Aerodynamics Of The Airplane

Aerodynamics
of the
Airplane
Hermciui Sch!ichthg and Erie lw c ro t
Translated by Heinrich J. Ramm

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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The Maple Press Company was printer and binder.
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 629.132'3 79-60
ISBN 0-07-055341-6

CONTENTS
Preface
Nomenclature
1 Introduction
1-1 Problems of Airplane Aerodynamics
1-2 Physical Properties of Air
1-3 Aerodynamic Behavior of Airplanes
References
Part 1 Aerodynamics of the Wing
vii
ix
1
1
2
8
22
23
2 Airfoil of Infinite Span
in Incompressible Flow (Profile Theory) 25
2-1 Introduction 25
2-2 Fundamentals of Lift Theory 30
2-3 Profile Theory by the Method of Conformal Mapping 36
2-4 Profile Theory by the Method of Singularities 52
2-5 Influence of Viscosity and Boundary-Layer Control
on Profile Characteristics 81
References 101
3 Wings of Finite Span in Incompressible Flow 105
3-1 Introduction 105
3-2 Wing Theory by the Method of ` ortex Distribution 112
3-3 Lift of Wings in Incompressible Flow 131
3-4 Induced Drag of Wings 173
3-5 Flight Mechanical Coefficients of the Wing 181
3-6 Wing of Finite Thickness at Zero Lift 197
References 206

Vi CONTENTS
4 Wings in Compressible Flow
4-1 Introduction
4-2 Basic Concept of the Wing in Compressible Flow
4-3 Airfoil of Infinite Span in Compressible Flow
(Profile Theory)
4-4 Wing of Finite Span in Subsonic and Transonic Flow
4-5 Wing of Finite Span at Supersonic Incident Flow
References
Part 2 Aerodynamics of the Fuselage and
the Wing-Fuselage System
5 Aerodynamics of the Fuselage
5-1 Introduction
5-2 The Fuselage in Incompressible Flow
5-3 The Fuselage in Compressible Flow
References
6 Aerodynamics of the Wing-Fuselage System
6-1 Introduction
6-2 The Wing-Fuselage System in Incompressible Flow
6-3 The Wing-Fuselage System in Compressible Flow
6-4 Slender Bodies
References
Part 3 Aerodynamics of the Stabilizers
and Control Surfaces
7 Aerodynamics of the Stabilizers
7-1 Introduction
7-2 Aerodynamics of the Horizontal Tail
7-3 Aerodynamics of the Vertical Tail
References
8 Aerodynamics of the Flaps and Control Surfaces
8-1 Introduction
8-2 The Flap Wing of Infinite Span (Profile Theory)
8-3 Flaps on the Wing of Finite Span and on the Tail Unit
References
Bibliography
Author Index
Subject Index
213
213
214
227
261
276
317
325
327
327
331
348
367
371
371
376
401
416
425
429
431
431
435
466
477
481
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 density of air (mass of unit volume)
g gravitational acceleration
cP, cv specific heats at constant pressure and constant volume,
respectively
y = cP/ci1 isentropic exponent
a = yp/,o speed of sound
µ coefficient of dynamic viscosity
v = µ/9 coefficient of kinematic viscosity
R gas constant
T absolute temperature (K)
t temperature (°C)
FLOW QUANTITIES
p pressure (normal force per unit area)
T shear stress (tangential force per unit area)
u, v, w velocity components in Cartesian (rectangular) coordinates
u, Wr, w.3 velocity components in cylindrical coordinates
V, U. velocity of incident flow
We velocity on profile contour
wt 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
=x/s,n=y/s,
z/s
Xf, Xr
xl, xp
A
AF
AH
Ay
b = 2s
bF
bH
A =b2/A
`4H, Ay
C
Cr, Ct
c11 =(2/A)foc2(y)dY
X = Ct/Cr
IF
cf
Xf=Cf/c
Tif
Cartesian (rectangular) coordinates: x = longitudinal axis,
y = lateral axis, z = vertical axis
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
wing area
fuselage cross-sectional area
area of horizontal tail (surface)
area of vertical tail (surface)
wing span
fuselage width
span of horizontal tail (surface)
aspect ratio of wing
aspect ratios of horizontal and vertical tails (surface),
respectively
wing chord
chord at wing root and wing tip, respectively
wing reference chord
wing taper
fuselage length
flap (control-surface) chord
flap (control-surface) chord ratio
flap deflection

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
"`LX = WX S/V,
any = W yCM/ V,
Z WZS/V
L
D
Y
Mx
M, My
MZ
Di
CL
CD
CMX
angle of attack (incidence)
angle of sideslip (yaw)
components of angular velocities in rectangular coordinates
during rotary motion of the airplane
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 wing data
F fuselage data
(W + F) data of wing-fuselage system
H data of horizontal stabilizer
V data of vertical stabilizer
f 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 the geometry of the aircraft for desired flow patterns
(indirect problem)
I

2 INTRODUCTION
Th Figure 1-1 Forces (loads) on an air-
plane in horizontal flight. L, lift; D,
drag; W, weight; Th, thrust; R,, re-
sultant of aerodynamic forces (result-
ant 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 aero-
dynamics 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)
1 1 b
R = 287 kg
K
(air) - )
(
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
Qy
Here y is the isentropic exponent, with
(1-2)
CP
y - cU
(1-3a)
= 1.405 (air) (1-3b)
cP 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
of
-1)+(7+1)PZ
Pi
e2 1 4
= ( - a)
el (7+1)+(7-1)Pi
7+1
= 5.93 (air) (1-4b)
7-1
where the indices 1 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 nor-
mal 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
_ -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 Viscosity viscosity
t Q
[°C]
[kg/m3
] [kg/ms]
[m2
/s]
-20 1.39 15.6 11.3
-10 1.34 16.2 12.1
0 1.29 16.8 13.0
10 1.25 17.4 13.9
20 1.21 17.9 14.9
40 1.12 19.1 17.0
60 1.06 20.3 19.2
80 0.99 21.5 21.7
100 0.94 22.9 24.5
(1-8a)
(1-8b)

6 INTRODUCTION
The decrease in the gravitational acceleration g(z) with increasing height z is
g(z) =
r,
2 go (1-9)
(ro + z)
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
a
(1-10)
go
0
+
r0
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 c nst (1-llb)
= o
9
?6
with n as the polytropic exponent (n <,y). From Eq. (1-11) we obtain by
differentiation and elimination of do/e,
dp n dT
(1.12a)
T n-- 1 T
dH 12b)
(1
BT .
The second relation follows from Eq. (1-8b). Finally, we have
dT n-1 9o
_ _ (1-13)
dH n R
Table 1-2 Reference values at the atmosphere layer boundaries, t
Hb
[km]
zb
[km]
Tb
[K]
Pb
[atm]
°b
[kg/rn3]
dT/dH
[K/km]
n
[-J
0 0 288.15 1 1.225 -6.5 1.235
11 11.019 216.65 2.234 10' 3.639 10' 0 1
20 20.063 216.65 5.403 10'2 8.803 10'2
+1 0.9716
32 32.162 228.65 8.567 10-3 1,322- 10-2
+2.8 0.9242
47 47.350 270.65 1.095 10-3 1.427 10-3
0 1
52 52.429 270.65 5.823 10-4 7.594 10-4
-2 1.062
61 61.591 252.65 1.797 10-4 2.511 10-4
-4 1.133
79 79.994 180.65 1.024 10-5 2.001 10'5
0 1
88.743 90 180.65 1.622 - 10-6 3.170 - 10'6
`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 R (H - Hb) (1-14)
Here it 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
11
n-1
Tb
- 1- nnl Ro (H H,)]
For the special case n = 1 (isothermal atmosphere), Eq. (1.15a) reduces to
r
P =expL- RTb
(H - Hb)
(1-15a)
(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
viscosity, are, after [2] ,
go = 9.8067 rn/s2
po = 1.0 atm
°o = 1.2250 kg/m3
To =288.15K
for the speed of sound and the
to=15°C
ao = 340.29 m/s
vo = 1.4607 - 10-5 m2 /s
(dT/dH)o = -6.5 K/km
kinematic
*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 7.846 - 10-1 8.217-10-1 0.9772 1.174
4 0.9097 6.085 - 10-1 6.688-10-1 0.9538 1.388
6 0.8647 4.660 - 10-1 5.389-10-1 0.9299 1.654
8 0.8197 3.518-10-1 4.292 10-1
0.9054 1.988
10 0.7747 2.615-10-1 3.376-10-1 0.8802 2.413
11.019 0.7519 2.234-10-1 2,971-10-1 0,8671 2.674
12 0.7519 1.915-10-1 2,546-10-1 0.8671 3.120
14 0.7519 1.399-10-1 1.860-10-1 0.8671 4.271
16 0.7519 1.022-10-1 1.359. 10-1 0.8671 5.846
18 0.7519 7.466 10-2
9.930. 10-2 0.8671 8.000
20 0.7519 5.457 - 10-2
7.258- 10-2 0.8671 1.095-101
20.063 0.7519- 5.403.10-2
7.186- 10-2 0.8671 1.106- 10'-
25 0.7689 2.516-10-2 3.272- 10-2 0.8769 2.474- 101
30 0.7861 1.181 10-2 1.503-10-2 0.8866 5.486-101
32.162 0.7935 8.567 10-3 1.080-10-2
0.8908 7.696 - 101
35 0.8208 5.671-10-3
6.909- 10-3 0.9060 1.236- 102
40 0.8688 2.834 - 10-3 3.262-10-3
0,9321 2.743- 102
45 0.9168 1.472 10-3 1.605 - 10-3 0.9575 5.819 - 102
47.350 0.9393 1.095 .10-3 1.165. 10-3 0.9692 8.170 - 102
50 0.9393 7.874 . 10-4 8.383 - 10-4 0.9692 1.136-103
52.429 0.9393 5.823-10-4 6.199- 10-4 0.9692 1.536 - 103
55 0.9218 4.219.10-4 4.578.10-4 0.9601 2.049.103
60 0.8876 2.217-10-4 2.497- 10-4 0.9421 3.645-103
61.591 0.8768 1.797 10-4 2.050- 10-4 0.9364 4,397- 103
65 0.8305 1.130 10'4 1.360-10-4
0.9113 6.340- 103
70 0.7625 5.448.10-5 7.146-10-5 0.8732 1.125 104
75 0.6946 2.458-10-5 3.538- 10-5 0.8334 2.100-104
79.994 0.6269 1.024-10-5 1.634-10-5
0.7918 4.161- 104
80 0.6269 1.023 - 10-5 1.632-10-5
0.7918 4.166-104
85 0.6269 4.071-10-6
6.494. 10-6 0.7918 1.047-105
90 0.6269 1.622 - 10-6
2.588 - 10-6 0.7918 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
Ma =
V
(1-16)
a
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
2.0
18
1.6
14
08
06
0.4
02
11<z<20
10
8
6
2
z=Okm
1 J
0 200 400 600 800 1000 1200 1400 1600 1800 2000 km /h 2400
V [km/h] --
Figure 1-3 Mach number Ma vs. flight velocity V and flight altitude z.
40
.106
36
32
16
12
8
4
0
z=Okm 1
1 _ Reference
length [m]
I I
i
I
i
i
i
i
V [km/h] - velocity V and flight altitude z.
3
4
5
6
7
8
.9
10
11
12
13
14
15
16
17
18
19
20
400 800 1200 1600 2000 km/h 2400'
Figure 1-4 Reynolds number Re vs. flight

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
where a is the angle between flight path and horizontal line.
Horizontal direction
Flight path
(1-18)
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 irI
low direction
Incident f
wz
Reference plane
Z f
3e
1-7t z
Figure 1-6 Systems of flight mechanical axes: airplane-fixed system, xf, yf, zf; experimental
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:
Drag:
Tangential force:
Lateral force:
Normal force:
Rolling moment:
Pitching moment:
Yawing moment:
L = cLA Wq
D = cDA wq
X=cxAwq
Y=cyAx,q
Z=czAwq
Mx = cmxA W sq
M= cMAwcuq
Mz = c Awsq
(1-21)
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 = Zf and tan Vyf (1-22)
Vxf
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:
CL = (a - ao)
deL
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:
C M C M O + dCM CL (1-26)
L
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
1 27
( - a)
Cµ CZ
CM _ dcM CMO
1 27b
( - )
CL dCL CL
Figure 1-7 Demonstration of location of
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 non-
dimensional coefficients for lift and pitching moment:
XN
CM = CMO - 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=
o
ap
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
acMx acmz
aS?
and
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,
V
(1-30)
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:
acMx
aQZ
and
acmz
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 acM
asp,,
and
asp,,

20 INTRODUCTION
The nondimensional pitching angular velocity
y
Dy
V
(1.32)
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 schemati-
cally 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
aCM
and
Angle of attack
W CM
a«
Figure 1-8 Schematic presentation of
quasi-stationary and nonsteady aerody-
namic 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 (aero-
elasticity).
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
at a 1
smLL =-=-=-
WT to Ma
(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 ede angle 27-
4)
25

26 AERODYNAMICS OF THE WING
C
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).
(Fig. 2-1b) is another important quantity. From these designated quantities the
following six geometric profile parameters may be formed:
t/c relative thickness (thickness ratio)*
hlc relative camber (camber ratio)*
xtlc relative thickness position
xh /c relative camber position
rN/c relative nose radius
2r 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
and Z= z
c 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 Aero-
dynamic 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.
Teardrop
63-
Z(s)
Mean camber
or skeleton
Z (0
a
b
C
69-
65-
66-
a a0
h
-0063
C
-0.068
h
=0,095
C
a-0.2
a=05
a=20
h
- = 0.055
c
Figure 2-2 Geometry of the most important NACA profiles. (a) Four-digit profiles. (b)
Five-digit profiles. (c) 6-series profiles.

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)
C
Z(s) = 4 hX(1 - X)
(2.4a)
(24b)
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:

1
t
1-2Xt
Teardrop:
2Xr c
b
Xt (2-7a)
1 h 1-2X
h
Skeleton: a= xh2 c
b= x2
jt
(2-7b)
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

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
fd 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 :
JAcp(x)
CL = dx
0
The pitching moment about the profile leading edge is
(2-10)
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,
1 c
CM=--fdcp(x)dx
0
2-2 FUNDAMENTALS OF LIFT THEORY
(2-12)
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.

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.

4dL
Pu
wo,
00-
P00
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
1P=pt - pu = 2 (wo,, + d w)2 - ° (w - A u')2 - 2Q u J w
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
L=
C
f.JpdA=b -1p dx
J (2-13a)
-
(A)
/4w
= 2 obwoo dx (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
.17= w d l (2-14a)
(1)

C' B C
I'= fdzvdx- fdzvdx=2 fdwdx (2-14b)
B,u C,[ 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

a
b
c
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-
face. (c) Circulation just sufficient to put rear
stagnation point on trailing edge. Smooth flow-
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

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
1' 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
starting vortex -r; a circulation +1' develops
consequently around 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 consider-
able 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

a 0 d IF c70 0'l-1
u c9x 7y
V
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
d z
= 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
F(z) = irlnz
2ir
where r is a clockwise-turning circulation.
(2-19)
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:
f (z) = f (x + y) (z, y) + i n (x, y) (2-20)
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)

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 (2-23)
az d = w(z)
d
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
b
v00
z plane
4a-C
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

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
vccs-
W) = uC' T i vt 2 - 4cc2
(2-25)
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,
= ITCV00
(2-26a)
(2-26b)
and the velocity distribution on the plate itself becomes, with and jtj < c/2,
u = w" cosy ± sing V c +
fl (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)]
r
2n

AC c - Pr - Pu = uu - ui = 2 sin 2a
2_
c + 2
P q00 woo
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)
the lift coefficient becomes
cL = bcq. = 21r sin a (2-30)
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
- 2rr
da
-050 -025 0
x
b
C
I
C
Py
11
Li s
G 0.5 x
Ic
C
(2-31)
Figure 2-9 Flow around an inclined flat plate. (a) Streamline pattern. (h) Pressure distribution
for angle of attack a = 10°. (c) Load distribution.

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
M _
C.u
- - sin2a
bc2
-
q.
4 (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_ 1
C
cL_4 (2-33)
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.
0.
0
t
cai0.
0.4
03
02
01
0
Theory
cL=2aa%
4 1
P
rofile Go 445-
1 Flat plate
Plate
J
Go 445
t
0° 2° 40 6° 8° 10°
a ---
12 ° 14°
Figure 2-10 Lift coefficient cL vs. angle of
attack a for a flat plate and a thin
symmetric profile. Comparison of theory,
Eq. (2-30), and experimental measure-
ments, after Prandtl and Wieselsberger
[47].

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

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.

IO
J
49
I I i
Thin profile..
71°
0.7
26
Flat plate
Q5
Q4
fl
2
a-Z1 °
t0°
0.1
20
18
16
1,4
12
1,0
08
06
04
021
01 1
0 0,02 004 005 008 010 072 074 016
b
0 °
a CD -
2 °
Flat plate
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)
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 is a symmetric Joukowsky profile, the thickness t of which is
determined by the location xo of the center of the mapping circle. The profile
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

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

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