Aerodynamics
of the
Airplane
Hermciui Sch!ichthg and Erie lw
c
Translated by Heinrich J. Ramm
ro t
AERODYNAMICS
OF THE AIRPLANE
Hermann Schlichting
Professor, Technical University of Braunschweig
and Aerodynamic Research Institute (A VA), Gottingen
Erich Truckenbrodt
Professor, Technical University of Munich
Translated by
Heinrich J. Ramm
Associate Professor, University of Tennessee Space Institute
McGraw-Hill International Book Company
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AERODYNAMICS OF THE AIRPLANE
Copyright © 1979 by McGraw-Hill, Inc. All rights reserved. Printed in the United States
of America. No part of this publication may be reproduced, stored in a. retrieval system,
or transmitted, in any form or by any means, electronic, mechanical, photocopying,
recording, or otherwise, without the prior written permission of the publisher.
1234567890 MPMP 7832109
Library of Congress Cataloging in Publication Data
Schlichting, Hermann, date.
Aerodynamics of the airplane.
Translation of Aerodynamik des Flugzeuges.
Bibliography: p.
Includes index.
1. Aerodynamics. I. Truckenbrodt, Erich,
date, joint author. H. Title.
TL570.S283313
79-60
629.132'3
ISBN 0-07-055341-6
CONTENTS
Preface
Nomenclature
1
1-1
1-2
1-3
2-1
2-2
2-3
2-4
2-5
3
3-1
3-2
3-3
3-4
3-5
3-6
ix
Introduction
Problems of Airplane Aerodynamics
Physical Properties of Air
Aerodynamic Behavior of Airplanes
References
Part 1 Aerodynamics of the Wing
2
vii
Airfoil of Infinite Span
in Incompressible Flow (Profile Theory)
Introduction
Fundamentals of Lift Theory
Profile Theory by the Method of Conformal Mapping
Profile Theory by the Method of Singularities
Influence of Viscosity and Boundary-Layer Control
on Profile Characteristics
1
1
2
8
22
23
25
25
30
36
52
81
References
101
Wings of Finite Span in Incompressible Flow
105
Introduction
Wing Theory by the Method of ` ortex Distribution
105
Lift of Wings in Incompressible Flow
Induced Drag of Wings
Flight Mechanical Coefficients of the Wing
Wing of Finite Thickness at Zero Lift
References
131
112
173
181
197
206
Vi CONTENTS
4
Wings in Compressible Flow
4-1
Introduction
4-2
4-3
Basic Concept of the Wing in Compressible Flow
Airfoil of Infinite Span in Compressible Flow
(Profile Theory)
Wing of Finite Span in Subsonic and Transonic Flow
Wing of Finite Span at Supersonic Incident Flow
References
4-4
4-5
213
213
214
227
261
276
317
Part 2 Aerodynamics of the Fuselage and
the Wing-Fuselage System
325
Aerodynamics of the Fuselage
327
5
5-1
5-2
5-3
6
6-1
6-2
6-3
6-4
Introduction
The Fuselage in Incompressible Flow
The Fuselage in Compressible Flow
7-1
7-2
7-3
8
8-1
8-2
8-3
331
References
348
367
Aerodynamics of the Wing-Fuselage System
371
Introduction
The Wing-Fuselage System in Incompressible Flow
The Wing-Fuselage System in Compressible Flow
Slender Bodies
References
Part 3 Aerodynamics of the Stabilizers
and Control Surfaces
7
327
Aerodynamics of the Stabilizers
Introduction
Aerodynamics of the Horizontal Tail
Aerodynamics of the Vertical Tail
371
376
401
416
425
429
431
431
References
435
466
477
Aerodynamics of the Flaps and Control Surfaces
481
Introduction
The Flap Wing of Infinite Span (Profile Theory)
Flaps on the Wing of Finite Span and on the Tail Unit
References
Bibliography
Author Index
Subject Index
481
486
506
517
521
527
537
PREFACE
Only a very few comprehensive presentations of the scientific fundamentals of the
aerodynamics of the airplane have ever been published. The present book is an English
translation of the two-volume work "Aerodynamik des Flugzeuges," which has already
appeared in a second edition in the original German. In this book we treat exclusively
the aerodynamic forces that act on airplane components-and thus on the whole
airplane-during its motion through the earth's atmosphere (aerodynamics of the
airframe). These aerodynamic forces depend in a quite complex manner on the
geometry, speed, and motion of the airplane and on the properties of air. The
determination of these relationships is the object of the study of the aerodynamics of
the airplane. Moreover, these relationships provide the absolutely necessary basis for
determining the flight mechanics and many questions of the structural requirements of
the airplane, and thus for airplane design. The aerodynamic problems related to
airplane propulsion (power plant aerodynamics) and the theory of the modes of
motion of the airplane (flight mechanics) are not treated in this book.
The study of the aerodynamics of the airplane requires a thorough knowledge of
aerodynamic theory. Therefore, it was necessary to include in the German edition a
rather comprehensive outline of fluid mechanic theory. In the English edition this
section has been eliminated because there exist a sufficient number of pertinent works
in English on the fundamentals of fluid mechanic theory.
Chapter 1 serves as an introduction. It describes the physical properties of air and
of the atmosphere, and outlines the basic aerodynamic behavior of the airplane. The
main portion of the book consists of three major divisions. In the first division (Part
1), Chaps. 2-4 cover the aerodynamics of the airfoil. In the second division (Part 2),
Chaps. 5 and 6 consider the aerodynamics of the fuselage and of the wing-fuselage
system. Finally, in the third division (Part 3), Chaps. 7 and 8 are devoted to the
problems of the aerodynamics of the stability and control systems (empennage, flaps,
and control surfaces). In Parts 2 and 3, the interactions among the individual parts of
the airplane, that is, the aerodynamic interference, are given special attention.
Specifically, the following brief outline describes the chapters that deal with the
intrinsic problems of the aerodynamics of the airplane: Part 1 contains, in Chap. 2, the
profile theory of incompressible flow, including the influence of friction on the profile
viii PREFACE
characteristics. Chapter 3 gives a comprehensive account of three-dimensional wing
theory for incompressible flow (lifting-line and lifting-surface theory). In addition to
linear airfoil theory, nonlinear wing theory is treated because it is of particular
importance for modern airplanes (slender wings). The theory for incompressible flow
is important not only in the range of moderate flight velocities, at which the
compressibility of the air may be disregarded, but even at higher velocities, up to the
speed of sound-that is, at all Mach numbers lower than unity-the pressure
distribution of the wings can be related to that for incompressible flow by means of
the Prandtl-Glauert transformation. In Chap. 4, the wing in compressible flow is
treated. Here, in addition to profile theory, the theory of the wing of finite span is
discussed at some length. The chapter is subdivided into the aerodynamics of the wing
at subsonic and supersonic, and at transonic and hypersonic incident flow. The latter
two cases are treated only briefly. Results of systematic experimental studies on
simple wing forms in the subsonic, transonic, and supersonic ranges are given for the
qualification of the theoretical results. Part 2 begins in Chap. 5 with the aerodynamics
of the fuselage without interference at subsonic and supersonic speeds. In Chap. 6, a
rather comprehensive account is given of the quite complex, but for practical cases
very important, aerodynamic interference of wing and fuselage (wing-fuselage system).
It should be noted that the difficult and complex theory of supersonic flow could be
treated only superficially. In this chapter, a special section is devoted to slender flight
articles. Here, some recent experimental results, particularly for slender wing-fuselage
systems, are reported. In Part 3, Chaps. 7 and 8, the aerodynamic questions of
importance to airplane stability and control are treated. Here, the aerodynamic
interferences of wing and wing-fuselage systems are of decisive significance.
Experimental results on the maximum lift and the effect of landing flaps (air brakes)
are given. The discussions of this part of the aerodynamics of the airplane refer again
to subsonic and supersonic incident flow.
A comprehensive list of references complements each chapter. These lists, as well
as the bibliography at the end of the book, have been updated from the German
edition to include the most recent publications.
Although the book is addressed primarily to students of aeronautics, it has been
written as well with the engineers and scientists in mind who work in the aircraft
industry and who do research in this field. We have endeavored to emphasize the
theoretical approach to the problems, but we have tried to do this in a manner easily
understandable to the engineer. Actually, through proper application of the laws of
modern aerodynamics it is possible today to derive a major portion of the
aerodynamics of the airplane from purely theoretical considerations. The very
comprehensive experimental material, available in the literature, has been included
only as far as necessary to create a better physical concept and to check the theory.
We wanted to emphasize that decisive progress has been made not through
accumulation of large numbers of experimental results, but rather through synthesis of
theoretical considerations with a few basic experimental results. Through numerous
detailed examples, we have endeavored to enhance the reader's comprehension of the
theory.
Hermann Schlichting
Erich Truckenbrodt
NOMENCLATURE
MATERIAL CONSTANTS
0
g
cP, cv
y = cP/ci1
a=
yp/,o
µ
v = µ/9
R
T
t
density of air (mass of unit volume)
gravitational acceleration
specific heats at constant pressure and constant volume,
respectively
isentropic exponent
speed of sound
coefficient of dynamic viscosity
coefficient of kinematic viscosity
gas constant
absolute temperature (K)
temperature (°C)
FLOW QUANTITIES
p
T
u, v, w
u, Wr, w.3
V, U.
We
wt
pressure (normal force per unit area)
shear stress (tangential force per unit area)
velocity components in Cartesian (rectangular) coordinates
velocity components in cylindrical coordinates
velocity of incident flow
velocity on profile contour
induced downwash velocity, positive in the direction of the
negative z axis
Lx
X NOMENCLATURE
q = (p/2)V2
q00 = (,o./2)U!
Re = VI/v
Ma=V/a
May, = U./ate,
Ma. cr
dynamic (impact) pressure
dynamic (impact) pressure of undisturbed flow
Reynolds number
Mach number
Mach number of undisturbed flow
drag-critical Mach number
Mach angle
displacement thickness of boundary layer
circulation
dimensionless circulation
vortex density
source strength
dipole strength
velocity potential
GEOMETRIC QUANTITIES
x,Y,z
Cartesian (rectangular) coordinates: x = longitudinal axis,
y = lateral axis, z = vertical axis
=x/s,n=y/s,
z/s
Xf, Xr
xl, xp
dimensionless rectangular coordinates
trigonometric coordinate; cos $ = q
coordinates. of wing leading (front) and trailing (rear) edges,
xo, x1oo, respectively
coordinates of quarter-point and three-quarter-point lines,
x25 , X75, respectively
b = 2s
wing area
fuselage cross-sectional area
area of horizontal tail (surface)
area of vertical tail (surface)
wing span
bF
fuselage width
A
AF
AH
Ay
span of horizontal tail (surface)
aspect ratio of wing
A =b2/A
`4H, Ay
aspect ratios of horizontal and vertical tails (surface),
respectively
C
wing chord
chord at wing root and wing tip, respectively
Cr, Ct
c11 =(2/A)foc2(y)dY wing reference chord
X = Ct/Cr
wing taper
IF
fuselage length
cf
flap (control-surface) chord
Xf=Cf/c
flap (control-surface) chord ratio
flap deflection
Tif
bH
NOMENCLATURE Xi
7
m = tan y/ tan
µ
E
V
N25
t
S = t/c
h
xt
Xh
Z(S)
Z(t)
dFmax
SF = dFinaxliF
17F=bFIb
D=2R
Zo
rH
EH
rv
sweepback angle of wing
leading edge semiangle of delta wing (Fig. 4-59)
parameter (Fig. 4-59); m < 1: subsonic flow edge, m > 1:
supersonic flow edge
twist angle
angle of wing dihedral
geometric neutral point
profile thickness
thickness ratio of wing
profile camber
(maximum) thickness position
(maximum) camber (height) position
skeleton (mean camber) line coordinate
teardrop profile coordinate
maximum fuselage diameter
fuselage thickness ratio
relative fuselage width
diameter of axisymmetric fuselage
wing vertical position
lever arm of horizontal tail (= distance between geometric
neutral points of the wing and the horizontal tail)
setting angle of horizontal stabilizer (tail)
lever arm of vertical tail (= distance between geometric
neutral points of the wing and the vertical tail)
AERODYNAMIC QUANTITIES (see Fig. 1-6)
WX, Wy, WZ
angle of attack (incidence)
angle of sideslip (yaw)
components of angular velocities in rectangular coordinates
during rotary motion of the airplane
"`LX = WX S/V,
any = W yCM/ V,
Z WZS/V
L
D
Y
Mx
M, My
MZ
Di
CL
CD
CMX
components of the dimensionless angular velocities
lift
drag
side force
rolling moment
pitching moment
yawing moment
induced drag
lift coefficient
drag coefficient
rolling-moment coefficient
Xii NOMENCLATURE
CM,CMy
CMZ
Cl
Cm
Cmf
Cif
CDi
CDp
(dcL/da)
cp =(p-pc,)/Q.
Cp pl
CP Cr
d Cp = (pi - pu)q
f = 2b/CL,o
k = 7r11/cLw
ae
ag = a
ai = wi/U,0
ao
OW =a+EH+aw
aw=w/UU
N
XN
Id XN
pitching-moment coefficient
yawing-moment coefficient
local lift coefficient
local pitching-moment coefficient
control-surface (hinge) moment coefficient
flap (control-surface) load coefficient
coefficient of induced drag
coefficient of profile drag
lift slope of wing of infinite span
pressure coefficient
pressure coefficient of plane (two-dimensional) flow
critical pressure coefficient
coefficient of load distribution
planform function
coefficient of elliptic wing
effective angle of attack
geometric angle of attack
induced angle of attack
zero-lift angle of attack
angle of attack of the horizontal tail
downwash angle at the horizontal tail location
aerodynamic neutral point
position of aerodynamic neutral point
distance between aerodynamic and geometric neutral points
angle of flow incident on the vertical tail
angle of sidewash at the station of the vertical tail
DIMENSIONLESS STABILITY COEFFICIENTS
Coefficients of Yawed Flight
acy/ao
acMX/a1
aCMZ/a 3
side force due to sideslip
rolling moment due to sideslip
yawing moment due to sideslip
Coefficients due to Angular Velocity
acylaQZ
acMXla QX
acMX/aQZ
acMZ/af?Z
acMZ l a X
aCL/a!?y
acJ/aQy
side force due to yaw rate
rolling moment due to roll rate
rolling moment due to yaw rate
yawing moment due to yaw rate
yawing moment due to roll rate
lift due to pitch rate
pitching moment due to pitch rate
NOMENCLATURE Xiii
INDICES
W
F
(W + F)
H
V
f
wing data
fuselage data
data of wing-fuselage system
data of horizontal stabilizer
data of vertical stabilizer
data of flaps (control surfaces)
CHAPTER
ONE
INTRODUCTION
1-1 PROBLEMS OF AIRPLANE AERODYNAMICS
An airplane moves in the earth's atmosphere. The state of motion of an airplane is
determined by its weight, by the thrust of the power plant, and by the aerodynamic
forces (or loads) that act on the airplane parts during their motion. For every state
of motion at uniform velocity, the resultant of weight and thrust forces must be in
equilibrium with the resultant of the aerodynamic forces. For the particularly
simple state of motion of horizontal flight, the forces acting on the airplane are
shown in Fig. 1-1. In this case, the equilibrium condition is reduced to the
requirement that, in the vertical direction, the weight must be equal to the lift
(W = L) and, in the horizontal direction, the thrust must be equal to the drag
(Th = D). Here, lift L and drag D are the components of the aerodynamic force R1
normal and parallel, respectively, to the flight velocity vector V. For nonuniform
motion of the aircraft, inertia forces are to be added to these forces.
In this book we shall deal exclusively with aerodynamic forces that act on the
individual parts, and thus on the whole aircraft, during motion. The most important
parts of the airplane that contribute to the aerodynamic forces are wing, fuselage,
control and stabilizing surfaces (tail unit or empennage, ailerons, and canard
surfaces), and power plant. The aerodynamic forces depend in a quite complicated
manner on the geometry of these parts, the flight speed, and the physical properties
of the air (e.g., density, viscosity). It is the object of the study of the aerodynamics
of the airplane to furnish information about these interrelations. Here, the following
two problem areas have to be considered:
1. Determination of aerodynamic forces for a given geometry of the aircraft
(direct problem)
2. Determination of
(indirect problem)
the geometry of the aircraft for desired flow patterns
I
2 INTRODUCTION
Th
Figure 1-1 Forces (loads) on an airplane in horizontal flight. L, lift; D,
drag; W, weight; Th, thrust; R,, resultant of aerodynamic forces (resultant of L and D); Rz , resultant of W
and Th.
The object of flight mechanics is the determination of aircraft motion for given
aerodynamic forces, known weight of the aircraft, and given thrust. This includes
questions of both flight performance and flight conditions, such as control and
stability of the aircraft. Flight mechanics is not a part of the problem area of this
book. Also, the field of aeroelasticity, that is, the interactions of aerodynamic
forces with elastic forces during deformation of aircraft parts, will not be treated.
The science of the aerodynamic forces of airplanes, to be treated here, forms
the foundation for both flight mechanics and many questions of aircraft design and
construction.
1-2 PHYSICAL PROPERTIES OF AIR
1-2-1 Basic Facts
In
fluid mechanics, some physical properties of the fluid are important, for
example, density and viscosity. With regard to aircraft operation in the atmosphere,
changes of these properties with altitude are of particular importance. These
physical properties of the earth's atmosphere directly influence aircraft aerodynamics and consequently, indirectly, the flight mechanics. In the following
discussions the fluid will be considered to be a continuum.
The density o is defined as the mass of the unit volume. It depends on both
pressure and temperature. Compressibility is a measure of the degree to which a
fluid can be compressed under the influence of external pressure forces. The
compressibility of gases is much greater than that of liquids. Compressibility
INTRODUCTION 3
therefore has to be taken into account when changes in pressure resulting from a
particular flow process lead to noticeable changes in density.
Viscosity is related to the friction forces within a streaming fluid, that is, to the
tangential forces transmitted between ambient volume elements. The viscosity
coefficient of fluids changes rather drastically with temperature.
In many technical applications, viscous forces can be neglected in order to
simplify the laws of fluid dynamics (inviscid flow). This is done in the theory of lift
of airfoils (potential flow). To determine the drag of bodies, however, the viscosity
has to be considered (boundary-layer theory). The considerable increase in flight
speed during the past decades has led to problems in aircraft aerodynamics that
require inclusion of the compressibility of the air and often, simultaneously, the
viscosity. This is the case when the flight speed becomes comparable to the speed of
sound (gas dynamics). Furthermore, the dependence of the physical properties of air
on the altitude must be known. Some quantitative data will now be given for
density, compressibility, and viscosity of air.
1-2-2 Material Properties
Density The density of a gas (mass/volume), with the dimensions kg/m3 or Ns'/m',
depends on pressure and temperature. The relationship between density e, pressure
p, and absolute temperature T is given by the thermal equation of state for ideal
gases
p =QRT
(1-la)
R = 287 kg (air)
K
(1 - 1 b)
where R is the gas constant. Of the various possible changes of state of a gas, of
particular importance is the adiabatic-reversible (isentropic) change in which pressure
and density are related by
p = const
(1-2)
Qy
Here y is the isentropic exponent, with
CP
y - cU
= 1.405 (air)
cP
(1-3a)
(1-3b)
and c are the specific heats at constant pressure and constant volume,
respectively.
Very fast changes of state are adiabatic processes in very good approximation,
because heat exchange with the ambient fluid elements is relatively slow and,
therefore, of negligible influence on the process. In this sense, flow processes at high
speeds can usually be considered to be fast changes of state. If such flows are
steady, isentropic changes of state after Eq. (1-2) can be assumed. Unsteady-flow
4 INTRODUCTION
processes (e.g., with shock waves) are not isentropic (anisentropic); they do not
follow Eq. (1-2).
Across a normal compression shock, pressure and density are related by
e2
el
=
of
-1)+(7+1)PZ
(7+1)+(7-1)Pi
7+1
7-1
where the indices
1
Pi
= 5.93 (air)
( 1 - 4a)
(1-4b)
and 2 indicate conditions before and behind the shock,
respectively.
Speed of sound Since the pressure changes of acoustic vibrations in air are of such
a high frequency that heat exchange with the ambient fluid elements is negligible,
an isentropic change of state after Eq. (1.2) can be assumed for the compressibility
law of air: p(e). Then, with Laplace's formula, the speed of sound becomes
(1-5a)
ao = 340 m/s (air)
(1-5b)
where for p/p the value given by the, equation of state for ideal gases, Eq. (1-la),
was taken. Note that the speed of sound is simply proportional to the square root
of the absolute temperature. The value given in Eq. (1-5b) is valid for air of
temperature t = 15°C or T = 288 K.
Viscosity In flows of an inviscid fluid, no tangential forces (shear stresses) exist
between ambient layers. Only normal forces (pressures) act on the flow. The theory
of inviscid, incompressible flow has been developed mathematically in detail, giving,
in many cases, a satisfactory, description of the actual flow, for example, in computing
airfoil lift at moderate flight velocities. On the other hand, this theory fails completely
for the computation of body drag. This unacceptable result of the theory of inviscid
flow is caused by the fact that both between the layers within the fluid and between
the fluid and its solid boundary, tangential forces are transmitted that affect the flow in
addition to the normal forces. These tangential or friction forces of a real fluid are
the result of a fluid property, called the viscosity of the fluid. Viscosity is defined
by Newton's elementary friction law of fluids as
(1-6)
Here T is the shearing stress between adjacent layers, du/dy is the velocity gradient normal to the stream, and u is the dynamic viscosity of the fluid, having the dimensions
Ns/m2. It is a material constant that is almost independent of pressure but, in gases,
INTRODUCTION 5
increases strongly with increasing temperature. In all flows governed by friction and
inertia forces simultaneously, the quotient of viscosity i and density Q plays an
important role. It is called the kinematic viscosity v,
(1-7)
and has the dimensions m2/s. In Table 1-1 a few values for density o, dynamic
viscosity p, and kinematic viscosity v of air are given versus temperature at constant
pressure.
1-2-3 Physical Properties of the Atmosphere
Changes of pressure, density, and viscosity of the air with altitude z of the
stationary atmosphere are important for aeronautics. These quantities depend on the
vertical temperature distribution T(z) in the atmosphere. At moderate altitude (up
to about 10 km), the temperature decreases with increasing altitude, the
temperature gradient dT/dz varying between approximately -0.5 and -1 K per 100
m, depending on the weather conditions. At the higher altitudes, the temperature
gradient varies strongly with altitude, with both positive and negative values
occurring.
The data for the atmosphere given below are valid up to the boundary of the
homosphere at an altitude of about 90 km. Here the gravitational acceleration is
already markedly smaller than at sea level.
The pressure change for a step of vertical height dz
is,
after the basic
hydrostatic equation,
dp = - Qg dz
(1-8a)
_ -ego dH
where H is called scale height.
Table 1-1 Density e, dynamic viscosity µ, and
kinematic viscosity v of air versus temperature t
at constant pressure p 1 atmosphere
Kinematic
Temperature
Density
t
Q
[°C]
-20
-10
0
10
20
40
60
80
100
[kg/m3
]
1.39
1.34
1.29
1.25
1.21
1.12
1.06
0.99
0.94
Viscosity
[kg/ms]
15.6
16.2
16.8
17.4
17.9
19.1
20.3
21.5
22.9
viscosity
[m2
/s]
11.3
12.1
13.0
13.9
14.9
17.0
19.2
21.7
24.5
(1-8b)
6 INTRODUCTION
The decrease in the gravitational acceleration g(z) with increasing height z is
r,
g(z) =
(ro + z) 2
(1-9)
go
with ro = 6370 km as the radius of the earth, and go = 9.807 m/s', the standard
gravitational acceleration at sea level. With Eq. (1-8) we obtain by integration
H = f g(z) dz =
+z
go
(1-10)
a
r0
0
For the homosphere (z < 90 km), the scale height is insignificantly different from
the geometric height (see Table 1-2).
The variables of state of the atmosphere can be represented by the thermal and
polytropic equations of state,
p = Q RT
(1-11a)
P
(1-llb)
9
with n
= c onst
?6
the polytropic exponent (n <,y). From Eq. (1-11) we obtain by
as
differentiation and elimination of do/e,
dp
n
dT
T
n-- 1
T
(1.12a)
BT dH
(1 . 12b)
The second relation follows from Eq. (1-8b). Finally, we have
_ _ n-1 9o
dT
dH
n
(1-13)
R
Table 1-2 Reference values at the atmosphere layer boundaries, t
Hb
[km]
0
11
zb
[km]
0
20
32
47
52
61
79
88.743
11.019
20.063
32.162
47.350
52.429
61.591
79.994
90
Tb
[K]
288.15
216.65
216.65
228.65
270.65
270.65
252.65
180.65
180.65
Pb
[atm]
1
2.234
5.403
8.567
10'
10'2
10-3
10-3
5.823 10-4
1.797 10-4
1.024 10-5
1.622 - 10-6
1.095
°b
dT/dH
n
[kg/rn3]
[K/km]
[-J
1.225
3.639
8.803
10'
10'2
1,322- 10-2
1.427
7.594
2.511
2.001
10-3
10-4
10-4
10'5
3.170 - 10'6
-6.5
0
+1
+2.8
0
-2
-4
0
1.235
1
0.9716
0.9242
1
1.062
1.133
1
`After "U.S. Standard Atmosphere" [2].
tHb, z b, Tb values at the lower boundary of the layer height; dTldH, n values in the layers.
INTRODUCTION 7
which shows that each polytropic exponent n belongs to a specific temperature
gradient dT/dH. Note that the gas constant* in the homosphere, up to an altitude
of H = 90 km, can be taken as a constant.
From Eq. (1-13) follows by integration:
T=Tb7Ln1
Here it
R (H - Hb)
(1-14)
has been assumed that the polytropic exponent and, therefore, the
temperature gradient are constant within a layer. The index b designates the values
at the lower boundary of the layer. In Table 1-2 the values of Hb, Zb, Tb, and
dT/dH are listed according to the "U.S. Standard Atmosphere" [2].
The pressure distribution with altitude of the atmosphere is obtained through
integration of Eq. (1-12a) with the help of Eq. (1.14). We have
Tb
-
11
1- nnl Ro (H
H,)]
n-1
(1-15a)
For the special case n = 1 (isothermal atmosphere), Eq. (1.15a) reduces to
P
r
=expL-
RTb
(H - Hb)
(1-15b)
In the older literature this relationship is called the barometric height equation.
Finally, the density distribution is easily found from the polytropic relation Eq.
Also given in Table 1-2 are the reference values Pb and eb at the layer
boundaries. For the bottom layer, which reaches from sea level to H= 11 km,
Hb = Ho has to be set equal to zero in Eqs. (1-15a) and (1-15b). The other sea level
values (index 0), inclusive of those for the speed of sound and the kinematic
viscosity, are, after [2]
,
go = 9.8067 rn/s2
to=15°C
po = 1.0 atm
ao = 340.29 m/s
°o = 1.2250 kg/m3
vo = 1.4607 - 10-5 m2 /s
To =288.15K
(dT/dH)o = -6.5 K/km
*The temperature gradient dT/dH determines the stability of the stratification in the
stationary atmosphere. The stratification is more stable when the temperature decrease with
increasing height becomes smaller. For dT/dH= 0 when n = 1, Eq. (1-13), the atmosphere is
isothermal and has a very stable stratification. For n = y = 1.405, the stratification is adiabatic
(isentropic) with dT/dH = -0.98 K per 100 in. This stratification is indifferent, because an air
volume moving upward for a certain distance cools off through expansion at just the same rate
as the temperature drops with height. The air volume maintains the temperature of the ambient
air and is, therefore, in an indifferent equilibrium at every altitude. Negative temperature
gradients of a larger magnitude than 0.98 K/100 m result in unstable stratification.
8 INTRODUCTION
Table 1-3 Barometric pressure p, air density o, temperature T, speed
of sound a, and kinematic viscosity v versus height z*
z [km]
T/To
p/po
Q/Po
I
a/ao
V/1'0
0
1.0
1.0
1.0
1.0
1.0
2
0.9549
0.9097
0.8647
0.8197
0.7747
0.7519
0.7519
0.7519
0.7519
0.7519
0.7519
0.75190.7689
0.7861
0.7935
0.8208
0.8688
0.9168
0.9393
0.9393
0.9393
0.9218
0.8876
0.8768
0.8305
0.7625
0.6946
0.6269
0.6269
0.6269
0.6269
7.846 - 10-1
6.085 - 10-1
4.660 - 10-1
8.217-10-1
6.688-10-1
5.389-10-1
3.518-10-1
2.615-10-1
2.234-10-1
1.915-10-1
1.399-10-1
1.022-10-1
4.292
0.9772
0.9538
0.9299
0.9054
0.8802
0,8671
0.8671
0.8671
0.8671
0.8671
0.8671
0.8671
0.8769
0.8866
0.8908
0.9060
0,9321
0.9575
0.9692
0.9692
0.9692
0.9601
0.9421
0.9364
0.9113
0.8732
0.8334
0.7918
0.7918
0.7918
0.7918
1.174
1.388
1.654
1.988
2.413
2.674
3.120
4.271
5.846
8.000
4
6
8
10
11.019
12
14
16
18
20
20.063
25
30
32.162
35
40
45
47.350
50
52.429
55
60
61.591
65
70
75
79.994
80
85
90
7.466
5.457 -
10-2
10-2
5.403.10-2
2.516-10-2
1.181
8.567
10-2
10-3
5.671-10-3
2.834 - 10-3
1.472 10-3
1.095 .10-3
7.874 . 10-4
5.823-10-4
4.219.10-4
2.217-10-4
10-4
1.797
1.130
3.376-10-1
2,971-10-1
2,546-10-1
1.860-10-1
1.359. 10-1
9.930. 10-2
7.258- 10-2
7.186- 10-2
3.272- 10-2
1.503-10-2
1.080-10-2
6.909- 10-3
3.262-10-3
1.605 - 10-3
1.165. 10-3
8.383 - 10-4
6.199- 10-4
4.578.10-4
2.497- 10-4
2.050- 10-4
10'4
5.448.10-5
2.458-10-5
1.360-10-4
7.146-10-5
3.538- 10-5
1.024-10-5
1.634-10-5
1.023 - 10-5
4.071-10-6
1.622 -
10-6
10-1
1.632-10-5
6.494. 10-6
2.588 - 10-6
1.095-101
1.106- 10'2.474- 101
5.486-101
7.696 - 101
1.236- 102
2.743- 102
5.819 - 102
8.170 - 102
1.136-103
1.536 - 103
2.049.103
3.645-103
4,397- 103
6.340- 103
1.125
104
2.100-104
4.161- 104
4.166-104
1.047-105
2,627-105
*After "U.S. Standard Atmosphere" [2].
The numerical values of pressure and density distribution are listed in Table 1-3, to
which the values for the speed of sound and the kinematic viscosity have been added.
More detailed and more accurate values are found in the comprehensive tables [2].
Finally, in Fig. 1-2, a graphic representation is given of the distributions of
pressure, density, temperature, speed of sound, and kinematic viscosity versus
altitude. Whereas pressure and density decrease strongly with height, kinematic
viscosity increases markedly.
1-3 AERODYNAMIC BEHAVIOR OF AIRPLANES
1-3-1 Similarity Laws
The question of the mechanical similarity of two flows plays an important role in
both the theory of fluid flows and the extensive testing procedures of fluid
INTRODUCTION 9
mechanics. That is, given are two fluids of different physical properties, in each of
which one of two geometrically similar bodies is located. Under what conditions are
the two flow fields about the two bodies similar-in other words, under what
conditions do they have a similar set of streamlines? Only in the case of
mechanically similar flow fields is it possible to draw conclusions from the
knowledge-which may have been obtained theoretically or experimentally-of the
flow field about one body on the flow field about another geometrically similar
body. To ensure mechanical similarity of flow fields about two geometrically
similar, but not necessarily identical, bodies (e.g., two airfoils) in different fluids of
different velocities, the condition must be satisfied that in each pair of points of
similar position, the forces acting on two fluid elements must be similar in direction
and magnitude. For the aerodynamics of aircraft, gravitation is of negligible
influence and will not be considered for the establishment of similarity laws.
Mach similarity law First, let us consider the case of a compressible, inviscid flow.
Here, except for inertia forces, only the elastic forces act on the fluid elements of a
homogeneous fluid. For mechanically similar flows, obviously the relative density
change caused by the elastic forces must be equal in the two flows. This leads to the
requirement that the Mach numbers of both flows, that is, the ratios of flow velocity
and sonic speed, should be equal. This is the Mach similarity law. The Mach number
V
Ma = a
(1-16)
Figure 1-2 Atmospheric pressure
p, air density o, temperature T,
speed of sound a, and kinematic
viscosity v, vs. height z. From
"U.S. Standard Atmosphere" [2].
10 INTRODUCTION
is, therefore, a first important dimensionless characteristic number of flow processes.
Since the effects of compressibility become noticeable for Ma > 0.3, as pointed out
above, the Mach similarity law needs to be considered only above this limiting
value. The fluid dynamic laws of an incompressible fluid can, therefore, be taken as
the laws for very small Mach numbers with the limiting case Ma -+ 0.
Reynolds similarity law Let us now consider the case of an incompressible, viscous
flow. Here, only inertia and viscous forces act on the fluid element. These two
forces are functions of the following physical quantities: approach velocity V,
characteristic body dimension 1, density o, and dynamic viscosity µ of the fluid. The
only possible dimensionless combination of these quantities is the quotient
Re -
°V i
V1
(1-17)
where Re is called the Reynolds number. The ratio p/Q = v has been introduced
above in Eq. (1-7) as the kinematic viscosity. This law was found by Reynolds in
1883 during investigations on the flow in pipes and is called the Reynolds similarity
law.
If velocity and body dimensions are not too small, as in aeronautics, the
Reynolds number is very large because of the very small values of v. This means
physically that the friction forces are much smaller than the inertia forces in such
cases. Inviscid flow (v -+ 0) corresponds to the limiting case Re --+ -0. The laws of
flow with small viscosity often correspond quite well to those without viscosity. On
the other hand, in many cases even a very small viscosity should not be neglected in
the theory (boundary-layer theory).
For compressible flow with friction, mechanical similarity requires that the
Mach and Reynolds similarity laws be satisfied simultaneously, which is very
difficult to accomplish in experimental investigations. The Mach similarity law and
the Reynolds similarity law govern decisively the whole realm of theoretical and
experimental fluid mechanics and particularly the laws of aeronautics.
To give a convenient survey of the Mach and Reynolds numbers occurring in
the aerodynamics of aircraft, the diagrams Fig. 1-3 and Fig. 1-4 have been drawn.
They show these two dimensionless characteristic quantities versus flight velocity
and flight altitude up to z = 20 km. Figure 1-3 shows that, at constant flight
velocity, the Mach number increases with altitude because the sonic speed decreases,
as was shown in Table 1-3. At an altitude of 10 krn, the speed of sound has
dropped to 300 m/s. At the same flight velocity, the Mach number at 10 km of
altitude is about 10% larger than at sea level. This fact is important for the
estimation of the aerodynamic properties of an airplane flying near the speed of
sound.
The Reynolds numbers in Fig. 1-4 are those for a reference length of l = 1 in,
where 1 may be the wing chord, fuselage length, or control surface chord. The
Reynolds numbers of the diagram must be multiplied by a factor that corresponds
to the reference length l in meters. Since the kinematic viscosity increases
considerably with increasing height (see Table 1-3), the Reynolds number decreases
INTRODUCTION 11
2,2
11<z<20
2.0
10
8
6
18
2
z=Okm
1.6
14
1J
08
06
0.4
02
0
400
200
600
1000
800
1200
1400
V [km/h] --
1600
1800
2000 km /h 2400
Figure 1-3 Mach number Ma vs. flight velocity V and flight altitude z.
z=Okm 1
40
.106
3
36
4
32
5
1 _ Reference
length [m]
6
I
I
7
8
i
.9
10
11
I
16
i
i
12
13
i
12
i
14
15
16
8
17
18
19
20
4
0
400
800
1200
V [km/h] -
1600
2000 km/h 2400'
Figure
1-4 Reynolds number Re vs. flight
velocity V and flight altitude z.
12 INTRODUCTION
sharply with increasing height for a constant flight velocity, making airplane drag a
particularly strong function of the height.
1-3-2 Aerodynamic Forces and Moments on Aircraft
Lift, drag, and lift-drag ratio Airplanes moving with constant velocity are subject to
an aerodynamic force R (Fig. 1-5). The component of this force in direction of the
incident flow is the drag D, the component normal to it the lift L.
Lift is produced almost exclusively by the wing, drag by all parts of the aircraft
(wing, fuselage, empennage). Drag will be discussed in detail in the following
chapters. It has several fluid mechanical causes: By friction (viscosity, turbulence)
on the surfaces, friction drag is produced, which is composed of shear-stress drag
and a friction-effected pressure drag. This kind of drag depends essentially on the
aircraft geometry and determines mainly the drag at zero lift. It is called form drag
or also profile drag. As a result of the generation of lift on the wing, a so-called
induced drag is created in addition (eddy drag), which depends strongly on the
aspect ratio (wing span/mean wing chord). An aircraft flying at supersonic velocity
is subject to a so-called wave drag, in addition to the kinds of drag mentioned
above. Wave drag is composed of a component for zero lift (form wave drag) and a
component caused by the lift (lift-induced wave drag).
The
inclination of the resultant R to the incident flow direction and
consequently the ratio of lift to drag depend mainly on wing geometry and incident
flow direction. A large value of this ratio LID is desirable, because it can be
considered to be an aerodynamic efficiency factor of the airplane. This efficiency
factor has a distinct meaning in unpowered flight (glider flight) as can be seen from
Fig.
1-5. For the straight, steady, gliding flight of an unpowered aircraft, the
resultant R of the aerodynamic forces must be equal in magnitude to the weight W
but with the sign reversed. The lift-drag ratio is given, therefore, after Fig. 1-5, by
the relationship
tall
E=D
(1-18)
where a is the angle between flight path and horizontal line.
Horizontal direction
Flight path
Figure 1-5 Demonstration of glide angle E.
INTRODUCTION 13
The minimum glide angle
EI,,
is
a very important quantity of flight
performance, particularly for glider planes. It is given by (L/D)max after Eq. (1-18).
The outstanding characteristic of the wing, in comparison to the other parts of the
aircraft, is its quite large lift-drag ratio. Here are a few data on LID for incompressible
flow: A rectangular plate of an aspect ratio A = b/c = 6 has a value of (L/D)max of
6-8. Considerably greater lifts for about the same drag are obtained when the plate
is somewhat arched. In this case (L/D)max reaches 10-12. Even more favorable
values of (L/D)max are obtained with wings that are streamlined. Particularly, the
leading edge should be well rounded, whereas the profile should taper out into a
sharp trailing edge. Such a wing may have an (L/D)m of 25 and higher.
Further forces and moments, systems of axes We saw that, for symmetric incident
flow, the resultant of aerodynamic forces is composed of lift and drag only. In the
general case of asymmetric flow, the resultant of the aerodynamic forces may be
composed of three forces and three moments. These six components correspond to
six degrees of freedom of the aircraft motion. We introduce two systems of axes,
depending on the flight mechanical requirements, to describe these forces and
moments (Fig. 1-6).
1. Airplane-fixed system: Xf, Y f, Zf
2. Experimental system: Xe, Ye, Ze
The origin of the coordinates is the same in the two systems and is located in the
symmetry plane of the aircraft. Its location in this plane is chosen to suit the specific
problem. For flight mechanical studies, the origin is usually put into the aircraft
center of gravity. For aerodynamic computations, however, it is preferable to put
the origin at a point marked by the aircraft geometry. In wing aerodynamics it is
advantageous to choose the geometric neutral point of the aircraft, as defined in
Sec. 3-1.
The lateral axes of the experimental system of axes xe, ye, ze and of the
system fixed in the airplane xf, yf, z f coincide so that ye = y f. The experimental
system is obtained from the airplane-fixed system by rotation about the lateral axis
by the angle a (angle of attack) (Fig. 1-6).
For symmetric incident flow, the aerodynamic state of the aircraft is defined
by the angle of attack a and the magnitude of the velocity vector. For asymmetric
incidence, the angle of sideslip 0* is also needed. It is defined as the angle between
the direction of the incident flow and the symmetry plane of the aircraft (Fig.
1-6).
Translator's note: According to the definition given by NASA, the angle of sideslip is the
angle between the direction of the incident flow and the symmetry plane of the airplane. The
angle of yaw is referred to a chosen direction, which may sometimes be the direction of the
airflow past the body, making the angle of yaw equal to the angle of sideslip. Under some
conditions, however, as in turning, a different reference direction may be used.
14 INTRODUCTION
Mze C)
Plane of
wz irI
Reference plane
Incident f low direction
Zf
3e
z
Figure 1-6 Systems of flight mechanical axes: airplane-fixed system, xf, yf, zf; experimental
1-7t
system, xe, ye, ze; angle of attack, a; sideslip angle, R; angular velocities, wX, wy, wz
Forces and moments in the two coordinate systems are defined as follows:
1. Aircraft-fixed system:
x f axis: tangential force Xf, rolling moment Mx f
yf axis: lateral force Yf, pitching mdment Mf (or Myf)
zf axis: normal force Zf, yawing moment Mzf
2. Experimental system:
Xe axis: tangential force Xe, rolling moment Mxe
Ye axis: lateral force Ye, pitching moment Me (or Mye)
ze axis: normal force Ze, yawing moment Mze
The signs of forces and moments are shown in Fig. 1-6.
It is customary to use lift L and drag D in addition to the forces and moments.
They are interrelated as follows:
L = -Z,,
D = -X,?
(for 1i = 0)
(1-19)
Further, because of the coincidence of the lateral axes yf = y,
Yf= Ye
Mf=Me =M
(1-20)
Dimensionless coefficients of forces and moments For the representation of
experimental results and also for theoretical calculations, it is expedient to
introduce dimensionless coefficients for the moments and forces defined in the
preceding paragraph. These coefficients are called aerodynamic coefficients of the
aircraft. They are related to the wing area AW, the semispan s, the reference wing
INTRODUCTION 15
chord cµ (Eq. 3-5b), and to the dynamic pressure q = O V'/2, where V is the flight
velocity (velocity of incident flow). Specifically, they are defined as follows.
Lift:
L = cLA Wq
Drag:
D = cDA wq
Tangential force:
X=cxAwq
Lateral force:
Y=cyAx,q
Normal force:
Z=czAwq
Rolling moment:
Mx = cmxA W sq
Pitching moment:
M= cMAwcuq
Yawing moment:
Mz = c
(1-21)
Awsq
A measurement that determines the three coefficients CL, cD, and cm as a
function of the angle of attack a is called a three-component measurement. The
diagram CL(CD) with a as the parameter was introduced by Lilienthal [1]. It is
called the polar curve or the drag polar. If all six components are measured, for
example, of a yawed airplane, such a test is called a six-component measurement.
Normally, the coefficients of forces and moments of aircraft depend considerably
on the Reynolds number Re and the Mach number Ma; in addition to the geometric
data. At low flight velocities, however, the influence of the Mach number on force
and moment coefficients is negligible.
1-3-3 Interrelation between the Aerodynamic Forces
and the Modes of Motion of the Airplane
Motion modes of the airplane After having discussed the aerodynamic forces and
the moments acting on the aircraft, its modes of motion may now be described
briefly. An airplane has six degrees of freedom, namely, three components of
translational velocity V, Vy, V, and three components of rotational velocity wx,
wy, wZ. They can be expressed, for instance, relative to the aircraft-fixed system of
axes x, y, z as in Fig. 1-6. The components of the aerodynamic forces, as
introduced in Sec. 1-3-2, and their dimensionless aerodynamic coefficients are
functions of these six degrees of freedom of motion.
The steady motion of an aircraft can be split up into a longitudinal and a
lateral motion. During longitudinal motion, the position of the aircraft plane of
symmetry remains unchanged. It is characterized by the three components of
motion
Vx, VZ, wy
(longitudinal motion)
The remaining three components determine the lateral motion
Vy, wx, wZ
(lateral motion)
16 INTRODUCTION
It is expedient for the analysis of the interrelation of aerodynamic coefficients
and components of motion to break down the general motion into straight flight, as
described by Vx and VV; yawed flight, described by Vy; and rotary motion about
the three axes. These rotary motions are, specifically, the rolling motion wx, the
pitching motion coy, and the yawing motion wZ. The quantities of angle of attack a
and angle of yaw !3,* which were introduced earlier (see Fig. 1-6), are then given by
tan a = Vxf
Zf and tan
Vyf
(1-22)
xf
The signs of a, a, o. , wy, and wZ can be seen in Fig. 1-6. At unsteady states of
flight, the aerodynamic forces also depend on the acceleration components of the
motion.
Forces and moments during straight flight The incident flow direction of an
airplane in steady straight flight is given by the angle of attack a (Fig. 1-6). -The
resultant aerodynamic force is fixed in magnitude, direction, and line of application
by lift L, drag D, and pitching moment M (Fig. 1.6). Let us now give some details
on the dimensionless aerodynamic coefficients introduced in Sec. 1-3-2. For
moderate angles of attack, the lift coefficient CL is a linear function of the angle of
attack a:
deL
CL = (a - ao) d«
(1-23)
where as is the zero-lift angle of attack and dcLlda is the lift slope. A further
characteristic quantity for the lift is the maximum lift coefficient CLmax, which is
reached at an angle of attack that depends on the airplane characteristics.
For moderate angles of attack and lift coefficients, the drag coefficient CD is
given by
CD = CDO + k, CL + k2cL
(1-24)
where CDO is the drag coefficient at zero lift (form drag). The constants kl and k2
depend mainly on the wing geometry.
For wings of symmetric profile without twist we have kl = 0, and thus
CD = CDO + k2 CL
(1-25)
This is the representation of the drag polar.
The pitching-moment coefficient cm is a linear function of the angle of attack
a and the lift coefficient cL, respectively:
CM
C M O + dCM CL
L
(1-26)
where cMo is the zero-moment coefficient and dcM/dcL is the pitching-moment
slope. The value of cMo is independent of the choice of the moment reference
*The angle R has been designated here as the angle of yaw. For the difference between
angle of yaw and angle of sideslip see the footnote on page 13.
INTRODUCTION 17
station, whereas dcM/dcL depends strongly on it. The quantity dcM/dcL is also
called the "degree of stability of longitudinal motion" (rotation about lateral axis).
The resultant of the aerodynamic forces of the airplane is completely determined
only when its magnitude, direction, and the position of its line of application are
known. These three data are obtained, for instance, from lift, drag, and pitching
moment. The position of the line of application of the resultant R, for example, on
the wing, can be defined as the intersection of the line of application with the
profile chord (Fig. 1-7a). This point is called the center of pressure or aerodynamic
center of the wing. With XA, the distance of the center of pressure from the
moment reference axis, we have
M=.AZ
For small angles of attack, the normal force with the negative sign is, in first
approximation, equal to the lift:
Z= -L
and by introducing the nondimensional coefficients,
xL
CM
Cµ
CZ
( 1 -27 a)
CM _
dcM
CMO
CL
dCL
CL
Figure
1-7 Demonstration of location of
( 1 -27b )
aerodynamic center (center of pressure). (a)
Aerodynamic center C. (b) Neutral point N.
In general, the reference wing chord is c = c.,.
18 INTRODUCTION
This relationship means that the position of the center of pressure generally
varies with the lift coefficient. The shift of the center-of-pressure position is given
by the term -CMO /CL
.
In agreement of theory with experiment, the pitching moment can generally be
described as the sum of a force couple independent of lift (zero moment) and a
term proportional to the lift:
M=M0 -xNL
In words, the pitching moment is the sum of the zero moment and of the
moment formed by the lift force and the distance XN between the neutral
point and the moment reference line (Fig. 1-7b). Again introducing the nondimensional coefficients for lift and pitching moment:
CM = CMO -
XN
CL
(1-28)
CA
Comparison with Eq. (1-26) yields, for the position of the neutral point
xN
dcM
cA.
dcL
(1-29)
which shows that the pitching-moment slope dcMldcL determines the position of
the neutral point. The terms dcL/da and dcM/da are designated as derivatives
,--of longitudinal motion.
Forces and moments in yawed flight When an aircraft is in stationary yawed flight,
the direction of the incident flow of the wing is determined by both the angle of
attack a and the angle of sideslip 1 (Fig. 1-6). Because of the asymmetric flow
incidence, additional forces and moments are produced besides lift, drag, and
pitching moment as discussed in the last section. The force in direction of the
lateral axis y is the side force due to sideslip; the moment about the longitudinal axis,
the rolling moment due to sideslip; and the moment about the vertical axis, the yawing
moment due to sideslip. The derivatives for 0 = 0,
(8C Y) 0=
ap
o
aCMZI
as
Q=0
are called stability coefficients of sideslip; in particular, acMZ/aa is called directional
stability. All three of these coefficients are strongly dependent on the wing
sweepback, besides other influences.
Forces and moments in rotary motion An airplane in rotary motion about the axes
x, y, z, as specified by the modes of motion of Sec. 1-3-3, is subject to additional
velocity components that are produced, for example, locally on the wing and that
change linearly with distance from the axis of rotation. The aerodynamic forces and
moments that are the result of the angular velocities wX, wy, wZ will now be
discussed briefly.
During rotary motion of the airplane about the longitudinal axis (roll) with
INTRODUCTION 19
angular velocity co, the lift distribution on the wing, for instance, becomes
antisymmetric along the wing span. The resulting moment about the x axis can
be called a rolling moment due to roll rate. It always counteracts the rotary motion
and is, therefore, also called roll damping. The asymmetric force distribution along
the span produces also a yawing moment, the so-called yawing moment due to roll
rate. Introducing the dimensionless coefficients according to Eq. (1-21), the stability
coefficients of sideslip
acmz
acMx
and
aS?
asp
are obtained.
The quantity .Q is the dimensionless angular velocity cw,. It is obtained from wX,
the half-span s, and the flight velocity V:
5Q,; = E. -I,
(1-30)
V
The rotary motion of an airplane about the vertical axis (yaw) produces
additional longitudinal air velocities on the wing that have reversed signs on the two
wing halves and that result in an asymmetric normal and tangential force
distribution along the wing span, which in turn produces a rolling and a yawing
moment. The yawing moment created in this way counteracts the rotary motion
and is called yawing or turning damping. The rolling moment is called rolling
moment due to yaw rate. Again by introducing nondimensional coefficients after Eq.
(1-21), further stability coefficients of yawing motion are formed:
acmz
acMx
and
aQZ
aQZ
Here the nondimensional yawing angular velocity is
(1-31)
The rotary motion of the aircraft about the lateral axis (pitch), Fig. 1-6,
produces on the wing an additional component of the incident velocity in the z
direction that is linearly distributed over the wing chord. This results in an
additional lift due to pitch rate and an additional pitching moment that counteracts
the rotary motion about the lateral axis. Therefore, it is also called pitch damping
of the wing. The magnitude of the pitch damping is strongly dependent on the
position of the axis of rotation (y axis). By using lift and pitching-moment
coefficients after Eq. (1-21), the following additional stability coefficients of
longitudinal motion are obtained:
aCL
asp,,
and
acM
asp,,
20 INTRODUCTION
The nondimensional pitching angular velocity
W y CM
Dy
(1.32)
V
is made dimensionless with wing reference chord
after Eq. (3-5b) contrary to
the rolling and yawing angular velocities Q,, and Qy , respectively, which were made
dimensionless with reference to the wing half-span.
Only the most important aerodynamic forces and moments produced by the
rotary motion of the aircraft have been discussed above. Omitted, for instance, were
detailed discussions of the side forces due to roll rate and yaw rate.
Forces and moments in nonsteady motion Besides the steady aerodynamic
coefficients discussed above, the nonsteady coefficients applicable to accelerated flight
have increasingly gained importance, particularly for flight mechanical stability
considerations. Nonsteady motions are more or less sudden transitions from one
steady state to another or time-periodic motions superimposed on a steady motion.
If the periodic motion is very slow (e.g., changes of angle of attack), the
aerodynamic forces can be computed from quasi-stationary theory; this means that,
for instance, the momentary angle of attack determines the forces. With periodic
motions of higher frequency, however, the aerodynamic forces are phase-shifted
(leading or lagging) from the motion. These conditions are demonstrated schematically in Fig. 1-8 for an airplane undergoing a periodic translational motion normal
to its flight path.
At nonsteady longitudinal motion, new aerodynamic force coefficients must be
used, for example, the derivatives
aCL
ac
and
aCM
a«
Angle of attack
Figure
1-8 Schematic
presentation of
quasi-stationary and nonsteady aerodynamic forces.
INTRODUCTION 21
W=O
w-i a
w
w=a
Figure 1-9 Propagation of sound waves from a sound source moving at the velocity w through a
fluid at rest. (a) Sound source at rest, w = 0. (b) Sound source moving at subsonic velocity,
w = a12. (c) Sound source moving at sonic velocity, w = a. (d) Sound source moving at
supersonic velocity, w = 2a; the sound waves propagate within the Mach cone of apex semiangle g.
where a= daldt is the timewise change of the angle of attack. The nonsteady
coefficients are important both for flight mechanics of the aircraft, assumed to be
inflexible, and for questions concerning the elastically deformable airplane (aeroelasticity).
Forces and moments in supersonic flight During the transition from subsonic to
supersonic flight, the aerodynamic behavior of an airplane undergoes a basic change.
This becomes obvious when the airplane is taken as the source of a disturbance that
moves through still air at a velocity V= w. Relative to this moving center
of disturbance, pressure waves emanate with the speed of sound a. A closer
investigation of this process shows the importance of the speed of sound-especially
the ratio of flight velocity to sonic speed, that is, the Mach number from Eq.
(1-16). In terms of fluid mechanics, the airplane can be considered as a sound
source. Figure 1-9a shows the propagation of sound waves from a sound source at
rest on concentric spherical surfaces. In Fig. 1-9b the sound waves, emitted at equal
time intervals, can be seen for a source that moves with one-half the speed of
sound, w = a/2. Figure 1-9c is the corresponding picture for w = a and finally, Fig.
1-9d is for w = 2a. In this last case, in which the sound source moves at supersonic
velocity, the effect of the source is felt only within a cone with the apex semiangle
µ, which is given by
22 INTRODUCTION
smLL =-=-=Ma
at
a
WT
to
1
(1-33)
This cone is called the Mach cone. No signals can be sent from the source to points
outside of the Mach cone, a zone called the zone of silence. No sound is heard,
therefore, by an observer who is being approached by a body flying at supersonic
speed. Physically, the process described is obviously identical to a sound source at
rest in a fluid approaching from the right with velocity w. We have to keep in mind,
therefore, the following characteristic difference: When the fluid velocity is smaller
than the speed of sound (w <a, subsonic flow), pressure disturbances propagate in
all directions of space (Fig. 1-9b). When the fluid velocity is greater than the speed
of sound, however (w > a, supersonic flow), pressure disturbances can propagate
only within the Mach cone situated downstream of the sound source (Fig. 1-9d).
Now, every point of the airplane surface can be considered as the source of a
disturbance (sound source) as in Fig. 1-9, in analogy to the previous discussion
where the whole airplane was taken as the sound source. It can be concluded,
therefore, that because of the different kinds of propagation of the individual
pressure disturbances as in Fig. 1-9b and d, the pressure distribution and
consequently the forces and moments on the various parts of the airplane (wing,
fuselage, control surfaces) depend decisively on the airplane Mach number, whether
the airplane flies at subsonic or supersonic velocities.
The above considerations show that subsonic flow has the characteristic
properties of incompressible flow, whereas supersonic flow is basically different. In
most cases, therefore, it will be expedient to treat subsonic and supersonic flows
separately.
REFERENCES
1. Lilienthal, 0.: "Der Vogelflug als Grundlage der Fliegekunst," 1889; 4th ed., Sandig,
Wiesbaden, 1965.
2. "U.S. Standard Atmosphere," National Oceanic and Atmospheric Administration and National
Aeronautics and Space Administration, Washington, D.C., 1962.
PART
ONE
AERODYNAMICS OF THE WING
CHAPTER
TWO
AIRFOIL OF INFINITE SPAN
IN INCOMPRESSIBLE FLOW
(PROFILE THEORY)
2-1 INTRODUCTION
In this chapter the airfoil of infinite span in incompressible flow will be discussed.
The wing of finite span in incompressible flow will be the subject of Chap. 3, and
the wing in compressible flow that of Chap. 4. More recent results and
understanding of the aerodynamics of the wing profile are communicated in
progress reports by, among others, Goldstein [19], Schlichting [56], and Hummel
[26].
Wing profile The wing profile is understood to be the cross section of the wing
perpendicular to the y axis. Accordingly, the profile lies in the xz plane and
depends, in the general case, on the spanwise coordinate y. The geometry of a wing
profile may be described, as in Fig. 2-la, by introducing the connecting line of the
centers of the inscribed circles as the mean camber (or skeleton) line, and the line
connecting the leading and trailing edges of the mean camber line as the chord. The
greatest distance, measured along the chord, is called the wing or profile chord c.
The largest diameter of the inscribed circles is designated as the profile thickness t
(Fig. 2-1b), and the greatest height of the mean camber line above the chord as the
maximum camber h (Fig. 2-1c). The positions of the maximum thickness and the
maximum camber are given by the distances xt (thickness position) and xh (camber
position). The radius of the circle inscribed at the profile leading edge is taken as
the nose radius rN; it is usually related to the thickness. The trailing ede4) angle 2725
26 AERODYNAMICS OF THE WING
Chord
Figure 2-1 Geometric terminology of lift-
ing wing profiles. (a) Total profile. (b)
Profile teardrop (thickness distribution).
(c) Mean camber (skeleton) line (camber
height distribution).
C
(Fig. 2-1b) is another important quantity. From these designated quantities the
following six geometric profile parameters may be formed:
t/c
hlc
xtlc
xh /c
rN/c
2r
relative thickness (thickness ratio)*
relative camber (camber ratio)*
relative thickness position
relative camber position
relative nose radius
trailing edge angle
For the complete description of a profile, the profile coordinates of the upper
and lower surfaces, zu(x) and zl(x), must also be known. A profile can be
considered as originating from a mean camber line z(s)(x) on which is superimposed
a thickness distribution (profile teardrop shape) z(t)(x) > 0. For moderate thickness
and moderate camber profiles, there results
zu,t(x) = z(s)(x) ± z(t)(x)
(2-1)
The upper sign corresponds to the upper surface of the profile, and the lower sign
to the lower surface.
*These quantities may be called in the text simply "thickness" and "camber" when a
misunderstanding is impossible.
AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 27
For the following considerations, the dimensionless coordinates
X=
x
c
and
Z=
z
C
are introduced. The origin of coordinates, x = 0, is thus found at the profile leading
edge.
Of the large number of profiles heretofore developed, it is possible to discuss
only a small selection in what follows. Further information is given by Riegels [501.
The first systematic investigation of profiles was undertaken at the Aerodynamic Research Institute of Gottingen from 1923 to 1927 on some 40
Joukowsky profiles [47]. The Joukowsky profiles are a two-parameter family of
profiles that are designated by the thickness ratio t/c and the camber ratio h/c (see
Sec. 2-2-3). The skeleton line is a circular arc and the trailing edge angle is zero (the
profiles accordingly have a very sharp trailing edge).
The most significant and extensive profile systems were developed, beginning in
1933, at the NACA Research Laboratories in the United States.* Over the years the
original NACA system was further developed [ 1 ] .
For the description of the four-digit NACA profiles (see Fig. 2-2a), a new
parameter, the maximum camber position xh/c was introduced in addition to the
thickness t/c and the camber h/c. The maximum thickness position is the same for all
*NACA = National Advisory Committee for Aeronautics.
Mean camber
or skeleton
Teardrop
Z (0
Z(s)
a
b
63-
a a0
h
C
-0063
69-
a-0.2
-0.068
C
65-
a=05
h
=0,095
C
66-
a=20
h
- = 0.055
c
Figure 2-2 Geometry of the most important
Five-digit profiles. (c) 6-series profiles.
NACA profiles. (a) Four-digit profiles. (b)
28 AERODYNAMICS OF THE WING
profiles xt/c = 0.30. With the exception of the mean camber (skeleton) line for
Xh = XhIC = 0.5, all skeleton lines undergo a curvature discontinuity at the location
of maximum camber height. The mean camber line is represented by two connected
parabolic arcs joined without a break at the position of the maximum camber.
For the five-digit NACA profiles (see Fig. 2-2b), the profile teardrop shape is
equal to that of the four-digit NACA profiles. The relative camber position,
however, is considerably smaller. A distinction is made between mean camber lines
with and without inflection points. The mean camber lines without inflection points
are composed of a parabola of the third degree in the forward section and a straight
line in the rear section, connected at the station X= m without a curvature
discontinuity.
In the NACA 6-profiles (see Fig. 2-2c), the profile teardrop shapes and the
mean camber lines have been developed from purely aerodynamic considerations.
The velocity distributions on the upper and lower surfaces were given in advance
with a wide variation of the position of the velocity maximums. The maximum
thickness position xtlc lies between 0.35 and 0.45. The standard mean camber line is
calculated to possess a constant velocity distribution at both the upper and lower
surfaces. Its shape is given by
Z(s)
=-
In 2[(l -X) In (1 -X) + X In X]
(2-3)
A particularly simple analytical expression for a profile thickness distribution, or
a skeleton line, is given by the parabola Z = aX(l - X). Specifically, the expressions
for the parabolic biconvex profile and the parabolic mean camber line are
Z(t) = 2 t X(1 - X)
(2.4a)
Z(s) = 4 h X(1 - X)
(24b)
C
Here, t is the maximum thickness and h is the maximum camber height located at
station X = 2
The so-called extended parabolic profile is obtained by multiplication of the
above equation with (1 + bX) in the numerator or denominator. According to
Glauert [17], such a skeleton line has the form
r
z(S) = aX(1- X)(l + bX)
(2-5)
Usually these are profiles with inflection points.
According to Truckenbrodt [49], the geometry of both the profile teardrop
shape and the mean camber line can be given by
,/-,) s-"
Z(X)
-a
X(1 - X)
1+bX
For the various values of b, this formula produces profiles without inflection
points that have various values of the maximum thickness position and maximum
camber position, respectively. The constants a and b are determined as follows:
AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 29
t
2Xr c
Teardrop:
1
1
a= xh2
Skeleton:
h
c
b
b=
1-2Xt
Xt
(2-7a)
1-2X
x2jt
(2-7b)
h
Of the profiles discussed above, the drop-shaped ones shown in Fig. 2-2 have a
rounded nose, whereas those given mathematically by Eq. (2-6) in connection with
Eq. (2-7a) have a pointed nose. The former profiles are therefore suited mainly for
the subsonic speed range, and the latter profiles for the supersonic range.
Pressure distribution In addition to the total forces and moments, the distribution
of local forces on the surface of the wing is frequently needed. As an example, in
Fig. 2-3 the pressure distribution over the chord of an airfoil of infinite span is
presented for various angles of attack. Shown is the dimensionless pressure
coefficient
Cp =
P -P.
q00
versus the dimensionless abscissa x/c. Here (p - p0,) is the positive or negative
pressure difference to the pressure po, of the undisturbed flow and q., the dynamic
pressure of the incident flow. At an angle of attack a = 17.9°, the flow is separated
Figure 2-3 Pressure distribution at various angles of attack a of an airfoil of infinite aspect ratio
with the profile NACA 2412 [12]. Reynolds number Re = 2.7 . 106. Mach number Ma = 0.15.
Normal force coefficients according to the following table:
a
- 1.70
2.8'
7.4°
13.9°
17.8'
-CZ
0.024
0.433
0.862
1..0,56
0.950
30 AERODYNAMICS OF THE WING
from the profile upper surface as indicated by the constant pressure over a wide
range of the profile chord.
The pressures on the upper and lower surfaces of the profile are designated as
pu and pl, respectively (see Fig. 2-3), and the difference d p = (p1- pu) is a
measure for the normal force dZ = A pb dx acting on the surface element dA = b dx
(see Fig. 2-5). By integration over the airfoil chord, the total normal force becomes
c
Z= -b
f
d p(x) dx
(2-9a)
0
= c2q.bc
(2-9b)
where cZ is the normal force coefficient from Eq. (1-21) (see Fig. 2-3). For small
angles of attack a, the negative value of the normal force coefficient can be set
equal to the lift coefficient cL :
CL =
JAcp(x) dx
(2-10)
0
The pitching moment about the profile leading edge is
M= -b f Ap(x) dx
(2-11a)
0
cMq.bc2
(2-11 b)
where nose-up moments are considered as positive. The pitching-moment coefficient
is, accordingly,
CM=-1
f
c
dcp(x)dx
(2-12)
0
2-2 FUNDAMENTALS OF LIFT THEORY
2-2-1 Kutta-Joukowsky Lift Theorem
Treatment of the theory of lift of a body in a fluid flow is considerably less
difficult than that of drag because the theory of drag requires incorporation of the
viscosity of the fluid. The lift, however, can be obtained in very good
approximation from the theory of inviscid flow. The following discussions may be
based, therefore, on inviscid, incompressible flow.* For treatment of the problem of
plane (two-dimensional) flow about an airfoil, it is assumed that the lift-producing
body is a very long cylinder (theoretically of infinite length) that lies normal to the
*The influence of friction on lift will be considered in Sec. 2-6.
AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 31
flow direction. Then, all flow processes are equal in every cross section normal to
the generatrix of the cylinder; that is, flow about an airfoil of infinite length is
two-dimensional. The theory for the calculation of the lift of such an airfoil of
infinite span is also termed profile theory (Chap. 2). Particular flow processes that
have a marked effect on both lift and drag take place at the wing tips of finite-span
wings. These processes are described by the theory of the wing of finite span (Chaps. 3
and 4).
Lift production on an airfoil is closely related to the circulation of its velocity
near-field. Let us explain this interrelationship qualitatively. The flow about an
airfoil profile with lift is shown in Fig. 24. The lift L is the resultant of the
pressure forces on the lower and upper surfaces of the contour. Relative to the
pressure at large distance from the profile, there is higher pressure on the lower
surface, lower pressure on the upper surface. It follows, then, from the Bernoulli
equation, that the velocities on the lower and upper surfaces are lower or higher,
respectively, than the velocity w. of the incident flow. With these facts in mind, it
is easily seen from Fig. 2-4 that the circulation, taken as the line integral of the
velocity along the closed curve K, differs from zero. But also for a curve lying very
close to the profile, the circulation is unequal to zero if lift is produced. The
velocity field ambient to the profile can be thought to have been produced by a
clockwise-turning vortex T that is located in the airfoil. This vortex, which
apparently is of basic importance for the creation of lift, is called the bound vortex
of the wing.
In plane flow, the quantitative interrelation of lift L, incident flow velocity w,,,
and circulation T is given. by the Kutta-Joukowsky equation. Its simplified
derivation, which will now be given, is not quite correct but has the virtue of being
particularly plain. Let us cut out of the infinitely long airfoil a section of width b
(Fig. 2-5), and of this a strip of depth dx parallel to the leading edge. This strip of
planform area dA = b dx is subject to a lift dL = (pl - pu) dA because of the
pressure difference between the lower and upper surfaces of the airfoil. The vector
dL can be assumed to be normal to the direction of incident flow if the small
angles are neglected that are formed between the surface elements and the incident
flow direction.
The pressure difference between the lower and upper surfaces of the airfoil can
be expressed through the velocities on the lower and upper surfaces by applying the
wo,
Figure 24 Flow around an airfoil profile with lift L. 1' = circulation of the airfoil.
32 AERODYNAMICS OF THE WING
4dL
Pu
wo,
00P00
Figure 2-5 Notations for the computation of lift
from the pressure distribution on the airfoil.
Bernoulli equation. From Fig. 2-4, the velocities on the upper and lower surfaces of
the airfoil are (w + J w) and (w - J w), respectively. The Bernoulli equation
then furnishes for the pressure difference
1 P=pt - pu = 2 (wo,, + d w)2
- ° (w - A u')2 - 2Q u
Jw
where the assumption has been made that the magnitudes of the circulatory
velocities on the lower and upper surfaces are equal, I d wji = JA wju = 1Aw1.
By integration, the total lift of the airfoil is consequently obtained as
C
L= f.JpdA=b
(A)
= 2 obwoo
-1
J- p dx
/4w dx
(2-13a)
(2-13b)
The integration has been carried from the leading to the trailing edge (length of
airfoil chord c).
The circulation along any line 1 around the wing surface is
wdl
.17=
(1)
(2-14a)
AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 33
B
C'
C
I'= fdzvdx- fdzvdx=2 fdwdx
B,u
C,[
(2-14b)
B
The first integral in the first equation is to be taken along the upper surface,
the second along the lower surface of the wing. From Eq. (2-13b) the lift is then
given by
L = o b zv, l'
(2-15)
This equation was found first by Kutta [35] in 1902 and independently by
Joukowsky [31] in 1906 and is the exact relation, as can be shown, between lift
and circulation. Furthermore, it can be shown that the lift acts normal to the
direction of the incident flow.
2-2-2 Magnitude and Formation of Circulation
If the magnitude of the circulation is known, the Kutta-Joukowsky formula, Eq.
(2-15), is of practical value for the calculation of lift. However, it must be clarified
as to what way the circulation is related to the geometry of the wing profile, to the
velocity of the incident flow, and to the angle of attack. This interrelation cannot
be determined uniquely from theoretical considerations, so it is necessary to look
for empirical results.
The technically most important wing profiles have, in general, a more or less
sharp trailing edge. Then the magnitude of the circulation can be derived from
experience, namely, that there is no flow around the trailing edge, but that the fluid
flows off the trailing edge smoothly. This is the important Kutta flow-off condition,
often just called the Kutta condition.
For a wing with angle of attack, yet without circulation (see Fig. 2-6a), the rear
stagnation point, that is, the point at which the streamlines from the upper and
lower sides recoalesce, would lie on the upper surface. Such a flow pattern would
be possible only if there were flow around the trailing edge from the lower to the
upper surface and, therefore, theoretically (in inviscid flow) an infinitely high
velocity at the trailing edge with an infinitely high negative pressure. On the other
hand, in the case of a very large circulation (see Fig. 2-6b) the rear stagnation point
would be on the lower surface of the wing with flow around the trailing edge from
above. Again velocity and negative pressure would be infinitely high.
Experience shows that neither case can be realized; rather, as shown in Fig.
2-6c, a circulation forms of the magnitude that is necessary to place the rear
stagnation point exactly on the sharp trailing edge. Therefore, no flow around the
trailing edge occurs, either from above or from below, and smooth flow-off is
established. The condition of smooth flow-off allows unique determination of the
magnitude of the circulation for bodies with a sharp trailing edge from the body
shape and the inclination of the body relative to the incident flow direction. This
statement is valid for the inviscid potential flow. In flow with friction, a certain
reduction of the circulation from the value determined for frictionless flow is
observed as a result of viscosity effects.
For the formation of circulation around a wing, information is obtained from
34 AERODYNAMICS OF THE WING
a
b
Figure 2-6 Flow around an airfoil for various
values of circulation. (a) Circulation l = 0: rear
stagnation point on upper surface. (b) Very large
circulation: rear stagnation point on lower sur(c) Circulation just sufficient to put rear
stagnation point on trailing edge. Smooth flowface.
c
off: Kutta condition satisfied.
the conservation law of circulation in frictionless flow (Thomson theorem). This
states that the circulation of a fluid-bound line is constant with time. This behavior
will be demonstrated on a wing set in motion from rest, Fig. 2-7. Each fluid-bound
line enclosing the wing at rest (Fig. 2-7a) has a circulation r = 0 and retains,
therefore, T = 0 at all later times. Immediately after the beginning of motion,
frictionless flow without circulation is established on the wing (as shown in Fig.
2-6a), which passes the sharp trailing edge from below (Fig. 2-7b). Now, because of
friction, a left-turning vortex is formed with a certain circulation -F. This vortex
quickly drifts away -from the wing and represents the -so-called starting or initial
vortex -T (Fig. 2-7c).
For the originally observed fluid-bound line, the circulation remains zero, even
though the line may become longer with the subsequent fluid motion. It continues,
however, to encircle the wing and starting vortex. Since the total circulation of this
fluid-bound line remains zero for all times according to the Thomson theorem,
somewhere within this fluid-bound line a circulation must exist equal in magnitude
to the circulation of the starting vortex but of reversed sign. This is the circulation
+T of the wing. The starting vortex remains at the starting location of the wing
and is, therefore, some time after the beginning of the motion sufficiently far away
from the wing to be of negligible influence on the further development of the flow
field. The circulation established around the wing, which produces the lift, can be
AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 35
replaced by one or several vortices within the wing of total circulation +1' as far as
the influence on the ambient flow field is concerned. They are called the bound
vortices.* From the above discussions it is seen that the viscosity of the fluid, after
all, causes the formation of circulation and, therefore, the establishment of lift. In
an inviscid fluid, the original flow without circulation and, therefore, with flow
around the trailing edge, would continue indefinitely. No starting vortex would
form and, consequently, there would be no circulation about the wing and no
lift
Viscosity of the fluid must therefore be taken into consideration temporarily to
explain the evolution of lift, that is, the formation of the starting vortex. After
establishment of the starting vortex and the circulation about the wing, the
calculation of lift can be done from the laws of frictionless flow using the
Kutta-Joukowsky equation and observing the Kutta condition.
*In three-dimensional wing theory (Chaps. 3 and 4) so-called free vortices are introduced.
These vortices form the connection, required by the Helmholtz vortex theorem, between the
bound vortices of finite length that stay with the wing and the starting vortex that drifts off
with the flow. In the case of an airfoil of infinite span, which has been discussed so far, the free
vortices are too far apart to play a role for the flow conditions at a cross section of a
two-dimensional wing. Therefore only the bound vortices need to be considered.
- --er-o
a
b
Figure 2-7 Development of circulation during set-
ting in motion of a wing. (a) Wing in stagnant
fluid. (b) Wing shortly after beginning of motion;
for the liquid line chosen in (a), the circulation
0; because of flow around the trailing edge, a
vortex forms at this station. (c) This vortex formed
by flow around the trailing edge is the so-called
1'
starting
vortex -r; a circulation +1'
consequently around the wing.
develops
36 AERODYNAMICS OF THE WING
2-2-3 Methods of Profile Theory
Since the Kutta-Joukowsky equation (Eq. 2-15), which forms the basis of lift
theory, has been introduced, the computation of lift can now be discussed in more
detail. First, the two-dimensional problem will be treated exclusively, that is, the
airfoil of infinite span in incompressible flow. The theory of the airfoil of infinite
span is also called profile theory. Comprehensive discussions of incompressible
profile theory, taking into account nonlinear effects and friction, are given by Betz
[5], von Karman and Burgers [70], Sears [59], Hess and Smith [23], Robinson
and Laurmann [51], Woods [74], and Thwaites [67]. A comparison of results of
profile theory with measurements was made by Hoerner and Borst [251, Riegels
[50], and Abbott and von Doenhoff [1].
Profile theory can be treated in two different ways (compare [73] ): first, by
the method of conformal mapping, and second, by the so-called method of
singularities. The first method is limited to two-dimensional problems. The flow
about a given body is obtained by using conformal mapping to transform it into a
known flow about another body (usually circular cylinder). In the method of
singularities, the body in the flow field is substituted by sources, sinks, and vortices,
the so-called singularities. The latter method can
also be applied to three-
dimensional flows, such as wings of finite span and fuselages. For practical purposes,
the method of singularities is considerably simpler than conformal mapping. The
method of singularities produces, in general, only approximate solutions, whereas
conformal mapping leads to exact solutions, although these often require considerable effort.
2-3 PROFILE THEORY BY THE METHOD
OF CONFORMAL MAPPING
2-3-1 Complex Presentation
Complex stream function Computation of a plane inviscid flow requires much less
effort than that of three-dimensional flow. The reason lies not so much in the fact
that the plane flow has only two, instead of three, local coordinates as that it can
be treated by means of analytical functions. This is a mathematical discipline,
developed in great detail, in which the two local coordinates (x, y) of
two-dimensional flow can
be combined to
a
complex argument. A plane,
frictionless, and incompressible flow can, therefore, be represented as an analytical
function of the complex argument z = x + iy :
F (z) = F (x + i y)
= 0 (x, y) + i'(x, y)
(2-16)
where 0 and q, the potential and stream functions, are real functions of x and y.
The curves 0 = const (potential lines) and qI = const (streamlines) form two
families of orthogonal curves in the xy plane. By taking a suitable streamline as a
solid wall, the other streamlines then form the flow field above this wall. The
velocity components in the x and y directions, that is, u and v, are given by
AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 37
u
a0
d IF
c9x
7y
V
c70
0'l-1
Jy
Jx
The function F(z) is called a complex stream function. From this function, the
velocity field is obtained immediately by differentiation in the complex plane,
where
dF
dz
= it - i V = w(z)
(2-17)
Here, w = u - iv is the conjugate complex number to w = u + iv, which is
obtained by reflection of w on the real axis. In words, Eq. (2-17) says that the
derivative of the complex stream function with respect to the argument is equal to
the velocity vector reflected on the real axis.
The superposition principle is valid for the complex stream function precisely as
for the potential and stream functions, because F(z) = c, F, (z) + c2 F2(z) can be
considered to be a complex stream function as well as Fl (z) and F2(z).
For a circular cylinder of radius a, approached in the x direction by the
undisturbed flow velocity u,,., the complex stream function is
F (z) = u (z +
a-)
(2-18)
For an irrotational flow around the coordinate origin, that is, for a plane
potential vortex, the stream function is
irlnz
F(z) =
(2-19)
2ir
where r is a clockwise-turning circulation.
Conformal mapping First, the term conformal mapping shall be explained (see [6] ).
Consider an analytical function of complex variables and split it into real and
imaginary components:
(2-20)
(z, y) + i n (x, y)
f (z) = f (x + y)
The relationship between the complex numbers z =.x + iy and _ + iri in Eq.
(2-20) can be interpreted purely geometrically. To each point of the complex z
plane a point is coordinated in the plane that can be designated as the mirror
image of the point in the z plane. Specifically, when the point in the z plane moves
along a curve, the corresponding mirror image moves along a curve in the plane.
This curve is called the image curve to the curve in the z plane. The technical
expression of this process is that, through Eq. (2-20), the z plane is conformally
mapped on the S plane. The best known and simplest mapping function is the
Joukowsky mapping function,
=z
ca
-21)
(2-21)
38 AERODYNAMICS OF THE WING
It maps a circle of radius a about the origin of the z plane into the twice-passed
straight line (slit) from -2a to +2a in the plane.
For the computation of flows, this purely geometrical process of conformal
mapping of two planes on each other can also be interpreted as transforming a
certain system of potential lines and streamlines of one plane into a system of those in
another plane. The problem of computing the flow about a given body can then be
solved as follows. Let the flow be known about a body with a contour A in the z
plane and its stream function F(z), for which, usually, flow about a circular cylinder
is assumed [see Eq. (2-18)]. Then, for the body with contour B in the plane, the
flow field is to be determined. For this purpose, a mapping function
= f (z)
(2-22)
must be found that maps the contour A of the z plane into the contour B in the
plane. At the same time, the known system of potential lines and streamlines about
the body A in the z plane is being transformed into the sought system of potential
lines and streamlines about the body B in the plane. The velocity field of the body B
to be determined in the plane is found from the formula
a
az d
= w(z)
d
(2-23)
F(z) and w(z) are known from the stream function of the body A in the z plane
(e.g., circular cylinder). Here dz/d = 1 If '(z) is the reciprocal differential quotient of
the mapping function = f(z). The sought velocity distribution i about body B
can be computed from Eq. (2-23) after the mapping function f(z) that maps body
A into body B has been found. The computation of examples shows that the major
task of this method lies in the determination of the mapping function = f (z),
which maps the given body into another one, the flow of which is known (e.g.,
circular cylinder).
Applying the method of complex functions, von Mises [71] presents integral
formulas for the computation of the force and moment resultants on wing profiles in
frictionless flow. They are based on the work of Blasius [71 J.
2-3-2 Inclined Flat Plate
The simplest case of a lifting-airfoil profile is the inclined flat plate. The angle
between the direction of the incident flow and the direction of the plate is called
angle of attack a of the plate.
The flow about the inclined flat plate is obtained as shown in Fig. 2-8, by
superposition of the plate in parallel flow (a) and the plate in normal flow (b). The
resulting flow
(c) = (a) + (b)
does not yet produce lift on the plate because identical flow conditions exist at the
leading and trailing edges. The front stagnation point is located on the lower surface
and the rear stagnation point on the upper surface of the plate.
U"
a
4a-C
b
v00
z plane
plane
Figure 2-8 Flow about an inclined flat plate. (a) Flat plate in parallel flow. (b) Flat plate in
normal (stagnation) flow. (c) Inclined flat plate without lift, (c) = (a) + (b). (d) Pure circulation
flow. (e) Inclined flat plate with lift (Kutta condition), (e) = (c) + (d).
39
40 AERODYNAMICS OF THE WING
To establish a plate flow with lift, a circulation P according to Fig. 2-8d must
be superimposed on (c). The resulting flow
(e) = (c) + (d) = (a) + (b) + (d)
is the plate flow with lift. The magnitude of the circulation is determined by the
condition of smooth flow-off at the plate trailing edge; for example, the rear
stagnation point lies on the plate trailing edge (Kutta condition). By superposition
of the three flow fields, a flow is obtained around the circle of radius a with its
center at z = 0. It is approached by the flow under the angle a with the x axis, a
being arctan
The complex stream function of this flow, taking Eqs. (2-18)
and (2-19) into account, becomes
F (z) = (u". - i v") z + (u"" + i v".) z + i
In z
(2-24)
For the mapping, the Joukowsky transformation function from Eq. (2-21) was
chosen. This function transforms the circle of radius a in the z plane into the plate
of length c = 4a in the plane. The velocity distribution about the plate is obtained
with the help of Eq. (2-23) after some auxiliary calculations as
vccsW) = uC' T i
r
2n
(2-25)
vt 2 - 4cc2
The magnitude of the circulation T is now to be determined from the Kutta
condition. Smooth flow-off at the trailing edge requires that there-that is, at
= +2a-the velocity remains finite. Therefore, the nominator of the fraction in Eq.
(2-25) must vanish for = 2a. Hence, because of 4a = c,
T = 4rravc,
(2-26a)
(2-26b)
= ITCV00
and the velocity distribution on the plate itself becomes, with
u = w" cosy ± sing V c +
fl
and jtj < c/2,
(2-27)
The + sign applies to the upper surface, the - sign to the lower surface. With w,,
the resultant of the incident flow, and a, the angle of attack between plate and
incident flow resultant, the flow components are given by um = w. cos a and
v., = w. sin a.
At the plate leading edge, t = -c/2, the velocity is infinitely high. The flow
around the plate comes from below, as seen from Fig. 2-8e. On the plate trailing
edge, t = +c/2, the tangential velocity has the value u = v cos a. At an arbitrary
station of the plate, the tangential velocities on the lower and upper surfaces have a
difference in magnitude zi u = uu - ul. At the trailing edge, v u = 0 (smooth
flow-off). The nondimensional pressure difference between the lower and upper
surfaces, related to the dynamic pressure of the incident flow qr, = (o/2)w',, is [see
Eq. (2-8)]
AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 41
ACP c - Pr - Pu = uu -2 ui = 2 sin 2a
woo
q00
2_
c+2
(2-28)
where uu and ul stand for the velocities on the upper and lower surfaces of the
plate, respectively. This load distribution on the plate chord is demonstrated in Fig.
2-9c. The loading at the plate leading edge is infinitely large, whereas it is zero at
the trailing edge. By integration, the force resulting from the pressure distribution
on the surface can be computed in principle [see Eq. (2-9)]. In the present case,
the result is obtained more simply by introducing Eq. (2-26b) into Eq. (2-15). With
L = prrbcw;, sin a
(2-29)
cL = bcq. = 21r sin a
(2-30)
the lift coefficient becomes
This equation establishes the basic relationship between the lift coefficient and
the angle of attack of a flat plate in two-dimensional flow. The so-called lift slope
for small a is
dCL
da
- 2rr
(2-31)
Py
I
11
Li
G
-050
b
-025
x
0
C
sx
Ic
0.5
C
C
Figure 2-9 Flow around an inclined flat plate. (a) Streamline pattern. (h) Pressure distribution
for angle of attack a = 10°. (c) Load distribution.
42 AERODYNAMICS OF THE WING
Figure 2-10 gives a comparison, based on Eq. (2-30), between theory and
experimental measurements for a flat plate and a very thin symmetric profile. Up to
about a = 6°, the agreement is quite good, although it is somewhat better for the
plate than for the profile. At angles of attack in excess of 8°, the experimental
curves lie considerably below the theoretical curve, a deviation due to the effect of
viscosity. When the angle of attack exceeds 12°, flow separation sets in. Flows
around profiles with and without separation are shown in Fig. 2-11. Naumann [42]
reports measurements on a profile over the total possible range of angle of attack, that
is, for 0° < a < 360°.
Without derivation, the pitching moment coefficient about the plate leading edge
(tail-heavy taken to be positive) is given by
-
C.u
M
bc2 q.
_
-
-
4
sin2a
(2-32)
From Eqs. (2.30) and (2-32), the distance of the lift center of application from the
leading edge at small angles of attack is obtained (see Fig. 2-9) as
XLCM_cL_4
1
(2-33)
C
Since lift and moment depend exclusively on the angle of attack, the center of
lift (= center of application of the load distribution in Fig. 2-9c) is identical to the
neutral point (see Sec. 1-3-3).
An astounding result of the just computed inviscid flow about an infinitely thin
I
0.
Theory
cL=2aa%
0.
4
0
1
P
1
t
rofile Go 445-
Flat plate
cai0.
0.4
Plate
03
J
02
Figure 2-10 Lift coefficient cL vs. angle of
attack a for a flat plate and a thin
symmetric profile. Comparison of theory,
Go 445
t
01
0
0°
2°
40
6°
a ---
8°
10°
12 °
14°
Eq. (2-30), and experimental measurements, after Prandtl and Wieselsberger
[47].
AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 43
a
Figure 2-11 Photographs of the flow about airfoil, after Prandtl and Wieselsberger [471. (a)
Attached flow. (b) Separated flow.
inclined flat plate is the fact that the resultant L of the forces is not perpendicular
to the plate, but perpendicular to the direction of the incident flow w.. (Fig. 2-9a).
Since only normal forces (pressures) are present on the plate surface in a frictionless
flow, it could appear to be likely that the resultant of the forces acts normal to the
plate, too. Besides the normal component Py = L cos a, however, there is a
tangential component P, = -L sin a that impinges on the plate leading edge. Together
with the normal component Py, the resultant force L acts normal to the direction
of the incident flow. For the explanation of the existence of a tangential
component P, in an inviscid flow-we shall call it suction force-a closer look at the
flow process is required. The suction force has to do with the flow at the plate
nose, which has an infinitely high velocity. Consequently, an infinitely high
44 AERODYNAMICS OF THE WING
underpressure is produced. This condition is easier to see in the case of a plate of
finite but small thickness and rounded nose (Fig. 2-12a). Here the underpressure at
the nose of the plate is finite and adds up to a suction force acting parallel to the
plate in the forward direction. The detailed computation shows that the magnitude
of this suction force is independent of plate thickness and nose rounding. It
remains, therefore, the value of S = L sin a in the limiting case of an infinitely thin
plate.
In real flow (with friction) around very sharp-nosed plates, an infinitely high
underpressure does not exist. Instead, a slight separation of the flow (separation
bubble) forms near the nose (Fig. 2-12b). For small angles of attack, the flow
reattaches itself farther downstream and, therefore, on the whole is equal to the
frictionless flow. The suction force is missing, however, and the real flow around an
inclined sharp-edged plate produces drag acting in the direction of the incident flow.
Also, this analysis shows that it is very important for keeping the drag small that
the leading edge of wing profiles is well rounded. Figure 2-13 shows (a) the polar
curves (CL vs. CD) and (b) the glide angles E = CD/CL of a thin sharp-edged flat plate
and of a thin symmetric profile. In the range of small to moderate angles of attack,
the thin profile with rounded nose has a markedly smaller drag than the sharp-edged
flat plate. Within a certain range of angles of attack, a is smaller than a (c < a) for
Px = 0
Figure 2-12 Development of the suction force S on the leading edge of a profile. (a) Thin,
symmetric profile with rounded nose, suction force present. (b) Flat plate with sharp nose,
suction force missing.
AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 45
20
IO
J
49
18
I
i
I
16
Thin profile..
Flat plate
71°
0.7
1,4
26
12
Flat plate
Q5
1,0
Q4
08
fl
2
06
a-Z1 °
04
t0°
021
0.1
01
0
0,02 004
005
008
010
CD -
072
074
0°
016
1
2°
4°
6°
B°
10°
12°
CC -Figure 2-13 Aerodynamic coefficients of a sharp-edged flat plate and a thin symmetric profile
for Re = 4 105, A = -, from Prandtl and Wieselsberger [47]. (a) Polar curves, CL vs. CD. (b)
a
b
Glide angle, E = CD/CL-
thin profiles; the resultant of the aerodynamic forces is inclined upstream relative to
the direction normal to the profile chord. This must be attributed to the effect of
the suction force.
2-3-3 Joukowsky Profiles
The Joukowsky transformation (mapping) function Eq. (2-21) is also particularly
suitable for the generation of thick and cambered profiles. In Sec. 2-3-1 it was
shown that this transformation function maps the circle z = a about the origin in
the z plane into the straight line = -2a to = +2a of the plane (Fig. 2-8a).
The same transformation function also allows generation of body shapes
resembling airfoils by choosing different circles in the z plane. These shapes may
have rounded noses and sharp trailing edges (Fig. 2-14). They are called Joukowsky
profiles, after which the transformation function is named. By choosing a circle in
the z plane as in Fig. 2-14a, the center of which is shifted by x0 on the negative
axis from that of the unit circle and which passes through the point z = a, a profile
is produced that resembles a symmetric airfoil shape. It encircles the slit from -2a
to +2a. This
a symmetric Joukowsky profile, the thickness t of which is
determined by the location xo of the center of the mapping circle. The profile
is
tapers toward the trailing edge with an edge angle of zero.
Circular-arc profiles are obtained when the center of the mapping circle lies on
the imaginary axis (Fig. 2-14b). When the center is set on +iyo and the
circumference passes through z = +a, the same mapping function produces a
46 AERODYNAMICS OF THE WING
Figure 2-14 Generation of Joukowsky profiles through conformal mapping with the Joukowsky
mapping function, Eq. (2-21). (a) Symmetric Joukowsky profile. (b) Circular-arc, profile. (c)
Cambered Joukowsky profile.
twice-passed circular arc in the plane. It lies between = -2a and = +2a. The
height h of this circular arc depends on yo. Finally, by choosing a mapping circle
the center of which is shifted both in the real and the imaginary directions (Fig.
2-14c), a cambered Joukowsky profile is mapped, the thickness and camber of
which are determined by the parameters x0 and yo, respectively.
As a special case of the Joukowsky profiles, the very thin circular-arc profile
(circular-arc mean camber) will be discussed.
Circular-arc profile In the circular-arc profile the mapping circle in the z plane is a
circle, as in Fig. 2-14b, passing through the points z = +a and z = -a with its center
at a distance yo from the origin on tie imaginary axis. The radius of the mapping
circle is R = a-4 + i with a 1 = yo /a. The circle K is mapped into a twice-passed
AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 47
profile in the plane, extending from = -2a to
length c = 4a and a camber height h/a = 2 E1 , or
+2a. This profile has a chord
(2-34)
It is easily shown that the profile in the plane is a portion of a circle for any
E1 . For small camber (E' < 1), the profile contour is given by
=
2 [1
-4
[1
- 4 (C )2]
(2-35)
C
This profile is also called a parabola skeleton. For small angles of attack, a G< 1, and
small camber, the lift coefficient becomes
cL = 27r (Cl +2 C
(2-36)
The lift slope dcL/da is again equal to 27r for small angles of attack, as in the case
of the inclined flat plate according to Eq. (2-31). For the zero-lift angle of attack
this equation yields ao = -2(h/c). The pitching-moment coefficient about the
profile leading edge becomes
CM = - 2
(a+4 h)
(2-37)
resulting in cMo = -ir(h/c) for the zero-moment coefficient when ao = -2(h/c). The
velocity distribution on the circular-arc profile is given for small camber and small
angles of attack by
WC-u'c,
1t4C
Y1-4(x)2±
(2-38)
The + sign applies to the upper profile surface, the - sign to the lower profile
surface. The second term, which is dependent on the camber, represents an elliptic
distribution over . The third term, which depends on the angle of attack a,
corresponds to the expression found for the inclined flat plate [Eq. (2-27)].
At the trailing edge, i = c/2, the velocity on the circular-arc profile is finite,
whereas in general its value becomes infinitely large at the leading edge, i _ -c/2.
Only for the angle of incidence a = 0 does the velocity remain finite at the leading
edge. This is the angle of smooth leading-edge flow (no flow around the leading
edge).* Velocity distributions, computed for this case, are shown in Fig. 2-15 for
*Translator's note: When the angle of attack of a thin profile (skeleton) is changed from
positive to negative values, the stagnation point moves from the lower surface to the upper
surface. Only at one angle of attack is the stagnation point exactly on the leading edge. This
angle is called the angle of smooth leading-edge flow (S.L.E.F.). Obviously, here, no flow rounds
the leading edge, which-in inviscid flow-would cause infinitely high velocities. Rather, the
S.L.E.F. is a smooth flow past the leading edge. Only for a flat plate is the angle of S.L.E.F.
equal to the angle of attack a = 0.
48 AERODYNAMICS OF THE WING
Y
015
X
h -005
!
\
C
0W
\
1
Figure 2-15 Velocity distribution of circular-arc profile with
025
Exact
--- Approximation
0
-100
-a75
-Q50
-025
0
S
0,25
too
0,75
0.50
/2
camber ratios h/c = 0.05 and
0.15 for smooth leading-edge
flow,
two circular-arc profiles of camber h/c = 0.05 and 0.15. For comparison, the
exactly computed distributions are also given. The agreement is very good for small
camber. For larger camber, some deviations can be seen.
Of particular interest is the largest velocity on the profile at a = 0. It occurs at
the profile center t = 0 and is obtained from Eq. (2-38) as
wCmax=wo,1 1 +4 k
(2-39a)
+ EL
L
(2-39b)
=Woo
1
7r
)
These equations allow a very simple estimation of the maximum velocity on a
very thin circular-arc profile with smooth leading-edge flow.
Inclined symmetric Joukowsky profile The symmetric Joukowsky profile may serve
as a further example. This profile is obtained from Fig. 2-14a when the mapping
circle passes through the point z = +a and is placed with its center on the negative
real axis at a distance x0 from the origin. The radius of the circle is
R=a+xo=a(l+E2)
with C2 =
-
xo
a
(2.40)
The unit circle and the mapping circle are tangent in z = a; that is, the tangents of
the two circles intersect under the angle zero. Since the angles remain unchanged in
conformal mapping, the trailing-edge angle of the Joukowsky profile is zero.* For a
The Joukowsky mapping function, Eq. (2-21), can be given in more general form in
various ways, leading to additional profile shapes that are obtained from mapping circles. For
example, when in Fig. 2-14a the mapping circle does not pass through the point +a on the real
axis but rather through a point located somewhat farther outside, the sharp trailing edge of the
normal Joukowsky profile is replaced by a rounded edge.
AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 49
very small thickness (E2 < 1), the profile chord length is c = 4a and the thickness
t _
C
3
4
/E2 = 1.299c2
(2.41)
The maximum thickness occurs at p = 1200, that is, at point c/4 from the leading
edge. The profile contour is given by
= 5E2(1-2C)V'-4(x)2
(2-42)
This profile shape is called the Joukowsky teardrop. The zero-lift direction of this
profile coincides with the profile chord (the i; axis).
The lift coefficient is
CL = 27r(1 + e2) sin a
=21r 1 +0.77
t
c
(2-43a)
a
(2-43b)
where the second expression is valid for small angles of attack. Accordingly, the lift
slope dcL /da increases somewhat with profile thickness.
The pitching-moment coefficient about the profile leading edge becomes
cm = -(rr/2)(1 + E2 )a, indicating that the lift force center of attack (neutral point)
lies at a distance c/4 from the profile nose. The velocity distribution on the contour
of the symmetric Joukowsky profile is obtained in a way similar to that for the
circular-arc profile. Presentation of the corresponding expression is omitted. In Fig.
2-16, pressure distributions on a symmetric Joukowsky profile of 15% thickness
ratio are presented for various lift coefficients. At an angle of attack a = 0 (CL = 0),
the pressure minimum occurs at approximately 15% chord behind the nose. When
the angle of attack increases, the minimum moves forward on the suction side and
farther back on the pressure side.
Cambered Joukowsky profiles The Joukowsky profile with a mean camber line
shaped like a circular arc is obtained by mapping an excentrically located circle with
its center at zo = x0 + iyo (see Fig. 2-14c). Further generalizations of the
Joukowsky mapping functions are given by von Karman and Trefftz [7], with
profile thickness, camber height, and trailing-edge angle as the parameters. The mean
camber line has the shape of a
circular
arc, however, as in the case of the
Joukowsky profiles, resulting in a considerable shift of the aerodynamic center. For
the elimination of this problem, Betz and Keune [7] developed suitable mapping
functions.
Experimental results Comprehensive three-component measurements on numerous
Joukowsky profiles have been reported in [47]. Figure 2-17 shows a comparison of
lift coefficients versus the angle of attack as obtained from theory and tests by Betz
[31 ] . The agreement is quite good in the angle-of-attack range from a = -10° to
50 AERODYNAMICS OF THE WING
\ cL°t00
015
Pressure side
c
0
1
50
-05
a
qg
a
Suction side
too
-1,0
_15
-20
i
-R5
0
01
05
Of
07
ad
09
10
X
Figure 2-16 Pressure distribution of an inclined symmetric Joukowsky profile, t/c = 0.15, for
various lift coefficients CL.
a= +10°; the small differences are caused by viscous effects. The moment curves
CM(CL) are in agreement with theory up to large thickness ratios in the case of
symmetric profiles; in the case of cambered profiles, however, the agreement is good
only for small thickness ratios. The theoretical and experimental pressure distributions are also in good agreement, as can be seen from Fig. 2-18.
Concluding remarks The disadvantage of using the method of conformal mapping to
determine aerodynamic properties of profiles lies in the necessity of first fording a
mapping function. The resulting profile shape must then be compared with the
desired shape. In general, it is not possible to know beforehand the proper mapping
function that is mapping the desired profile shape on the circle. To a first
approximation, this problem can be solved as shown by Theodorsen and Garrick
[66] ; see also Ringleb [32]. The methods for the treatment of profile theory by
means of conformal mapping will not be discussed further, because the method of
singularities, which will be discussed next, has proved to be more suitable and
allows simpler computation of velocity distributions over a given profile. Furthermore, the method of singularities has the marked advantage over the method of
AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 51
6
Lift
I
1.
Re=105
Theory/
Expe riment
i
122
Drag
0
-02
I
f
-04
!
, -8°
-12°
0°
12°
B°
4°
°
Figure 2-17 Lift and drag for plane flow
around a cambered Joukowsky profile,
after Betz [311. Profile after Fig. 2-18.
conformal mapping that it can be applied to three-dimensional problems (wings of
finite span) whereas conformal mapping is strictly limited to two-dimensional
problems. The great value of the method of conformal mapping remains nevertheless, because this method allows one to establish exact solutions for the velocity
distribution on certain profiles that then can be compared with approximate
solutions as obtained, for instance, by the method of singularities. For the design
zo
Lower surface
05
I
0
a=s°
Experiments
25
E
0
ll
07
U ppe r
--- Theory
surfaceRe-;0
Figure 2-18 Comparison of theoretical and experimental pressure distribu-
I
0.2
tions of an inclined cambered Jou-
!
0.9
00
05
0.6
07
0B
02
t0
kowsky profile resulting in the same
lift, after Betz [31].
52 AERODYNAMICS OF THE WING
problem, that is, the problem of determining the profile shape for a given pressure
distribution, Eppler [13] has developed a procedure that uses conformal mapping.
2-4 PROFILE THEORY BY THE METHOD
OF SINGULARITIES
2-4-1 Singularities
The method of conformal mapping was applied in Sec. 2-3 to the computation of
velocity distributions about a given wing profile. Another means of computing the
aerodynamic properties of wing profiles is the method of singularities (see Keune
and Burg [33]). This consists of arranging sources, sinks, and vortices within the
profile. Through superposition of their flow fields with a translational flow, a
suitable body contour (profile) is produced. The flow field within the contour has
no physical meaning. For the creation of a symmetric profile in a symmetric
incident flow field (teardrop profile), only sources and sinks are required, whereas
for the creation of camber, vortices must be added within the profile. This
procedure is shown schematically in Fig. 2-19.
These sources, sinks, and vortices are termed singularities of the flow. In most
cases it is necessary to distribute the singularities continuously over the profile
chord rather than discretely.
It is expedient to treat the very thin profile (skeleton profile) first. For such
profiles the skeleton theory (Sec. 24-2) produces all essential results for their lift.
For representation of the skeleton profile, only a vortex distribution is needed. The
symmetric profile of finite thickness (teardrop profile) in symmetric flow. (angle of
attack zero) is produced by source-sink distributions (teardrop theory). In this case
the displacement flow about the profile is obtained (Sec. 2-4-3). The cambered
2-19 The singularities
method. (a) Cambered profile of
finite thickness with angle of attack a. (b) Symmetric profile of
Figure
finite thickness in symmetric flow,
a = 0. (c) Very thin profile with
angle of attack.
AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 53
C
Figure 2-20 The skeleton theory. (a) Arrangement of the vortex distribution on
the skeleton line. (b) Arrangement of the
vortex distribution on the chord (slightly
cambered profile). (c) Circulation distribution along the chord (schematic).
profile of finite thickness is essentially the product of superposition of a mean
camber line (skeleton line) with a teardrop profile (Sec. 244).
2-4-2 Very Thin Profiles (Skeleton Theory)
Fundamentals of skeleton theory As was stated above, the very thin profile
(skeleton profile) is obtained by superposition of a translational flow with that of a
distribution of plane potential vortices. This theory has therefore been termed the
theory of the lifting vortex sheet. It was first developed by Birnbaum and
Ackermann [8] and by Glauert [171, and later expanded in several treatises,
particularly by Helmbold and Keune [22, 32], Allen [3], and Riegels [49].
For the following discussion a coordinate system as shown in Fig. 2-20a is used.
Accordingly, the profile chord coincides with the x axis. The coordinate system
origin lies on the profile leading edge. The mean camber line is given by z(s')(x).
From Fig. 2-20a, the mean camber line is seen to be covered with a continuous
vortex distribution. With the assumption that the skeleton profile has only a slight
camber and, therefore, rises only a little above the profile chord (x axis), the vortex
distribution can be arranged on the chord instead of the mean camber line (Fig.
2-20b). The mathematical treatment of the problem is considerably simplified in
this way.
The vortex strength of a strip of width dx of the vortex sheet is, from Fig.
2-20b,
dr = k (x) d x
(244)
54 AERODYNAMICS OF THE WING
Here, k is the vortex density (vortex strength per unit length) or the circulation
distribution. By applying the law of Biot-Savart, the velocity components in the x
and z directions, respectively, that are induced by the vortex distribution at station
x, z are
C
U
1
(x, z) =
fk(x')
-
z
(x - x')2
0
+ -`
d x'
(2-45a)
C
w(x z) _ -
1
x-x
fl- (x')
dx'
(245b)
0
For slightly cambered profiles, the velocity components on the skeleton line are
approximately equal to the values on the profile chord (z = 0). The velocity
components on the chord are obtained through limit operations as z -> 0 of Eqs.
(2-45a) and (2-45b)
U (X) _
k (X)
(2-46a)
1
2n
W (X)
fk(X')
dX1
(2-46b)
0
The dimensionless quantities
X= X
C
an d
Z(s) = z (s)
(2-4 7)
C
were introduced in Sec. 2-1, with c being the chord length.
The velocity component u is proportional to the vortex density. The upper sign
is valid for the profile upper surface, the lower sign for the lower surface. When
crossing the vortex sheet, the velocity component u changes abruptly by an amount
du=uu - ul=k
(248)
The integral for the velocity component w has a singularity at X= X.*
The distribution of the vortex density on the chord is determined by the
kinematic flow condition, which requires that the skeleton line is a streamline.
Specifically, a translational velocity U. is superimposed on the vortex distribution
that forms the angle of attack a with the chord (Fig. 2-20).
The kinematic flow condition can also be formulated by the requirement that
the velocity components normal to the mean camber line must disappear. Within
the framework of the above approximation, it is sufficient to satisfy this condition
on the chord instead of the mean camber line, resulting in
*It is necessary to take the Cauchy principal value
f
(Y-e
lir n {
111
1
.. d X' ; j.... d Y' j
+e
AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 55
U00
ra
-
d7I' (X)
l1
1
dX
+w(X) = 0
(249)
This equation relates the angle of attack a and the ordinates of the camber Zisi to
the induced normal velocities w.
The velocity distribution on the profile surface and the vortex density are
related by
U(X) = U,,,, + 26(X) = Uc,, _ J- k(X)
(2-50)
This relationship is valid for small angles of attack according to Eq. (246a).
The Kutta condition, Sec. 2-2-2, requires that the velocities on the profile
upper and lower surfaces be equal at the trailing edge. It is required, therefore, that
in Eq. (2-50),
for X= 1
k=0
(2-51)
The total circulation around the profile is determined from the distribution of
the vortex density as
T = fk(x)dx=cjk(x)dx
0
(2-52)
0
The pressure difference between the lower and upper surface is obtained by
means of the Bernoulli equation:
Pi-PuU.Au=oUUk
With Eq. (2-48), the dimensionless pressure coefficient takes the form
dcP(X) = Pi -Pu = 2 k(X)
q
U.
(2-53)
with q. _ U,2o/2 being the dynamic pressure of the incident flow. Consequently,
the distribution of the vortex density produces directly the load distribution over
the profile chord. From Eq. (2-10), the lift coefficient CL = L/q..bc is expressed by
(I
CL
= AJ cp (X) LAX
0
(2-54a)
i
2
UC'
.f k(X) dX
(2-54b)
0
The latter relationship may also be found from the interrelation of lift and
circulation after the Kutta-Joukowsky equation (2-15) for w = U.. Equation
(2-12) yields the pitching-moment coefficient relative to the profile leading edge,
cm = M/q.bc2 (tail-heavy = positive):
c,,r = - f dc1,(X) X dX
0
(2-55a)
56 AERODYNAMICS OF THE WING
C M = - U fk(x)xdx
(2-55b)
0
Computation of the mean camber line from the distribution of circulation Determining the shape of the mean camber line and the angle of attack from a given
distribution of circulation k(X) requires two steps. First, from Eq. (2-46b), the
distribution of the induced downwash velocity w(X) is obtained along the profile
chord. Then, this distribution is introduced into the kinematic flow condition, Eq.
(2-49), and the following expression for the shape of the mean camber line is
obtained by integration over X:
x
Z() (X) = a X -}- f w (X) d X + C
(2-56)
0
These two steps may be combined into one equation by introducing Eq.
(246b) into Eq. (2-56) and integrating over X. The angle of attack and the
integration constant C are determined in such a way that the ordinates of the mean
camber line disappear on the leading and trailing edges, resulting in
i
Z(.4) (X)
=aX-
(' k(X) in
1
2nJ
Uoo
X_ X, dX'
X'
(2-57)
i
0
for the mean camber line and
X=
27C
f
0
U 00
in 1 g,
d X'
(2-58)
for the angle of attack as measured from the chord.
In the case of a constant distribution of circulation along the profile chord,
k = 2UOOC, Eqs. (2-57) and (2-58) yield, for the mean camber line and the angle of
attack,
Z(s)(X)
C [(1 -X)ln(1 -X) +X1nX] with a=0
(2-59)
The maximum camber height is h/c = (In 2/7r)C = 0.221 C and lies at 50% chord.
This mean camber line is found in NACA profiles of the 6-series (see Fig. 2-2c;
a = 1.0). The lift coefficient is obtained from Eq. (2-54b) as
CL = 4C =
In 2 c
(2-60)
Following up on the investigations of Birnbaum and Ackermann, Glauert [171
proposed the following Fourier series expansion for the circulation distribution in
the two-dimensional airfoil problem:
k (r) = 2 U,,,, (A0 tan `
' A.,, sinn
(2-61)
AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 57
Here
X =j-(1 + cos cp)
(2-62)
so that on the leading edge X = 0 and cp = ir, and on the trailing edge X = 1 and
cp = 0. Each term in Eq. (2-61) satisfies the Kutta condition, Eq. (2-51).
By introducing the expression for the distribution of circulation, Eq. (2-61),
into the equation for the induced downwash velocity, Eq. (2-46b), the simple
relationship *
?1' (1p)
UI
- - (A0 + - i
N
1
(2-63)
A!, COS n cp J
is found after integration.
The interrelation of the Fourier coefficients of Eq. (2-63), the shape of the
mean camber line, and the angle of attack are obtained with the help of Eq. (2-49)
as
N
A 0 --r- ,4 A cos n 92 = a n-i
dZ(s)(X)
(2-64)
dX
With a given distribution of the circulation, this is a differential equation for the
mean camber line Z(s)(X).
The first two terms in Eq. (2-61) represent particularly simple mean camber
lines: The distribution of circulation of the first standard distribution becomes
k = A0 kl =. 2 Uoo A0 tan
E
= 2 U,,. A0 V
1
19
X
X
(2-65)
The distribution k is shown in Fig. 2-21a. The induced downwash velocity is
determined from Eq. (2-63) to be w/U,,, = -A0, leading to
Further, from the kinematic flow condition, Eq. (2-64), it follows that the profile
inclination dZ(s)/dX must be constant. This is possible only when Z(s) = 0, and,
therefore,
A0 =a
(2-66)
It has thus been shown that the first normal distribution represents flow about
the inclined flat plate.
The second normal distribution is given by
lc= A1krf=2U... A1sin cp=4U, A1VX(1 -X)
*Note that the following relation is valid according to Glauert [ 17 1:
:z
1r
z
0
cosncp'
cosrp - cosrp
,
sing. rp
sin (P
(2-67)
58 AERODYNAMICS OF THE WING
6
5
41
3
a
2
b
2
X
0
02
0V
06
0.8
02
"0
ZO
04'
06
08
10
Figure 2-21 The first and the second normal distributions; circulation distribution by Eq.
(2-61). (a) The inclined flat plate. (b) The parabolic skeleton at zero angle of incidence.
This is an elliptic distribution (Fig. 2-21 b). The induced downwash velocity is obtained
from Eq. (2-63) as
I = - cosgq =-(2X - 1)
and with Eq. (2-56), the shape of the mean camber line is given by
Z(') =A 1.X(1 - X) = 4 c X(1 - X)
with a = 0
(2-68)
This is a parabolic mean camber line with camber height h/c = Al /A0 . The results
obtained for the inclined flat plate and the parabolic camber without angle of
attack agree with the exact solutions found by the method of conformal mapping
for small angles of attack, Secs. 2-3-2 and 2-3-3, respectively.* In particular, the
relationships for lift and pitching-moment coefficients are also valid.
Computation of the aerodynamic coefficients Equations will now be presented that
allow one to compute the aerodynamic coefficients directly from a given mean
camber line. The lift coefficient is obtained from Eq. (2-54b) after lintegration# with
the help of Eqs. (2-61) and (2-62) for the distribution of circulation as
CL = 7r(2Ao +A,)
(2-69)
In the same way, the pitching-moment coefficient relative to the leading edge is
obtained from Eq. (2-55b) as
c111
=-
(2A0+2A1+A2)
(2-70)
4
This equation was first presented by Munk [41].
*Note that /c = X - z
T In this process, most of the integrals over cp disappear as a result of the orthogonality
conditions of the trigonometric functions.
AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 59
The angle of attack for zero lift (CL = 0) is obtained by setting 2A0 = -A1,
and the zero-lift moment coefficient becomes cm,) = -(7r/4)(A1 +A2). Consequently, the pitching-moment coefficient can also be written as
1
CM = CMo - 4 CL
From Eq. (1-29), the neutral-point location
is
(2-71)
given by -dcM/dcL = XN/c.
Consequently, the distance of the neutral point from the leading edge becomes
XN
C
(2-72)
4
which is independent of the shape of the mean camber line.
The Fourier coefficients are found through Fourier analysis:
Z
A.0
1 f d'P)
dg
:Z
d92
rdZ'a)
All
J0 dX
0
cosn q7 dq7
(n > 1)
(2-73)
The integrals can be transformed through integration by parts into terms in which
the camber line coordinates Z(S) replace the camber line inclination dZ(S)/dX. By
introducing the coefficients A0 and Al into Eq. (2.69), the relation
da =
21r
(2-74)
is obtained for the lift slope, independent of the camber line shape, and the lift
coefficient from Eq. (1-23) is
CL = 21r(a -a0)
(2-75)
The equations for ao and cMo are given in Table 2-1.
On the profile leading edge, X = 0, that is, cp = 7r, in general the vortex density
and consequently the velocity are infinitely large (Eq. 2-61). There is an angle of
attack, however, for which the velocity remains finite on the leading edge. In Sec.
2-3-3, the designation of angle of smooth leading-edge flow was introduced for this
angle of attack. This angle as can be determined from Eq. (2-61) by setting A0 = 0.
The expressions for as and for the lift coefficient for smooth leading-edge flow are
also presented in Table 2-1.
If there is flow around the leading edge, the velocity is infinitely high,
streaming either from below to above, or vice versa. The strong underpressures near
the leading edge produce a force acting upstream on the leading edge, called suction
force in Sec. 2-3-2. The suction force coefficient c8 = S/q.bc can be expressed by
1
7
rk(X) w(X) dX
0
Introducing Eqs. (2-61) and (2-63) into this equation yields
c,s.= 2.-;r A2
(2-76)
a
-
P
K
N N'
N
O
I
0
N
8
v0
I
GN
K
oA
2
K
K
L
cl
+
4-
o
0
I..I Wyy
w O
V b
O
o
a.
O
y
..i
60
fl.
r
tv
z
o
w
O
.yr O
N
¢
0
Wrr
p b
O
a Gn
AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 61
and with CL of Eq. (2-69) and CLS = 7rA1,
1
Cs = 21r (CL - BLS)
z
(2-77)
Consequently, the suction force is zero for smooth leadinb edge flow, but grows
with the square of (CL - cLS).
For a given distribution of circulation k(X), the coefficient AID in Eq. (2-61) is
obtained by the limit operation
A o = 2U--
lmo[k(X)V ]
In the integral formulas of Table 2-1 for the computation of the various
coefficients, only the. distribution of the mean camber coordinates Z(s')(p) appear
besides certain trigonometric functions of gyp. In addition, simple quadrature formulas
are given for the numerical evaluation of the integrals. Accordingly, the profile
coordinates Zm = Z(Xm), at the stations Xm are multiplied with once-for-all-computed
coefficients Am, ... , F,,,, and the sums are then formed of these products (see
Table 2-2).
In Table 2-3 a few results are presented that can easily be verified. Case (a)
refers to a uniformly cambered skeleton line from Eq. (2-6); case (b) refers to an
asymmetrically cambered line from Eq. (2-5).
For the case of a simple parabolic mean camber (Xh = Z), the numerical values
are
ao=-2-
CMo=-irh
cLS = 4tr h
US = 0
c
(2-78)
C
The profiles with fixed aerodynamic centers according to the discussion in Sec.
1-3-2 are obtained from the above skeleton family by setting cMo = 0. From Table
2.3, case (b), it follows immediately that b = - s . This camber line has an inflection
point (S shape). The case b = 0 is again the simple parabola skeleton.
Table 2-2 Coefficients A, B, C, D, E, F for the computation of the
aerodynamic coefficients of Table 2-1 for N = 12 (after Riegels f49, 50] )
m
Xm
A.
B.
1
0.9830
0.9330
0.8536
0.7500
0.6294
0.5000
0.3706
0.2500
0.1465
0.0670
0.0170
0.6440
0
0.2357
0
0.1726
0
0.1726
0
0.2357
0
0.6439
-4.8919
2
3
4
.3
6
7
8
9
10
11
0
-0.5690
0
-0.2249
0
-0.1324
0
-0.0976
0
-0.0848
C.
0.6864
0.1667
0.3333
0.2887
0.2387
0.3333
0.0601
0.2887
-0.3333
0.1667
-1.8017
D.
I
-7.9370
-0.2267
-1.0790
-0.1309
-0.4210
0
-0.1402
0.1309
0.0318
0.2267
0.1197
'
E.
Fm
-2.4032
15.6333
0
-0.2357
0
-0.0462
0
2.0944
0
1.1224
0
0
1.12'24
0
0.2357
2.0944
0
0.0462
0
2.4032
0
15.6333
62 AERODYNAMICS OF THE WING
Table 2-3 Aerodynamic coefficients of uniformly and asymmetrically
cambered skeleton lines
(a) Skeleton from
Eq. (2-6)
Coefficient
Zero-lift angle
a0
Zero-moment coefficient
cM0
Angle of incidence for
S.L.E.F.*
as
h
it h Xh(3 - 2Xh)
2c
1-Xh
1 h 1 - 2Xh
2 c Xh(1 -Xh)
h
CLS
*
- 8 (4+3b)
1
C 1-XI,
Lift coefficient for
S.L.E.F.
(b) Skeleton from
Eq. (2-5)
'r
-
32a
(8+7b)
1
- -gab
2a(2+b)
1
cXh(1-Xh)
*Smooth leading-edge flow.
In the NACA systematic listing, various skeleton line shapes are used (see Sec.
2-1).
Four-digit NACA profiles In Fig. 2-22, zero-lift angles of attack and zero moments
are plotted versus the maximum camber height (crest) location. Test results [1] are
also shown for comparison with theory. Because of the slight effect of profile
thickness in the range of thickness ratios 0.06 < t/c < 0.15, a mean curve of
experimental data is shown. The plotted bars represent three data points each for
cambers h/c = 0.02, 0.04, and 0.06. The agreement of theory and experiment is
4
4
t
Z
'
C
3
2
`,,yam
3
I
JL
a
0
02
11T
a4
Xh--
0.6
08
02
04'
Xh---
a6
0,8
Figure 2-22 Zero-lift angle of attack as and zero-moment coefficient cMo of NACA skeleton
lines. Comparison of theory and experiment from NACA Repts. 460, 537, and 610. Curve 1,
four-digit skeleton lines. Curve 2, five-digit skeleton lines.
AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 63
satisfactory. As a result of friction, the deviations increase somewhat with a
downstream shift of the camber crest.
Five-digit NACA profiles In Fig. 2-22, results for zero-lift angle and zero moment
are presented for the skeleton lines without inflection points. Test results from [1]
are also shown. The influence of the profile thickness is again negligibly small. The
plotted test data are the results for values of CLS = 0.3, 0.45, 0.6, and 0.9.
Agreement between theory and experiment is better than in the case of the
four-digit NACA profiles.
NACA 6-profiles The skeleton lines of the NACA 6-series have been established
from purely aerodynamic considerations. Preestablished are the resultants of the
pressure distributions on the lower and on the upper profile surfaces (Fig. 2-23a).
The corresponding skeleton lines are presented in Fig. 2-2c.
For the aerodynamic coefficients, the following expressions are established:
as =
4n
1
{a
[1
-(1 - a) ln(1 - a) + 1 a- lnaJ
1
as = CIS - 2 CLS
0110 = -
1
!E q-
(2-79a)
(2-79b)
+ 4a2
+a)
(2-79c)
1
Zero-lift angles of attack and zero-moment coefficients for CLS = 1 are given in
Fig. 2-23b and c versus the quantity a. These results are compared with test results
of NACA Rept. 824 and show satisfactory agreement.
Bent plate (flap, wing, control surface) Another valuable application of the skeleton
theory is found in the calculation of the aerodynamic coefficients of the flap wing.
By replacing the flap wing by a skeleton line, the bent plate, Fig. 2-24, is obtained.
This problem was attacked first by Glauert [18] .
With the assumption of a small deflection angle
rj
the ordinates of the
skeleton line Z(s) = Zf, relative to an imaginary chord connecting the leading edge
with the trailing edge of the deflected flap, are
(0<X<Xf)
Zf=AfX,?f
Zf=(1 -Xf)(1 -
7f
(Xf<X< 1)
(2-80)
where Af= cf/c is the flap chord ratio (see Sec. 3-1-1).
Since the profile inclinations are constant within the ranges of Eq. (2-80),
integrations for the determination of aerodynamic coefficients can easily be
performed. It is expedient to introduce in addition the following relationship for
the position of the station:
Xf= 1 -Xf=
(1 +cos cpf)
2
(2-81)
64 AERODYNAMICS OF THE WING
The change of the zero-lift angle with flap deflection is measured relative to the
fixed portion of the profile (wing, stabilizer) rather than relative to the imaginary
chord. The change of angle of attack (= change of the zero-lift angle with flap
deflection) is then given by
a«° = - I (sin cpf + p f)
rlf
2(
Xf(l - Xf) + arcsin
S
)
(2-82a)
a
,05
i1
V0.
CLS=1
a
0`
0
0.2
0.4' X_ 06
08
.10
0.8
%0
10i
O
a
0
63-series
o 66-series
I
65-series
0
0 66-series
0.2
0,4'
a__
06
0,3
I
0
C
02
0.4'
a--
06
0.8
to
Figure 2-23 Aerodynamic coefficients for skeleton lines of the NACA 6-profiles at the lift
coefficient of the smooth leading-edge flow CLS = 1.0. Comparison of theory [Eqs. (2-79a)(2-79c)] and experiment, after NACA Rept. 824. (a) Pressure distribution d cp; (b) Zero-lift
angle ao . (c) Zero pitching-moment coefficient cMo
AIRFOIL OF INFINITE SPAN 1N INCOMPRESSIBLE FLOW (PROFILE THEORY) 65
Figure 2-24 Coordinates used in the skeleton
theory of airfoils with flaps.
The term aao l arjf is frequently called the flap or control-surface efficiency, because
it is a direct measure of the lift change caused by the flap (control surface). The
flap efficiency vanishes for X f = 0 and amounts to -1 for X f = 1, that is, when the
whole profile is being deflected as a flap.
The change of the zero-moment coefficient (moment change at constant lift)
becomes
a ° _ - 2 sin pf{1 + cos pf) _ -2 af(1 - Af)3
(2-82b)
f
The results of the above formulas are shown in Fig. 2-25. The theoretical
relationship between the aerodynamic flap coefficients and the flap chord ratios is
well supported by measurements. In Fig. 2-25, the test results of simple cambered
flaps by Gothert [21 ] are added. The deviations are again due to friction
effects.
1.0
1.0
i
08
0.8
0.8
I
06
}
e1 p-ac
ro
I
02
09
a 0I
0
0,Z
12
f
Xf= C
i
08
0.6
1D
b
O1
0
1
OZ
1
1
09
Cf
06
(18
ZO
f= C -+
Figure 2-25 Aerodynamic coefficients of a flap wing. (a) Angle of attack derivative. (b)
Pitching-moment derivative.
cambered flap.
(
) Theory from Eq. (2-82a, b). (---) Tests on a simple
66 AERODYNAMICS OF THE WING
The aerodynamics of flaps and control surfaces will be discussed in more detail
in Sec. 8-2-1.
Computation of the velocity distribution on the skeleton line The problem of
computing the distribution of circulation and consequently the velocity distribution
will now be treated for a given skeleton line shape at a given angle of attack. By
introducing Eq. (2.49) into Eq. (2-46b), the equation defining the circulation
distribution becomes
1
dX'
U°°dX
This is an integral equation for
a - dZ(s)/dX. It
X - X,
(2-83)
0
the vortex density k with given values of
was first solved by Betz [4]. By taking into account the Kutta
condition Eq. (2-51) and Eq. (2-50), the velocity distribution about the skeleton
profile (see also Fuchs [16]) is given by
1
= 1 + Ir- % (a +
U(
±'X' d$'
l 1Y-X
('
1
0
(2-84)
For the case of the uncambered profile, Z(s) = 0, the already known result for
the inclined flat plate is valid; see Eq. (2-65). To evaluate the quadrature formula
for the velocity distribution, Riegels [49] makes the Fourier substitution
n
Z(s)
_
I a cosv 9)
(2-85a)
V-1
X = f (1 + cos (P)
.(2-85b)
Introducing these expressions into Eq. (2-84) makes elementary evaluation of the
integrals possible.* The velocity distribution of the skeleton profile is then
U
c.
U00
an
L+,
2
cos v 99 - 1
va
sin 92
(2-86)
where the upper sign is valid for the upper side, the lower sign for the lower side of
the skeleton profile. Numerical evaluation of this equation by. means of simple
quadrature formulas is treated in [49] (see also [28] ). The first term of Eq. (2-86)
represents the velocity distribution of the inclined flat plate. For the parabolic skeleton
Z(s)
_h
sine cp =
2 c (1 - cos 2cp)
[see Eq. (2-68)] , a1 = h/c, a2 = -h/c, a3 = a4 =
= 0. Therefore, Eq. (2-86) yields,
for a = 0,
U(T)
U0
-I =1
=1-!-2u cos2q
sin?
`See the footnote on page 57.
4 h-sine
C
AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 67
a
result already obtained in Eq. (2-67),
taking into account Eq.
(2-50).
Velocity and pressure distributions for the skeleton lines of the four- and
five-digit NACA profiles, Figs. 2-2a and 2-2b, are given in [1 ] . In Fig. 2-26 some
pressure distributions are presented for the angle of smooth leading-edge flow. For
the skeleton lines of the NACA 6-series, pressure distributions have been presented
in Fig. 2-23a.
Pressure distribution for given lift coefficient and moment coefficient The problem of
approximating a given skeleton line by superposition of an inclined flat plate and a
parabolic skeleton in such a way that lift and zero-moment coefficients of
approximation and given skeleton are equal can be solved with the help of the
above-introduced Fourier series expansion. In this case the Fourier coefficients from
Eqs. (2-69) and (2-70) become
Ao =
102
2 CL + 22 emo
1
0..4'
0.6
and
03
Al
=-44 cMo
>.0
Figure 2-26 Theoretical pressure distribution of NACA skeleton lines from NACA Rept. 824 at
smooth leading-edge flow. (a) Four-digit NACA profile with h/c = 1.0. (b) Five-digit NACA
profile with CLS = 1.0.
68 AERODYNAMICS OF THE WING
coefficients
These
are introduced into Eq. (2-61), and the resultant pressure
distribution, taking into account Eq. (2-53), is obtained as
dcp(X) = cLho(X) + cMo4h1 (X)
with
ho(X) = z
1/i XX and hl(X) =-(l - 4X)
7r
_
X`I
U 11
(2-87)
(2-88)
The distributions ho(X) and h1 (X) are shown in Fig. 2-27.
Bent plate (flap wing) Finally, the pressure distribution of the bent flat plate (flap
wing) will be mentioned. For the zero-lift angle of attack, CL = 0, the result is
(8c
l =
2
77fp 10
l
In
1-
±
cos('Pf
T)
97)
- 2sincpftan
(2-89)
2
J
In Fig. 2-28, the pressure distribution according to this equation is presented
for the flap chord ratio Xf = 4 . The cross-hatched area represents the load on the
flap, the determination of which as well as that of the flap moment (control-surface
moment) will be discussed in Sec. 8-2-1.
2-4-3 Symmetric Profiles of Finite Thickness
in Chord-Parallel Incident Flow
(Teardrop Theory)
Fundamentals of teardrop theory The term teardrop profile means a symmetric
profile of finite thickness. With the method of singularities, a teardrop profile is
obtained through superposition of a source-sink distribution along the profile chord
with a translational flow (Fig. 2-29). Let a continuous source-sink distribution be
given along the profile chord, the source strength per unit length of which is q(x).
This source distribution induces the velocity component u(x) in the x direction and
produces the velocity component w(x) in the z direction (Fig. 2-29). Let z(t)(x) be
the equation of the upper surface of the teardrop with the coordinate origin on the
4
3
Z
I
0
,-,'I
-1
0`
I
0.Z
04
X
08
08
9.0
Figure 2-27 The functions ha and h, for the pressure
distribution on the chord at given lift and moment
coefficients [Eqs. (2-87) and (2-88)].
AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 69
6
2
0
f
A x i s o f t h e fl ap
2
-
t
Figure 2-28 Theoretical pressure distribution of the
folded plate (flap wing) of Fig. 2-24 at zero lift [Eq.
0
all
0.6'
X --r
08
to
(2-89)]
0.25.
.
Chord ratio of flap and wing af= cf/c =
leading edge. Then, the relation between source distribution and teardrop shape is
obtained easily by applying the continuity equation to the area element ABCD in
Fig. 2-29, with the result
(U + u) zW + Z q dx =
(U00
+ u + du dx) (ZM +
d
(r)
dx dx)
From this the source distribution in linear approximation is obtained as
{(U"
q (x) = 2 x
=2U
}d
,U)
z('))
(x)
Figure 2-29 Basic elements of teardrop theory. q(x) = source-sink distribution.
(2-90a)
(2-90b)
70 AERODYNAMICS OF THE WING
For teardrops of moderate thickness, the induced velocities u can be disregarded as
compared with U., with the exception of the vicinity of the stagnation point.
In the case of thin profiles it can be assumed that the velocity components on
the teardrop contour are approximately equal to the values on the profile chord. In
analogy to Eq. (2-46), the components of the induced velocity on the chord are
obtained as
f q (X')
it X )
dX'
(2-9 la)
0
w(X) = j 1 q(X)
(2-91b)
To obtain a closed profile contour, the total strength of the source-sink
distribution must be zero (closure condition):
C
f q(x)dx=0
(2-92)
xm0
In computing the source-sink distribution for a given teardrop shape z(t)(x), the
closure condition is automatically fulfilled because of Eq. (2-90).
With the profile chord length c, the dimensionless quantities
X=x
Z(t) =
and
z(t)
(2-93)
C
will now be introduced.
The kinematic flow condition, namely, that the profile contour is a streamline,
is
_
d z<<'
uw (x)
TX-
U
It can be verified immediately that this condition is identical to Eq. (2-90b).
Computation of the velocity distribution on the teardrop profile The problem of
determining the velocity distribution on the profile contour may be treated first.
For thin profiles, the velocity distribution on the contour is little different from
that on the chord, except for a small range on the profile nose. The velocity
distribution on the chord is obtained by introducing Eq. (2-90b) into Eq. (2-91a):
U (X) = Uc 1
L
dX'
dZit)
1
1
dX
;
(2-94)
0
According to Riegels [49], the velocity distribution on the contour is then
established by division with
x(X) _
l
/ dTO 2
dX
)
(2-95)
AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 71
Here, l/;-.(x) is called the Riegels factor. Since l/u(x) is zero at the leading edge, it
is assured that the velocity goes to zero at the front stagnation point, and the
velocity distribution on the contour is finally
TWc(X)
U"
- x(X)
1
i
dZ"' dX'
1 + rc j dX' X - X'
1
(2-96)
0
In this way, the computation of the velocity distribution for a given teardrop
profile has been reduced to a quadrature formula.
By disregarding the Riegels factor, the velocity distribution on a simple
biconvex parabola profile becomes
W
C -1+ b[i 7-
U co
; (1 -2X)In
(2-97)
Here, 6 is the thickness ratio t/c. The highest velocity occurs at X =1 with the
value u.../U. = (4/rr)& Likewise, the velocity distributions for the extended
parabola profiles from Eq. (2-6) can be computed (see Truckenbrodt [49] ).
For the evaluation of the quadrature formula, Riegels [49] introduces the
Fourier series
,Z
X = j(1 + cosrp)
Z(i) = I- 16, sine 99
V-1
(2-98)
in analogy to Eq. (2-85). When this expression for Z(t) is inserted into Eq. (2-96),
the velocity distribution on the contour assumes the simple form
WC (m)
U00
_
1
1
Y. ((P)
,Z
+v-1 v
by sinv
sin
(2-99)
J
The numerical evaluation of the equation by means of simple quadrature formulas is
treated in [49].
From Eq. (242), the contour of the thin symmetric Joukowsky profile* is
given by
Z(t) =
2
= 2e
sin p(1 -cos gyp)
(2-100a)
X(1 - X)3
(2-100b)
where
3
4
t
- 0. 77
5
an d
xt =
1
4
The Fourier coefficients are bl = E, bz = -c/2, and b3 = b4 =
the velocity distribution on the contour is given by
Note that /c = X - s = z cos p and rl =Z().
= 0. Consequently,
72 AERODYNAMICS OF THE WING
We
Uoo
1+s(1-2cosq,)
+
V'-
cos 92 - cos 2 T
(2-101)
sing,
(
Figure 2-30 shows the velocity distributions computed by Eq. (2-101) for various
thickness ratios. Within the accuracy of the plot, complete agreement of the
approximate and the exact solutions by conformal mapping is obtained. See Fig.
2-16 for CL = 0. Because the trailing-edge angle of the Joukowsky profiles is zero,
the velocity distribution has no rear stagnation point on the trailing edge.
For the four- and five-digit NACA profiles from Fig. 2-2a, the theoretical
velocity distributions may be found in [1] . A few distributions at various thickness
ratios S = t/c are shown in Fig. 2-31. They were computed by the procedure of
Theodorsen and Garrick [66]. Values computed by the singularities method deviate
only slightly from them. The teardrop shapes of the NACA 6-series, Fig. 2-2c, were
established from given velocity distributions, which were determined mainly by the
location of the maximum velocity. In Fig. 2-32, the velocity distributions are shown
for the four profiles of Fig. 2-33 with a thickness ratio 6 = 0.12. Figure 2-33 gives a
comparison between theoretical and experimental pressure distributions on the
NACA profile NACA 0010 and shows good agreement.
The maximum velocity on a profile is of considerable importance for the
D=r-OZO
C
0.95
0>0
U
005
08
0.6
Figure 2-30 Velocity distribution on
the profile contour for symmetric
Joukowsky profiles of various thickness ratios tic in chord-parallel flow.
Figure 2-31 Velocity distribution on
the profile contour for four- and
five-digit symmetric NACA profiles in
smooth leading-edge flow. (NACA
Rept. 824.)
AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 73
T-
12
63 -012
1.1
65-012
0.9
0,60
02
0.
X---
as
O.5
20
Figure 2-32 Velocity distribution on the profile contour for the NACA 6-profiles in chord- parallel
flow. Profile contours from Fig. 2-33. Thickness ratio tic = 0.12.
behavior of the profile at high subsonic velocities (critical Mach number, Sec. 4-3-2).
In Fig. 2-34, the ratio of the maximum velocity difference to the constant
translational velocity is given against the thickness ratio of most of the teardrop
profiles discussed above. Accordingly, the maximum velocity depends heavily on the
thickness distribution for an otherwise unchanged thickness ratio. The elliptic
profile produces the smallest velocity difference, the Joukowsky profile the
largest.
1,0
i
o Upper surfac e u
0,6
Lower surfac e I
-0.2
Theory
-as
cc=00
-10
I
'0
U.
ti
0,2
041
X
06'
C _u
Profile NACA 0010
as
1,0
Figure 2.33 Comparison of theoretical and
experimental pressure distributions for the
symmetric profile NACA 0010 in chordparallel flow.
74 AERODYNAMICS OF THE WING
I
03
J
Joukow sky profile
NACA profiles
00-
63-
65-
64-
65
of
E lliptic p rofile
0
0.05
Qf t
015
02
C
Figure 2-34 Maximum perturbation velocity Umax of
teardrop profiles in chord-parallel flow vs. thickness
ratio S = t/c.
Computation of the teardrop profile from a given velocity distribution In analogy
to the Birnbaum-Glauert series expansion for the distribution of circulation in the
case of skeleton theory [Eq. (2-61)], the source distribution will now be
represented in the form of a trigonometric series (see, e.g., Allen [3] ):
± r B sinn
q(,p) = 2 U. (B0 tan
n-1
(2-102)
The relation between x and co is given in Eq. (2-62). The closure condition for the
profile contour, Eq. (2-92), is satisfied when
2Bo + B1 =0
(2-103)
By introducing Eq. (2-102) into Eq. (2-91a), the induced velocity in the x direction
is obtained as
u(9)
co
N
= B0± ,b-1
(2-104)
The profile contour is determined by introducing Eq. (2-102) into Eq. (2-90b) and
integrating along X:
Z<o =
I
Bo sin (p(1 - cosrp) -
- 6 B3(2sin2g9 -
1
12
sin4q,)
B`(3sinT - sin39 )
(2-105)
The first term represents the Joukowsky profile, as can be verified by
comparison with Eq. (2-100). Profile shape and velocity distribution are interrelated
by Eq. (2-96), which must now be interpreted as the integral equation for the
profile inclination dZ(t) fdX. Following Betz [4] and Fuchs (16], the solution is
AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 75
f
1
___
d0t)
1
dX
it, (X')
U0
Co
X'(1-X')
d11'
X(1-X) X- 1'
(2-106)
where
. =x
U".
We
Uro
-1
denotes the induced velocity distribution on the chord, and r is defined in Eq.
(2-95). Since the needed profile inclination is a term of this equation, Eq. (2-106)
can be solved only through iterations (see Truckenbrodt [68] ). The publications by
Eppler [13] and Strand [63] on this subject should also be mentioned.
The simplest case of a constant induced velocity u on the chord leads to an
elliptic teardrop profile. For a linear distribution of the induced velocity u/U.,
u
Z') =
(2-107a)
U=c(1-bXU00
[4 - b(1 + 2X)] X(1 - X)
(2-107b)
For b = 0 the profile is elliptic; for b = 3 , the Joukowsky profile [Eq. (2-100)] is
obtained.
2-4-4 Inclined Profile of Finite Thickness
Computation of the velocity distribution on the profile contour The general case of
a cambered profile of finite thickness will now be treated, after the case of the very
thin profile (skeleton) discussed in Sec. 2-4-2 and the case of the symmetric profile
of finite thickness in chord-parallel flow discussed in Sec. 2-4-3. The general case is
obtained essentially by superposition of these two previously discussed cases.
A cambered profile of finite thickness can be composed of a skeleton line
Z(S) = z(S)/c and a teardrop profile Z(t) = z(t)lc (Fig. 2-1). In Eq. (2-1), the profile
ordinates are given as
Zu,1= Z(S) ± Z(t)
(2-108)
where the upper sign is valid on the upper surface, the lower sign on the lower
surface. With Z, the ordinates on the upper surface, and Z1, those on the lower
surface, Eq. (2-108) can be broken down into
Z(S)
= (Zu + Z1)
and
Z(t) = z (Zu - Z1)
(2-109)
z
The velocity distribution of the general profile is the sum of those of skeleton
and teardrop. A third distribution has to be added, however, which is produced
through the inclination of the teardrop profile. Riegels [49] computed this
contribution. The velocity contribution caused by the inclination of the teardrop
profile may be interpreted as the influence of an additional camber and an
76 AERODYNAMICS OF THE WING
additional angle of attack, as was shown in [68]. Accordingly, the following
contributions have to be added to the geometric skeleton (mean camber) line Z(s):
dZ(s)
= a [1/_ X x Z(1) +
da
=- 2 a `r d0i)
d9;
2(1
- X) (
c')
d92 )
-v-o ]
(2 - 11O a)
(2 - 11 Ob )
1_0
The equations for the computation of velocity distribution and aerodynamic
coefficients that were derived by the method of singularities in the cases of skeleton
and teardrop profiles can be evaluated conveniently through numerical quadrature
formulas. Details for the computations are found in [49].
The result of a sample computation from the above outlined method for the
computation of velocity distributions on profiles is presented in Fig. 2-35 for the
profile NACA 66 (215)-216, a = 0.6. A theoretical velocity distribution from
Theodorsen and Garrick [66] (conformal mapping) is also shown, and for
comparison with the theory, a measurement from [1] is added. The agreement of
the two theoretical curves with the test data is good. Note the appreciable
agreement of these pressure distributions for the inclined symmetric Joukowsky
profile, obtained by the method of singularities, with those of Fig. 2-16, computed
by the method of conformal mapping.
Lift and pitching moment The computation of the aerodynamic coefficients for the
skeleton profile was discussed in Sec. 24-2. The results in that section are also valid
for the inclined profile of finite thickness if the influence that is introduced by the
inclination of the teardrop profile in Eq. (2-110) is taken into account. The
resulting relationships for lift slope and neutral-point location are given in Table 2-1.
The other aerodynamic coefficients (zero-lift angle of attack, zero-moment
coefficient, angle of attack, and lift coefficient of the smooth leading-edge flow) are
equal for the profile of finite thickness and the skeleton profile.
For the Joukowsky profile in Eq. (2-100) the lift slope is
dCL
da
gN
2_x
dcL
L
I a) = ,.on (i -...0. j f _V)
i
=4
.
(2-111a)
(2 - 111 b)
in agreement with the solution from the method of conformal mapping, Eq. (2-33).
For the NACA profiles of Fig. 2-2, these formulas show that the lift slope
always increases with thickness, whereas the neutral point lies behind the c14 station
of the profile. The test data of the lift slope from [1 ] lie in all cases below the
theoretical values. When the profile thickness increases, the lift slope of the older
NACA 00-series with a relatively large trailing-edge angle decreases (Fig. 2-2b),
whereas it shows the theoretical trend in the newer NACA 6-series with a smaller
trailing-edge angle (Fig. 2-2c). In all cases, the lift slope is smaller when the surface
AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 77
1.
o
O
CL-O.23
°
O
a - toy o
U pper surface
I
1.
o/
1
Lower surface
\O
0
Theory of Theodorsen
---Theory of Riegels
C
Measurements at Re = 6
10 6
a
0.2
0.41
as
08
Lower surface
Figure 2-35 Velocity distribution on the profile contour of profile NACA 66 (215)-216,
a = 0.6. Lift coefficient cL = 0.23. Comparison of theory and experiment.
is rough than when it is smooth. This behavior shall be taken up again in Sec. 2-5.
A similar comparison shows disagreement of measurement and theory for the
neutral-point location of the older NACA series.
Using a procedure by Martensen [28], Jacob [28] extends the singularities
method by arranging vortices on the profile contour instead of the profile chord.
The investigation of Lan [37] and the comments by Maskew and Woodward [40]
should be mentioned. Furthermore, profile theories of higher order for incompressible flow are found in, among others, Keune [22, 32] and Lighthill [39]. An
essential contribution to the theoretical and experimental investigations of fluid
mechanical behavior behind blunt profiles has been made by Tanner [65]. Base
pressure and base drag play an important role in this case.
78 AERODYNAMICS OF THE WING
2-4-5 Special Problems of Profile Theory
Airfoil in curved flow So far, it had been assumed in all considerations of profile
theory that the wing moves in straight incident flow. When investigating the
interference of the airplane parts with each other, the case is encountered, however,
where the wing lies in a curved incident flow field. The aerodynamic problems of
such a wing can also be treated with the skeleton theory.
A flat plate in curved flow behaves approximately as a cambered skeleton in
parallel flow. The variable angles of attack a '(X) along a profile in curved flow can
be replaced by changed equivalent skeleton line inclinations (a-dZ(S)/dX), as
shown in Fig. 2-36. With this procedure in mind,
dZ(S>
a - a(X)
dX =
must be substituted in the formulas of the skeleton theory (Sec. 24-2). By using
the expressions of Table 2-1, the mean angle of attack a is obtained as
n
-fa'(g7)(1+cosg7)dq7
(2-112a)
0
This is, to express it again in a somewhat different way, the angle of attack that
produces in straight flow the same lift CL = 27ra as the variable angle-of-attack
distribution. The zero-moment coefficient becomes then
n
cJI0 = -' f a' ((p) (cosg7 + cos2g7) d 97
(2-112b)
0
At a constant angle-of-attack distribution a'(X) = a, the above equations produce
the relationships for the inclined plate: cz = a, cM0 = 0. Approximate expressions
for a and CMo have been derived by Pistolesi [45] and Multhopp (see Chap. 5
[40]), respectively. Using only local a' values on a few characteristic stations of the
chord, they arrived at the expressions
a = a;;
--- ai
a
(2-113a)
,Plate
a'j
b
o
X
Figure 2-36 Airfoil in curved flow-sche-
matic explanation. (a) Variation of a' on
c
plate. (b) Angle of incidence distribution
a'(x). (c) Skeleton profile Z(S).
AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 79
Figure 2-37 Distribution of the induced downwash angle on the extended wing chord for the
inclined flat plate.
z
C:ft o = 16 (o
+ 2 c 0 - 3 ai ors )
(2-113b)
where 4o, a'so, a'7s, and ale mean the local angles of attack a' at the stations
X= 0, 0.5, 0.75, and 1.0, respectively. Hence, the value of the angle of incidence at
the a -chord station should determine the lift when the angle-of-incidence distribution is variable. Compare the formulas of Weissinger [73],
Theorem of Pistolesi The two approximation formulas of Pistolesi and Multhopp
are of particular importance when the continuously distributed circulation on the
wing chord is replaced by a single vortex at the 4 -chord station, as is customary for
simplified treatment of the wing of finite span (lifting-line theory, Sec. 3-34). The
Pistolesi approximation for the mean angle of incidence is in agreement with the
exact solution, as seen in Fig. 2-37. The Multhopp approximation for the zero
moment produces the right value cMo = 0 for the flat plate, because a' = -a'
and aso = 3aiDo when the plate is replaced by a single vortex at the i -chord station.
The velocity near-field of the profile So far, the velocity distributions have always
been determined on the profile surface. For certain aerodynamic problems of
airplanes, for example, for investigations into the influence of the wing on the
incident flow of the fuselage or horizontal stabilizer, knowledge of the velocity field
off the wing is required. This matter will now be discussed to some extent.
First, let the wing be replaced by its skeleton profile; that is, it is represented
by its vortex distribution, Eq. (2-44). Of main interest are the z components w of
the velocities because they produce a change of the effective angles of incidence on
the airplane components that lie before and behind the wing. The induced velocity
component w will be considered only in the wing plane, that is, on the x axis.
80 AERODYNAMICS OF THE WING
The distributions of the induced upwash and downwash velocities along the x
axis are given by Eq. (2-46b):
f
1
1
TV (X) _ - 2n
X-(XX,
dX'
(2-114)
0
Here k(x) is the circulation distribution along the profile chord and X = x/c, the
dimensionless coordinate in direction of the profile chord. The total circulation of
the wing 1' is found from the circulation distribution by integration [see Eq.
(2-52)]. Evaluation of Eq. (2-114) over the profile chord, 0 < X< 1, was described
earlier using the Birnbaum-Glauert substitution for the circulation distribution [Eqs.
(2-61) and (2-63)].
Equation (2-114) can also be used to compute the induced downwash velocity
before and behind the profile. A singular point of the integrand no longer exists in
this case, however. The previous substitution of Eq. (2-62) now must be replaced by
X = (1 ± cosh 99)
(2-115a)
X' _ (1 + Cos (p')
(2-115b)
The lower sign applies to points before the wing and the upper sign to those behind
the wing. By introducing Eqs. (2-115) and (2-61) into Eq. (2-114) and integrating,
the downwash angle distribution aw = w/UU is obtained as
aw(X) = -A0 (1 - V
1) +r2' An [1 - 2X (1
X
-
gX
(2-116)
forX>1 andX<0.*
In the circulation distribution Eq. (2-61), the term with A0 represents the flat
plate inclined by the angle a, with A0 = a. The term with A 1 represents the
parabola skeleton with camber h/c in a flow parallel to the chord with A 1 = 4h/c.
For the first case, the distribution of the downwash angles along the profile chord is
plotted in Fig. 2-37. For comparison is shown the induced downwash angle
distribution that is obtained when the wing is substituted by a single vortex of total
circulation 1', located in the center of action XA of the distributed circulation, that
is, at XA = a of the inclined plate. At large distances before and behind the wing,
the distributions of the induced downwash angles of the continuous vortex
distribution (lifting wing) and the single vortex (lifting line) are in agreement.
It is noteworthy that in the case of the inclined plate, the induced downwash
velocities at station X = a are equal for the lifting surface and for the lifting line
(Pistolesi theorem).
In the case of flow around thick profiles, the main interest is directed toward
the induced velocity u in the x direction before and behind the profile. From Eq.
*Note that, according to Jaeckel [301, the following relation applies:
n
1
f
0
cosn cp'
± cosh 97 - cos T'
d
,
-(
1)
n+i coshn cp - sinhn rp
sink T
AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 81
U.
2-38 Distribution of induced longitudinal velocity on the
Figure
extended wing chord for a symmetric Joukowsky profile (6 =
tic = thickness ratio).
(2-9la), this component is obtained by introducing the source distribution from Eq.
(2-102). The evaluation is analogous to that of w(x) and produces for u(x)/U00 an
equation analogous to Eq. (2-116), where the coefficients A7z of the circulation
distribution must be replaced by -Bn of the source distribution. The latter satisfy
the closure condition Eq. (2-103). In Fig. 2-38, evaluation of such a computation is
shown for a symmetric Joukowsky profile.
Although it is not possible in this book to cover the problems of unsteady flow
that are of importance to airplane aerodynamics, the fundamental study of Wagner
[72] on unsteady wing lift in plane flow should not be overlooked.
2-5 INFLUENCE OF VISCOSITY AND
BOUNDARY-LAYER CONTROL
ON PROFILE CHARACTERISTICS*
So far, all the discussions of this chapter have been based on the assumption of
inviscid flow of an incompressible fluid. Now, a few data will be given on the effect
of viscosity and the control of the boundary layer close to the wall. The effect of
compressibility on the aerodynamic coefficients of a wing profile will be treated in
detail in Chapter 4.
2-5-1 Effect of Reynolds Number on Lift
The most important quantity characterizing viscosity effects is the Reynolds number
[Eq. (1-17)]. For a given profile geometry, this nondimensional quantity determines
decisively the aerodynamic coefficients of a wing.
The great importance of the Reynolds number as well as of turbulence on the
profile performance is demonstrated in the summary report of Schlichting [52].
*The authors are indebted to K. O. Arnold, who contributed considerably to this section
in the original German version of the book.
82 AERODYNAMICS OF THE WING
Investigations on wing profiles in the critical Reynolds number range are reported
by Kraemer [34].
First, the influence of the Reynolds number on the lift and its interplay with
the geometric profile parameters will be discussed. Then, some information on
the profile drag will be given that, as was pointed out earlier, cannot be determined
with the theory of inviscid fluids.
The Reynolds numbers of the wings that are of interest in modern aeronautics
are of the order of Re = U.c/v = 106 to 5 - 107, except for model airplanes and
certain glider planes for which they lie between 105 <Re < 106. In the former
Reynolds number range, the boundary layer on conventional profiles is turbulent
over most of its length. This is not true, of course, for so-called laminar profiles. In
the range of Reynolds numbers of Re > 106 , but even down to Re = 105 , the lift
as computed from potential theory is in satisfactory agreement with experimental
results when the angle of attack is small to moderately large. This fact can be seen,
for example, in Fig. 2-10 for the inclined flat plate and for the profile Go 445 at
Re = 4 - 105, and in Fig. 2-17 for the Joukowsky profile at Re = 105. In these
cases the flow is attached to the wing; that is, no boundary-layer separation occurs.
Likewise, the pressure distributions on the profile, determined from potential theory,
agree well with experiments in this range of angles of attack and Reynolds numbers;
see Fig. 2-18 for a Joukowsky profile, Fig. 2-33 for a symmetric NACA profile at
zero angle of attack, and Fig. 2-35 for a cambered NACA profile with angle of
attack.
Lift slope For the profile NACA 2412, Fig. 2-39 gives the lift coefficient CL against
the angle of attack a from Jacobs and Sherman [29]. Figure 2-39a shows that for
the range from Re = 8 - 104 to 3 - 106 , no important effect of Reynolds number
on the lift can be expected as long as the profile is not too much inclined (a< 80).
Since the cL(a) curve is linear in this a range, the Reynolds number influence can
be described simply by the lift slope dcL/da. This kind of presentation is used in
Fig. 2-40 for a few four- and five-digit NACA profiles. These measurements show a
slight increase of the lift slope dcL/da with the Reynolds number for Re < 3 - 106;
beyond this Reynolds number, up to Re = 10', practically no change occurs.
In addition, the lift slope depends on both the profile thickness and the
trailing-edge angle. It decreases with increasing thickness ratio t/c in the four- and
five-digit NACA profiles, whereas the opposite behavior is found in the NACA 6series, namely, an increase of dcL/da with increasing thickness ratio tic.
Conversely, increasing the trailing-edge angle always results in a reduction of the
lift slope. The quotient x = (dcL /da)exp /(dcL /da)theor is plotted in Fig. 241 as a
function of the trailing-edge serniangle T (see Fig. 2-1 b). The quotient % goes to 1
when the trailing-edge angle approaches zero (r = 0). When r increases, the quotient
x declines to about 0.8 for smooth surfaces, and to 0.7 for rough surfaces (see also
Hoerner and Borst [251). The deviations of the measured lift slopes from the theoretical values are caused by the boundary layer and the wake near the trailing edge.
The difference in boundary-layer thickness on the upper and the lower profile
surfaces-thicker above, thinner below-is equivalent to an additional negative
AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 83
18
1.6
Profile NACA 2412
ZA
Re-8.3.10``
-3.3.10 5
a
-6.6.105
o
-9.3.106
v
-3.1 105
0.4
o
0.2
0
-02
-04
F
-8°
a
IT
-4°
00
40
15°
12°
1q0
2400
-RO°
a
001
0.02
0.03
0.04
0.05 0.06
cpp
b
Figure 2-39 Lift and drag measurements on the NACA 2412 profile at various Reynolds
numbers. (a) Lift coefficient cL vs. angle of attack a. (b) Polar curves CL vs. CDp.
camber; compare also Pinkerton [44]. The boundary layers change the Kutta
condition, too, in that the rear stagnation point shifts from the trailing edge to the
profile upper surface.
At extremely small profile Reynolds numbers, Re < 10$ , such as occur in
free-flight airplane models (see Schmitz [57] ), often no linear relationship exists
between lift coefficient CL and angle of attack a, even for very small angles of
attack. In this case the measured CL values deviate strongly from theory over the
whole angle-of-attack range, because the flow is widely separated from the profile.
Conversely, at larger Reynolds numbers, Re > 10', the separation that is caused by
8
NACA 0072
0
n
4412
o
n
4415
o
n
I
f
I
23012
23015
Y
I
.5
Figure 2-40 Reynolds number in-
fluence on lift slope dcL/da for
4
5
5
8
10 6
2
3
Re ---
4
S
6
107
four- and five-digit NACA profiles
with smooth surfaces.
84 AERODYNAMICS OF THE WING
1.
. X4 C4 00
a4
0
63
n
,,
o
"
0.04
64
65
66
008
012
Q16
020
024
tan r -,
W
10
Figure 2-41 Comparison of the lift
b
04
t
0
004
008
j
a1z
ale
tan ,r -
E
020
024
slope from theory and experiment for
NACA profiles of various trailing-edge
where x = (dCL/da)exp/
(dcL1da)theor.'(a) Smooth surface. (b)
angles
028
27-,
Rough surface.
a steep pressure rise on the suction side of the profile occurs only at larger angles of
incidence, a = 5-200, depending on the profile shape. The lower value of a is valid
for thin profiles. As soon as local separation occurs on the wing, the lift slope
decreases. The deviation from the linear characteristic of the theory grows larger
with an extention of the range of separated flow on the profile until finally, at large
a, the flow is almost entirely separated on the suction side, and the lift drops, as
demonstrated in Fig. 2-1 lb. The phenomenon of separation from the wing, which is
to be discussed later in detail, has a decisive effect on the maximum lift coefficient
CL max- This coefficient is of great aeronautical importance (in take-off and la nding).
Maximum lift The aerodynamic problems of maximum lift are summarized by,
among others, Nonweiler [43], Schlichting [54], and Smith [60]. The maximum
lift of a profile depends decisively on the flow conditions in the boundary layer on
the suction side. At very small Reynolds numbers, the boundary layer is completely
laminar and separation occurs near the profile nose (leading-edge stall) because of
the strong pressure rise on the suction side immediately downstream of the leading
edge. The location of the separation point is almost independent of the Reynolds
number. The maximum lift is, therefore, independent of the Reynolds number in
this range. Only at a certain larger Reynolds number, the value of which depends on
the profile geometry, do the flow characteristics change. The laminar boundary
layer still separates; transition to turbulent flow now takes place in the separated
flow, however, leading, in general, to reattachment of the turbulent boundary layer
farther downstream. In this way, a laminar separation bubble forms between the
points of laminar separation and turbulent reattachment. The reattachment point
moves upstream with increasing Reynolds number until it finally reaches the
separation point, that is, until the length of the separation bubble becomes zero.
The maximum lift increases strongly with Reynolds number as a result of the
AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 85
superposition of two effects: In the first place, lift is gained at a fixed angle of
attack because of the reduction of the separation length, and then the wing can be
set at a higher angle of attack before the flow ultimately separates.
At very high Reynolds numbers, a natural transition from the laminar to the
turbulent boundary layer occurs before the point of laminar separation. This
transition point travels upstream with increasing Reynolds number and the length of
the turbulent boundary layer, and consequently the boundary-layer thickness
increases. As a result of this process, the circulation around the profile is
diminished; that is, the maximum lift may again decrease somewhat at high
Reynolds numbers. As an example for the CLmax behavior, the maximum lift of
profiles of the NACA 6-series is plotted in Fig. 2-42 against Reynolds number for
various thickness ratios t/c and camber heights h/c according to Loftin and Smith
[29] ; see also Fig. 2-39. In the range Re > 106 of interest to aeronautics, profiles
of moderate thickness (t/c - 0.12) produce the largest lift. The influence of camber
is reflected in an increase of CLmax with h/c because the critical, separation
promoting pressure rise on the profile suction side is occurring at larger angles of
attack for increased h/c. The most important geometric parameter affecting
separation at large angles of attack, and thus affecting the maximum lift, is the
shape of the profile nose, because this shape determines decisively the pressure
distribution in the vicinity of the leading edge. The measurements by Nonweiler
[43] of Fig. 2-43 convey some insight into these relationships through curves that
show CLmax values for a fixed Reynolds number (Re = 6
106) as a function of the
thickness ratio t/c. The nose radius is characterized by the profile ordinate z1 at
x = 0.05c. Accordingly, the nose radius has no effect on CLmax for very thin
profiles, whereas for profiles of moderate thickness, CLmax increases considerably
with zi It.
A similar parameter, namely, the ordinate zo Ic of the profile suction side at
station x/c = 0.0125, has been used by Gault [77]. It allows delineation of ranges
of the various separation processes as a function of Reynolds number in a universal
diagram. This presentation, Fig. 2-44, is based on measurements on about 150
1.B
1.6
'
AA CA 64 -409
64,-41Z
o
64z 415
Ii
o
I
(
1
#A CA 64r - 012
I
I
n
j
o
64, A 272
o
64,-412
(
i
Re--
6 1075 6
9 106
2
3
4 55 810'
Re
b
Figure 2-42 Maximum lift coefficient of profiles of the NACA 6-series vs. Reynolds number. (a)
a
Effect of thickness ratio. (b) Effect of camber ratio.
86 AERODYNAMICS OF THE WING
18
_Z
t
!
l
- 06S
r
0,6
1.6
QSS
0.50
Qcf3
0,35
1.0
030
t
0.8
020
0.10
06
009
0
008
t 212
016
0.20
Figure 2-43 Maximum lift coefficient
at Reynolds number Re = 6 - 106 vs.
thickness ratio tic and nose radius in
terms of z, It. z, = z (x/c = 0.05).
After [43].
02'9
S
profiles with smooth surfaces at low wind-tunnel turbulence. It shows that profiles
with sharp leading edges, or with very small nose radii (zo/c < 0.009), have, at all
Reynolds numbers, a specific separation characteristic that is termed thin-airfoil
stall. Even at small angles of attack a, separation of the flow over the thin leading
edge occurs directly at the profile nose, followed by reattachment. The velocity
profile of the boundary layer at the point of reattachment is neither typically
laminar nor typically turbulent. Not before the boundary layer approaches the
trailing edge is a fully turbulent flow pattern established (see McCullough and Gault
[77] ). Reattachment occurs more and more downstream when the angle of attack
increases, leading to a growing separation range and consequently a gradually
diminishing lift slope. As soon as the flow is detached on the whole suction side, CL
decreases continuously with increasing a (see also Young and Squire [77] ).
0.036
0.032
2
0.028
0.024
0.020
0.076
Figure 2-44 Separation from profiles
vs. Reynolds number and nose radius
[in terms of zo /c, with z0 = z (x/c =
0.072
0.0125)], after [77]. (1) Separation
from a thin profile. (2) Laminar
separation from profile nose. (3)
Combination of laminar and turbu-
0.008
0.004
I
I
0
4
6
8 70 6
2
4
Re
6
8 707
Z
4
lent separation. (4) Turbulent separation.
AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 87
A basically different separation characteristic is found on wings of moderate
thickness having a moderately large nose radius, the leading-edge curvature of which
is, however, still relatively large.* The steep pressure rise behind the profile nose
then leads to separation of the laminar boundary layer at larger angles of attack.
The transition to turbulent flow takes place, however, in the separated flow that
results in reattachment- farther downstream. A laminar separation bubble is formed,
the extent of which decreases with increasing angle of attack because the transition
point, and with it the turbulent reattachment point, move closer to the separation
point, which moves likewise toward the leading edge. Eventually, the laminar
boundary layer separates on the very nose where the contour curvature is too large
for the transition to cause reattachment. This process, known as the leading-edge
stall, is characterized by a sudden sharp lift drop (see Crabtree (11] and Tani
[64] ).
On the other hand, on most thick profiles (tlc > 0.15), that is, at large nose
radii, flow reattachment occurs behind the laminar separation point, even at large
angles of attack. In this case the maximum lift is determined by two processes that
influence each other. These are the expansion of the laminar separation bubble from
the nose, and the turbulent separation that starts at the trailing edge and moves
upstream with increasing angle of attack (combined leading-edge and trailing-edge
stall). The variation of the lift cL(a) depends on the predominance of one or the
other of these two separation processes. The separation bubble may disappear
entirely on very thick, strongly cambered profiles and at very high Reynolds
numbers. The reason for this is that the Reynolds number is then large enough for a
natural transition to turbulent flow upstream of the station of strong pressure rise.
The turbulent boundary layer separates only a short distance upstream of the
trailing edge (trailing-edge stall). This separation point moves upstream continuously
with growing angle of attack, and the lift does not drop abruptly after passing
CLmax but very gradually, similarly to the case of the thin profile. The profile shape
of optimum lift coefficient at flow without separation can be computed following a
procedure of Liebeck [381.
Pressure distribution In Fig. 2-45, pressure distributions on profiles of the NACA 6-
series are presented in the range of the maximum lift at a Reynolds number
Re = 5.8 106 according to McCullough and Gault [77]. Separation from thin
profiles (NACA 64A006) is characterized by a very slight underpressure near the
leading edge. This underpressure is even reduced with an a increase, whereas the
separation range (cP = const) grows from the profile nose downstream. Conversely,
very strong suction peak exists on profiles of larger thickness ratio for
a< acL max The laminar separation bubble is too short to be noticeable in the
a
pressure distribution, if it exists at all. The NACA 631-012 profile causes laminar
separation at the nose, resulting in an abrupt collapse of the high underpressure on
*Translator's note: Remember that the term "nose radius" does not necessarily imply a
circular nose. The definition of nose radius is of the kind found in Figs. 2-43 and 244. The
curvature can, therefore, be relatively large locally on the nose, even if the radius in the above
sense is not small.
88 AERODYNAMICS OF THE WING
245 Measured pressure distribution at
Reynolds number Re = 5.8 106 on profiles of
NACA 6-series with various separation characteristics in the range of maximum lift. M. Separation
from thin profile. (2) Laminar separation from
Figure
0.2
0.6
0.4.
x/c
0.8
10
profile nose. (3) Turbulent separation.
the leading edge and an immediate flow separation over the entire suction side. This
in turn results in the steep lift drop when the angle of attack for CLmax is
exceeded. As soon as turbulent separation has been established, as is the case on the
NACA 633-018 profile, the suction peak at the leading edge remains, even when a is
larger than acLmax The separated range expands from the trailing edge farther and
farther upstream, causing the lift to decrease continuously.
The separation characteristics of a given profile may be different for the various
Reynolds numbers, as shown in Fig. 2-46 for the example of the pressure
distribution on the profile NACA 4412 at the angle of attack c= 16°
(see
Pinkerton [29] ). For Re = 1 - 105 and 4.5 - 105 , the pressure distribution is similar
to that of the profile NACA 64A006 (Fig. 245); that is, separation has the same
character on thin profiles, although only at larger angles of attack. The separated
range decreases with increasing Reynolds number in this case. According to Fig.
2-44, for the thin profile at Re < 106, there are only two possibilities, namely,
laminar separation or turbulent separation near the trailing edge. Transition from
one behavior to the other requires that the profile is made thicker when
the Reynolds number is reduced. When the Reynolds number is raised to 1.8 - 106,
a laminar separation bubble 0.005c long forms on the NACA 4412 profile, and at
x/c = 0.40 turbulent separation sets in. Finally, at Re = 8.2 106 , the flow is
attached over the whole profile. A further increase in Reynolds number has
practically no influence on the pressure distribution, which agrees quite well with
theory as long as no separation occurs (see Cooke and Brebner [10] ). Note,
AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 89
Profile NACA 4412
+I
Re - 1 105;cL-0.98
o
o
-4
o
---
-4.5.105;
- 1.15
6.106 ;
- 136
- 8.2.106;
- 1.67
-
1.
Theory
-2
.
-1
0
1
0
0,2
0.4
0.5
x/c --
0. B
1.0
Figure 2-46 Effect of Reynolds number on pressure distribution on profile NACA 4412 at a large
angle of attack (a= 16°).
however, that the theoretical curve of Fig. 2-46 is obtained from a modified theory
from Pinkerton [44] and not from pure potential theory.
The influence of the boundary layer on the pressure distribution on a profile as
a function of angle of attack is presented in Fig. 2-47. Figure 2-47a gives the
distribution at moderate angles, which is susceptible of computation after the
a
b
c
Figure 2-47 Change of pressure distribution on a wing profile with angle of attack [671. (a)
Attached flow, medium angle of attack. (b) Beginning of separation from trailing edge
CL = CL max- (c) Separation from leading edge with enclosed vortex (bubble).
90 AERODYNAMICS OF THE WING
methods of potential theory. At larger angles of attack, separation sets in first on
the upper surface of the profile near the trailing edge (Fig. 2-47b). From there, it
travels upstream with increasing angle of attack. At the same time a wake forms in
which a vortex (bubble) is embedded. At very large angles of attack, beyond the
maximum of the lift coefficient, the wake shifts upstream to the wing nose (Fig. 247c).
The flow reattaches again further downstream.
A comprehensive listing of experimental data on the lift problem is found in
Hoerner and Borst [25].
Based on studies of Preston [61], Spence [61] makes some recommendations
about the theoretical inclusion of the friction effect into the aerodynamics of the
wing profile. Theoretical determination of the pressure distribution for separated
incompressible flow about profiles of almost any shape is possible using a
computational method of Jacob [271, but the abrupt leading edge separation and
reattachment of the flow cannot be obtained directly by this method.
2-5-2 Effect of Reynolds Number on Drag
When the lift coefficient is small, the profile drag is caused essentially by friction.
Its value depends on the position of the transition point and hence the lengths of
laminar and turbulent stretches. The local velocities increase with angle of attack,
leading to a slight rise of the profile-drag coefficient CDp. A further contributing
factor is the increasing length of the turbulent boundary layer with a simultaneous
shrinking of the length of the laminar layer. In the CLmax range, the profile drag
rises
steeply because of the strong increase in pressure drag caused by local
separation. The Reynolds number has a very strong influence on the magnitude of
the profile drag because both the pressure drag and the friction drag decrease with
increasing Reynolds number (see Fig. 2-39b).
The dependence of the minimum drag coefficient CDmin on the Reynolds
number [29] is plotted in Fig. 249 for several four-digit NACA profiles. Laminar
separation causes quite high values of the minimum profile drag CDmin for small
Reynolds numbers (Re < 5 - 105 ). Symmetric profiles produce minimum drag at
CL = 0, cambered profiles at the angle of smooth leading-edge flow. The value of
CDmin decreases strongly when the Reynolds number grows. As soon as fully
attached flow is established, the trend of the CDmin curve is similar to that of the
friction drag of the flat plate (see Fig. 2-48). In this range of Reynolds numbers
(Re > 8 - 105), the minimum drag coefficient is raised more and more above the
value of friction drag when the profile thickness grows (Fig. 2-49a). The same
behavior is found for the camber (Fig. 2-49b).
Peculiarities of the drag appear at laminar profiles (see Wortmann [75] ). As an
example, three-component measurements on the NACA 662-415 profile are plotted
in Fig. 2-50 for various Reynolds numbers (after [29] ). Over a limited range of
small lift coefficients, the profile drag is constant, independent of the angle of
attack. It is lower than that of a normal profile if the Reynolds number is large
enough to prevent laminar separation. When the Reynolds number grows, CDp
decreases; at the same time the dip in the drag curve, that is, the lift range for
91
92 AERODYNAMICS OF THE WING
e NACA 0009
s
a
a
NA CA 0012
"
2412
o
4412
0
0
6412
o
0012
0015
°
0019
--Flat plate
---Flat plate
I
2
105
34
6 810
3
2
6 810710
4
1
3
5 8106
4
2
3
4
6 8l07
Re---b
Re -Figure 249 Minimum drag of four-digit NACA profiles vs. Reynolds number. (a) Effect of
a
thickness ratio. (b) Effect of camber ratio.
minimum drag, becomes narrower. When the angle of attack is increased, the
pressure minimum shifts toward the nose and, in general, the transition point jumps
upstream abruptly, causing a very strong increase in profile drag. This process is
observed at reduced a when Re increases and at last, at very large Reynolds
numbers, the dip in the drag curve disappears completely. A normal polar curve
with an elevated cD min takes over (see [50] ).
Computational determination of profile drag The profile drag of lifting wings can
be determined theoretically by means of boundary-layer theory as long as the flow
1.8
1.6
Re-1.106
-2.90 6
1
-
°
3.106
o
-6 10 6
-9 906
o
011
Profile NACA 5o-415
0.4
Tyr-
c
4
0.2
I
I
j
;
0
-0.2
-04
-8°
-4°
0°
40
8°
12°
a--
16°
200
24°
-0.008
0
0.008
0,016
0.1CM ,CDp
Figure 2-50 Three-component measurements on the laminar profile NACA 662 -415 at various
Reynolds numbers.
AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 93
is
fully attached. Pretsch
[48]
and Squire and Young [62] were the first
investigators to publish such methods, which were later improved by Cebeci and
Smith [9]. The profile drag (= pressure drag plus friction drag) is obtained from the
velocity distribution in the wake at large distance from the body in the form
+ro
Dp=nb fu(Uu)dy
(2-117)
Here, b is the span of the wing profile, y is the coordinate normal to the incident
flow direction, and u(y) is the velocity distribution in the wake. By defining the
profile drag coefficient CDp by Dp = cDpbc(Q/2)UU and introducing the momentum
thickness 62,x, the drag of both sides of the profile with a fully turbulent boundary
layer is given as
CD p = 2
6200
(2-118a)
c
0.148
SRe
1
U
(35d()108
c
(2-118b)
Here Re =
is the Reynolds number and U(x) is the velocity distribution over
the profile as obtained for potential flow. The second relationship [Eq. (2-118b)]
is derived from the findings of boundary-layer theory (see Schlichting [55] ). For a
plate in parallel flow, there is U(x) = U. = const.
For some symmetric wing profiles in chord-parallel flow, the coefficients for
the profile drag from [62] are summarized in Fig. 2-51. The profile thickness varies
from t/c = 0 (flat plate) to t/c = 0.25 and the Reynolds number ranges from
Re = 106 to 108. The profile drag is strongly dependent on the location of the
laminar-turbulent transition point xtr, which varies from xtr./c = 0 to 0.4. The
increase in profile drag with thickness must be attributed essentially to a rising
pressure drag.
Truckenbrodt
[48]
extended the drag formula, Eq. (2-118b), to contain
explicitly the profile shape instead of the velocity distribution of potential flow.
Application of this method to a large number of NACA profiles produces the simple
relationship between the profile-drag coefficient and the thickness ratio t/c,
CDp = 2Cft 1 + C c
(2-119)
Here c ft is the drag coefficient of the flat plate with a fully turbulent boundary
layer. The constant C lies between C = 2 and 2.5 (see also Scholz [48] ).
The above statements apply to the profile drag at zero lift. The CD values
determined in this way agree, in general, satisfactorily with experiments.
A comprehensive presentation of experimental data on the drag problem is
found in Hoerner [24]. Truckenbrodt [69] summarized the decisive findings on
drag of wing profiles. Progress in the development of profiles of low drag has been
reported by Wortmann [76].
y
N
h
/,V/F
/ZZ /l/
U
dU3000L
tb
N
N
-
a
n
C
O
d
11
-Y
dpi 0001.
H
QO
N.Z
N
-- CiO_OO b
OO
ft
> 4-
AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 95
2-5-3 Boundary-Layer Control on the Wing
A change of the flow in the very thin wall boundary layer may, under certain
conditions, alter considerably the entire flow pattern around the body. A number of
methods have been developed for boundary-layer control that, in some instances,
have obtained importance for the aerodynamics of the airplane. The basic principles
of boundary-layer control will be explained briefly in this section. In most cases,
boundary-layer control is considered in the following contexts: elimination of
separation for drag reduction or lift increase, or only change of the flow from
laminar to turbulent, or maintaining of laminar flow. The various methods that have
been investigated mainly experimentally, but also theoretically in some instances,
can be highlighted as follows: boundary-layer acceleration (blowing into the
boundary layer), boundary-layer suction, maintaining of laminar flow through
proper profile shaping (laminar profile). A comprehensive survey of this field is
given by Lachmann [36].
Boundary-layer acceleration A first possibility of avoiding separation is given by
introducing new energy into the slowed-down fluid of the friction layer. This can be
done either by discharging fluid from the body interior (Fig. 2-52a) or, in a simpler
way, by taking the energy directly from the main flow. This method consists of
injecting fluid of high pressure into the decelerated boundary layer through a slot
(slotted wing, Fig. 2-52b). In either case, the velocity in the wall layer increases
through energy addition and thus the danger of separation is removed. For practical
applications of the method of fluid ejection as in Fig. 2-52a, particular care is
required in designing the slot. Otherwise, the jet may disintegrate into vortices
shortly after its discharge. More recently, extensive tests [46] have led to the
method of discharging a jet at the trailing edge of the wing, which has proved to be
b
c
Figure 2-52 Various arrangements for boundary-layer
control. (a) Blowing. (b) Slotted wing. (c) Suction.
96 AERODYNAMICS OF THE WING
very successful in raising the maximum lift (jet flap). The same benefit has been
gained from blowing into the slot of a slotted wing.
A slotted wing (see Fig. 2-52b) functions in the following way: On the front
wing (slat) A-B, a boundary layer forms. The flow through the slot carries this layer
out in the free stream before it separates. At large angles of attack, the steepest
pressure rise and hence the greatest danger of separation occurs on the suction side
of the slat. Starting at C, a new boundary layer is formed that may reach the
trailing edge without separation. Hence, by means of wing slats, separation can be
prevented up to much larger angles of attack, so that much larger lift coefficients
can be obtained. In Fig. 2-53, polar diagrams (lift coefficient CL vs. drag coefficient
cD) are given of a wing without and with a wing slat and with a rear flap. In
the slot between main wing and rear flap (Fig. 2-52b), the processes are the same, in
principle, as those in the front slot. The lift gain from a front slat and a rear flap is
considerable. Further information on this item will be given in Chap. 8.
Boundary-layer suction Boundary-layer suction is applied for two purposes: to
avoid separation and to maintain laminar flow (see Schlichting [53] and Eppler
[15] ). In the first case, the slowed-down portions of the boundary layer in a region
of rising pressure are removed by suction through a slot (Fig. 2-52c) before they
can cause flow separation. Behind the suction slot, a new boundary layer is formed
that, again, can overcome a certain pressure rise. Separation may never take place if
the slots are suitably arranged. This principle of boundary-layer removal by suction
JA
20
o
z
2s°
2 2°
1.6
r
ac - 9
5.s °
27 °
ae
-1.7°
I
a.u
I
-7.5 °
0
9°
I
-12°
-02
0
I
I
01
02
CD
I
1
03
'
Fig ure 2-53 Po l ar cu rv es
foil with slat and flap.
of
an
AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 97
5
103cQ10
2
5
3_
10-2
2
5
2
t.
2
05
90-3
a2l
S
2
10-
10°
a
5
105
2
S
106
2
Rep U.C
s
107
2
s
108
V
Figure 2-54 Drag (friction) coefficients of flat plate in parallel flow with homogeneous suction;
cQ = (-v0)/U.. = suction coefficient; -u,, = constant suction velocity. Curves
1,
2, and 3
without suction. 1, Laminar; 2, transition laminar-turbulent; 3, fully turbulent; 4, most effective
suction.
was checked out for a circular cylinder by Prandtl as early as 1904 and has been
investigated by Schrenk [58] for wing profiles.
In the second case, suction is applied for the reduction of friction drag of wings
(see Goldstein [20] ). This is. accomplished if suction causes a downstream shift of
the laminar-turbulent transition point. For this purpose, it turned out to be more
favorable to apply areawise-distributed (continuous) suction, for example, through
porous walls rather than through slots. In this way the disturbances by the slots
were avoided, which could have led to premature transition. That the flow can be
kept laminar through suction may be seen from the fact that the friction layer
becomes thinner when suction is applied and, therefore, has less of a tendency to
turn turbulent. Also, the velocity profile of a laminar boundary layer with suction
has a shape, compared with that of a layer without suction, that makes transition to
turbulence less likely even when the boundary-layer thickness is equal in both cases.
Of particular interest is the drag law of the plate with homogeneous suction, as
given in Fig. 2-54, because it is characteristic for the drag savings gained through
suction-maintained laminar flow. In comparison, the drag law of the plate with a
turbulent boundary layer (without suction) is added as curve (3). The drag savings that
may actually be achieved cannot yet be derived. First, the limiting suction coefficient
must be known, which is necessary to keep the boundary layer laminar-even for large
Reynolds numbers. This minimum suction coefficient was determined as
CQcr = 1.2 - 10-4
up to the highest Reynolds numbers. This remarkably small value is included in Fig. 2-54
as "most favorable suction" (curve 4). The difference between curves 3 "turbulent"
and 4 "most favorable" suction represents the optimum drag savings. In the Reynolds
number range Re = 106 to 108, they amount to about 70-80% of the fully turbulent drag.
98 AERODYNAMICS OF THE WING
It should be understood, however, that this saving does not take into account
the power needed for the suction. Even when taking this power into account,
however, the drag savings are still considerable.
Ackeret et al. [2] were the first investigators to prove experimentally that it is
possible to hold the boundary layer laminar by suction. Some of their test results
on a wing profile are given in Fig. 2-55. This wing profile was provided with a large
number of slots. The considerable savings in drag, even including the blower power
needed for the suction, is obvious. The favorable theoretical results about drag
savings by maintaining laminar flow have been confirmed completely through
investigations of Jones and Head [20] on wings with porous surface.
Boundary layer with blowing Another very efficient means of influencing the
boundary layer is the tangential ejection of a thin jet at a separation point. This
method has been applied very successfully to wings with trailing-edge flaps. By
ejecting a thin jet at high speed at the nose of the deflected flap, flow separation
from the flap can be avoided and hence lift can be increased considerably. The
underlying physical principles are demonstrated in Fig. 2-56. At large deflections,
the effectiveness of the flap as a lift-producing element is markedly reduced by flow
separation (Fig. 2-56a). The lift of a wing with deflected flap does not reach at all
the value that is predicted by the theory of inviscid flow. Flow separation from the
flap and a resulting loss in lift may be avoided, however, by supplying the boundary
layer with sufficient momentum. This is accomplished by a thin jet of high speed,
tangential to the flap, introduced near the flap nose into the boundary layer (Fig.
2-56b). The lift gain that can be realized through blowing is shown in Fig. 2-56c as
T-MTC1 ?'ZT
Suction slots
0,8
1
J,
Zy-
6
Re = >2
With out suction
cL= 0 9
With suction
1
CL=0.16
03
cL =
V 'q
02
cL=0.Z3I
01
i
15
1.5
2
Re ------ +
3
S- 10
0
2
COp
.?
S
7 03
Figure 2-55 Reduction of drag coefficient of wing profiles by suction through slots, after
Pfenninger [2]. (a) Optimum drag coefficient of wing with suction vs. Reynolds number. (b)
Profile-drag polar.
AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 99
Potential
theoretical
pressure
distribution
Lift-gain through
blowing
Pressure distribution
for separated flow
Figuue 2.56 Flap wing with blowing at
the flap nose for increased maximum
lift. (a) Flap airfoil without blowing,
separated flow. (b) Flap airfoil with
blowing, attached flow.
(c) Pressure
distribution.
c
the difference between the two pressure distributions. The effect of blow jets and
jet flaps is discussed in more detail in Sec. 8-2-3. A synopsis of the increase of
maximum lift of wings through boundary-layer control has been written by
Schlichting [54]
.
Maintaining laminar flow through shaping Closely related to maintaining laminar
flow through suction is maintaining a laminar boundary layer through proper
shaping of the body. The goal is the same, namely, to reduce the friction drag by
shifting the transition point downstream. Doetsch [12] was the first to demonstrate
experimentally that considerable drag reductions can be obtained in the case of a
wing profile whose maximum thickness is sufficiently far downstream (laminar
profile). By shifting the maximum thickness downstream, the pressure minimum,
and thus the laminar-turbulent transition point of the boundary layer, is also shifted
downstream because, in general, the boundary layer remains laminar in the range of
decreasing pressure. Only after the pressure rises does the flow turn turbulent. These
conditions are shown in Fig. 2-57 by comparing a "normal wing" of a maximum
thickness position of 0.3c and a laminar profile with a maximum thickness position
of 0.45c. In the former case the pressure minimum lies at 0.1c, in the latter case at
0.65c. The drag diagram indicates that, in the Reynolds number range from 3 - 106
100 AERODYNAMICS OF THE WING
3
Z5
I
a
2`
9
-1
9
_L
55
10 6
8
2
Re-
u
c
3
4
5
6
8
101
-
v
t
NA CA 0009
0.3 C
t
NACA 66-009
0.45 c
12
Velocity maximum
H
NACA 66-009
NACA 0009
V
X
as
Q8
Figure 2-57 Drag coefficients and velocity distribution of laminar profile,
after [1]. (a) Drag coefficients: 1,
laminar; 2, fully turbulent; 3, transito tion laminar-turbulent. (b) Velocity
(pressure) distributions.
to 107, the drag of the laminar profile is only about one-half that of the normal
profile. The aerodynamic properties of such laminar profiles have been investigated
in much detail in the United States [1]. Practical application of laminar profiles is
impeded particularly by the extraordinarily high demand on surface smoothness
necessary to ensure that the conditions for maintaining laminar flow are
not lost with surface roughness. The studies of Wortmann [75] and Eppler
[14, 15] on the development of laminar profiles for glider planes should be mentioned.
AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 101
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104 AERODYNAMICS OF THE WING
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66. Theodorsen, T. and I. E. Garrick: General Potential Theory of Arbitrary Wing Sections,
NACA Rept. 452, 1933; 411, 1931. Gebelein, H.: Ing.-Arch., 9:214-240, 1938. Kochanowsky, W.: Jb. Lufo., 1:52-58, 1937, 1:82-89, 1938, 1:72-80, 1940. Mangler, W. and A.
Walz: Z. Angew. Math. Mech., 18:309-311, 1938. Wittich, H.: Jb. Lufo., 1:52-57, 1941.
67. Thwaites, B. (ed.): Uniform Inviscid and Viscous Flow Past Aerofoils, in "Incompressible
Aerodynamics-An Account of the Theory and Observation of the Steady Flow of
Incompressible Fluid Past Airfoils, Wings, and Other Bodies," pp. 112-205, Clarendon Press,
Oxford, 1960.
68. Truckenbrodt, E.: Die Berechnung der Profilforrn bei vorgegebener Geschwindigkeitsverteilung, Ing.-Arch., 19:365-377, 1951. Riegels, F.: Z. Angew. Math. Mech.,
24:273-276, 1944.
69. Truckenbrodt, E.: Die entscheidenden Erkenntnisse uber den Widerstand von Tragfliigeln,
Jb. WGLR, 54-66, 1966; Tech. Sci. Aer. Spat., 97-111, 1967. Riegels, F.: Jb. WGL, 44-55,
1952.
70. von Karman, T. and J. M. Burgers: General Aerodynamic Theory-Perfect Fluids, in W. F.
Durand (ed.), "Aerodynamic Theory-A General Review of Progress," div. E, Springer, Berlin,
1935, Dover, New York, 1963.
71. von Mises, R.: Zur Theorie des Tragflachenauftriebes, Z. Flug. Mot., 8:157-163, 1917;
11:68-73, 87-89, 1920. Blasius, H.: Z. Math. Phys., 58:90-110, 1910.
72. Wagner, H.: ITber die Entstehung des dynamischen Auftriebs von Tragfliigeln, Z. Angew.
Math. Mech., 5:17-35, 1925. Forsching, H. W.: "Grundlagen der Aeroelastik," pp. 149-373,
Springer, Berlin, 1974. Kiissner, H. G.: Lufo., 13:410-424, 1936.
73. Weissinger, J.: Theorie des Tragfliigels bei stationarer Bewegung in reibungslosen, inkompressiblen Medien, in S. Fliigge (ed.), "Handbuch der Physik, vol. VIII/2, Stromungsmechanik II,"
pp. 385-437, Springer, Berlin, 1963.
74. Woods, L. C.: "The Theory of Subsonic Plane Flow," (Cambridge Aeronautics Series, III), pp.
301-425, Cambridge University Press, Cambridge, 1961.
75. Wortmann, F. X.: Ein Beitrag zum Entwurf von Laminarprofilen fur Segelflugzeuge and
Hubschrauber, Z. Flugw., 3:333-345, 1955, 5:228-243, 1957. Speidel, L.: Z. Flugw.,
3:353-359, 1955. Stender, W.: Luftfahrt., 2:218-227, 1956.
76. Wortmann, F. X.: Progress in the Design of Low Drag Aerofoils, in G. V. Lachmann (ed.),
"Boundary Layer and Flow Control-Its Principles and Application," pp. 748-770,
Pergamon, Oxford, 1961.
77. Young, A. D. and H. B. Squire: A Review of Some Stalling Research-Appendix: Wing
Sections and Their Stalling Characteristics, ARC RM 2609, 1942/1951. Gault, D. E.: NACA
TN 3963, 1957. Goradia, S. H. and V. Lyman: J. Aircr., 11:528-536, 1974. Kao, H. C.: J.
Aircr., 11:177-180, 1974. McCullough, G. B. and D. E. Gault: NACA TN 2502, 1951.
CHAPTER
THREE
WINGS OF FINITE SPAN
IN INCOMPRESSIBLE FLOW
3-1 INTRODUCTION
For an airfoil of infinite span, the flow field is equal in all sections normal to the
airfoil lateral axis. This two-dimensional flow has been treated in detail by profile
theory in Chap. 2. For an airfoil of finite span as in Fig. 3-1, however, the flow is
three-dimensional. As in Chap. 2, incompressible flow is presupposed.
3-1-1 Wing Geometry
The wing of an aircraft can be described as a flat body of which one dimension
(thickness) is very small in relation to the other dimensions (span and chord). In
general, the wing has a plane of symmetry that coincides with the plane of
symmetry of the aircraft. The, geometric form of the wing is essentially determined
by the wing planform (taper and sweepback), the wing profile (thickness and
camber), the twist, and the inclination or dihedral of the left and right halves of the
wing with respect to each other (V form) (see Fig. 3-1). In what follows, the
geometric parameters that are significant in connection with the aerodynamic
characteristics of a lifting wing will be discussed.
For the description of wing geometry, a coordinate system in accordance with
Fig. 3-1 that is fixed in the wing will be established with axes as follows:
x axis, wing longitudinal axis, positive to the rear
y axis, wing lateral axis, positive to the right when viewed in flight direction, and
perpendicular to the plane of symmetry of the wing
z axis, wing vertical axis, positive in the upward direction, perpendicular to the xy
plane
105
106 AERODYNAMICS OF THE WING
b
Figure 3-1 Illustration of wing geometry. (a)
c
X
Planform, xy plane. (b) Dihedral (V form), yz
plane. (c) Profile, twist, xz plane.
It is expedient to select the position of the origin of the coordinates as suitable
for each case. Frequently it is advisable to place the origin at the intersection of the
leading edge with the inner or root section of the wing (Fig. 3-1), or at the
geometric neutral point [Eq. (3-7)]. The wing planform is given in the xy plane; the
twist, as well as the profile, in the xz plane; and the dihedral in the yz plane.
The largest dimension in the direction of the lateral axis (y axis) is called the.
span, which will be designated by b = 2s, where s represents the half span.
Frequently the coordinates will be made dimensionless by reference to the half-span
s, and abbreviated notations
(3-la)
(3-1 b)
(3-1c)
are here introduced.
The dimension in the direction of the -longitudinal axis (x axis) will be
designated as the wing chord c(y), dependent on the lateral coordinate y. The wing
chord of the root or inner section of the wing (y = 0) will be designated by Cr, and
the corresponding dimension for the tip or outer section by ct. In Fig. 3-2, the
geometric dimensions are illustrated for a trapezoidal, a triangular, and an elliptic
planform.
For a wing of trapezoidal planforr (Fig. 3-2a), an important geometric
parameter is the wing taper, which is given by the ratio of the tip chord to the root
chord:
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 107
A special case of the trapezoidal wing is the triangular wing with a straight trailing
edge, also designated as a delta wing (Fig. 3-2b).
The wing area A (reference area) is understood to be the projection of the wing
on the xy plane. For a variable wing chord, the area is obtained by integration of
the wing chord distribution c(y) over the span b = 2s; that is,
3
A
fc(y)dy
-3
Quarter-point line
NZ3
.\
V4 CU
a
b
b =2S
Quarter-point line
C
b=2s
Figure 3-2 Geometric designations of
wings of various planforms. (a) Swept-back
wing. (b) Delta wing. (c) Elliptic wing.
108 AERODYNAMICS OF THE WING
From the wing span b and the wing area A, there is obtained, as a measure for the
wing fineness (slenderness) in span direction, the aspect ratio
!1=
b2
(3-a)
b
Cm
(3 4b)
As mean chord and reference wing chord, especially for the introduction of
dimensionless aerodynamic coefficients, the quantities
A
Cm
(3-Sa)
b
s
C
fc2(y)ciy
"`
(3-5b)
A
fS
are used, where the ratio
1. For the trapezoidal planform, it may be easily
demonstrated that the reference chord c. is equal to the local chord at the position
of the center of gravity of the half wing; that is, cP, = c(yc) (Fig. 3-2a and b). The
sweepback of a wing is understood to be the displacement of individual wing cross
sections in the longitudinal direction (x direction). Representing the position of a
wing planform reference line by x(y), the local sweepback angle of this line is
tan92(y) =
(3-6)
If x(y) represents the connecting line of points of equal percentage rearward
position, measured from the leading edge at the y section under consideration, then
this fact is designated by giving the percentage number as an index of the value x.
Accordingly, the position of the quarter-chord line is designated by x25(y). For the
sake of simplicity, the index will be omitted in the case of the sweepback angle of
the quarter-chord-point line. For aerodynamic considerations, furthermore, the
geometric neutral point plays a special role. Its coordinates are given by
8
XV95 = A
,lc(y)
x25 (y) dy
y:V25 ` 0
(3-7)
For a symmetric wing planform, the geometric neutral point may be demonstrated
to be the center of gravity of the entire wing area, whose quarter-chord-point line is
overlaid by a weight distribution that is proportional to the local wing chord. The
rearward distance of the geometric neutral point of a wing with a swept straight
quarter-chord-point line is equal to the rearward distance of the quarter-chord point
of the wing section at the planform center of gravity of the half-wing. Since, for a
trapezoidal wing, the wing chord at the center of gravity of the half-wing is equal to
the reference chord c1,, the geometric neutral point for this wing lies at the cu/4
point (see Fig. 3-2a and b).
Of particular importance is the delta wing, a triangular wing with a straight
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 109
trailing edge (Fig. 3-2b). For the geometric magnitudes of this wing, especially
simple formulas are obtained:
_
b
b
Crrt-2 Cr
3
Cu
4
tancp
Cm
3
Cr
-N25 - 2
(3-8)
For a wing of elliptic planform as in Fig. 3-2c, the geometric quantities become
A=
7r
bcr
!1 =
4 b
r
C'm
Cr
7r
=
= 0.785
c
Cr
-
3 7r
= 0.848
(3-9)
4
4
A further geometric magnitude related to the wing planform is the flap
(control-surface) chord cf(y). The flap-chord ratio is defined as the ratio of flap
chord (control-surface chord) to wing chord:
Xf =
Cf(Y)
(3-10)
(Y)
For the description of the whole wing, data on the relative positions of the profile
sections are required at various stations in span direction. They are required in
addition to the knowledge of wing planforms and wing profiles. The relative
displacement in longitudinal direction is specified by the sweepback, the displacement in the direction of the vertical axis by the dihedral, and the rotation of the
profiles against each other by the twist.
In what follows, the geometric twist e(y) is defined as the angle of the profile
chord with the wing-fixed xy plane (Fig. 3-3).* For aerodynamic reasons, in most
cases the twist angle is larger on the outside than on the inside. The dihedral
determines the inclination of the left and the right wing-halves with respect to the
*In addition to the geometric twist, there is an aerodynamic twist, characterized by a twist
angle measured against the profile zero-lift direction instead of the profile chord.
2
X
Figure 3-3 Illustration of geometric V;gist.
110 AERODYNAMICS OF THE WING
xy plane. Let z(s)(x, y) be the coordinates of the wing skeleton surface. Then the
local V form at station x, y is given by
tanv (x, j) =
8z(a)(x, y)
ay
(3-11)
The partial differentiation is done by holding x constant. If the wing is twisted, it
must be specified in addition at which station xp(y) the angle v is to be measured.
According to Multhopp [61], the aerodynamically effective dihedral has to be taken
approximately at the three-quarter point xp = x75 .
3-1-2 Shapes of Actual Wings
To convey a concept of the various wing shapes that have actually been used in
airplanes, the profile thickness ratio 5 = t/c, the aspect ratio A = b2 fA, and the
sweepback angle of the leading edge pf of some airplanes are plotted in Fig. 3-4 against
the flight Mach number. The plots show a clear trend of profile thickness and aspect
ratio in the transition from subsonic to supersonic airplanes.
The profile thickness ratio decreases sharply with increasing Mach number,
reaching values of tic = 0.04 for supersonic airplanes. The aspect ratios are
particularly large in the subsonic range for long-distance airplanes but considerably
smaller for maneuverable fighter planes. In the supersonic range, the implementation
of larger aspect ratios is no longer required for aerodynamic reasons. In this range,
therefore, design considerations have led to aspect ratios as small as A = 2. The
sweepback angle is close to zero at low Mach numbers but increases to pf -- 45,0 at
high subsonic speeds. In the supersonic range, airplanes with both relatively large
sweepback 6pf 60°) and small sweepback ('pf ~ 30°) are found. Truckenbrodt
[86] has shown to what extent the geometric wing data of Fig. 34 have been
determined by a decisive understanding of the drag of wings.
3-1-3 Lift Distribution
The lift distribution over the span is defined in analogy to Eq. (2-9b) as
dL = cl(y)c(y)q dy
(3-12)
Here the local lift coefficient has been introduced in analogy to Eq. (2-10) as
cl(y) ~ -cn(y).* The lift distribution of a wing in symmetric incident flow is shown
in Fig. 3-5b. Finally, in Fig. 3-6 there is also shown the distribution of measured
local lift coefficients cl over the span of a rectangular wing at various angles of
attack.
By integrating Eq. (3-12) over the span, the total lift L and further, with Eq.
(1-21), the lift coefficient are determined as
*To distinguish between the coefficients of the total forces and moments, the indices of
which are always expressed in capital letters, lowercase letters will'be used for the indices of the
coefficients of local forces and moments.
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 11 1
s
L
1
CL = Aq = A
ct(Y)c(y) dy
(3.13)
-s
Only single wings will be treated in this book. Wing systems such as, for
example, biplane and tandem arrangements or ring wings (tube-shaped cylindrical
surfaces) will not be considered.
reports, more recent results and the understanding of the
In progress
aerodynamics of the wing are presented for certain time periods, among others, by
Schlichting [72, 74], Sears [781, Weissinger [97], Gersten [20], Blenk [7], Ashley
et al. [21, Kuchemann [491, and Hummel [35]. The very comprehensive
compilation of experimental data on the aerodynamics of lift of wings of Hoerner
and Borst [311 must also be mentioned.
(12
.40
`I-
0.1
\
I
i
I
i
I
I
I
I
\
7
.
I
11
,
4
1
'
1
1
Z5
b)
.
I
'I
+
0
750
t/c = 0.15
}
1
c)
L
C.
05
L
Macr
F
1.0
1.5
2.0
25
Ma
3.0
Figure 3-4 Most important geometric wing
data of actual airplanes vs. Mach number.
Evolution from subsonic to supersonic airplanes. (a) Profile thickness ratio 6 = t1 c.
(b) Aspect ratio A. (c) Sweepback angle of
wing leading edge of Macr = drag-critical
Mach number (see Sec. 4-3-4).
112 AERODYNAMICS OF THE WING
Figure 3-5 Illustration of lift distribution of
wings. (a) Geometric designations. (b) Lift
distribution over span.
3-2 WING THEORY BY THE METHOD
OF VORTEX DISTRIBUTION
3-2-1 Fundamentals of Prandtl Wing Theory
The creation of lift of a wing is tied to the existence of a lifting (bound) vortex
within the wing (Fig. 3-7). This fact has been explained in Sec. 2-2 by means of
Fig. 2-4. The position of the bound vortex on the wing planform is described in
Sec. 2-3-2 for the inclined flat plate. Accordingly, it is expedient to position the
vortex on the one-quarter point of the local wing chord. An unswept wing in
symmetric incident flow is therefore represented by a bound vortex line normal to
the incident flow direction.
Profile 60 420
C
'4
° -0.4°
0.60
5.4°
O.B9
17.1 °
9.21
°
0.2
--
1
0.4
0.6
0.9
N
1.0
Figure 3-6 Distribution of local lift coefficients
for a rectangular wing of aspect ratio A = 5 and
profile Go 420. Reynolds number Re = 4.2 101; Mach number Ma = 0.12.
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 113
Figure 3.7 Vortex system of a
wing of finite span.
Since the pressure differences between lower- and upper-wing surfaces decrease
to zero toward the wing tips, producing a circulation around the wing, the flow
field of a wing of finite span is three-dimensional. This pressure equalization at the
wing tips, shown schematically in Fig. 3-8b, causes an inward deflection of the
streamlines above the wing and an outward deflection below the wing (Fig. 3-8a). In
this way, streamlines that converge behind the wing have different directions. They
form a so-called surface of discontinuity with inward flow on the upper surface,
outward flow on the lower surface (Fig. 3-8c). The discontinuity surface tends to
roll up farther downstream (Fig. 3-8d), forming two distinct vortices of opposite
C
d
e
f
r
F
°)
r
Figure 3-8 Evolution of the free vortices behind
a wing of finite span.
114 AERODYNAMICS OF THE WING
sense of rotation. Their axes coincide approximately with the direction of the
incident flow (Fig. 3-8e and f). These two vortices have a circulation strength P.
Thus, behind the wing there are two so-called free vortices that originate at the
wing tips (Fig. 3-7). Far downstream, these two vortices are connected by the
starting vortex, the evolution of which was explained in Sec. 2-2-2. The bound
vortex in the wing, the two free vortices, originating at the wing tips, and the
starting vortex together form a closed vortex line in agreement with the Helmholtz
vortex theorem. The vortices produce additional velocities in the vicinity of the
wing, the so-called induced velocities. They are, as a result of the sense of rotation
of the vortices, directed downward behind the wing. They play an important role in
the theory of lift.
The
starting vortex need not be taken into account in steady flow for
treatment of the flow field in the vicinity of the wing. This is understandable when
it is realized that the wing has already moved over a long distance from its start
from rest. In this case the vortex system consists only of the bound vortex in the
wing and the two infinitely long, free vortices. These form again an infinitely long
vortex line shaped like a horseshoe, open in the downstream direction. This vortex
is called a horseshoe vortex.
The very simplified vortex model of Fig. 3-7, having one bound vortex of
constant circulation, is still insufficient for quantitative determination of the
aerodynamics of the wing of finite span. A further refinement of the two simple
free vortices originating at the wing is necessary. The above-mentioned pressure
equalization at the wing tips causes the lift, and consequently the circulation, to be
reduced more near the wing tips than in the center section of the wing. At the very
wing tips even complete pressure equalization occurs between upper and lower
surfaces. The circulation drops to zero. The actual circulation distribution is. similar
to that shown in Fig. 3-9; it varies with the span coordinate, T =r(y). The variable
circulation distribution T (y) in Fig. 3-9 can be thought of as being replaced by a
step
distribution. At each step a free vortex of strength d T is shed in the
downstream direction. In the limiting case of refining the steps to a continuous
circulation distribution, the free vortices assume an areal distribution (vortex sheet).
A strip of this vortex sheet of width dy has the circulation strength d .P =
(dr/dy) dy. Thus the slope of the circulation distribution T(y) of the bound
vortices determines the distribution of the vortex strength in the free vortex
sheet.
It was Prandtl [69] who for the first time gave quantitative information on the
three-dimensional flow processes about lifting wings based on the above discussed
mental picture. Earlier, Lanchester had investigated this problem qualitatively (see
von Karman [90] ).
Lift and induced drag From the Kutta-Joukowsky theorem [see Eq. (2-15)], the
lift dL of a wing section of width dy and its circulation T (y) are related by
dL = oVT(y) dy
(3-14)
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 115
tiT
y
Figure 3-9 Wing with variable circulation distribution over the span.
The total lift is obtained by integration as
s
L = , V F (y) dy
(3-15)
fs
As the most important consequence of the formation of free vortices, the airfoil of
finite span undergoes a drag even in frictionless flow (induced drag), contrary to the
airfoil of infinite span. Physically, the induced drag can be explained by the roll-up of
the discontinuity sheet into the two free vortices: During every time increment a
portion of the two free vortices has to be newly formed. To this end, work must be
done continually; this work appears as the kinetic energy of the vortex plaits. The
equivalent of this work is expended in overcoming the drag during forward motion
of the wing.
On the other hand, the formation of induced drag may also be understood by
means of the Kutta-Joukowsky theorem as follows: The downstream-drifting free
vortices produce a downwash velocity wi behind and at the wing, after Biot-Savart.
At the wing the incident flow velocity of the wing profile is therefore the resultant
of the incident flow velocity V and this induced downwash velocity wj.
Accordingly, the resultant incident flow direction at the wing is inclined downward
by the angle al against the undisturbed incident flow direction, with
(3-16)
In general, wi << V and hence aj
sin aZ ~ tan ai.
116 AERODYNAMICS OF THE WING
From the Kutta-Joukowsky theorem (Sec. 2-2-1), the resultant dR of the
aerodynamic forces at the wing cross section y (Fig. 3-10) stands normal to the
resultant incident flow direction. Hence, normal to the undisturbed flow direction
there is a lift component dL = dR cos a1 -- dR and parallel to the undisturbed flow
direction a drag component dD1= dR sin ai Maj. The latter is the induced
drag of the wing cross section y, which, with Eq. (3-16), becomes
dDi = a1dL = dL V
(3-17)
Hence, the total induced drag is obtained through integration over the wing span
from y = -s toy = + s, and by noting Eq. (3-14), as
Di = fr)wi)dY
(3-18)
where wr(y) is the distribution of the induced downwash velocity that is variable in
the general case.
The distribution of the induced downwash velocity along the span is obtained
by applying the Biot-Savart law to the semi-infinitely long free vortex behind the
wing. The contribution of the vortex strip dy' at station y' to the downwash
velocity at the location of the lifting line y (Fig. 3-9) is
dwi (y, y') =
1
dl'(y')
47r y - y'
dl' dy'
4n dy' y - y'
=
1
with (y - y') being the distance of the point under consideration (control point) y
from the location y' of the free vortex line. From this, the induced velocity at the wing
is found by integration over the area of the free vortices as*
wt (y) =
4 or
y J dyz'
- .!
(3-19)
z
From this equation, the induced downwash velocity wi at the location of the lifting
line can be computed when the circulation distribution F(y) is known. Finally, the
induced drag can be determined from Eq. (3-18).
It should be mentioned here that the induced downwash velocity wm very far
behind the wing has twice the value of the downwash velocity wi at the wing from
*At station y' = y, the integrand has a singularity. The analysis shows that the integral has
to be evaluated through the Cauchy principal value. Hence, the range y - e < y' < y + e must
be excluded during integration and the limit operation
y-E
lim
must be conducted.
I
s
f ... d y" -r f ... d y'
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 117
- - -Zero-lift direction
v. Direction of undisturbed incident flow
---.-Effective incident flow direction
Figure 3-10 Evolution of the induced drag.
Eq. (3-19). This is obvious from the fact that far downstream the free vortices can
be taken as being infinitely long vortex lines, leading to
w. (y) = -2 iv; (y)
(3-20)
The velocity w. is taken as positive in the direction of the positive z axis (see
Fig. 3-21).
Prandtl's integral equation of the circulation distribution The above considerations
will now be applied to the derivation of an equation for the determination of the
spanwise circulation distribution for a given wing of finite span.
The change of the incident flow direction that results from the downwash
velocity induced by the free vortices was explained in Fig. 3-10. This change of
flow incidence, at equal geometric angles of attack a, is responsible for the reduced
lift at the cross section y of a finite-span wing in comparison with the lift at the
same cross section of an infinitely long wing.
For a span element dy of a finite-span wing, Eq. (3-12) yields for the lift:
dL = cl(y)
V 2 c(y) dy
(3-21a)
V2
(3-21b)
2
=
czoo ae(Y)
2
c(Y) dy
Here c(y) is the wing chord at station y (Fig. 3-9) and cz(y) = cl.cxe(y) is the local
lift coefficient of the area element dA = c(y) dy; cxe(y) is termed the effective angle
of attack (Fig. 3-10) and cl. = (dcz/da). is called the lift slope for the airfoil of
infinite span. The latter value is close to 21r, from the theory of thin profiles (see
Chap. 2). For the inclined flat plate, cl. is exactly equal to 27T. Equation (3-21) is
based on the concept that a profile cross section of a wing of finite span behaves
like that of a wing of infinite span (plane flow) at an angle of incidence ae
The geometric angle of attack a(y), measured from the zero-lift position, the
effective angle ae(y), and the induced angle ai(y) [Eq. (3-16)] are related by
.
118 AERODYNAMICS OF THE WING
(3-22)
a(y)=cz (y)+a1( )
as shown in Fig. 3-10. The effective angle of attack ae is obtained from Eq. (3-21b)
with the help of Eq. (3-14), and the induced angle of attack from Eq. (3.19) with
ai = wt/ V as
Cie =
2T(y)
(3-23a)
Vc(y)ciC*
S
1
PLC (y) -
(' d F
d y'
(3-23b)
4nV J dy' y - y
-s
Introducing Eq. (3-23) into Eq. (3-22) yields the following basic equation for the
determination of the circulation distribution:
«(1')
-
2T (y)
Vc(y)c
S
+ i
4n V
dr
d y'
dy
y -y
-s
(3-24)
This is Prandtl's integral equation for the circulation distribution of a wing of finite
span as first published by Prandtl in 1918 [691. It is a linear integral equation for
the circulation distribution P(y), where I' depends linearly on the angle of attack
a. The profile coefficient cl. is known from profile theory (Chap. 2).*
With given wing geometry [chord distribution c(y) and angle-of-attack distribution a(y)], the circulation distribution can be determined from Eq. (3-24). This is
the so-called direct problem of wing theory. Conversely, if the circulation
distribution '(y) is known, either the angle-of-attack distribution (twist angle) a(y)
can be computed from Eq. (3-24) when the chord distribution c(y) is given, or the
chord distribution c(y) when the angle-of-attack distribution a(y) is given. This is
the so-called indirect problem of wing theory. In either case, from the circulation
distribution I'(y) the lift is obtained from Eq. (3-15) and the induced drag from
Eq. (3-18).
From a mathematical viewpoint, the direct problem is considerably more
difficult than the indirect problem, because in the former case an integral equation
has to be solved while in the latter case only a quadrature has to be performed.
Elliptic circulation distribution A particularly simple solution of Eq. (3-24) that is
of great practical importance is found for the elliptic circulation distribution along
the span. In this case the circulation becomes
T (Y) = ro
1 - (S )2
(3-25)
where ro is the circulation at the wing center y = 0 (Fig. 3-11). From Eq. (3-15),
the lift becomes
*If the profile coefficient
cl. = cLa,
27r.
cj. is known over
the span,
it may be replaced by
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 119
L = 4 obVP0
(3-26)
The induced downwash-velocity is obtained from Eq. (3-19). Execution of the
integral yields for points within the span, lyl < b/2,
Wi(J)
To
(3 -27a)
=2b
= const
ai (y) = L
;rgb2
(3-27b)
This remarkable result shows that, for elliptic circulation distribution, the induced
downwash velocity w1, and consequently the induced angle of attack aj, are
constant over the span (Fig. 3-11).
By introducing Eqs. (3-25) and (3-27a) into Eq. (3-18), the induced drag is
obtained with To from Eq. (3-26) as
DZ
=
(3 -28a)
LZ
_ irgb2
(3-28b)
Here, q = (p/2) V2 is the dynamic pressure resulting from the velocity V. The
induced drag is proportional to the square of the lift and inversely proportional to
the dynamic pressure and the square of the span. Comparison of Eqs. (3-28b) and
(3-27b) confirms the relationship DZ = a1L, given in Eq. (3-17).
The geometry of the corresponding wing is obtained in a particularly simple
way when starting from the wing without twist, a(y) = a = const. Since, from Eq.
(3-27b), the induced angle of attack an(y) = const, Eq. (3-22) shows that the
effective angle of attack along the span must also be constant: ae(y) = const.
Figure 3-11 Elliptic circulation distribution
with the corresponding elliptic wing planform and the constant induced downwash
WI = const
velocity over the wing span.
120 AERODYNAMICS OF THE WING
From Eqs. (3-23a) and (3-25), it follows that the chord is distributed elliptically
over the span:
2
c(y) = c, ri
s
(3-29)
The elliptic wing planform is shown in Fig. 3-11.x` Thus it has been demonstrated
that an elliptic wing without twist has an elliptic circulation distribution. From Eq.
(3-21), it also has a constant local lift coefficient c1(y) over the span.
Coefficients Finally, the most important results for the induced angle of attack
[Eq. (3-27b)] and for the induced drag [Eq. (3-28b)] will also be expressed
through the dimensionless coefficients of lift and induced drag. They are defined as
follows :
L = cLgA
(3-30a)
DI = cDigA
(3-30b)
with A being the wing planform area. Consequently, Eqs. (3-27b) and (3-28b) yield
(3-3 la)
=
C
(3-31 b)
Here A = b2 /A is the aspect ratio of the wing from Eq. (3.9). The important result
for the coefficient of the induced drag of Eq. (3-31b) is compared in Fig. 3-12 with
test results for a wing of aspect ratio A. = 5. The theoretical curve for the induced
drag agrees quite well over the whole CL range with the polar curve of the measured
data. The difference between the two curves is about constant over the whole CL
range. It is caused by the effect of friction that has been neglected in the above
theory. Figure 3-12 suggests splitting up the drag coefficient into a component that
is nearly independent of the lift coefficient and a component that is dependent on
the lift coefficient. The former is called the coefficient of profile drag CDp, the
latter the coefficient of induced drag CD1. They are related by
CD = CDp + CDi
(3-32a)
2
= CDp + 7111
(3-32b)
For the geometric angle of attack, Eqs. (3-22) and (3-3 la) yield
a= CYe
+
c
7tl1
CL
CL
+7r1
(3-33a)
(3.33 b)
'The elliptic wing consists of two ellipse halves, the large axis of which is the c/4 line.
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 121
1#
1,2
cps
=20°
4=5
CD
22.x°
1,0
Re=2. 7. 106
0.8
87°
N,4C4 24'12
02
0
-0.2
-041
004
0,08
0.19
CD -
016
a20
024
Figure 3-12 Measured polar curve of a wing of
aspect ratio A = 5 and theoretical curve for the
induced drag, CDi = cL/rrd .
From Eq. (3-21), ae = CL /CL 00 because the constant local lift coefficient c1(y) and
the total lift coefficient CL are equal in this case. The latter equation allows one to
determine the lift slope of the wing of finite span as a function of the aspect ratio.
From Eq. (3-33b) it follows:
dCL
_
CL
(3-34a)
1+CL-
da
7r A
CL
(3-34b)
A +2
CL-
with cL = dcL/da and cL00 = 21r. Equation (3-34b) expresses the degree of
reduction of lift slope and consequently also of lift because of the finite aspect
ratio when the angles of attack are equal. In Fig. 3-13 this ratio of lift slopes is
presented as a function of the aspect ratio.
As will be shown later in more detail, the formulas for induced drag and lift
slope found here for the elliptic wing are valid for other wing shapes in good
approximation. This is true particularly for the rectangular wing, as shown by Betz
[5] ; see Figs. 3-32 and 3-57.
.
Prandtl's transformation formulas The above-derived results on the effect of aspect
ratio on lift and drag have been checked experimentally by Betz and Wieselsberger
[99]. For comparison of the polar curves of two wings of aspect ratios Al and :'12
at equal angles of attack, Eq. (3-32b) with CDP2 = CDpI yields
CD2 = CD1
+
d
IT
1
!3.2
-
1
<11)
(3-35)
122 AERODYNAMICS OF THE WING
Re
t
a
r
1
8
6'
A
10
12
Figure 3-13 Ratio of the lift slope of wings
of finite and infinite aspect ratios vs. aspect
ratio, cL°, = 27r.
N.
In Fig. 3-14a the measured polar curves are plotted for a number of rectangular
wings with aspect ratios :11 = 1, 2, ... , 7. Figure 3-14b shows the result of the
transformation of these polars to the aspect ratio A2 = 5 from Eq. (3-35). The
transformed curves fall well on one curve, confirming experimentally the validity of
Eq. (3-35). In Fig. 3-14b the theoretical polar curve of the reduced drag for A. = 5
is also included. On the other hand, comparison of the lift curves CL (a) of two
wings of aspect ratios Al and r12 of equal lift coefficient yields, with Eq. (3-33b),
OZ-2 = at
+ L (1
A2
- 1)i
(3-36)
1
11
1
1 L44 .1 1
.
A=S
°
o
+CP
110
0
i
---l1=9
H
!
°
2
_
ace
212
0 g6
CD - -
a20
Figure 3-14 Demonstration of the experimental verification of the transformation formula for
the drag, from [991. (a) Measured polars for rectangular wings of aspect ratios A = 1-7. (b)
Polar curves transformed to A1= 5 and comparison with the theory of induced drag. Eq. (3-35).
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 123
For the wings of Fig. 3-14, the lift curves were converted to the aspect ratio
A,2= 5. Again, the converted curves fall together, confirming experimentally the
validity of Eq. (3-36).
The two equations (3-35) and (3-36) can therefore
for the
transformation of measured drag polars CD(CL) and lift curves CL (a) at aspect ratio
be
used
!11 to those of a wing with a different aspect ratio 112 if both wings have the same
profile. These equations are called, therefore, transformation formulas of the wing of
finite span.
3-2-2 Integral Equation for Circulation
Distribution from Wing Theory
Vortex system of the lifting surface To simplify the problem, it was assumed in
Sec. 3-2-1 that the circulation representing the wing was concentrated on one line
(lifting-line theory); see Fig. 3-7. This concept is a fairly good approximation for a
real wing only when its chord is much smaller than its span (wing of large aspect
ratio). When the chord is no longer much smaller than the span, it is necessary to
replace the concept of a lifting line by that of a distribution of lifting vortices over
the wing chord. Such a continuous vortex distribution over the wing chord was the
basis for the skeleton theory (Sec. 2-4-2). In the preceding section, the free vortices
were assumed to be distributed on the surface. By applying this concept of a
continuous circulation distribution logically to the wing of finite span, a vortex
distribution on the surface results that varies in chord and span direction (lifting
surface). An outline of this lifting-surface theory will now be derived. This theory is
of practical importance particularly for wings of small aspect ratio, for swept-back
and delta wings, and for yawed wings. This vortex distribution on the surface can
be taken to be a distribution of singularities in the sense of Sec. 2-4-2. During the
further development of wing theory, instead of vortex distributions, dipole
distributions will be used occasionally; see, for example, Prandtl [69 (1936)].
After the fundamental publication of Prandtl on wing theory using vortex
distributions, Blenk [69] further developed this theory by extending the twodimensional Birnbaum-Ackermann theory, Chap. 2 (8], to three dimensions.
The distribution of vortex strength over a given surface can be accomplished in
various ways. Let the wing surface have an arbitrary shape, and let a rectangular
wing-fixed coordinate system be chosen whose y axis is normal to the incident flow
direction.
A first possible approach to the replacement of the wing by a vortex
distribution is to cover this surface with two areal vortex distributions k,(x, y) and
ky(x, y), as in Fig. 3-15. The former distribution consists of vortex lines parallel to
the x axis, the latter of those parallel to the y axis. The ky vortices are of the kind
that was previously applied to the two-dimensional wing theory (see Fig. 2-20); the
kX vortices, however, resemble the free vortices in the vortex sheet behind the wing
(see Fig.. 3-9). Only the ky vortices contribute to the lift of the wing when the
incident flow is in the x direction. The vortex distributions kx(x, y) and ky(x, y)
124 AERODYNAMICS OF THE WING
Direction of incident flow
Figure 3-15 Wing with areal vortex distribution.
kx = vortex density of vortex lines in the x
direction, ky = vortex density of vortex lines in
the y direction.
cannot be chosen arbitrarily; rather, they must produce velocities induced by the
vortex sheet that satisfy the condition of irrotationality au/ay - av/ax = 0.
According to Eq. (246a), in the vicinity of the vortex sheet (z - 0) the
perturbation velocities are
u=+-k. v=+I,kx
(3-37)
where the upper sign is valid above, the lower sign below the vortex sheet. Hence
akx+aky
ax
ay
This relationship is called the condition of source-free vortex distribution.
The connection between circulation and rotation (Stokes's theorem) yields
- wy, which is another formulation of the spatial vortex
kx wx and ky
conservation law.
A second possible way to represent a wing by a vortex distribution consists, as
suggested by Glauert [231, of replacing the wing by so-called elementary wings of
infinitesimal span dy and of chord c(y) (Fig. 3-16). Each elementary wing occupies
its special location within the wing boundaries as defined by the wing geometry.
The vortex system of each elementary wing consists of a number of vortex lines,
one behind the other, parallel to the y axis, which is equivalent to a series
arrangement of horseshoe vortices as introduced in Sec. 3-2-1. This representation
was given for arbitrary wing planforms by Truckenbrodt [841, among others. In
Fig. 3-17, this concept is again demonstrated by the example of a yawed swept-back
wing. Note that the free vortices of the individual horseshoe vortices have been
drawn separately in this picture, but only for clarity; actually, all of them are
located on two parallel lines of distance dy. The circulation distribution density of
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 125
Direction of incident flow
I
dy
0
Figure 3-16 Substitution of a lifting wing by
X
an elementary wing of span dy and chord
b
c(y).
the elementary wing in direction of the chord (x direction) is k(x) per unit length.
In the terminology of Fig. 3-15, k corresponds to ky of this figure. From Eq.
(2-44), it follows that the circulation of a surface element of the elementary wing
with span dy and chord dx is
clI'(x, y) = Ic (x, y) cl x
(3-38)
and the total circulation of the bound vortex of the elementary wing at the wing
section y becomes
Xr
I'(y) = fkJ x
(3-39)
Xf
where xf(y) and x,.(y) designate the x coordinates of the front and rear edges of the
section, respectively. The same circulation is found in the two free vortices
originating at the trailing edge of the elementary wing.
1
Elementary wing
d
Figure 3.17 Vortex system of a
b=2s
l!
yawed wing, from 184].
126 AERODYNAMICS OF THE WING
Within the framework of linear wing theory, that is, limitation to small profile
camber of the individual wing sections and to small angles of attack, it can be
assumed that bound and free vortices of all elementary wings lie in the same plane
(xy plane). This assumption was also made for the profile theory in Sec. 2-4-2.
Equation for the determination of the circulation distribution To establish an
equation for the computation of the circulation distribution, an expression first
must be developed for the requirement that the lifting surface carrying the vortices
is a stream surface, that is, that the normal component of the resultant velocity is
equal to zero on this surface. This is the so-called kinematic flow condition. In Fig.
3-18 a wing cross section y of the lifting surface (skeleton surface) z(S)(x, y) = z(x,
y) is sketched. It is located in a flow field of incident flow velocity U. that forms
the geometric angle of attack ag(y) = aF + £(y) with the chord.* Here aF is the
angle of attack, measured from the x axis, and c(y) is the twist angle.
The kinematic flow condition becomes, in analogy to Eq. (249),
_
u
I-XF
1
az(x, y)
ax
+ u' (x, y) = 0
(3-40)
J
where w(x, y) is the velocity in the z direction induced by the total vortex system
at the point x, y of the xy plane (w > 0 in the direction of the positive z axis). The
brackets contain the term describing the angle between the incident flow direction
and the skeleton tangent. Equation (3-40) must be satisfied in all points x, y of the
lifting surface.
Furthermore, as a next step, the induced velocity w(x, y) on the lifting surface
must be determined from the given vortex distribution k(x, y). To simplify the
problem, the induced velocity is computed, however, at the projection of the lifting
surface on the xy plane that is identical with the vortex sheet. The induced velocity
w(x, y) at an arbitrary point of the xy plane is obtained by first determining the
contribution of one horseshoe vortex of one elementary wing (Fig. 3-19). The total
induced velocity w(x, y) is then the result of integrating first over one elementary
wing in the x direction and consecutively in the y direction over the total number
*Contrary to Sec. 3-2-1, the incident flow velocity is designated now by U,o instead of V.
Section y
Zero-lift direction
aF ,Bx
Skeleton line z(s) = z(x)
Figure 3-18 Illustration of the
kinematic flow condition of wing
theory.
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 127
H
Bound vortex element
dI'-k (z; y)d.s'
Figure 3-19 Explanation of the determination of
the induced velocity w(x, y) of the horseshoe
vortex of an elementary wing.
of elementary wings. Execution of this integration yields the following result, as
shown in detail in [84] :
8
w(x, y)
=-
2
Em
47t E-+o
G(x,
6
y; y) -
r G(x' Y,
j
J,) dy'
(y - y 'l)2
(3-41)*
-s
with the kernel function
xrV)
G(x, y; y') = f.k(xi, y') (1
x - x'
-}-
,J
}1 (x
xf(y)
dx'
(3-42a)
- X')2 - (y - y')2
X
G (x, y; y) = 2 f k (x', y) d x'
(3-42b)
Xf(y)
For the derivation of Eqs. (3-41) and (3-42), th-. Biot-Savart theorem must be
applied in such a way that the point in which the induced velocity w(x, y) is to be
computed is first positioned outside of the vortex sheet (z * 0). It is then shifted
into the vortex sheet (z -+ 0). On the right-hand side of Eq. (3-4 1), the first term
represents the self-induction of the elementary wing in the section y' = y
(downwash), whereas the second term represents the external induction of all other
elementary wings in the sections y' = y (upwash). Finally, introducing Eq. (3-41)
into the kinematic flow condition Eq. (3-40) yields
U'° I aF
2x
47c
1--G(x,y;y)
o
a
li
-s
(y - y')'
dy
(3-43)
G(x, y; y') is related to k(x, y) through Eq. (3-42). Equation (3-43) is an integral
equation for the circulation distribution k(x, y) of the lifting surface in which the
angle of attack aF and the wing shape z(x, y) are given quantities. To satisfy the
dy'
f ...dy'±
r...dy'
128 AERODYNAMICS OF THE WING
Kutta condition for the wing, the vortex density k(x, y) at the trailing edge
x = xr(y) must disappear [see Eq. (2-51)] . After having determined the vortex
density k(x, y) from Eq. (3-43), the resultant of the pressure distribution of lower
and upper surface at the point x, y, from Eq. (2-53), takes the form
-
d cP (x, y) _ P1-Pu = 2 k(x,y)
q00
U03
(3-44)
Here, q,c = o U. /2 is the dynamic pressure of the incident flow.
As in the case of the Prandtl wing theory (Sec. 3-2-1), the wing geometry (twist
and camber) can be established with Eq. (3-43) when the wing area and vortex
distribution k(x, y) are given quantities. The indirect problem requires quadratures as
in Eqs. (3.42) and (3-43). When the wing geometry (planform and angle of attack)
is given, Eq. (3-43) produces the vortex distribution on the wing surface. This direct
problem leads to an integral equation for the vortex distribution k(x, y), the
solution of which poses considerable mathematical difficulties. Approximation
methods need to be applied, therefore, which can be laid out in various ways.
A first possibility for obtaining an approximate solution is given by imposing
beforehand the vortex distribution k(x, y) in the direction of the wing span y.
By selecting for k(y) an expression of m terms, the first of which may, for instance,
represent the elliptic distribution, the integral equation Eq. (3-43) can no longer be
satisfied on the whole lifting surface, but only on m sections in the chord
direction.
A second possibility for the establishment of approximate solutions consists of
imposing beforehand the vortex distribution k(x, y) in the direction of the wing chord
x, for example, using the Birnbaum normal distribution of Eq. (2-61). If one selects
for k(x) an expression of n terms, then the integral equation can be satisfied only
on n lines along the span. Such procedures have been established for n = 1 (first
normal distribution) by Weissinger [95], for n = 2 (first and second normal
distributions) by Multhopp [62] and Truckenbrodt (84], and for n = 5 by Wagner
[91] and also by Kulakowski and Haskell [12].
A third possibility consists of imposing beforehand distributions with m terms
over the span and simultaneously distributions with n terms over the chord. In this
case the integral equation can be satisfied at (m - n) points suitably distributed over
span and chord. Such a procedure was applied by Blenk [69]. More recently, the
so-called panel procedure was developed [46] (see Sec. 6-3-1).
Previously, Falkner [14] presented a procedure in which discrete vortices were
arranged in both the chord and span directions. Also, the work of Jones [39] and
Lan [511 must be mentioned.
Velocity potential The induced velocity field of the vortex system of a wing can also
be defined by means of a spatial velocity potential 0 (x, y, z). Here the velocity
components induced by the vortex system are
co
00
0x
0y
IV =
co
cz
(3-45)
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 129
According to Truckenbrodt [84], the potential is
S
O(x, Y, Z) =
with
1
47r
-8
z
G(x,y,z;y')dy'
(y- y')',+z'
xr(Y)
G (x, y, z; y') =
k (x', y') 11 +
x-x
(x-x')2+(y-y')2± z2
(3-46)
d x'
(3.47)
xf(Y)
This expression for the velocity potentials of a lifting surface had been presented
earlier in similar form by von Karman [89] and Burgers [69] ; see also [84].
The potential is discontinuous at the lifting vortex sheet and in the free vortex
sheet behind the wing. Closer investigation shows that it changes abruptly when
crossing the. vortex sheet from the upper to the lower surface. This step of the
potential above (index u) and below (index 1) the vortex sheet is given at the lifting
surface [xf(y) <x <xr(y)] by
z
Ou (x, y)
f k (x', y) dx'
- 0r (x, y) =xf(y)
(3-48a)
and the free vortex sheet [x > xr(y)] by
x rr(Y)
0u(x, y) - 0(x, y) = f k(x', y) dx' = T(y)
(3-48b)
xf(Y)
Very far upstream and very far downstream of the wing, the function 0, in terms
of I' from Eq. (3-39) becomes
(3-49a)
0 (- oo, y, z) = 0
zf
8
0 (+ oc, y, z) =
2iz
ry)
(y - y')'2 + z2
d y'
(349b)
-s
Equation (3-49b) represents the two-dimensional potential of the induced velocity
field in the yz plane far behind the wing (potential in the Trefftz plane [69] ).
Acceleration potential For the treatment of the problem of the lifting surface by
means of the Laplace potential equation there is available, besides the method of the
velocity potential just discussed, the method of the acceleration potential. This was
first published by Prandtl [69 (1936)].
The method of the acceleration potential has been applied to the circular plate by
Kinner [44] and to the elliptic plate by Krienes [47] .
3-2-3 Integral Equation for the Circulation Distribution
from the Extended Lifting Line Theory
The lifting-surface theory of Sec. 3-2-2 can be transformed into a simpler theory of the
kind given in Sec. 3-2-1 by replacing the continuously distributed circulation along the
130 AERODYNAMICS OF THE WING
chord by a vortex line, arranged at a suitably chosen station on the local chord
theory). Let x' = xc(y') be the location of this lifting-vortex line
which, from the results of Sec. 2-3-2 for the inclined flat plate, is expediently
(lifting-line
placed on the quarter-point line (Fig. 2-37). Then the function G(x, y; y') of Eq.
(3.42a) becomes
i+
G(x. y; y') = I'(y')
x -x'
c
(x -x') zT (y
- 02
(3-50a)
Here I'(y') is the total circulation around the wing section y'. Furthermore, for
y' = y and x > xc this function becomes
G(x, y; y) = 21r(y)
(3-50b)
The kinematic flow condition [Eq. (3-40)] can be satisfied in this case at one point
of the chord only. This control point has the coordinate xp(y). Expediently, it is
placed on the three-quarter-chord station, measured from the leading edge
(three-quarter point, theorem of Pistolesi), see Sec. 2-4-5. Hence, the expression in
parentheses on the left-hand side of Eq. (3-43) becomes
az(' y)
ex
_ a (y)
(3-51)
where a(y) is the measured angle of attack relative to the zero-lift direction (Fig.
3-18).
By introducing Eqs. (3-51) and (3-50) into Eq. (3-43), the integral equation
for the circulation distribution from the extended lifting-line theory i§ obtained
as
U" (y)
4n
lim
s
r(y)
J
1
(y
(Y')
- YT
+
_XP - xC
dy,
(x -xc )2 ; (y - y')2
(3-52)
Compared with the simple lifting-line theory discussed in Sec. 3-2-1, Eq. (3-52)
has the great advantage that it is also applicable to yawed and swept-back wings.
This extended lifting-line theory is also called the three-quarter-point method. It was
developed in detail and applied particularly by Weissinger [95]. Also Reissner [95]
was engaged in the establishment of a solid foundation for this lifting-line
theory.
For the swept-back wing a vortex arrangement as in Fig. 3-20 is obtained. In
Fig. 3-20a the replacement of the wing by a system of elementary wings and in Fig.
3-20b the equivalent vortex system according to Prandtl's concept (Fig. 3-9) are
demonstrated.
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 131
a
b
Figure 3-20 Vortex system of a swept-back wing (lifting-line theory). (a) Substitution of the wing
by elementary wings. (b) Bound and free vortices according to Prandtl (see Fig. 3-9).
In Prandtl's lifting-line theory and in the three-quarter-point method described
above, the wing is replaced by just one lifting line. Wieghardt [101 ] proposed the
arrangement of several lifting lines in series. This method can be designated as a
multiple points method. Scholz [77] developed this method in more detail and
applied it especially to the cambered rectangular wing.
3-3 LIFT OF WINGS IN INCOMPRESSIBLE FLOW
3-3-1 Methods of Wing Theory
The theoretical basis for this section was laid in Sec. 3-2. For practical applications,
the computational methods discussed below (simple and extended lifting-line
theories, lifting-surface theory) proved to be particularly convenient and may be
characterized as follows: The simple lifting-line theory applies only to wings with
straight c14 lines in symmetric flow, that is, to unswept wings. It gives good results
for larger aspect ratios (A > 3) and allows the determination of lift distributions
over the span from which total lift, rolling moment, and induced drag, but not
pitching moment, may be computed. The extended lifting-line theory (three-quarterpoint method) applies to wings of any planform and aspect ratio. Thus, it applies to
swept-back and yawed wings. It gives the lift distribution over the span from which
total lift, rolling moment, induced drag, and, approximately, pitching moment are
obtained. The lifting-surface theory, like the extended lifting-line theory, applies to
any wing and aspect ratio, but gives lift distributions over the span and over the
chord from which total lift, rolling moment, induced drag, and also pitching
moment, and thus the neutral-point position of the wing, are found. Accurate
knowledge of the neutral-point position is particularly important for swept-back
wings.
132 AERODYNAMICS OF THE WING
Summaries and detailed presentations on the methods of wing theory in
incompressible flow are given by Betz (61, von Karman and Burgers [88], Robinson
and Laurmann [701, Thwaites (82], Weissinger [96], von Karrnan [89], Flax [15],
Hess and Smith [28], and Landahl and Stark [52]. The development of the
lifting-line theory as a "singular perturbation problem" is due to van Dyke [87] ; see
also the references on page 111. Extensions of wing theory to include nonlinear
angle-of-attack effects and the behavior of wings near the ground (ground effects)
are found, for example, in [8, 19, 21, 40] and [2, 81, 100], respectively. Although
it is not possible in this book to treat the questions of nonsteady flow that are
important for airplane aerodynamics, the references [2, 50, 52, 53] shall be
mentioned in this connection. Problems of flexible wings are discussed in [221.
Studies on design aerodynamics have been prompted by Kuchemann and
accomplished for swept-back wings in particular [3].
3-3-2 Computation of Total Lift
Basic formulas The local lift coefficient c1(y) of a wing section y is obtained
through integration of the pressure distribution over the wing chord in analogy to
Eq. (2-54) as
xrr(Y)
ci(y)
c(Y)J v cp
(x,
y) dx
(3-53)
xf(Y)
The total lift coefficient
CL = L/Aq
of the
wing is
thus obtained with
as
CL
=
Aff dc, dx dy
(3-54a)
(A)
= A1
cr(y)c(Y) dy
(3-54b)
-s
Compare also with Eq. (3-13). By using Eq. (3-54a), the total lift is obtained
through integration of the pressure distribution over the wing chord. With Eq.
(3-15), it may also be obtained from the Kutta-Joukowsky theorem. Here the
circulation distribution has to be taken from Eq. (3-39), the distribution of vortex
density from Eq. (3-44). Below, a further expression for the total lift will be
derived by applying the momentum law. As in Fig. 3-21, a cylindrical control
surface is arranged about the wing. The axis of the cylinder runs in the direction of
the incident flow velocity U.,. The two base surfaces I and II of the cylindrical
control surface are assumed to be very far upstream and downstream of the wing,
respectively. The diameter of the control cylinder is chosen large enough to make
pressure and velocity on the cylindrical surface equal to the values per, and U. of
the undisturbed flow on surface 1, respectively. In computing the lift from the
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 133
I
Jr
r-- Ii
Z
a
Vortex sheet
Fig re 3-21 Computation of lift by means of the momentum law and of the induced drag by
the energy law.
momentum law, it can be assumed that the free vortex sheet is parallel to the
incident flow direction far downstream of the wing.*
The fluid mass permeating an area element dy dz of surface II per unit time is
o U. dy dz. It produces, together with the velocity w induced by the wing, a
momentum component in the z direction of magnitude o U.w. dy dz. Since the
induced velocity on surface I is zero, the integral of the momentum over the surface
II represents the force exerted normal to the incident flow direction to the wing,
that is, the lift
L = -.Q U, f f w, dy dz
(3-55)
(II)
Now, the identity of Eqs. (3-55) and (3-15) will be shown for the not-rolled-up
vortex sheet. The field of the induced velocities very far downstream of the wing
can be described by means of a two-dimensional velocity potential P(y, z) [see Eq.
(3-49b)], where w. = ao/az. By introducing this expression into Eq. (3-55),
integration over z yields
+00
L
= -2 Uo-Y=-f 00
[O]Z 00 N dy
(3-56a)
8
_ e U. f I'(y) dy
(3 -56b)
s
On the boundaries y = ±0o and z = ±-, the values 0 vanish, whereas in the vortex
sheet, at z = ±0, the potential in the z direction from Eq. (348b) changes abruptly
by the amount d O(y, 0) = 0,,(v, 0) - 01(y, 0) = T (y). The integration limits
y = ±o may be replaced by y = ±s = ±b/2 because d0 (y, 0) = 0 outside of the
wing span. Introduction into Eq. (3-56a) yields Eq. (3-56b), in agreement with Eq.
(3.15). The total lift thus depends only on the circulation distribution over the wing
span. It is thus immaterial whether the circulation distribution is created by wing
*Kraemer [791 points out the decisive significance of the inclination of the free vortex sheet
for computation of the induced drag by means of the momentum law.
134 AERODYNAMICS OF THE WING
planform (aspect ratio, sweepback, taper), wing twist, or camber of the wing
surface.
Certainly, Eq. (3-55) is also valid for the rolled-up vortex sheet as in Fig. 3-8.
Now let bo = 2so be the distance between the two free vortices of circulation
strength To, whereby the circulation distribution along the span is symmetric (Fig.
3-22). The induced velocity w. at a point of the yz lateral plane very far behind
the wing (x - °°) becomes, from the Biot-Savart law,
w (X, Y)
so -y
2n [(so + y)2 + z2 + (So -A' + z2
so + y
To
Introducing this expression into Eq. (3-55) and integrating twice yield
L = n Uro I'ob0
(3-57)
By taking into account the Kutta-Joukowsky lift theorem, this formula states that
the lift of a wing of span b = 2s and of variable circulation distribution T (y) is
equal to the lift of a wing of span bo and over the span constant circulation
distribution To. Comparison of Eqs. (3-56b) and (3-57) yields the distance between
the two free vortices:
S
ho = r
0
f I'(y) dy
(3-58)
0
This relationship can also be interpreted as a statement that the vortex moment
about the longitudinal axis (x axis) remains constant during roll-up. For the right
Figure 3-22 Wing with rolled-up vortex sheet.
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 135
wing-half, the moment of the not-rolled-up vortex sheet about the x axis is equal to
- foS (d I'/dy) v dy and that of the rolled-up vortex sheet is equal to To bo /2. By
equating these two terms and integrating by parts, Eq. (3-58) is obtained directly.
Numerical data for bo/b are given in Fig. 7-17.
Introduction of the dimensionless lift distribution For the following computations
it is advisable to use the dimensionless quantities
Y(77) =
r
bU,,.
CI(77)C(77)
2b
(3-59a)
(3-59b)
with 77 = y/s [Eq. (3-1b)]. The relationship Eq. (3-59b) is obtained from Eqs. (3-12)
and (3-14). The significance of the linear wing theory of Sec. 3-2 is expressed by
the fact that the circulation distributions 71 (77) and y2 (77), resulting from two given
angle-of-attack distributions al (77) and a2 (77), can be superimposed linearly:
a (77) = al (17) + °L2 (77)
(3-60a)
Y(W = YIN) + Y2(77)
(3-60b)
The total lift coefficient is obtained from Eq. (3-56b) or Eq. (3-54b) as
i
cLq
=11 rY(?7)d77
(3-61a)
-1
1
dcL
dx
=.11fYu(77)d
(3-61b)
-1
The lift slope is obtained by computing the circulation distribution of the wing
without twist yu for a = 1. The aspect ratio of the wing A is given by Eq. (3-4a).
The zero-lift angle ao of a symmetric wing without twist is understood,
according to Sec. 1-3-3, to be the angle of attack that produces the total lift zero.
It can be determined as follows: For a given angle-of-attack distribution ag(y)
(measured from a wing-fixed reference plane), a circulation distribution yg(77), and,
with Eq. (3-61a), a total lift coefficient CLg are computed, from which ao and CLO are
obtained as
ao = - d a
cLg
(3-62a)
L
1
CLo= i'l f yo (ii) d77 = 0
(3-62b)
1
It is expedient to represent the circulation distribution of the twisted wing for
an arbitrary angle of attack by superposition of the distribution of the wing without
twist -yz, and a zero-distribution yo of the twisted wing for which the total lift is
136 AERODYNAMICS OF THE WING
zero. Consequently, the circulation distribution of the twisted wing at given angle of
attack a = const is given by
(3-63)
Y(77) = aYu(y1) + Yo (77)
The zero distribution yo (77) is obtained from
(3-64)
70 (77) _ 'Yg(77) + ao 7u (7?)
Through procedures similar to those applied for the lift, integration over 77 for a
known circulation distribution y(rl) produces other simple relationships for the
lateral distance of the lift center of a wing-half, for the lift force of a wing-half, and
also for the rolling moment about the x axis. They are summarized in Table 3-1.
Introduction of a Fourier series Computation of the integrals for the coefficients of
lift and rolling moment turns out to be. particularly simple when the circulation
distribution is expressed as a Fourier polynomial of the form
Al
y
2 S' a. sin,uzg
(3 -65a)
Et=1
Table 3-1 Compilation of the formulas for the aerodynamic coefficients of wings
of finite span*
Symmetric lift distribution
M+ 1
L
CL = A
R.
iT/i
M
2
Z yn
M + 1 n=1
I ! y(n) d -n
sun an
11 I Av yv
v=1
M+ I
2
f y(n)n dry
3'L
_
0
S
f y(n) do
I Bvyv
V=1
M+1
}
Gr A v'Yv
2
o
v=1
Antimetrict lift distribution
M-1
Cl =
(A2)12)q.
2 !1 / Y(77) dr
-
/If
2
V=1
Cvyv
M-1
M
cMX =
M
A
sq .
-A f -y(i7)i7 do
1 yn sin
- 2(M + 1) n=1
2
I
v=1
Dvyv
*Lift coefficient cL, lateral distance of the lift center of a wing-half rlL, lift coefficient of
a wing-half cL, rolling-moment coefficient cMX (sign convention from Fig. 1-6). Coefficients are
given in Table 3-2.
tFor an explanation of antimetric, see p. 190.
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 137
cos 0 _
with
(3-65b)
This procedure was first introduced by Trefftz [69] and Glauert [23]. The first
term in Eq. (3-65a) represents the elliptic circulation distribution y = 2a1 sin L _
2a,-\I1 --r7 as treated previously in Sec. 3-2-1.
After execution of the integrations over -1 < rl < I and 0 < 6 <ir, respectively,
the coefficients of lift and rolling moment are obtained with dr?
sin 6 dO as
CL = nAa1
(3-66a)
cMx=- 2Aa2
7Z
(3-66b)
The coefficients aµ result from Fourier analysis:
a,2 =
1 ry
sing 0di
(3-67a)
0
M
1
sin,cl0,,
where the integral
is
(3-67b)
evaluated by a summation formula. Here the symbols
yn = y(in) = y(rln) signify the circulation values at the stations
rln = Cos $n
7rn
with On =
M+ 1
(3-68)
By introduction into Eqs. (3-66a) and (3-66b), simple quadrature formulas are
obtained with p = 1 and p = 2 for the lift coefficient CL and the rolling-moment
coefficient cMx (Table 3-1). In addition,-quadrature formulas are also given for the
lateral distance of the lift center of a wing-half r7L = YL is and for the lift
coefficient of a wing-half cL . Table 3-2 contains the coefficients for the formulas
for practical application of the last column of Table 3-1.
Basic equations The starting equation for the Prandtl lifting-line theory has been
given as Eq. (3-22),
a (y) = ae (y) -i- ai (y)
(3-69)
where a(y) is the angle of attack relative to the zero-lift direction as in Fig. 3-23.
By introducing the dimensionless circulation distribution y(rr) from Eq. (3-59a)
with r7 = y/s, and further the dimensionless planform function
f(r7) =
2b
C1oC
1
ir
b
c(17)
(3-70a)
(3 -70b)
138 AERODYNAMICS OF THE WING
Table 3-2 Coefficients A, B, C, D for the computation of the aerodynamic
coefficients of a wing of finite span of Table 3-1, M = 7 and M = 15
B
1
2
i
3
4
1
2
3
15
4
6
7
8
!
0.9239
0.7071
0.3827
0.0000
0.3006
0.5555
0.7256
0.9808
0.9239
0.8315
0 . 7071
0.5556
0.3827
0.1951
0.0000
+
D.
C,,
0.3260
0.4952
0.8593
0.2777
0.:3927
0.2769
0.3951
0.2702
0.0317
0.0766
0.1503
0.2182
0 . 2777
0.3265
0.3628
0.3852
0.1964
0.0751
0.1389
0.1813
0 . 1965
0.1811
0.1395
0.0733
0.0078
0.0797
0.1438
0.2285
0 . 2622
0.3495
0.3266
0.4546
0.0751
0.1388
0.1814
0.3927
0.2777
0.1964
0.1814
0.1388
0-0751
the effective or, respectively, the induced angle of attack becomes*
ae(7)) = f(0 Y(n)
(3-71 a)
1
f dy
1
2n
-1
dn'
(3-71 b)
d'1' 77 - 91'
The formula for ai(n) can also be written, through integration by parts, as
cc i(77) _
lim
2
2n E--o
1
Y(r1) -
r
d,,"
(3-71c)
-1
By introducing Eqs. (3-71a) and (3-71b) into Eq. (3-69), the Prandtl integral
equation for the dimensionless circulation distribution 'y(n) is obtained in the form
(77) = f (@)) Y (11) T
i
2n
1
f,
d7j
ds7 n-17
(3 -72a)
*See footnote on page 139.
Section y
Zero-lift direction
Chord
Figure 3-23 Wing section y: a(y), angle of attack
against zero-lift direction; ag(y), geometric angle of
attack against wing chord; a,, (y), zero-lift angle,
a=ag-a0
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 139
In abbreviated form, the integral equation of the simple lifting-line theory can be
written:
x (l) = ai (')) H -
f(?1) Y (1))
(3-72b)
Schmidt et al. [76] deal with the mathematical formulation of the simple
lifting-line theory and present comprehensive results.
Solution with Fourier polynomials A convenient method of solving Eq. (3-72) for
the circulation distribution consists of expressing the circulation distribution as a
Fourier polynomial such as Eq. (3-65) (M= n). By introducing Eq. (3-65a) into Eq.
(3-71b), first the induced angle of attack is obtained*:
Al
I n an
sinnfi1
(3-73)
sing
n=1
After introduction of Eqs. (3-65a) and (3-73) into Eq. (3-72b), the following
equation is obtained, defining the Fourier coefficients an:
M
a (i) sin?5 = Z a [2 / (6) sin 6 H- n] sin n t
(3-74)
n=1
Here the distribution of the angle of attack a(6) and the wing planform f(6) are
given beforehand. The coefficients al, a2i ... , am are determined by satisfying Eq.
(3-74) at M points t51 , 62, ... , 6M along the span. This results in a system of M
linear equations for al to am. Lotz [56] simplified this procedure by introducing
Fourier polynomials for the functions a(6) sin 6 and f(i3) sin 6. After the Fourier
coefficients an have been determined, the circulation distribution is obtained from
Eq. (3-65a) and the distribution of the local lift coefficients from Eq. (3-59b).
Weinig [93] suggests that the theory of the lifting line be solved by comparison
with the corresponding grid flow.
Wing of elliptic planform The elliptic wing has been treated in Sec. 3-2-1. There it
was shown that an elliptic wing without twist has an elliptic circulation distribution
over the span. The elliptic wing with twist may be computed from the above
formulas very easily as, among others, Schmidt [69] has shown.
For the elliptic wing c = c, 1 - 712 = Cr sin 6 and A = 4b/?TCr [Eq. (3-9)], and
thus from Eq. (3-70),
2sin0/(0) _ 'A
(3-75a)
CL cc
k
*Note that, according to [23 ]
,
z
r cos no,
1
7t j Cos'0 - cos 0'
0
(3 -7 5b)
d6'
sin?09
sin
140 AERODYNAMICS OF THE WING
Hence, Eq. (3-74) for the wing with elliptic planforrn becomes
M
a(O) sine
n=1
(k+n)aasinnO
(3-76)
and the coefficients a, can be computed directly through a Fourier analysis as
an
_
2f a
79
1
k+
sink sinnO d6
7r
(3-77)
0
This solution will now be discussed for a few particularly simple angle-of-attack
distributions. Setting
sin m&
(3-78)
sin,&
the corresponding circulation distribution is obtained with a,, = 0 for n
m and
with am = rm J(k + m) for n = m as
Y(6)=2k
75
(3-79)
gin,
For a wing with the aspect ratio A = 6, that is, k = 3, the results for m = 1, 2, and
3 are presented in Fig. 3-24. Here m = 1 gives the constant angle-of-attack
a
b
-oz
0.2
Figure 3-24 Lift distribution according to the simple lifting-line theory of an elliptic wing at
various twists (Eq. (3-78)]; aspect ratio A = 6. (a) Wing planform. (b) m = 1: wing without
twist. (c) m = 2: linear angle-of-attack distribution. (d) m = 3: parabolic angle-of-attack
distribution.
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 141
7
2
',
C
2
I
I
A3
4(
t
If
7
6'
Figure 3-25 Lift slope of elliptic wings
vs. aspect ratio; cL = 2n. (1) Simple
lifting-line theory, Eq. (3-80b). (2) Extended lifting -line theory, Eq. (3-98).
(- o -) Exact solutions according to Kinner [44] and Krienes [47].
distribution (wing without twist), m = 2 gives the linear angle-of-attack distribution
such as, for instance, is encountered in a rolling motion, and m = 3 gives the
parabolic angle-of-attack distribution (symmetric twist, CL = 0).
Circulation distribution and coefficients of lift and rolling moment of the
elliptic wing with twist are obtained from Eq. (3-66a) and also from Eq. (3-66b) by
introducing the corresponding coefficients a7z according to Eq. (3-77). The lift
coefficient thus becomes
CL
k+ i
.
IT
r()S1fl2CZz
(3-80a)
0
dcL
,cal.
da
k+1
(3-80b)
For the wing without twist, a = const, the coefficient of lift slope is obtained
in agreement with Eq. (3-34a). It is presented in Fig. 3-25 as a function of the
aspect ratio A l . Also shown are the results based on the extended lifting-line theory
that will be treated further in Sec. 3-3-4, and the exact solution for the elliptic
wing.
From Eqs. (3-80a) and (3-80b) the zero-lift angle is obtained with Eq. (1-23) as
+1
2 f a("q) Y1 -?12d.1
a0=--
(3-81)
For approximate computations, the relationships of the elliptic wing can be
applied to other wing shapes.
Quadrature method of Multhopp The simplest and most used method for the
computation of the lift distribution of unswept wings according to the simple
Here, the wing planform is an exact ellipse, which, e.g., becomes a circular disk for
A = 4/ir.
142 AERODYNAMICS OF THE WING
lifting-line theory is that of Multhopp [60]. This method will be briefly sketched
now: Starting from the expressions for the circulation distribution [Eq. (3-65)] and
for the Fourier coefficients [Eq. (3-67)] in connection with Eq. (3-68), the
summation expression, Eq. (3-67b), is introduced into Eq. (3-73). The induced angle
of attack at the discrete stations
nv=costg
+1
with t%, =
(v= 1,2,...,M)
(3-82)
... , X)
(3-83)*
is then obtained in the form
1I
ai(77v) = a:, = b,q y, - 2:' b,,t y,,
(v = 1, 2,
n=1
with the universal coefficients
__
b"v
bvn
=
M+ 1
4 sin tv
(3-84a)
1 -(-1)v-n
sin 6 n
2(M + 1)
(cos 6v - Cos $n) 2
(v
n)
(3-84b)
By introducing expression (3-83) for the induced angle of attack into the
integral equation for the circulation distribution Eq. (3-72b), the following system
of equations is obtained for the values of -y,,:
(bvv +.fv)7v
b, y.
aY +
(v = 1, 2,
... , M)
(3-85)
9t=1
This is a system of M linear equations for the M circulation values 7v = -y(rv) with
v = 1, 2, ... , M. In Eq. (3-85), the following relationships apply:
av = a(rly)
fv =
2b
C100 0v
with cv = c(nv)
(3-86)
For M= 7 and M= 15, the universal coefficients are compiled in Tables 3-3
and 3-4. The values
for (v -nf = 2, 4, ... are equal to zero. For the numerical
solution of the system of equations, it is significant that the system of M equations
can be split up into two systems of (M + 1)12 or (M - 1)/2 equations, respectively,
which can be solved conveniently by iteration. By splitting an arbitrary angle-ofattack distribution into its symmetric and antisymmetric contributions, the
procedure of the numerical solution can be further simplified. For a continuous
behavior of wing chord c(r1) and angle of attack a(77), usually M = 15 points along the
span are sufficient for all practical purposes. For discontinuous angle-of-attack
distributions, as found for flap deflections, Multhopp recommends that one split off
*Here, the prime () on the summation sign indicates that the term n = v is to be omitted
in the summation.
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 143
and b,,,, for the computation
Table 3-3 Universal coefficients
of circulation distributions, for M = 7, according to Eq. (3-84)2
4
1
3
2
(7)
(5)
(6)
771,
0.9239
0.3827
0.7071
0.0000
b,.,,
5.2262
2.1648
2.8284
2.0000
1.0180
1.0972
0.0973
0.0180
0.0560
0.7887
0.7887
0.0560
V
n
2(6)
4(4)
6(2)
bm
j
-
ft
1.8810
0.1464
0.0332
-
0.8398
0.8536
0.0744
-
1(7)
3(5)
5(3)
7(1 )
,
aAfter Multhopp [60].
the discontinuity stations before applying the above computational procedure; see
Chap. 8.
Equations (3-85) are valid for unswept but otherwise arbitrary wings of
sufficiently large aspect ratio (Ai > 3) and also for arbitrary angle-of-attack
distributions.
Further results of the simple lifting-line theory In Fig. 3-25 the dependence of the
lift slope on the aspect ratio is shown for a wing of elliptic planform. This result is
approximately valid also for wings of different-for instance, trapezoidal-planforms.
To demonstrate the effect of the aspect ratio on the lift distribution, the
circulation distributions over the span were computed for three rectangular wings
with c = const and aspect ratios A = 6, 9, and 12. When A increases, the circulation
distribution approaches more and more a rectangular distribution. Figure 3-26
demonstrates this fact. Illustrated are the local lift coefficients cl with reference to
the total lift coefficient cL along the span. For A °° (plane problem), cl/CL = 1, and
for very small aspect ratios (A -+ 0) the lift distribution is elliptic. This can easily be
seen from Eq. (3-72b), which for ii --> 0 goes to «(r7) = al(r7) because f(ry) = 0.
Hence, for a = const, aZ = const, meaning, from Sec. 3-2-1, that the circulation
distribution is elliptic.
To show the effect of wing taper on the lift distribution, Fig. 3-27 illustrates
the circulation distribution for four trapezoidal wings without twist of aspect ratio
A.= 6 and tapers X = ct/c,. = 0, a , a , and 1. The taper has a strong effect on the
distribution of the local lift coefficients along the span. This can be seen in Fig.
3-28 in which the curves Cl/CL are shown. The strongly tapered wings have, near the
wing tip, local lift coefficients that are considerably larger than the total lift
coefficient CL. This fact is significant for the flow-separation characteristics of such
rU O
v
O
Q
O
O
O
Oo
O
Gli
G
C
.--i
CC
*
r
N
0
00 0
000*-i_ 000
O0
O
O O0
1-4 X0 Cfl - *1 =
Cq
.-1
0O
O rNMN M C Cl
CO
0
OO
M
0 0 .-i , C C c;
eM
M -4 GV 00 M cp [ M
N
c
00
dJ
OS
L
l
CC'
NM O
In
ocici06666
Cl
M
O
O M 1
CO CO 00 -1 dl Cl - 0
+ GV CV Cl 0 0 0 0
CR
0
OeO.-+000
L^
of
O
v
t M rd ----M ,-,i
.p
0
N
..
E-
C
O CC O CV O M ++
N
O
LeD CO N ZC
O .- cfl c .-+ O O
(30
00_+-1000
eM
O
CO
00
O
eM
-
ay XO N0
d
G
OV O O O
,
G
S.
O
CO
00
ODCQ0000
GV CJ 00000
00
_
G
c-
OC
0
00
O
O
O
I
O
1
Mto 1-40000
x 0 0 0 0 0 0
'4CplO
4
-
C
OO O eH 0 -- LO
LID -4 10
000CO CO M 1-4O
I
CI
O
--
4CO00 00ad+
Q
C
144
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 145
1//
-1
12
12
10
0
A-6
9
03
12
02
0,1
05
0.3
97
0.6
00
09
to
Figure 3-26 Lift distribution ci/cL of rectangular wings without twist of aspect ratios e1= 6, 9,
12; also limiting curves for A. -- 0 and A -* -; cL,,, = 21r.
wing shapes at high lift coefficients. With increasing angle of attack, separation
begins approximately at the station of maximum local lift coefficient, hence on
strongly tapered wings close to the wing tips, but on rectangular wings in the
middle of the wing.
3-3-4 Extended Lifting-Line Theory
Method of Weissinger The method of the extended lifting-line theory, as explained
in its basic aspects in Sec. 3-2-3, has been developed into computational procedures
for practical applications by Weissinger [951. The basic equation for the
a
=0
oas
a
0.50
70
02
0,1
of
0.2
0.3
0.4
05
06
07
OF
09
10
Figure 3-27 Circulation distribution y of a trapezoidal wing without twist of taper ? = 0, a, ;,
1; aspect ratio a = 6; cL. = 27r.
146 AERODYNAMICS OF THE WING
Z=O
12
0.25
0.5
1.D
I
06
05
02
0
01
0,2
0,3
0//
0.5
07
D.6
1J-
0.6
0.9
119
Figure 3-28 Distribution of the local lift coefficients cl/cL over the span for trapezoidal wings
without twist of tapers X = 0, 1,
1, 1; aspect ratio ,1 = 6; cj,,, = 27r.
4 2
determination of the circulation distribution using this procedure is Eq. (3-52). With
the dimensionless space coordinates , ri from Eqs. (3-la) and (3-1b) and the
dimensionless circulation distribution y from Eq. (3-59), Eq. (3-52) takes the form
a(r!) = i2Zlim
e0
4
E
Y(77) -
I
ti (5
P!
1'lY(i1') dl]'
(3-87)
-
(3-88a)
(?1-71)
J
with
' - tc
(gyp _ ' )2+ ( -
(3-88b)
S ( , 7p 17) = 2
As shown in Fig. 3-29,
711)2
t,(77') is the position of the lifting line at a distance c14
from the leading edge and tp = tp(rj) the position of the control points. As
explained in Sec. 3-2-3, the control points are arranged at three quarters of the local
wing chord; thus xp = xC + c12. This choice of the position of the control points
(three-quarter points) results from two-dimensional skeleton theory for which
the position of the control point
cL m = 21r. To introduce another value for cL
can be changed by setting (see [831 ):
)+01.C(')
27r
(3-89a)
2
P(77)
5(77)
= E p(77) -
(Tn) =
2n
(3-89b)
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 147
By introducing, according to [95] , the function
K (71' i1,) = K ($p )1
_
?7')
71
(3-90)
Eq. (3-87) becomes
f
1
cc (27) = 2x:(71) +
2c
K(rl, -77')
dr1'
(3-91)
-1
where a1(77) is taken from Eq. (3-71 c).* The kernel function K(r7, T?') in Eq. (3-90)
has been selected to be regular at r7' = r7, whereas the integrand in Eq. (3-87) is
singular at this point. By a simple computation it can be shown that
1
K(r1, 1) _
_
2(sep
1
262(17)
(3-92)
The integral equation of the extended lifting-line theory now takes the following
form, in analogy to Eq. (3-72b) of the simple lifting-line theory:
a (n) = 2 [at (77) + z (ti7)l
(3-93)
1
(3-94)
where
The kernel function K(77, 77') depends exclusively on the geometry of the wing
planform [83].
Wing with elliptic planform In Sec. 3-3-3 the wing with elliptic planform was
treated by using the simple lifting-line theory. Now this wing shape will be
computed using the extended lifting-line theory. A result of the simple lifting-line
*2ai is the induced downwash angle far behind the wing, it - -.
Lifting line (quarter-point line) c(r,) Figure 3-29 Sketch for the extended liftingline theory.
----- Line of the control points tp (77)
148 AERODYNAMICS OF THE WING
theory, namely, that the elliptic wing without twist has an elliptic circulation
distribution along the span, will be taken to apply here, too. Then the following
study shows the difference in the total lift as determined from the simple and from
the extended lifting-line theories, respectively. Since in the present case of an
elliptic wing the circulation distribution along the span after the simple theory is
assumed to apply, the kinematic flow condition can be satisfied only at one point
of the three-quarter-point line. Following Helmbold (27], the three-quarter point of
the wing half-span section will be chosen. The kinematic flow condition thus
becomes, from Eq. (3-40),
c +'XE'(4, 0) = 0
(3-95)
Here tp = xp/s is the dimensionless distance of the control point from the c/4 line.
We shall not perform the calculation in detail, but the induced downwash angle
at the wing. middle section becomes, according to Glauert [23], for elliptic
circulation distribution,
a.u(p,0)_-1 1-{-2
E ai
SAP+1
SE
?L
(3-96)
Here E is the complete elliptic integral of the second kind with module 1/ /p + 1,
and cx = CL/7rA is the induced angle of attack introduced earlier. To simplify the
computation, an approximate expression can be given for Eq. (3-96) that no longer
contains the elliptic integral (see [27] ). With Eq. (3-95), this expression becomes
2
a = .1
1 -{-
z sep
z
2
CL
7r "l
(3-97)
The position of the three-quarter points is obtained from Eq. (3-89b) with
0, and further with A = 4b/7rc,. from Eq. (3-9) and k = 7rA/cL- from Eq.
(3-75b)* as
P
_
CLao
27r
Cr_ 2
b
Irk
By introducing this expression into Eq. (3-97), the lift slope is found to be
da
k-+1 +1
(3-98)
In Fig. 3-25 the lift slope after this formula is presented for ci = 21r, that is,
k = /1/2, versus the aspect ratio. For comparison, the curve according to the simple
lifting-line theory [Eq. (3-80b)] is also shown. The difference between the two
theories is similar to that for the rectangular wing of Fig. 3-32.
Equation (3-98) for the extended lifting-line theory evolves from Eq. (3.80b) for
the simple theory by formally replacing k by k2 + 1. In an analogous way, the
Fourier coefficients for the circulation distribution of the twisted wing can be
modified to comply with the extended lifting-line theory. Thus Eq. (3-77) takes the form
`See footnote on page 118.
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 149
z
an =
1
yk2+10, ; n
c.. (t$) sin to sin -10 d 0
2
(3-99)
0
The usefulness of this formula has been confirmed by numerous examples.
The rolling-moment coefficient cMX is obtained in closed form by introducing
into Eq. (3-66b) the value for a2 from Eq. (3-99) and observing that 77 = cos 6, Eq.
(3-65b), as
=1
cMX
This is
k'
Tc
J
a (,1) i1 y 1 - 1]2 d )j
(3-100)
-1
a quite simple equation for the determination of the rolling -moment
coefficient.
Quadrature methods For the numerical evaluation of Eq. (3-93), Weissinger [95]
presented a refined quadrature method analogous to that of the simple lifting-line
theory (method of Multhopp). This method will not be presented here; instead,
reference is made to [95]. Comprehensive sample computations using the Weissinger
method have been conducted by de Young and Harper [103].
Further results of the extended lifting-line theory In Fig. 3-30 the circulation
distribution versus the span at a= 1 is demonstrated for the rectangular wing
without twist of aspect ratio 1= 6. For comparison, the curve using the simple
lifting-line theory is also given. This figure shows that the extended theory produces
a smaller lift than the simple theory for the same angle of attack. Furthermore, Fig.
3-31 illustrates the lift distribution c1/CL of the same wing. The extended lifting-line
theory produces a somewhat less full distribution curve than the simple lifting-line
theory. Both of these statements are typical for the extended lifting-line theory.
The lift slope of rectangular wings after the extended and after the simple
lifting-line theory are compared in Fig. 3-32. The difference between the curves is
Figure 3-30 Circulation distribution of the rectangular wing without twist of aspect ratio A = 6
for a = 1; cL. = 27r. (1) Simple lifting-line theory. (2) Extended lifting-line theory.
150 AERODYNAMICS OF THE WING
000
0,2
02
0,1
03
0, 41
0.7
0.5
0, 5
88
0.9
10
1
Figure 3-31 Lift distribution cllcL of the rectangular wing without twist of aspect ratio A1= 6;
c',. = 2ir. (1) Simple lifting-line theory. (2) Extended lifting-line theory:
rather small for large aspect ratios A. It is considerable, however, for small values of
.1. The limiting values of dcL/da for :1 --> 0 of the simple [Eq. (3-101a)] and the
extended [Eq. (3-101b)] lifting-line theory* are
dcL
zA
doL
z
da
2
(A ->-
0)
( 3 - 101 a)
(A -- . 0)
1
(3-101b)
The two limiting values are also indicated in Fig. 3-32; see also Fig. 3-25.
' For A -> 0, a(r1) = ai(r1) in the simple lifting-line theory; for the extended theory,
however, a(77) = 2ai(n) because K(n, ra') = 0.
2
2
3
1
5
6
7
d
9
10
17
72
Figure 3-32 Lift slope dcLldca of rectangular wings vs. aspect ratio A; cL = 27r. (1) Simple
lifting-line theory. (2) Extended lifting-line theory.
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 151
In Fig. 3-33, results for a trapezoidal wing, a swept-back wing, and a delta wing
with aspect ratios between A = 2 and 3 are presented. The geometric data for these
three wings are compiled in Table 3-5. Figure 3-33 gives the circulation distribution
for the wing without twist at a = 1. For the trapezoidal wing, the curve using the
simple lifting-line theory has been added. In this case, too, it lies above the curve
for the extended lifting -line theory. For all three wings, results are shown of the
lifting-surface theory, which will be discussed in Sec. 3-3-5. Agreement between the
extended lifting-line theory and the lifting-surface theory is good. The values for the
lift slope and the neutral-point displacement, together with additional aerodynamic
coefficients yet to be discussed, are compiled in Table 3-5.
Transition from extended to simple lifting-line theory It should be shown that the
extended lifting-line theory may be transformed into the simple lifting-line theory
for large aspect ratio. In performing this limit operation, according to Truckenbrodt
[83], the control-point line p(r?) for the kinematic flow condition of the extended
lifting-line theory must be shifted toward the lifting line t1(r7), tp -- 1, or 5. -* 0
(Fig. 3-29). Thus the kinematic flow condition becomes
« , (a > 0, n) + oc (n) = 0
(A = large)
(3-102)
where S(ri) is defined by Eq. (3-89b). The dimensionless induced downwash velocity
according to Biot Savart of a lifting line normal to the incident flow becomes, for a
control point p = xp/s = 8 that lies very close to the lifting line,
-a,,(a - 0, n)
= ai (77) + i 2 (77)
(3-103)
The first term of the right-hand side signifies the contribution of the free vortex,
the second term that of the bound vortex. Since, from Eqs. (3-89b) and (3-70a),
r5 (?7) = 1/f(r7), it follows from Eq. (3-102) that
C(n) = ai(n) + f('r1)Y(17)
(3-104)
10
08
I aTrapezoidal wing
L _11
bSwept-back wing
3
3
i
02
0
92
0.#
06'
09
10 0
0.2
09
,o0
71
Figure 3-33 Circulation distribution of three wings without twist of Table 3-5; a = 1; cL = 27r.
(a) Trapezoidal wing; cp = 0; A = 2.75; x = 0.5. (b) Swept-back wing; yp = 50°; A = 2.75;
= 52.4°; A = 2.31; a = 0. Curve 1, simple lifting-line theory of
X = 0.5. (c) Delta wing;
Multhopp. Curve 2, extended lifting-line theory of Weissinger. Curve 3, lifting-surface theory of
Truckenbrodt.
6
OA
f
4
O
,4n
110 r. 4 Ci
cq
M
CN GV
C
e^
O
ci
C
O C
O
Z
xU
8
Cc
Cpl v G
co cc
C O ri
{
-
cd
H
C
V-4
LO
Ln 00
O
O
t
Z
I
oN
01
G
(
H
H
-
~
N
H
0
M
O
1
L^.
N
!:7
C
L^
O
O
00
c7
N
N
tr.
GU
H
C1
N
C?
C
O O
O
CO
N
to
00
00 oc
M CJ O C
I
I
O
00
c
Or
N
O
O
GV
C
w O
O .C
V
N
Cs
II
a' , U
.
%V
y
N
v O
6)
152
Ow
G)
Cl-
Q
ry
C]
e,..
~
;
C)
cn
y
=
`
"..
z
<U
FNr
U
d
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 153
Thus it has been shown that the extended lifting-line theory is reduced to the
simple lifting-line theory for very large aspect ratios.
This limit operation can be applied to the yawed wing according to [831 and leads
to the earlier-stated theory of the yawed wing of Weissinger [94]. The limit transition
from the extended to the simple lifting-line theory is also treated in [83]. Based on
this, Laschka and Wegener [83] developed a quadrature procedure for computing the
lift distribution of swept-back wings of large aspect ratio.
3-3-5 Lifting-Surface Theory
General formulation of the procedure The integral equation for the computation of
the circulation distribution according to the wing theory was presented as Eq.
(3-43). With the dimensionless areal coordinates l;, r1 of Eqs. (3-la) and (3-1b), the
dimensionless kernel function, and the local angle of attack of the cambered lifting
surface, that is, with
G
9
(3 -105a)
b U00
a (5, 7) = aF - a
(3-105b)
Eq. (3-43) becomes
1
2;r
lim
9($''7; ?1')
2 g
d.,
(3-106)
'
6
For the areal distribution of the vortex density k(x, y), the following product
expression according to Truckenbrodt [84] is introduced:
k (:2', /)
=
1
c
S
U.
(3-107)
c., (I) h,, (x)
)r=(J
Here the cn(y) represent the spanwise distribution and the hn(x) the chordwise
distribution of the lift. Introducing Eq. (3-107) into Eq. (3-42a) yields
Xr(Y ')
with
H" (x.11; y')
(3-108)
S c,, (y') c(y') H,, (.r.
G (x,
C h,,(x') (1 +
CV) J
1 (:L
xf(v')
-
x- )
(
.i'')=
'/
tl:a
(3-109)
i/')=
According to the skeleton theory of Sec. 2-4-2, the distribution function over the
wing chord is expressed as
z
sing
(n = 0, 1...)
(3-110)
154 AERODYNAMICS OF THE WING
where, from Eq. (2-62),
X- 1( 1 + cos P)
c(y)
Here c(y) is the chord and x f the position of the wing leading edge at section y.
For the values n = 0, 1, and 2, the distributions are given in Fig. 3-34; see Fig.
2-27. The functions ho and h 1 of Eq. (3-110) have been normalized to produce the
local lift and moment coefficients relative to the c/4 point through integration over
the chord after introduction into Eq. (3-107) [see Eqs. (2-54) and (2-55)]. The
result is
cr(y) = co (Y)
(3-112a)
Cm (y) = 14C, (Y)
(3-112b)
The explicit expression for the functions H (x, y; y'), which are dependent only on
the wing planform, is obtained by introducing the distribution functions hn of Eq.
(3-110) into Eq. (3-109).
Writing the function G of Eq. (3-108) in dimensionless form and considering
Eq. (3-105a) leads to
977') Zn=0H.
.4
with
(3-113a)
77; 77') fn (q')
(3-113b)
2b
Introducing this function g(, n; rl') into Eq. (3-106) finally yields
1 I lim
y7) =
x(
11=0 e-+0
?E H. (, 7J
Hn ,
r1) f>a (77)
/)'
d f7'
(3-114)
-1
41
3
m=0
2
1
0
_2
02
X-
Of
06'
0
10
Figure 3-34 The functions h 0 , h 1, and 11 2 for the lift
distribution vs. wing chord, from [84); see Eq. (2-88).
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 155
This is a system of integral equations for the (N+ 1) functions f,(n), (n = 0, 1,
... ,
N). Choosing (N+ 1) distribution functions by satisfying the kinematic flow
conditions on (N+ 1) control-point lines along the span, (N+ 1) distribution
functions can be determined. After having determined the functions fo(ri),fl (r7), and
so on, from this system of equations, the lift distribution is obtained from*
cr(17)c(77)
(3-115a)
= fo (77) = Y('?)
2b
and the moment distribution (moment coefficient relative to the c/4 point) from
C.(2bc(?7)
= 4 /.(n)
(3-115b)
The resultant of the pressure distribution of the lower and upper surfaces (load
distribution) follows in analogy to the expression Eq. (3-107) and the relationship
Eq. (3-44) as
A cp =
=
P1
qC0
N
2b
h" (
C07) n
(3-116)
) A 07)
In the following section this procedure will be explained through numerical
execution.
Method of Multhopp and Truckenbrodt Multhopp [62] and Truckenbrodt [84]
independently developed methods for the numerical evaluation of the method
outlined above. In either publication the two distribution functions ho and h1 mark
the basic approach. Multhopp puts the two control-point lines at 34.5 and 90.5% of
the local chord. Truckenbrodt prefers positions of the control-point lines on the
trailing edge and the c/4 line of the wing. A comparison of the best known
lifting-surface theories is given in [18].
The explanation of the computational procedure now to be given follows
closely [84]. As already stated, only two distribution' functions over the chord are
chosen, limiting the correlation functions of the method to Ho. and H1. With new
designations,
97;97')
(3-117a)
77') =70,97;97')
(3-117b)
Ho( ,97;97')
4H,
Eq. (3-114) becomes
1
1
77) = 27 lim
e-;0
2
s
2 i(
p, 77;
77)Y(/) -
-J
1
7(fir,,9) ;97)it(97)
97
(97 - 97 )"
Y(1')
d77'
,
' )2
(3-118)
-1
*fa (n)
(3-59).
is identical to the dimensionless circulation distribution y(r7) = T /b U. from Eq.
156 AERODYNAMICS OF THE WING
where fo and fl are replaced by y and µ as in Eqs. (3-115a) and (3-115b). This
equation must be satisfied for two values of gyp, namely, for
p = t25 = I
and
tp = tioo = to
(3-119)
Here 25(ri) stands for the c/4 line and tioo(?7) for the trailing edge.
The two functions y(77) and 4(r1) of Eq. (3-118) are now to be determined. The
angle-of-attack distribution a(p, rl) is given directly by the wing geometry, and the
kernel functions i and j are given indirectly as functions of the wing planform. Only
the angle-of-attack distribution values on the c/4 line and on the trailing edge are
required in Eq. (3-118).
Wagner [91] expanded the described lifting-surface method to more than the
two distribution functions over the span ho and hl. Accordingly, the number of
control-point lines must be increased. In selecting five distributions ho through h4,
in [91 ] the control-point lines are laid on the leading edge, the one-quarter,
one-half, and three-quarter point lines, and on the trailing edge.
Quadrature methods The numerical solution of Eq. (3-118) is accomplished through
an extended quadrature method, following Multhopp's procedure for the lifting-line
theory. Because of the considerable extent of the computations, use of an electronic
computer is necessary. Further possible solution procedures for the equations of
lifting-surface theory are reported by, among others, Kulakowski and Haskell [12],
Cunningham [12], and Borja and Brakhage [9]. The panel method of Kraus and
Sacher [46] should also be mentioned.
Lift distribution After having obtained the values y(rl) and µ(r1) by solving the
system of equations, the lift distribution along the span follows from Eq. (3-.115a).
The load distribution over the wing chord is consequently derived from Eq. (3-116)
[compare also Eqs. (2-87) and (2-88)] as
zi e22 (X,11)
= C(1b7) [ho(X)
y (?7)j
(3-120)
where the functions ho and h1 are taken from Fig. 3-34.
Lift, rolling moment The total lift coefficient, the lateral distance of the lift center
of a wing-half, the lift coefficient of a wing-half, and the rolling-moment coefficient
may be determined from the formulas of Table 3-1.
Pitching moment The local pitching moment about the local c/4 point is given by
Eq. (3-115b). In Fig. 3-35, x25(y) designates the c/4 line and x1(y) the line of the
local aerodynamic center. It follows, then, that the moment of a wing section y
about the c/4 point is dM= -dx1 dL. By setting dM = c,,go,c? dy and dL =
c1gc.,c dy, the distance between the local aerodynamic force and the local c/4 point
becomes, with the help of Eqs. (3-115a) and (3-115b),
Jx1(17)
cm(17)
_ x1(rl)-x25(?7) _
4(77)
c(11)
c1(17)
c(17)
7(77)
(3-121)
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 157
Section y
Figure 3-35 Computation of the pitching mo-
ment. xl(y) = position of the local aerody-
X25
namic centers. x,, (y) = local c/4 line, N = neutral point of total wing.
- XI
The pitching moment of the whole wing is obtained from the contributions of
the individual sections to the moment about the y axis dM = xl dL, resulting in
- f xl(y) dL = -
8
4
131 =
-S
f [x25 (Y) + J x1(v) ] dL
-S
Hence the pitching-moment coefficient cM =117/q.,Acu, with CA as the reference wing
chord [Eq. (3-5b)], becomes
1
r
cal = __ i f Cy(71)
-t
x250)
C
(77)
Cu
dC/1
(3-122)
Finally, the neutral-point position of the whole wing is obtained from Eq. (1-29).
Results of wing theory and comparison with tests The examples computed in this
section include rectangular, trapezoidal, swept-back, and delta wings.
Earlier, in Fig. 3-33, circulation distributions of a trapezoidal wing, a
swept-back wing, and a delta wing, all without twist, were presented for several
computational methods.* The geometric data of these three wings are compiled in
*Computation of the lift distribution of delta wings has also been treated by, among
others, Garner [17].
158 AERODYNAMICS OF THE WING
Table 3-5. From Fig. 3-33 it was concluded that the difference between the
extended lifting-line theory and the lifting-surface theory is quite small. In Fig.
3-36, the lift distribution of three wings without twist is illustrated in the form
clc/CLCm versus the span coordinate. In this kind of presentation, the computational
results are practically identical. The lift slopes dcL /da of these three wings, based
on various theories, are compiled in Table 3-5.
Neither the simple nor the extended lifting-line theory allows determination of
the local neutral-point position because these methods require that the local neutral
point be fixed on the lifting line (c/4 line). Application of wing theory according to
Eq. (3-121) is required for local neutral-point determination. In Fig. 3-37, the local
neutral-point positions over the span are plotted for the three wings of Fig. 3-36;
see also Table 3-5. The local neutral points of the unswept wing lie before the c/4
line over the whole span. On the other hand, the local neutral points of both of the
swept-back wings lie behind the c/4 line near the wing root and before the c/4 line
in the range of the wing tips. The resulting total wing neutral points and the
geometric neutral points according to Eq. (3-7) are also shown in Fig. 3-37. The
distance between aerodynamic and geometric neutral points is very large, particularly on the delta wing. The numerical data for this displacement are compiled in
Table 3-5. Comparisons between theoretically and experimentally determined local
neutral points of swept-back wings have been published by Hickey [29].
Additional test results on a series of delta wings from [85] are shown in Fig.
3-38. They have aspect ratios from 1 to 4. Lift slope dcL/da and neutral-point
displacement J xN/cu are plotted against the aspect ratio. Here, too, agreement
between theory and experiment is good.
In Fig. 3-39, the theoretical lift distribution over the span of a delta wing is
compared with measurements of Kraemer [85]. Agreement is very good for angles
of attack up to about a = 5°. Flow separation from the outer parts of the wings
b Swept-back wing
72
70
0.8
10
0.2
02
0.
00
0.8
7.00
02
0#
49S
0.8
100
02
00
00
t8
10
Figure 3-36 Lift distribution clc/cLcm of three wings without twist of Table 3-5 and Fig. 3-33,
cL, = 2n; cm = Alb = mean wing chord. Curve 1, simple lifting-line theory of Multhopp. Curve
2, extended lifting-line theory of Weissinger. Curve 3, lifting-surface theory of Truckenbrodt.
Curve 3a, lifting-surface theory of Wagner (five-chord distributions).
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 159
c Delta wing
b Swept-back wing
a Trapezoidal wing
Local neutral-point position
Figure 3-37 Local neutral-point positions of three wings without twist of Table 3-5; cl = 27r;
lifting-surface theory. Curve 1 according to Truckenbrodt, curve 2 according to Wagner.
N25 = geometric neutral point of the whole wing; N = aerodynamic neutral point of the whole
wing.
Figure 3-38 Lift and neutral-point positions
of delta wings of various aspect ratios with
taper
2
3
X = e;
cp = 0.68cy.
Comparison
of
theory and experiment from Truckenbrodt.
Profile NACA 0012. (a) Lift slope. (b) Neutral-point displacement; J x N = distance of
aerodynamic neutral point N from geometric
neutral point N2$.
160 AERODYNAMICS OF THE WING
12
Measurements
10
Azt
L
N 0S
-
' T
I
Theory
0.0
0.2
0
02
a
2y/
oe
--
0.8
10
Figure 3-39 Lift distribution c1c/2bcti of a
delta wing of aspect ratio ,i= 2.3; profile
NACA 65A005 according to measurements
of Kraemer; comparison with lifting-surface
theory of Truckenbrodt [84].
causes strong deviations of the measured lift distribution from theory for the large
lift coefficients.
The local neutral-point positions are compared with theory in Fig. 3-40. Here
again, satisfactory agreement is found. For the same wing, the measured pressure
distributions for a few sections along the span are compared in Fig. 3-41 with
theory according to Eq. (3-120). In general, the agreement is satisfactory. The
340 Local neutral-point
positions of a delta wing of aspect
= 2; comparison of theory
ratio
[84] and measurements [NACA
TN 1650]. Profile NACA 0012.
Figure
V
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 161
=0396
U0.
Itl
NA CA 0012
0
0,2
a#
x
01
as
10
C
rj=060//
7f=0,0
1
Theory
IL
T_
92
1-
Of
-x
I
I
06
09
2
10
04'
x
as
09
70
C
0
02
04t
x
06'
OR
10
0
02
04f
x
0.e
08
C
1.0
Figure 3-41 Pressure distribution
over wing chord for the delta wing
of Fig. 3-40; theory [84] and
measurements [NACA TN 1650].
cL = 0.585; profile NACA 0012.
deviations between theory and experiment can be partially explained by the fact
that theory is valid for infuiitely thin profiles and, therefore, does not account for
the profile thickness.
Now, comparisons of experiment and theory will be made for unswept wings
(rectangular wings). Figure 3-42 illustrates lift slope versus aspect ratio. The
theoretical curve has been computed according to the multipoint method of Scholz
[77] ; it is in agreement with the curve for the extended lifting-line theory in Fig.
3-32. The test points from several sources follow the theoretical curve well. In Fig.
3-43, the neutral-point positions for the same series of rectangular wings are plotted
against the aspect ratio. The neutral-point shifts considerably upstream of the c/4 line
when the aspect ratio A is reduced. Also included are measurements on rectangular
plates that are in good agreement with theory.
Results for a series of swept-back wings of constant chord are presented in Fig.
3-44. For both lift slope and neutral-point position, the measurements are in good
agreement with theory. Note particularly that the lift slope of the swept-back wing,
especially with a large aspect ratio, is considerably smaller than that of the unswept
wing, p = 0. This reduction of the lift slope through sweepback can be assessed
particularly well by considering the swept-back wing of infinite aspect ratio. Figure
3-45 depicts a span section b of an unswept and of a swept-back wing of infinite
span. The section of the unswept wing produces the lift
L = I U!bccL--a
Let the swept-back wing with sweepback angle
be inclined to make, in the plane
of the incident-flow direction U., the angle of attack a equal to that of the
162 AERODYNAMICS OF THE WING
Gottingen
measurements
0
Profile Clark Y
after Zimmermann
Flat plate
°
I aft
Profile Go 409'f Wi:
r
ter
Flat plate
Figure 3-42 Lift slope of rectangular
wings of various aspect ratios A ;
comparison of theory and experiment.
Theory from Scholz (multiple-points
theory) [77]. Measurements from
Wieghardt, Scholz, and NACA Rept.
Profile
NACA 0015
after Scholz
1
V
I
Z
7
3
7
A
431.
unswept wing. Then, in the plane normal to the leading edge, the angle of attack is
a* = a/cos cp. For the lift of the swept-back wing, only the velocity component
normal to the leading edge, U. cos gyp, is effective. Thus, the cross-hatched surface
portion of the swept-back wing has a lift
L* =
2
2
'
(U cos cp) bCCLoo
°L
cos
99
2
U2 bcc'L a cos O
M
Hence, the lift coefficient of the swept-back wing is CL = L*/bc(p/2)U! _
CL a cos gyp, whereas that of the unswept wing is (CL ),,= O = CL oo a. The two lift
slopes are thus related by
C11CL
dC
da
(da
T-0
COST
(3-123)
0.25
0.20
0.05
b
1
2
3
A ---
5
5
Figure 3-43 Neutral-point position of rectangular plates; comparison of measurements and theory, from Scholz [771.
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 163
q
Theory
/ SP=O
xA
Measure ments
N
SP
to
Nzs
S
Figure 3-44 Lift and neutral-point
position of swept-back wings of con-
stant chord and various aspect ratios; sweep-back angle p = 45°; comparison of theory and measure-
ments from Truckenbrodt. Profile
NACA 0012. (a) Lift slope. (b)
Neutral-point displacement.
cas9'
Figure 3-45 Geometry and velocity components explaining the lift of swept-back wings of infinite span.
164 AERODYNAMICS OF THE WING
This relationship has been confirmed experimentally by Jacobs [37]. It is also valid,
to good approximation, for the pressure distribution along the chord.
To show the effect of the sweepback angle on the lift slope, Fig. 3-46
illustrates, for swept-back wings of constant chord, the lift slope dcL /da versus
sweepback angle and aspect ratio according to de Young and Harper [1031. For
large aspect ratios 11, the decrease in lift slope with increasing sweepback angle is
considerably stronger than for small aspect ratios. For r1 00, the cos .p law of Eq.
(3-123) is also shown for comparison.
The sweepback angle also strongly affects the circulation distribution over the
span. This is apparent in Fig. 3-47, which demonstrates the circulation distribution
along the span of a rectangular wing (cp = 0) and a swept-back wing (gyp = 45°). The
maximum value of the lift distribution of the swept-back wing is found at the outer
wing portion. Sweepback causes a shift of the station of maximum local lift from
the middle toward the outer end. Hence, the separation tendency of the swept-back
wing is increased at large angles of attack compared with the. unswept wing. In this
respect, sweepback produces unfavorable effects similar to a strong taper of an
unswept wing (see Fig. 3-28).
Let us deviate from the wing theory discussed here. A swept-back wing theory
has been developed by Kuchemann [48] that is not based on the Bimbaum normal
distribution over the chord. This method, partially empirical, takes into account the
wing thickness and the boundary layer, and also certain nonlinear effects. It
therefore agrees very well with test results.
A cylindrical body in a flow that is inclined against its generatrix (yawed
cylinder) may be subject to- complex three-dimensional flow processes in the
boundary layer. These are of considerable importance to the aerodynamic properties
of swept-back wings. At larger lift coefficients, both yawed and swept-back wings
undergo a strong pressure drop toward the rearward wing tip on the suction side
2Z
A cc
0
t
l
a
8
6
3
2
I
1
-50°
-900
-30°
200
-10°
0°
10
20°
300
M°
30°
Figure 3-46 Lift slope of swept-back wings of constant chord vs. sweepback angle p and aspect
ratio, from [103] ; extended lifting-line theory. Curve forA = oo: cbs cp law from Eq. (3-123).
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 165
Figure 347 Circulation distribution and distribution of the local lift coefficients over the span
for two wings of constant chord; aspect ratio it = 5, sweepback V = 0 and p = 45°; cL, = 27r;
1; lifting-surface theory of Truckenbrodt [84]
.
near the wing nose, as shown in Fig. 3-48. In this figure, the isobars for the suction
side of a yawed, inclined wing are seen. The fluid, decelerated in the boundary
layer, follows this pressure gradient and consequently a strong cross flow in
direction of the rearward wing sets in. Measurements of Jones [37] and Jacobs [37]
have shown that, therefore, a marked thickening of the boundary layer is caused on the
rearward wing tip and, as a consequence, a premature flow separation results. In
airplanes with swept-back wings, this departure of the boundary layer toward the
outside causes separation to occur first at the outer portion of the wing, in the
Figure 3-48 Evolution of cross flow in the
boundary layer of a yawed wing (swept-
back wing). Curves of constant pressure
(isobars) on suction side of the wing.
166 AERODYNAMICS OF THE WING
range of the aileron. This in turn causes the feared "roll-off" toward the stalled
wing. This initiation of separation at the outer portion of the wing, and thus the
undesirable "roll-off," can be avoided by providing the wing with a boundary-layer
fence (stall fence). This is a thin sheet-metal wall on the suction side of the front
wing portion that prevents cross flows in the boundary layer. Liebe [13] describes
the improvements in stall behavior by this provision. The work of Queijo et al. [13]
includes results of comprehensive measurements on the improvement of the
aerodynamic properties of a wing by means of boundary-layer fences. Compare also
the basic studies of Das [13].
Poisson-Quinton [68] makes a contribution to the theoretical and experimental
investigations on the problem of the aerodynamics of folding wings (wings with
adjustable sweepback angle).
For wings of small aspect ratio, an essential simplification of wing theory is
feasible, according to a proposal first made by Jones [36]. The basic concept of
this theory is that the perturbation velocities in the x direction in the flow field
about a slender wing are small compared to those in the transverse directions (y and
z directions). The potential equation is then reduced to that of a two-dimensional
flow in the yz plane (slender-body theory). In connection with this theory, the
method of Lawrence [54] for the computation of the lift distribution of wings of
small aspect ratio and the treatment of very strongly swept-back wings according to
[59] should be mentioned. The application of slender-body theory to wings of
extremely large thickness (covering of the wing contour with singularities) has been
attacked by Hummel [341.
3-3-6 Nonlinear Wing Theory
The wing theory treated
so
far establishes
a linear correlation between lift
coefficient and angle of attack. It is designated, therefore, linear wing theory. It is
known from experimental investigation that for wings of very small aspect ratio,
A < 1, lift coefficients cL are considerably larger than those obtained from linear
theory when plotted against the angle of attack. Figure 3-49 illustrates this behavior
for rectangular wings of aspect ratios A = 0.2, 0.5, 1.0, and 5.0 as compiled by
Gersten (21]. The dashed theoretical curves represent linear theory as discussed
earlier. Although linear theory produces the right lift slope (dcL/da)a=o even for
small aspect ratios, strong deviations of the measurements from linear behavior are
already obvious for small angles of attack.
All wing theories discussed so far are based on the concept that bound and free
vortices lie in the same plane. A linear relation between lift and angle of attack is
the necessary consequence.
This much simplified vortex model must be abandoned for a theoretical
explanation of the nonlinear relation between lift and angle of attack. A first trial
in this direction was made by Bollay [8]. He used a vortex model similar to Fig.
3-50a in which the free vortices no longer lie in one plane but rather are shed in the
downstream direction from the wing tips under the angle a/2 with the wing plane.
Bollay assumes that the bound vortices are constant over the span. Gersten [21]
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 167
0.8
0.6
0.2
a0
0.8
A= 05
o_ .
06
2
0
0.00
o
I
_j
r
.
0
00,
1
0.6
2
/000,
e
0.2
cj
I
A=10
cow
2
0.8
4
47
Figure 3-49 Measured lift coefficients CL vs.
11=5.0
0.2
a
0
8°
72 °
16°
20°
24°
angle of attack a for rectangular wings of aspect
ratios A = 0.2, 0.5, 1.0, and 5.0. Curve 1, linear
theory of Scholz. Curve 2, nonlinear theory of
Gersten.
refined this vortex model by prescribing a variable circulation distribution over the
span (Fig. 3-50b). The CL(a) curves based on this theory are given in Fig. 3-49 as
solid lines. They are in very good agreement with this theory (see Winter [102] ).
By the same theory, pitching moment, induced drag, and lift distribution along the
span have also been determined. Agreement between tests and theory is good in
these cases, too. Furthermore, the nonlinear theory has been extended by Gersten
to arbitrary wing shapes. It represents an extension of the lifting-surface theory of
Sec. 3-3-5 to the nonlinear angle-of-attack range. The cL(a) curves as determined
168 AERODYNAMICS OF THE WING
a
Ya
b
C'
z
Figure 3-50 Vortex model of nonlinear wing theory. (a) Vortex model of Bollay. (b) Vortex
model of Gersten.
from this theory and the comparison with test data are shown in Fig. 3-51 for a
swept-back and a delta wing.
It is known from test results that the aerodynamic coefficients of wings of
small aspect ratio are strong functions of the wing leading-edge design. This is true
particularly for swept-back and delta wings with sharp leading edges which, even at
very small angles of attack (a = 3°), promote flow separation from the leading edge
of the kind shown in Fig. 3-52. Starting at the wing tips, two vortex sheets form on
the two leading edges that roll up into free vortices when floating downstream.
This process was first discussed by Legendre [55] and has been treated in
°
Nonlinear
theory
o
I
i Linear
theory
IL
0
1.0r
Figure 3-51 Lift coefficient of sweptback wings with sharp leading edge and
small aspect ratio vs, angle of attack.
(- - -) Linear theory from Eq.
0,2
12'
16
090
24
) Nonlinear theory of
(3-101b). (
(o)
Measurements.
(a) SweptGersten.
back wing .1 = 1, X = 1, tp = 45°. (b)
Delta wing .4 = 0.78, X = 8.
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 169
Figure 3-52 Bursting of the free vortices of a delta wing according to Hummel. Aspect ratio
r = 0.78, taper x = 0.125. (a) Vortex formation shown schematically. (b) a = 20°, (3 = 0°, no
bursting. (c) a = 30°, g = 0°, bursting of the vortices at large angles of attack. (d) a = 20°,
0 = -10°, bursting of one vortex of yawed wing. (e) a = 20°, p = 0°, bursting of one vortex
through an artificial pressure rise.
numerous other publications [4, 10, 33, 55, 98]. The roll-up of vortex sheets has
been studied theoretically by Roy [71] and by Mangler and Smith [58, 80]. Roy
established details through numerous flow-pattern photographs. Under certain
circumstances, a striking change in the structure of the rolled-up vortex sheets can
be observed that can be termed bursting of the vortices. Figure 3-52b-e shows
smoke pictures of this phenomenon from Hummel [33]. The bursting of vortices
occurs (1) at large angles of attack in symmetric incident flow (Fig. 3-52c), (2) at
170 AERODYNAMICS OF THE WING
the vortex of the upstream-turned side of yawed wings (Fig. 3-52d), and (3) when
an obstruction is placed into the vortex flow (Fig. 3-52e). Naturally, the bursting of
vortices has a strong effect on the aerodynamic properties of the delta wing;
compare [4, 33]. These processes affect lift and pitching moment as well as drag.
Further investigations of nonlinear effects on wings of small aspect ratios,
especially on delta wings, are reported in [19, 32, 57, 67]. A very recent survey of
the aerodynamic properties of slender wings with a sharp leading edge has been
given by Parker [661.
3-3-7 Maximum Lift of Wings
of this chapter, the fluid was considered to be
incompressible and inviscid when establishing the theory of lift. The wing theory
based on this concept is in good agreement with measurements as long as the angle
of attack is small to moderate; see, for example, Figs. 3-38, 3-39, 3-42-3-44, and
3-49. Only in the range of large angles of attack does the effect of friction have
In the previous sections
significance for the lift. In particular, the maximum lift of a wing is not only
determined by its geometry, but it is also considerably affected by friction.
Determination of the maximum lift of a wing by strictly theoretical methods is not yet
possible. Cooke and Brebner [11] report on flow separation from wings in general
terms. Schlichting [73] presents the aerodynamic problems of maximum lift of wings
in comprehensive form.
From measurements it is known that the maximum lift coefficient is strongly
dependent on the geometric profile parameters (thickness, camber, nose radius) and
on the Reynolds number. In Sec. 2-5-1 this relationship was discussed briefly; see,
for example, Figs. 2-39 and 242-2-44: These previously reported results should be
supplemented by the statement that the maximum lift of an unswept wing is
essentially a problem of two-dimensional flow. A large aspect ratio of unswept
wings of finite span cannot have an important effect on flow separation and
consequently on the maximum lift because in this case the flow over the major
portion of the wing deviates only a little from plane flow. Quite different are the
conditions for wings of small aspect ratio. Here the flow around the wing tips
reaches to the middle of the wing. For strongly swept-back wings, which includes
delta wings, the flow conditions are particularly complex because the leading edge
acts in a similar way as the tips of an unswept wing. For these kinds of wings, even
the attached flow is much harder to assess than that for unswept wings, because the
flow directions in the boundary layer may deviate from that of the outside flow
(departure of the boundary layer to the wing tips, boundary-layer fence).
Contrary to unswept wings, the flow over strongly swept-back wings without
twist separates locally first at the wing tips because the lift load has its maximum
there (see Fig. 347).
When the angle of attack increases, the separated region expands inward in span
direction. This behavior is discussed in more detail in [26]. A very comprehensive
compilation of material on this behavior of swept-back wings at large angles of
attack and high Reynolds numbers has been given by Furlong and McHugh [161.
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 171
The effect of the aspect ratio and the sweepback angle on the maximum lift
coefficient will now be examined using some test results.
In Fig. 3-53, results are plotted for the maximum lift coefficient of rectangular
wings and swept-back wings of constant chord (p = 45°). The Reynolds numbers of
these measurements are Re 106. Figure 3-53a confirms that the maximum lift
coefficient for A > 2 is almost independent of the angle of attack. For very small
aspect ratios, CL max is somewhat larger than for large aspect ratios. Particularly
noteworthy in Fig. 3-53b is, for aspect ratios A < 2, the strong increase to values of
a - 30° in the angle of attack for which the maximum lift coefficient is obtained.
In Fig. 3-54 curves are given for the lift coefficients of a series of delta wings
plotted against the angle of attack. When the aspect ratio <A1 decreases, the lift slope
becomes considerably smaller, while the maximum lift coefficient and the
corresponding angles of attack increase. The lift slopes dcL/da of these wings have
been presented earlier in Fig. 3-38. Maximum lift coefficients CLmax for these and
additional delta wings are plotted in Fig. 3-55 against the aspect ratio. Comparison
7,4
7.2
10
1
0s
7
3
X
J 0.6
0.4
02
a
0
35°
30°
25°
Figure 3-53 Maximum lift coefficients of rectangular wings (gyp = 0) and swept-back wings of
constant chord (gyp * 0), Reynolds number Re
106. (a) Maximum lift coefficient CLmax Vs-
aspect ratio A. (b) angle of attack a for CL max
vs. aspect ratio .i. Curve 1, p = 0°; profile NACA
70°
0015, from Bussmann and Kopfermann [25].
2, p = 45°; profile NACA 0012, from
Truckenbrodt [85 ]. Curve 3, tp = 0°; 6 - 0.10,
'.
Curve
f
13
9°
0
Z
3
J1-=
4
6
mean values of various measurements. Curve 4,
p = 35°; 6 - 0.10, mean values of various measurements.
172 AERODYNAMICS OF THE WING
1.2
1.0
0O
0.5
x
-A-0
.
'I
i
83
1,61
i
- 02
'
1. 38
3.S4
-0.6
0°
10°
a
20°
30'
00
3. if
Figure 3-54 Lift coefficients CL vs. angle of attack « for delta wings of various aspect ratios .1;
taper X = s, thickness ratio 5 = 0.12, Reynolds number Re - 7 101, from Truckenbrodt [85 1.
with Fig. 3-53a shows that the increase in CLmax for small aspect ratios
is
considerably larger than for rectangular and swept-back wings. Also, Fig. 3-55b
shows a strong increase of CYCLmax at small aspect ratios in agreement with Fig.
3-53b. Experimental studies on the separation characteristic of delta wings have
been carried out by Truckenbrodt and Feindt [85] by means of simple wake
measurements.
Figure 3-55 Maximum lift coefficients of delta wings, Reynolds
number Re 106. (a) Maximum
lift coefficient cLmax vs. aspect
ratio .4. (b) Angle of attack a
for CL,max vs aspect ratio A.
Curve 1, delta wing; A = 0; pro-
file NACA 0012, from Lange
and Wacke [25]. Curve 2, delta
wing; A = 8; profile NACA 0012,
from Truckenbiodt [85 1. Curve
3, mean values of various measurements.
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 173
3-4 INDUCED DRAG OF WINGS
3-4-1 Drag of Wings of Finite Span
The total drag of a wing of finite span, Sec. 1-3-2, is composed of profile drag and
induced drag:
D = Dp + Di
(3-124)
The profile drag Dp is created by friction effects. It is almost independent of
the wing aspect ratio. A procedure for the theoretical determination of the profile
drag has been developed in Sec. 2-5-2. Experimentally, the profile drag can be
determined through wake measurements (momentum-loss measurements). The
induced drag D= exists only at a wing of finite span. It is created by the flow
processes at the wing tips and can be determined from the laws of inviscid flow. In
Sec. 3-2-1 it has been demonstrated that the induced drag is proportional to the
square of the lift.
The drag coefficient cD = D/Aq. of a wing with elliptic lift distribution is,
from Eq. (3-32b):
CD = CDp + CDi
(3-125a)
2
= CDp + _L
(3-125b)
where A, from Eq. (3-4a), is the wing aspect ratio.
There are two methods available for the determination of the induced drag.
They differ in their physical concepts. In the first method, the induced drag is
found from the pressure forces that act on the wing itself. In the second method,
the induced drag is obtained from energy considerations. The latter approach allows
the determination of the induced drag of only the whole wing. Conversely, the first
method produces, within the framework of simple lifting-line theory, the local
distribution of the induced drag. Truckenbrodt [86] summarizes the state of the art
of the drag of wings. Basic considerations to the drag problem stem from Jones
[38]. Also, the comprehensive compilation of experimental data of the wing
aerodynamics by Hoerner [30] must be mentioned.
3-4-2 Computation of Induced Drag
Application of the Kutta-Joukowsky theorem The induced drag of an unswept wing
of finite span from the Prandtl lifting-line theory, Eq. (3-18), is
s
Di = 2 f I (y) u- (y) d y
(3-126)
s
Here F(y) is the circulation distribution and wi(y) is the distribution of the induced
downwash velocity over the span -s <y <s, Eq. (3-19).
Now it shall be shown that Eq. (3-126) is also valid for arbitrary wing
174 AERODYNAMICS OF THE WING
planforms. Following Fig. 3-16, let the wing be replaced by so-called elementary
wings of the infinitesimal span dy and wing chord c(y). The vortex system of an
elementary wing (see Fig. 3-17) consists of a number of horseshoe vortices of width
dy arranged in series, one behind the other. In Fig. 3-56, two horseshoe vortices are
drawn that originate at the stations x1, y1 and X2, Y2 of the wing. Their respective
widths and circulation strengths are dy 1i dye and dr1, dr2 . Horseshoe vortex dl'1
induces at station X2, y2 the upward velocity d2 w21i whereas horseshoe vortex d r2
induced at station x1, yl the upward velocity d2w12. In analogy to the
Kutta-Joukowsky theorem [see Eq. (3-14)], the lifting-circulation elements dr'1 and
d r2 produce forces normal to the upward flow that are the result of the
upward-flow velocities d2w12 and d2w21, respectively. These forces represent
contributions to the induced drag. The vortex system of Fig. 3-56 produces the
partial induced drag
d4Di = -odT1 d2w12 dy1
-
dl'2 d2w21 dye
(3-127)
where the sense of rotation of the circulation elements has been taken into account.
Because dr = k dx, the induced upward velocities of the exterior induction
(y1 =Y2) are found from Eq. (341) with Eq. (3-50a), as long as yl Oy2, as
d 2w12 =
d2w21
4
_ i
4
x1 - x2
d r2
1
(y1
-- 2
2
i
dr1 Ji +
(y2 - y1)-
,/(x1
V (XI
- x2)2 + (Y1 - y2) 2
x2 - x1
dye
(3-128a)
1 dy1
(3-128b)
r (x2 - x1)2 + (y2 - y1)2
44
Figure 3-56 Explanatory sketch for computation of induced drag.
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 17/5
It can be seen that the second terms in the square brackets differ by their signs
only. When introducing Eq. (3-128) into Eq. (3-127), these terms do not contribute
to the induced drag, leading to:
d4D;
=
-
ar dr
2
(y 2 1 T'2
dyi dye
(3-129)
From these considerations there follows immediately that the location of the
lifting-circulation elements in the x direction does not affect the drag. Hence, the
relationship found earlier [Eq. (3-126)] for the unswept wing is valid for the total
induced drag of a wing of arbitrary planform. Since the induced downwash velocity
wj(y) from Eq. (3-19) depends only on the circulation distribution over the span,
the total value of the induced drag also depends only on the circulation distribution
over the span. It is independent of the arrangement of the elementary horseshoe
vortices in the. chord direction (flight direction). This result was realized very early
by Munk [63, 64] and is known as the Munk displacement theorem. Thus, it is
immaterial for the magnitude of the induced drag whether the circulation
distribution is caused by the wing planform (aspect ratio, sweepback, taper), by a
wing twist, or by camber of the wing surface.
Application of the energy law Although the total lift of a wing can easily be
computed by using the momentum law (Sec. 3-3-2), computation of the induced
drag by means of the momentum law is considerably more difficult because the
inclination of the vortex sheet has to be considered.* However, when using the
energy law, the inclination of the free vortex sheet (Fig. 3-21), relative to the
incident flow direction, can be disregarded. Since the induced velocities on surface I
(Fig. 3-21) are zero, the mass o U. dy dz permeating the area element dy dz of
surface II
per unit time undergoes an energy increase dEu =
wi,) dy dz. Here v. and w. are the induced velocities in the y and z directions,
respectively, and the area integral over dEn is the work done by the induced drag
U0D1 per unit time. Hence, after division by U.,
DZ
2
r f (vim
+
dy dz
(3-130)
(.III)
This relationship is valid for both not-rolled-up and rolled-up vortex sheets behind the
wing.
The equivalence of Eqs. (3-130) and (3-126) will now be shown for the notrolled-up vortex sheet. The induced velocity field very far behind the wing with
components v.(y, z) and
z) can be expressed through the two-dimensional
velocity potential 0(y, z) as
*Kraemer [791 conducted a more detailed study into the application of the momentum law
to the computation of the induced drag; see Sears [79].
176 AERODYNAMICS OF THE WING
ao
(3-13 la)
v00
ay
1VC0 = 7a
and
(3-131b)
Here 0(y, z) satisfies the potential equation
a-0
32
ay=+
ati2
(3-132)
=
Introduction of Eq. (3-131) into Eq. (3-130) and integration by parts, the first
integral with respect to y, the second with respect to z, yield, with Eq. (3-132),
e
2
['p
f [(P
];°c0 dz + f
=-oo
az
v=-00
I
rods
where surface II has been extended to infinity. Since the values of 0, a 013y, and
aO/az vanish at the boundaries y = ± oo and z = ± oo, whereas the potential
according to Eq. (3-48b) changes abruptly in the z direction for z = ±0 by the
amount Pu(y, 0) - 01(y, 0) = F(y), with Eq. (3-13lb) the induced drag becomes
4
Di = -
f
J r(y) woo (y) dJ
2
(3-133)
-S
The integration limits y = ± -o can be replaced by y = ±s, because 0(y, 0) = 0
beyond the wing span. Now, by introducing Eq. (3-20) with w. (y) = -2wi(y) into
Eq. (3-133), Eq. (3-126) is finally obtained, as was to be proved.
Equation (3-130). is valid for the not-rolled-up vortex sheet. Kaufmann [411 showed
that the same induced drag is obtained for the rolled-up vortex sheet, where it must
be assumed, however, that the cores of the two free vortices have finite velocities.
Practical computation of the induced drag From Eq. (3-126) the formula for the
coefficient of induced drag is obtained with y = T/bU.. and ai = wi/U. from Eq.
(3-71b), and with r7 = y/s as
I
CDi = q _
Z1 f Y(71)ai(77) d11
(3-134)
-1
the aspect ratio of the wing from Eq. (34). By expressing the
circulation distribution y by a Fourier polynomial as in Eq. (3-65a), the result of
where
;1
is
the integration becomes, with ai from Eqs. (3-73) and (3-65b),
CD i = :z /l' n a;j
(3-135a)
9Z=1
=
2
CL
?lT
- .rul na ,
zll
ya=3
(3-135b)
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 177
coefficient CL was determined from the
In the second relationship, the lift
coefficient al of Eq. (3-66a). In Eq. (3-135b) the first term represents the value for
the elliptic circulation distribution [see Eq. (3-31b)] . Since the second term is
always positive, the important theorem follows that the induced-drag coefficient for
elliptic circulation distribution is a minimum. This theorem is true for fixed aspect
ratio and for fixed lift coefficient CL. It was proved first by Munk [63].
A summation formula for the coefficient of induced drag can be derived in a
way similar to that which led to the summation formulas for the lift-related
coefficients of Table 3-1. It has the form
Dt =
3f
-rA
(3-136)
S' yn Biz: sin 0
where the values for c have to be computed from Eq. (3-83).
Equation .(3-134) for the induced drag will now be applied to trapezoidal wings
with symmetric twist. This example explains the relationship between twist and
induced drag. In Sec. 3-3-2 it was shown that the circulation distribution of a
symmetrically twisted wing can be put together from that of a wing without twist
and a zero distribution. In the same way as the circulation distribution was split up
in Eq. (3-63), the induced angle of attack of the twisted wing can be split up:
'%i(0 = ix xi'(71) + aio(71)
(3-137)
of the
twisted wing leads, with
Thus Eq. (3-134) for the induced drag
a = (da/dcL )CL , to
2
L
CDt=C2
7t11+C1CL+Co
(3-138)
1
with
C2 = 1,12 (L)2 r yu iu c'1
(3-139a)
1
1
C1 =
da j'
dcL J (Yunio
Yoaiu) tdi 7
(3-139b)
1
Co =CDio =1`i f Yomio d?1
(3-139c)
When the wing has no twist, the induced drag is determined just by the term C2.
The wing with twist requires, in addition to this term, a term C, that is
proportional to CL, and a term Co that is independent of cL. Here, the first term
represents a linear twist, the latter a quadratic twist. As can easily be seen by
comparison with Eq. (3-31b), the constant C2 is unity for an elliptic circulation
distribution. For the wing without twist the coefficient C2 signifies physically,
therefore, the ratio of the induced drag to its minimum value for elliptic circulation
distribution.
As an example, the induced drag of a trapezoidal wing with twist is given in
178 AERODYNAMICS OF THE WING
01
11-9
'6
I
V
b___V
001
a
7
0
02
0.o
06,
08
70
04C
06
08
10
C
-0.02
100
0
I
-0.0110
0.2
02
10
08
d
Figure 3-57 Induced drag of symmetrically twisted tapered wing of various aspect ratios i and
various tapers A from Eq. (3-138) (lifting-surface theory). (a) Wing planform with linear twist.
(b) Induced drag of wing without twist, from Eq. (3-139a). (c), (d) Twist contribution to the
induced drag from Eqs. (3-139b) and (3-139c).
Fig.
3-57. It is based on
symmetric linear twist with as(p) _ 1iilai. The
corresponding circulation distribution has been computed from the lifting surface.
Figure 3-57b indicates that the induced drag of the wing without twist has a
minimum for a taper X ~ 0.45 at all aspect ratios A. The value of this minimum is
only a little different from that of the elliptic wing (C2 = 1). For delta wings
(X = 0) and rectangular wings (), = 1), cDi is in many instances considerably larger
than for elliptic wings. The contribution that is independent of the lift, Fig. 3-57c,
is always positive. The sign of the contribution that is linearly dependent on lift,
Fig. 3-57d, depends on the value of the taper. When the taper X -- 0.45, this
contribution is zero for all aspect ratios. The reason is found in the nearly elliptic
circulation distribution over wings of this taper without twist. Furthermore, by
means of Eq. (3-135b), it can be shown that C1 = 0 for elliptic wings with arbitrary
twist and that
M
CDio = Co = ?s!1 f na,,
n-2
(3-140)
is the contribution of the induced drag caused by the twist for zero lift.
Investigations related to the establishment of the local drag distribution along
the span are compiled in [1].
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 179
3-4-3 Tangential Force and Suction Force
Tangential force Earlier, in Sec. 1-3-2, the wing-fixed components of the aerodynamic force, normal force, and tangential force were introduced together with the
flow-field-fixed components of lift and drag. For small angles of attack, the normal
force is almost equal to the lift, whereas the tangential force deviates considerably
from the drag, even for small angles of attack. The tangential force T is taken as
positive in the direction from the wing leading edge to the trailing edge. When
limiting the angle of attack to small or moderately large values (see Fig. 1-7a), the
tangential force coefficient CT = T/Aq becomes, with T= -X,
CT = CD -CLa
(3-141)
where CD is the coefficient of total drag as composed from profile drag and induced
drag from Eq. (3-125). By introducing CD from Eq. (3-125b) and a= (aa/aCL)CL
into Eq. (3-141), the tangential force coefficient for elliptic circulation distribution
becomes
CT = CDP -
(ddCccL
-
i) CL
(3-142)
With dcL/da from Eq. (3-80b) for the simple lifting-line theory and from Eq. (3-98)
for the extended lifting-line theory, Eq. (3-142) yields
CT=CDpCT = CDP -
(3-143a)
k2 _+1 2
n-A
CL
(3-143b)
where k = nA/cL . In Fig. 3-58, the difference (CDP - CT)/CL from Eq. (3-143) is
shown against the aspect ratio. Accordingly, for large aspect ratios this difference,
and thus the tangential force, are independent of the aspect ratio.
Figure '3-59 illustrates the dependence of the tangential force coefficient on the
lift coefficient for wings of various aspect ratios A. The profile drag coefficient had
been taken to be CDP 0.05. It is remarkable that the coefficient of the tangential
force assumes negative values when the lift coefficient CL > 0.5. In this case the
0,5
N
Q0.2
%1
-1
0
L
j
1
3
7
Figure 3-58 Tangential force coefficient
CT vs. aspect ratio .1. (1) Based on the
simple lifting-line theory, Eq. (3-143a).
(2) Based on the extended lifting-line
theory, Eq. (3-143b). cDp = coefficient
of profile drag; cLoo = 2a.
180 AERODYNAMICS OF THE WING
A-12
11
6
I12
CDi
CD
3
0.8
CT
Figure 3-59 Lift coefficient cL vs.
tangential force coefficient CT for
wings of various aspect ratios A,
from Eq. (3-143b). For comparison,
0.2
V
-0.15
-0.10
-0.05
I
0.05
0.70
0.15
0..2
kCD'6+
the drag polars cD(cL) are also
shown.
CT; CD --+
tangential component of the resultant of the aerodynamic forces
is directed
upstream along the wing chord. The drag polar curves CD(CL) are also included in
Fig. 3-59.
Suction force The discussions about the drag of wings of infinite span of Sec. 24-2
have shown that the flow around the leading edge of an inclined profile produces a
suction force in an inviscid fluid (Fig. 2-12a). This is the result of the strong
underpressure in the vicinity of the leading edge. Now, the suction force on wings
of finite span will be examined. The suction force is a part of the induced drag [Eq.
(3-124)], with the total drag being split into profile drag and induced drag.
Equation (3-125) is therefore the expression for the drag coefficient with suction
force. It has been pointed out in Sec. 24-2 that no suction force exists for very
sharp leading edges. In this case, the flow around the leading edge causes local
separation, eliminating the strong underpressure that results in a suction force.
Rather, the resultant force of the pressure distribution over sharp-edged noses acts
normal to the wing surface and, therefore, has the component La in the incident flow
direction. Thus the drag coefficients with and without suction force are, respectively,
CD = CDp + CDi
(3-144a)
CDp +CLa
(3-144b)
CD
The difference of the drag coefficients of Eqs. (3-144a) and (3-144b) yields the
suction force coefficient cS = S/Aq., *:
CS = CLOY
da
dCL
(3-145a)
CDi
1
2
TA) CL
*The suction force is considered positive when acting upstream.
(3-145b)
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 181
Comparison with Eq. (3-142) yields
CS = CDp -CT
(3-146)
Consequently, the quantity of Fig. 3-58 is a direct measure of the suction force. In
particular, cS = cL /27r for wings of very large span. This result is in agreement with
Eq. (2-77), remembering that the lift coefficient for smooth leading-edge flow cLS is
zero for symmetric profiles.
In conclusion, a few experimental results on wings of small aspect ratio
according to Hansen [44] will be presented, confirming the above considerations.
Figure 3-60 shows polar curves for a slender circular disk. To show the effect of the
suction force, the leading edge of the disk was formed in several ways, as can be
seen from Fig. 3-60. The suction force increases with the leading-edge nose radius.
The theoretical curves for the drag coefficient with and without suction force from
Eqs. (3-144a). and (3-144b) are added in this figure. The tests show the expected
result, namely, that the measured drag coincides with the theoretical curve,
including suction forces, when the leading edge is well rounded. When the leading
edge is very sharp, however, the measured drag lies close to the theoretical curve
without suction force. All measurements with differently formed leading edges lie
between the two theoretical curves.
3-5 FLIGHT MECHANICAL COEFFICIENTS
OF THE WING
3-5-1 Contributions of the Wing to Stability
The methods for the computation of the aerodynamic forces on a wing have been
discussed in detail in Secs. 3-2-3-4. This section will show how these methods can
be applied to the determination of the flight mechanical coefficients of the wing. A
survey on these coefficients has been given previously in- Sec. 1-3-3.
The flight mechanical coefficients are determined by the motion of the wing
0.5
0.4
0.3
i
Figure 3-60 Drag polars of circular wings of
Diskl
various degrees of leading-edge rounding, accord-
ing to measurements of Hansen [44]. Theory
according to Kinner [44]. The drag at CL = 0 has
been subtracted from the measured values. Disks
j
-Q1
0
I and II: cDp = 0.012; disk III: cDp = 0.008.
Curve 1, with suction force from Eq. (3-125);
OR
0.09
0.06
CD-.-
0.08
0.70
0.72
CD = CDp + 0.274cL. Curve 2, without suction
force from Eq. (3-144b); cD = cDp + 0.55cL.
182 AERODYNAMICS OF THE WING
and the wing geometry. In the following discussions, only those coefficients will be
considered that are significant for airplane stability. The coefficients that determine
maneuverability will be treated later in Chap. 8.
In addition to the wing, the other parts of the airplane (fuselage, empennage)
contribute, sometimes considerably, to these flight mechanical coefficients. These
contributions will be discussed later, too. In the present section, only the
contributions of the wing will be discussed.
The flight mechanical coefficients of the wing depend on numerous geometric
parameters of the wing, such as wing planform (aspect ratio, taper, sweepback),
twist, and dihedral (see Sec. 3-1-1). The dependence of the flight mechanical
coefficients on wing geometry is too varied to attempt a complete description of all
these interrelations. In some cases the contribution of the wing to the stability
coefficients of the whole airplane is small. Further investigations will be restricted
to the cases in which the wing makes an essential contribution. Reference will be
made to the summary reports of Betz [61, Schlichting [72], and Multhopp [61].
Of the two axis systems of Fig. 1-6, the experimental system will be used.* The
coefficients are defined in Eq. (1-21).
The motion of the airplane can be divided into a longitudinal motion and a
lateral motion, as has been explained in Sec. 1-3-3. During longitudinal motion, the
position of the plane of symmetry of the airplane does not change. This motion is
characterized by three parameters: flight velocity IT, angle of attack a, and pitching
angular velocity w,, (Fig. 3-61). The lateral motion is defined by sideslip angle
rolling angular velocity wX, and yawing angular velocity wZ (Fig. 3-61). The
stability coefficients are understood to be the changes of force and momentum
coefficients with the above motion parameters.
3-5-2 Stability Coefficients of Longitudinal Motion
Straight flight For longitudinal motion, the resultant aerodynamic force may be
represented by lift, drag, and pitching moment. Their dependence on the angle of
attack (see Fig. 3-61a) has been discussed in the previous section. The two most
important coefficients are lift slope dcLIda and pitching-moment slope dcMfdcL. The
latter determines the position of the aerodynamic neutral point of the wing by Eq.
(1-29). The lift slope dcLlda is presented for various wing. shapes in Figs. 3-25, 3-38,
3.42, 3-44, 3-46, and Table 3-5. The neutral-point positions of various wing forms
can be obtained from Figs. 3-37, 3-38, 3-43, 3-44, and Table 3-5. The flight
mechanical computations for the neutral-point position require great accuracy. The
neutral-point position depends strongly on the individual planform. In general,
therefore, it is required that for its determination the lift distribution should be
computed by using the lifting-surface theory (see Sec. 3-3-5).
Pitching motion Pitching motion is actually a nonsteady motion. In general it proceeds
slowly enough, however, that it can be treated as "quasi-steady." When the wing
*In what follows, the index e of forces and moments and their coefficients, which
indicates the axis system used, will be omitted.
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 183
Straight flight
\ cc
x
Figure 3-61 The motion modes of the wing. (a) Straight flight. (b) Yawed flight. (c)-(e) Rotary
motions: rolling, pitching, yawing.
(Fig. 3-62) performs a rotary motion with angular velocity coy about the lateral axis
through xS (Fig. 3-61 d), a vertical additive velocity VZ = wy(x - xs) is produced
that varies linearly over the wing chord. Together with the incident flow velocity V,
the rotary motion in chord direction produces an additive angle-of-attack distribution a(x) = VZ/V of magnitude
n: (x) = °y (x
xs)
(3-147)
This angle-of-attack distribution produces an additive lift distribution, the integra-
tion of which leads to an additive lift and an additive pitching moment. These
quantities are designated lift due to pitch rate and pitch damping. Both depend
linearly on wv. It is expedient, therefore, to introduce the coefficients acL /a 2 y as
lift due to pitch rate and acM/aQy as pitch damping, where Qy = wycjV is the
dimensionless pitching angular velocity and c,1 is the wing reference chord,
introduced earlier by Eq. (3-5b). These coefficients depend only on the wing
geometry and the position of the axis of rotation.
Now it will be explained how these two quantities can be determined and, in
184 AERODYNAMICS OF THE WING
Figure 3-62 Explanatory sketch for aerodynamic
coefficients of the pitching wing.
particular, how their values change with the position of the axis of rotation xs.*
It is evident that there is an axis of rotation x0 for which the lift due to
pitch rate is zero. For a rectangular wing, this axis of rotation lies at a distance 4c
from the leading edge, according to the Pistolesi's theorem (see Sec. 2-4-5). The
pitch damping, however, cannot be zero for any position of the axis of rotation.
For the computation of the lift due to pitch rate, the angle-of-attack distribution of
Eq. (3-147) is rewritten in the form
a (x) = V (x - xo) + y (xo - XS)
(3-148)
By setting xS = x0 in this equation, the second term becomes zero, whereas, by
definition, the first term produces zero lift due to pitch rate, (aCL /a Shy) = 0. The
contribution of the first term to the pitch damping will be expressed by
(aCMlaQy). The second term in Eq. (3-148) represents a constant angle of attack
and gives the total lift due to pitch rate as
acL
dcL xo - xs
a Sty s
du
c.
(3-149)
Here dcL/da is the lift slope of the wing. Equation (3-149) shows that the lift due to
pitch rate is a linear function of the position of the axis of rotation (see Fig. 3-63a).
The moment of the pitching motion is obtained from Eq. (1-28) as
CM = CMO -
XN -XS
C
CL
JU
which leads with Eq. (3-149) to the pitch damping:
GcM
(aQJ S
-
dclyi
(0 2,)O
xy - XSxo - xs dcL
da
cry
Cu,
(3-150)
*For flight mechanical computations, the axis of rotation coincides with the lateral axis
through the airplane's center of gravity.
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 185
This equation shows that the pitch damping depends parabolically on xs. In
particular, it is immediately obvious that for x s = xN and for xS = xo the
pitch damping has the same value, namely, (acM/a Qy )o , as in Fig. 3-63h.
To be able to compute the pitch damping from Eq. (3-150) for an arbitrary
position of the axis of rotation x5, the determination of (acM/a-2y)o and of xo/cN,
is required, whereas the coefficients dcL/da and xN/cL are known from earlier
discussions.
For xs = xN, Eqs. (3-149) and (3-150) yield
X0
xN
cA
cu
a rnz
as o
-
do acL
'
dcL (BSZyiN
(a cat
(3-15la)
(3-151b)
r
Thus the problem of determining the lift due to pitch rate and the pitch damping
for an arbitrary
position of the
axis
of rotation has been reduced to the
computation of the two coefficients (aCL/a.Qy)N and (acM/aQy)N for the position
of the axis of rotation in the neutral point. These latter two coefficients are
obtained from the lift and pitching moments as determined from lifting-surface wing
theory for the angle-of-attack distribution corresponding to Eq. (3-147):
a (x) =
x
xN
CA
DY
(3-152)
In Table 3-6, numerical data on the positions of the axis of rotation for zero lift
due to pitch rate and of the corresponding pitch damping are compiled for a
trapezoidal wing, a swept-back wing, and a delta wing (Table 3-5). Compare also
Garner [62] and Gothert and Otto [24] .
In the case of airplanes with a separate horizontal tail, the contribution of the
wing to the lift due to pitch rate is small compared with that of the tail surface. An
act
aay
a
acM
i9ay
b
Figure 3-63 Lift due to pitch rate (a) and pitch damping
(b) vs. position of axis of rotation xs. xN = neutral-point
position; x,, = axis of rotation for vanishing lift due to
pitch.
186 AERODYNAMICS OF THE WING
Table 3-6 Position of the axis of rotation for zero-lift
due to pitch rate and corresponding pitch damping
for a trapezoidal wing, a swept-back wing, and a
delta wing*
irapezoiaai
wing
y
2.°
CAI
a r_
BS?,) o
swept-oacK
wing
Delta wing
0.533
0.485
0.604
-0.358
- 0.498
- 0.285
*The distance d x0 is measured relative to the geometric
neutral point N25 , the position of which is given in Table 3-5.
Table 3-6 is based on data from Table 3-5.
accurate computation is therefore not required. On the other hand, in the case of
all-wing airplanes, whose total pitch damping is almost completely produced by the
wing, a more accurate computation may be required, depending on the specific case.
3-5-3 Stability Coefficients of Lateral Motion
Yawed flight During steady yawed flight, the incident flow condition is determined
by the sideslip angle a (Fig..3-61b) in addition to the angle of attack a. Because of
the asymmetric incident flow, in addition to lift, drag, and pitching moment,
additive forces and moments are created, namely, the side force due to sideslip Y,
the rolling moment due to sideslip M, and the yawing moment due to sideslip MZ
(see Fig. 1-6). They vary linearly with R for small angles of sideslip. The derivatives
of the dimensionless coefficients with respect to the sideslip angle are, therefore,
independent of the sideslip angle. They are termed stability coefficients of lateral
motion. All three coefficients for a wing are strongly dependent on the sweepback
angle and the dihedral angle.
First, the wing without dihedral will be treated, followed by a discussion of the
effect of the dihedral angle.
A fundamental treatment of the yawed wing was first given by Weissinger [94].
The resulting theory can be designated as simple lifting-line theory in the sense of
Sec. 3-3-3. In this theory it is assumed that free vortices are shed only from the
edge and that these are parallel to the incident-flow direction. The
inclination of the free vortex strips against the wing axis of symmetry is of
trailing
secondary effect on the results of the Weissinger theory. In this theory, Weissinger
[94] introduced a correction factor taking into account the effect of the wing end
flaps on the rolling moment due to sideslip. Later Gronau [25] made comprehensive computations of the rolling moment due to sideslip and the yawing
moment due to sideslip, mainly for swept-back and delta wings, using the method
of the extended lifting-line theory (Sec. 3-3-4). Here, too, the effect of the free
vortex strip inclination has only approximately been taken into account.
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 187
Test results from various sources for the rolling moment due to sideslip of
rectangular, swept-back, and delta wings against the aspect ratio are shown in Fig.
3-64. For comparison, theoretical curves from [25] are added. Agreement between
measurements and theory is good. With decreasing aspect ratio, the rolling moment
due to sideslip decreases strongly. This presentation reveals further that sweepback
causes a strong increase in the rolling moment due to sideslip. This means that the
rolling moments due to sideslip of swept-back and delta wings, in particular, are
strongly dependent on the lift coefficient (see also Kohlman [45] ). Figure 3-65
gives the corresponding plots for the yawing moment due to sideslip.
A wing in asymmetric flow (yawed wing) corresponds aerodynamically to a
wing of asymmetric planform (see Fig. 3-17). Based on this concept, its circulation
distribution can be computed from the extended lifting-line theory (Sec. 3-3-4), or
the lifting-surfaces theory (Sec. 3-3-5). However, the required computation effort is
considerably greater than for symmetric incident flow because of the asymmetry of
the wing planform.* The result of such a computation is the circulation distribution
over the span, measured normal to the incident flow direction. With it, the total lift
and the neutral-point positions are obtained .from the formulas of Sec. 3-3-2. In this
way the rolling moment about the experimental x axis is also determined.
The rolling moment due to sideslip, accordingly, is proportional to the total
lift. The circulation distribution for the three wings without twist examined earlier
has been computed by this method at three different angles of sideslip. In Fig. 3-66,
the circulation distributions for
0° and a = 10° have been presented over the
span coordinate, measured normal to the direction of the incident flow. For all
three wings, the circulation distribution changes very little with the angle of sideslip.
*Only after electronic computers became available has this procedure gained practical
value.
7,4
7.2
'
1.0
I
2
0.8
I
I
i
I
4
5
6
A
Figure 3.64 Rolling moment due to sideslip of rectangular wings, swept-back wings, and delta
wings vs. aspect ratio A; theory of Gronau. Measurements: curve 1, rectangular wing (gyp = 0°),
(o) from Bussmann and Kopfermann, (v) from NACA Rept. 1091. Curve 2, swept-back wing of
constant chord (gyp = 45°), (.) from Gronau, (o) from NACA TN 1669, (v) from Jacobs. Curve
3, delta wing (X = 8), (s) from Gronau, (0) from Lange and Wacke.
188 AERODYNAMICS OF THE WING
7.
I
3
0.2
7
2
1
4
5
8
Figure 3-65 Yawing moment due to sideslip of rectangular wings, swept-back wings, and delta
wings vs. aspect ratio e; theory of Gronau. Measurements: curve 1, rectangular wing (gyp = 0°),
from Bussmann and Kopferrnann. Curve 2, swept-back wing of constant chord ('p = 45°), from
Gronau. Curve 3, delta wing (A = 8), from Gronau.
It should be mentioned that this behavior is typical for wings without dihedral. The
locations of the neutral points for three angles of sideslip are inscribed into the
wing planforrns. The coefficients of the rolling moment due to sideslip and the
coordinates of the neutral points are compiled in Table 3-7. The yawing moment
due to sideslip is caused by the difference in drag of the two wing-halves. It consists
of a contribution from the profile drag and one from the induced drag. The latter
a
b
c
08
as
fl-10° I
j
-o
04
02
0.2
0
-1.0
-0.5
0
0
0.5
1.0
-10
-0,5
0
0.5
10
?7
Figure 3-66 Circulation distribution of three yawed wings without twist in sideslipping, based
on the lifting-surface theory of Truckenbrodt. Angle of attack a = 1, measured in the section
parallel to the incident flow direction, y = r/Vb. Geometric data of the wings from Table 3-5.
(a) Trapezoidal wing: V = 0'; A = 2.75; A = 0.5. (b) Swept-back wing: p = 50°; A = 2.75;
X = 0.5. (c) Delta wing: , = 5 2.4°; = 2.31; 1\ = 0.
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 189
Table 3-7 Coefficients of the rolling moment due-to
sideslip and position of the neutral point for a = 0°,
50, and 100 for a trapezoidal, a swept-back, and a
delta wing*
Trapezoidal
wing
1 acMX
CL
ap
0°
50
s
100
yi9
s
Swept-back
wing
Delta wing
0.111
0.717
0.580
0.219
0.221
0.223
0.781
0.794
0.814
1.027
1.024
1.018
00
0
0
0
5°
-0.010
-0.020
-0.060
-0.123
-0.050
-0.102
100
*Distances are measured in the wing-fixed coordinate
system from the leading-edge station of the wing middle (root)
section. Table 3-7 is based on Table 3-5 (see Fig. 3-66).
contribution is proportional to the square of the lift, precisely like the induced
drag. The side force due to sideslip of a wing without dihedral can be determined
approximately by considering that the profile drag of a yawed wing acts parallel to
the direction of the incident flow, but the induced drag acts in the direction of the
wing axis of symmetry. Consequently, in asymmetric incident flow, only the
component of the profile drag cy = CDP sin j3 acts in the direction of the wing-fixed
lateral axis. Hence the side force slope is
acY
ao
= CDP
(3-153)
Wing with dihedral The dihedral of a wing is understood to be the inclination of
the left and the right wing-halves relative to the xy plane (Fig. 3-61b). The dihedral
angle is designated as v; in the general case v may vary along the span. The stability
coefficients of yawed flight acy/a ji, acM/ao, and acMZ/a j3 of the wing are strong
functions of the dihedral. For the total airplane, the contributions of the wing to
the side force due to sideslip acy/a(3 and to the yawing moment due to sideslip are
relatively small. Conversely, the contribution of the wing to the rolling moment due
to sideslip of the total airplane is of decisive significance. Selection of the wing
dihedral is governed exclusively by the requirement of a flight mechanically
favorable value of the rolling moment due to sideslip.*
The aerodynamic effect of the dihedral in yawed flight is due to the angle of
attack, increased by the amount .da of the leading half-wing, and the angle of
attack decreased by d a of the trailing half wing. This angle d a can be determined
The value of the rolling moment due to sideslip of the total airplane depends on the
vertical position of the wing relative to the fuselage in addition to the dihedral.
190 AERODYNAMICS OF THE WING
as follows: From Fig. 3-67a and b, the lateral component of the incident flow
V, = V sin a produces on either half-wing a component normal to the wing of amount
V,, = ± V. sine
Together with the component VX = V cos (3 of the incident flow, the additive
angle-of-attack change becomes
Ja=
y"`
z
= ± tan fi sin'v
(3-154a)
= +(3v
(3-154b)
The second relationship is valid for small angles of sideslip and small dihedral angles.
The exact establishment of the dihedral angle from a given wing geometry must be
based on Eq. (3-11).
The lift distribution of a wing with dihedral during yawed motion may thus be
determined by adding the geometric angle-of-attack distribution of the wing without
dihedral to the antimetric* twist from Eq. (3.154) (see also Fig. 3-67). As in Fig.
3-67, the lift (L/2 +A L/2) acts on the leading wing-half, the lift (L/2 -J L/2) on
the trailing wing-half. L is the lift for symmetric incident flow and AJ L/2 is the
additive lift of one wing-half in yawed motion.
For the determination of the aerodynamic forces of the two wing-halves, it has
to be realized that, as Fig. 3-67 demonstrates, the resultant incident flow direction
is deflected up by the angle J a on the leading wing-half but deflected down by the
same angle Aa on the trailing wing-half. These angle-of-attack changes are relative to
the angles of symmetric incident flow. The resultant aerodynamic forces on the two
wino halves undergo the same direction changes. The exact determination of the side
force due to sideslip, of the rolling moment due to sideslip, and of the yawing
moment due to sideslip requires computation of the lift distribution on the given
wing for the antimetric angle-of-attack distribution in Eq. (3-154).
Approximate expressions for the aerodynamic quantities of the yawed wing
with dihedral giving an explicit account of their dependence on the dihedral angle
and the total lift coefficient can be gained, however, through the following estimations:
The side force due to sideslip resulting from the dihedral is, from Fig. (3-67b),
Y=2
2
sin v= dLv
with 2 = 2 V2 2 da
da
v
being the additive lift of one wing-half, where, from Eq. (3-154b), d a = v(i.
Consequently, the coefficient of the side force due to sideslip becomes
dap
=
(dCa)y1,2
(3-155)
The coefficient (dcL /da) of this equation can be determined exactly only by
computation of the lift distribution on a wing with antimetric twist as in Fig. 3-67c.
*Translator's note: The word antimetric, found repeatedly in the text, has been coined by the
authors to avoid an inconvenient expression like "acting or pointing in opposite directions but
being of equal magnitude."
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 191
C
da
-7
7
t
57
i
da=-v.
r
Figure 3-67 Aerodynamics of a wing with dihedral
in sideslipping. (a) Wing planform, xy plane. (b)
Dihedral yz plane. (c) Additive antimetric angle-ofattack distribution due to the dihedral ]a = ± vg. (d)
Incident flow resultant and aerodynamic-forces resultant of the two wing halves.
As an approximation, however, it may be assumed that this coefficient is equal to
that of a wing without twist of aspect ratio A/2. Equation (3-155) reveals, then,
that the coefficient of the side force is proportional to the square of the dihedral
angle and independent of the total lift coefficient. Introduction into Eq. (3-155) of
the lift-slope value for an aspect ratio A/2 from the extended lifting-line theory of
Eq. (3-98) yields, for the unswept wing,
192 AERODYNAMICS OF THE WING
n!1
a cy
Vk2+4+2
where k = 7rA/ct
v4
(3-156)
..
Measurements that confirm the above formula are reported in the summary
account [721.
The rolling moment due to the dihedral is (see Fig. 3-67b)
Mx=2d2 LYL
where YL designates the distance of the center of the additive lift of the half-wing,
:dL/2, from the wing root. For the rolling-moment coefficient cMx = Mx/qAs results,
corresponding to the above discussion,
acMx
_
(dCL)()v
( 3-157)
Here, (nL )v = YL IS is the dimensionless distance of the center of the additive lift of
the half-wing from the wing root. This equation demonstrates that the coefficient of
the rolling moment due to yaw as a result of the dihedral is proportional to the
dihedral angle and independent of the total lift coefficient. To a good approximaWith this value, the following
31r=0.424.
tion, (iL)v can be set equal to
approximate relationship for the unswept wing is obtained. Here (dcL /da)v from
Eq. (3-98) for one-half of the aspect ratio A 12 is again introduced.*
acMx
4
8f3
3n
n!1
1/k2
+ 4 +2
V
(3-158)
The additive yawing moment due to sideslip resulting from the dihedral is very
small in general. Its sign is such that it tends to turn the leading half-wing further
upstream. This comes about because, as shown in Fig. 3-67d, the resultant
aerodynamic force at the leading half-wing is being turned toward the front and at
the trailing half-wing toward the rear. Measurements are given in [72].
Rolling motion A linearly variable vertical velocity VZ = Wxy is obtained when the
wing executes a rotary motion about the longitudinal axis as in Fig. 3-68a (see also
Fig. 3-61c). Superposition with the incident flow velocity V results, from Fig. 3-68b
and c, in an additive antimetric angle-of-attack distribution
._l a (?7)
= 77-x
(3-159)
where Qx = coxs/V is the dimensionless angular rolling velocity. This angle-of-attack
distribution produces an antimetric lift distribution along the span and consequently
a moment about the x axis that always tends to inhibit the rotary motion. This
moment is designated rolling moment due to roll rate or roll damping. The
*Through evaluation of Eq. (3-100) with Eq. (3-154), Eq. (3-158) may be established as
solution for the elliptic wing.
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 193
b
V
Figure 3-68 Aerodynamics of the rolling wing. (a) Wing planform. (b) Additive antimetric
angle-of-attack distribution A s = rDX. (c) Resultant incident flow direction and resultant
aerodynamic force of the two wing-halves.
asymmetric force distribution along the span furthermore produces a yawing
moment, the so-called yawing moment due to roll rate. These two moments are
proportional to the dimensionless rolling angular velocity QX , making their
In determining the
coefficients acMX/aQX and acMZIaQX independent of
aerodynamic force of the two wing-halves from Fig. 3-68c, it should be noted that
.,
relative
to the symmetric incident flow direction, the resultant incident flow
direction of the downward-turned wing-half is deflected upward by the angle A a
and that of the upward-turned wing-half deflected by the same angle Aa downward.
Consequently, the local aerodynamic forces on the two wind halves undergo the
same directional changes.
For the determination of the roll damping of a given wing, the antimetric
circulation distribution ya(r1) over the span has to be established following a
procedure for the computation of the lift distribution of Sec. 3-3. Hence the roll
damping is, from Table 3-1,
i
OCM.X
8Q
=
-:li yar/ all
(3-1oua)
with
(3-160b)
is independent of the total lift
coefficient of the wing. The roll-damping coefficients of the three wings (trapezoidal, swept-back, delta) examined earlier are found in Table 3-5.
Accordingly, the roll-damping coefficient
aCMX/8Q.(
A simple approximate formula for the roll damping of unswept wings is
obtained by setting a =,n in Eq. (3-100):
acMX
t
QX
'
7cA
V1c2 ; 4
(3-161)
2
194 AERODYNAMICS OF THE WING
of this approximate formula
not recommended for wings of strong
sweepback. A more accurate computation should be made. Schlottmann [75]
Use
is
demonstrates the theoretical determination of the roll damping of slender wings by
a nonlinear theory and experimental confirmation of the computed results.
The yawing moment due to roll rate tends to turn the downward-turning
wing-half forward. This behavior can be understood as follows: On the downward-
moving wing-half, the resultant incident flow direction
is
turned upward and
consequently the resultant aerodynamic force turns forward. On the upward-moving
wing-half, the resultant aerodynamic force consequently turns rearward. On a
section y of an unswept wing, the force dD' = dD1 - dL d a = A(al -.i a) is thus
obtained in direction of the undisturbed incident flow.* Here dDi = dLat from Eq.
(3-17). Integration produces the induced yawing moment due to roll rate:
+8
MZ= f (a1-da)ydL
-8
With dL = e V T dy from Eq. (3-14) and 'y = T /b V and Al a from Eq. (3-159), the
coefficient of the yawing moment is determined as
cMZ = A f y (a4 - S2x-l) ?7 d
The total circulation y is composed of the contribution of the wing in
symmetric incident flow ys and of the contribution ya created by the rotary motion
for as = rl that is, y = ys + SQ X`ya . Correspondingly, the induced angle of attack
becomes ai = ais + Qxaia . Introduction of these relationships into the above
equation yields
1
ac1Z
aslx = 11J L(aia - ?7) Ys -t- atsYai 17 d r1
(3-162)
For wings without twist of elliptic circulation distribution, a simpler evaluation
of the integral is possible. The circulation distribution is obtained from Eq. (3-65),
specifically, ys with p = 1 and ya with p = 2, 3, ... , M. Correspondingly, the
induced angles of attack are found from Eq. (3-73), at3 with n = 1 and at,, with
n = 2, 3, ... , M. By taking into account Eq. (3-65b) with 'r = cos 6 and
drl = - sin t$ dzg, the integration over 0 c 6 c n yields the relationship
a em'
as2x
= 4 Aa1(6a2 - 1)
(3-163)
where al = cL/7rA and a2 = -(2/7rA)(acMX/aQX) from Eqs. (3-66a) and (3-66b).
By introducing Eq. (3-161), the following approximate formula for the
coefficient of the yawing moment due to Toll rate is finally found:
*The profile drag is not taken into account.
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 195
acMZ
i
a QX
4
yk'2 ± 4 - 1
}"kz _+4 + 2
L
(3-164)
Thus the coefficient of the yawing moment. due to roll rate is proportional to the
total lift coefficient.
Yawing motion Motion of the airplane about the vertical axis produces additive
longitudinal velocities, of reversed signs on the two wing-halves (Fig. 3-69; see also
Fig. 3-61e). An asymmetric lift distribution over the span results, creating a yawing
moment and a rolling moment. This yawing. moment counteracts the rotary motion
and is termed, therefore, yaw-damping or turn-damping of the wing. It is very small
compared with that of the whole airplane, and therefore its computation is omitted.
The rolling moment created by the yawing motion is termed rolling moment
due to yaw rate or turning rolling moment. The turning rolling moment tends to turn
the forward-moving wing-half upward.
The turning rolling moment can be computed in the following way: Through
the rotary motion with angular velocity wZ from Fig. 3-69b, a linear distribution of
the longitudinal velocity is generated along the span:
V. (y) = V - a.zJ
(3-165)
To ensure that the wing is a strearnlayer of this inhomogeneous flow field, the
kinematic flow condition
V. (x, y) + w(x, v) = 0
(3-166a)
.y
c
a;
,y
Figure 3-69 Yawed wino. (a) Wing planform. (b)
Velocity distribution. (c) Resultant incident flow
z
yx
direction of the two wing-halves.
196 AERODYNAMICS OF THE WING
(3-166b)
must be satisfied at each point of the wing surface. Here V, is the component of
the longitudinal velocity Vx normal to the wing chord; thus, V, = a Vx (Fig. 3-69c).
In a homogeneous flow field, Vx = V, the kinematic flow condition becomes
w/V + a = 0. Comparison with Eq. (3-166b) demonstrates that the inhomogeneous
flow is equivalent to a homogeneous flow with the mathematical angle of attack
ab = a
y =all - Qj)
(3-167)
where nZ = w2s/V. Consequently, the circulation distribution for inhomogeneous
flow can be computed by using the computation procedures of Sec. 3-3, but by
applying an angle-of-attack distribution as in Eq. (3-167). The resulting circulation
distribution
is
lb = b Vyb. A wing strip of width dy thus produces a lift
dL = 0 Vx I'b dy = 0V(1 -0zrt)Tb dy, and the rolling moment becomes
8
S
Mx = - f ydL = -of V-,rby dy
-s
-8
And further, the coefficient of the rolling moment cMx = Mx/qAs is found as
1
cMx = -A f (1
- Ox 77) Yb77 d?7
-1
The circulation distribution yb at the angle of attack ab, from Eq. (3-167),
may be composed as follows:
Yb =
(3-168)
a -2zYa
Here yu is the circulation distribution for a = 1, and ya that for as = rl [see Eq.
(3-160)]. For the sake of simplicity, let a wing without twist a = const, be considered.
Introduction of Eq. (3-168) into the equation for the rolling-moment coefficient yields
f
1
1
acm,
8S2z - A f yu?72 d + A yak d
-1
-1
do L
cz
(3-169)
This equation demonstrates that the coefficient of the rolling moment due to
yaw rate is proportional to the angular velocity Q, and the total lift coefficient CL .
For a wing without twist of elliptic circulation distribution, the following
approximation formula is obtained with Eq. (3-98) and after evaluation of the
integrals similar to those of Eqs. (3-162) and (3-163):
cMx
8z
t
4
1 + k2
t
Vk2 + 44
T1
CL
(3-170)
2
This expression is nearly independent of the aspect ratio. For the three wings that
have been examined (trapezoidal, swept-back, and delta, Table 3-5), the coefficients
of the rolling moment due to sideslip are listed in Table 3-8.
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 197
Table 3-8 Coefficients of the rolling moment due to yaw rate for
a trapezoidal, a swept-back, and a delta wing based on Table 3-5
1 acM"
CL &Q
Trapezoidal wing
Swept-back wing
Delta wing
0.410
0.443
0.378
For an accurate computation of the rolling moment due to yaw rate, it must be
realized that the rotary motion of the wing causes the free vortices to be shed into
a lateral flow. Hence the portions of the free vortices that lie on the wing produce
an addition to the lift and thus to the rolling moment due to yaw rate. A detailed
computation reveals that the coefficient of the rolling moment due to yaw rate for
wings of small aspect ratio (ii < 3) depends considerably on the position of the axis
of rotation.
3-6 WING OF FINITE THICKNESS
AT ZERO LIFT
3-6-1 Displacement Problem of the Wing
The theory of the wing of infinite span as discussed in the previous sections of this
chapter was based on the assumption of a very thin profile (skeleton). For the
theory of the wing of finite span, the extension from the skeleton theory to the
theory of the inclined wing of finite thickness (profile teardrop) has long been
available (Sec. 2-4-4). A similar extension for the wing of finite span and finite
thickness is still lacking. However, for the wing of finite span and finite thickness of
the wing profiles (symmetric profiles), there does exist a computational method that
allows the determination of the displacement effect of the wing and thus of the
pressure distribution on the surface of such wings, provided that the lift is zero. It
represents, therefore, a teardrop theory for wings of finite span; note the
publications of Keune [42] and Neumark [65] and compare also [43]. The method
of singularities is used in which the body within the flow field is replaced by a
system of sources and sinks. The fundamentals of this method have been established
in Sec. 2-4-3 for two-dimensional flow and applied to the problem of an airfoil in
plane flow.
For the assessment of the effect of compressibility in both two-dimensional and,
in particular, three-dimensional flow, it is important to know the maximum
perturbation velocity on the wing. The computational procedures, treated in the
following sections, for the velocity disturbance on wings of finite span and finite
thickness are of significance, therefore, for the aerodynamics of the wing of high
subsonic velocities.
198 AERODYNAMICS OF THE WING
3-6-2 Method of Source-Sink Distribution
Source system of the wing of finite span For the computation of the threedimensional flow field about a slender body resembling a wing of finite span and
finite thickness, a distribution of three-dimensional sources and sinks is established
in the plane of the surface A (wing planform plane). An area element dx dy carries
the source strength
d2Q (x, y) = q (x, y) d x d y
(3-171)
when q(x, y) designates the source strength per unit area. The source strength q(x, y)
must satisfy the so-called closure condition
f f q (x, y) d.x d y= 0
(3-172)
(A)
in order to form a closed body shape. Compare also the corresponding expressions
for the plane case, Eq. (2-92).
Velocity distribution on the wing contour Superposition of the velocity field,
produced by the source distribution, with a translational flow of velocity U., the
direction of which lies in the source plane (Fig. 3-70), produces a closed stream
surface that can be interpreted as the contour of the wing of finite thickness;
compare again Sec. 2-4-3. Let u, v, and w be the velocities induced by the source
distribution (perturbation velocities) and let z(t)(x, y) = z(x, y) be the shape of the
wing contour, symmetric to the xy plane. Then the condition for tangency of the
velocity resultant on the entire contour is
az
w = ( U + u) ax + v
y
az
=U00 ax
( 3 - 173a)
(3-173b)
This is the kinematic flow condition. Since, for slender bodies, the velocities u and
v are small compared to the incident flow velocity U., except for the immediate
Figure 3-70 Flow around a wing of
1
-b-2S-
finite span and finite thickness at zero
lift.
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 199
vicinity of the leading edge and the wing tips, it is sufficient to work with the
simplified form, Eq. (3-173b). This is, formally, the same relationship as in the
plane case. Both for the kinematic flow condition and for the computation of the
pressure distribution on the contour, the velocity components u, v, w are required
on the contour. For slender bodies, it is sufficient, however, to compute the
velocity components in the wing plane, as in the teardrop theory of plane flow.
This simplifies the problem considerably.
The source distribution of Eq. (3-171) constitutes a three-dimensional source.
Thus, the velocity potential of the source distribution q(x', y') at a point x, y, z is
obtained as
1
4d
ff
q(x'.y')dx'dy'
N
(3-174)
(x - x')2 -± (y - y')s -L
(A)
where the integration is performed over the wing area A covered by sources. The
corresponding velocity components are found from Eq. (3-45). At a point x, y of
the wing plane z = 0, they become*
it (x, y, 0) = 4
1
ff q (x',
y')
(x - x') dx' d y'
-
(x - x')2 + (y -y')
23
(3-175a)
(A)
V (x,
y, 0)
4-
['f
(y-y")dy'dx'
q (x',
(3-175b)
y'(x - x')2 _L (y - y')23
.(A)
(3-175c)
w(x, y, 0) = ± 1 q(x, y)
The upper sign is valid for z > 0, the lower for z < 0. Hence, the induced velocities
normal to the xy plane are discontinuous across the source layer (wing plane).
Introduction of Eq. (3-175c) into the kinematic flow condition Eq. (3-173b)
yields
q (x, y) = 2 U,n
az
-x
(3-176)
Consequently, the source strength is proportional to the slope of the contour in the
zx plane [see also Eq. (2-90h)]. The formulas obtained by properly introducing Eq.
(3-176) into Eqs. (3-175a) and (3-175b) describe the velocities added at the location
of the wing by flow displacement (profile teardrop) of the wing. Presentation of
these formulas is omitted. For the wing of infinite span (plane problem), Eq. (2-94)
yields
*Because of the singular points of the integrands in Eqs. (3-175a)-(3-175c), integration of
Eq. (3-175a) must be conducted first over x' and then over y'. For Eq. (3-175b) the reverse
order of integration is necessary. If the integration is to be performed in a different order,
however, the Cauchy principal value must be taken for the second integration in either
equation.
200 AERODYNAMICS OF THE WING
Z
U P1
1
Uro
is
(' az dx'
J ox, x - x'
(3-177)
0
Also, in this case vpl = 0.
It should be stated here that the velocity differentials (perturbation velocities)
of wings of finite span in Eqs. (3-175a), (3-175b), and (3-176) are proportional to
the profile thickness ratio S = tic, in analogy to the two-dimensional profile theory
(Eq. (3-177)]. The above linear theory is sufficiently accurate for all practical
purposes up to about S = 4.
The resultant velocity on the contour is
We=j(U"-f-v2 U.+It
(3-178)
when quadratic terms in u and v are neglected.
3-6-3 Results of the Teardrop Theory
for Wings of Finite Span
Rectangular wing of finite span In the simple case of a rectangular wing c(y) = c of
constant profile over the span, the contour is represented by z(x, y) = z(x) for
-s <y <s. Hence, introduction into Eqs. (3-176) and (3-175a) and integration over
y yield
u
UPI
Uco
Uo
,
1
du
(3-179)
Uro
with
du
U11
_
1
1
2x
f
az f 2 _
s -Fy
ax'
(x - x')2 + (s ± y)=
dx'
(x -. x')1 ± (s - y)s x - x
I
0
It is easily verified that the quantity J u is negative in general. This means that
the perturbation velocities on the contour of a finite wing are. reduced in
comparison with those on the infinitely long wing. For the wing middle (root)
section, y = 0, the perturbation velocity becomes
du
__
U"
t
I
7L
f
0
az
ax' (
_-_
s
1
dx'
Y(x-x')2-82 x - X'
,r
(3-180)
When the aspect ratio is small, it follows with Z = z/c and X = x/c that
u
TIT 11
i azZAInA (A.=small)
.z
a Y=
(3-181)
For A = 0 this reduces to u = 0.
On the parabola profile Z = 26X(1 -X) of thickness ratio 5 = t/c, the
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 201
maximum (perturbation) velocity at station x = c/2 of the middle section y = 0 is,
from Eqs. (3-179) and (3-180),
U ax = 7T 5A sinh-1
Thus, for the plane case,
(3-182
1
it follows that Umax pi/U., = 46/7r, in agreement
A1.
with Eq. (2-97).
Results for the rectangular and the parabolic profiles are given in Fig. 3-71.
Figure 3-71a shows the maximum perturbation velocity, which lies at X= 0.5, as a
function of the aspect ratio for the two sections r7 = 0 and 77 = 0.5. In Fig. 3-71b
the maximum perturbation velocities are depicted for various aspect ratios over the
span coordinate. In conclusion, it should be stated that the maximum perturbation
velocity on the wing of finite span becomes noticeably smaller than on the wing of
infinite span only for aspect ratios A < 2.
Elliptic wing The theory of Sec. 3-6-2 for the computation of the perturbation
velocities and the above example for the rectangular wing were based on
approximate methods valid for small thickness ratios 8. One example of an exact
solution will now be given.
A wing of elliptic planform and elliptic profile as in Fig. 3-72 is a general
ellipsoid of which the two axial ratios are very different. Let a 1, b, , c1 be the three
semiaxes of the ellipsoid. Then
(3.183a)
(3.183b)
*The trigonometric functions will be given as sinh-' rather than arcsinh.
0.2
a
i
0.5
1.5.
1.0
20
2.5
3.0
0.2
0.4
0.5
0,8
1,0
/1
Figure 3-71 Maximum perturbation velocities on rectangular wings with parabolic profile at zero
lift. For infinite span Umax p1 = (4/71)g U. at station x/c = X = 0.5. (a) Dependency on aspect ratio
A. (b) Dependency on span coordinate n = y/s.
202 AERODYNAMICS OF THE WING
tr -2c,
For
Figure 3.72 The geometry of the elliptic wing
with elliptic profile (general ellipsoid).
general ellipsoid, the velocity
a
distribution on the contour can be
determined in closed form. The pressure distribution on the surface of the ellipsoid
in a flow parallel to the x axis is, from Maruhn, Chap. 5 [40],
Ic ) 2
=1
A2
1
a
((x
ai)
2
z
a,)'
2
\
z
\ci\ci/
+\bi! \bi)
2
(3-184)
where c p = (p -p.)1(Q12)U.' is the dimensionless pressure coefficient and A =
A(bl /a1 , cl /bl ), a quantity that depends on the two axis ratios of the ellipsoid.
Equation (3-184) demonstrates that the pressure minimum and thus the
velocity maximum lie at x = 0. This velocity maximum is constant along the y axis.
From Eq. (3-184), cp min = 1 -A 2 = 1 - (Umax/Uco)2, with Umax = UC, + Umax
being the maximum velocity on the contour. Hence, the maximum perturbation
velocity becomes
Q.1n11X
iTCr
_ A (), i1) - 1
(3-185)
In Fig. 3-73, the ratio Umax/Umax pl is plotted against the aspect ratio A for
various thickness ratios 5, where Umax p1 = 5U... The curve S ->0 represents linear
1,0
0.5
X
co
E
0.2
0.5
1.0
7.5
2,0
0.5
3,0
3.5
V0
4.5
5.0
A
Figure 3-73 Maximum perturbation velocity on an elliptic wing with elliptic profile as in Fig.
3-72 for zero lift, at x = 0, -s <Y s +s. For the plane problem it is umaxpl = 5U..
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 203
theory. The curves for the other values of 5 show the deviations of the exact
solutions from linear theory.
Swept-back wing Another example is the swept-back wing of constant chord. The
wing of infinite span (see Fig. 3-74) is considered first. Its sweep-back angle is cp and
the profile z(x, y) = z(xr) is constant over the span, where xr is the x coordinate of
the middle (root) section. The wing sections at large distance from the plane of
symmetry are always in quasi-two-dimensional flow. Its velocity distribution can be
determined by assuming an incident flow normal to the leading edge of the
magnitude U. cos gyp. The result is a perturbation velocity in the x direction that,
for the swept-back wing, is smaller by the factor cos p than that for the unswept
wing (plane problem):
(3-186)
oo) = up, cos p
Now the velocity distribution on the middle section is to be computed. From
Fig. 3-74, x - x' = x - x' - ly' I tan gyp. Introduction of this relationship into Eq.
(3-175a), together with Eq. (3-176), yields
u(Xr)
U
_
1
C
0
az
8x'
f
Xr- Xr - y' taal qi
(xr - Xr- j tan 92)2 + y'2'
tly' dxr
U
The integration requires special caution [see footnote to Eq. (3-175)]. The result
for the middle section is
u xr = cosrp
U0.
UPI
z 2x tanh-1 (sinT)
This relationship was first published by Neumark [65]. The first term represents the
velocity distribution on a wing section far away from the wing plane of symmetry
Figure 3-74 Geometry of the swept-back wing
of infinite span.
204 AERODYNAMICS OF THE WING
50
-.N,
section of swept-back wings of infinite span with
o ti..
00,
0.2
Figure 3-75 Velocity distribution at the middle (root)
INN
N
Xr-
O..Y
0..5
0.8
10
generalized parabolic profiles from Eq. (2-6) at zero
lift. Sweepback angle p = -45°, 0°, + 45°. (a) Relative
thickness position Xt = 0.2. (b) Relative thickness
position Xt = 0.3. (c) Relative thickness position Xt =
0.5.
as in Eq. (3-186). The second term represents the change in velocity distribution
caused by wing folding. In the case of backward sweepback (gyp >0), the
perturbation velocity in the front part of the middle section is reduced, and in the
rear part of the middle section it is increased.
The above equation has been extended to generalized parabolic profiles. The
result is presented in Fig. 3-75 for profiles with relative thickness positions
Xt = xt/c = 0.2, 0.3, and 0.5. The curves for the sweepback angles cp = -45°, 0°,
and +45° show a very considerable influence of sweepback on the velocity
distribution over the middle section. The maximum perturbation velocities are
shown once more separately in Fig. 3-76 over the sweepback angle.
Figure 3-76 Maximum perturbation velocity at middle (root) section of swept-back wings of constant
chord and infinite span vs. sweepback angle gyp; see
Fig. 3-75.
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 205
if
-0
0. 70
1.2
023
1.0
-
/
'
71
050 '
1
I
1
075'i 0.90
0.2
ti
1.00
0
-0.2
0,2
0X
-
Figure 3-77 Velocity distribution of swept-back wing
of constant chord, with aspect ratio .1 = 2.0 and
sweepback angle V= 53° at zero lift for several
sections along the span, according to Neumark. Wing
profile: parabolic profile Xt = 0.5.
rnaxO)
(4/ir)(8 cos VU°,) = maximum perturbation velocity
of swept-back wing of infinite span at section y = -.
For a swept-back wing of constant chord and finite span, corresponding
computations have been made by Neumark [65]. The velocity distribution u of a
wing of aspect ratio A1= 2 and sweepback angle cp = 53° is illustrated in Fig. 3-77
for various sections along the span. It is related to the maximum perturbation
velocity of the swept-back wing of infinite span at a section far away from the wing
root [Eq. (3-186)]. For the same wing, the lines of constant velocity (isobars) are
drawn on the wing planform in Fig. 3-78. This figure demonstrates particularly well
that, as a result of the sweepback, the maximum perturbation velocity increases
Figure 3-78 Isobars of a swept-back wing of
aspect ratio .1= 2, with sweepback angle
p = 53° at zero lift, from [651. Curves
u (x, y) lu p, max(Y _ oo) = const; see Fig.
3-7 7 .
206 AERODYNAMICS OF THE WING
considerably in the vicinity of the wing middle (root) section, and that the velocity
maximum at this middle section has shifted far back.
Investigations on the pressure distribution over the middle section of a
lifting swept-back wing of infinite span have been conducted by Kuchemann and
Weber [48], and those on a swept-back wing with an arbitrary symmetric profile by
Weber [92].
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in "Wing Theory," pp. 169-297, Cambridge University Press, Cambridge, 1956.
71. Roy, M.: On the Rolling-up of the Conical Vortex Sheet Above a Delta Wing, Prog. Aer.
Sci., 7:1-5, 1966; Rech. Aer., 56:3-12, 1957; Z. Angew. Math. Phys., 9:554-569, 1958:
Z. Flugw., 7:217-227, 1959; Comp. Rend. Acad. Sci. (Paris), 234:2501-2504, 1952.
72. Schlichting, H.: Neuere Beitrage der Forschung zur aerodynamischen Fli gelgestaltung
(Umriss, Verwindung, Rumpfeinfluss), Jb. Lufo.,1:113-132, 1940.
210 AERODYNAMICS OF THE WING
73. Schlichting, H.: Aerodynamische Probleme des Hochstauftriebes, Z. Flugw., 13:1-14,
1965.
74. Schlichting, H.: Einige neuere Ergebnisse aus der Aerodynamik des Tragflugels, Jb. WGLR,
11-32, 1966; Rev. Roum. Sci. Tech., Ser. Mec. App., 13:191-213, 1968.
75. Schlottmann, F.: Stations a and instationare Rollmomentenderivativa schlanker Fliigel in
Rollbewegung, Z. Flugw., 22:331-344, 1974.
76. Schmidt, H., A. Kupper, H. Schubert, K. Bausch, and H. Sohngen: Mitteilungen fiber
Ergebnisse der Prandtlschen Tragflugeltheorie, Lufo., 15:219-274, 560-562, 1938. Filotas,
L. T.: J. Aircr., 8:835-836, 1971. Jaeckel, K.: Lufo., 16:47-52, 1939; 17:47-53, 81,
1940.
77. Scholz, N.: Beitrage zur Theorie der tragenden Flache, -Ing.-Arch., 18:84-105, 1950;
Forsch. Ing.-Wes., 16:85-91, 1949/1950; J. Aer. Sci., 16:637-638, 1949. Byrd, P. F.:
Ing.-Arch., 19:321-323, 1951.
78. Sears, W. R.: Some Recent Developments in Airfoil Theory, J. Aer. ScL, 23:490-499,
1956.
79. Sears, W. R.: On Calculation of Induced Drag and Conditions Downstream of a Lifting
Wing, J. Aircr., 11:191-192, 1974. Kraemer, K.: Z. Angew. Math. Mech., 48:193-202,
1968.
80. Smith, J. H. B.: Improved Calculations of Leading-Edge Separation from Slender, Thin,
Delta Wings, Proc. Roy. Soc. A, 306:67-90, 1968.
81. Thomas, F.: Aerodynamische Eigenschaften von Pfeil- and Deltafliigeln in Bodenniihe, Jb.
WGL, 53-61, 1958. Ackermann, U.: Jb. WGLR, 104-109, 1962. Braunss, G. and W.
Lincke: Z. Flugw., 10:282-285, 1962. van der Decken, J.: Jb. DGLR, 59-76, 1969.
Gersten, K.: Abh. Braunschw. Wiss. Ges., 12:95-115, 1960. Gersten, K. and J. van der
Decken: Jb. WGLR, 108-125, 1966. Hummel, D.: Z. Flugw., 21:425-442, 1973.
82. Thwaites, B. (ed.): "Incompressible Aerodynamics-An Account of the Theory and
Observation of the Steady Flow of Incompressible Fluid Past Aerofoils, Wings, and Other
Bodies," pp. 255-368, Clarendon Press, Oxford, 1960.
83. Truckenbrodt, E.: Beitrage zur erweiterten Traglinientheorie, Z. Flugw., 1:31-37, 1953;
5:259-264, 1957. Laschka, B. and F. Wegener: Z. Flugw., 7:39-45, 1959.
84. Truckenbrodt, E.: Tragflachentheorie bei inkompressibler Stromung, Jb. WGL, 40-65,
1953; Z. Angew. Math. Mech., 32:277, 1952; 33:165-173, 1953. Niemz, W.: Jb. WGL,
130-133, 1956. Rogmann, H.: Z. Angew. Math. Mech., 42:356-358, 1962.
85. Truckenbrodt, E.: Theoretical and Experimental Investigations on Swept and Delta Wings
in Incompressible Flow, J. Aer. Sci., 21:637-638, 1954; Z. Flugw., 2:185-201, 1954.
Kraemer, K.: Jb. WGL, 179, 1961; Z. Flugw., 10:297-305, 1962. Truckenbrodt, E. and E.
G. Feindt: Z. Flugw., 6:97-102, 1958.
86. Truckenbrodt, E.: Die entscheidenden Erkenntnisse fiber den Widerstrand von Tragfliigeln,
Jb. WGLR, 54-66, 1966; Tech. Sci. Aft. Spat., 97-111, 1967. Riegels, F.: Jb. WGL,
44-55, 1952.
87. van Dyke, M.: Lifting-Line Theory as a Singular-Perturbation Probletn, Arch. Mech.- Stos.
16:601-614, 1964; J. App. Math. Mech., 28:90-102, 1964. Kerney, K. P.: AIAA J.,
10:1683-1684, 1972.
88. von Karman, T. and J. M. Burgers: General Aerodynamic Theory-Perfect Fluids, in W. F.
Durand (ed.), "Aerodynamic Theory-A General Review of Progress," div. E, Springer,
Berlin, 1935, Dover, New York, 1963.
89. von Karman, T.: Neue Darstellung der Tragfliigeltheorie, Z. Angew. Math. Mech.,
15:56-61, 1935; "Collected Works," vol. III, pp. 171-178, Butterworths, London, 1956.
Fuchs, R.: Ing.-Arch., 10:48-63, 302, 1939.
90. von Karman, T.: Lanchester's Contributions to the Theory of Flight and Operational
Research, J. Roy. Aer. Soc., 62:80-93, 1958; "Collected Works," vol. V, pp. 213-234, von
Kirman Institute, Rhode-St. Genese, 1975.
91. Wagner, S.: Beitrag zum Singularitatenverfahren der Tragflachentheorie bei inkompressibler
Stromung, Ing.-Arch., 36:403-420, 1967/1968; J. Aircr., 6:549-558, 1969. Jordan, P. F.:
Jb. WGLR, 192-210, 1967.
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 211
92. Weber, J.: The Calculation of the Pressure Distribution over the Surface of Two-
Dimensional and Swept Wings with Symmetrical Aerofoil Sections, ARC RM 2918,
1953/1956; 2993, 1954/1957; 3026, 1955/1957; 3098, 1957/1959.
93. Weinig, F.: Beitrag zur Theorie des Tragfliigels endlicher insbesondere kleiner Spannweite,
Lufo., 13:405-409, 1936; 14:434-437, 1937; NACA TM 1151, 1947.
94. Weissinger, J.: Der schiebende Tragflugel bei gesunder Stromung, Jb. Lufo., 1:138-181,
1940; Lil.-Ges. Lufo., S2:5-51, 1938/1939; ZWB Lufo., TB 10, no. 7, 1943. Fuchs, R.:
Lil.-Ges. Lufo., S2:65-82, 1938/1939. Jaeckel, K.: Z. Angew. Math. Mech., 33:65-66,
1953. Weinig, F.: Lufo., 13:45-54, 1937.
95. Weissinger, J.: Uber eine Erweiterung der Prandtlschen Theorie der tragenden Linie, Math.
Nachr., 2:45-106, 1949; NACA TM 1120, 1947. Reissner, E.: Proc. Nat. Acad. Sci.
U.S.A., 35:208-215, 1949.
96. Weissinger, J.: Der Tragflugel endlicher Spannweite, in S. Fligge (ed.), "Handbuch der
Physik, vol. VIII, Stromungsmechanik II," pp. 402-431, 434-437, Springer, Berlin, 1963.
97. Weissinger, J.: Neuere Entwicklungen in der Tragflugeltheorie bei inkompressibler
Stromung, Z. Flugw., 4:225-236, 1956.
98. Werle, H.: Sur 1'Eclatement des Tourbillons d'Apex d'une Aile Delta aux Faibles Vitesses,
Rech. Aer., 74:23-30, 1960.
99. Wieselsberger, C.: Experimentelle Priifung der Umrechnungsformeln, in L. Prandtl, C.
Wieselsberger, and A. Betz, "Ergebnisse der Aerodynamischen Versuchsanstalt zu Gottingen," vol. 1, pp. 50-53, Oldenbourg, Munich, 1921. Betz, A.: in L. Prandtl,
"Gesammelte Abhandlungen zur angewandten Mechanik, Hydra- and Aerodynamik," p.
389 (footnote), Springer, Berlin, 1961.
100. Widnall, S. E. and T. M. Barrows: An Analytical Solution for Two- and Three-Dimensional
Wings in Ground Effect, J. Fluid Mech., 41:769-792, 1970. Kida, T. and Y. Miyai: Aer.
Quart., 27:292-308, 1976.
101. Wieghardt, K.: Uber die Auftriebsverteilung des einfachen Rechteckfliigels fiber die Tiefe,
Z. Angew. Math. Mech., 19:257-270, 1939; NACA TM 963, 1940. Ginzel, I.: Jb. Lufo.,
1:238-244, 1940.
102. Winter, H.: Stromungsvorgange an Platten and profilierten Korpern bei kleinen Spannweiten, Forsch. Ing.-Wes., 6:67-71, 1935; NACA TM 798, 1936. Flachsbart, 0.:. in L.
Prandtl, C. Wieselsberger, and A. Betz, "Ergebnisse der Aerodynamischen Versuchsanstalt
zu Gottingen," vol. IV, pp. 96-100, Oldenbourg, Munich, 1932.
103. Young, J. De and C. W. Harper: Theoretical Symmetric Span Loading at Subsonic Speeds
for Wings Having Arbitrary Plan Form, NACA Rept. 921,. 1948. Young, J. De: J. Aer.
Sci., 22:208-210, 1955.
CHAPTER
FOUR
WINGS IN COMPRESSIBLE FLOW
4-1 INTRODUCTION
The theory of the wing in incompressible flow was discussed in Chap. 2 for the
two-dimensional problem (infinite span) and in Chap. 3 for the three-dimensional
problem (finite span). In this chapter the wing in compressible flow will be treated.
Both subsonic and supersonic flows will be considered, depending on whether the
flow velocities are lower or higher than the speed of sound, respectively. The
connection between these two kinds of flow is formed by the transonic flow. Flows
with very high supersonic velocities, so-called hypersonic velocities, are designated as
hypersonic flows.
The influence of compressibility must be taken into account at Mach numbers
higher than Ma 0.3. Compressible flow is of great significance for flow about
wings, because the Mach numbers in aeronautics are, in general, considerably higher
than 0.3.
The discussions of this chapter will be organized similarly to those of Chaps. 2
and 3, in that first the airfoil of infinite span in compressible flow (profile theory)
and then the wing of finite span in compressible flow will be treated. From gas
dynamics it is known that the compressible flows of subsonic and supersonic
velocities are basically different; the same is true, naturally, for wing flows (see Fig.
1-9). For the theoretical treatment of compressible wing flow, the so-called linear
theories will be applied predominantly, because they yield results that can be
interpreted easily and thus allow the establishment of general validity and of practical
conclusions. For the theoretical considerations, mainly inviscid flow will be assumed
as in Chaps. 2 and 3.
Besides the textbooks on gas dynamics listed in Section II of the Bibliography,
213
214 AERODYNAMICS OF THE WING
basic questions of compressible wing flow are treated by Taylor [94], Prandtl [73,
74], von Karman [1011, Howarth [34], Robinson and Laurmann [78], Sears [83],
Kuo and Sears (47], Heaslet and Lomax [30], Jones and Cohen [39] , and Garrick
[26]. Furthermore, more recent results and understanding of the theory of the
aerodynamics of wings in compressible flow are presented in progress reports for
certain time intervals by, among others, von Karman [102, 104], Ashley et al. [31,
Kuchemann [45], Schlichting [81], Landahl and Stark [48], and Hummel [351.
Problems of experimental wing aerodynamics are treated by Frick [24]. In this
connection, the comprehensive compilations of experimental data on the aerodynamics of lift and drag by Hoerner and Borst [32], and Hoerner [31 ] should be
mentioned.
4-2 BASIC CONCEPT OF THE WING
IN COMPRESSIBLE FLOW
4-2-1 Temperature Effects in Compressible Flow
It is a peculiarity of all compressible flows that their aerodynamic processes are
always coupled with thermodynamic processes. The pressure changes in the flow, in
general, are connected to temperature changes that may be determined from the
equation of state [Eq. (1-la)] .
Stagnation flow An outstanding station in the flow about a body is, according to
Fig. 4-1, the front (upstream) stagnation point, at which the velocity is zero. The
flow quantities at the stagnation point will be designated by the index 0. The
pressure in the undisturbed flow of velocity w is p., the density p., and the
temperature T.. At the stagnation point, the velocity is wo = 0, and the pressure,
density, and temperature are po, °o, and To, respectively. A pressure increase
dp = po --p,c, takes place on the streamline incident on the stagnation point, which
causes
a temperature increase 4T = To - T.. The pressure coefficient at the
stagnation point is obtained with steady, that is, isentropic compression as
cpo
P-0
1}
Poo W
2
_
y
.,..
y Maw
{(1+7;'M)_
-1
1
-(4=1-)
O°
The dependence of the pressure coefficient at the stagnation point cpo on Ma
is shown in Fig. 4-2a. For moderately high Mach numbers of the incident flow,
woo
P., T.
Figure 4-1 Temperature rise through compression.
WINGS IN COMPRESSIBLE FLOW 215
3.0
3.0
2.5
Isentropic
(continuo
Isentropic
(continuous)ir
'0.
184 o
Z.0
1.5
15
1/
With shock wave
(discontinuous)
With shock wave
(discontinuous)
Approximation
1.0
0
I
I
10L
0
J
Z
1
3
a
Z
1
3
Ma
Figure 4-2 Compressible stagnation flow of air with y = 1.405. (a) Pressure coefficient at
b
stagnation point. (b) Temperature ratio.
particularly in the subsonic range, Eq. (4-1) is reduced by binomial expansion to
1 + 4Ma,., as also shown in Fig. 4-2. Agreement of the approximation with
cP o
the exact formula, Eq. (4-1), is quite good up to Ma. = 1. For Mao, -p 0, Eq. (4-1)
becomes the well-known formula for the stagnation pressure of incompressible flow,
po - p. = (o ./2)w2 , which is -the basis for velocity measurements with the Prandtl
impact-pressure tube (pitot tube). Such a tube measures the pressure difference
(Po -p.) in compressible flow as well.
At an incident flow of supersonic velocity, the pressure changes from p, to po
discontinuously through a shock wave located somewhat upstream of the stagnation
point (Fig. 4-2a). The pressure change can be determined in this case by first
computing the pressure jump across the shock wave from the equations of the
normal shock. In the subsonic flow behind the shock wave, the pressure change is
isentropic. The result of this computation is
c+1
PO
y
(y-k1)2Ma00-
17-1
2[2yMa-(y-1)J
2
yMa;0
()
At very high Mach numbers, May, -* -, the pressure coefficient approaches a finite
value, which for air of y = 1.405 is CP0 max = 1.84. It can be seen from Fig. 4-2a
that for supersonic incident flow, the pressure increase at the stagnation point with
unsteady compression (which describes the physical reality) is considerably smaller
than that obtained from the computation of steady compression.
Temperature The pressure increase of the stagnation point is always tied to a
temperature increase. It is obtained from the energy equation as
216 AERODYNAMICS OF THE WING
T =T 00 T 2c
woo-
(4-3a)
0
TTo
1+y
,, 1
Ma.2o
(4-3b)
00
This temperature increase at the stagnation point is shown in Fig. 4-2b. It is equally
valid for steady and for unsteady compression. It increases with the square of the
velocity, and therefore, reaches appreciable values in the supersonic incident flow
range. Note that a temperature increase according to Eq. (4-3) occurs not only at
the stagnation point and its vicinity but, approximately, everywhere along a solid
wall. In a thin layer (friction or boundary layer) close to the wall, the kinetic
energy of the moving gas is transformed into heat through viscous effects (see Fig.
4-3). This results in heating of the wall by an amount d T = T - T,,, which can be
represented approximately by a relationship similar to Eq. (4-3). It can be realized,
therefore, that a "heat cushion"
is
found over the entire surface of a body
immersed in a flow of high velocity. In the immediate vicinity of the stagnation
point this heating is produced by compression, and on the remaining portion of the
surface by friction.
4-2-2 Friction Drag on a Flat Plate
in Compressible Flow
In Sec. 2-5-2 the friction drag of wing profiles in incompressible flow was
discussed. Particularly, in Fig. 248 the influence of the Reynolds number on the
drag of a flat plate in chord-parallel incident flow was demonstrated. The insight
gained then will now be extended to the case of compressible flow.
Wall flow The compressible boundary layer is decisively affected by the heat
transfer between the wall and the streaming fluid. Here, the case of the wall
without heat transfer (adiabatic wall) is of particular importance.
The laminar boundary layer of compressible flow can be treated theoretically,
but theoretical studies dealing with the turbulent compressible boundary layers are
still limited to semiempirical theories of the type of the Prandtl mixing-length
hypothesis, in which, however, additional assumptions must be made. Drag
coefficients of the flat plate at zero incidence over Reynolds and Mach numbers are
given in Fig. 4-4 in comparison with measurements. Agreement between computation and measurement is not satisfactory in all cases. However, some uncertainty of
measurement at high Mach numbers should be taken into account. Also, in Fig. 4-5,
W.
TW
W.
Figure 4-3 Heating of a solid wall
through friction; W = velocity boundary
layer: T = temperature boundary layer.
WINGS IN COMPRESSIBLE FLOW 217
100
80
i
60
I
Turbulent
Laminar
V0
1
i
I
zo
10
1328
i ff
e
I
f
da, .
6
/
J
I
!
i
I
B/ps/GS
Lld
V
OO
I
`
-
1. 69
Q6
173
290
Ow
V 2.00
s 12
oZ.19
02
2
9
8 10
I
2
6
'hd
T
6 a 105
V
I
1
48
02
Ada m
le
2
Q10 y
_
4,
6 8106
Y
1
Umc
Re -
T -12 71!
6 8107
L
-7
6 8108
M
71
E
4'
6
8 109
vo,
Figure 4-4 Skin-friction coefficient of the flat plate at zero incidence vs. Reynolds number and
Mach number for laminar and turbulent flow in the boundary layer and for adiabatic walls, from
van Driest.
the ratio of the drag coefficients at compressible and incompressible flow are
presented against the Mach number up to very high Mach numbers. The decrease of
friction drag is very pronounced at high Mach numbers. Curve 1 of the two
theoretical curves is valid for the adiabatic wall, curve 2 for the wall with heat
transfer. Measurements of several authors are in good agreement with theory. For
completeness, the friction coefficients of the flat plate at zero incidence are also
given for compressible laminar flow.
4-2-3 Similarity Rules for Wing Theory
at Compressible Flow
Velocity potential (linearization) For slender body shapes (wings) in an incident
flow of velocity U.. in the direction of the x axis (longitudinal axis), the local
velocities differ only a little from U. in direction and magnitude. The total flow can
then be separated into a basic flow and a superimposed perturbation flow by setting
Z%=Ucc
2G
tr
TV=v'.
(4-4)
where it, v, w are the perturbation velocities caused by the wing, for which it is
required that
c<U
V
U,
it,
U
By retaining only the largest terms (linearization), the potential equation of
compressible flow for such a flow problem takes the form
(1 -
a2
a2 (1)
`20
aX2
r?/-
cZ-
=0
(4-5)
218 AERODYNAMICS OF THE WING
Dhawan
(5
C Coles (plane flow)
Brinich, Diaconis (i xial
flow)
Chapman, Kester xial
flow)
Seiff (axial flow)
e
°
o
2
e7 Lobb, Winkler, Pen h
ac
a
Q.
0
vas
c
E
v
(plane flow)
3
o Hill (axial flow)
E
Of
0
03
02
Theory of Wilson, without
-- Theory of van Driest, with
heat transfer
0.1
0
1
2
3
6
S
9
10
ma".
Figure 4-5 Ratio of the skin-friction coefficients of the flat plate at zero incidence for
compressible and incompressible flow vs. Mach number; Reynolds number Be - 10'. Comparison of theory and experiment.
where Ma = U/a is the local Mach number. This equation is valid for subsonic,
transonic, and supersonic flow.* It is nonlinear in the velocity potential. The
components of the perturbation velocities are obtained from Eq. (4-5) as
U
ao
ax
00
-
v
ay
u'
(4-6)
bz
In analyzing the linearized Eq. (4-5) further, the first term requires particular
attention because it changes sign when passing the speed of sound (Ma = 1) and
thus changes the mathematical character of the differential equation. By retaining
only the linear terms in u/U., the local Mach number Ma in Eq. (4-5) may be
expressed by the Mach number of the incident flow Ma. = U../a as follows:
Mat=[1+2(1-}-72 1Ma)
i-]
MaM
(4-7)
For pure subsonic and pure supersonic flow, the simplified potential equation
(1 - Ma")
a2ax'
+
a2
aY2
+
a2o
azs
=0
(111a.. > 1)
(4-8)
is obtained by replacing Ma by Ma. in first approximation.
This differential equation for 0 is now linear. For pure subsonic flow, it is of
*For Ma = 0, Eq. (4-5) reduces to the well-known relationship of incompressible flow. It is
not valid for hypersonic flow; see Sec. 4-3-5.
WINGS IN COMPRESSIBLE FLOW 219
the elliptic type, as is the equation for incompressible flow. For pure supersonic
flow, however, it is of the hyperbolic type.
When the undisturbed flow velocity is equal to the speed of sound (May, = 1),
a transonic flow results whose velocity field may include stations of Ma = 1. In this
case, from Eqs. (4-5) and (4-7), it follows that
- 7± i ao aw0 +
u ax ax,
This differential
+ "" = 0
a20
ay-
az21
(1Y1a,,
(4-9)
equation for 0 is nonlinear. Therefore, the computation of
transonic flow fields is considerably more difficult than computations of pure
subsonic and pure supersonic fields. The potential equations derived above [(4-8)
and (4-9)] will now be used to derive similarity rules for three-dimensional wing
theory at subsonic, supersonic, and transonic flows.
Similarity rules for subsonic and supersonic flow For subsonic flows, similarity rules
can be derived from Eq. (4-8) according to Prandtl [73], Glauert [27], and
Gothert [28], the application of which greatly simplifies the computation of
compressible potential flows. These procedures can be applied similarly to
supersonic flows; see Ackeret [1]
.
Of the various possible derivations of these
similarity rules, the so-called streamline analogy will be applied.
The similarity rules for a wing in subsonic or supersonic incident flow
(Ma.. > 1) are obtained through a transformation of the potential equation, Eq.
(4-8). This transformation is such that the Mach number of the undisturbed flow no
longer appears explicitly in the transformed- potential equation. To this end, a
transformed reference flow is established for the given flow in a suitable way. The
variables of the reference flow are designated by a prime:
X, = X
z' =
y' = cl y
0 = C20'
C1Z
UO'C = U11-
(4-10)
By introducing these terms into Eq. (4-8), the factor cl > 0 is determined in such a
way that the Mach number is eliminated, resulting in
cl = 1 - 1V1a200 (Maw < 1)
(4-1 la)
cl = 111a 00 -
(4-11b)
1
(?V1a,,. > 1)
These cases can be combined to
cl =
j1
- Mall
(4-12)
With Eq. (4-11) for the transformed reference flow, the following differential
equations are obtained for the velocity potential:
a201
a x'3
°20'
a x'2
+
C20'
020'
oy y
a z,2
a2 0'
ay,2
a-45'
az12
=0
(Ma. <1)
(4-13)
=0
(Max > 1)
(4-14)
220 AERODYNAMICS OF THE WING
The transformed equation for subsonic flow is identical to the potential
equation for incompressible flow, and the transformed equation for supersonic flow
is identical to the linear potential equation, Eq. (4-8), at Mach number May, = f.
This transformation shows that computations of subsonic flows of any Mach number
can be reduced to computations of flow at Ma,, = 0, and computations of
supersonic flows to those at Ma. = \[2-.
The transformation factor c2 in Eq. (4-10) remains undetermined for the time
being. Its value will be given later.
Application of the transformation formulas, Eq. (4-10), to wings of finite span
will now be treated. The coordinate system x, y, z of Fig. 4-6 will be used with its
x axis parallel to the incident (undisturbed) flow. Equation (4-10) describes the
procedure for determining the transformed wing from a given wing of a given Mach
number where the flow field about the transformed wing is to be computed,
according to the above rules, for subsonic flow at Maw = 0 and for supersonic flow
at Ma., _ Nf2-. The transformed wing according to Eq. (4-10) is then obtained from
the given wing by decreasing or increasing, respectively, the dimensions in the
directions normal to the incident flow direction (y and z directions) by the factor
cl of Eq. (4-12).
.For the wing planform, the following relationships between the transformed
(primed symbols) and the given data are thus obtained:
(4-15a)
7.' = 1
Taper:
Aspect ratio:
Sweepback angle:
A
;z
A' = A 1 - Ma.N
cot 92' = cotg2
1 - MaLI
(4-15b)
(4-15c)
Figure 4-6 Wing geometry. (a) Wing plan-
form; x = ct/c,, taper; ,1 = b2 /A, aspect
ratio; A = wing planform area; 0 = sweepback angle. (b) Profile section y = const;
zC(x) = profile contour; h/c = relative camber; t/c = relative thickness; cu = angle of
attack.
WINGS IN COMPRESSIBLE FLOW 221
a,
Given wing
lU.
b Transformed wing
tTa
i, i-j'2'
Figuze 4-7 Application of subsonic and
M¢ Z.O-ff
supersonic similarity rules to the example
of a tapered, swept-back wing. (a) Given
wing, to be computed for Mach numbers
Ma,, = 0.7, 0.9, 1.1, and 2. (b) Transformed wing for these Mach numbers.
From Eqs. (4-15b) and (4-15c), the remarkable relationship
<1' tan rp' _ A. tan T
(4-15d)
is obtained, where it is immaterial to which -of the planform contour lines -the
sweepback angle is referred, for example, the leading edge or the trailing edge.
In Fig. 4-7 the transformation of the wing planform is explained through the
example of a swept-back wing. The crosshatched wing planform in Fig. 4-7a is the
shape of the given wing, the flow field of which is to be determined for the various
Mach numbers Ma. = 0.7, 0.9, 1.1, and 2.0. The corresponding transformed wing
planforms are shown in Fig. 4-7b, where at Maw < 1 the transformed wings are to
.
be computed for incompressible flow (Ma.. = 0), and at Maw, > I for Ma. _
In Fig. 4-8, the wing planform transformation as given by Eqs. (4-15a)-(4-15c)
is explained in more detail. Here, A'/i1 and cot cc'/cot p are plotted versus Ma...
Again, the given wing planform, which is to be computed for the various Mach
numbers, has been crosshatched. The open wing planforms represent the trans-
222 AERODYNAMICS OF THE WING
Figure 4-8 Illustration of the application of subsonic and supersonic similarity rules; aspect ratio A' and sweepback
angle gyp' of the transformed wing vs.
Mach number-
formed wings for the corresponding Mach numbers. When the given Mach numbers
respectively, the transformed and the given wings are
are Ma = 0 and Ma. =
identical. Figure 4-8 shows that, in the subsonic range, an increase of Ma results in
a decrease of the aspect ratio whereas the sweepback angle increases. For Ma. -* 1,
the aspect ratio of the transformed wing approaches !1' - 0 and the sweepback
angle gyp' -+ 900. In the supersonic range Ma. >,,/2-, the aspect ratio of the
transformed wing increases with Ma,. whereas the sweepback angle decreases. In the
limit of very large Ma., the aspect ratio of the transformed wing A1' -)- - and the
sweepback angle cp' - 0. The remarkable result is found that for large Mach numbers
the three-dimensional wing flow field is converted into a two-dimensional field.
The Prandtl-Glauert-Gothert-Ackeret rule is also applicable to asymmetric
incident flow (yawed wings); see Truckenbrodt [281. For the profile cross section
and angle of attack of Fig. 4-6b, Eqs. (4-10) and (4-12) lead to the following
expressions:
Camber :
Thickness ratio:
Angle of attack:
This shows that for Ma,. <
h'
h
C'
C
t
t
C
C
11 MaL I
a' = a V 11 -MaLI
(4-16a)
(4-16b)
(4-17)
the transformed wing has less camber, is thinner,
it
and has a smaller angle of attack than the given wing; conversely, for May, >
has more camber, is thicker, and has a larger angle of attack.
After the effect of the transformation, Eq. (4-10), on the wing geometry has
been discussed, the relationship between the pressure distributions of the given and
the transformed wing must be studied.
WINGS IN COMPRESSIBLE FLOW 223
The dimensionless pressure coefficients cp = (p - p4/(p U42) assume, within
the framework of linearization, the approximate form
cP
u
= -2 U=
2
Um ax
2"
c'2
a-
2
aO-
U ax,
U11
(4-18a)
(4-18b)
where the velocities of the incident flow U. of the given and transformed flow
must be equal.
This leads with Eq. (4-10) directly to
(4-19)
cP = c2 cp
The still-unknown transformation factor c2 is determined from the kinematic
flow conditions for the two wings (streamline analogy). These are, within the
framework of linearized theory,
W = UCC aZx
az'
w = U00 arc
(4-20)
where w and w' are the z components of the perturbation velocity on the profile
contour zC and zc, respectively (Fig. 4-6b). Because w = aO/az and w' = aO'/az',
we find with Eq. (4-10):
1
C2
(4-21)
11 - Ma_ 1
The meaning of the subsonic and supersonic similarity rules can now be
summarized as follows: From the given wing and the incident flow Mach number,
the transformed wing is found by multiplying the dimensions of the given wing in
the y and z directions and its angle of attack by the factor cl = I(1 -Mam)I,
whereas the dimensions in the x direction remain unchanged. For subsonic
velocities, the flow about the transformed wing is computed from the incompressible equations; for supersonic velocity, however, it is computed from the
compressible equations for Ma,, = V2-. If the incident flow velocities are equal for
both wings, the pressure coefficients are related by
C
P
= P - Pm
el"
(version I)
(4-22)
11- Mal l
With regard to practical applications, it is expedient to choose a transformation in
which only the dimensions in the y direction (wing planform) are distorted, whereas
the dimensions in the z direction (profile and angle of attack) remain unchanged.
Such a transformation is obtained from the above version I by removing the
distortion in the z direction according to Eqs. (4-16a), (4-16b), and (4-17). Thus,
q00
from Eq. (4-22), the pressure coefficient is changed, within the limits of the
linearized theory, by the factor
pressure coefficient becomes
11 - Mam 1, that is, cP = cp
I 1 --Ma'. 1, and the
224 AERODYNAMICS OF THE WING
Cp =
P-P°°
Cp
(version II)
I1 -Ma.
q°°
(4-23)
This relationship is shown in Fig. 4-9. Thus, the following version is obtained for
the subsonic and the supersonic similarity rule.
From the given wing and Mach number, a transformed wing is formed by
multiplyin the dimensions of the given wing in the y direction with the factor
Cl = I(1 -Ma ,',)I, whereas the dimensions in the x and z directions remain
unchanged. For the transformed wing thus obtained, the incompressible flow field is
computed when the given incident flow Mach number lies in the subsonic range.
When the Mach number lies in the supersonic range, however, the flow field about
the transformed wing is computed from compressible equations at Ma. = N f2-. For
equal incident flow velocities U. of given and transformed wings, the pressure
coefficients are interrelated through Eq. (4-23). From the subsonic and supersonic
similarity rules, the following generally valid relationships for the aerodynamic
coefficients are obtained: Let the function
cr = S' fl
(A' ;
x'
A'; cot q,' ;-;
C
'
y,
(4-24)
s
describe the dependency of the pressure coefficient on the geometric wing data at
Ma,, = 0 or Ma = f . Then the corresponding dependency of the geometric wing
data at an arbitrary Mach number is obtained, because of Eqs. (4-15) and (4-22), in
the form:
cP=
a
1-Ma"I
f2(A;AVI1-Mat1;cotgpVj1-Ma`,I;x;y (4-2 5a)
C
00
S
Here S stands for the relative thickness t/c, the relative camber height h/c, or the
angle of attack. This equation can be written in a simpler form:
6
cp =
/3 (A; A tancp; A
Ji
V
I 1 - .Ma2
00 I; x; '-)
C
I
s
(4-25b)
From this formula for the pressure distribution, the lift coefficient is obtained in
corresponding form by integration over the wing surface:
CL
S
Fi(A; A tanfp; A
1- Mat00I)
(4-26)
111 -Mu0l
Here 5 stands for the angle of attack or for the relative camber height. By going to
the limiting case of the airfoil of infinite span (X = 1, p = 0, A -* 00), the subsonic
similarity rule transforms into the well-known Prandtl-Glauert rule of plane flow.
A formula analogous to Eq. (4-26) for the drag coefficient (wave drag) that is
valid, however, only for supersonic flow (see the discussions of Sec. 4-5-5) is given as
CD =
-
Ma7-1
F2 (A, A tan q9, A I Ma - 1)
(4-27)
For wings with zero angle of incidence, S is the wing thickness ratio t/c. In this
case, the drag coefficient at zero lift CD = C- DO is proportional to 5 2 .
WINGS IN COMPRESSIBLE FLOW 225
Figure 4-9 Illustration of the applicaV2
c
3
tion of subsonic and supersonic similarity rules (version II): transformation of
the pressure coefficients.
The outstanding value of the above formulas lies in their describing the Mach
number effect in a simple way. They can, however, also be used to great advantage
for the classification of test results.
Transonic similarity rule For flows of velocities near the speed of sound (transonic
flows), a similarity rule can be derived after von Karman [103] that is related to
those for subsonic and supersonic flows. For wings in a flow field of sonic incident
velocity (Ma.. = 1), it is obtained from the potential equation, Eq. (4-9).
Contrary to the similarity rules for subsonic and supersonic flows, for which
the dependency of aerodynamic coefficients from the geometric wing parameters
and the Mach number was investigated, only the dependency of the aerodynamic
coefficients on the geometric parameters must now be studied, because Ma. =
const = 1.
The problem can be posed in the following way: Given is a wing with all
geometric data (planform and profile) at an angle of attack zero. What, then, is the
geometry of a reference wing, also in an incident flow field of Maw, = 1, that has an
affine pressure distribution equal to that of the given wing? To answer this
question, the following transformation is introduced into Eq. (4-9) [see Eq. (4-10)] :
X, = x
y' = C3 f
z' = C3 Z
.0 = C40'
Uc'o = Uc-
(4-28)
where the quantities without primes refer to the given wing, those with primes to
the reference wing.
Introducing Eq. (4-28) into Eq. (4-9) yields, with
C3 = C4
(4-29)
the following nonlinear differential equation for the velocity potential of the
transformed flow:
226 AERODYNAMICS OF THE WING
-y+ i a 0l a201
U,, ax' ax'
+
(a2 0'
ay'2
ay 0')
+
=0
az'
(Ma. = Ma' = 1)
(4-30)
For an additional relationship between the constants c3 and c4, the kinematic flow
conditions, Eq. (4-20), for both wings have to be established.
For chord-parallel incident flow, this relationship is
azC aZC =
C3C
axl ax'
S
(4-31)
$'
where 6 = tic is the thickness ratio of the wing profile, which has been assumed to
be symmetric.
Hence, with Eq. (4-29):
(6
C4 = (a,)y
C3 -
13
(4-32)
The distortion of the geometric data of the wing planform is given by the
factor c3 in Eq. (4-28). Hence, the following transformations are valid:
2' = 7.
Taper:
Aspect ratio:
Angle of sweepback:
Al
=
cot cp'
(4-33a)
s
113
6,
a 1/3
A
(4-33b)
cot cp
(4-33c)
As an example for the transonic similarity rule, the transformation for a swept-back
wing is presented in Fig. 4-10.
Transformation of the pressure distribution is obtained in analogy to Eqs.
(4-18) and (4-19) merely by replacing c2 by c4i that is, cp =C4c,. With C4
according to Eq. (4-23), it follows that
2/3
CP
(4-34)
C
If the pressure distribution is to be related to the geometric parameters, Eq. (4-34),
considering Eqs. (4-33a)-(4-33c) leads to
CP = 62/3 f
/
z,
;1 tan!p, A61/3..C
i
(4-35)
Hence it is shown that the pressure coefficient from the transonic similarity rule is
proportional to 5 213 , whereas it is proportional to 6 according to the subsonic and
supersonic similarity rules of Eq. (4-25).
From Eq. (4-35) the following expression is found for the drag coefficient,
CD = 55I3 F (, ;1 tang,
-16113)
(4-36)
showing that the drag coefficient is proportional to 51", whereas it is proportional
to 62 according to Eq. (4-27).
WINGS IN COMPRESSIBLE FLOW 227
b
Figure 4-10 Application of the transonic
similarity rule for sonic incident flow to the
example of a trapezoidal swept-back wing.
(a) Thickness ratio S = tic = 0.05. (b) Thickness ratio b' = t'/c' = 0.10.
The formulas for the airfoil of infinite span (X = 1, cp = 0, A - 00) will be given
in Sec. 4-3-4 in extended form (Mao 1 instead of Ma. = 1).
4-3 AIRFOIL OF INFINITE SPAN
IN COMPRESSIBLE FLOW
(PROFILE THEORY)
4-3-1 Survey
Now that a basic understanding of the compressible flow over wings (slender bodies)
has been established in Sec. 4-2, the airfoil of infinite span will be discussed. On the
basis of the similarity rules of Sec. 4-2-3, it turns out to be expedient to study pure
subsonic and supersonic flows (linear theory) first, that is, flows with subsonic and
supersonic approach velocities (Ma.. 1), Secs. 4-3-2 and 4-3-3. The validity range
of linear theory for Ma < 1 is limited by the critical Mach number Mar', for the
drag of Sec. 4-3-4. Later, transonic flow (nonlinear theory) will be discussed, at
which the incident flow of the wing profile has sonic velocity (Ma.. ~ 1). Lastly, in
Sec. 4-3-5, a brief account of hypersonic flow will be given, characterized by
incident flow velocities much higher than the speed of sound (Ma. > 1).
4-3-2 Profile Theory of Subsonic Flow
Linear theory (Prandtl, Glauert) The exact theory of inviscid compressible flow
leads to a nonlinear differential equation for the velocity potential for which it is
228 AERODYNAMICS OF THE WING
quite difficult to establish numerical solutions in the case of arbitrary body shapes.
For slender bodies, however, particularly for wing profiles, this equation can be
linearized in good approximation, Eq. (4-8). For such body shapes, explicit
solutions are therefore feasible. In these cases, the physical condition has to be
satisfied that the perturbation velocities caused by the body are small compared
with the incident flow velocity. This condition is satisfied for wing profiles at small
and moderate angles of attack. Linear theory of compressible flow at subsonic
velocities leads to the Prandtl-Glauert rule. It allows the determination of
compressible flows through computation of a subsonic reference flow. As discussed
in Sec. 4-2-3, this subsonic similarity rule (version II) consists essentially of the
following.
For equal body shapes and equal incident flow conditions, the pressure
differences in the compressible flow are greater by the ratio 1 J 1 -Ma;, than those
in the incompressible reference flow. Here, Ma. = U.Jam, is the Mach number, with
U. the incident flow velocity and a the speed of sound. Hence, the pressure
distribution over the body contour from Eq. (4-23) becomes
1
f,,
P (x) - p. = y 1 - Mci U'inc(x) - P- ]
(4-37)
Here the quantities of compressible flow are left without index, those of the
incompressible reference flow have the index "inc."
For the dimensionless pressure coefficient, the formula of the translation from
incompressible to subsonic flow is obtained as
eP
P
-
Poo
q00
1
V1-Maz
00
Cpinc
(version II)
(4-38)
Here it has been assumed that profile contours and angles of attack of compressible
flow and of the incompressible reference flow are equal; that is,
Zinc(X) = Z(X)
(4-39a)
ainc = a
(4-39b)
where X = xlc and Z = zlc are the dimensionless profile coordinates according to
Eq. (2-2).
An experimental check of Eq. (4-38) is given in Fig. 4-11 for the simple case of
a symmetric profile of 12% thickness in chord-parallel flow. Agreement between
theory and experiment is very good in the lower Mach number range. At higher
Mach numbers some differences are found. In Fig. 4-11, the values of the local
sonic speed (Ma = 1) are included, showing that sonic speed is first reached locally
at Ma. = 0.73.
The lift, obtained by integration of the pressure distribution over the profile
chord, increases with the transition from incompressible to compressible flow as
1/-../l -Ma, because of Eq. (4-38).
The expression for the lift coefficient is given in Table 4-1, which also contains
the transformation formulas for the other lift-related aerodynamic coefficients. For
WINGS IN COMPRESSIBLE FLOW 229
0.6
0.4
vL
02
0'
081
0.6
0.4
0.2
0
0.2
0.6
Od
04
'Y /C ----
1.0
Theory ----Measurement
0.2
0,4
x/c
0.6
0.8
1.0
Figure 4-11 Pressure distributions of the profile NACA 0012 at chord-parallel incident flow for
several subsonic Mach numbers May,. Theory according to the subsonic similarity rule, Eq. (4-38);
measurements from Amic [88] ; Ma = 1 (wc= a) signifies points where the speed of sound is
reached locally.
incompressible flow, the determination of neutral-point position, zero-lift angle,
zero-moment coefficient, and angle of attack and angle of smooth leading-edge flow
has been discussed in Sec. 2-4-2. For lift slope and neutral-point position of the
skeleton profile, the values found for the inclined flat plate are valid, namely,
(dcL/sla}inc- 2rr and ( N/c)inc = lift
, respectively.
In Fig. 4-12, the theoretical
slopes are plotted against the incident flow
Mach number.
Since, according to Eq. (4-37), the pressure distributions over a body at various
Mach numbers are affine to the incompressible pressure distribution, it follows
immediately that the position of the resultant aerodynamic force in the subsonic
range (as long as no shock waves are formed) is equal to that in incompressible
flow. Also, the drag in the subsonic range is determined by the same processes as in
incompressible inviscid flow; that is, it is equal to zero.
Comparison with test results In Fig. 4-13, the most important results of the subsonic
similarity rule are compared with measurements of Gothert [88]. For 5 symmetric
230 AERODYNAMICS OF THE WING
Table 4-1 Aerodynamic coefficients of a profile in subsonic incident flow
based on the subsonic similarity rule (version II)*
Pressure distribution
cp
Lift
CL
- cpinc
1
yl - 1ti7aN
l
dCL
d
Lift slope
Zero-lift angle
ao
Pitching moment
cM
Ma;
dcL
1
y'1
cL in c
- lilaN \
2r
` inc
j/1
-
M(12
o inc
1
CMinc
1
1
1
Angle of smooth
leading-edge flow
- Maro
CMO inc
'inc
s
«s
Lift coefficient of smooth
1
CLs
leading-edge flow
1 -Ma;, cLsinc
*« _ «inc, It/c = (h/c)inc For aerodynamic coefficients for incompressible flow, see
Table 2-1.
14
0.2
0.4
0.6
Ma co
o.d
j
Figure 4-12 Theoretical lift slope at subsonic incident
flow according to the subsonic similarity rule.
WINGS IN COMPRESSIBLE FLOW 231
0.14
20
0.18
0,06
P1001- 6101tert
0.15
0.12
0,09
0.04
0 12
0.02
0
41 CJ 0
0
1-0.02
-0--
t
C
-0-X
-0.Aq
0.15
-0.08
-0.10
1
0 0.3
0.5 0.6
0.7
a
0.8
0.9
-0.120
0..3
0.5 0.6
0.7
0.6
0.85
0.9
b
Figure 4-13 Lift slope (a) and neutral-point position (b) of NACA profiles of various thickness
tic vs. Mach number, for subsonic incident flow, from Multhopp; measurements from Gothert;
neutral-point position as distance from the c/4 point.
wing profiles of thickness ratios t/c = 0.06, 0.09, 0.12, 0.15, and 0.18, lift slopes
are plotted in Fig. 4-13a and neutral-point positions in Fig. 4-13b, both against the
Mach number of the incident flow. For comparison, the theory with (dcL/da)lnc =
5.71 is drawn as a straight line in Fig. 4-13a.*
In the lower Mach number range, agreement between theory and measurement
is very good, with the exception of the profile of 18% thickness. The theoretical
curve follows the experimental data up to a certain Mach number, which shifts
toward Ma. = 1 with decreasing profile thickness. The differences between theory
and experiment beyond this Mach number are caused by strong flow separation.
This fact can also be seen in the presentation of the drag coefficients of the same
profiles in Fig. 4-14a.
According to the present linear theory for very thin profiles, the neutral-point
position should be independent of Mach number. The experimental results of the
profiles of Fig. 4-13b show, however, a considerable dependence of the neutralpoint position on the Mach number when the profile thickness increases.
For the same symmetric profiles that have just been discussed with regard to
lift slope and neutral-point position, the dependence of the drag (= profile drag) on
the angle of attack a and on the Mach number of the incident flow Ala. is
demonstrated in Fig. 4-14. The behavior of the curves for the drag coefficient
cDp(Maa,), with t/c as the parameter, is characterized by the near independence of
CDp from the Mach number in the lower Mach number range, whereas a very steep
`Presented in double-logarithmic scale is dcL/d« vs. (1 -Ma;0).
232 AERODYNAMICS OF THE WING
0.05
0.04
I
0.03
cc 0°
A
0,02
r- 01800
I
__j
C
Q
009 0
O
tj
0.01
0.009
0.008
0.007
0.006
0.005
a
I
o
0.5,
L
0.6
0.7
0.8
May, -
0.85
0.9 00.3
0.5 0.6
0.7
00
085
09
Ma.
a
b
Figure 4-14 Profile drag of NACA profiles of various thickness vs. Mach number, for subsonic
incident flow, from measurements of Gothert. (a) Symmetric incident flow, a= 0°. (b)
Asymmetric incident flow, a = 4°.
drag rise occurs when approaching Ma = 1. This drag rise results from flow
separation, caused by a shock wave that originates at the profile station at which
the speed of sound is locally exceeded. The associated incident flow Mach number
In the case of chord-parallel
is designated as drag-critical Mach number
incident flow (a = 0) the drag rise and, therefore, Ma.,,, occur closer to Ma = 1
for thin profiles than for thick ones (Fig. 4-14a). For a profile with angle of attack
(a * 0), the profile thickness has a negligible influence on the drag rise, as seen in
Fig. 4-14b. As would be expected, the drag rise shifts to smaller Mach numbers with
increasing angle of attack of the profile. The effect of the geometric profile
parameters of relative thickness ratio, nose radius, and camber on the trend of the
curves cDP(Ma,o) is shown in Fig. 4-15.
Attention should be called to the test results reported by Abbott and von
Doenhoff, Chap. 2 [1 J , and by Riegels, Chap. 2 [50].
In summary, it can be concluded from the comparison of theory and
experiment that the subsonic similarity rule (Prandtl-Glauert rule) is always in good
agreement with measurements before sound velocity has been reached locally on the
profile, that is, when no shock waves and corresponding separation of the flow can
occur. Since these two effects are not covered by linear theory, the drag-critical
Mach number is at the same time the validity limit of linear profile theory.
Determination and significance of the critical Mach number Ma., will be discussed
in detail in Sec. 4-3-4.
Higher-order approximations (von Karman-Tsien, Krahn) From the derivation of the
linear theory (Prandtl, Glauert), it can be concluded that the deviations of this
approximate solution from the exact solution are increasing when the Mach number
approaches Ma = 1. The same is shown in the pressure-distribution measurements of
WINGS IN COMPRESSIBLE FLOW 233
Fig. 4-11. Several efforts have been made, therefore, to improve the Prandtl-Glauert
approximation. Steps in this direction have been reported by von Karman and Tsien
[96], Betz and Krahn [7], van Dyke [99], and Gretler [29]. By the von
Karman-Tsien formula, the computation of a compressible flow about a given
profile is reduced to the determination of an incompressible flow about the same
profile. The result is given here without derivation:
Cpinc
cp _
(4-40)
/1 -Maro T 2 (1 - Y'1- Ma's, )cpinc
It can be seen immediately that this equation becomes the Prandtl-Glauert formula
for small values of cpinc According to von Karman-Tsien, the
underpressures assume larger values and the overpressures smaller values than
according to Prandtl-Glauert. In Fig. 4-16, the von Karman-Tsien rule and the
[Eq. (4-38)]
Prandtl-Glauert rule are compared with measurements on the profile NACA 4412.
Obviously, for the higher Mach numbers the von Karman-Tsien rule is in markedly
better agreement with experiment than the Prandtl-Glauert rule.
At the stagnation point of a profile, both theories give the pressure coefficients
too high, whereas the Krahn theory, which will not.be discussed here, describes the
behavior at this point accurately. Also, for Maw - 1, Eqs. (4-38) and (4.40) lose
validity, as would be expected from the assumptions made in their derivation. The
relationship for the critical pressure coefficient cpcr (Ma.) is shown in Fig. 4-16 as
a limiting curve (see Sec. 4-3-4, Fig. 4-28).
0.04
0.03
L? 02
c. 0.01
--4
0.009
0.005
0.007
0.4
0.005
0005
'G
0.004
003!
0 e3
1
I
i
I
c5 06
i
0.7
0
i
:%3500.3
05
.s
0.7
0
1
I
0.d 003 0.5 co
I
i
/7 7
b
Figure 4-15 Profile drag of NACA profiles vs. Mach number for subsonic incident flow, from
measurements of Gothert. Profile thickness t/c = 0.12; cL = 0. (a) Effect of relative thickness
position xt/c. (h) Effect of nose radius rN/c. (c) Effect of camber h/c; relative camber position
x,,,/c = 0.35.
234 AERODYNAMICS OF THE WING
2,0
c. =0°
X = 0,275
X-_20
- = 0,30
C
C
1.5
i
T cp cr
Cpcr
t 1.2
1.2
i
R
o °
q
1
0
0.8
0.8
°
0.4
0
0
0,4
0.2
0.4
0.6
Aboo -
1.0
0L
0
0.2
0.4
016
Mao---0.
0.8
1.0
b
a,
Figure 4-16 Comparison of measured pressure coefficients in subsonic flow with theory. (1) von
Karman-Tsien, Eq. (4-40); (2) Prandtl-Glauert, Eq. (4-38), measurements from [89].
4-3-3 Profile Theory of Supersonic Flow
When a slender body with a sharp leading edge is placed into a supersonic flow field
streaming in the direction of the body's longitudinal axis (Fig. 4-17), the leading
edge of this body assumes the role of a sound source in the sense of Fig. 1-9d. As a
consequence, Mach lines originate at the sharp leading edge, upstream of which the
incident parallel flow remains undisturbed. Only downstream of these Mach lines is
the flow disturbed by the body. As an example of this behavior, the flow pattern
about a convex profile in supersonic incident flow is shown in Fig. 4-18. The Mach
lines, at which the pressure changes abruptly, have been made visible by the
Schlieren method. The incident flow velocity can be determined quite accurately,
with Eq. (1-33), from the angle of the Mach lines that originate at the profile
leading edge.
Linear theory (Ackeret) In analogy to the case of subsonic incident flow of Sec.
4-3-2, inviscid compressible flow about slender bodies (wing profiles) can be
Figure 4-17 Supersonic flow over a sharp-edged
wedge.
WINGS IN COMPRESSIBLE FLOW 235
Figure 4-18 Supersonic flow over a biconvex profile, Schlieren picture. Mach waves originate at
the leading and trailing edges.
computed by a linear approximation theory in the case of supersonic incident flow
as well. The linearized potential equation, Eq. (4-8), is valid both for subsonic and
supersonic flows. It was Ackeret [1 ] who laid the foundation for this linear theory
of supersonic flow. The essential concept of this linear theory is expressed by the
requirement that the perturbation velocity u in the x direction is a function only of
the inclination of the profile contour area elements with respect to the incident
flow direction, of the velocity U, and of the Mach number Maw :
u(x) = -
L(X)
Maw -
1
U.
with w(x) = 3(x)UU
(4-41)
according to the kinematic flow condition (3 > 0: concave; 0 < 0: convex).
The inclinations of the contour on the upper and lower surfaces against the
incident flow direction, 6u and zg1, respectively, are given for slender profiles of
finite thickness and pointed nose (see Fig. 4-19) as
t9
u, t = + a -
dxz
(4-42)
where a is the angle of attack of the chord and z(x) is the profile contour.
In linear approximation, the dimensionless pressure coefficient becomes cp =
-2u/U,.,, [see Eq. (4-18)], leading with Eq. (4-41) to
(x)
p - pec = 26(x)
(443a)
cP
Q
Maw - 1
2
_
2
- + 1Vla -1
a-
dz(x)
(4-43 b)
dx
Here the upper sign applies to the upper surface of the profile, the lower sign to the
lower surface. Equation (4-43) confirms the supersonic similarity rule (version II) as
236 AERODYNAMICS OF THE WING
Figure 4-19 Geometry and incident
flow vector used in the profile theory
at supersonic velocities.
derived in Sec. 4.2-3 [see Eq. (4-25)]. For the further evaluation of Eq. (443), it is
expedient to separate the profile contours again, as in the case of the incompressible
flow in Chap. 2, into the profile teardrop and the mean camber (skeleton) line [see
Eq. (2-1)]
.
Z=
C
= Z(s) ± Z(t)
X= X
and
(444)
Here, as previously in Eq. (2-2), the coordinates have been made dimensionless with
the profile chord c. Again, the upper sign applies to the upper surface of the
profile, the lower sign to the lower surface.
For the pressure difference between the lower and upper surfaces of the profile
(load distribution), Eq. (4.43b) yields with Eq. (4-44):
ZJcP(`Y)
= Pt-Pu =
40o
4
a00-1 \
___
dX)
a
(4.45)
The aerodynamic coefficients are easily obtained from the pressure distribution
through integration. The lift coefficient is, from Eq. (2-54a),
i
cL=
4
dc,(X)dX =
x
(4-46)
JJ
0
It is a remarkable result that the lift coefficient depends only on the angle of
attack a and not at all on the profile shape; that is, the zero-lift direction coincides
with the profile chord (x axis). The moment coefficient, referred to the profile
leading edge (nose up = positive), becomes, from Eq. (2-55a)*:
1
c=-
J c,(X) X dX = o
4
a
M1 2
PO dX
-{-
(4-47)
.1
The lift-related aerodynamic coefficients are compiled in Table 4-2. They include
the lift slope dcLlda and the neutral-point position xNlc = -dcMldcL, of which the
dependence on the incident Mach number Ma,c > 1 is demonstrated in Fig. 4-20a
and b. For comparison, the dependencies for the skeleton profile in subsonic
incident flow, Mam < 1, are also shown (see Table 4-1). These results are identical
to those of the inclined flat plate. For Ma. - 1, both linear theories presented here
fail, because the assumptions made are no longer valid. This is true particularly for
' The integral of the second equation is obtained through integration by parts.
WINGS IN COMPRESSIBLE FLOW 237
the lift slope, as can be seen from Fig. 4-33. The location of the neutral point is at
xN/c = a for subsonic flow and at xN/c = a for supersonic flow. This marked shift
toward the rear when the flow changes from subsonic to supersonic velocities
should be emphasized.
In addition to lift, drag is produced in supersonic frictionless flow. It is called
wave drag. The two forces are expressed by
c
C
L=b
D=b
(JPr - JPu)dx
(J PA +JpuJu)dx
0
0
where JP/(X) = pl(x) -p. and d pu(x) = pu(x) -p. are the pressures on the lower
and upper surfaces of the profile, respectively, and
and zit are the profile
inclinations from Eq. (4-42). By using the pressure coefficients from Eq. (4-43b) and
evaluating the integrals under the, assumption that the profiles are closed in front
and in the rear, the lift coefficient CL is obtained as in Eq. (4.46), and the drag
coefficient CD becomes*
2
May
CD
2 az +
f(--)y
>/
dX
(4-48a)
2d
+J1dX
*Note that, also in subsonic flow, the wing of finite span has a drag that is proportional to the
square of the lift (induced drag, see Sec. 3-4-2).
Table 4-2 Aerodynamic coefficients of a profile in supersonic incident
flow based on the linear theory (Ackeret)
Pressure distribution
c,
_ -Tr-
2
00
Lift slope
Neutral-point position
Zero-lift angle
dCL
da
llla - 1
CIO
-
zN
C
NO
dZ)
-
4
=
a
i
2
=0
`
1
Zero moment
4
Z131 d X
citito
dcD
dcL
i
Wave drag
4
CDo
VMa
- i.
dZ:1, 2
L\dX)
/dZ'c,
2]
'td
) j
238 AERODYNAMICS OF THE WING
Supersonic flow
Subsonic flow
Maoo<1
Ma00>1
r
Pr andt/-Gloue rt
2n
Acke ret
Incompressib le
4
2
a 0
02
04
0,6
0.8
70
12
74
16
18
2.0
Mao,
Profile leading edge
b 0
0,2
0,4
0,6
0,8
10
72
74
76
78
20
Mao, ----- 00,6
Figure 4-20 Aerodynamic forces of
the inclined flat plate at subsonic
I
Co
02
0,4
0,6
08
10
72
14
16
78
Mao,
2,0
and supersonic flows. (a) Lift slope
dcL/da. (b) Position of the resultant
of the aerodynamic forces xN. (c)
Drag coefficient CD.
Replacing a by CL as in Eq. (446), and Zu, I by Z (s) and Z(') as in Eq. (4-44),
results in
cD =
Mad
- 1 c+
i
Ma-1
i
Z 2 dX + r ()y dX (448b)
f (iE-)
n
.1
1
0
It should be noted that the total wave drag is composed of three additive
contributions. The first contribution is proportional to CL and independent of the
profile geometry. It is plotted in, Fig. 4-20c against the incident flow Mach number.*
The second and third contributions are independent of the lift coefficient and
proportional to the square of the relative camber and the relative thickness,
respectively. Consequently, it can be seen directly that the flat plate is the so-called
best supersonic profile, because the second and third contributions are equal to zero
in this case.
The formulas for the drag rise dcDldcL and for the zero drag CD at CL = 0
have been listed once more separately in Table 4-2. A simple explanation of the
wave drag will be given for the subsequently discussed case of the inclined flat plate.
*See footnote on page 237.
WINGS IN COMPRESSIBLE FLOW 239
Results of linear theory The physical understanding of the last section was applied
for the first time by Ackeret [1 ] to a quite simple computation of the flow over a
flat plate in a flow of supersonic velocity U. at a small angle of incidence a.
According to Fig. 4-21, the streamline incident on the plate leading edge forms with
the plate a corner of angle a that is concave on the lower side of the plate and
convex on the upper side. Consequently, an expansion Mach line originates on the
upper side and a compression Mach line on the lower side. At the trailing edge, the
compression line is above, the expansion line below the plate. Behind the plate the
velocity is again equal to U. and the pressure equal to p., as it is ahead of the
plate. Consequently, there is a constant underpressure pu on the entire upper
surface and a constant overpressure pl on the lower surface. The pressure coefficient
cp(x) = const follows from Eq. (4-43b) with a* 0 and z(x) = 0. The characteristic
difference in the pressure distributions for supersonic and subsonic incident flow is
explained in Fig. 4-22. From Fig. 4-22a, at subsonic velocity the pressure
distribution produces a force-resultant N normal to the plate, and in addition, the
flow around the sharp leading edge produces a suction force S directed upstream
along the plate (see Sec. 3-4-3). The resultant of the normal force N and the suction
force S is the lift L, which acts normal to the incident flow direction U,,. The
resultant aerodynamic force has no component parallel to the incident flow
direction; in other words, the drag in the frictionless subsonic flow is equal to zero.
For the case of supersonic flow, Fig. 4-22b, the force N resulting from the
pressure distribution also acts normal to the plate. However, because there is no
flow around the leading edge, no suction force parallel to the plate exists here. The
normal force N in inviscid flow therefore represents the total force. Separation into
components normal and parallel to the incident flow direction establishes the lift
L = N cos a N and the wave drag D = N sin a La. There is another physical
explanation for the existence of drag at supersonic incident flow, namely, that for
the production of the pressure waves (Mach lines) originating at the body during its
motion, energy is expended continuously.
As a further example of the pressure distribution on profiles in supersonic flow,
a biconvex parabolic profile and an infinitely thin cambered parabolic profile, given
by the equations
Z(t)=2 CtX(l -X)
(4-49a)
Expansion 4pu
Y
PJ,
P/
A
Compression Gyp/
Expansion
*'
Figure 4-21 Inclined plate in supersonic
incident flow.
240 AERODYNAMICS OF THE WING
Figure 4-22 Pressure distribution and forces on an inclined flat plate in compressible flow. (a)
Subsonic incident flow (Ma. < 1). (b) Supersonic incident flow (Mao, > 1).
Z(S)=4kX(l-X)
(4-49b)
are compared in Fig. 4-23. Both profiles are in chord-parallel incident flow, a= 0°.
Consequently, from Eq. (4-46), CL = 0 for either profile. The pressure distributions,
as computed from Eq. (4-43), are given in Fig. 4-23. The zero moment of the
teardrop profile is equal to zero, whereas that of the skeleton profile is turning the
leading edge down (nose-loaded). The lift-independent share of the wave drag is
obtained from Eq. (448b) as
(4-50a)
CDO =
(4-50b)
These expressions show that the zero-drag coefficients are proportional to the
squares of the thickness ratio t/c and the camber h/c, respectively. In Fig. 4-24, the
a
b
Figure 4-23 Pressure distribution at
I
CM) eo
supersonic incident flow for parabolic profiles at chord-parallel incident flow. (a) Biconvex teardrop
profile. (b) Skeleton profile.
WINGS IN COMPRESSIBLE FLOW 241
.Expansion line
I
C
d
e
f
Figure 4-24 Pressure distribution on profiles at supersonic incident flow. 1, lower surface; ii,
upper surface. (a) Inclined flat plate. (b) Parabolic skeleton at angle of attack at = 0°. (c)
Biconvex profile at a = 0°. (d) Circular-arc profile, a = 0° . (e) Biconvex profile, a 0°. (f)
Circular-arc profile, a T 0°.
pressure distributions of an inclined flat plate (Fig. 4-24a), a parabolic skeleton (Fig.
4-24b), a symmetric biconvex profile, and a circular-arc profile at angle of attack
a = 00 (Fig. 4-24c and d), as well as at a T 0° (Fig. 4-24e and f), are compared.
Further, a few data should be given about the dependence of wave drag on the
relative thickness position for double-wedge profiles and parabolic profiles. The
242 AERODYNAMICS OF THE WING
geometry of parabolic profiles was given by Eq. (2-6). In Table 4-3 the results are
compiled, and in Fig. 4-25 the contribution to the wave drag that is independent of
CL is plotted against the relative thickness position. For a relative thickness position
xt = 0.5, the wave drag of the double-wedge profile is
cDo=
t)
4
V Maw
-1
(4-51)
c
Thus, the drag of this double-wedge profile is lower by a factor a than that of the
parabolic profile (Xt = 0.5). The double-wedge profile (Xt = 0.5) is the profile of
lowest wave drag for a given thickness. Data on additional profile shapes are found
in Wegener and Kowalke [21].
Information on the remaining aerodynamic coefficients, namely, zero-lift angle
and zero moment, is compiled in Fig. 4-26 for skeleton profiles of all possible
relative camber positions. The geometric data of the skeleton line were given in Eq.
(2-6). For comparison, the coefficients for subsonic velocities are also shown. The
zero-lift angle and the zero moment are plotted against the relative camber position
in Fig. 4-26a and b, respectively. In either case the basically different trends at
subsonic and supersonic velocities are obvious.
Higher-order approximations (Busemann) The above-stated linear profile theory for
supersonic flow, characterized by a local pressure difference (p -p'.) proportional
to the local profile inclination 0 was later extended by Busernann [10] to a
higher-order theory by adding terms of d2 and X93. The pressure coefficient of the
extended theory changes Eq. (4-43a) into
cp(x) =
Ma;, -1
[1
+K$(x)]
(4-52a)
Table 4-3 Wave drag at supersonic incident flow for double-wedge profiles
and parabolic profiles (see Fig. 4-25)
Designation
Double-wedge profile
Parabolic profile
Side view
r)
.1
2 Xt
for (I)
Contour C
8
(1-` )for(I1)
i - Xt.
- 2 Xr
r)
X(1-X)
(1 - 2 Xt) X
Wave drag
Vja
-1
ODo
62
1
1
xt(1- X-)
:3X2 (1 - Xt)Z
WINGS IN COMPRESSIBLE FLOW 243
25
2
20
1
N
0
U V
15
r6
3
Figure 4-25 Wave drag at supersonic flow vs.
relative thickness position for double-wedge
0
08
0.6
0.4
0.2
Xt.
1.
profile (1) and parabolic profile (2), from
[211 (see Table 4-3).
-
with
h' -
1
(31.a. 20
4
- 2)2 + yMal
(4-52b)
(1VIa 0 - 1)3/2
The aerodynamic coefficients can be determined from Eq. (4-52), but no details will
be given here. For the lift-independent contribution, an additional term is obtained
that is proportional to (t/c)3 for symmetric profiles. Theoretical drag values,
computed using this theory of second-order approximation, are compared in
Fig. 4-27 with measurements by Busemann and Walchner [10] . Good agreement is
obtained.
6
5
,fa,j 1
I
0,
0.2
0.4
0.6
as
10
00,
Xh
a
0.2
0..4
Xh
0.6
00
1.0
b
Figure 4-26 Aerodynamic coefficients of cambered skeleton profile at subsonic and supersonic
flows. (a) Zero-lift angle a, . (b) Zero-moment coefficient cm,.
244 AERODYNAMICS OF THE WING
Test
i'
04
/
------- Theory
X11
8°
i
0.2
1
o t/c=0,0885
I
-`ot/C=0
.t/c=0 1670
.
c
6°
4°
1
8°
1250
,
4
I
0
v
Z
;0 °
d =0
-220
-02
\
-4°
4°
00
7-
-0.4
46\
-8° N.
-8°
-6°
-20
-06
-08
0,05
01
Q75
02
cD --
Q25
03
035
04
Figure 4-27 Drag polars cL(cD) of circular-arc profiles of several thickness ratios t/c at Mach
number Ma°o = 1.47, from measurements of Busemann and Walchner; comparison with
second-order approximation theory of Busemann.
With greater accuracy than by the above-illustrated theory of second-order
approximation, the supersonic flow about thin profiles can be determined by the
method of characteristics. Compare, for instance, the publications of Lighthill [51,
52].
4-3-4 Profile Theory of Transonic Flow
Both approximation theories for subsonic and supersonic flows discussed in Secs.
4-3-2 and 4-3-3 fail when the incident flow velocity approaches the speed of sound.
In this case the flow becomes of the mixed type; that is, both subsonic and
supersonic velocities exist in the flow field. At certain points the flow therefore
passes the speed of sound. In transonic flow fields of this kind, shock waves are
formed in most cases, and theoretical treatment is made much more difficult.
Drag-critical Mach number First, the limiting Mach number should be established up
to which the theory of subsonic flow of Sec. 4-3-2 is still valid. In the case of a
wing profile at subsonic incident flow velocity (Ma.. < 1), Fig. 4-13a demonstrated
that the lift slope can no longer be described by the linear theory at higher subsonic
Mach numbers. The results on the neutral-point position of Fig. 4-13b, and in
particular those on the drag coefficients of Figs. 4-14 and 4-15, confirm this fact,
which is caused by flow separation on the profile. Depending on the profile shape
(thickness ratio, camber ratio, nose ratio) and the angle of attack, a critical Mach
number Maw, cr can be established up to which no significant flow separation occurs.
This will be designated as the drag-critical Mach number. It can be defined, for
WINGS IN COMPRESSIBLE FLOW 245
instance, as the Mach number at which the drag coefficient CD is higher by
d CD = 0.02 than at May, = 0.6.
The physical reason for flow separation at higher subsonic Mach numbers is
that shock waves are formed when sonic velocity is reached locally on the profile
and exceeded over a certain range. The critical Mach number Ma.,, is understood,
therefore, to be the Mach number of the incident flow at which sonic velocity is
reached locally on the profile. The critical pressure coefficient at the critical Mach
number Ma., is cpcr. The critical Mach number Ma,ocr is obtained by setting for
cpcr the highest underpressure cpmin that occurs at the body. For slender bodies,
Cpmin is small and Mao,cr is close to unity. In this case, based on streamline theory
of compressible flow, neglecting higher-order terms, cpcr becomes
1 -Ma o, cr
2
c
p Cr
=-7+
.
1
Ma;o cr
( 4 - 53 a)
(4-53b)
= Cpmin
From Eq. (4-38), Cpmin is a function of Mach number. Introducing Eq. (4-38) into
Eqs. (4-53a) and (4-53b) yields
(1 -Ma200cr)3/2
Mat
+1
2
(Cpmin )inc
(4-54)
. Cr
In Fig. 4-28, cpcr from Eq. (4-53a) is shown versus Ma, as curve 1. For a given
wing profile, Mao, cr is determined by the intersection of curve i according to Eq.
(4-53b) with curve 3 according to Eq. (4-38); see also Fig. 4-16. More simply,
Ma, can be obtained by starting from Eq. (4-54). This relationship is given as
curve 2.
,
The value of cpmin depends strongly on the profile shape and the angle of
attack. It is obtained from the velocity distribution of potential flow with
Cpmin = -2umax/Uoo. The maximum pressures for various profiles in incompressible
flow are plotted in Fig. 2-34 against the thickness ratio. The critical Mach numbers
for chord-parallel flow are shown in Fig. 4-29 for several profiles as functions of
0.
0.
'
I
2-A
03
.12
1a
1
02
Cpmin
mm inc
0.1
Fire 4-28 Illustration of determination of
drag-critical Mach number Ma-cr of a wing
'
1
O5
0.8
I
0.7
Maoocr
08
-0.
1.0
profile; y = 1.4. Curve 1 from Eq. (4-53), curve
2 from Eq. (4-54), curve 3 from Eq. (4-38).
246 AERODYNAMICS OF THE WING
1,0
0.8
Joukowsky profile
0.2
0
405
0.15
0,10
0,20
0.25
Figure 4-29 Drag-critical Mach number Mao,cr
t
of several profiles at chord-parallel incident
c
flow; see Fig. 2-34.
profile thickness S = t/c and relative thickness position Xt = xtlc. As would be
expected, the critical Mach number decreases sharply with increasing thickness ratio
for all profiles.
Physical behavior of transonic profile flow When a wing profile is exposed to an
incident flow velocity high enough to form areas of local supersonic velocity in its
vicinity, shock waves are formed in the ranges where the velocity is reverted from
supersonic to subsonic. In these shock waves, pressure, density, and temperature
change very strongly. The strong pressure rise in the shock wave frequently leads to
flow separation and consequently to a complete change of the flow pattern. This
effect causes a strong increase in the drag (pressure drag).
To demonstrate these processes, the pressure distribution on a wing profile is
given in Fig. 4-30a for various Mach numbers from measurements in reference [89].
The pressure distribution is steady for Mach numbers at which the maximum
velocity on the profile contour is everywhere smaller than the local sound speed,
we <a. In the present case, this holds up to Maw 0.6. Up to Ma. ~ 0.6 the
pressure rise at the rear end of the profile is as steady as the pressure drop is in
front. For higher Mach numbers, Ma. > 0.7, at which the sonic velocity is exceeded
locally, we > a, the pressure rise behind the pressure minimum occurs unsteadily in
a shock wave. The height of the pressure jump increases with Mach number. This
abrupt pressure rise is very undesirable with respect to the boundary layer, which
tends to separate even at a steady pressure rise. In most cases, the shock wave
causes separation of the flow from the wall and thus a strong drag rise, as is obvious
from the curve of the drag coefficient versus Mach number of Fig. 4-30b; see also
Figs. 4-14 and 4-15.
In Fig. 4-31, a Schlieren picture and an interferometer photograph from Holder
[33] are shown of a wing of angle of attack a = 8° in a flow field of Ma.. = 0.9.
The formation of the shock wave and a strong separation immediately behind the
shock are clearly noticeable.
WINGS IN COMPRESSIBLE FLOW 247
The flow pattern in the transonic velocity range, which is, in general, quite
complicated, is displayed schematically in Fig. 4-32 for a biconvex profile in
symmetric incident flow. Pressure distributions and streamline patterns are given
over a range of increasing Mach number. Figure 4-32a represents the incompressible
case, Fig. 4-32b the subsonic case in which the "sonic limit" has not yet been
exceeded anywhere. Figure 4-32c-e demonstrates the formation of the shock wave
after the "sonic limit" of the pressure distribution (critical pressure) has been
passed. Figure 4-32f and g represents the typical pressure distribution of supersonic
flow that was previously shown in Fig. 4-24.
The formation of shock waves in the transonic range also has a strong effect on
the lift. This is demonstrated schematically in Fig. 4-33, in which the solid curve
represents a typical measurement of the relation between lift coefficient and Mach
number, whereas the dashed line corresponds to the linear theory according to Fig.
4-20a. For a better understanding of the measured lift curve, the positions of the
shock wave and the velocity distributions on the profile for the points A, B, C, D,
and E are shown in Fig. 4-34. At Mach number Ma = 0.75 (point A), a shock
wave does not yet form because the velocity of sound has not been exceeded
Q 78
pcr=01527pp I
a
02
04
Qs
x/c---
08
10
006
004
Figure 4-30 Measurements on a wing profile
at subsonic incident flow from (891, angle of
attack a = 0°. (a) Pressure distribution at
00
A11a°°cr0,7
b
02
04
Mac r
0.6
08
;0
various Mach numbers. (b) Drag coefficient vs.
Mach number.
248 AERODYNAMICS OF THE WING
a1
Figure 4-31 Flow about a wing profile at Mach number Ma,. = 0.9. Angle of attack a = 8°,
from Holder. (a) Schlieren picture. (b) Interferometer photograph.
WINGS IN COMPRESSIBLE FLOW 249
Sonic limit
b
Shock wave
Local supersonic flow
F..
////// //M
U-L"11 ii77.1
e
f
c
C
g
C)
Q
U,
Figure 4-32 Pressure distribution and flow patterns of a biconvex profile in the transonic range
(schematic).
G
05
10
Ma.
75
ZO
Figure 4-33 Lift coefficient of wing
vs. Mach number. Solid curve: typical trend of measurements. Dashed
curve: theory according to Fig.
4-20a.
t
250 AERODYNAMICS OF THE WING
a
i=
b
Velocity distribution
on profile
Position of shock wave
Z2
u
10
A
Wake flow
U
,75
w°°
08 ,
31x0.6
/
I
O4
-j
I
0
0,5
70
x/c--«
1,6
14
U
12
781
Shock wave
--..
B
f
3d
/
10 -',-- -
'
08
0.6
04
0
05
10
x/c --
1,6
14
u
12
7,89
-- -
_
3
C
08,
06
04
0
0.5
10
0.5
70
X/C-.-
1.6
u
14
1.2
298
-,.
u
10
0.8
D
r
06
04
0
-
X/C --
2. 0
U
1.6
1. 4
810 -- 1.2
14
u
08
E
06
04
0
05
x/c---
70
Figure 4-34 Transonic flow over a wing profile at various Mach numbers; angle of attack
a = 2°, from Holder. The points A, B. C, D, and E correspond to the lift coefficients of Fig.
4-33. (a) Position of shock wave. (b) Velocity distribution on profile.
WINGS IN COMPRESSIBLE FLOW 251
significantly on either side of the profile. Up to this Mach number, the flow is
subsonic and the lift follows the linear subsonic theory (Prandtl, Glauert). At
Mac, = 0.81 (point B), the velocity of sound has been exceeded significantly on the
front portion of the profile upper surface. A shock wave at the 70% chord is the
result. The lower surface is still covered everywhere by subsonic flow. Up to point
B, the lift increases with Mach number. At Mach number 0.89 (point C), the
velocity of sound is also exceeded over a large portion of the lower surface. A
shock wave therefore forms on the lower surface near the trailing edge. This changes
the velocity distribution over the profile considerably, resulting in a marked lift
reduction. At Mach number Ma. = 0.98 (point D), the two shock waves on the
upper and lower surfaces are considerably weaker than at Ma. = 0.89 and are
located at the trailing edge. The lift, therefore, is again larger than at point C.
Finally, at Ma. = 1.4 (point E), pure supersonic flow has been established with a
velocity distribution typical for supersonic flow. The magnitude of the lift now
corresponds to the linear supersonic theory (Ackeret).
All tests indicate that the processes in the shock wave are markedly affected by
the friction layer. This interaction between shock wave and boundary layer is,
besides other effects, particularly complicated because the behavior of the boundary
layer changes with Reynolds number, but on the other hand, the shock wave
depends strongly on the Mach number. Above a certain shock strength, the pressure
rise in the shock causes boundary-layer separation which, in addition to the drag
rise
already discussed, leads to strong vibrations as a result of the nonsteady
character of this flow. This phenomenon is also called "buffeting" in aeronautics;
see, for example, Wood [109]. Both the Mach numbers of sudden drag rise and of
buffeting are influenced by the profile shape and the angle of attack a (see Fig.
4-35). The so-called buffeting limit restricts the Mach number range for safe airplane
operation. By increasing the incident flow Mach number to supersonic velocities, the
shock moves to the wing trailing edge and the buffeting effects disappear again. For
very thin and slightly inclined profiles, this state can be reached without the shock's
gaining sufficient strength to excite buffeting while it is moving over the profile.
The individual phases of the flow in Fig. 4-35a are explained by the pressure
distributions of Fig. 4-35b.
Because of the complicated flow processes above the critical Mach number, a
strictly theoretical determination of the buffeting limit is not possible. However,
Thomas and Redeker [109] developed a semiempirical method for the determination of the buffeting limit; see Sinnott [84]. A comprehensive experimental
investigation of this problem, which is most important for aeronautics, has been
reported in detail by Pearcey [69] and Holder [33].
Similarity rule
for transonic profile flow So far, analytical determinations of
transonic flows with shock waves have succeeded only in a few cases. In some cases,
however, a steady transition through the sonic velocity (without shock waves) has
also been observed. In this latter case, transonic flows can be treated theoretically
by means of an approximation method. They lead to similarity rules for pressure
distribution and drag coefficient (Sec. 4-2-3) that are in quite good agreement with
252 AERODYNAMICS OF THE WING
A... C Attached flow
Flow separated at the shock
Shock at the trailing edge
D
£
x/c
Figure 4-35 Behavior of a wing in the transonic velocity range (schematic), from Thomas. (a)
Buffeting limit vs. Mach number. (b) Pressure distributions at several Mach numbers.
measurements. It can be shown that the transonic similarity rule remains valid even
when the flow includes weak shock waves.
Between pressure distribution and drag coefficient of wing profiles of various
thickness ratios t/c and at various transonic Mach numbers of the incident flow
(Mi -- 1), the following expressions are valid according to reference [103], and
extend Eqs. (4-35) and (4-36):
cp
,
x t
,
C
where
Mam
(7+ 1) 1/3
erp
(t/c)5/3
t
CD
x
(t/c)2 /3
,
Mao,
ynoo
r + 1) 1/3
(7
Mat1
- [(7±1)C
moc
C
v
(4-55)
(4-56)
(4-57)
Here, cp is called the reduced pressure coefficient, and ED is the reduced drag
coefficient. For the special case Maw = 1 (sonic incident flow), mc, = 0 from Eq.
(4-57). From this it follows immediately that the pressure coefficient cp is
proportional to (t1c)213 in this case and the drag coefficient proportional to (t/c)s/3
[see Eqs. (4-35) and (4-36), respectively].
WINGS IN COMPRESSIBLE FLOW 253
Malavard [103] checked the similarity rules, Eqs. (4-55)-(4-57), in comprehensive experiments. He clearly verified the transonic similarity rule for pressure
distribution and drag coefficient of symmetric biconvex profiles of thickness ratios
t/c = 0.06-0.12 at chord-parallel flow of incident Mach numbers of Ma. _
0.775-1.00. Plotting of the drag coefficient CD against the Mach number in Fig.
4-36a shows the well-known strong drag rise near Ma. = 1 and, moreover, the
strong increase of this rise with the thickness ratio t/c.
Theories for the computation of transonic profile flows The transonic profile flow
with shock waves can be treated only by nonlinear theory, in contrast to the linear
theories of subsonic and supersonic profile flows. There exist numerous trials and
methods for the solution of this task. A survey of the more recent status of
understanding of theory and experiment for transonic flow is given by Zierep
[111]. So far, the hodograph method, the integral equation method, the parabolic
method, and the method of characteristics have been applied to computations.
Guderley uses mainly the hodograph method, Oswatitsch generally prefers the
integral equation method. The many publications quoted in [63, 66, 79, 84-87,
111 ] show that no generally valid solution has been found for the computation of
the pressure distribution of wings on which shock waves form at transonic incident
flow. More recent progress has been discussed at the two Symposia Transsonica
[67].
Supercritical profiles For wing profiles operating at high subsonic flight velocities,
the. drag-critical Mach number Mao, according to Figs. 4-14a and 4-29 can
be shifted to higher values by reducing the profile thickness ratio or by lowering
,
010
5
0,08
4
0,10
10,01
008
00
0. 06
0,02
0
a
I
i
0
08
09
70
Mao,--
1.1
72
-12 -10 -08 -06 -04 -02 0
02 04 06 Qd
10
moo
Figure 4-36 Drag measurements on symmetric profiles in the transonic velocity range at
chord-parallel incident flow, from Malavard. (a) Drag coefficient CD vs. Mach number Ma.. for
symmetric profiles of various thickness ratios t/c. (b) Reduced drag coefficient cD from Eq.
(4-56) vs. reduced Mach number h,,. from Eq. (4-57) for symmetric profiles of various thickness
ratios t/c.
254 AERODYNAMICS OF THE WING
the profile lift coefficient.* Profiles at which the critical pressure coefficient cp
Cr
from
Eq. (4-53a) has not yet been exceeded or has just been reached on the suction side
(profile upper side) are termed subcritical profiles. On them no shock waves form, and
therefore no shock-induced flow separation occurs. Through suitable profile design,
local areas of supersonic flow can be created on the profile in which recompression to
subsonic flow occurs steadily or in weak shock waves only. On these profiles the
pressure rise in the recompression zone is gradual and therefore does not cause flow
separation. Transonic profiles designed according to the stated criterion are termed
supercritical profiles.
A few more statements should be made about the evolution from subcritical to
supercritical wing profiles. In many designs the product of lift-to-drag ratio and Mach
number must be optimized. This request may roughly be transferred to the aim to
achieve for a given profile thickness ratio at the design Mach number the highest
possible lift at fully attached flow conditions. By starting with the pressure
distribution la in Fig. 4-37 found on the suction side of the conventional NACA
64A010 profile a gain in lift first may be obtained by further upstream and
downstream extension of the minimum suction pressure just along its critical value
*The feasibility of increasing the drag-critical Mach number by sweeping back the wing will
be discussed in Sec. 4-4-4.
Figure 4-37 Pressure distributions of various wing profiles. (a) Suction side (upper surface). (b)
Pressure side (lower surface). (1) Conventional profile NACA 64A010 at Mao, = 0.76, a = 1.20,
measurements of Stivers [651. (2) Roof-top profile. (3) Supercritical profile of thickness ratio
t/c = 0.118 with "rear loading," from Kacprzynski [65). Theory: Ma".=0,75, cL = 0.63.
Measurements: Ma = 0.77, cL = 0.58.
WINGS IN COMPRESSIBLE FLOW 255
Figure 4-38 Comparison of the contour of a supercritical profile with a conventional profile
(NACA 641 A212), thickness ratio t/c = 0.12.
according to curve 2a. Such profiles are called "roof-top profiles." In the range of the
profile nose, a strong acceleration of the flow is required, which is accomplished by
increasing the nose radius. The onset of the recompression needed to match the
pressure at the profile trailing edge (pressure at the rear stagnation point in inviscid
flow) must be chosen to allow establishment of a pressure gradient over the rear
portion of the profile that does not cause flow separation. Chordwise linear
recompression according to curve 2a has been found to be good in practical
applications. A further marked increase in lift is obtained by admitting a local
supersonic flow field on the profile suction side, which means choosing pressure
distributions exceeding the critical pressure coefficient. That kind of flow implies a
further increase in nose radius, and, in addition, a flattening of the upper surface. In
this case, an essentially shock-free or weak shock pressure distribution along the
profile chord, allowing recompression without separation, curve 3a, is of decisive
importance. The pressure distribution over the rear portion of the pressure side of
conventional profiles is little different from that on the suction side (curves 1 a and
l b). Thus, the rear portion of such profiles contributes little to the lift. A larger
difference in the pressure distribution of upper and lower side, curves la and 3b, is
obtained through changing the profile lower contour between the range of maximum
thickness and the trailing edge such that a reduced local thickness is obtained. This
change means, according to Fig. 4-38, the establishment of a corresponding profile
camber. Measures of that kind are known as "rear loading." At such profile designs,
caution is necessary to avoid flow separation in the recompression region, precisely as
it was required on the suction side.
A comparison of the geometries of a subcritical and supercritical profile with
"rear loading" and thickness ratios tlc = 0.12 is shown in Fig. 4-38. Systematic
investigations on profiles with shock-free recompression from subsonic to supersonic
flow have been made by Pearcy [69]. The first design intended to produce
shock-free supercritical profiles, so-called quasi-elliptic profiles, was conducted by
Niewland [65] and confirmed in the wind tunnel (Fig. 4-39). Since then, a number
of generally applicable design methods for supercritical profiles have been
developed, and profile families have been checked out successfully in the wind
tunnel [4, 54, 55].
4-3-5 Airfoil of Infinite Span in Hypersonic Flow
By taking into account the similarity rules of Sec. 4-2-3, specific profile theories
have been developed for flow about wing profiles (slender bodies) that depend on
256 AERODYNAMICS OF THE WING
Figure 4-39 Pressure distribution of a quasi-elliptic symmetric shock-free supercritical profile in
chord-parallel flow, from Niewland, Ma. = 0.786. Measurements: o NPL, 4 NLR.
the values of the incident flow Mach number. For May, < 1 the subsonic flow is
described in Sec. 4-3-2, for Maw > 1 the supersonic flow in Sec. 4-3-3, and for
Ma., = i the transonic flow in Sec. 4-3-4. For very high Mach numbers of incident
flow, that is, Ma. > 1, the theory of supersonic flow does not lead to satisfactory
results. For this case of incident flow with hypersonic velocity (Ma., > 4), a few
statements on a profile theory of hypersonic flow will be made. First, the following
considerations will be based on a slender profile, pointed in front.
Theory of small deflections in hypersonic flow Through a concave deflection by the
angle > 0, a compression flow is produced that can be computed according to the
theory of the oblique shock. Conversely, an expansion flow is formed behind a
convex deflection by the angle < 0 that can be treated as a Prandtl-Meyer corner
flow. The fluid mechanical quantities before and behind the deflection will be
marked by the indices 1 and 2, respectively. The deflection angle is assumed to be
small 161 << 1, which means that the velocities before and behind the deflection
differ only by a small perturbation velocity. The range of Mach numbers of the
hypersonic flow considered here is Mal > 1 and Mat > 1. The pressure coefficients
cp =J p/q i of the pressure change A p = P2 - PI , relative to the dynamic pressure
before the deflection q1 = (ol /2)Ul, are obtained as [53]
992 > 0
y(Mal
0)2
1-
] + y-1Ma1
2
lg y
($ > 0)
+92 < 0
(4-58a)
(t < 0)
(4-58b)
WINGS IN COMPRESSIBLE FLOW 257
In either case, the pressure coefficient at small deflections of a hypersonic flow
is given as
cp = 62f(Ma1 6)
(4-59)
where Mal t5 is the similarity parameter of hypersonic flow. The parameter will be
discussed later in more detail in connection with the hypersonic similarity rule.
For large values of Ma l 19 > 1, the expressions
cP = (y + 1)62
(4-60a)
(Ma1 $ -> 00)
2752
7(Ma1 6)2
-Ma1
19 >
2
7-1
(4-60b)
are valid. The latter formula indicates that after deflection, vacuum (p2 = 0) is
obtained for values of -Ma1 3 > 2/(y - 1). In Fig. 4-40, the pressure coefficient in
relation to the square of the deflection angle cP/02 is plotted as a function of the
hypersonic similarity parameter Mal 6 by curves 1 and 2. For comparison, the
supersonic approximation of Eq. (443a) for high Mach numbers is
9
a-i 7
(4-61a)
VMi
2
(supersonic approximation)
(4-61b)
t52
shown as curve 3. This approximation agrees better with the expansion flow than
with the compression flow. The deviations are too large, however, to adopt this
approximation as the pressure equation for hypersonic flow with small deflections.
Inclined flat plate in hypersonic flow By setting 6 = ±a in Eqs. (4-58a) and
(4-58b), a being the angle of attack, the pressure distributions on the lower and
upper surfaces of an inclined flat plate in hypersonic flow can be easily computed.
They are constant over the chord. The lift is then obtained from the resultant
pressure distribution of the lower and upper surfaces. The lift coefficient is obtained as
CL =
cp
a2F(Ma a)
(4-62a)
S
2
3
i
Figure 4-40 Pressure coefficients at hypersonic
u,
i
2
3
flow (y = 1.4). (1) Expansion: lower sign, from
---- -- S
Eq. (4-58b). (2) Compression: upper sign, from Eq.
5
Ma, 15
(4-58a). (3) Supersonic approximation from Eq.
(4-61).
258 AERODYNAMICS OF THE WING
CL = (y + 1)a2
(Ma -+ 00)
(4-62b)
In Fig. 4-41, this result is presented for various Mach numbers of the incident
flow Mat =Ma according to Linnel [53]. It can be seen that the lift coefficient
for a fixed angle of attack decreases sharply with increasing Mach number and that
the hypersonic theory deviates from the supersonic theory. The curves for Ma = 0
(incompressible flow) and Ma = -- mark the limiting cases.
Hypersonic similarity rule Specific similarity rules were established in Sec. 4-2-3 for
subsonic, transonic, and supersonic flows. With their help, flows about geometrically
similar bodies can be related to each other. Such a similarity rule also exists for hypersonic flow. It was first presented by Tsien [98] and proved to be completely general
by Hayes [98]. The relation between pressure coefficient and deflection angle and
Mach number is expressed in Eq. (4-59). For symmetric incident flow, the
deflection angle is proportional to the thickness ratio t/c. In this case the Mach
number Mal becomes the incident flow Mach number Ma,,. Hence, in analogy to
Eqs. (4-35) and (4-36), the following expressions are obtained for the pressure and
drag coefficients:
cp = 82f 5 Ma.,
)
(4-63)
(4-64)
Hypersonic flow over a blunt profile The flow pattern in the vicinity of the nose of
a body in hypersonic incident flow is sketched in Fig. 4-42. Keeping in mind the
Figure 4-41 Lift coefficient of the flat plate vs.
angle of attack « for various Mach numbers
(y = 1.4). Hypersonic theory for small angles of
I
00
2°
4°
60
8°
a
10°
attack according to Linnell. (-) Hypersonic
theory, Eq. (4-62a), Ma -: cL = (y + 1)a2.
(-- -) Theory based on Eq. (4-46), Ma -} 0:
cL = 2ira.
WINGS IN COMPRESSIBLE FLOW 259
Figure 4-42 Sketch of a hypersonic flow. Zone A: boundary layer with friction and rotation.
Zone B: inviscid layer, but with rotation.
important fact that the leading edge of every body is somewhat-even if very
little-rounded, it is obvious that a stagnation point always exists on the nose, and
therefore a detached shock wave is formed upstream of the stagnation point in
which the approaching hypersonic flow is abruptly reduced to subsonic flow. As a
result, extremely high temperatures are produced near the stagnation point, which
may lead to dissociation and ionization of the gas and thus to deviations from the
properties of ideal gases. The thermic equation of state [Eq. (1-1)] is no longer
valid, for instance, and the specific heat capacity cp does not stay constant either.
The dependence of the temperature rise that occurs near the stagnation point
after passage of the shock wave on the Mach. number is presented in Fig. 4-43 for
air. The dashed line is valid for the ideal gas (see Fig. 4-2b) and the solid curves for a
24000
/
20000
00
160000
12000°
P. =
12i
10 'Atm
e00o°
10 -2
10_4
4000°
i
I
1
I
010
4
B
i
I
12
16
20
24
Figure 4-43 Temperature rise behind normal shock vs. Mach number (temperature before the
shock: T. = 222 K). Curve 1: real gas for several values of the static pressure per,. Curve 2:
ideal gas (y = 1.4).
260 AERODYNAMICS OF THE WING
real gas at several values of the static pressure p. of the incident flow. Because of
dissociation, the temperature rise at high Mach numbers is considerably smaller for
real gases than for ideal gases.
At larger distances from the stagnation point the shock wave closely approaches
the body contour. It is strongly curved, therefore, particularly near the stagnation
point (Fig. 4-42). On the body contour itself, a friction (boundary) layer (range A)
forms because of the viscosity, the thickness of which is now of the same order of
magnitude, however, as the distance between shock wave and the outer edge of the
boundary layer (range B). The formation of the boundary layer is governed by the
pressure distribution on the body, which, at hypersonic incident flow, is determined
mainly by the shape of the shock wave. This, in turn, depends on the body contour
and its boundary layer. There prevails, consequently, a very strong interaction
between friction layer and shock wave in hypersonic flow.
Another difficulty contributes to the problem. Since the shock wave is curved,
the entropy increases in the shock wave are different for each streamline. These
increases depend on the shock-wave inclination at the respective stations. Therefore,
the flow behind the curved shock is no longer isentropic. This means that the flow
behind the shock is no longer irrotational and that the separation into a rotational
friction layer and an irrotational outer flow, customary in boundary-layer theory, is
no longer possible. On the contrary, the total flow field between shock wave and
body contour is now rotational. The friction effects, however, are of significance
only in the zone next to the wall, zone A of Fig. 4-42, whereas zone B represents
an inviscid, but not irrotational, layer. An important characteristic of hypersonic
flow is its small lateral extent. Therefore the flow quantities vary strongly in the
lateral direction, whereas they vary only little in direction of the incident flow.*
The computations of the flow about a body with a blunt leading edge, and
particularly the computation of the shock-wave shape and of the pressure
distribution on the body, are very difficult, even when friction is disregarded,
because the flow field contains, side by side, zones of hypersonic, supersonic, and
subsonic flow.
In the special case (Ma. - co, y -; 1), the incident flow would remain
undisturbed up to the body contour and then be deflected in direction of the
contour. Thereby a portion of the horizontal momentum would be transmitted to
the body wall and thus produce the body drag. This special case is termed
Newtonian flow because Newton based his theory for the drag of arbitrary bodies
on this concept. It leads to the following expression for the pressure coefficient:
cP = 2 sine :g
(Newtonian approximation)
(4-65)
with a being the deflection angle.t This relationship serves as a rough approximation for the front portion of the body, whereas the above momentum consideration
*The opposite trend is found in transonic flow, in which the changes of the flow quantities
are small in the lateral and strong in the longitudinal direction.
1 This formula and its comparison with measurements will be discussed in more detail in
Sec. 5-3-3.
WINGS IN COMPRESSIBLE FLOW 261
does not give an answer for the rear body portion. In this context the expression
aerodynamic shadow is used.
The methods for the exact computation of hypersonic flows are very lengthy
and can be handled only with modern electronic computers. Investigations in this
field are still in progress, and many aerodynamic problems-particularly those
including the deviations from the properties of ideal gases-are not yet completely
solved.
Monographs in book form on hypersonic flow are listed in Section II of the
Bibliography. Compare also Schneider [82].
4-4 WING OF FINITE SPAN IN SUBSONIC
AND TRANSONIC FLOW
4-4-1 Application of the Subsonic Similarity Rule
It has been shown in Sec. 4-2-3 that the computation of flow about a wing of finite
span with incident flow Mach number Ma. < 1 can be reduced to the determination of the incompressible flow for a wing of finite span by means of the subsonic
similarity rule (Prandtl, Glauert, Gothert). The corresponding problem for the airfoil
of infinite span (profile theory) was discussed in Sec. 4-3-2. Computation of
incompressible flows was treated in detail in Chap. 2 for the airfoil of infinite span
and in Chap. 3 for the wing of finite span. The methods of wing theory for
incompressible flow therefore have a significance that reaches far beyond the area of
incompressible flow.
The second version of the subsonic similarity rule of Sec. 4-2-3 is the starting
point for further discussions. In what follows, the reference wing in incompressible
flow that is coordinated to the given wing at given Mach number will be designated
by the index "inc." Thus, the transformation formulas for the wing planform
according to Eqs. (4-10) and (4-15) are
(4-66)
Coordinates:
Xinc = x,yinc
Span:
bins = b
Wing chord:
cinc =C
(4-67b)
Taper:
Ainc = X
(4-68a)
Aspect ratio:
Ainc= A
Sweepback:
cot cpinc
I - Ma
1 -1VIax
1 - Maco
= cot p I
- Mc
(4-67a)
(4-68b)
(4-68c)
The geometric transformation for a trapezoidal swept-back wing in straight flight
and in yawed flight for Mach number Ma. = 0.8 is presented in Fig. 4-44.
For unchanged profile (h/c)inc = h/c, (t/c)inc = t/c, and unchanged angles of
attack ainc = a, the pressure coefficient of the given wing cp is obtained according
to Eq. (4-23) from that of the transformed wing cpinc as
262
WINGS IN COMPRESSIBLE FLOW 263
CP
_
- y1-Maw
Cpinc
(version II)
(4-69)
Compare Figs. 4-8 and 4-9 for the Mach number range 0 <Ma.. < 1. In the case of
airfoils of infinite span, the subsonic similarity rule is no longer valid for Maw, = 1
(see Sec. 4-3-2). Approximately, however, it may be applied to May, = 1 in the case
of wings of finite span. More details will be given later. Attention should be drawn
to the panel method of Kraus and Sacher [44], which includes the influence of
compressibility.
4-4-2 Inclined Wing at Subsonic Incident Flow
General formulas The local lift coefficient of a wing section is obtained through
integration of the pressure distribution over the wing chord according to Eq. (2-10).
By taking into account Eq. (4-69), the transformation formulas for the local lift
coefficient and, accordingly, for the local pitching-moment coefficient are thus given as
C1(y)
Cm
Clinc(Yinc)
=
1 - Ma00
(a -- CYinc)
CM inc(Yinc)
()
=
V
1-Maro
(4-70a)
(4-70b)
For incompressible flow, the wing theory of Sec. 3-3-5 produces the dimensionless
lift distribution yinc (?yinc) and the dimensionless pitching-moment distribution from
Eqs. (3-115a) and (3-115b). By introducing Eqs. (4-67a), (4-67b), (4-70a), and
(4-70b), the dimensionless distributions for subsonic flow become
clc
7=2b=yinc
µ
_CmC =
2b
(4-71a)
(« _ «inc)
(4-71 b)
1uinc
These equations show that the dimensionless lift and moment distributions remain
unchanged during transition from incompressible to compressible flow. It should be
noted, however, that the distributions y and yin, and p and pi,,, belong to
different planforms (Fig. 4-44).
The transformation of the coefficients of total lift and pitching moment, taking
into account Eqs. (4-66), (4-67a), (4-67b), and (4-69), results in
CL =
CLinc
(4-72a)
M(a
Cal =
-
CMinc
(a = «inc)
(4-72b)
1 1 - M)Ia
coefficient of induced drag in incompressible flow for elliptic lift
distribution is, from Eq. (3-31b), CDi inc =CL inc/T-line Introducing CL inc and iliac
The
into the above transformation formulas yields the relationship
264 AERODYNAMICS OF THE WING
CDi
_
2
CL
(4-73)
7TA
Hence, the formula for the coefficient of induced drag in relation to the lift
coefficient is independent of the Mach number. The transformation formulas for the
remaining aerodynamic coefficients are compiled in Table 4-4.
Elliptic wing Simple closed formulas for the lift slope as a function of the Mach
number can be established for wings with elliptic planform. For incompressible
flows, computations follow Eq. (3-98) of the extended lifting-line theory. Applying
the subsonic similarity rule yields
dcL
da
2rA
1/(1-1YIa2)A-'
+4±2
00
(4-74)
Table 4-4 Transformation formulas for the aerodynamic coefficients of an
inclined wing of finite span in subsonic flow (Prandtl, Glauert, Gothert),
a - «inc
1
Pressure distribution
cp
Lift
CL
Lift slope
1/1-Ma.
1
- Maw cL inc
dCL
da
1-
Zero-lift angle
co
= aoinc
Pitching moment
CM
Neutral-point position
cpinc
doL
1
Maw da inc
1
- Maw
_
xN
r
- cp
c,,
chin c
inc
1
Zero-pitching moment
y 1 - Ma's
CMo inc
1
Rolling moment
CMX
Ma-
cMx inc
i
Induced drag
1
cDi
1-Ma;,
cDi inc
WINGS INCOMPRESSIBLE FLOW 265
3,0
Figure 4-45 Ratio of lift slopes at subsonic
and incompressible flow for elliptic wings of
various aspect ratios A vs. Mach number of
incident flow according to Eq. (4-74).
70
aZ
19
IM
70
48
Mo.
from which the limiting values
dcL
da
dCL
da
2
A
`
(A - 0)
2z
j/1
(A - °°)
- Ma.
(4-75a)
(4-75b)
are obtained. Equation (4-75a) is identical to Eq. (3-101b). For very small aspect
ratios, the dependence of the lift slope on the Mach number thus disappears.
Equation (4-75b) is identical to the expression of the plane problem from Table
4-1
For the case Maw = 1, the lift slope becomes
dcL_?zl
da
2
(May,=1)
(4-75c)
Contrary to the airfoil of infinite span (A = °°), for which (dcL Ida). _ 00, the lift
slope of wings of finite span has finite values. The significance of this result will be
investigated more closely in Sec. 4-4-4.
The ratio of the lift slopes for Ma,,
0 and Maw = 0 is shown in Fig. 445 for
the Mach number. This figure shows that the
compressibility influence on the lift slope becomes smaller when the aspect ratio
is reduced. This fact was first pointed out by Gothert [28].
several aspect ratios against
Wings without twist The aerodynamic coefficients will be computed for the same
wings for which the lift distribution was determined in Sec. 3-3. These were a
trapezoidal, a swept-back, and a delta wing, with aspect ratios between A= 2 and
A1= 3. These three given wings are depicted in the upper boxes of Fig. 4-46. The
geometric data for the wings are compiled in Table 3-4. The second and third rows
of boxes show the wings transformed with the subsonic similarity rule for
Ma. = 0.4 and 0.8, respectively. The lift distributions of these wings have been
266 AERODYNAMICS OF THE WING
Figure 4-46 Planforms of given and transformed wings for the examples of lift distribution at
subsonic incident flow. Given wings: see Table 34. (a) Trapezoidal wing: 0 = 0° , A= 2.75,
X = 0.5. (b) Swept back wing: o = 50°, n = 2.75, X = 0.5. (c) Delta wing: p = 52.4°, .i = 2.3 1,
computed according to the wing theory for incompressible flow of Sec. 3-3-5.
The results of these computations for the lift distribution of the wing without
twist (a = 1) are presented in Fig. 4-47. The lower figures give the dimensionless lift
distributions y according to Eq. (4-71a) for Mach numbers Ma. = 0 and Ma, = 0.8.
The curves for Ma. = 0 are identical to curve 3 of the distributions in Fig. 3-33. In
the upper figures, the local neutral points and the total neutral points N are plotted
on the wing planform. At the upper part of Fig. 4-48, the lift slopes are plotted
against the Mach number; at the lower part, the neutral-point displacements with
respect to the geometric neutral point. The points for Ma. = 1, shown as open
circles, are theoretical values of an approximation method that will be explained in
Sec. 4-4-4. They agree with Eq. (4-75c) for trapezoidal and delta wings. In addition,
in all six diagrams, measurements by Becker and Wedemeyer [5] are included. The
measured lift slopes agree well with theory in all cases. In general, the dependence
on Mach number of the neutral-point positions is given satisfactorily by theory.
0)
d
o
O
0
0
pO
eo'
LA
c
267
268 AERODYNAMICS OF THE WING
I
a
5
b
C
5
4
4
-'I
't
2
1
0
0.2
0,4
0.6
0.8
1.0
0
62
0.4
0,6
0.6
40
Mao,,
0,
1
0.05
X0.10
ti
015
0.111
0
0.2
0,4
0.6
0,8
Mao.-
025
1.
0.30
0.4
0.6
0.8
Ma.. --
1.0
0
0.2
0.4
0.6
08
1,0
Mcz..-
Figure 4-48 Lift slopes and neutral-point displacements for the three wings of Fig. 4-46 vs.
Mach number. (
) Subsonic similarity rule (wing theory, Sec. 3-3-5); approximation theory
for Ma,. = 1; Sec. 4.4-3. (- - -) measurements from Becker and Wedemeyer, profile thickness
6 = 0.05. (a) Trapezoidal wing. (b) Swept-back wing. (c) Delta wing.
Certain discrepancies between theory and experiment of the neutral-point positions
can be explained mainly by the effect of the finite .profile thickness disregarded in
the theory; compare also Fig. 4-13b. It is noteworthy that the neutral point of the
trapezoidal wing shifts considerably upstream under the compressibility influence.
However, this theoretical result is only partially confirmed by measurements,
because shock waves form when the drag-critical Mach number is exceeded. On the
two other wings, the swept-back and the delta wings, the neutral points are
displaced toward the rear.
No more detailed statements are needed on the induced drag, since, as shown
by Eq. (4-73), the quotient cDi/cL is independent of Mach number and thus equal
to that of incompressible flow (see Table 3-4).
Further results on the aerodynamic coefficients of delta wings of various aspect
ratios are compiled in Fig. 4-82, together with results for supersonic incident
flow.
Data for the compressibility effects on the flight mechanical coefficients at
subsonic incident flow, for example, of the rolling, pitching, and yawing wing, are
found in Kowalke [5] and Krause [5].
WINGS IN COMPRESSIBLE FLOW 269
4-4-3 Inclined Wing at Transonic Incident Flow
It has been shown in Sec. 4-34 that the aerodynamic coefficients of a wing profile
undergo strong changes during transition from subsonic to supersonic flow, that is,
at transonic flow. The linear approximation methods for incident flows of subsonic
and supersonic velocities for the airfoil of infinite span fail when sonic velocity,
Ma -- 1, is approached (see Fig. 4-33). For wings of finite span, however,
physically plausible results may be obtained at Ma,, = 1. In this case, the same
limiting values are obtained for the lift-related coefficients (see, e.g., Fig. 4-82), by
approaching Ma. = 1 both from subsonic and from supersonic incident flow.
Now, for the lift problem at May, = 1, a few results will be presented that have
been obtained according to the method of Truckenbrodt [95] ; compare also the
publications of Mangler [58], Mangler and Randall [581, and Spreiter [85].
For tapered swept-back wings, the lift slope and the neutral-point position are
shown in Fig. 4-49 as functions of the geometric parameter cr/a for several values of
crlao . The wing geometry is seen in Fig. 4-49a, the lift slope in Fig. 4-49b, and the
neutral-point position in Fig. 449c. It is noteworthy that for crla > 1 [i.e., when
the trailing edge of the inner (root) section lies farther back than the leading edge
of the outer (tip) section], the lift slope is equal to iiA/2 for all wing shapes in
agreement with Eq. (4-75c). For cr/a < 1 (i.e., when the trailing edge of the root
section lies farther upstream than the leading edge of the tip section), the lift slope
is smaller than iiA/2. The neutral point for cr/a > 1 lies at xN/a = 3 (see Fig.
4-49c). For delta wings (do = a = Cr), XN/cr =!.. For cr/a < 1, the neutral point
0.81
I Cr
id0
04!
07
2
06
a4l
a
0
e2-
04
75-77-2-777
b
Figure 4-49 Aerodynamic coefficients of inclined swept-back wings at sonic incident flow
Ma. = 1, from Truckenbrodt, (a) Wing geometry. (b) Lift slope. (c) Neutral-point position.
270 AERODYNAMICS OF THE WING
shifts upstream. The linear theory for Maw, = 1 also allows computation of the
pressure distribution on the wing surface. Here, for uncambered wings, wing areas of
which the local span remains constant in the chord direction (Fig. 4-50a), or decreases
(Fig. 4-50b), do not contribute to the lift (d cp = 0).
Finally, a few test results [16] are given in Fig. 4-51 for the lift of delta wings
at Mach numbers close to unity. The lift slopes dcL/da are plotted against the
parameter A2(Ma', - 1), which results from the similarity transformation of
compressible flow [see Eq. (4-26)]. The pronounced peak in the theoretical curve
of dcL /da at Ma. = I is not fully confirmed through measurements. In the
subsonic and supersonic range, theory is well represented by the measurements.
Further experimental results on wings in transonic flow are found in Frick [24].
4-4-4 Wing of Finite Thickness at Subsonic
Incident Flow
Pressure distribution In this section, the wing of finite span at incident flow of
subsonic velocities will be investigated. at zero lift (displacement problem). The
pressure distribution of such a wing of finite thickness is of particular interest with
regard to the determination of the drag-critical Mach number at high subsonic
incident flow. The concept of critical Mach number has already been explained in
Sec. 4-34. The incident flow velocity of the critical Mach number is the lower limit
for the formation of the shock waves, which change the entire flow pattern
considerably and, in particular, lead to a strong drag rise (see Figs. 4-14 and 4-15).
The pressure distribution Of a three-dimensional wing with symmetric wing
profiles at subsonic incident flow is obtained from that of the transformed wing of
Eq. (4-69) as
Cp inc
e00 U2
2
°°
1-Mao
(6 = 5inc)
(4-76)
where Cp inc is the pressure distribution of the transformed wing for which the
pressure distribution of incompressible flow is to be computed. The computational
Figure 4-50 Pressure distribution on wings without camber at Ma,, = 1. The white areas do not
contribute to the lift, _o cp = 0, because for them (a) the span is constant in the chord direction,
(b) the span decreases in the chord direction.
WINGS IN COMPRESSIBLE FLOW 271
I
1.
L inear theory
A=3
l ip
Profile NACA
63 X 002
63 A 094
0.4
61AX5
0.2
L
FTI
-5
-6
0,6
I
'
l
l
3
7
.7
1.05
1.X1
Figure 4-51 Lift slope of delta wings of various thicknesses; aspect ratio A = 3, from [16].
Comparison with linear theory.
method was given in Sec. 3-6. The transformation of the wing planform follows
Eqs. (4-66)-(4-68); the thickness ratio 5 = t/c remains unchanged (version II of the
subsonic similarity rule of Sec. 4-3-2).
Drag-critical Mach number On three-dimensional wings, contrary to the plane
problem, frequently the wing leading or trailing edges are not perpendicular to the
incident flow direction. The simplest cast of that kind is the swept-back wing of
constant chord .and infinite span. This case has been treated previously for
incompressible flow in Sec. 3-6-3. The sweepback has a significant influence on the
magnitude of the critical Mach number, because only the velocity component
normal to the leading edge determines the maximum perturbation velocity on the
contour of such wings of finite thickness. From Eq. (4-53a), the critical pressure pcr
of the wing in incident flow normal to its leading edge is obtained after
multiplication with n U!,,/2 and with Mao cr = U. la. as
PCT
2,
x
000 pro
y+1 (
l
CL;ro
)
By introducing now, in agreement with the above statement, U= Cr cos .p as the
effective velocity instead of U0cr, and again adopting the dimensionless notation,
the critical pressure coefficient of the swept-back wing becomes
Cp Cr =
Per -Pcc
200
7
r2oo er
2
1 -Ma2mcr cost
y+1
Ilfa2
Cr
(4-77)
272 AERODYNAMICS OF THE WING
Here, as in Eq. (4-76), the pressure coefficient of the swept-back wing is referred to
the dynamic pressure of the incident flow. The relation cpcr(Ma.i.) is shown in
Fig. 4-52 for p = 00 (see curve 1 of Fig. 4-28) and for cp = 45°. To determine the
critical Mach number of the incident flow Ma.,,, the curve cp min is drawn in Fig.
4-52 up to its intersection with the curve Cpcr (see Fig. 4-28).
Swept-back airfoil of infinite span For the determination of the pressure difference
(p - p.) of a swept-back wing, it should be observed that (p is proportional
to the dynamic pressure of the effective velocity
cost tp. It is also
proportional to the thickness ratio or the angle of attack, respectively, determined
in the plane of the effective incident flow; that is, it is proportional to (t/c) cos gyp.
It follows that in incompressible flow,
Pinc -P- = (Pinc -P-)w=o COS cp
.
Referred to the dynamic pressure of the incident flow (p /2)UU, the relation
between the pressure coefficients becomes (cpmin)inc = COS Vinc(Cpmin)inc,,p-oWith Eqs. (4-76) and (4-68c) it is
s
Cpmin
(Cpmin)inc,cp=o
1
with cos Pinc =
1-
cos (p
Q°°
1 '- Mac. COS 2
cp
By substitution, finally,
/
cosq7
Cpmin .V1
- Mci' cosaT (Cputin)inc,,p=o
(4-78)
The above-explained procedure has been applied to an example in Fig. 4-52. Chosen
were two airfoils of infinite span, one unswept and one with a sweepback angle of
45°. For the unswept airfoil, (Cpmin)inc, V =o = -0.2 has been assumed, resulting in
a critical Mach number
0.83. The effect of the sweepback is seen in
a shift of the critical Mach number to a considerably larger value of
1.13. This shift is caused by three effects. First, the curve cp cr is shifted to the
right because of the sweepback; second, by the sweepback, Cpmin at Ma. = 0 is
o.
a
04
I
cpcr,
Y
cp;mi n
Figure 4-52 Determination of drag
Of
I
1
+
Ma,ocr\i Ma00cr
0
0.2
0.4
0.6
40
mam1
0.9
7.2
1.4
1.6
critical Mach number Mao, cr for an
unswept and a swept-back airfoil of
infinite span. (cp
o ,Mac = o
-0.2.
WINGS IN COMPRESSIBLE FLOW 273
2,0
U
1.8
16
1,c
=60°
30 °
15
0,8
a
0,6
-02
-01
-03
-06
-0.5
-0..q
(cpmin)inc,,p=0
800
Cos f0
/i
I
i
.
Um
zoo
c
'
Figure 4-53 Drag-critical Mach number of the
incident flow of swept-back airfoils of infinite
b
I
span. (a) Effect of pressure coefficient. (b)
Effect of thickness ratio (biconvex parabolic
0°
2
Ma,o cr
shape).
reduced; and third, the rise of cprnin with Mach number is much weaker for a
swept-back wing than for an unswept wing.
An extension of Fig. 4-52 is given in Fig. 4-53a, where the critical Mach
numbers of swept-back airfoils of infinite span are presented relative to
(cpmin)inc, =o For a biconvex parabolic profile,(cprnin)inc, =o =-2(urnaxIUU)inc =
-(8/ir)(t/c). Corresponding to the example shown in Fig. 4-53a, the sweepback angle
has been evaluated in Fig. 4-53b as a function of the critical Mach number and for
several thickness ratios. For S = t/c = 0, this function is
Ma cr =
1
cos
(A -> cc, 5 -. 0)
(4-79)
274 AERODYNAMICS OF THE WING
Thus, sweepback may raise the drag-critical Mach number of very thin profiles
considerably above unity.
Middle (root) section of the swept-back wing The discussions about the effect of
wing sweepback presented so far are valid only for the straight airfoil of infinite
span (see Fig. 4-52). For folded wings (Fig. 3-74), the favorable sweepback effect
(raising of the drag-critical Mach number) is not realized fully in the vicinity of the
root section. The middle portion of the wing performs somewhat as if it were
unswept. For the computation of the critical Mach number of the middle section of
the folded swept-back wing, the following procedure has to be applied: For
incompressible flow, the velocity distribution over the root section is given by Eq.
(3-187). The maximum velocity over the root section produces the largest
underpressure (Cpmin)inc = -2(UmaxlU-)inc. The value Of (urnax/U-)inc of a
parabolic profile is plotted in Fig. 3-76 against the sweepback angle ipinc- Conversion
of (Cpmin)inc into Cpmin for the various Mach numbers is given by Eq. (4-76),
where the sweepback angle also has to be transformed according to Eq. (4-68c). The
critical Mach number is then obtained as the intersection of the curves cp min and
cpcr of Fig. 4-52, where for the root section the curve cpcr for p = 0 has to be
taken. The result of this computation is presented in Fig. 4-54, for sweepback
angles p = 0, 45, and -45° and for several relative thickness positions Xt. The
dashed curve for ep = ±45° shows the values for the straight swept-back wing. They
are valid for sections of the folded wing at large distances from the root. It is
clearly seen that the swept-back wing (gyp = +45°) has the most favorable critical
Mach number of the root section for relative thickness positions of about 30%,
12
p=t45°
10
14
0
i
-45°
+45°
2
1
02
0.2
Xt
0.4
x t=02
0.3
0.5
_. - -
i__9
-
06
o.4
10
Xt
Figure 4-54 Drag-critical Mach numbers for middle (root) and outer (tip) sections of folded
swept-back wings of various relative thickness
positions; thickness ratio 6 = tic = 0.1. (1) Root
section. (2) Tip section.
WINGS IN COMPRESSIBLE FLOW 275
whereas the swept-forward wing (gyp = -45°) is most favorable for relative thickness
ratios of about 70%. These results show that the critical Mach number of the
middle section of folded swept-back wings is, in general, considerably lower than
that of the tip section. It follows that the favorable sweepback effect of the straight
swept-back wing cannot be fully realized by folded wings.
Investigations of the drag-critical Mach number of folded swept-back wings were
made by Neumark [64]. He also studied the influence of finite aspect ratios on the
critical Mach number, but no marked differences with the airfoil of infinite span
were found; see Fig. 3-71.
Experimental results Raising of the drag-critical Mach number by sweepback has
found practical applications of great importance for airplane design. As has
previously been shown in Sec. 4-3-2, increasing the critical Mach number produces a
shift of the compressibility-caused drag rise to higher Mach numbers (Fig. 4-14a). It
must be expected, therefore, that sweepback causes a shift to higher Mach numbers
of the strong rise of the profile-drag coefficients with Mach number, cDp(Ma.).
This fact was first realized by Betz in 1939 and has been checked experimentally by
Ludwieg [57]. A few of his measurements are plotted in Fig. 4-55. The polars for
an unswept and for a swept-back trapezoidal wing (cp = 45°) show the following:
The profile drag (CL = 0) of the unswept wing is several times larger at Ma = 0.9
than at Maw = 0.7. Thus the drag-critical Mach number of this wing lies between
Maw, = 0.7 and May, = 0.9. For the swept-back wing, however, the profile drag at
Maw, = 0.9 is only insignificantly higher than at Ma. = 0.7. In other words, the
critical Mach number of this wing lies above Mae, = 0.9. Another example of this
important swept-back wing effect is demonstrated in Fig. 4-56. Here, from [71],
CDp is shown versus Ma,. for an unswept and a swept-back wing (p = 45°). The
sweepback effect is manifested by a shift of the onset of the drag rise from about
Maw, = 0.8-0.95. This favorable sweepback effect has been exploited by airplane
designers since World War II. The presentation of Fig. 3-4c, namely, sweepback
angle versus flight Mach number, shows very clearly that the sweepback angle of
airplanes actually built increases markedly when Mach number Ma. = 1 is approached.
Thick wing at sonic incident flow The subsonic similarity rule of Sec. 4-4-3 leads to
useful results in computing the lift for incident sonic flow (Maw, = 1). It fails,
however, in the computation of the displacement effect of a finitely thick wing at
sonic incident flow. The reason is that the pressures on the wing become infinitely
high. Compare, for example, [70] for an account of this difference between the lift
problem and the thickness problem in the limiting case Mae, -} 1. To obtain useful
information on the thickness problem at Ma. = 1, nonlinear approximation
methods have to be applied. The transonic similarity rule (see Sec. 4-3-4) is
particularly well suited for classification and systematic presentation of test results
on wings of finite span; see Spreiter [103]. Further information on the theory of
transonic flow of wings is found in publications by Keune [43] and Pearcey [69] and
in reference [68] on the equivalence theorem of wings of small span in transonic flow
of zero incidence.
276 AERODYNAMICS OF THE WING
a=12,4°
f
Z/
0or
Maw =0.7
38
°
.4
4 Maw = as
9,B°
"Oo
12,0
174
Maw =0. 9
0.
22 °
= 0°
1 1.80
.4°
0
S
5,6°
-0
-04
'
a
0.1
cD.
VT
430
0,2
0.2
0.1
b
0.
CD
Figure 4-55 Polars, lift coefficient CL, and drag coefficient cD at high subsonic incident flow;
Mach number Ma. = 0.7 and 0.9, for a straight and a swept-back wing of profile Go 623, from
Ludwieg. (a) Straight wing, b = 80. mm, Cr = 22.5 mm; Re = Uo°c,.1v = 3.0 105 at Ma = 0.7,
= 3.5 - 105 at Ma o° = 0.9. (b) Swept-back wing, p = 45°, b' = 57 mm, Cr = 32 mm; Re =
Uocr/v=4.2 105 atMao,=0.7,=5.0 - 10' at Ma.
0.9.
4-5 WING OF FINITE SPAN AT SUPERSONIC
INCIDENT FLOW
4-5-1 Fundamentals of Wing Theory at Supersonic Flow
Mach cone (influence range) There is an essential physical difference between flows
of subsonic and supersonic velocities, namely, that the disturbances of a sound
point source in the former flow propagate in all directions, but in the latter flow
only within a cone that lies downstream of the sound source (Fig. 1-9b and d). This
so-called Mach cone has the apex semiangle a, which, by Eq. (1-33), is given by
sing =
1
11J. a 00
and
tang =
1
VMaL-1
(4-80)
with Ma. = U. 1a.. The state of affairs just discussed may also be interpreted (see
Fig. 4-57) that a given point in a supersonic flow, U. > ate, can influence only the
space within the downstream cone, whereas it can itself be influenced only from the
space within the upstream cone. Application of this basic fact of supersonic flow on
a wing of finite span is demonstrated in Fig. 4-58. The flow conditions at a point x,
WINGS IN COMPRESSIBLE FLOW 277
0.10
Q0.
t
0,06
=45
004
0.0
0
12
Ma
Figure 4-56 Profile drag coefficients vs. Mach
number for an unswept and a swept-back wing
(gyp = 45°), t/c = 0.12, A = 4.
y, z = 0 on the wing can be influenced only from the crosshatched area A' of the
wing that is cut out of the wing by the upstream cone. When the Mach line M.L.
lies before the wing leading edge, as in Fig. 4-58, the area between this Mach line
and the leading edge also contributes to the influence on point x, y, z = 0.
Downstream, the influence range is bounded by the two Mach lines through the
point x, y, z = 0.
Subsonic and supersonic edge The conditions of Fig. 4-57 find an important
application in oblique incident flow on a wing edge. If, as in Fig. 4-59a, a Mach line
lies before the wing edge, the component v,, of the incident flow velocity U.
normal to the edge is smaller than the speed of sound a.. Such an edge is termed
subsonic edge. Conversely, if, as in Fig. 4-59b, the Mach line lies behind the wing
edge, then v,, is larger than ate, . In this case, the edge is termed supersonic edge.
With p as the Mach angle and y as the angle of the edge with the incident flow
direction (Fig. 4-59), the expression
m
= tang
tan
- tan y Xa., - 1
(4-81)
Figure 4-57 Upstream cone and downstream cone of a point
in supersonic flow. µ = Mach angle.
278 AERODYNAMICS OF THE WING
Figure 4-58 Wing in supersonic incident
flow. A'= influence range.
allows one to determine whether the edge is subsonic or supersonic. Thus the edges
are characterized as follows.
Subsonic edge:
vn < a.
p>y
m<1
(4-82a)
Supersonic edge:
vn > a
p<y
m>1
(4-82b)
The special case y = 00 (m = 1) is a subsonic edge for all supersonic Mach numbers,
and the case y = 900 (m = oc) is a supersonic edge. The concept of subsonic and
supersonic edges is of significance not only for the leading edge, but also for the
trailing and side edges. This fact is explained in' Fig. 4-60. Here, the subsonic edges
are drawn as dashed lines, the supersonic edges as solid lines. For the same wing
planform, the Mach lines for three different Mach numbers are drawn. At the lowest
U00
Uoo
U00
\\
vn=Uco
a
b
Figure 4-59 Concept of subsonic and supersonic edges. (a) Subsonic edge (0 < m < 1). (b)
Supersonic edge (m > 1).
WINGS IN COMPRESSIBLE FLOW 279
,
01
Mach line
11
Figure 4-60 Example for the explanation of subsonic and supersonic edges of swept-back wings.
Dashed lines: subsonic edges; solid lines: supersonic edges. (a) Subsonic leading edge and subsonic trailing edge. (b) Subsonic leading edge and
supersonic trailing edge. (c) Supersonic leading
edge and supersonic trailing edge.
Mach number (Fig. 4-60a), all edges are subsonic, at the highest Mach number (Fig.
4-60c), the leading and trailing edges are supersonic, but the side edges are still
subsonic. Distinction between subsonic and supersonic edges is conditioned by the
difference in flow patterns in the vicinity of the edges. In Fig. 4-61, the various
types of flow patterns are sketched, which are the sections normal to the leading
and trailing edges, respectively. In close vicinity to the section plane, the flow may
be considered to be approximately two-dimensional. The basically different
character of subsonic and supersonic flows over an inclined flat plate was
demonstrated in Fig. 4-22. Based on this figure, Fig. 4-61 shows the subsonic
leading edge, at which flow around the leading edge is incompressible according to Fig. 2-9a. An .essential characteristic of this flow is the formation
of an upstream-directed suction force on the nose (see Fig. 4-22a). Figure 4-61b
shows the subsonic trailing edge with smooth flow-off according to the Kutta
condition (see Sec. 2-2-2). At such a trailing edge, the pressure difference between
the lower and upper surfaces is equal to zero (Fig. 4-22a). Complete pressure
equalization between the lower and upper surfaces is achieved. In Fig. 4-61c and d,
the supersonic leading edge and the supersonic trailing edge, respectively, are shown.
In both cases, neither flow around the edge nor smooth flow-off is achieved, but
Mach lines originate at the edges along which the flow quantities change unsteadily.
Between the lower and upper surfaces, a finite pressure difference exists (see Fig.
4-22b).
Finally, the pressure distributions over a wing section are shown schematically
for the three different cases of Fig. 4-60. For the section with subsonic leading and
280 AERODYNAMICS OF THE WING
b
vn < a,.
Figure 4-61 Typical flow patterns at subsonic and
supersonic edges (see Fig. 4-59). (a) Subsonic leading
edge, vn < a., flow around edge. (b) Subsonic trailing
edge, vn < a-, smooth flow-off (Kutta condition). (c)
Supersonic leading edge, un > a-, with Mach lines. (d)
Supersonic trailing edge, vn > a,o, with Mach lines.
4-62a,, the pressure distribution is similar to that of
incompressible flow, as would be expected. The rear Mach line, however, causes a
break in the pressure distribution. In the case of the section with supersonic leading
and trailing edges (Fig. 4-62c), the pressures at the leading and trailing edges have
finite values. The front Mach line again produces a break in the pressure
trailing
edges,
Fig.
distribution.
4-5-2 Method of Cone-Symmetric Supersonic Flow
Fundamentals Before the general theory of the three-dimensional wing in supersonic
incident flow is treated in the following sections, a simple special case will be
discussed first that has great significance, particularly for wings of finite span.
Consider the flow about a triangular plane surface. In Fig. 4-63, two Mach lines
originate at the apex A0 of the triangle, where, in this example, the right-hand edge
of the triangle is a subsonic edge, the left-hand edge a. supersonic edge. Further, the
flow conditions are studied on a ray originating at the triangle apex. The flow
conditions at point A 1 of this ray are determined exclusively by the area that is cut
out of the triangle by the upstream cone of A1, supplemented-if applicable-by the
area between the Mach line M.L. and the wing leading edge (influence range of A 1).
The flow conditions at A2 likewise are determined exclusively by the influence
range of A2. The two influence ranges of Al and A2 are geometrically similar, and
the flow conditions in Al and A2 must be equal. It follows that the flow properties
WINGS IN COMPRESSIBLE FLOW 281
Cp
b
Cp
Figure 4-62 Pressure distributions
over the wing chord (schematic)
for a section of an inclined sweptback wing. (a) Subsonic leading
and trailing edges. (b) Subsonic
M. L.
leading and supersonic trailing
edge. (c) Supersonic leading and
trailing edges.
(pressure, density, velocity, and temperature) are constant on the whole ray through
A0. This statement is valid for any ray through A0. The flow field thus described is
called a cone-symmetric (conical) flow field, according to Busemann. It is a
requirement for the above considerations that the edges of the triangular area be
straight lines; they are two special rays of the cone-symmetric flow field.
Figure 4-63 Cone-symmetric flow over
triangular flat plate at supersonic flow.
a
282 AERODYNAMICS OF THE WING
a
l
i
I
I
j
Figure 4-64 Examples of the application of cone-symmetric flows. (a) Triangular wing of finite
thickness at zero lift. (b) Triangular flat plate with angle of attack. (c) Rectangular flat plate
with side edges.
A few examples of the application of such cone-symmetric flows are given in
Fig. 4-64. Figure 4-64a shows a delta wing with a double-wedge profile in sections
normal to the incident flow direction. This is an example of a wing of finite
thickness at zero lift. Figure 4-64b depicts the triangular flat plate with angle of
attack (lift problem). The flow over the side edge of an inclined rectangular plate is
seen in Fig. 4-64c. In the triangular part of the plate surface, limited by the Mach
line M.L., the flow conditions are constant on each of the rays through the corner
A0. On the remaining part of, the surface, the flow field is constant because here, in
sections normal to the plate leading edge, the flow is two-dimensional and
supersonic (see Fig. 4-22b).
For the cone-symmetric flow just discussed, the three-dimensional potential
equation, Eq. (4-8), assumes a simplified form. By choosing for the cone-symmetric
flow the coordinate system according to Fig. 4-65, the perturbation potential
0 (x, y, z) = x f (77, C)
with
7=
y
x
and
z
S=-
x
(4-83a)
(4-83b)
Figure 4-65 Cone-symmetric flow at supersonic
velocity.
WINGS IN COMPRESSIBLE FLOW 283
satisfies the condition that the velocity components from Eq. (4-6) are constant on
the rays through the cone apex A. By introducing Eqs. (4-83a) and (4-83b) into Eq.
(4-8), the following differential equation of second order for f(77, ) is obtained,
772)
21
02f
- 2 -?7
a>7z
+ (tan 2y
77 as
- ") -ta- = 0
.:.,
(4-84)
where tan p = 1 / Ma. - 1. This equation for the new function f depends only on
the two space variables 77 and in the plane normal to the incident flow direction
(x direction) (see Fig. 4-65). In the lateral planes (x = const), the v and w
components form a quasi-plane flow. Application of the cone-symmetric supersonic
flow was restricted at first to wings with straight edges. Later it was extended to
"quasi-cone-symmetric" flows, see [30].
Classification. of ranges The application of this method will be demonstrated for
one wing at various Mach numbers by means of Fig. 4-66. The chosen example, a
pointed swept-back wing without twist, is shown in Fig. 4-66. In Fig. 4-66a, it has
subsonic leading edges only, in Fig. 4-66b only supersonic leading edges. In range I
of Fig. 4-66a, the flow is cone-symmetric with the wing apex A as the cone center.
In the remaining crosshatched zones, no cone symmetry exists with reference to the
centers B and C, since on the Mach lines through B and C the pressure cannot be
constant because of range I. In Fig. 4-66b, the pressure is constant over the entire
range II, as will be shown later. In range III, there is cone-symmetric flow, the cone
Uooj
A
i /ioho
Figure 4-66 Flow types of inclined wings of
finite span at supersonic incident flow; example
of a tapered swept-back wing. M.L. = Mach line.
(b)
(a) Wing with subsonic leading edge, u >
Wing with supersonic leading edge, µ < -y. Without hatching = pressure is constant. Single hatching = pressure distribution is cone-symmetric.
Cross-hatching = pressure distribution is not conesymmetric.
284 AERODYNAMICS OF THE WING
tip of which is the wing apex Ao, since the pressure is constant on the Mach lines
from point A because of range II. Also, range IV is covered by cone-symmetric flow
with reference to point B. In the crosshatched zones, however, the flow is not
cone-symmetric. Now, some information will be given on the pressure coefficients in
the various ranges (Table 4-5). The values are referred to the constant pressure
coefficient of the inclined flat plate, according to Eq. (443):
cPP, _
P_ _
eW
U
2
'/Ma
f
-2
(4-85)
L%
1
Table 4-5 Basic solutions for the pressure distribution of the inclined flat
surface in supersonic incident flow (cone-symmetric flow) for ranges 1, II,
III, and IV of Fig. 4-66*
CpICPpl
na
U
m
E' (m)
A 0<m<1
II
-
m>1
III
A
m>1
IV
B
in > 1
Unwept leading
edge (m - oc)
Swept-back
leading edge
1
-
t
1 ___V
in
1
mz - 1
in
2
m2-1
1
2
arc cos
arc cos
1-t
z
1
m2-t2
1 + 2t m + 1
Vrraz-1
in -
i
are cos (1 + 2t)
0<m--I
B
x
*cppl from Eq. (4-85); m from Eq. (4-81). Range I, wing with subsonic leading
edge, n from Eq. (4-86); 11, wing with supersonic leading edge, range before the Mach
line, t from Eq. (4-90); III, wing with supersonic leading edge, range behind the Mach
line, t from Eq. (4-90); IV, wing with supersonic leading edge and side edge, t from Eq.
(4-92).
tE' (mn) - f V1 - (1 - m''-') sin2,p dip; E'(0) = 1.
0
WINGS IN COMPRESSIBLE FLOW 285
C
cp P1
10
Figure 4-67 Inclined wing with subsonic
leading edge (0 < m < 1). (a) Wing planform (triangular wing). (b) Pressure distri-
bution on a section normal to the flow
direction, m = 0.6.
The index pl designates the plane problem. The upper sign will be used for the
upper side, the lower sign for the lower side.
Wing with subsonic leading edge Without going into the details, the computed
pressure distributions in sections through the wing, normal to the incident flow
direction (0 < f < 1), are tabulated in Table 4-5; see [20, 77] for a wing with
subsonic leading edge (range I in Fig. 4-66a). In the present case, m assumes the
values 0 < m < 1. From Fig. 4-67a, the following relation applies to
Range I:
On the wing,
77 =
y
x
cot
runs from -1 to + 1, where
edges.
In Fig. 4-67b, the
pressure
tan
tan r
-1 and
(4-86)
1 are the leading
distribution is shown. On the two edges,
c, is infinitely large, as would be expected for flow around a sharp subsonic
leading edge (see Figs. 4-61a and 4-62a and b). The mean value of the pressure over
the width (span) is
286 AERODYNAMICS OF THE WING
Cp
2 E'(na)CPpl
2
c,(0)
(4-87)
Wing with supersonic leading edge The simplest case of a wing with supersonic
leading edge is the inclined flat plate in incident flow normal to the leading edge.
This problem has been treated before in Sec. 4-3-3 as a plane problem [see Fig.
4-22b and Eq. (4-85)].
The pressure distribution of the swept-back flat plate, the leading edge of which
forms the angle y with the incident flow direction (Fig. 4-68) is obtained by
considering that only the component of the incident flow velocity normal to the
leading edge, that is, U. sin 'y, is affecting the lift (see Fig. 3-45). In the section
normal to the leading edge, the plate angle of attack a* = a/sin y. Here a is the
angle of attack in the plane of the velocity U.. Consequently, the pressure
distribution of the swept-back inclined flat plate becomes
CP _ p - poo
_
2asiny
Mao, sine y
Q00 Ua
2
-1
(4-88)
00
The swept-back plate, 'like
the unswept plate, has a constant pressure
distribution over the wing chord. The ratio of the pressure coefficients of swept-back
and unswept plates becomes, with Eqs. (4-81) and (4-85),
P_ Cppl
n7,
1/,rn2
-1
(4-89)
where m > 1, according to the assumptions made. It is noteworthy that cP/cpp1 > 1,
which signifies that the swept-back plate produces a higher lift per unit area than
the unswept plate, presupposing that the angles of attack, measured in the incident
flow direction, are equal. For y = 7r/2, that is, m = oo, cp/cppl = 1, as would be
expected. For y = p, that is, m = 1, cp/Cp p1 = o. In this case, the Mach line falls on
the leading edge, and thus the incident flow component normal to the leading edge
is equal to the speed of sound. Linear supersonic theory therefore fails.
These results for two-dimensional flow about a swept-back flat plate can be
applied to the wing of finite span. To that end, an inclined delta wing with
ML
`
\v
Figure 468 Swept-back flat plate with supersonic leading edge.
WINGS IN COMPRESSIBLE FLOW 287
a
Figure 4-69 Inclined wing with supersonic
leading edge (in > 1). (a) Wing planform
(triangular wing). (The hatched area A' is
explained on page 293.) (b) Pressure distri-
bution on a section normal to the flow
1
in
t
direction, in = I.S.
supersonic leading edge (m > 1), according to Fig. 4-69a, may be considered. Here,
m is given by Eq. (4-81), and the following relationships apply to
Ranges 11 and III:
t
=tan y' _ ?/ cot,u, _ '! Va - 1
00
x
tan /c
(4-90)
X
The straight lines t = const are rays through the wing apex, where t runs from 0 to
m > 1; t = ±1 represents the Mach line, t = ±m the leading edge. On this wing, the
ranges II and III of Fig. 4-66b must be distinguished. The pressure is constant and is
given by Eq. (4-89), between the Mach line and the leading edge, that is, in range II
(1 < t < m). Details of the computation for range III (0 < t < 1) will not be given
here.
In Table 4-5, formulas are listed for the basic solutions in ranges II and III at
cone-symmetric supersonic incident flow. Figure 4-69b gives the pressure distribu-
tion in a section normal to the flow direction. Note that the pressures on the
portions of the surface that lie before the Mach lines originating at the apex are
larger than in the case of a leading edge normal to the incident flow. Conversely,
the pressures are considerably smaller behind these Mach lines. The mean value of
cp over the span is
Cp = Cppl
(4-91)
288 AERODYNAMICS OF THE WING
Wing with a supersonic leading edge and supersonic side edge So far, the wing with
a supersonic leading edge has been treated. Now, for a further basic solution, the
wing with a supersonic leading edge and a supersonic side edge will be discussed. A
side edge is defined as an edge that is parallel to the incident flow in the planform
(Fig.
4-70). From point B of the
side edge,
a wedge-shaped range IV of
cone-symmetric flow is formed rearward (see Fig. 4-66b). This range is bounded by
the side edge of the wing and the two Mach lines issuing from A and B. The
boundary conditions for the pressure distribution in range IV are cp = 0 on the side
edge and cp = crII = const on the Mach line. By using the coordinate system 2, y
of Fig. 4-70a, the following relationship applies to
cotu=
Range IV:
VXa'-1
(4-92)
z
where t = 0 represents the side edge and T= -1 the Mach line. The relationship for
the pressure coefficient is given in Table 4-5.
A particularly comprehensive compilation of basic solutions is found in Jones
and Cohen [39].
Superposition principle Determination of the lift distributions at supersonic flow
over an arbitrary wing shape is not yet possible by means of the basic solutions of
a
i
Leading edge
i'
\0
B
//
I'll
Side edge (tip)
X T__7
21
t-0
Figure 4-70 Inclined wing with supersonic leading edge and side edge. (a)
x
Swept-back
wing.
wing.
(b)
Rectangular
WINGS IN COMPRESSIBLE FLOW 289
Figure 4-71 The superposition principle at supersonic velocities. Wing
AED : basic; ABCD: given wing.
Table 4-5. In those ranges of the wing that are covered by the Mach cones of
several disturbance sources, for example, the crosshatched zones in Fig. 4-66, the
basic solutions cannot be immediately applied. For these areas, a solution can be
found, however, with the help of a simple superposition procedure, which will be
sketched briefly.
Sought is the lift distribution of a tapered swept-back wing without twist,
ABCD in Fig. 4-71. To this end, the wing is complemented to a wing with a sharp
tip AED for which the basic solution of the lift distribution is known from Table
4-5. To obtain the given wing ABCD from this initial wing AED, a disturbance
source is thought to be placed on point B. Two Mach lines under the angle p with
the side edge BC issue from this point. The left-hand Mach line intercepts the
trailing edge of the given wing at point F. In the range ABFD of the given wing, no
change in lift distribution is caused by the disturbance source B. Now, the following
solution has to be added to the solution of the wing AED to obtain the solution for
the given wing ABCD: For the range BEF, a solution is to be found with the
following characteristics (so-called compensation wing). In the partial range BEC,
the lift of the compensation wing has to be equal but opposite to that of the wing
AED so that the total lift disappears in the former after superposition (lift
extinction). In the partial range BCF, the compensation wing must not have a
normal velocity component to keep the angle of attack, of this range unchanged
after superposition. The details for the computation of such compensation wings
cannot be discussed here. A comprehensive listing of the most important
compensation wings and their velocity distributions is found, however, in Jones and
Cohen [39]. For the fundamentals of the theory, compare also Mirels [62]. The
above method may be applied to a simple example like that given by Fig. 4-72.
4-5-3 Method of Singularities for Supersonic Flow
in Sec. 4-5-2, the method of cone-symmetric flow was applied to the computations
of flows about wings in supersonic incident flow. This method is limited to the
treatment of special cases, such as wings without twist and with straight edges.
Wings of arbitrary planform with twist cannot be treated using this method. For
them, the method of singularities is available.
290 AERODYNAMICS OF THE WING
a
.11
Cp P1
C
1
1
0
CP2- - CP p I
d.
Figure 4-72 Application of the superposition principle to the inclined rectangular flat plate. (a) Given wing. (b)
Basic wing (infinitely wide plate). (c),
(d) Compensation wings 1 and 2. (e)
Procedure for determination of the
e
pressure distribution.
'-M. L.
A detailed presentation of this method and of its applications is found in Jones
and Cohen [39] and Heaslet and Lomax [30] ; see also the basic contribution of
Keune and Burg [42].
The basic features of the method of singularities for incompressible flow have
been explained in Secs. 3-2 and 3-6. An analogous procedure has been developed for
supersonic flows. The equation for the velocity potential of three-dimensional
incompressible flow O(x, y, z) is given for Ma00 > 1 in Eq. (4-8).
Vortex distribution It has been shown in Sec. 3-2-2 that a solution of the potential
equation for a wing with lift in incompressible flow can be obtained by means of a
vortex distribution in the xy plane. By designating the vortex element at station
x'y' (Fig. 3-17) by k(x', y'), Eqs. (3-46) and (3-47) yield for the contribution of
this element to the velocity potential
d2 0 (x, y, z ; x', y,) =
with
L;
(x',
4 y')
r=V(x
(y - 01 . z-' (1
I x - x') d x' d y'
-x')s±(y- y')2Tz2
(Ma,. = 0)
WINGS IN COMPRESSIBLE FLOW 291
By applying the supersonic similarity rule Eq. (4-10) with Eq. (4-12), the
corresponding solution for supersonic incident flow becomes
d20(x, y, z; x', y') =
with
2k(x', y')
4n
z
(y - y')2 +
x -xdx'dy'
r
(4-93)
r = V (x - x')2 - (Ma2 - 1) [(y - y')2 + z2]
The analogous formula for a source distribution is Eq. (4-101).
For the transition to the potential of supersonic flow, the term in the incompressible equation that is formed by multiplication with the i in the brackets
must be eliminated because it is real in the entire space and, therefore, physically
impossible in supersonic flow. The term with l 1r in the potential equation of
incompressible flow becomes, in the potential equation of the supersonic flow, a
term that is real only within the Mach cone. Because a point P is affected by two
disturbances in supersonic flow but by only one in subsonic flow, as demonstrated
in Fig. 4-73b, the factor before the vortex element k has, for supersonic flow, twice
the value of that for incompressible flow.
In order to obtain now the total potential at a point x, y, z, the contributions
of the vortex elements have to be integrated in the x y' plane. Here, only the
downstream cones of the vortex elements are taken into account; the upstream
cones remain unused. Hence, the potential of the vortex distribution, see Eq. (3-46),
becomes
+8
'(x,y,z)=
r
z
y'), + z2 G(x' y, z; y') dy'
(4-94)
a
l~iguze 4-73 The effect of a sound point
source at subsonic and supersonic velocities.
292 AERODYNAMICS OF THE WING
with the kernel function
2:,(y')
G(x, y, z; y') = 2
k(x', y') (x - x') dx'
J
(1Yla2-1)[(y-y')2+z2]
{x-x')9_
(4-95)
In Eq. (4-94), the integration has to be conducted over the width of the
upstream cone in span direction (see Fig. 4-74). Integration of Eq. (4-95) has to be
conducted over x' in the upstream cone of the point x, y, z from the leading edge
to the Mach cone xo (y'), given by
xo(y') = x - f(Ma0 - 1) [(y - y')2 + z2]
(4-96)
Corresponding to Eq. (3-45), the velocity components in the x and z directions at
the wing location z = 0 are obtained from Eqs. (4-94) and (4-95) [compare also
Eqs. (3-37) and (3-41)] as
u (x, y, 0) = ± , k (x, y)
(4-97)
+3
1
4a
lim
,0
2
G(--, y; y)
r 0(x,Y;Y') d
T (y - 01
-
(4-98)
3
with the kernel function
z,(y')
,
j
k(x', y') (x - x') dx'
(x - x')2 - (Ma's - 1) (y - y')'
(4-99a)
Xf(Y')
(4-99b)
G
Xf(Y)
The equation for the vortex density k(x, y) is obtained from the kinematic
flow condition, which for the wing without twist with z = 0 and aF = a is given
from Eq. (3-40) by
U0 a + w(x, y) = 0
(4-100)
xa ')<
Figure 4-74 The integration range for the velocity
potential of a wing at supersonic incident
flow velocity from Eqs. (4-94) and (4-95).
WINGS IN COMPRESSIBLE FLOW 293
By introducing Eq. (4-98) into Eq. (4-100), the latter equation becomes an integral
equation for the determination of the vortex density k(x, y) in which the wing
shape must be given. Solution of this integral equation is quite difficult, as in the
incompressible case; see [15, 18, 30].
From the velocity component u, the pressure difference between the lower and
upper sides of the wing is obtained in the form of the pressure coefficient from Eq.
(344).
A relatively simple solution for the method of singularities was outlined in the
early days of aerodynamics in a few examples by Prandtl [75] and Schlichting [80].
Source distribution It has been shown in Sec. 3-6-2 that the potential equation for
incompressible flow with Ma. = 0 can be solved through a source distribution on
the wing surface. The method of source distributions for supersonic flow has been
developed into a computational procedure by Evvard [18] ; see also Puckett [76].
The source element q(x', y') at the station x', y' contributes, from Eq. (3-174),
to the perturbation velocity potential the amount
d
2O
(x, y, z ; x',
y,)
q( x', y)'
dx'dy'
4n
r
(Ma = 0)
where, again,
r=
(x - x')2 + (y - y')2 + z2
The corresponding solution at supersonic incident flow becomes, with Eq.
(4-12) and the supersonic similarity rule, Eq. (4-10),
d2O(x,y,z;x',y')=-4
1
2q(x', y')
r
dx, dy
(4-101)
where r is given by Eq. (4-93). It can be proved that this expression is a solution of
the potential equation, Eq. (4-8). The square root has real values only within the
two Mach cones of the point x', y', z' = 0 (upstream and downstream cones, see
Fig. 4-57) with the apex semiangle u, where tang = 1 / Maw, -1. For physical
reasons, however, the source element produces a contribution to the potential of
only the points x, y, z that lie in the downstream cone of the source element.
Equation (4-101) contains an additional factor of 2, however, for reasons that were
explained for Eq. (4-93).
The total potential at the point x, y, z is obtained by integration over the
contributions of source elements in the x 'y' plane, considering only the downstream
cones of the source elements. The upstream cones are not considered. Hence
-,
1
rr
a(x'.v')dx'dy'
(4-102)
- ti [(Y - V')2 ; ='J
(A')
Here, A' is the influence range (integration range) of the point x, y, z. It is shown
for z = 0 in Fig. 4-58. For z # 0, the influence range is bounded by a hyperbola
(see Fig. 4-74).
294 AERODYNAMICS OF THE WING
The velocity components in the x and z directions at the wing location z = 0
are obtained from Eqs. (3-45) and (4-102) as
is (x, y)
1
a
2n ax
Jf
q (x', y') dx' d y'
y(x - x')2 - (111ta
(A)
w (x, y)
2 q (x, y)
-
1) (y
(4-103)
y')2
(4-104)
where the upper sign is valid for z > 0 and the lower for z < 0. The partial
differentiation with respect to x in Eq. (4-103) requires particular precautions
because the integrand goes to infinity on the boundaries of the integration ranges
formed by the Mach lines, and these boundaries depend on x and y. Those integrals are
best solved by the method of finite constituents of divergent integrals of Hadamard.*
The pressure coefficient of supersonic flow becomes the same as in incompressible and subsonic flow [Eq. (4-18)] :
cp (x, y)
2 u U,)
(4-105)
Equation (4-103) is suitable immediately in the given form for the computation
of the velocity distribution on a wing of finite thickness at supersonic flow.
(displacement problem) (see Sec. 4-5-5 for a specific discussion).
The method of source distribution will now be applied to the inclined wing at
supersonic flow (lift problem); the inclined wing with subsonic leading edge cannot
be treated by the discussed method of source distribution without complications,
because in this case flow around the leading edge is present. Instead of the source
distribution, the dipole distribution according to [30] and a vortex distribution of
the kind described above are therefore preferable. A method will be given later,
however, by which a wing with subsonic leading edge can be computed after all by
the source method. A simple application of the source distribution method is the
computation of the inclined wing with supersonic leading edge. Since the incident
flow component normal to the leading edge is larger than the speed of sound and,
consequently, there is no flow around the leading edge (Fig. 4-61c), the solution for
the lower and upper sides of a wedge profile with linearly growing thickness is at
the same time the solution for the inclined flat surface (see Fig. 4-64a and b). The
starting point for further consideration is the velocity potential of the source
distribution of Eq. (4-102). For an inclined wing, source distributions of different
signs have to be arranged in the wing plane on the upper and lower wing surfaces.
Thus, a pressure discontinuity is produced at the wing that results in lift. Further
discussion needs to be conducted for the upper half-space, z > 0, only. The upper
source distribution corresponds to the potential P(x, y, z). Then, the velocity
components of the perturbation flow are computed with Eq. (3-45). The source
strength from Eq. (4-104) is
q (x, y) = 2 w (x, y)
(4-106)
`Translator's note: See M. A. Heaslet and H. Lomax in W. R. Sears (ed.), "General Theory of
High Speed Aerodynamics," Princeton University Press, Princeton, N.J., 1954, for a discussion of
Hadamard's method.
WINGS IN COMPRESSIBLE FLOW 295
For the solution of the problem the following conditions must be satisfied: For the
supersonic leading edge, the flow in the range before the wing is undisturbed. For
the wing with subsonic leading edge, the flow is undisturbed before the Mach lines.
Thus, in these two ranges 0 = 0.
On the wing, the kinematic flow condition must be satisfied, namely,
U" a (x, y) -{- w (x, y) = 0
(4-107)
where a(x, y) is the angle-of-attack distribution. Thus, from Eq. (4-106), the source
distribution of the wing becomes
q (x, y) = -2 U. L-4 (x, y)
(4-108)
For the wing with subsonic leading edge, an upwash range with the local
streamline inclination X(x, y) lies between the Mach lines and the wing leading
edge. In analogy to Eq. (4-108), it follows that
q (x, y)
2 U,. (x, y)
(4-109)
In this upwash range, no pressure discontinuity can exist in the z direction,
however, requiring that u(x, y) = v(x, y) = 0. Introducing Eqs. (4-108) and (4-109)
into Eq. (4-102) yields
0(x' y, z)
= "',
[ff
a (x', y)
V (X
dx,dy,
- x')2 - (Ma', - 1) [(y - y')2 + z"]
(R W)
A(x', y') dx- dy'
+
(R u)
V(x - x')2 - (Ma;o - 1) [(y - 02 + Z2]
(4-110)
Here, Rw is the integration range on the wing and Ru that of the upwash zone.
These ranges may be explained now through three examples: In Fig. 4-69, a delta
wing with two supersonic leading edges is shown. In this case, the range R,, does
not exist, whereas-the range R w is identical to the hatched range A'. In Fig. 4-75, a
wing with a supersonic and a subsonic leading edge is sketched. As has been shown
Figure 4-75 Application of the singularities method of Eward to the computation of lift
distributions of wings at supersonic incident flow. (a) A supersonic and a subsonic leading edge,
from Evvard. (b) Two subsonic leading edges, from Etkin and Woodward.
296 AERODYNAMICS OF THE WING
by Evvard [18] , only the integral over the range RW is left for the potential at the
point P(x, y, 0), because the integrals over the ranges R,, and R'yy just cancel each
other. The wing with two subsonic leading edges is shown in Fig. 4-75b. In this
case, the above Evvard theorem, applied twice, leads to the conclusion that,
approximately, only the hatched ranges R'W contribute to the integral Eq. (4-110);
see Etkin and Woodward [17], Hancock [18], and Zierep [18]. Application of the
Evvard procedure is always feasible for wings with supersonic trailing edges. The
flows with subsonic trailing edges, however, require consideration of the vortex
sheet behind the wing. A contribution to the solution of this problem was made by
Friedel [25].
4-5-4 Inclined Wing in Supersonic Flow
Before reporting on a general computational procedure for the determination of the
lift distribution on wings of finite span in supersonic incident flow, first two
particularly simple wing shapes will be treated, namely, the rectangular wing and the
triangular wing (delta wing). Fundamentally, these two wings can be computed by
the relatively simple method of cone-symmetric flow of Sec. 4-5-2. For arbitrary
wing shapes, however, the method of singularities discussed in Sec. 4-5-3 must be
used.
Rectangular wing The simple rectangular wing is obtained by setting 7 = rr/2 in Fig.
4-70. Thus, from Eq. (4-81), m = °°. During transition from the swept-back leading
edge of Fig. 4-70a to the unswept leading edge of Fig. 4-70b, the Mach line
originating at point A disappears because point A is no longer a center of
disturbance. Hence, range II of constant pressure distribution now embraces the
entire surface outside of range IV. The solution for the edge zone of the rectangular
wing (range IV) is obtained from Table 4-5 for m ->. = as
--P
cppl
=
i
fl
arccos. (1
2 t)
(4-111)
with t from Eq. (4-92). This pressure distribution is shown in Fig. 4-76. It was first
investigated by Schlichting [80]. From Fig. 4-76 it can be seen that the lift of the
edge zone is only half as high as that of an area of the same size in plane flow. This
solution allows a simple determination of the total lift of a rectangular wing. The
lift slope becomes
dcL
da
=
4
M2.--1
1
` 2AVMa -1
i
(4-112)
This formula is applicable as long as the two edge zones do not overlap, that is, for
A Afa', - 1 > 2 (Fig. 4-77a). They overlap for 1 < A Ma. - 1 < 2 (Fig. 4-77b).
The Mach lines from the upstream corners intersect the wing trailing edge. For
A v1_1 -Maw, < 1, they intersect the side edges and are reflected from them as
shown in Fig. 4-77c. The pressure distribution in the ranges affected by two Mach
cones (simple overlapping) may be gained by superposition (see Sec. 4-5-2).
WINGS IN COMPRESSIBLE FLOW 297
a
1-1
0.5
Figure 4-76 Inclined rectangular plate at supersonic incident flow. (a) Planform. (b) Pressure
distribution at the wing edge, from Eq. (4-111).
The lift slope of the rectangular wing is seen in Fig. 4-78a, where Eq. (4-112) is
valid even up to !1 Ma, - 1 = 1. A detailed explanation thereof will be omitted
here. In Fig. 4-78b and c, the neutral-point positions and the drag coefficients are
also shown. Finally, the pressure distribution over the wing chord and the lift
distribution over the span are given in Fig. 4-79 for a rectangular wing of aspect
ratio A1= 2.5; in Fig. 4-79a the Mach number Mac. = 1.89, and in Fig. 4-79b
a
b
c
Figure 4-77 Inclined rectangular plate of finite span
at supersonic incident flow for several Mach numbers. (a) -I -,a> 2. (h) 1 < J
-1 < 2.
(c) .I\AZ -1 < 1.
298 AERODYNAMICS OF THE WING
01/
a.
0
0.5
ZO
20
15
AMaZ-1-
0.5
24
3.0
2.5
30
-
0.4'
N
N
0.5
1.0
1,5
Z0
2.5
f
O
2.0
k
15
0.5
C
0
05
7.0
1.5
A Ma.j-1
2.0
2.5
3,0
Figure 4-78 Aerodynamic forces on inclined
rectangular wings of various aspect ratios at
supersonic incident flow. (a) Lift slope. (b)
Neutral-point position. (c) Drag coefficient.
Ma = 1.13. It can be shown easily that a wing with A Maw - 1 = 1, as at
Ma. = 1, has an elliptic circulation distribution. The influence of the profile
thickness of an inclined rectangular wing has been investigated, in the sense of a
second-order theory, by Bonney [8] ; compare also Leslie [50).
Delta wing As a further example, the delta wing will be discussed. This includes
wings with subsonic and supersonic leading edges, depending on the Mach number
(Figs. 4-67 and 4-69).
Wings with subsonic trailing edges are entirely described by range I, as can be
WINGS IN COMPRESSIBLE FLOW 299
concluded from Fig. 4-66a. The corresponding pressure distribution has already been
given in Table 4-5 and in Fig. 4-67. In terms of the mean value of the pressure over
the span from Eq. (4-87), the total lift is obtained by integration over the wing area as
L=2
U.2,4 J cp
p,
where J cp pi = cp pt 1 - cp pi u is the mean pressure difference between the lower and
upper surfaces of the unswept plate. With Acppi = 4a/ Ma. - 1, the lift slope of the
delta wing with subsonic leading edge becomes
dcL
d
»z
2z
ro-
E' (art) 1/1t1ci'
2:-r
tiny
(4-113a)
1
(0 < 972. < 1)
(4-113b)
forMa,>1 and0<m<1.
ucc
Figure 4-79 Pressure distribution over the chord and lift distribution over the span for the
inclined rectangular plate of aspect ratio .1= 2.5 at supersonic incident flow. (a)
t -1=4:Maa,=1.89.(b)
Yla;e-1=3:Ma«,=1.13.
300 AERODYNAMICS OF THE WING
One result of Eq. (4-113b) should be emphasized: For very slender wings (y
very small), m approaches zero for any Mach number, and because E'(0) = 1,
dcL
= 27r tan y
(4-114a)
dot
(y-+0, 11-0)
=
(4-114b)
Al
2
with tan y =A /4. Thus, the lift slope of very slender wings is independent of the
Mach number when Ma. > 1. The same result was found in Eq. (4-75a) for
Ma,o < 1. This is the so-called slender-body theory of Jones [371. For Ma = 1,
again m = 0, and in this case Eq. (4-113) is also valid, in agreement with Eq.
(4-75c). Thus it has been shown that the lift slope at Ma = 1 has the value
dcL/da = 7rA/2 for arbitrary aspect ratios A, whether Ma. = 1 is reached from the
subsonic or from the supersonic range (see Fig. 4-51).
The neutral point lies in the surface center of gravity because the pressure
differences, averaged in the lateral direction, are constant in the longitudinal
direction. Thus, the neutral point lies at
xN
Cr
=2
(4-115)
3
The drag of a wing with subsonic leading edge is composed of the partial force
La, which depends on the pressure distribution on the wing, and the suction force
S, which is produced by the flow around the leading edge (see Sec. 3-4-3). Thus,
the drag is given by
D=La-S
(4-116)
The contribution La is known from the above discussion. The suction force S
can be determined from the vortex density k(x, y) in the vicinity of the leading
edge. This relationship for plane incompressible flow is given in Eq. (2-76).
Determination of the suction force for compressible flow with subsonic leading edge
is treated, for example, in [37] and [77]. For a delta wing with subsonic leading
edge (m < 1), the drag coefficient without suction force CD = CLa becomes
.
D =
CD
= 2 E' (m)
27vaa
nz
E'' (m) )/Ma2c
-1
2
CL
re el
(4-117)
Here it has been taken into account that, from Eq. (4-81),
m = tan y Mad - 1 = 4 Ma=w - 1
(4-118)
The coefficient of the suction force is determined from [77] as
2
GS =
V1
- nag
(4-119)
According to Eq. (4-116), this quantity is to be subtracted from the drag coefficient
from Eq. (4-117) to obtain the total drag (reduced drag + wave drag + suction
force). Hence
WINGS IN COMPRESSIBLE FLOW 301
2
CD =
{2 E'(7n)
-
-
(4-120)
The wing with supersonic leading edge is composed of ranges II and III of Fig.
4-66b only. The corresponding pressure distributions have been given previously in
Table 4-5 and in Fig. 4-69b.
By taking the mean value of the pressure over the width from Eq. (4-91), the
lift slope of a delta wing with supersonic leading edge becomes
dcL
-
da
4
Ma - 1
(m > 1)
(4-121)
00
Hence, the lift slope of a delta wing with supersonic leading edge is equal to that of
the plane problem (Table 4-2).
Likewise, the neutral point of a delta wing with supersonic leading edge lies in
the area center of gravity because the pressure difference,. averaged laterally, is
constant in the longitudinal direction. Thus the neutral-point position is the same as
that of a delta wing with subsonic leading edge, namely, xN/c,. = 3 [Eq. (4-115)] .
The total drag (induced + wave drag) is
D=La
Since there is no flow around the leading edge, no suction force is created. The drag
coefficient becomes, therefore,
4a2
CD
-1
2
= 4 Y Mao -
(4-122a)
VMa
(4-122b)
2
CL
=7m nA
(4-122c)
in agreement with the flat plate of infinite span (see Table 4-2). Equation (4-122c)
is obtained with Eq. (4-118).
The ratio of the lift slopes of a delta wing from Eqs. (4-113) and (4-121) and
that of an inclined flat plate of Table 4-2, with
dcL
4
da
Ma;, -1
is plotted in Fig. 4-80 against m [Eq. (4-118)]. The lift slope of a delta wing with a
subsonic leading edge (0 <m < 1) is considerably smaller than that of a delta wing
with a supersonic leading edge (m > 1). The theoretical results for the lift slope of
delta wings in the entire Mach number range are compiled in Fig. 4-82a. The values
for Ma., < 1 have been established from the linear theory of subsonic incident flow
(Sec. 44-2), those for supersonic incident flow from the above formulas, which are
also linear. The curve for it = - corresponds to the plane problem in Fig. 4-20a.
The neutral-point positions of a delta wing for the entire Mach number range
302 AERODYNAMICS OF THE WING
0.2
Figure 4-80 Lift slope of a delta wing at
0S
2
an y
supersonic incident flow. Subsonic leading
edge: 0 <rn < 1. Supersonic leading edge:
tan,u
rn > 1.
1.5
1
are presented in Fig. 4-82b for several aspect ratios A. The curve for t1=
corresponds to the plane problem in Fig. 4-20b.
The drag coefficient of delta wings is given in Fig. 4-81, where curve la
represents the case Ma < 1 without suction force, Eq. (4-117), and curve lb the
case with suction force. Curve 2 is the case m > 1 from Eq. (4-122c).
In incompressible flow it is customary to designate the contribution CD =
cL/ rA, caused by the velocity field induced behind the wing, as induced drag. Such
a contribution is made to the drag in compressible flow, too, and it is logical to call
it induced drag also (Fig. 4-81, curve 3). Subtracting this drag from the total drag at
supersonic velocities, the wave drag (Fig. 4-81) is obtained. For practical purposes,
separate determination of the induced drag has no particular significance. Only the
sum of induced drag and wave drag is required; see Schlichting [80].
The drag coefficient of delta wings without twist at various aspect ratios A is
shown in Fig. 4-82c versus the Mach number. The curve for A = 00 corresponds to
6
5
y
I'
Suction
force
cs
Wav e drag
lay
JR, 3
1b
2
I
2
__
3
1
Figure 4-81 Drag of delta wings at supersonic
incident flow vs. m from Eq. (4-118). in < 1:
I
I
Induced drag
i
0,5
tan
to
1
15
2
subsonic leading edge. in > 1: supersonic leading
edge. Curve la, from Eq. (4-117). Curve lb, from
Eq. (4-120). Curve 2, from Eq. (4-122). Curve 3,
induced drag from Eq. (3-134).
Z,1t
e
-oo
/
r
A-CO
5
Supersonic
-
leading edge
-3
3
=2
2
aI
=1
1
A=0
a
02
0.6
0,4'
oe
to
Ma
12
46
1 4f
18
20
0.25
0<A<oo
020
=2
N25
cr
3
3cr
Ao4
0,10
jU
0D5
b
A
b
02
Oaf
0.6
08
10
12
1.6
1.6
2.0
Ma,,
0.5
A-1
0.4
-1
A-2
=3
2
I
I
A =3
l
0.1
c
L_
02
1
06
aOo
!
Ole
f
10
t
I
1
12
1#
i
16
18
20
Mao
Figure 4-82 Aerodynamic forces of inclined delta wings of various aspect ratios at subsonic and
supersonic flows. (a) Lift slope. (b) Neutral-point position. (c) Drag coefficient (with suction
force).
303
304 AERODYNAMICS OF THE WING
the plane problem in Fig. 4-20c. Note that the aspect ratio has a strong effect on
the lift-related drag at subsonic incident flow. Conversely, this effect is negligible for
supersonic incident flow.
For airplane design, wing forms with large aspect ratio do not offer an advantage
at supersonic flight velocities (see Fig. 3-4b).
A survey of the pressure distributions over the wing chord and the lift
distributions over the span is found in Fig. 4-83 for delta wings with subsonic and
supersonic leading edges. The lift distributions (ctc) are illustrated in Fig. 4-84 for
several values of m. It is noteworthy that the lift distributions over the span are
elliptic for all wings with subsonic leading edge, 0 < m < 1. For wings with
supersonic leading edge, m > 1, the lift distribution approaches a triangular form at
very high Mach numbers (m - co).
Systematic measurements to check the three-dimensional wing theory at
supersonic incident flow have been published by Love [56] for delta wings with
rounded and sharp-edged noses. In these measurements the aspect ratio A lies
Figure 4-83 Pressure distribution over
the wine chord and lift distribution
over the wing span of delta wings at
supersonic incident flow. (a) Subsonic
leading edge, 0 < m < 1. (b) Supersonic leading edge, in > 1.
WINGS IN COMPRESSIBLE FLOW 305
Z0
1.8
Y\
\
1.6
7n-l
\
7Th-1.5
\
14
0<7T
1
1.2
U
0.8
0.6
O.u
\
0.2
Figure 4-84 Lift distributions over the span of
delta wings at supersonic flow for several values
0
0.2
0,6
0.4
Y
7I ° s
0.8
7.0
of m from Eq. (4-118). 0 < m < 1: subsonic
leading edge. m > 1: supersonic leading edge.
between 0.7 and 4, the profile thickness is S = t/c = 0.08, and the relative thickness
position Xt = xt/c = 0.18; the Mach numbers are Ma. = 1.62, 1.92, and 2.40.
The results for the lift slope are given in Fig. 4.85. As the abscissa, the
parameter in was chosen. The ordinate for the range of subsonic leading edges
(rn < 1) is the quantity cot 'y (dcL/da) = (4/i1) do /da ; for the range of supersonic
leading edges (m > 1), the quantity (dcL/da) Ma;, -1 is the ordinate. Test results
for the 22 wings at the 3 different Mach numbers lie quite close to one curve,
confirming the validity of the supersonic similarity rule of Sec. 4-2-3. The measured
curve follows the theoretical curve fairly well. The deviations between theory and
measurements at m = 0 and m = I are understandable, because m -- 0 means
transonic flow (Ma
1), and in = 1 signifies transition from a subsonic to a
supersonic leading edge.
The analogous plotting of the drag coefficients is given in Fig. 4-86. Only the
values for rounded noses are included. Here also, the measured drag coefficients lie
near one single curve, again confirming the supersonic similarity rule. In the range of
subsonic leading edges the curve of the measured drag coefficients lies, at the lower
values of in, between the theoretical curves with and without suction force.
Finally, in Fig. 4-87, the measured neutral-point positions are plotted. Here,
too, the supersonic similarity rule finds a satisfactory confirmation. The neutral
points of wings with rounded noses lie somewhat more upstream than those with
8
ft.
4
dcL
a
dCL
da
M0 1
(m>1)
A
da
(m<1)
j
`
Zit
6
Xt 0.18
4
a
'fr
°
d
Maw I I
04
1.62
2
°4
1.92
240
Mac, 1
0.5
1.0
2,0
1.5
2.5
7n= 4 Ma00-1
Figure 4-85 Measured lift slopes for delta wings at supersonic incident flow, from Love.
0 < m < 1: subsonic leading edge. m > 1: supersonic leading edge.
0,4
-CD- 1
A cL
'
-c-L'
)
i
(m,> 1)
(72<1)
o
0,3
00
°
o ice/
Without suction
force
-V
02
.gg
----- -
0--s-
Theory
- ---
/
j
o
°
\
0.1
q>L
c5'G08
M
With suction
Xt- 0'8
forc e
V
M17
00
i
0.5
I'D
15
2.0
2.5
Figure 4-86 Measured drag coefficients due to lift of delta wings in supersonic incident flow,
from Love. 0 < in < 1: subsonic leading edge. in > 1: supersonic leading edge.
306
WINGS IN COMPRESSIBLE FLOW 307
0."
Maw
Jr
0
0
4
1. 5Z
1.92
I
2.40
I
t
0.3
2°
cl
Theory
6 0 -0-0
00 0
0
ca e
8-0,08
0,1
Xt- 0..18
PMa-1
0
I
I
0.5
1.0
I
I
1,5
2.0
2.5
Figure 4-87 Measured neutral-point positions for delta wings at supersonic incident flow, from
Love. 0 < m < 1: subsonic leading edge. m > 1: supersonic leading edge.
sharp-edged noses. The measured neutral-point position moves slightly upstream and
increases with Mach number, although, from the linear theory, it should be
independent of Mach number.
Swept-back wing Lift slopes of swept-back wings with constant wing chord (taper
X = 1) are given in Fig. 4-88 with A cot y as the parameter (zi = aspect ratio,
y = sweepback angle measured from the wing longitudinal axis). The lift slope is
referred to that of the plane problem do /da). = 4/ Maw, - 1 and depends on the
parameter rn = tan 7/tan p = tan y Ma. - 1 and on the purely geometric quantity
. Ai cot y, and may be written as
cot
(
The fact that the lift slopes depend only on these three parameters can be
realized by setting tan z = cot y in the supersonic similarity rule Eq. (4-26) and
observing that A Ma. - 1 /A tan cp = tan y/tan p = m [see Eq. (4-81)] . Under :low
conditions rendering the leading edge of the present wing shapes subsonic, the lift
slopes-in a way similar to that shown for delta wings (Fig. 4-80)-deviate
considerably from those of the plane problem. Conversely, when the leading edge of
the present wing shapes is supersonic, the lift slopes are almost equal to those of
308 AERODYNAMICS OF THE WING
1.5
A coty-6
5
3
I
f
0,5
7b
1.5
7.0
ms
tarry
tan
y
2.0
2.5
=tang Ma- 1
Figure 4-88 Lift slope of swept-back wings (taper X = 1) at supersonic incident flow, from
155]. 0,< m < 1: subsonic leading edge. m > 1: supersonic leading edge.
the plane problem. For a better illustration, the wing planforms are sketched in Fig.
4-88 for A = 3. However, the diagram applies to other values of A, too. The figure
does not include rectangular wings, because the chosen presentation is not
applicable to the case of y = ir/2. The lift slopes of the rectangular wing were given
earlier in Fig. 4-78a.
Arbitrary wing planforms So far, results have been presented for the linear wing
:ieory at supersonic incident flow for the unswept rectangular wing, the delta
(triangular) wing, and the swept-back wing. In this section, a few results will be
given for a trapezoidal wing, a swept-back wing, and a delta wing; see Fiecke [21 ] .
The theoretical lift slopes of these three wings are given in Fig. 4-89 for the Mach
number range from Ma. = 0 to, May, = 2.5. For the same Mach number range, the
drag coefficients of these three wings are presented in Fig. 4-90. Two curves each
apply to the subsonic range and to the supersonic range with subsonic leading edge.
The dashed curve applies to the values with suction force, the solid curve to those
without. The former are described by the well-known formula for the induced drag
CD = C2L/1rA. The drag without suction force is found from CD = CLa = cL(da/dcL),
where the values of dcL/da are taken from Fig. 4-89. It can be expected that the
suction force is fully effective on a well-rounded profile nose and that the dashed
lines represent the drag coefficients. Conversely, the suction force is negligible on
thin profiles with sharp noses, as used in most cases on supersonic airplanes, and
thus the upper curve applies. In Fig. 4-91, the neutral-point positions of these three
wings are shown schematically against the Mach number. The typical behavior
during transition from subsonic to supersonic velocities is seen, namely, that the
neutral point moves considerably downstream when a Mach number of unity is
WINGS IN COMPRESSIBLE FLOW 309
=3
5
I
1
1
T
1 A oo
A
2
l
I
0.5
Z5
1.0
2.0
2.5
Figure 4-89 Lift slope vs. Mach number for a
trapezoidal, a swept-back, and a delta wing of
aspect ratio .s = 3, from [21 ].
0. 50
0.25
;rA
0
7n
0.5
1.0
0.75
1.5
2.0
Maw
2.5
`711, -1
fAL
0
0.5
-25
7.0
2.0
2.5
Maoo
0.7
Figure 4-90 Drag coefficient due to lift vs.
025
Mach number for a trapezoidal, a sweptback, and a delta wing of aspect ratio
.1 = 3, from [21 ]. Dashed curve: with suc-
0
0.5
7.0
Ma00
1.5
2.0
2.5
tion force. Solid curve: v'ithout suction
force.
310 AERODYNAMICS OF THE WING
Ma
Figure 4-91 Neutral-point position
vs. Mach number for a trapezoidal,
a swept-back, and a delta wing,
from (21]. (0) Neutral-point position
for Ma < 1. (.)
Neutral-
point position for Ma. > 1.
exceeded. This means an increase in longitudinal stability of the airplane during
transition from subsonic to supersonic flight.
Finally, a brief account will be given of the experimental confirmation of linear
wing theory. In Fig. 4-92, the lift slopes dcL/da are plotted over the Mach number
for four different wings (rectangular, trapezoidal, triangular, and swept-back). For
the subsonic range, the theoretical curves were determined according to Sec. 4-4-2,
for the supersonic range, from Friedel [251. The measured lift slopes are in good
agreement with theory, except for the immediate vicinity of Ma. = 1. Additional
details of a three-component measurement in the subsonic and supersonic ranges of
the trapezoidal wing of Fig. 4-92b are illustrated in Fig. 4-93. The curves CL(a) of
Fig. 4-93a show clearly that the linear range and the coefficient of maximum lift CL
are considerably larger in supersonic than in subsonic flow. Also, the pitchingmoment curves CL(cm) in Fig. 4-93c confirm that the linear range is markedly larger
for Ma. > 1 than for Mar < 1. In this connection, the publications [59, 76, 90]
are noted; they are concerned with the computation of twisted wings and flight
mechanical coefficients of wings at supersonic velocities.
4-5-5 Wing of Finite Thickness
in Supersonic Flow
General statements In the previous sections, the inclined wing of finite span in
supersonic flow was treated (lift problem). Now, the special case of a wing of finite
WINGS IN COMPRESSIBLE FLOW 311
thickness with zero lift (displacement problem) will be discussed in more detail. Of
interest here are the pressure distribution over the wing contour and the resulting
wave drag. The latter is a strong function of the profile thickness, as was discussed
for the plane problem in Sec. 4-3-3. The most general method of determining the
pressure distribution of wings of finite thickness at zero lift is the source-sink
method of von Karman [100]. The fundamentals of this method for the wing with
supersonic incident flow were furnished in Sec. 4-5-3. The basic concept of this
method is to cover the planform area of the given wing with a source distribution
q(x, y) in the xy plane. From this, the x component of the velocity on the wing
surface u(x, y) is obtained from Eq. (4-103) and the z component w(x, y) from Eq.
(4-104). By describing the wing contour by z(t)(x, y) = z(x, y), the kinematic flow
condition is expressed by Eq. (3-173b). Introducing this into Eq. (4-104) yields the
source distribution Eq. (3-176). Introducing this result into Eq. (4-103) furnishes
the pressure, coefficient cp = -2u/U. as
az (x ' , y ' )
ax,
e:r,
(A')
'7
X
,1
"I /
(4-124)
y')2
V(x - x')"- --
Here, A' is the influence range of the point x, y, as indicated in Fig. 4-58 by
cross-hatching. The pressure distribution for a given wing contour z(x, y) can thus
be determined.
Subsonic leading edge
Supersonic leading edge
S
SBT
t
SBT
Theory
l1=2,75
0
JO
0
a
Th eory
Am .Pw
o
C
!Supersonic leading edge Subsonic leading edge Supersonic leading edge
Subsonic leading edge
5
SBT
O
SBT
Theory
OO
/1=2.75
=2%S
0
Theory
3 0
1
/V' M cc,)
Figure 4-92 Experimental confirmation of linear wing theory at subsonic and supersonic
incident flow. Lift slope vs. Mach number: measurements from Becker and Wedemeyer [51,
Stahl and Mackrodt [90] . Theory for supersonic flow from Friedel [25]. SBT = Slender-body
theory, Sec. 4-4-3.
I
_3ti
.p
--
p
u
I
I
0
0
70
a
{
N
l
NNKN,X
Al
(
h.
p
°0
0
0 0
U
f3
312
WINGS IN COMPRESSIBLE FLOW 313
Wave drag The coefficient of wave drag of the wing at zero lift is obtained through
integration of the pressure distribution over the wing area A as
f
2
CDO = A
c, (x, y) a2 dx dy
(4-125)
(A)
This formula is applicable to sharp-edged profiles only. The dependence of the drag
coefficient on profile thickness ratio, taper, aspect ratio, sweepback angle, and Mach
number of the incident flow is given according to the supersonic similarity rule by
Eq. (4-27). This relationship is of great value for a systematic presentation of
theoretical and experimental results.
Rectangular wing For the wing of rectangular planform and spanwise constant
profile z(x, y) = z(x), introducing Eq. (4-124) into Eq. (4-125) and integrating twice
yield (see Dorfner [15] )
1
4
CDO=
YMa'
00
r0,Y)
Z - dX
(A'> 1)
- 1 0J
(4-126a)
1
4
VMa.
-1
1
+
2
Wx
3
cl X
0
dZ
nA.' J dX
T-1'
dZ
dX'
(X - X')2 - A
X - X'
dX' dX
(^1' < 1) (4-126b)
A'
Note that, for A' = A Ma. - 1 > 1, the drag. formula for the rectangular wing of
finite span is identical to that of the rectangular wing of infinite span (see Table
4-2).
For a convex parabolic profile Z = z/c = 26X(1 -X) with X = x/c, the
integration yields
CDO
1
CDo-
=
2n
(4-127a)
(A'_>_1)
L4 arcsin A' - A' ()l1
L
- A'2 - (6 - A'2) cosh-1 A)
-L ]
(A' < 1)
(4-127b)
where CDO- is given by Eq. (4-50a). The numerical evaluation is given in Fig. 4-94.
Delta wing A few results will be added for delta (triangular) wings. Delta wings
with double-wedge profiles have been computed by Puckett [761, those with
biconvex parabolic profiles by Beane [76]. Coefficients of the wave drag at zero lift
for double-wedge and biconvex parabolic profiles of 50% relative thickness position
are shown in Fig. 4-95 as a function of the parameter m = Maw -1-4/4. For the
double-wedge profile, CDO
is expressed by Eq. (4-51). For supersonic leading edges
314 AERODYNAMICS OF THE WING
0.6
8
X00.4
}
Figure 4-94 Drag coefficient (wave drag)
at zero lift for rectangular wings at super-
02
t0
05
2.0
45
2.5
30
A Ma, 1
sonic incident flow vs. Mach number.
Biconvex parabolic profile cDo . from
Eq. (4-50a).
(m > 1), cDo /eD o 00 is almost independent of Mach number, whereas it changes
strongly with Mach number for subsonic leading edges (m < 1). Both curves have
pronounced breaks at m = 1, that is, when the Mach line coincides with the leading
edge. The curve for the double-wedge profile has another break at m = 2 , that is,
when the Mach line is parallel to the line of greatest thickness.
In Fig. 4-96, a number of measurements on delta wings with double-wedge
profiles and 19% relative thickness position are plotted from [56]. Similar to Fig.
4-86, different representations have been chosen for m < 1 and m > 1. At the kind
of presentation chosen here, these measurements on 11 wings at Mach numbers
Ma. = 1.62, 1.92, and 2.40 fall very well on a single curve- Hence, the supersonic
similarity rule of Eq. (4-27) has been confirmed again. The theoretical curve from
I
jr
0.
0.2
00
02
0.4.
0.6
017
1.0
Lan j-,
ia n
1.2
1.4
1.6
1.6
2.0
?.2
4 'M -1
Figure 4-95 Drag coefficient (wave drag) at zero lift for delta wing (triangular wing) vs. Mach
number. Profile I: double-wedge profile cDoo,, from Eq. (4-51). Profile II: parabolic profile,
cDo,,, from Eq. (4-50a). 0 < in < 1: subsonic leading edge. in > 1: supersonic leading edge.
WINGS IN COMPRESSIBLE FLOW 315
,\
10,
4 CD o
I
(m mil)
Mm
l1
Jo
I
1
7
Theory
fI
a
a
/
5.
i
Theory
2.5
M¢
1.62
192
2.40
1l
<
4
d-0.08
Xt°0,18
4
m-A
Figure 4-96 Measured drag coefficients (wave drag) at zero lift for delta wings at supersonic
incident flow, from Love, Theory from Puckett. Double-wedge profile of 18% relative thickness
position. 0 < m < 1: subsonic leading edge. m > 1: supersonic leading edge.
Puckett [76] for the relative thickness position Xt = 0.18 shows a high peak at
m = 1 that is not confirmed by measurements, as would be expected because the
incident flow velocity at the leading edge is just sonic. Comparison between theory
and experiment suffers from the uncertainty in the determination of the friction
drag, which has to be subtracted from the measured values.
The treatment of the thickness problem of a delta wing with sonic leading edge
has been compared with transonic flow theory by Sun [93].
Swept-back wing The wave drag coefficients of swept-back wings of constant chord
are illustrated in Fig. 4-97. The corresponding information for the lift slope was
given in Fig. 4-88. The wing has a double-wedge profile, of which the drag
coefficient in plane flow CDO is obtained from Eq. (4-51). The curves show a
pronounced break at m = 1, that is, when the Mach line and leading edge fall
together. It should be noted that, according to [15],
CDO
CDO,o
in
f -1
for rn > I --
!i cot f
(4-128)
is obtained in the range of the supersonic leading edge if the Mach line originating
at the apex (line g) intersects the trailing edge.
316 AERODYNAMICS OF THE WING
2,
1.
I
F
1
A cotJ-1
1
05
0
0. .
1.0
ton
12 s ton;
-
1.5
2.0
2.5
Figure 4-97 Drag coefficients (wave drag) at zero lift of swept-back wings (taper X = 1) at
supersonic incident flow, from [49]. 0 < m < 1: subsonic leading edge. m > 1: supersonic
leading edge. Dashed curve (g) from Eq. (4-128).
Arbitrary wing planforms To conclude this discussion, the total drag coefficient at
zero lift (wave drag + friction drag) of the three wings (trapezoidal, swept-back, and
delta) treated earlier (Figs. 4-89-4-91) is plotted in Fig. 4-98 against the Mach
number. These three wings have double-wedge profiles with a thickness ratio
t/c = 0.05 and an aspect ratio A = 3. Within the Mach number range presented, the
0,03
A=3
4
Wave
drag
Friction drag
(Re
107)
4-98 Total drag coefficient
(wave drag + friction drag) vs. Mach
number for a trapezoidal, a swept-back,
and a delta wing of aspect ratio e = 3.
Double wedge profile tic = 0.05, Yt/c =
Figure
0.50, from [21 ] .
WINGS IN COMPRESSIBLE FLOW 317
wave drag is two to three times larger than the friction drag. The latter has been
determined from Fig. 4-4 for Reynolds numbers Re 107. Since the wave drag at
supersonic incident flow is proportional to (t/c)2, this contribution, and thus the
total wing drag at zero lift, can be reduced considerably by keeping t/c small. This
fact is taken into account in airplane design by choosing extremely small thickness
ratios for supersonic airplanes; compare Fig. 3-4a.
Concluding remarks In addition to the references included in the text, attention
should be directed toward summary reports and reports dealing with various
theories on the aerodynamics of the wing in supersonic flow [6, 11, 19, 22, 23, 40,
51, 92, 105-107]. The special case of the aerodynamics of the wing of small aspect
ratios, first studied by Jones [37], has been investigated comprehensively as the
"slender-body theory" for both lift and drag problems [2, 13, 14, 41, 108]. The
aerodynamics of slender bodies is treated in Sec. 6-4. The influence of vortex
shedding at the lateral wing edges of rectangular wings, and the leading-edge
separation on swept-back and delta wings at supersonic flow, are treated in [12, 72,
91 ], based on the understanding of incompressible flow. Based on a suggestion of
Jones, questions concerning the minimum wing drag have been investigated by
several authors [36, 61, 97, 1101. In this connection, the investigations on the
design aerodynamics of wings at high flight velocities, promoted mainly by
Kuchemann, play an important role [9, 38, 46, 60].
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WINGS IN COMPRESSIBLE FLOW 321
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75. Prandtl, L.: Theorie des Flu- zeugtragfligels im zusammendriickbaren Medium, Lufo.,
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322 AERODYNAMICS OF THE WING
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11-32, 1966.
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Nonlifting Airfoils at High Subsonic Speeds, NACA Rept. 1217, 1955; 1359, 1956. Rotta,
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WINGS IN COMPRESSIBLE FLOW 323
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1:93-108,1965.
PART
TWO
AERODYNAMICS OF THE FUSELAGE
AND THE WING-FUSELAGE SYSTEM
CHAPTER
FIVE
AERODYNAMICS OF THE FUSELAGE
5-1 INTRODUCTION
5-1-1 Geometry of the Fuselage
Whereas the main function of the airplane wing is the formation of lift, it is the
main function of the fuselage to provide space for the net load (payload). It is
required, therefore, that the wing at given lift and the fuselage at given volume have
the least possible drag. Consequently, the fuselage has, in general, the geometric
shape of a long, spindle-shaped body, of which one dimension (length) is very large
in comparison with the other two (height and width). The latter two dimensions are
of the same order of magnitude. In Fig. 5-1, a number of idealized fuselage shapes
are compared. In general, the plane of symmetry of the fuselage coincides with that
of the airplane. The cross sections of the fuselage in the plane of symmetry and
normal to the plane of symmetry (planform) have slender, profilelike shapes. The
most important geometric parameters of the fuselage that are of significance for
aerodynamic performance will now be discussed.
In analogy to the description of wing geometry, a fuselage-fixed rectangular
coordinate system as in Fig. 5-1 will be used, with
x axis: fuselage longitudinal axis, positive in rearward direction
v axis: fuselage lateral axis, positive toward the right when looking in flight
direction
z axis: fuselage vertical axis, positive in upward direction
327
328 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
Figure 5-1 Geometric nomenclature
for fuselages. (a) General fuselage
shape.
Skeleton; angle of attack
Z}
(b), (c)
Fuselage teardrop
with noncircular cross sections. (d)
zF(x)
Fuselage teardrop with circular cross
sections (axisymmetric fuselage). (e)
e
Fuselage
line.
mean
camber
(skeleton)
In general, it is expedient to place the origin of the coordinates on the fuselage
nose. For axisymmetric fuselages, utilization of cylinder coordinates as in Fig. 5-ld
is frequently preferable, where r stands for the radius and $ for the polar angle.
The main dimensions of the fuselage are the fuselage length 1F, the maximum
fuselage width bFinax, and the maximum fuselage height hFmax (Fig. 5-1). The
fuselage cross sections in the yz plane are usually oval-shaped (Fig. 5-1 b and c). The
simplest case is the fuselage with circular cross sections as in Fig. 5-1d, with
bF max - hF max - dF max , where dF max is the maximum fuselage diameter. From
these four main dimensions, the following relative quantities can be formed:
dFinax
-S
fuse age
1
Finax _ S*
+1,;
is kn ess is +'Io
fi1Se age W I'dt h rat 10
1
IF
hFmax
= bF*
fuselage height ratio
_
f use1abe
Q cross-sect'ion Idt'io
1F
hFinax
bFmax
F
AERODYNAMICS OF THE FUSELAGE 329
The first three quantities are measures of the slenderness or fineness ratio of the
fuselage. For the fuselage of circular cross section, 5F = SF = SF* and XF = 1.
A more detailed description of fuselage geometry can be given by introducing
the fuselage mean camber line. As shown in Fig. 5-la, this line is defined as the
connection of the centers of gravity of the cross-sectional areas AF(x). The line
connecting the front and rear endpoints of the skeleton line is designated as the
fuselage axis; it should coincide with the x axis. The fuselage skeleton line zF(x) as
shown in Fig. 5-le lies in the fuselage symmetry plane. The largest distance of the
skeleton line from the fuselage axis is designated as fF.
In analogy to the wing shape, Sec. 2-1, a general fuselage shape as shown in
Fig. 5-la can be thought of as being composed of a skeleton line ZF(x) on which
cross sections AF(x) are distributed. The body with this cross-section distribution is
also termed a fuselage teardrop. In the case of noncircular cross sections of the
fuselage, fuselage teardrops are characterized by the distributions hF(x) and bF(x)
as in Fig. 5-lb and c. In the case of circular fuselage cross sections, the fuselage
teardrop is determined uniquely by the distribution of the radii R(x) (Fig. 5-1d).
The geometric parameters of a wing (teardrop and skeleton) can be selected first for
the required aerodynamic performance. For fuselages this procedure is possible only
to a very limited degree, because the fuselages must satisfy important requirements
that may not be compatible with the aerodynamic considerations. For theoretical
investigations on the aerodynamic properties of fuselages, the profile teardrops
discussed in Sec. 2-1 are well suited.
The ellipsoid of revolution of Fig. 5-2a is a simple fuselage configuration for
subsonic velocities. Another simple fuselage of axial symmetry that is used
particularly for supersonic flight velocities is the paraboloid of revolution with a
pointed nose as shown in Fig. 5-2b.* To accommodate jet engines, fuselage
configurations with blunt tails may be chosen. Among the design parameters not
only fuselage length and diameter play an important role, but also fuselage volume
and surface area. Volume and surface area of axisymmetric fuselages are given by
IF
VF =
JR2(x) dx
(5-1a)
0
*The axis of rotation is parallel to the tangent at the vertex.
IF
=i
'Fo
I
Figuae 5-2 Special axisymmetric fuselages. (a)
Ellipsoid of revolution. (b) Paraboloid of
revolution.
330 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
lF
SF = 27r f R(x) ds
(5-1 b)
0
where s is the path length along the fuselage contour and ll is the associated length
of a meridional section measured on the fuselage contour.
Finally, a few data are given here for the volume of the ellipsoid of rotation
and the paraboloid of rotation (1F = lFo) of Fig. 5-2, respectively:
VF =
a3lFAFinax
VF = is1FAFinax
(ellipsoid)
(5-2a)
(paraboloid)
(5-2b)
Here, 1F is the fuselage length and AFinax is the maximum fuselage cross-sectional
area, also called the frontal area.
5-1-2 Forces and Moments on the Fuselage
The following sections will be devoted to a detailed discussion of fuselage
aerodynamics. To give a feeling for the magnitudes of the forces and moments
acting on the fuselage, a typical measurement on a fuselage will be presented first.
In Fig. 5-3, some results of a three-component measurement on an axisymmetric
fuselage by Truckenbrodt and Gersten [50] are plotted. Here, the following
dimensionless coefficients have been introduced for the components of the resultant
force (lift and drag) and for the pitching moment:
Lift:
LF = CLF VF 3 q00
Drag:
DF = CDF VF
Pitching moment:
MF=cmFVFq
11 3
q.
(5-3)*
where q _ (9/2) U! is the dynamic pressure of the incident-flow velocity U. and
VF is the fuselage volume. Figure 5-3 shows the lift coefficient cLF, the drag
coefficient cDF, and the pitching-moment coefficient cMF plotted against the angle
of attack a. The position of the axis of reference for the pitching moment is
indicated in Fig. 5-3. In the range near a = 0, the lift coefficient changes linearly
with angle of attack a. At larger angles of attack, CLF grows more than linearly.
This lift characteristic CL(a) is very similar to that of a wing of very small aspect
ratio (see Fig. 3-49). The drag coefficient CDF is approximately proportional to the
square of the angle of attack, similar to that of the wing. In the range of large
angles of attack, the pitching-moment coefficient depends almost linearly on the
angle of attack.
Forces and moments, in addition to those discussed above, act on the fuselage
*Fusela?e volume is introduced in this case as a quantity of reference in compliance with
the theory of fuselages (see Sec. 5-2-3). The drag coefficient is frequently referred to the surface
SF or the frontal area AFinax of the fuselage.
AERODYNAMICS OF THE FUSELAGE 331
0.6
00
0.2
0
-0.2
-0.4
-0.6
-6°
00
6°
f2°
2/f0
19°
-*a
Moment reference point
.90°
Figure
5-3 Three-component
mea-
surements CLF, cDF, and cMF vs.
angle of attack on an axisymmetric
fuselage. Reynolds number Re = 3
106. Theory
(5-34).
for cMF from Eq.
as a result of the turning and sideslip motions of the airplane, as has been discussed
for the wing in Sec. 3-5.
The summary reports of Munk [41] , Wieselsberger [58], Goldstein [141,
Thwaites [47], Howarth [22], Heaslet and Lornax [17], Brown [5], Ashley and
Landahl (4], Hess and Smith [181, and Krasnov [28] deal with the questions of
flow over a fuselage in incompressible, and, to some extent also in compressible
flow. Also, the survey of Adams and Sears [1 ] must be mentioned. Furthermore,
the comprehensive compilations of experimental data on the aerodynamics of drag
and lift of fuselages of Hoerner [19] and Hoerner and Borst [20] should be
pointed out.
5-2 THE FUSELAGE IN INCOMPRESSIBLE FLOW
5-2-1 General Remarks
Now that some experimental results have been given. the theory of flow over
fuselages will be presented. Fuselage theory can be established, similar to profile
theory, by two different approaches.
The first approach consists of the establishment of exact solutions of the
three-dimensional potential equation, which can be done successfully in only a few
cases. The second approach is the so-called method of singularities, in which the flow
pattern about the fuselage is formed by arranging sources, sinks and, if necessary,
332 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
dipoles on the fuselage axis. This procedure is fairly simple for bodies of revolution
(see von Karman [54] and Keune and Burg [26] ). An extention of this method for
the computation of the flow over fuselages consists of arranging ring-shaped source
distributions on the body surface (see Lotz [34], Riegels [32], and Hess [18] ). By
this method, body shapes can be treated whose cross sections deviate somewhat
from circles.
First, the fuselage in axial flow will be discussed, then the fuselage in oblique
flow.
5-2-2 The Fuselage in Axial Flow
Pressure distribution by the method of source-sink distribution The method of
source-sink distribution for bodies of revolution in axial flow was first presented in
detail by Fuhrmann [13). The flow over such a body can be represented, as in Fig.
5-4, by a distribution q(x) of three-dimensional sources on the body axis that is
superimposed by a translational flow U.. Compare the discussions of the plane
problem (profile teardrop) of Sec. 2-4-3.
The connection between the source distribution q(x) and the fuselage contour
R(x) can be established intuitively through application of the continuity equation to
the volume element ABCD of Fig. 5-4:
(U""
-{-u)nR2-}-gdx=(U.. +u+dx
dx)v(R-{-RdX2
Hence, it follows the source distribution
(5-4a)
dx [(U.+u)R2]
q(x)
_
(R2)
40!L
= U`'° dAF
0O
dx
U00
dx
Figure 5-4 Fuselage theory at axial flow. q (x) = source-sink distribution.
(54b)
AERODYNAMICS OF THE FUSELAGE 333
Except for the vicinity of the stagnation point, u << U. for slender bodies, to which
the second relationship applies. The quantity AF(x) = rrR2 (x) is the local fuselage cross
section. The closure condition for a closed fuselage contour [see Eq. (2-92)] is automatically satisfied by the expression for the source distribution if AR = 0 at the nose and
at the tail. For the induced velocity components in the axial and radial directions,
u(x , r) =
IF
q( x)
1
4:s
(x - x) dx
( x - x')2 ±
(5-5a)
r23
0
(x , r)
W,
=
r
4 7v
IF
f
q(x') dx'
(x - x')2 4 r2
(5-5b)
3
0
To determine the velocity distribution on the surface of slender fuselages, the values
of the induced velocities u and w, for small values of r are needed. Special caution is re-
quired in establishing these values, because on the fuselage axis even the induced
velocities are singular. Expansion of u and w, for small values of r leads, under the assumption that dq/dr is steady in the vicinity of the point x' = x, to the following expressions:
it(x,r 0)_-lim
2 (1 -In4 n Ego
2s dq(x)
r ) dX
w,.(x,r-a 0 ) =
-{-
x-e
q(x') dx'
r (x - z')
,J
0
IF
dx
z+e
q(x)
1
(5-6a)
(x - x')2
(5-6b)*
2n r
lim(rwr )=Uc R dR
(5-6c)
dx
2-4
Equation (5-6c) is obtained by introducing Eq. (54b) into Eq. (5-6b). These
equations show that the two components of the induced velocity become infinitely
large on the fuselage axis (r = 0). This constitutes a basic difference to the plane
case (profile theory); see Eq. (2-91). The radial velocity component is given, from
Eq. (5-6b), by the limiting value (boundary condition) on the fuselage axis. Hence,
the induced velocities on the surface of the slender body [r = R(x)] are finally
obtained from Eqs. (5-6a) and (5-6b) and by introducing the expression Eq. (5-4b)
for the source distribution q(x) as
U, (x)
1
Uro
4 Ego
EM 2 1 - In
1
1
(
d2(R2)
d x2
x-e
2E
d2
IF1
d(R2)
(R=)
dx2
IF
d(R-)
dx'
-i
dx'
J dx' (x - x')2 x- +Je dx' (x - x')2
0
In R (x)
(5-7a)-
IF
`The validity of Eq. (5-6b) is immediately established from the continuity equation.
tAt stations of the fuselage contour at which the curvature d ZR/dx2 is unsteady, it must
be set:
da (R2)
1
2 C(
d2 (R2)
dx2 )1_0
d2 (R2)
1
( dx2 )x+oJ
334 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
_
IV,(x)
d R (x)
(5-7b)
dx
U00
The latter equation is equivalent to the kinematic flow condition, which states that
the direction of the velocity vector on the surface is tangent to the surface.
Pressure distribution From the Bernoulli equation, the pressure distribution on the
surface of the body is obtained as
Cp =
1'
4
(WC)2
°D = i
[2
(.J±_)2+ (wr)2]
U
where WC = (U. + u)2 + w2 is the velocity on the fuselage contour.
As in wing profile theory, the quadratic terms of the induced velocities may be
disregarded. Thus, the first approximation of the equation becomes
cp(x) = -2 Ux)
(first approximation)
(5-8)
U.
A more accurate formula for the pressure distribution is obtained by retaining
the term wY because, by Eq. (5-7b), w,, is proportional to the slope of the contour
dR/dx, the influence of which is thus taken into account more effectively. A second
approximation is thus obtained as
cp(x)
2
u(x)
Um
- rdR(x)12
dx
(second approximation)
(5-9)
Equations (5-8) and (5-9), together with Eq. (5-7a), allow the determination of
the relationship between the pressure coefficient and the fuselage thickness ratio
SF = dFinaxllF. This relationship is found as
cp(x) = Lf(x) + g(x) ln5F] 82F
w ith
1
g (x)
IF'
R21W,.,
(5-10a)
d2(R2)
(5 - 10b)
dx2
where the functions f(x) and g(x) depend only on the fuselage form but not the
thickness ratio.
Examples A few examples of this method of source-sink distribution will now be
discussed. The induced velocity u(x) of an ellipsoid of revolution of thickness ratio
5F = dFinax/lF is obtained from Eq. (5-7a) with X = x/lF as
I
U
U
14X
+ In
SFJ S2
2
F
(5-11 a)
The pressure distribution of an ellipsoid of revolution of thickness ratio
SF = 0.1 is given in Fig. 5-5. Both the first approximation from Eq. (5-8) and the
second approximation from Eq. (5-9) are shown. For comparison, the exact solution
is given, and will be discussed in the next section. The second approximation agrees
AERODYNAMICS OF THE FUSELAGE 335
-0.0
Poraboioi01111
2
-0.06
\
-0.04
/
3-
\\
E//rp soid (Il
0
I
0.02
,
{
0.04
0
0.2
0.6
0, 4
0.8
1.0
IF
U.
L IM
01
-a
- - -- --- - II
Figure 5-5 Pressure distribution on bodies of
r evo luti o n (ellipso id , para b o l o id) at i n c ompres sible axial flow. Fuselage thickness ratio 6F =
0.1. (1) Exact solution from Eq. (5-14) or
Lessing. (2) Second approximation, Eq. (5-9).
(3) First approximation, Eq. (5-8).
well with the exact solution over the entire contour. The first approximation
deviates at the front and rear portions.
For the maximum perturbation velocity at the ellipsoid of revolution that
occurs at station X = 2, Eq. (5-1 la) yields
Ua'` _ - (1
-}- In
f) SF
(ellipsoid)
(5-11 b)
This value is plotted against the fuselage thickness ratio in Fig. 5-6. Here, too, the
exact solution is shown for comparison. At large thickness ratios the values of the
0z
Ellipse profile
Paraba/oid
/
0.1
02
03
SF
Figure 5-6 Maximum perturbation velocity of bodies of revolution in axial flow vs. thickness
ratio SF. (1) Exact solution from Eq. (5-15) or Lessing, respectively. (2) Approximation from Eq.
(5-11b) or (5-12b), respectively.
336 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
exact solution are larger than those of the approximation solution of the source-sink
method with the source distribution on the axis. Also included in Fig. 5-6 is the
perturbation velocity for the plane problem of the elliptic profile as in Fig. 2-34. In
this case umax/U = 5 (= SF). The comparison of the curves for the elliptic profile
and the ellipsoid demonstrates by how much the maximum perturbation velocity at
the body of revolution is smaller than that at the wing profile of the same thickness
ratio.
For the induced velocity of a paraboloid of rotation [see Eqs. (2-6) and
(2-7a)] , Eq. (5-7a) yields
u(X)
Uro
= 2 [1 - 6X(1 - X)] [3 + -- InX (1 - X) -f- 2ln5F] 62V
(5-12a)
The corresponding pressure distribution (second approximation) for 5F7=0.1 is
shown as curve 2 of Fig. 5-5. The maximum perturbation velocity, lying again at
X = 1, is obtained from Eq. (5-12a) as
umaX
(3 -E- 2 InsF) SF
Uc
(5-12b)
(paraboloid)
This value is represented by curve 2 of Fig. 5-6.
The computations discussed so far are based on source distributions on the
fuselage axis. Results of Lessing [32] for distributions of source rings on the body
surface are included in Figs. 5-5 and 5-6 as curves 1. These results can be considered
to be "exact." The considerable improvement of the theory based on surface
distribution over that based on axial distribution is obvious in Fig. 5-6.
The pressure distribution for a body of revolution, composed of a half-ellipsoid
of revolution and a matching infinitely long cylinder, is given in Fig. 5-7. For
evaluation of Eq. (5-7a) at the station of the curvature discontinuity, x/lp = 2 , the
-001
0
of
02
0.3
Eliip roid-
04
Os
0.5
0,7
98
0.9
10
1,1
Cylinder
Figure 5-7 Pressure distribution on an axisymmetric half-body (dFinax/lF = 0.1) in axial flow
(source distribution on the axis).
AERODYNAMICS OF THE FUSELAGE 337
-0oe1
-a.06
05
X=--i
041
U
r-f/ip,roid
06
to
0,7
-Cylinder
H
0,321 IF
04'O'-IF.
IF
Figure 5-8 Pressure distribution on a body of revolution with cylindrical center section.
6F = dFinax/lF = 0.09 (source distribution on the axis).
footnote to this equation must be observed. In this way, the specifically marked
value of cn is obtained.* Finally, Fig. 5-8 shows the pressure distribution of a body
of revolution composed of a frontal half-ellipsoid of revolution, a rear half-paraboloid
of revolution, and a matching cylindrical center section. For the marked points at the
stations of curvature discontinuity, the comment that was made for Fig. 5-7 applies.
A body of revolution of the airship kind has been studied particularly by
Fuhrmann [13]. The flow pattern produced by a slender body of revolution
(so-called streamlined body) is illustrated in Fig. 5-9. Its generating source-sink
distribution is indicated in the upper picture. The theoretical pressure distribution is
in excellent agreement with measurements.
Exact solutions A few more data will be given on the exact solutions for fuselages
in axial incident flow. Such exact solutions of the spatial potential equation can be
found in closed form for a few cases only.
The general ellipsoid in axial incident flow was first investigated by Tucker-
mann [36] and Zahm [36] and later, more explicitly, by Maruhn [36]. The
pressure distribution on the surface of the ellipsoid, Fig. 5-10, in incident flow
parallel to the x axis is given in [36] as
(r)2
]-L
-
[(__)2sin2 t9 - ()'cos2]Jo
`The dashed curves of Fig. 5-7 are the results of Eq. (5-7a) without consideration of the
footnote given there.
338 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
a
U.
U_
uo
R4
-o- Measurement
a8 b8
----- Theory
R
J
I
0
i
PaW
12
Re= UHF =1.3.106
V
nn
0
ai
b
U2
0
Of
!25
x/IF-
06
07 M
49
to
Figure 5-9 Streamline pattern and pressure distribution of a body of revolution in axial flow
(from Fuhrrnann, body III). (a) Streamline pattern. (b) Pressure distribution on the body
surface.
where a, b, and c are the semiaxes of the ellipsoid. The origin of the coordinates
lies at the center of the ellipsoid. The quantity A is a function of the two axis
ratios a/c and b/c; it is presented in Fig. 5-11, from (36].
The special case of an ellipsoid of revolution in axial incident flow is obtained
from Eq. (5-13) for b = c as
c -I-A2.
p
-
1L1
!Z
-2d-
U
)
(5-14)
\a
AZ
_2b
Figure 5-10 Geometry of a general
ellipsoid.
AERODYNAMICS OF THE FUSELAGE 339
C =Z
1,
125
120
110
105
969
Figure 5-11 Coefficient A for the determination of the pressure distribution on a general
100
0
as
2.5
)0
ellipsoid in
axial incident flow, from Eq.
(5-13), vs. the two axis ratios a/c and b/c.
Here, bla = 5F is the thickness ratio of the body of revolution. The evaluation of
Eq. (5-14) is included in Fig. 5-5 as the exact solution. The pressure minimum
cpmin = 1 -A2 lies at x = 0. Hence, the maximum perturbation velocity becomes
UmaxA
(5-15)
U
where
z
A=
2 2
ao
with ao =
2 SF
(tanh-1
y1 - cSF
a
/1
- -1aF
SF)
(5-16)
The relation between umax/Ue and 5F is shown in Fig. 5-6 as the exact solution
for the ellipsoid of revolution. For small values of 5F, the three equations above
yield the relationship Eq. (5-11b) that was derived by means of the singularities
method.
Effect of viscosity So far in this chapter, the fluid has been assumed to be inviscid
and incompressible. The effect of compressibility on the aerodynamic properties of
a fuselage will be treated in the following sections. First, a few data will be given on
the effect of viscosity in incompressible flow (effect of Reynolds number). At
moderately large Reynolds numbers (Re > 105), the effect of viscosity on the
pressure distribution on bodies of revolution in axial incident flow is quite small.
This can be seen, for instance, from Fig. 5-9, in which the pressure distribution
computed for inviscid flow is compared with measurements. The slight deviations of
the pressure distribution as obtained from potential theory from that found in
viscous flow is responsible for the pressure drag of the body of revolution. In
addition there is a friction drag, which is produced by the wall shear stress.
The effect of friction on the flow about fuselages is determined from
340 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
boundary-layer computations, quite similar to the case of wing profiles. In the latter
case the boundary layers are two-dimensional, whereas in the case of fuselages with
circular cross sections in axial incident flow, the boundary layers are axisymmetric.
The computational procedures for the latter are very similar to those for the
two-dimensional boundary layers, both laminar and turbulent.
These boundary-layer computations for a given body produce distributions of
boundary-layer thicknesses (momentum thickness and displacement thickness) and
of a form parameter of the boundary-layer profiles over the contour. They
determine drag and position of the separation point. Young [59] and Scholz [59]
conducted comprehensive computations of the drag of bodies of revolution. They
found that the contribution of the wall shear stress to the drag of bodies of
revolution is, in general, equal to that of the flat plate in parallel incident flow of
equal surface area and equal Reynolds number with reference to the body length.
For fully turbulent flow, the body drag due to friction DFf may be obtained
approximately from the flat plate drag Dp from the formula
DFf = Dp(l + CSF)
(5-17)
with c 0.5. Here Dp is the drag of the flat plate in parallel incident flow of the
same surface area SF and the same length IF as those of the body of revolution.
Hence, Dp = CfSFgo,, where the coefficient c f for smooth surfaces can be taken
from Fig. 2-48. Further data on fuselage drag are found in Hoerner [19].
5-2-3 The Fuselage in Asymmetric Incident Flow
General remarks Now the asymmetric inviscid flow about an inclined fuselage as in
Fig. 5-12 will be considered.
First, it is important to state that, in inviscid flow, only a moment, not a
resultant force, is acting on the -inclined fuselage. This is caused by the
underpressures on the upper side of the body nose and the lower side of the tail
and, conversely, the overpressures on the lower side of the nose and the upper side
of the tail. This pressure distribution results in a moment MF that attempts to turn
the fuselage nose up (unstable moment). At small angles of attack a, this moment is
proportional to the angle of attack.
The fuselage-wing interaction changes the magnitude of this moment greatly
(see Chap. 6). However, the moment of the fuselage alone will be treated here, first
in inviscid flow and later with consideration of friction. It should be mentioned that
Figure 5-12 Inviscid flow about an inclined fuselage.
AERODYNAMICS OF THE FUSELAGE 341
01
0.2
03
0.41
0.5
aC
-
0.5
0.7
0.8
0,9
10
Figure 5-13 Coefficient k for the computation of the moment of an inclined general ellipsoid of
Eq. (5-18b), from Vandrey.
the effect of friction on the aerodynamic properties of the fuselage is considerable.
The moment in inviscid flow can be obtained from simple momentum considerations. Computation of the pressure distribution on the fuselage surface requires
application of potential theory. As in the case of the fuselage in axial incident flow,
exact solutions and approximate solutions to the singularities method are known.
Finally, the effect of friction can be determined with the help of boundary-layer
theory.
Fuselage moment by the momentum method of Munk An early account of the
computation of the moment of an inclined fuselage was given by Munk [41]. It is
based on an application of the momentum law. The momentum far behind a body
moving at constant velocity in an inviscid fluid remains unchanged and no resultant
force acts on the body, but this does not exclude the existence of a free force
couple. According to the Munk theory, lift and pitching moment (free force couple)
of a fuselage at an angle of incidence a and at free stream velocity U. are
LF = 0
(5-18a)
MF = 2kga, VFa
(5-18b)
Here, q,o = (o/2)UU is the dynamic pressure of the incident flow, VF is the body
volume, and k is a factor describing the ratio of the volume of the fluid quantity
moving with the body to the body volume. Values of k for general ellipsoids have
been given by Zahm '[36] and presented graphically by Vandrey [40] The
coefficients k for general ellipsoids of volume VF = s 7rabc are given in Fig. 5-13 as
a function of the axis ratios c/a and b/c. Accordingly, the coefficient k for slender
ellipsoids of revolution (b = c and c/a < 0.2) differs little from unity. Thus, from
.
Eq. (5-18b), the moment of slender bodies of revolution is obtained from the
simple approximation formula
342 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
Mp=2q. VFa
(5-19)
Note that the unstable moment of the aerodynamic forces acting on an inclined
slender fuselage of revolution is proportional to the angle of attack a and the body
volume VF [Eq. (5-1a)].
Pressure distribution by the method of dipole distribution The flow field of an
inclined body of revolution can be computed by the singularities method. In the
simplest approach, a distribution of spatial dipoles as in Fig. 5-14 is arranged on the
body axis.* The axes of the dipoles are parallel to the z axis. The potential of the
dipole distribution is
IF
m(x') dx'
i' (x - x')2 ± r23
r cos
4n
J
t/
0
cos 6 m (x)
2n
r
(5-20a)
(5-20b)t
where m(x) stands for the dipole strength. The second relationship results from the
expansion of the potential for small distances r from the axis, as required for slender
bodies. The velocity components in axial, radial, and circumferential directions,
respectively, are obtained from Eq. (5-20b) as
a0
cos l drn(x)
2n r
dx
1
8x
U',,,=
ao
1
ar
2n
1
a4)
?1
e6
=
cos i
r1
sin 6
2n r"
(5-21a)
(5 - 21 b)
(x)
()
912 (X)
( 5 - 21 c)
The dipole strength is determined from the kinematic flow condition, which
demands that, on the body, the velocity component normal to the surface is zero.
*For asymmetric incident flow, the method of source distributions on the surface with
nonaxisymmetric distribution has been successful.
Note that this expression for the potential of a very long body of revolution is identical
to the potential of the dipole distribution of a circular cylinder.
Figure 5-14 Illustration for the theory of a fuselage of revolution at asymmetric incident flow.
AERODYNAMICS OF THE FUSELAGE 343
z
zf
Teardrop R(r)
Skeleton line zp(x)
a/z) t zF ; fuselage axis
.cc(,-)
U" =a(.)-cos 0'
Figure 5-15 Illustration for the theory of a cambered fuselage with angle of attack.
For a body with a cambered skeleton line as in Fig. 5-15, which is a generalization
of Fig. 5-14, the kinematic flow condition becomes*
a (x) U,,,, cos 3 + wr (X) = 0
for
r=R
(5-22)
Here a(x) is the local angle of attack of the skeleton line referred to the incident
flow direction of V. as given by
a(x)
=a- dzF(x)
dx
(5-23)
where a is the angle of attack of the fuselage axis and zF(x) is the skeleton line of
the fuselage. Introduction of wr from Eq. (5-21b) into Eq. (5-22) yields, for the
dipole distribution,
m (x) = 2n U,,. a (x) R2 (x) = 2 U,,. a (x) AF(x)
(5-24)
where AF(x) is the cross-sectional area of the fuselage.
Pressure distribution The inclination of the fuselage causes a pressure distribution
on the body surface that, from Eq. (5-8), is given in first approximation as cp(x,
6) = -2u(x, 6)1U.. Introducing Eq. (5-24) into Eq. (5-21a) yields
cP (x, t) = - 2
cos z$
d
B (x) d x [x (x) R2 (x))
(5-25a)
If the angle of attack is constant along the fuselage axis, this equation takes the
simpler form
cp(x, 6) = -4a cos t5
d dxx)
fu(x) = const]
(5-25b)
Cc
An example of these pressure distributions is given in :gig. 5-16 by means of
ellipsoids of revolution of thickness ratios 5F = dFinaxllF = 0.1 and 0.2 and angle
"Here, the dipole distribution can be left on the body axis, as in the case of the plane
skeleton theory (see Sec. 2-4-2).
344 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
Ex act
1.0
5 F=01
05
.- -1.0
19,
1
1
Exact
-1.5
-20
02
OB
1.0
Figure 5-16 Pressure distribution result-
ing from asymmetric incident flow on
ellipsoids of revolution of thickness ratios
5F = 0.1 and 0.2 from Eq. (5-26). Exact
solution of Eq. (5-33).
of attack a = const. The following expression for the pressure distribution is easily
found :
cP_-2acost$ 1-2X
SF
(5-26)
YJ_
Lift distribution The lift distribution is obtained from the pressure distribution by
integration. A fuselage portion of length dx is supported by the lift force dLF of
magnitude
In
dLF
P (x) d x f cP cos 0 d t$
(5-27)
0
Observing Eq. (5-25a) and integrating over t3 yields the lift distribution,
dLF
= 2zq. dx [x (x) R2 (x)]
(5-28)
This relationship has been derived by Multhopp [40] from momentum considerations.
AERODYNAMICS OF THE FUSELAGE 345
Equation (5-28) shows directly that the total lift force of a closed body
vanishes, because
1
LF =
dx = 27rq. [u(x)R (x)] 1
fF dLF
dx
Z
(5-29a)
0
if R(x) = 0 at the nose and tail of the body [see Eq. (5-18a)].
Pitching moment The pitching moment of the fuselage at constant angle of attack
a(x) = const is obtained from Eq. (5-28) through integration by parts as
IF
MF = - f
IF
dd Fx dx = 27rq.a 1 R2(x) dx = 2q., VFa
(5-29b)
0
0
where VF is the fuselage volume from Eq. (5-la). In this way, the Munk
approximation formula for the moment of slender bodies of revolution has also
been obtained by means of the singularities method. Because LF = 0, the fuselage
moment is independent of the location of the reference axis. It is a so-called free
moment.
As an example, Fig. 5-17 illustrates, for theory and experiment, the lift
distribution from [16] of an inclined ellipsoid of revolution of thickness ratio
The theoretical lift distribution is obtained from Eq. (5-28) with
6F=-!.
a(x) = const as
dLF
dh = 2nq (1 - 2X) dFinax«
(5-30)
included in Fig. 5-17 as the solid line (line 1). The
measurements agree well with theory in the front portion of the fuselage, but some
This approximation is
deviations are found for the rear portion. For comparison, see Sec. 5-2-2.
The above. discussions apply to bodies of revolution. To determine the moments
of bodies of noncircular cross sections at constant angle of attack, it should be
realized that, essentially, only the fuselage width distribution bF(x) determines the
20
Figure 5-17 Lift distribution of an ellipsoid of
revolution of thickness ratio 6F = -',. (1) Approximation theory from Eq. (5-30). (2)
Exact theory (inviscid). (3) Theory with friction, from Hafer [16].
346 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
moment caused by the inclination. Equation (5-29b) can therefore be applied to
fuselages of noncircular cross sections by substituting DF/2 for R and introducing a
correction factor k*. This leads to
MF = 2k*q VFa
(5-31a)
IF
with
VF
= 4 r bF(x) dx
(5-31 b)
0
Here VF is the volume of a body of revolution that has the body width for its
diameter. The correction factor can be determined by comparing Eq. (5-31a) with
the exact equation Eq. (5-18b) for general ellipsoids. Because VF = (b/c)VF, we
have k* = kc/b, where k is given in Fig. 5-13. The values of k* thus computed are
presented in Fig. 5-18 as functions of fuselage width ratio 5* = bFinax/lF and the
cross-section ratio of the fuselage XF = hFinax/bFinax (see Fig. 5-1). It follows,
therefore, that the factor k* is almost unity for slender fuselages of all practical
cross-section ratios
AF. Thus, the above discussion has shown that for the
computation of the moment of slender fuselages of noncircular cross sections, Eq.
(5-29b) may be used in good approximation if the radius R is replaced by the
semiwidth bF/2.
The moment of the fuselage of variable angle of attack a(x) is obtained by
using the semiwidth bF/2 in Eq. (5-28) instead of R. Hence, integration over the
fuselage length yields for the pitching moment, from Eq. (5-29b),
IF
MF = q. 2 f a(x)bF(x) dx
(5-32)
0
This equation is applicable to the fuselage with cambered skeleton line from Eq.
(5-23) and to fuselages in curved flow as encountered during rotation about the
lateral axis [see Eq. (3-147)]. Furthermore, this relationship is important for the
10
AF-Z
0.8
-rI
3
J
02
I
t
I
0
Q1
0,2
2
02
0,¢
Figure 5-18 Coefficient k* for the computation
fuselage of
noncircular cross sections, from Eq. (5-31a).
0 of the moment of an inclined
AERODYNAMICS OF THE FUSELAGE 347
computation of the fuselage moment when a wing is attached to the fuselage (see
Chap. 6).
The above considerations on the lift distribution and on the moment furnish,
accordingly, the side force distribution and the yawing moment due to sideslip for a
yawed fuselage.
Exact solutions A few data will now be given on the exact solutions for inclined
ellipsoids of revolution. Maruhn [36] determined the pressure distribution of the
inclined ellipsoid of revolution at small angles of attack as
eP = (cp)a-o ; 2 B
a
b
b
a,
1 - I - COs ?
N
(5-33)
1 - C1 - (a)-] (a )-
Here, bla = 5F is the fuselage thickness ratio. The quantity B is defined as
B4 _ a2
8
2
where ao is given by Eq. (5-16). The angle-of-attack-dependent pressure distribution
for this exact solution is shown in Fig. 5-16 for 6F = 0.2. In the vicinity of the
nose and the tail, the exact solution gives somewhat smaller values of the pressure
coefficient than the approximate solution by the method of singularities. This
means that the correction factor k* for the moment in Eq. (5-31a) is somewhat
smaller than unity. In Fig. 5-17, too, the exact solution for the lift distribution is
included as curve 2. Near the nose and the tail, the exact solution deviates
somewhat from the approximation solution. In the vicinity of the nose, the
measurements agree quite well with the exact solution. Larger differences remain,
however, near the tail. They are caused by viscosity effects to be treated in the next
section.
The values of the moment from the exact solution have already been given in
Eq. (5-1 gb). From Eq. (5-31a), the theoretical moment coefficient cMF = AfFlg.. VF
is obtained as
cMF = 2k*a
(5-34)
This theoretical value, with k* = 0.95, is compared in Fig. 5-3 with a measurement.
The moment slope dcMFlda from this theory is considerably steeper than that of
the measurement. This difference is caused by viscosity effects. Viscosity effects are
also responsible for the deviation of the measured lift from zero, as seen in Fig. 5-3.
Viscosity effects Qualitatively, viscosity affects the flow over the inclined fuselage
(Fig. 5-12) in such a way that the pressure on the tail section is reduced, because
the inviscid outer flow is forced outward by the boundary layer. Consequently, the
negative lift of the tail section is somewhat smaller than the positive lift of the nose
section. Overall, therefore, viscosity effects cause a positive lift, which is also termed
friction lift. This fact may be seen in Fig. 5-17 for the lift distribution. The friction
348 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
lift changes the moment; specifically, it creates an additive nose-down moment with
reference to the lateral axis through the fuselage center. Hafer [16] has described a
method for the approximate computation of the viscosity effect by means of
boundary-layer theory. Accordingly, the boundary-layer displacement thickness 51 is
determined along the fuselage surface at axial incident flow. A rather strong growth
of the boundary-layer thickness near the tail is found, as sketched in Fig. 5-19.
Consequently, to compute the pressure distribution, the local fuselage radius R(x) in
Eq. (5-25a) must be replaced by the radius [R(x) + 51(x)] , that is, the radius R(x)
enlarged by the displacement thickness 51. The lift distribution is obtained from the
pressure distribution corrected for viscosity by integration. In Fig. 5-17, the lift
distribution, computed in this way from [16], is also shown. Through the
correction for viscosity, better agreement is reached with the measurements,
particularly in the vicinity of the tail. Lift, pitching moment, and neutral-point
position are determined through further integrations. In Fig. 5-20, the lift slope
dcLFlda, the moment slope dcMFlda, and the neutral-point position xNF/IF are
plotted against the inverse fuselage thickness ratio 1F/dFinax for several axisymmetric
fuselages. These measurements were taken by Truckenbrodt and Gersten [501.
Curves 1 are those from the inviscid theory, curves 2 from the viscous theory
of Hafer [161. The latter theory agrees quite well with measurements.
5-3 THE FUSELAGE IN COMPRESSIBLE FLOW
5-3-1 Similarity Rules for Fuselage Theory
of Compressible Flow
Velocity potential (linearization) For slender fuselages under a small angle of
incidence, the magnitude and direction of the local velocities are only a little
different from the velocity of the incident flow U,,. It is expedient, therefore, to
split up the total flow into a basic, undisturbed flow and a superimposed
perturbation flow:
U = U" + it
w o = Wo
W,. = u'r
(5-35)
where u, w,., w6 are the perturbation velocities with
it 4 U00
Wr < U00
iao < Um
IF
Figure 5-19 Viscosity effect on the flow about fuselages. S, (x) = displacement thickness of the
boundary layer.
AERODYNAMICS OF THE FUSELAGE 349
a5
2,5
04
20
k
.3
3
0
Z
02
X
0-7
a1
05
D
0
6
a
to
a
72
6
1k
F
B
b
dFmax
12
10
14
IF
dFinax
c
dFinax
Figure 5-20 Effect of viscosity on the aerodynamic coefficients of inclined axisymmetric
fuselages. Position of the moment reference point x0 is different for all fuselages:
0.251F < x,, <0,51,F. (I) Theory without friction. (II) Theory with friction, from Hafer. (a) Lift
2/ 3
slope, cLF = LF/q . VF
. (b) Pitching-moment slope cMF =
(c) Neutral-point
position xNF
By retaining only the largest terms (linearization), the potential equation of
compressible flow becomes, in analogy to Eq. (4-5),
(1 - Mat) a2 +
0X'-
020
ar2
+
1
ao
+
r ar
1 a20
r2
a,&2
=0
(5-36)
Here Ma = U/a is the local Mach number. Equation (5-36) applies to subsonic,
transonic, and supersonic flows. The components of the perturbation velocity become
ao
zc= ax
Wr= ar
Iva= ir ao
(5-37)
1?6
The relationship between the local Mach number Ma and the Mach number of the
incident flow Maw, = U. /a. is given by Eq. (4-7).
For purely subsonic and purely supersonic flows, Ma can be replaced
approximately by Ma,. Hence, the following linear differential equation for the
potential is obtained in analogy to Eq. (4-8):
(i-1Vda200
)a2
ax-
cr2
r ar
t °-
rz a79-
= 0 (Maw
1)
(5-38)
In analogy to Eq. (4-9), the equation for transonic flow becomes
y
1 a0 a2(
a-(
ax ax2
er2
U,-,
1
ao
ar
+ 1- a20
a.8=
= 0 (Max = 1)
(5-39)
Contrary to Eq. (5-38), this differential equation for the potential is nonlinear. in
analogy to the case of the wing of finite span, the potential equations derived
above, Eqs. (5-38) and (5-39), will now be applied to the development of similarity
rules for subsonic, transonic, and supersonic flows.
350 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
Subsonic and supersonic similarity rules The similarity rules for subsonic and
supersonic flows are obtained through a transformation of the potential equation
[Eq. (5-38)]. To this end, the given compressible flow is transformed into a flow,
the potential equation of which no longer contains the Mach number. This
transformation is accomplished, in analogy to Eq. (4-10), by setting
x' = x
r' = clr
0 = c20'
aV _
U' = U00
(5-40)
where the primes signify the transformed quantities. The factor cl is determined in
such a way that the Mach number Ma no longer appears in the transformed
potential equation. The factor c2 is obtained by applying the streamline analogy
(kinematic flow condition). These factors cl and c2 are given by the expressions
Eqs. (4-12) and (4-21) derived earlier. The transformed potential equations are, in
analogy to Eqs. (4-13) and (4-14),
a20'
a20'
ax ,"- -l- are,
a20'
ax'2
a20'
ar'2
+
1 a0'
r ar ,
r'
a20'
+ r2 a2, = 0
1
=0
r'2 a6'2
L9 r'
(Macc < 1)
(
M a,,,, >
1)
(5-41)
( 5 - 42 )
The transformed potential equation for subsonic flow is identical to the
potential equation (Ma = 0). The transformed potential equation for supersonic
flow is identical to the linear potential equation Eq. (5-38) for Ma =-\/2-. These
transformations show that the computation of subsonic flows of any Mach number
can be reduced to the computation of the flow at Ma = 0 and the computation of
supersonic flows of any Mach number to that at Ma = s. This is the
Prandtl-Glauert-Gothert-Ackeret rule for fuselages. It can be formulated in the
following way, corresponding to version I for wings of finite span (Sec.
4-2-3).
From the given fuselage and the given Mach number, a transformed fuselage is
obtained by a distortion of its dimensions in the y and z directions and of its angle
of attack by the factor ci = [1 -Ma2,1. Its dimensions in the x direction remain
unchanged. For the fuselage, transformed in this way, the incompressible flow has
to be computed if the given Mach number is subsonic. If the given Mach number is
has to be computed.
supersonic, however, the compressible flow for Ma =
The transformation formulas of the geometric quantities of the fuselage are
Thickness ratio:
d=
Camber ratio:
lF =
Angle of attack:
a' =
I
dF
F
(5-43a)
I ZF
(5-43b)
Maw, I
F
I 1 -Ma2, j a
(5-44)
Hence, when the velocities of the incident flow of the given and the transformed
fuselages are equal, the pressure coefficients are related by
AERODYNAMICS OF THE FUSELAGE 351
P
cP
Po
=
c'P
(5-45)*
Ma
The geometric transformation of Eq. (5-43a) is illustrated in Fig. 5-21, in which
the transformed thickness ratio is plotted against the Mach number. The hatched
body is the given body the flow over which is computed for different Mach
numbers. The transformed bodies belonging to the given Mach numbers are drawn
without hatches. The flow about these transformed bodies has to be computed as
incompressible flow when Ma < 1, and as flow at Maw, =
when Ma. > 1.
Applications of this rule will be discussed in Secs. 5-3-2 and 5-3-3.
Transonic similarity rule The similarity rules explained above apply only to
subsonic and supersonic flows. Now, a similarity rule for fuselages at transonic flow
(Ma. = 1) of axial incidence will be given. This similarity rule was first formulated
by von Karman [56]. A more detailed presentation of this similarity rule was later
given by Keune and 0swatitsch [27]. The following simplified derivation should be
sufficient.
By starting with the nonlinear potential equation, Eq. (5-39), the problem may
be formulated as follows: Given is an axisymrnetric fuselage of revolution at
May, = 1. Then, what is the pressure distribution over an affine reference fuselage at
the same incident flow Mach number Mam = 1? In analogy to Eq. (4-28), the
following transformation is introduced:
x'=x r'=car 0=c4P' U,= U.
(546)
*The validity of this transformation formula for the pressure distribution reaches beyond
the framework of the first approximation of Eq. (5-8), as has been shown, e.g., by
Truckenbrodt [49 ]. It applies to the second approximation of Eq. (5-9) as well.
L U.
CIO
I
to
Figure 5-21 The application of the
0
1
IT
Maw
2
3
Prandtl-Glauert-Ackeret rule to fuselages. Thickness ratio 5' of the transformed fuselage vs. Mach number.
352 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
Again, the quantities with primes signify the reference fuselage, those without
primes the given fuselage. Substituting Eq. (5-46) in Eq. (5-39) yields, in analogy to
Eq. (4-29), C3 = C4. To establish another relationship between the constants c3 and
c4, the radial velocity component w,. is derived from the boundary condition Eq.
(5-6c):
lim (r' iv'.) = U R'
lim (r -w,.) = U B dR
dx
r
r->0
d
d a:'
(5-47)
Be cause
of the affinity of the two fuselages, R'= (SF/SF)R , with 5F and SF being
the fuselage thickness ratios. With W. = aO/ar and w' = W/ar',
C4
(IF
2
_
and
5F
5F
(5-48)
F
Finally, the relationship between the pressure distributions cp and c, of the two
fuselages remains to be determined. Because cp = -2u/U,. = -(2/U,.)aO/ax, this
relationship is obtained immediately as
S
Cp = C4 c 7, _ (SF
2
(5-49)
CV,
This is the well-known von Karman similarity rule for bodies of revolution at
transonic incident flow.
As was first shown by Oswatitsch, a correction term to this formula can be
determined, leading to
cp = cP(SF ) + 2g(x)5F In SF
2
I
F
(5-50)
Here, g(x) is given by Eq. (5-1Ob).
5-3-2 The Fuselage in Subsonic Incident Flow
Computational procedures In Sec. 5-3-1 it was shown that at Mach numbers
Ma < 1, the computation of the flow about a fuselage may be reduced to the
determination of the incompressible flow for a fuselage that is suitably transformed.
The computation of the incompressible flow over a fuselage was discussed in detail
in Sec. 5-2. The starting point for further consideration is the subsonic similarity
rule. By assigning the index "inc" to the reference fuselage that corresponds to the
given fuselage at a given Mach number, the transformation formulas for the
geometric data of the fuselage become, from Eqs. (5-40), (543), and (5-44),
Coordinates:
xinc = x
rinc = r
Fuselage radius:
Rinc = R
1 -Mao,
Fuse lage l engt h :
1
'Finc = 6F
Angle of attack:
Oinc = a
t,inc = d
(5-51)
(5-52a)
( 5 - 52b )
Finc = lF
Thickness ratio:
1 -Mao
1 -Mao,
l -Ma o
(5-52c)
(5-52d)
AERODYNAMICS OF THE FUSELAGE 353
The transformation formula for the pressure coefficient is, from Eq. (5-45),
1
Cp
(5-53)
= 1 _Mao, Cpinc
This computation procedure will now be applied to fuselages in axial and inclined
incident flow at subsonic velocities.
The fuselage at axial incident flow The pressure coefficient for incompressible flow
from Eq. (5-10a) can be given in the form
Cp inc = [f(x) +g(x) In SFinc] SFinc - (Cp)Ma,o =o
where the functions f (x) and g(x) are independent of the thickness ratio of the
fuselage. Introducing Eqs. (5-52c) and (5-53) into the above equation yields
Cp = (Cp) Ila
a°O°
i d 2AF
n dzZ
In
1 - Ala
(5-54)
when taking into account that, from Eq. (5-l0b), g(x)SF = -(1 /n)(d 2AF/dx2) with
AF = nR2 as the fuselage cross section. From Eq. (5-54), it can be seen that the
influence of compressibility on the pressure distribution is taken into account by a
term additive to the pressure distribution at incompressible flow. It is proportional
to the second derivative of the distribution of the fuselage cross section. Since, in
general, this derivative is negative, the additive term represents an increase in the
negative perturbation pressure. The similarity rule of Sec. 5-3-1 is thus confirmed,
namely, that the computation of subsonic flow of arbitrary Mach number,
0 <Ma, < 1 may be reduced to the computation for Ma. = 0.
The pressure distributions for the paraboloid of revolution of thickness ratio
5F = 0.1 are shown in Fig. 5-22 for several Mach numbers. Marked changes of the
pressure distribution because of the compressibility effect are found only near the
fuselage center section (see Krause [29]).
Drag-critical Mach number The critical Mach number of the incident flow Ma,, cr at
which the velocity of sound is reached locally on the body is obtained, from Eq.
(4-53b), from the lowest pressure on the body cpmin. In Fig. 5-23, determination of
the drag-critical Mach number for paraboloids of revolution is demonstrated for
several thickness ratios SF. As the figure shows, the intersections of the curves
Cpmin versus Ma., of the various paraboloids of revolution from Eq. (5-54) with the
curve from Eq. (4-53b) have to be established. For comparison see also Fig. 4-28.
The critical Mach number, determined in this way, is plotted in Fig. 5-24 against
the fuselage thickness ratio. The critical Mach numbers of ellipsoids of revolution
are included in this figure. They are somewhat larger than those of the paraboloids.
Comparison of these critical Mach numbers of bodies of revolution with those of
wing profiles of Fig. 4-29 shows that, for the same thickness ratio (5F = 6), the
critical Mach number for three-dimensional flow is considerably larger than for
plane flow.
The drag-critical Mach number is of significance for the drag rise at high
subsonic Mach numbers; compare Fig. 4-14 for wing profiles. Finally, the drag
354 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
Mom o8
A /-a06
.0
o02
aoa
1
1
1
1
I
i
aoa
of
X-fF oW
4dFinax
to
Figure 5-22 Pressure distribution on a paraboloid
of revolution of thickness ratio 6F = 0.1 in axial
flow for several Mach numbers.
X
IF
coefficients at axial incident flow, from Gothert [15] are presented in Fig. 5-25 for
a few relatively thick fuselages as a function of the Mach number (see Krauss [30] ).
These measurements show that the drag rise for fuselages lies at higher Mach numbers
than for wing profiles of the same thickness, as would be expected from theory.
The fuselage in asymmetric flow The pressure distribution due to the angle of
attack is given for incompressible flow by Eq. (5-25) when the index inc is added
to all quantities. Introducing Eqs. (5-51)-(5-53) into this equation yields the
pressure distribution at compressible flow as
cP (x, t3) _ -2
d
B (x) d x [x (x) R2 (x)]
cos
(5-55)
Figure 5-23 Determination of the drag-critical Mach
number Ma..cr of paraboloids of revolution of
thickness ratio SF at axial incident flow. Curve i
from Eq. (5-54) and Fig. 5-6. Curve 2 from Eq.
(4-53b).
AERODYNAMICS OF THE FUSELAGE 355
S
0
005
015
010
020
025
i00
Z95
Ellipsoid
!90
Paraboloid
H_ _
Figure 5-24 The drag-critical Mach number Maa,cr
of paraboloids and ellipsoids of revolution
vs.
thickness ratio bF and axial incident flow.
q
By comparison with Eq. (5-25a), it is apparent that the pressure distribution due to
the angle of attack is independent of the Mach number. It follows that the
relationships of Sec. 5-2-3 for the lift distribution, the lift, and the moment in
incompressible flow apply directly to compressible subsonic flow.*
Studies of the computation of the pressure distribution on fuselages of arbitrary
cross section shapes, for both subsonic and supersonic flows, have been conducted,
for example, by Hummel [23]. A nonlinear second-order theory is given by Revell
[421
5-3-3 The Fuselage in Supersonic Flow
Fundamentals The essential difference between subsonic and supersonic flows has
already been explained by Fig. 1-9. Furthermore, the specific problems of the wing
-
*This is true also for supersonic incident flow, as will be shown in Sec. 5-3-3.
015
0
1
an
2
GIV
3
6
F
"-
-
0q t4L
0..113
» - 0387
6Finax
1
'
Z
0353
IF
co;
002
0
Oa
0.3
04
05
No.
06
07
08
0S
Figure 5-25 Drag coefficients of bodies
of revolution in axial incident flow vs.
Mach number, from measurements of
Gothert. c.DF refers to the frontal area.
356 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
of finite span at supersonic velocities were discussed in Sec. 4-5. The essential
physical difference between flows of subsonic and supersonic velocities lies in the
fact that, at the latter, a given point can affect only the space enclosed by the
downstream cone. This point itself can be affected only by disturbances within the
upstream cone. The application of these fundamental facts of supersonic flow to a
fuselage is explained in Fig. 5-26. The flow at a station x, r can be influenced only
by the crosshatched range cut out of the fuselage by the upstream cone of apex
semiangle M. The Mach angle u is related to the approach Mach number Ma,. by Eq.
(4-80). The upstream cone to the point (x, r) intersects the fuselage axis (x axis) at
the point
xo = x - r cot4u = x - r
am
-1
(5-56)
In the following discussions, the length x0 will be termed "influence length."
The Mach cone generated by the fuselage nose is also sketched in Fig. 5-26. The
supersonic flow about a circular cone (fuselage nose tip) in axial incident flow
represents the simplest case of a cone-symmetric supersonic flow, which has been
discussed previously in Sec. 4-5.
Now the. slender body of revolution at flows without (axial) and with a small
angle of incidence will be treated. Either case can be computed approximately with
the method of singularities (source-sink and dipole distributions, respectively). This
method has been presented previously in Sec. 5-2 for incompressible flow. Another
possibility is the application of the method of characteristics. Besides the linear
theory of supersonic flow over fuselages, which will be presented below in detail,
nonlinear theories of higher order have been developed by, for example, van Dyke
[51 ] and Lighthill [33]. Comprehensive presentations concerning the fundamentals
of the aerodynamics of fuselages in compressible flow are found in the pertinent
publications on gas dynamics, listed in, Section II of the Bibliography.
The fuselage at axial incident flow The axisymmetric fuselage in axial incident flow
of supersonic velocity can be treated by means of the source-sink method in a way
similar to that which has been explained for incompressible flow (Sec. 5-2-2). This
Figure 5-26 Fuselage theory at supersonic
incident flow.
AERODYNAMICS OF THE FUSELAGE 357
method was developed by von Karman and Moore [55]. The relationship between
the source distribution q(x) and the fuselage contour R(x) can be established
through the same considerations as in the case of incompressible flow; that is, here,
too, Eq. (5-4b) is valid. The procedure for translating the source-sink method of
incompressible flow into that of supersonic flows has been treated in detail for the
wing in Sec. 4-5-3 and can be applied to the fuselage.
The potential 0 (x, r) of the flow induced by the linear source distribution q(x)
on the x axis is given [see also Eq. (4-102)] as
70
i
(
ir=-2
g(Z) dx'
)
V(x
(5-57)
- x')= - (Ma"
0
Here x0 is the influence length from Eq. (5-56). The velocity components are
obtained for the entire space in the well-known way as
2G =
(5-58)
2C,
ax
or
In executing these differentiations it should be noted that the upper limit x = x0 of
the integral in Eq. (5-57) depends on x and r, and that. for x = x0, that is, on the
Mach cone, the denominator of the integrand vanishes.
To determine the velocity distribution on the fuselage surface, the values of the
induced velocities are needed for small radial distances r; see Eqs. (5-6a) and (5.6b).
Equations (5-57) and (5-58) yield
U'-E
2E
q(x') dx'
1 dq(x)
J
(x - x')2
(5-59a)
zv,(x,r-* 0)=
I q(x)
2z r
(5-59b)
The final form of the induced velocity components is obtained by introducing Eq.
(5-4b) into Eqs. (5-59a) and (5-59b) as
U
1
')
3
Uro
-
1 - In IF
lim
EAU
1
d2 (B2)
d
w,(x)
L' cc
_
dR(x)
dx
X-E
2E) d2(R2)
In (R(x)
IF
dx +
]iMa
JC d(R2)
-
dx'
dx' (x - x')-
1)
1 d (R2)
E
dx
(5-60)
(5-61)
To determine the pressure distribution from the induced velocities, the formulas
of the incompressible flow are directly applicable, that is, Eq. (5-8) for the first
approximation and Eq. (5-9) for the second approximation. In analogy to Eq.
358 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
(5-10a), the dependence of the pressure distribution on the body thickness ratio 8F
can be found for fuselages in axial incident flow. By observing Eqs. (5-60) and
(5-61), this dependence is obtained as
(5-62)
cp(x) = [fi(x) + 91(x) ln(SF Maw, - 015F"
The functions fl (x) and gl (x) depend on the fuselage geometry. They do not
depend, however, on the thickness ratio of the fuselage. Consequently, Eq. (5-62)
for the pressure distribution may be written in the following form:
Cp =: (Cp)Ma0.=y5 -
1 ddx
1 Ma - 1
(5-63)
This equation is analogous to Eq. (5-54) for subsonic incident flow.
It may be seen from Eq. (5-63) that, at supersonic incident flow, the
compressibility effect on the pressure distribution is given by a term additive to the
pressure distribution at Ma = N/2-. This confirms the similarity rule of Sec. 5-3-1,
stating that the computation of a supersonic flow of arbitrary Mach number can be
reduced to the computation at Ma _
The above computational procedure for
the pressure distribution of fuselages in supersonic axial flow will be explained now
by means of a few examples. The supersonic flow over the nose tip of a
cone-shaped body was treated early by Taylor and Maccoll [46], Tsien [48], and
Busemann [6]. Results for a blunt-body nose in supersonic incident flow have been
published by Holder and Chinneck [21] and van Dyke [521.
In Fig. 5-27, the pressure coefficient for Ma,,, =
(second approximation) is
presented for the paraboloid of revolution of thickness ratio 6F = 0.1. The
.
functions fi and gl of Eq. (5-62) become in this case
fi(X) = -4(22X2 - 16X + 1) - 8(6X2 - 6X + 1)ln(1 -X)
g1(X)=-8(6X2-6X+1)
(5-64)
with X= x/1F. For comparison, the pressure coefficient from the method of
characteristics is also shown. Agreement of these two computational methods is very
good. Furthermore, the pressure distribution for incompressible flow (Ma = 0)
from Fig. 5-5 is added. It is noteworthy that, in supersonic flow, the pressure
minimum lies behind the middle of a body that is symmetric to X = 0.5.
Furthermore, it should be noted that for the same shape of the body cross sections,
the pressure distribution in the axisymmetric case shows a completely different
character than in the plane case, as may be verified by comparison with Fig. 4-23a.
As in the case of a wing, the pressure distribution over the total surface of an
axisymmetric fuselage in supersonic incident flow results in a force in flow direction
that is different from zero. As in the case of the wing, this force is termed wave
drag. It is caused by the Mach waves originating at the body. Computation of the
wave drag may be done either with the help of the momentum law or through
direct integration of the pressure distribution over the surface. Only the latter
computational procedure will be described below.
Integration of the pressure distribution over the surface (component of the
AERODYNAMICS OF THE FUSELAGE 3 59
-Q08
-006
M aw; 0
-004
I
I
I
0.02
1
Figure 5-27 Pressure distribution on a paraboloid
of revolution of thickness ratio 6p = 0.1 at Ma . =
004
0
08
0.2
to
f and Ma = 0 in axial flow. Curve 1, singularities method; second approximation from Eqs.
.(5-62) and (5-64). Curve 2, linear method of
characteristics.
pressure force in the x direction) yields the wave drag of the body of revolution in
axial incident flow as
IF
DF = 21r f (p -pc)R
dR
dx
IF
dx = q f cp
dAF
dx
dx
0
0
To establish the effect of the Mach number on the wave drag,
substituted for cp in Eq. (5-65). Integration by parts yields
DF = (DF)Ma. =
1
2
2n q-
dAF
(IF
dx
2
In
2
Maw - 1
For R = 0 or dRldx = 0 at the fuselage tail, the wave drag is independent of the
Mach number because dAF f dx = 27rR(dR/dx) = 0. Its value becomes equal to that
for Ma. _. In this case the wave-drag coefficient of the fuselage with reference
to the frontal area AF is obtained from Eq. (5-66) as
DF
CDF =
q-AFinax
- (CDF)Ma = f = number SF
(5-67)
Here the "number" depends on the fuselage geometry, but not on the thickness
ratio. Consequently, the coefficient of the wave drag, referred to the frontal area, is
*In this section, the drag of the fuselage as obtained in inviscid flow (wave drag) is
designated as DF. Because of viscosity effects (friction), a contribution DF must be added to
this drag [see Eq. (5-17)].
360 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
proportional to 52.* Evaluation of the above equations for the paraboloid results
in
cDF = s 5F = 10.6752
(paraboloid)
(5-68)
where cp(x) of Eq. (5-62) has been substituted, using the expressions of Eq. (5-64).
The coefficients of the wave drag of truncated paraboloids of Wegener and Kowalke
[11] are given in Fig. 5-28. For the paraboloid cut off in the middle (IFIlFO =
it becomes
2),
CDF = s 52 = 4.6752
(paraboloid tip)
(5-69)
This drag coefficient does not include the contribution made by the suction
pressure on the blunt tail surface (so-called base pressure). For paraboloids of
thickness ratios 5F = 0.1 and 0.2, the drag coefficients as determined by the
method of source distributions are compared in Fig. 5-29 with those from the linear
method of characteristics. The deviations of the coefficients from the two methods
are very small for 5F = 0.1. They are no longer negligible for 5F = 0.2, however. By
substituting in Eq. (5-65) the expression for cp(x) of Eq. (5-62), a formula for the
wave drag of a general pointed body of revolution is obtained that depends on the
*In this connection it should be remembered that the wave-drag coefficient of the wing of
finite span, referred to the planform area, is likewise proportional to the thickness ratio (Sec.
4-3-3).
12
F M;10.1
Z
0.2
10
8
16
Z
0
Q5
06
07
F
Q8
09
1.0
1Fo
Figure 5-28 Coefficients of wave drag for
truncated paraboloids of revolution vs. thickness ratio 8F = dFinaxllF and Mach number,
from Wegener and Kowalke.
AERODYNAMICS OF THE FUSELAGE 361
05
I
0.
6F-02
03
6F=0.1
0.1
-- 2
Figure 5-29 Coefficients of wave drag
for paraboloids of revolution of thickness ratios 5F = 0.1 and 0.2. Comparison of the singularities method (1),
0
2.5
ao
10
from Eq. (5-68), and the method of
characteristics (2).
body geometry. von Karman and Moore [55] and Ward [57] established the
following equation; see the derivation [5]
IF
AF' (1F) f AF in
DF = 2 U!
1 - lF
dx
\
0
IF IF
-2f
1
0
-
f AF(x')AF11 (x) In
0
[AF(lF)] 2 In
R(lF)
2lF
x -xl
lF
z
Maw
-
dx' dx
(5-70)
being the fuselage crosswhere AF = dAF/dx and AF = d 2AF/dx2 , with
sectional area. With this formula, the wave drag at given body geometry may be
determined through relatively simple quadratures (see [9] ).
362 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
Das [8]
discusses some basic questions about the connection between the
various theories for the computation of the wave drag of fuselages, about the ranges
of their applicability and the limitations in their accuracies. Both the various
theories (linear, nonlinear) and the test results are compared. The summary report
on wave drag of fuselages by Morris [39] and the investigations on the base drag of
bodies with blunt tails of Tanner [45] should be mentioned here. The computation
of the friction drag of bodies of revolution in supersonic flow has been treated by
Young [601.
The evaluation of drag measurements for the determination of the wave drag
includes considerable uncertainties, because the measured total drag is composed of
friction drag and, if the tail is blunt, base drag, besides the wave drag. Measurements
in which these three contributions were determined individually have been
conducted by Chapman and Perkins [10] and Evans [101. In Fig. 5-30, the test
results of [10] for a truncated paraboloid are plotted as drag coefficients against
the Mach number. The comparison of these measurements with theory was
accomplished by adding to the measured base drag the theoretical friction drag from
Fig. 4-5 and the wave drag from Fig. 5-28. Agreement of the drag coefficients
computed in this way with the measurements is quite good. It should be mentioned,
however, that there are cases of larger differences between measurements and
020
Measurement
Re =3
- Theory
10'
0.15
Wave drag
U
Frictio n drag
005
Base drag (measurement)
0
125
175
1,50
200
MGM
+
dFinax
j
Figure 5-30 Measured drag coefficients of a truncated paraboloid of
revolution in axial flow (dFinax/
IF=0.07) at supersonic velocities,
from Evans. Comparison with
theory. Curve 1, base drag. Curve 2,
base and friction
08-IF
IF
total drag.
drag. Curve 3,
AERODYNAMICS OF THE FUSELAGE 363
4
3
x
M
E
LL.
13
U.
IF
0.71
4
3
Theor y
2
0.05
00#
0.03
70
Figure 5-31 Drag coefficients (pressure
drag without base drag) of slender fuselages vs. Mach number Ma,,, from measurements of [3] (body contour shown
with increased ordinates). (1) Optimum
body, from Haack and Sears, dFinax/IF =
of revolution,
(3)
Cylindrical
body,
dFinax/IF = 0.091.
dFinax/IF = 0.08. (4) Cylindrical body
with contraction, dFinax/IF= 0.08.
0.086.
is
2.0
2,S
3,0
Ma,. -
40
(2)
Paraboloid
theory. Additional test results are given in Fig. 5-31, namely, the coefficients of the
pressure drag CDF of four slender fuselages in axial incident flow plotted against the
Mach number Ma.. These drag coefficients do not include the base drag. Fuselage i
is a body of minimum wave drag for a given volume and a given length, from Haack
[43] and Sears [43]. Fuselage 2 is a paraboloid of revolution. Fuselages 3 and 4
have cylindrical tail sections. For fuselages 2 and 3, the theoretical values of Eq.
(5-70) are also shown.
Another optimum fuselage configuration with pointed nose and blunt tail was
specified by von Karman [55]. Also, Das [7] concerned himself with the
determination of optimum shapes of a fuselage with regard to its drag at supersonic
flow. A compilation of additional test results and of comparisons with theory is
found in Fiecke [11 ] . Miles [38] derived a linear theory for the computation of
the wave drag of fuselages at supersonic incident flow.
The flow picture of Fig. 5-32 gives a more profound insight into the flow about
a fuselage in supersonic flow of axial incidence. In particular, it shows clearly the
bow wave and the tail wave at a Mach number of Ma. = 3.5.
364 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
Figure 5-32 Shadowgraph picture of a fuselage at Mach number Mao, = 3.5.
Body of revolution with a blunt nose in hypersonic incident flow In Sec. 4-3-5, the
profile with a blunt nose in hypersonic incident flow was treated. For the
computation of the pressure distribution on the body surface, Newton's approximation, Eq. (4-65), was furnished as the simplest expression. This relationship, which
was established for plane flow, can be applied likewise to axisyrnmetric flow as
present in the case of fuselages. The pressure distribution on a half-body consisting
of a cylinder with a matching spherical nose pertaining to such a hypersonic flow is
plotted in Fig. 5-33. According to Newton's concept of momentum transfer from
the flow particles to the body, the pressure distribution would be given by Eq.
(4-65). The real flow does not correspond to this concept, and Eq. (4-65) cannot
properly represent the pressure distribution. Nevertheless, a very good approxima1.0
as
0.6
a4
Figure 5-33 Pressure distribution of a
half-body with spherical nose, from Lees.
1
-01 ZOL
0.4
0.6
1.2
s
R
1.6
2.0
2.4
2.B
(o) Mao,, = 5.8, Re = UooR/v. = 1.2
105. (o) Mao, = 3.8, Re = U.RR/vo, _
) Modified Newtonian
1.4 - 105. (
approximation, from Eq. (5-71).
AERODYNAMICS OF THE FUSELAGE 365
tion for the pressure distribution is obtained, at least near the stagnation point, by
substituting in Eq. (4-65) the actual value at the stagnation point for the factor 2.
Thus, the so-called modified Newton formula is obtained:
cp = op mas Sln2 t.
(5-71)
This relationship is also given in Fig. 5-33, showing very good agreement. It should
be emphasized, however, that Eq. (5-71) is an empirical relationship.
The fuselage in asymmetric incident flow The fuselage in asymmetric incident flow
of supersonic velocity can be treated by means of a dipole distribution on the body
axis, similar to the method presented in Sec. 5-2-3 for incompressible flows. The
adaptation of the dipole distribution of incompressible flow to supersonic flow
follows the rules explained for the axial incident flow. The potential 0 (x, r, t9) of a
line
distribution of three-dimensional dipoles m(x) on the x axis becomes, in
analogy to Eq. (5-20a),
TO
r (Ma' - 1) cos z
m(x') dx'
2,r
*
(x - x')'2 - (Ma' - 1) r23
0
Here, x0 is the influence length from Eq. (5-56) and Fig. 5-26. The expansion of
0 (x, r, 6) for small radial distances r yields
0 (x, r -* 0, t$) =
cos
27r
m (x)
r
(5-73)
in agreement with Eq. (5-20b) for incompressible flow. Consequently, the velocity
components determined from Eq. (5-21) for supersonic flow are identical to those
for incompressible flow. Furthermore, the kinematic flow condition of Eq. (5-22),
and hence the determining equation for the dipole distribution Eq. (5-24), applies
directly to supersonic flows. Finally, it follows that the formula for the pressure
distribution at incompressible flow, Eq. (5-25a), is also valid for any supersonic
Mach number of incident flow. Since it has been found that Eq. (5-55) for the
pressure distribution at subsonic incident flow is identical to Eq. (5-25a), the
remarkable result is obtained that, over the entire Mach number range, the pressure
distribution due to the angle of attack of the fuselage, and the lift distribution, the
lift, and the moment, can be determined from the formulas for incompressible flow.
For instance, the lift of a fuselage, truncated in the rear, at supersonic incident flow
is, from Eq. (5-29a),
LF = 2aq.AFt
(5-74)
where AFt is the cross-sectional area of the fuselage tail.
The sign f signifies, according to Hadamard, that only the finite part of this integral has
to be taken.
366 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
12
dFinax
10
Um
F
09
ios
Measurement,
Figure 5-34 Lift coefficient CLF = LF/
AFrnax4- of a slender body of revolution with blunt tail vs. angle of attack a,
from 153]. Body thickness ratio dFmax/
0.#
0.Z
IF=0.10, Mach number Mac, =1.97,
a
Reynolds number Re = UO,lF/v - 106 ,
linear theory from Eq. (5-74).
e
All computational methods for the lift of fuselages treated so far lead to a
linear dependence of the fuselage lift on the angle of attack. At larger angles of
attack, however, the lift increases more than linearly with angle of attack. As an
example, in Fig. 5-34 the lift coefficient CLF of a slender body of revolution with a
blunt tail is plotted against the angle of attack for Mach number Ma., -- 2. Compare
also Fig. 5-3 for the case of incompressible flow. This lift characteristic much
resembles that of a wing of extremely small aspect ratio (see Sec. 3-3-6). The
nonlinearity is caused by viscosity effects. At larger angles of attack, the flow
separates on the upper and lower surfaces of the fuselage because of cross flow over
the body. Subsequently, the flow rolls up and, as in the case of the flow over the
side edges of a wing of small aspect ratio, free vortices form that are shed from the
body under an angle different from zero (see Fig. 3-S0a). The formation of the
vortex sheet on slender bodies at large angles of attack is sketched in Fig. 5-35 for a
rectangular wing and for a delta wing of small aspect ratio, and for a slender
fuselage. Details of the flow about slender bodies at large angles of attack and the
theoretical determination of the nonlinear lift characteristic are treated in [2, 24,
35, 37, 51 ]
.
For transonic flow about bodies of revolution, generally valid solutions are not
yet available. However, the investigations of Keune and 0swatitsch [25, 271,
Spreiter [441, Fink [12], and Krupp and Murman [311 must be mentioned here.
a,
b
c
Figure 5-35 Vortex formation on wings of small aspect ratio and on slender bodies, leading to
nonlinear lift characteristics. (a) Rectangular wing. (b) Delta wing. (c) Fuselage.
AERODYNAMICS OF THE FUSELAGE 367
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Flugw., 1:2-8, 1953. Sherman, A.: NACA Rept. 575, 1936.
51. van Dyke, M. D.: First- and Second-Otder Theory of Supersonic Flow Past Bodies of
Revolution, J. Aer. Sci., 18:161-178, 216, 1951; J. Fluid Mech., 1:1-15, 1956. Broderick,
J. B.: Quart. J. Mech. App. Math., 2:98-120, 1949. Moore, F. K.: J. Aer. Sci., 17:328-334,
383, 1950.
52. van Dyke, M. D.: The Supersonic Blunt-Body Problem-Review and Extension, J. Aer. Sci.,
25:485-496, 1958. Traugott, S. C.: J. Aerosp. Sci., 27:361-370, 1960.
53. Voellmy, H. R.: Experimentelle Untersuchungen an verschieden stark konvergenten,
schlanken Rotationskorpern bei massig hohen Uberschallgeschwindigkeiten, Mitt. Inst. Aero.
ETH Zurich, Mitt. 24, 1958.
54. von Karman, T.: Berechnung der Druckverteilung an Luftschiffkorpern, Abh. Aer. Inst. TH
Aachen, 6:3-17, 1927; "Collected Works," vol. II, pp. 253-273, Butterworths, London,
1956; NACA TM 574, 1930. Moran, J. P.: J. Fluid Mech., 17:285-304, 1963.
55. von Karman, T. and N. B. Moore: Resistance of Slender Bodies Moving with Supersonic
Velocities, with Special Reference to Projectiles, Trans. Amer. Soc. Mech. Eng., 54:303310,1932; "Collected Works," vol. 11, pp. 376-393, Butterworths, London, 1956. von Kirmin.
T.: Volta-Kongress Rom, 222-276, 1935; "Collected Works," vol. III, pp. 179-221, Butterworths, London, 1956. Stetter, H. J.: Z. Angew. Math. Mech., 37:145-146, 1957.
56. von Karmin, T.: The Similarity Law of Transonic Flow, J. Math. Phys., 24:182-190, 1947;
"Collected Works," vol. IV, pp. 327-335, Butterworths, London, 1956.
57. Ward, G. N.: Supersonic Flow Past Slender Pointed Bodies, Quart. J. Mech. App. PMlath.,
2:75-97, 1949. Berndt, S. B.: Z. Angew. Math. Mech., 35:362, 1955. Fraenkel, L. E.: ARC
RM 2954, 1952/1955. Kahane, A. and A. Solarski: J. Aer. Sci., 20:513-524, 1953.
58. Wieselsberger, C.: Airplane Body (Non Lifting System) Drag and Influence on Lifting
System, in W. F. Durand (ed.), "Aerodynamic Theory-A General Review of Progress," 1935,
div. K, Springer, Berlin, Dover, New York, 1963.
370 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
59. Young, A. D.: The Calculation of the Total and Skin Friction Drag of Bodies of Revolution
at Zero Incidence, ARC RM 1874, 1939. Cebeci, T., G. J. Mosinskis, and A. M. 0. Smith:
J. Aircr., 9:691-692, 1972. Granville, P. S.: Dav. Tay. Mod. Bas. Rept. 849, 1953. Scholz,
N.: Jb. Schiffb., 45:244-263,.1951.
60. Young, A. D.: The Calculation of the Profile Drag of Aerofoils and Bodies of Revolution at
Supersonic Speeds, Jb. WGL, 66-76, 1953; ARC RM 2204, 1945.
CHAPTER
SIX
AERODYNAMICS OF THE WINGFUSELAGE SYSTEM
6-1 INTRODUCTION
6-1-1 General Remarks on the Interactions
among Parts of the Airplane
The aerodynamic coefficients of the major components of the airplane-wing,
fuselage, empennage-are quite well established through theory and systematic
measurements. The aerodynamics of the wing was treated thoroughly in Chaps. 2-4.
The findings established there apply accordingly to the empennage (vertical
stabilizer and rudder, and horizontal stabilizer and elevator; see Chap. 7). The
aerodynamics of the fuselage was the subject of Chap. 5. When these individual
parts are assembled into a complete airplane, however, their interaction (interference) plays a very important role in the formation of aerodynamic forces. In
many cases these interference effects are of the same order of magnitude as the
contributions of the individual parts to the aerodynamic forces of the airplane as a
whole. For this reason, consideration of these interactions is indispensible to the
study of the aerodynamics of the airplane. The physical processes behind the
aerodynamics of the interactions, are, of course. much harder to conceive than
those of the aerodynamics of the individual parts. Consequently, the theoretical
study of the interference problem has been attacked much later and is, even today,
not yet established to the extent of that of the individual parts. The theory of
interference aerodynamics is available to a large extent for inviscid flow only.
371
372 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
Most important of the numerous interference effects among the various airplane
components are the interactions between the wing and the fuselage and between the
wing and the empennage. The interference between the wing and the fuselage is felt
mainly in a changed lift distribution over these parts. The effect of the wing on the
empennage, on the other hand, lies mainly in a changed incident flow direction of
the empennage caused by the induced velocity field of the wing.
A further important interference effect is the so-called ground effect, which is
created during flight near the ground. Hereby for equal lift, the lift slope is
increased and the induced drag is usually reduced. This problem has been treated in
detail in theory and experiment; see the references cited in Sec. 3-3-1.
In this chapter, only the interaction between wing and fuselage will be
investigated. The interference problems related to the empennage will be treated in
Chap. 7 together with the aerodynamics of the empennage.
6-1-2 Geometry of the Wing-Fuselage System
For a better understanding of the aerodynamics of the wing-fuselage system to be
discussed below, the geometry of such a system will be discussed first. The
geometry of the wing has been described in Sec. 3-1 (Figs. 3-1 and 3-2), that of the
fuselage in Sec. 5-1 (Fig. 5-1). The geometry of the wing-fuselage system is
illustrated in Figs. 6-1 and 6-2. Figure 6-1 gives the plan view and the side view of a
wing-fuselage system, Fig. 6-2 the rear view of two wing-fuselage systems. The
position of the wing relative to the fuselage is defined by the wing rearward
position e, the wing high position zo, and the angle of wing setting 60. As shown in
Fig. 6-1, the wing rearward position e is the distance between the geometric neutral
\
Wing chord Fuselage axis
o
If
10
ct
FZ,>/o
\.
V
N25p_
41
e
Figure 6-1 Geometry of a wing fuselage system (side view and plan view).
AERODYNAMICS OF THE WING-FUSELAGE SYSTEM 373
z
7
a!
bFo
- b -2s
Figure 6-2 Geometry of wingfuselage systems (rear view). (a)
High-wing system without dihedral. (b) Mid-wing system
with dihedral.
point of the wing (Sec. 3-1) and the fuselage nose. According to Fig. 6-2a, the wing
high position zo is the distance between the wing and the fuselage axis. Its values
are
High-wing airplanes:
zo > 0
Mid-wing airplanes:
zo = 0
Low-wing airplanes:
zo < 0
A typical mid-wing airplane with dihedral is sketched in Fig. 6-2b. The angle of
wing setting co is, from Fig. 6-1, the angle between the chord of the wing root
section and the fuselage axis. When the wing penetrates the fuselage, the portion of
the wing shrouded by the fuselage requires special explanation. In the case of a
swept-back trapezoidal wing, it is advantageous to replace the portion of the wing
shrouded by the fuselage by a rectangular wing section. This rectangle is formed by
the length of the root section to and by the mean fuselage width in the range of the
wing bFo . For conventional wing-fuselage systems, bFo is almost equal to the
maximum fuselage width bFinax, according to Fig. 5-1. The wing thus defined will
be termed the "substitute wing," whereas the wing from which it has been derived
will be termed the "original wing."
Another important geometric parameter of a wing-fuselage system is the ratio
of fuselage width bFo and wing span b:
rIF
= bFo
b
(relative fuselage width)
6-1-3 Aerodynamic Coefficients
It is advantageous and generally customary to refer the aerodynamic coefficients of
a wing-fuselage system to the geometric quantities of the original wing. A summary
of the aerodynamic coefficients of the wing has been given by Eq. (1-21). These
definitions of the aerodynamic coefficients are applicable directly to the wingfuselage system when the forces and moments of the wing-fuselage system are
374 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
substituted. The reference axes and the signs of forces and moments are shown in
Fig. 1-6.
To
convey a feeling for the magnitude of the interference effects on
wing-fuselage systems, a few test results are given in Figs. 6-3 and 6-4. In Fig. 6-3a
the lift coefficient CL is shown plotted against the angle of attack a for a simple
mid-wing system of a rectangular wing and an axisymmetric fuselage, and for the
wing alone. In the range of moderate angles of attack, the fuselage does not
noticeably affect the trend of the CL(a) curve. The coefficient of maximum lift
CL max, however, is markedly reduced by the fuselage. This can be understood by
realizing that the flow about the wing of a mid-wing airplane is strongly disturbed
by the fuselage, leading to premature flow separation. The lift coefficient CL versus
the pitching moment coefficient cm for the wing alone and the wing-fuselage system
is plotted in Fig. 6-3b. Here the fuselage causes a strong increase in the
pitching-moment slope dcM/dcL. The inclined fuselage alone has a pitching moment
that tends to turn it into a crosswind position (see Sec. 5-2-3), and this pitching
moment obviously is greatly increased by the effect of the wing.
In Fig. 6-4, the rolling-moment coefficient cMX is plotted against the angle of
i
1.2
Figure 6-3 Lift and pitching
moment of a mid-wing system
and of the wing alone, from
Molier and Trienes. Fuselage:
ellipsoid of revolution of axis
ratio 1: 7. Wing: rectangle of
0
aspect
-0,2
- 04
.80
ratio
el = 5,
profile
NACA 23012. (a) Lift coefficient cL vs. angle of attack a.
0°
8°
16
PX ° -0.08
0
-0.04
CM
0.04
0.08
Lift coefficient CL vs.
pitching-moment coefficient cM.
(b)
AERODYNAMICS OF THE WING-FUSELAGE SYSTEM 375
0.
0.0
Wing + fus elage
0.0.
Wing
0.0
ac'B,5°
- 0.0
-0.0
0"'"300
Figure 6-4 Rolling moment due to side-20°
-10
0
P_
sideslip
70
.10°
slip of a high-wing system and of the
wing alone, from M6ller; fuselage and
wing of Fig. 6-3.
for a high-wing system also consisting of a rectangular wing and an
axisymmetric fuselage. The difference between the trends of the curves cMx(3) for
the wing alone and for the wing-fuselage system is quite large. The effect of the
fuselage of a high-wing airplane consists of a strong increase in the rolling moment
due to sideslip acm,/aa. This effect is caused by the cross flow over the fuselage.
The interference effects shown in Figs. 6-3 and 6-4 can be treated theoretically.
Other interference problems, particularly those of the drag of wing-fuselage systems,
are hardly accessible to theoretical determinations. Therefore, in these cases
experimental studies are indispensible [11, 15].
Summary reports about the interactions between the wing and fuselage in
incompressible, and to some extent in compressible flow, have been published by
Wieselsberger [511, Muttray [34], Schlichting [39, 41], Ferrari [6], and Lawrence
and Flax [26], as well as Ashley and Rodden [2]. Surveys of the aerodynamics
of slender bodies have been given by Adams and Sears [1] and Gersten
[7].
376 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
6-2 THE WING-FUSELAGE SYSTEM
IN INCOMPRESSIBLE FLOW
6-2-1 Fluid Mechanical Fundamentals of the
Wing-Fuselage Interference
In the next section the quantitative computation of the interactions between wing
and fuselage will be discussed, but a few physical explanations will be given here
first. When putting together a wing and a fuselage, a flow about the wing-fuselage
system results, with the fuselage lying in the flow field of the wing and the wing in
the flow field of the fuselage. Thus, an aerodynamic interference exists between the
fuselage and the wing, in that the presence of the fuselage changes the flow about
the wing and the presence of the wing changes the flow about the fuselage.
Consequently, the computation of the flow about a wing-fuselage system can be
accomplished by first computing the flows about the wing and fuselage separately,
and then adding the interference effects of the wing on the fuselage and the
fuselage on the wing. These interference effects are obtained by satisfying the
kinematic flow condition (zero normal component of the velocity on the surface of
the wing-fuselage system).
The, flow field of a wing-fuselage system at subsonic velocity in symmetric,
incident flow (angle of sideslip 13 = 0) is illustrated in Fig. 6-5. Figure 6-5a shows
the flow about the fuselage as affected by the wing. Along the fuselage axis,
additive velocities normal to the fuselage axis are induced by the wing, which are
directed upward before the wing and downward behind it. In the range of the
wing-fuselage penetration, the flow is parallel to the wing chord, corresponding to a
constant downwash velocity along the wing chord. The fuselage is therefore in a
curved flow with an angle-of-attack distribution a(x) varying along the fuselage axis
as shown in Fig. 6-5a. This angle-of-attack distribution, induced by the
wing, shows that the fuselage is subjected to an additive nose-up pitching moment.
The effect of the fuselage on the flow about the wing is sketched in Fig. 6-5b.
The component of the incident flow velocity normal to the fuselage axis
U. sin a. x
generates additive upwash velocities in the vicinity of the
fuselage. The effect on the wing of these induced velocities normal to the plane of
the wing is equivalent to an additive symmetric angle-of-attack distribution over the
wing span (twist angle).
The flow field of a wing-fuselage system at asymmetric incident flow is shown
schematically in Fig. 6-6. The flow about the wing-fuselage system with the angle of
can be thought to be divided into an incident flow parallel to the plane of
symmetry of the velocity U. cos a U. and an incident flow normal to the plane
of symmetry of the velocity U. sin 03 U43. The latter component of the incident
flow generates a cross flow over the fuselage as illustrated in Fig. 6-6b, c, and d for
a high-wing, a mid-wing, and a low-wing system, respectively. This cross flow over
the fuselage results in an additive antimetric* distribution of the normal velocities
along the span that is equivalent to an antimetric angle-of-attack distribution a(y).
sideslip
*See footnote on page 190.
AERODYNAMICS OF THE WING-FUSELAGE SYSTEM 377
Upwash
Downwash
1
X
b
y
Figure 6-5 Symmetric flow about a wing-fuselage system (schematic). (a) Flow in the airplane
plane of symmetry and angle-of-attack distribution «(x) on the fuselage axis. (b) Flow in a
plane normal to the fuselage axis and angle-of-attack distribution cx(y) over the wing span.
The
lift distributions
over
the wing span generated by this angle-of-attack
distribution have reversed signs for high-wing and low-wing airplanes. The rolling
moment (rolling moment due to sideslip), as affected by this antimetric lift
distribution, is zero for the mid-wing airplane, positive for the high-wing airplane,
and negative for the low-wing airplane. These findings are confirmed by the test
results of Fig. 6-4, which show that the rolling moment due to sideslip Zc,L IaQ of
a high-wing airplane is larger than for the wing alone.
The effect of the fuselage on the wing in yawing motion may be interpreted,
therefore, as the effect of a positive dihedral of the wing on the high-wing airplane,
and as that of a negative dihedral on the low-wing airplane.
378 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE
SYSTEM
b
Figure
d
over
a
6-6 Asymmetric flow
wing-fuselage system
(schematic). (a) Wing planform.
(b)
High-wing
airplane
with
angle-of-attack distribution a(y).
(c) Mid-wing airplane. (d) Lowwing
airplane
with
attack distribution a(y).
angle-of-
AERODYNAMICS OF THE WING-FUSELAGE SYSTEM 379
6-2-2 The Wing-Fuselage System in Symmetric
Incident Flow
Total lift of a wing-fuselage system The first attempt at a theoretical description of
the interference of a wing-fuselage system was made by Lennertz [27]. First, only
the lift distribution on the wing and fuselage of such a system will be investigated.
For simplicity, let the fuselage be an infinitely long circular cylinder as shown in
Fig. 6-7, whereas the unswept wing has an infinite span. For the portion of the
wing not shrouded by the fuselage, let the lift distribution over the span be known
and thus the circulation distribution r(y). The vortex system of the wing can be
composed, from Fig. 3-20a, of horseshoe vortices of width dy and vortex strength
T, as shown in Fig. 6-7b. To determine the lift of this arrangement generated at the
fuselage, the kinematic flow condition must be satisfied on the fuselage surface,
thus making the fuselage surface a stream surface. In a cross section normal to the
fuselage axis far behind the wing, the flow in the yz plane is two-dimensional. The
kinematic flow condition can here be satisfied by means of the reflection principle;
that is, for every free vortex outside of the fuselage, a vortex reflected with respect
to a circle has to be placed into the fuselage that has the same vortex strength but
y
Figure 6-7 Determination of the total
lift
of a
wing-fuselage system. (a)
Rear view. (b) Plan view with vortex
system. (c) Circulation distribution in
span direction.
380 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
the opposite sense of direction of rotation. The reflected vortex belonging to the
free vortex at station y is located at a distance YF = R2ly from the fuselage axis,
where R is the radius of the fuselage cross section.* Thus, a circulation distribution
is obtained on the fuselage as demonstrated in Fig. 6-7c.
The lift of the wing portion not shrouded by the fuselage L!, is obtained
through integration of the circulation distribution over the span from Eq. (3-15) as
L w= 2 e U... f r(y) d y
(6-2)
y=R
Analogously, the lift of the fuselage becomes, with dyF = -(R2 /y2) dy and
r(yF) = r (y) for the bound vortex,
S
R
r'' dyF 2Q Ucc f r(Y)
LF = 2,o U f r (YF)
=
yF'-R 1S
y=R
dy
y2
(6-3)
The total lift of the wing-fuselage system follows from Eqs. (6-2) and (6-3) as
S
L(w+F) =Lw +LF= 2eUo, rr(Y) 1+ Rz) du
y
.1
y=R
(6-4)
For numerical evaluation of this equation, an assumption must be made about
the circulation distribution r(y). The simplest case is a constant circulation
distribution T (y) = To = const. Here, Eqs. (6-2) and (6-3) yield for the ratio of
fuselage lift to wing lift and for the ratio of fuselage lift to total lift:
LF
W
RS
1P,
LF
?7F
L(W+F)
1 + r1F
(6-5)
The latter ratio is presented in Fig. 6-8 versus the relative fuselage width 17F = R/s
as curve 1. Lawrence and Flax [26] and Luckert [32] have shown that curve 1 of
Fig. 6-8 may also be applied, in very good approximation, to different lift
distributions. Curve 2 of Fig. 6-8, from Spreiter [44], applies to wings of small
aspect ratio (cf. Sec. 6-4). The result of Eq. (6-3) for the lift of fuselages may also
be obtained from the integral of the pressure over the body surface or by means of
the momentum theorem.
The above considerations fail to give information about the distribution of the
lift of the fuselage over its length. This problem will be treated in the following
section.
Lift distribution of the fuselage To determine the lift distribution over the fuselage
length under the influence of the wing, the corresponding considerations for the
fuselage alone of Sec. 5-2-3 may be applied. It was shown there that the lift
*It can easily be seen that the flow pattern of the two counter-rotating vortices at y and
YF and at (y + dy) and (yF + dyF), respectively, contains, as a streamline, the circle of radius R
about the origin.
AERODYNAMICS OF THE WING-FUSELAGE SYSTEM 381
01
--0,3
0.2
77F =
R
0.4
0.5
Figure 6-8 Ratio of the fuselage lift Lp to
the total lift of a. wing-fuselage system
L(yy+F) vs. relative fuselage width 77F =
R/s. Curve 1, theory from Lennertz (r =
const). Curve 2, theory from Spreiter
(slender-body theory).
distribution over the fuselage length for a fuselage as shown in Fig. 6-9 is given by
Eq. (5-28) as
clxF
= 4'00 y d [,x(x)bF(x))
(6-6)
Here dLF is the lift force of a fuselage section of length dx, bF(x) is the local
fuselage width, a(x) is the local angle of attack of the fuselage axis, and
q. = oUU/2 is the dynamic pressure of the incident flow. To compute the lift
distribution of the fuselage alone, the angle of attack in this equation has to be
S
Figure 6-9 The lift distribution of an
IF
inclined fuselage.
382 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
taken as a(x) = a = const. For the lift distribution over the fuselage under the
effect of the wing, the angle of attack has to be expressed as
ti(x) = aoo 1i' aw(x)
(6-7)
where ax,(x) represents the upwash and downwash angles induced by the wing at
the location of the fuselage (see Fig. 6-5a). For bF(O) = 0 = bF(IF), the total lift of
the fuselage under the influence of the wing is obtained from Eq. (6-6), in
agreement with Eq. (5-29a), as LF = 0. As was shown in Sec. 5-2-3, this relationship
is valid for inviscid flows.
To compute the pitching moment at a variable angle-of-attack distribution a(x),
Eq. (5-32) is already available. This pitching moment is independent of the position
of the reference axis because it
is
a free moment. The above method for the
computation of the wing-fuselage interference was developed by Multhopp [32].
The computation of the lift distribution over the fuselage length from Eq. (6-6) and
of the pitching moment from Eq. (5-32) requires the determination of the
distribution along the fuselage axis of the angle of attack
induced by the
wing. This is a problem of wing theory that has already been treated in Sec. 24-5
for the two-dimensional case and basically in Sec. 3-2 for the three-dimensional case. A
comprehensive presentation of the computational procedures for the induced
velocity fields of wings will be given in Chap. 7.
The fundamentals of the method for the computation of the lift distribution
and of the pitching moment can be understood from the simple case of a
wing-fuselage system with a wing of infinite span, as shown in Fig. 6-10. The
induced angle of attack of the inclined flat plate is given by Eq. (2-116) with
A0 = a and A = 0 for n > 1 [see also Eq. (2-66)]. Hence, Eq. (6-7) yields, for
the local angle of attack,
a(X) = a V XA,-1
for X>1 and X < 0
(6-8a)
where X = x/c is the dimensionless distance from the plate leading edge. This
distribution is shown in Fig. 6-l Ob. Within the range of the wing, 0 < X < 1, there
is a,(x) = -a and thus
a(X) = 0
for 0 < X < 1
(6-8b)
The local angle of attack a(x) from Eqs. (6-8a) and (6-8b) is discontinuous at
the wing leading edge: The quantity a(x) drops abruptly from an infinitely large
positive value to zero. At this station, daldx has an infinitely large negative value,
requiring special attention when determining the lift distribution from Eq. (6-6).
For clearness in the computation of the lift distribution, the discontinuity of the
a(x) curve has been drawn in Fig. 6-10b as a steep but finite slope. With the local
angle-of-attack change thus established, the lift distribution of Fig. 6-10c is
obtained.* It has a large negative contribution in the form of a pronounced peak
For a blunt fuselage nose and tail, Eq. (6--6) gives finite values for dLF/dx, contrary to
the exact values dLF/dx = 0.
AERODYNAMICS OF THE WING-FUSELAGE SYSTEM 383
Figure 6-10 Computation of
the lift distribution on the
fuselage of a wing-fuselage
system. (a) Geometry of the
wing-fuselage system. (b) Angle-
of-attack distribution a(x). (c)
Lift distribution dLF/dx.
directly before the wing leading edge. This is caused by the large negative value of
da/dx close to the wing nose. The magnitude of this negative contribution is easily
found when one realizes that for the fuselage section from the fuselage nose to a
station shortly behind the wing leading edge, the lift force must be zero according
to Eq. (5-29a), because bF = 0 at the fuselage nose and a = 0 shortly behind the
wing leading edge. Accordingly, the positive contribution LFI and the negative
contribution LF2 are equal.
On the other hand, the lift distribution of the wing alone (without fuselage
interference) has a strongly pronounced positive peak in the vicinity of the wing
leading edge. Actually, this positive lift peak of the wing is reduced by the negative
lift peak of the fuselage LF2 mentioned above. Hence, a lift distribution over the
fuselage is obtained, including the shrouded wing area, given as the solid curve of
Fig. 6-10c.
Finally, this analysis shows that the total lift of the fuselage in the
wing-fuselage system is approximately equal to the lift of the shrouded wing
portion.
An example of this computational procedure and a comparison with measure-
ments is given in Fig. 6-11. The fuselage is an ellipsoid of revolution of axis
ratio I : 7 that is combined with a rectangular wing of aspect ratio A = 5 in a
mid-wing arrangement. Curve 1 shows the theoretical lift distribution from Eq.
(6-6). It is in quite good agreement with the measurements in the ranges before and
384 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
o
r
r
2
3
o
0
2
1
-Z
i
-1
3
4
Figure 6-11 Lift distribution on the
fuselage of a wing-fuselage system
(mid-wing airplane). Fuselage: ellipsoid of revolution of axis ratio 1 : 7.
Wing: rectangle of aspect ratio A =
5. Measurements from [41]; theory:
curve 1 from Multhopp, curve 2 from
Lawrence and Flax, curve 3 from
IF
curve 2, from Adams and Sears.
behind the wing. No result is obtained by this computational procedure within the
range of the wing. The measured lift distribution shows a pronounced maximum in
the vicinity of the wing leading edge. Curve 2 represents' an approximation theory
of Lawrence and Flax [26], which will be discussed later; it is in satisfactory
agreement with the measurements in the range of the wing. Curve 3 will also be
explained later.
The -influence - of the wing - shape -- on- the - wing-11 selage - interference can - be
assessed best by means of the angle-of-attack distribution induced on the fuselage
axis. For unswept wings, Fig. 6-12 illustrates the effect of the aspect ratio on the
distribution of the angle of attack. All the wings have an elliptic planform. The
angle-of-attack distribution has been computed using the lifting-line theory. For an
elliptic circulation distribution its value becomes, Eq. (3-97),
= x/s and the coordinate origin x = 0 lies on the c/4 line. Because
= 8X/irf with X = x/l, and with the relationship between CL and a. of Eq.
where
(3-98), Eqs. (6-9) and (6-7) yield
AERODYNAMICS OF THE WING-FUSELAGE SYSTEM 385
(A)2
a (x)
a
=1--
4
+ Ya _- X
2 a+
X
2
(6-10)
1+
In Fig. 6-12, a/a. is shown versus X.* Hence, in the range before the wing, the
upwash angles become markedly smaller when the aspect ratio A is reduced. In the
range behind the wing, however, the downwash angles increase with decreasing
aspect ratio. At the 4 c point, all curves have the value a = 0, as should be expected
because of the computational method used (extended lifting-line theory = threequarter-point method).
The effect of the sweepback angle on the distribution of the angle of attack is
shown in Fig. 6-13 for a wing of infinite span, constant chord, and unswept middle
section. This latter section represents the shrouding of the wing by the fuselage as
shown in Fig. 6-1. The induced angle-of-attack distribution on the x axis is obtained
from the lifting-line theory according to Biot-Savart as
aw (x) = -
r
with
U.c
r
x-}
xa+gb,sin9
2vU.. x xcosrp+sF.sin T
= 7 acc cos
(6-1 la)
(6-11b)
where r is the circulation of the lifting line, cp is the sweepback angle, and SF is the
semiwidth of the unswept middle section. The relationship between the circulation
*For this illustration, the coordinate origin has been laid on the leading edge.
tHere, the coordinate origin lies at the c/4 point of the root section.
9.
16
10,
-2
-1
0
1
2
Figure 6-12 Distribution of the local angle of attack on the fuselage axis of wing-fuselage systems
with wings of several aspect ratios A.
386 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
3
I
I
r
r
SF
14
D
00
ooo.ol
Q
Z
-1
X
2
3
C
Figure 6-13 Distribution of the local angle of attack on the fuselage axis of wing-fuselage systems
with swept back wings of infinite span and with rectangular middle portion (lifting-line theory).
Solid curves, of = sp/c = 0. Dashed curves, ep = 0.5.
r and the angle of attack ate, of a swept-back wing of infinite span is expressed by
Eq. (6-1 lb), because
cL = 2r/ U.c and cL = 27ra. cos p from Eq. (3-123).
Consequently, Eq. (6-1 la) maybe written in the form
a (X)
cos 97 X + XZ + QF sin 97
a00
2X X cos g)+ aF sin T
(6-12)
with X = x/c and QF = sF/c. The angle-of-attack distributions computed by this
equation are plotted in Fig. 6-13 for sweepback angles cp = 0, +45, and -450, and
for (Yp = 0 and 0.5.* From Fig. 6-13 it can be seen that the upwash before the
wing is reduced in the case of a backward-swept wing and the downwash behind the
wing is increased. In the case of a forward-swept wing, the reverse occurs. As would
be expected, introduction of the rectangular middle section reduces the effect of
sweepback. The distribution of the induced angle of attack on the fuselage axis for
the swept-back wing without a rectangular middle section (sF = 0) is given, from
Eq. (6-1 la), as
() _ - 2' U0,rx cos 9
06W x
1
'-x
'
sin
(6-13)
I xi
Since aw = -T/27rU.x for the unswept wing, Eq. (6-13) shows that the effect of
the sweepback angle on the induced downwash angle may be expressed by a factor.
The procedure discussed so far for the determination of the wing influence on
the angle-of attack distribution of the fuselage does not give any information about
*Compare the footnote on page 385.
AERODYNAMICS OF THE WING-FUSELAGE SYSTEM 387
the distribution in the range of the wing, as may be seen from Fig. 6-11. Lawrence
and Flax [26] developed a method allowing determination of the angle-of-attack
distribution over the entire fuselage length, including the shrouded wing section.
The basic concept of this method is indicated in Fig. 6-14. Contrary to the previous
approaches, which were based on an undivided wing, now the fuselage is taken as
being undivided and the wing as divided. Consequently, the effect of the two partial
wings on the fuselage is determined, whereby both the x component and the z
component of the induced velocity must be taken into account.
The first contribution to the lift distribution is generated by the longitudinal
velocity components u(x) because they determine the pressure distribution on the
fuselage surface by cp = -2u/U.. The induced velocities on the surface z = R cos 6
can be expressed by
_ z (z-c) = UUR cos 6 do
u = z (a az
-u z=0
dx
8x z=o
1
Here it has been taken into consideration that au/az = 8w/ax, because the flow is
irrotational, and further that the simple relationship daw/dx = da/dx follows from
Eq. (6-7). The second contribution to the lift distribution is generated by the
b -2s
I
U-
9
Figure 6-14 Computation of the lift distribution on the fuselage of a wing-fuselage system
according to the theory of Lawrence and Flax.
388 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
upwash velocities on the fuselage axis resulting from the vortex system of the two
wing parts. The corresponding pressure distribution is obtained from Eq. (5-25a).
Thus, the resulting pressure distribution on the fuselage is
C (x, $) _ -4 cos 6 dx [a(x)R(x)]
Introduction of this expression into Eq. (5-27) and integration over 0 <
(6-14)
< 27r
yield the total lift distribution
dd F = 4 1rq R (x)
dx [a(x)R(x) ]
(6-15)
Note the difference from Eq. (5-28). For the case a = const (fuselage alone), the
equations are identical. Lawrence and Flax [26] have evaluated Eq. (6-15) assuming
that the circulation distribution is constant on either wing part. This result is given
as curve 2 of Fig. 6-11. For the fuselage portions before the' wing and within the
wing range, agreement of this approximation theory with measurement is quite
good. For the fuselage portion behind the wing, the deviations from measurement
are considerable. Therefore, a correction for this range has been given by Adams
and Sears [1], shown as curve 3. It should be mentioned in this connection that the
computational procedure of Multhopp [32] leads to nearly the same results.
Lift distribution of the wing Since the effect of the wing on the fuselage has been
discussed, the effect of the fuselage on the lift generation on the wing will now be
investigated more closely. A typical test result on this problem is shown in Fig.
6-15. For a mid-wing system consisting of a rectangular wing and an axisymmetric
5.6°
0.
f
1.
CC
71.4'
-7y
I
I
0
0.2
0.6
0.4
7-3
---Wing alone
0.8
-Mid-wing airplane
1.0
Figure 6-15 Measured lift distributions on
the span for a mid wing system and for the
wing alone at several angles of attack, from
[41]. Fuselage: ellipsoid of revolution of
axis ratio 1:7. Wing: rectangle of aspect
ratio A = 5. The curves for the mid-wing
airplane include only the fuselage lift within
the wing range.
AERODYNAMICS OF THE WING-FUSELAGE SYSTEM 389
Figure 6-16 Induced angle-ofattack distribution of a wing-fuse-
lage system. The fuselage is an
infinitely long circular cylinder;
a
R = co /2. Curve 1, angle-of-attack
distribution a(x) = a,,. = const
over the entire fuselage length.
Curves 2 and 3, angle-of-attack
distributions a(x) before and behind
-0
We-
the wing are constant,
a(x) = 0 within the wing range.
Curve 2 for the upswept wing,
curve 3 for the swept-back wing
= 45°. Curves 2 and 3 give the
distribution of the induced angle
of attack on the a -point line of
the wing.
fuselage, and for the wing alone, distributions of the local lift coefficients over the
span are shown. These data have been extracted from comprehensive pressure
distribution measurements of Moller [15] on wing-fuselage systems. In the case of
the wing-fuselage system, the lift coefficients refer to the wing portion shrouded by
the fuselage. The lift distributions on the wing portions outside of the fuselage at
three different angles of attack are consistently little affected by the fuselage.
However, within the fuselage range, a considerable drop in the lift coefficient
occurs. This reduced wing lift within the fuselage range has previously been
discussed in connection with Fig. 6-10c.
For the theoretical determination of the influence of the fuselage on the lift
distribution of the wing, the additive angle-of-attack distribution from Fig. 6-5b has
to be determined that is the result of the cross flow over the fuselage. Figure 6-16
shows as curve 1 the additive angle-of-attack distribution induced by an infinitely
long fuselage of circular cross section. Outside the fuselage, the induced angle of
attack J a = w/U for mid-wing systems is given by
da(y)
a00
=
R2
y
(y
>R)
(6-16a)
where R is the radius of the circular cylinder. For the range -R <y < +R, J a is
determined from the velocity component in the z direction on the fuselage surface,
resulting in
390 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
da
a = - (1 -2)
(0 <y <R)
(6-16b)
The angle-of-attack distribution thus determined has a very sharp peak of
d a/a = +1 on the fuselage side wall, whereas the value d a/a = -1 is reached on
the fuselage axis, that is, the local angle of attack a = a + d a = 0 on the axis. By
using this angle-of-attack distribution of the wing according to curve 1 of Fig. 6-16,
the fuselage influence on the wing is greatly overrated because it is based on the
assumption that the angle of attack of the fuselage is a = a within the wing range,
too. Multhopp [32] computed lift distributions with angle-of-attack distributions of
this kind. Compare also Liess and Riegels [32] and Vandrey [47].
A better approximation for the fuselage influence on the wing is obtained
under the assumption that the wing turns the flow within the wing range parallel to
the fuselage axis, that is, that a = 0 in this range. The corresponding distribution of
the induced angles of attack over the span can be determined by arranging a dipole
distribution on the fuselage axis that is dependent on x. This procedure has been
given for the fuselage alone in Sec. 5-2-3. With r cos 6 = z and r2 = y2 + z2 and
with m from Eq. (5-24), Eq. (5-20a) yields for 4 a = w/U = (aq/az)/U. in the
wing plane z = 0,
IF
a(x') R2(x')
1
da(x, y) =
(x -x')2 -1-y2 a
2
dx'
(6-17)
0
valid for y >R. For an infinitely long fuselage of constant width whose angle of
attack is constant before and behind the wing and zero (a = 0) within the wing
range, the result is
4a (x, y)
MOO
- 1 R2 - l4 - x 2 y2
0u - x)3 + y3
2
x
11x2 +. y3
(6-18)
Here, l0 is the wing chord at the fuselage side wall. The distribution of the induced
angle of attack, computed with Eq. (6-18), is shown in Fig. 6-16 as curves 2 and 3
for an unswept wing and for a swept-back wing with p = 450, respectively. The
computed values are valid for the a c point of the wing. Comparison of curves 2 and
3 with curve 1 demonstrates that this refined computational method leads to a
considerably smaller fuselage influence.
Neutral-point position of wing-fuselage systems Besides the changes of the lift
distributions of fuselage and wing, the change of the neutral-point position is of
particular importance for flight mechanical applications (see Sec. 1-3-3). The
distance of the neutral point from the moment reference axis is generally given by
xN = -dM/dL. Hence, for the wing-fuselage system it becomes
XN
M(W+F)
(6-19a)
(W +F)
dMw
dLw
dMF
dLw
(6-19b)
AERODYNAMICS OF THE WING-FUSELAGE SYSTEM 391
where M(W+F) is the pitching moment and L(W+F) is the total lift of the
wing-fuselage system. The pitching moment of the wing-fuselage system may be
composed of the contributions of the fuselage MF and of the wing MW. The
fuselage contribution can be computed as described previously. The wing contribu-
tion will be taken to be the moment of a wing with rectangular middle section
(substitute wing). Since the fuselage influence on the wing is generally small, it can
often be disregarded (see Hafer [11]). The lift of the wing-fuselage system L(w+F)
given approximately by the lift of the wing alone Lw, as was shown earlier.
Because M(W +F) = MW + MF and L(w+F) LF, Eq. (6-19b) is obtained. The first
term gives the neutral-point position of the wing with rectangular middle section,
which can be determined through computation of the lift distribution of such a
wing according to the lifting-surface method. The second contribution gives the
neutral-point displacement caused by the fuselage including the influence of the
is
wing on the fuselage.
It is advantageous to refer the neutral-point position of the wing-fuselage
system to the position of the neutral point of the wing alone, that is, of the original
wing (Fig. 6-1). As reference chord, that of the original wing is chosen likewise. The
neutral-point displacement of the wing-fuselage system from the aerodynamic
neutral point of the wing alone becomes, from Eq. (6-19b),
(A XN)(W+F)
_ (A 4N')W +
(A XN)F
CA
CA,
CA
(6-20)
Here (A xS)W is the neutral-point displacement because of the planform change of
the wing (introduction of the rectangular middle section into the range shrouded by
the fuselage) and (A xN)F is the neutral-point displacement because of the fuselage.
Obviously, the first contribution can be of real importance for only swept-back and
delta wings. By considering, as a first approximation, the displacement of the
geometric neutral point only, the neutral-point displacement of the swept-back wing
of constant chord becomes
(A x125)W
4
C
!1'ij
tan rp
(6-21)
with 'qF as the relative fuselage width from Eq. (6-1).
The second contribution in Eq. (6-20), that is, the neutral-point displacement
due to the fuselage, is obtained from the fuselage moment MF by the relationship
(AXN)F
CA
_
-
1
1 dMr dam
Acu qro daa; dCL
(6-22)
where dcL/dca is the lift slope of the wing (see Sec. 3-5-2).
The neutral-point displacement caused by the fuselage of Eq. (6-22) depends
mainly on the following geometric parameters, as intuitively plausible: wing rearward
position, fuselage width ratio, and sweepback angle. In Figs. 6-17-6-19, a few
computational results from Hafer [11] on the influence of these parameters are
presented and compared with measurements.
The neutral-point displacement due to the wing rearward position for an
392 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
-0.16
4
ti
0.08
7
o Mid-wing
0 High-wing
A Low-wing
of
02
0-7
airplane
0,5
0.4
0.5
0.7
e
IF
Figure 6-17 Neutral-point shift of wing-fuselage systems due to the fuselage effect vs. the wing
rearward position, from Hafer. Fuselage: ellipsoid of revolution of axis ratio 1: 7. Wing: rectangle
of aspect ratio A = 5.
unswept wing is given in Fig. 6-17 as a function of the widely varied wing rearward
position. The fuselage causes an upstream displacement of the neutral point
(destabilizing fuselage effect) that increases with the rearward wing position. The
wing high position, also varied in these measurements, has no marked effect.Agreement between theory and experiments is good.
Figure 6-18 illustrates the effect of the sweepback angle on the neutral-point
-0,
-0.16
1-0,08
Theory
k
A=0.2
(
0
0.04
0
V
01i9
_Z L'
-10°
0°
10u
k=0.2
Zllu
.V°
40 q
9
F
Figure 6-18 Neutral-point shift of wing-fuselage systems due to the fuselage effect vs. sweepback
angle of the wing, from Hafer. Fuselage: ellipsoid of revolution of axis ratio 1: 7. Wing: aspect ratio
A = 5; taper a = 1.0 and 0.2.
AERODYNAMICS OF THE WING-FUSELAGE SYSTEM 393
Figure 6-19 Neutral-point shift of
wing-fuselage systems due to the
fuselage effect vs. wing rearward
position, from Hafer. (I) Sweptback wing; A = 2.75; X = 0.5; p =
50°. (II) Delta wing: A = 2.33;
X=0.125. Curve 1, fuselage with
pointed nose. Curve 2, fuselage
611F
with rounded nose.
position caused by the fuselage. The measurements are for wing-fuselage systems
with wings of constant chord (A = 1) and with trapezoidal wings (A = 0.2). The
neutral-point displacement becomes markedly smaller when the sweepback angle
increases. It is noteworthy that the neutral-point displacement is almost zero for
strong sweepback (gypcz:l 45°). Here, too, agreement between theory and measurement
is quite good. The first theoretical studies about the effect of the sweepback angle
on the neutral-point displacement
caused by the fuselage was conducted by
Schlichting [401.
Finally, in Fig. 6-19, results are given on the influence of the wing rearward
position of a swept-back wing and a delta wing. The swept-back wing has the aspect
ratio A = 2.75, the taper A = 0.5, and the sweepback angle of the quarter-point line
cp = 50°. The neutral-point position of this wing has been shown in Fig. 3-37b. The
delta wing has the aspect ratio A = 2.33 and the taper A = 0.125. The results of
Fig. 6-19 are given for two different fuselage shapes, namely, a pointed and a
rounded fuselage front portion. For either wing, in agreement with Fig. 6-17, a
considerable increase in the neutral-point displacement is caused by the fuselage
when the wing is moved rearward. Here, too, agreement between theory and
measurement is good. Important contributions to the interference between a
swept-back wing and a fuselage are also due to Kuchemann [24].
Drag and maximum lift of wing-fuselage systems The interference effect of
wind fuselage systems on drag and maximum lift lies mainly in the altered
separation behavior when wing and fuselage are put together. These effects are
hardly accessible to theoretical treatment, however, and their study must be limited
to experimental approaches. The first summary report hereof comes from Muttray
[34] ; compare also Schlichting [38]. Very comprehensive experimental investiga-
tions on the interaction of wing and fuselage, particularly concerning the drag
394 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
problem, have been conducted by Jacobs and Ward [15] and by Sherman [151.
For drag and maximum lift of a wing-fuselage system, the low-wing arrangement is particularly sensitive, because the fuselage lies on the suction side of the
wing, strongly influencing the onset of separation at larger lift coefficients. Through
careful shaping of the wing-fuselage interface by means of so-called wing-root
fairings, the flow can be favorably affected in this case, that is, the onset of
separation can be shifted to larger angles of attack.
The investigations of Jacobs and Ward [15] and of Sherman [15] cover a
comprehensive program on two different 'fuselages (circular and rectangular cross
sections) and two wings of different profiles (symmetric and cambered). Varied
were the wing rearward position, the wing high position, and the wing setting angle.
Included in the study was the effect of wing-root fairings.
The drag of a wing-fuselage system depends predominantly on the wing high
position, and very little on its rearward position and its setting angle. In Fig. 6-20,
the lift coefficient CL is plotted against the coefficient of the form drag
2
CL
_
CDe -
CD
-
(6-23)
of several wing-fuselage systems. The coefficient of the form drag is obtained as the
difference of the coefficients of total drag and induced drag. These wing-fuselage
systems are a mid-wing airplane with round fuselage and low-wing airplanes with
round and square fuselages. For comparison, the wing alone is added as curve 1. A
strong drag increase above a certain lift coefficient is characteristic for wing-fuselage
systems. It is the result of the onset of separation caused by the fuselage. This
7
1.
2
I
i
--
-
4
I
0.2
0
Figure 6-20 Lift coefficients of wing-fuselage
systems vs. drag coefficients, from Jacobs and
j
-0.2
-04
0
Ward. CDe = coefficient of the form drag
I
0.02
0.04
I
0.06
cDe
I
0.06
from Eq. (6-23). Fuselages with circular and
i
0.10
0.12
0.14
square cross sections, wing profile NACA
0012.
AERODYNAMICS OF THE WING-FUSELAGE SYSTEM 395
1.e
Wing
L-7-
I
___L - Wing + fuselage
1,2
0,4
1,6
.o
Wing
1
_
X
M
E
J
1
T
4
Q
ge
Mid-wing airplane
t
-Q8
-0.4
0
04
0.8
L
Figure 6-21 Maximum lift coefficients
of wing-fuselage systems, from [38).
Fuselages with circular cross sections,
wing profile NACA 0012. (a) Maximum
lift coefficient vs. wing rearward position, zo /10 = 0. (b) Maximum lift coefficient vs. wing high position, e0 /10 = 0.
phenomenon is most pronounced in the low-wing system with round fuselage, curve 3,
where separation begins very early at CL = 0.6. Here fuselage side wall and wing upper
surface form an acute angle that particularly promotes boundary-layer separation.
Considerably more favorable than the low-wing airplane is the mid-wing airplane,
curve 2, because here the wing is attached to the fuselage at a right angle. By going
from a round to a square fuselage, the conditions may be further improved, as
shown by curve 4 for the low-wing airplane.
Theoretical results on the pressure distribution at the wing-fuselage interface are
given by Liese and Vandrey [47] for the case of a symmetric wing-fuselage system
(mid-wing) in symmetric incident flow (CL = 0).
The maximum lift of wing-fuselage systems depends on both the wing high
position and the wing rearward position. A survey of the CLmax values for several
high and rearward positions is given in Fig. 6-21. From Fig. 6-21a, the maximum
lift coefficient CLmax decreases with increasing rearward position. In the most
favorable case, CLmax of a wing-fuselage system is equal to that of the wing alone.
With regard to the wing high position, the mid-wing arrangement is least favorable,
as shown by Fig. 6-21b (compare also Fig. 6-3). From this value for the mid-wing
arrangement, CLmax increases when the wing is shifted to both high- and low-wing
positions.
396 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
6-2-3 The Wind Fuselage System in Asymmetric
Incident Flow
Rolling moment due to sideslip of a wing-fuselage system In asymmetric incident
flow of a wing-fuselage system, the lateral component of the flow about the
fuselage creates an additive antimetric distribution of the angle of attack of the
wing as discussed in Sec. 6-2-1 and demonstrated in Fig. 6-6. It has reversed signs
for high-wing and low-wing airplanes, and it is zero for mid-wing airplanes. This
antimetric angle-of-attack distribution generates an antimetric lift distribution at the
wing and thus a rolling moment due to sideslip. This additive rolling moment due to
sideslip caused by the fuselage also has reversed signs for high-wing and low-wing
airplanes.
For a theoretical assessment of the influence of the fuselage on the lift
distribution of the wing, the antimetric angle-of-attack distribution as shown in Fig.
6-6 must be determined as caused by the cross flow over the fuselage with velocity
U,. sin 0 ~ U43. This angle-of-attack distribution d a = w/U for an infinitely long
fuselage with circular cross section (radius R) becomes
= - 2R2
(Y >yo)
Z3)2
(y2
(6-24)
where the fuselage cross section is given as in Fig. 6-22 as yo + zo = R2 . Within the
range of the fuselage, that is, for yo < y < +yo , d a has to be taken as being zero,
Ace= 0.
For the wing without dihedral, following Fig. 6-2a, z has to be replaced in Eq.
(6-24) by zo (z = zo). Thus the angle-of-attack distribution may be expressed by the
dimensionless coordinates yls = rt and zo/s = o with T1F = R/s as the relative
fuselage width. The angle-of-attack distributions computed by this method are
shown in Fig. 6-22 for two values of o . They have a very pronounced maximum
near the fuselage axis (at 77 = 0.578"0), which, however, in some cases lies within
the fuselage, and thus does not contribute to the lift distribution.
To determine the angle-of-attack distribution of a fuselage of finite length, a
consideration equivalent to that of Sec. 6-2-2 [see Eq. (6-17)] leads to
IF
d a(x, y, z)
2o
f
0
x2
yzR(x')
(
)
+ Z2 5
dx'
(6-25)
Y
As will be shown later, it is sufficient in most cases, however, to assume an
infinitely long fuselage.
In Fig. 6-23, the rolling moments due to sideslip acm la¢ of a low-wing, a
mid-wing, and a high-wing fuselage system from measurements of M611er [15] are
plotted against the lift coefficient CL. For comparison, the values for a wing
without dihedral and for a wing with a dihedral of v = 30 are also shown. The
fuselage causes a parallel shift of the curve for the wing alone. Thus the fuselage
influence is reflected in a contribution to the rolling moment due to sideslip,
independent of the lift coefficient, corresponding to the contribution of the
18
16
14
12
10
0.2
0,4
0,6
17 -W
1.0
1018
Figure 6-22 Additive angle-of-attack distribution of wing-fuselage systems at asymmetric
incident flow. Fuselage of circular cross
section.
0
Oa
0.1,
OR
.01101
.000
L
I
-0.04
Figure 6-23 Coefficient of the rolling
moment due to sideslip acM /a(3 vs. lift
coefficient CL of wing-fuselage systems,
i
OHM
'
from Nloiler. Fuselage: ellipsoid of revolu-
-0.12
'L
-0.1
02
tion of axis ratio 1:7. Wing: rectangle
i
0,4
CL
06
1.0
A = 5. L = low-wing airplane, M = midairplane, 11 = high-wing airplane,
wing
W = wing alone (v = angle of dihedral).
397
398 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
dihedral at the wing alone. Figure 6-23 shows that the effect of the fuselage on the
rolling moment due to sideslip may be replaced by that of an "effective dihedral"
of the wing. Here the high-wing airplane has a positive effective dihedral, the
low-wing airplane a negative effective dihedral.
This fact is taken into account in airplane design: In order to obtain approximately
the same rolling moment due to sideslip for different wing high positions, the low-wing
airplane is given a considerably larger geometric dihedral than the high-wing airplane.
Following the above procedure, Jacobs [16] determined theoretically the
fuselage influence on the rolling moment due to sideslip for an infinitely long
fuselage. In Fig. 6-24, results are plotted of his computations for the additive rolling
moment due to sideslip d (acMX/a p) as a function of the wing high position zo /R
Here the coefficient of the rolling moment due to sideslip is defined as
Mx =
with s being the semispan of the wing. These theoretical results
are compared. with measurements by Bamber and House [16] and by Moller [15].
Theory and measurements are carried to large wing high positions at which wing
.
and fuselage no longer penetrate each other. Agreement between theory and
measurement is very good. A closed formula may be obtained for the rolling
moment due to sideslip caused by the fuselage by introducing into Eq. (3-100) the
angle-of-attack distribution from Eq. (6-24) with z = zo or yls = rl, zo/s = fl, and
R/s = nF, respectively:
+1
7rA
4+2
8
co r 1F
7J2
7
1
i - n2
(rl2+ C0)2
d
77
(6-26)
770
all
r
a
o
2
i
i
i
2R
i
-0.
2
1
Figure 6-24 Additive rolling moment due to sideslip vs. wing high
position, from Bamber and House
and from Moller. Theory from
-0.06
-
2
-008
Jacobs [the theoretical curves have
been corrected considering the ex-1
I
2
tended lifting-line theory (cL,0 =
Relative fuselage width 'nF =
1:7.5; aspect ratio A = 5.
21r) ]
.
AERODYNAMICS OF THE WING-FUSELAGE SYSTEM 399
7
ZO
R
7.0
±0,8
60
±0.6
2R
2s
5
±0.4
Z30
20
±0.2
10
Figure 6-25 Effective dihedral angle veff for a
0.0
0.04
012
wing-fuselage system of np = R/s and wing
high position z0 /R for fuselages of circular
0,20
0,16
cross sections. Theory from Eq. (6-28).
T1F
Here, r70 = 7F - o is the coordinate on the fuselage surface and k = 7r:1/c'
A/2. For a simplified integration in Eq. (6-26), Multhopp [32] gave the value of
unity to the square root in the integrand and changed the upper integration limit
from unity to 2/7r. For o < (2/ir)2, this leads to
a
"
"A
k z T4
i
2
. (77F
1
Z + arcsin F - :z Co
(6-27)
T?F
which is valid for fuselages with circular cross sections and wing high positions
-R <za <R.
Comparison of Eqs. (6-27) and (3-158) yields the following expression for the
effective dihedral, corresponding to the additive rolling moment due to sideslip
caused by the fuselage:
eff -
2 1_
3 nF
B
0
1-(
LO
),
-r-
aresin R - 777F R
(6-28)
In Fig. 6-25, the computed effective dihedral angle is plotted against the relative
fuselage width 71F for several wing high positions zo /R. The effective dihedral angle
increases strongly with increasing relative fuselage width 77F and increasing wing high
position. For instance, for 71F = 0.12 and z0 /R = ± 1, its values are Veff = +3 and
-3°, respectively.
Multhopp [32] conducted that kind of computation for fuselages of elliptic
cross sections. Computations of the rolling moment due to sideslip for other
fuselages have been conducted by Maruhn [16] . Some of his results are presented in
Fig. 6-26. Fuselages with angular cross sections produce a particularly large rolling
moment due to sideslip. All the theoretical results discussed so far are valid for
400 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
io
bF
F
0.18
0.16
0.14
0,08
0.05
Figure 6-26 Additive rolling moment
due to sideslip of wing-fuselage systems
vs. wing high position for several
/A17
0.04
0,02
0
7
22
114
OB
bF
98
to
1.2
1.,
shapes of the fuselage cross section,
from Maruhn [the theoretical curves
have been corrected considering the
extended lifting-line theory (c' , =
2ir)]. Wing: ellipse A = 3.8. Relative
fuselage width 'RF = 6. Fuselage crosssection ratios hF/bF = 1.0 and 1.5.
infinitely long fuselages. Braun and Scharn [16] computed the effect of fuselages of
finite lengths.
Yawing moment due to sideslip and side force due to sideslip of a wing-fuselage
system The wing-fuselage arrangement has a quite small effect on the yawing
moment due to sideslip. Essentially, the right value for the yawing moment due to
of a wing-fuselage system can be obtained by adding the stabilizing
contribution of the wing (Sec. 3-5-3) to the destabilizing contribution of the
sideslip
fuselage (Sec. 5-2-3). Figure 6-27 shows the yawing moment due to sideslip of three
different wing-fuselage systems (low-wing, mid-wing, and high-wing arrangements)
from measurements of Moller [15]. For comparison, the wing alone and the
fuselage alone are also shown. Obviously, no substantial interference exists.
Furthermore, it should be noted that, for the yawing moment of the entire airplane,
the usually destabilizing contribution of wing and fuselage is much smaller than the
stabilizing contribution of the vertical tail assembly (see Chap. 7). The interference
of wing and fuselage is more pronounced, however, for the side force due to
sideslip. Figure 6-28 shows the side force due to sideslip, again from measurements
of Moller [15], for the three wing-fuselage systems of Fig. 6-27. Note that at
CL = 0.2, the side force due to sideslip for the high-wing and the low-wing airplanes
is about twice as large as for the mid-wing airplane. Also, the coefficient of the side
AERODYNAMICS OF THE WING-FUSELAGE SYSTEM 401
0.1
--
F1
0.12
Figure 6-27 Yawing moment due
0
to sideslip of wing-fuselage systems
vs. lift coefficient. Measurements
from Moller. Fuselage: ellipsoid of
revolution 1:7. Wing: rectangle
A = 5. L = low-wing airplane, M =
1-7 1
W
0
02
CL
0.6
0.4
0.8
mid-wing airplane, H = high-wing
airplane, W = wing alone, F = fuse-
1.0
lage alone.
force due to sideslip of the high-wing and the low-wing airplanes depends strongly
on the lift coefficients. The larger values of acy/aa and their dependence on the lift
coefficient for low-wing and high-wing planes find their explanation in the induced sidewash. Puffert [16] and Gersten and Hummel [8] studied these phenomena theoretically.
6-3 THE WING-FUSELAGE SYSTEM
IN COMPRESSIBLE FLOW
6-3-1 The Wing-Fuselage System in Subsonic
Incident Flow
Fundamentals The following discussions on the flow about a wing-fuselage system
at subsonic velocities will be limited to the case of straight flight. The effect of
0.2,
0.
I, y
0.05
W
0.2
CL
0.4
-
0.5
0.8
1.0
Figure 6-28 Side force due to sideslip
vs. lift coefficient. Measurements
from Moller (system as in Fig. 6-27).
402 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
compressibility on the flow about a wing has been explained by means of the
Prandtl-Glauert-Gothert rule in Sec. 4-4-2 for wings, and for fuselages in Sec. 5-3-2.
This rule renders feasible the determination of a subsonic flow (Ma., < 1) about
wings and fuselages by means of a transformation to incompressible flow. By this
means the incompressible flow will be computed for a transformed wing and a
transformed fuselage. The transformation of the geometric quantities for wing and
fuselage is given by Eqs. (4-66), (4-67a), (4-67b), (4-68a)-(4-68c), (5-51), (5-52a),
and (5-52b), where the quantities for incompressible flow are marked by the index
"inc" and those for the compressible flow are given without the index. These
quantities are as follows:
Coordinates:
1 -Ma.
xinc = x, yinc =Y
(6-29)
Zinc = Z
1 _-Ma".
Span:
binc = b
1 -Ma;,
Wing chord:
Cinc = C
(6-30b)
Taper:
zinc = a
(6-30c)
(6-30a)
A inc=A 1-Mal
Aspect ratio:
Sweepback:
cot Pinc = cot P
Fuselage width:
bF inc = bF
Fuselage length:
IF inc =1F
(6-30d)
1 -Mat
(6-30e)
l -Mat
(6-3 la)
(6-31b)
By computing the incompressible flow for the transformed wing-fuselage system at
the angle of attack of the compressible flow, the transformation of the pressure
coefficient from Eq. (4-69) becomes
1
-MaCpinc
CP
1
(«inc = a)
(6-32)
Through an analogous transformation, the lift coefficients and the pitching-moment
coefficients of wino fuselage systems are obtained.
The discussions about the incompressible flow over wing-fuselage systems of
Sec. 6-2-2 led to the conclusion that the lift slope of the wing-fuselage system is
little different from that of the wing alone if the relative width of the fuselage is
small to moderately large. Consequently, the relationship Eq. (4-74) for the wing
alone applies directly to the wing-fuselage system, or
dC
2-rA
,
(da)(W+F)
V(1
- Ma") Ac
4- 2
(6-33)
*For the fuselage, a different transformation formula of the pressure coefficient was given
by Eq. (5-53), where the angle of attack was transformed according to Eq. (5-52d). However,
within the framework of the linear lift theory, the Eq. (6-32) for the fuselage is equivalent to
Eqs. (5-52d) and (5-53).
AERODYNAMICS OF THE WING-FUSELAGE SYSTEM 403
The dependence of the neutral-point position on the Mach number of a
wing-fuselage system follows immediately from the relationships just stated, because
XN/C = -dCM/dCL, as
XN(W+F)
XN(W +F) inc
CA
CA
(6-34)
Here XN(W +F) inc is the neutral-point position of the wing-fuselage system at
incompressible flow as transformed according to Eqs. (6-29)-(6-31). As an example,
the lift slopes and neutral-point positions are presented in Fig. 6-29 against the
Mach number. These measurements of Schneider [42] are compared with
theoretical results. The lift slope is little affected by the fuselage; the neutral-point
displacement, however, shows a considerable fuselage influence. Results for
wing-fuselage systems with a rectangular and a delta wing are also available in [42].
Investigation of the wing-fuselage system by means of the panel method As a result of
utilizing efficient computers, methods have become more useful that are based on
singularities distributed on the body surface, thus satisfying exactly the kinematic
boundary conditions. Such generalizations have the advantage that geometric
restrictions in the body shape are essentially eliminated. Based on the computational
procedure for the displacement flow of Smith and Hess [13], the simultaneous
treatment of the displacement flow and the lift flow of wing-fuselage systems has
been presented independently by Kraus and Sacher [22] and Labrujere et al. [25].
In the method of Kraus and Sacher, the displacement flow is generated through a
s
0
Nas
01
1
i
0
-02
-
o
-0.3
±
OS
a
I
_041
S
0.6
07
0.8
Mc -
99
70
OS
b
0.5
07
0.3
0.9
70
He
Figure 6-29 Lift slope (a) and neutral-point shift (b) due to the fuselage effect vs. Mach number
for axisymmetric fuselage with swept-back wing. Measurements from Schneider; theory for
o) Wing alone, A = 2.75, A = 0.5, p = 52.4°.
incompressible flow from Sec. 6-2-2. (o
(v - -- n) Wing and fuselage, 1F/d = 12.5, e/d = 7.25, b/d = 5.0. (c - - - - - c) Wing and
fuselage, IFId = 10, e/d = 6.5, b/d = 3.33.
404 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
source-sink distribution on the surface of the body, the abrupt change of the
potential of the circulation flow through a vortex distribution within the body. The
total potential results from the superposition of the individual contributions. The
defining equations for the as yet unknown singularities are established by means of
the kinematic boundary condition, to be satisfied on the surface. The flow
conditions for the displacement problem are expressed by the requirement that no
flow is to penetrate the body surface, that is, that the velocity normal to the body
surface is zero. For the lift problem (circulation flow), this condition requires
smooth flow-off at the trailing edges of the lift-producing surfaces. The distribution
of the singularities and the perturbation potentials are obtained from the solutions
of the equations defining the singularities. Thus the total potential and the velocities
and pressure coefficients are obtained in the entire flow field and on the body
surface. The pressure coefficients are computed with all three of the components of
the perturbation velocity.
A suitable approach to the solution of the defining equations is found in the
panel method. The singularities, first assumed to be distributed continuously, are
now assumed to be constant on small flat surfaces (panels) and thus are accessible
to analytic integration of their defining equations. For the displacement flow, panels
covered with a constant source-sink density as in Fig. 6-30a are distributed on the
surface. For the circulation flow, panels on the inner surface are used, and on the
edges of these panels a vortex filament is laid of constant vortex strength. This leads,
as in Fig. 6-30b, to the well-known picture of lifting surfaces consisting of vortex
ladders composed of individual horseshoe vortices forming elementary wings.
On each surface panel, carrying a singularity density assumed to be constant, a
control point lies in its center of gravity at which the kinematic flow condition is to
be satisfied. Hence there exist as many control points as surface panels with
singularity densities individually assumed to be constant but as yet unknown in
magnitude. To each vortex ladder (consisting of several inner panels with a vortex
filament of constant circulation strength on their edges) a control point is assigned
at the trailing edge of the wing in which the Kutta condition is satisfied. Thus there
are as many Kutta control points as vortex ladders with unknown total circulation
strength (the circulation distribution within each of the vortex ladders on the
individual panels is assumed a priori to be a Birnbaum distribution).
The requirement of the defining equation that the kinematic boundary
condition has to be satisfied at all control points for all perturbation potentials of
all panels leads to a system of linear equations of a form similar to that derived for
the lifting-surface method (see [221). This, in most cases very extensive,. system of
linear equations is solved through iteration by means of a Gauss-Seidel procedure.
By this means the singularity strength and thus the velocity and the pressure
distribution at the control points are obtained. A typical result of a computation of
a wing-fuselage system is shown in Fig. 6-31 by means of the pressure distributions
on a few selected fuselages and wing sections in comparison with measurements by
Schneider [42]. The effect of compressibility has been taken into account through
the Gothert rule (see (22]). Further methods of general validity for the
computation of the aerodynamics of wing-fuselage systems at subsonic flow have
been developed by Woodward [521, Giesing et al. [91, and Korner (211.
AERODYNAMICS OF THE WING-FUSELAGE SYSTEM 405
b
Figure 6-30 Application of the panel method of Kraus and Sacher to a wing-fuselage system. (a)
Outer surface (source-sink distribution). (b) Inner surface (vortex distribution).
6-3-2 The Wing-Fuselage System in Supersonic
Incident Flow
General remarks Numerous contributions to the aerodynamics of the wing-fuselage
interference at supersonic velocities have been published. However, the establish-
ment of a simple, generally valid method for its computation, like the method
already available for incompressible flow (Sec. 6-2), has not yet been devised.
Summary presentations have been given by Lawrence and Flax [26], Ferrari [6],
Pitts et al. 137], and Ashley and Rodden [2].
406 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
Section A
Section 1
-0.4
cp cr
Swept-back wing
-Q2 1
-e1= 6.0 450 = 30° X = 0.33
Profile RAE 101.9%, mod.
U.S.
0
a2
Relative fuselage width
L.S.
Section C
77F=D/b=0.11
0
02
Q.
06
08
t0
X/C
Section 2
Section E
AA_
;, 1-useiage stations
08
a
b
w
X/C
C
U.S. = upper side
L.S. = lower side
Figure 6-31 Wing-fuselage system at subsonic incident flow, c = 8°, Ma. = 0.7. Measurements
from Schneider, theory from Kraus. (a) Geometry. (b) Pressure distributions, wing sections. (c)
Pressure distributions, fuselage sections.
The earlier theories on the wing-fuselage interferences were based on the
assumptions made for the theories of the wing and of the fuselage and were limited
to specific wing-fuselage systems. The first work of this kind came from Kirkby and
Robinson [5]. Here a wing of large aspect ratio, attached to a conical body, is
treated by means of the stripe method. This method does not take into account the
lift loss at the wing-fuselage interface and the effect of the wing on the fuselage due
to the large ratio of wing span to fuselage diameter. According to Cramer [4], these
two contributions cancel each other to a large extent and the total lift is obtained
relatively well. Fundamental investigations in the field of wing-fuselage interference
at supersonic velocities have been conducted by Ferrari [5]. He was concerned with
the problem of a rectangular wing of large aspect ratio on a cylindrical fuselage with
a pointed nose. The solution is accomplished through an iteration procedure. After
having determined separately the potential functions for the wing and for the
fuselage, these functions are combined and corrected step by step in such a way
that the boundary conditions are satisfied exactly for one part of the system only,
either for the wing or for the fuselage, whereas the boundary conditions of the
other part are disregarded. This procedure converges after a few steps.
Browne et al. [3] investigated a delta wing with conical fuselage (wing and
fuselage apex coincide) by the method of cone-symmetric flows. Results are given
for wings with subsonic and supersonic leading edges. Although this method does
not offer extensive practical applications, its exact solutions are valuable for comparisons with approximation solutions. The results of Lock [28] illustrate this fact.
A rectangular wing with a cylindrical fuselage has been treated by Morikawa
AERODYNAMICS OF THE WING-FUSELAGE SYSTEM 407
[311 and Nielsen [37]. By applying the Laplace transform, an exact solution is
obtained in the form of a series. Morikawa offers only an approximate solution, but
Nielsen succeeded in retransforming the solution. The computational procedure of
Woodward [52], applicable to subsonic and supersonic flows as well, should be
mentioned.
Now it will be shown that the essential relationships may be gained through
simple, physically plain considerations without lengthy mathematical derivations; see
also Schrenk [43].
In analogy to incompressibly flow, the wing-fuselage interference for supersonic
flow will be analyzed by first discussing the effect of the wing on the fuselage and
then the effect of the fuselage on the wing.
Lift distribution
of the fuselage As has been shown in Sec. 5-3-3, the lift
distribution at supersonic incident flow of the fuselage alone may be determined
from the relationship for incompressible flow, by setting a(x) = a. = const in Eq.
(5-28), as
dLF
= 27ra q d(R2)
dx
In Fig.
(6-35)
6-32a, the supersonic flow for the simple wing-fuselage system of an
axisymmetric fuselage and a rectangular wing in mid-wing position is demonstrated,
schematically. The absence of an influence of the wing on the fuselage portion
before the wing in supersonic flow marks the important difference between
supersonic incident flow and incompressible flow. Hence, the lift distribution for
the front portion of the fuselage in a wing-fuselage system is identical to that of the
fuselage alone as given by Eq. (6-35). Thus the lift of the fuselage front portion
becomes
LFf = 27ra.qRa
(6-36)
For the remaining portion of the fuselage, a simple survey will be given of the
lift distribution created in addition to Eq. (6-35) by the wing on the fuselage. To
simplify the problem, a wing of infinite span has been assumed in Fig. 6-32. In this
case the wing generates perturbation velocities only in the range between the Mach
lines m, and m2 originating at its leading edge and its trailing edge. Under the
simplifying assumption that the lift distribution of the wing is unchanged in the
fuselage range, no additive lift force, caused by the wing, acts on the rear fuselage
portion either. Hence the fuselage feels an additive lift force only within the range
between the Mach lines ml and m2.
This lift due to the wing influence is caused by the velocities induced by the
wing in the x direction, that is, u(x), and in the z direction, that is, w(x). In Fig.
6-32b and c, the distribution of the induced velocities u(x) and w(x) is given. Their
variation on the fuselage surface at z = 0 and 6 = 90°, respectively, is marked by
the dashed curve, and that at y = 0 and 6 = 00, respectively, by the dash-dotted
curve. These two curves are merely displaced from each other in the longitudinal
408 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
m7
U(X)
e=go
-
b
C
(dLF)
MF2
dx
e
x
Figure 6-32 Computation of the inter-
ference of wing-fuselage systems at
dLF
supersonic incident flow. (a) Geometry
of the wing-fuselage system. (b) Distri-
n
bution of the longitudinal velocity
u(x). (c) Distribution of the vertical
i
dx
velocity w(x). (d) Lift distribution due
to the longitudinal velocity. (e) Lift
distribution due to the vertical veloc-
f
ity. (f) Resultant lift distribution.
direction. The maximum values of the induced longitudinal and vertical velocities
are given in Eqs. (441a) and (4-41b) as
2c =
a°°
VMa5-1
w = - aco UN
Um
(6-37a)
(6-37b)
The solid curves signify the mean values of the induced velocities over the
circumference. They are essential for the computation of the lift distribution of the
AERODYNAMICS OF THE WING-FUSELAGE SYSTEM 409
fuselage. The following expressions from Ferrari [5 ] are obtained for 0 < x <xo , if
in this range R(x) = Ro = const:
Ft (x)
U".
`i-
1-
x0 Y
Y
w(x)
(arcsin
UM
x
(6-38a)
co
-
x
1
x
(6-38b)
Here xo =R0 '/Ma w, -1. Corresponding formulas are valid for c <x< c -F-x0.
From these mean values of the induced velocities, two contributions are
obtained to the lift distribution of the fuselage in the wing range, namely,
(dLF/dx)1 from the longitudinal velocity u(x), and (dLFldx)2 from the vertical
velocity a,,, U + w(x). For the case of a constant fuselage radius R(x) = Ro within
the wing range 0 <.x < c + xo , these contributions are
(.F)
u(x)
= 8q.Ro U
dx
dLF
dx
= 2nq.Ro
2
(6-39a)
fix)
ddx U2
(6-39b)
The qualitative trend of these two contributions is shown in Fig. 6-32d and e.
The resulting lift distribution as the sum of these two contributions is presented in
Fig.
6-32f. Integration of the contribution of Eq. (6-39b) results in LF2 = 0,
because, according to Fig. 6-32c, w(x) is equal to zero before and behind the wing.
From Fig. 6-32b, the lift force is obtained through integration of Eq. (6-39a),
because LF1 = LF, as
LF = 8 q
,
°L°°
VMa' -1
Roc
(6-40)
This simple relationship for the lift of the fuselage due to the wing applies to the
cases where the intersection of the front Mach line ml with the fuselage surface at
y = 0 lies before the wing trailing edge (see Fig. 6-32a). There is another
interpretation of this result of Eq. (6-40), namely, that the fuselage lift due to wing
effects is equal to the lift in plane flow of the wing portion AA = 2Roc shrouded
by the fuselage. Also, the case has been investigated by Ferrari [6] where the front
Mach line intersects the fuselage upper edge behind the wing trailing edge. In this
case the computation of the additive fuselage lift is considerably more difficult than
explained above.
In Fig. 6-33 the lift distribution of the fuselage under the influence of the wing
is shown for an example of Cramer (4]. The theoretical curve has been computed
by Ferrari [5]. At the Mach number Mac, = 2 used in this study and the chosen
geometry of the wino fuselage system, the Mach line from the leading edge of the
wing intersects the upper edge of the fuselage behind the wing trailing edge.
Therefore, contrary to Fig. 6-32f, the lift distribution reaches far beyond the wing
trailing edge. Agreement between theory and measurement is good.
410 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
0
CL,, a8
0
Theory
Measurement
-0
2
j c-7.4Ro
3
4
S
7
6
a
RO
t-- _,
,Z
Figure 6-33 Lift distribution on the fuselage for a wing-fuselage system (mid-wing airplane) at
supersonic velocities, from Cramer. Mach number Ma. = 2, angle of attack a = 8°. Theory from
Ferrari, wing chord c = 1.4R0 , wing span b = 8R 0 .
As stated earlier, the above theoretical results apply to the case of wings of
very large span. The effect of the wing aspect ratio can be seen in Fig. 6-34, where,
from [6], the additive lift distributions of the wing are plotted for wing-fuselage
systems with wings of several aspect ratios. The Mach number is also Ma. = 2. When
the aspect ratio decreases, the additive lift force decreases considerably, as would be
expected.
For the test series of Fig. 6-34, the ratio of the fuselage lift LF and the lift of
4-4.3
e
'
I
.8
2
1 0S
2
7
Figure 6-34 Effect of wing aspect ratio on the lift distribution on the fuselage for wing-fuselage
systems (mid-wing airplane) at supersonic velocities, from Ferrari. Mach number Ma. = 2, angle of
attack a. = 8°.
AERODYNAMICS OF THE WING-FUSELAGE SYSTEM 411
1.0
11-asti
0.8
06
I
Figure 6-35 Ratio of fuselage lift LF to wing
4.3
0,
0.2
lift L'W vs. relative fuselage width nF for
wing-fuselage systems at supersonic velocities, from Ferrari (system as in Fig. 6-34).
Mac. z
OCm °B°
0,2
0,4
0.6
0.8
Mach number Ma = 2, angle of attack
a, = 8°. Curve 1, theory from Lennertz.
1.0
'7F --
Curve 2, slender-body theory.
the wing portion not shrouded by the fuselage, L'' , is plotted in Fig. 6-35. The
data are compared with theoretical curves: Curve 1 reflects the theory of Lennertz
[27] from Eq. (6-5), valid also for supersonic flow. Curve 2 gives the theory of
wing-fuselage systems with wings of small aspect ratio of Sec. 6-4. The two
theoretical curves are not very different. The measured data agree quite well with
the theoretical curve (2).
The lift distribution leads to the pitching moment from Eq. (5-32). By
introducing the two contributions of the lift distribution from Eqs. (6-39a) and
(6-39b), the following two contributions to the pitching moment, referred to the
middle of the wing, are obtained, where the contribution of the fuselage front
portion has been disregarded:
C+xo
MF1= -8q".Po 1 X -
2G
C1x
2
(6-41a)
U,,
U
2;zq,,. '. p20 C
(6-41b)
C+x
MF2 = 2zq.Po
E(x)
L
tlx
(6-42a)
0
_ -27rq,c Plc
(6-42b)
Here x is measured from the wing leading edge. From Fig. 6-32e, MF2 is a free
moment. The total moment (without the fuselage front portion), referred to the
middle of the wing, thus becomes
MF=
4,-z q,
R2c
(6-43)
Note that this additive moment is independent of the Mach number.
The neutral-point displacement due to the wing influence on the fuselage with
reference to the neutral point of the wing alone, XJ.T W = c/2, becomes
412 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
MF
- _ L (W+F)
(,XN)(W+F) _
For small relative fuselage widths qF = 2R0 /b, L(w+F) may be approximated by
Lw. For the wing of large aspect ratio, from Eq. (4-46), this leads to
L W = q0,a,,
4
VM a'
-1
be
Hence with A = b/c > 1, the approximate expression is obtained:
(d xN)(W +F)
C
=
Ir
4
211
?IF
Ma a,
-
1
(644)
This stabilizing neutral-point displacement due to the effect of the wing on the
fuselage counteracts the destabilizing contribution of the fuselage front portion.
The above results on the effect of the wing are valid for the unswept wing of
large aspect ratio, that is, for wings with supersonic leading edges. For wings with
small aspect ratio, the discussions of Sec. 6-4 should be examined.
Lift distribution of the wing The effect of the fuselage on the lift distribution of
the wing at supersonic velocities can be determined approximately by the method
applied in Sec. 6-2-2 to incompressible flow. The additive angle-of-attack distribution, caused by the cross flow over the fuselage as in Fig. 6-5b, creates additive lift
locally on the wing. Under the assumption of an infinitely long fuselage, the
additive angle-of-attack distribution for a given fuselage cross-section shape is the
same as in incompressible flow, because the velocity of the cross flow of the
fuselage is considerably lower than the speed of sound. Equations (6-16a) and
(6-16b) give the distribution of the induced angle of attack for a fuselage of circular
cross section (radius R) with a wing in mid-wing position. The computation of the
approximate lift distribution along the span for the given angle-of-attack distribution
may be conducted very easily with the so-called stripe -method.* Hence, the local
lift coefficient becomes
CI(y) `
with d a{y) from Eq. (6-16a).
_
4a,,,
M a., _1
h
AM(y) )
a
(6-45)
An example for a wing-fuselage system of an axisymmetric fuselage and a
rectangular wing is given in Fig. 6-36. It shows the lift distribution plotted against
the span for the Mach number Ma. = 2 at an angle of attack a,. = 8°. Curve 1
reflects the theory of the stripe method, Eq. (6-45), and curve 2 a theory of Ferrari
[6]. Both theories agree quite well with the measurements, except for the stripe
method in the immediate vicinity of the fuselage. For comparison, the theory for
*The stripe method is a procedure whereby the local lift coefficient is set proportional to
the local angle of attack based on the lift slope of the plane problem, which from Eq. (4-46) is
given for supersonic velocities by (dc1/da),,, = 4 f Ma«, -1.
AERODYNAMICS OF THE WING-FUSELAGE SYSTEM 413
0.
0.6
Mmco = 2
i
oGo,=B
05
t
0.4
4
0.2
0,1
L
M
I
0.2
0.4
0.5
0.8
1.0
7 y
Figure 6-36 Lift distribution on the wing due
to the fuselage effect for a wing-fuselage
system
(mid-wing airplane) at supersonic
velocities. Curve 1, theory, stripe method,
from Eq. (6-45). Curve 2, theory from
Ferrari. Curve 3, measurements from Ferrari.
Curve 4, theory, wing alone.
the wing alone is added as curve 4. Obviously, the influence of the fuselage on the
lift distribution of the wing is rather large.
The above results on the effect of the fuselage on the lift distribution of the
wing apply to wings of large aspect ratios. For wings of small aspect ratios, reference
should again be made to Sec. 6-4.
Wave drag The problem of the determination of the wave drag of wing-fuselage
systems at supersonic velocities has been attacked by Vandrey [48], Lomax and
Heaslet [30], Jones [181, and Keune and Schmidt [19]. Also, the experimental
investigations of Schneider [42] should be mentioned.
6-3-3 The Wing-Fuselage System in Transonic
Incident Flow
The following discussions on the interference in wing-fuselage systems in transonic
flow will be restricted mainly to the drag problem. The drag of wing-fuselage
systems near Ma = 1 is generally larger than the sum of the drags of the wing
414 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
alone and the fuselage alone. Here the wave drag at zero lift is the major factor.
Figure 6-37 shows drag measurements by Whitcomb [50] on wing-fuselage
systems in the Mach number range from Ma,0 = 0.85 to Ma00 = 1.1 with CL = 0.
The tested models are shown in Fig. 6-37a, their total drag in Fig. 6-37b, and the
drag remaining after subtraction of the friction drag in Fig. 6-37c. The curve for the
fuselage alone (model 1) shows a strong drag rise near Mao, = 1. The simple
combination of wing and fuselage (model 2) produces a particularly large drag in
the transonic range. Whitcomb [50] showed that by contracting the fuselage within
the wing range, the drag in the transonic range may be greatly reduced (model 3).
This contraction of the fuselage has to be chosen such that the wing-fuselage system
and the original fuselage (model 1) have approximately equal distributions of the
cross-sectional areas normal to the fuselage axis. This rule for the distribution of
cross-sectional areas of a wing-fuselage system is called the "area rule." Figure 6-38
shows the application of this rule to an airplane, where Fig. 6-38a gives the plan
view of the airplane, Fig. 6-38b the contour of an axisymmetric body of equal
cross-sectional area distribution AF(x), and Fig. 6-38c the variation of this
cross-sectional area along the fuselage axis dAFldx. In Fig. 6-38c, the case without
a
b
/.169
0.020
2
0. 015
I
i
0. 012
OSM
1
,
C
0.072
Figure 6-37 Drag coefficients of wingfuselage systems and axisymmetric fuse-
0.004E
0.84
ON
0.92
OX
Maw
1,90
74)
1.0
1J2
lages in the transonic Mach number range,
from Whitcomb. (a) Geometry. (b) Total
drag coefficients CD at zero lift. (c)
Coefficients of wave drag.
AERODYNAMICS OF THE WING-FUSELAGE SYSTEM 415
A
Figure 6-38 The area rule for transonic
flow. (a) Airplane planforrn. (b) Distribu-
tion of the cross sections AF(x) of the
equivalent body of revolution. (c) Variation of the cross-sectional area distribution
along the fuselage dAFldx.
area contraction is drawn as a solid line, the case with area contraction as a dashed
line. The fuselage area contraction has been chosen for as smooth a dAF/dx
variation as possible.
For the experimental proof, Whitcomb [50] also tested a fuselage whose
cross-sectional area distribution is equal to that of the wing-fuselage system without
contraction (model 4 of Fig. 6-37). This model indeed has the same drag rise as
model 2 in the transonic range. The theoretical basis of this phenomenon has been
studied by Jones [18] and Oswatitsch [35], as well as by Keune and Schmidt [19].
Finally, it may be seen in Fig. 6-39 that the advantage of the area rule is limited to
0
2
i
f
0.7
0
0.8
7
0.9
1,0
47
1.3
1.4
Ma,,
Figure 6-39 Drag coefficients of wing-fuselage systems at zero lift in the transonic Mach number
range; measurements from Jones. Curve 1, without fuselage contraction. Curve 2, with fuselage
contraction, computed for Ma. = 1. Curve 3, with fuselage contraction, computed for Ma. = 1.2.
416 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
the transonic Mach number range. This figure gives the drag coefficients of 3
wing-fuselage systems in the Mach number range from Ma = 0.8 to Ma = 1.4.
Model 1 is the fuselage without contraction, whereas models 2 and 3 are fuselages
with two different contractions. The contraction of model 2 has been chosen for
largest drag reduction at Ma = 1, whereas that of model 3 is for lowest drag at
Ma = 1.2. These tests show that contraction according to the area rule yields
favorable results only in the transonic range. In the supersonic range, the results are
even less favorable than for fuselages without contraction.
In this connection, the comprehensive experimental studies should be mentioned that Schneider [42] conducted on wing-fuselage systems with three different
wings (rectangular, swept-back, and delta wings). A computation of the pressure
distribution on wing-fuselage systems at an incident flow of Ma = 1 and a
comparison with measurements have been conducted by Spreiter and Stahara [45].
Compare also the computational methods in [201.
.
6-4 SLENDER BODIES
In the previous sections of this chapter wing-fuselage systems with wings of large to
moderately large aspect ratios have been discussed. Now systems with wings of small.
aspect ratios will be treated. Here the slender triangular wings (delta wings) with
large sweepback play a special role. With flight velocities having increased from
subsonic to supersonic speed ranges over the past decades, this kind of slender body
(Fig. 6-40) has become most important. They are characterized by aerodynamic
coefficients that are largely independent of the Mach number but depend, to a large
extent nonlinearly, on the angle of attack (see Secs. 3-3-6 and 5-3-3).
The theory for lift computation developed by Munk [33] for slender fuselages
and by Jones [17] for wings of low aspect ratio has been extended by Ward [49]
and Spreiter [44] to wing-fuselage systems with wings of low aspect ratio; see also
Jacobs [44]. The basic thought underlying this theory is the fact that changes in
the perturbation velocities about slender bodies are small in the x direction (fuselage
axis, wing longitudinal axis) compared with those of the perturbation velocities in
dF
b
i,z
ix
ix
Figure 6-4U Slender bodies: wing, tuselage,
wing fuselage system.
AERODYNAMICS OF THE WING-FUSELAGE SYSTEM 417
Z,
V"
Figure 6-41 Theory of wing-fuselage systems with wings of small aspect ratio. (a)
Sketch of the wing-fuselage system. (b)
Cross section x = const. (c) Conformal mapping of cross section x = const of b.
C
31
the y and z directions normal to the x direction. This causes the potential equation,
Eq. (4-8), to be reduced to that of two-dimensional flows in the yz plane:
a20
aft
-}-
a2 = 0
az2
(6-46)
where v = a0/ay and w = aOlaz are the induced velocities in the lateral plane.
Since Eq. (6-46) is valid for both incompressible and compressible flows, the results
given below can be applied to both subsonic and supersonic incident flows.
The potential equation, Eq. (6-46), is to be solved for each cross section
x = const (Fig. 6-4la), which can be accomplished by conformal mapping, for
instance. The flow about a wing-fuselage system (Fig. 6-41b) can therefore be
determined from the flow about a flat plate at normal incidence (Fig. 6-41c).
Some results from Spreiter [44] and Ward [49] will now be discussed; see also
Ferrari [6] and Haslet and Lomax [12].
Pressure distribution For wino fuselage systems consisting of a delta wing and an
infinitely long body of circular cross section, pressure distributions for two sections
1 and 2 normal to the axis are shown in Fig. 6-42. The load distribution on the
wing is
418 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
[R() [('
R
,
J cp = 4Nc, tany
s(x)
1
yy
1+
4
ls(z)/ J -
y
s(X)'+Y
s
z
for R2 <y2 < s2 (6-47a)
J
and that on the body is
1 - (s x) /4
Jcp =4.xrotanyAI+
4
for 0 <y2 < R2
(647b)
(y
For the wing alone, Eq. (6-47a) yields, with R = 0,
4N,,,, tan y
J Cp =
1 - (S(x) )3
(648)
In Eqs. (647a)-(648), y is the leading-edge semiangle of the wing, s(x) = x tan y is
the local half span, and R(x) is the body radius. The load-distribution curve in Fig.
Figure 6-42 Load distribution over the span for
a wing-fuselage system with a delta wing
(slender-body theory). Curves 1 and 2 for the
wing-fuselage system. Curve 1' for the wing
-4
-2
0
2
4
6
R
alone.
AERODYNAMICS OF THE WING-FUSELAGE SYSTEM 419
70
i1
H
y
6
y=R
C
6
Wing + fusela
4
I
Wing al one
=0
Wing + fus elage
Figure 6-43 Load distribution in the
longitudinal direction on the middle
section (y = 0) and on the section at
t arry
I
2
.3
R -tarry'
6
the wing root (y = R) for a wingfuselage system with a delta wing
(slender-body theory).
6-42 shows that the influence of the fuselage on the pressure distribution is greater
at the front portion of the wing than at the rear portion. For comparison, the load
distribution of the wing alone is also drawn for cross section 1 (curve 1 ').
In Fig. 6-43 the load distribution in the longitudinal direction along the
wing-root section y =R is shown for the wing-fuselage system of Fig. 6-42. The
influence of the body is seen in a somewhat smaller load decline in the axial
direction than for the wing alone. The load distribution for the middle section (y = 0)
is also given.
A procedure for computing the pressure distribution on slender bodies with
arbitrary planform and cross-section shape is given by Hummel [14].
Lift distribution In Fig. 6-44, the lift distribution is shown versus the span of the
wing-fuselage system of Figs. 6-42 and 6-43. The relative body width is ?7F = 3. The
effect of the body on the lift distribution is considerable. An example of the lift
distribution over the body length is shown in Fig. 6-45. Note that the fuselage
contributes to the lift only in the range of the wing. Close to the wing nose, the
fuselage lift increases strongly; at the wing trailing edge, it drops abruptly to zero.
Total lift In Fig. 6-8, curve 2, the ratio of body lift LF and total lift L(W+F) of a
delta wing and an infinitely long body of circular cross section, according to this
theory, was plotted as a function of the relative body width r?F. Comparison of
curves 2 and 1 in Fig. 6-8 shows that the slender-body theory yields almost the
same values of LFIL(w+F) as the theory of Lennertz [27], which is valid for
arbitrary aspect ratios. We can conclude, therefore, that the values of LF/L(w+F)
from the slender-body theory can also be used for wing-body systems with wings of
larger aspect ratios.
The total lift for wing-body systems from Fig. 6-44 is
L(w+F) =
-R2)2
(6-49)
420 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
2R
2s
IT
Figure 6-44 Lift distribution over the span
y
for a wing fuselage system with a delta wing,
77F = a (slender-body theory). Curve 1,
wing + fuselage. Curve 2, wing + flattened
fuselage. Curve 3, wing alone.
8
1
4
y
n
0
LT
4
5
R tan7
r
Figure 6-45 Lift distribution of the
fuselage for a wing-fuselage system with
a delta wing (slender-body theory).
AERODYNAMICS OF THE WING-FUSELAGE SYSTEM 421
Hence, when referring the lift coefficient CL to the wing area A = crs and the
dynamic pressure of the incident flow q., setting A = 4s/c, and riF = R/s, the lift
slope becomes
dCL
da
_
?r
A (1 _ r1F)2
(6-50a)
2
(W +F)
(dcL)rri
W
(6-50b)
The second relationship applies to the wing alone (riF = 0); see Eq. (3-101b). In
Fig. 6-46, the ratio of the total lift to the lift of the wing alone, that is,
L(w+F)/L w, is plotted as curve 1 against the relative fuselage width ?7F- With
increasing i7F, the ratio L(w+F)/Lw decreases strongly and becomes zero for
77F = 1.
At a fuselage that is pointed in front, this finite front portion of the fuselage
produces a lift additive to that of the infinitely long front portion from Eq. (6-6):
LF f = 2na. q.Ro
(6-51)
This means an increase of the lift slope over the value of Eq. (6-50a), and the lift
slope becomes
dcL
da (W+F) = 2 A (1 - r1F + 77F)
(6-52)
The ratio L(w+F)/Lw for this case is plotted against the relative fuselage width 17F
as curve 2 in Fig. 6-46.
Neutral-point position Finally, in Fig. 6-47, a few results are shown on the shift of
the neutral point as caused by the fuselage. For the wing-fuselage system of Fig.
1
0.
2
a,
U-
Q
0.
0.2
0,4
77F
0.6
0.3
1.0
Figure 6-46 Ratio of total lift to wing lift for
wing-fuselage systems with a delta wing
(slender-body theory). Curve 1, infinitely long
fuselage. Curve 2, fuselage of finite length
(with fuselage nose).
422 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
Figure 6-47 Neutral-point shift of wingfuselage systems with a delta wing (slender-
body theory), from Spreiter.
02
0.4
0.6
0.8
Curve
1,
wing + fuselage. Curve 2, wing + flattened
fuselage. Curve 3, substitute wing (with
rectangular middle portion).
1.0
'1F-
6-44, the shift of the neutral point caused by the fuselage, relative to the neutral
point of the wing alone (dxN/cF,)(W+F), is plotted as curve I of Fig. 6-47 against
the relative fuselage width. From the theory for small aspect ratios, the neutral
point of the wing alone lies at a distance 3c, from the wing nose. When ?7F increases,
the neutral point moves rearward by an amount
"XN
Cµ
2
_
(W+F)
2+ 1F
(1
(6-53)
)2
For 7?F = 1, the shift of the neutral point becomes (d xN/cµ)(yy+F) = that is, in
2; trailing
this case the neutral point of the wing-fuselage system is located at the wing
edge, as can easily be understood from inspection of Fig. 6-45. In Fig. 6-47, curve
2, the neutral-point shift is given for a "flat" fuselage (height zero). The difference
to curve 1 is relatively small. For the case of a flat fuselage, the lift distribution
over the span is
also shown in Fig. 6-44 as curve 2. For comparison, the
neutral-point shift for a wing with rectangular middle section (substitute wing) is
given in Fig. 6-47, curve 3.
At last the case of a fuselage with a front portion of finite length will be
discussed. The moment of the fuselage front portion, relative to the axis through the
wing neutral-point, is given from Eqs. (6-6) and (5-32) as
if
(R2 (x) dx + (xNyy
MFf =
- lf)Ro
(6-54)
0
Here, if is the length of the fuselage front portion from Fig. 6-48, and xNW is the
distance of the wing neutral point from the fuselage nose. The distance (xNW - l f)
is easily determined as 3cr(2 - 371F). Evaluation of Eq. (6-54) for a fuselage with a
parabolic nose yields
AERODYNAMICS OF THE WING-FUSELAGE SYSTEM 423
1-
MFf = 2:cc q R0
3
f
I
71F +
Y/
(6-55)*
where cu = 3 Cr.
For the wing-fuselage system with a fuselage front portion of finite length, the
neutral-point shift relative to that of the wing alone is
M(W +F) °°
lJ X jy
L(W
+F) °o
+MFf
I LFf
The index - refers to wing-fuselage systems with infinitely long fuselages, where
L(w+F) is computed from Eq. (6-49) and M(w+F)- _ -J/ XNL(W+F)OO With JxN
from Eq. (6-53). The values of LFf and MFf are given by Eqs. (6-51) and (6-55),
respectively.
For a fuselage with elliptic nose section, the factors of if/Cr must be replaced by 1.
0.
-=0
If
.Cr
1s
0.,
L
0
a
'
i
2Ro
-- 0,1
-0.2
7.0
-0,3
-0,4
- 050
0.2
0.4
0.6
'1F
0.8
1,0
Figure 6-48 Neutral-point shift of wingfuselage systems with a delta wing and a
fuselage of finite length (slender body
theory), from Eq. (6-56).
424 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
By substitution, the final result is
2
JC N
(W +F)
=
2F
1
4
1 - 71F + 'OF 2
2 - 571, + 471, -
8 if
5 Cr
(6-56)
The shift of the neutral point according to Eq. (6-56) is plotted in Fig. 6-48 as a
function of the relative body width r1F for various lengths of the fuselage front
portion lf/cr. These plots show that the shift of the neutral point d XN is positive
(stabilizing) for small values of l flcr, as in the case of an infinitely long fuselage
(Fig. 6-47). At larger values of Zflcr, however, the unstable contribution of the
fuselage front portion is predominant, making v XN negative.
dFinax
--
---
I
F
..
2
o
3
0
b
I
! '
--E-I
I
I
I
I
'Theory for
wing alone
1
1
C
-70'
0°
J-
75°
10,
20°
25'
30'
S'
40°
cc
Figure 649 Lift coefficient vs. angle of attack cL(c) for slender bodies, from measurements of
Otto. (a) Wing alone, ,i = 1. (b) Fuselages 1, 2, 3 alone: dFinax/IF = 0.10, 0.10, 0.05. (c)
Wing-fuselage systems with fuselage 2, Re = U.lF/v = 7.5
106
.
AERODYNAMICS OF THE WING-FUSELAGE SYSTEM 425
Test results Finally, some test results will be presented that show the nonlinear lift
characteristics CL(a) for slender bodies. In Fig. 6-49 the lift coefficients for three
wings, three fuselages, and three wing-fuselage systems in incompressible flow are
presented from Otto [36]. The lift coefficients of the fuselages (Fig. 6-49b) are
referred to the wing area. For the wings alone, the results of linear theory for
slender bodies according to Eq. (6-50b) are also shown. In all three cases (wing,
fuselage, wing-fuselage system), the deviation from linear theory is considerable.
Corresponding investigations on slender conical wino fuselage systems in supersonic
incident flow have been reported by Stahl 146]. Measurements on the vortex
system of inclined wing-fuselage systems have been conducted by Grosche [10] .
The design of slender, integrated airplanes for supersonic flight has been
proposed-by, among others, Kuchemann [23]. Design questions for airplanes in the
transonic flight mode are discussed by Lock and Bridgewater [29].
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20:85-98, 1953.
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4:431-472, 1972.
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Flow, J. Aer. Sci., 15:443-452, 1948.
4. Cramer, R. H.: Interference between Wing and Body at Supersonic Speeds-Theoretical and
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NAVORD Rept. 3146, 1952.
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1949.
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"Aerodynamic Components of Aircraft at High Speeds," Sec. C, Princeton University Press,
Princeton, N.J., 1957.
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,
Interferenzeinfliisse an schiebenden Fliigel-Rumpf-Anordnungen mit Pfeil- and Deltafliigein,
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Aerodynamics for Multiple Interfering Wings and Bodies, J. Aircr., 9:693-702, 1972. Hua,
H. M.: J. Aircr., 10:422-426, 1973; 11:366-368, 1974; 12:916, 1975. Kalman, T. P., W. P.
Rodden, and J. P. Giesing: J. Aircr., 8:406-413, 1971.
10. Grosche, F.-R.: Wind Tunnel Investigation of the Vortex System near an Inclined Body of
Revolution with and without Wings, AGARD CP 71, 2:1-13, 1971; Z. Flugw., 18:208-217,
1970.
11. Hafer. X.: Untersuchungen zur Aerodynamik der Flugel-Rumpf-Anordnungen, Jb. WGL,
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Flugw., 1:2-8, 1953. Truckenbrodt, E. and K. Gersten: Z. Flugw., 5:204-216, 1957.
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R. Sears (ed.), "General Theory of High Speed Aerodynamics," Sec. D, pp. 249-275,
Princeton University Press, Princeton, N.J., 1954; NACA Rept. 1185, 1954.
426 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
13. Hess, J. L. and A. M. O. Smith: Calculation of Potential Flow About Arbitrary Bodies,
Prog. Aer. Sci., 8:1-138, 1967. Carmichael, R. L.: AGARD CP 71, 4:1-3, 1971. Smith, A.
M. O. and J. L. Hess: Douglas Aircr. Rept. ES 29988, 1958.
14. Hummel, D.: Berechnung der Druckverteilung an schlanken Flugkorpern mit beliebiger
Grundriss- and Querschnittsform in Unter- and Uberschallstromung, Jb. DGLR, 158-173,
1968.
15. Jacobs, E. N. and K. E. Ward: Interference of Wing and Fuselage from Tests of 209
Combinations in the NACA Variable-Density Tunnel, NACA Rept. 540, 1935. Moller, E.:
Jb. Lufo., 1:336-365, 1942; ZWB Lufo. TB 11, no. 5, 1944. Sherman, A.: NACA Rept.
575, 1936.
16. Jacobs, W.: Berechnung des Schiebe-Rollmomentes fur Flugel-Rumpfanordnungen, Jb.
Lufo., 1:165-171, 1941. Bamber, M. J. and R. O. House: NACA TN 703, 1939; 730, 1939.
Braun, G. and H. Scharn: Jb. Lufo., 1:246-262, 1942. Maruhn, K.: Jb. Lufo., 1:263-279,
1942. Puffert, H. J.: Z. Flugw., 3:323-331, 1955.
17. Jones, R. T.: Properties of Low-Aspect-Ratio Pointed Wings at Speeds Below and Above the
Speed of Sound, NACA Rept. 835, 1946; "Collected Works," NASA TM X-3334, pp.
369-375, National Technical Information Service, Springfield, Va., 1976. Ashley, H. and M.
Landahl: "Aerodynamics of Wings and Bodies," pp. 99-123, Addison-Wesley, Reading,
Mass., 1965.
18. Jones, R. T.: Theory of Wino Body Drag at Supersonic Speeds, NACA Rept. 1284, 1956;
Adv. Aer. Sci., 1:34-51, 1959;AIAA J., 10:171-176, 1972; "Collected Works," NASA TM
X-3334, pp. 609-623, 625-644, 657-664, National Technical Information Service, Springfield, Va., 1976.
19. Keune, F. and K. Oswatitsch: Aquivalenzsatz, Ahnlichkeitssatze fur schallnahe Geschwindigkeiten and Widerstand nicht angestellter Korper kleiner Spannweite, Z. Angew. Math. Phys.,
7:40-63, 1956; Z. Flugw., 1:137-145, 1953. Keune, F. and W. Schmidt: Jb. WGL,
150-155, 1956; Z. Angew. Math. Mech., 36:301-303, 1956. Keune, F.: Jb. WGL, 176-186,
1955; Z. Flugw., 5:121-125, 1957. Keune, F., H. Riedel, and H. Emunds: Z. Flugw.,
20-257-261, 1972.
20. Klunker, E. B. and P. A. Newman: Computation of Transonic Flow About Lifting
Wing-Cylinder Combinations, J. Aircr., 11:254-256, 1974. Rohlfs, S. and R. Vanino: Z.
Flugw., 23:239-245, 1975.
21. Korner, H.: Berechnung der potentialtheoretischen Stromung um Flugel-RumpfKombinationen and Vergleich mit Messungen, Z. Flugw., 20:351-368, 1972.
22. Kraus, W. and P. Sacher: Das Panelverfahren zur Berechnung der Druckverteilung von
Flugkorpern im Unterschallbereich, Z. Flugw., 21:301-311, 1973. Kraus, W.: NASA-TT
F-14117, 1972; VKILect. Ser. 87, 1976.
23. Kuchemann, D.: Aircraft Shapes and Their Aerodynamics for Flight at Supersonic Speeds,
Adv. Aer. Sci., 3:221-252, 1962; Prog. Aer. ScL, 6:271-353, 1965; Jb. WGLR, 85-93,
1964. Baals, D. D., A. W. Robins, and R. V. Harris, Jr.: J. Aircr., 7:385-394, 1970. Kane,
E. J. and W. D. Middleton: AGARD CP 71, 3:1-14, 1971.
24. Kuchemann, D.: Some Remarks on the Interference between a Swept Wing and a Fuselage,
AGARD CP 71, 1:1-9, 1971. Kuchemann, D. and J. Weber: ARC RM 2908, 1953/1956.
25. Labrujere, T. E., W. Loeve, and J. W. Slooff: An Approximate Method for the Calculation
of the Pressure Distribution on Wing-Body Combinations at Subcritical Speeds, AGARD CP
71, 11:1-15, 1971.
26. Lawrence, H. R. and A. H. Flax: Wing-Body Interference at Subsonic and Supersonic
Speeds-Survey and New Developments, J. Aer. Sci, 21:289-324, 328, 1954; 22:208, 1955.
Flax, A. H.: J. Aer. Sci., 20:483-490, 1953. Lawrence, H. R.: J. Aer. Sci., 20:541-548,
1953.
27. Lennertz, J.: Beitrag zur theoretischen Behandlung des gegenseitigen Einflusses von
Tr agflache and Rumpf, Z. Angew. Math. Mech., 7:249-276, 1927; Z. FZug. Mot., 18:11-13,
1927; in W. F. Durand (ed.), "Aerodynamic Theory-A General Review of Progress," div. K.
pp. 152-157, Springer, Berlin, 1935, Dover, New York, 1963.
AERODYNAMICS OF THE WING-FUSELAGE SYSTEM 427
28. Lock, R. C.: Theoretical Pressure Distribution at Zero Lift at Supersonic Speeds for Slender
Delta Wings Having Fuselages of Circular Cross Section, Aer. Quart., 12:95-130, 1961.
Jones, J. G.: Aer. Quart., 11:51-70, 1960.
29. Lock, R. C. and J. Bridgewater: Theory of Aerodynamic Design for Swept Winged Aircraft
at Transonic and Supersonic Speeds, Prog. Aer. Sct, 8:139-228, 1967. Gustavsson, A. and
R. Vanino: Z. Flugw., 23:257-262, 1975. Lock, R. C.: in K. Oswatitsch (ed.), "Symposium
Transsonicum I," pp. 276-287, Springer, Berlin, 1964. Lock, R. C. and E. W. E. Rogers:
Adv. Aer. Sci., 3:253-275, 1962.
30. Lomax, H. and M. A. Heaslet: Recent Developments in the Theory of Wing-Body Wave
Drag, J. Aer. Sci., 23:1061-1074, 1956; .NACA Rept. 1282, 1956. Bonner, E.: I. Aircr.,
8:347-353, 1971. Ferri, A. and J. H. Clarke: J. Aer. ScL, 24:1-18, 1957. Ferri, A., J. H.
Clarke, and L. Ting: J. Aer. Sci., 24:791-804, 1957. Graham, M. E.: J. Aer. ScL,
24:142-144, 1957. Licher, R. M.: T. Aer. Sci., 23:1037-1043, 1956.
31. Morikawa, G. K.: Supersonic Wing-Body Lift, J. Aer. Sci., 18:217-228, 503-504, 1951;
Quart. App. Math., 10:129-140, 1952.
32. Multhopp, H.: Zur Aerodynamik des Flugzeugrumpfes, Lufo., 18:52-66, 1941; NACA TM
1036, 1942. Liess, W. and F. Riegels: Jb Lufo., 1:366-373, 1942. Luckert, H. J.: Can. Aer.
J, 1:205-217, 1955. Weber, J., D. A. Kirby, and D. J. Kettle: ARC RM 2872, 1951/1956.
33. Munk, M. M.: The Aerodynamic Forces on Airship Hulls, NACA Rept. 184, 1924.
34. Muttray, H.: Die gegenseitige Beeinflussung der Einzelteile am Flugzeug ohne laufende
Schraube, Ringb. Luft., I A 4, 1937; Lufo., 2:33-39, 1928; 11:131-139, 1934.
35. Oswatitsch, K.: The Area Rule, App. Mech. Rev., 10:543-545, 1957. Oswatitsch, K. and F.
Keune: Z. Flugw., 3:29-46, 1955.
36. Otto, H.: Calculation of Nonlinear Lift and Pitching Moment Coefficients for Slender
Wing-Body Combinations, J. Aircr., 11:489-491, 1974; Z. Flugw., 22:187-200, 1974;
DLR-FB 73-66, 1973.
37. Pitts, W. C., J. N. Nielsen, and G. E. Kaattari: Lift and Center of Pressure of
Wing-Body-Tail Combinations at Subsonic, Transonic, and Supersonic Speeds, NACA Rept.
1307, 1957. Flax, A. H.: J. Aircr., 11:303-304, 1974. Nicolai, L. M. and F. Sanchez: J.
Aircr., 10:126-128, 1973. Nielsen, J. N.: Thesis, Cal. Inst. Tech., 1951.
38. Schlichting, H.: Neuere Beitrage der Forschung zur aerodynamischen Fliigelgestaltung
(Umriss, Verwindung; Rumpfeinfluss), Jb. Lufo., 1:113-132, 1940.
39. Schlichting, H.: Die Stabilitatsbeiwerte des Flugzeuges unter Beriicksichtigung der Interferenz
von Flugel, Rumpf and Leitwerk, Sonderheft: Flugmech. Probleme, Akad. Lufo. 2/43 g,
3-23, 1943.
40. Schlichting, H.: Calculation of the Influence of a Body on the Position of the Aerodynamic
Centre of Aircraft with Swept-Back Wings, ARC RM 2582, 1947/1952.
41. Schlichting, H.: Monograph on the Aerodynamics of Mutual Interference between the
Components of the Airplane, Nat. Res. Coun. Can., Tech. Transl. TT-92, 1949; Inst. Stro.
THBraunschw. 46/5, 1946.
42. Schneider, W.: Experimentelle Bestimmung von Flugel-Rumpf-Interferenzen im Machzahlbereich Ma = 0,5 bis Ma = 2,0, DLR FB 67-93, 1967; AVA 70 A 23, 1970, 70 A 40, 1970;
AGARD CP 35, 1968.
43. Schrenk, 0.: Angenaherte Berechnung der gegenseitigen Beeinflussung zwischen Flugel and
Rumpf im Uberschallbereich, Z. Angew. Math. Phys., 1:202-209, 1950.
44. Spreiter, J. R.: The Aerodynamic Forces on Slender Plane- and Cruciform-Wing and Body
Combinations, NACA Rept. 962, 1950; J. Aer. Sci., 19:571-572, 1952. Cambell, G. S.: J.
Aer. Sci., 25:60-62, 1958. Jacobs, W.: Jb. WGL, 168-171, 1955.
45. Spreiter, J. R. and S. S. Stahara: Aerodynamics of Slender Bodies and Wing-Body
Combinations at M. = 1, AIAA J., 9:1784-1791, 1971.
46. Stahl, W.: Untersuchungen an schlanken, kegligen Rumpf-Flugel-Kombinationen in Uberschallstromung, insbesondere hinsichtlich Volumenverteilung and Wolbung, Z. Flugw.,
18:461-473, 1970.
47. Vandrey, F.: Zur theoretischen Behandlung des gegenseitigen Einflusses von Tragfiiigel and
428 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
Rumpf, Jb. Lufo., 1:158-166, 1938; Lufo., 14:347-355, 1937; Jb. Lufo., 1:367-370, 1940.
Liese, J. and F. Vandrey: Jb. Lufo., 1:326-335, 1942.
48. Vandrey, J. F.: Der gegenseitige Einfluss zwischen einem geraden Fliigel and einem Rumpf
bei Nullauftrieb im Uberschallgebiet, Z. Flugw., 4:44-46, 1956.
49. Ward, G. N.: Supersonic Flow Past Slender Pointed Bodies, Quart. J. Mech. App. Math.,
2:75-97, 1949. Miles, J. W.: J. Aer. Sci., 19:287, 1952. Stocker, P. M.: Aer. Quart.,
3:61-79, 1951. Yang, H. T.: AIAA J., 10:1535-1536, 1972.
50. Whitcomb, R. T.: A Study of the Zero-Lift Drag-Rise Characteristics of Wing-Body
Combinations Near the Speed of Sound, NACA Rept. 1273, 1956.
51. WieseLsberger, C.: Airplane Body (Non Lifting System) Drag and Influence on Lifting
System, in W. F. Durand (ed.), "Aerodynamic Theory-A General Review of Progress," div. K,
Springer, Berlin, 1935, Dover, New York, 1963.
52. Woodward, F. A.: Analysis and Design of Wing-Body Combinations at. Subsonic and
Supersonic Speeds, J. Aircr., 5:528-534, 1968; NASA CR 2228, 1973. Bradley, R. G. and
B. D. Miller: J. Aircr., 8:400-405, 1971.
PART
THREE
AERODYNAMICS
OF THE STABILIZERS
AND CONTROL SURFACES
CHAPTER
SEVEN
AERODYNAMICS OF THE STABILIZERS
1
7-1 INTRODUCTION
7-1-1 Function of the Stabilizers
and Control Surfaces
The main parts of an airplane are the wing, the fuselage, and the tail unit or
empennage. The aerodynamics of the wing has been discussed in detail in Chaps.
2-4, and that of the fuselage alone and the interaction between the wing and
fuselage in Chaps. 5 and 6, respectively. Now, in Chaps. 7 and 8, the aerodynamics
of the stabilizing and control surfaces will be discussed. Generally, an airplane has
(see Fig. 7-1) a horizontal tail consisting of a horizontal stabilizer (tail plane) with
an elevator, a vertical tail consisting of a vertical stabilizer (fin) with a rudder, and
two ailerons.
A primary purpose of the tail unit is the stabilization of the airplane. This
means that the airplane should have the tendency to return to a stationary flight
attitude after a small disturbance. This process should take place "by itself"; that is,
the aerodynamic forces should move the airplane back to the original attitude
without application of the control surfaces.
Another equally important function of the tail unit is control of the airplane.
Whereas the horizontal stabilizer and the elevator control the motion about the
lateral axis, the vertical stabilizer with the rudder and the ailerons control that
about the vertical and longitudinal axes (Fig. 1-6). The control of the airplane
requires establishment of an equilibrium of the moments about the three axes. Here,
431
432 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
rH
-
Elevator
Figure 7-1 The geometry of the tail surfaces (empennage).
in addition to the moments of the aerodynamic forces, those of the inertia forces
play a role.
As has already been pointed out in Sec. 1-3-3, the motion of the airplane about
the lateral axis is termed longitudinal motion, that about the vertical and
longitudinal axes lateral motion (side motion). Consequently, the horizontal
stabilizer and the elevator stabilize and control the longitudinal motion. The vertical
stabilizer and the rudder stabilize and, together with the ailerons, control the lateral
motion.
Generally, each of the three control assemblies has the form of a wing with a
control surface as shown in Fig. 2-24. It consists of a fixed and a movable part. The
fixed part is termed a fin or vertical stabilizer at the vertical tail and a horizontal
stabilizer or tail plane at the horizontal tail. The movable part is the control surface.
It is termed a rudder at the vertical tail and an elevator at the horizontal tail. In
Fig. 7-1, the horizontal stabilizer and the elevator and the vertical fin and the
rudder are indicated by hatches. The changes of the moments required for control
are effected by deflections of the control surfaces. At the horizontal and vertical
tail assemblies, the moments may also be controlled by a stabilizer adjustment
(stabilizer trim). The horizontal tail of many airplanes does not have a separate
stabilizer and elevator. Here, the change of the moment about the lateral axis is
achieved by displacement of the entire horizontal surface.
The aerodynamic effect of the horizontal tail is illustrated in Fig. 7-2 for an
airplane with and without a horizontal tail. The lift coefficient is plotted against
both the angle of attack and the moment coefficient. According to Fig. 7-2a, the
contribution of the horizontal tail to the total lift is relatively small. Figure 7-2b
gives the moment curves for several setting angles EH at the tail plane. Comparison
AERODYNAMICS OF THE STABILIZERS 433
with the curves for the airplane without a horizontal tail shows that at all setting
angles EH of the tail plane, the horizontal tail causes a considerable increase in the
stability coefficient aCM/acL, as defined-in Sec. 1-3-3. A change in the tail-plane
setting angle EH causes a parallel shift of only the moment curve CM(CL).* When
the moment reference axis passes through the center of gravity at a steady flight
attitude, the equation cm = 0 describes the moment equilibrium about the lateral
axis. Figure 7-2b shows that this condition can always be satisfied by choosing the
proper setting angle EH of the tail plane for a given lift coefficient. The results on
wing-fuselage-tail systems at subsonic, transonic, and supersonic incident flows
reported by Pitts et al. [26] should be pointed out. Schlichting [34] gives a
summary of the importance of the interference among wing, fuselage, and tail unit
for the stability coefficients of the airplane.
The aerodynamics of the tail units will be treated in two parts: The problems
concerning the, tail surfaces without deflection of the control surfaces (stabilization)
will be covered in Chap. 7, those concerning the effect of the control surfaces
(control) and of the flaps (lift increase) will be discussed in Chap. 8.
7-1-2 Geometry of the Tail Surfaces
The geometry of the horizontal and the vertical tails may be described basically like
that of a wing (see Sec. 3-1). In general, the horizontal tail has a symmetric
planform and the vertical has an asymmetric side elevation (Fig. 7-1).
The planform of the horizontal tail is defined, in analogy to Sec. 3-1, by the
following main quantities:
*The conclusion should not be drawn that the fin setting angle does not affect the
longitudinal stability, because the determination of the degree of stability must always be
related to an equilibrium state.
a
b
10
0.8
02
Without
'r, ccWith
I
Me m9
horizon tal
tail
0
=0z
-'F°
0°
'f°
8°
a
12°
16°
20°-0,3
-02
-0.1
0
03
cM
Figure 7-2 Wind tunnel measurements on an airplane (Messerschmitt model Me 109) with and
without empennage. EH= setting angle of the tail plane (see Fig. 7-6a). (a) Lift coefficient vs. angle
of attack. (b) Lift coefficient vs. pitching-moment coefficient.
434 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
Span of the horizontal tail bH
Area of the horizontal tail AH
Aspect ratio of the horizontal tail AH = b%I/AH
Setting angle of the tail plane (Fig. 7-6a) £H
Deflection of the elevator (Fig. 7-6a) rlx
The position of the horizontal tail relative to the airplane is given by the lever
arm rH of the horizontal tail, defined as the distance between the geometric neutral
points of the horizontal tail and the wing. The geometric neutral point is defined in
Sec. 3-1.
For some airplanes, the high position of the horizontal tail relative to the wing
plays some role.
For the aerodynamic effects
of the horizontal tail, the following two
dimensionless quantities, which express size and location of the horizontal tail
relative to the wing quantities, are particularly important; area ratio AH/A and
relative tail-surface distance rHlcµ. Here, A is the wing area and cµ is the reference
wing chord according to Eq. (3-5b).
For a large number of airplanes, the area ratio lies between AH/A -- 0.15 and
0.25 and the relative tail distance between rH/cµ 2 and 3.
The side elevation of the vertical tail is described by the following quantities
(Fig. 7-1):
Height of the vertical tail h v
Area of the vertical tail A V
Deflection of the rudder 71V
The location of the vertical tail relative to the airplane is given by the lever arm
ry of the vertical tail, defined as the distance between the geometric neutral points
of the vertical tail and the wing. A general definition of the aspect ratio of the
vertical tail is not feasible because of the great variety of tail-surface shapes and the
various positions of the vertical tail relative to the fuselage and to the horizontal tail
(see Sec. 7-3-2).
For the aerodynamic effect of the vertical tail the following two dimensionless
quantities are important, as for the quantities for the horizontal tail: area ratio
AV/A and relative tail-surface distance ry/s, where s = b/2 is the wing semispan.
Approximately, A VIA = 0.1-0.2 and r y/s = 0.5 -1.0.
On many newer airplanes, the horizontal tail has been eliminated so that the
airplane has only a vertical tail as shown in Fig. 7-3. Such an airplane is termed an
all-wing airplane (flying wing). Here the function of the elevator (control about the
lateral axis) has been assigned to a control surface (elevator) of width bH.
Besides the most commonly used central arrangement of the vertical tail as
shown in Figs. 7-1 and 7-3, various other arrangements are also found. For instance,
Fig. 7-4a shows two fins (vertical tail surfaces) at the tips of the horizontal tail.
Figure 7-4b illustrates a tail surface with large dihedral (V tail surface), combining
the functions of both the horizontal and the vertical tails.
AERODYNAMICS OF THE STABILIZERS 43 5
Figure 7-3 The geometry of the empennage of
an all-wing airplane.
For the aileron, to be discussed in more detail in Chap. 8, the aileron span SA
as shown in Figs. 7-1 and 7-3 is important in addition to the aileron chord ratio.
7-2 AERODYNAMICS OF THE HORIZONTAL TAIL
7-2-1 Contribution of the Horizontal Tail to the
Aerodynamics of the Whole Airplane
Airplane in straight flight The lift acting on the horizontal tail adds considerably to
the pitching moment of the whole airplane because of. its large lever arm compared
bH
a
Figure 7-4 Various forms of horizontal tail. (a) Horizontal tail with two fins. (b) Horizontal tail
with large dihedral.
436 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
to the wing chord (see Fig. 7-1). Let LH be the lift of the horizontal tail andr r'
the distance of this lift force from the moment reference axis (usually the lateral
axis through the wing center of gravity). Then, from Fig. 7-5, the contribution of
the horizontal tail to the pitching moment of the whole airplane is
MH = -rHLH
(7-1)
where the nose-up pitching moment is taken as positive. Here the contribution of
the tangential force of the horizontal tail to the pitching moment has been
disregarded because of the small high position of the tail surface relative to the
fuselage axis. For the contributions of the horizontal tail to the lift LH and to the
pitching moment MH, dimensionless coefficients are introduced through
LH = CIHAHgH
(7-2a)*
MH = cMHAcAq00
(7-2b)
Here qH is the dynamic pressure at the location of the tail surface. It is, in general,
smaller than the dynamic pressure of the undisturbed flow qi, because of the effect
of the wing on the tail surface. The moment coefficient of the tail surface referred
to the wing quantities is obtained from Eqs. (7-1)-(7-2b) as
qH AH rH
CMH - -CIH q A C.
CIH =
with
dclH
daH
(aH - aCH 77H
(7-3a)
(7-3b)
al7f,
The lift coefficient of the horizontal tail CIH depends on, in addition to the
geometric data, its angle of attack aH and the elevator deflection i7H (see Fig. 7-6a).
The term dclHldcH represents the lift slope of the horizontal tail without
interference, and (aaH/a7IH)TIH the change in the direction of the horizontal tail for
zero lift caused by the elevator deflection. For the plane problem of the airfoil with
control surface (flap), this coefficient has been given as a function of the
control-surface chord ratio; for additional information see Chap. 8.
Generally, the incident flow direction of the horizontal tail is considerably
different from that of the wing because the tail surface is strongly influenced by the
wing and fuselage and lies in the wing downwash (interference). The incident flow
directions of the wing and horizontal tail differ, as shown in Fig. 7-6a, by the
downwash angle a, = w/U.., induced by the wing and fuselage at the location of
*Note that the index
I
has been chosen for the lift coefficient with reference to the
tail-surface quantities.
Figure 7-5 Contribution of the horizontal tail to
the pitching moment (schematic). C.G. = center
of gravity of the airplane. W = weight of the
airplane.
AERODYNAMICS OF THE STABILIZERS 437
a
Incident flow direction
of horizontal tail
rH
rH
b
-L
LoHLH
0XNH
rH N
Figure 7-6 Aerodynamics of a horizontal tail in straight flight. N25 = geometric neutral point,
N = aerodynamic neutral point, (N25 )H = geometric neutral point of horizontal tail. (a) Incident
flow direction of the horizontal tail, aH = a + eH + a W. (b) Aerodynamic forces on the wing and
horizontal tail.
the tail surface. Here, w < 0 means downwash and w > 0 means upwash. The angle
of attack of the horizontal tail thus becomes
aH = a -{
(7-4)
EH + Y-W
where off is the setting angle of the horizontal tail relative to the wing chord and a
is the angle of attack of the wing. Hence the contribution of the horizontal tail to
the pitching moment at zero elevator deflection becomes
CMH = - d«H (a + LH + aw) qH AH rH
(77H = 0)
(7-5)
k
where rH is the distance of the neutral point of the horizontal tail from the
moment reference axis of the airplane. The change of the moment with the angle of
attack at fixed setting angle of the horizontal tail (stability coefficient) is then
obtained as
acMH
( aa
yaap qH AH"H
dd1H
a0t) q,0 A CA
daH (
)eH=const
The quantity
a'%H_1
oa
,
I
afxw
G-x
`For simplicity it has been assumed that the ratio of the dynamic pressures gHJq, is
independent of the angle of attack a.
438 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
is termed the efficiency factor of the horizontal tail. For the moment change with
setting angle of the horizontal tail at constant angle of attack, Eq. (7-5) yields
dCIH qH AH r'y
CMH
( raeH )'%=Const
dag q. A CA
Comparison of Eqs. (7-6) and (7-8) shows that the moment change with angle
of attack (stability contribution of the horizontal tail) depends on the interference
between the wing and the horizontal tail. It is proportional to the efficiency factor
of the horizontal tail, aaH/aa = (1 +
The efficiency factor of the
horizontal tail is generally considerably less than unity, as will be shown more
accurately later. The moment change with setting angle of the horizontal tail
(control), however,
is
not affected by the interference if the ratio qH/q is
disregarded.
To establish the contribution of the horizontal tail to the lift of the whole
airplane, it is advantageous to define the lift coefficient of the horizontal tail, in
analogy to Eq. (7-2b), as
(7-9)
LH = CLgAq,o
In analogy to Eq. (7.5),
CLH
daH (a + EH + aw) qH AH
(7-10)
Here the comments made in connection with Eq. (7-5) apply also to the derivatives
of C1H with respect to a and EH.
In the investigations made so far of the contribution of the horizontal tail to
the pitching moment and the lift of the whole airplane, the respective coefficients
have been established as functions of the angle of attack of the airplane and the
setting angle of the horizontal tail. For some problems it is more favorable,
however, to establish the contribution of the tail surface to the angle of attack and
to the pitching moment as a function of the lift coefficient of the whole airplane
and of the setting angle of the horizontal tail.
The lift coefficient of the whole airplane is composed of that of the airplane
without the horizontal tail CL OH, and the contribution of the horizontal tail CLH,
that is, CL = CLOH + CLH. Hence the lift slope of the whole airplane, without
consideration of the effect of the tail surface on the wing at fixed setting angle of
the horizontal tail, is obtained from Eq. (7-10) as
dCL
dc1H
1+
aaw
4H AH
(7-11)
as ) qr A
M),-H=const
(doc)OH
The sought change of the angle of attack with the lift coefficient of the whole
dag (
airplane is given by the reciprocal value of the right-hand side of Eq. (7-11).
The change of the angle of attack with the setting angle of the horizontal tail
EH at constant lift coefficient of the whole airplane becomes
AERODYNAMICS OF THE STABILIZERS 439
_
as
8a)
aCLH
(aCL EH=const ( aEg )a=const
(a EH cL=const
(7-12)*
Here the second factor is given by Eq. (7-10).
Like the wing alone, the whole airplane has a neutral point, that is, a point on
which that portion of the lift force of the whole airplane acts that is proportional
to the angle of attack (compare Sec. 1-3-3). As shown in Fig. 7-6b, let the distance
of the neutral point of the whole airplane from the neutral point of the airplane
without the tail unit be designated as xNH. This distance is identical to the
neutral-point displacement caused by the horizontal tail. This neutral-point
displacement XNH, as shown in Fig. 7-6b, can be determined from the moment
equilibrium about the neutral point NOH of the airplane without the tail unit
XNHL = rHNLH. Here rHN is the distance of the neutral point of the horizontal tail
from the neutral point of the airplane without a horizontal tail. The result is
(aCLH\
as
XNH = aCL
eA-cons t
rHN
( as )eK=const
Introducing Eqs. (7-10) and (7-11) into this equation finally leads to the
neutral-point displacement caused by the horizontal tail,
XNH
Cµ
dCIH
aaw
dag ( i
aa )
dCL
+
( da )oH
+
dCIH
dag (
+
qH AH
q. A
aaw
rHN
qH AH cu
( 7 -1 3)
aa) q= A
In this equation, the first fraction on the right-hand side determines the percentage
of the tail-surface lever arm by which the airplane neutral point is shifted rearward
relative to the neutral point of the airplane without a horizontal tail. In many cases,
the second term of the sum of the denominator can be disregarded in comparison
with the first term. The neutral-point position of the whole airplane is obtained
from the neutral-point position of the wing alone (Chaps. 3 and 4), from the
neutral-point displacement caused by the fuselage (including the wing-fuselage
interference, Chap. 6), and from the neutral-point displacement caused by the tail
surface as given above.
The change of the moment coefficient of the horizontal tail as a function of
the lift coefficient from Eq. (1-29) is obtained immediately from the neutral-point
displacement caused by the horizontal tail in the form
*This equation follows from consideration of the total differentials
da = aCL dcL + aH dEH
with acL/aEH = BCLH/aEH.
and
dCL = aa« dcti + a H dEH
440 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
3CMH
aCL ej,t=const
XNH
(7-14)
CA
Finally, the change of the moment with the setting angle of the horizontal tail may
be determined for a fixed lift coefficient of the whole airplane. This is a free
moment because the total lift remains constant. As shown in Fig. 7-6b,
MH = -rHNLH, where LH is the lift of the horizontal tail caused by the change of
the setting angle of the horizontal tail, and rHN is the distance between the neutral
points of the horizontal tail and of the whole airplane. Thus, observing Eq. (7-10)
and with rHN = rHN - XNH, the following relationship is obtained:
aCMH
aCH
__ _ dCIH qH AH rHN
cL = const
daH q.. A
Cµ
(7-15)
This relationship is also valid for
The two coefficients of
Eqs. (7-14) and (7-15) can be taken from Fig. 7-2b, the first as the difference of
the slopes of the curves CM(CL) with and without tail surface and the second from
the curves with different setting angles EH of the horizontal tail.
To evaluate the above equations for the contribution of the horizontal tail to
the lift and the moment, attention must be paid to the ratio qH/q,,. Special
attention must be paid, however, to the angle of incidence caH of the horizontal tail,
because it depends strongly on the interference between the wing and the tail
surface. The angle of incidence of the horizontal tail aH is decisively affected by
the induced downwash angle a,,< 0 caused by the wing at the location of the
horizontal tail [see Eq. (7-4)]. In Fig. 7-7a, the change of aH with the rearward
position of the tail surface is shown under the assumption of a horizontal tail chord
parallel to the wing chord (EH = 0). At the wing trailing edge, aH = 0 because here
the kinematic flow condition requires that a + a11, = 0. With increasing distance
b
U.
-0-
TH
Figure 7-7 Angle of incidence of the horizontal tail. (a) In straight flight at angle of attack a. (b)
In pitching motion with angular velocity wy.
AERODYNAMICS OF THE STABILIZERS 441
from the wing trailing edge, aH increases and assumes a constant value at a large
distance that is considerably smaller than a. The distribution of a11, and thus of aH
behind the wing can be computed with wing theory. This matter will be discussed
below.
The lift slope of the horizontal tail dclH/daH for a horizontal tail free of
interference may be determined with wing theory.
Airplane in pitching motion So far only the influence of the airplane angle of
attack on the aerodynamics of the horizontal tail has been considered. In addition,
however, the rotational motion of the airplane about the lateral axis is particularly
important for the aerodynamics of the horizontal tail. During rotation of the
airplane about the lateral axis with angular velocity wy, an angle-of-incidence
distribution aH of the horizontal tail is created as shown in Fig. 7-7b that increases
linearly with distance from the axis of rotation. This angle of incidence at the
location of the horizontal tail (three-quarter point) becomes, with the distance from
the axis of rotation rH,
aH
with
Sly
=
V VC
x
(7-16a)
u
wycu
(7-16b)
V
as the dimensionless pitching angular velocity. By introducing this expression for aH
into Eq. (7-3b) and the resulting formula into Eq. (7-3a), the change of the moment
coefficient with the pitching angular velocity is obtained, with rH -- rH, as
aCMH
any
dc1H qH AH
daH q. A
(7-17)
This coefficient is termed the contribution of the horizontal tail to the pitch
damping. Comparison of this formula with Eq. (7-6) shows that the contribution of
the horizontal tail to the stability is proportional to (AH/A)(rJlcu), and that to the
damping is proportional to (AH/A)(rHlc2)2.
7-2-2 The Horizontal Tail in Incompressible Flow
The horizontal tail without interference The further discussions on the aerodynamics of the horizontal tail of this section will deal first with incompressible
flow and then with compressible flow at subsonic and supersonic velocities. The
horizontal tail without interference from fuselage and wing will be treated first,
followed by an account of the effect of the wing on the horizontal tail.
For the horizontal tail in incompressible flow without interference, the
three-dimensional wing theory of Chap. 3 can largely be applied. Of the
aerodynamic coefficients, first the lift slope de1H/daH for small and moderately
large aspect ratios AH is required. In Fig. 7-8 a few theoretical curves are given for
the lift slope of the horizontal tail as a function of the aspect ratio AH. A
442 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
Figure 7-8 Lift slope of a horizontal tail
without interference for incompressible
flow vs. aspect ratio of the tail surface
dN
AH (lifting-surface theory).
rectangular, a swept-back, and an elliptic wing are described. The elliptic wing
follows, from Eq. (3-98), the simple formula
dCIH
da$
_
_H
kg +
i+
i
with k$ ==;ig
CL
2
Ax
(7-18)
Further information on the lift slope and comparisons with measurements have
been given in Sec. 3-3. There, the neutral-point position can also be found, which is
required for the determination of the tail-surface lever arm.
The above data for the lift slope can be applied to a horizontal tail without a
vertical tail surface and also to a horizontal tail with a single vertical tail.
For a horizontal tail with two fins, as shown in Fig. 7-9, the lift slope is
considerably larger because of the end-plate effect. Theoretical investigations on
wings with end plates have been conducted by Mangler [221. The effect of end
plates on the lift slope can be taken into account approximately by introducing,
besides the geometric aspect ratio /1H, a so-called effective aspect ratio A*. For a
horizontal tail with end plates, these two values Aff and AH are related by the
empirical formula
1 = AB 1 -{- 21 bCV Ccv
V H
a
(7-19)
Measurements on the effect of end plates were first published by Prandtl and
Betz [27]. In Fig. 7-9, the lift slopes dc1H/doaf, based on those measurements, are
given as a function of the effective aspect ratio A * . The solid curve applies to the
rectangular wing of Fig. 7-8.
Effect of the fuselage on the horizontal tail The interference of the wing and
fuselage with the horizontal tail consists of a reduction of the dynamic pressure at
the location of the tail surface and also in an altered incident flow direction of the
tail surface. The reduction in dynamic pressure is caused mainly by the boundary
layer at the wing-fuselage interface, and the change in incident flow direction of the
AERODYNAMICS OF THE STABILIZERS 443
tail surface by the induced velocity field of the wing-fuselage system. Whereas the
induced velocity field can be reasonably well determined theoretically, the dynamic
pressure reduction must be found experimentally.
It is desirable that the value of the ratio qH/q, be as close to unity as possible
and that it be essentially independent of the angle of attack of the airplane. Both
requirements can be satisfied through suitable selection of the horizontal tail relative
to the wing and the fuselage; compare Hafer [13].
Now, the influence of the fuselage on the horizontal tail will be discussed first.
The arrangement of a horizontal tail on the fuselage corresponds basically to a
wing-fuselage system as treated in Sec. 6-2. There is the difference, however, that
the fuselage usually does not extend behind the tail surface. It is very difficult to
establish a general procedure for the computation of the influence of the fuselage
on the tail plane because of the many different arrangements of the horizontal tail
(high, mid, low surface) and the various shapes of the tail of the
fuselage.
Therefore, a review of some test results on this influence must suffice.
Koloska [13] reports three-component measurements on fuselage-tail surface
systems. The tail surfaces were rectangular of aspect ratio AH = 2 and 1.2, attached
to a partial fuselage. The lift slopes as affected by the fuselage, dc1H/daH, are
considerably smaller than those for the horizontal tail without interference as shown
in Fig. 7-8. In Fig. 7-10, values of dc1H/daH under the influence of the fuselage are
given as a function of the aspect ratio of the horizontal tail AH and the relative fuselage
width bF/bH. Accordingly, to give an example, at an aspect ratio AH = 2 and a relative
fuselage width bF/bH = 0.3, the fuselage effect reduces the lift slope by about 20%.
Effect of the wing on the horizontal tail The effect of the wing on the tail surface
consists essentially of a change of the angle of incidence of the horizontal tail
AH
3
3
AH= Ly = 3
2
S°
i8
3+`
6
°
Theory
c
bH
V-1
rcH'
1
!,
!
1
Figure 7-9 Measured lift slope of horizontal tail with end plates, from [27],
Profile 90535
i
0
b
2
3
A*H
vs. effective aspect ratio of the tail
1
6
surface A -, from Eq. (7-19). Theoretical curves from Fig. 7-8 for AH = ' H-
444 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
9
i3
<bF/b H
Z2
O
122
Q3
0.4
O'S'
0
1
2
3
AH
9
5
Figure 7-10 Lift slope of the horizontal tail as
affected by the fuselage vs. aspect ratio of the
horizontal tail fox several relative fuselage
widths bF/bH, from Koloska.
because of the induced downwash velocity behind the wing. The relationship
between the angle of incidence of the horizontal tail aH and that of the wing a is
given by Eq. (7-4); the change of the angle of incidence of the horizontal tail with
the angle of attack of the airplane is given by Eq. (7-7). In general, the coefficient
0) and, for a given wing, depends only on the
aax,/aa is negative
position of the tail surface. The coefficient aaH/aa acts as an efficiency factor of
the horizontal tail [Eq. (7-6); see also Eq. (7-13)]. Its value is usually between 0
and 1 and signifies that the downwash reduces the stabilizing effect of the
horizontal tail.
The aim of the remainder of this section is the determination of this efficiency
factor as a function of geometric and aerodynamic, data of the wing and of the
position of the horizontal tail relative to the wing.
The induced downwash velocity is generated by the vortex system of the wing
(bound and free vortices). Figure 7-11 illustrates schematically the vortex system of
a given circulation distribution. Figure 7-11a shows the free, not yet rolled-up
vortex sheet, whereas in Fig. 7-1 lb the free vortex sheet is rolled up into two single
vortices at a certain distance behind the wing.
A plane vortex sheet as in Fig. 7-1 la is unstable and tends to roll up into two
single vortices (see also Figs. 3-8 and 3-22). From the known vortex system of a
wing, the field of the induced downwash velocities is obtained with the Biot-Savart
law. The vortex system of a given wing is obtained from the lift distribution as
described in Sec. 3-3. In Fig. 7-12, the induced downwash and upwash angles on the
longitudinal axis (.x axis) are shown for an elliptic wing without twist. The induced
downwash angle a,, is referred to the induced angle of attack at = CL11TA of the
wing by Eq. (3-3 la). The ratio a,^,/al is dependent on the angle of attack of the
AERODYNAMICS OF THE STABILIZERS 445
r1
Figure 7-11 The vortex system behind a wing (schematic). (a) Not-rolled-up vortex sheet. (b)
Rolled-up vortex sheet.
wing. The relative downwash angle a11,/aj is given as a function of the dimensionless
longitudinal coordinate = x/s. The solid curve represents the induced downwash
angle of the total vortex system (bound and free vortices) from Eq. (3-96), and the
dashed curve represents the contribution of the free vortices. The difference
between the solid and the dashed curves is the contribution of the bound vortices.
This latter contribution becomes meaningless for t > 1. For such distances of the
tail surface, the induced downwash angle is determined predominantly by the
contribution of the free vortices, so that
a,, = -2ai
(7-20)
3
2
I
i
I
i
i
I
I
I
I
-
r
-
I
l
t
I
I
i
Z'
I
I
I
I
I
-2s
Free vortices
'
i
I
; Bound vortices
-2
CL
7rA
-3
Total vortex system
-9
-1,2
-1.0
-08
0.2
at
06
0.8
1,0
1.2
Figure 7-12 Induced downwash angle ai,,, on the x axis of a wing of elliptic planform, from [37].
Contribution of free and bound vortices.
446 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
To obtain the induced downwash angle at the location of the horizontal tail,
the position of the horizontal tail relative to the vortex sheet must also be known.
Here it must be realized that, in general, the vortex sheet behind the wing lies
neither in the wing plane nor in a plane parallel to the incident flow direction. Its
shape is curved as shown in Fig. 7-13. Its distance from the wing chord z = 0 is given by
zI(x, y). Because of the kinematic flow condition, the vortex sheet at the wing trailing
edge xr is tangent to the wing plane; farther downstream it is deflected more and more
upward from the wing plane. Its position may be easily determined from the equation
x
zl (x, y) = f [a -l
(7-21)*
a,,, (x, y) ] dx
X.
whose validity is obvious from Fig. 7-13. The location of the wing trailing edge is
given by x,.(y). Once the position of the vortex sheet is found, the distance of the
horizontal tail from the vortex sheet, needed to determine the induced downwash
angle at the location of the horizontal tail, is given by (z - zl ).
Now, by means of theoretical results and measurements, we shall discuss the
influence of the wing shape and of the lift distribution on the distribution of the
downwash angle behind the wing.
For the not-rolled-up vortex sheet with a given circulation distribution
T(y) = bU,7(y), the downwash angle at z = zl is obtained from lifting-line theory
by the Biot-Savart law from Eqs. (341), (3-50a), and (3-50b) as
1 lim
aw
2 rv
+1
4
E
y M - jr (,7y-01")
77/)2
1
+
l
d
L
I (S - ei)2 + (71 - 17 /)Z
(7-22)
*To evaluate the integral, the induced downwash aN, < 0 should be considered to be
constant.
Um
Figure 7-13 Position of the vortex sheet behind the wing (schematic).
AERODYNAMICS OF THE STABILIZERS 447
T= consf
at d z CL
111
I
b-as
3,5
IrW
1
t
Z
CL
a',1td
l(yf
c
2.5
Approximation to 2
3
j
40
0
0.2
a#
0.6
08
10
1.'
1.2
16.
18
20
Figure 7-14 Downwash angle in the vortex sheet ( = 1', ) for 77 = 0 (plane of symmetry of the
airplane) behind unswept wings, from Truckenbrodt, computed by lifting-line theory. Curve 1,
constant circulation distribution. Curve 2, elliptic circulation distribution. Curve 3, parabolic
circulation distribution.
Here t = x/s, rl = y/s, and = z/s are the dimensionless coordinates, and i = t(n')
gives the location of the lifting line in the wing from Fig. 3-29. For unswept wings
the coordinate origin lies on the lifting line and t' = 0. For the numerical evaluation
of this equation, a quadrature procedure has been developed by Multhopp [25].
Other computational methods and results have been published by Glauert [111,
Lotz and Fabricius [21 ], and Helmbold [211.
The effect of the lift distribution on the downwash distribution in the plane of
symmetry of the wing (77 = 0) and in the vortex sheet = si is shown in Fig. 7-14
for the rectangular, elliptic, and parabolic lift distributions. The downwash angle ati
is referred to the induced angles of attack ai in the middle of the wing (r1= 0),
whose values are also given in the figure. Hence, for all three lift distributions the
ratio a,,/cat = 2 far downstream of the wing. This result, which has been given in
Eq. (7-20), is obtained by setting t -o in Eq. (7-22) and comparing with Eq.
(3-71c). The curves of Fig. 7-14 demonstrate that the kind of lift distribution over
the wing span has a considerable influence on the values of the downwash angle at
small distances from the wing. For a constant circulation distribution the downwash
is expressed by the simple formula
0) = 1
ti
-
1
L
W = const)
(7-23)
where cL/27r%1= aj(0). This formula is obtained from Eq. (7-22), but also directly
from the horseshoe vortex by means of the Biot-Savart law. For the elliptic
circulation distribution the downwash angle becomes, according to Glauert [11],
448 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
-aw(,0)1-{-
V
zll
79
(T=To 1-rte)
where E is the complete elliptic integral of the second kind with the module
2 + 1. In the present case, ai(0) = CL/TtJl. The downwash angle at some
1/
distance behind the wing is given by an approximation formula of Truckenbrodt
[25] as
-a. ($, 77) -- tai (q) rt
ti
(2
+
1
(7-25a)
CL
-r/1
(5 >0)
2) CL
(7-25b)
45 n"1
This last expression applies to elliptic circulation distributions with cap = CL/irA. The
result of this formula is added in Fig. 7-14 as an approximation.
The effect of the wing planform on the distribution of the downwash angle
over the span at a distance = 1 behind the wing is shown in Fig. 7-15. The three
wings have an aspect ratio A = 6 and taper ratios X = 1.0, 0.6, and 0.2. This figure
shows that the shape of the wing planform decisively affects the distribution of the
downwash angle over the span. Hence the effectiveness of the horizontal tail
is much smaller for a highly tapered trapezoidal wing than for a rectangular
wing.
The solid curves were determined by a computational procedure of Multhopp
[25], whereas the dashed curves were computed using the approximation formula
Eq. (7-25a).
Figure 7-16 shows the effect of the sweepback angle on the distribution of the
downwash angle behind the wing. For simplicity, constant circulation distribution
over the span has been assumed for all those sweepback angles. The distribution of
the downwash angle over the longitudinal axis shows that the downwash is much
I---- b42s--i
v+
"_1K
3
I
,z
i
_'Ellipse
- Ellipse
---r-Eclipse -
X32
I
b
02
0.
0.6
08
100
c
l
0.2
i
!
9'/
0,6
08
10 0
0.2
0
0.6
08
10
Figure 7-15 Downwash-angle distribution over the span in the vortex sheet at distance = x/s = 1
behind the wing, for 3 unswept wings of aspect ratio A = 6, computed by simple lifting-line
theory. (a) Rectangular wing. (b) Trapezoidal wing of taper A = 0.6. (c) Trapezoidal wing of taper
X = 0.2. Solid curves, exact solution from Multhopp. Dashed curves, approximate solution from
Truckenbrodt.
AERODYNAMICS OF THE STABILIZERS 449
8
Z r= const
0°
-4s°
-
b= 2s
0
D..s
10
za
IS'
Figure 7-16 Distribution of the downwash angle aw on the x axis behind swept-back wings of
constant circulation distribution.
greater at a backward-swept wing than at a forward-swept wing. The Biot-Savart law
leads to the following simple formula for the downwash distribution:
0)
= 11
-{-
tanc)2 -{- 1 +
tangy)]
I
1
2nd.
(7-26)
where CL/27rzl = a=(0). Systematic measurements on the downwash of swept-back
wings have been conducted by Trienes [40] by the probe surface method. Note also
the investigations of Silverstein and Katzoff [38] and of Alford [2].
The results obtained so far were based on the flow with a not-rolled-up vortex
sheet. A few data will now be given of the influence of the vortex sheet roll-up on
the downwash at the location of the horizontal tail. As has been described by Fig.
7-11b and in more detail in Sec. 3-2-1, the vortex sheet rolls up into two single
vortices at some distance behind the wing. They have the circulation To of the root
section of the wing, and, from Eq. (3-58), are apart by bo far behind the wing. In
Fig. 7-17, the ratio bo/b is plotted against the aspect ratio for a rectangular wing
according to Glauert [11]. For an elliptic circulation distribution the ratio is
constant:
IT
b
4
(elliptic circulation distribution)
(7-27)
For rectangular wings, bo/b increases from this value when the aspect ratio A
becomes larger. For very large A, it approaches unity asymptotically, which is the
value
of the constant circulation distribution. A simpler computation of the
downwash at a rolled-up vortex sheet is possible by considering a horseshoe vortex
as in Fig. 7-17 of strength TO whose free vortices have the distance bo. This quite
idealized picture of the roll-up process has not been fully confirmed by
measurements of Rohne [16], as seen from Fig. 7-18. Here, the ratio bo /b and the
distance ao behind the wing at which the rolling-up process has been completed
450 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
7-----I-
10
r=cnnst
0.8
X0
Rectangular wing
TTF
Elliptic wing
r(y)
r
Figure 7-17 Aerodynamics of the rolled-
up vortex sheet behind a wing (schematic). Ratio b0 /b vs. aspect ratio of the
wing A. Rectangular wing from [I I ] .
6
A-
have been plotted against the lift coefficient. The measured ratio bo lb is noticeably
larger than the theoretical value of Fig. 7-17. A summary report on early downwash
measurements is given by Flugge-Lotz and Kuchemann [8].
Studies of the physical explanation of the roll-up process were first made by
Kaden [16] and Betz [161, somewhat later by Kaufmann [16] and Spreiter and
Sacks [39]. More recently, additional insight has been gained, to some extent,
through the use of efficient computers [3, 4, 12, 30, 421. To convey a feeling for
al
I
o Rectangular wing
c
T
Trapezoidal wing
1.0
b0
b=25
1019
b
0.7 0
d
10
0.5
15
CL
Figure 7-18 Measurements of the aerodynamics of the rolled-up vortex sheet behind a wing, from
Rohne. (a) Vortex system. (b) Tested wings (profile Go 387). (c) Distance ao at which the
rolling-up process is completed. (d) Distance bo between the two rolled-up vortices. Dashed
straight line, theory according to Fig. 7-17.
AERODYNAMICS OF THE STABILIZERS 451
the magnitude of the effect of the wing on the horizontal tail, the efficiency factor
of the horizontal tail from Eq. (7-7) is plotted in Fig. 7-1.9 against the aspect ratio.
These values apply to very large distances of the tail surface from the wing Q.-* -0) and
for wings with elliptic circulation distributions. With the value for the lift slope of Eq.
(3-98), the efficiency factor of the horizontal tail for not-rolled-up vortex sheets becomes
aaH
as
-1+
aaw
as
=
1/112 +4
142
-2
+4+2
(` ' °O)
(7-28a)
(y > cc)
(7-28b)
For the rolled-up vortex sheet (horseshoe vortex) it is
YA2+4-2 (bo)
8a
a«
=
z
1//1z+4+2
-1
-
with bo lb = it/4 from Eq. (7-27). At small wing aspect ratios, the efficiency factor
of the horizontal tail is relatively small; it increases strongly with A.
All the results on downwash obtained so far apply to control points in the
vortex sheet. The horizontal tail lies, depending on the angle of attack of the
airplane, in, above, or below the vortex sheet. Outside the vortex sheet the
downwash is always smaller than in the sheet. This will be shown by the following
examples. Before pursuing this matter, however, the position of the vortex sheet
(Fig. 7-13) will be discussed. With the help of Eq. (7-21), the position of the vortex
sheet is obtained from the distribution of the downwash behind the wing. In Fig.
7-20 the position of the vortex sheet in the root section r7 = 0 behind the wing is
shown for an elliptic wing. The distance between vortex sheet and the wing plane is
proportional to the angle of attack of the wing. For the downwash angle outside of
the vortex sheet, the following equation is obtained for a given circulation
distribution by generalization of Eq. (7-22) according to lifting-line theory:
+1
w
(77, ) =
1
_
where r =
f ()
i
)- - (b
(
[[I:/;(I::;:]2
\i +
(77
{7 -77')2+( -C1)2
\S - J)2 + ( - 771)2
r3
_i d ?7'
(7-29)
+ (S - J 1 )z
1.0
Figure 7-19 Efficiency factor of the horizontal tail aaH/aa in incompressible flow
02
vs. aspect ratio of the wing for rolled-up and
not-rolled-up vortex sheets. Computation
from lifting line theory for elliptic circulation distribution at a very large distance
behind the wing
452 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
Figure 7-20 Position of the vortex
0
sheet behind elliptic wings of several
aspect ratios A (see Fig. 7-13).
Z
1
r/cr
The quantities used in this equation are defined in connection with Eq. (7-22).
Equation (7-29) is converted into Eq. (7-22) by
According to Multhopp
[25], the change in downwash with distance from the vortex sheet is given by
a Lcx , - I C -- C11 Idly
8a,n
(7-30)
d.?71
Thus the curves of the downwash angle o
against the distance from the vortex
sheet have, in general, a break at the station of the vortex sheet. Experimental
results of this kind for unswept and swept-back wings are plotted in Fig. 7-21, from
Trienes [40]. They have been obtained by the probe surface method, which is
c-450 300 190 00 -30°
-04
-37
02
b
0.2
0.4
06
08
Oaf
0
-
02
0.4
- aa-W/ as
0,6
----
0e
4
Cr
i.
b=2s
Figure 7-21 Downwash distribution outside the vortex sheet; measurements of Trienes by means
of the probe surface method. xH = s = rearward position, and H = aH/s the relative high position
of the horizontal tail, a,,, = downwash angle as averaged over the probe surface. (a) For an upswept
trapezoidal wing. (b) For a swept-back wing of constant chord. Hatched area = probe surface.
AERODYNAMICS OF THE STABILIZERS 453
described in [40] and, therefore, are mean values of the downwash angle a,, over
the span of the horizontal tail surface. These experimental results confirm that the
downwash angle has a peak value in the vortex sheet. Finally, in Fig. 7-22,
theoretical downwash distributions from Glauert [11] are included for the
transverse plane far behind the elliptic wing. They show that, for any high position, downwash prevails within the wing span range and upwash outside this
range.
To compute the downwash in the vortex sheet, as pointed out above, a
quadrature method based on lifting-line theory has been given by Multhopp [25].
An extension of this quadrature method for the computation of the downwash
outside the vortex sheet has been developed by Gersten [10] for both the theories
of the lifting line and of the lifting surface.
The induced downwash velocity according to lifting-surface theory is obtained
from the velocity potential of Eq. (3-46), where w = aO/az, as
4-3
IV (x, y, z) =4 z
Gl (x, y. z;
_ y)) 3_
y,)
[(yy
8
-
-
+S
4
G2(x,y,z;y')
+ (zz
(y
_S
-y)2
_ 11) 2
-
)2]'
d y,
-
2
+(z-21)2dy'
(7-31)
a
rry
Upwash
0.
1.6
y
o,z
X
z
0.3
y!,
-04
Downwash
-0.8
' 1=QS
-12
CL
aL
0,1
JCI
f
03
02
-1.6
7-22 Theoretical downwash
and upwash angle distributions over
the span outside the vortex sheet for
an elliptic wing, from Glauert.
Figuze
0.2
0,6
ae
1,0
12
16
454 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
Here, G1 is the expression of Eq. (3-47), and
G2 (=C, y, 2; Y')
=
zr(y')
Z- 1'r'
,
') (, - ')x dx'
V (X - x')2 + (y - y')2 + (z
Xf(y')
-
21)23
(7-32)
In analogy to the lifting-surface method of Sec. 3-3-5, Gersten [10] based the
evaluation of Eq. (7-31) on two fundamental functions for the vortex density k. In
this way he succeeded in developing a relatively simple computational procedure to
determine the downwash.
Stabilization by the horizontal tail (neutral-point displacement) This discussion of
the downwash will now be concluded with a simple reflection on the displacement
of the neutral point of the airplane caused by the horizontal tail xNH (see Fig. 7-6).
The analytical expression for this quantity has been given by Eq. (7-13). Let the
wing and horizontal tail be of elliptic planform and the distance between the two
neutral. points be rffN. The aerodynamic coefficients in Eq. (7-13) have already
been discussed in detail. The lift slope of the airplane without horizontal tail
(dcL/da)OH is taken to be equal to that of the wing according to Eq. (3-98). The
lift slope of the horizontal tail without interference has been given in Eq. (7-18)
and the efficiency factor of the horizontal tail (1 +
in Eq. (7-28a). Under
the assumption that qH/q = 1, introduction of these expressions into Eq. (7-13)
yields, after some intermediate steps,
AH
a waH A
xNH
1 + awaH
AH rHN
(7-33)
A
Here
aw =
A
d2+4+2
4g
ag =
!l$+4±2
(7-34a)
(7-34b)
Equation (7-33) expresses a remarkably simple relationship between the neutralpoint shift caused by the horizontal tail and the four geometric parameters: aspect
ratio of the wing A and of the tail surface AH, respectively; ratio of the areas of
horizontal tail and wing AHIA; and distance between the neutral points of the tail
surface and the wing rHJ1. This relationship is shown in Fig. 7-23. In this diagram is
also shown the neutral-point displacement that would be obtained without
interference. It is computed, for simplicity, by the stripe method, in which the lift
slopes of wing and horizontal tail are set equal to 27r. This case is obtained from
Eq. (7-33) with aw = aH = 1 as
_ A
(7-35)
xNH ` A +HAH rHN
AERODYNAMICS OF THE STABILIZERS 455
016
Stripe
method
NW
Wing
........ ,.
.
2 xNHI
0.12
T a il
,""i
I
f
s ur ace
/
A-6;Ay=61
00
A=12;11y=6
002
Figure 7-23 Neutral-point displacement caused
by the horizontal tail of wing-horizontal tail
0
02
0.1
AH/A
03
systems vs. the area ratio AHIA, from Eq. (7-33).
Stripe method from Eq. (7-35).
The difference between this curve and the others indicates the interference effect
of the wing on the horizontal tail with respect to the neutral-point displacement, including the influences of the finite aspect ratios of wing and tail
surface.
Stability at nose-high flight attitude (stall) When an airplane gets into the nose-high
flight attitude, safety requires that the pitching-moment curves in this range still be
stable (aCMl aCL < 0). For many wing shapes, for example, swept-back wings of
large aspect ratio, this condition is not fulfilled. There are a number of measures,
such as, for example, boundary-layer fences and slat wings, that lead to a wing stall
behavior ensuring that no nose-up (tail-heavy) pitching moment (pitch up) can
occur. Particular attention must be paid to the effect of the downwash as changed
by the partial flow separation from the wing on the horizontal tail. Besides the wing
planform, the position of the horizontal tail relative to the wing plays an important
role, and particularly the high position of the tail surface. Furlong and McHugh [9]
give a detailed report on this problem.
Severe stability problems can arise, particularly for swept-back-wing airplanes
with a tail surface in extreme high position (T fin) at very large angles of attack.
Here the horizontal tail lies in the separated flow of the wing, and its incident
flow has a very low velocity. This leads to an unstable action and an almost
complete loss of maneuverability. Then the angle of attack increases more and
more until, eventually, at a very large angle of attack, a stable flight attitude
is again established. Because of the lack of control effectiveness, it is impossible to change this extreme flight attitude, and the airplane is in danger of
crashing. This flight attitude is termed "super-stall" or "deep stall." Byrnes et
al. [6] have studied this problem in detail.
456 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
7-2-3 The Horizontal Tail in Subsonic
Incident Flow
The effect of compressibility on the aerodynamic coefficients had been determined
by means of the Prandtl-Glauert-Gothert rule for the wing in Sec. 4-4 and for the
wing-fuselage system in Sec. 6-3-1. In the same way, this effect can be determined
for the horizontal tail. Through a transformation, the subsonic similarity rule allows
one to reduce the compressible subsonic flow about the whole airplane to
incompressible flow. Here the incompressible flow is computed for a transformed
airplane as shown by an example in Fig. 7-24 for Ma = 0.8. The transformation of
the geometric data is given in Eqs. (6-29)-(6-31). For the geometric data on the
horizontal tail, Eqs. (6-30a)-(6-30e) apply accordingly. For the transformation of
the distance of the tail surface from the wing, the relationship rHinc = rH has to be
added, observing Eq. (6-29). The same relationship as for the wing alone applies to
the dependence of the lift slope of the horizontal tail without interference on the
Mach number Ma.. Hence, with Eq. (4-74), the relationship
dc1H
dxH
2nAH
V(1
- Ma') A'2 + 4 -F 2
(7-36)
is obtained, which is shown in Fig. 4-45. By computing the incompressible flow for
the transformed airplane at the angle of attack of the subsonic flow, that is, for
«inc = a, the induced downwash angle in the vortex sheet becomes
aw(S, n) = aw inc(inc, Thnc)
(7.37a)
= - 2ai inc
(7-37b)
( -* °°)
This relationship allows one to determine in a very simple manner the downwash
field of compressible flow from that of incompressible flow. A simple approximation formula for the downwash of incompressible flow at some distance behind the
wing has been given by Eq. (7-25b). With the above transformation and with Eq.
Lla
yi nc
Figure 7-24 The Prandtl-Glauert rule at subsonic incident flow velocities. (a) Given airplane. (b)
Transformed airplane.
AERODYNAMICS OF THE STABILIZERS 457
23
1.5
0.6
OZ
08
110
Figure 7-25 Effect of Mach number on the
downwash angle at the longitudinal axis
behind a wing of elliptic circulation distribution, from Eq. (7-38).
Ma.
(4-72a), this formula can be reduced to subsonic flow. For elliptic lift distribution
there results
- aw =
L
J_451 (1 _ Mat00 )J rcll
(7-38)
In Fig. 7-25 the downwash angles so computed for
= 1, 1.5, and 2 have been
12
plotted against the Mach number Maw, .
As a further result, in Fig. 7-26 the efficiency factors of the horizontal tail
from Eq. (7-28a) are plotted against the Mach number for several aspect ratios. The
analytical expression is
caXE
- 1 + 8a, _
8a
'am
A2 (1
- Moo) + 4 - 2
VA2(1-Mat)+4+2
oo)
(7-39)
0.7r
Q6F-
02
0,1
Figure 7-26 Efficiency factor of the horizontal tail vs. Mach number for elliptic wings
of various aspect ratios A, from Eq. (7-39) for
Ma,,,
458 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
This figure indicates the remarkable result that the efficiency factor decreases
strongly with increasing Mach number at all aspect ratios A. For Ma = 1, the
efficiency factor of the horizontal tail becomes zero at all aspect ratios, a result in
agreement with slender-body theory (see also Sacks [32] ). Finally, in Fig. 7-27, the
efficiency factor of the horizontal tail acH/aa for a delta wing of aspect ratio
A= 2.31 is given for several Mach numbers as a function of the tail surface
distance. Accordingly, the efficiency factor changes only a little with Mach number
in the range 0 < Ma., <0.8.
7-2-4 The Horizontal Tail in Supersonic
Incident Flow
Fundamentals The influence on the horizontal tail of the forward airplane
components (wing and fuselage) is, at supersonic incident flow, generally greatly
different from that at subsonic incident flow. This difference is a result of the
limited influence zones at supersonic incident flow as shown in Fig. 7-28. The flow
at a point of the horizontal tail can be affected only by the parts of the airplane
lying within the upstream cone of this point. This cone is, from Fig. 4-58, the Mach
cone of the generating semiangle p, located upstream of the control point under
consideration and with axis parallel to the incident flow direction. The relation
between incident flow Mach number and Mach angle is given by Eq. (4-80). The
upstream cone cuts out of the airplane the influence zone that affects the
horizontal tail (see also Fig. 4-58). This influence zone is marked in Fig. 7-28 for
two Mach numbers (Mach lines ml and ?n2, respectively). The influence zone
shrinks with increasing Mach number; that is, it would be expected that the effect
on the horizontal tail, particularly of the upstream-lying wing, decreases with
increasing Mach number. Furthermore, Fig. 7-28 demonstrates that the distance
between the horizontal tail and the wing is of paramount importance for the
magnitude of the interference. At constant Mach number, u = const, the horizontal
os
0.1
Figure 7-27 Effect of Mach number on the
1.0
0.8
06
0.4
cr/J
02
0
efficiency factor of the horizontal tail behind
a delta wing of aspect ratio A = 2.31.
AERODYNAMICS OF THE STABILIZERS 459
mz
I,-,
\
I I- \ I
mz
u,
Figure 7-28 Effect of wing and fuselage on the horizontal
tail at supersonic velocity.
tail is less affected when it is close to the wing than when it is farther away. To
establish computational methods for the determination of the downwash at the
location of the horizontal tail, those for incompressible flow must be modified to
take into account whether, as in Fig. 7-28, the influence zone of the horizontal tail
encloses, at the respective Mach number, only a part of the wing (ml) or the whole
wing (m2).
As a first step, the physical character of the downwash field generated by a
wing in supersonic incident flow will be discussed qualitatively by means of Fig.
7-29. Here a rectangular wing is sketched with its circulation distribution as in Fig.
4-79a. It generates downwash and upwash velocities only within the two Mach
cones originating at the two forward corners. In the middle part of the wing of
width b* the flow is purely two-dimensional, and according to Fig. 4-21 does not
generate a downwash behind the wing. Thus the triangular zone I of Fig. 7-29
remains without downwash (a.,, = 0). From the triangular surface zones at the wing
tips in which the circulation drops off, free vortices are shed downstream as in
incompressible flow. Thus downwash velocities (ati < 0) are in zone II behind the
wing. Conversely, upwash velocities (a,,> 0) prevail in the two zones III that
contain the outer halves of the two Mach cones. In the entire range IV before and
beside the wing, outside of the Mach cones aw = 0.
The horizontal tail without interference in supersonic flow According to Sec. 7-2-1,
the contribution of the horizontal tail to the pitching moment and to the lift of the
whole airplane depends on the lift slope of the tail surface dcjH/daH and on the
efficiency factor aaH/aa = 1 + aa,/aa. First, a few data will be given on the lift
slope dclH/doH of the horizontal tail without interference. They may be taken from
Sec. 4-5-4, in which the theory of wings of finite span at supersonic incident flow
460 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
i
y
I
W>0
W.0
Figure 7-29 Induced downwash and
upwash fields in the vicinity of a
rectangular wing in supersonic incident flow (schematic).
was discussed. For a horizontal tail of rectangular planform as in Eq. (4-112), the
lift slope becomes
d clH
4
-j
1
1
2 AH Maw --1
(7-40)
if AH Ma;, - 1 > 1. The first factor represents the lift slope in plane flow, the
second the correction for the finite aspect ratio of the horizontal tail. This
relationship is illustrated in Fig. 4-78a.
Influence of the wing on the horizontal tail in supersonic incident flow For
quantitative assessment of the qualitative findings about the downwash at supersonic
flow, first the simple case of a wing with constant circulation distribution over the
span will be investigated. In this case, for supersonic flow the effect of the wing on
its vicinity can also be described by means of a horseshoe vortex, whose bound
vortex lies on the wing half chord. The effect of the two free vortices is restricted,
however, to the range within the Mach cones originating at the wing tips. Only the
downwash on the x axis will be computed for this arrangement. This can be done
by means of the results for the horseshoe vortex at incompressible flow according
to Eq. (7-23), which may be applied to supersonic flow by referring to the
corresponding discussion of Sec. 4-5. Thus, the distribution of the downwash angle
on the x axis behind the wing becomes
0) = cL
,
(Mad
- 1)
(7-41)
5
where cL121rA = aj(0). The downwash distribution according to this equation is
shown in Fig. 7-30 for several Mach numbers. These curves demonstrate that, as has
AERODYNAMICS OF THE STABILIZERS 461
already been discussed in connection with Fig. 7-29, no downwash at all exists on
the middle section over a certain stretch closely behind the wing (down to
l;a = Ma.. - 1). For large distances, > 0, first the downwash increases strongly
and then reaches the asymptotic value ati,, = -2a, = CL /trzl for
which is the
value for incompressible flow (see Fig. 7-14).
To show more accurately an induced velocity field of a free vortex at
supersonic flow, the velocity distribution will now be considered in a Mach cone
originating, as shown in Fig. 7-31, at the tip of a semi-infinite wing. This flow was
first studied by Schlichting [33]. In Fig. 7-31c the streamline pattern is shown in a
lateral plane x = const, normal to the Mach cone axis. Here the cone shell is a
singular surface because it is formed completely by Mach lines. The streamline
pattern within the Mach cone consists partially of closed streamlines encircling the
vortex filament and partially of streamlines entering the cone on one side and
leaving it on the other. Near the cone axis, the flow is comparable to that in the
vicinity of a vortex filament in incompressible flow. The distribution of the
downwash velocity over the Mach cone diameter for the plane z = 0 is obtained
according to [33] as
w=
TO
2ny
1-
xtanu
(7-42)
y
This distribution is shown in Fig. 7-31d, where x tang =R is the radius of the
Mach cone at the distance x. Because w = TQ/21ry in the potential vortex, it can be
concluded from Eq. (7-42) that, at supersonic flow, the distribution of the induced
velocity near the axis y = 0 deviates only a little from that at incompressible flow.
Both distributions are given in Fig. 7-3 Id.
Lagerstrom and Graham [17] gave an exact solution for the downwash field of
the inclined plate of semi-infinite span. They obtained it by means of the
cone-symmetric flow (Sec. 4-5-2) by first establishing the solution for the laterally
cut-off plate of infinite chord, which is
`CCW
as
= -1 (t <0)
(743a)
Figure 7-30 Downwash at the longi-
tudinal axis of a wing of constant
0.5
10
15
2.0
2.5
30
circulation distribution (horseshoe vortex) at supersonic velocities of several
Mach numbers Ma,o, from Eq. (7-41).
462 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
LT
a
y
i
f--R
Section
R --=cons
z
I
y
c
Figure 7-31 Velocity distribution within the Mach
cone of a free vortex at supersonic flow. Semi-
d
infinitely long wing of constant circulation distribution, from Schlichting. (a) Circulation distribution.
(b) Wing planform and Mach cone. (c) Streamline
pattern of the section x = const. (d) Downwash and
-R
upwash velocities in the plane z = 0, Solid curve,
from Eq. (7-42). Dashed curve, plane potential
vortex.
=t
t
i
+
O
arcta n
2
t (i - t) _ I
(0 < t <+ 1
1-2t
)
(7-43b)
Here, as in Fig. 7-32, t = y/x tan µ = y/R.
This solution leads to that for the downwash field of the laterally cut-off flat plate
of finite chord by superposition. In Fig. 7-32, the distribution of the downwash factor
aa,,,/aa in the plane of the plate is shown for several distances x/c behind the plate.
There are downwash velocities within the inner half of the Mach cone, upwash
velocities within the outer half. The curve for x/c = 1 applies on the inner half to
points immediately behind the trailing edge, whereas, from Eq. (7-43a), aa,1J/aa = -1
for points on the surface. At a very large distance (x - 00), the following expressions
are obtained:
2
ac
It
aa
aaW
as
__
2
1c
(-Ro <y<0)
1-
(7-44a)
(y > 0 and y < -Ro)
(7-44b)
Here Ro = c tan p is the radius of the Mach cone at the wing trailing edge.
The downwash field of the rectangular wing of finite chord and finite span is
AERODYNAMICS OF THE STABILIZERS 463
t
y
\
0
I
-'
Upwash
0.
t=-1 / =- Ro
t °9
A--- R
ya
2
f
C
0.4
l5
f
I
02
10
-Q2
5
2
10
I
!
C
-06
2
Downwash
at
-110
-0.2
0.2
0C
08
1.0
Figure 7-32 Distribution of the downwash factor behind a semi-infinitely long flat plate of chord c
at supersonic velocities for several distances x/c, from [ 17] .
obtained from the above solution by superposition. In Fig. 7-33, the downwash
factor aa,,/aa for the middle section according to Laschka [18] is plotted against
the distance x/c and with /1 Ma,,, - 1 as the parameter. Here the downwash factor
shows the same trend as seen in Fig. 7-30. For A Ma., - 1 < 2, Mach lines
originating at the 2 forward corners intersect each other on the wing. Thus there is
t
y/7z
I
A d -1 a 1.0
z
:
rT
3
zs
,
-
2.0
(1T7
4
,l'/C - -
Figure 7-33 Distribution of the downwash factor on the
longitudinal axis behind rectangular wings at supersonic
incident flow for various values of the parameter
=1
Mao, -1, from [ 18] . Asymptotic values for x
from Eq. (7-45).
464 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
no zone behind the wing in which the downwash is zero. At a very large distance
behind the wing (x -), there is, for y = 0,
a=2N
1- 1-
2
A Ma's - 1
(A - , I M - 1 > 2)
For A Ma;. - 1 < 2 the result is
(7-45)
a rectangular
wing of aspect ratio Al = 2, the downwash factor aax,/aa is given in Fig. 7-34 at
several distances x/c as a function of the Mach number Ma.. The very strong
influence of the Mach number on the efficiency factor of the horizontal tail is
obvious.
Experimental studies about the downwash behind the rectangular wing at
supersonic velocities have been conducted by Davis [1] and by Adamson and
Boatright [ 1 ]
.
The above theoretical results have been obtained with the lifting-surface theory.
Mirels and Haefeli [24] developed a lifting-line theory that has been applied to both
rectangular and delta wings. The results of this lifting-line theory agree with the
lifting-surface theory at some distance behind the wing, as would be expected.
Another computational method for the downwash, applying dipole distributions, has
been given by Lomax et al. [20]. By using this method, comprehensive
computations of examples have been conducted on delta wings with subsonic leading
edges. Likewise, delta wings with subsonic leading edges have been treated by
Robinson and Hunter-Tod [29] and by Ward [411. Some results for delta wings with a
supersonic leading edge are found in Lagerstrom and Graham [17].
The results presented so far in Figs. 7-32-7-34 apply to the conditions on the
vortex sheet (z = 0). In conclusion, a few data will now be given for the downwash
factor outside the vortex sheet. In Fig. 7-35, aaw/aa is plotted against the vertical
coordinate for several values of the parameter
Ram, - 1. As for incompressible
1.
I
IN
r
09
t
77
d
C
0.2
Figure 7-34 Distribution of the downwash
factor on the longitudinal axis behind a
rectangular wing of aspect ratio A = 2 at
0
1.0
1,2
14
16
18
Mam
20
2,2
1,4
supersonic incident flow for several Mach
numbers Ma., from [18].
AERODYNAMICS OF THE STABILIZERS 465
13
1.2
1,1
1,
0.9
0.8
A Maw-1=Z.O
30l
0.
0.3
02
y=0
X=W
0.1
III
0
-02
0
0.2
0.
06
Figure 7-35 Downwash factor in the root section
0.8
1.0
1.2
14'
(y = 0) behind rectangular wings vs. the high
position at supersonic velocities, from [18 ].
flow (Fig. 7-21), the downwash factor decreases strongly with increasing distance
from the vortex sheet. Corresponding results for delta wings are found in
[20].
Now a computational method that is analogous to that for incompressible flow
will be briefly described. The transformation from incompressible to supersonic flow
has been explained in Sec. 4-5. Accordingly, Eq. (7-31) for the downwash velocity
is also valid for supersonic flow if the function G1 is replaced by the function G of
Eq. (4-95) and the function G2 , corresponding to Eq. (7-32), by
G2 (x, y, z; y) = -
2co(a -
M '
1) lc (x ' , ii)
' (x - x' ) d x '
(x - x')2 - (Maw - 1) [(y - y')2 + zz]3
(7-46)
Xf(y')
Here xo(y') is the location of the Mach line according to Eq. (4-96). Laschka [18]
suggests that one compute G in Eq. (4-95) and G2 in Eq. (7-46) by taking the
vortex density k as a constant over the chord x and as a variable over the span y,
that is, k(x, y) = k(y). Thus G1 and G2 can be integrated in closed form. For the
determination of the downwash velocity w in Eq. (7-31), only an integration over
the span coordinate remains to be done. Ferrari [7] gives a summary survey of the
downwash in compressible flow.
The integral
integrals.
must be evaluated after the Hadamard method of finite parts of divergent
466 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
7-3 AERODYNAMICS OF THE VERTICAL TAIL
7-3-1 Contribution of the Vertical Tail to the
Aerodynamics of the Whole Airplane
The airplane in sideslipping flight The function and the geometry of the vertical tail
have already been described in Sec. 7-1. As shown in Fig. 7-36, the vertical tail at
asymmetric incident flow of the airplane of sideslip angle 0 is subject to a side force
Yv. Because of its large lever arm, this side force generates the predominant portion
of the yawing moment due to sideslip of the whole airplane. Moreover, the vertical
tail also contributes to the side force due to sideslip and the rolling moment due to
sideslip of the airplane. The contribution of the vertical tail to the yawing moment
,due to sideslip of the airplane is
MZ v
= -rVYv
(7-47)
where, from Fig. 7-36, r'y is the distance of the side force vector of the vertical tail
from the moment reference axis that generally coincides with the vertical axis
through the airplane center of gravity.
In analogy to Eqs. (7-2a) and (7-2b) for the horizontal tail, dimensionless
coefficients may be introduced for the side force Yy and the yawing moment MZ y
of the vertical tail by
Incident flow direction
of the vertical tail
YV = c1VAVgV
(7-48a)
Mz V = CMz VAsq
(7-48b)
Figure 7-36 Incident flow direction
of the vertical tail. C.G. = center of
gravity of the airplane.
AERODYNAMICS OF THE STABILIZERS 467
Here qV is the dynamic pressure at the location of the vertical tail, which is
generally smaller than the dynamic pressure of the undisturbed flow q. because of
the interference of wing and fuselage with the vertical tail. The coefficient of the
yawing moment of the vertical tail, referred to the wing quantities, is obtained from
Eqs. (7-47)-(7-48b) as
gVAVrV
CMzV=-C1vq A s
with
ctV
dcjv
day
(7-49a)
v - aay
(7-49b)
The lift coefficient of the vertical tail ctv depends on the angle of attack ay
(angle of sideslip 1iv) and the rudder deflection 77V of the vertical tail, in addition
to its geometric data. The term dctvldav stands for the lift slope of the
interference-free vertical tail and (aav/arlv)riv stands for the change in the zero-lift
direction of the vertical tail caused by the rudder deflection.
In some cases the incident flow direction of the fin (3v is considerably different
from that of the airplane 0 because of the interference of wing and fuselage with
the vertical tail. The two incident flow angles differ, as shown in Fig. 7-36, by the
sidewash angle av = v/U. induced by the wing and fuselage at the location of the
vertical tail:
(7-50)
3V=0+3v
Hence, for a rudder deflection of zero, the contribution of the vertical tail to the
yawing moment is given as
rs
dc, V
CMzV =
day (0 + v)
A
(7-51)
(77V = 0)
Q moment with the angle of sideslip
It follows, then, that the change in yawing
(contribution of the vertical tail to the directional stability, Sec. 1-3-3) becomes
aCMZV
ao
dcty
daV
aLv
1+
qv Av rv
a3 q A
(7-52)*
s
The quantity
a0V = I +
a
(7-53)
is designated as the efficiency factor of the vertical tail. From Eq. (7-52) it follows
that the contribution of the vertical tail to the directional stability is proportional
to the efficiency factor.
To establish the contribution of the vertical tail to the side force of the whole
airplane, the coefficient of side force of the vertical tail is defined, in analogy to
Eq. (7-9) for the horizontal tail, as
*Here for simplicity it has been assumed that the ratio qv/q. is independent of the sideslip
angle g.
468 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
Yy = cyyAq,0
(7-54)
In analogy to Eq. (7-52), the contribution of the vertical tail to the side force
due to sideslip becomes
acyy _ dc,V
as
day
A
1 +a0u
30) qv Ay
q.
(7-55)
Hence, the contribution of the vertical tail to the side force due to sideslip, too, is
proportional to the efficiency factor of the fin. Generally, the vertical tail also
contributes to the rolling moment due to sideslip because the point of application
of the vertical tail side force lies, in most cases, considerably above the airplane's
longitudinal axis.
The airplane in yawing motion Besides the sideslipping considered so far, the rotary
motion of the airplane about the vertical axis (yawing motion) is also of great
importance to the aerodynamics of the vertical tail. A rotary motion about the
vertical axis with angular velocity o
9V =
with
Px =
generates a sideslip angle at the vertical tail
w y
= Qz
sr
( 7 - 56a)
CUzS
(7-56b)
V
as the dimensionless angular sideslip velocity. By introducing this expression for py
into Eq. (7-51) considering Eq. (7-50), the change in the coefficient of the yawing
moment with the angular sideslip velocity becomes
acMzv
as2z
_ _ dcly qv Ay rv
day , qc A
s
2
(7-57)
This coefficient is termed the contribution of the vertical tail to the sideslip or yaw
damping. Comparison of this formula with Eq. (7-52) shows that the contribution
of the vertical tail to the directional stability
in terms of the geometric
quantities, proportional to (A y/A)(r'y/s) and that to the yaw damping is
is,
proportional to (A y/A)(r'y/s)2 . The following discussions will be limited to
incompressible flow.
7-3-2 The Vertical Tail without Interference
To evaluate the above equations, the lift slope dcl y/day must be known for the
interference-free vertical tail. Basically, this can be computed with the methods of
three-dimensional wing theory. Since the shapes of the vertical tails are in most
cases quite asymmetric, this task is particularly complicated. Hence, wind tunnel
measurements are indispensable for the acquisition of these aerodynamic quantities
of the vertical tail. An attempt has been made in Fig. 7-37 to represent the
measured lift slopes of single-fin assemblies with partial fuselages as a function of a
uniquely defined aspect ratio ply = bv/A V. The meaning of A y and bV is obvious
AERODYNAMICS OF THE STABILIZERS 469
° 9 Circular
cross section of
the fuselage
Rectangular
0
Without
. a With
0,5
10
1
horizontal tail
2.0
15
Av
is'
Figure 7-37 Measured lift slopes of
an interference-free vertical tail with
partial fuselages from DVL measurements and Koloska [ 13 ].
from the sketch in Fig. 7-37. The aspect ratios Ay he between 1 and 2. Fuselages
of round and rectangular cross sections and with horizontal and vertical tail edges
were investigated as well as systems with and without horizontal tails. The ratio of
the fuselage
height hF to the span b y of the vertical tail was limited by
hF/b y = 0.35 and 0.5. Curve 1 of Fig. 7-37 shows the theoretical trend for the lift
slope as in Fig. 3-32. It represents approximately the test points for vertical tails
with circular fuselages and with horizontal tails. For such vertical tails, the lift slope
follows the relationship
dcry
day
_
2r11y
4V+ 4 + 2
(7-58)
Curve 2, lying considerably lower than curve 1, represents a vertical tail with
fuselages of rectangular cross section and without horizontal tail surfaces. Between
these curves he, as curve 3, the results for systems of circular fuselages without
horizontal tails and those of rectangular fuselages with horizontal tails. Additional
measurements for vertical tail assemblies with two fins are given in [35].
Theoretical studies of the lift slope of a vertical tail with a horizontal tail have been
conducted by Rotta [31].
It is almost impossible to give generally valid data for the aerodynamic
coefficients of vertical tails at compressible flow.
7-3-3 Effect of the Wing-Fuselage System
on the Vertical Tail
Fundamentals As has been shown in Sec. 7-2-2, the effect of the wing on the
horizontal tail at symmetric incident flow lies essentially in the downwash of the
470 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
wing. The fuselage and the relative position of wing and fuselage (wing high
position) contribute little to the interference. In all cases, however, the effectiveness
of the horizontal tail is reduced by the wing and fuselage.
Considerably different conditions prevail for the effect of the wing and fuselage
on vertical tails at asymmetric incident flow. Schlichting and Frenz [35, 36]
showed that vertical tails are markedly affected only by a combination of wing and
fuselage. This interference results in an increase or in a reduction of the
effectiveness of the vertical tail, depending on the high position of the wing. This
influence on the vertical tail
is
caused
physically by the quite asymmetric
circulation distribution over the span of wing-fuselage systems. This asymmetry,
explained by Fig. 6-6, causes a rolling moment due to sideslip. This fact has been
discussed in Sec. 6-2-3. In Fig. 7-38, this antirnetric circulation distribution along
the span is illustrated for a sideslipping high-wing airplane. The lift increase of the
leading wing-half and the lift decrease of the trailing wing-half generate a pressure
drop on the upper side of the wing toward the advancing wing-half. This pressure
drop leads to an induced flow, as explained in Fig. 7-38, which revolves around the
wing. This velocity induced at the wing is effective at the vertical tail as an induced
a.
b
C
-u
Figure 7-38 Evolution of induced sidewash of a
wing-fuselage
tv
system
in yawed flight. (a),
(b)
Geometry (high-wing airplane). (c) r(y) = circulation distribution, rg(y) = circulation distribution at
symmetric incident flow. (d) Induced velocity field
at the location of the vertical tail.
AERODYNAMICS OF THE STABILIZERS 471
lateral velocity of about the same magnitude. Figure 7-38d shows immediately that,
for conventional positions of the vertical tail, the incident flow angle of the vertical
tail is decreased by the lateral velocity v, that is, that the effectiveness of the
vertical tail is reduced. As in Fig. 6-6d, the sign of the induced lateral velocity is
reversed for the low-wing airplane from that of the high-wing airplane. This results,
for the same relative positions of fuselage and vertical tail, in an increased
effectiveness of the vertical tail. In consequence of its evolution, the lateral velocity
induced by the fuselage-wing interference is proportional to the sideslip angle Q and
independent of the angle of attack a. Thus the resultant velocity in the y direction
at the location of the vertical tail is
Vy = U. +
Vg -IL
3vf
(7-59)
where f3U00 is the lateral velocity due to the sideslip angle, vg is the induced lateral
velocity at symmetric incident flow, and 13vR is the additional induced lateral
velocity due to sideslip as in Fig. 7-38d. The effective sideslip angle of the vertical tail is
(7-60)
Hence the efficiency factor of the vertical tail is
U
a = 1 -f-
(7-61)
because vg is independent of 0. Because vR/U =
Eq. (7-61) is identical to
Eq. (7-53). Equation (7-61) shows that only the lateral velocity due to sideslip is
required to determine the efficiency factor of the vertical tail.
For an experimental confirmation of the above considerations, a few test results
on the efficiency factor are plotted in Fig. 7-39b for the wing-fuselage-vertical tail
system of Fig. 7-39a. From measurements of the yawing moment due to sideslip
with (wT7) and without (oT7) vertical tail, a mean efficiency factor of the vertical
tail has been established by Jacobs [14] in the form
acMz
aCMZ
apv
ag
WV
a13
oTf
aCMZ
a13 IV
where (acMZ/a1i)y is the contribution of the vertical tail to the yawing moment.*
This experimentally determined efficiency factor is given in Fig. 7-39b as a function
of the high position of the vertical tail. The result is
for the low-wing airplane:
a
>1
(stabilizing)
'This has been determined as the difference of the measurements on fuselage and vertical
tail and of the fuselage alone.
472 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
1,6
airplane
Mid-win 9
airplane
1.2
.
a--1°
0
-Q2
b
a.
Low-wing
airplane
Mid-wing
airplane
'Low-wing
0_-r
0
High-wing
airplane
High-win 9
airplane
CL .= 0
06-02
0.2
zV
c
S
0
0.2
Oq
zV- -
0.6
Figure 7-39 Efficiency factors of the vertical tail for high-wing, mid-wing, and low-wing airplanes
at several high positions of the fin. (a) Geometry. (b) Measured efficiency factors from Jacobs. (c)
Theoretical efficiency factors from Jacobs.
and for the high-wing airplane:
(destabilizing)
1
Thus the above conclusions have been confirmed.
Theoretical determination of sidewash Computation of the distribution of the
induced sidewash velocity for a known circulation distribution can basically be done
like that of the downwash, namely, with the help of the Biot-Savart law. A few
qualitative considerations may be noted first. In Fig. 7-40, a symmetric and an
asymmetric circulation distribution are compared. Because the circulation distributions have been taken as constant, the symmetric distribution of Fig. 7-40a
produces one horseshoe vortex, and the antimetric distribution of Fig. 7-40b two
horseshoe vortices turning in opposite directions. It is immediately obvious that in
a
b
r
y
y
r
i
r
Figure 7-40 Determination of
T v
i2r -i'
-w
-v
the induced sidewash. (a)
Symmetric circulation distribution. (b) Antimetric circulation
distribution.
AERODYNAMICS OF THE STABILIZERS 473
the middle plane, y = 0, a downwash velocity -w is obtained for the symmetric
circulation distribution but a sidewash ±v for the antimetric distribution, having
reversed signs on the upper and lower sides. The latter results essentially from the
counterclockwise-turning "double vortex," shed in the middle. However, this highly
idealized vortex model is insufficient to determine the induced sidewash quantitatively.
The computation of the induced sidewash must be based on a variable
circulation distribution T (y), for example, like that for the sideslipping wingfuselage system of Fig. 7-38. The sidewash velocity very close to the vortex sheet is
obtained in analogy to Eq. (2-46a) as
vu,, =
(7-62)
(z =Z1)
1 dr
where the upper sign applies above the vortex sheet and the lower sign below. The
validity of this equation can also be checked by inspecting Fig. 7-38c and d. There,
the slope of the circulation distribution is shown for y = 0, and the sign of the
sidewash velocity v is indicated. The induced sidewash angle av = vl U,,, is obtained
from Eq. (7-62) by introducing the dimensionless circulation distribution 7 =
T/bUU and the dimensionless coordinate in the span direction 17 =y/s as
y=f
(7-63)
G = J 1)
d_y
By introducing the expression 7(r?) = 7g('7) + R7,8(77) for the circulation distribu-
tion, where yg is the distribution in straight flight and !370 the additive circulation
for sideslipping flight, Eq. (7-63) yields, for the efficiency factor of the vertical tail in
the vortex sheet,
aP = i
d
( = j1)
(7-64)
The above derivation shows that Eqs. (7-62)-(7-64) are valid for any distance
behind the wing for a not-rolled-up vortex sheet.
From Eq. (7-64) it is seen that the efficiency factor changes abruptly in the
vortex sheet. The quantity dyp/dr? is obtained from the circulation distribution of
the sideslipping wing-fuselage system. The yp distribution for the high-wing airplane
is illustrated in Fig. 7-41a. The determination of the induced sidewash outside the
vortex sheet has been studied by Jacobs and Truckenbrodt [141. By applying the
Biot-Savart law, the induced sidewash angle for a given circulation distribution 7(77)
is obtained from lifting-line theory as
-r 1
1f
2
')'n/'
(aryl
-1
y
-/51)s]2 (
Y
(Y - 77,) (C - S1)
s
r
pp
7/i
3
(7-65)
with r as in Eq. (7-29). For unswept wings and a very large distance (
Jacobs
[14] gave a simple procedure for the evaluation. The solution for arbitrary wing
474 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
c
0.4
0.2
-0.2
1
-10
-0.5
0
0.5
10
15
a13v
aj3
b
(rr'
.,I-
Y
-
-02
0
02
06
y
Figure 741 Sidewash factors of a wing-fuselage system from [14], computed with simple
lifting-line theory. (a) Additive circulation distribution of the high-wing system, b/2R = 7.5;
i = 5. (b) Streamline pattern of the induced velocity field. (c) Distribution of
the sidewash factor over the vertical position in the root plane y = 0. (d) Distribution of the
rectangular wing,
sidewash factor over the span for several high positions.
planforms has been studied by Gersten [101. For large distances behind the wing it
suffices to use the values for - o.
In conclusion, results of a few sample computations will be reported. In Fig.
741 the induced sidewash field is given for a high-wing system. Figure 7-41a
illustrates the geometry and the additive circulation distribution 'yQ due to the
sideslipping. Figure 7-41b represents the streamline pattern of the induced velocity
field very far behind the wing, and Fig. 7-41c gives the distribution of the sidewash
factor a3,/ap as a function of the distance from the vortex sheet for the middle
plane 7 = 0. This figure demonstrates the discontinuity of the sidewash factor at the
vortex sheet
i and the strong drop with distance from the vortex sheet. Figure
7-41d gives the distribution of the sidewash factor in the span direction for several
distances from the vortex sheet.
In Fig. 7-42 for a high-wing and for a low-wing airplane the curves of constant
local efficiency
factor of the vertical tail a1V/aa = const are shown for the
transverse plane at the location of the vertical tail. The total efficiency factor of the
AERODYNAMICS OF THE STABILIZERS 475
vertical tail is obtained from this through integration over the vertical tail height.
The field of the curves as v/ag = const is independent of the angle of attack of the
airplane. There is, however, a dependence of the efficiency factor of the vertical tail
on the angle of attack because, with a change of the angle of attack, the vortex
sheet is displaced relative to the vertical tail (see Fig. 7-20). This influence is quite
noticeable, as may be seen by comparing the cases CL = 0 and CL = 1 in Fig. 7-42.
For the system of wing, fuselage, and vertical tail of Fig. 7-39a, Jacobs [14] applied
this method to determine the efficiency factors theoretically (Fig. 7-39c). The
agreement with measurements in Fig. 7-39b is satisfactory.
The problem area of the interaction of wing, fuselage, and vertical tail at sideslipping
has been investigated by Puffert [28] The concepts established for the induced sidewash
have been translated into that for the rolling wing by Michael [23 ] and by B obbitt [5 ] .
.
7-3-4 Interaction of the Vertical Tail
and the Horizontal Tail
The flow conditions at the vertical and horizontal tails are affected not only by the
fuselage and wing but also considerably by their mutual interaction. Of special
a.
cL=O
Vortex sheet
CL=7
105
1.1
0.
0.95
1.05
1.1
1.2
OB
15
0
095
Figure 7-42 Local efficiency factors of the vertical tail. Curves apv/ag = const, from [14],
b12R = 7.5. Wing of rectangular planform ii = 5. (a) High-wing airplane. (b) Low-wing airplane.
476 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
a
TV
b
Figure 7-43 Interference between vertical and horizontal tails. Circulation distribution and free
vortex sheet of a sideslipping vertical and horizontal tail system, from Laschka [191.
interest here are the conditions at the tail unit at sideslipping and rolling. A tail
unit at which the middle section of the horizontal tail lies over the root of the
vertical tail will be considered to demonstrate this fact.
On a
vertical tail
in an incident flow of sideslip angle 0, a circulation
distribution is generated that does not drop to zero at the root section but rather has
a finite value because of the end-plate effect of the horizontal tail. A circulation
discontinuity results now in the shedding of a single vortex that turns in a direction
opposite to that of the rest of the free vortices. This vortex in turn induces at the
horizontal tail a downwash exceeding the counteracting induction effect of the
continuous free vortex sheet. The resulting circulation distribution at the horizontal tail
has, as shown in Fig. 7-43b, a discontinuity in the middle of the horizontal tail; it is
antimetric and generates a rolling moment due to sideslip that is reversed from that
of the vertical tail (see Fig. 7-43, from Laschka [19]).
To reduce the load induced on the horizontal tail by the sideslipping vertical
tail, a positive dihedral may be provided. This increases, however, the total rolling
moment due to sideslip. On the other hand, the rolling moment due to sideslip of
the tail unit
dihedral.
may be reduced by providing the horizontal tail with a negative
By extending and applying a suitable panel method as described in Sec. 6-3-1
for the wing-fuselage system, the pressure distributions, and thus the acting forces
AERODYNAMICS OF THE STABILIZERS 477
and moments, can also be determined for the whole airplane; compare, for example,
[15]
.
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34. Schlichting, H.: Die Stabilitatsbeiwerte des Flugzeuges unter Beriicksichtigung der Interferenz von Flilgel, Rumpf and Leitwerk, Sonderheft: Flugmech. Probleme, Akad. Lufo.
2/43 g, pp. 3-23, 1943.
35. Schlichting, H. and W. Frenz: Uber den Einfluss von Fliigel and Rumpf auf das
Seitenleitwerk, Jb. Lufo., 1:300-314, 1941. Staufer, F.: Jb. Lufo., I:383-391, 1940;
1:294-299, 1941.
36. Schlichting, H. and W. Frenz: Systematische Sechskomponentenmessungen fiber die
gegenseitige Beeinflussung von Fli gel, Rumpf and Leitwerk, ZWB TB 11, no. 6,
1944.
37. Schulz, G.: Der Abwind auf der L'angsachse des Fliigels bei Betzscher Zirkulationsverteilung,
Lufo., 19:367-373, 1942.
38. Silverstein, A. and S. Katzoff: Design Charts for Predicting Downwash Angles and Wake
Characteristics Behind Plain and Flapped Wings, NACA Rept. 648, 1939. Silverstein, A., S.
Katzoff, and W. K. Bullivant: NACA Rept. 651, 1939.
39. Spreiter, J. R. and A. H. Sacks: The Rolling Up of the Trailing Vortex Sheet and Its Effect
on the Downwash Behind Wings, J. Aer. Sci., 18:21-32, 72, 1951.
AERODYNAMICS OF THE STABILIZERS 479
40. Trienes, H. and E. Truckenbiodt: Systernatische Abwindmessungen an Pfeilfliigeln,
Ing.-Arch., 20:26-36, 1952.
41. Ward, G. N.: Calculation of Downwash Behind a Supersonic Wing, Aer. Quart., 1:35-38,
1950.
42. Williams, G. M.: Viscous Modelling of Wing-Generated Trailing Vortices, Aer. Quart.,
25:143-154, 1974.
CHAPTER
EIGHT
AERODYNAMICS OF THE FLAPS
AND CONTROL SURFACES
8-1 INTRODUCTION
8-1-1 Function of the Flaps and Control Surfaces
As has been explained in Sec. 7-1, the tail surfaces of an airplane serve a twofold
purpose, namely, to stabilize and to control the airplane. In general, the tail surfaces
consist of a fixed part, the stabilizer, termed a fin on the vertical tail and a
(horizontal) tail plane on the 'horizontal tail, and a movable part, the control
surface, termed an elevator on the horizontal tail and a rudder on the vertical tail.
There is another set of control surfaces attached to the wing, termed ailerons; see
Figs. 7-1 and 7-3. The tail surfaces, with the control surfaces fixed, serve to stabilize
the airplane. The corresponding aerodynamic problems have been discussed in detail
in Chap. 7. The airplane is controlled by deflection of the control surfaces. Control
about the lateral axis is accomplished with the elevator, that about the vertical axis
and the longitudinal axis with the rudder and the ailerons.
The geometry of the tail surfaces and of the ailerons is that of an airfoil with a
flap (flap-wing) as shown in Fig. 8-1 (see also Fig. 2-24). The aerodynamic effect of
the control surfaces consists of an additive lift produced by their deflection. This
lift, acting on the tail surfaces or the wing, respectively, controls the airplane. The
aerodynamic forces acting on the control surfaces generate a moment that, referred
to the control-surface axis of rotation, is termed control-surface moment or hinge
moment. The effect of the control surface should be strong enough to generate an
additive lift that, for a given control-surface deflection, is as large as possible. At the
481
482 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
same time, however, the hinge moment should be as small as possible so that the
forces needed for the operation of the control surfaces also remain small. A control
surface in the form of a simple flap as shown in Fig. 8-1 has relatively large hinge
moments. Efforts have therefore been made to reduce the moments required to
move the control surfaces. This has been accomplished by means of so-called
control-surface balances, as shown in Fig. 8-2. The most important types of
aerodynamic control-surface balances are the inner balance (nose balance) as shown
in Fig. 8-2a, the balance tab as shown in Fig. 8-2b, and the outer balance (horn
balance) as shown in Fig. 8-2c. In all cases of control-surface balance, it is
important that the lift increase caused by the control-surface deflection (controlsurface effectiveness) should, if possible, not be reduced by the control-surface
balancing.
The airfoil with control surface of Fig. 8-1 may serve two purposes: first, to
control the airplane, and second, to be used as a landing device. In the latter case,
its effect is to increase the maximum lift of the airplane, thus holding down the
landing speed. This lift increase is usually accompanied by a drag increase. In Fig.
8-3, several designs of such landing flaps are shown. In the arrangements of Fig.
8-3a-e, the flaps are attached to the rear end of the wing, whereas in Fig. 8-3f and
g, flaps are shown in front of the wing (slat, nose flap). Some of these arrangements
are also employed as take-off assistance to reduce take-off distance.
Finally, a few more forms of flaps may be mentioned, namely, the system of a
Axis of rotation
Axis of rotation
Balance tab
Axis of rotation
Horn
-
Ru dd er
Axis of rotation
Horn
Elevator
C
Figure 8-2 Various forms of aerodynamic control-surface balances. (a) Inner balance (nose
balance). (b) Balance tab. (c) Outer balance (horn balance).
AERODYNAMICS OF THE FLAPS AND CONTROL SURFACES 483
a-<
d
Figure 8-3 Several control-surfaces and
flaps. (a) Cambered flap. (b) Slot flap. (c)
Double-section wing. (d) Fowler flap. (e)
Split (spreader) flap. (f) Slat. (g) Nose
flap.
brake flap (air brake) on the upper and lower sides of the wing (see Fig. 8-28).
They have the shape of a rectangular plate and are set normal to the flight
direction. It is the function of the air brakes in their extended position to increase
strongly the drag of the airplane, thus reducing considerably the speed and
generating a steeper glide angle (brake effect).
8-1-2 Geometry of the Flaps and Control Surfaces
For the aerodynamics of the wing with control surface (flap), the most important
geometric parameters as shown in Fig. 8-1 are as follows:
Control-surface angle (flap angle): rlf
Control-surface chord ratio (flap chord ratio): a f = cf/c
These quantities have already been given for the whole wing with control
surface (flap wing) in Sec. 2-4-2 and in Fig. 2-24. If the control surface does not
extend over the whole span, as in, for example, the aileron in Fig. 8-4a, the span of
the control surface bA = 2SA becomes another important geometric quantity. On
the horizontal tail plane and the fin, the control surface usually extends over the
whole span of the horizontal tail bH and the height of the vertical tail h y,
484 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
b
Axis of
rotation
by
Axis of rotation
Figure 8-4 Geometry of the control surface. (a) Ailerons.
(b) Elevator. (c) Rudder.
respectively (Fig. 8-4b and c). In many cases the control-surface chord ratio of is
varied along the span. In this case it is preferable to use the control-surface area
ratio A f/A' instead of the control-surface chord ratio cf/c, where Af is the
control-surface area and A' is the wing area within the span range of the control
surface.
8-1-3 Aerodynamic Coefficients of the Flaps
and Control Surfaces
The following aerodynamic coefficients are introduced for the wing with control
surface:
Lift:
L = cLAgr
(8-1)
Pitching moment:
M = cMAcq.
(8-2)
Control-surface moment:
Mf = cm fA fcfgc,
(8-3)
Here the lift coefficient and the pitching-moment coefficient are referred to the
geometric quantities of the wing, as in the case of the wing without control surface
[see Eq. (1-21)].. The control-surface moment M' (flap moment, hinge moment) is
referred to the axis of rotation of the control surface; its sign can be seen from Fig.
8-1. The coefficient of the control-surface moment c,,. f is referred to the geometric
AERODYNAMICS OF THE FLAPS AND CONTROL SURFACES 485
quantities of the control surface. These three aerodynamic coefficients depend on
the angle of attack a and the control-surface angle r1f.
As an example of measurements, the lift coefficient CL of a simple flap wing is
plotted against the angle of attack in Fig. 8-5a for several flap angles r7f. The flap
deflection 17f causes, corresponding to Fig. 2-24, an additive camber and thus, at
constant angle of attack, an increase in lift. The curves cL(a) for several angles 17f
are parallel to each other. The dependence of the lift coefficient on a and 17f for
small angles may be expressed as
cL=
asa+a
f
= a« a - a
(8-4a)
r?f
(8-4b)
f 77f
where as/ar? f indicates the change in the zero-lift direction of the wing because of
the flap deflection (flap effectiveness) [see Eq. (7-3b)] . The coefficient as/ar?f
depends strongly on the control-surface chord ratio. Data on this effect have been
given in Fig. 2-25a for a flap wing of infinite span.
In Fig. 8-5b, the lift coefficient CL is plotted against the moment coefficient
cm for several flap angles 77f. The flap angle causes a parallel shift of the moment
curves. The dependence of the moment coefficient cm on CL and 17f for small values
of these parameters may be expressed as
CM .
acM
aCL
CL +
acM
(8-5)
17f
1.
f -20°
16
nf = 20 °
(
1.0
11°
I
50
0°
0.
{
i
{
E
Ay
/I X /V
-5- 1
F)
i 10
l
-11;16°
{
i
I
i
7
20° '
I
-1.0
-15
a
=25°
-5° 0° 5°
15°
25° -0.4
b
-0.2
0.2
0.4
CM
Figure 8-5 Aerodynamic coefficients of a simple rectangular flap wing from measurements of Gothert [18]. (a) Lift coefficient CL vs. angle of attack a for several flap angles
rrf. (b) Lift coefficient CL vs. pitching-moment coefficient cMfor several flap angles 77f.
486 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
Here, acM/art f gives the change of the zero moment with the flap deflection. This
coefficient depends strongly on the flap chord ratio. Data on the wing of infinite
span have been given in Fig. 2-25b.
Frequently it is advantageous to specify the location of the aerodynamic center
of the additional forces generated by the flap deflection. This point is termed the
flap neutral point. The distance of the flap neutral point from the neutral point of
the wing without flap deflection (= neutral-point displacement) is obtained from
Eqs. (8-5) and (8-4) as
(d XN)f
aCM/a7lf
C
aCL/ar?f
(s-6)
where acL/arlf may be taken from Eq. (8-4a).
In Fig. 8-6, cL is shown as a function of the control-surface moment coefficient
Cm f for several values of r7f. Here, too, a linear relationship applies of the form
Cmf
aCmf
aCL CL +
aCmf
77f
(8-7)
a7jf
The dependence of this coefficient on the flap chord ratio will be discussed in Sec.
8-2. The condition cm f = 0 determines a certain coordination of r7f and CL and thus
also of r?f and a for self-setting of the free control surface.
8-2 THE FLAP WING OF INFINITE
SPAN (PROFILE THEORY)
8-2-1 The Flap Wing in Incompressible Flow
The flap wing as a bent plate The fundamentals of the theory of the flap wing of
infinite span in incompressible flow have been given in Sec. 2-4-2. In its simplest
form, the wing with a deflected flap is replaced by a bent plate as shown in Fig.
2-24, on the chord of which, according to Glauert [16], a vortex distribution is
arranged.
For the coefficients of the flap effectiveness, the expressions of Eq. (2-82)* are
a77fa = - 2 ( f(1 - Xf) + aresin )
a 77f
= -21 - Xf)3
(8-8a)
(8-8b)
In Fig. 8-7 these theoretical coefficients have been given against the flap chord ratio
Xf. The problem of the single-bent plate has been solved by Keune [24] with the
method of conformal mapping. The most important result of this study is the
confirmation of Glauert's approximate solution for small flap angles. For larger flap
*The index 0 has been omitted.
AERODYNAMICS OF THE FLAPS AND CONTROL SURFACES 487
1.
77f °GO°
I
170
7!
I
IN
1,
I
II
5° 0°
l-IL
11 "Nl
i
5 -11°I
I
I
I
1
I
-0
i
!
I
-7,0
1
i,
I
I
-0.2
02
0.4
Cmf
Figure 8-6 Lift coefficient CL vs. control-surface coefficient cm f of a simple flap wing (aspect ratio
A = 3.5; flap chord ratio Xf = 0.5), from measurements of Gothert [18 ].
V.°
j
0.7
os
The ory
Mea surem ent
Th eory
I
Me asurem ent
Gt4
03
1
Cambered flap
Slot flap
Cambered flap
-- Slot fla p
72
Double-section wing
0.
S p l it
1
fl ap (spreader
I
flap)
7
a
0,2
Af
cf
=C
0.3
0.4
05
0.60
b
0.1
0.2
0.3
0.4
Xf cf
= C _
0.6
0.5
Figure 8-7 Flap effectiveness of several designs: theory and measurements. (a) Angle-of-attack
change due to flap deflection as/arlfe vs. flap chord ratio Xf. (b) Pitching-moment change due to
flap deflection acM/anfe vs. flap chord ratio Xf.
488 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
angles, the deviations are more pronounced. In Fig. 8-7 these results are added to
the results of comprehensive test series on wings of various flap shapes. The measured
coefficients have been taken from test series for small flap angles. The coefficients thus
obtained have been designated as aoz/arlfe and acM/an fe. Comparison of theory
and experiment shows that the measured values are smaller than the theoretical ones
for both the change in the angle of attack and the change in the moment. The
curve for the wing with a split flap (spreader flap) shows the largest deviation from
the theoretical curve. For larger flap deflections, the flap effectiveness declines. This
trend is shown in Fig. 8-8 by assigning an effective flap angle rife to each geometric
flap angle r?f. This coordination applies approximately to the moment change as
well.
The differences between the theoretical curves and the measurements in Fig.
8-7a and b cannot be fully explained by the influence of the profile thickness. They
should essentially be due to friction effects. For theoretical studies of the flap wing,
it is advisable to apply empirical corrections to the coefficients of the flap wing as
obtained from profile theory. This is accomplished simply by multiplying the effect
of the camber on the coefficients as/ar1f and acM/arlf with an empirical factor x.
Then the adjusted coefficients assume the form
f=
acm
(8-9a)
-
acM
ar f
(8-9b)
17f X-1
Here the terms with the index x = 1 are the theoretical values from Eqs. (8-8a) and
(8-8b). In Fig. 8-9, these coefficients for x = 0.75 are also shown; they agree
satisfactorily with the measurements of Fig. 8-7,
In Fig. 8-10, the theoretical values for the position of the flap neutral point
from Eq. (8-6) are plotted against the flap chord ratio with aCL/aa = 27r. In this
figure, the distance between the
/0
neutral point and the leading edge,
flap
A -/
00
Cambered flap
--Slot flap
-10°
-
Double-section wing
-----Split flap (spreader flap)
-Z0°
-70°
00
100
r? f
200
,300
400
°
V[/°
Figure 8-8 Correlation between the
effective flap deflection rife and the
geometric flap deflection 17f for several flap designs (see Fig. 8-7).
AERODYNAMICS OF THE FLAPS AND CONTROL SURFACES 489
7,
Theory
0.
x =10
..75
/10
0.
Theory
a=1,0
0.75
0,2
Figure 8-9 Reduction of flap effectiveness
from Eqs. (8-9a) and (8-9b). (a) Change of
angle of attack due to flap deflection. (b)
0
0.2
0.4
0.5
10.6
1.0
Change of pitching moment due to flap
deflection.
xNf = c/4 + (d xN)f, is given, where c14 is the position of the wing neutral point. It
is noteworthy that, for small flap chords, the flap neutral point lies at c/2. This is
in consequence of the fact that the deflection of even a small flap strongly affects
the pressure distribution on the front portion of the wing.
Computation of the flap loading (control-surface loading) and of the flap
moment (control-surface moment) requires that the pressure distribution on the
deflected flap be known. The theoretical pressure distribution on a bent plate is
illustrated in Fig. 2-28, whereas Fig. 8-11 gives the experimentally determined
pressure distribution on a wing with split flap from Schrenk '[40] (see also Seiferth
[41]).
The aerodynamic force on the flap (flap loading), the knowledge of which is
important for computation of the structural strength of the flap, is obtained from
the pressure distribution on the flap as
Lf = bf fl_p)dxcjfbfcfq .
(cf)
(8-10)
490 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
0.5
Lf
0.7
0o
0
0.2
0.6
0,4
08
T0
Figure 8-10 Position of the flap neutral point
vs. the flap chord ratio for incompressible
flow.
where cif is the coefficient of flap loading (control-surface loading). The dependence
of the coefficient of flap loading on the lift coefficient and on the flap angle is
given, in analogy to Eq. (8-7), as
cif
acifCL
L
+
a f of
(8-11)
f
The theory of the bent plate yields the coefficients
acif
_
acL
ac77ff =
2
af
g
[aresin
-
- Af) ]
0 - Xf)
(8-12a)
(8-12b)
In Fig. 8-12 the two coefficients have been plotted against the flap chord ratio Xf.
The flap moment (control-surface moment) of a wing portion of width bf,
referred to the control-surface axis of rotation, is
Mf= -bf f(p,-p)(x-x1) dx
Figure 8-11 Pressure distribution on a wing with a slot flap, from Schrenk.
(8-13)
AERODYNAMICS OF THE FLAPS AND CONTROL SURFACES 491
J.
Lf
2
cf
c
2.
1,
acIf
aT?f
1.0
acI f
acL
a
0
Xf -
0,2
0.4
0.6
06
1,0
Figure 8-12 Flap loading; theory from Glauert.
Curve 1, change of the coefficient of flap loading
with lift coefficient. Curve 2, change of the
coefficient of flap loading with flap angle.
where xf is the position of the axis of rotation as shown in Fig. 8-1 and c f is the
flap chord. The theory of the flap wing (bent flat plate, Sec. 2-4-2) yields the.
following relationships for the control-surface moment coefficient cmf:
ac m f
= - 21rXf
1 2 [( 3 - 2Xf)
f(1 - Af) - (3
f) arcsin
ac,,
-0
'
ac ,n f
4
1- f
r?f
W
Af
a
[arcsin
-
X_ f(1 - Af) ]
/]
(8-14a)
( 8 -14b)
In Fig. 8-13, these coefficients are plotted against the flap chord ratio Af. Test
results for simple cambered flaps are also shown. They lie considerably below the
theoretical curves. These differences are caused by the influences of the profile
thickness and, particularly, of the friction.
To reduce the control-surface moment Mf, several forms of control-surface
balance arrangements have already been shown in Fig. 8-2. Of these, only the inner
balance and the balance tab can be considered two-dimensional problems. At the
inner balance, the control-surface moment is decreased by moving the axis of
rotation rearward. Then, in deflecting the control surface, a control-surface "nose"
protrudes from the profile, forming a contour that is hardly accessible to
computation. To determine the aerodynamic coefficients of the flap wing with inner
balance, mainly experimental studies have to be applied, such as, for example, those
published by Gothert [181. The aerodynamic coefficients of a flap wing with
balance tab were first treated by Perring [16] using the theory of the multiple-bend
plate. A comparison of his theoretical results with measurements is given by Gothert
(18]. In this case the effect of friction is particularly strong.
The flap wing as a wing profile Several investigators have studied theoretically not
only the flap wing of finite thickness but also the effect of flap arrangement and
492 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
0.
1
4
Theory
eo
Measurement
a 01,0
ae
f MCS
77f
C
O
Theory
Measu rement
Figure 8-13 Coefficient of control-surface
moment vs. flap chord ratio Xf; theory from
Gothert. (a) Change of the coefficient of
control-surface moment with lift coefficient. (b) Change of the coefficient of the
control-surface moment with flap deflection.
0.2
b
0.2
0,4
c
0.6
0.e
1,0
Af = c
flap shape and, particularly, that of a slot between the fixed airfoil and the movable
flap. In particular, the publications of Allen [2], Flugge-Lotz and Ginzel [13],
Keune [241, and Jacob and Riegels [22] should be pointed out. The results of
these studies have been presented systematically by Gothert [18] within the
framework of an experimental study. Furthermore, comprehensive test results on
flap wings have been reported by Wenzinger [49] and by Keune [24]. Summary
accounts of these studies are found in [7] and [45] ; compare also [1, 35].
Theoretical investigations on the behavior of the boundary layer of flap wings and
comparisons with measurements have been conducted by Goradia and Colwell [171.
8-2-2 The Flap Wing in Compressible Flow
Lift and moment The theory of the flap wing of infinite span in compressible flow
may be derived approximately from the profile theory of compressible flow as given
in Sec. 4-3. There solutions were obtained for subsonic incident flow using the
subsonic similarity rule (Prandtl, Glauert) and for the supersonic incident flow using
the supersonic similarity rule (Ackeret). The following formulas apply for fixed flap
chord ratios Xf ='Afinc
For subsonic incident flow (Maw < 1),
as
a??f
_
cmm
_ amf
(8-15a)
inc
AERODYNAMICS OF THE FLAPS AND CONTROL SURFACES 493
acm
-
1isl
1
(8-15b)
}/1 -MaL (t'aC?7.f Ji nc
Here the terms marked by "inc" are those of the incompressible flow from Eqs.
(8-8a) and (8-8b) and as shown in Fig. 8-7.
For supersonic incident flow (Ma > 1),
a77f
as
a77f
_ - Af
(8-16a)
acb,
2
a 77f
VMa- -- 1
- Xf( 1
- Tf)
( 8 -1 6 b)
In Fig. 8-14, the changes in zero-lift angle and zero moment caused by the flap
deflection are given as a function of the flap chord ratio.
By using the above coefficients, the position of the flap neutral point can be
com uted with Eq. (8-6), where it has to be considered that acL/aa= 27r/
_-a" for Maw, < 1 and acL/aa= 4/ Ma;.- 1 for Ma.. > 1. The position of
7the flap neutral point is given in Fig. 8-15 against the flap chord ratio af. Here the
1
relationships xNf = c/4 + (A XN)f applies for Ma,0 < 1 and xNf = c/2 + (A XN)f for
At supersonic velocities the flap neutral point lies much farther back than at,
subsonic velocities, as should be expected. The following expressions are obtained
for the coefficients of the flap moment (control-surface moment) at subsonic
incident flow (Ma,, < 1):
aCyyi f
aCL
`
aC,n
f
acL
(8 -17a)
inc
f
1.
08
0
/
M¢c
0.2
0.6
0.4
fRaco-,
0.8
,.0
Cf
c
Figure 8-14 Aerodynamic coefficients of a flap wing at subsonic and supersonic incident flows. (a)
Change of the zero-lift angle with flap deflection. (b) Change of the zero moment with flap
deflection.
494 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
1.0
0,8
XNf
- c
Md.-1
2
Ma
0,2
Figure 8-15 Position of the flap neutral
01
point vs. the flap chord ratio for compressible flow (subsonic and supersonic veloci-
f
0.2
0.4
0,6
0,8
1,0
ties).
aCmf
al?f
acm f
-Ma.
i
aff
(8-17b)
inc
Again, the coefficients marked inc are those of incompressible flow from Eq.
(8-14) and Fig. 8-13. Corresponding relationships are found for the coefficients of
flap loading.
For supersonic velocities (Maw > 1), the Ackeret rule yields (see Sec. 4-3-3)
acmf
1
aCL
2
(8-18a)
aCmf
2
8 18b
Ma;. - 1
The coefficients of flap loading are determined immediately as clf = 2Cm f by
realizing that the pressure distribution over the flap chord is constant.
aI7f
8-2-3 Take-off and Landing Devices*
General remarks As has been mentioned in Sec. 8-1, the take-off and landing
devices on the wing serve to increase the maximum lift coefficient. A great variety
of arrangements are utilized to increase the maximum lift. The older kinds of
take-off and landing devices consist of flaps and balance tabs attached to the wing
trailing edge or the wing nose (Fig. 8-3). More recently, devices have frequently
been used that increase -the lift through boundary-layer control by suction or
ejection. A brief account of this method has been given in Sec. 2-5-3. A
comprehensive survey of the various methods for the increase in maximum lift is
included in Lachmann [28].
The effect of take-off and landing devices on the lift characteristic CL(a) of a
*The assistance of K. O. Arnold in preparing this section is gratefully acknowledged.
AERODYNAMICS OF THE FLAPS AND CONTROL SURFACES 495
wing is presented schematically in Fig. 8-16. Curve 1 gives the values without flap
deflection. Curve la shows the increase in the coefficient CLmax by boundary-layer
control at the wing nose. Curve 2 gives the values with flap deflection, and curve 2a
again the increased values of CLmax through boundary-layer control at the nose.
Curves 3 and 3a give the corresponding data when, in addition, the boundary layer
at the flap nose is controlled as well. The summary report about theoretical and
experimental studies on boundary-layer control by Carriere et al. [8] should be
mentioned. Earlier, a paper on the properties of flap wings was given by Young
[55]
.
Flaps The simplest method of increasing CLmax is the deflection of a cambered flap
as shown in Fig. 8-17a. This effect is obtained because the flap deflection increases
the effective camber of the wing, resulting in a lift augmentation that may be
considerable. As an example, Fig. 8-17a shows CL against the angle of attack for
several flap deflections. The increase in CLmax depends on the flap chord ratio X f;
the highest values are usually obtained for Xf = 0.20-0.25 [7].
A quite simple landing device in terms of design is the split flap as shown in
Fig. 8-3e. This is a flat plate lying against the lower side of the wing and turning
about its forward edge. The lift curves cL(a) of Fig. 8-17b for several flap angles qf
are similar to those of the cambered flap (compare Fig. 8-17a). The effectiveness of
the split flap is, according to Gruschwitz and Schrenk [19], due not only to an
increased camber but also to a reduction of the static pressure on the suction side
of the profile. In Fig. 8-18, the pressure distribution is shown for a wing with
7f
-60°
i
3
?f =40°
ZS
10
7
01
'7f
0
S°
10'
7S°
a-
20°
IS'
30°
Figure 8-16 Effect of flap deflection
and boundary-layer control on the lift
of a flap wing (schematic). Explanations in the text.
496 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
2.4
0
15°
a = 30°
1,6
o = 45°
60°
0001,
0
00
a
16 °
32°-16°
b
a-
Figure 8-17 Lift coefficients of flap wings vs. angle of attack a for several flap deflections T ?f.
Profile NACA 23012, Reynolds number Re = 6 105, from [491. (a) Simple cambered flap, flap
chord ratio Xf = 0.2. (b) Split flap, Xf = 0.2.
qr
01.1
-4
-5
Figure 8-18 Pressure distribution on a wing with
deflected split flap, from [19). Curve 1, without flap
-6
deflection. Curve 2, with flap deflection.
AERODYNAMICS OF THE FLAPS AND CONTROL SURFACES 497
deflected split flap. Because of the flow around the sharp trailing edge of the
deflected plate, a strong low-pressure range is formed in the wake of the flap,
having an effect up to the upper side of the wing.
Basically, the CLmax value increases with Reynolds number. In Fig. 8-19, the
results on the effect of the Reynolds number on the value of CLmax are given, both
for a wing without a flap and one with a 600 deflection of a split flap. Young [54]
reports on the separation characteristics of flap wings. Flaps extending over only a
portion of the wing span will be treated in Sec. 8-3.
The effectiveness of the simple cambered flap is limited by the flow separation
occurring at large deflection 77f right behind the flap nose. By boundary-layer
control at the station of greatest danger of separation, the lift-increasing effect of
the cambered flap can be improved, as shown schematically in Fig. 8-16.
Boundary-layer control by suction or ejection requires a considerable design and
construction effort and will be discussed later in more detail. On the other hand,
the slotted flap as shown in Fig. 8-3b, first suggested by Betz [6] and by Lachmann
[27] , represents a simple design for natural boundary-layer control. The slotted flap
functions in such a way that the air, flowing through the slot from the lower to the
upper side, carries the boundary layer, formed on the wing, into the free flow
before separation can occur. Starting at the flap nose, a new boundary layer forms
that can again grow over a larger distance before separation.
The maximum lift coefficient CLmax depends on the separation processes at the
main wing in front of the flap as discussed in detail in Sec. 2-5-1. The most
unfavorable flow conditions occur shortly behind the profile nose of the wing and
at large angles of attack, a -_ CLmax). Here, the pressure increase that follows the
3
I
!
!
i
I
!
2.4
0.6
°
NACA64, -412
e
NACA 64,3 -418
NACA 23012
With
04
!
j--- Without
split fla p '
Figure 8-19 Change of maximum lift coeffi-
cient with Reynolds number for a wing
Z.0
Re
3.0
4.0 5.0-10
without and with a split flap. Flap chord ratio
Xf= 0.20, flap angle r?f= 60°, from [71.
498 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
suction peak usually leads to boundary-layer separation at the wing leading edge
(see Fig. 2-44). By boundary-layer control, similar to that of the trailing-edge flap,
separation can be shifted to larger angles of attack. The extension of the linear
range of the cL(a) curve of Fig. 8-16 leads to a considerable additional lift gain.
Another effective arrangement for the increase of the maximum lift is the slat
(flap before the wing leading edge) as shown in Fig. 8-3f, whose characteristics have
already been discussed in Sec. 2-5-3. A polar curve of it is given in Fig. 2-53. Figure
8-20 shows the lift coefficient plotted against the angle of attack for a wing without
and with a slat. In agreement with profile theory, the slat does not generate a
noticeable change of the profile camber, because this would cause a parallel shift of
the CL(a) curves without and with slat. Because of natural boundary-layer control,
the maximum lift coefficient of a wing with a slat is reached at very large angles of
attack.
An effect similar to that of the slat is produced by the so-called nose flap, first
proposed by Kruger [44]. Here, the increase of a(CLmax) results from a different
effect, namely, the shape of the profile nose, responsible for the separation process,
which is changed favorably by the flap deflection (see also Fig. 2-44).
In addition to the conventional landing devices on the trailing edge discussed so
far, the double-section wing as shown in Fig. 8-3c and the Fowler flap as shown in
Fig. 8-3d must be mentioned. The former is a simpler design of the slotted flap.
The latter consists of a flap that is driven out rearward and deflected. A
simultaneous camber and area increase is thus accomplished.
Frequently, several landing devices are utilized in combination to establish a
maximum lift that is as large as possible. As an example, Fig. 8-21 gives the lift
coefficient of the profile Go 819 with a slat and a double-section flap against the
angle of attack. The favorable effect on the boundary layer of the flow through the
slot between the slat and the main wing is clearly indicated by comparison with the
measurement when the nose slot is closed. In this latter case, the cLmax values for
J
Figure 8-20 Lift coefficient CL(a) of a wing with slat,
from (48]. Profile Clark Y, Reynolds number Re =
6 - 105. Curve 1, without slat. Curve 2, with slat.
AERODYNAMICS OF THE FLAPS AND CONTROL SURFACES 499
J
is
Z.0
as
Nose slot closed
0
Nose slot open
i
9,06
°
0 819
-05
-10
-S°
0°
5°
70-
7y a
20°
25°
a-
Figure 8-21 Lift coefficient CL(Q) for
the profile Go 819 slat and double-slot
flap, from Wuest [53].
all measured flap angles are lower by dCLma,X ~ 0.6; also, the flow separation leads
to a larger lift drop than for the open nose slot.
Comprehensive data on the maximum lift coefficient of wings with and without
landing devices are given in [32, 33, 461.
Suction In an effort to increase further the maximum lift of wings, suction was
studied quite early (see Betz [4] ).
The suction intensity is defined by a dimensionless suction coefficient as
cQ = AQ,,
(8-19)
Here Q is the volume removed per unit time, A is the wing area, and U. is the
incident flow velocity. The maximum lift can be increased considerably by slot
suction. Comprehensive tests on this method were conducted by Schrenk [4]. The
most effective method, particularly for thick profiles, was found to be slot suction
with a flap wing. Lift coefficients up to about CL = 4 may be obtained, as shown in
Fig. 8-22 for a thick profile with flap and suction. Here the coefficients of suction
are about cQ = 0.01-0.03 and the suction pressures cp = (p - p,,,)/q. = -2 to -4,
where Q stands for the total flow volume removed, p for the pressure in the suction
slot, and q,. _ (g. /2)UU for the dynamic pressure of the incident flow. The effect
of suction lies in its keeping the flow essentially attached to the flap. The greatest
danger of separation is near the flap nose. If the decelerated boundary layer at this
500 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
//
30
c U .Joz
/
F
O
/
al
J
IP
77f'95°
17f -,70.
Suction
10
of
c
0
1.0
2,0
Y0
.10
-cp ;
Figure 8-22 Lift coefficients of flap
wings with slot suction, from
Schrenk.
station is removed strongly enough by suction, the flow over the entire trailing-edge
flap may be kept attached. After favorable wind.tunnel. results had been obtained,
for flap profiles with suction, the Aerodynamische Versuchsanstalt Gottingen (AVA)
conducted the first flight tests of the suction effect in the early 1930s. The possible
gain in lift for fully attached flap flow (CQ = cQL) over the lift of uncontrolled
flow (cQ = 0) may be seen in Fig. 8-23. This diagram shows CL as a function of
deflection at several angles of attack of the wing. Note that the lift for
potential flow is reached when the suction is just strong enough for complete
prevention of separation. Arnold [4] studied the computation of the required
amount cQL . More recently, both slot suction and continuous suction through
flap
perforated walls have been applied, the latter at the trailing-edge flap as well as at
the wing nose. Further developments of suction procedures have been summarized
by Regenscheit [36] and Schlichting [36].
The continuously distributed suction has been studied theoretically by
Schlichting and Pechau [381. Flight tests by Schwarz [38] and by Schwarz and
Wuest [38] confirm the feasibility of nose suction.
Ejection The boundary layer may be controlled by ejection as well as by suction
for increased maximum lift. This method has been applied most successfully to the
wing with a trailing-edge flap. By tangential ejection of a thin jet of high velocity at
the nose of the deflected flap, flow separation from the flap can be prevented and
the lift can be increased. Critical for the effectiveness of ejection is, according to
Williams [51], the dimensionless momentum coefficient
ci _
e.A
(8-20)
AERODYNAMICS OF THE FLAPS AND CONTROL SURFACES 501
where the index j refers to the conditions in the jet and the index C- to those of the
incident flow.
Comprehensive studies on the lift increase of flap wings with ejection have been
conducted by Thomas [43]. In Fig. 8-24, a typical result of these measurements is
given, namely, the gain in the lift coefficient JCL against the momentum coefficient
c1 for several flap angles 77f. The curves A CL versus c1 clearly show two ranges: first,
a very steep increase at small momentum coefficients; and second, a considerably
smaller increase at large momentum coefficients. The first range is that of
boundary-layer control. It extends to the momentum coefficient that just suffices to
produce complete flow attachment back to the flap trailing edge, thus completely
preventing separation. The second range of considerably smaller lift gain with the
momentum coefficient is the range of supercirculation. Here, the "hard jet" (of very
high momentum) acts similarly to an extended mechanical flap.
In Fig. 8-25, the lift coefficient of a wing at fixed flap deflection is plotted
against the angle of attack for several momentum coefficients cy. The ejection has a
similar effect as an increased camber (flap deflection). Flow separation sets in at
smaller angles of attack, however, than without ejection. Inspecting Fig. 8-16 shows
that an additional lift gain can be generated by combination with a boundary-layer
Theory cp -0
Z.
or
a=70°
70
j
Zs
7.2
o81
--- -
r Jam
05,
m
rfl
0
00
75°
30°
K.
?7f
60°
Figure 8-23 Lift increase due to slot
suction at the trailing-edge flap for
completely attached flap flow, from
Arnold. (- - -) Measurements with) Measurements with
out suction. (
suction.
502 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
6.0
5.0
U
NACADO1o
'
-
77f -900
f
w" I 75
60
a --5°
45°
!
2.0
150
I0
0
Figure 8-24 Flap wing with ejection, lift increase
AcL vs. momentum coefficient ci for various flap
0.4
0.2
0.8
0.6
cj - Qj vj s/y°, c
angles ref at constant angle of attack a = -5°,
from Thomas.
2.
A0.131
0,053
2
Q037
2.0
0.018
7.7
1.5
°
1.0
0,75
0.5
0.25
0
-10
-5
If
5°
10°
200
Figure 8-25 Lift coefficient of a wing
with ejection over the trailing-edge
flap, from Williams, profile tic = 0.08.
Flap deflection 77f= 45°, flap chord
ratio Af = 0.25.
AERODYNAMICS OF THE FLAPS AND CONTROL SURFACES 503
control at the wing nose, either by suction or by ejection (see Gersten [15] ). Even
the flow is completely attached, a further increase in lift may be
accomplished by stronger ejection on the flap. This is the result of supercirculation
and the jet reaction force. This problem area has been summarized by PoissonQuinton [34] and by Williams [511; see also [28]. Levinsky and Schappelle [29]
developed a method aimed at maintaining potential flow through tangential ejection
when
on flap wings.
Jet flaps Effects very similar to those generated by a solid trailing-edge flap are
obtained by ejecting a high-speed jet under a certain angle nj near the wing trailing
edge. This method, illustrated in Fig. 8-26, is termed a jet flap. The vertical
component of the reaction force of the jet is supplemented by an induced lift that
may be many times larger than the jet reaction (supercirculation). This effect has
been studied by many experiments [34, 521.
In Fig. 8-26, the theories of Spence [42] and Jacobs [9] are compared with
experiments on a symmetric profile with jet flap. The figure shows the dependence
of the lift slopes acL jaa and acL /ar7j on the momentum coefficient c1 as defined by
Eq. (8-20). Here the momentum coefficients cj are much larger than in Fig. 8-25.
Up to values of about ci = 0.1, the jet acts on the boundary layer; for larger values
of cj it essentially causes the circulation to increase (supercirculation). Either lift.
slope increases strongly with increasing cp For cj = 4, the lift slope acL/aa has
about twice the value of that without ejection (c1 = 0). The agreement of theory
1
dcL
1,
1
J/
Theor
: rY
/
e
4
//
+
I
2
6
I
4
5
Figure 8-26 Profile with jet flap, comparison of theory and experiment for lift
slopes acL/aa and acL/ary. Theory from
Spence and Jacobs. Measurements from
Dimmok [52]. (c) nj = 31°. (o)
58°.
504 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
and experiment is good. Helmbold [20] studied the theory of the wing of finite
span with jet flap.
A comprehensive wing theory for the wing of finite span with jet flap has been
developed by Das [9]. An example of this theory and a comparison with
experiments is given in Fig. 8-27 for a swept-back wing with a jet flap spanning the
entire trailing edge. Agreement between theory and experiment is good. Murphy and
Malmuth [9] report on the computation of the aerodynamics of the jet flap wing in
transonic flow. The jet flap wing. near the ground has been studied by Lohr [30].
The aerodynamic problems of the maximum lift have been summarized by
Schlichting [37]. Questions of the practical application of the jet effect to the
generation of high lift on wings with and without flap are discussed in the
summarizing paper of Korbacher [251,
Air brakes, spoilers The aerodynamic effect of air brakes has been investigated
repeatedly (see Arnold [3] ). In particular, various positions of the brakes on the
lower and upper sides of the wing have been studied. Figure 8-28 shows the result
of three-component measurements for a wing with air brakes over the entire span.
The polar curves illustrate the very large drag increase. Compared with the wing
alone, the drag coefficient is about 20 times larger.
Devices of a similar kind mounted only on the upper side of the wing are also.
termed spoilers. By extending them on only one side of the wing, they can be used
U.
7
V.
3.
lo,
C'
2.8
A
cj=2
14
72
/
0.8
/
i
i * X0.2
---
I
Theory
Measurerne+
0
-8°
40
00
8°
72°
a ----
Z0°
t
Figure 8-27 Lift coefficient of a
swept-back wing with jet flap; comparison of theory and measurement
from Das [9 ]. Aspect ratio .4 = 3.5,
sweepback angle p= 45°, jet angle
rj = 30°.
AERODYNAMICS OF THE FLAPS AND CONTROL SURFACES 505
_F Z-T
--yr c
U.,c/4-I
*11
S+P
-0.8
-1G
-20'
-70'
70'
20'
30'
26
WO
S+p
P
-0,8
0,7
02
0S
03
CD
8-28 Three-component measurements on a rectangular wing with air brake,
from Reller [ 3 ]. Aspect ratio A = 5.1; flaps
Figure
-08
extend over the entire span. WO, wing
-76
-0.08
008
0.16'
0.2'
without flap; S, flap on suction side; P, flap
on pressure side.
506 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
for control about the vertical and longitudinal axes. The flow separation from the
wing caused by the spoiler leads to a strong, one-sided lift loss and thus to a rolling
moment. Wing tunnel test results on spoilers and a few computations on the effect
of the spoiler are found in [11, 21, 23, 50] .
8-3 FLAPS ON THE WING OF FINITE SPAN
AND ON THE TAIL UNIT
8-3-1 Flaps on the Wing in Incompressible Flow
Computational methods The aerodynamics of the flap wing of infinite span (plane
problem) has been discussed in the previous section. Now the effect of a flap
(control surface) on a wing of finite span will be treated. A further geometric
parameter, the span of the flap, is added (see Figs. 7-1, 7-3, and 8-4a). Furthermore,
in many cases the flap chord ratio varies over the flap span (see Fig. 8-1). To
determine the lift distribution, a wing with a deflected flap is equivalent to a wing
with an additional angle-of-attack distribution over the span (twist). For a flap
covering only a portion of the span, this additional angle-of-attack distribution is
discontinuous. The angle-of-attack distribution that is equivalent to a given flap.
deflection is obtained from the theory of the flap wing of infinite span as
af(n) _ -
aQ_17f
r?f
(8-21)
where aa/ar f is the local flap effectiveness from Eqs. (8-8a) and (8-9a) and from
Figs. 8-7a and 8-9a. If the flap chord ratio Xf varies over the span, it is a
function of the span coordinate 77 =y/'s.
According to the procedure for the computation of the lift distribution on
wings of Sec. 3-3, the additive circulation distribution caused by the flap deflection
can be determined for such an angle-of-attack distribution. Special attention should
be paid to the station of discontinuity in the angle of attack.
The case of a symmetric angle-of-attack distribution corresponds to a landing
flap at the wing or an elevator at an all-wing airplane as shown in Fig. 7-3. The
antimetric angle-of-attack distribution corresponds to the ailerons (Figs. 7-1 and
7-3).
Following simple lifting-line theory (Sec. 3-3-3), Multhopp (Chap. 3, [60] )
developed a method for handling the discontinuity in the angle-of-attack curve. In
Fig. 8-29, a result of this method for a trapezoidal wing of aspect ratio A = 2.75
and taper X = 0.5 is shown as curve 1. The station of discontinuity in the
angle-of-attack distribution of lies at r70 = 0.5. In Fig. 8-29a it is symmetric, in Fig.
8-29b it is antimetric. According to Fig. 8-29, the symmetric flap deflection at the
wing outside generates a considerable lift, even in the wing middle section. The
circulation distributions according to extended lifting-line theory (three-quarterpoint method, Sec. 3-34) are also shown in Fig. 8-29 as curves 2. As should be
expected, extended lifting-line theory gives a smaller lift than simple lifting-line
AERODYNAMICS OF THE FLAPS AND CONTROL SURFACES 507
21
V
7
- 170
/_"
Of
0.
'0
a0
0.4
of
- , -rIo
770
/IZ11.
r"o
b 07
n_ 1
22
n. 3
n. 4
as
n_ B
11 7
/10
2R
L
Figure 8-29 Circulation distribution over the span due to a discontinuous angle-of-attack
distribution for a trapezoidal wing of aspect ratio A= 2.75; taper X = 0.5. Curve 1, simple
lifting line theory. Curve 2, extended lifting-line theory. (a) Symmetric angle-of-attack distribution.
(b) Antimetric angle-of-attack distribution.
theory. A computational method for the lift distribution on wings with flaps, based
on lifting-surface theory (Sec. 3-3-5), is given in [46]. This method requires the
availability of the angle-of-attack distributions caused by the flap deflection on the
c/4 line (il) and on the trailing edge (i;,). They are, considering Eq. (8-21),
4 ac3c.
a,
elf
«f( r,
a«
2 acrn
anf
2
ae f
of
(8-22a)
)17f
(8-22b)
where the coefficients as/ary and ac,n /ary from Eqs. (8-8a) and (8-8b) and from
Eqs. (8-9a) and (8-9b), respectively, are known from the profile theory of the flap
wing and depend only on the control-surface chord ratio.* An improved method for
describing the effect of the angle-of-attack discontinuity has been given by Hummel
[46]. Lift distributions of wings with deflected flaps (angle-of-attack distribution
with a break) have been computed by Bausch [5] from simple lifting-line theory for
a wing of elliptic planform. For a wing with a trapezoidal planform, corresponding
computations have been published by Richter [5]. A large number of computations
*Since these equations contain local coefficients, the coefficient CM of Eq. (8-5) has here
been written as c,n.
508 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
have been conducted by de Young [10], who applied extended lifting-line theory;
however, he did not exclude the station of discontinuity in his computations.
Investigations, applying lifting-surface theory, have been conducted by Truckenbrodt
and Gronau [46] on delta wings with deflected flaps.
A summary of American tests on wings of finite span with flaps that extend
only over a portion of the span is given in [14]. It includes the separation
characteristics of such wings; compare the publications [31, 541.
Results of a few sample computations of the lift distribution of wings with flap
and control-surface deflections will be given in the following section.
Landing flaps, elevators For the wing of elliptic planform, the change in the mean
zero-lift angle caused by the flap deflection is obtained according to Sec. 3-3-3. For
a sectionwise-constant, symmetric angle-of-attack distribution, Eq. (3-81) yields,
after integration,'
as - -1 +
2
(arccos 77, - 770
1 - rlo)
(8-23)
Here the flap (control surface), having a constant flap chord ratio, extends from
-r?o to +rro. The relationship between of and the flap angle rrf is given by the
theory of the two-dimensional flap wings of Eq. (8-21). The coefficient as/aaf is.
shown in Fig. 8-30 as a function of the flap span. This result is obtained by both
simple and extended lifting-line theories.
A further example, in which Truckenbrodt and Gronau [46] applied liftingsurface theory, is shown in Fig. 8-31. It deals with a delta wing of aspect ratio
A1* = 2b*/cr = 2 equipped with a flap that is symmetrically deflected. The. flap
chord ratio Xf = cf/c, however, varies between Af = s at the wing root and Af = 1 at
the wing tips. The local flap effectiveness was obtained by introducing Eqs. (8-9a)
and (8-9b) into Eqs. (8-22a) and (8-22b). The changes of the mean zero-lift angle
a«/arlf and of the mean zero-moment coefficient acM/anf were computed first.
10f
0.e
0.6
0.2
0,2
0.4
PO
0A
0.8
1.0
Figure 8-30 Change of the mean zero-lift
angle due to flap deflection for an elliptic
wing with various forms of the flap, from
Bausch.
a
?7f
.
Oe
Theory
.°
Measure ments
0
OZ
0°
0°
4°
8°
12°
16°
20°
24°
Mc
-5°
Theory
;, ,
Measureme
°
I
is
8-31 Measured aerodynamic coefficients of a delta wing
with symmetrically deflected flap
Figure
nf5°
I
-02
. 0.16 -0.72 -0.09 -0.04
C
H
0
cM
0.04
0.G08
0.12
0.16
extending over the entire trailing
edge. Aspect ratio A * = 2, profile
NACA 0012; comparison of theory (;, = 0.75) and experiment,
from Truckenbrodt and Gronau.
(a) Geometry. (b) Lift coefficient
vs. angle of attack. (c) Lift coefficient vs. pitching-moment coefficient.
Figure 8-31b and c shows the good agreement of the theoretical results CL(a) and
cyl(a) with measurements at small flap deflections.
Ailerons In Fig. 8-32, the rolling-moment coefficients are given for a wing of
elliptic planform and antimetric control-surface deflection. Figure 8-32a gives the
510 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
2
2
,
I
of
i
1
I
0
4
2
a
6
10
8
12
A
1.0
0.8
1
1o
Figure 8-32 Rolling-moment coeffi-
cient vs. flap deflection for an elliptic
wing, from Bausch. (a) Flap extending
over the entire half-span; curve 1,
b
0.2
0,4
0,6
0.8
extended lifting-line theory; curve 2,
simple lifting-line theory. (b) Effect of
1.0
the flap span.
r10
rolling moment of the ailerons plotted against the aspect ratio with each aileron
extending over the entire half-span. The extended lifting-surface theory of Eq.
(3-100) yields
acMX
4
orA
3n Vkz + 4 + 2
(8-24a)
aaf
where k = irA/cL.. A/2. For comparison, this coefficient according to simple
lifting-line theory is added. The rolling moment of the ailerons for the case of an
aileron extending over only a part of the wing half-span is shown in Fig. 8-32b. In
this case, Eq. (3-100) yields
(acMX
f
sax
1 -170 3
(8-24b)
asf ino =0
where (acMxlaaf)no _o is given by Eq. (8-24a) and Fig. 8-32a. For the delta wing of
Fig. 8-3la, the theoretical coefficients of the aileron rolling moment of
antimetrically deflected ailerons extending over the entire half-span are compared in
Fig. 8-33 with measurements. Agreement between theory and experiment is good
for small and moderate angles of attack.
AERODYNAMICS OF THE FLAPS AND CONTROL SURFACES 511
Aileron investigations and comprehensive experimental results are summarized
in [12, 45].
8-3-2 Flaps on the Wing in Compressible Flow
The flap wing of finite span in compressible flow may be treated according to the
theory of the wing of finite span as discussed in Secs. 4-4 and 4-5.
Subsonic incident flow At subsonic velocities, the subsonic similarity rule (Prandtl-
Glauert) of Sec. 4-4-1 applies. It requires the determination of a wing, to be
computed for incompressible flow, that is transformed from the given geometry of
the wing of finite span at compressible flow. These transformation formulas for the
geometries of the wings are found as Eqs. (4-66)-(4-68). The influence of
compressibility on the aerodynamic coefficients of the wing is obtained from the
transformation formulas Eqs. (4-69)-(4-72). Here, the angle-of-attack distribution
due to the flap deflection remains unchanged and is determined with lifting-surface
theory from Eq. (8-22). Accordingly, Eqs. (8-15a) and (8-15b) give the changes of
the angle of attack and of the momentum coefficient with the flap deflection.
However, these equations for the incompressible reference flow now have to be
evaluated for the transformed wing planform from Eq. (4-15). In Fig. 8-34, the
results of sample computations for wings of finite span with deflected flaps are
shown. They are the three wings discussed several times previously, namely, a
trapezoidal, a swept-back, and a delta wing; see Table 3-4.
Supersonic incident flow The computation of the aerodynamic effect of a flap on a
wing of finite span at supersonic velocities is in some respect simpler than at
subsonic velocities. This becomes obvious from Fig. 8-35, which shows a rectangular
Figuae 8-33 Measured roiling-moment coef-
ficients of a delta wing as shown in Fig.
8-31a, with flaps extending over the entire
half-span for several
I
l
f 10°
of
15°
±20°
angles of attack a.
Comparison of theory (% = 0.75) and measurements from Truckenbiodt and Gronau.
512 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
a
ZA
b
c
0.8
10.6
f
t
0
24
1
0.8
0,4
0
0.2
0.4
0.6
0.8
Ma°,--
7.00
0.2
0.4
0.6
0.6
Mao,
1.00
0.2
0.4
0.6
06
1.0
mat,
I-
Figure 8-34 Change of the zero-lift angle and the zero-moment due to flap deflection for several
wings with flaps extending over the entire trailing edge vs. Mach number for subsonic incident
flow, from Kowalke [26] ; lifting surface theory, % = 1. (a) Trapezoidal wing: A = 2.75,1 = 0.5,
yp = 0°, Af=
(b) Swept back wing: .i = 2.75, A = 0.5, P = 50°, xf= s. (c) Delta wing:
A = 2.31, X = 0,'P = 52.4°, Xf= Cr/8 = cont,
wing and a delta wing with flaps of constant chord extending over the entire trailing
edge. When the flap is being deflected, an additive lift is generated only on this flap
that is equal to the lift of a rectangular wing of span b and of the flap chord cf.
The lift of the wing lying before the flap is not changed by the flap deflection.
To compute the lift caused by flap deflection, the results for the rectangular
wing of Sec. 4-5-4 may be recalled. From Eq. (4-112), the lift coefficient produced
by the flap and referred to the total wing area A is given as
acL
a77f
_ Af
1^
4
A Maw-1
1
Cf
2
b }'Magi - 1
(8-25)
which is valid for cf c b Ma;° - 1, but independent of the wing shape.
For the rectangular wing of Fig. 8-35, the change of the zero-lift angle caused
by the flap deflection can easily be determined. Because aa/arrf = -(acL/ark f)1
(acL/aa), Eqs. (8-25) and (4-112) yield
as
arjf
_
2A Ma-00
1-Af
2A}IMac-1-1
(8-26)
where 1 f = c f/c = A fIA is the control-surface chord ratio. In this equation, the
fraction on the right-hand side, which is always greater than unity, gives the
AERODYNAMICS OF THE FLAPS AND CONTROL SURFACES 513
J
Figure 8-35 Aerodynamics of the flap wing at supersonic
incident flow. (a) Rectangular wing with flap extending
over the entire trailing edge. (b) Delta wing with flap
extending over the entire trailing edge.
correction of the value for the two-dimensional flap wing, as can be verified by
comparison with Eq. (8-16a).
The pressure distribution on the flap of a wing in supersonic incident flow may
also be established quite easily. Figure 8-36 shows a flap design in which the
right-hand-side edge of the flap is an "outer edge," the left-hand edge an "inner
edge," both of which are parallel to the incident flow direction. When the flap is
deflected, Mach lines originate at either upstream edge. In the case of no
intersection of these Mach lines on the flap, the pressure distribution in zone 1 is
77
t=1
tr-1
t=1
Figure
8-36 Pressure distribu-
tion due to flap deflection on a
t=-1 0
1
t=-1
0
rectangular flap at supersonic
incident flow.
514 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
that for plane flow. The resultant pressure coefficient on the upper and the lower
side is, therefore, with Eq. (4-85) and Table 4-5,
4'77 f
Cpl =C pp1 =
,
)/Maw
(8-27a)
-1
The flow in zones 2 and 3 is cone-symmetric. For zone 2, Eq. (4-111) yields
cp2 =
1
arccos (1 + 2t)cppl
(8-27b)
7c
For zone 3, Tucker and Nelson [47] found the expression
cp3 =
i arccos (-t) Cppl
(8-27c)
In these expressions t = y/x tan p = (y/x) -NIMa, - 1, and y is measured from the
upstream corners of the flap. In Fig. 8-36, the pressure distributions are shown for a
section x = const. On the side of the inner edge, the flap deflection causes, within
the range of the Mach cone, a lift on the undeflected wing that is equal to the lift
loss at the adjacent portion of the flap.
Furthermore, Fig. 8-37 shows a flap arrangement with a swept-back outer edge
of the flap such as, for example, is found in delta wings. In Fig. 8-37a, the outer.
edge is a subsonic edge. If the two Mach lines originating at the two, upstream flap
corners do not intersect on the flap, zone 1 has again, as in Fig. 8-36, the pressure
distribution of plane flow. In the case of the subsonic edge (m < 1) of Fig. 8-37a,
the pressure distribution of zone 4 is of the kind given in Fig. 4-67 for a delta wing
with a subsonic leading edge. In the case of the supersonic edge (Mac, > 1) of Fig.
8-37b, where the Mach cone from the right-hand upstream corner lies entirely on
the flap, the pressure distributions of zones 5 and 6 are of the kind given in Fig.
a
b
Flap
l
I
Figure 8-37 Pressure distribution due to flap deflection on a trapezoidal flap at supersonic incident
flow, from Tucker and Nelson. (a) Subsonic outer edge, m = 4. (b) Supersonic outer edge, m = 1 .
AERODYNAMICS OF THE FLAPS AND CONTROL SURFACES 515
1.0
0.8
2
0.6
a
04
bf
3
7
-
0.2
0
3
2
1
bf
cf
7
->
Figure 8-38 Lift due to flap deflection at supersonic incident flow.
Curve 1, inner flap. Curve 2, tip
flap. Curve 3, full-span flap.
4-69 for a delta wing with a supersonic leading edge. The pressure in zone 6 is
constant, Eq. (4-89):
Cps -
Y1L
n
CP p1
(8-28)
ynv -1
with m = tan 7/tan µ. The pressure distributions in zones 4 and 5 have been
determined by Tucker and Nelson [47].
Finally, a few data will be given. in the following two figures on the lift
produced by the flap deflection and on the position of its center of application.
Figure 8-38 gives the total lift of three rectangular flaps. Flap 1 has two inner edges
(inner flap), flap 2 an inner and an outer edge (tip flap), and flap 3 two outer edges
bf.
(full-span flap). Shown in this figure is the ratio of the total lift produced by the
flap to the lift of the two-dimensional flap wing as a function of the quantity
Ma', - 1 /c f. Flap 1 does not cause any lift loss compared with the
two-dimensional flap wing; Eq. (8-25) applies to flap 3. The lift of flap 2 is the
arithmetic mean of those of flaps 1 and 3. Figure 8-39 shows the position of the
lift force of the flap (flap neutral point). Here, xf is the distance of the flap neutral
point from the axis of rotation. For flap 1, the flap neutral point lies at the flap
half-chord. It shifts forward for flaps 2 and 3.
The rolling moment due to aileron deflection can be computed very easily by
realizing that the lift force at antimetrically deflected flaps acts, in very good
approximation, on the half-span of the flap.
Further information on rectangular flaps is found in Schulz [471. Flaps on
rectangular, delta, and swept-back wings have been investigated by Lagerstrom and
Graham [47]. Flaps with outer (horn) balances have been studied by Naylor [47].
8-3-3 Control Surfaces on the Tail Unit
in this section, a brief discussion will be given of the aerodynamic forces generated
by the control-surface deflection of the tail unit and their effect on the force and
516 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
0.
r
O'N
2
0,4
3
0.3
f
0.2
bf
C,
t
r
i
i
0.1
3
2
1
bf
Maw -1
cf
Figure 8-39 Position
of the flap
neutral point for flap designs of Fig.
--_-
8-38.
moment equilibrium of the whole airplane. For the case of zero control-surface
deflection, the contributions of the horizontal tail and the vertical tail, respectively,
to the aerodynamic forces of the whole airplane have been given in Secs. 7-2-1 and
7-3-1.
Elevator For the contribution of the horizontal tail with deflected elevator to the
pitching moment of the whole airplane, Eqs. (7-3a) and (7-3b) yield
Ms - -
CMH
do E
H Ca
aax
a'7H
77H
4'H AH rH
q. A cg
Here, from Fig. 7-5, rH is the distance of the lift force of the horizontal tail from
the moment reference axis of the airplane.
The change in the moment caused by the elevator deflection at constant angle
of attack is thus obtained as
(acMH)
a77H a=const
= dcrH a«H qa Ax rH
dcH a77Hgoo A Cµ
(8-29)
Here, the quantity rH of the previous equation has been replaced by the lever arm
,rH, which is the distance of the flap neutral point from the moment reference axis
of the horizontal tail.
For the two-dimensional flap wing in incompressible flow, the position of the
flap neutral point is given in Fig. 8-15. The change in the pitching moment caused
by the elevator deflection at constant lift coefficient (zero-moment coefficient) is
obtained in analogy to Eq. (7-15) by substituting -(acH/3r1H)77H for EH as
_delHaoH4HAHrHN
aCMg
a77H
cL=const
das a?7H q. A
(8-30)
Cu
Here rHN is the distance of the neutral point of the elevator from the neutral
point of the whole airplane (see Fig. 7-6b).
AERODYNAMICS OF THE FLAPS AND CONTROL SURFACES 517
Rudder The contribution of the vertical tail with a deflected rudder to the yawing
moment of the whole airplane becomes, from Eqs. (7-49a) and (7-49b),
cMZv
-`dcty
day
aav
Qv
arty
(V)
gvAv r'v
q0 A s
Here, r'y from Fig. 7-36 is the distance of the side force of the vertical tail from the
moment reference axis of the airplane.
The change in the yawing moment caused by the rudder deflection is thus given
as
acMZV
an v
= dcrv aav qv AV r'v
day anv q. A
s
(8-31)
Here the quantity ry of the previous equation has been replaced by the lever arm
r 'y , which is the distance of the flap neutral point from the moment reference axis
of the vertical tail.
Rudder moments Information on the rudder moments of the airfoil of infinite span
for incompressible flow is found in Sec. 8-2. The control-surface moments of the
elevator and rudder and also of the ailerons cannot, in general, be computed with
sufficient accuracy, because for the control-surface moments the transformation,
from the airfoil of infinite span (plane problem) to the wing of finite span is not
possible in a reliable way. The control-surface moments for control surfaces with
balance provisions of Fig. 8-2 (inner balance, outer balance, balance tabs) are
particularly difficult to determine because they are greatly affected by the boundary
layer as well as by' inviscid flow problems. The control-surface moments must
therefore be determined largely through wind tunnel and flight tests (see. Stiess
[18]). Some wind tunnel measurements on the control-surface moments of tail
surfaces with inner and outer balances were reported by Schlichting and Ulrich
[39].
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520 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
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Sci., 4:619456, 1962. Butler, S. F. J. and J. Williams: Aer. Quart., 11:285-308, 1960.
Davidson, I. M.: J. Roy. Aer. Soc., 60:25-50, 1956. Dirnmock, N. A.: Aer. Quart.,
8:331-345, 1957. Hirsch, R.: Aircr. Eng., 29:366-375, 1957; 30:11-19, 1958. Stratford, B.
S.: Aer. Quart., 7:45-59, 85-105, 169-183, 1956. Williams, J.: Z. Flugw., 6:170-176,
1958. Williams, J. and A. J. Alexander: Aer. Quart., 8:21-30, 1957.
53. Wuest, W.: Messungen an einem Fliigelprofil mit Nasenabsaugung im Vergleich zu einem
Profil mit Nasenspalt, AVA 62-03, 1962.
54. Young, A. D.: A Review of Some Stalling Research, ARC RM 2609, 1942/1951.
55. Young, A. D.: The Aerodynamic Characteristics of Flaps, ARC RM 2622, 1947/1953.
BIBLIOGRAPHY
1. Books and handbooks-Contributions to the aerodynamics of the airplane
Abbott, I. H. and A. E. von Doenhoff: "Theory of Wing Sections, Including a Summary of
Airfoil Data," Dover, New York, 1959.
Alexandrow, W. L.: "Luftschrauben" (transl. from the Russian), Verlag Technik, Berlin, 1954.
Ashley, H. and M. T. Landahl: "Aerodynamics of Wings and Bodies," Addison-Wesley, Reading,
Mass., 1965.
Belotserkovskii, S. M.: "The Theory of Thin Wings in Subsonic Flow" (transl. from the
Russian), Plenum, New York, 1967.
Bonney, E. A.: "Engineering Supersonic Aerodynamics," McGraw-Hill, New York, 1950.
, M. J. Zucrow, and C. W. Besserer: "Aerodynamics, Propulsion, Structures, and Design
Practice (Principles of Guided Missile Design)," Van Nostrand, Princeton, N.J., 1956.
Carafoli, E.: "Tragfldgeltheorie, inkompressible Fliissigkeiten (transl. from the Romanian),
Verlag Technik, Berlin, 1954.
,
D. Mateescu, and A. Nastase: "Wing Theory in Supersonic Flow," Pergamon, Oxford,
1969.
Clancy, L. J.: "Aerodynamics," Pitman, London, 1975.
Dommasch, D. 0., S. S. Sherby, and T. F. Connolly: "Airplane Aerodynamics," 4th ed.,
Pitman, New York, 1967.
Donovan, A. F. and H. R. Lawrence (eds.): "Aerodynamic Components of Aircraft at High
Speeds," vol. VII of T. von Karman, H. L. Dryden, and H. S. Taylor (eds.), "High Speed
Aerodynamics and Jet Propulsion," Princeton University Press, Princeton, N.J., 1957.
, H. R. Lawrence, F. Goddard, and R. R. Gilruth (eds.): "High Speed Problems of Aircraft
and Experimental Methods," vol. VIII of T. von Karman, H. L. Dryden, and H. S. Taylor
(eds.), "High Speed Aerodynamics and Jet Propulsion," Princeton University Press, Princeton,
N.J., 1961.
Durand, W. F. (ed.) : "Aerodynamic Theory-A General Review of Progress," Springer, Berlin,
1934-1936, Dover, 1963.
Frankl, F. 1. and E. A. Karpovich: "Gas Dynamics of This Bodies" (trans!. from the Russian),
Interscience, London, 1953.
521
522 BIBLIOGRAPHY
Fuchs, R., L. Hopf, and F. Seewald: "Aerodynamik," vol. I. "Mechanik des Flugzeuges," 1934;
vol. II. "Theorie der Luftkrafte," 2nd ed., 1935, Springer, Berlin.
Glauert, H.: "The Elements of Aerofoil and Airscrew Theory," Cambridge University Press,
Cambridge, 1947. "Die Grundlagen der Tragfliigel- and Luftschraubentheorie" (German
transl. by H. Holl), Springer, Berlin, 1929.
Grammel, R.: "Die hydrodynamischen Grundlagen des Fluges," Vieweg, Braunschweig, 1917.
Houghton, E. L. and A. E. Brock: "Aerodynamics for Engineering Students (SI Units)," 2nd
ed., Arnold, London, 1970. Houghton, E. L. and R. P. Boswell: "Further Aerodynamics
for Engineering Students (Metric and Imperial Units)," Arnold, London, 1969.
Krasnov, N. F.: "Aerodynamics of Bodies of Revolution" (transl., 2nd Russian ed.), American
Elsevier, New York, 1970.
Kiichemann, D.: "The Aerodynamic Design of Aircraft," Pergamon, Oxford, 1978.
Lanchester, F. W.; "Aerial Flight," Constable, London, 1907. "Aerodynamik" (German transl.
by C. Runge and A. Runge), Teubner, Leipzig, 1911.
Martynov, A. K.: "Practical Aerodynamics" (transl. from the Russian), Pergamon, Oxford, 1965.
McCormick, B. W., Jr.: "Aerodynamics of V/STOL Flight," Academic, New York, 1967.
Miele, A. (ed.): "Theory of Optimum Aerodynamic Shapes," Academic, New York, 1965.
Miene Thomson, L. M.: "Theoretical Aerodynamics," 4th ed., Macmillan, London, 1966.
Pope, A.: "Basic Wing and Airfoil Theory," McGraw-Hill, New York, 1951.
Proll, A.: "Grundlagen der Aeromechanik and Flugmechanik," Springer, Vienna, 1951.
Rauscher, M.: "Introduction to Aeronautical Dynamics," Wiley, New York, 1953.
Riegels, F. W.: "Aerodynamische Profile, Windkanal-Messergebnisse, theoretische Unterlagen,"
Oldenbourg, Munich, 1958. "Aerofoil Sections" (English transl. by D. G. Randall),
Butterworths, London, 1961.
Robinson, A. and J. A. Laurmann: "Wing Theory" (Cambridge Aeronautics Series II),
Cambridge University Press, Cambridge, 1956.
Schmidt, H.: "Aerodynamik des Fluges, Eine Einfi. hrung in die mathematische Tragflachentheorie," De Gryter, Berlin, 1929.
Schmitz, F. W.: "Aerodynamik des Flugmodells, Tragflugelrnessungen," 4th ed., Lange,
Duisburg, 1960.
Sears, W. R. (ed.): "General Theory of High Speed Aerodynamics," vol. VI of T. von Kirmin,
H. L. Dryden, and H. S. Taylor (eds.), "High Speed Aerodynamics and Jet Propulsion,"
Princeton University Press, Princeton, N.J., 1954.
Theodorsen, T.: "Theory of Propellers," McGraw-Hill, New York, 1948.
Thwaites, B. (ed.): "Incompressible Aerodynamics-An Account of the Theory and Observation
of the Steady Flow of Incompressible Fluid Past Aerofoils, Wings, and Other Bodies,"
Clarendon, Oxford, 1960.
von Mises, R.: "Theory of Flight," Dover, New York, 1960. "Fluglehre" (German version by K.
Hohenemser), 6th ed., Springer, Berlin, 1957.
Weinig, F.: "Aerodynamik der Luftschraube," Springer, Berlin, 1940.
Weissinger, J.: "Theorie des Tragfliigels bei stationarer Bewegung in reibungslosen, inkom-
pressiblen Medien," in S. Fliigge and C. Truesdell (eds.), "Handbuch der Physik," vol.
VIII/2. "Stromungsmechanik II," pp. 385-437, Springer, Berlin, 1963.
Woods, L. C.: "The Theory of Subsonic Plane Flow'" (Cambridge Aeronautics Series III),
Cambridge University Press, Cambridge, 1961.
II. Books and handbooks-Aerodynamics of fluid mechanics (selection)
Albring, W.: "Angewandte Stromungslehre," 4th ed., Steinkopff, Dresden, 1970.
Batchelor, G. K.: "An Introduction to Fluid Dynamics," Cambridge University Press,
Cambridge, 1967.
Becker, E.: "Gasdynamik," Teubner, Stuttgart, 1965. "Gas Dynamics" (English transl by E. L.
Chu), Academic, New York, 1968.
Betz, A.: "Konforme Abbildung," 2nd ed., Springer, Berlin, 1964.
Chang, P. K.: "Separation of Flow," Pergamon, Oxford, 1970.
,
BIBLIOGRAPHY 523
Chernyi, G. G.: "Introduction to Hypersonic Flow" (transl. from the Russian), Academic, New
York, 1961.
Cox, R. N. and L. F. Crabtree: "Elements of Hypersonic Aerodynamics," Academic, New York,
1965.
Curie, N. and H. J. Davies: "Modern Fluid Dynamics," vol. I. "Incompressible Flow," 1968; vol.
II. "Compressible Flow," 1971, Van Nostrand Reinhold, London.
Currie, I. G.: "Fundamental Mechanics of Fluids," McGraw-Hill, New York, 1974.
Dorfner, K.-R.: "Dreidimensionale Uberschallprobleme der Gasdynamik," Springer, Berlin, 1957.
Dryden, H. L., F. D. Murnaghan, and H. Bateman: "Hydrodynamics," Dover, New York, 1956.
Duncan, W. J., A. S. Thom, and A. D. Young: "An Elementary Treatise on the Mechanics of
Fluids (SI Units)," 2nd ed., Arnold, London, 1970.
Eskinazi, S.: "Vector Mechanics of Fluids and Magnetofluids," Academic, New York, 1967.
Ferrari, C. and F. G. Tricomi: "Transonic Aerodynamics" (transl. from the Italian), Academic,
New York, 1968.
Ferri, A.: "Elements of Aerodynamics of Supersonic Flows," Macmillan, New York, 1949.
Flugge, S. and C. Truesdell (eds.): "Handbuch der Physik" ("Encyclopedia of Physics"), vols.
VIII/1, VIII/2, IX. "Stromungsmechanik I, 11, III" ("Fluid Dynamics I, II, III,"), Springer,
Berlin, 1959, 1960, 1963.
Forsching, H. W.: "Grundlagen der Aeroelastik," Springer, Berlin, 1974.
Goldstein, S. (ed.): "Modern Developments in Fluid Dynamics-An Account of Theory and
Experiment Relating to Boundary Layers, Turbulent Motion and Wakes," vols. I and II,
Dover, New York, 1965.
Guderley, K. G.: "Theorie schallnaher Stromungen," Springer, Berlin, 1957. "The Theory of
Transonic Flow" (English transl. by J. R. Moszynski), Pergamon, New York, 1962.
Hayes, W. D. and R. F. Probstein: "Hypersonic Flow Theory," 2nd ed., vol. I. "Inviscid Flows,"
1966; vol. II. "Viscous and Rarefied Flows" (in preparation), Academic, New York.
Hilton, W. F.: "High-Speed Aerodynamics," Longmans, Green, London, 1952.
Hoerner, S. F.: "Fluid-Dynamic Drag-Practical Information on Aerodynamic Drag and
Hydrodynamic Resistance," 3rd ed., Hoerner, Midland Park, N.J., 1965.
and H. V. Borst: "Fluid-Dynamic Lift-Practical Information on Aerodynamic and
Hydrodynamic Lift," Hoerner, Brick Town, N.J., 1975.
Howarth, L. (ed.): "Modern Developments in Fluid Dynamics-High Speed Flow," vols. I and II,
Clarendon, Oxford, 1964.
Karamcheti, K.: "Principles of Ideal-Fluid Aerodynamics," Wiley, New York, 1966.
Keune, F. and K. Burg: "Singularitatenverfahren der Stromungslehre," Braun, Karlsruhe, 1975.
Kuethe, A. M. and C.-Y. Chow: "Foundations of Aerodynamics-Bases of Aerodynamic Design,"
3rd ed., Wiley, New York, 1976.
Lachmann, G. V. (ed.): "Boundary Layer and Flow Control-Its Principles and Application,"
vols. I and II, Pergamon, Oxford, 1961.
Liepmann, H. W. and A. Roshko: "Elements of Gas Dynamics," Wiley, New York, 1957.
Loitsyanskii, L. G.: "Mechanics of Liquids and Gases" (transl., 2nd Russian ed.), Pergamon,
Oxford, 1966.
Miles, E. R. C.: "Supersonic Aerodynamics-A Theoretical Introduction," Dover, New York,
1950.
Miles, J. W.: "The Potential Theory of Unsteady Supersonic Flow," Cambridge University Press,
Cambridge, 1959.
Milne-Thomson, L. M.: "Theoretical Hydrodynamics," 5th ed., Macmillan, London, 1968.
Oswatitsch, K.: "Grundlagen der Gasdynamik," Springer, Vienna, 1977. "Gas Dynamics"
(English transl., 1st ed., by G. Kuerti), Academic, New York, 1956.
(ed.): "Symposium Transsonicum I," Springer, Berlin, 1964; Oswatitsch, K. and D. Rues
(eds.): "Symposium Transsonicum II," Springer, Berlin, 1976.
Pai, S.-I.: "Introduction to the Theory of Compressible Flow," Van Nostrand, Princeton, N.J.,
1959.
Prandtl, L. and Tietjens, 0.: "Hydro- and Aeromechanik," vol. I, 1929; vol. 11, 1944, Springer,
Berlin.
524 BIBLIOGRAPHY
"Hydro- and Aeromechanics," vols. I and II (English transl. by L. Rosenhead and J. P. den
Hartog), Dover, New York, 1957.
, K. Oswatitsch, and K. Wieghazdt (eds.): "Fiihrer durch die Stromungslehre," 7th ed.,
Vieweg, Braunschweig, 1969. "Essentials of Fluid Dynamics" (English transl., 3rd ed.),
Blackie, London, 1952.
Sauer, R.: "Einfiihrung in die theoretische Gasdynamik," 3rd ed., Springer, Berlin, 1960.
"Nichtstationare Probleme der Gasdynamik," Springer, Berlin, 1966.
Schlichting, H.: "Grenzschicht-Theorie," 5th ed., Braun, Karlsruhe, 1965. "Boundary-Layer
Theory" (English transl. by J. Kestin), 7th ed., McGraw-Hill, New York, 1979.
Shapiro, A. H.: "The Dynamics and Thermodynamics of Compressible Fluid Flow," vol. 1, 1953,
vol. II, 1954, Ronald, New York.
Shepherd, D. G.: "Elements of Fluid Mechanics," Harcourt, Brace, World, New York, 1965.
Truckenbrodt, E.: "Stromungsmechanik-Grundlagen and technische Anwendungen," Springer,
Berlin, 1968.
Truitt, R. W.: "Hypersonic Aerodynamics," Ronald, New York, 1959.
van Dyke, M.: "Perturbation Methods in Fluid Mechanics," Academic, New York, 1964.
von Karman, T.: "Aerodynamics-Selected Topics in the Light of Their Historical Development," Cornell University Press, Ithaca, N.Y., 1954. "Aerodynamik-Ausgewahlte Therrien
im Lichte der historischen Entwicklung" (German transl. by F. Seewald), Interavia, Genf,
1956.
, H. L. Dryden, and H. S. Taylor (eds.): "High Speed Aerodynamics and Jet Propulsion,"
vols. I-XII, Princeton University Press, Princeton, N.J., 1954-1964.
von Mises, R. and K. O. Friedrichs: "Fluid Dynamics," Springer, New York, 1971.
Ward, G. N.: "Linearized Theory of Steady High-Speed Flow, Cambridge University Press,
Cambridge, 1955.
White, F. M.: "Viscous Fluid Flow," McGraw-Hill, New York, 1974.
Wieghardt, K.: "Theoretische Stromungslehre, Eine Einfiihrung," Teubner, Stuttgart, 1965.
Zierep, J.: "Theoretische Gasdynamik," 3rd ed., Braun, Karlsruhe, 1976.
III. Collected treatises and general survey papers*
Betz, A.: Lehren einer fiinfzigjahrigen Stromungsforschung, Z. Flugw., 5:97-105, 1957.
Dryden, H. L.: Gegenwartsprobleme der Luftfahrtfoischung, Z. Flugw., 6:217-233, 1958.
FIAT Review of German Science: "Naturforschung and Medizin in Deutschland, 1939-1946,"
vol. 5, pt. 3, A. Walther (ed.), "Mathematische Grundlagen der Stromungsmechanik," 1947;
vol. 11, A. Betz (ed.), "Hydro- and Aerodynamik," 1947.
Jones, R. T.: "Collected Works," NASA TM X-3334, National Technical Information Service,
Springfield, Va., 1976.
Kiichemann, D., P. Carriere, B. Etkin, W. Fiszdon, N. Rott, J. Smolderen, I. Tani, and W. Wrist
(eds.): "Progress in Aeronautical Sciences," Pergamon, Oxford, 1961.
Prandtl, L.: "Gesammelte Abhandlungen zur angewandten Mechanik, Hydro- and Aerodynamik," vols. I-III, Springer, Berlin, 1961.
and A. Betz: "Vier Abhandlungen zur Hydrodynamik and Aerodynamik (Fliissigkeit mit
kleiner Reibung; Tiagfldgeltheorie, I. and U. Mitteilung; Schraubenpropeller mit geringstem
Energieverlust)," Kaiser Wilhelm Institut, Gottingen, 1927.
,
C. Wieselsberger, and A. Betz: "Ergebnisse der Aerodynamischen Versuchsanstalt zu
Gottingen," vol. I, 4th ed., 1935; vol. II,1923;vol. 111, 1935; vol. IV, 1932, Oldenbourg, Munich.
Taylor, G. I.: "Scientific Papers," vol. I, 1958; vol. II, 1960; vol. III, 1963; vol. IV, 1971,
Cambridge University Press, Cambridge.
van Dyke, M., W. G. Vincenti, and J. V. Wehausen: "Annual Review of Fluid Mechanics," Annual
Reviews, Palo Alto, Calif., 1969-1979.
*Note the special survey papers listed in the individual chapters.
BIBLIOGRAPHY 525
von Karman, T.: "Collected Works," vols. I-IV, 1902-1051, Butterworths, London, 1956; vol. V,
1952-1963, von Karman Institute, Rhode-St. Genese,1975.
: Supersonic Aerodynamics-Principles and Applications, J. Aer. Sci., 14:373-409, 1947;
"Collected Works," vol. IV, pp. 271-326, Butterworths, London, 1956.
Some Significant Developments in Aerodynamics Since 1946, J. Aerosp. Sci., 26:129144, 154, 1959; "Collected Works," vol. V, pp. 235-267, von Karrnan Institute, Rhode-St.
Genese, 1975.
IV. Yearbooks, irregular periodicals, and journals
AFITAE (AFITA), Association Franraise des Ingenieurs et Techniciens de l'Aeronautique et de
I'Espace, Paris: Technique et Science Aeronautiques, 1950-1961; Technique et Science
Aeronautiques et Spatiales, 1962-1967; l'Aeronautique et 1'Astronautique, 1968-.
AGARD, Advisory Group for Aerospace (Aeronautical) Research and Development, Neuilly-surSeine, Paris: Agardographs, Reports, Conference Proceedings, Lecture Series, 1952-.
AIAA (IAS), American Institute of Aeronautics and Astronautics (Institute of the Aeronautical
Sciences), New York: Journal of the Aeronautical Sciences, 1934-1958; Journal of the
Aerospace Sciences, 1958-1962; Aeronautical Engineering Review, 1942-1958; Aerospace
Engineering, 1958-1963; Astronautics and Aerospace Engineering, 1963; Astronautics and
Aeronautics, 1964-; AIAA Journal, 1963-; Journal of Aircraft, 1964-; Journal of
Spacecraft and Rockets 1964-; Journal of Hydronautics, 1967-.
AIDA, Associazione Italiana di Aerotecnica, Rome: L'Aerotecnica, 1920-.
ARC, Aeronautical Research Council, London: Reports and Memomoranda, 1909-; Current
Papers, 1950-.
ARL, Aeronautical Research Laboratory, Melbourne: Technical Reports, Notes, Annual Reports,
1939-.
DFVLR (AVA/DVL/DFL), Deutsche Forschungs- and Versuchsanstalt fur Luft- and Raumfahrt,
Porz-Wahn, K61n: Berichte 1953-1964 (AVA), 1955-1964 (DVL), 1956-1963 (DFL);
DLR-Mitteilungen, 1964-1975; DLR-Forschungsberichte, 1964-; Jahresberichte, 1969-.
DGLR (WGLR/WGL), Deutsche Gesellschaft fur Luft- and Raumfahrt, K61n: Jahrbiicher,
1912-1936, 1952-1961 (WGL), 1962-1967 (WGLR), 1968- (DGLR); Zeitschrift fur
Flugtechnik and Motorluftschiffahrt, 1910-1933; Zeitschrift fur Flugwissenschaften, 19531976, in cooperation with DFVLR; Zeitschrift fur Flugwissenschaften and Weltraumforschung, 1977-, in cooperation with DFVLR.
Dt. Akad. Lufo., Deutsche Akademie der Luftfahrtforschung: Schriften, Mitteilungen, Jahrbncher, 1938-1944.
FFA, Flygtekniska Forsoksanstalten (The Aeronautical Research Institute of Sweden), Stockholm: Reports, Memoranda, 1945-.
LGL, Lilienthal-Gesellschaft fur Luftfahrtforschung: Luftwissen, 1934-1944 (eds.: Reichsluftfahrtministerium).
NASA (NACA), National Aeronautics and Space Administration (National Advisory Committee
for Aeronautics), Washington, D.C.: NACA Rept., TN, TM, 1915-1958; NASA CR, SP,
TM, TN,TT,1959.
NLL, National Luchtvaartlaboratorium, Amsterdam: Reports, Technical Notes, 1921-.
NRCA, National Research Council of Canada, Ottawa: Canadian Aeronautical Journal,
1955-1961; Canadian Aeronautics and Space Journal, 1962-.
ONERA, Office National d'Etudes et des Recherches Aerospatiales, Chatillon-sous-Bagneux,
Paris: La Recherche Aeronautique, 1950-1963; La Recherche Aerospatiale, 1963-; Notes
Techniques, 1951-.
RAE, Royal Aircraft Establishment, Farnbourough: Reports, Technical Notes, 1909.
RAeS, The Royal Aeronautical Society, London: Journal of the Royal Aeronautical Society,
1897-1967; The Aeronautical Journal, 1968-; The Aeronautical Quarterly, 1949/1950-;
Data Sheets (ESDU), 1965-.
526 BIBLIOGRAPHY
VKI, von Karman Institute for Fluid Dynamics, Rhode-St. Genese, Brussels: Lecture Series,
1962.
ZWB, Zentrale fur wissenschaftliches Berichtswesen der Luftfahrtforschung, Berlin-Adlershof:
Forschungsberichte, Untersuchungen and Mitteilungen, 1933-1945; Jahrbiicher der
deutschen Luftfahrtforschung, 1937-1942 (Jb. 1943 as preprint); Ringbuch der Luftfahrttechnik, 1937-; Luftfahrtforschung, 1928-1943.
AUTHOR INDEX
Abbott, I. H., 27, 36, 62, 63, 67, 72, 76, 100,
101, 492, 517, 521
Ackeret, J., 27, 42, 43, 45, 49, 98, 101, 103,
219, 232, 239, 317
Ackermann, U., 132, 210
Ackermann, W., 53,101, 123
Adams, M. C., 293, 310, 313, 317, 318, 321,
331, 363, 367, 369, 375, 388, 425
Adamson, D., 464, 477
Albring, W., 521, 522
Alexander, A. J., 503, 520
Alexandrow, W. L., 521
Alford, W. J., Jr., 449,477
Alksne, A., 253, 322
Allen, H. J., 53, 74, 101, 366, 367, 492, 517
Alway, G. G., 128, 155, 185, 209
Arnic, J. L., 229, 322
Anderson, S. B., 499, 500, 517
Angelucci, S. B., 366, 368
Anliker, M., 317, 323
Arnold, K. 0., 363, 367, 499, 500, 504, 505
517
Ashill, P. R., 178, 206
Ashley, H., 111, 132, 206, 214, 331, 367, 375,
405, 416, 425, 426, 521
Baals, D. D., 425, 426
Babaev, D. A., 317, 318
Bagley, J. A., 132, 206
Bamber, M. J., 398, 399, 400, 401, 426
Barrows, T. M., 132, 211
Bartlett, G. E., 169, 170, 206
Batchelor, G. K., 522
Bateman, H., 523
Bauer, F., 255, 317
Bausch, K., 139, 210, 485, 487, 491, 492, 507,
517, 518
Beane, B., 293, 310, 313, 321
Becker, E., 266, 268, 311, 317, 522
Behrbohm, H., 289, 317, 320
Belotserkovskii, S. M., 521
Bera, R. K., 300, 317, 319, 361, 367
Berndt, S. B., 361, 369
Besserer, G. W., 521
Betz, A., 33, 36, 49, 51, 52, 66, 74, 75, 101,
102, 121, 132, 182, 206, 233, 317, 443,
449, 450, 477, 478, 497, 499, 500, 517,
518, 522, 524
Bhateley, I. C., 170, 209
Bilanin, A. J., 450, 477
Birnbaum, W., 53, 101, 123
Black, J., 169, 170, 206
Bland, S. R., 132, 208
Blasius, H., 38, 104
Blenk, H., 111, 114, 118, 123, 128, 129, 132,
206, 209
Bloom, A. M., 450, 477
Boatright, W. B., 464, 477
Bobbitt, P. J., 475, 477
Bollay, W., 132, 166, 206
Bonner, E., 413, 427
Bonney, E. A., 298, 317, 521
Borja, M., 156, 206
Borst, H. V., 36, 90, 102, 111, 207, 214, 319,
331, 367, 523
Bradley, R. G., 170, 209, 404, 407,428
Brakhage, H., 156, 206
Braun, G., 398, 399, 400, 401, 427, 447, 452,
453, 478
527
528 AUTHOR INDEX
Braunss, G., 132, 210
Brebner, G. G., 88, 101, 132, 170, 206, 317,
320
Bridgewater, J., 255, 320, 425, 427
Brock, A. E., 522
Broderick, J. B., 356, 366, 369
Brown, C. E., 169, 206, 293, 310, 313, 317,
318, 321, 331, 361, 367
Browne, S. H., 406,425
Bryer, D. W., 169, 170, 206
Buford, W. E., 366, 368
Bullivant, W. K., 449, 47 8
Burg, K., 52, 102, 197, 208, 253, 290, 296,
310, 311, 318, 319, 323, 332, 368, 523
Burgers, J. M., 36, 104,114, 118, 123, 128,
129,132, 209, 210
Busemann, A., 242,243, 318, 358, 367
Bussmann, K., 171, 172, 186, 187, 207
Butler, S. F. J., 503, 520
Byrd, P. F., 131, 161,210
Byrnes, A. L., 455, 477
Cahill, J. F., 492, 495, 497, 518
Cahn, M. S., 254, 255, 321
Campbell, G. S., 380, 416, 417, 427
Carafoli, E., 317, 318, 521
Carmichael, R. L., 403, 426
Carriere, P., 495, 518
Cebeci, T., 93, 101, 340, 370
Chang, P. K., 522
Chapman, D. R., 362, 367
Chaudhuri, S. N., 164, 206, 208
Chen, A. W., 87, 102
Cheng, H. K., 169, 206, 317, 318
Chernyi, G. G., 523
Chinneck, A., 358, 368
Chow, C.-Y., 523
Clancy, L. J., 521
Clarke, J. H., 413, 327
Clarkson, M. H., 298, 320
Cleary, J. W., 358, 369
Cohen, D., 214, 289, 290, 317, 319, 323
Cole, J. D., 253, 320
Colwell, G. T., 492, 518
Conolly, T. F., 521
Cook, W. L., 499, 500, 517
Cooke, J. C., 88, 101, 170, 206
Cooper, G. E., 499, 500, 517
Cox, R. N., 523
Crabtree, L. F., 87, 101, 523
Cramer, R. H., 406, 409, 425
Crowk, A. E., 492, 506, 518
Crown, J. C., 253, 321
Cunningham, A. M., Jr.,128,
128,156, 206
Curie, N., 523
Currie, I. G., 523
Das, A., 166, 170, 206, 209, 317, 318, 362,
363, 367, 504, 518
Davidson, I. M., 503, 520
Davies, H. J., 523
Davis, T., 464, 477
Deffenbaugh, F. D., 366, 368
Diesinger, W. H., 293, 296, 318
Dimmock, N. A., 503, 520
Doetsch, H., 29, 99, 101, 171, 172, 186, 187,
207, 485, 487, 491, 492, 517, 518
Dommasch, D. 0., 521
Donaldson, C. duP., 450,477
Donovan, A. F., 521
Dorfner, K.-R., 293, 313,315, 318, 523
Drougge, G., 317, 320
Dryden, H. L., 521, 523, 524
Duncan, W. J., 523
Durand, W. F., 521
Edwards, R. H., 169, 206
Egginton, W. J., 486, 492, 518
Eichelbrenner, E., 495, 518
Emerson, H. F., 271, 318
Eminton, E., 361, 367
Emunds, H., 413, 415, 426
Eppler, R., 52, 75, 100, 101
Erickson, J. C., Jr., 503, 519
Eskinazi, S., 523
Etkin, B., 296, 318
Evans, A. J., 362,367
Evans, M. R., 450,477
Evvard, J. C., 293, 296, 318
Fabricius, W., 447, 478
Fage, A., 450,477
Falkner, V. M., 128, 155, 185, 207, 209
Feindt, E. G., 52,102, 153, 158,171, 172, 210
Fell, J., 298, 320
Fenain, M., 317, 318
Ferrari, C., 363, 369, 375, 405, 406, 409, 417,
425, 465,477, 523
Ferri, A., 413, 427, 523
Fiecke, D., 242, 243, 308, 309, 310, 316, 318,
363, 367
Filotas, L. T., 139, 210
Fink, M. R., 366, 367
Fink, P. T., 168, 169, 209
AUTHOR INDEX 529
Fischel, J., 506, 511, 518
Fitzhugh, H. A., 251, 253, 322
Flachsbart, 0., 167, 211
Flax, A. H., 132, 207, 317, 318, 375, 380, 384,
387, 388, 405, 407, 426, 427
Graham, E. W., 317, 319, 515, 520
Graham, M. E., 413, 427, 461, 463, 464, 477
Grammel, R., 522
Granville, P. S., 340, 370
Fliigge, S., 523
Gronau, K.-H., 171, 172, 186, 187, 207, 499,
507, 508, 520
Grosche, F.-R., 425
Gruschwitz, E., 495, 496, 518
Guderley, K. G., 225, 253, 275, 323, 523
Gullstrand, T. R., 253
Gustavsson, A., 425, 427
Gyorgyfalvy, D., 100, 101
Fliigge-Lotz, I., 49, 101, 450, 477, 492, 518
Forsching, H. W., 81, 104, 132, 208, 214, 318,
523
Foster, D. N., 492, 518
Fowell, L. R., 317, 318
Fraenkel, L. E., 361, 369
Frankl, F. I., 521
Frenz, W., 470, 478
Frick, C. W., 214, 270, 318
Friedel, H., 296, 310, 311, 318
Friedman, L., 406, 425
Friedrichs, K. 0., 524
Fuchs, D., 504, 505, 517, 522
Fuchs, R., 74, 101, 114, 118, 123, 128, 129,
132, 153, 186, 209, 210, 211
Fuhrmann, G., 332, 337, 367
Fulker, J. L., 255, 320
Furlong, G. C., 170, 207, 455, 477, 508, 518
Garabedian, P., 255, 317
Garcia, J. R., 254, 255, 321
Garner, H. C., 128, 132, 155, 157, 170, 178,
185, 206, 207, 209
Garrick, I. E., 50, 72, 76, 104, 214, 318
Gault, D. E., 85, 86, 87, 104, 508, 519
Gebelein, H., 50, 72, 101, 104, 114, 118, 123,
128, 129, 132, 209
Geissler, W., 332, 336, 368
Gerber, N., 310, 320
Germain, P., 317, 323
Gersten, K., 111, 132, 166, 207, 210, 275, 319,
330, 348, 369, 375, 391, 401, 425, 453,
454, 474, 477, 503, 518
Giesing, J. P., 178, 206, 404, 425, 477
Gilruth, R. R.; 521
Ginzel, I., 131, 211, 317, 319, 492,518
Gispert, H.-G., 233, 317
Glauert, H., 28, 53, 56, 57, 63, 101, 102, 124,
137, 148, 207, 219, 318, 447, 449, 450,
453, 477, 486, 491, 518, 522
Goddard, F., 521
Goldstein, S., 25, 97, 98, 102, 331, 367, 523
Gonor, A. L., 317, 320
Goradia, S. H., 85, 86, 87, 104, 492, 518
Gothert, B., 219, 222, 229, 265, 318, 322, 354,
367, 506, 518
G6thert, R., 65, 102, 185, 207, 485, 487, 491,
492, 517, 518
Gretler, W., 129, 141, 208, 233, 319
Haack, W., 363, 369
Hackett, J. E., 450, 477
Haefeli, R. C., 464, 478
Hafer, X., 345, 348, 367, 375, 391, 425, 443,
469, 477
Hagermann, J. R., 511, 518
Hallstaff, T. H., 293, 296, 318
Hancock, G. J., 178, 206, 293, 296, 318
Hansen, H., 253, 321
Hansen, M., 129, 141, 181, 208
Hantzsche, W., 233, 317
Harder, K. C., 363, 369
Harmon, S. M., 310, 320
Harper, C. W., 149, 164, 170, 207, 211
Harris, R. V., Jr., 425, 426
Haskell, R. N., 128, 156, 206, 293, 310, 313,
321
Hay, J. A., 486, 492, 518
Hayes, W. D., 233, 258, 323, 523
Head, M. R., 97, 98, 102
Heaslet, M. A.,214, 283, 290, 293, 294, 319,
331, 366, 367, 369, 413, 417, 425, 427,
464, 465, 478
Heimbold, H. B., 53, 77, 93, 102, 103, 148,
207, 447, 478, 504, 518
Hensleigh, W. E., 455,477
Hess, J. L., 36, 102, 132, 207, 331, 332, 367,
403,426
Hewitt, B. L., 155, 207
Heyser, A., 506, 518
Hickey, D. P., 158, 207
Hilton, W. F., 523
Hirsch, R., 503, 520
Hodes, I., 406, 425
Hoerner, S. F., 36, 82, 90, 93, 102, 111, 171,
172, 173, 186, 187, 207, 214, 319, 331,
340, 367, 523
Holder, D. W., 251, 319, 358, 368
Hopf, L., 74, 101, 522
530 AUTHOR INDEX
Hosakawa, I., 253, 269, 322, 366, 369
Hosek, J. J., 293, 310, 313, 321
Hough, G. R., 128, 208, 232
Houghton, E. L., 522
House, R. 0., 398, 399, 400, 401, 426
Howarth, L., 331, 368, 523
Hua, H. M., 404, 425
Hubert, J., 442, 478
Hucho, W.-H., 132, 207
Hueber, J., 139, 209
Hummel, D., 25, 102,111, 132, 166, 169, 170,
207, 208, 210, 214, 319, 355, 368, 375,
401, 419, 425, 426, 499, 507, 508, 519
Hunter-Tod, J. H., 464, 478
Hiirlimann, R., 170,207
Imai, I., 233, 323
Jacob, K., 66, 77, 90, 102, 495, 518
Jacobs, E. N., 82, 85, 88, 90, 102, 330, 348,
369, 375, 389, 393, 394, 396, 398, 400,
426
Jacobs, H., 504, 505, 517
Jacobs, W., 164, 165, 171, 172, 186, 187, 207,
208, 380, 398, 399, 400, 401, 416, 417,
426, 427, 473, 474, 475, 477, 503, 504,
518
Jacquignon, H., 503, 519
Jaeckel, K., 53, 80, 101, 102, 123, 139, 153,
186, 210, 211
James, R. M., 36, 102, 361, 367
Jameson, A., 255, 317
Jaquet, B. M., 166, 207
Jaszlics, I., 169, 170, 206
Jeffreys, L, 310, 320
Jen, H., 450,477
Johnson, W. S., Jr., 293, 310, 313, 321
Jones, A. L., 492, 506, 518
Jones, 1. G., 317,_320,4063 427
Jones, M., 97, 98, 102
Jones, R. T., 164, 165, 166, 173, 208, 214,
289,290,300,317,319,323,413,415,
416, 426, 524
Jones, W. P., 128, 208
Jordan, P. F., 128, 142, 143, 144, 156, 181,
208, 209, 210, 449, 450, 477
Joukowsky, N., 33, 49, 51,102
Jungclaus, G., 28, 53, 61, 66, 70, 71, 75, 90,
102, 103, 104, 486, 492, 518
Kaatari, G. E., 405, 407, 427, 433, 478
Kacprzynski, J. J., 254, 255, 321
Kaden, H., 449, 450, 477
Kahane, A., 242, 243, 318, 361, 369
Kainer, 1. H., 293, 310, 313, 317, 321, 323
Kalman, T. P., 178, 206, 404, 477
Kandil, 0. A., 132, 208
Kane, E. J., 425, 426
Kao, H. C., 85, 86, 87, 104
Kaplan, C., 233, 317
Karamcheti, K., 523
Karpovich, E. A., 521
Katzoff, S., 449, 478
Kaufmann, W., 53, 101, 123, 176, 208, 317,
320, 449, 450, 477
Kawasaki, T., 317, 320
Kelly, H. R., 366, 368
Kerney, K. P., 132, 210, 503, 518
Kestin, J., 524
Kettle, D. J., 380, 382, 388, 390, 399, 427
Keune, F., 49, 50, 52, 53, 77, 101, 102, 197,
208, 275, 290, 317, 319, 321, 332, 337,
351, 366, 367, 368, 413, 415, 426, 427,
486, 492, 518, 523
Kida, T., 132, 211, 504, 519
Kiel, G., 489, 519
Kinner, W., 181, 208
Kirby, D. A., 380, 382, 388, 390, 399,427
Kirkby, S., 406, 409, 425
Kirkpatrick, D. L. I., 168, 169, 209
Klunker, E. B., 253, 317, 318, 321, 416,426
Knepper, D. P., 251, 253, 322
Knoche, H.-G., 506, 518
Kochanowsky, W., 50, 72, 76, 104
Kohler, M., 27, 42, 43, 45, 49,103
Kohlman, D. L., 169, 170,187, 206, 208
Koloska, P., 443, 469, 477
Kolscher, M., 507, 517
Kopfermann, K., 171, 172, 186, 187, 207
Korbacher, G. K., 503, 504, 519
Korn, D., 255, 317
Korner, H., 404, 426
Koster, H., 358, 3-63,-367
Kowalke, F., 242, 243, 308, 309, 310, 316,
266, 268, 311, 317, 318, 363, 369, 512,
519
Kraemer, K., 82, 102, 133, 153, 158, 171, 172,
210
Krahn, E., 232, 233, 317
Kramer, M., 29, 99, 101, 102
Krasnov, N. F., 331, 368, 522
Kraus, W., 128, 156, 208, 263, 320, 403, 404,
426
Krause, F., 266, 268, 311, 317, 353, 368
Krauss, E. S., 354, 368
Kreuter, W., 142, 143, 144, 209
AUTHOR INDEX 531
Krienes, K., 129, 141, 208
Kriesis, P., 132, 166, 206
Kriiger, W., 498, 519
Krupp, J. A., 253, 320, 366, 368
Krux, P., 317, 318
Kdchemann, D.,111, 132, 164, 166, 206, 208,
214, 317, 320, 322, 332, 368, 393, 425,
426, 450, 477, 524
Kuethe, A. M., 523
Kulakowski, L. J., 128, 156, 206
Kunen, A. E., 170, 209
Kuo, Y. H., 214, 320
Kupper, A., 139, 210, 485, 487, 491, 492, 517,
518
Kiissner, H. G., 81, 104, 132, 208
Kutta, W., 33,102
Labrujere, T. E., 155, 207, 403, 426
Lachmann, G. V., 36, 88, 95, 101, 102, 494,
497, 503, 519, 523
Lagerstrom, P. A., 461, 463, 464, 477, 515,
520
Laidlaw, W. R., 166, 208
Laitone, E. V., 233, 323, 358, 369, 406, 409,
425
Lamar, J. E., 317, 320
Lamar, J. R., 128, 155, 185, 209
Lamb, O. P., 492, 506, 518
Lambourne, N. C., 169,170, 206
Lamla, E., 233, 317
Lampert, S., 304, 314, 320
Lan, C. E., 77, 102, 128, 208, 232
Lance, G. N., 317, 323
Lanchester, F. W., 522
Landahl, M. T., 111, 132, 206, 208, 214, 253,
317, 322, 331, 367, 416, 426, 521
Lange, G., 171, 172, 186, 187, 207
Laschka, B., 132,146,147, 151, 153, 208,
210, 463, 464, 465, 476, 477, 478
Laurmann, J. A., 36, 103, 132, 209, 214, 322,
522
Lawford, J. A., 166, 207
Lawrence, H. R., 166, 208, 375, 380, 384, 387,
388, 405, 426, 521
Lawrence, T., 316, 320
Leelavathi, K., 366, 367
Lees, L., 242, 243, 318
Legendre, R., 168, 169, 209
Lehrian, D. E., 128,132, 170, 207
Leiter, E., 289, 293, 296, 318, 320
Lennertz, J., 379, 411, 419,426
Leslie, D. C. M., 298, 320
Lessing, F., 332, 336, 368
Levinsky, E. S., 503, 519
Licher, R. M., 413, 427
Liebe, H., 375, 391, 425
Liebe, W., 166, 206
Liebeck, R. H., 87, 102
Liepmann, H. W., 523
Liese, J., 390, 395, 428
Liess, W., 380, 382, 388, 390, 399, 427
Lighthill, M. J., 77, 103, 244, 317, 320, 356,
368
Lilienthal, 0., 15, 22
Lincke, W., 132, 210
Lindsey, W. F., 234, 246, 247, 322
Linnel, R. D., 256, 320
Lipowski, K., 310, 311, 322
Lissaman, P. B. S., 504, 519
Littell, R. E., 234, 246, 247, 322
Lock, R. C., 255, 320, 406, 425, 427
Loeve, W., 403, 426'
Loftin, L. K., Jr., 82, 85, 88, 90, 102
Lohr, R., 503, 504, 518, 519
Loitsyanskii, L. G., 523
Lomax, H., 214, 25 3, 283, 290, 293, 294, 319,
321, 331, 367, 413, 417, 425, 427, 464,
465, 478
Lord, W. T., 317, 320
Losch, F., 492, 518
Lotz,.I., 139, 209, 332, 368, 447,478
Love, E. S., 304, 314
Luckert, H. J., 380, 382, 388, 390, 399, 427
Ludwieg, H., 170, 209, 275, 320
Lusty, A. H., Jr., 317, 320
Lyman, V., 85, 86, 87, 104
Maccoll, J. W., 214, 322, 358, 369
McCormick, B. W., Jr., 522
McCullough, G. B., 85, 86, 87, 104
McDonald, J. W., 132,206
McHugh, G. C., 170, 207
McHugh, J. G., 455, 477, 508, 518
Mackrodt, P. A., 310, 311, 322
McLean, F. E., 317
Maddox, S. A., 170, 209
Magnus, R., 253, 320
Maki, R. L., 170, 207
Malavard, L., 225, 253, 275, 323
Malrnuth, N. D., 503, 504, 518,519
Malvestuto, F. S., 293, 310, 313, 321
Mangler, K. W., 50, 72, 75, 100, 101, 104, 128,
132, 155,166,169, 185, 206, 209, 269,
317, 318, 320, 442, 478
Margolis, K., 293, 310, 313, 321
Marshall, F. J., 366, 368
532 AUTHOR INDEX
Martensen, E., 66, 77, 102
Martin, J. C., 310, 320
Martynov, A. K., 522
Maruhn, K., 337, 338, 341, 347, 368, 398, 399,
400,401,426
Mascheck, H.-J., 503, 504, 518
Maskell, E. C., 317, 320, 503, 518
Maskew, B., 77, 103, 202
Mateescu, D., 521
Mattioli, G. D., 114, 118, 123, 128, 129, 132,
209
Maurer, F., 506, 518
Mello, J. F., 366, 368
Michael, W. H., Jr., 169, 206, 475, 478
Middleton, W. D., 425, 426
Miele, A., 317, 320, 5 22
Miles, E. R. C., 523
Miles, J. W., 363, 368, 416, 417, 428, 523
Miller, B. D., 404, 407, 428
Milne-Thomson, L. M., 522,523
Mirels, H., 166, 209, 289, 320, 464, 478
Miyai, Y., 132, 211, 504, 519
Naylor, D., 515, 520
Nelson, R. L., 515, 520
Neumark, S., 197, 203, 205, 209, 275, 321
Newman, P. A., 253, 321, 416, 426
Nickel, K., 53, 66, 74, 101, 123, 175, 209, 497,
518
Nicolai, L. M., 405, 407, 427
Nielsen, J. N., 405, 407, 427, 433, 478
Niemz, W., 124, 125, 127, 128, 129, 153, 154,
155, 160, 161, 210
Nieuwland, G. Y., 254, 255, 321
Nixon, D., 253, 321
Nonweiler, T., 84, 85, 103, 499, 519
Nostrud, H., 233, 253, 321, 323
O'Hare, W. M., 511, 518
Orrnsbee, A. I., 87, 102
Osbome, J., 251, 253, 322
Oswatitsch, K., 225, 253, 275, 321, 323, 351,
366, 368, 413, 415, 426, 427, 523, 524
Otto, H., 185, 207, 425, 427
Moller, E., 171, 172, 186, 187, 207, 330, 348,
369, 375, 389, 391, 393, 394, 396, 398,
400,425,426
Mook, D. T., 170, 209
Moore, F. K., 356, 366, 369
Moore, K. C., 317, 320
Pai, S.-I., 523
Panico, V. D., 361, 367
Pappas, C. E., 170, 209
Parker, A. G., 170, 209
Moore, N. B., 357, 361, 363, 369
Moore, T. W. F., 87, 101
Pearcey, H. H., 132, 206, 251, 255, 275, 321
Pechau, W., 84, 99, 103, 500, 519
Moran, J. P., 332, 357, 361, 363, 369
Morikawa, G. K., 407, 427, 458, 478
Morris, D. N., 362, 368
Mosinskis, G. J., 93,101, 340, 370
Muter, W., 332, 337, 367
Multhopp, H., 78,128, 142,143, 144, 155,
182, 185, 209, 317, 319, 341, 344, 368,
380, 382, 388, 390, 399, 427, 447, 452,
453, 478
Munk, M. M., 58, 103,175,177, 209, 317,
318, 331, 341, 368, 416, 427
Murman, E. M., 253, 320, 366, 368
Murnaghan, F. D., 523
Murphy, W. D., 503, 504, 518, 519
Muttray, H., 33,49,51, 102, 375, 398, 427,
Peckham, D. H., 169, 170, 206
Perkins, E. W., 362, 366, 367
Perring, W. G. A., 486, 491, 518
Petersohn, E., 450, 477
Petrikat, K., 497, 519
Pfenninger, W., 98, 101
Piercy, N. A. V., 49, 101
Pike, J., 317, 320
Pinkerton, R. M., 82, 83, 85, 88, 89, 90, 102,
450,477
103
Piper, E. R. W., 49, 101
Pistolesi, E., 78, 103
Pitts, W. C., 405, 407, 427, 433, 478
Pleines, W., 499, 518
Pohlhamus, E. C., 170, 209, 275, 317, 321
Poisson-Quinton, P., 95, 103, 166, 209, 495,
503,518,519
Naeseth, R. L., 511, 578
Nagaraja, K. S., 164, 206, 208
Nash, J. F., 77, 104
Nastase, A., 317, 318, 521
Naumann, A., 42, 103, 506, 518
Nayfeh, A. H., 132, 208
Pope, A., 522
Powell, B. J., 255, 320
Prandtl, L., 27, 33, 42, 43, 45, 51, 102, 103,
114, 118, 123, 128, 129, 132, 209, 214,
219, 293, 443, 478, 523, 524
Preston, J. H., 49, 90, 103
Pretsch, J., 93, 103
AUTHOR INDEX 533
Pritchard, R. E., 317, 320
Probstein, R. F., 523
Proll, A., 522
Puckett, A. E., 293, 310, 313, 321
Puffert, H. J., 398, 399, 400, 401, 427, 475,
478
Queijo, M. J., 166, 206
Rakich, J. V., 358, 369
Ramaswamy, M. A., 363, 369
Randall, D. G., 269, 320
Ras, M., 98, 101
Raspet, A., 100, 101
Rauscher, M., 522
Redeker, G., 169, 170, 207, 251, 323
Regenscheit, L. B., 96, 103, 500, 519
Reissner, E., 130,145, 149, 211
Reller, E., 504, 505, 517
Rennemann, C., Jr., 363, 369
Revell, J. D., 355, 368
Ribner, H. S., 293, 310, 313, 321
Richter, G., 229, 322 '
Richter, W., 447, 452, 453, 478, 507,517
Riedel, H., 413, 415, 426
Riegels, F. W., 27, 28, 36, 53, 60, 61, 66, 70,
74, 75, 76, 77, 90, 92, 93, 102, 103,
104, 173, 210, 332, 336, 368, 380, 382,
388, 390, 399, 427, 492, 518, 519, 522
Ringleb, F., 50, 53, 77, 102
Roberts, R. C., 293, 310, 313, 321
Robins, A. W., 425,426
Robinson, A., 36,103, 132, 166, 209, 214,
285, 300, 322, 406, 409, 425, 464, 478,
522
Rodden, W. P., 178, 206, 375, 404, 405, 425,
477
Roe, P. L., 317, 320
Rogallo, F. M., 506, 520
Rogers, E. W. E., 255, 320, 425,427
Rogmann, H., 124, 125, 127, 128, 129, 153,
154, 155, 160, 161,210
Rohlfs, S., 253, 321, 416, 426
Rohne, E., 449, 450, 477
Roshko, A., 523
Rossner, G., 50, 53, 77, 102, 114, 118, 123,
128,129, 132, 209
Rossow, V. J., 450, 478, 486, 492, 518
Rothmann, H., 355, 368
Rott, N., 317, 319
Rotta, J., 253, 322, 469, 478
Roy, M., 87, 101, 163, 169, 209
Rubbert, P. E., 253, 322
Ruden, P., 489, 519
Rues, D., 253, 321
Sacher, P., 128, 156, 208, 263, 320, 403, 404,
426
Sacks, A. H., 450, 458, 478
Sanchez, F., 405, 407, 427
Sann, B., 42, 103
Sato, J., 253, 321
Sauer, R., 524
Schappelle, R. H., 503,519
Scharn, H., 398, 399, 400, 401, 426, 447, 452,
453,478
Schindel, L. H., 366, 368
Schlichting, H., 25, 81, 84, 93, 96, 99, 103,
111, 170, 182, 192, 209, 210, 214, 293,
302, 322, 375, 384, 388, 393, 395, 427,
433,469,470,478,500,504, 517, 519,
524
Schlottmann, F., 194, 210
Schmidt, H., 114, 118, 123, 128, 129,132,
139, 209, 219, 522
Schmidt, W., 363, 369, 413, 415, 426
Schmitz, F. W., 83, 103, 522
Schneider, W., 261, 322, 403, 404, 413, 416,
427
Scholz, N., 93, 103, 131, 161, 210, 275, 321,
340, 370
Schrenk, 0., 33, 49, 51, 84, 99, 101, 102, 103,
403, 404, 413, 416, 427, 442, 478, 489,
495,496,499, 500, 504, 517, 518, 519
Schroeder, H.-H., 317, 318, 363, 367
Schubert, H., 139, 210
Schultze, E., 158, 207
Schulz, G., 445, 478, 515, 520
Schwarz, F., 500, 519
Sears, W. R., 59, 97, 103, 111, 133, 175, 210,
214, 317, 320, 322, 363, 369, 375, 388,
425, 522
Sedney, R., 317, 319
Seewald, F., 74, 101, 522
Seibold, W., 506,518
Seiferth, R., 27, 42, 43, 45, 49, 103, 489, 519
Sells, C. C. L., 233, 319
Shapiro, A. H., 524
Shepherd, D. G., 524
Sherby, S. S., 521
Sherman, A., 82, 85, 88, 90, 102, 330, 342,
369,375,389, 393, 394, 396, 398, 400,
426
Shortal, J. A., 498, 520
Silverstein, A., 449, 478
534 AUTHOR INDEX
Simmons, L. F. G., 450, 477
Sinnott, C. S., 251, 253, 322
Sivells, J. C., 508, 518
Slooff, J. W., 403, 426
Sluder, L., 464, 465, 478
Srnetana, F. 0., 251, 253, 322
Smith, A. M. 0., 36, 84, 93, 101, 102, 132,
207, 331, 332, 340, 367, 370, 403, 426
Smith, C. W., 170, 209
Smith, H. A., 82, 85, 88, 90,102
Smith, J. H. B., 169, 209, 210, 317, 318, 319
Snedeker, R. S., 450, 477
Sohngen, H., 139, 210, 492, 518
Solarski, A., 361, 369
Spee, B. M., 254, 255, 321
Speidel, L., 90, 100, 104
Spence, B. F. R., 128, 155, 185, 209, 504, 518
Spence, D. A., 90, 103, 503, 519
Spoonner, S. H., 508, 518
Spreiter, J. R., 214, 225, 253, 269, 275, 283,
290, 293, 294, 319, 322, 323, 366, 369,
380,416,417,427,450,478
Squire, H. B., 85, 86, 87,93,94, 103, 104,
170,207
Squire, L. C., 317, 322
Srinivasan, P. S., 169, 170, 207
Stack, J., 229, 234, 246, 247, 322
Taylor, G. L, 214, 322, 358, 369, 524
Taylor, H. S., 521
Theodorsen, T., 50, 72,104, 522
Thom, A. S., 523
Thomas, F., 132, 210, 251, 323, 501, 519
Thwaites, B., 36, 89, 104,132, 210, 331, 369,
522
Tietjens, 0., 523
Ting, L., 413; 427
Toll, T. A., 492, 511, 519
Tolve, L. A., 455, 477
Traugott, S. C., 358, 369
Trefftz, E., 49, 101, 114,118, 123, 128, 129,
132, 209
Tricomi, F. G., 5 23
Trienes, H., 330, 348, 369, 375, 391, 425, 449,
452, 453, 479
Trilling, L., 169, 170,206
Truckenbrodt, E., 28, 53, 61, 66, 70, 71, 75,
76,93, 103, 104, 124,125,127, 128,
129, 146, 147, 151,154, 154, 155 ,15 8,
160, 161, 171, 172,173, 210, 219, 222,
265, 269, 319, 322, 330, 348, 351, 369,
375, 391, 425, 447, 449, 452, 453, 473,
474,475, 477, 478, 479, 499, 507, 508,
Stahara, S. S., 253, 322, 366, 369, 416, 427
Stahl, W., 310, 311, 322, 425, 427
Stanbrook, A., 317, 322
520,524
Truitt, R. W., 524
Stanewsky, E., 253, 321
Stark, V. J. E., 132, 208, 214, 320
Tsien, H. S., 232, 233, 258, 317, 323, 358, 369
Tucker, W. A., 515, 520
Tuckermann, L. B., 337, 338, 341, 347, 368
Staufer, F., 469, 470, 478, 485, 487, 491, 492,
517, 518
Steger, J. L., 253, 321
Stender, W., 90, 100, 104
Stetter, H. J., 357, 361, 363, 369
Stevens, J. R., 132, 206
Stewart, H. J., 293, 310, 313, 317, 321, 322
Stiess, W., 485, 487, 491, 492, 517,518
Stivers, L. S., Jr., 27, 36, 62, 63, 67, 72, 76,
100, 101, 254, 255, 321
Stocker, P. M., 416, 417, 428
Strand, T., 75, 104, 317, 323
Strassl, H., 497, 519
Stratford, B. S., 503, 520
Streit, G., 501, 519
Subramanian, N. R., 366, 367
Sullivan, R. D., 450,477
Sun, E. Y. C., 315, 322
Szabo, I., 129, 141, 208
Tani, I., 87, 104
Tanner, M., 77, 104, 362, 369
Tsakonas, S., 132, 208, 214, 320
Ulrich, A., 517, 519
Ursell, F., 317, 318
van der Decken, J., 132, 210
Vandrey, J. F., 332, 336, 341, 344, 368, 390,
395, 413, 427, 428
van Dyke, M. D., 132, 210, 233, 323, 356, 358,
366, 369, 524
Vanino, R., 416, 425, 426, 427
Vidal, R.J., 169, 170, 206
Vincenti, W. G., 317, 323, 524
Viswanathan, S., 363, 369
Voellmy, H. R., 366, 369
Voepel, H., 504, 505, 517
von Baranoff, A., 458,478
von Doenhoff, A. E., 27, 36, 62, 63, 67, 72, 76,
100, 101, 229, 322, 492, 517, 521
von Karmar_. T., 36, 49,104,114, 129, 132,
210, 214, 225, 232, 233, 253, 275, 311,
AUTHOR INDEX 535
von Kirman, T. (Cont.), 323, 332, 351, 357,
361, 363, 369, 521, 524, 525
von Mises, R., 38, 104, 522, 524
Wacke, 171, 172, 186, 187, 207
Wagner, H., 81, 104
Wagner, S., 128, 156, 210
Walchner, 0., 242, 243, 318
Walz, A., 49, 50, 72, 101, 104, 486, 492, 518
Wanner, A., 504, 505, 517
Ward, G. N., 317, 318, 323, 361, 369, 416,
417, 428, 464, 479, 524
Ward, K. E., 330, 348, 367, 375, 389, 393,
394, 396, 398, 400, 426
Watson, E. J., 128, 155, 185, 209
Watson, J. M., 506, 518
Weber, J., 164, 166, 206, 207, 208, 211, 317,
320, 323, 380, 382, 388, 390, 393, 399,
426, 427
Wedemeyer, E., 266, 268, 311, 317
Wegener, F., 146, 147, 151, 153, 210, 242,
243, 308, 309, 310, 316, 318, 363, 367
Wehausen, J. V., 524
Weick, F. E., 498, 520
Weinberger, W., 486, 492, 518
Weinel, E., 332, 368
Weinig, F., 139, 153, 186, 211, 497, 519, 522
Weissinger, J., 36, 79,104, 111, 130-,132, 141,
142, 143, 144,149, 153, 186, 209, 211,
Widnall, S. E., 132, 211, 214, 317
Wieghardt, K., 131, 211,524
Wieland, E., 132, 207
Wieselsberger, C., 27, 33,42,43,45,49,51,
102,103, 121, 122, 211, 331, 369, 375,
428, 524
Wilby, P. G., 255, 320
Williams, G. M., 450, 479
Williams, J., 500, 503, 520
Winter, H., 167, 211
Wittich, H., 28, 50, 53, 61, 66, 70, 71, 72, 75,
76,103,104
Wolhart, W. D., 166, 207
Wood, C. J., 251, 323
Wood, M. N., 503, 520
Woods, L. C., 36,104, 522
Woodward, F. A., 77, 103, 202, 296, 318, 404,
407,428
Wortmann, F. X., 90,93,100,104
Wuest, W., 499, 500, 518, 520
Wurzbach, R., 449, 450, 477
Yang, H. T., 416, 417, 428
Yoshihara, H., 253, 317, 321, 323
Young, A. D., 85, 86, 87, 93, 94, 103, 104,
149, 164, 170, 207, 211, 340, 362, 370,
495,49-7, 498, 508, 520, 523
Young, J. de, 149, 164, 211, 508, 518
522
Wellmann, J., 317, 318, 363, 367
Wendt, H., 233, 317
Wentz, W. H., Jr., 169, 170, 206
Wenzinger, C. J., 492, 496, 506, 520
Werle, H., 169, 211
Whitcomb, R. T., 414, 415, 428
White, F. M., 5 24
Zahm, A. F., 337, 338, 341, 347, 368
Zienkiewicz, H. K., 293, 310, 313, 321
Zierep, J., 253, 323, 524
Ziller, F., 114, 118, 123, 128, 129, 132, 209
Zimmer, H., 253, 321
Zucrow, M. J., 521
SUBJECT INDEX
Acceleration potential, 129
Aerodynamic center (center of pressure), 17
Aileron:
geometry of, 431, 484
rolling moment of, 510, 515
Airfoil theory, 123, 131, 153, 263, 269, 280,
288, 290, 453
nonlinear, 166
[See also Wing (airfoil) ]
Angle of attack (incidence), 13, 16
of fuselage, 376, 382, 384, 387
of horizontal stabilizer, 437, 440, 443
of smooth leading-edge flow, 60
of wing, 56, 117, 376, 389, 396,412
Angle of incident flow, 78
Area rule, 414
Atmosphere, 5, 8
Balance tab, 482, 491
Blowing [see Ejection (blowing)]
Boundary-layer control, 81, 95
Boundary-layer fence, 166, 455, 494
Brake flap (air brake), 483, 504
Buffeting, 251
Bursting of vortex, 169
Cambered flap, 483, 487, 495
Center of pressure [see Aerodynamic center
(center of pressure) ]
Characteristics, method of, 244, 358, 360
Circular wing, 181
Circular-arc profile, 46
Circulation, 33
Circulation distribution:
over profile, 54, 56
over wing, 114, 117, 123, 126, 129, 136,
140, 298, 379
Circulation (lift) distribution:
constant (rectangular), 447, 448, 460
elliptic, 118, 263, 444, 447, 449, 453
parabolic, 447
Closure condition, 70, 198, 333
Coefficients, aerodynamic:
definition of, 14, 330, 436, 485
effect of friction (viscosity) on, 81, 170,
347
Compression shock (bow wave), 245, 246, 250,
259
Conical flow, 280, 461
Control surface, balance, 482, 491
[See also Flap (control surface)]
Coordinate systems, 13, 105, 327
Delta wing, 106, 108
drag of, 152, 178, 268, 302, 305, 308, 313,
316
lift of, 152, 157, 168, 171, 266, 269, 301,
305, 308
lift distribution of, 151, 158, 266, 304, 419
neutral point of, 152, 158, 266, 269, 301,
307, 308, 393
pressure distribution of, 160, 285, 287, 304,
417
suction force on, 300
Density, 3
Dipol distribution, 123, 342, 365, 390
Direct problem, 1, 118, 128
537
538 SUBJECT INDEX
Double-section flap, 498
Double-section wing, 483, 487, 498
Drag, 12, 14
(See also Induced drag; Profile drag (friction
Fuselage (Cont.):
pitching moment of, 330, 340, 345, 346,
348
(See also Ellipsoid; Paraboloid)
drag); Wave drag]
Ejection (blowing), 95, 98, 500
Elementary wing, 124, 126, 130, 174, 379
Elevator, 432, 484, 508, 516
Ellipsoid, 201, 329, 334, 337, 343, 345, 347,
353, 374, 392, 397, 401
Elliptic wing, 109, 119, 120
downwash and upwash of, 383, 384, 444,
453
drag of, 119, 178
lift of, 118, 121, 146, 264
lift distribution of, 141
perturbation velocity of, 202
End plate, 442
Energy law, 175
Fin (see Stabilizer)
Flap (control surface), 63, 109, 481, 491
angle of attack, change by, 64, 96, 486, 492,
493, 508, 512
control-surface moment of, 484, 486, 490,
493,517
geometry of, 481, 483
lift of, 484
loading of, 489, 494
moment change by, 65, 486, 493, 512
neutral point of, 486, 488, 493, 516
pressure distribution on, 489, 496, 513
rolling moment of, 509
Flap, double-section, 498
Flap wing, 96, 482
[See also Lift (lift slope), of flap-wing
system]
Flap with trailing edge blowing, 98
Fowler flap, 483, 498
Fuselage:
in curved flow, 346, 376
drag of, 330, 354, 358, 360, 362
geometry of, 327, 363
lift of, 330, 348, 365, 380
lift distribution of, 344, 380
neutral point of, 348
perturbation velocity on, 335
(See also Induced velocity)
pressure distribution on, 332, 334, 343,
347, 352, 353, 354, 358
Glide angle, 12
Gottingen profile system, 27
Ground effect, 132, 371, 504
High-wing airplane, 373, 375, 378, 395, 396,
400, 470, 472, 474
Hinge moment (see Flap, control-surface
moment of)
Horizontal tail, 432, 433
and vertical tail, control-surface balance of,
482, 491
dynamic pressure ratio of, 437
efficiency (downwash) factor of, 438, 444,
451, 457, 462
geometry of, 434
lift of, 436, 438, 441, 443, 456, 459
neutral-point shift caused by, 439, 454
pitch damping of, 441
pitching moment of, 436, 437, 440
Horn balance, 482
Indirect (design) problem, 1, 118, 128
Induced angle of attack, 115, 117, 119, 138,
139, 142
Induced drag, 114, 119, 152, 173, 175, 176,
264, 301
Induced velocity (source, dipole), 80, 199, 293,
333, 357
Induced velocity (vortex):
downwash, 57, 80, 115, 119, 291, 444, 453,
456
sidewash, 472
Influence zone (line), 277, 283, 292, 295, 356,
458
Interference:
of fuselage-horizontal tail system, 442
of vertical-horizontal tail system, 475
of wing-fuselage system, 371, 376, 405,413
of wing-fuselage-vertical tail system, 467,
470
of wing-horizontal tail system, 436, 443,
456,458
Jet flap, 503
Joukowsky profile, 45, 46, 48, 72, 246
SUBJECT INDEX 539
Kinematic flow condition, 54, 70, 126, 198,
235, 292, 379
Kutta (flow-off) condition, 33, 40, 66, 128,
279
Kutta-Joukowsky lift theorem, 30, 134
Laminar flow, maintenance of, 96, 97, 99
Laminar profile, 99
Landing device, 482, 494
Landing flap, 482, 508
Lateral motion, 15, 181, 186,432
Lift (lift slope), 12, 14, 16, 110, 135
of flap-wing system, 485, 486, 492, 494
of fuselage, 330, 348, 365, 380, 393, 402
of smooth leading-edge flow, 60, 230
of stabilizer: horizontal, 436, 438, 441, 456,
459
vertical, 469
of wing: compressible, 224, 229, 230, 237,
249, 264, 269
incompressible, 30, 41, 49, 55, 58, 60,
81, 84, 114, 132, 136, 156, 166,
170
of wing-fuselage system, 374, 379, 382,419
Lift distribution (circulation distribution):
of fuselage, 330, 344, 380, 387, 407, 409
of wing, 110, 135, 263, 269, 388, 412, 419,
506
Lifting-line theory:
simple, 131, 137, 151, 446, 451, 506
extended, 129, 131, 145, 151, 506
Lifting-surface theory, 153, 507
(See also Airfoil theory)
Longitudinal motion, 15, 181, 432
Low-wing airplane, 373, 378, 394, 396, 400,
471, 474
Mach cone, 22, 276
Mach number, 9
drag-critical, 227, 232, 244, 271, 274, 353
Maximum lift, 84, 96, 170, 393, 494, 497
Method of characteristics, 52, 244, 358, 360
Mid-wing airplane, 373, 374, 378, 394, 395,
396, 400, 472
Momentum law, 132, 175, 341
Multhopp's quadrature method, 141
Multiple-points method, 131
Munk displacement theorem, 175
NACA profiles, 27, 62, 67, 72, 76, 82, 228,
230, 233, 271
Neutral point:
of fuselage, 348
of horizontal tail, 439, 454
of wing: geometric, 108
aerodynamic (general), 18
compressible, 230, 237, 264, 269
incompressible, 42, 59, 60, 157
of wing-fuselage system, 390, 421
Nonlinear lift effects, 166, 330, 366, 4.25
Normal force, 14
Nose balance, 482
Nose flap, 483, 498
Panel method, 403
Parabolic profile (biconvex), 28, 47, 58, 62, 66,
71, 200, 204, 239, 242, 246, 247, 253, 313
Paraboloid, 329, 336, 353, 358, 360, 362
Perturbation velocity, 72, 200, 336
Pistolesi's theorem, 79, 80
Pitch:
damping, 19, 183, 441
lift due to, 183
motion, 16, 182,441
Pitching moment:
of flap-wing system, 484, 48S
of fuselage, 345, 348
of horizontal tail, 436
of wing, 14, 18
compressible, 230, 264
incompressible, 55, 58, 156
of wing-fuselage system, 374, 3 82
Plate, flat:
in chord-parallel flow, 90, 97, 216
inclined (with angle of attack): compressible,
229, 238, 239, 257, 286, 461
incompressible, 38, 57, 78
Polar curve (drag), 15, 120, 121, 181, 275, 394
Prandtl wing theory, 112, 117, 138
transformation formulas for, 121
Pressure distribution (pressure coefficient):
on flap, 68, 489, 496, 513
on fuselage, 332, 334, 343, 347, 352, 353,
354, 358, 364
on wing: compressible, 214, 223, 224, 226,
228, 230, 235, 237, 241, 246, 257,
258, 260, 261, 270, 285, 294, 311
incompressible, 28, 55, 67, 72, 87, 128,
155
on wing-fuselage system, 402, 406, 417
Pressure equalization, wing, 113
Profile:
computation of: skeleton (mean camber)
line of, 56
540 SUBJECT INDEX
Profile, computation of (Cont.):
teardrop of, 74
with fixed aerodynamic center, 61
friction effect on, 81
geometry of, 26
supercritical, 253
[See also Circular-arc profile; Joukowsky
profile; NACA profiles; Parabolic
profile (biconvex); Wedge profile]
Profile drag (friction drag):
of fuselage, 330, 354
of wing, 90, 92, 97, 120, 173, 216, 230,
253, 275
of wing-fuselage system, 394
Profile theory:
based on: conformal mapping, 36
singularities method, 52
skeleton theory, 53, 486
teardrop theory, 68
hypersonic, 255, 260
incompressible, 25
subsonic, 227, 230, 232
supersonic, 234, 242
transonic, 244, 253
Rectangular wing:
downwash and upwash of, 385, 448, 449,
459,462
drag of, 178, 275, 297, 313
lift of, 149, 161,166, 171, 296, 311, 374
lift distribution of, 143, 149, 297, 412
neutral point of, 161, 297, 392
perturbation velocity on, 201
pressure distribution on, 296
Reference wing chord, 108
Reynolds number, 10, 81, 90
Riegels factor, 70
Roll damping, 19, 192
Roll motion, 16, 192
Rolling moment:
of wing, 14, 136, 149, 156, 264, 396
of wing-fuselage system, 374, 396
due to sideslip, 18, 375, 396, 466
due to yaw rate, 19, 192
Roll-up of vortex, 134, 168, 444, 449
Rudder, 432, 484, 517
Separation of flow, 42, 83, 88, 96, 168, 170,
244, 246, 366, 394, 455, 498
Side force, 14, 466
due to roll rate, 20
due to sideslip, 18, 186, 190, 400, 466
Side force (Cont.):
due to yaw rate, 20
Sideslip:
angle of, 13, 16, 471
definition of, 13
Sideslipping (yawed) flight, 16, 18, 186, 466
Similarity rule:
hypersonic, 258, 364
subsonic, 219, 233, 261, 350, 402, 456,
492, 511
supersonic, 219, 350,492
transonic, 225, 251, 351
Singularities method:
for fuselage, 331, 342, 356, 365
for wing, 52, 123, 197, 289
for wing-fuselage system, 403
Slat (flap), 96, 455, 483,498
Slender body, theory of, 265, 300, 311, 416,
458
Slot flap, 483, 487, 490, 497, 498
Slotted wing, 96
Sonic incident flow, 269, 275
Sound, speed of, 4, 332
Source-sink distribution, 198, 293, 311, 356
Split (spreader) flap, 483, 487, 495
Spoiler, 504
Stabilizer, 481
horizontal (tail plane), 431, 432
vertical (fin), 431, 432
Stagnation point, 214, 259, 365
Stall fence (see Boundary-layer fence)
Starting vortex, 34
Straight flight, 16, 182, 435
Streamline analogy, 219
Subsonic edge, 277, 514
Subsonic incident flow, 263, 270, 285, 352,
402, 456
Substitute wing, 373, 385
Suction, 96, 499
Suction force, 43, 59, 96, 180, 300, 308
Supercirculation, 503
Superposition principle, 288
Supersonic edge, 277, 286, 288, 514
Supersonic flight, 21
Supersonic incident flow, 276, 296, 310, 355,
405, 458
Super-stall, 455
Swept-back wing, 108
downwash and upwash of, 385, 448, 452
drag of, 152, 270, 275, 308, 315, 316
drag-critical Mach number of, 271, 274, 353
lift of, 152, 161, 164, 168, 171, 266, 269,
307, 308
lift distribution of, 151, 157, 164, 266
SUBJECT INDEX 541
Swept-back wing (Cont.):
neutral point of, 152, 158, 161, 308, 392,
393
velocity distribution of, 203
Vortex (wing) (Cont.):
free, 113, 115, 131, 166,460
horseshoe, 114, 124, 126, 379
starting, 114
Vortex sheet, 53, 114, 123, 134, 169, 290, 366,
444, 449, 453, 464, 486
Tail plane (see Stabilizer)
Tail surface (see Horizontal tail, and vertical
tail)
Take-off assistance, 482, 494
Tangential force, 14, 179
Temperature increase:
through compression, 215, 259
through friction, 216
Three-quarter-point method, 130, 146, 385
Trailing edge:
angle, 25, 82
ejection, 98
Transformation, geometric, 220, 261, 350, 352,
402, 456, 511
Transonic (incident) flow, 219, 226, 269, 413
Trapezoidal wing, 106
downwash of, 448, 452
drag of, 152, 308, 316
lift of, 152, 266, 269, 308
lift distribution on, 143, 151, 158, 266
neutral point of, 152, 158, 266, 269, 308
Wave drag:
of fuselage, 358, 360
of wing, 224, 226, 237, 258, 301, 313
of wing-fuselage system, 413
Wedge profile, 242, 313
Wing (airfoil):
aspect ratio of, 107, 121
dihedral (V shape) of, 105, 109, 189, 373,
398, 399
pressure equalization on, 113
reference chord of, 108
taper of, 106, 107
twist of, 105, 135, 177
(See also Airfoil theory; Delta wing; Elliptic
wing; Rectangular wing; Swept-back
wing; Trapezoidal wing)
Wing in curved flow, 78
Wing, lifting (with angle of attack), of finite
thickness (displacement), 68, 197, 270,
310
Wing-fuselage system:
Unsteady motion, 20
Velocity distribution on contour, 66, 70, 71,
75, 198, 292, 333
Velocity near-field of profile, 79
Velocity potential:
of fuselage, 342, 348, 357, 365
of slender bodies, 417
of wing: compressible, 217, 218, 225, 293
incompressible, 128, 199
drag of, 393, 413
geometry of, 371
lift of, 374, 379, 382, 393, 402, 410, 419,
425
neutral point of, 380, 403, 411, 421
pitching moment of, 374, 382,.411
pressure distribution over, 402, 417
rolling moment of, 374, 396
side force on, 400
yawing moment of, 400
Vertical tail, 432, 433
dynamic pressure ratio of, 467
efficiency (sidewash) factor of, 467, 471,
473
geometry of, 434
side force (lift) of, 466
yawing moment of, 466
Viscosity, 4
Vortex density [see Vortex strength (circulation
distribution) ]
Vortex strength (circulation distribution), 53,
123, 153
Vortex (wing) :
bound (lifting), 31, 35, 80, 113, 131, 166
bursting of, 169
Yaw (turning) damping, 19,468
Yawed (sideslipping) flight, 18, 186, 466
angle of, 467, 471
Yawing moment, 14
due to roll rate, 19, 193
due to sideslip, 18, 186, 400, 466
Yawing motion, 19, 195, 468
Zero moment, 16
compressible, 230, 237, 264
incompressible, 60, 76
Zero-lift angle, 16
compressible, 230, 264
incompressible, 60, 135, 141, 237