0.15%c
Δy
369
thickness, t
camber
leading edge
radius
chord line
chord, c
Fig. 7.1 Airfoil nomenclature
of the pressure increase or decrease can be influenced by the radius of curvature. A
small radius of curvature causes a rapid acceleration or deceleration resulting in
a sharp pressure gradient. A large radius of curvature results in smaller pressure
gradients. In addition to the effect of the curvature, the boundary conditions at the
leading and trailing edge also have a large effect on the velocity distribution. A
stagnation point should be present near the leading edge (velocity is zero). At the
trailing edge the Kutta condition should be satisfied. In theory this should result in
a second stagnation point at the trailing edge for any airfoil having a finite trailing
edge angle. In practice we see that the flow velocity at the trailing edge takes a value
closer to the freestream velocity.
To study the variation of pressure distribution about an airfoil we first define the
major parameters that characterize each airfoil shape. These parameters are shown in
Fig. 7.1. The main dimensions of the airfoil are governed by its maximum thickness
(tmax ) and its chord length, c. Their ratio is often referred to as the thickness-to-chord
ratio and abbreviated with t/c. The mean camber line of the airfoil is formed by the
locus of points halfway between the upper and lower surface measured perpendicular
to the camber line itself. The most forward and rearward points of the camber line are
the leading and trailing edge, respectively. The chord line is a straight line connecting
these points. The camber is the distance between the chord line and the camber line as
measured perpendicular to the chord line. Finally, the curvature of the leading edge
can often locally be approximated by a circular arc with a particular radius. We call
this the leading edge (or nose) radius of the airfoil, R. The leading-edge sharpness is
often represented by the value of Δy, which is also defined in Fig. 7.1 and expressed
as a fraction of the chord length.
We first look at two different symmetric airfoils under 6◦ angle of attack. We
evaluate the pressure distribution over these airfoils by using a vortex panel method
in combination with a separate boundary-layer model based on the Von Kármán’s
integral equation (6.89).1 The two airfoils under consideration are the NACA 0006
and NACA 0018. These airfoils have their maximum thickness at 30 % and have a t/c
of 6 and 18 %, respectively. The airfoils and their associated pressure distributions
1 The program that is used for this evaluation is Xfoil version 6.94. XFOIL carries out a vortexpanel analysis of subsonic isolated airfoils and has the option to include a boundary layer. More
information can be found on http://web.mit.edu/drela/Public/web/xfoil/.
370
7 Airfoil Aerodynamics
(a)
-2.0
-1.5
-1.0
c p -0.5
NACA 0006
M = 0.2000
Re = 10E6
α =6
c l = 0.6821
cmc/4 = 0.014
(b)
-2.0
-1.5
-1.0
c p -0.5
0.0
0.0
0.5
0.5
1.0
1.0
NACA 0018
M = 0.2000
Re = 10E6
α =6
cl = 0.7020
cmc/4 = 0.000
Fig. 7.2 Comparison of pressure distribution between two airfoils of different thickness as simulated by Xfoil version 6.94. a NACA 0006. b NACA 0018
have been plotted in Fig. 7.2. The aerodynamic coefficients of each of the airfoils are
displayed next to the pressure distribution.
Let us start with the similarities between the two airfoils. First of all, they are
both symmetrical, which means their camber is identical. Classical airfoil theory
tells us that the lift curve slope for a symmetrical airfoil should equal dcl /dα = 2π.
Since we know that for this airfoil cl = 0 at α = 0 we expect cl to be close to 0.66
for α = 6◦ . Comparing that to the value shown in Fig. 7.2a, b we see that the lift
coefficient for both airfoils are close to this value, meaning that both airfoils generate
almost the same amount of lift. Secondly, we look at the moment coefficient about
the quarter-chord point. Assuming classical airfoil theory this value should be zero
for both airfoils. We see that this is indeed the case for the NACA 0018, but that the
NACA 0006 displays a small nose-up pitching moment about the quarter chord.
We have established that both profiles have a similar overall effect on the lift
and moment coefficient of the airfoil. Yet their pressure distributions are somewhat
different. The thinner profile of the NACA 0006 causes a much higher suction peak
over the leading edge of the airfoil. This large suction peak is followed by a relatively sharp adverse pressure gradient that gradually lessens in magnitude further
downstream. The thick airfoil is much more blunt. The radius of curvature at the
nose of this airfoil is therefore larger resulting in a lower suction peak. The lower
suction peak is followed by a shallower adverse pressure gradient. The magnitude of
the suction peak can directly be translated to the magnitude of the supervelocities.
A high suction peak yields high supervelocities that can become supersonic at relatively low freestream Mach numbers. At M = 0.2 the critical pressure coefficient is
−16.3 according to Eq. (3.42).
In the previous chapter we have seen that the value of the adverse pressure gradient in combination with Reynolds number determines whether or not the boundary
layer separates. Even though that information cannot be extracted from the analysis
carried out above, the shape of the pressure distribution can tell us qualitatively where
we should expect flow separation to start. For the thinner airfoil the large adverse
pressure gradient near the leading edge will become so large with increasing angle of
7.2 Pressure Distribution About Airfoils
371
attack that the boundary layer is likely to separate from the surface close to the leading edge (even though the boundary layer itself is relatively fresh). Leading edge stall
is therefore to be expected on this profile. The thicker airfoil shows a more shallow
adverse pressure gradient which is likely to separate the flow starting at the trailing
edge, where the boundary layer shape factor (H ) has its highest value [see (6.63)
on p. 305 for the definition of the shape factor]. Typically when H is between 2.4
and 2.6 boundary layer separation occurs. With increased angle of attack this point
is likely to move forward leading to a relatively gentle onset of stall. These deductions on the stall behavior of the airfoil are purely conjectural based on the pressure
distribution. However, experimental results from Ref. [1] support these conclusions.
We will elaborate more on the separation of the boundary layer in low-speed and
high-speed conditions in Sects. 7.5 and 7.6, respectively.
The effect of compressibility on the local pressure coefficient has been discussed
qualitatively in Sect. 3.2. The Prandtl-Glauert compressibility correction magnifies
the value of the pressure coefficient. It has been shown that this correction (or derivatives thereof) can accurately predict the value of the pressure coefficient by simply multiplying this correction factor to the pressure coefficient obtained from an
incompressible-flow calculation (see Chap. 3). However, when Mach numbers are
increased beyond Mcrit the compressibility correction loses its accuracy. In Fig. 7.3
it is shown why this correction cannot be applied in transonic conditions. The same
airfoil (NACA 0012) is evaluated with two different prediction tools for the same
freestream conditions (a constant angle of attack and varying Mach number). The
prediction in Fig. 7.3a is based on a vortex panel method (Xfoil) with an embedded
compressibility correction. We see that the pressure distributions at low Mach numbers are merely amplified. The graphs in Fig. 7.3b are generated using an Euler solver
(b) −1.5
M = 0.74
M = 0.68
M = 0.2
−1
pressure coefficient, C p (~ )
pressure coefficient, C p (~)
(a) −1.5
−0.5
0
0.5
NACA 0012
M = 0.74
M = 0.68
M = 0.2
−1
−0.5
0
0.5
NACA 0012
1
1
0
0.2
0.4
0.6
position, x/c (~)
0.8
1
0
0.2
0.4
0.6
0.8
1
position, x/c (~)
Fig. 7.3 Comparison between results from two prediction tools for the flow over a NACA 0012
airfoil with α = 2.1 and Rec = 10 million. a Vortex panel method + compressibility correction
(Xfoil v. 6.94). b Euler approximation (MES v. 3.04)
372
7 Airfoil Aerodynamics
(MSES2 ). The graphs of Fig. 7.3b show a distinct change in pressure distribution
when the Mach number is M = 0.68 or higher. We see how the adverse pressure gradient of Fig. 7.3a is replaced by a more gradual increase in negative C p in Fig. 7.3b.
The steep drop in (negative) C p is indicative of a shock wave, which is also absent in
Fig. 7.3a. This comparison demonstrates the limitation of compressibility corrections
in the transonic regime. We will discuss this change in pressure distribution in more
detail in Sect. 7.4.
7.3 Laminar-Flow Airfoils
Having looked at the basic characteristics of airfoils we now wonder: what airfoil
performs best? Even though this answer might seem simple enough, there exists
no one-size-fits all airfoil that performs best for all applications. First of all, what
do we quantify as airfoil performance? This question cannot be answered quite
straightforwardly. There are a few key performance parameters of an airfoil: its
maximum lift coefficient, its lift-curve slope, its lift-do-drag ratio at the design lift
coefficient(s), and its moment coefficient at the design lift coefficient(s). The plural
form of the lift coefficients already indicates that airfoils are often designed for a range
of operating conditions (read: angle of attack and Mach number). Therefore, finding
the best airfoil might eventually result in an airfoil that finds the best compromise in
performance parameters over the specified range of operating conditions [45].
To maximize airplane range and endurance the lift-to-drag ratio of the airfoil is
of relatively high importance. The first efforts to maximize this ratio during the preWWII period concentrated on creating large domains of laminar boundary-layer flow
over the upper and lower side of the airfoil. At the National Advisory Committee for
Aeronautics (NACA) a series was developed for precisely this reason: minimize the
drag by generating a favorable pressure coefficient over the upper and lower surface
such that the point of transition would be shifted more and more aft. This resulted
in the NACA 6-series airfoils that are described in detail in Ref. [1]. These airfoils
were designed with a specific lift coefficient in mind. In addition, the position of
minimum pressure was specified, indicating the chordwise location of the onset of
the adverse pressure gradient. We compare a NACA 6-series airfoil to a NACA 4series airfoil in Fig. 7.4. Two airfoils of identical thickness are evaluated3 at identical
lift coefficients, Mach numbers, and Reynolds number. Their pressure coefficient
and friction coefficient distribution are shown in Fig. 7.4a, b, respectively.
Let us evaluate the results presented in Fig. 7.4. First of all, we would like to
emphasize the similarity between the two airfoils: they have the same thickness and
2 MSES is a viscous-inviscid analysis program that couples the numerical solution of the Von Kár-
mán equation (6.89) in the boundary layer to the numerical solution of the steady state conservative
Euler equations (2.183) outside the boundary layer. More information can be found in Refs. [10, 15].
3 This prediction is carried out by Xfoil 6.94 using an e N method for transition prediction with
Ncrit = 10.
7.3 Laminar-Flow Airfoils
373
Pressure Coefficient, C p (~)
(b)
−0.5
0
NACA 2412
6
x 10
2
lower surface
0
0.2
0.4
0.6
position, x/c (~)
0.8
NACA 661 212
0.5
6
x 10
−3
f
upper surface
4
0
0
−3
f
Friction Coefficient, c (~)
0.5
−0.5
Friction Coefficient, c (~)
Pressure Coefficient, C p (~)
(a)
1
upper surface
4
lower surface
2
0
0
0.2
0.4
0.6
0.8
1
position, x/c (~)
Fig. 7.4 Predicted pressure and shape factor distribution for cl = 0.3, M = 0.2, and Rec = 6×106 .
a NACA 2412 at α = 0.4◦ . b NACA 661 212 at α = 0.9◦
both of them produce the same section lift coefficient. If each airfoil would be used
in a separate wing, the structural weight of each wing for a given planform shape
would be virtually identical. Even though the integrated difference of the pressure
coefficient between upper and lower surface is identical for both wings, their pressure
distributions are quite different. The thickest point of the 2412 airfoil is located
(by default) at 30 % chord. On the upper surface we see the start of the adverse
pressure gradient at approximately x/c = 0.2. Transition to a turbulent boundary
layer occurs around x/c = 0.45. On the lower surface the adverse pressure gradient
is slightly steeper and transition occurs closer to the leading edge than on the upper
surface. Notice that in this simulation transition can be seen in the jump in the friction
coefficient. This indicates where the boundary layer has transitioned from laminar to
turbulent. In this particular simulation the step change in displacement thickness that
is caused at the transition point can also be seen in the pressure distribution curve by
means of a little bump.
The thickest point of the 661 212 has been shifted beyond the 50 % chord. This
creates an almost flat plateau in the pressure distribution over the suction side of the
airfoil up the 60 % chord. On the lower surface there is a favorable pressure gradient
pressure up to x/c = 0.6. Transition occurs on both sides beyond the 60 % chord.
Notice the effect of the favorable pressure gradient on the lower side: transition is
shifted more aft. We know that the average friction coefficient, C f is correlated to
the integral of the local friction coefficient, c f , over the chord length. The total area
under the two lines in the friction-coefficient diagram are therefore representative
for the total amount of friction drag of the profile. Based on that knowledge we can
directly see that the total friction drag produced by the 661 212 is considerably lower
374
7 Airfoil Aerodynamics
.024
Section drag coefficient, cd (~)
Fig. 7.5 Experimental
comparison of drag
coefficient between two
airfoils at Re = 6 × 106
(data from Ref. [1])
NACA 2414
NACA 661212
.020
.016
with roughness
.012
.008
smooth
.004
drag bucket
0
-1.6
-1.2
-0.8
-0.4
0
0.4
0.8
1.2
1.6
Section lift coefficient, cl (~)
than the friction drag produced by the 2412. The predicted drag coefficient is 0.0031
or 31 drag counts4 for the 661 212 against 51 counts for the 2412.
Comparison with experimental data of Ref. [1] shows that drag coefficient is
somewhat underestimated. In Fig. 7.5 the experimental wind tunnel measurements
are presented for a Reynolds number of 6 million. It can be seen that at cl = 0.3 the
drag coefficients is around 34 counts for the 661 212 versus 63 counts for the 2412.
We notice a 45 % decrement in drag when switching from a 2412 to a 661 212. The
lift-to-drag ratio of the airfoil is increased with more than 80 % from 48 to 88.
As can be seen from Fig. 7.5, the natural-laminar-flow (NLF) airfoil has a limited
range of lift coefficients where the friction drag is low. As a matter of fact, the
designation of the airfoil already shows this:
position of minimum pressure
thickness in percent chord
661 212
design lift coefficient in tenths
low-drag range of lift coefficient in tenths
The design lift coefficient is 0.2 and we can expect a low drag coefficient (also known
as the drag bucket) between cl = 0.1 and cl = 0.3. Outside of the drag bucket the drag
of the NLF airfoil is either close to, or higher than the drag coefficient of the NACA
2412. In addition, when the skin is roughened (using NACA standard roughness) the
drag coefficient at cl = 0.3 increases to 95 cts while the drag of the 2412 increases to
100 cts, making the difference between the two airfoils very small. This illustrates the
sensitivity of both profiles to skin roughness. In practice, roughness can stem from
irregularities on the skin, rivets, or waviness in the skin surface. In addition, steps
4 One drag count equals a drag coefficient variation of 0.0001.
7.3 Laminar-Flow Airfoils
375
in the surface due the presence of a leading-edge high-lift device or an inspection
hatch are also detrimental for laminar boundary layers. Each of them triggers a much
earlier onset of transition, increasing the drag dramatically.
In addition to the higher drag outside of the drag bucket, the NACA 661 212 also
has a lower maximum lift coefficient than the NACA 2412. The sharper nose causes
an early onset of leading-edge stall, with a maximum lift coefficient of 1.45. The nose
of the 2412 has a larger radius and therefore generates a maximum lift coefficient
of 1.7 [1].
7.4 Supercritical Airfoils
When the thrust of engines increased with the commercial introduction of the jet
engine in the 1950s, it quickly became clear that close to the speed of sound the
drag of the airplane increased exponentially (drag divergence). The Mach number
at which drag divergence starts is often defined as the Mach number at which the
numerical value of the slope of the curve of cd versus M is 0.10 [19]:
Mdd =
∂cd
∂M
cl =constant
= 0.10
(7.1)
This exponential increase in drag was attributed to the formation of strong shock
waves on the wing surface in combination with separation at the shock foot. It
was found that the natural laminar flow airfoils produced a significant amount of
wave drag whenever they were exposed to Mach numbers beyond the critical Mach
number (see Fig. 7.6a). This limited the maximum Mach number of early jet fighters
(b) −1.5
−1
c l = 0.7
c l = 0.5
c l = 0.3
−0.5
Cp, crit
p
pressure coefficient, C (~)
−1
p
pressure coefficient, C (~)
(a) −1.5
Cp, crit
−0.5
0
c l = 0.7
c l = 0.5
c l = 0.3
0.5
0
0.5
NASA SC(2) 0412
NACA 661 212
1
1
0
0.2
0.4
0.6
position, x/c (~)
0.8
1
0
0.2
0.4
0.6
0.8
1
position, x/c (~)
Fig. 7.6 Predicted pressure distribution for cl = 0.3, cl = 0.5, and cl = 0.7 at M = 0.75,
Rec = 20 × 106 . a NACA 661 212. b NASA SC(2) 0412
376
7 Airfoil Aerodynamics
that used NACA 6-series airfoil sections. One effort to tackle this problem was
to investigate whether airfoil shapes could be developed that would have a higher
critical Mach number in combination with lower wave drag in supercritical (i.e.
Mlocal > 1) conditions. In the United Kingdom it was H.H. Pearcey that investigated
new airfoil shapes that would allow for supersonic flow on the upper surface of the
airfoil without a strong terminating shock wave [35]. In the Unites States under the
supervision of R.T. Whitcomb a new family of supercritical airfoils was developed
specifically designed to produce low wave drag at high Mach numbers [17]. The
pressure distribution of one of these airfoils is shown in Fig. 7.6b.
If we compare the pressure distributions between the NLF airfoil and the supercritical (SC) airfoil of identical thickness, we see a large difference on both the upper
and lower surface. In the graph for the NLF airfoil we already recognize the presence
of a shock wave at cl = 0.3 (remember that this cl is still the low-speed design range
for this airfoil). We recognize the shock wave by the sharp increase in C p close to
x/c = 0.6. If we look at the SC airfoil we see that the suction over the upper surface
is much lower at cl = 0.3. Most of the lift for this airfoil stems from the difference in
pressure on the aft part of the airfoil. The concave shape of the lower surface causes
an increase in pressure on the lower surface which makes a large contribution to the
total lift coefficient. We call this aft loading and it is one of the means to lower the
required supervelocities on the upper surface of the airfoil for a given lift coefficient
(see Problem 7.2). When the lift coefficient is increased to 0.5 the shock wave on
the NLF airfoil increases in strength (larger pressure rise). On the SC airfoil we also
see the formation of a shock wave around x/c = 0.35. However, the latter shock
wave is much weaker than the one over the NLF airfoil as can be observed from the
smaller pressure increase. Finally, at cl = 0.7 the shock over the NLF airfoil has
shifted forward, while its strength has remained approximately the same. However,
the trailing-edge C p has now become substantially more negative, indicating that
the boundary layer has separated from the trailing edge. On the SC airfoil we see
that the shock has moved aft and has increased in strength. There is no indication of
boundary layer separation.
In Fig. 7.7 the corresponding curves for the drag coefficients are shown. Notice
that the drag coefficient (on the vertical axis) has been expressed in terms of drag
counts. The drag coefficient consists of two fundamental components: the friction
drag coefficient, cd f , and the pressure drag coefficient, cd p . In this analysis the pressure drag coefficient has been found by integrating the pressure over the section and
projecting the resulting force vector onto the direction of the flow.5 If we compare
the drag coefficient development between the two airfoils, we immediately see that
the NLF airfoil shows a sharp increase in drag coefficient between cl = 0.3 and
cl = 0.5. Since the friction drag is almost constant, this increase in drag is almost
solely due to the formation of a strong shock wave on the upper surface. The SC
airfoil shows almost a constant value for the total drag coefficient over this range. At
cl = 0.5 the drag of the SC airfoil is therefore 37 % lower than for the NLF airfoil.
At cl = 0.7 the shock has become significantly stronger, which results in an increase
5 This calculation procedure for obtaining the pressure drag is often called a “near-field analysis”.
7.4 Supercritical Airfoils
377
(a) 180
140
d
df
cd
p
d
120
100
80
60
40
d
c
df
140
c
dp
120
100
80
60
40
20
0
0.3
c
160
c
drag coefficient, c (cts)
d
drag coefficient, c (cts)
160
(b) 180
c
20
0.4
0.5
0.6
0
0.3
0.7
0.4
lift coefficient, c (~)
0.5
0.6
0.7
lift coefficient, c (~)
l
l
Fig. 7.7 Predicted drag coefficients at M = 0.75, Rec = 20 × 106 . Note cd p indicates pressure
drag, while cd f indicates friction drag. a NACA 661 212. b NASA SC(2) 0412
in pressure drag. However, the flow remains attached and the pressure drag does not
rise as fast as for the NLF airfoil.
For a given airfoil shape, the thickness ratio (t/c) and lift coefficient are often
the most important parameters that influence the drag divergence Mach number.
Figure 7.8 shows how for supercritical and NACA airfoils the thickness ratio influences the drag-divergence Mach number. According to this graph, for supercritical
airfoils, the drag-divergence Mach number decreases linearly according to the following statistical relation:
t
for cl = 0.5
Mdd = 0.92 − 1.16
c
0.88
Drag Divergence Mach Number, MDD (-)
Fig. 7.8 Effect of thickness
ratio on drag divergence
Mach number for NACA and
supercritical airfoils (after
Ref. [37])
G-8050-46
c = 0.5
l
0.84
State-of-the-art
Supercritical Airfoils
64A406
0.80
0.76
(7.2)
Conventional
Airfoils
0.72
NA Rockwell
NASA
NACA
Northrop
0.68
66-210
64A211
64A412
65-215
0.64
0.04
0.06
0.08
0.10
0.12
0.14
Thickness Ratio, t/c (-)
0.16
0.18
378
7 Airfoil Aerodynamics
The decrease in drag divergence Mach number can be explained by the fact that the
thicker airfoils create higher supervelocities over the upper and lower surface than
a thin airfoil. A thicker airfoil will therefore encounter critical conditions at a lower
free stream Mach number than a thin airfoil. In general, this also results in an earlier
onset of shock-induced boundary-layer separation and therefore drag divergence.
In Fig. 7.9 we show an example of the effect of lift coefficient on the drag divergence Mach number. The drag divergence Mach number in this graph is defined as
the point on the line where ∂cd /∂ M = 0.10. We can observe that with increasing lift
coefficient the drag divergence Mach number decreases. This can be explained by
the fact that the increase in lift coefficient is a result of higher supervelocities on the
suction side of the airfoil, which causes an earlier formation of the shock wave and
associated shock-induced separation. A rudimentary equation to compute the drag
divergence Mach number that includes both the thickness-to-chord ratio and the lift
coefficient is given by Korn:
Mdd + t/c + 0.10 = κ
(7.3)
where κ is a technology factor that amounts to 0.95 for supercritical airfoil sections.
Torenbeek [44] derived a modified version of Korn’s equation based on empirical
data of second generation supercritical airfoils [17]:
Mdd + t/c + 0.10cl1.5 = M with M = 0.935
(7.4)
It should be emphasized that a different airfoil shape can highly influence the dependency of the drag divergence Mach number to the lift coefficient. It was already
shown by Göthert in 1944 that careful modification of a NACA 0012 could yield a
constant drag divergence Mach number of 0.78 for a lift coefficient ranging from 0
to 0.4 [16]. Blackerby and Johnson show in Ref. [7] that changing the forward 12 %
of the airfoil can have a profound effect on the drag divergence Mach number.
.040
Section drag coefficient, cd (~)
Fig. 7.9 Effect of lift
coefficient on drag
divergence Mach number for
the Lockheed C141A airfoil
at 38.9 % of the semi span
(data from Ref. [7])
.038
d
= 0.10
.036
cl = 0.55
.034
.032
cl = 0.50
.030
cl = 0.45
.028
.026
.70
.72
.74
.76
.78
Mach number, M (~)
.80
7.4 Supercritical Airfoils
379
7.4.1 Shock-Free Supercritical Airfoil
What we now know as a supercritical airfoil was still to be discovered in the early
1960s. In a 1963 paper on the development of low-drag SC airfoils, Pearcey states
the questions that needed to be answered at the time [35]: “Just how much can the
velocity be allowed to exceed sonic locally without incurring strong, drag-producing
shock waves, and how does this in turn depend on the velocity distribution and
section shape?” Keeping in mind that numerical tools such as those employed in
this chapter were unavailable at this time and that the analytical methods to predict
subsonic pressure distributions were inadequate for transonic flow, these questions
could only be answered through a thorough experimental investigation. The goal of
this investigation was to find a section shape that would allow for supersonic flow to
develop over the top surface but without the drag-producing shock wave. One of the
key findings of this investigation was a clear elaboration on the effect of the expansion
waves generated at the leading edge of the airfoil. We will follow this elaboration
(Ref. [35]) to demonstrate the complex interaction of expansion and compression
waves in the supersonic domain of the flow.
We first consider a generic airfoil in high subsonic conditions (Fig. 7.10). The
flow over the airfoil accelerates to beyond M = 1, which introduces a supersonic
flow domain on the upper side of the airfoil. The line separating this supersonic
bubble from the subsonic domain is called the sonic line. This sonic boundary forms
a constant pressure surface and attaches to the airfoil close to the leading edge (at
the point where the flow accelerates locally beyond M = 1). We assume that right
behind this line an expansion wave (left-running Mach wave) is generated by a small
disturbance. This wave travels through the supersonic flow field until it reaches
the sonic line. Since expansion waves cannot travel in the subsonic flow domain
outside the sonic line, it is reflected towards the surface. Because the sonic line is
a constant-pressure surface the expansion wave is reflected as a compression wave
(reflection in unlike sense). The supersonic flow in between the expansion wave
and the compression wave experiences an increment in Mach number (ΔM) due to
Compression
Shock Wave
Expansion
Sonic
boundary
M
- M
M
Fig. 7.10 Notional reflection of a single expansion wave emanating close to the leading edge (after
Ref. [35])
380
7 Airfoil Aerodynamics
the expansion wave. The flow behind the reflected compression wave experiences a
decrement in Mach number of equal strength. The resulting Mach number behind
this compression wave would therefore be equal to the Mach number in front of the
first expansion wave.
When the compression wave reaches the airfoil surface it is being reflected. If the
surface is flat the compression wave is reflected as a compression wave (reflection in
like sense). However, if the surface has sufficient convex curvature, the flow tangency
condition requires the streamline to bend towards the surface and the incident compression wave is reflected as an expansion wave. Alternatively, the surface can have
exactly the right amount of convex curvature such that there is no reflection from the
wall. To decelerate the flow it is beneficial to reduce the convex curvature such that
the reflected wave is also a compression wave. This has been implicitly assumed in
the notional sketch of Fig. 7.10. The flow behind this second compression wave has
a lower Mach number than the flow in front of the first expansion wave (−ΔM in
Fig. 7.10). The compression waves therefore aid in the reduction of the Mach number
further downstream over the airfoil. Because this compression process is isentropic
it is termed isentropic recompression.
In reality the single expansion wave should be replaced by an infinite number of
expansion waves that emanate from the convex surface close to the leading edge and
possibly also further downstream on the airfoil. This expansion fan is reflected from
the sonic boundary as a compression fan. The expansion waves and compression
waves are intersecting. This is visualized in Fig. 7.11 where the expansion waves are
shown as characteristics. We see the expansion characteristics as dashed lines and
the compression characteristics as solid lines. In the lower graph the Mach number
distribution is shown, where ω is the Prandtl-Meyer angle corresponding to the local
Mach number of the flow. In this graph the upper dashed line shows the Mach
number distribution if only the expansion waves would be taken into account. Due to
the reflection of the compression waves from the sonic line and subsequently from
the crest, it can be seen how the Mach number is lowered.
In Ref. [39] it is described that the formation of a shock is the result of a coalescence
of the recompression characteristics. If the reflected compression waves coalesce,
they merge to form a shock wave. If even a short segment of the profile is straightsided (instead of convex) a shock will be formed as the result of a coalescence of
the compression waves reflected from this segment. In Fig. 7.12 we see two airfoils
and their pattern of characteristics. The airfoil in Fig. 7.12a has been designed to be
shock free. In other words, the recompression waves emanating from the sonic line
do not coalesce in the physical plane. This can only happen if the shape of the sonic
line and the flow deflection are carefully tailored to each other [39]. If only a small
disturbance travels downstream along the reflected Mach waves it will result in their
coalescence: a shock (Fig. 7.12b). This shows how carefully the airfoil geometry
must be arranged in order to obtain shock-free supersonic flow.
Pearcey demonstrated experimentally that even subsonic flows that locally accelerate up to M = 1.4 can be decelerated isentropically without the formation of a
shock. To achieve these properties the airfoil had to have the following properties [35]:
7.4 Supercritical Airfoils
381
Sonic
boundary
Expansion from convex surface
First reflected compression
} Subsequent reflections
} Limiting characteristic
and reflection
Free-stream
P
direction
Crest
Local Mach No.
Simple wave expansion,
(i) from sonic
point
(ii) from
crest
MT.E
(Mcrest -1)
Surface distribution
1.0
Fig. 7.11 Sketch of Mach-wave pattern in a region of supersonic flow over an airfoil and the Mach
number distribution resulting from the supersonic position of the first simple wave, ω1 , and the
compressive effect, ω2 (after Ref. [33])
(a)
(b)
Fig. 7.12 Pattern of characteristic waves over two airfoils (after Ref. [39]). a Shock-free. b Coalescence causes shock
(a) a sharp suction peak needs to be present at the leading edge during subsonic flow
conditions at a particular angle of attack
(b) behind the suction peak rapid deceleration has to occur in subsonic flow, and
(c) behind the leading edge the curvature distribution of the upper surface has to be
such that the expansion waves in the supersonic region, the sonic line, and the
reflected compression waves are formed behind the suction peak.
382
7 Airfoil Aerodynamics
(a)
(b)
CP
CP
-1.0
-1.0
CP*
CP*
-0.5
-0.5
0
inviscid theory
0
0.5
0.5
1.0
1.0
x/c
x/c
Fig. 7.13 Pressure distribution about a two different shock-free airfoils. a M = 0.709, α = 3.6
(modified from Ref. [8]). b M = 0.765, cl = 0.58, Rec = 21 × 106 (after Ref. [23])
Due to the sharp suction peak and high adverse pressure gradient in the subsonic
pressure distribution these airfoils are often termed ‘peaky’ airfoils. Pearcey showed
that the significant amount of isentropic recompression could result in shock-free airfoils or airfoils with a weak normal shock wave terminating the supersonic domain.
Other researchers (e.g. [8, 23, 39]) demonstrated the potential and limitations of
shock-free supercritical airfoils. Figure 7.13 shows the pressure distribution for two
of such airfoils. In both graphs the critical C p has been indicated with a dashed line.
We see that both airfoils show a smaller adverse pressure gradient allowing for a
larger supersonic bubble. According to Ref. [32] one can expect that for given values
of lift coefficient and thickness ratio, the design Mach number for shock-free airfoils
will be generally somewhat lower for the peaky designs than for the “nonpeaky”
designs of Fig. 7.13. However, this does not necessarily mean that these nonpeaky
airfoils also have a higher drag divergence Mach number. On the contrary, they have
been shown to be more sensitive to off-design conditions than peaky airfoils.
We have now shown that shock-free airfoils have the possibility of producing
relatively high lift coefficients (around cl = 0.6) without producing any wave drag.
In theory this is an ideal solution. However, in practice these airfoils are hardly used
on high-subsonic transport aircraft. One of the main reasons for this is their sensitivity
to changes in the flow condition. These could stem from a change in Mach number,
angle of attack, or even Reynolds number. This is illustrated for the Garabedian-Korn
airfoil of Fig. 7.13b in Ref. [21] where the airfoil is predicted6 to be shock free for
M = 0.751 and cl = 0.625 while for M = 0.750 and cl = 0.629 it is predicted to
have two shocks. In reality the displacement effect of the boundary layer also plays
6 The prediction was made by assuming the flow to be inviscid (Euler solution).
7.4 Supercritical Airfoils
383
an important role in whether the shock-free condition can be attained. Given the fact
that the displacement thickness is dependent on the Reynolds number it is very well
possible that an airfoil shows to be shock-free in the wind tunnel while having a
weak shock during flight for the same Mach number and lift coefficient.
7.4.2 Supercritical Airfoils with Shocks
The ‘peaky’ airfoil sections of Pearcey proved to be able to produce shock-free
pressure distributions. Even with increases in Mach number and lift coefficient these
airfoils proved to produce relatively weak shock waves. At NASA the work of Pearcey
had not remained unnoticed and under the leadership of Richard Whitcomb a series
of supercritical airfoils were developed (of which an example is shown in Fig. 7.6b).
The goal was to develop an airfoil which would have acceptable drag characteristics
at Mach numbers significantly beyond the critical Mach number. The first airfoil
was a 13.5 % thickness airfoil (NACA 64A series, see Ref. [1]) that exhibited a slot
close to the trailing edge (see Problem 7.5). With a design lift coefficient of 0.65, the
drag-divergence Mach number increased from M = 0.67 for the airfoil without slot
to M = 0.79 for the slotted airfoil [47].
Even though the slotted airfoil performed well in terms of drag rise postponement,
there were a number of disadvantages associated with this concept. It introduced
additional structural complexity in the design as well as increased friction drag.
In addition, it was found that the curvature of the lower surface near the slot was
extremely critical and close to dimensional tolerance [17]. Therefore, work proceeded
to find airfoil shapes that had the same high-speed characteristics as the slotted airfoil
but consisted of a single element. From an aerodynamic point of view the design of
the upper surface was to be such that the compression and expansion waves would
balance each other to get an airfoil with a flat pressure distribution even though there
would be continuous curvature over the upper surface.
There are two primary factors that influence the balance of these waves: the
leading edge radius and the upper surface curvature up to the mid chord. A strong
expansion is required over the leading edge of the airfoil such that these waves can be
reflected back as compression waves. This is done by making the leading edge thicker
(blunter). The leading edge of a typical transonic airfoil has a radius of curvature
which is twice as large as for NACA 6-series airfoils of the same thickness-to-chord
ratio. The rounded nose of the airfoil is beneficial at low speeds too, because it
reduces the suction peak at the nose and, hence, delays stall, thereby improving takeoff and landing characteristics [4]. Secondly, the curvature of the mid-chord region
must be fairly small to reduce the strength of the expansion waves that emanate
from this convex surface (see Fig. 7.11). The curvature should be such that the emanating expansion waves and reflected compression waves are balanced. Hence the
flattened upper surface of many supercritical airfoils. Finally, the lower surface aft
of the mid chord shows distinct cambering to generate the aft loading. This results
in an increased nose-down pitching moment and also reduces the available space for
384
7 Airfoil Aerodynamics
P
sonic line
Q
M<1
M>1
M<1
weak shock wave
G
rounded nose
relatively flat central upper surface
A
boundary layer thickening
F
B
NASA SC(2) 0412
boundary layer edge (not to scale)
concave lower surface to increase local pressure
Fig. 7.14 Typical geometry and characteristics of a supercritical airfoil
the (flap) structure. Figure 7.14 shows the geometric characteristics of a 12 % thick
supercritical airfoil.
In addition to the geometric characteristics, Fig. 7.14 also shows key aerodynamic
properties of a supercritical airfoil. A notional boundary layer thickness has been
drawn for reference. We see that the sonic line, which separates the two flow domains,
forms a continuous boundary around the supersonic bubble. We see that the shock
wave is slightly curved. The tip of the shock (point P) is in the supersonic domain.
The curved part of the shock (oblique with respect to the local flow direction) causes
the flow decelerate to a Mach number higher than unity, causing a small supersonic
region behind this part of the shock. The flow decelerates isentropically to subsonic
conditions in this region. The sonic line therefore attaches to the shock at point Q.
Below point Q the shock is normal to the local flow direction and the flow decelerates
to subsonic conditions. If we move towards the foot of the shock (point F) we see
that the shock is formed by the coalescence of compression waves that smear the
pressure increase over a finite interval. Due to this pressure increase, the boundary
layer thickens (as is explained in Sect. 6.4.3).
Most modern jet transports employ supercritical airfoils or use some form of supercritical geometry in their airfoil design. For example, all the Airbus aircraft employ
supercritical airfoils, as do the latest Boeing aircraft (777, 787). Other Boeing aircraft (e.g. 737-700 and 767) also include airfoils that are modified using supercritical
airfoil technology [4].
Example 7.1 Consider an aircraft that performs an accelerated climb from M = 0.50
to M = 0.77 while the lift coefficient of the aircraft remains constant. At 70 % semispan the wing has a DSMA-523a supercritical airfoil with a thickness-to-chord ratio
7.4 Supercritical Airfoils
385
of 11 %, developed by McDonnel Douglas (see below). To investigate the aerodynamic phenomena at that spanwise location, you are asked to perform a twodimensional aerodynamic analysis for a constant lift coefficient of cl = 0.85. This
includes the pressure distributions over the airfoil at the Mach numbers M = 0.50,
M = 0.73 and M = 0.77 and the corresponding moment coefficients and drag
coefficients.
DSMA−523a
Solution:
First we need to chose a suitable solver for this question. Since we know that the
flow about a supercritical airfoil in high-subsonic conditions will be mixed (both
subsonic and supersonic domains) we choose a numerical approximation of the Euler
equations. In this case we use a finite-volume approximation of the Euler equations
on an unstructured grid. A detailed description of this approximation can be found
in Ref. [9]. Since these calculations are based on the assumption that the flow is
inviscid, we will not be able to deduce any viscous effects such as separation. In
addition, the exact shock position is most likely to be somewhat inaccurate due to
the lacking shock wave-boundary layer interaction (SWBLI). However, it does give
a good insight into the development of the shock wave and its position and gives
a reasonable prediction for high Reynolds-number flows, such as found over highsubsonic transport aircraft. It is therefore found to be a suitable tool to determine
the pressure distributions about this airfoil. The results for the prescribed conditions
are found in Fig. 7.15. Note that these plots show the input values above the graph:
Mach number and angle of attack. The output in terms of aerodynamic coefficients is
displayed within the plot. To keep the lift coefficient constant at each Mach number
we performed several iterations where we changed the angle of attack, until the result
showed a lift coefficient close to the target of cl = 0.85.
Pressure Coefficient, Cp (~)
M = 0.50, α = 2.05deg
M = 0.73, α = 0.25deg
M = 0.77, α = −0.60deg
−2
−2
−2
−1.5
−1.5
−1.5
−1
−1
−0.5
−0.5
0
0
c = 0.854
l
= 0.00336
c
d
cm, c/4 = −0.131
0.5
1
−1
CP*
0
0.5
1
chordwise position, x/c (~)
1
0
c = 0.852
l
= 0.00638
c
d
cm, c/4 = −0.163
0.5
0
CP *
−0.5
0.5
c = 0.857
l
= 0.00384
c
d
cm, c/4 = −0.203
0.5
1
chordwise position, x/c (~)
1
0
0.5
1
chordwise position, x/c (~)
Fig. 7.15 Development of pressure distribution with constant lift coefficient and increasing Mach
number over a DSMA-523a airfoil. Note this is the result of an inviscid calculation, hence cd = cd p
386
7 Airfoil Aerodynamics
Let us analyze the changing pressure distribution of Example 7.1. First we consider
the effect of the increasing Mach number on the pressure distribution over the airfoil
while the lift coefficient remains constant (see Fig. 7.15). In the left graph we see
the expected peaky behavior of the supercritical airfoil at low-speed conditions. The
suction peak at the leading edge is followed by a relatively sharp adverse pressure
gradient, due to the lack of curvature on the top side of the airfoil. We also identify
the aft loading, which is caused by the concave shape of the lower side of the airfoil
over the aft 40 %c. When the freestream Mach number goes beyond the critical
Mach number, a plateau is formed in the pressure distribution. This represents the
supersonic domain of the flow and is terminated by a sharp pressure increase: the
shock wave. As we increase the freestream Mach number even more the shock
wave starts to move backwards towards the trailing edge, while the crest of the
pressure distribution lowers. Notice that by only increasing the Mach number by
0.05, we now have more than 85 % of the top surface emersed in supersonic flow. Also
note that the pressure distribution over the bottom side of the airfoil hardly changes.
The flow is fully subsonic there and we only distinguish a magnification of the
pressure coefficient between M = 0.5 and M = 0.73 due to compressibility effects.
Furthermore, we see that the drag coefficient decreases when the Mach number
increases from 0.73 to 0.77. This large decrease is rather unrealistic and is attributed
to the absence of viscosity in the flow solver. Finally, we would like to point out
the rapid change in moment coefficient about the quarter chord point. In subsonic
conditions we would expect the moment coefficient to stay constant about this point.
It is clear from the values for cm, c/4 that the nose-down pitching moment increases
substantially. This phenomenon is often referred to as Mach tuck or tuck under. To
compensate for this and balance the aircraft around its center-of-gravity, we need to
apply a larger nose-up pitching moment. For a conventionally configured airplane
this is achieved by providing more downforce on the horizontal stabilizer, which
increases the trim drag.
Example 7.2 Consider an aircraft that is flying at a constant Mach number of M =
0.732. The airfoil at 70 % semispan is again the DSMA-523a supercritical airfoil.
A coordinated turn (no side-slip) at constant Mach number and altitude requires a
change in lift coefficient of the wing, and therefore on the wing section. To investigate
the aerodynamic phenomena, you are asked to do a two-dimensional analysis. This
includes the calculation of the pressure distribution about the DSMA-523a airfoil at
a constant Mach number of M = 0.73 and the following lift coefficients: cl = 0.5,
cl = 0.9, and cl = 1.1. In addition, you are asked to calculate the moment coefficient
about the quarter chord and the wave drag coefficient.
Solution:
Having done the previous example (Example 7.1), we use the same numerical approximation of the steady Euler equations, keeping in mind that we assume the flow to be
inviscid. We evaluate the airfoil at the prescribed lift coefficients. However, since the
lift coefficient is an output of the calculation (by summing the pressure difference
7.4 Supercritical Airfoils
387
Pressure Coefficient, Cp (~)
M = 0.73, α = −1.50deg
M = 0.73, α = 0.30deg
M = 0.73, α = 1.00deg
−2
−2
−2
−1.5
−1.5
−1.5
−1
−1
−0.5
Cp
−0.5
0
0
cl = 0.498
c = 0.0018
d
cm, c/4 = −0.178
0.5
1
−1
0
0.5
chordwise position, x/c (~)
1
−0.5
0
cl = 0.889
c = 0.00754
d
cm, c/4 = −0.161
0.5
1
*
0
0.5
cl = 1.08
c = 0.0163
d
cm, c/4 = −0.163
0.5
1
chordwise position, x/c (~)
1
0
0.5
1
chordwise position, x/c (~)
Fig. 7.16 Development of pressure distribution with increasing lift coefficient and constant Mach
number over a DSMA-523a airfoil. Note this is the result of an inviscid calculation, hence cd = cd p
between the suction and pressure surface) we need to do this iteratively by varying
the angle of attack in the input file. The result is displayed in Fig. 7.16.
In the example above we can distinguish the effect that changing lift coefficient
has on the pressure distribution over the wing. We see that for a relatively low lift
coefficient, the main contribution to the lift coefficient comes from the aft loading of
the airfoil, which, in turn, produces quite a large nose-down pitching moment about
the quarter-chord point. When the lift coefficient increases (through the change in
angle of attack), we see that a plateau is formed in the pressure distribution. This
plateau rises and becomes longer when the lift coefficient is even further increased. At
the same time we see that the shock wave becomes stronger, yielding more wave drag
with higher lift coefficient. A quick analysis of the results shows that the average
lift curve slope of this airfoil at M = 0.73 is 14.1(1/rad), which is more than
twice the value of what we expect from this airfoil at low speeds, which is around
2π = 6.28 (1/rad). This demonstrates that in the present conditions this airfoil is
much more responsive to changes in angle of attack, than at low speeds. In practice
this means that atmospheric gusts can have a relatively large impact on the pressure
distribution over the wing and can hence introduce large fluctuations in wing lift. In
the cabin we feel this as if the aircraft is driving on a road dotted with potholes. We
will see in Chap. 8 that the introduction of sweep reduces the lift-curve slope of the
aircraft and, therefore, partially compensates for this large sensitivity.
The results shown in Examples 7.1 and 7.2 are predictions of an inviscid Euler
code. In Fig. 7.17 a Mach and cl sweep is shown about a supercritical airfoil. Both
experimental results obtained in a wind tunnel (Rec = 14.5 × 106 ) and numerical results obtained by an Euler code with boundary layer approximation (viscousinviscid prediction) are shown. If we look at the wind tunnel test results we see
a similar behavior as in the results of the aforementioned examples. With increasing Mach number and constant lift coefficient the shock wave moves aft while the
minimum pressure coefficient decreases somewhat. When the Mach number is kept
388
7 Airfoil Aerodynamics
inviscid calculation:
-2.0
Cp
-2.0
Cp
-2.0
Cp
M =0.499
CL =0.770
-1.5
-1.5
M =0.690
CL =0.927
-1.5
-1.0
-1.0
-1.0
-0.5
-0.5
-0.5
0
20
0.5
40 60
x/c (%)
80
0
0
20
0.5
1.0
40 60
x/c (%)
80
M =0.732
CL =0.898
20
40
60
80
0.5
1.0
1.0
const.
const. CL
-1.0
M =0.732
CL =0.645
-0.5
Cp 0
Mach number sweep
Sym
Ref.
RC x 10
Windtunnel test data
14.5
Douglas / Garabedian
14.5
20
Lift
coefficient
sweep
60
80
-1.0
1.0
-0.5
Cp 0
DSMA 671
40
0.5
-6
0.5
1.0
20
40 60 80
x/c (%)
M =0.732
CL=0.499
Fig. 7.17 Mach number and cl sweep for a 14 % thick supercritical airfoil (after Ref. [22])
constant while the lift coefficient is increased, a shock wave starts to form. The
plateau in the lift distribution rises with increasing lift coefficient, while the shock
position remains almost constant.
Finally, we note that the prediction of the pressure distribution of the viscousinviscid flow solver is quite accurate. Both position of the shock and the supercritical
C p is accurately predicted. We see in the top right graph of Fig. 7.17 an Euler prediction (inviscid) for two conditions: constant α and constant cl . This means that the
equations are being solved respectively for the same α and cl as in the wind tunnel
experiment. We see that the Euler prediction is unable to properly capture the position
of the shock wave. Due to the absence of the displacement thickness of the boundary
layer, the higher effective curvature of the upper side of the airfoil generates a higher
supercritical Mach number which pushes the shock wave more aft. For the same
lift coefficient (constant cl ) and higher Mach number the airfoil is positioned under
a lower angle of attack, resulting in an underprediction of the supercritical Mach
number. For the evaluation at constant angle of attack, the prediction is even more
off: the supercritical speeds are over predicted and the shock wave is too far aft. This
combination results in an over estimation of the lift coefficient. The large discrepancy
between the Euler prediction and the measured data is evidence that the inclusion
of the boundary layer is important to accurately capture the pressure distribution
7.4 Supercritical Airfoils
389
(a) .8
0
Section lift coefficient, cl (~)
.6
-2°
-1°
0°
1°
2°
4°
6°
8°
10°
12°
.4
.2
0
-.2
-.4
NACA
-.6
.2
.4
.3
(b)
Section drag coefficient, cd (~)
.5
.6
.7
.8
.9
1.0
Mach number, M (~)
.06
0
-2°
-1°
0°
1°
2°
4°
6°
.04
.02
NACA
0
.2
.3
.4
.5
.6
.7
.8
.9
1.0
Mach number, M (~)
Fig. 7.18 The measured variation of section properties with Mach number of a NACA 64A010
airfoil for various angles of attack [18]. a Lift coefficient. b Drag coefficient
about supercritical airfoils. The results of Examples 7.1 and 7.2 can therefore not be
perceived as an accurate prediction, but merely serve as examples.
The change in lift and drag coefficient with Mach number is shown in Fig. 7.18
for a modified NACA 6-series airfoil of 10 % thickness. In Fig. 7.18a we observe
that initially the low-subsonic lift coefficient is amplified by the compressibility
effect, in accordance with the compressibility corrections that were introduced in
Chap. 3. However, at higher Mach numbers, the formation of shock waves break the
amplification trend and with ever higher Mach number, shock-induced separation
(see also Sect. 7.6) causes the lift coefficient to drop. At the same time, the drag
coefficient due to the presence of the shock wave and shock-induced separation starts
to rise. With increased angle of attack, the onset of the drag rise occurs at ever lower
Mach numbers, showing that the drag-divergence Mach number is a function of both
390
7 Airfoil Aerodynamics
Mach number and angle of attack. This behavior is not unique to this specific airfoil
but applies to most supercritical airfoils. The higher the required lift coefficient, the
higher the angle of attack needs to be to achieve that. With higher angle of attack,
higher supervelocities on the upper surface of the airfoil are encountered, resulting
in an earlier onset of shock waves and shock-induced separation.
7.4.3 Sonic Rooftop Airfoils
Whereas on supercritical airfoils the supersonic region on the upper surface of the
airfoil is terminated with a weak shock, so-called ‘sonic rooftop’ airfoils lack a
supersonic region all together. The ‘sonic rooftop’ refers to the sonic flow over the
first part of the airfoil, similar to the supersonic region on a supercritical airfoil. A
sonic rooftop airfoil is designed for transonic speeds, but such that the flow at the
design condition just reaches sonic speeds over the flat region of the upper surface.
This implies that a terminating shock wave is absent and wave drag amounts to zero
in the design condition. The drag divergence Mach number for sonic rooftop airfoils
is higher than for supercritical airfoils. However, this does come at an expense.
Sonic rooftop wings are thinner and, consequently, hold less fuel. They also require
a heavier structure to transfer the bending moment in the wing. Furthermore, the
nose radius is smaller which decreases their maximum lift coefficient at low speeds.
In Fig. 7.19 we compare two airfoils of similar but different geometry. One is
designed as an SC airfoil with a design lift coefficient of 0.635, while the second
airfoil is a sonic rooftop airfoil with a design lift coefficient of 0.500. Both airfoils
have a design Mach number of 0.72. Their respective pressure distributions are shown
in Fig. 7.19a, b. The sonic rooftop profile of the pressure distribution is clearly visible
in Fig. 7.19b. If we compare the two drag components of each airfoil we see that the
friction drag is predicted to be the same, while pressure drag of the SC airfoil is
10 drag counts higher than for the sonic rooftop airfoil. The lift to drag ratio of the
SC airfoil in the design condition is cl /cd = 78, while for the sonic rooftop airfoil
this value is estimated to be cl /cd = 70. Looking at Fig. 7.19c we see that the drag
divergence Mach number is higher for the sonic rooftop airfoil than for the SC airfoil.
Furthermore, we distinguish in this figure the relative geometry of both airfoils which
corresponds to the description of the previous paragraph.
7.4.4 Effect of Trailing Edge Geometry
One important characteristic of the pressure distribution over the upper surface of
a supercritical airfoil is the strong adverse pressure gradient that exists over the aft
part of the airfoil. We know from Chap. 6 that the boundary layer is more susceptible
to separate under a strong adverse pressure gradient. Therefore transonic airfoils
are often designed to reduce this adverse pressure gradient by incorporating two
7.4 Supercritical Airfoils
(a) -2
M = 0.72,
c = 0.636,
l
391
= 1.206 deg
cm = -0.082
(b) -2
cd = 0.0027, cd = 0.0054
p
M = 0.72,
= 0.015 deg
c = 0.500, cm = -0.078
l
cd = 0.0017, cd = 0.0054
p
f
f
-1
-1
CP, crit
CP
Cp, crit
Cp
0
0
1
1
(c) c
sonic rooftop: Mdesign = .72 cl, design = 0.5
d
t/c = 0.110
supercritical: Mdesign = .72 cl, design = .635 t/c = 0.126
.012
.010
Mdd
supercritical
.008
sonic rooftop
.6
.7
.75
M
Fig. 7.19 Comparison between a supercritical and a sonic rooftop airfoil of similar but different
geometry (after Ref. [33]). Note cd f indicates friction drag while cd p indicates pressure drag. a
Supercritical airfoil. b Sonic rooftop airfoil. c Drag coefficient versus Mach number
Finite angle
V2
a
V1
At point a: V1 = V2 = 0
Cusp
a
V1
V2
At point a:= V1 V2 0
Fig. 7.20 Trailing edge shapes relating to the Kutta condition (after Ref. [3])
geometric measures: a cusp trailing edge, and a finite thickness of the trailing edge.
We will discuss these two measures subsequently.
In Fig. 7.20 we show two generic trailing edges. The first trailing edge has a finite
angle, while the upper and lower surface of the second trailing edge are parallel
at point a. In the theoretical analysis of incompressible, inviscid flow the Kutta
condition is applicable (see for example Ref. [3]). This simply implies that the flow
smoothly leaves the trailing edge. If we apply this condition to the trailing edge with
finite angle, we see that it can only be satisfied for the upper and lower surface when
the velocities there are zero. This implies a stagnation point (C p = 1) located at
point a. If we consider the cusped trailing edge, we notice that the Kutta condition
can be satisfied while the upper and lower velocity are larger than zero C p < 1.
392
7 Airfoil Aerodynamics
The only condition that applies is that upper and lower velocity are equal. This
implies that C p < 1 for a cusp trailing edge.
In practice we do not have inviscid conditions, and the presence of the boundary
layer allows trailing-edge pressure coefficients to be lower than 1 (as is shown in
many of the figures in this chapter). However, we still see that airfoils with cusped
trailing edges allow for lower trailing edge pressure coefficients without separation
than airfoils with a finite trailing-edge angle. Therefore, cusped trailing edges have
a smaller adverse pressure gradient over the aft part of the airfoil, which postpones
separation. Many supercritical airfoils therefore employ cusped trailing edges (e.g.
SC(2)-0412 and DSMA-523a that have featured in this chapter).
One of the major disadvantages of having a cusped trailing edge is the lack of
physical space in the trailing edge to make a structure. In this region of the wing
often flaps or ailerons are positioned. Due to the diminishing trailing edge thickness, the second moment of area is relatively small. To give the trailing edge its
required stiffness (to prevent aeroelastic deformation) a relatively heavy structure
is required. Obviously, this introduces weight penalties, which eventually translate
into an increase in induced drag (see Problem 7.6). To remedy this problem, aerodynamicists investigated the effect of cusped trailing edges with a finite thickness.
It was shown that blunt trailing edges reduce the adverse pressure gradient on the
upper surface by utilizing off-surface pressure recovery [41]. In practice this means
that the pressure coefficient of the lower surface could be lower than the pressure
coefficient on the upper surface.
Let us try to imagine what the effect of such a blunt trailing edge would be. First
of all, we foresee an increase in base drag: the low pressure a the trailing edge acts
on an aft-facing surface creating additional pressure drag. In subsonic conditions we
therefore expect an increase in drag coefficient for a given lift coefficient (hence,
a reduction of the lift-to-drag ratio). However, with increasing Mach number and
the formation of a shock wave, the boundary layer is exposed to an ever increasing
adverse pressure gradient. Due to the lower adverse pressure gradient of the airfoil
with blunt trailing edge, the momentum thickness will be smaller (see also Example 6.4) reducing the total momentum loss in the boundary layer. So we have two
effects at transonic conditions: an increase in pressure drag due to the blunt base of
the airfoil, and a decrease in pressure drag due to the reduced momentum thickness
at the trailing edge. Which of these two phenomena dominates depends on the relative thickness of the trailing edge compared to the airfoil chord. Investigations at
NASA showed that the optimum trailing-edge thickness varied with the maximum
thickness of the airfoil and was somewhat below 0.7 %c [17]. Figure 7.21 shows the
effect of a blunt trailing edge on the drag coefficient development of an 11 % thick
supercritical airfoil at cl = 0.7. It becomes clear from this figure that at subsonic
conditions the drag coefficient due to the blunt trailing edge is higher, while at transonic conditions (between M = 0.68 and M = 0.79) the blunt trailing edge reduces
the drag coefficient.
7.5 Low-Speed Stall
393
.018
10
y/c (%)
sharp
.016
blunt
-10
drag coefficient, cd (~)
0
10
20
30
40
50
60
70
80
90
100
x/c (%)
.014
.012
.010
(t/c)TE in percent:
0 (sharp)
1.0 (blunt)
.008
.006
.60
.64
.68
.72
.76
.80
.84
Mach number, M (~)
Fig. 7.21 Effect of blunt trailing edge on drag coefficient for an 11 % thick airfoil at cl = 0.7 (after
Ref. [17])
7.5 Low-Speed Stall
When an aircraft goes into a stall, the flow over the wing starts to separate from the
upper wing surface and creates a wake of turbulent air. With increasing angle of attack
this separated region becomes larger and larger, which reduces the rate at which lift
increases. At a particular angle of attack, the rate at which the lift rises with angle of
attack (dL/dα) reduces to zero. At this point the aircraft experiences its maximum
lift coefficient (C L max ). Increasing the angle of attack even further reduces the total
lift of the airplane. At the same time, the wake becomes larger, which increases
the drag. As the lift over the wing decreases, the moment balance on the aircraft
is disturbed. Many aircraft are designed such that the horizontal tailplane provides
a large enough nose down pitching moment to reduce the angle of attack once the
wing stalls. During this motion the airplane starts to lose altitude and increases speed.
While the angle-of-attack decreases, the flow remains separated. At an angle of attack
well below the stall onset angle-of-attack the flow finally re-attaches to the wing and
the airplane recovers from the stall. The latter effect is called stall hysteresis. This
behavior is graphically shown in Fig. 7.22. We see the hysteresis effect on both the
lift coefficient (Fig. 7.22a) and pitching moment coefficient (Fig. 7.22b).
The maximum (trimmed) lift coefficient is an important parameter for airplane
performance. It is directly related to the required take-off field length and the landing
394
7 Airfoil Aerodynamics
(b)
Moment coefficient, Cm (~)
(a)
Lift coefficient, CL (~)
CL, max
CL, max
0
25
Anlge of attack
CL, max
0
(deg)
25
Anlge of attack
(deg)
Fig. 7.22 Typical stall behavior of a passenger aircraft during flight. Data obtained from Ref. [12]
for a Dornier 328. a Lift coefficient. b Moment coefficient
length of the aircraft. Therefore, many aircraft employ high-lift devices to increase
the lift coefficient of the wing such that they can have a lower stall speed and therefore
meet field length requirements. Typical high-lift devices are slotted flaps at the trailing
edge and flaps, slats, or drooped noses at the leading edge. Fighter aircraft also employ
their high-lift devices during combat maneuvers, which often occur in the transonic
speed realm. At any lift coefficient the high-lift devices are automatically positioned
such that maximum lift-to-drag ratio is achieved. This is an important parameter for
the minimum turn radius or maximum turn rate that can be achieved.
Back to the maximum lift coefficient at low speeds and why it is important to treat
this in a text on transonic aerodynamics. In the following subsections we discuss the
various types of stall and the effect of Reynolds number and Mach number on the
maximum lift coefficient of an airfoil. Furthermore, we present the aerodynamics of
high-lift devices. We also show that transonic effects can often be the limiting factor
for the maximum lift coefficient that can be achieved, even at low subsonic speeds.
7.5.1 Qualification of Low-Speed Stall
The mechanism of stall is universal in all cases. Be it low-speed or high-speed, stall
is always caused by a separated boundary layer. To find out why a boundary layer
separates it is instructive to review the reverse effect: attached flow. We know that
at small angles of attack a fluid tends to nicely follow the contour of an airfoil. But
why is this familiar concept so straightforward?
The tendency of a fast moving fluid to be attracted to a nearby surface is called
the Coandă effect. It was named the effect after Henri Coandă. Coandă had become
interested in the phenomenon after identifying that hot turbine air was attracted to
nearby surfaces. The Coandă effect is a result of fluid entrainment. This means that
when a jet of air moves in ambient air the low pressure of the jet pulls ambient air
inwards towards the jet. When a nearby wall obstructs the surrounding fluid to be
7.5 Low-Speed Stall
395
pulled inwards, the jet moves to the body instead. Gasses experience the same effect
around an airfoil. The flow is accelerated due to the presence of the body. Its pressure
reduces (Bernoulli) and the flow ‘sticks’ to the body. The further away from the body,
the lower the velocity increase. This principle has been illustrated in Sect. 6.2.1. The
fact that the flow attaches to the body can therefore be attributed to the lower pressure
of the fluid around it. When this pressure rises due to an adverse pressure gradient in
the streamwise direction, the velocity of the air around the airfoil decreases and the
Coandă effect diminishes. At the same time, the boundary-layer momentum reduces
and reverse flow within the boundary layer marks the beginning of boundary layer
separation (Chap. 6). When the angle of attack increases further, the boundary layer
separates from the surface and becomes a shear layer between a turbulent wake and
the laminar outer flow: the wing has stalled.
We distinguish three types of stall: leading-edge stall, trailing-edge stall, and thinairfoil stall. The latter one is merely an artifact of low-Reynolds number flows and,
contrary to what its name suggests, not only pertains to thin airfoils. At a certain
angle of attack the laminar boundary layer separates due to the high adverse pressure
gradient right after the nose of the airfoil. Further downstream the laminar boundary
layer transitions to a turbulent boundary layer and re-attaches to the surface. The
bubble of reverse flow that is formed between the separation and re-attachment point
is called the laminar separation bubble. When increasing the angle of attack, this
bubble expands downstream. Beyond a certain angle of attack the boundary layer
does not reattach and a turbulent wake is present over the entire aircraft. This means
an instant loss in lift and a large increase in drag. Most aircraft that are designed for
high-subsonic speeds actually fly at higher Reynolds numbers, making thin-airfoil
stall almost never seen in practice. However, scaled versions of these aircraft which
are tested in at a lower Reynolds number in the wind tunnel could develop thin-airfoil
stall. Thin-airfoil stall can be prevented by forcing transition to occur prior to the
point where the boundary-layer separates from the surface.
Leading-edge stall occurs when the boundary layer is not capable of negotiating
the large adverse pressure gradient that follows the suction peak on the nose. Before
the stall angle of attack is reached as small laminar separation bubble forms just aft
of the suction peak. This bubble may span as little as 1 % of the chord length and
forms behind the suction peak under the steep adverse pressure gradient. The separated laminar boundary layer transitions over the bubble to turbulent and reattaches.
Increasing the angle of attack moves this bubble forward and shortens it until the
laminar boundary layer is incapable of reattachment. The result is that flow separates
and that a wake is created starting at the nose of the airfoil (see also Fig. 6.23b).
Alternatively, when a turbulent boundary layer is capable of reattachment it could
still separate at a short distance behind the bubble due to the adverse pressure gradient. When either of these two events happens the suction peak over the leading edge
collapses and a relatively constant pressure coefficient over the airfoil surface results.
In the lift curve of the airfoil we see a step decrease in lift over a very small increment
in angle of attack. A wake over the entire airfoil is created which creates a sudden
increase in drag. In Fig. 7.23 leading-edge stall is shown for a 10 %-thick airfoil.
396
7 Airfoil Aerodynamics
angle of attack, α (deg)
lift coefficient, c l (~)
2.0
-8
-4
0
4
8
12
drag coefficient, cd (~)
16
20
0
.01
.02 .03
.04 .05
1.6
1.2
0.8
0.4
0
-12
-8
Cp
-4
0
Fig. 7.23 Leading edge stall: notional lift and drag curves (top), typical pressure distributions
(middle), and upper-surface stream line (bottom). Data based on Ref. [28] for Re = 4.0 × 106 ,
M = 0.17
Trailing-edge stall, on the other hand, is more gradual. Once the boundary layer
has survived the steep adverse pressure gradient behind the leading edge, it progressively grows thicker and gets a higher shape factor. This means that the momentum
of the boundary layer close to the wall is diminishing. At a certain angle of attack
the boundary layer starts to separate at the trailing edge (usually when H is between
2.4 and 2.6), creating a small wake. Increasing the angle of attack further increases
the adverse pressure gradient over the aft part of the airfoil and moves the separation
point upstream. At the same time a laminar separation bubble could occur closer to
the leading edge of the airfoil. With increasing angle of attack the size of the wake
from the rear separation point increases and the pressure coefficient behind the separation point remains relatively constant. Increasing the angle of attack even further
could merge the two separated regions creating a complete wake over the airfoil.
Figure 7.24 visualizes the interaction between separation, pressure distribution and
lift coefficient.
In the previous discussion we have treated leading-edge and trailing edge stall as if
they were two independent types of stall. In practice often mixed types of stall occur.
In such cases a short laminar separation bubble occurs after which the boundary layer
reattaches. Further downstream the turbulent boundary layer separates indicating
trailing-edge stall. The laminar separation near the leading edge can have a profound
effect on the thickness of the boundary layer as is noted in [30]. The increase in the
7.5 Low-Speed Stall
397
angle of attack, α (deg)
lift coefficient, c l (~)
2.0
-8
-4
0
4
8
12
16
drag coefficient, cd (~)
20
0
.01
.02 .03
.04 .05
1.6
1.2
0.8
0.4
0
-8
Cp -4
0
Fig. 7.24 Trailing edge stall: notional lift and drag curves (top), typical pressure distributions
(middle), and upper-surface stream line (bottom). Data based on Ref. [31] for Re = 6.3 × 106 ,
M = 0.15
extent of an existing bubble increases the initial thickness of the turbulent boundary
layer and, therefore, increases the tendency for the latter to separate. In this way the
laminar separation bubble promotes the onset of turbulent stall near the trailing edge.
If we compare Figs. 7.23 and 7.24 we see that the difference in stall behavior is
a function of the airfoil shape. Important parameters that determine whether leading
or trailing-edge stall occurs include the thickness-to-chord ratio, the leading-edge
radius, and the leading-edge camber. Typically, blunt and thick airfoils are more
likely to suffer from trailing-edge stall, while sharp and thin airfoils suffer from
leading-edge stall. In addition, the Reynolds number plays an important role whether
the leading-edge or trailing-edge stall prevails.
7.5.2 Reynolds Effect on Maximum Lift Coefficient
With increasing wing size and flight speed, the Reynolds number grows. With increasing Reynolds number the (turbulent) boundary layer at a given location on the airfoil
reduces in thickness according to (6.58). In general, increasing the Reynolds number increases the maximum lift coefficient for a given type of stall (leading-edge or
trailing-edge). The thinner boundary layer is capable of negotiating a larger adverse
pressure gradient without separating. There is, however, a more important Reynolds
398
7 Airfoil Aerodynamics
Fig. 7.25 Stall curves for a
NACA 641 -012 airfoil at
various angles of attack
(after Ref. [6])
1.6
trailing-edge stall
l
lift coefficient, c (~)
1.2
leading-edge stall
0.8
thin-airfoil stall
0.4
0
0
Reynolds number:
0.7 million
3.0 million
9.0 million
5
10
15
20
angle of attack, α (deg)
number effect that is instrumental for the maximum lift coefficient: the position of
the transition region. An increase in Reynolds number could also imply a switch
from leading-edge stall to trailing-edge stall. In Fig. 7.25 this is demonstrated for a
moderately thick airfoil, which’ stall behavior is evaluated at three different Reynolds
numbers. Note that this airfoil shows thin-airfoil stall at low Reynolds numbers even
though it is not considered a thin airfoil. This confirms that thin-airfoil stall is not
limited to thin airfoils. We explain the various Reynolds-number-related effects on
stall in the subsequent paragraphs.
As the Reynolds number increases the position where the boundary layer transitions from laminar to turbulent shifts forward (see Sect. 6.5.4). As the region of
transition has a large effect on the boundary layer development over the airfoil it has
an impact on when (at what angle of attack) and where (right after re-attachment
or at the trailing edge) the turbulent boundary layer separates. For a low Reynolds
number the transition point lies downstream of the suction peak. If a small laminar
separation bubble is present the boundary layer transitions from laminar to turbulent over the bubble and reattaches. The reattached boundary layer at the end of the
separation bubble has a significantly higher thickness, which still grows in thickness
downstream of the reattachment point. Increasing the Reynolds number shifts the
transition region forward, which reduces the size of the separation bubble and consequently reduces the thickness of the boundary layer downstream of the bubble. The
downstream boundary layer therefore separates at a higher angle of attack, which
increases the maximum lift coefficient. When the transition region moves in front of
the separation point, the effect of a higher Reynolds number is limited to the thickness decrease of the turbulent boundary layer, which could increase the maximum
lift coefficient even further.
7.5 Low-Speed Stall
1.4
low-drag airfoils
conventional airfoils
max
(~)
1.3
Maximum lift coefficient, CL
Fig. 7.26 Effect of Mach
number on airplane
maximum lift coefficient of
several propeller aircraft
(after Ref. [40]). Note data
not at constant Reynolds
number
399
1.2
1.1
P-63A
1.0
P-51B
YP-80A
0.9
P-39N
0.8
F8F-3
0.7
0.6
0.2
P-38F
0.3
0.4
0.5
0.6
0.7
0.8
Mach number, M (~)
7.5.3 Mach Effect on Maximum Lift Coefficient
The Mach number can have a significant effect on the maximum lift coefficient
that can be attained. Even at low-subsonic conditions this statement holds up. In
Ref. [40] it is shown through a series of flight tests how the airplane maximum
lift coefficient is affected both by Reynolds and by Mach number. In Fig. 7.26 the
maximum lift coefficients of six different propeller-powered aircraft are graphed as
a function of the Mach number. Three aircraft used modified airfoils designed to
promote a laminar boundary layer over a large part of the upper surface during cruise
conditions (the ‘low-drag airfoils’). The other three aircraft relied on NACA 4 and 5
series airfoils (the ‘conventional airfoils’). We see a distinctly different maximum lift
behavior between the airplanes having low-drag airfoils and the airplanes having the
conventional airfoils. In the subcritical domain the uniform lowering of the maximum
lift coefficient is attributed to the steepened adverse pressure gradient resulting from
compressibility effects.
At Mach numbers beyond the critical Mach number the maximum lift coefficient
is governed by the suction peak. In Fig. 7.27 it can be seen that increasing the Mach
number reduces the suction peak for both types of airfoils. In both cases we see
that the suction peak is widening with increasing Mach number. Furthermore, the
low-drag airfoil develops a shock wave beyond M = 0.63. The pressure coefficient
in front of the shock remains below the dash-dotted line, which corresponds to a
maximum local Mach number. The minimum pressure coefficient always appears
to stay below the dashed line, which also corresponds to a maximum local Mach
number. We shall investigate why this happens.
400
7 Airfoil Aerodynamics
NACA 4-series airfoil
NACA 6-series airfoil
-4
Cp
-4
crit
-2
-2
Cp
Cp
M = 0.40
0
0
CL = 1.23
CL = 0.96
max
max
Cp *
M = 2
-4
-4
-2
-2
Cp
Cp
M = 0.50
0
0
CL = 1.06
CL = 0.93
max
Cp
-2
Cp
Cp
max
M =
γ+3
2
crit
0
-2
Cp
M = 0.63
0
CL = 0.77
CL = 1.08
max
max
-2
-2
Cp
Cp
M = 0.68
0
CL = 0.68
max
0
CL = 1.03
max
Fig. 7.27 Typical pressure distributions about NACA 4-series airfoils (left) and NACA 6-series
airfoils (right) at maximum lift coefficient (after Ref. [40])
The same theoretical boundary has been shown in Fig. 6.9 where the dashed
line shows a theoretical limit in local Mach number for supersonic flow over a
convex surface embedded within
subsonic flow. It has been shown that a characteristic
√
Mach number of M ∗ = 2 is the highest Mach number for which Mach waves
can exist for which the curvature is monotonic.7 This corresponds to a situation
where the streamline curvature is still continuous and the flow is isentropic and
7 A monotonic curvature means that the Mach wave is either entirely convex or entirely concave.
If Mach wave curvature is not monotonic it means an inflection point is present somewhere on the
Mach wave.
7.5 Low-Speed Stall
401
uniform [27]. Via Eq. (4.10b) we calculate that this corresponds to a maximum local
Mach number of:
2
≈ 1.58 for γ = 1.4
(7.5)
M=√
3−γ
We can compare this maximum Mach number to the maximum Mach number that
can be achieved in the throat of a supersonic convergent-divergent channel. In that
case the flow is said to be choked. Whatever we do upstream or downstream of the
throat, the Mach number can never exceed M = 1. In our case we have a similar
condition only this time we consider external flow and limit the freestream Mach
number to values below unity. The theoretical value of 1.58 is limited to flow over
convex surfaces that provide monotonic curvature to the Mach waves. The absolute
maximum is attained when discontinuities in streamline curvature are allowed.
In
√
this case the first discontinuity in streamline curvature starts at M ∗ = 2. This
corresponds to theoretical accelerations of infinity at the point of discontinuity. The
theoretical maximum that can be attained is referred to as the Ringleb solution [36]:
M=
4
≈ 2.50
3−γ
(7.6)
For practical cases (7.5) is “the most likely maximum value to be approached in
most cases,” according to Laitone [24]. Substituting (7.5) in (6.17) and rewriting this
for C p yields a theoretical minimum local pressure coefficient as a function of the
freestream Mach number:
γ
γ−1
3−γ
γ−1 2
2
M∞
1+
−1
(7.7)
C p M ∗ =√2 =
γ M∞
γ+1
2
This minimum has been indicated in Fig. 7.27 with the dashed line. Note that the
minimum pressure comes very close to this value but never exceeds it. This confirms
the statement that (7.7) is a practical limit on the minimum pressure coefficient
and that (7.5) is a limiting Mach number. This analysis also demonstrates that even
at speeds well below the speed of sound, transonic effects dictate maximum lift
coefficient.
The theoretically obtained minimum pressure coefficient has been compared to
experimental data obtained from a series of wind tunnel tests by Axelson [5]. As
can be seen in Fig. 7.28 at Mach numbers beyond 0.5, considerably lower pressure
coefficients can be obtained than those predicted by (7.7). An alternative prediction
by Mayer comes closer to the measured values. Mayer based his predictions on a
series of wind tunnel tests between M = 0.4 and M = 2.2 [29]. He proposed a
statistical relation between the minimum value of the pressure coefficient and the
freestream Mach number:
−1
(7.8)
C pmin = 2
M∞
402
7 Airfoil Aerodynamics
leading edge
−3.5
min
(~)
−4
ahead of shock
−3
−2.5
−2
-1
Cp = 2
min M
−1.5
8
Pressure coefficient, Cp
Fig. 7.28 Minimum
pressure coefficients in
transonic flow (after Ref. [5])
−1
Mlim=
2
3−γ
−0.5
Mlim=
γ+3
2
0.5
0.6
0
0.3
0.4
0.7
0.8
8
Free−stream Mach number, M
0.9
(~)
It can be shown (see Problem 7.9) that this results in a static pressure ratio of approximately 0.30 between the suction peak and the free stream. In other words, it is
predicted to achieve 70 % of pure vacuum under the suction peak. At Mach numbers
higher than 0.5 it can be seen from Fig. 7.28 that Mayer’s prediction comes closer to
the measured values published in [5], but still underestimates in certain cases.
If we return to Fig. 7.27 we observe that at M = 0.63 and M = 0.68, a shock
wave is present for the low-drag airfoil. In addition, the dash-dotted line indicates a
theoretical maximum local Mach number that cannot be exceeded in front of a shock
wave without creating separated flow [24]:
M=
γ+3
≈ 1.48 for γ = 1.4
2
(7.9)
In Ref. [24] it is demonstrated that for constant stagnation conditions ahead of the
shock and a Mach number of 1.48 the maximum absolute value of p2 (static pressure
behind the shock) is achieved. In other words, for this limiting Mach number the
maximum magnitude of static pressure recovery is achieved (although this is usually
less than the free stream static pressure). If we imagine the local velocity to increase
beyond this Mach number and form a shock wave further downstream, the static
pressure recovery would be less ( p2 would be lower). Therefore, the higher subsonic
free stream static pressure would push the normal shock wave back upstream until
the maximum absolute static pressure would be produced. Substituting (7.9) in (6.17)
and rewriting this for C p yields a theoretical limiting local pressure coefficient in
front of the normal shock wave:
⎤
⎡
γ
γ−1
2 2
γ−1 2
2 ⎣
1+
M∞
Cp
=
− 1⎦
(7.10)
γ+3
2
M=
γ M∞
γ+1
2
2
7.5 Low-Speed Stall
403
This is confirmed by the graphs for M = 0.63 and M = 0.68 for the NACA
6-series airfoil in Fig. 7.27. We observe that the pressure coefficient is increasing
between the suction peak and the shock. This indicates that the local Mach number
is reducing from a value close to 1.58 to a√lower value in front of the shock.
It is demonstrated in [11] that M = (γ + 3)/2 is the only Mach number for
which a curved streamline can become straight when passing through a normal shock
wave. Consequently, it would be the most likely Mach number in front of a shock
wave whenever boundary layer separation occurs at the foot of the shock. If we
combine these observations we can conclude that on convex profiles the maximum
Mach number in front of a normal shock wave never exceeds 1.48 and that it is
likely that this shock causes boundary layer separation. Evidence of the accuracy of
this predicted pressure coefficient is presented in Fig. 7.28. The experimental data
consistently shows that the value of M = 1.48 ahead of the shock is never exceeded
in any of the experimental tests.
7.5.4 High-Lift Devices
Most high-subsonic aircraft have leading-edge and trailing-edge devices to allow for
a higher lift coefficient and hence a lower stall speed. Many contemporary aircraft
use slats or leading-edge flaps to increase the stall angle of attack, while trailingedge flaps are used to increase the camber of the wing and extend the local chord.
The combined effect can increase the maximum lift coefficient by more than 100 %.
The reader is referred to [33] or [38] for a general discussion on the aerodynamics
of high-lift devices. In this section we limit ourselves to the transonic effects that
accompany the flow over multi-element airfoils.
The idea behind high-lift devices is to increase the maximum lift coefficient.
This can only be achieved if the suction over the upper surface is magnified without
boundary layer separation. The slots between the individual components play a pivotal role to achieve this [38]. However, the same aerodynamic limitations in terms of
Reynolds and Mach number effects, as discussed previously, exist for multi-element
airfoils. Increasing the Reynolds number has a beneficial effect on the maximum lift
coefficient, while the maximum local Mach number limits the minimum pressure
coefficient and therefore the lift coefficient. The main difference with single-element
airfoils is the magnitude of the suction peak. For well-designed multi-element airfoils, the highest suction peak occurs on the slat and can easily exceed C p = −20 at
a theoretical Mach number of zero. As we now know, the transonic effects therefore
occur at much lower freestream conditions sometimes as low as M = 0.2. This
means that the stall speed of an aircraft in landing configuration at sea level, Vs0 ,
could in fact be limited by the maximum supersonic Mach number over the leading
edge of the slat (see Problem 7.10).
Figure 7.29 shows the airfoil of the A320 wing in landing configuration with slat
and flap deployed. The dash-dotted lines indicate the edge of the various boundary
layers that are formed on this multi-element airfoil. We can see how the wake of
404
7 Airfoil Aerodynamics
boundary-layer flow wake flow
slat wake
slat wake
wing wake
wing boundary layer
flap boundary-layer separation
shock-wave-boundarylayer interaction
transition
transition
cove separation
cove separation
supersonic bubble
SLAT
WING
FLAP
Fig. 7.29 A320 airfoil in landing configuration (modified from Ref. [13])
the slat merges with the boundary layer of the wing. This is termed boundary layer
confluence. This happens again over the upper surface of the flap where the wake of
the wing surface merges with the flap boundary layer. In this example, there exists
reversed flow over the flap, which indicates the onset of stall. Over the slat we also
observe a local bubble of supersonic flow that is terminated with a shock wave.
Under the shock wave the flow transitions from laminar to turbulent resulting in a
thickening of the boundary layer. We can imagine how, with increasing Mach number,
the strength of the shock wave increases, which eventually results in boundarylayer separation at the shock foot (see also Sect. 7.6.1). It is this phenomenon that
effectively limits the maximum lift coefficient as a function of the freestream Mach
number.
To substantiate these claims, consider Fig. 7.30. Here we have a typical multielement airfoil consisting of a slat and a double-slotted flap system. The graph shows
the minimum pressure coefficients measured at the suction peak on the leading edge
of the slat. In addition, the statistical limit by Mayer (7.8) as well as the theoretical
limit of (7.7) are shown. Note that the theoretical limit predicts a lower pressure
coefficient than the statistical prediction by Mayer on this low-subsonic domain.
The symbols indicate various combinations of flap and slat deflection. We√clearly
see that the boundary is formed by the limiting Mach number of M = 2/ 3 − γ,
regardless of the flap or slat setting. We also observe a decrement in minimum
pressure coefficient before the local Mach number reaches the limiting value. In the
cases where the slat is deployed the decrement in lift coefficient between M∞ = 0.19
and M∞ = 0.28 for this particular airfoil is between 0.3 and 0.4 [46]. This causes
a 10 % decrease in maximum lift coefficient due to the transonic “choking” of the
flow over the slat leading edge.
7.6 High-Speed Stall
405
δf
δs
1
δf
(~)
−24
Mlim=
−22
Cp
2
3−γ
=
2
-1
δs
δf
−18
10
35
10
−16
10
35
-
-
35
10
-
35
-
-
-
-
min
M
2
8
Pressure coefficient, Cp
min
−26
−20
−14
1
δf
2
−12
−10
0.18
0.2
0.22
0.24
0.26
8
Free−stream Mach number, M
0.28
0.3
(~)
Fig. 7.30 Minimum pressure coefficients at the leading-edge suction peak for Re = 6.9 × 106
(after Ref. [46])
7.6 High-Speed Stall
High-speed stall is often characterized by the onset of aerodynamically induced
vibrations. In general, the term buffet is used for broadband vibrations induced by
pressure fluctuations in the flow. In the present context buffet refers to transonic
buffet. Transonic buffet is primarily caused by separation of the boundary layer
that occurs at the foot of the shock wave [33]. Shock-induced vibrations are typical
to high-subsonic commercial jet transports. There is a relatively weak aeroelastic
interaction because the frequency of the shock-induced vibrations can be several
orders of magnitude beyond the wing natural frequencies. However, buffet has a
profound effect on the maximum lift capability of the wing at high (transonic) Mach
numbers. Airworthiness regulations stipulate that there needs to be a margin of 30 %
between the design-lift coefficient and the lift coefficient at which buffet onset occurs
for any operating Mach number. This margin is defined by a 1.3 g maneuver or a
vertical gust. In practice, this limits the cruise lift coefficient in the design condition.
If we plot the buffet-onset lift coefficient against the Mach number we have the buffet
onset boundary. Figure 7.31 schematically demonstrates the buffet onset boundary
and the required margins in terms of ΔC L and ΔM∞ .
The buffet-onset boundary presents limitations to the maximum lift coefficient of
the airplane at each Mach number. With respect to our very simple Breguét equation (1.1), we see that ideally we would like to maximize the combination of M∞
and C L , while keeping the C D as low as possible. Or, vice versa, reduce the fuel burn
(and consequently the operational cost) for a given range. The buffet boundary of a
given wing could therefore be a constraint on the commercial viability of an aircraft.
406
7 Airfoil Aerodynamics
total separation (stall)
shock
attached flow
Lift coefficient, CL (~)
shock
total separation
CL, max
buffet boundary
boundary for
critical freestream
conditions
ΔCL
CL, max
CL = const.
design point
subsonic
ΔM∞
transonic
supersonic
1.0
Freesetream Mach number, M∞ (~)
Fig. 7.31 Separation boundary at various Mach numbers and lift coefficients (after Ref. [42])
In addition, this combination of Mach number and lift coefficient effectively limits
the altitude an airplane can reach at a given Mach number when flying in steady,
symmetric, horizontal flight (see Problem 7.11).
7.6.1 Flow Separation at the Shock Foot
Transonic buffet is an off-design condition for high-subsonic aircraft. It occurs when
the supervelocities over the top side of the airfoil become relatively large and the
shock wave that terminates the supersonic domain increases in strength. Such large
supervelocities can occur when the aircraft encounters a gust (i.e. an increase in
freestream Mach number) or when it performs a maneuver and requires a higher lift
coefficient. The mechanics of transonic buffet can entirely be related to the shock
boundary layer interaction as discussed in Sect. 6.4.3. If we re-examine Fig. 6.40 we
see the separation bubble under the shock. With increasing shock strength the size
of this bubble increases until it merges with the small separated region close to the
trailing edge.
The shock-induced separation was studied by Pearcey in the early 1950s [34]. He
systematically investigated the effect of flow parameters such as incidence angle (or
angle of attack) and freestream Mach number on the shock-induced separation. In
Fig. 7.32 we see schlieren images of the flow about a 6 % thick airfoil (RAE 104)
under an angle of attack of 2◦ . In the schlieren image the density gradients in the
flow are visualized. The shadow pattern is a light-intensity representation of the
expansions (low density regions) and compressions (high density regions) which
characterize the flow.
7.6 High-Speed Stall
407
Fig. 7.32 Schlieren images of the flow around a 6 % thick RAE 104 airfoil at 2◦ angle of attack
(from Ref. [34]). a M = 0.84. b M = 0.86. c M = 0.88. d M = 0.90
Let us progressively analyze the schlieren images in Fig. 7.32. At M = 0.84 we
see an the shock wave located at around the 60 % chord. The boundary layer is still
attached. At M = 0.86 the shock wave moves aft and its strength grows (the pressure
ratio over the foot of the shock is beyond 1.4). A separation bubble under the shock
occurs. However, the boundary layer remains attached and the trailing edge pressure
is unaffected. Increasing the Mach number to M = 0.88 causes the boundary layer to
fully separate from the shock foot. This decreases the trailing edge pressure substantially. Because the trailing edge pressure on the upper and lower surface are equal
(assuming no off-surface pressure recovery), this also affects the development of the
flow on the lower surface. The lower trailing edge pressure induces higher supervelocities on the lower surface. If the Mach number is increased to M = 0.90 we see
that also on the lower surface a shock has formed. In addition, the severe separation
7 Airfoil Aerodynamics
Section lift coefficient, cl (~)
408
0.6
( p2 )upper surface = psonic
0.5
pTE = ( p2 )lower surface
0.4
a
bc
d
0.3
0.2
0.1
0
0.5
0.6
0.7
0.8
0.9
1.0
Freestream Mach number, M (~)
Fig. 7.33 Development of the section lift coefficient of the 6 % thick RAE 104 airfoil at α = 2
(after Ref. [34]). Note p2 indicates the pressure downstream of the shock, PTE is the trailing edge
pressure, while psonic corresponds to the pressure for sonic flow at points downstream of the shock,
i.e. psonic = 0.528 pt
on the upper surface creates an inclined ramp for the flow outside the boundary layer
resulting in a more oblique shock wave terminating the supersonic domain.
The effect of shock-induced separation on the lift coefficient is shown in Fig. 7.33.
The letters a through d refer to the respective images in Fig. 7.32. We see that the
maximum lift coefficient at a constant angle of attack is reached around M = 0.87.
When the boundary layer fully separates from the surface the lift coefficient starts to
decrease rapidly. Most of this decrease is caused by the increase in suction over the
lower surface due to the decrease in trailing-edge pressure when the boundary layer
has separated. With increasing Mach number, the shock wave on the lower surface
moves faster towards the trailing edge than the upper shock wave creating more
suction over the lower surface and hence reducing the lift coefficient. A minimum
is obtained when the lower shock reaches the trailing edge. Increasing the Mach
number even further results in a higher value of the lift coefficient that corresponds
with both shocks having reached the trailing edge.
In Fig. 7.34 we schematically repeat this development of the shock pattern over
a generic airfoil of low thickness. The same effects as in the wind tunnel are seen
as can be verified by comparison to Fig. 7.32. In Fig. 7.34 the strength of the shock
has been indicated by the thickness of the shock line. Note how the shock strength
is highest at the surface, while its strength decrease further away from the body.
Intuitively, we can verify this because the supervelocities are highest close to the
surface. Furthermore, the relative size of the shock wave increases with higher Mach
number. At M = 1.4 the shocks at leading and trailing edge even exceed the bounds
of this figure. Finally, the notional point of separation is shown along with the growth
of the wake over the transonic regime. Note how the wake grows up to M = 0.9 and
decreases again above that value.
409
Section lift coefficient, cl (~)
7.6 High-Speed Stall
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.6
Freestream Mach number, M (~)
Fig. 7.34 Notional development of section lift coefficient over a symmetric airfoil at α = 2
Fig. 7.35 Schlieren images of the flow around a 6 % thick RAE 104 airfoil at M = 0.75 (from Ref.
[34]). a α = 2.7◦ . b α = 3.7◦ . c α = 4.7◦ . d α = 5.7◦
410
7 Airfoil Aerodynamics
Let us follow Pearcey [34] in his investigation into the behavior of shock-induced
separation at constant Mach number and increasing angle of attack. In Fig. 7.35 we
see schlieren images of the same airfoil as in Fig. 7.32. The Mach number has been
fixed at M = 0.75 while the angle of attack increases progressively from α = 2.7◦
through α = 5.7◦ . It is important to note that the transition has been artificially
induced at the nose and that the shock wave interacts with a turbulent boundary
layer at any of the aforementioned angles of attack. We see that somewhere between
α = 2.7 and α = 3.7 the boundary starts to separate at the shock foot. The shock is
still quite close to the leading edge and the boundary layer reattaches downstream.
With increasing angle of attack the shock moves aft and the separation bubble expands
rapidly towards the trailing edge. This causes the trailing edge pressure to decrease
and the shock to halt before it starts to move forward. At even higher angles of attack
the shock moves further forward together with the separation point.
In Fig. 7.36 the position of the shock foot is plotted for a range of angles of attack
and Mach numbers. We distinguish uniform behavior at the various Mach numbers:
initially the shock moves aft up to the point that the separation bubble reaches the
trailing edge. Beyond that angle of attack the shock wave and the separation point
move forward. With decreasing Mach number the angle of attack at which shockinduced separation starts becomes progressively higher. Simultaneously, the shock
position at which separation starts becomes closer to the leading edge. It becomes
difficult to judge at what Mach number the separation ceases to be induced by the
shock instead of being induced by the adverse pressure gradient behind the suction
peak, as seen at low speeds.
Let us take a look at the effect of shock-induced separation on the lift coefficient.
In Fig. 7.37 we see the lift coefficient development over the RAE 104 airfoil at two
Mach numbers. In 7.37a we see that lift coefficient remains close-to constant beyond
the point where the separation bubble reaches the trailing edge. For M = 0.80 we
expect the separation to start at a lower lift coefficient. This is indeed the case as
can be seen in Fig. 7.37b. However, here we have somewhat of a different case.
xsh
(~)
c
Locus of points for the
divergence of pTE
1.0
Upper-surface shock position,
Fig. 7.36 Shock position as
a function of angle of attack
for a 6 % thick RAE 104
airfoil with fixed transition at
the nose (after Ref. [34])
0.8
M
0.95
0.6
0.90
0.4
0.85
0.2
0.80
0.70
0.75
6
8
Angle of attack,
(deg)
0
0
2
4
10
7.6 High-Speed Stall
411
(a)
(b) 1.4
1.2
Section lift coefficient, cl (~)
RAE 104
Section lift coefficient, cl (~)
1.0
0.8
0.6
0.4
0.2
1.0
0.8
0.6
0.4
0.2
0
-0.2
0
2
4
6
Angle of attack,
8
(deg)
10
0
2
4
6
Angle of attack,
8
10
(deg)
Fig. 7.37 Lift versus angle of attack for a 6 % thick RAE 104 airfoil at two transonic Mach numbers
(after Ref. [34]). a M = 0.70. b M = 0.80
The lift coefficient increases even when the shock-induced separation covers the
upper surface from shock foot to trailing edge. Such behavior is attributed to offsurface pressure recovery and occurs beyond the angle of attack where the pressure
coefficient at the trailing edge falls below the critical pressure coefficient. When
this is the case, the pressure coefficient at the trailing edge has no effect anymore
on the development of the flow over the lower surface. With increasing angles of
attack, the pressure coefficient on the lower surface begins to rise again. At the same
time, the pressure coefficient on the upper surface decreases even further below the
critical pressure coefficient allowing the shock wave to move backwards resulting in
an increase in suction over the upper surface. The combined effect is an increase in
lift coefficient beyond the angle of attack of shock-induced separation. This effect
is dependent on the presence of off-surface pressure recovery and therefore partially
on the shape of the airfoil trailing edge.
7.6.2 Transonic Buffet
In the previous section we have examined the effect of separation at the shock foot
at transonic speeds. We know from experience with low speed stall that separation
and the associated wake cause pressure fluctuations that are transmitted to the airfoil.
These fluctuations in pressure can be felt as structural vibration. Apart from these
fluctuations the separated flow is considered to be steady (e.g. invariant with time). In
transonic conditions, however, the separated boundary layer can also cause a periodic
motion of the shock wave over the surface of the airfoil. Such fluctuations usually
have a frequency on order of 102 Hz and generate large fluctuations in pressure (see
Fig. 7.38). These large fluctuations can be felt as a strong vibration in the airframe
412
7 Airfoil Aerodynamics
(a)
(b)
2.0
2.0
-Cp
1.5
1.5
1.0
1.0
.5
-Cp
.5
0
= 3.5o
-.6
M∞ = 0.723
x/c
R = 4.2·106
.2
= 3.5o
-.6
M∞ = 0.723
R = 4.2·106
-1.0
-1.0
0
0
.4
.6
.8
1.0
0
.2
x/c
.4
.6
.8
1.0
Fig. 7.38 RA16SC1 airfoil. Envelope of the pressure distributions at buffet onset (after Ref. [25]).
a Calculation. b Experiment
(a)
(b) Strong shock induces
Vs
seperation
M1
M
M
M1
Vs
Wake deflected
upwards
Weak shock
(attached flow)
(c)
(d)
Vs
M1>1.3
Strong shock
Weak shock
M
M
Wake
M1>1.3
Vs
Fig. 7.39 Shock wave development over a 14 % thick circular arc airfoil with increasing subsonic
freestream Mach number (after Ref. [14]). a M1 < 1.2 (no separation). b M1 > 1.2 start periodic
flow. c 1/2 cycle after (b). d M1 > 1.3 shock too strong for reattachment
structure. This phenomenon is termed transonic buffet and its mechanics is further
explained in this section.
Before we look at a realistic case of transonic buffet on a supercritical airfoil
it is instructive to consider the shock motion over a circular arc airfoil. The shock
development over a 14 % thick circular arc airfoil at symmetric flow conditions
is presented in Fig. 7.39. In Fig. 7.39a we see shocks of equal strength appearing at
both the upper and lower surface. In Fig. 7.39b we assume that a pressure disturbance
upstream of the shock on the upper surface causes the shock to move more forward.
The instantaneous shock Mach number (Ms ) is given by:
Ms = M1 +
1 dxs
a dt
(7.11)
7.6 High-Speed Stall
413
where xs is the location of the shock and dxs /dt = Vs is the velocity of the shock
wave with respect to the airfoil. As the shock moves forward Ms increases beyond the
Mach number at which separation is triggered. So as the shock moves forward it also
shifts the separation point forward creating an asymmetric wake. At the same time, the
trailing edge pressure decreases which moves the shock on the lower surface closer
to the trailing edge. In moving backward, Ms decreases which prevents the boundary
layer from separating. As the upper shock moves forward over the convex surface it
encounters a slower supersonic region. Therefore, its strength weakens as it moves
upstream. The flow attaches at a point where the shock becomes too weak to initiate
separation. On the lower side the reverse is happening. As the shock moves closer
to the trailing edge its strength is increasing and at some point it initiates separation.
This causes the wake to move to deflect downward as can be seen in Fig. 7.39c. This
process is now repeating itself on the lower surface. The deflected wake essentially
causes the shock on the lower surface to move forward, while the shock on the
upper surface starts to move backward. Even though there is a time lag between the
downward deflection of the wake and the motion of the shock wave, the flow can
maintain a self-sustained periodic motion. Finally, when M1 > 1.3 (Fig. 7.39d) the
shock is strong enough at all times to cause separation. No reattachment can take
place and the periodic motion of the shock wave ceases.
Close to the surface of the airfoil we can approximate the flow as one dimensional.
We follow Ref. [26] to relate the oscillating shock position to the oscillating strength
of the shock wave. The latter one is given as the ratio of static pressure over the
shock wave. We denote the properties of the flow ahead of the shock wave with a
subscript 1 and behind the shock with subscript 2. The shock strength would then
be quantified as p2 / p1 . We assume that the flow has a varying velocity, u, a varying
static pressure, p, and a varying speed of sound, a, in the streamwise (x) direction.
The displacement of the shock in time is given by Δx. The pressure and velocity in
front and behind the shock can be written as follows:
dP1
Δx + p̃1
dx
p2 = P2 + p̃2
dU1
Δx
u 1 = U1 − u s + ũ 1 +
dx
u 2 = U2 − u s + ũ 2
p1 = P1 +
(7.12)
(7.13)
(7.14)
(7.15)
where p̃ and ũ are perturbation pressures and velocities, respectively, and P and U
are the steady-state pressures and velocities, respectively. We know that the pressure
ratio over the shock wave is given by the Rankine-Hugoniot relation:
2γ
p2
=1+
p1
γ+1
u 21
a12
−1
(4.14)
Substitution of (7.14) yields a long expression of p2 / p1 as a function of each of the
velocity components in (7.14). We omit this expression for brevity. We assume that
414
7 Airfoil Aerodynamics
1
the velocities of ũ, u s , and dU
dx Δx are an order of magnitude smaller than U1 . We
therefore only retain the multiples of U1 leading to the following expression:
ũ 1
2γ
us
dU1 Δx
p2
2
M1 − 1 + 2M1
=1+
−
+
p1
γ+1
a1
a1
dx a1
(7.16)
where we substituted M1 = U1 /a1 . We identify the steady-state part of the RankineHugoniot equation with an additional term between brackets encompassing the velocity terms introduced by the shock motion. We rewrite (7.16) therefore accordingly:
P2
4γ
p2
M1
=
+
p1
P1
γ+1
dM1
us
ũ 1
Δx −
+
dx
a1
a1
(7.17)
where P2 /P1 can be obtained by employing (4.14) for u 1 = U1 . If we assume
that the shock oscillates periodically over a distance Δx0 we can write the shock
displacement, velocity and perturbation quantities as follows:
Δx = Δx0 eiωt , u s = iωΔx0 eiωt , ũ 1 = ū 1 eiωt
(7.18)
where ū 1 is the time average of the perturbation velocities ahead of the shock. Substituting (7.18) in (7.17) yields an expression for the time dependency of the shock
strength:
dM1
iω
P2
4γ
ū 1
p2
Δx0 eiωt
M1
−
=
+
+
(7.19)
p1
P1
γ+1
dx
a1
a1
If we assume that the disturbances in the supersonic region upstream of the shock
are negligible (i.e. ũ 1 ≈ 0) the relation reduces to:
P2
4γ
p2
M1
=
+
p1
P1
γ+1
dM1
iω
Δx0 eiωt
−
dx
a1
(7.20)
Let us try to interpret the contents of (7.20). On the RHS it requires the input of the
shock oscillation: amplitude (Δx0 ) and frequency (ω in rad/s). In addition, it requires
an estimate for the change in upstream Mach number with displacement (dM1 /dx)
at the neutral position of the shock. Finally, it requires an estimate of the steady state
properties of the shock: P2 /P1 and M1 .
Example 7.3 Assume we have a NACA 64A006 airfoil at M = 0.90 which has an
induced shock oscillation (this experiment is described in Ref. [43]). The steady-state
Mach number in front of the shock has been determined from quasi-steady conditions
and is M1 = 1.18 and dM1 /dx = 1.7. A periodic shock motion is induced with a
frequency of 120 Hz and an amplitude of Δx0 /c = 0.05. The static temperature in
front of the shock wave is 290 K. Calculate the following:
(a) The steady-state pressure ratio, P2 /P1
(b) The amplitude of the pressure ratio
(c) The phase shift in radians between pressure jump and shock position.
7.6 High-Speed Stall
415
Solution:
To calculate the steady shock jump in static pressure we employ (4.14):
P2
2γ 2
M1 − 1 = 1.46
=1+
P1
γ+1
This means that the time-averaged pressure ratio is 1.46 over this shock. To find the
amplitude of the pressure ratio we employ part of (7.20):
4γ
M1
amplitude =
γ+1
2
dM1
dx
−
iω
a1
2
Δx0 = 0.380
Finally, we determine the face shift between the shock position and the pressure
jump. We plot one period for the shock position and pressure jump, respectively, in
Fig. 7.40. We see that the pressure jump is lagging the shock position. We calculate
the phase shift, ϕ, as follows:
Shock displacement,
xs/c (~)
ϕ = tan
−1
ω dx
a1 dM1
= 0.91 (rad)
0.05
0
−0.05
0
1
2
3
4
5
6
7
8
6
7
8
Shock pressure jump, p2/p1 (~)
Time, t (ms)
2
1.8
1.6
1.4
1.2
1
0
1
2
3
4
5
Time, t (ms)
Fig. 7.40 Oscillating shock position and corresponding pressure jump according to (7.18) and
(7.20), respectively
416
7 Airfoil Aerodynamics
The phase shift between shock position and pressure jump amounts to 1.21 ms.
For clarity, the oscillating shock position and oscillating shock jump are shown in
Fig. 7.40.
In the previous example it becomes clear that the shock pressure jump periodically
alters between p2 / p1 = 1.83 and p2 / p1 = 1.09. With the knowledge in mind that
the pressure magnitude of the pressure jump is dominant for boundary layer separation, we can deduce from this simple example that when the shock is weakest it is
conceivable that the boundary layer remains attached. Vice versa, it also conceivable
that when the shock strength passes a certain threshold the boundary layer fully separates. Pearcey suggested separation to occur when p2 / p1 > 1.4. The limiting Mach
number of (7.9) predicts separation to occur at p2 / p1 = γ + 1 (see Problem 7.13).
The shock motion in the previous example was induced by deflecting a control
surface (c f = 0.25c) with an amplitude of 1◦ . The deflection of the control surface
caused an oscillation in the wake and an oscillation in the back pressure of the shock
wave. When separated and attached flow follow each other rapidly we see similar
behavior: an oscillating wake causing an oscillating shock wave (see also Fig. 7.39).
The amplitude and phase shift of the pressure jump are largely dependent on the
sensitivity of the upstream Mach number (Ms ) to the shock position as well as on the
frequency, f . It can be shown (see Problem 7.14) that when dM1 /dx → 0 the phase
shift tends to π/2 or 90◦ . Furthermore, when the frequency increases the amplitude
of the shock decreases as is demonstrated in Ref. [43]. This results in a frequency at
which the amplitude of the static pressure jump is maximum (see Problem 7.15).
Example 7.4 One of the prime parameters responsible for the phase shift is the
value of dM1 /dx. Calculate this value for an SC(2) 0412 supercritical airfoil around
M = 0.775 for cl = 0.5 at a Reynolds number of 20 million.
Solution:
A reasonable quick way to investigate the change of Mach number ahead of the shock
with the position of the shock is to evaluate the pressure distribution. This has been
plotted for three values of the Mach number (M = 0.77, M = 0.775, and M = 0.78)
in Fig. 7.41. We see how the small changes in Mach number have a relatively large
effect on the shock position. From the jump in pressure coefficient we can calculate
the Mach number ahead of the shock wave. First we rewrite the pressure jump in
terms of the pressure coefficient using (3.11):
2 C +2
γ M∞
p2
p2
=
2 C +2
p1
γ M∞
p1
Furthermore, we rewrite (4.14) to calculate M1 :
M1 =
γ+1
2γ
p2
−1 +1
p1
7.6 High-Speed Stall
417
Fig. 7.41 Steady solution
for the pressure distribution
about an SC(2) 0412 airfoil
at M = 0.77, M = 0.775,
and M = 0.78
−1
p
pressure coefficient, C (~)
−1.5
−0.5
0
0.5
1
0
0.2
0.4
0.6
0.8
1
position, x/c (~ )
Table 7.1 Change in shock position with Mach number
M
C p1
C p2
p2 / p1
0.77
0.775
0.78
−0.935
−0.962
−0.987
−0.418
−0.357
−0.299
1.35
1.43
1.50
M1
xs /c
1.140
1.169
1.197
0.558
0.596
0.627
We tabulate the values for the pressure coefficient, pressure jump, M1 , and the shock
position at each Mach number (Table 7.1).
Based on the values of M1 and xs /c at the three conditions we make an estimate
for dM1 /dx according to the following formula:
1 1.197 − 1.169 1.169 − 1.140
0.747 + 0.916
dM1
≈
+
= 0.831
=
d(x/c)
2 0.627 − 0.596 0.596 − 0.558
2
In the previous example the Mach number in front of the shock wave was still
relatively low (≈1.17). Separation would therefore be unlikely. However, the example
does demonstrate how one could calculate the change in shock strength with its
position. With modern CFD tools unsteady separation from the shock foot can be
predicted. In Fig. 7.42 the estimated flow field about a supercritical airfoil is shown
at four instances within one oscillation. A numerical approximation of the Unsteady
Reynolds-Averaged Navier-Stokes equations is used for these predictions. Errors on
the frequency prediction are between 9 and 17 %, which demonstrates that there is
still room for improvement. However, the numerical simulations do give us insight
into the mechanism of the shock-boundary layer interaction.
The shock motion on the upper surface of the supercritical airfoil of Fig. 7.42 is
slightly different from the case of the biconvex circular arc airfoil of Fig. 7.39. In this
case, we only have a single shock wave on the upper surface and no shock is formed
418
7 Airfoil Aerodynamics
(a)
Mach number, M 0.1 0.2 0.4 0.5 0.7 0.8 1.0 1.1 1.3 1.4
(b)
Mach number, M 0.1 0.2 0.4 0.5 0.7 0.8 1.0 1.1 1.3 1.4
(c)
Mach number, M 0.1 0.2 0.4 0.5 0.7 0.8 1.0 1.1 1.3 1.4
(d)
Mach number, M 0.1 0.2 0.4 0.5 0.7 0.8 1.0 1.1 1.3 1.4
Fig. 7.42 Mach number and flow streamline predictions for the RA16SC1 airfoil at developed
buffet conditions, M∞ = 0.732, α = 4◦ , and Re = 107 (from Ref. [20]; printed with permission
of the authors). a Most downstream position of the shock. b Middle position as the shock is
moving upstream. c Most upstream position of the shock. d Middle position as the shock is moving
downstream
on the lower surface during the cycle. The onset of buffet is shown to start after the
onset of flow separation. In other words, the flow has already fully separated from the
shock foot and oscillation only starts when the angle of attack is further increased.
7.6 High-Speed Stall
419
As the separated region grows, this high-pressure bubble pushes the shock upstream.
When the shock moves upstream it becomes stronger due to a combination of effects.
We have discussed the effect of a moving shock on the incident Mach number above.
Iovnovich and Raveh [20] call this the dynamic effect. In addition, they mention
the wedge effect, which alludes to the shock becoming more oblique due to the
formation of a wedge-like separation region at its foot. The shock strengthens as
it moves forward as is evidenced by the increase in inclination angle (Fig. 7.42b).
Finally, as the shock moves upstream and encounters more curvature (closer to the
leading edge) the region of expansion reduces and the shock strength weakens. In [20]
this is referred to as the curvature effect. In this example, the boundary layer does
not fully reattach at its most forward position (see Fig. 7.42c). Because the Kutta
condition is not satisfied, the circulation around the airfoil decreases substantially
leading to lower accelerations ahead of the shock. This is another contribution to
further weakening of the shock wave. When the flow reattaches the shock moves
downstream until separation starts again and the cycle is repeated.
7.7 Summary
In the quest for higher speeds in the late 1940s and early 1950s the wing aerodynamics
proved to be the limiting factor. The increase in wave drag that they produced at
Mach numbers close to the speed of sound proved to be prohibitive to achieve the
desired top speeds. This sparked research into new airfoil shapes that would yield a
comparatively low drag coefficient at the specified lift coefficient and Mach number.
Originally, laminar-flow airfoils were proposed for this purpose. Their favorable
pressure gradient over the first half of the airfoil allowed for a laminar boundary
layer up to 50 or 60 % of the airfoil upper surface. Even though these airfoils proved
to be suited for subsonic application, they developed strong shock waves at relatively
low Mach numbers resulting in excessive wave drag. New airfoils were developed
that could decelerate the flow over the upper surface without producing a strong
shock. The supercritical airfoil was born.
Shock-free supercritical airfoils are designed to isentropically decelerate the
supersonic flow on the upper surface of the airfoil to subsonic conditions. The local
Mach number on the upper surface is dominated by the interaction of compression
and expansion waves. This interaction is highly dependent on the shape of the sonic
line separating the supersonic bubble from the subsonic outer flow as well as on the
leading edge radius and the upper surface curvature of the airfoil. A slight change
in either of these parameters can cause the compression waves to coalesce and form
a shock wave. To avoid this sensitivity, many supercritical airfoils are designed to
allow for a (weak) shock wave to terminate the supersonic bubble. The pressure
distribution about these airfoils are characterized by a flat plateau over the first half
of the airfoil, followed by a sharp drop (normal shock) and a steep adverse pressure
gradient towards the trailing edge. The lower surface often has increased camber
420
7 Airfoil Aerodynamics
near the trailing edge. This increases the pressure on the aft part of the lower surface
contributing to a higher lift coefficient for a given angle of attack. A cusped trailing
edge is often used to allow for off-surface pressure recovery and a reduction of the
adverse pressure gradient preventing trailing-edge stall. In addition, a finite trailing edge thickness has been demonstrated to have a similar effect. Both measures
contribute to lower profile drag at the design condition of the airfoil.
The stall of airfoils starts with the separation of the boundary layer. In low speed
conditions it has been shown that a limiting Mach number of 1.58 exists on the top
surface of the airfoil that effectively “chokes” the flow over continuously convex
profiles in conditions where M∞ < 1. This effectively limits the maximum lift
coefficient that can be achieved by profiles in subsonic conditions. This Mach number
limit is extended to flows over multi-element airfoils (employing flaps and slats). The
limiting Mach number can reduce the achievable lift coefficient at Mach numbers as
low as 0.25. In addition, a theoretical maximum Mach number of 1.48 exists in front
of a normal shock wave. At this Mach number the highest static pressure behind the
shock can be achieved for given stagnation conditions. Boundary layer separation is
likely to occur at values somewhat below this Mach number. The relatively strong
shock causes the boundary layer to separate at its foot. If the resulting separation
bubble merges with the separated boundary layer from the trailing edge we have
high-speed stall. The interaction between separated boundary layer and shock wave
can cause shock oscillation. The resulting high-frequency fluctuation in pressure
distribution causes vibrations. These vibrations are referred to as transonic buffet.
The buffet boundary forms a limiting line in the Mach-C L space of a wing that
effectively limits the achievable lift coefficient at a specified Mach number.
Problems
Supercritical Airfoils
7.1 Calculate the increase in drag divergence Mach number when we reduce a 10 %
thick supercritical airfoil down to 8 % for the same lift coefficient of cl = 0.5.
7.2 Explain why introducing aft-loading on an airfoil can reduce the required supervelocities on the upper surface of the airfoil for a given lift coefficient.
7.3 In graph below the relationship between the drag coefficient and the Mach number of two aircraft are shown: the DC-10-30, and the MD-11, which was the successor
of the DC-10. For each lift coefficient the graphs for both aircraft display an exponential increase at the so-called drag-divergence Mach number. Although the drag
divergence Mach numbers are similar for both aircraft, the DC-10 suffers from drag
creep (a steady increase of the drag coefficient with Mach number) while the MD-11
does not.
7.7 Summary
421
CD x 104
DC-10-30
MD-11
300
CL = 0.5
250
CL = 0.4
CL = 0.3
200
0.60
0.70
0.80
0.90
Mach number, M (~)
Modified from Ref.[33]
(a) Explain the aerodynamic cause for drag divergence.
(b) Explain the aerodynamic cause for drag creep.
(c) Explain how the addition of aft loading on the MD-11 (the DC-10 did not have
any) resulted in less drag for a lift coefficient of 0.5.
7.4 (a) What are the differences between a modern supercritical airfoil and a
so-called “sonic rooftop” profile in an aerodynamic sense? Refer in your answer
to the critical pressure coefficient.
(b) Indicate the differences in a sketch, drawing the two profiles on top of each other.
7.5 Below the first supercritical airfoil with slot is shown (from Ref. [47]). Can
you explain why the addition of a slot resulted in an increase in the drag-rise Mach
number?
7.6 Thin trailing edges require a heavier rib structure to be properly supported.
Argue why such an increase in structural weight can eventually lead to an increment
in (induced) drag.
7.7 The lower rear surface of the wing of the Airbus A340 exhibits a strong camber
(see figure below). What are its specific advantages and disadvantages?
422
7 Airfoil Aerodynamics
Photo R.Slingerland
7.8 The Lockheed P-38 Lightning (below) was in many ways innovative. An unfavorable characteristic was that in straight high-speed dives the aircraft experienced
a nose-down pitching tendency (termed tuck-under or Mach tuck) above M = 0.65.
This tendency increased the dive angle and thereby the speed even more, leading to
structural break-up of the aircraft. Only after the addition of dive recovery flaps this
problem was solved. Explain why the nose had the tendency to drop at these Mach
numbers.
Photo USAF
Low-Speed Stall
7.9 Show that the Mayer limitation on the minimum pressure distribution corresponds to a static pressure ratio of pmin / p∞ ≈ 0.30.
7.7 Summary
423
7.10 Consider an airplane with a 1-g stall speed of 66 m/s at 0 m ISA conditions.
This stall speed is achieved by deploying slats and flaps. Assume that the suction
peak over the slat amounts to C p = −24.
(a) Calculate the stall Mach number at 0 m ISA conditions.
(b) Calculate the minimum allowed pressure coefficient at 0 m ISA using (7.7).
(c) Calculate the stall speed at 1,655 m and Mach number at 1,655 m altitude.
Assume the lift coefficient and wing loading are the same at both altitudes.
(d) Calculate the minimum allowed pressure coefficient at 1,655 m altitude.
(e) Based on your calculations, do you think the maximum lift coefficient at 1,655 m
is still the same as at sea level?
High-Speed Stall
7.11 Prove that for an airplane in steady, symmetric, and horizontal flight the buffetonset boundary essentially limits its maximum operational altitude.
7.12 An airplane is cruising at its maximum cruise lift coefficient in steady horizontal
flight at a speed of 250 m/s.
(a) Calculate the smallest turn radius it should be able to attain without encountering
buffet at n = 1.3.
(b) Calculate the associated bank angle to reach n = 1.3
7.13 Derive from (7.9) and (4.14) that the pressure jump at which the boundary
layer separates at the shock foot amounts to p2 / p1 = 1 + γ.
7.14 Consider a shock oscillation on the upper surface of an airfoil. Assume M1 ,
f , and Δx0 are as in Example 7.3 and that the chord of the airfoil measures 1m.
Furthermore, assume that the static temperature upstream of the shock is constant
and amounts to 300 K.
(a) Calculate the phase shift in milliseconds between the shock position, xs and the
pressure jump p2 / p1 for the following cases:
• dM1 /dx = 5 (1/m)
• dM1 /dx = 2 (1/m)
• dM1 /dx = 0.5 (1/m)
• dM1 /dx = 0 (1/m)
(b) Calculate the corresponding amplitude of the pressure jump in each of the above
cases.
(c) Plot the phase shift in radians and the amplitude as a function of dM1 /dx.
7.15 Consider a shock oscillation on the upper surface of an airfoil. Assume M1 =
1.18, dM1 /dx = 1.5, and c = 1 m. Furthermore, assume that the static temperature
upstream of the shock is constant and amounts to 300 K. The amplitude of the shock
424
7 Airfoil Aerodynamics
Lift coefficient,
CL (~)
CL (~)
CL (~)
CL (~)
CL (~)
CL (~)
0.9
0.8
0.7
0.6
0.5
0.4
0.8
0.7
0.6
0.5
0.4
0.3
0.7
0.6
0.5
0.4
0.3
0.2
0.6
0.5
0.4
0.3
0.2
0.1
0.5
0.4
0.3
0.2
0.1
0
0.4
0.2
0.3
0.3
0.2
0.2
0.1
0.1
0
M=0.50
0
0.1
0
1
0
1
1
2
2
3
3
4
+0.2
3
4
5
R
1
2
3
4
5
M=0.75
R
1
2
3
3
4
5
(°)
R
2
2
3
4
4
4
(°)
R
(°)
5
(°)
5
(°)
6
(°)
P,TE
0
6
+0.2
-0.2 C
P,TE
(°)
5
4
5
angle of attack,
5
(°)
-0.2 C
R
+0.2
-0.2 C (~)
P,TE
0
2
R
0
3
1
R
Pressure coefficient trailing edge,
-0.2 C (~)
M=0.50
P,TE
0
1
2
R
M=0.60
0
0
1
0
M=0.75
0
M=0.70
M=0.80
0
0.1
M=0.85
0
0
+0.2
(~)
1
-0.2 C
M=0.60
(°)
R
2
(~)
3
P,TE
0
5
+0.2
-0.2
4
M=0.80
R
1
2
3
4
(°)
0
5
+0.2
(~)
M=0.70
R
1
2
3
CP,TE (~)
1
2
4
(°)
5
M=0.85
(°)
R
3
4
5
Fig. 7.43 Figure belonging to Problem 7.16
is measured for four frequencies: f = 10, 30, 60, and 80 Hz. The corresponding
amplitudes measure Δx0 = 0.105, 0.102, 0.063, and 0.056, respectively.8
(a) Calculate the phase shift in degrees between the shock position, xs and the
pressure jump p2 / p1
(b) Calculate the corresponding amplitude of the static pressure jump in each of the
above cases.
(c) Plot the phase shift in radians and the amplitude as a function of the frequency, f .
7.16 In the graphs of Fig. 7.43 the lift curves and trailing edge pressure coefficients
are drawn for the outboard wing of a given swept wing. Determine the buffet onset
boundary and plot this in an M versus C L graph.
8 The data presented in this problem is based on an experiment described in [43]. In this experiment
the shock movement was introduced trough a periodic change in angle of attack around α = 3.0 of
a NLR 7301 airfoil at M = 0.70.
References
425
References
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(1959)
2. Anderson, J.: Modern Compressible Flow with Historic Perspective, 3rd edn. McGraw Hill,
New York (2003)
3. Anderson, J.: Fundamentals of Aerodynamics, 5th edn. McGraw Hill, New York (2010)
4. Anon.: The supercritical airfoil. NASA TF-2004-13 DFRC (2004)
5. Axelson, J.A.: Estimation of transonic aircraft aerodynamics to high angles of attack. AIAA
paper 75-996, Los Angeles (1975)
6. van den Berg, B.: Reynolds number and Mach number effects on the maximum lift and the
stalling characteristics of wings at low speed. NLR TR 69025 U, Amsterdam (1969)
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8. Boerstoel, J.W., Uijlenhoet, R.: Lifting aerofoils with supercritical shockless flow. In: Proceedings of ICAS (1970)
9. Carpentieri, G.: An adjoint-based shape-optimization method for aerodynamic design. Ph.D.
thesis, Delft University of Technology (2009)
10. Drela, M., Giles, M.B.: Viscous-inviscid analysis of transonic and low Reynolds number airfoils. AIAA J. 25(10), 1347–1355 (1987). doi:10.2514/3.9789
11. Emmons, H.W.: The theoretical flow of a frictionless, adiabatic, perfect gas inside of a twodimensional hyperbolic nozzle. NACA TN 1003 (1946)
12. Fischenberg, D., Jategaonkar, R.V.: Identification of aircraft stall behavior from flight test data.
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Flight Testing”. RTO MP-11, Madrid, Spain (1998)
13. Flaig, A., Hilbig, R.: High-lift design for large civil aircraft. In: AGARD Conference Proceedings CP-515, pp. 31.1–31.12. AGARD (1993)
14. Gibb, J.: The cause and cure of periodic flows at transonic speeds. In: Proceedings of ICAS,
pp. 1522–1530. ICAS-88-3.10.1 (1988)
15. Giles, M.B., Drela, M.: Two-dimensional transonic aerodynamic design method. AIAA J.
25(9), 1199–1206 (1987). doi:10.2514/3.9768
16. Göthert, B.: Widerstandsanstieg bei Profilen im Bereich hoher Unterschallgeschwindigkeiten,
Deutche Versuchsanstalt fur Luftfahrt, E.V. Untersuchungen und Mitteilungen Nr. 1167 (1944)
17. Harris, C.D.: NASA supercritical airfoils; a matrix of family related airfoils. NASA TP2969,
Langley (1990)
18. Hemenover, A.D.: Tests of the NACA 64-010 and 64A010 airfoil sections at high subsonic
Mach numbers. NACA-RM-A9E31. Moffett Field (1949)
19. Hoak, D.E., Anderson, R., Goss, C.R.: USAF Stability and Control Datcom. Air Force Wright
Aeronautical Laboratories, Wright Patterson Air Force Base, Ohio (1978)
20. Iovnovich, M., Raveh, D.E.: Reynolds-averaged Navier-Stokes study of the shock-buffet instability mechanism. AIAA J. 50(4), 880–890 (2012). doi:10.2514/1.J051329
21. Jameson, A., Ou, K.: 50 years of transonic aircraft design. Prog. Aerosp. Sci. 47(5), 308–318
(2011). doi:10.1016/j.paerosci.2011.01.001
22. Johnson, D.A., Spaid, F.W.: Supercritical airfoil boundary-layer and near-wake measurements.
J. Aircr. 20(4), 298–305 (1983)
23. Kacprznski, J., Ohman, L.H., Garabedian, P.R., Korn, D.G.: Analysis of the Flow Past a Shockless Lifting Airfoil in Design and Off-Design Conditions. NRCC LR-554, Ottawa (1971)
24. Laitone, E.: Local supersonic region on a body moving at subsonic speeds. In: Oswatitish,
K. (ed.) Symposium Transsonicum, vol. I, pp. 57–70. Springer, Heidelberg (1964)
25. LeBalleur, J.C., Girodroux-Lavigne, P.: Viscous-inviscid strategy and computation of transonic
buffet. In: Zierep, J., Oertel, H. (eds.) Symposium Transsonicum III. IUTAM, pp. 49–63.
Springer, Berlin (1988)
26. Lee, B.: Self-sustained shock oscillations on airfoils at transonic speeds. Prog. Aerosp. Sci.
37(2), 147–196 (2001). doi:10.1016/S0376-0421(01)00003-3
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7 Airfoil Aerodynamics
27. Lin, C.C., Rubinov, S.I.: On the flow behind curved shocks. J. Math. Phys. 27(2), 105–129
(1948)
28. Maki, R., Hunton, L.: Investigation at subsonic speeds of several modifications to the leadingedge region of the NACA 64A010 airfoil section designed to increase maximum lift. NACA
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29. Mayer, J.P.: A limit pressure coefficient and an estimation of limit forces on airfoils at supersonic
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at low speed. NACA TN 2502. Moffett Field (1951)
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section for general aviation applications. NASA TN-D-7428. Hampton (1973)
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and design. Annu. Rev. Fluid Mech. 5, 119–150 (1973)
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Transonic Flow Past Aerofoils. National Physics Laboratory R&M 3108 (1955)
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3, 277–322 (1963)
36. Ringleb, F.: Exakte Loesungen der Differntialgleichungen einer adiabatischen Gasstroemung.
Zeitschrift fur angewandte Mathematik und Mechanik 20(4), 185–198 (1940)
37. Roskam, J.: Airplane Design, Part 6: Preliminary Calculation of Aerodynamic, Thrust and
Power Characteristics. DARcorp, Lawrence (2006)
38. Smith, A.M.O.: High-lift aerodynamics. J. Aircr. 12(6), 501–530 (1975)
39. Sobieczky, H., Seebass, A.R.: Supercritical airfoil and wing design. Annu. Rev. Fluid Mech.
16, 337–363 (1984)
40. Spreiter, J.R., Steffen, P.J.: Effect of Mach and Reynolds numbers on maximum lift coefficient.
NACA TN 1044 (1946)
41. Standish, K.J., van Dam, C.P.: Aerodynamic analysis of blunt trailing edge airfoils. J. Sol.
Energy Eng. 125(4), 479 (2003). doi:10.1115/1.1629103
42. Stanewsky, E., Basler, D.: Mechanism and Reynolds Number Dependence of Shock-Induced
Buffet on Transonic Airfoils. In: Zierep, J., Oertel, H. (eds.) Symposium Transsonicum III.
IUTAM. Springer, Berlin (1988)
43. Tijdeman, H., Seebass, A.R.: Transonic flow past oscillating airfoils. Annu. Rev. Fluid Mech.
12, 181–222 (1980)
44. Torenbeek, E.: Elements of aerodynamic wing design. Advanced Aircraft Design, Chap. 10.
Wiley, Chichester (2013)
45. Viken, J., Wagner, R.D.: Design limits of compressible NLF airfoils. In: 29th Aerospace Sciences Meeting. AIAA-91-0067, Reno (1991). doi:10.2514/6.1991-67
46. Vogelaar, H.: Wind-tunnel investigation on the two-dimensional F-29 model 12-1 airfoil section
with various high lift devices at various Mach and Reynolds numbers. NLR TR 83059C (1983)
47. Whitcomb, R.T., Clark, L.R.: An Airfoil Shape for Efficient Flight at Supercritical Mach
Numbers. NASA TM X-1109. Langley (1965)
Chapter 8
Aerodynamics of Swept Wings
Abstract In this chapter we discuss the aerodynamics of swept wings in transonic
flow. To demonstrate the merits of swept-back wings, simple sweep theory is presented. It is shown why a swept wing can experience local supersonic flow while still
being in subcritical conditions, thereby postponing the onset of strong shock waves
and drag divergence. It is also shown how the wave drag coefficient of a swept wing
can be estimated based on the wave drag coefficient of an unswept wing. This method
can be used to show the favorable effect of wing sweep on the wave drag coefficient
over the transonic Mach number range. Apart from these advantages the chapter also
presents the adverse effects of wing sweep. It is shown how the chordwise pressure
distribution changes over the center section and tip section of a swept wing of finite
span. It is explained what shape modifications (both in airfoil and in planform) can be
applied to reduce the form drag of the center section of an aft-swept wing. In addition,
viscous effects on swept wings are detailed. In particular, the transition mechanisms
of attachment-line instabilities and crossflow vortices are explained. It is also shown
why aft-swept wings are susceptible to tip stall and how a controlled form of stall
can lead to a stable leading-edge vortex. Finally, the aeroelastic implications of
(aft-)swept wings are detailed: a change in aerodynamic twist due to wing bending
and a resulting reduction in control surface effectiveness. This chapter contains 7
examples and closes with 24 practice problems.
8.1 Introduction
In the previous chapter we have seen transonic flow about airfoils. We now add a
third dimension to the airfoil and investigate the flow over finite wings in transonic
conditions. If we look at airplanes that fly in the transonic regime, there is one
distinct feature that separates these aircraft from their subsonic counterparts: they
exhibit wing sweep. On a first glance the use of wing sweep is not very obvious. For
a given aspect ratio, it lengthens the structural span of the wing, which means that
a heavier structure is required. In addition, wing weight is even more increased to
resist the aerodynamic twist that is incurred during bending (see Sect. 8.5). Moreover,
the control surfaces and high-lift devices on the trailing edge of a swept wing are
© Springer Science+Business Media Dordrecht 2015
R. Vos and S. Farokhi, Introduction to Transonic Aerodynamics,
Fluid Mechanics and Its Applications 110, DOI 10.1007/978-94-017-9747-4_8
427
428
8 Aerodynamics of Swept Wings
less effective. In view of these drawbacks one might wonder why virtually all highsubsonic aircraft still exhibit a swept-back wing.
The answer to this question lies in the higher drag divergence Mach number.
The swept wing simply allows an aircraft to fly at a higher Mach number without
the exponential increase in wave drag. For supersonic aircraft, a higher wing sweep
reduces the maximum value of the drag coefficient when traversing from the subsonic
to the supersonic domain. For high-subsonic aircraft wing sweep is a complimentary
measure to the supercritical airfoil. Both are designed to push the boundary of drag
divergence to a higher Mach number. However, we shall discover in this chapter that
the introduction of wing sweep triggered a string of changes to the wing design to
counter the adverse effects that were alluded to in the first paragraph.
In this chapter we mainly focus on the effect of sweep on the aerodynamic properties of the wing. Alternative texts, such as Anderson [4], present the effects and
associated aerodynamic theory of finite wings in subsonic flow. We presume that
the reader is familiar with this theory and therefore try to extend his/her knowledge to wings that are swept. Section 8.2 presents the aerodynamic rationale behind
swept wings and discusses the inviscid theory of infinite swept wings. In Sect. 8.3
we present the effect of wing sweep on the streamline curvature over infinite wings
as well as over finite wings. In Sect. 8.4 we revisit the flow in the boundary layer over
a swept wing. We shall discover that different transition mechanisms exist and that
the spanwise flow of the boundary layer can be a dangerous source of separation.
Finally, in Sect. 8.4 we address the adverse aeroelastic effects that are encountered
on swept-wing aircraft.
8.2 The Advantages of Wing Sweep
The effect of sweep on drag reduction has long been known in transonic/supersonic
aircraft, well before the invention of the supercritical airfoil and development of
the area rule. It was in fact in the Fifth Volta Conference in 1935 in Rome, Italy
where Adolf Büsemann outlined the theory of swept wings and their aerodynamic
characteristics in supersonic flow. Büsemann argued that it was the normal component
of flow to the wing leading edge that dictated its high-speed performance. He thus
argued that swept wings experience a delayed compressibility effect and as a result
postpone drag-divergence behavior observed in unswept wings in transonic flow.
Schlichting [57] shows the drag polar of a wing with zero-sweep and another wing
with 45◦ of sweep at transonic Mach numbers in Fig. 8.1. Side-by-side comparison
of drag at constant lift in Fig. 8.1a, b shows the dramatic reduction in drag as a result
of wing sweep.
Note that the effect of sweep, as depicted by Schlichting, also introduces the effect
of aspect ratio since both wings have equal structural span. Therefore, we may ask for
8.2 The Advantages of Wing Sweep
429
(a)
(b)
1.0
M∞=0.7
0.8
M∞=0.7
0.6
M =0.9
0.4
CL
M ∞=0.9
M∞
Λ = 45
0.2
0.0
-0.2
-0.4
0.0
0.1
0.2
0.3 0.0
0.1
0.2
CD
0.3
CD
Fig. 8.1 The effect of sweep on drag polar in transonic speeds (data from [57]). a Λ = 0◦ .
b Λ = 45◦
Fig. 8.2 The drag polar of a
wing with variable sweep but
with fixed aspect ratio at
Mach 0.925 using NACA
65-110 profiles in planes
normal to leading edge and
A = 3 (after Ref. [2])
0.06
0
30
0.04
45
Sweepback
Angle
CD
0.02
0
-0.2
0
0.2
0.4
0.6
CL
the distinct contribution of sweep as compared to the contribution of aspect ratio to
drag reduction shown in Fig. 8.1. Adler [2] compared the effect of sweep on wings of
constant aspect ratio at Mach 0.925, as shown in Fig. 8.2. Here we have the isolated
effect of the sweep (A = 3) on wing drag polar at a transonic Mach number. We
note a significant drop in drag with the sweep angle; and in particular we note a
significant reduction in C D0 and C D,min with increasing sweep as shown in Fig. 8.2.
430
8 Aerodynamics of Swept Wings
oblique wing
straight wing
8
V
V
8
v
8
8
u

u
w’
V
8
u
Voblique
8
w
V =0
u’
Fig. 8.3 Oblique flow and straight flow over a wing section with constant airfoil
8.2.1 Theory of Wing Sweep
Now that we have seen the favorable effect of wing sweep in transonic conditions,
let us investigate the effect of wing sweep on the flow over a wing following Ringleb
[51]. In Fig. 8.3 we see two straight wings of infinite span. The left wing is swept
such that the airfoil is exposed to oblique flow. In other words, the direction of the
freestream velocity vector is under an angle Λ with respect to the chordwise direction
of the airfoil. The airfoil on the right is in straight flow. Both wings experience the
same freestream velocity, V∞ .
We investigate the velocity at a random point somewhere in the flow field over
both airfoils. The velocity over the oblique wing is influenced by the curvature of
the airfoil, resulting in velocities u and w. In addition, the spanwise component of
the freestream velocity, v∞ remains constant. This component does not encounter
any curvature and therefore remains constant over the wing. The total velocity can
therefore be written as:
2
2
= u 2 + w2 + v∞
(8.1)
Voblique
For the wing in straight flow we have a similar expression without the cross-flow
component:
2
= u 2 + w2
(8.2)
VΛ=0
u∞
Because the velocity, V∞ is related to u ∞ according to V∞ = cos
Λ the velocity
components scale with the same factor:
u
cos Λ
w
w =
cos Λ
u =
(8.3a)
(8.3b)
8.2 The Advantages of Wing Sweep
431
We rewrite (8.2) as follows:
2
=
VΛ=0
u 2 + w2
cos2 Λ
(8.4)
We can now relate the two velocities to each other (see Problem 8.1):
Voblique
V∞
2
= sin2 Λ +
Vstraight
V∞
2
cos2 Λ
(8.5)
The pressure coefficient in incompressible conditions at any point in the flow can be
calculated according to:
V 2
(6.11)
Cp = 1 −
V∞
Substitution of (8.5) in (6.11) results in the following relation between straight wing
pressure coefficient, C p, Λ=0 , and the oblique wing pressure coefficient, C p, oblique
(see Problem 8.1):
(8.6)
C p, oblique = C p, Λ=0 cos2 Λ
We can extend the result of (8.6) to the lift coefficient and pitching-moment coefficient of the airfoil by employing (3.30) and (3.35), respectively. This results in the
following relations:
cl, oblique = cl, Λ=0 cos2 Λ
(8.7a)
cm 0 , oblique = cm 0 , Λ=0 cos Λ
(8.7b)
2
The derivation shown above is known as the simple sweep theory. Let us briefly
reflect on the results that have been obtained. We have now shown that the supervelocity as well as the absolute value of the pressure coefficient at any point in the
flow about the oblique airfoil is lower than at the same point in the flow about the
straight airfoil. If we remember our discussion on airfoil aerodynamics in Chap. 7,
we can directly deduce that the formation of supersonic flow is delayed to higher
freestream Mach numbers for the airfoil in oblique flow. The critical Mach number therefore increases. The same argument can be applied to the drag-divergence
Mach number. Due to the lower supervelocities over the airfoil in oblique flow, a
higher freestream Mach number is required to yield strong shock waves that induce
separation at the shock foot. Therefore, drag divergence is postponed to a higher
freestream Mach number (this will be elaborated in Sect. 8.2.2). If we go back to
our discussion on the relation between shape and pressure distribution (Sect. 6.2) we
understand why the result of our analysis makes sense. If we assume that the flow
over the airfoil continues along the direction of the freestream velocity vector, we an
imagine that the flow ‘sees’ an airfoil with a smaller thickness-to-chord ratio. The
smaller thickness-to-chord ratio induces lower supervelocities, reducing the absolute
value of the pressure coefficient, which is in line with the findings of (8.6). A simple
432
8 Aerodynamics of Swept Wings
alternative for the swept wing would be to simply reduce the thickness of a straight
(λ = 0) wing. However, this would result in a heavier structure to carry aerodynamic
loads on the wing (see Problem 8.4).
To demonstrate that simple sweep theory can be used to estimate the pressure
distribution over swept wings once the pressure distribution over a straight wing is
known, we consider Fig. 8.4 from Abbott and von Doenhoff [1]. This figure shows
an experimental setup of a straight wing which is positioned at three different sweep
angles in a wind tunnel: Λ = 0◦ , Λ = 20◦ and Λ = 40◦ . The pressure is measured
over the center cross section at each sweep angle. The resulting pressure coefficient
is scaled back to the equivalent pressure coefficient of the straight wing by using
(8.6). The result is plotted on the right of Fig. 8.4. This shows that the simple sweep
theory indeed gives a good relation between the pressure distributions of the swept
and straight wing. The discrepancies between the distributions could be explained by
the effect of the wind tunnel walls, which force the stream lines at the wall to remain
parallel to the wall. We will see in Sect. 8.3 that in reality the flow over a swept wing
causes the streamlines to curve over the planform. The wind tunnel walls prevent
this curving, which influences the pressure distribution.
So far we have only considered the effect of sweep on infinite wing sections. In
Fig. 8.4 we already see that effects at the boundary have an effect on the pressure
distribution. In practice, swept wings have a finite span and are often tapered (meaning
the tip chord is smaller than the root chord). A tapered wing does not have a uniform
sweep angle. The local sweep angle depends on the chordwise position and varies
between the leading edge and the trailing edge. It is therefore impossible to use the
simple sweep theory directly on a swept wing that is also tapered. Therefore, we need
to choose a reference sweep line if we want to predict the pressure distribution about
an airfoil in a swept wing. From a practical point of view it has been demonstrated
that using the quarter-chord sweep line (Λc/4 ) gives good results in predicting the
Cp
80 mm
-0.5
Profile R-4009
ß=40o
20o
x
0
M∞
Section of
400
measurement
0.5
α=-2o
1.0
ß=0o; M∞= 0.44
+0.5
ß=20o; cos (20o)M∞= 0.44
ß=40o; cos (40o)M∞= 0.44
+1.0
80
Fig. 8.4 Comparison between pressure distribution over a straight wing and a swept wing at
Λ = 20◦ and Λ = 40◦ for constant normal Mach number (after Ref. [1])
8.2 The Advantages of Wing Sweep
433
(a) -2.8
(b)
Λ=60°
Pressure coefficient, Cp (~)
-2.4
Λ=45°
-2.0
-1.6
η=0.815
-1.2
Experiment, η=0.815
Theory - chordwise (64A010)
Theory - streamwise (64A007)
cl=0.4
-0.8
Pressure coefficient, Cp (~)
-2.4
-2.0
-1.6
η=0.50
-1.2
Experiment, η=0.50
Theory - chordwise (64A010)
Theory - streamwise (64A005)
cl=0.26
-0.8
-0.4
-0.4
0
0
0.4
0.4
0.8
0.8
0
.2
.4
.6
.8
1.0
Chordwise position, x/c (~)
0
.2
.4
.6
.8
1.0
Chordwise position, x/c (~)
Fig. 8.5 Comparison of theoretical section pressure distributions with experimental loadings on
finite wing panels (after Ref. [30]). a 45◦ swept wing with NACA 64A010 sections normal to the
quarter chord line. b 60◦ swept wing with NACA 64A010 sections normal to the quarter chord line
drag-divergence Mach number. This is a line that connects the quarter-chord points on
the chord line of each individual airfoil in the wing. Figure 8.5 shows the comparison
between the theoretical prediction and the experimentally obtained results for the
same lift coefficient. The solid theoretical line has been obtained by calculating the
chordwise pressure distribution about the airfoil (in this case NACA 64A010) and
applying (8.6). We see that it matches closely to the measured pressure distribution.
The dashed line is a prediction based on an equivalent “sheared” wing. The difference
between these two approaches becomes clear in the next section.
8.2.2 Effect on Drag Divergence Mach Number
In the previous discussion we have assumed that airfoils are defined perpendicular
to the wing LE. Hence we have assumed an unswept wing that is rotated about a
pivoting axis in order to orient its airfoils at an angle with respect to the freestream
flow (hence have them experience a lower velocity component than the freestream
434
8 Aerodynamics of Swept Wings
Λ
Λ
oblique wing
sheared wing
Airfoil perpendicular to LE
(t/c)
perp
Airfoil parallel to freestream
> (t/c)
parallel
Fig. 8.6 Generation of a swept wing from a straight wing: rotating or shearing (sketch: G. La
Rocca)
by a factor cos ΛLE ). However, it is often industry practice to design a swept wing by
shearing the airfoil in parallel planes to the freestream rather than pivoting the wing.
The difference between the two cases is demonstrated in Fig. 8.6. In the case of a
sheared wing, the effective flow component (u ∞ ) does not see the selected airfoil, but
a similar one with a higher thickness-to-chord ratio (actually with the same thickness
but shorter chord). The theory that is presented in the previous section then applies
to the airfoil perpendicular to the leading edge. By shearing the wing forward or
backwards we always have the same airfoil in streamwise direction. This allows us
to investigate the effect of wing sweep on the drag divergence Mach number for an
airfoil at constant angle of attack.
Let us first compare the sheared wing to the oblique (or “yawed”) wing in terms
of the lift coefficient. If we consider the airfoil to be thin we can replace it by an
infinite vortex sheet [4]. For an unswept wing, the vortex vectors are perpendicular
to the oncoming flow resulting in the following value of the two-dimensional lift
coefficient:
c
γ(x)dx
(8.8)
L Λ=0 = ρ∞ V∞
0
For a swept wing the vorticity vector, γ is inclined at an angle π/2 − Λ. Therefore,
the two-dimensional lift vector, L , can be written as:
c
L sheared = ρ∞ V∞
γ(x) cos Λdx
(8.9)
0
We therefore conclude that the lift coefficient of the sheared airfoil relates to the lift
coefficient of the unswept airfoil as:
cl, sheared = cl, Λ=0 cos Λ
(8.10)
8.2 The Advantages of Wing Sweep
435
Note that this result is different from what we found for the oblique wing, i.e. (8.7a).
This makes sense because for the oblique wing we evaluated the lift coefficient of the
reference airfoil of the unswept wing. This airfoil is not being deformed in streamwise
direction when the wing is being sheared, while it is stretched when the wing is being
yawed. Depending on the way a wing is designed, one of the two prediction methods
should be used to evaluate the effect of wing sweep on the aerodynamic coefficients.
Now that we have defined this alternative form of the sweep theory, let us return
to the drag-divergence Mach number. The Mach number at which drag divergence
starts is strongly dependent on the Mach number in front of the shock wave. As we
know from Chap. 7, the wave drag is directly proportional to the momentum loss
over the shock wave, which, in turn, depends on the static pressure increase over
the shock. To make a first estimate of the drag divergence Mach number, Mdd , we
can rely on a crude but simple empirical relationship between the drag divergence
Mach number (defined here as the Mach number for which ∂C D /∂ M = 0.05) and
the critical Mach number of a finite swept wing [59]:
Mdd = Mcrit 1.02 + 0.08(1 − cos Λc/4 )
(8.11)
To find the critical Mach number we employ a modified procedure to the one
outlined in Chap. 3 (Fig. 3.4). We consider a nonlifting swept wing at zero angle
of attack. First we modify the relation between the critical pressure coefficient and
the critical Mach number, (3.42). We replace the critical freestream Mach number,
Mcrit , by Mcrit cos Λ in accordance with simple sweep theory. We obtain the following
relation:
⎡
⎤
γ
γ−1 2
2 Λ γ−1
1
+
M
cos
2 ⎣
crit
2
C pcrit =
(8.12)
− 1⎦
2
γ+1
γ Mcrit
2
Secondly, we modify the two-dimensional compressibility (see Sect. 3.2) corrections by following Ref. [39]. We assume that the freestream flow direction is aligned
with the x-axis in the orthogonal coordinate system xyz. Furthermore, we consider
a wing parallel to the xy-plane that is sheared over an angle Λ about the z-axis
(Fig. 8.7). Note that the airfoils of the wing remain unchanged in shape and lateral
position of the wing (see RHS of Fig. 8.6). We define a second orthogonal coordinate
system (ξηζ) that is fixed to the wing. In this coordinate system, the ξ axis is parallel
to the local chord line, while η is parallel to the leading edge of the wing and runs
in spanwise direction. Any point in the x y-plane can be related to its ξη-coordinates
by the following relations:
ξ = x cos Λ − y sin Λ
(8.13a)
η = x sin Λ + y cos Λ
ζ=z
(8.13b)
(8.13c)
436
8 Aerodynamics of Swept Wings
Fig. 8.7 Coordinate system
of sheared wing (after
Ref. [39])
V∞
y
Λ
ξ
c
η
x
We can subsequently write the full potential equation (3.2) in the rotated coordinate system for two-dimensional flow:
1−
Φξ2
a2
Φξξ − 2
Φz2
Φξ Φ z
Φ
+
1
−
Φzz = 0
ξz
a2
a2
(8.14)
Critical conditions appear when (8.14) changes from elliptic to hyperbolic. It is of
parabolic form when the discriminant equals zero:
Φξ Φ z
a2
2
− 1−
Φξ2
a2
1−
Φz2
a2
=0
(8.15)
This can be simplified to read:
1−
Φξ2 − Φz2
a2
=0
(8.16)
The velocity components in ξ and z direction can be written as:
Φξ = Vξ = V∞ cos Λ + û
Φz = Vz
(8.17a)
(8.17b)
Note that we have used Vξ and Vz as the velocity components parallel to the ξ and
z-axis, respectively. Substituting this in (8.16) results in the following expression for
the critical condition:
2
sin2 Λ = a 2
Vξ2 + Vz2 = a 2 or V 2 − Vη2 = V 2 − V∞
(8.18)
Note that in this equation it is implied that v (spanwise velocity component tangent to the leading edge, V∞ sin Λ) does not change over the wing; an assumption
of simple-sweep theory. When examining the critical condition (8.18) we observe
that only the velocity component normal to the direction of sweep ( Vξ2 + Vz2 )
8.2 The Advantages of Wing Sweep
437
must be sonic. The total velocity, V , may exceed sonic speed at critical conditions:
2 sin2 Λ ≥ a 2 . Pure subsonic flow over a wing exists until the velocV 2 = a 2 + V∞
ity along the peak suction line becomes sonic. For unswept wings this is also the
end of the subcritical regime. However, when the wing is swept, subcritical flow
exists until the component perpendicular to the peak-suction line becomes sonic.
For swept wings this marks the end of the subcritical regime. In other words, a swept
wing can be in subcritical conditions even when local supersonic flow exists. While
being in the subcritical regime, the flow cannot have any shock waves. This intuitively demonstrates the effect of wing sweep on the drag divergence boundary: the
subcritical domain is extended, such that (strong) shocks occur at higher freestream
Mach numbers.
To investigate the effect of the sweep angle on the compressibility factor, β (3.18),
let us return to (8.14). In the limit of small perturbations, where we neglect the squares
and products of small parameters, i.e. the second order terms, and for freestream Mach
numbers that exclude transonic and hypersonic flows (8.14) may be written as:
β Φξξ + Φzz = 0 with β = 1 −
2
2
Vξ
a
2
(8.19)
2 sin2 Λ. We can
We subsequently assume that Vz Vξ such that Vξ2 = V 2 − V∞
2
therefore rewrite for β :
2 sin2 Λ
V − V∞
(8.20)
β2 = 1 −
a2
Remember that for isentropic flow the following holds:
2
a 2 = a∞
−
γ−1 2
2
(V − V∞
)
2
(3.8)
Substitution of (3.8) in (8.20) and subsequent manipulation yields the following
expression:
2
cos2 Λ + VV∞ − 1
2
β 2 = 1 − M∞
(8.21)
2
γ−1
V
1 − 2 M∞ V∞ − 1
We can further simplify this expression by acknowledging
that the term in the
2
(V
/V
M
denominator will be close to unity because γ−1
∞
∞ ) − 1 1. In
2
many cases the velocity will be known for incompressible flow. The velocity term
can then be replaced by the incompressible pressure coefficient, −(C p ) M∞ =0 =
(V /V∞ )2 − 1. The final expression for β 2 then becomes:
2
cos2 Λ − (C p ) M∞ =0
β 2 = 1 − M∞
(8.22)
438
8 Aerodynamics of Swept Wings
If we look at this expression, we recognize the Prandtl-Glauert compressibility correction for the special case where (C p ) M∞ =0 = 0 and Λ = 0. The rule (8.22) is due
to Johanna Weber and is laid out in Ref. [39]. We can relate the pressure coefficient
at non-zero Mach numbers to the pressure coefficient at M∞ = 0 by following the
procedure leading up to (3.29) on p. 86. We then obtain the following expression
relating both pressure coefficients:
(C p ) M∞ =
C p M =0
β
∞
=
2
1 − M∞
(C p ) M∞ =0
cos2 Λ − (C p ) M∞ =0
(8.23)
The compressibility factor, β, has been extended to three-dimensional wings [39].
It has been shown that β takes on different values in the center, sheared, and tip
regions. Further elaborations and refinements of the method have resulted in complex
analytical expressions for the velocity distributions over swept wings [6]. In the
present context we limit ourselves to the theoretical case of infinite sheared wings
with a symmetric airfoil at zero angle of attack.
Based on (8.23) we can construct a graph (similar to Fig. 3.6) where the pressure
coefficient and freestream Mach number are related via (8.12) and (8.23). Such a
graph is presented in Fig. 8.8 for an incompressible pressure coefficient of −0.5.
We first turn our attention to the curves relating the critical pressure coefficient
to the critical Mach number (according to (8.12)). The thickened line relates the
critical pressure coefficient to the critical Mach number for an unswept wing and is
the same as in Fig. 3.4. When the wing sweep is applied Fig. 8.8 indicates that this
line moves to the right, indicating that for a given critical pressure coefficient, the
critical Mach number is increased with sweep angle. We now turn to the second set of
−0.8
Λ=0
30
45
60
−0.6
Λ=0
Cp
(C p )
M =0
−0.2
0
0.5
45
60
= -0.5
Cp
0.6
0.7
0.8
Free−Stream Mach Number, M
0.9
8
−0.4
30
8
Pressure Coefficient, C p (~)
−1
(~)
Fig. 8.8 Effect of wing sweep on the critical pressure coefficient
1
crit
8.2 The Advantages of Wing Sweep
439
curves that show the effect of the freestream Mach number on the pressure coefficient.
With increasing sweep angle we observe that the compressibility correction becomes
weaker. In other words, the change in pressure coefficient with Mach number is
lessened. Finally, the intersection between the two curves of the same sweep angle
marks the critical Mach number. We see that the critical Mach number increases
from M∞ = 0.68 when Λ = 0 to M∞ = 0.87 when Λ = 60◦ .
We have now established the relation between the critical Mach number, the sweep
angle, and the incompressible pressure coefficient. If we set the right-hand sides of
(8.12) and (8.23) equal to each other we can deduce the critical Mach number for a
given incompressible pressure coefficient and sweep angle. If we do this for several
values of the incompressible pressure coefficient, we can make a chart where we relate
the critical Mach number to the sweep angle for a given value of the incompressible
pressure coefficient. Such a chart is presented in Fig. 8.9. We can use this chart in
combination with (8.11) if we would like to determine how much sweep is required
for a given drag-divergence Mach number. The following example demonstrates the
use of the chart.
Example 8.1 Many transonic aircraft cruise at or slightly beyond their dragdivergence Mach number. Assume we desire a cruise Mach number of 0.85 and
have an airfoil which has a peak suction pressure coefficient of C p = −0.35 at low
Mach number and zero angle of attack. Calculate the sweep angle of the airfoil such
that drag divergence occurs at M = 0.85.
(C p )
M =0
8
60
-1.0 -.90 -.80
-.95 -.85 -.75
-.70
-.60 -.55 -.50 -.45
-.65
-.35
-.30
50
Sweep Angle, Λ (deg)
-.40
-.25
-.20
40
-.15
-.10
30
-.05
20
10
0
0.5
0.6
0.7
0.8
0.9
Critical Mach Number, M
crit
1
(~)
Fig. 8.9 Critical Mach number for two-dimensional flow based on Weber’s compressibility
correction (8.22)
440
8 Aerodynamics of Swept Wings
Solution:
To solve this problem, we need to combine the chart of Fig. 8.9 and Eq. (8.11). We
use Λ = Λc/4 . Because both the chart and the formula contain critical Mach number
and the sweep angle, the equation cannot be solved in a closed expression. Therefore,
we need to solve this problem iteratively. We start with a first guess for the critical
Mach number, making sure it is somewhat below the drag-divergence Mach number:
Mcrit = 0.8. We find a corresponding sweep angle of Λ = 32◦ (see Fig. 8.10). We
then employ (8.11):
Mdd = 0.8 [1.02 + 0.08(1 − cos 32)] = 0.826 < 0.85
Our first guess yields a drag-divergence Mach number that is a little too low. We try
Mcrit = 0.82, which gives a corresponding sweep angle of Λ = 36◦ (see Fig. 8.1).
With (8.11) we find:
Mdd = 0.8 [1.02 + 0.08(1 − cos 32)] = 0.849 ≈ 0.85
We have now found a sweep angle that yields a drag divergence Mach number that
is very close to the desired cruise Mach number. The iterative solution is illustrated
in Fig. 8.10.
60
(C p )
= -0.35
8
M =0
40
30
initial guess
first iteration
Sweep Angle, Λ (deg)
50
20
10
0
0.5
0.6
0.7
0.8
0.9
1
Critical Mach Number, Mcrit (~)
Fig. 8.10 Graph of sweep angle versus critical Mach number belonging to Example 8.1
8.2 The Advantages of Wing Sweep
441
8.2.3 Effect on Wave Drag Coefficient
At speeds close to the drag-divergence Mach number we would like to calculate the
change in drag coefficient with Mach number. We therefore use a modified version of
the transonic similarity laws of Chap. 3. First step is to construct the wave-drag versus
Mach number graph for the unswept wing. We can do this by employing Fig. 3.21
for a given value of thickness-to-chord ratio, τ , and aspect ratio, A. Based on this
graph, we read a value for the maximum value of the wave drag coefficient, C Dw, max ,
and the corresponding Mach number (MC Dw, max , Λ=0 ) as well as the drag-divergence
Mach number. The latter one can be obtained by using the definition of Mdd from
(7.1): ∂C D /∂ M = 0.10.
Now, for a wing with sweep these values need to be converted. We consider a
wing that is pivoted about a hinge point. In the theoretical case of such an oblique
wing, the drag divergence Mach number would scale with cos Λc/4 , similar to the
velocity. In real life, as we will see in the subsequent sections, the flow over the wing
is fairly three-dimensional. Therefore, the assumption that the airfoil only sees the
normal component with respect to the quarter chord line is an oversimplification.
The actual drag divergence Mach number should be somewhere between the one for
the airfoil and the one for the oblique wing [5]:
Mdd for airfoil < Mdd for swept wing <
Mdd for airfoil
cos Λc/4
(8.24)
Following the same reasoning we can state that the effective Mach number, Meff , can
be related to the freestream Mach number according to:
Meff = M∞ (cos Λc/4 )n with 0 < n < 1
(8.25)
For the case of ideal conditions n = 1 [33]. Torenbeek [66] uses n = 1/2. We will use
the latter approximation to estimate the drag divergence Mach number of the swept
wing and the Mach number at which the maximum drag coefficient is obtained:
(Mdd )Λ=0
(Mdd )oblique = cos Λc/4
(MC Dw, max )Λ=0
(MC Dw, max )oblique = cos Λc/4
(8.26a)
(8.26b)
The drag wave-drag coefficient of the oblique wing can be correlated to the unswept
wing according to [31]:
(C Dw, max )oblique = (C Dw, max )Λ=0 (cos Λc/4 )2.5
(8.27)
We can now construct the line of M versus C Dw . We have the combination of Mach
number and wave-drag coefficient at the peak value. In addition, we have the position
442
8 Aerodynamics of Swept Wings
Table 8.1 Table belonging to Example 8.2
M 2 −1
τ 2/3
M
−2.35
−1.81
−1.24
−0.64
0
0.33
0.67
1.37
2.10
0.80
0.85
0.90
0.95
1.00
1.025
1.05
1.10
1.15
C Dw
τ 5/3
C Dw
0
0.10
0.58
1.96
3.02
3.08
3.10
3.12
3.13
0
0.00009
0.0053
0.0179
0.0278
0.0283
0.0285
0.0287
0.0288
of the drag divergence Mach number where ∂C Dw /∂ M = 0.10. It is advisable to
draw this curve in the same graph as for the unswept wing, such that the latter can
be used as a guide to construct the curve for the oblique wing. We will clarify this
procedure in the following example.
Example 8.2 Consider a wing with the following characteristics:
A = 4.0
Sw = 144 sq in.
Λc/4 = 45◦
Swet = 288 sq in.
λ = 0.60
NACA 65A006
c̄ = 6.125 in.
t/c = τ = 0.06
Determine the variation of the wing’s wave drag coefficient with Mach number
between M = 0.80 and M = 1.4.
Solution:
To solve this problem, we first determine the variation of the wave drag coefficient
with Mach number for an unswept wing. We rely on the relation shown in Fig. 3.21
where the zero-lift drag coefficient for round-nose airfoils is related to Mach number,
thickness ratio, and Mach number. We need to calculate the following quantity:
Aτ 1/3 = 1.568
We tabulate the values of (M 2 − 1)/τ 2/3 as a function of the Mach number. We
subsequently use Fig. 3.21 and manually interpolate between the lines of Aτ 1/3 =
1.5 and Aτ 1/3 = 2.0 to find the value of C Dw /τ 5/3 at Aτ 1/3 = 1.568 at each
Mach number. Finally, we calculate the corresponding value of C Dw . The values
have been tabulated in Table 8.1. The values of (M 2 − 1)/τ 2/3 higher than 1.0 have
been estimated based on the extrapolation of the curves of Fig. 3.21.
The relation between the wave drag coefficient and the Mach number has been
plotted in Fig. 8.11 with the dashed line. The markers on the line correspond to the
points from the table. From Fig. 8.11 we can deduce the following quantities:
443
0.03
unswept
w
Wing wave drag coefficient, CD (~)
8.2 The Advantages of Wing Sweep
0.025
experiment
0.02
dCD /dM = 0.10
w
swept
0.015
Mdd
0.01
M
dd
theory
0.005
0
0.8
0.9
1
1.1
1.2
Mach number, M (~)
1.3
1.4
Fig. 8.11 Graph of drag coefficient versus Mach number belonging to Example 8.2
(Mdd )Λ=0 ≈ 0.885
C Dw, max = 0.0288
(MC Dw, max )Λ=0 = 1.125
We subsequently use (8.26a), (8.26b), and (8.27) to calculate the drag-divergence
Mach number, the peak drag coefficient, and the Mach number at which the peak
drag coefficient occurs, respectively:
(Mdd )Λ=0
= 1.05
(Mdd )oblique ≈ cos Λc/4
(C Dw, max )oblique = (C Dw, max )Λ=0 (cos Λc/4 )2.5 = 1.34
)Λ=0
(MC D
(MC Dw, max )oblique = w, max
= 0.0121
cos Λc/4
To construct the curve that relates the wave drag coefficient of the swept wing to
the Mach number we now have the following information: (1) at M = 1.05 the
slope of the line is ∂C Dw /∂ M = 0.10; (2) at M = 1.34 we know the maximum
drag coefficient is C Dw = 0.0121 and that the slope ∂C Dw /∂ M = 0. With this
information we can construct the C Dw curve for the swept wing. The curve for the
unswept wing can be used as a guide for constructing the curves between the two
points. The result of this exercise is shown by the solid curve in Fig. 8.11.
The approximation that is demonstrated in the previous example is close to the
experimentally obtained prediction indicated by the dash-dotted grey line in the graph
of Fig. 8.11. We notice that the prediction of the drag divergence Mach number is too
high, and that the drag coefficient beyond M = 1 is underestimated by this method.
444
8 Aerodynamics of Swept Wings
On the other hand, this analytic treatment can yield a quick estimation without the
requirement of using computational fluid dynamics.
The minimum drag (drag at zero-lift) of a finite wing is dependent on many
factors: aspect ratio, sweepback angle, taper ratio, airfoil section family, thicknessto-chord ratio, twist distribution, dihedral, Reynolds number, and Mach number. We
would like to know what the effect is of each of these factors on the transonic drag
performance of the wing. To reduce the complexity of this problem, we follow Ref.
[35] and experimentally investigate the effect of sweep, thickness-to-chord ratio, and
Mach number on the minimum drag coefficient, C Dmin . We do this for one family of
airfoils: NACA 65A0XX, where XX represents the value of the thickness-to-chord
ratio. Furthermore, we set the aspect ratio to 4 and the taper ratio to 0.6 and we
consider wings without dihedral or twist. Finally, we evaluate the drag properties
at Reynolds numbers higher that 10 million. From the experimental results we can
now plot the minimum drag as a function of the three independent variables. This is
shown in Fig. 8.12. Note that this graph is constructed based on experimental results
from various wind tunnel tests as described in Ref. [35].
The graph in Fig. 8.12 shows both the effect of sweep and thickness-to-chord
ratio on the minimum drag coefficient in the transonic realm. If we consider the
unswept wing (Λc/4 = 0◦ ) we see an increase in drag coefficient by a factor of about
20 between M = 0.7 and 1.0 for the airfoil of 12 % thickness. By decreasing the
thickness-to-chord ratio we observe three effects: the drag-divergence Mach number
increases, the drag peak decreases, and the absolute value of the slope dC Dmin /dM
decreases. This can all be explained based on the two-dimensional aerodynamics:
lower thickness leads to lower supervelocities and therefore weaker shock waves.
A lower thickness-to-chord ratio is therefore a powerful means to reduce excessive
drag in the transonic domain.
We now turn to the effect of the sweep angle. Let us consider the wings with the
thickest profiles (t/c = 12 %). By comparing the lines at Λc/4 = 0◦ and Λc/4 = 60◦ ,
we see a reduction of the peak drag coefficient of 80 % when sweeping the wing
backwards over 60◦ . The effect of sweepback is lessened for the thinner wings. For
a wing with 9 % thickness the reduction in peak drag coefficient is around 60 %.
Apart from the absolute value of the drag peak, we also distinguish a backwards
shift in drag peak and the drag divergence Mach number. Both can be explained if
we remind ourselves that the critical conditions on the wing are determined by the
velocity component perpendicular to the local isobars on the wing. On a swept wing
of infinite span these isobars run parallel to the local sweep angle of constant-chord
points. Due to the increase in sweep angle we therefore postpone the Mach number
at which critical conditions occur and simultaneously push back the drag-divergence
Mach number as well as the peak in drag coefficient.
Figure 8.12 gives a quantitative view on how the transonic drag behavior can
be altered by modifying either the thickness-to-chord ratio or the sweep angle. In
practice, considerations on wing weight, fuel tank volume, aeroelastic characteristics, control characteristics, and low-speed performance dictate which combination of these two parameters yields the best compromise. Depending on the type of
8.2 The Advantages of Wing Sweep
.3
.4
.5
.6
.7
.8
.9
445
1.0 1.1 1.2 1.3 1.4 1.5 1.6
0
NACA 65A0XX Sections
Parallel to stream
AR=4
λ=0.6
CL=0
Re ≥10x10^6
15
30
0° Twist
0° Dihedral
Maximum thickness ratio
12%
45
60
.12
9%
6%
4%
Minimum drag coefficient, CD, min
.11
.10
.09
.08
.07
.06
0
eep
Sw
.05
15
gle
an
.04
at
.03
rd,
ho
rc
e
art
qu
30
.02
45
Λ c/4
.01
(°)
60 .2
.3
.4
.5
.6
.7
.8
.9
0
1.0 1.1 1.2 1.3 1.4 1.5 1.6
Mach number, M (-)
Fig. 8.12 Effect of sweep and thickness ratio on minimum drag coefficient (after Ref. [35])
aircraft this could yield thick wings with relatively much sweep (e.g. high-subsonic
transports) or thin wings with zero sweep (e.g. supersonic interceptors).
If one knows the drag-coefficient behavior of a given airfoil section (2-D) as a
function of Mach number and lift coefficient, Ref. [32] gives an alternative method
to calculate the drag-coefficient versus Mach number for a 3-D wing. An important
446
8 Aerodynamics of Swept Wings
assumption for this method is that the spanwise loading of the wing is neglected
and the airfoil in the wing experiences an average lift coefficient. In this method
it is assumed that the change in airfoil section wave drag coefficient due to Mach
number and lift coefficient is known, i.e. cdw = f (MΛ=0 , clΛ=0 ). Such data could,
for example, be obtained from experimental tests on wing profiles or from twodimensional airfoil analysis. In addition, one should also know how the section
minimum drag coefficient, cd0 , is affected by the sweep angle. Reference [75] shows
that the profile drag coefficient of a swept NACA 64-012 airfoil section can be
approximated as follows:
√
cd0 (Λ) = cd0 , Λ=0 cos Λ for 106 < Re < 108
(8.28)
If cd0 (Λ) and cdw (M, cl ) quantities are known, the wing drag can be estimated
according to [32]:
(8.29)
C D = cd0 cos Λ + cdw cos3 Λ + C Di
The induced drag, C Di , is a function of the lift coefficient and spanwise loading of
the wing and is independent of the Mach number in subsonic conditions. The vortex
induced drag coefficient of a finite wing is inversely proportional to wing aspect
ratio, A, in low subsonic Mach numbers and directly proportional to the square of
lift coefficient:
C Di =
C L2
πAφ
(8.30)
The symbol φ is the span efficiency factor due to non-elliptic wing loading.1 For
elliptical lift distribution φ = 1 (minimum induced drag coefficient). In all other
cases φ is less than one. Plotting (8.29) constitutes the wing drag polar. The data
of Ref. [60] shown in Fig. 8.13a supports the dominance of parabolic behavior of
C D versus C L and the inverse relationship with aspect ratio at low subsonic Mach
numbers (e.g., M∞ = 0.5). The lowest wing drag coefficient is seen to occur at the
highest aspect ratio. Now, examine the drag polar of the same wing, but at a high
subsonic Mach number of M∞ = 0.875 in Fig. 8.13b. Due to the dominant wave
drag, the drag polar behavior seems to be turned upside down. The high aspect ratio
wing (e.g., A = 9) shows the highest drag and vice versa. In this high-subsonic case
we see that for high-aspect ratio wings a small variation of lift coefficient results in a
large variation of C D (note the higher scale of drag coefficient in 8.13b versus 8.13a).
This behavior is due to the appearance of shocks on the wing and subsequent flow
separations causing a loss of lift and rise in drag. Here the wing drag is not dominated
1 The Oswald factor, e, of a full airplane is not the same as the span efficiency factor. However, the
two are related according to 1e = X π A + φ1 . X is the coefficient that relates the profile and parasite
drag to C L2 . For more information on this relation, the reader is referred to Ref. [53].
8.2 The Advantages of Wing Sweep
447
(b) 0.09
A=9
0.08
5
0.07
0.06
CD
0.05
3
0.04
(a)
0.03
0.03
A= 2
3
5
0.02
0.02
∞
CD
0.01
0
2
9
0.01
0
0.1
0.2
0.3
0.4
CL
0.5
0.6
0.7
0
0
0.1
0.2
0.3
CL
Fig. 8.13 The effect of wing aspect ratio on drag polar of an unswept wing with NACA 0012 airfoil
at freestream Mach numbers of 0.5 and 0.875 (after Ref. [60]). a M∞ = 0.5. b M∞ = 0.875
by the contribution of vortex lift to drag, but rather by the shock stall behavior of the
airfoil section and the associated compressibility drag. The low aspect-ratio wings
(2 and 3) have reduced drag compared to the higher aspect ratio wings (5 and 9).
This change in characteristics is associated with the delayed and less rapid rise of
drag as the aspect ratio is decreased [60].
The wing drag coefficient of a 6.4 aspect ratio wing composed of NACA airfoil
section [65(112) −213] for three constant-lift coefficient cases in the range of transonic
Mach numbers is shown in Fig. 8.14. The figure shows the critical Mach number for
the three constant C L cases. A seemingly constant plateau before Mcrit is followed by
a rapid and dramatic (nearly 7-fold) rise in drag up to sonic flight, where seemingly
another plateau is reached. The reader should note that the classical data shown in
Figs. 8.13 and 8.14 pre-date the supercritical airfoil era.
Implicitly, C D is dependent on the lift coefficient, C L , and Mach number, M
through the section wave drag coefficient, cdw . To correlate these wing properties to
the airfoil properties we use simple sweep theory and the assumption that the lift
coefficient of the airfoil section can be correlated to the average wing lift coefficient [32]:
448
0.16
0.14
Drag coefficient, CD (~)
Fig. 8.14 Transonic drag
characteristics of a 6.4
aspect-ratio-wing for several
lift coefficients (after Ref.
[69]). Note the drag
coefficient includes the
contribution of
vortex-induced drag
8 Aerodynamics of Swept Wings
AR=6.4
NACA Profile
65(112) - 213
CL = 0.4
0.12
0.2
0.10
0
0.08
0.06
0.04
Mcrit
0.02
0
0.6
0.7
0.8
0.9
1.0
1.1
Mach number, M∞ (∼)
CL
cos2 Λ
MΛ=0 = M cos Λ
clΛ=0 =
(8.31a)
(8.31b)
For each combination of C L and M, one can now find a value for the section drag
coefficient, cd (M, C L ) through the drag data of the section. We can subsequently
find cdw as follows:
(8.32)
cdw = cd − cd0
In Fig. 8.15 the comparison between measured drag data and the estimation of (8.28)
through (8.32) is shown for three wings with moderate sweep angle (30◦ and 35◦ ).
The two-dimensional data on which the estimation is based, stems from experimental
tests on airfoils and is therefore considered to be reliable. We see that the prediction
of the drag-divergence Mach number is within 10 % of the measured drag-divergence
Mach number. Even though the correlation is not very good, we do see the effect of
the wing’s lift coefficient on the drag-divergence Mach number and also the shape
of the estimated drag curve matches well with the experimentally obtained results.
In Ref. [32] it can be seen that for wings of higher sweep angle the correlation
between estimation and experimental results deteriorates (see Problem 8.6). With the
estimation of (8.29) it is possible to find a drag-divergence Mach number well beyond
1. In practice, it is shown in [32] that there exists a limit in drag-divergence Mach
8.2 The Advantages of Wing Sweep
449
(a)
(b)
Drag coefficient, CD (~)
.08
CL=.6
.06
.04
.4
CL=.6
0
.4
0
.02
0
0
.2
.4
.6
.8
1.0
1.2
0
.2
.4
.6
.8
1.0
1.2
Mach number, M∞ (~)
Mach number, M∞ (~)
(c)
(d)
Drag coefficient, CD (~)
.08
.06
.04
CL=.6
.02
0
.4
0
0
.2
.4
CL=.4
0
.6
.8
1.0
1.2
Mach number, M∞ (~)
0
.2
.4
.6
.8
1.0
1.2
Mach number, M∞ (~)
Fig. 8.15 Drag coefficient versus Mach number for three wings based on Eq. (8.29) (after Ref.
[32]). Solid line represents experimental data while dashed line represents the prediction according
to (8.29). Note λ is the taper ratio of the wing. a Λc/4 = 35, A = 3.5, λ = 0.2. b Λc/4 = 35,
A = 5.1, λ = 0.71. c Λc/4 = 30, A = 7.4, λ = 0.38. d Λc/4 = 35, A = 10, λ = 0.5
number of about 0.95. Clearly, more advanced methods (e.g. CFD) are required to
accurately predict the drag behavior of arbitrary wing shapes at high Mach numbers.
8.3 Inviscid Flow over Swept Wings
It has now been established that wing sweep has a favorable effect on the highsubsonic drag characteristics of wings. For the remainder of this chapter we will
illuminate some of the challenges associated with swept-wing design. In this section
we will start by examining the inviscid flow over a swept wing. Remember that
the flow over a body can essentially be divided in two categories: inviscid flow and
viscous flow. As stated before, viscous forces are only important in the boundary
layer and in the thin region around the shock wave. Outside these regions, the flow
can essentially be treated as being inviscid. We investigate how sweep influences
the inviscid flow field. We first present the effect of wing sweep on the streamline
curvature over infinitely swept wings. Subsequently, we show how the presence of a
450
8 Aerodynamics of Swept Wings
wing tip and a wing center section affects the streamline curvature. Furthermore, we
will demonstrate how the three-dimensional wing can be adapted in order to reduce
the adverse root and tip effects that are caused by wing sweep.
8.3.1 Flow over Infinite Swept Wings
If we assume that the inviscid crossflow velocity over an oblique wing is constant and
amounts to v∞ , then we can calculate the flow direction over the wing by comparing
the local flow component in chordwise direction to the flow component in spanwise
direction (crossflow component). The angle between these two values indicates the
direction of the flow:
v ∞
(8.33)
Λlocal = tan−1
u
If we know the local sweep angle of the streamline in the plan view of the wing, we
can also determine the path of the streamline over the wing. The following example
shows how this path can be determined.
Example 8.3 Consider a NASA SC(2)-412 airfoil at M = 0.2 and Re = 20 × 106
at a lift coefficient of cl = 0.5. Assume that this airfoil is in oblique flow where
Λ = 45◦ .
(a) Determine the pressure distribution about this airfoil.
(b) Determine velocity distribution, V /V∞ , on the upper and lower surface of oblique
wing in streamwise direction.
(c) Determine the path of the streamlines in plan view over the upper and lower
surface of the oblique wing.
Solution:
To determine the pressure distribution over the airfoil we choose to use a twodimensional Euler solver with an uncoupled boundary-layer solver.2 The pressure
distribution is shown in Fig. 8.16a.
Based on the C p distribution we calculate the velocity distribution about the airfoil
by using (3.11) and assuming that the flow is incompressible:
Voblique
= 1 − Cp
u∞
Furthermore, since v∞ is constant we can calculate the velocity distribution as follows:
2
2
+ v∞
V = Voblique
2 The solver that is used is MSES. See Refs. [21, 26] for details on this viscous-inviscid solver.
8.3 Inviscid Flow over Swept Wings
451
(b) 1.4
M = 0.2
c l = 0.5
Re = 20M
1.3
local velocity, V /V (~)
p
−1
1.2
8
pressure coefficient, C (~)
(a) −1.5
−0.5
0
0.5
NASA SC(2) 412
1.1
1
0.9
0.8
0.7
1
0
0.2
0.4
0.6
0.8
chordwise position, x/c (~)
1
0
0.2
0.4
0.6
0.8
streamwise position, x/c’ (~)
1
Fig. 8.16 Predicted pressure and velocity distributions about NASA supercritical airfoil. a Pressure
distribution. b Velocity distribution
The resulting velocity distribution is shown in Fig. 8.16b. Note that we denote the
streamwise chord length with c , where c = c/ cos Λ.
We employ (8.33) to calculate the local sweep angle of the stream line. Since both
u and v∞ are known, this is a straight-forward operation. To calculate the path of a
particle, we subsequently assume that a particle on the stream line follows a straight
line between two grid points. This line has an angle Λlocal, i between xi and xi+1 .
We calculate the displacement in y direction as follows:
yi+1
yi
xi+1 − xi
=
+
tan Λlocal, i
c
c
c
In addition, we assume that y1 = 0. We now have a displacement yi at every chordwise airfoil point, xi . We can now plot the stream line path in the coordinate system
of the airfoil. This has been shown in Fig. 8.17. However, we have rotated the coordinate system of the airfoil over an angle Λ, such that the streamlines can easily be
compared to the freestream direction of the flow.
If we examine the streamline pattern over the swept wing of Fig. 8.17, we observe
that the upper and lower streamlines do not follow the same path. While the lowersurface streamline stays close to the streamline direction of the free stream, the
upper-surface streamline clearly bends in the negative y-direction. If this wing were
aft-swept, the upper-surface streamline would therefore bend towards the wing root.
Even though we are aware of the fact that simple-sweep theory is not a very good
representation of the real flow conditions, the example above intuitively shows why
inviscid streamlines over a swept wing curve inboard or outboard depending on the
local pressure distribution.
452
8 Aerodynamics of Swept Wings
lower-surface streamline
upper-surface streamline
u
8
v
8
y


V
c
8
x
8
V
8
V
8
V

c’
Fig. 8.17 Plan view of inviscid stream lines on upper and lower surface of oblique wing according
to simple sweep theory
Fig. 8.18 Plan view of
stream lines over a
symmetric wing with zero
lift (after Ref. [39])
Sheared wing
of infinite span
Tip region of
sweptback wing
Centre region of
sweptback wing
In Fig. 8.18 a sketch of the streamline pattern over a finite swept wing is presented.
This wing has a symmetric airfoil profile, no twist, and is positioned in the flow at
zero angle of attack. In other words, the streamline pattern on the upper and lower
surface is purely caused by the thickness distribution of the airfoil and is identical
on the lower and upper side of the foil. If we zoom in on the “infinite” portion of the
wing (far away from root and tip), we indeed see the same streamline paths as for
8.3 Inviscid Flow over Swept Wings
453
le
ad
in
g
ed
ge
isobars
fluid particle
direction of motion
ili
tra
ng
0
0.
ge
ed
)
=
ak
Cp
pe
5
n
0. .0
io
0 .5
ct
-0 0 (su
.
-1
.5
-0
direciton of steepest
pressure gradient
Fig. 8.19 Notional isobar pattern over an infinitely long swept wing
the upper surface of the airfoil in Example 8.3. We see that the streamlines ahead of
the wing curve outboard. Over the leading edge of the airfoil the flow experiences
the most inboard curvature, while further downstream this is reduced.
The curvature of the streamlines over infinitely swept wings can be explained by
the three-dimensional pressure distribution over the symmetric wing. If we would
map the pressure distribution onto the wing’s surface, we would have a contour plot
of the isobars. The isobars would all be aligned with the sweep angle of the wing
(see Fig. 8.19). Any particle flowing over the leading edge is now being subjected
to a pressure gradient that is highest in the direction perpendicular to the isobars.
Therefore, the path of the fluid particle ahead of the suction peak is curved in the
direction of the strongest pressure gradient: inboard on an aft-swept wing. Behind
the suction peak there exists an adverse pressure gradient that is again highest perpendicular to the isobars. This gradient has a component which is in the outboard
direction. Therefore, the particle that is experiencing this adverse pressure gradient
is pulled more outboard on an aft-swept wing. This results in the s-shaped path that
a fluid particle describes over a swept wing under inviscid conditions.
8.3.2 Flow over Finite Swept Wings
In Fig. 8.20 we show the spanwise lift distribution over two plain semi-wings of finite
span (aspect ratio 8) without any taper, twist or camber. These results have been
computed by Matrics V, a viscous-inviscid flow solver based on the full potential
equation (2.199) for the inviscid solution [67, 68]. The first wing has a sweep angle
of 35◦ (aft-swept) while the second wing has a sweep angle of −35◦ (forwardswept). Both wings are at a lift coefficient of C L = 0.5. As a reference, we have
454
0.8
Local lift coeffient, cl (~)
Fig. 8.20 Spanwise
distribution of lift and drag
coefficient for a wing of
A = 8, no twist, no taper,
NACA 0012 section. Results
obtained with Matrics V [67,
68]. The following
conditions apply: C L = 0.5,
M = 0.2, Re = 20 × 106 .
8 Aerodynamics of Swept Wings
0.6
0.4
elliptical lift
Λ= 35 deg
Λ= −35 deg
0.2
0
0
0.2
0.4
0.6
0.8
1
Spanwise location, y/(b/2) (~)
also plotted the elliptical lift distribution for a wing of C L = 0.5 We see that the lift
distribution of the aft-swept wing is quite different from the forward swept wing. The
center of lift on the aft-swept wing is shifted more outboard. We also notice that the
lift distribution over the forward-swept wing is much closer to the ideal elliptical lift
distribution. This results in a higher span efficiency factor for the forward-swept wing
(φ = 0.97) than for the aft-swept wing (φ = 0.86). The change in lift distribution
can be explained by the wing sweep. As the center of the aft-swept wing is ahead
of the outboard wing, it effectively generates upwash over the outboard wing that
increases the effective angle of attack of the outboard wing. Vice versa, the tip of
the forward-swept wing is ahead of the inboard wing creating more upwash over
the inboard wing. The effective angle of attack over the inboard wing is therefore
increased leading to a higher local lift coefficient. The sweep angle of the wing
therefore has an impact on the lift-induced drag of the wing. It should be noted that
for both wings a change in taper, twist distribution or camber distribution can result
in lift distribution very close to the elliptical lift distribution.
Let us return to Fig. 8.18 where at the junction of two wing halves (termed center
section, also referred to as root), the requirement of symmetry forces the streamline
to be straight. In other words, there exists no net pressure gradient perpendicular to
the freestream direction at the root. For this statement to be true, the isobars should
be exactly perpendicular to the freestream direction at the root. At the root, the
isobars are therefore completely unswept. This means that the swept isobaric pattern
of Fig. 8.19 should somehow blend with the unswept isobars at the root. In Fig. 8.21
we see that the isobars indeed curve backwards at the root and become completely
orthogonal to the flow direction over the center section. At the tip, there is no strict
flow symmetry requirement. Therefore, the isobars do curve forward but show some
variation. This variation is caused by the three-dimensionality of the flow around
the tip. We see that the isobaric pattern on the wing with aspect ratio 1.05 is in fact
dominated by the presence of root and tip. The wing with aspect ratio 2 has a larger
portion where the isobars are aligned with the sweep angle of the wing. The effect
of root and tip on the pressure distribution over a swept wing are often termed the
root effect and tip effect, respectively.
8.3 Inviscid Flow over Swept Wings
455
A=2
p
C
=-
0
0.
-0 15
.
-0 20
.2
M5
in
.
-0
.2
5
-0
.2
0
-0
.1
5
-0
.1
0
-0
.0
5
0
.1
0
.0
5
-0
-0
.2
0
.1
5
-0
-0
p
C
=-
0.
1
-0 5
.2
0
-0
.2
M5
-0 in.
.2
5
A=1.05
Fig. 8.21 Isobar pattern over two 53◦ swept wings of different aspect ratio and zero lift (after
Ref. [39])
The fact that the isobars are curved at root and tip has a profound effect on the
chordwise pressure distribution at these two stations. The pressure distribution on
two spanwise locations along a swept wing at transonic Mach number is shown in
Fig. 8.22. The nomenclature on the figure indicates that a 9 % thick (symmetrical)
A=6
cC
t / c = 0.09
30
cT / cC = 1/3
M0 = 0.9
α= 0
cT
-0.6
-0.4
experiment
CFD
y=0
y/b/2 = 0.5
Cp*
Cp*
Cp -0.2
0
0.2
0.4
0.6
x/c
0.8
1.0
0.2
0.4
0.6
1.0
x/c
0.2
Fig. 8.22 Comparison between CFD and experiment on pressure distributions at two spanwise
locations on a swept wing at Mach 0.9 (after Ref. [29])
456
8 Aerodynamics of Swept Wings
Total velocity, V/V∞ (~)
(a) 1.15
Aspect ratio
Aspect ratio
1.05
2
Exp.
1.10
Centre
1.05
1.0
0
0.2
0.4
0.6
Chordwise station, x/c (~)
1.0
t/c=0.12 at 0.3c
(b) 1.20
Basic section RAE 101
t/c = 0.12
A = 2.2
40°
Centre-section
Mid-semispan
Exp.
Tip-section
Total velocity, V/V∞ (~)
1.15
Inf. sheared wing
Centre, with ϕ=48° Calc.
Tip
1.10
1.05
1.00
0
0.2
0.4
0.6
0.8
Chordwise station, x/c (~)
Centre
Mid-semispan
1.0
Tip
Fig. 8.23 Velocity distributions over swept wing sections (modified from Ref. [39]). a Velocity
distributions at center and tip sections of two 53◦ swept-back wings of constant chord. b Velocity
distributions at three spanwise stations on a 50◦ swept-back wing of constant chord with modified
section shapes
profile is used with a taper ratio of 1/3rd, the wing sweep angle at the mid-chord
is 30◦ , the aspect ratio of the wing is 6 and the (geometric) angle of attack is zero.
The wing is in Mach 0.9 freestream. The symbol C ∗p indicates the critical pressure
coefficient and thus the demarcation between locally subsonic and locally supersonic
flows on the wing at two spanwise locations (note that y = 0 is the wing root and
8.3 Inviscid Flow over Swept Wings
457
y/b/2 = 0.5 is at the mid point of the semispan). The appearance of shocks in both
spanwise locations where the flow transitions from supersonic to subsonic is evident
in the data. The computational method is based on the Transonic Small Disturbance
(TSD) equation. Although in general the trend in pressure distribution is captured, the
CFD prediction misses the peak suction and the chordwise location of the shock foot.
By comparing the two pressure distributions in Fig. 8.22 we see that the shock
wave near the root of the wing is positioned much farther aft than at 50 % of the semi
span. The root effect causes a backward shift in suction peak, a more aft position of the
shock wave, and a steeper adverse pressure gradient behind the shock. Furthermore,
the magnitude of the suction peak is also reduced. As is evident in Fig. 8.23a, the
curved isobars reduce the suction over the front part of the airfoil and increase the
suction over the aft part of the airfoil. This results in a net drag force that can
be attributed to the root effect. The tip experiences the opposite effect. The forward
sweeping isobars induce additional leading-edge suction over the tip of the wing. This
results in a force in opposite direction to the freestream (a thrust force). In addition,
we see in Fig. 8.23a that the suction peak is higher. In high-subsonic conditions, this
means higher supercritical velocities near the leading-edge of the tip and therefore
an early onset of shock formation and shock-induced separation.
The adverse effects at the root and tip can be remedied by modifying the wing
shape such that the chordwise pressure distribution at these locations is almost identical to the one at the mid point of the semispan. This can be achieved by modifying
planform and/or airfoil parameters. In Fig. 8.23b it can be seen how the thickness
distribution of the airfoils can be modified in order to achieve the same shape of
the pressure distribution at root, mid section, and tip. We see that the thickest point
on the root airfoil has been moved forward to move the suction peak more forward.
Analogously, the thickest point on the tip airfoil has been moved rearwards. We now
have three different airfoils at the three aforementioned spanwise stations. Because
they are part of a three-dimensional swept wing, they can be designed such that the
shape of the pressure distribution in the design condition is the same. By increasing
the thickness of the root airfoil and the tip airfoil, the pressure distribution can be
exactly matched to the pressure distribution of the section at the mid semispan. A
more forward position of the thickest point and an increase of the thickness-to-chord
ratio are indeed two characteristics of root airfoils of modern high-subsonic wings.
A more aft-position of the thickest point on the tip airfoil has a negative effect
on the maximum lift coefficient due to the sharper leading edge that results. To
remedy the tip effect we can therefore also choose to modify the tip planform shape
of the wing. Küchemann and Weber [39] introduced a parabolically curved leading
edge leaving the streamwise section shape unaltered. The curved part of the leading
edge extends inboard by 25 % of the tip chord. The idea behind such a shape is to
increase the leading-edge thrust not by increasing the suction peak, but by reducing
the positive pressure near the leading edge. Such a Küchemann tip does not straighten
the isobars completely but increases their spacing near the leading edge significantly
(see Fig. 8.24). A Küchemann tip can easily be designed by following the guidelines
presented on the right-hand side of Fig. 8.24.
458
8 Aerodynamics of Swept Wings
-0.3
min.
53°
Pe
ak
-0.2
-su
P1
cti
on
-0.1
P1
lin
A
e
B
Cp=0
C
53°
P2
A’ A simple geometric
construction of the
curved tip as the
envelope of the
lines AA’, BB’, etc
B’ where A, B, and A’, B’
are equal intercepts
on P1P 2 and P2P3
respectively
C’
A=2
ϕ =53°
Aerofoil section
with t/c=0.12
at 30% c.
Conventional
rounded tip
Tip with curved
leading edge
P3
P3
Fig. 8.24 Effect of planform modifications to isobar pattern at the wing tip (after Ref. [39])
So far, we have discussed modifications to root and tip such that the velocity
distribution due to thickness is constant over the span of the wing. All the previous examples concern untwisted wings without camber and in symmetric flow
conditions (zero angle of attack). To produce lift, the wing is usually exposed to
an angle of attack (unsymmetric flow conditions). Figure 8.25 shows the pressure
distribution at 5 different spanwise stations of a 45◦ swept wing of aspect ratio 3.
This wing has a symmetric airfoil section and exhibits no twist. We observe that
the pressure distribution around the midspan of the semi-wing remains constant
(Fig. 8.25d). However, if we move inboard, we see a gradual change in pressure distribution. The suction peak reduces, and close to the center section the suction peak
has shifted back to about 20 % of the local chord length. Furthermore, the upper
surface experiences more suction over the aft part of the airfoil. If we focus on the
lower surface, we observe that ahead of the airfoil’s thickest point, a higher pressure
is present near the center section than further outboard. These three changes in pressure distribution have the combined effect that the pressure-induced drag (form drag)
around the center airfoil increases compared to the airfoils positioned more outboard.
In other words, the application of sweep on a lifting wing introduces additional form
drag near the midsection.
8.3 Inviscid Flow over Swept Wings
459
(b) -0.6
Pressure coefficient, Cp (~)
Pressure coefficient, Cp (~)
(a) -0.6
η=0.07
-0.4
-0.2
0
0.2
0
0.2
0.4
0.6
0.8
1.0
η=0.14
-0.4
-0.2
0
0.2
0
0.4
(d) -0.6
Pressure coefficient, Cp (~)
Pressure coefficient, Cp (~)
(c) -0.6
η=0.27
-0.4
-0.2
0
0.2
0
0.2
0.2
0.4
0.6
0.8
Chordwise position, x/c (~)
0.6
0.8
1.0
Chordwise position, x/c (~)
Chordwise position, x/c (~)
1.0
η=0.41
η=0.61
-0.4
-0.2
0
0.2
0
0.2
0.4
0.6
0.8
1.0
Chordwise position, x/c (~)
Fig. 8.25 Pressure distribution at various spanwise stations over a 45◦ swept wing of aspect ratio
y
y
y
y
y
3 (after Ref. [70]). a b/2
= 0.07. b b/2
= 0.14. c b/2
= 0.27. d b/2
= 0.41 and b/2
= 0.61
We have already seen that a modification of the thickness distribution of the root
can have an effect on the pressure distribution over the airfoil at zero lift. It was shown
by Weber et al. that this modification only works for wings at low angles of attack. At
angles of attack of 4◦ and upwards the isobars are distorted near the center because of
the change in chordwise lift in that region [70]. To bring the isobars near the root more
forward at elevated angles of attack, Küchemann [37] proposes a different solution:
increase the incidence and add negative camber to the wing root. These combined
measures ensure the desired value of the lift coefficient at the center section and a
chordwise lift distribution that matches to the one on the outboard wing. Küchemann
proposes an elegant method based on a vortex representation of the wing to predict
the required twist and camber distribution that matches a predefined chordwise and
spanwise lift distribution. The interested reader is referred to Refs. [37, 38] for further
reading. In Fig. 8.26 we show the result of wind tunnel tests on a 45◦ swept wing of
aspect ratio 3. As can be seen in the figure, the center airfoil features a positive twist
angle of 2.8◦ , which gradually reduces to zero at the mid-semispan. The graphs in
Fig. 8.26a, b show the chordwise lift distribution at various spanwise stations for an
angle of attack of 2.3◦ and 4.4◦ , respectively. For comparison, the dashed line shows
the lift distribution for the center airfoil without any twist or camber. The effect of
460
8 Aerodynamics of Swept Wings
center section of wing with
-1.2% camber and 3.1º twist
(b) 1.4
basic symmetrical section, t/c = 0.12
1.0
basic wing
0.8
η
0
0.05
0.24
0.51
0.6
wing with
camber and
twist
0.4
α =2.3°
0.2
0
0.2
0.4
0.6
0.8
Chordwise station, x/c (~)
1.0
Change in pressure distribution to to lift, -Δ Cp (~)
Change in pressure distribution to to lift, -Δ Cp (~)
(a)
α =4.4°
1.2
basic wing
1.0
η
0.8
0
0.05
0.24
0.51
wing with
camber and
twist
0.6
0.8
0.6
0.4
0.2
0
0.2
0.4
1.0
Chordwise station, x/c (~)
Fig. 8.26 Chordwise lift distributions between on a symmetrical wing and on a wing with camber
and twist. Both wings are untapered, have a 45◦ sweep angle, and an aspect ratio of 3 (after Ref.
[70]). a α = 2.3. b α = 4.4
the modification is striking: the lift distribution at the center section matches closely
to the more outboard sections. This indicates that the suction peak of the pressure
distribution has shifted forward considerably. These conclusions hold for both angles
of attack that are shown, indicating that the result of this modification is independent
of the overall lift coefficient of the wing.
In summary, we have seen two modifications to the wing root to ensure straight
and parallel isobars up to the center section of a swept wing. The first modification
requires the thickest point of the root airfoil to move forward and the thickness-tochord ratio to grow. This ensures that supervelocities due to thickness are constant
over the span. The second modification is an increase in local incidence angle of
the root airfoil in combination with a reduction in camber. This results in a constant
chordwise lift distribution with span and ensures that the spanwise lift distribution
meets a designated shape. Both modifications gradually merge with the baseline
airfoil within one chord length of the wing. Each of these modifications can be seen
in the wing of the Airbus A310 aircraft, which is depicted in Fig. 8.27. We can see
that the airfoil in the outboard wing has its thickest point at 28 % of the local chord.
In addition, the thickness to chord ratio in the wing outboard of the kink gradually
decreases from 11.8 % at the kink to 10.8 % at the tip. The incidence angle of the
airfoil at the kink is +1.5◦ while at the tip the incidence angle is reduced to −3◦
8.3 Inviscid Flow over Swept Wings
461
28%
28%
22%
28%
28%
11,8/6%
2
+1°5
+3°
0
1
1
7006
+5°
2
2
4096
11,8%
15%
2017
3400
11,6%
+1°5
1
10,8%
-3°
8380
3
4
5
6
7
8
9
10
11
12
13
Meters
Fig. 8.27 Example of a wing for a high-subsonic transport aircraft (after Ref. [7])
to postpone stall at the tip3 (see Sect. 8.4.3). If we turn our attention to the root airfoil,
we see its thickness has grown to 15 %, its thickest point has moved forward to 22 %,
the incidence angle is raised to +5◦ , and clearly some negative camber has been
applied. All of these modifications are there to account for the three-dimensional flow
effects of swept wings and to make sure that the form drag of the wing is minimized.
Additional benefits include a reduction in wing weight and a larger internal wing
volume.
8.4 Viscous Flow over Swept Wings
In Sect. 1.5 we demonstrated that it is important to include viscosity in the governing
flow equations in transonic conditions. We showed in Sect. 8.3 that the flow outside
of the boundary layer tends to curve inboard over an aft-swept wing. We will show
in this section that the boundary-layer flow has the tendency to move outboards
under an adverse pressure gradient over an aft-swept wing. This so-called boundarylayer crossflow has an effect on boundary-layer transition and flow separation. The
characteristics of a three dimensional boundary-layer over a swept wing, along with
the implications for transition and separation are discussed in Sects. 8.4.1, 8.4.2, and
8.4.3, respectively.
8.4.1 Three-Dimensional Boundary Layer over Swept Wings
In the early 1950s when wind tunnel tests on swept wings commenced, it was
observed that the transition process in boundary layers over swept wings differed
from that over unswept wings. Where transition over unswept wings is dominated
by the Tollmien-Schlichting waves (see Sect. 6.4.2), it was noted that the transition
3 A negative incidence angle of the wing is often termed “washout”. This wing would therefore
have +3◦ of washout at the tip.
462
Fig. 8.28 Notional sketch of
swept-wing boundary-layer
profile (after Ref. [56])
8 Aerodynamics of Swept Wings
yt
wt
ut
inflection
point
resulting
component
crossflow
component
tangential
component
zt
wall shear
xt
mechanism over swept wings was remarkably different. The combination of pressure gradient and wing sweep deflects the streamlines inboard (see Fig. 8.18). In the
pressure-recovery region, closer to the trailing edge, the streamlines are bent outboard
again under influence of the local pressure gradient. Outside the boundary layer, the
high momentum of the flow causes a relatively small deflection of the streamline
due to the presence of the pressure gradient. However, inside the boundary layer,
the momentum of the flow is considerably reduced due to friction with the surface.
Therefore, the adverse pressure gradient causes a larger deflection of the streamlines
inside the boundary layer resulting in boundary-layer crossflow.
In Fig. 8.28 a notional boundary-layer profile is shown over a swept wing. The
axes system is oriented such that the xt -axis is aligned with the inviscid streamline
outside of the boundary layer. The boundary-layer profile spirals upwards creating
an angle between the wall shear vector and the inviscid stream line. The projections
of the boundary-layer profile on the xt yt -plane and z t yt -plane are also shown. The
shape of the boundary layer on the xt yt -plane looks familiar to us. However, in
the z t yt -plane the cross-flow profile looks entirely different. At the wall and at the
boundary-layer edge, the crossflow component is zero. Somewhere in the middle,
the crossflow component reaches a maximum. The notional sketch of Fig. 8.28 is
somewhat misleading as quantitative results show that the value of maximum crossflow velocity is typical around 5–10 % of the boundary-layer-edge velocity [9]. The
crossflow profile itself shows an inflection point (where the curvature changes from
concave to convex).
8.4 Viscous Flow over Swept Wings
463
The profile of a cross-flow boundary layer is influenced by the pressure
gradient along the inviscid streamline outside the boundary layer. In Fig. 8.29
the three-dimensional boundary-layer development of a laminar boundary layer is
depicted, after Arnal and Casalis [9]. Here, xM represents the position along the
streamline where the velocity outside the boundary layer (u e ) reaches its maximum value (i.e. the position of the suction peak). From the leading edge to the
chordwise location xM , the crossflow (w) is directed towards the concave part
of the external streamline. It subsequently increases rapidly due to the external
flow acceleration under the negative pressure gradient. The larger the magnitude
of the pressure gradient, the more crossflow results. As xM is approached, the
intensity of the pressure gradient decreases, resulting in a decrease of the crossflow amplitude. At x > xM , the pressure gradient becomes positive, the curvature
of the external streamline changes sign and the cross-flow velocity close to the
wall reverses its direction resulting in an S-shaped profile. If the positive pressure
gradient is strong enough, the crossflow velocity profile can be completely reversed.
In the same region, an inflection point appears in the streamwise boundary-layer
profile as a result of the adverse pressure gradient.
In Fig. 8.30 the boundary-layer flow over a swept wing with engine nacelle is
visualized by means of oil flow. What we see here is essentially the direction of the
shear-stress vector at the surface of the wing. Note that the location of the shock
wave is distinctly visible. It extends from a spanwise position close to the engine
all the way to the tip. The dark area behind part of the shock wave is caused by
shock-induced separation. We know that the pressure coefficient in this area is lower
than for the attached flow over the inboard wing. This results in a strong spanwise
pressure gradient that has a large effect on the boundary-layer flow over the inboard
wing. The shear stress vector at the surface at some instances is rotated over more
than 90◦ . Clearly, the shockwave over the inboard wing causes a kink in the she
shear-stress vector. The steep pressure gradient that is induced by the shock wave
causes a large reduction of the boundary-layer mean-flow velocity in u-direction. The
∂p
<0
inviscid streamline ∂x
suction peak
xM
∂p
>0
∂x
u
w
β0
wall shear stress direction
Fig. 8.29 Laminar boundary layer development on a swept wing. xM is the location of the inviscid
streamline inflection point. β0 is the angle between the wall shear stress vector and the inviscid
streamline (after Ref. [9])
464
8 Aerodynamics of Swept Wings
Fig. 8.30 Boundary layer cross flow visualized on a swept wing with fluorescent oil on the right
wing and mini-tufts on the left wing (Photo Douglas Aircraft Company)
presence of the strong spanwise pressure gradient subsequently causes an increase
in crossflow resulting in the observed kink in shear-stress vector. Note that if we look
at the pattern inboard of the engine (where there exists no shock), the shear-stress
vector shows a more gradual curvature. In this area it is subjected to a much lower
adverse pressure gradient than the flow outboard of the engine.
8.4.2 Transition
In Sect. 6.4.2 the transition of a laminar boundary layer to a turbulent boundary layer
was presented. We showed that flow over a flat plate transitions from laminar to
turbulent due to the amplification of Tollmien Schlichting (TS) waves. In addition, in
Sect. 7.3 we demonstrated that laminar-flow airfoils benefit from a favorable pressure
distribution to limit this amplification and postpone transition to more than 50 %c
over the suction-side of an airfoil. On swept wings it is more difficult to define a
8.4 Viscous Flow over Swept Wings
465
TS instability
Crossflow instability
V∞
Λ
Leading-edge contamination
Attachment-line instability
Centrifugal instability
TS instability
Crossflow instability
Fig. 8.31 Notional sketch of instability mechanisms that might dominate in different parts of
a three-dimensional swept wing boundary layer (courtesy of Tempelmann [63], modified from
Ref. [19])
wing geometry that supports natural laminar flow over a substantial part of the wing
surface. Numerous efforts have been undertaken to achieve laminar flow over swept
wings and other lifting surfaces such as the horizontal or vertical tail. However, the
vast majority of these endeavors have not been implemented in production aircraft.
So, why is postponing transition so difficult on swept-wing aircraft? To answer this
question, we need to investigate the fundamental instability mechanism in threedimensional boundary layers. Apart from the TS waves, three additional instability
mechanisms can be identified: attachment-line instability, cross-flow instability, and
centrifugal instability. Each of these mechanisms can cause the boundary layer to
transition from laminar to turbulent. The region in which each of the mechanisms is
important is shown in Fig. 8.31.
The attachment-line instability refers to the instability that occurs at the leadingedge attachment line. The leading-edge attachment line is an imaginary line that
connects the maximum pressure points on the leading-edge of the wing. For an
airfoil, this would be a stagnation point but when sweep is added, the flow does
not fully come to a standstill but starts moving outboard along the leading edge.
Laminar-flow flight experiments with swept-wing aircraft in the 1950s and 1960s
had demonstrated unexpected early transition near the leading edge. Pfenniger [47]
demonstrated that the stability of the attachment line boundary layer can be correlated to the momentum-thickness Reynolds number at the attachment line, Reθal .
He showed that when Reθal > 240 the attachment line is inherently unstable and
causes transition. When 90 < Reθal < 240, the boundary layer is stable but susceptible to exterior disturbances that can cause transition. For example, disturbances
can be caused by local surface waves or a turbulent boundary layer stemming from
the wing/fuselage junction (referred to as leading-edge contamination). When Reθal
is less than 90–100, the boundary layer is stable, resistant to exterior disturbances,
and does not propagate turbulence along the wing span. To calculate the momentum thickness at the attachment line, Saric and Reed [54] propose the following
approximate relation (for incompressible flow) based on [47]:
466
8 Aerodynamics of Swept Wings
r/c = 1.8%
NASA SC(2) 0412
r/c = 0.8%
NACA 661 212
Fig. 8.32 Estimated airfoil radii of two airfoils (belonging to Example 8.4)
0.404
Reθal = √
1+ε
V∞r sin2 Λle
v cos Λle
(8.34)
where r is the leading-edge radius of the airfoil, v is the dynamic viscosity of the fluid,
and ε is the thickness-to-chord ratio of an equivalent ellipse, matching the leading
edge geometry. The value of ε can be computed as a function of the leading-edge
radius and the maximum thickness of the airfoil, tmax [12]:
ε=
2r/c
tmax /c
(8.35)
A quick assessment of (8.34) reveals that in order to reduce Reθal one needs to
have a small leading-edge radius and/or a low sweep angle. The following example
illustrates this.
Example 8.4 Consider a sheared wing without taper and twist and with a chord of
6.0 m. We consider two airfoils for this wing that are both oriented in the streamwise
direction: a NACA 661 212 and a NASA SC(2) 0412 (see Fig. 8.32). The first airfoil
has a leading-edge radius of r/c = 0.008, while the second airfoil has r/c = 0.018.
Compute for each airfoil, the maximum sweep angle for which attachment-line disturbances are damped, i.e. Reθal < 100. Assume that V∞ = 100 m/s and that sea-level
conditions apply.
Solution:
We first calculate the value for the dynamic viscosity at sealevel:
ν0m ISA =
μ 18.1 × 10−6 [kg/(sm)]
=
= 14.91 × 10−6 [kg/m2 ]
ρ 0m ISA
1.225 [kg/m3 ]
Subsequently, we employ (8.35) to compute that ε is 0.13 and 0.30 for the NACA
661 212 and a NASA SC(2) 0412, respectively. Using these values, we calculate
the value of the attachment-line momentum thickness using (8.34) for a range of
leading-edge sweep angles. If we plot this relation for both of the airfoils and add
a line for the transition boundary we can find the leading-edge sweep angles for
which attachment-line transition occurs. This is shown in Fig. 8.33. We can therefore
8.4 Viscous Flow over Swept Wings
467
conclude that for the given conditions, the maximum leading-edge sweep angle for
the SC(2) 0412 amounts to only 18◦ , while the 661 212 allows a sweep angle of 26◦ .
A different criterion is proposed by Poll [49]. For compressible flows, he defines
a Reynolds number based on the length scale ζ, and the boundary-layer-edge crossflow velocity along the attachment line, ve :
Reζal =
we ζ
ν
(8.36)
where ν is the kinematic viscosity (μ/ρ) and ζ is defined as follows:
ζal =
ν
du (8.37)
dx x=0
Poll demonstrates that when Reζal < 245 attachment line turbulence decays. This corresponds to an attachment-line momentum-thickness Reynolds number of 100. For
compressible flows, the dynamic and kinematic viscosity are dependent on the flow
temperature through Sutherland’s equation (2.81). Since the temperature changes in
the boundary layer, Sturdza [62] proposes to use an intermediate temperature, T̄ :
T̄ = Te + 0.1(Tw − Te ) + 0.6(Tr − Te )
(8.38)
Fig. 8.33 Attachment-line
transition of two airfoils as a
function of leading-edge
sweep angle (belonging to
Example 8.4)
attachment−line momentum−thickness
Reynolds number, Reθ
where Tr is the recovery temperature, which in adiabatic conditions equals the wall
temperature, Tw . Te is the inviscid edge temperature of the boundary layer. If we
write the cross-flow component as we = V∞ sin Λ, the Reynolds number based on
¯ ζal , can then be expressed as follows:
this intermediate temperature, Re
200
150
2)
SA
50
A
AC
12
04
NA
transition boundary
100
(
SC
212
66 1
N
0
0
5
10
15
20
25
Leading−edge sweep angle, Λle (deg)
30
468
8 Aerodynamics of Swept Wings
¯ ζal = V∞ sin Λ
Re
du ν̄ dx
x=0
(8.39)
Assuming the boundary-layer flow remains laminar along the attachment line,
crossflow instabilities are a second cause for the early onset of transition. The type
of crossflow instability that occurs within the boundary layer depends on the receptivity of the boundary layer to external disturbances such as freestream turbulence,
acoustic waves, or roughness. Receptivity is defined as the mechanism by which
freestream disturbances enter the boundary layer and create the initial conditions for
unstable waves [55]. For low levels of turbulence (Tu < 0.2 %), crossflow transition
is dominated by stationary waves, while for higher turbulence levels (Tu > 0.2 %)
transition is caused by traveling waves [19]. The inflection point in the crossflow
boundary-layer profile (see Fig. 8.28) causes crossflow vortex structures, which have
their rotational axis within a few degrees aligned with the inviscid streamline. These
crossflow vortices all rotate in the same direction and create a “cat’s eye” structure
when viewed in the streamwise direction. In Fig. 8.34 the notional vortex structure
of these vortices is shown along with a photo showing the cross flow vortices. The
y
(a)
z
(b)
Fig. 8.34 Cross flow vortices over a swept wing. a Notional sketch of “cat’s eye” structure of crossflow vortices (after Ref. [50]). b Naphthalene surface patterns showing crossflow vortices (Photo
from Ref. [54]; reprinted by permission of the American Institute of Aeronautics and Astronautics,
Inc.). Flow is from left to right
8.4 Viscous Flow over Swept Wings
469
breakdown of these vortices results in a serrated transition pattern over the span of
the wing as is clear from Fig. 8.34b.
Experiments show that surface roughness represents the key parameter responsible for the initiation of stationary crossflow vortices. Increased surface roughness
hastens the onset of crossflow vortices and moves the transition region upstream [19].
The receptivity to a two-dimensional surface protuberance and especially acoustic
disturbances is very weak.
The effect of turbulence levels on crossflow transition is more complicated than
for Tollmien-Schlichting waves in a two-dimensional boundary layer. Remember
that for TS waves there exists a logarithmic relation between the turbulence level and
the critical amplification factor, N (see Sect. 6.5.4). In contrast to two-dimensional
boundary layers, up to moderate values of turbulence intensity, freestream turbulence
affects transition only indirectly. It attenuates the growth of the stationary vortices and
retards transition. Only at values of Tu > 0.2 %, depending on the roughness height,
do traveling modes become dominating. The stationary vortex structure is washed
away by traveling crossflow waves, which results in an earlier onset of transition and
a more diffuse transition pattern [54]. A further increase of freestream turbulence
then accelerates transition. An airplane in flight is likely to see turbulence levels
below 0.1 %, while turbulence levels in the wind tunnel can be higher than 0.2 %.
Since the crossflow transition mechanism is different for both situations, it is difficult
to use wind-tunnel tests for the prediction of crossflow transition on full-scale wings.
Computationally expensive simulation methods such as direct numerical simulation
(DNS) can be used to include the effect of receptivity on the crossflow instability
mechanisms and transition.
The final transition mechanism in three-dimensional boundary layers is the centrifugal instability. Centrifugal flow occurs when streamlines are curved by the presence of convex or concave surfaces, as was shown in Fig. 6.4. If, in an inviscid circular
flow, |r V | decreases with an increase in r the Rayleigh circulation criterion is satisfied and the flow is unstable. In this case the instability is in the form of steady,
streamwise-oriented, counter-rotating vortices as shown in Fig. 8.35. These vortices
are commonly called Görtler vortices after H. Görtler who first described the formation of these vortices over concave walls in his doctoral dissertation in 1940 [27].
The Görtler instability is an important boundary-layer instability that, under some
conditions, leads the flow through a transition to turbulence. It is known that a Görtler
instability can cause transition on the wall of a supersonic nozzle in a boundary layer
that would otherwise be laminar. Moreover, the Görtler vortex structure exists in a
laminar boundary layer over a concave surface such as turbine-compressor blades.
If we return to Fig. 8.31, we see that Görtler instabilities are indicated as a possible
cause for transition over the concave region on the lower surface of the wing. However, it has been shown that beyond a very small sweep angle, the principal instability
on a swept wing with concave curvature is a crossflow instability and not a Görtler
instability. This reduces the concern about Görtler vortices in swept-wing flows [55]
but makes them important for unswept wings such as found on sail planes.
470
8 Aerodynamics of Swept Wings
Fig. 8.35 Notional
representation of Görtler
vortices over a concave wall
(after Ref. [27])
8.4.3 Separation
The oil flow pattern in Fig. 8.30 is aligned with the local shear vector on the wing’s
surface. This image shows how the flow inside the boundary layer moves outboard.
We should recall that the boundary layer thickness at any point on the surface is a
function of the distance that the flow inside the boundary layer has traveled from
the leading edge (see the discussion on p. 305). With that notion, we can easily
deduce that the boundary layer over the outboard wing is thicker than what would
be expected based on two-dimensional boundary-layer properties. Conversely, the
crossflow in the boundary layer causes the thickness of the boundary layer at the
root to decrease compared to its two-dimensional counter part. Also recall that a
thicker boundary layer has less momentum close to the wall and therefore tends to
separate more easily than a thin boundary layer when an adverse pressure gradient
is present. We can therefore deduce that the root of a swept wing is relatively stall
resistant, while the outboard wing is more susceptible to stall. Since most aft-swept
wings are also tapered, this effect is amplified due to the lower Reynolds number
at the tip compared to the root. In Fig. 8.36 we show that for a 45◦ aft-swept wing
of aspect ratio 3 this stall behavior can indeed be observed. The airfoil close to the
y
= 0.91) stalls when the wing reaches an angle of attack of 17◦ . The more
tip ( b/2
inboard profiles stall at a progressively higher angle of attack. Even at an angle of
attack of 27.5◦ the two most inboard profiles show no sign of stall yet.
Due to the addition of wing sweep to postpone adverse compressibility effects,
we have now encountered a serious problem at low subsonic Mach numbers. The
early onset of tip stall could cause the outboard wing to lose lift. As the tips of an
aft-swept wing are located behind the airplane’s center of gravity, this effectively
creates a nose-up pitching moment. This pitching moment, in turn, causes the angle
of attack to increase, aggravating the stall even further! This unstable stall behavior
was a serious threat to the first generation of swept-wing aircraft such as the F-86
Sabre and MiG-15. If the onset of separation starts asymmetrically (one tip stalling
before the other), a large rolling moment results. This phenomenon is called roll off
and was another threat to early jet fighters with swept wings.
To prevent tip stall on swept wings, various measures can be taken. The MiG-15
featured full-chord fences to stop the cross flow of the boundary layer and start a fresh
8.4 Viscous Flow over Swept Wings
471
1.4
45°
20’’
0.14
1.2
0.27
0.41
0.61
0.83
1.0
Lift coefficient, CL (~)
η=0
BE
38’’ Span
Aspect ratio 2
59” Span
Aspect ratio 3.04
0.91
0.8
0.6
y/c=
0
0.2
0.4
0.6
0.9
0.41
0.61
0.83
1.28
1.35
0.4
η=
0
0.14
0.27
Calculated
0.91
0.2
0
0
0
0
0
0
0
5°
10°
15°
20°
25°
Angle of attack, α (~)
Fig. 8.36 Local lift coefficients over a 45◦ swept wing of aspect ratio 3 (after Ref. [71])
boundary layer outboard of each of the fences (see Fig. 8.37). We can easily imagine
that such fences cause interference drag at high subsonic conditions and are therefore
not ideal. Therefore, over the course of several decades several more subtle add-ons
were invented to prevent tip stall while keeping the drag penalty at high-subsonic
speeds acceptable. Examples include small fences, leading-edge snags, vortilons,
and vortex generators. The interested reader is referred to [46] for examples of these
devices and their effect on high-lift performance of various transport and military
aircraft. In addition, Ref. [46] also presents modifications to the airfoil shape of the
outboard wing to postpone tip stall.
Whether wing sweep causes a higher or lower maximum lift coefficient depends
on multiple factors. The aspect ratio and the thickness-to-chord ratio are among them.
An important parameter is the sharpness of the airfoil. The leading-edge sharpness
can be represented by the value of Δy (see Fig. 7.1) and expressed as a fraction of
the chord length. For untwisted wings of a constant airfoil section and aspect ratio
larger than 4, it is shown in Ref. [31] that when Δy > 2 %c, the maximum lift
coefficient decreases with increasing sweep angle. In other words, the maximum
lift coefficient of wings with blunt leading edges deteriorates with increasing sweep
472
8 Aerodynamics of Swept Wings
Fig. 8.37 Example of a swept-wing aircraft featuring wing fences: MiG-15 (Photo USAF)
angle. However, when Δy < 2 %c the maximum lift coefficient tends to increase
with increasing sweep angle. This is attributed to the formation of a leading-edge
vortex which forms when the flow separates from the leading edge. Leading edge
separation can lead to the formation of a separation bubble and the spanwise pressure
gradient converts this bubble into a stable leading-edge vortex. The high velocity in
the leading edge vortex causes a low static pressure, which acts on (part of the) wing’s
upper surface. The leading-edge vortex is therefore contributing to the wing’s lift and
can therefore increase the maximum lift coefficient of the wing. At the same time,
since the suction peak at the leading edge has diminished, the leading-edge vortex
also increases the drag of the wing (see Problem 8.16).
It is shown in Ref. [48] that the formation of a leading-edge vortex is dependent on
a combination of leading-edge sweep angle, airfoil sharpness, angle-of-attack, and
Reynolds number. In Fig. 8.38, the oil-flow pattern of two wings is shown. For the
sharp-nosed wing (Fig. 8.38a) we observe a streamline pattern that is indicative of
the presence of a leading edge vortex. In this particular example, a continuous flow
structure exists running from the leading edge at the root to the trailing edge at the
tip. The lines of tip separation and leading-edge separation have joined in this case.
If we compare this to the blunt-nosed wing of Fig. 8.38b, we see a different pattern.
There is no indication of the presence of a vortex but the flow is starting to separate
near the wing tip. However, as is shown in Ref. [48], this wing produced a part-span
vortex at angles of attack beyond 20◦ , demonstrating that the critical angle-of-attack
to produce a leading-edge vortex increases with increasing nose radius.
8.4 Viscous Flow over Swept Wings
473
(a)
Surface oil flow
Kink induced by
vortex burst
Attachment
line
Secondary
seperation
line
(b)
Surface oil flow
Short seperation
bubble
Seperation
line forming
Fig. 8.38 Surface oil-flow about two wings of A = 3.39, λ = 1, Λ = 45◦ , α = 14◦ , and
Re = 2.1 · 106 . Both wings feature a RAE 101 airfoil (t/c = 10 % at 30 %c) with modified leading
edge (after Ref. [48]). a Leading edge radius = 0.03 %c. b Leading edge radius = 3.0 %c
As we know from the design of high-subsonic airfoils, many supercritical airfoils
have a relatively blunt nose. It is therefore unlikely that on moderately swept wings
with blunt airfoils a leading-edge vortex occurs before trailing-edge separation starts.
Many high-subsonic aircraft that employ these airfoils therefore experience a reduction in maximum lift coefficient due to the introduction of wing sweep. On the other
hand, many fighter aircraft have much thinner (symmetric) wings with a sharper
leading edge. When sufficient wing sweep is present, the formation of the vortex can
positively increase their maximum lift coefficient. There are also many examples of
474
8 Aerodynamics of Swept Wings
Fig. 8.39 Examples of highly-swept surfaces to create a leading-edge vortex. a Northrop YF-17
featuring leading-edge root extensions ahead of the main wing (Photo USAF). b Fokker 50 featuring
a dorsal fin ahead of the vertical tail (Photo Aleksandrs Samuilovs)
fighter aircraft that feature leading-edge root extensions (also known as “strakes”).
These are essentially highly swept wings without a distinct profile but with a sharp
leading edge (see Fig. 8.39a). These surfaces are added to create a stable vortex and
increase the maximum lift coefficient of the airplane and the corresponding stall
angle-of-attack. This is also true for the highly-swept dorsal fins that are sometimes
added to vertical tails of low-subsonic and high-subsonic aircraft (see for example
Fig. 8.39b). They allow for a higher sideslip angle, increase the maximum side-force
coefficient of the fin, and avoid the “rudder lock” problem.
8.5 Aeroelasticity of Swept Wings
In this final section of this chapter (and of this textbook!), we are turning our attention
to another Achilles heel of aircraft with swept wings: aeroelasticity. Aeroelasticity
is the common term that describes the interaction between aerodynamic, elastic and
inertial forces as described by the Collar triangle [18] in Fig. 8.40. Static aeroelasticity is the interaction between aerodynamic and elastic forces. Examples include
wing bending due to aerodynamic loading or wing twisting due to control-surface
deflection. Flutter is an example of dynamic aeroelasticity. The inertial forces play
an important role in dynamic aeroelasticity as they affect the natural frequency of
the structure. Since the early days of flying, flutter, wing divergence, and control
effectiveness have been primary problems for airplane designers [22]. It can be easily imagined that with the increase in aircraft size, their flexibility has increased
considerably. Combine this with high dynamic pressures and compressibility effects
in the transonic realm and aeroelasticity becomes a very important and complex
phenomenon.
As many passengers of commercial aircraft have experienced, atmospheric turbulence can lead to uncomfortable vibrations inside the aircraft. These vibrations are
often caused by atmospheric instabilities and are usually a combination of a rigidbody oscillation and elastic vibrations. Atmospheric turbulence takes the shape of
8.5 Aeroelasticity of Swept Wings
475
Fig. 8.40 Collar’s
aeroelastic triangle (after
Ref. [18])
Inertial Forces
vibration
stability and control
Dynamic Aeroelasticity
Elastic Forces
Aerodynamic Forces
static aeroelasticity
discrete gusts that can come from any direction. As the reader knows, the loads on the
aircraft are directly coupled to the local velocity. Any change in local velocity (both
in magnitude and direction) immediately changes the loads on the aircraft. Aerodynamically induced vibrations can induce a cyclic loading on the structure which
can accelerate the growth of micro-cracks in metal parts. This is known as metal
fatigue and is related to the operational life span of the aircraft. Fatigue occurs due
to cyclic stresses on the structure. Reducing these stresses can be achieved by adding
more material to the structure. If a structure is reinforced to reduce the aeroelastic
effects on the structure, this is referred to as an aeroelastic weight penalty (or simply
aeroelastic penalty) [73].
In this section we discuss four different aeroelastic effects that swept wings in
high-subsonic conditions (might) experience: static deformation, reduced controlsurface effectiveness, structural divergence, and flutter. In Fig. 8.41 we show a qualitative relationship between the wing sweep and the three critical aeroelastic speeds.
We see that aft-swept wings are most sensitive to a change in aileron reversal speed
and we therefore place some more emphasis on this topic than on divergence and
flutter. No attempt is made to present an in-depth physical model for aeroelastic
behavior. For an extensive treatment of this subject, the reader is referred to texts
on aeroelasticity (e.g. [14, 72, 73]). The intention is to present the reader with the
aeroelastic implications of having a swept-wing.
Fig. 8.41 Notional effect
of wing sweep on critical
speeds
Speed
Divergence speed
Bending-torsion flutter speed
Aileron reversal speed
Sweep forward
Sweep back
476
8 Aerodynamics of Swept Wings
8.5.1 Static Deformation
Static aeroelastic effects occur when elastic and aerodynamic forces result in a
strained aircraft structure. This can result in quite large displacements. Inertia plays
no roll in static deformation. One can imagine that for the same surface area, a high
aspect-ratio wing experiences a larger static deformations than a low-aspect-ratio
wing. There exists a profound difference between the shape of the wing when it is
manufactured (often referred to as the jig shape or 0-g shape) and the shape it takes
during flight (1-g shape). Due to the pressure distribution over the wing, the structure
deforms in two ways: it twists and it bends.
If we consider a typical monocoque primary wing structure, the two spars and
stiffened upper and lower skins form a box. This is the load-carrying structure of the
wing that transfers the aerodynamic loads to the fuselage. When the box experiences
a torque it will twist about its elastic axis. This is an imaginary axis, with position
dependent on the material choice and material distribution in the wing box. If the
resultant force on a wing section does not coincide with the position of the elastic
axis, a torque is introduced and the section will twist. The amount of twist depends
on the torsional stiffness of the wing and the distance between the resultant aerodynamic force and the elastic axis. To avoid twist in the structure, one can design the
box structure such that (in the design condition) the distance between the resultant
aerodynamic force and the elastic axis is zero. Such a measure is an example of
aeroelastic tailoring.
Even though the structural twist (or geometric twist) in a wing can be limited
by modifying the torsional stiffness distribution over the wing section, the wing will
still bend under aerodynamic loading. For an unswept wing, bending only affects the
effective dihedral angle4 but has a negligible effect on the effective angle of attack
of the wing. The effective angle of attack is the local angle of attack that the wing
or wing section experiences. When a wing deforms, its orientation with respect to
the flow changes resulting in a change in effective angle of attack. On swept wings,
pure bending (no torsion) leads to aerodynamic twist. The aerodynamic twist angle
at an arbitrary spanwise location is defined as the zero-lift angle of attack of the local
airfoil section. An aerodynamic twist distribution can thus be created by changing
the camber of the wing along the span. In the present context, we look at aerodynamic
twist that is caused by bending of an aft-swept wing. This is schematically shown
in Fig. 8.42 for a wing of infinite torsional stiffness about its elastic axis but a finite
bending stiffness. When the wing bends due to aerodynamic loading, it bends about
an imaginary bending axis. As we can see in Fig. 8.42, the bending axis is oriented at
approximately a right angle with respect to the elastic axis. This implies that when
the wing bends, the rear spar and front spar displace exactly the same amount along
the local bend axis. If we project these displacements on the rear view of the wing,
we can observe that at a given spanwise station the rear spar is elevated more than the
front spar. This results in a negative incidence angle of the wing sections outboard
of the root reducing their effective angle of attack.
4 The dihedral angle is the upward angle of the wing measured with respect to the horizontal.
8.5 Aeroelasticity of Swept Wings
477
fuselage side
lea
din
ge
fro
nt
dg
e
Top view
spa
r
ela
stic
rea
ben
axi
din
ga
s
xes
r sp
trai
ar
ling
edg
e
Side view
Rear view
1-g shape
fuselage side
r
rear spa
ar
front sp
0-g (jig) shape
Fig. 8.42 Aerodynamic twist due to upward wing bending under 1-g loading assuming infinite
torsional stiffness and finite bending stiffness of the wing box (after [46])
Example 8.5 Assume that we have a simplified representation of a swept wing,
shown in Fig. 8.43. The wing box structure can be approximated by an EulerBernoulli beam with flexural stiffness EI (E being the material stiffness, I being the
second moment of area). The lift, L applies exactly on the elastic axis of the beam
and is located in the middle of the wing. Calculate the angle of attack due to bending,
α, at the tip of the wing. Assume the following values: Λ = 30◦ , EI = 106 Nm2 ,
b = 10 m, and L = 10 kN.
Solution:
The geometric twist (θ) at the tip can be calculated by comparing the displacement,
w, of the leading edge (LE) and trailing edge (TE), respectively:
α = sin
−1
wTE − wLE
c
(8.40)
To calculate these displacements at the tip, we first need to calculate the displacement,
w and rotation, ϕ at the point where the lift force attaches. We use the following
478
8 Aerodynamics of Swept Wings
b/2
Λ
b/4
fro
n
t sp
ar
c
rea
r sp
L
ar
l1
l2
l3
Fig. 8.43 Figure belonging to Example 8.5
equations from beam theory [25]:
Ll13
3EI
Ll 2
ϕ(l1 ) = 1
EI
w(l1 ) =
(8.41a)
(8.41b)
where l1 = b/(4 cos Λ). In this example we assume the total lift to be concentrated in
a single attachment point. Beyond this discrete attachment point, the wing therefore
remains straight (no bending). The displacement at the leading and trailing edge can
be calculated as follows:
wLE = w(l1 ) + l2 ϕ(l1 )
(8.42a)
wTE = w(l1 ) + (l2 + l3 )ϕ(l1 )
(8.42b)
where l3 = c tan Λ. Substituting this in (8.40) results in the following expression:
α = − sin−1 [ϕ(l1 ) tan Λ]
(8.43)
We can express this equation in terms of the wing properties, b, Λ, EI and L:
Fig. 8.44 Spanwise lift
distribution on Boeing B-47
(after Ref. [72]). Note that c
indicates local chord and c̄
the mean aerodynamic chord
479
Local lift, cl c/c (~)
8.5 Aeroelasticity of Swept Wings
Rigid wing
Flexible wing
0
0.2
0.4
0.6
0.8
1.0
Spanwise position, η (~)
α = − sin−1
tan Λ
Lb2
1 + cos(2Λ) 8EI
(8.44)
We can now substitute the values of each of the wing properties and find a tip twist
of αtip = −2.76◦ .
Airplanes with high aspect-ratio wings and high sweep angles can have a considerable amount of aerodynamic twist under 1-g loading. The Boeing B-47 (1947)
was one of the first airplanes that featured such a high-aspect-ratio swept-back wing.
In addition, its engines were podded in nacelles attached to the lower surface of the
wing by means of pylons (see Fig. 1.11). Since it had not been designed with the
aerodynamic twist in mind, its spanwise lift distribution differed considerably from
the lift distribution produced by the undeformed wing (Fig. 8.44). Because we know
that the induced drag of a wing is dependent on its lift distribution (elliptical lift
distribution yields the lowest induced drag), we can easily deduce that the deformed
swept wing experienced a higher drag and a lower bending moment at the root than
the undeformed wing.
The discrepancy between the jig-shape and the in-flight shape of the swept wing
should be considered when the aircraft is designed. The desired in-flight shape of
the wing does not correspond to the jig-shape of the wing. The wing should be
designed such that under 1-g loading, it takes on the desired shape. This requires
a more in-depth analysis of the elastic and geometric coupling under 1-g loading.
Moreover, bending due to lift has a significant effect on the bending moment due
to maneuvering and gusts when the load factor, n > 1. To arrive at the optimal
design under 1-g loading a structural design loop can be employed where first the
desired wing shape is determined and then the wing structure is designed that would
deform to this shape under 1-g loading [3]. The B-52 is an early example of a highsubsonic aircraft (maximum operating Mach number, MMO = 0.93) that utilized the
difference between jig shape and the 1-g shape. Because the B-52 had a wing sweep
angle of 35◦ and an aspect ratio of 8.5, the bending of the wing resulted in a change
in angle of attack of −4.7◦ at the tip of the wing [15]. However, the designers at
480
8 Aerodynamics of Swept Wings
(b)
-1.5
Pressure coefficient, Cp (~)
Pressure coefficient, Cp (~)
(a)
Ma=0.80
CL=0.45
Re=50·106
-1.0
-0.5
0.0
η=0.1
η=0.3
η=0.5
η=0.7
η=0.9
0.5
1.0
0.0
0.2
0.4
0.6
0.8
1.0
-1.5
Ma=0.80
CL=0.47
Re=50·106
-1.0
-0.5
0.0
η =0.1
η =0.3
η =0.5
η =0.7
η =0.9
0.5
1.0
0.0
Chordwise position, x/c (~)
0.2
0.4
0.6
0.8
1.0
Chordwise position, x/c (~)
Fig. 8.45 Difference in section pressure distribution on B-52 wing between flexible and rigid
conditions (after Ref. [45]). a Flexible wing. b Rigid wing (jig shape)
Limit up
Jig
position
nlimit, up=2.0g
14.16 ft
4.52 ft
Limit down
Fig. 8.46 Bending of the wing of a B-52 under aerodynamic loading (modified from Ref. [15])
Boeing had designed for such an aerodynamic twist and the wing assumed a shape
that resulted in a more favorable pressure distribution than would be expected from
the rigid wing design (Fig. 8.45). Figure 8.46 shows the difference between the jig
position of the wing and the shape under aerodynamic loading. The deformed wing
features reduced outboard loading as can be deduced from Fig. 8.45.
Apart from the wing, also the empennage suffers from static aeroelastic effects.
Due to the flexible attachment of the tailplane to the fuselage and due to the flexibility
of the fuselage itself, the effectiveness of the tailplane is reduced. As an example,
Fig. 8.47 shows how the lift-curve slope (C L αh ) of the horizontal tail plane of an
Airbus A300 depends on the Mach number at various altitudes. We see that for a
rigid connection, the effectiveness increases with Mach number due to the compressibility effect. However, when the flexibility of the tailplane and the fuselage are
taken into account, we see that particularly at low altitudes and high Mach numbers
(i.e. high dynamic pressure, 21 γ pM 2 ) the effectiveness of the horizontal tailplane
decreases. This reduced effectiveness directly influences the longitudinal stability of
the aircraft. If C L αh reduces more than the lift-curve slope of the wing-fuselage combination, C L αwf , the longitudinal stability of the aircraft reduces. Aeroelastic effects
are therefore not limited to just the change in wing shape; they also influence the
stability and controllability of the aircraft.
8.5 Aeroelasticity of Swept Wings
∂CLT
∂αT
481
3.5
Rigid structure
3.0
H=
2.5
9000 m
6000 m
3000 m
2.0
0m
1.5
Tailplane, attachment
and fuselage empennage
1.0
0.5
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Mach number, M (~)
Fig. 8.47 Influence of fuselage flexibility on tailplane lift-curve slope (after Ref. [58])
8.5.2 Reduced Control-Surface Effectiveness
When a wing is loaded during flight, the lift applies close to the elastic axis and
therefore does not typically cause twist in the wing. However, when an aileron is
deflected, say downward, the pressure distribution over the airfoil is altered, such
that the section’s center of pressure shifts substantially backwards. This creates a
significant torque about the elastic axis and can result in aileron-induced (geometric)
twist. When the aileron is deflected downward, the outboard wing will twist leadingedge down. This results in a lower effective angle of attack of the section and therefore
reduces the increment in lift that was initially created by the aileron deflection. When
the wing has a relatively low torsional stiffness, the lift decrease due the aileroninduced twist can be higher than the increase in lift due the aileron deflection itself.
When this is the case, roll control on the aircraft is reversed. In other words, when
the pilot puts in right aileron, the airplane rolls to the left. When this happens we
speak of aileron reversal.
Reduced aileron effectiveness is typically found on high-aspect-ratio swept-back
wings. Reduced aileron effectiveness is more profound in swept wings because the
aileron-induced bending of the wing also results in aerodynamic twist. A high-aspect
ratio, aft-swept wing therefore experiences both geometric and aerodynamic twist
upon aileron deflection. The speed at which the aileron effectiveness reduces to zero
is called the aileron reversal speed. Introducing 45◦ sweep in the wing requires the
torsional stiffness to increase with 40 % with respect to the unswept wing in order to
maintain the same aileron-reversal speed [16]. The aileron reversal speed is dependent
on the altitude and Mach number of the airplane. The following derivation is presented
482
8 Aerodynamics of Swept Wings
b
M∞
Λ
Top View
flexible
attachement
b
k
s /2
T
c/4
ec
A
k
aerodynamic center
of semi wing
ela
cf
stic
L
2s
m
2
axi
fla
p
Section AA
m
2
s
L
2
c
elastic axis
α
ξ
ec
cf
A
c
0.25c
c
Fig. 8.48 Flexibly attached non-tapered rigid wing with full-span plain flap
to determine the effect of Mach number and sweep angle on the effectiveness of the
control surface.
Consider an airplane with a swept wing of aspect ratio A, surface area S, and
sweep angle, Λ (see Fig. 8.48). Assume the wing has a symmetric section and a
trailing-edge control surface over its entire span. When this control surface deflects
downwards, the camber of the wing is increased and a lift force and aerodynamic
moment is generated. The lift force and aerodynamic moment apply at the aerodynamic center of each wing half, which is assumed to be located at the quarter-chord
line of the wing at 50 % of the structural semi-span, i.e. at bs /4. Since we consider
a semi-wing, the lift force and aerodynamic moment are also halved. If we focus
on section AA in Fig. 8.48 we have defined an aerodynamic moment vector, m/2,
perpendicular to the freestream direction. The semi-wing is modeled as a rigid component that is constrained in twist (θ) and dihedral (Γ ) by two torsional springs
of stiffness k T and k, respectively. These springs represent the torsional and bending stiffness, respectively. We express the effectiveness of the control surface as the
change in lift coefficient with control surface deflection, C L δ , where δ is defined
perpendicular to the hinge line.
To start our calculations, we first need to determine the span and aerodynamic
chord of the wing. They are defined follows:
8.5 Aeroelasticity of Swept Wings
483
b=
√
AS
c = S/b
(8.45a)
(8.45b)
The structural span and structural chord of a single wing can then be calculated,
respectively:
b
cos Λ
cs = c cos Λ
bs =
(8.46a)
(8.46b)
When the semi-wing bends or twists, its angle of attack α, changes. If we assume
that the effective angle of attack, dihedral angle, Γ , and the twist angle, θ, are small,
we can establish the following relation:
α = θ cos Λ − Γ sin Λ
(8.47)
The twist angle and dihedral angle can be related to the lift (L/2) and pitching moment
(m/2) about each semi wing. As can be seen from Fig. 8.48, the pitching moment can
be divided into a component that exerts torsion on the wing (m T = m cos Λ) and a
component that induces bending of the wing (m b = m sin Λ). These two components
of the pitching moment in combination with the lift force can be related to the twist
angle and dihedral angle as follows:
L
mT
+ ecs = k T θ
2
2
bs L
mb
+
= kΓ
−
2
4 2
(8.48a)
(8.48b)
Traditional wing structures consist of a box structure with stringers and ribs. The box
consists of an upper and lower skin panel together with a front and an aft spar. When
loaded in bending, the upper skin is being compressed, while the lower skin is put
in tension. The flanges of the spars transfer shear stresses. When loaded in torsion,
the upper and lower skin together with the two spar flanges transfer shear stresses.
By placing ribs in the box structure, the torsional stiffness of the wing box can
be significantly increased. This reduces the amount of twist that is produced under
loading. Let us consider a semi-wing box that is rectangular shaped and which has
a height (h) that is equal to the maximum thickness of the airfoil, a width, w, and
a length, bs /2. The upper and lower parts of the box are formed by the wing skins
with thickness tskin . The front and aft part of the box are formed by the spar flanges
of thickness tflange . Furthermore, we assume that the elastic axis of the box is located
in the geometric center of the box. Intuitively, we know that when the box grows in
size, or when the thicknesses increases, the box becomes stiffer. The bending and
torsional stiffness of the box are related to the second moments of area of the box’
cross section (I x , I y , and J , respectively). For the box dimensions presented above,
484
8 Aerodynamics of Swept Wings
the following moments of inertia can be obtained (see Problem 8.19):
wh 2 tskin
3
w2 htflange
Iz =
3
J = I x + Iz
Ix =
(8.49a)
(8.49b)
(8.49c)
The second factor that determines the box’ stiffness is the Young’s modulus (E) of
the material that is used. The shear modulus of the material (G) is related to the
Young’s modulus via the Poisson ratio of the material (ν):
G=
E
2(1 + ν)
(8.50)
If we assume a box without any ribs, we can calculate the flexural rigidity as the
product of the Young’s modulus and the moment of inertia about the x-axis. The
torsional rigidity is the product of the polar moment of inertia (J ) and the shear
modulus:
flexural rigidity = EI
torsional rigidity = GJ
If we model the wing box as a rigid component that attaches to a set of flexible springs,
the flexural and torsional rigidity need to be mapped onto the spring stiffnesses. For
the flexural rigidity, we do this as follows. We assume a flexible beam of length bs /2
with flexural rigidity EI that is subjected to a uniformly distributed positive load
L/bs [N/m]. Under this load, the beam bends upward over a distance Δ at the tip.
Secondly, we consider a rigid bar of length bs /2 that is connected to a torsion spring
of stiffness k [Nm/rad], which is subjected to the same uniformly distributed load
L/bs [N/m]. This bar rotates such that the tip displacement in vertical direction is
also Δ. With this comparison we can show that the spring stiffness is related to the
flexural rigidity as follows (Problem 8.20):
k=
8EI x
bs
(8.51)
We do the same for the torsional spring that represents the torsional rigidity of the box.
The flexible beam with torsional rigidity GJ is subjected to a uniformly distributed
moment m/bs [Nm/m]. It can be shown that under such a moment distribution the
twist distribution is quadratic with a maximum twist at the tip of θtip = m T bs /8GJ.
The average twist amounts to θaverage = m T bs /24GJ. We set the average rotation to
the rotation of the rigid bar that is connected to the spring and deduce that the spring
stiffness is as follows (Problem 8.21):
8.5 Aeroelasticity of Swept Wings
485
kT =
12GJ
bs
(8.52)
The force and moment can be expressed as a function of the Mach number perpendicular to the leading edge and the freestream static pressure ( p∞ ) as follows:
2 S
clξ
clα
p∞ γ M ∞
L
= f (A)
α+
ξ
2
2
2
2
2 1 − M∞
(8.53a)
2 Sc
cm ξ p∞ γ M∞
m
= f (A)
ξ 2
2
2 2 1 − M∞
(8.53b)
Here, we have implicitly assumed that the moment coefficient due to angle of attack
about the quarter-chord point equals zero. We recognize in (8.53) the lift coefficient
due to angle of attack (clα ) and the change in lift and moment coefficient due to
flap deflection (clξ and cm ξ , resp.). These three derivatives pertain to the airfoil, not
the wing. The lift coefficient on a sheared airfoil is related to the lift coefficient of
the unswept airfoil according to (8.10). The angle of attack and the deflection angle
of the flap in spanwise direction are assumed to be affected by the sweep angle as
follows:
clα = clα |Λ=0 cos Λ
(8.54a)
clξ = clξ |Λ=0 cos Λ = clδ cos Λ
(8.54b)
cm ξ = cm ξ |Λ=0 cos Λ = cm δ cos Λ
(8.54c)
In the above equations we define δ as the deflection angle of the flap perpendicular
to the hinge line. The angles ξ and δ are interrelated according to:
tan ξ = tan δ cos Λ
(8.55)
2 ) and the
We also recognize in (8.53) the dynamic pressure (q = 21 p∞ γ M∞
Prandtl-Glauert compressibility correction. Finally, to obtain the forces of the threedimensional wing, the surface area (S) and the structural chord (cs ) are inserted in
these equations, as well as a factor ( f ) that depends on the aspect ratio derived from
Helmbold’s equation:
A
(8.56)
f (A) =
√
2 + A2 + 4
To make the following derivations more transparent, we lump a number of
re-occurring parameters into a single constant K :
K =
2
A
p∞ γ M ∞
S
√
2
2 + A2 + 4 4 1 − M∞
(8.57)
486
8 Aerodynamics of Swept Wings
We can now combine the previous equations into a set of three linear equations
in α, θ and Γ :
kT
θ = − cm ξ + clξ e cs ξ
K
k
bs
bs
ξ
clα α − Γ = cm ξ c sin Λ − clξ
4
K
4
α − θ cos Λ + Γ sin Λ = 0
clα ecs α −
(8.58a)
(8.58b)
(8.58c)
We convert this to the following notation:
⎞
⎤⎛ ⎞ ⎛
− cos Λ sin Λ
0
α
⎣ clα bs
0
− Kk ⎦ ⎝ θ ⎠ = ⎝ cm ξc sin Λ − clξ b4s ⎠ ξ
4
kT
Γ
− cm ξ + clξ e cs
clα ecs − K
0
⎡
1
(8.59)
This system of equations can be solved for α, θ and Γ . The wing lift (L) due
to flap deflection can subsequently be calculated by employing (8.53a). The flap
effectiveness can hence be calculated according to:
C L δ = lim
δ→0
L(δ) − L(0)/q S
δ
(8.60)
The following example illustrates the effect of wing sweep on the flap effectiveness.
Example 8.6 Consider the wing of Fig. 8.48. Assume this wing has a planform area
of 93.5 m2 and an aspect ratio of 8.4. Furthermore, assume it features a box structure
of width w = 0.5cs and height h = 0.12cs . The geometric center of this box
is located at 40 % of the structural chord (i.e. e = 0.15). The thicknesses of the
box are tskin = 1 cm and tflange = 5 mm. The Young’s modulus of the material is
70 × 109 N/m2 and the Poisson ratio is 0.31. The flap measures 20 % of the chord, i.e.
c f = 0.20c. Assume the wing is sheared with the following sweep angles: Λ = 0,
Λ = 15◦ , Λ = 30◦ , and Λ = 45◦ .
(a) Make a topview of the four wings.
(b) Plot the flap effectiveness (C L δ ) as a function of the Mach number ranging from
M = 0 to M = 0.9. Assume a static pressure of 26.4 kN/m2 (which is the
atmospheric pressure at an altitude of 10 km).
Solution:
First, we draw the top views of the wing to get a feeling for the relative dimensions of
the wing. This is done on the left-hand side of Fig. 8.49. Subsequently, we estimate the
following values for the sectional aerodynamic derivatives from handbook methods
[1, 52] for an airfoil with c f = 0.20c:
clα = 2π [1/rad] clδ = 3.45 [1/rad] cm δ = −0.64 [1/rad]
(8.61)
8.5 Aeroelasticity of Swept Wings
487
5
Sweep angle, Λ (deg):
δ
0
Aileron effectivenss, CL (1/rad)
4.5
15
30
45
Sweep angle, Λ (deg):
4
0
3.5
3
15
2.5
2
1.5
30
1
45
0.5
0
0
0.2
0.4
0.6
0.8
Mach number, M∞ (~)
Fig. 8.49 Aileron effectiveness of four different wings at 10 km altitude (belonging to Example 8.6)
For calculation purposes, we assume δ = 1◦ and calculate ξ for every sweep angle
according to (8.55). We subsequently solve (8.59) for various combinations of M and
Λ to find the values for Γ , θ and α. For each combination of M and Λ we substitute
the value of α along with the value of ξ into (8.53a) and using (8.60) to obtain a
value for C L δ . The result is plotted in the graph on the right-hand side of Fig. 8.49.
Let us briefly reflect on the result of Example 8.6. What we have simulated is the
effect of wing sweep on the full-span flap effectiveness of a wing. In shearing the
wing backwards, both the structural and aerodynamic properties of the wing change.
With increasing sweep angle the structural chord decreases and the structural length
increases. With decreasing structural chord comes a decrease in flexural and torsional
rigidity. Aerodynamically, the wing becomes less sensitive to changes in angle of
attack or changes in flap angle. For the unswept wing (Λ = 0) we observe that
the flap effectiveness is continuously increasing with Mach number. This trend is
attributed to the compressibility effect. We will see in Problem 8.22 that the torsional
flexibility reduces the aileron effectiveness of the unswept wing and that this effect
is magnified with sweep angle. A nose-down pitching moment is generated due to
the flap deflection, which results in a nose-down twist of the wing.
When the sweep angle increases the flap effectiveness decreases. Even at low
speeds, one can observe a decrease in flap effectiveness due to wing sweep. This
aerodynamic effect is independent of the flexibility of the wing. It demonstrates that
flaps become less efficient on a swept wing, compared to a straight wing. At increased
Mach number, the bending of the flexible wing starts to affect the angle of attack
of the section. As soon as the wing bends upwards, the angle of attack decreases
and the amount of lift is decreased. This results in a decrease in flap effectiveness
488
8 Aerodynamics of Swept Wings
with Mach number. For a 30◦ swept wing, the flap effectiveness has halved at a
Mach number of 0.85. This demonstrates the detrimental effect of wing sweep on
the effectiveness of a flap on a flexible wing. In practice, swept wings therefore
have reduced aileron effectiveness or even aileron reversal at combinations of speed
and altitude that result in a high dynamic pressure. To remedy this problem and
allow for roll control during the entire flight envelope, many high-subsonic aircraft
either employ outboard spoilers or inboard ailerons close to the wing root (known
as high-speed ailerons).
In our example, we have made a gross simplification of a wing to demonstrate
the effect of wing sweep on flap effectiveness on a flexible wing. We have assumed
an untapered wing of constant cross section. In practice, most wings are tapered,
resulting in an increase in both flexural and torsional stiffness towards the root. In
addition, the control surface usually spans only a part of the trailing edge where we
have assumed a full-span flap. Also, we implicitly assumed a constant lift distribution over the wing, where in reality the lift distribution would be closer to being
elliptical. Furthermore, the Prandtl-Glauert compressibility correction is known to
be inaccurate at Mach numbers approaching unity, which overestimates the forces
and resulting displacements. Finally, we made a gross simplification of the structure
(e.g. no ribs) and subsequently mapped the flexural and torsional stiffness onto the
spring stiffness of two torsional springs. Any of the aforementioned assumptions
has an effect on the results. However, with this gross simplification of an airplane
semi wing we have demonstrated in a very simple manner the adverse effect of wing
sweep and Mach number on flap effectiveness. In the following example we will
demonstrate how Mach number and altitude dictate the flap effectiveness for a wing
of given geometry.
Example 8.7 We would like to investigate the effect of altitude on the full-span flap
effectiveness of the wing of Example 8.6 with Λ = 30◦ . For this wing, plot the
flap effectiveness as a function of Mach number for Mach numbers between 0 and
0.9 and for altitudes of 0, 3, 6, and 9 km. Assume international standard atmosphere
conditions. In addition, plot the relation between the flap altitude and Mach number
at which the flap effectiveness is C L δ = 1.5, 1.0, 0.5 and 0 per radian. In the latter
plot, also indicate where control reversal occurs.
Solution:
To make these two plots, we follow the same approach as in Example 8.6. However,
this time we keep the sweep angle constant at 30◦ . To estimate the pressure ( p) at
each altitude (h) we employ the following equation for the troposphere [53]:
p = p0m ISA 1 +
λh
T0m ISA
−g
Rλ
(8.62)
The following constants are used: p0m ISA = 101325 N/m2 , T0m ISA = 288.15 K,
g = 9.81 m/s2 , λ = −0.0065 K/m, and R = 287 J/kg/K. For a given value of the
8.5 Aeroelasticity of Swept Wings
489
9000
2.5
4000
0.85
0.6
0.55
3000
0.25
9
0.4
(1/rad) =
)=
0

m
3
5000
L
(k
6
1
6000
C
Altitude, h (m)
h
1.5
1.15
1.75
7000
1.45
8000
2

Aileron effectivenss, C (1/rad)
L
Sweep angle, deg
2000
0.5
1000
0
0
0.2
0.4
0.6
Mach number, M (~)
0.8
0
0
0.2
0.8
Mach number, M (~)
Fig. 8.50 Effect of altitude on flap effectiveness (belonging to Example 8.7)
altitude (up to 11 km), we can now find the corresponding pressure. Using the same
wing properties as in Example 8.6, we can hence make the two required graphs. They
are shown in Fig. 8.50.
Again, we briefly reflect on the result of Example 8.7. It can be seen in the
graph on the left-hand side of Fig. 8.50 that the flap effectiveness increases with
increasing altitude and decreases with increasing Mach number. The lower pressure
at higher elevations results in a lower dynamic pressure, which results in a smaller
loss in control-surface effectiveness. Vice versa, at lower altitudes the effectiveness
is decreased. If we shift our attention to the second graph in Fig. 8.50, we could
imagine that this graph could be used to define an envelope of altitude and Mach
number for which a predefined value of control power is guaranteed. For example,
if C L δ > 1.15, we could use the corresponding line in the graph to bound the flight
envelope of the airplane in terms of Mach number and altitude. In that case, the
airplane would be restricted to fly below M = 0.35 at sea level (ISA). However, at
9 km (ISA) it would be allowed to fly M = 0.61.
This notional example demonstrates the practical implications of reduced controlsurface effectiveness. The addition of wing sweep makes it more difficult to fulfill
control power requirements. If for a given wing geometry the control power is insufficient due to aeroelastic effects, the structure needs to be made stiffer. This can be
done by adding more material to the structure, which increases its mass. This increase
in mass to reduce adverse aeroelastic effects is an example of the aeroelastic penalty.
We have qualitatively shown that the stability and control characteristics of large
transonic transport aircraft can be significantly altered due to aeroelastic effects.
Simple control laws corrected for small aeroelastic effects obtained from handbooks
or wind tunnel tests do not suffice [22]. To determine the stability and control derivatives (e.g. C L αh and C L δ ) and how they change with Mach number, the fluid-structure
490
8 Aerodynamics of Swept Wings
interaction needs to be taken into account. This can, for example, be done by coupling a finite-element (FE) model of the aircraft structure to a numerical fluid model
(CFD).
8.5.3 Structural Divergence
Wing divergence is characterized as the continuous increase in wing twist due to
aerodynamic loads until the structure fails. On straight wings, divergence speeds
can be an important boundary in the flight envelope of aircraft. However, rearward
swept wings do not suffer from divergence problems because the bend-twist coupling
results in a converging effect rather than a diverging effect (see also Fig. 8.41). On
the other hand, for forward-swept wings the opposite is true: bending of the wing
results in higher angles of attack. Therefore, forward swept wings have much lower
divergence speeds than straight wings of the same flexural and torsional stiffness.
The following factors are favorable for increasing the divergence speed on a forwardswept wing: high wing stiffness, especially in bending, forward position of flexural
axis and high wing-taper in planform [16]. The X-29 (Fig. 8.51) used bend-twist
coupled composite laminates in the upper and lower wing skins that induced a nosedown twist in the wing when it bent upward. This reduced the effective angle of attack
and eliminated structural divergence of the wing within its flight envelope [28].
Fig. 8.51 X-29 with forward swept wing and supercritical airfoils [8]
8.5 Aeroelasticity of Swept Wings
491
8.5.4 Flutter at High Subsonic Mach Numbers
Dynamic aeroelastic effects occur when elastic, inertial and aerodynamic forces interact in such a manner that an oscillation is produced [73]. Dynamic aeroelastic effects
are not exclusive to transonic flow. However, the increasing influence of compressibility results in particularly strong pressure and density changes in transonic flow
when instabilities occur. This results in severe loads on the boundaries between fluid
and structure (e.g. wing) and strong pressure waves in the flow [42]. Damped oscillations are forced vibrations that are caused by instabilities in the flow that feed energy
to the structure. These forced vibrations can have different causes, e.g. cyclic separation of the boundary layer due to shockwave impingement (buffet, see Sect. 7.6) or
atmospheric turbulence. The result of this oscillatory loading is a forced vibration in
the structure. If the amplitude of the vibration grows in time towards an asymptote,
we speak of a limit cycle oscillation (LCO). If the amplitude grows unbounded, we
speak of flutter. Contrary to damped oscillations, flutter is an unbounded dynamic
form of aeroelasticity. At the flutter speed (V f ), any infinitesimally small disturbance
results in an unbounded vibration of the wing that eventually leads to a failure of
the structure. Flutter can allude to a pure bending motion of the wing, a pure torsion
motion, or a coupling of both motions. Probably the most dangerous type of aircraft
flutter results from coupling between the bending and torsional motions of a relatively
large aspect ratio wing. In this section we discuss flutter and limit-cycle oscillations
of swept wings. The reader is referred to other texts that explain the mechanisms of
flutter in more detail [14, 73].
We further explain the mechanism of bend-torsion flutter by considering the forces
acting on a two-dimensional (rigid) wing section of mass (m) and inertia (I y ) as
shown in Fig. 8.52. In this example, sc is the distance between the elastic axis and
the center of mass referred to as the static unbalance of the section. The structural
stiffness is represented by a linear spring system consisting of a torsion spring and
a tension/compression spring. The section is able to plunge (h) and pitch (α) about
the elastic axis. The aerodynamic moment (M y ) is considered about the elastic axis,
while the inertial force (m ḧ) and moment (I α̈) act at the center of mass of the section.
L
α
sc
c
My
undeflected airfoil center line
elastic
axis
kT
x
h
k
center of
mass
Fig. 8.52 Notional sketch of aerodynamic, structural, and inertial forces acting on a twodimensional wing section
492
8 Aerodynamics of Swept Wings
We can imagine that when this system is in oscillation, each of the forces and moments
shown in Fig. 8.52 are somehow dependent on the position or acceleration of the
section. The only sources of damping or instability in the system are the lift and the
aerodynamic moment.
If we only consider the plunging motion, we notice that it has an effect on the
angle of attack of the wing section. For example, when the outboard wing heaves
downwards, it experiences an increase in local angle of attack, which induces an
upward lift force. This lift force, in turn works opposite to the direction of motion and
therefore decelerates the downward motion. This is what is what we call aerodynamic
damping. The damping force depends on the rate of change in position. If we again
consider a downward motion, there is no aerodynamic force that depends on the
displacement (heave) of the outboard section, the damping is entirely due to the
motion of the wing. In other words, the aerodynamic stiffness due to bending is (close
to) zero. Whether the aerodynamic stiffness due to torsion is positive or negative
depends on the value of e, the nondimensional distance between the aerodynamic
center and the elastic axis. If the aerodynamic center of the wing lies ahead of
the elastic axis, the torsional stiffens is negative. This implies that the increase in
twist causes an aerodynamic moment that increases the twist even further. Vice versa,
when the aerodynamic center is behind the elastic axis, there is positive aerodynamic
torsional stiffness. It can be shown that the quasi-steady aerodynamic moment is
also dependent on α̇, resulting in damping or amplification of the torsional motion
depending on the speed.
The flutter speed is defined as the speed at which this system becomes critically
damped, i.e. when the aerodynamic damping goes to zero. The flutter speed therefore
depends on a variety of factors including the static unbalance, the location of the
elastic axis, the location of theaerodynamic center, the relative mass ratio (m/ρc2 ),
the radius of gyration (r = I y /m) about the elastic axis, and the ratio of the
uncoupled natural frequencies of the bending and torsional motion (ωh /ωα ).
Typical wings of high-performance aircraft have large density ratios and positive static unbalances. The frequency of the bending-torsion flutter lies somewhere
between ωh and ωα , with ωh being the smaller of the two. Increasing the density
ratio increases the flutter speed. The most critical condition is therefore encountered
at sea-level rather than at high altitude. Increasing the static unbalance or decreasing
the radius of gyration both reduce the flutter speed. This has led to the concept of
mass balancing. Placing additional mass ahead of the elastic axis reduces the static
unbalance. The torsional inertial loads become smaller and therefore the torsional
natural frequency increases resulting in an increase in flutter speed. However, even
when the mass and elastic axis are aligned with the aerodynamic center on the quarter
chord of the wing, flutter can still occur due to negative damping terms [73]. It can
be shown that for a given position of the aerodynamic center, a displacement of the
elastic axis closer to the leading edge also results in a an exponential increase in the
flutter speed.
The Mach number has an effect on the flutter speed. In the lower transonic
regime, the flutter speed reduces when the Mach number increases towards unity.
This results in a dip in the flutter speed versus Mach number graph, which we term
8.5 Aeroelasticity of Swept Wings
4
2
5.6
m
b 2 = 100
4.8
20
Reduced flutter speed, Vf /bωα (~)
Fig. 8.53 Predicted relation
between flutter speed, V f ,
and Mach number for
various mass ratios.
Predictions based on
potential flow theory from
Refs. [24, 43]. Note a is the
speed of sound and ωα the
wind-off natural frequency
of the wing in torsion
493
4.0
10
3.2
“transonic dip”
5
a
bωα = 1
2.4
1.6
1/2
0.8
c = 2b
0
0
0.8
1.6
2.4
Mach number, M∞ (~)
3.2
the transonic dip. When the Mach number increases towards unity the flutter speed
also increases again and continues to increase in the supersonic domain. References
[24, 43] were among the first to publish analytically obtained results based on potential flow calculations that showed the transonic dip for high relative mass ratios.
The relation between flutter speed (V f ) and Mach number is shown in Fig. 8.53. We
clearly observe the transonic dip in flutter speed for high relative mass ratios. The
quantitative accuracy of these curves can be questioned due to the nonpotential flow
below M∞ = 1 and the erratic behavior of the aerodynamic coefficients near the
speed of sound. However, the curves give a good indication of the compressibility
effect on the flutter speed.
Let us further discuss the effect of Mach number on dynamic response of the
wing. With increasing Mach number the section becomes more sensitive to changes
in angle of attack (remember the Prandtl-Glauert compressibility correction that was
discussed in Chap. 3). This amplifies the aerodynamic forces when there is a change in
angle of attack. Furthermore, the formation of a shock wave and the associated mixed
subsonic-supersonic flow have a profound effect on the energy exchange between
494
8 Aerodynamics of Swept Wings
airstream and wing and hence on the flutter speed. This is caused by the strong
dependence of the shock location on the angle of attack. During the oscillatory motion
of the wing, the shock oscillates over the profile or even temporarily vanishes [64].
We also saw this effect when we discussed the periodic shock-induced separation
that causes transonic buffet (Sect. 7.6). Generally speaking, the flutter frequencies
of the coupled bending-torsion mode are one to two orders of magnitude smaller
than for buffet of full-scale high-aspect ratio wings. Therefore, there exists a smaller
phase shift in the shock location with respect to the angle of attack. A representative
relation between shock position and angle of attack has previously been shown in
Figs. 7.16 and 7.17.
Bendiksen [11] performed a numerical investigation into the (de)stabilizing effect
of shocks waves on wings in the transonic domain. It was shown that the destabilizing
effect of shocks becomes noticeable when the shocks reach a position between 40
and 50 % chord. The most destabilizing shock location is 75 %c, which generally
coincides with the Mach number at the bottom of the transonic dip. With further
Mach number increases, the shock approaches the trailing edge and the influence of
the shock changes from destabilizing to strongly stabilizing, resulting in the post-dip
rise in the flutter speed. The mean position of the shock also relates to the location
of the aerodynamic center. The aerodynamic center shifts progressively rearwards
from the 25 % chord below the critical Mach number towards the 50 % chord in
supersonic conditions. At supersonic Mach numbers no flutter exists when the center
of mass is ahead of the midchord. Finally, the shock-boundary layer interaction can
play an important role as the boundary layer might separate at the shock foot when
the section’s angle of attack becomes large enough. This can effectively change a
bending-torsion instability into a limit-cycle oscillation. In Fig. 8.54 the effect of
Mach number on the flutter speed of a swept-wing wind tunnel model is shown
0.5
0.45
0.4
Experiment
Doublet lattice
Euler
0.35
0.3
0.25
ct
0.2
ar
Λ=45º
0.1
b
te
r-c
0.15
ho
rd
ela
lin
sti
e
ca
xi
s
Euler + viscosity
qu
Flutter speed coefficient, Vf /bωα µ (~)
Fig. 8.54 Variation of flutter
speed with Mach number for
AGARD I-Wing 445.6 and
comparison to theory (after
Refs. [40, 74]). Note
μ̄ = m/ρV with V being the
volume of a conical frustum
having streamwise root
chord (cr ) as lower base
diameter, streamwise tip
chord (ct ) as upper base
diameter, and panel span (b)
as height
0.05
cr
0
0.5
0.6
0.7
0.8
0.9
1.0
Mach numer, M (~)
1.1
1.2
8.5 Aeroelasticity of Swept Wings
495
along with aerodynamic predictions. Clearly, a flutter dip can be observed in the
experimental results around the speed of sound.
Aircraft with high aspect ratio wings are more prone to flutter than low-aspect
ratio wings due to the larger absolute displacement and rotation of the wing for a
given aerodynamic load [10]. However, due to the coupling between wing bending
and aerodynamic twist, aft-swept wings generally have a higher flutter speed than
their unswept counterparts (as was shown in the notional graph of Fig. 8.41). The
interaction between aerodynamic, elastic and inertial forces can still result in structural oscillation. Since the natural frequencies (in torsion and bending) are a direct
function of the stiffness of the fluid-structure system, they change with the velocity
of the aircraft. On an aft-swept wing, the aerodynamic bending stiffness is introduced
due to the coupling of bending and aerodynamic twist. Whenever the wing heaves
downwards, the effective angle of attack is increased to provide an upward lift force
trying to restore the wing to its neutral position. An aft-swept wing therefore has
a positive aerodynamic stiffness, which contributes to a higher overall stiffness of
the system. Vice versa, a forward-swept wing typically reduces the overall stiffness
of the system. Experimental and theoretical investigations have demonstrated that
aft-swept wings have a tendency to reduce the amplitude of the vibration due to
the coupling of wing bending and aerodynamic twist. Due to this coupled effect the
vibration beyond the flutter speed does not grow unbounded immediately, but results
in a limit cycle oscillation [10].
Flutter is not only limited to wings. Tail flutter can also be a potential risk in
transonic flow. Moreover, a coupling between fuselage, wing, engines, and empennage can even flutter. For example, the location of a wing-mounted engine has a
major effect on the flutter behavior of the aircraft. Flutter grows unbounded when
the energy that is extracted from the airstream is larger than the energy that is dissipated by damping in the fluid-structure system (whether this is just the wing or the
full aircraft). It occurs when the aerodynamic forces couple with a natural vibration
mode of the structure to produce a periodic motion in the structure that leads to
failure if the damping goes to zero. As mentioned previously, the speed at which
this occurs is referred to as the flutter speed and forms an important boundary of
the flight envelope. The flutter motion of wings is usually a coupling between the
bending and torsion modes. If an aircraft reaches its flutter speed, there can be a
coupling between wing bending, wing torsion, rigid-body translation of the fuselage, and rigid-body rotations of the fuselage. The interaction between the various
elastic and rigid-body motions demonstrates the complexity of the flutter motion.
Adding the fact that transonic aerodynamics is challenging from a modeling point of
view makes predicting flutter speeds and flutter behavior one of the most challenging
problems in the aeronautical sciences.
It was already acknowledged in the early 1950s that flutter performance can be
improved significantly if a rapidly responding closed-loop control system is integrated in the aircraft structure (this is often referred to as aeroservoelasticity). The
flutter frequencies are generally too high for pilots to respond to and should therefore work autonomously [14]. Commercial aircraft can therefore be equipped with
a flight control system that has a gust load alleviation system [44]. For example,
496
8 Aerodynamics of Swept Wings
the Boeing 787 uses advanced in-flight controls to allow the aircraft to sense and
respond to atmospheric turbulence (gusts), creating a smoother ride [34]. The alleviation systems rely on sensors (usually accelerometers and rate gyros) that are placed
close to the center of gravity of the aircraft. Air data sensors are used to inform the
control system about the aircraft’s speed and attitude. Control input is then given to
the control surfaces (e.g. ailerons and spoilers) such as to balance any gust-induced
motion of the aircraft. Such a vibration-suppression system could potentially also be
used to fly beyond the flutter speed of the aircraft. This has only been successfully
demonstrated on wind tunnel models [73].
8.5.5 Prediction and Detection of Aeroelastic Phenomena
Aeroelasticity phenomena range among the most complicated to be modeled in an
efficient and accurate fashion. Both the airflow and the structure should be modeled.
For structural modeling Finite Element (FE) models are most commonly used. For
incompressible flows, fluid modeling is often successfully done by relying the doublet
lattice method. As the name suggests, this is a method based on the potential flow
equations. The doublet lattice method is not capable of capturing transonic effects
such as the formation of a shock or the oscillation thereof. Euler methods can therefore
be employed as is done in Refs. [10, 11]. As can be seen from Fig. 8.54 the Euler
prediction gives a conservative prediction of the transonic dip. If shock-boundary
layer interaction is to be taken into account, one can rely on a coupled Euler-boundary
layer solver, such as proposed in Ref. [40]. Alternatively, one can employ the unsteady
RANS equations in the entire flow domain. Progressively, the aforementioned models
have more degrees of freedom (DoF), which could easily be up to several million
for the RANS calculations. The interaction at every time step between the finite
element model and the fluid model, therefore, results in an exponential increase in
the DoF of the total system. Solving the linear systems that result from the combined
models is computationally very expensive and therefore hardly used for full-scale
three-dimensional aeroelasticity calculations.
In order to reduce the computational cost of dynamic aeroelastic calculations,
so-called reduced order models (ROMs) can be employed. They can reduce the DoF
for a given system with several orders of magnitude and therefore accelerate the
process significantly [20]. The dynamic response of a structure can generally be
captured by (the superposition of) shape functions. For example, the first bending
mode could be described by a displacement of the elastic axis according to a 1cosine function between 0 and π/2. The magnitude of the displacement of all the
nodes on the elastic axis would then simply be a function of the amplitude of the
cosine function. If more complicated mode shapes exist, various mode shapes can be
superimposed and their respective amplitudes need to be determined. To find which
mode shapes should be used, a finite element model of the airplane can be employed.
By periodically exciting the structure, one can find the mode shapes that dominate in
each part of the airplane. Although it might feel counter-intuitive, a similar approach
8.5 Aeroelasticity of Swept Wings
497
can be used for the equations that describe the fluid flow (i.e. unsteady Euler or
unsteady RANS equations). The equations of motion are then changed from the time
domain (where all variables are a function of time, t) to the frequency domain by
means of a Fourier transformation.5 By doing this, the aerodynamic properties can
be transformed to a truncated Fourier series that can generally be reduced to contain
just three, two, or even one Fourier term. This essentially reduces the system to
a multi-DoF damped spring-mass model with a forcing function [13]. Using such
an approximation of the fluid-structure system, we can relatively quickly determine
whether a structure will be prone to flutter. However, due to the many simplifications
the resulting prediction is not as accurate as we would like it to be. An overview of
various numerical methods to analyze the static and dynamic aeroelasticity is given
by Livne in Ref. [41].
Flutter predictions can also be done by wind tunnel experiments or by flight testing full scale aircraft. Which of these measures is most effective depends on a variety
of factors including the anticipated margin of safety from flutter, the Mach number
range, and the number of different mass and structural configurations to be analyzed
[14]. The last argument is particularly applicable in the case of military aircraft
that can carry a variety of external weapons or stores. Testing each configuration
during flight tests is expensive and time-costly. Even though present-day analytic
predictions are still underdeveloped, they can be used to indicate which combination of stores/weapons can potentially be flutter sensitive. By test-flying only these
configurations the number of flight tests can be reduced significantly [20].
To determine the dynamic aeroelastic characteristics of a transonic aircraft, wind
tunnel tests can be carried out on a dynamically scaled model of the aircraft. A
dynamically scaled model of the aircraft is constructed such that the (dynamic) elastic deformations under aerodynamic load are proportional to that of the full-scale
aircraft, although the time required for a certain motion is different and requires
mathematical factors for interpretation of the results. Even though these tests are
not sufficiently reliable to determine the absolute flutter speed of the full-scale aircraft, it helps the main certification calculations by validating unsteady aerodynamic
methodologies. In addition, it is used for performing parametric studies, studying
new configurations, investigating interference and compressibility effects, or studying the effect of advanced control algorithms on the aeroservoelasticity (e.g. gust
response) of the aircraft [17]. Figure 8.55 shows two wind tunnel experiments in a
transonic wind tunnel. In Fig. 8.55a a semispan model of a wing is mounted to the
wind tunnel wall. This model has been dynamically scaled such that the relative displacements of the wing under aerodynamic loading are close to the full-scale wing.
With this model the amount of bend-twist coupling under aerodynamic loading can
be determined and the dynamic aeroelastic response can be measured. In Fig. 8.55b
a model of the Boeing 747 is shown. Similar to the wing of Fig. 8.55a, this model is
also dynamically scaled. However, it is also strung to the side and upwards to mimic
free-flight conditions. These strings allow the aircraft to perform rigid-body motions,
which often occur when the aircraft experiences limit-cycle oscillations.
5 For more information on Fourier transformations, the reader is referred to [61].
498
8 Aerodynamics of Swept Wings
Fig. 8.55 Flutter tests in the NASA Langley transonic wind tunnel facility [17]. a Semi-span model
of Cessna citation (Photo NASA). b Scale model of Boeing 747 (Photo NASA)
A first step in determining the flutter characteristics of a full-scale aircraft is to do
a ground vibration test (Fig. 8.56) to determine the structural modes and the dynamic
response to various forcing frequencies. The information from these tests can in turn
be used to adjust the finite element models and update the flutter calculations [36].
In addition, techniques exist to determine the control surface free play. Free play
results from the slop that exists in the control surface actuation system and can result
in control-surface flutter or a lower flutter speed of the wing.
Flight flutter tests are carried out to determine the dynamic response of the aircraft
at the edges of its flight envelope. This is done in a progressive manner, starting at
conditions (speed and altitude) away from the suspected flutter boundary and subsequently moving closer to the boundary once this point has been shown to be free of
flutter. The response of the aircraft to excitations is measured by interferometers that
are positioned on the aircraft. The response to a pulse input (induced by stick/pedals
or explosive charges) as well as the response to a frequency sweep is investigated.
The excitations for a frequency sweep can be induced by harmonic oscillations in the
control surfaces, the addition of oscillating vanes, or the oscillation of inertial loads
within the fuselage of the aircraft. The response to the frequency sweep is recorded
and curve-fitted to determine for example the damping of the induced vibrations. The
damping values can be compared to values of previous tests, leading to a damping
trend with speed and altitude. From the damping trend a potential flutter speed can be
obtained [73]. When the gradient of damping ratio with the air speed is fairly steep,
a relative small increment in speed can result in the onset of flutter. This is what is
referred to as hard flutter and is of greatest concern during flight flutter testing. If
the damping ratio trend approaches the critical speed with a shallow gradient, this is
known as soft flutter.
8.6 Summary
499
Fig. 8.56 Ground vibration test of a Lockheed Jetstar at NASA Dryden Flight Research Laboratory.
Photo from Ref. [36], courtesy of NASA
8.6 Summary
Many high-subsonic transport aircraft feature swept-back wings. In this chapter we
have demonstrated that wing sweep is a powerful design measure to postpone the
drag-divergence Mach number, and reduce the peak in drag coefficient during transonic conditions. This can be attributed to the lower supervelocities over a wing
section in oblique flow compared to a wing section in straight flow. The lower supervelocities postpone the onset of shock waves and shock-induced separation. It has
been shown that the sweep angle can be correlated to the critical Mach number
and that a swept wing can experience supersonic flow over its surface, while still
being in subcritical conditions. These characteristics have resulted in a wide-spread
application of swept wings in both military and civil aircraft.
Even though wing sweep is a powerful addition to the supercritical airfoil, it
is not without its challenges. We showed that swept wings cause curved inviscid
streamlines creating inviscid crossflow over the wing. At the root of the wing, the
streamlines are forced to be aligned with the free stream, which causes additional
form drag. Several shape modifications are described to reduce this root effect. At
the tip we see a forward shift of the suction peak on aft-swept wings causing an early
onset of shock waves and shock-induced separation. To remedy this tip effect many
aircraft feature rounded tips or tailored airfoil modifications.
Also the boundary-layer flow is affected by the sweep angle of the wing. Due
to the strong effect of viscosity, the boundary-layer mean flow is not aligned with
the inviscid streamlines creating boundary-layer crossflow. Depending on the state
of the boundary layer, the amount of wing sweep, and the local pressure gradient,
500
8 Aerodynamics of Swept Wings
the angle between the shear vector at the wall and the inviscid streamline can be
as high as 90◦ . Due to the inflection point in the crossflow boundary-layer profile,
crossflow instabilities occur in the form of (stationary) crossflow vortices. These
vortices are responsible for the early onset of boundary-layer transition on airfoils that
would otherwise have remained laminar. In addition, on an aft-swept wing the mean
boundary-layer flow tends to move outboard under an adverse pressure gradient. This
results in a thinner boundary layer near the root, making it more resistant to stall,
while a thicker boundary-layer is formed over the outboard wing and at the wing
tip. The tip therefore has a lower stall angle-of-attack, which can lead to premature
tip stall, one of the most dangerous forms of stall on swept-wing aircraft due to the
resulting destabilizing pitching moment.
Finally, we presented the effect of wing sweep on some fundamental aeroelastic properties of the wing. We showed that swept wings exhibit aerodynamic twist
whenever they bend under aerodynamic loading. A swept wing should therefore
be designed for 1-g loading. Furthermore, we showed that aft-swept wings reduce
the effectiveness of trailing-edge control surfaces. Due to the coupling between wing
bending and aerodynamic twist on an aft-swept wing, the effectiveness of control surfaces deteriorates. At the reversal speed, the control-surface effectiveness has reduced
to zero and a further increase in velocity causes control reversal. For high-aspect ratio
wing with ailerons this is known as aileron reversal, a potentially dangerous characteristic when flying at high dynamic pressures. On the other hand, aft-swept wings
generally have a higher flutter and divergence speed as compared to their unswept
counterparts. In low transonic conditions the flutter speed decreases rapidly with
Mach number due to destabilizing oscillating shock waves. Once the shocks move
closer to the trailing edge, they become stabilizing and the flutter speed increases
rapidly with Mach number in the upper-transonic and supersonic domain.
Problems
The Advantages of Wing Sweep
8.1 Show that …
(a) …(8.5) can be obtained by combining (8.1) through (8.2).
(b) …(8.6) can be derived from (8.5) and (6.11).
8.2 The R-4009 profile of Fig. 8.4 has a peak pressure coefficient of −0.60. Calculate
the drag divergence Mach number of an infinite-span wing consisting of this airfoil…
(a) …when it is swept over 20◦ .
(b) …when it is swept over 40◦ .
8.6 Summary
501
8.3 Write a computer program to generate a graph similar to the one in Fig. 8.9
with on the x-ordinate the drag divergence Mach number instead of the critical
Mach number. You need to use (8.12), (8.23), and (8.11). Use this graph to answer
Problem 8.2. Do you get the same answers?
8.4 Consider a typical lift distribution over a straight, unswept wing. Argue why the
weight of this wing increases when the thickness-to-chord ratio reduces.
8.5 In Fig. 7.18 the lift and drag coefficient of a NACA 64A010 are shown as a
function of Mach number. We would like to estimate the drag coefficient as a function
of lift coefficient and Mach number of a swept, untwisted wing of Λ = 25◦ . We
assume that this wing has an aspect ratio of 6 and a span efficiency factor of φ = 0.85.
Assume cd0 is independent of the sweep angle and that we may use (8.29) to predict
the wing’s drag coefficient as a function of Mach number and lift coefficient.
(a) Combine Fig. 7.18a, b and tabulate the values of the lift coefficient and drag
coefficient at each Mach number in a spreadsheet.
(b) Use this spreadsheet to draw the drag coefficient (cd ) of the airfoil as a function
of the Mach number for the following lift coefficients: cl = 0, 0.2, 0.4, and 0.6.
(c) Use your spreadsheet and (8.29) to draw the drag coefficient (C D ) of the wing
as a function of the Mach number for the following lift coefficients: C L = 0,
0.2, and 0.4.
(d) Repeat (c) for a wing of the same aspect ratio and span loading but with Λ = 15◦
and Λ = 35◦ , respectively. Can you discover a trend in drag-divergence Mach
number at each of the lift coefficients?
Drag coefficient, CD (~)
8.6 In the figure below you can see the drag-curve for a 45◦ swept wing of aspect
ratio 5.5 and taper ratio of 0.6. The wing features a NACA 64A010 airfoil. The
estimation of the drag coefficient is based on (8.29). Why is the drag-divergence
Mach number for the wing so much lower than would be expected from elementary
swept-wing theory?
Measured
Estimated
CL=.3
0
0.2
0.4
0.6
0.8
1.0
Mach number, M (~)
1.2
502
8 Aerodynamics of Swept Wings
Inviscid Flow over Swept Wings
8.7 (a) Explain what is meant by the ‘root effect’ of swept wings.
(b) List four modifications that should be made to the geometry of the wing root in
order to reduce the root effect.
8.8 Consider the planform of the A321 wing in the picture below. Explain why the
tip has been rounded.
Photo A.Pingstone
8.9 In the pictures below an early design of the Boeing 7E7 is shown (top) as well
as the production version (bottom). One of the striking features is the highly-swept
“shark-fin” vertical tail. The tip has been rounded-off and there is a curved contour
at the base where the trailing edge blends in with the fuselage, both at the leading
and trailing edge.
(a) Do the tip and base fin shapes have any aerodynamic significance and why (not)?
(b) Explain why you think the production version of the 787 had a more conventional
(straight) fin leading edge?
8.6 Summary
503
7E7
787
8.10 In the figure below the airfoils of the Airbus A310 are shown at three spanwise
locations. The airfoil at the wing root does not look like a supercritical airfoil. Explain
why its shape is so different from the airfoils in the outboard sections.
Parallel
to Datum
Datum
Wing tip
Planform crank
Wing root
Parallel
to Datum
After Ref. [23]
8.11 In the sketches below the pressure distribution is shown for M = 0.78 and C L =
0.27 for the wing profile as applied to the North American Sabreliner, a business jet
of the 1950s. During the 1970s another company came up with a modification of the
inboard wing that eliminated the inboard shock which reduced high speed drag. In the
bottom left figures you may notice the difference in pressure distribution before and
after the modification (indicated by “Mark Five”). Sketch what profile modification
yields the aforementioned change in pressure distribution at cross section A (data
from Ref. [65]).
504
8 Aerodynamics of Swept Wings
A
Spanwise upper surface pressure
distribution for the basic Sabreliner wing
-.3
-.5
M=0.78
CL=0.27
-.7
-.8
-.8
Shock wave
-.5
-.3
-.1
0
.1
A
A
Spanwise upper surface pressure distribution
for the Mark Five wing
-.9
-.8
-.7
-.6
-.5
M=0.78
CL=0.27
-.4
-.3
-.2
-.1
Shock wave
0
-.1
A
Pressure coefficient upper surface, Cp,upper (~)
Pressure distribution at AA
-1.0
Modified airfoil
-0.8
Original airfoil
-0.6
M = 0.78
CL= constant
-0.4
-0.2
0
0.2
0.4
0.1
0.2
0.3 0.4 0.5 0.6 0.7 0.8 0.9
Chordwise position, x/c (~)
1.0
8.6 Summary
505
8.12 In Sect. 6.6.2 modifications to the wing-pylon-nacelle area of the Convair 990
have been presented. It was stated that the strong drag increase merely stemmed from
the inboard side of the pylons. Why would the inboard of the pylon produce more
interference drag than the outboard side?
Viscous Flow over Swept Wings
8.13 Let us consider a starboard wing with a mean aerodynamic chord of 6.0 m,
a taper ratio of 0.25, an aspect ratio of 5, and a quarter-chord sweep angle of 30◦
(akin to the Airbus A330). Furthermore, assume the wing has a SC(2) 0412 airfoil
(see Example 8.4), oriented in streamwise direction. Assume this wing is subjected
to air at a velocity of 0.8 times the local speed of sound at an altitude of 10 km and
that (8.34) may be used to compute the attachment-line momentum thickness.
(a) Calculate the span, b of the wing in meters.
(b) Calculate the tip-chord length (ctip ) and the root-chord length (croot ) in meters.
(c) Calculate the sweep angle of the leading edge (Λle ).
(d) Calculate the density, the viscosity, and the local speed of sound for air at an
altitude of 10 km ISA.
(e) Define the relation between nose radius (r ) and spanwise position (y).
(f) Plot the relation between attachment-line momentum-thickness Reynolds number versus spanwise position. Also mark the border where the attachment line
switches from being stable to being susceptible to disturbances and the border
where the attachment line switches from being susceptible to disturbances to
being inherently unstable.
(g) Based on these findings, what conclusions can you draw?
8.14 Near the end of WWII the Germans applied the swept-wing concept to attain
higher speeds. An example is the Junkers, Ju-287, which was a jet-powered bomber.
This aircraft had the peculiar feature that the wings were swept forward.
(a) Explain the general principle of a swept wing.
(b) Why do you think a forward swept wing was chosen, rather than an aft swept
wing?
(c) What is an important drawback of a forward swept wing compared to and aft
swept wing?
506
8 Aerodynamics of Swept Wings
Local lift coefficient, cl, local (~)
8.15 A plain wing (without camber and twist) with Λc/4 = 45◦ having constant
profile has a calculated spanwise distribution of the local lift coefficient as indicated
in the figure below. The profile, considered perpendicular to the quarter-chord line,
has a maximum lift coefficient clmax of 1.5. At what wing lift coefficient do you expect
the initiation of tip stall and unstable behavior?
1.4
CL,wing
1.2
1.0
0.90
0.8
0.75
0.6
0.50
0.45
0.4
0.30
0.2
0.15
0
0
0.5
1
Spanwise position, η (~)
8.16 When the suction peak of a swept wing diminishes due to the formation of a
stable leading-edge vortex, explain why/how this increases the profile drag of the
wing.
Aeroelasticity of Swept Wings
8.17 Consider the wing of Example 8.5. We want to investigate the effect of the
geometric wing parameters on the aerodynamic twist.
(a) Calculate the aerodynamic twist when the wing span increases with 10 %.
(b) Calculate the aerodynamic twist when the flexural rigidity increases with 10 %.
8.6 Summary
507
(c) Calculate the aerodynamic twist when the sweep angle increases with 3◦
(d) To which of the aforementioned parameters is the aerodynamic twist most
sensitive?
8.18 Consider the wing of Example 8.5 but with Λ = 45◦ . We want to know the
sensitivity of the aerodynamic twist to the wing parameters.
(a) Calculate the sensitivity in [degree/m] of the aerodynamic twist w.r.t. the span
of the wing, i.e. dθ/db.
(b) Calculate the sensitivity in [degree/(Nm2 )] of the aerodynamic twist w.r.t. the
flexural stiffness of the wing, i.e. dθ/d(E I ).
(c) Calculate the sensitivity in [degree/degree] of the aerodynamic twist w.r.t. the
sweep angle of the wing, i.e. dθ/dΛ.
(d) To which of the aforementioned parameters is the aerodynamic twist most sensitive? In other words, if each of the aforementioned parameters is increased with
1 %, which has the largest effect on the aerodynamic twist of the wing?
8.19 Below a generic cross section of a wing box is shown. Calculate the second
moment of area about the x and y axis, respectively. Do they match with (8.49)?
What assumptions were made?
z
tflange
h
tskin
x
w
8.20 Below, two beams of the same dimensions are shown that are both subjected
to the same uniform lift distribution. The upper beam is completely rigid and rotates
about a hinge point. The lower beam is clamped and bends under the uniform loading.
Both beams have an identical tip deflection of Δ, where Δ bs /2. Show that the
spring stiffness, k can be expressed as in (8.51).
l
Δ
Γ
k
l
Δ
EI
b /2
s
8.21 Below, two beams of the same dimensions are shown that are both subjected
to the same uniform moment distribution. The upper beam is completely rigid and
rotates about a hinge point. The lower beam is clamped and twists under the uniform
loading. The average twist of the flexible beam equals the rotation of the rigid beam.
508
8 Aerodynamics of Swept Wings
(a) plot the twist distribution in the flexible beam as a function of the spanwise
position, y. Assume m̄/GJ = 1 and bs /2 = 1.
(b) calculate the average rotation in the flexible beam and indicate this in the plot
of (a).
(c) Show that the spring stiffness, k T can be expressed as in (8.52).
kT
mT
m
GJ
bs /2
8.22 Implement the steps outlined in Example 8.6 in a computer program.
(a) Reproduce the plot of Fig. 8.49.
(b) For the same input parameters, plot (as a function of Mach number) the angle of
attack, the twist angle, and the dihedral angle of the wing for each of the sweep
angles. Can you explain their behavior?
8.23 Use the computer program developed in Problem 8.22 to make the following
assignments.
(a) Change the aspect ratio to 2 and 10, respectively. Keep all other parameters as in
Example 8.6. Plot the relation between Mach number and C L δ for sweep angles
of 0, 15, 30, and 45◦ . Also plot the change in α, θ, and Γ . Based on your results,
describe the effect of aspect ratio on the flap effectiveness.
(b) Change the Young’s modulus of the material to 35 and 140 GPa, respectively.
Keep all other parameters as in Example 8.6. Plot the relation between Mach
number and C L δ for the aforementioned sweep angles. Also plot the change in
α, θ, and Γ . Based on your results, describe the effect of material stiffness on
the flap effectiveness.
(c) Keep all parameters as in Example 8.6. Plot the relation between Mach number
and C L δ for sweep angles of 0, −15, −30, and −45◦ . Also plot the change in
α, θ, and Γ . Based on your results, describe the effect of forward sweep on flap
effectiveness. What adverse aeroelastic effect do you observe?
8.24 Implement the steps outlined in Example 8.7 in a computer program.
(a) Reproduce the plots of Fig. 8.50
(b) For the same input parameters, plot (as a function of Mach number) the angle
of attack, the washout angle, and the dihedral angle of the wing for each of the
altitudes. Can you explain their behavior?
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509
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Appendix A
Isentropic Flow Table
Table A.1 displayes the isentropic flow properties for γ = 1.4.
Table A.1 Isentropic flow properties for γ = 1.4
M
Tt /T
pt / p
ρt /ρ
A/A∗
T /T ∗
p/ p ∗
ρ/ρ ∗
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0.22
0.24
0.26
0.28
0.3
0.32
0.34
0.36
0.38
0.4
0.42
0.44
0.46
1.20E+00
1.20E+00
1.20E+00
1.20E+00
1.20E+00
1.20E+00
1.20E+00
1.19E+00
1.19E+00
1.19E+00
1.19E+00
1.19E+00
1.18E+00
1.18E+00
1.18E+00
1.18E+00
1.17E+00
1.17E+00
1.17E+00
1.16E+00
1.16E+00
1.16E+00
1.15E+00
1.89E+00
1.89E+00
1.89E+00
1.88E+00
1.88E+00
1.87E+00
1.87E+00
1.86E+00
1.85E+00
1.84E+00
1.83E+00
1.82E+00
1.81E+00
1.79E+00
1.78E+00
1.76E+00
1.75E+00
1.73E+00
1.71E+00
1.70E+00
1.68E+00
1.66E+00
1.64E+00
1.58E+00
1.58E+00
1.57E+00
1.57E+00
1.57E+00
1.57E+00
1.56E+00
1.56E+00
1.55E+00
1.55E+00
1.54E+00
1.53E+00
1.53E+00
1.52E+00
1.51E+00
1.50E+00
1.49E+00
1.48E+00
1.47E+00
1.46E+00
1.45E+00
1.43E+00
1.42E+00
1.00E+00
1.00E+00
1.00E+00
1.00E+00
1.00E+00
1.00E+00
1.00E+00
1.01E+00
1.01E+00
1.01E+00
1.01E+00
1.01E+00
1.01E+00
1.02E+00
1.02E+00
1.02E+00
1.02E+00
1.03E+00
1.03E+00
1.03E+00
1.04E+00
1.04E+00
1.04E+00
1.00E+00
1.00E+00
1.00E+00
1.00E+00
1.01E+00
1.01E+00
1.01E+00
1.02E+00
1.02E+00
1.03E+00
1.03E+00
1.04E+00
1.05E+00
1.06E+00
1.06E+00
1.07E+00
1.08E+00
1.09E+00
1.10E+00
1.12E+00
1.13E+00
1.14E+00
1.16E+00
1.00E+00
1.00E+00
1.00E+00
1.00E+00
1.01E+00
1.01E+00
1.01E+00
1.01E+00
1.02E+00
1.02E+00
1.02E+00
1.03E+00
1.03E+00
1.04E+00
1.05E+00
1.05E+00
1.06E+00
1.07E+00
1.07E+00
1.08E+00
1.09E+00
1.10E+00
1.11E+00
2.89E+01
1.45E+01
9.67E+00
7.26E+00
5.82E+00
4.86E+00
4.18E+00
3.67E+00
3.28E+00
2.96E+00
2.71E+00
2.50E+00
2.32E+00
2.17E+00
2.04E+00
1.92E+00
1.82E+00
1.74E+00
1.66E+00
1.59E+00
1.53E+00
1.47E+00
1.42E+00
(continued)
© Springer Science+Business Media Dordrecht 2015
R. Vos and S. Farokhi, Introduction to Transonic Aerodynamics,
Fluid Mechanics and Its Applications 110, DOI 10.1007/978-94-017-9747-4
513
514
Appendix A: Isentropic Flow Table
Table A.1 (continued)
M
Tt /T
pt / p
ρt /ρ
A/A∗
T /T ∗
p/ p ∗
ρ/ρ ∗
0.48
0.5
0.52
0.54
0.56
0.58
0.6
0.62
0.64
0.66
0.68
0.7
0.72
0.74
0.76
0.78
0.8
0.82
0.84
0.86
0.88
0.9
0.92
0.94
0.96
0.98
1
1.02
1.04
1.06
1.08
1.1
1.12
1.14
1.16
1.18
1.2
1.22
1.24
1.12E+00
1.13E+00
1.14E+00
1.15E+00
1.16E+00
1.18E+00
1.19E+00
1.20E+00
1.22E+00
1.23E+00
1.25E+00
1.26E+00
1.28E+00
1.30E+00
1.31E+00
1.33E+00
1.35E+00
1.37E+00
1.39E+00
1.41E+00
1.43E+00
1.46E+00
1.48E+00
1.50E+00
1.53E+00
1.55E+00
1.58E+00
1.60E+00
1.63E+00
1.66E+00
1.69E+00
1.72E+00
1.75E+00
1.78E+00
1.81E+00
1.85E+00
1.88E+00
1.92E+00
1.95E+00
1.38E+00
1.34E+00
1.30E+00
1.27E+00
1.24E+00
1.21E+00
1.19E+00
1.17E+00
1.15E+00
1.13E+00
1.11E+00
1.09E+00
1.08E+00
1.07E+00
1.06E+00
1.05E+00
1.04E+00
1.03E+00
1.02E+00
1.02E+00
1.01E+00
1.01E+00
1.01E+00
1.00E+00
1.00E+00
1.00E+00
1.00E+00
1.00E+00
1.00E+00
1.00E+00
1.01E+00
1.01E+00
1.01E+00
1.02E+00
1.02E+00
1.02E+00
1.03E+00
1.04E+00
1.04E+00
1.15E+00
1.14E+00
1.14E+00
1.13E+00
1.13E+00
1.12E+00
1.12E+00
1.11E+00
1.11E+00
1.10E+00
1.10E+00
1.09E+00
1.09E+00
1.08E+00
1.08E+00
1.07E+00
1.06E+00
1.06E+00
1.05E+00
1.05E+00
1.04E+00
1.03E+00
1.03E+00
1.02E+00
1.01E+00
1.01E+00
1.00E+00
9.93E-01
9.87E-01
9.80E-01
9.73E-01
9.66E-01
9.59E-01
9.52E-01
9.46E-01
9.39E-01
9.32E-01
9.25E-01
9.18E-01
1.62E+00
1.60E+00
1.57E+00
1.55E+00
1.53E+00
1.51E+00
1.48E+00
1.46E+00
1.44E+00
1.41E+00
1.39E+00
1.36E+00
1.34E+00
1.32E+00
1.29E+00
1.27E+00
1.24E+00
1.22E+00
1.19E+00
1.17E+00
1.14E+00
1.12E+00
1.10E+00
1.07E+00
1.05E+00
1.02E+00
1.00E+00
9.77E-01
9.54E-01
9.31E-01
9.09E-01
8.87E-01
8.65E-01
8.43E-01
8.22E-01
8.01E-01
7.81E-01
7.60E-01
7.41E-01
1.41E+00
1.40E+00
1.38E+00
1.37E+00
1.35E+00
1.34E+00
1.33E+00
1.31E+00
1.30E+00
1.28E+00
1.26E+00
1.25E+00
1.23E+00
1.22E+00
1.20E+00
1.18E+00
1.17E+00
1.15E+00
1.13E+00
1.12E+00
1.10E+00
1.08E+00
1.07E+00
1.05E+00
1.03E+00
1.02E+00
1.00E+00
9.83E-01
9.67E-01
9.50E-01
9.34E-01
9.18E-01
9.01E-01
8.85E-01
8.69E-01
8.54E-01
8.38E-01
8.22E-01
8.07E-01
1.05E+00
1.05E+00
1.05E+00
1.06E+00
1.06E+00
1.07E+00
1.07E+00
1.08E+00
1.08E+00
1.09E+00
1.09E+00
1.10E+00
1.10E+00
1.11E+00
1.12E+00
1.12E+00
1.13E+00
1.13E+00
1.14E+00
1.15E+00
1.15E+00
1.16E+00
1.17E+00
1.18E+00
1.18E+00
1.19E+00
1.20E+00
1.21E+00
1.22E+00
1.22E+00
1.23E+00
1.24E+00
1.25E+00
1.26E+00
1.27E+00
1.28E+00
1.29E+00
1.30E+00
1.31E+00
1.17E+00
1.19E+00
1.20E+00
1.22E+00
1.24E+00
1.26E+00
1.28E+00
1.30E+00
1.32E+00
1.34E+00
1.36E+00
1.39E+00
1.41E+00
1.44E+00
1.47E+00
1.49E+00
1.52E+00
1.56E+00
1.59E+00
1.62E+00
1.66E+00
1.69E+00
1.73E+00
1.77E+00
1.81E+00
1.85E+00
1.89E+00
1.94E+00
1.98E+00
2.03E+00
2.08E+00
2.14E+00
2.19E+00
2.24E+00
2.30E+00
2.36E+00
2.43E+00
2.49E+00
2.56E+00
(continued)
Appendix A: Isentropic Flow Table
515
Table A.1 (continued)
M
Tt /T
pt / p
ρt /ρ
A/A∗
T /T ∗
p/ p ∗
ρ/ρ ∗
1.26
1.28
1.3
1.32
1.34
1.36
1.38
1.4
1.42
1.44
1.46
1.48
1.5
1.52
1.54
1.56
1.58
1.6
1.62
1.64
1.66
1.68
1.7
1.72
1.74
1.76
1.78
1.8
1.82
1.84
1.86
1.88
1.9
1.92
1.94
1.96
1.98
2
2.02
1.99E+00
2.03E+00
2.07E+00
2.11E+00
2.15E+00
2.20E+00
2.24E+00
2.29E+00
2.33E+00
2.38E+00
2.43E+00
2.48E+00
2.53E+00
2.58E+00
2.64E+00
2.70E+00
2.75E+00
2.81E+00
2.87E+00
2.93E+00
3.00E+00
3.06E+00
3.13E+00
3.20E+00
3.27E+00
3.34E+00
3.41E+00
3.49E+00
3.56E+00
3.64E+00
3.72E+00
3.81E+00
3.89E+00
3.98E+00
4.07E+00
4.16E+00
4.25E+00
4.35E+00
4.44E+00
1.05E+00
1.06E+00
1.07E+00
1.08E+00
1.08E+00
1.09E+00
1.10E+00
1.11E+00
1.13E+00
1.14E+00
1.15E+00
1.16E+00
1.18E+00
1.19E+00
1.20E+00
1.22E+00
1.23E+00
1.25E+00
1.27E+00
1.28E+00
1.30E+00
1.32E+00
1.34E+00
1.36E+00
1.38E+00
1.40E+00
1.42E+00
1.44E+00
1.46E+00
1.48E+00
1.51E+00
1.53E+00
1.56E+00
1.58E+00
1.61E+00
1.63E+00
1.66E+00
1.69E+00
1.72E+00
9.11E-01
9.04E-01
8.97E-01
8.90E-01
8.83E-01
8.76E-01
8.69E-01
8.62E-01
8.55E-01
8.48E-01
8.41E-01
8.34E-01
8.28E-01
8.21E-01
8.14E-01
8.07E-01
8.00E-01
7.94E-01
7.87E-01
7.80E-01
7.74E-01
7.67E-01
7.60E-01
7.54E-01
7.47E-01
7.41E-01
7.35E-01
7.28E-01
7.22E-01
7.16E-01
7.09E-01
7.03E-01
6.97E-01
6.91E-01
6.85E-01
6.79E-01
6.73E-01
6.67E-01
6.61E-01
7.21E-01
7.02E-01
6.83E-01
6.65E-01
6.47E-01
6.29E-01
6.12E-01
5.95E-01
5.78E-01
5.62E-01
5.46E-01
5.31E-01
5.16E-01
5.01E-01
4.86E-01
4.72E-01
4.59E-01
4.45E-01
4.32E-01
4.20E-01
4.07E-01
3.95E-01
3.84E-01
3.72E-01
3.61E-01
3.50E-01
3.40E-01
3.29E-01
3.20E-01
3.10E-01
3.00E-01
2.91E-01
2.83E-01
2.74E-01
2.66E-01
2.57E-01
2.50E-01
2.42E-01
2.35E-01
7.92E-01
7.77E-01
7.62E-01
7.47E-01
7.33E-01
7.18E-01
7.04E-01
6.90E-01
6.76E-01
6.63E-01
6.49E-01
6.36E-01
6.23E-01
6.10E-01
5.98E-01
5.85E-01
5.73E-01
5.61E-01
5.49E-01
5.38E-01
5.26E-01
5.15E-01
5.04E-01
4.94E-01
4.83E-01
4.73E-01
4.62E-01
4.52E-01
4.43E-01
4.33E-01
4.24E-01
4.14E-01
4.05E-01
3.97E-01
3.88E-01
3.79E-01
3.71E-01
3.63E-01
3.55E-01
1.32E+00
1.33E+00
1.34E+00
1.35E+00
1.36E+00
1.37E+00
1.38E+00
1.39E+00
1.40E+00
1.41E+00
1.43E+00
1.44E+00
1.45E+00
1.46E+00
1.47E+00
1.49E+00
1.50E+00
1.51E+00
1.52E+00
1.54E+00
1.55E+00
1.56E+00
1.58E+00
1.59E+00
1.61E+00
1.62E+00
1.63E+00
1.65E+00
1.66E+00
1.68E+00
1.69E+00
1.71E+00
1.72E+00
1.74E+00
1.75E+00
1.77E+00
1.78E+00
1.80E+00
1.82E+00
2.63E+00
2.70E+00
2.77E+00
2.85E+00
2.93E+00
3.01E+00
3.09E+00
3.18E+00
3.27E+00
3.37E+00
3.47E+00
3.57E+00
3.67E+00
3.78E+00
3.89E+00
4.01E+00
4.13E+00
4.25E+00
4.38E+00
4.51E+00
4.65E+00
4.79E+00
4.94E+00
5.09E+00
5.24E+00
5.41E+00
5.57E+00
5.75E+00
5.92E+00
6.11E+00
6.30E+00
6.50E+00
6.70E+00
6.91E+00
7.13E+00
7.35E+00
7.58E+00
7.82E+00
8.07E+00
(continued)
516
Appendix A: Isentropic Flow Table
Table A.1 (continued)
M
Tt /T
pt / p
ρt /ρ
A/A∗
T /T ∗
p/ p ∗
ρ/ρ ∗
2.04
2.06
2.08
2.1
2.12
2.14
2.16
2.18
2.2
2.22
2.24
2.26
2.28
2.3
2.32
2.34
2.36
2.38
2.4
2.42
2.44
2.46
2.48
2.5
2.52
2.54
2.56
2.58
2.6
2.62
2.64
2.66
2.68
2.7
2.72
2.74
2.76
2.78
2.8
4.54E+00
4.65E+00
4.75E+00
4.86E+00
4.97E+00
5.08E+00
5.20E+00
5.31E+00
5.43E+00
5.56E+00
5.68E+00
5.81E+00
5.94E+00
6.08E+00
6.21E+00
6.35E+00
6.50E+00
6.64E+00
6.79E+00
6.95E+00
7.10E+00
7.26E+00
7.43E+00
7.59E+00
7.76E+00
7.94E+00
8.12E+00
8.30E+00
8.48E+00
8.67E+00
8.87E+00
9.06E+00
9.27E+00
9.47E+00
9.68E+00
9.90E+00
1.01E+01
1.03E+01
1.06E+01
1.75E+00
1.78E+00
1.81E+00
1.84E+00
1.87E+00
1.90E+00
1.94E+00
1.97E+00
2.01E+00
2.04E+00
2.08E+00
2.12E+00
2.15E+00
2.19E+00
2.23E+00
2.27E+00
2.32E+00
2.36E+00
2.40E+00
2.45E+00
2.49E+00
2.54E+00
2.59E+00
2.64E+00
2.69E+00
2.74E+00
2.79E+00
2.84E+00
2.90E+00
2.95E+00
3.01E+00
3.06E+00
3.12E+00
3.18E+00
3.24E+00
3.31E+00
3.37E+00
3.43E+00
3.50E+00
6.55E-01
6.49E-01
6.43E-01
6.38E-01
6.32E-01
6.26E-01
6.21E-01
6.15E-01
6.10E-01
6.04E-01
5.99E-01
5.94E-01
5.88E-01
5.83E-01
5.78E-01
5.73E-01
5.68E-01
5.63E-01
5.58E-01
5.53E-01
5.48E-01
5.43E-01
5.38E-01
5.33E-01
5.29E-01
5.24E-01
5.19E-01
5.15E-01
5.10E-01
5.06E-01
5.01E-01
4.97E-01
4.93E-01
4.88E-01
4.84E-01
4.80E-01
4.76E-01
4.71E-01
4.67E-01
2.27E-01
2.20E-01
2.14E-01
2.07E-01
2.01E-01
1.94E-01
1.88E-01
1.83E-01
1.77E-01
1.72E-01
1.66E-01
1.61E-01
1.56E-01
1.51E-01
1.47E-01
1.42E-01
1.38E-01
1.34E-01
1.29E-01
1.26E-01
1.22E-01
1.18E-01
1.14E-01
1.11E-01
1.07E-01
1.04E-01
1.01E-01
9.78E-02
9.49E-02
9.20E-02
8.92E-02
8.65E-02
8.38E-02
8.13E-02
7.88E-02
7.65E-02
7.42E-02
7.19E-02
6.98E-02
3.47E-01
3.39E-01
3.32E-01
3.25E-01
3.17E-01
3.10E-01
3.04E-01
2.97E-01
2.90E-01
2.84E-01
2.78E-01
2.71E-01
2.65E-01
2.60E-01
2.54E-01
2.48E-01
2.43E-01
2.37E-01
2.32E-01
2.27E-01
2.22E-01
2.17E-01
2.12E-01
2.08E-01
2.03E-01
1.99E-01
1.94E-01
1.90E-01
1.86E-01
1.82E-01
1.78E-01
1.74E-01
1.70E-01
1.67E-01
1.63E-01
1.59E-01
1.56E-01
1.53E-01
1.49E-01
1.83E+00
1.85E+00
1.87E+00
1.88E+00
1.90E+00
1.92E+00
1.93E+00
1.95E+00
1.97E+00
1.99E+00
2.00E+00
2.02E+00
2.04E+00
2.06E+00
2.08E+00
2.10E+00
2.11E+00
2.13E+00
2.15E+00
2.17E+00
2.19E+00
2.21E+00
2.23E+00
2.25E+00
2.27E+00
2.29E+00
2.31E+00
2.33E+00
2.35E+00
2.37E+00
2.39E+00
2.42E+00
2.44E+00
2.46E+00
2.48E+00
2.50E+00
2.52E+00
2.55E+00
2.57E+00
8.33E+00
8.59E+00
8.86E+00
9.14E+00
9.44E+00
9.73E+00
1.00E+01
1.04E+01
1.07E+01
1.10E+01
1.14E+01
1.17E+01
1.21E+01
1.25E+01
1.29E+01
1.33E+01
1.37E+01
1.42E+01
1.46E+01
1.51E+01
1.56E+01
1.61E+01
1.66E+01
1.71E+01
1.76E+01
1.82E+01
1.88E+01
1.93E+01
2.00E+01
2.06E+01
2.12E+01
2.19E+01
2.26E+01
2.33E+01
2.40E+01
2.48E+01
2.55E+01
2.63E+01
2.71E+01
(continued)
Appendix A: Isentropic Flow Table
517
Table A.1 (continued)
M
Tt /T
pt / p
ρt /ρ
A/A∗
T /T ∗
p/ p ∗
ρ/ρ ∗
2.82
2.84
2.86
2.88
2.9
2.92
2.94
2.96
2.98
3
3.05
3.1
3.15
3.2
3.25
3.3
3.35
3.4
3.45
3.5
3.55
3.6
3.65
3.7
3.75
3.8
3.85
3.9
3.95
4
4.05
4.1
4.15
4.2
4.25
4.3
4.35
4.4
4.45
1.08E+01
1.10E+01
1.13E+01
1.15E+01
1.18E+01
1.20E+01
1.23E+01
1.26E+01
1.28E+01
1.31E+01
1.38E+01
1.46E+01
1.54E+01
1.62E+01
1.71E+01
1.80E+01
1.90E+01
2.00E+01
2.10E+01
2.21E+01
2.33E+01
2.45E+01
2.57E+01
2.70E+01
2.84E+01
2.98E+01
3.13E+01
3.28E+01
3.45E+01
3.62E+01
3.79E+01
3.97E+01
4.16E+01
4.36E+01
4.57E+01
4.78E+01
5.01E+01
5.24E+01
5.48E+01
3.57E+00
3.64E+00
3.71E+00
3.78E+00
3.85E+00
3.92E+00
4.00E+00
4.08E+00
4.15E+00
4.23E+00
4.44E+00
4.66E+00
4.88E+00
5.12E+00
5.37E+00
5.63E+00
5.90E+00
6.18E+00
6.48E+00
6.79E+00
7.11E+00
7.45E+00
7.80E+00
8.17E+00
8.55E+00
8.95E+00
9.37E+00
9.80E+00
1.03E+01
1.07E+01
1.12E+01
1.17E+01
1.22E+01
1.28E+01
1.34E+01
1.40E+01
1.46E+01
1.52E+01
1.59E+01
4.63E-01
4.59E-01
4.55E-01
4.51E-01
4.47E-01
4.44E-01
4.40E-01
4.36E-01
4.32E-01
4.29E-01
4.20E-01
4.11E-01
4.02E-01
3.94E-01
3.86E-01
3.78E-01
3.70E-01
3.62E-01
3.55E-01
3.48E-01
3.41E-01
3.34E-01
3.27E-01
3.21E-01
3.15E-01
3.09E-01
3.03E-01
2.97E-01
2.91E-01
2.86E-01
2.80E-01
2.75E-01
2.70E-01
2.65E-01
2.60E-01
2.55E-01
2.51E-01
2.46E-01
2.42E-01
6.77E-02
6.56E-02
6.37E-02
6.18E-02
5.99E-02
5.81E-02
5.64E-02
5.47E-02
5.31E-02
5.15E-02
4.78E-02
4.44E-02
4.12E-02
3.83E-02
3.56E-02
3.31E-02
3.08E-02
2.86E-02
2.67E-02
2.48E-02
2.31E-02
2.16E-02
2.01E-02
1.87E-02
1.75E-02
1.63E-02
1.53E-02
1.43E-02
1.33E-02
1.25E-02
1.17E-02
1.09E-02
1.02E-02
9.58E-03
8.98E-03
8.42E-03
7.90E-03
7.42E-03
6.96E-03
1.46E-01
1.43E-01
1.40E-01
1.37E-01
1.34E-01
1.31E-01
1.28E-01
1.26E-01
1.23E-01
1.20E-01
1.14E-01
1.08E-01
1.03E-01
9.73E-02
9.23E-02
8.76E-02
8.32E-02
7.90E-02
7.51E-02
7.14E-02
6.78E-02
6.45E-02
6.14E-02
5.84E-02
5.56E-02
5.29E-02
5.04E-02
4.80E-02
4.58E-02
4.36E-02
4.16E-02
3.97E-02
3.79E-02
3.62E-02
3.45E-02
3.30E-02
3.15E-02
3.01E-02
2.88E-02
2.59E+00
2.61E+00
2.64E+00
2.66E+00
2.68E+00
2.71E+00
2.73E+00
2.75E+00
2.78E+00
2.80E+00
2.86E+00
2.92E+00
2.98E+00
3.05E+00
3.11E+00
3.18E+00
3.24E+00
3.31E+00
3.38E+00
3.45E+00
3.52E+00
3.59E+00
3.66E+00
3.74E+00
3.81E+00
3.89E+00
3.96E+00
4.04E+00
4.12E+00
4.20E+00
4.28E+00
4.36E+00
4.44E+00
4.53E+00
4.61E+00
4.70E+00
4.78E+00
4.87E+00
4.96E+00
2.80E+01
2.88E+01
2.97E+01
3.07E+01
3.16E+01
3.26E+01
3.36E+01
3.46E+01
3.56E+01
3.67E+01
3.96E+01
4.26E+01
4.59E+01
4.94E+01
5.32E+01
5.72E+01
6.15E+01
6.61E+01
7.10E+01
7.63E+01
8.19E+01
8.78E+01
9.42E+01
1.01E+02
1.08E+02
1.16E+02
1.24E+02
1.33E+02
1.42E+02
1.52E+02
1.62E+02
1.73E+02
1.85E+02
1.98E+02
2.11E+02
2.25E+02
2.40E+02
2.55E+02
2.72E+02
(continued)
518
Appendix A: Isentropic Flow Table
Table A.1 (continued)
M
Tt /T
pt / p
ρt /ρ
A/A∗
T /T ∗
p/ p ∗
ρ/ρ ∗
4.5
4.55
4.6
4.65
4.7
4.75
4.8
4.85
4.9
4.95
5
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
6
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9
7
7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
5.73E+01
5.99E+01
6.26E+01
6.54E+01
6.83E+01
7.13E+01
7.45E+01
7.77E+01
8.11E+01
8.46E+01
8.82E+01
9.58E+01
1.04E+02
1.13E+02
1.22E+02
1.32E+02
1.43E+02
1.54E+02
1.66E+02
1.79E+02
1.93E+02
2.07E+02
2.22E+02
2.39E+02
2.56E+02
2.75E+02
2.94E+02
3.14E+02
3.36E+02
3.59E+02
3.83E+02
4.09E+02
4.36E+02
4.64E+02
4.94E+02
5.25E+02
5.58E+02
5.93E+02
6.29E+02
1.66E+01
1.73E+01
1.80E+01
1.88E+01
1.96E+01
2.04E+01
2.13E+01
2.21E+01
2.31E+01
2.40E+01
2.50E+01
2.71E+01
2.93E+01
3.16E+01
3.42E+01
3.69E+01
3.97E+01
4.28E+01
4.60E+01
4.95E+01
5.32E+01
5.71E+01
6.12E+01
6.56E+01
7.02E+01
7.51E+01
8.03E+01
8.58E+01
9.16E+01
9.77E+01
1.04E+02
1.11E+02
1.18E+02
1.26E+02
1.34E+02
1.42E+02
1.51E+02
1.60E+02
1.69E+02
2.38E-01
2.33E-01
2.29E-01
2.25E-01
2.21E-01
2.18E-01
2.14E-01
2.10E-01
2.07E-01
2.03E-01
2.00E-01
1.93E-01
1.87E-01
1.81E-01
1.76E-01
1.70E-01
1.65E-01
1.60E-01
1.55E-01
1.51E-01
1.46E-01
1.42E-01
1.38E-01
1.34E-01
1.31E-01
1.27E-01
1.24E-01
1.20E-01
1.17E-01
1.14E-01
1.11E-01
1.08E-01
1.06E-01
1.03E-01
1.00E-01
9.80E-02
9.56E-02
9.33E-02
9.11E-02
6.54E-03
6.15E-03
5.78E-03
5.43E-03
5.11E-03
4.81E-03
4.53E-03
4.27E-03
4.02E-03
3.79E-03
3.58E-03
3.19E-03
2.84E-03
2.54E-03
2.27E-03
2.03E-03
1.83E-03
1.64E-03
1.48E-03
1.33E-03
1.20E-03
1.08E-03
9.79E-04
8.87E-04
8.04E-04
7.30E-04
6.63E-04
6.03E-04
5.49E-04
5.01E-04
4.57E-04
4.18E-04
3.82E-04
3.50E-04
3.21E-04
2.94E-04
2.70E-04
2.48E-04
2.28E-04
2.75E-02
2.63E-02
2.52E-02
2.41E-02
2.31E-02
2.21E-02
2.12E-02
2.03E-02
1.95E-02
1.87E-02
1.79E-02
1.65E-02
1.52E-02
1.40E-02
1.29E-02
1.20E-02
1.11E-02
1.02E-02
9.50E-03
8.82E-03
8.19E-03
7.62E-03
7.09E-03
6.60E-03
6.16E-03
5.75E-03
5.37E-03
5.02E-03
4.69E-03
4.39E-03
4.12E-03
3.86E-03
3.62E-03
3.40E-03
3.19E-03
3.00E-03
2.83E-03
2.66E-03
2.51E-03
5.05E+00
5.14E+00
5.23E+00
5.32E+00
5.42E+00
5.51E+00
5.61E+00
5.70E+00
5.80E+00
5.90E+00
6.00E+00
6.20E+00
6.41E+00
6.62E+00
6.83E+00
7.05E+00
7.27E+00
7.50E+00
7.73E+00
7.96E+00
8.20E+00
8.44E+00
8.69E+00
8.94E+00
9.19E+00
9.45E+00
9.71E+00
9.98E+00
1.02E+01
1.05E+01
1.08E+01
1.11E+01
1.14E+01
1.17E+01
1.20E+01
1.23E+01
1.26E+01
1.29E+01
1.32E+01
2.89E+02
3.08E+02
3.28E+02
3.48E+02
3.70E+02
3.93E+02
4.18E+02
4.43E+02
4.70E+02
4.99E+02
5.29E+02
5.94E+02
6.66E+02
7.46E+02
8.34E+02
9.30E+02
1.04E+03
1.15E+03
1.28E+03
1.42E+03
1.58E+03
1.75E+03
1.93E+03
2.13E+03
2.35E+03
2.59E+03
2.85E+03
3.14E+03
3.45E+03
3.78E+03
4.14E+03
4.53E+03
4.95E+03
5.41E+03
5.90E+03
6.43E+03
7.01E+03
7.62E+03
8.29E+03
(continued)
Appendix A: Isentropic Flow Table
519
Table A.1 (continued)
M
Tt /T
pt / p
ρt /ρ
A/A∗
T /T ∗
p/ p ∗
ρ/ρ ∗
7.9
8
8.1
8.2
8.3
8.4
8.5
8.6
8.7
8.8
8.9
9
9.1
9.2
9.3
9.4
9.5
9.6
9.7
9.8
9.9
10
6.67E+02
7.07E+02
7.49E+02
7.93E+02
8.40E+02
8.88E+02
9.38E+02
9.91E+02
1.05E+03
1.10E+03
1.16E+03
1.23E+03
1.29E+03
1.36E+03
1.43E+03
1.51E+03
1.58E+03
1.66E+03
1.75E+03
1.84E+03
1.93E+03
2.02E+03
1.80E+02
1.90E+02
2.01E+02
2.13E+02
2.25E+02
2.38E+02
2.51E+02
2.65E+02
2.80E+02
2.95E+02
3.11E+02
3.27E+02
3.44E+02
3.62E+02
3.81E+02
4.01E+02
4.21E+02
4.42E+02
4.64E+02
4.87E+02
5.11E+02
5.36E+02
8.90E-02
8.70E-02
8.50E-02
8.31E-02
8.12E-02
7.94E-02
7.77E-02
7.60E-02
7.44E-02
7.28E-02
7.13E-02
6.98E-02
6.83E-02
6.69E-02
6.56E-02
6.43E-02
6.30E-02
6.18E-02
6.06E-02
5.94E-02
5.82E-02
5.71E-02
2.10E-04
1.94E-04
1.79E-04
1.65E-04
1.53E-04
1.41E-04
1.31E-04
1.21E-04
1.12E-04
1.04E-04
9.66E-05
8.97E-05
8.34E-05
7.76E-05
7.22E-05
6.73E-05
6.27E-05
5.85E-05
5.46E-05
5.10E-05
4.77E-05
4.46E-05
2.36E-03
2.23E-03
2.10E-03
1.99E-03
1.88E-03
1.78E-03
1.68E-03
1.59E-03
1.51E-03
1.43E-03
1.36E-03
1.29E-03
1.22E-03
1.16E-03
1.10E-03
1.05E-03
9.96E-04
9.48E-04
9.02E-04
8.59E-04
8.19E-04
7.81E-04
1.35E+01
1.38E+01
1.41E+01
1.44E+01
1.48E+01
1.51E+01
1.55E+01
1.58E+01
1.61E+01
1.65E+01
1.68E+01
1.72E+01
1.76E+01
1.79E+01
1.83E+01
1.87E+01
1.91E+01
1.94E+01
1.98E+01
2.02E+01
2.06E+01
2.10E+01
9.00E+03
9.76E+03
1.06E+04
1.15E+04
1.24E+04
1.34E+04
1.45E+04
1.57E+04
1.69E+04
1.82E+04
1.96E+04
2.11E+04
2.27E+04
2.44E+04
2.62E+04
2.81E+04
3.02E+04
3.23E+04
3.47E+04
3.71E+04
3.97E+04
4.24E+04
Appendix B
Normal Shock Table
Table B.1 shows the normal shock table for γ = 1.4.
Table B.1 Normal shock table for γ = 1.4
M1
M2
T2 /T1
p2 / p1
ρ2 /ρ1
pt2 / pt1
Pt2 / p1
s/R
1
1.02
1.04
1.06
1.08
1.1
1.12
1.14
1.16
1.18
1.2
1.22
1.24
1.26
1.28
1.3
1.32
1.34
1.36
1.38
1.4
1.42
1.44
1
1.00E+00
1.00E+00
1.00E+00
9.99E-01
9.99E-01
9.98E-01
9.97E-01
9.96E-01
9.95E-01
9.93E-01
9.91E-01
9.88E-01
9.86E-01
9.83E-01
9.79E-01
9.76E-01
9.72E-01
9.68E-01
9.63E-01
9.58E-01
9.53E-01
9.48E-01
1.8929
1.94E+00
1.98E+00
2.03E+00
2.08E+00
2.13E+00
2.19E+00
2.24E+00
2.29E+00
2.35E+00
2.41E+00
2.47E+00
2.53E+00
2.59E+00
2.65E+00
2.71E+00
2.78E+00
2.84E+00
2.91E+00
2.98E+00
3.05E+00
3.12E+00
3.19E+00
0
9.96E-06
7.67E-05
2.49E-04
5.69E-04
1.07E-03
1.79E-03
2.74E-03
3.96E-03
5.45E-03
7.23E-03
9.31E-03
1.17E-02
1.44E-02
1.75E-02
2.08E-02
2.45E-02
2.86E-02
3.30E-02
3.77E-02
4.27E-02
4.81E-02
5.38E-02
1
0.9805
0.962
0.9444
0.9277
0.9118
0.8966
0.882
0.8682
0.8549
0.8422
0.83
0.8183
0.8071
0.7963
0.786
0.776
0.7664
0.7572
0.7483
0.7397
0.7314
0.7235
1
1.01E+00
1.03E+00
1.04E+00
1.05E+00
1.06E+00
1.08E+00
1.09E+00
1.10E+00
1.12E+00
1.13E+00
1.14E+00
1.15E+00
1.17E+00
1.18E+00
1.19E+00
1.20E+00
1.22E+00
1.23E+00
1.24E+00
1.25E+00
1.27E+00
1.28E+00
1
1.05E+00
1.10E+00
1.14E+00
1.19E+00
1.25E+00
1.30E+00
1.35E+00
1.40E+00
1.46E+00
1.51E+00
1.57E+00
1.63E+00
1.69E+00
1.74E+00
1.81E+00
1.87E+00
1.93E+00
1.99E+00
2.06E+00
2.12E+00
2.19E+00
2.25E+00
1
1.03E+00
1.07E+00
1.10E+00
1.13E+00
1.17E+00
1.20E+00
1.24E+00
1.27E+00
1.31E+00
1.34E+00
1.38E+00
1.41E+00
1.45E+00
1.48E+00
1.52E+00
1.55E+00
1.59E+00
1.62E+00
1.65E+00
1.69E+00
1.72E+00
1.76E+00
(continued)
© Springer Science+Business Media Dordrecht 2015
R. Vos and S. Farokhi, Introduction to Transonic Aerodynamics,
Fluid Mechanics and Its Applications 110, DOI 10.1007/978-94-017-9747-4
521
522
Appendix B: Normal Shock Table
Table B.1 (continued)
M1
M2
T2 /T1
p2 / p1
ρ2 /ρ1
pt2 / pt1
Pt2 / p1
s/R
1.46
1.48
1.5
1.52
1.54
1.56
1.58
1.6
1.62
1.64
1.66
1.68
1.7
1.72
1.74
1.76
1.78
1.8
1.82
1.84
1.86
1.88
1.9
1.92
1.94
1.96
1.98
2
2.02
2.04
2.06
2.08
2.1
2.12
2.14
2.16
2.18
2.2
2.22
2.32E+00
2.39E+00
2.46E+00
2.53E+00
2.60E+00
2.67E+00
2.75E+00
2.82E+00
2.90E+00
2.97E+00
3.05E+00
3.13E+00
3.21E+00
3.28E+00
3.37E+00
3.45E+00
3.53E+00
3.61E+00
3.70E+00
3.78E+00
3.87E+00
3.96E+00
4.05E+00
4.13E+00
4.22E+00
4.32E+00
4.41E+00
4.50E+00
4.59E+00
4.69E+00
4.78E+00
4.88E+00
4.98E+00
5.08E+00
5.18E+00
5.28E+00
5.38E+00
5.48E+00
5.58E+00
1.79E+00
1.83E+00
1.86E+00
1.90E+00
1.93E+00
1.96E+00
2.00E+00
2.03E+00
2.07E+00
2.10E+00
2.13E+00
2.16E+00
2.20E+00
2.23E+00
2.26E+00
2.30E+00
2.33E+00
2.36E+00
2.39E+00
2.42E+00
2.45E+00
2.48E+00
2.52E+00
2.55E+00
2.58E+00
2.61E+00
2.64E+00
2.67E+00
2.70E+00
2.73E+00
2.75E+00
2.78E+00
2.81E+00
2.84E+00
2.87E+00
2.90E+00
2.92E+00
2.95E+00
2.98E+00
9.42E-01
9.36E-01
9.30E-01
9.23E-01
9.17E-01
9.10E-01
9.03E-01
8.95E-01
8.88E-01
8.80E-01
8.72E-01
8.64E-01
8.56E-01
8.47E-01
8.39E-01
8.30E-01
8.22E-01
8.13E-01
8.04E-01
7.95E-01
7.86E-01
7.77E-01
7.67E-01
7.58E-01
7.49E-01
7.40E-01
7.30E-01
7.21E-01
7.12E-01
7.02E-01
6.93E-01
6.84E-01
6.74E-01
6.65E-01
6.56E-01
6.46E-01
6.37E-01
6.28E-01
6.19E-01
3.26E+00
3.34E+00
3.41E+00
3.49E+00
3.57E+00
3.65E+00
3.72E+00
3.81E+00
3.89E+00
3.97E+00
4.05E+00
4.14E+00
4.22E+00
4.31E+00
4.40E+00
4.49E+00
4.58E+00
4.67E+00
4.76E+00
4.86E+00
4.95E+00
5.05E+00
5.14E+00
5.24E+00
5.34E+00
5.44E+00
5.54E+00
5.64E+00
5.74E+00
5.85E+00
5.95E+00
6.06E+00
6.17E+00
6.27E+00
6.38E+00
6.49E+00
6.60E+00
6.72E+00
6.83E+00
5.98E-02
6.61E-02
7.28E-02
7.98E-02
8.71E-02
9.46E-02
1.03E-01
1.11E-01
1.19E-01
1.28E-01
1.37E-01
1.46E-01
1.56E-01
1.66E-01
1.76E-01
1.86E-01
1.97E-01
2.07E-01
2.18E-01
2.30E-01
2.41E-01
2.53E-01
2.65E-01
2.77E-01
2.89E-01
3.02E-01
3.14E-01
3.27E-01
3.40E-01
3.54E-01
3.67E-01
3.81E-01
3.94E-01
4.08E-01
4.22E-01
4.36E-01
4.51E-01
4.65E-01
4.80E-01
0.7157
0.7083
0.7011
0.6941
0.6874
0.6809
0.6746
0.6684
0.6625
0.6568
0.6512
0.6458
0.6405
0.6355
0.6305
0.6257
0.621
0.6165
0.6121
0.6078
0.6036
0.5996
0.5956
0.5918
0.588
0.5844
0.5808
0.5774
0.574
0.5707
0.5675
0.5643
0.5613
0.5583
0.5554
0.5525
0.5498
0.5471
0.5444
1.29E+00
1.31E+00
1.32E+00
1.33E+00
1.35E+00
1.36E+00
1.37E+00
1.39E+00
1.40E+00
1.42E+00
1.43E+00
1.44E+00
1.46E+00
1.47E+00
1.49E+00
1.50E+00
1.52E+00
1.53E+00
1.55E+00
1.56E+00
1.58E+00
1.59E+00
1.61E+00
1.62E+00
1.64E+00
1.66E+00
1.67E+00
1.69E+00
1.70E+00
1.72E+00
1.74E+00
1.75E+00
1.77E+00
1.79E+00
1.80E+00
1.82E+00
1.84E+00
1.86E+00
1.87E+00
(continued)
Appendix B: Normal Shock Table
523
Table B.1 (continued)
M1
M2
T2 /T1
p2 / p1
ρ2 /ρ1
pt2 / pt1
Pt2 / p1
s/R
2.24
2.26
2.28
2.3
2.32
2.34
2.36
2.38
2.4
2.42
2.44
2.46
2.48
2.5
2.52
2.54
2.56
2.58
2.6
2.62
2.64
2.66
2.68
2.7
2.72
2.74
2.76
2.78
2.8
2.82
2.84
2.86
2.88
2.9
2.92
2.94
2.96
2.98
5.69E+00
5.79E+00
5.90E+00
6.01E+00
6.11E+00
6.22E+00
6.33E+00
6.44E+00
6.55E+00
6.67E+00
6.78E+00
6.89E+00
7.01E+00
7.13E+00
7.24E+00
7.36E+00
7.48E+00
7.60E+00
7.72E+00
7.84E+00
7.96E+00
8.09E+00
8.21E+00
8.34E+00
8.46E+00
8.59E+00
8.72E+00
8.85E+00
8.98E+00
9.11E+00
9.24E+00
9.38E+00
9.51E+00
9.65E+00
9.78E+00
9.92E+00
1.01E+01
1.02E+01
3.01E+00
3.03E+00
3.06E+00
3.08E+00
3.11E+00
3.14E+00
3.16E+00
3.19E+00
3.21E+00
3.24E+00
3.26E+00
3.29E+00
3.31E+00
3.33E+00
3.36E+00
3.38E+00
3.40E+00
3.43E+00
3.45E+00
3.47E+00
3.49E+00
3.52E+00
3.54E+00
3.56E+00
3.58E+00
3.60E+00
3.62E+00
3.64E+00
3.66E+00
3.68E+00
3.70E+00
3.72E+00
3.74E+00
3.76E+00
3.78E+00
3.80E+00
3.82E+00
3.84E+00
6.10E-01
6.01E-01
5.92E-01
5.83E-01
5.75E-01
5.66E-01
5.57E-01
5.49E-01
5.40E-01
5.32E-01
5.23E-01
5.15E-01
5.07E-01
4.99E-01
4.91E-01
4.83E-01
4.75E-01
4.68E-01
4.60E-01
4.53E-01
4.45E-01
4.38E-01
4.31E-01
4.24E-01
4.17E-01
4.10E-01
4.03E-01
3.96E-01
3.89E-01
3.83E-01
3.76E-01
3.70E-01
3.64E-01
3.58E-01
3.52E-01
3.46E-01
3.40E-01
3.34E-01
6.94E+00
7.06E+00
7.18E+00
7.29E+00
7.41E+00
7.53E+00
7.65E+00
7.77E+00
7.90E+00
8.02E+00
8.15E+00
8.27E+00
8.40E+00
8.53E+00
8.66E+00
8.79E+00
8.92E+00
9.05E+00
9.18E+00
9.32E+00
9.45E+00
9.59E+00
9.72E+00
9.86E+00
1.00E+01
1.01E+01
1.03E+01
1.04E+01
1.06E+01
1.07E+01
1.09E+01
1.10E+01
1.12E+01
1.13E+01
1.15E+01
1.16E+01
1.18E+01
1.19E+01
4.94E-01
5.09E-01
5.24E-01
5.39E-01
5.54E-01
5.70E-01
5.85E-01
6.00E-01
6.16E-01
6.32E-01
6.47E-01
6.63E-01
6.79E-01
6.95E-01
7.11E-01
7.27E-01
7.44E-01
7.60E-01
7.76E-01
7.93E-01
8.09E-01
8.26E-01
8.42E-01
8.59E-01
8.76E-01
8.92E-01
9.09E-01
9.26E-01
9.43E-01
9.60E-01
9.77E-01
9.94E-01
1.01E+00
1.03E+00
1.05E+00
1.06E+00
1.08E+00
1.10E+00
0.5418
0.5393
0.5368
0.5344
0.5321
0.5297
0.5275
0.5253
0.5231
0.521
0.5189
0.5169
0.5149
0.513
0.5111
0.5092
0.5074
0.5056
0.5039
0.5022
0.5005
0.4988
0.4972
0.4956
0.4941
0.4926
0.4911
0.4896
0.4882
0.4868
0.4854
0.484
0.4827
0.4814
0.4801
0.4788
0.4776
0.4764
1.89E+00
1.91E+00
1.93E+00
1.95E+00
1.97E+00
1.98E+00
2.00E+00
2.02E+00
2.04E+00
2.06E+00
2.08E+00
2.10E+00
2.12E+00
2.14E+00
2.16E+00
2.18E+00
2.20E+00
2.22E+00
2.24E+00
2.26E+00
2.28E+00
2.30E+00
2.32E+00
2.34E+00
2.36E+00
2.39E+00
2.41E+00
2.43E+00
2.45E+00
2.47E+00
2.50E+00
2.52E+00
2.54E+00
2.56E+00
2.59E+00
2.61E+00
2.63E+00
2.66E+00
(continued)
524
Appendix B: Normal Shock Table
Table B.1 (continued)
M1
M2
T2 /T1
p2 / p1
ρ2 /ρ1
pt2 / pt1
Pt2 / p1
s/R
3
3.02
3.04
3.06
3.08
3.1
3.12
3.14
3.16
3.18
3.2
3.22
3.24
3.26
3.28
3.3
3.32
3.34
3.36
3.38
3.4
3.42
3.44
3.46
3.48
3.5
3.52
3.54
3.56
3.58
3.6
3.62
3.64
3.66
3.68
3.7
3.72
3.74
3.76
1.03E+01
1.05E+01
1.06E+01
1.08E+01
1.09E+01
1.10E+01
1.12E+01
1.13E+01
1.15E+01
1.16E+01
1.18E+01
1.19E+01
1.21E+01
1.22E+01
1.24E+01
1.25E+01
1.27E+01
1.28E+01
1.30E+01
1.32E+01
1.33E+01
1.35E+01
1.36E+01
1.38E+01
1.40E+01
1.41E+01
1.43E+01
1.45E+01
1.46E+01
1.48E+01
1.50E+01
1.51E+01
1.53E+01
1.55E+01
1.56E+01
1.58E+01
1.60E+01
1.62E+01
1.63E+01
3.86E+00
3.88E+00
3.89E+00
3.91E+00
3.93E+00
3.95E+00
3.96E+00
3.98E+00
4.00E+00
4.01E+00
4.03E+00
4.05E+00
4.06E+00
4.08E+00
4.10E+00
4.11E+00
4.13E+00
4.14E+00
4.16E+00
4.17E+00
4.19E+00
4.20E+00
4.22E+00
4.23E+00
4.25E+00
4.26E+00
4.27E+00
4.29E+00
4.30E+00
4.32E+00
4.33E+00
4.34E+00
4.36E+00
4.37E+00
4.38E+00
4.39E+00
4.41E+00
4.42E+00
4.43E+00
3.28E-01
3.23E-01
3.17E-01
3.12E-01
3.06E-01
3.01E-01
2.96E-01
2.91E-01
2.86E-01
2.81E-01
2.76E-01
2.71E-01
2.67E-01
2.62E-01
2.58E-01
2.53E-01
2.49E-01
2.45E-01
2.40E-01
2.36E-01
2.32E-01
2.28E-01
2.24E-01
2.20E-01
2.17E-01
2.13E-01
2.09E-01
2.06E-01
2.02E-01
1.99E-01
1.95E-01
1.92E-01
1.89E-01
1.85E-01
1.82E-01
1.79E-01
1.76E-01
1.73E-01
1.70E-01
1.21E+01
1.22E+01
1.24E+01
1.25E+01
1.27E+01
1.28E+01
1.30E+01
1.32E+01
1.33E+01
1.35E+01
1.37E+01
1.38E+01
1.40E+01
1.42E+01
1.43E+01
1.45E+01
1.47E+01
1.48E+01
1.50E+01
1.52E+01
1.54E+01
1.55E+01
1.57E+01
1.59E+01
1.61E+01
1.62E+01
1.64E+01
1.66E+01
1.68E+01
1.70E+01
1.72E+01
1.73E+01
1.75E+01
1.77E+01
1.79E+01
1.81E+01
1.83E+01
1.85E+01
1.87E+01
1.11E+00
1.13E+00
1.15E+00
1.17E+00
1.18E+00
1.20E+00
1.22E+00
1.23E+00
1.25E+00
1.27E+00
1.29E+00
1.30E+00
1.32E+00
1.34E+00
1.36E+00
1.37E+00
1.39E+00
1.41E+00
1.43E+00
1.44E+00
1.46E+00
1.48E+00
1.49E+00
1.51E+00
1.53E+00
1.55E+00
1.56E+00
1.58E+00
1.60E+00
1.62E+00
1.63E+00
1.65E+00
1.67E+00
1.68E+00
1.70E+00
1.72E+00
1.74E+00
1.75E+00
1.77E+00
0.4752
0.474
0.4729
0.4717
0.4706
0.4695
0.4685
0.4674
0.4664
0.4654
0.4643
0.4634
0.4624
0.4614
0.4605
0.4596
0.4587
0.4578
0.4569
0.456
0.4552
0.4544
0.4535
0.4527
0.4519
0.4512
0.4504
0.4496
0.4489
0.4481
0.4474
0.4467
0.446
0.4453
0.4446
0.4439
0.4433
0.4426
0.442
2.68E+00
2.70E+00
2.73E+00
2.75E+00
2.77E+00
2.80E+00
2.82E+00
2.85E+00
2.87E+00
2.90E+00
2.92E+00
2.95E+00
2.97E+00
3.00E+00
3.02E+00
3.05E+00
3.08E+00
3.10E+00
3.13E+00
3.15E+00
3.18E+00
3.21E+00
3.23E+00
3.26E+00
3.29E+00
3.32E+00
3.34E+00
3.37E+00
3.40E+00
3.43E+00
3.45E+00
3.48E+00
3.51E+00
3.54E+00
3.57E+00
3.60E+00
3.63E+00
3.65E+00
3.68E+00
(continued)
Appendix B: Normal Shock Table
525
Table B.1 (continued)
M1
M2
T2 /T1
p2 / p1
ρ2 /ρ1
pt2 / pt1
Pt2 / p1
s/R
3.78
3.8
3.82
3.84
3.86
3.88
3.9
3.92
3.94
3.96
3.98
4
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
5
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
6
6.1
6.2
6.3
6.4
6.5
6.6
6.7
1.65E+01
1.67E+01
1.69E+01
1.70E+01
1.72E+01
1.74E+01
1.76E+01
1.78E+01
1.79E+01
1.81E+01
1.83E+01
1.85E+01
1.94E+01
2.04E