Aerodynamics
of the Airplane Hermciui Sch!ichthg and Erie lw
c
Translated by Heinrich J. Ramm
ro t
AERODYNAMICS OF THE AIRPLANE
Hermann Schlichting Professor, Technical University of Braunschweig and Aerodynamic Research Institute (A VA), Gottingen
Erich Truckenbrodt Professor, Technical University of Munich
Translated by
Heinrich J. Ramm Associate Professor, University of Tennessee Space Institute
McGraw-Hill International Book Company New York St. Louis San Francisco Auckland Beirut Bogota
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This book was set in Press Roman by Hemisphere Publishing Corporation. The editors were Lynne Lackenbach and Judith B. Gandy; the production supervisor was Rebekah McKinney; and the typesetter was Wayne Hutchins. 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 79-60 629.132'3 ISBN 0-07-055341-6
CONTENTS
Preface Nomenclature 1 1-1
1-2 1-3
2-1
2-2 2-3 2-4 2-5
3 3-1
3-2 3-3 3-4 3-5 3-6
ix
Introduction Problems of Airplane Aerodynamics Physical Properties of Air Aerodynamic Behavior of Airplanes References
Part 1 Aerodynamics of the Wing 2
vii
Airfoil of Infinite Span in Incompressible Flow (Profile Theory) Introduction Fundamentals of Lift Theory Profile Theory by the Method of Conformal Mapping Profile Theory by the Method of Singularities Influence of Viscosity and Boundary-Layer Control on Profile Characteristics
1
1
2 8
22
23
25 25
30 36 52 81
References
101
Wings of Finite Span in Incompressible Flow
105
Introduction Wing Theory by the Method of ` ortex Distribution
105
Lift of Wings in Incompressible Flow Induced Drag of Wings Flight Mechanical Coefficients of the Wing Wing of Finite Thickness at Zero Lift References
131
112 173 181
197
206
Vi CONTENTS
4
Wings in Compressible Flow
4-1
Introduction
4-2 4-3
Basic Concept of the Wing in Compressible Flow Airfoil of Infinite Span in Compressible Flow (Profile Theory) Wing of Finite Span in Subsonic and Transonic Flow Wing of Finite Span at Supersonic Incident Flow References
4-4 4-5
213
213 214
227 261
276 317
Part 2 Aerodynamics of the Fuselage and the Wing-Fuselage System
325
Aerodynamics of the Fuselage
327
5 5-1
5-2 5-3
6 6-1 6-2 6-3 6-4
Introduction The Fuselage in Incompressible Flow The Fuselage in Compressible Flow
7-1 7-2 7-3
8 8-1 8-2 8-3
331
References
348 367
Aerodynamics of the Wing-Fuselage System
371
Introduction The Wing-Fuselage System in Incompressible Flow The Wing-Fuselage System in Compressible Flow Slender Bodies References
Part 3 Aerodynamics of the Stabilizers and Control Surfaces 7
327
Aerodynamics of the Stabilizers Introduction Aerodynamics of the Horizontal Tail Aerodynamics of the Vertical Tail
371
376 401
416 425
429
431 431
References
435 466 477
Aerodynamics of the Flaps and Control Surfaces
481
Introduction The Flap Wing of Infinite Span (Profile Theory) Flaps on the Wing of Finite Span and on the Tail Unit References
Bibliography Author Index Subject Index
481
486 506 517 521
527 537
PREFACE
Only a very few comprehensive presentations of the scientific fundamentals of the aerodynamics of the airplane have ever been published. The present book is an English translation of the two-volume work "Aerodynamik des Flugzeuges," which has already appeared in a second edition in the original German. In this book we treat exclusively the aerodynamic forces that act on airplane components-and thus on the whole airplane-during its motion through the earth's atmosphere (aerodynamics of the airframe). These aerodynamic forces depend in a quite complex manner on the
geometry, speed, and motion of the airplane and on the properties of air. The determination of these relationships is the object of the study of the aerodynamics of the airplane. Moreover, these relationships provide the absolutely necessary basis for determining the flight mechanics and many questions of the structural requirements of
the airplane, and thus for airplane design. The aerodynamic problems related to airplane propulsion (power plant aerodynamics) and the theory of the modes of motion of the airplane (flight mechanics) are not treated in this book. The study of the aerodynamics of the airplane requires a thorough knowledge of aerodynamic theory. Therefore, it was necessary to include in the German edition a
rather comprehensive outline of fluid mechanic theory. In the English edition this section has been eliminated because there exist a sufficient number of pertinent works in English on the fundamentals of fluid mechanic theory. Chapter 1 serves as an introduction. It describes the physical properties of air and of the atmosphere, and outlines the basic aerodynamic behavior of the airplane. The main portion of the book consists of three major divisions. In the first division (Part 1), Chaps. 2-4 cover the aerodynamics of the airfoil. In the second division (Part 2),
Chaps. 5 and 6 consider the aerodynamics of the fuselage and of the wing-fuselage system. Finally, in the third division (Part 3), Chaps. 7 and 8 are devoted to the problems of the aerodynamics of the stability and control systems (empennage, flaps, and control surfaces). In Parts 2 and 3, the interactions among the individual parts of the airplane, that is, the aerodynamic interference, are given special attention. Specifically, the following brief outline describes the chapters that deal with the intrinsic problems of the aerodynamics of the airplane: Part 1 contains, in Chap. 2, the profile theory of incompressible flow, including the influence of friction on the profile
viii PREFACE
characteristics. Chapter 3 gives a comprehensive account of three-dimensional wing theory for incompressible flow (lifting-line and lifting-surface theory). In addition to linear airfoil theory, nonlinear wing theory is treated because it is of particular importance for modern airplanes (slender wings). The theory for incompressible flow
is important not only in the range of moderate flight velocities, at which the compressibility of the air may be disregarded, but even at higher velocities, up to the speed of sound-that is, at all Mach numbers lower than unity-the pressure distribution of the wings can be related to that for incompressible flow by means of
the Prandtl-Glauert transformation. In Chap. 4, the wing in compressible flow is treated. Here, in addition to profile theory, the theory of the wing of finite span is discussed at some length. The chapter is subdivided into the aerodynamics of the wing at subsonic and supersonic, and at transonic and hypersonic incident flow. The latter
two cases are treated only briefly. Results of systematic experimental studies on simple wing forms in the subsonic, transonic, and supersonic ranges are given for the qualification of the theoretical results. Part 2 begins in Chap. 5 with the aerodynamics of the fuselage without interference at subsonic and supersonic speeds. In Chap. 6, a rather comprehensive account is given of the quite complex, but for practical cases very important, aerodynamic interference of wing and fuselage (wing-fuselage system). It should be noted that the difficult and complex theory of supersonic flow could be treated only superficially. In this chapter, a special section is devoted to slender flight articles. Here, some recent experimental results, particularly for slender wing-fuselage
systems, are reported. In Part 3, Chaps. 7 and 8, the aerodynamic questions of importance to airplane stability and control are treated. Here, the aerodynamic interferences of wing and wing-fuselage systems are of decisive significance. Experimental results on the maximum lift and the effect of landing flaps (air brakes) are given. The discussions of this part of the aerodynamics of the airplane refer again to subsonic and supersonic incident flow. A comprehensive list of references complements each chapter. These lists, as well
as the bibliography at the end of the book, have been updated from the German edition to include the most recent publications. Although the book is addressed primarily to students of aeronautics, it has been
written as well with the engineers and scientists in mind who work in the aircraft industry and who do research in this field. We have endeavored to emphasize the theoretical approach to the problems, but we have tried to do this in a manner easily understandable to the engineer. Actually, through proper application of the laws of modern aerodynamics it is possible today to derive a major portion of the aerodynamics of the airplane from purely theoretical considerations. The very comprehensive experimental material, available in the literature, has been included only as far as necessary to create a better physical concept and to check the theory.
We wanted to emphasize that decisive progress has been made not through accumulation of large numbers of experimental results, but rather through synthesis of theoretical considerations with a few basic experimental results. Through numerous detailed examples, we have endeavored to enhance the reader's comprehension of the theory. Hermann Schlichting Erich Truckenbrodt
NOMENCLATURE
MATERIAL CONSTANTS 0
g cP, cv y = cP/ci1
a=
yp/,o
µ v = µ/9 R T
t
density of air (mass of unit volume) gravitational acceleration specific heats at constant pressure and constant volume, respectively isentropic exponent speed of sound coefficient of dynamic viscosity coefficient of kinematic viscosity gas constant absolute temperature (K) temperature (°C)
FLOW QUANTITIES p T
u, v, w u, Wr, w.3
V, U. We wt
pressure (normal force per unit area) shear stress (tangential force per unit area) velocity components in Cartesian (rectangular) coordinates velocity components in cylindrical coordinates velocity of incident flow velocity on profile contour induced downwash velocity, positive in the direction of the negative z axis Lx
X NOMENCLATURE
q = (p/2)V2
q00 = (,o./2)U! Re = VI/v
Ma=V/a May, = U./ate,
Ma. cr
dynamic (impact) pressure dynamic (impact) pressure of undisturbed flow Reynolds number Mach number Mach number of undisturbed flow drag-critical Mach number Mach angle displacement thickness of boundary layer circulation dimensionless circulation vortex density source strength dipole strength velocity potential
GEOMETRIC QUANTITIES x,Y,z
Cartesian (rectangular) coordinates: x = longitudinal axis, y = lateral axis, z = vertical axis
=x/s,n=y/s, z/s Xf, Xr
xl, xp
dimensionless rectangular coordinates trigonometric coordinate; cos $ = q coordinates. of wing leading (front) and trailing (rear) edges, xo, x1oo, respectively coordinates of quarter-point and three-quarter-point lines, x25 , X75, respectively
b = 2s
wing area fuselage cross-sectional area area of horizontal tail (surface) area of vertical tail (surface) wing span
bF
fuselage width
A
AF
AH Ay
span of horizontal tail (surface) aspect ratio of wing A =b2/A `4H, Ay aspect ratios of horizontal and vertical tails (surface), respectively C wing chord chord at wing root and wing tip, respectively Cr, Ct c11 =(2/A)foc2(y)dY wing reference chord X = Ct/Cr wing taper IF fuselage length cf flap (control-surface) chord Xf=Cf/c flap (control-surface) chord ratio flap deflection Tif bH
NOMENCLATURE Xi
7 m = tan y/ tan
µ
E V
N25
t S = t/c
h
xt Xh Z(S)
Z(t) dFmax
SF = dFinaxliF 17F=bFIb
D=2R Zo
rH
EH rv
sweepback angle of wing leading edge semiangle of delta wing (Fig. 4-59) parameter (Fig. 4-59); m < 1: subsonic flow edge, m > 1: supersonic flow edge twist angle angle of wing dihedral geometric neutral point profile thickness thickness ratio of wing profile camber (maximum) thickness position (maximum) camber (height) position skeleton (mean camber) line coordinate teardrop profile coordinate maximum fuselage diameter fuselage thickness ratio relative fuselage width diameter of axisymmetric fuselage wing vertical position lever arm of horizontal tail (= distance between geometric neutral points of the wing and the horizontal tail) setting angle of horizontal stabilizer (tail) lever arm of vertical tail (= distance between geometric neutral points of the wing and the vertical tail)
AERODYNAMIC QUANTITIES (see Fig. 1-6)
WX, Wy, WZ
angle of attack (incidence) angle of sideslip (yaw) components of angular velocities in rectangular coordinates during rotary motion of the airplane
"`LX = WX S/V, any = W yCM/ V,
Z WZS/V L D Y
Mx M, My
MZ Di CL
CD CMX
components of the dimensionless angular velocities lift drag side force rolling moment pitching moment yawing moment induced drag lift coefficient drag coefficient rolling-moment coefficient
Xii NOMENCLATURE
CM,CMy CMZ Cl
Cm Cmf Cif CDi CDp
(dcL/da) cp =(p-pc,)/Q. Cp pl CP Cr
d Cp = (pi - pu)q f = 2b/CL,o k = 7r11/cLw ae
ag = a ai = wi/U,0 ao
OW =a+EH+aw aw=w/UU N XN Id XN
pitching-moment coefficient yawing-moment coefficient local lift coefficient local pitching-moment coefficient control-surface (hinge) moment coefficient flap (control-surface) load coefficient coefficient of induced drag coefficient of profile drag lift slope of wing of infinite span pressure coefficient pressure coefficient of plane (two-dimensional) flow critical pressure coefficient coefficient of load distribution planform function coefficient of elliptic wing effective angle of attack geometric angle of attack induced angle of attack zero-lift angle of attack angle of attack of the horizontal tail downwash angle at the horizontal tail location aerodynamic neutral point position of aerodynamic neutral point distance between aerodynamic and geometric neutral points angle of flow incident on the vertical tail angle of sidewash at the station of the vertical tail
DIMENSIONLESS STABILITY COEFFICIENTS
Coefficients of Yawed Flight acy/ao acMX/a1 aCMZ/a 3
side force due to sideslip rolling moment due to sideslip yawing moment due to sideslip
Coefficients due to Angular Velocity acylaQZ acMXla QX acMX/aQZ acMZ/af?Z
acMZ l a X aCL/a!?y
acJ/aQy
side force due to yaw rate rolling moment due to roll rate rolling moment due to yaw rate yawing moment due to yaw rate yawing moment due to roll rate lift due to pitch rate pitching moment due to pitch rate
NOMENCLATURE Xiii
INDICES W
F (W + F)
H V
f
wing data fuselage data data of wing-fuselage system data of horizontal stabilizer data of vertical stabilizer data of flaps (control surfaces)
CHAPTER
ONE INTRODUCTION
1-1 PROBLEMS OF AIRPLANE AERODYNAMICS An airplane moves in the earth's atmosphere. The state of motion of an airplane is determined by its weight, by the thrust of the power plant, and by the aerodynamic forces (or loads) that act on the airplane parts during their motion. For every state of motion at uniform velocity, the resultant of weight and thrust forces must be in
equilibrium with the resultant of the aerodynamic forces. For the particularly simple state of motion of horizontal flight, the forces acting on the airplane are shown in Fig. 1-1. In this case, the equilibrium condition is reduced to the requirement that, in the vertical direction, the weight must be equal to the lift (W = L) and, in the horizontal direction, the thrust must be equal to the drag (Th = D). Here, lift L and drag D are the components of the aerodynamic force R1 normal and parallel, respectively, to the flight velocity vector V. For nonuniform motion of the aircraft, inertia forces are to be added to these forces. In this book we shall deal exclusively with aerodynamic forces that act on the individual parts, and thus on the whole aircraft, during motion. The most important parts of the airplane that contribute to the aerodynamic forces are wing, fuselage,
control and stabilizing surfaces (tail unit or empennage, ailerons, and canard surfaces), and power plant. The aerodynamic forces depend in a quite complicated manner on the geometry of these parts, the flight speed, and the physical properties of the air (e.g., density, viscosity). It is the object of the study of the aerodynamics of the airplane to furnish information about these interrelations. Here, the following two problem areas have to be considered:
1. Determination of aerodynamic forces for a given geometry of the aircraft (direct problem) 2. Determination of (indirect problem)
the geometry of the aircraft for desired flow patterns I
2 INTRODUCTION
Th
Figure 1-1 Forces (loads) on an airplane in horizontal flight. L, lift; D, drag; W, weight; Th, thrust; R,, resultant of aerodynamic forces (resultant of L and D); Rz , resultant of W and Th.
The object of flight mechanics is the determination of aircraft motion for given aerodynamic forces, known weight of the aircraft, and given thrust. This includes questions of both flight performance and flight conditions, such as control and stability of the aircraft. Flight mechanics is not a part of the problem area of this book. Also, the field of aeroelasticity, that is, the interactions of aerodynamic forces with elastic forces during deformation of aircraft parts, will not be treated. The science of the aerodynamic forces of airplanes, to be treated here, forms the foundation for both flight mechanics and many questions of aircraft design and construction.
1-2 PHYSICAL PROPERTIES OF AIR 1-2-1 Basic Facts In
fluid mechanics, some physical properties of the fluid are important, for
example, density and viscosity. With regard to aircraft operation in the atmosphere,
changes of these properties with altitude are of particular importance. These physical properties of the earth's atmosphere directly influence aircraft aerodynamics and consequently, indirectly, the flight mechanics. In the following discussions the fluid will be considered to be a continuum. The density o is defined as the mass of the unit volume. It depends on both
pressure and temperature. Compressibility is a measure of the degree to which a fluid can be compressed under the influence of external pressure forces. The compressibility of gases is much greater than that of liquids. Compressibility
INTRODUCTION 3
therefore has to be taken into account when changes in pressure resulting from a particular flow process lead to noticeable changes in density. Viscosity is related to the friction forces within a streaming fluid, that is, to the tangential forces transmitted between ambient volume elements. The viscosity
coefficient of fluids changes rather drastically with temperature.
In many technical applications, viscous forces can be neglected in order to simplify the laws of fluid dynamics (inviscid flow). This is done in the theory of lift of airfoils (potential flow). To determine the drag of bodies, however, the viscosity has to be considered (boundary-layer theory). The considerable increase in flight
speed during the past decades has led to problems in aircraft aerodynamics that require inclusion of the compressibility of the air and often, simultaneously, the viscosity. This is the case when the flight speed becomes comparable to the speed of sound (gas dynamics). Furthermore, the dependence of the physical properties of air
on the altitude must be known. Some quantitative data will now be given for density, compressibility, and viscosity of air.
1-2-2 Material Properties Density The density of a gas (mass/volume), with the dimensions kg/m3 or Ns'/m', depends on pressure and temperature. The relationship between density e, pressure p, and absolute temperature T is given by the thermal equation of state for ideal gases
p =QRT
(1-la)
R = 287 kg (air) K
(1 - 1 b)
where R is the gas constant. Of the various possible changes of state of a gas, of particular importance is the adiabatic-reversible (isentropic) change in which pressure and density are related by
p = const
(1-2)
Qy
Here y is the isentropic exponent, with CP
y - cU
= 1.405 (air) cP
(1-3a)
(1-3b)
and c are the specific heats at constant pressure and constant volume,
respectively.
Very fast changes of state are adiabatic processes in very good approximation, because heat exchange with the ambient fluid elements is relatively slow and, therefore, of negligible influence on the process. In this sense, flow processes at high
speeds can usually be considered to be fast changes of state. If such flows are steady, isentropic changes of state after Eq. (1-2) can be assumed. Unsteady-flow
4 INTRODUCTION
processes (e.g., with shock waves) are not isentropic (anisentropic); they do not follow Eq. (1-2). Across a normal compression shock, pressure and density are related by e2
el
=
of
-1)+(7+1)PZ
(7+1)+(7-1)Pi 7+1
7-1 where the indices
1
Pi
= 5.93 (air)
( 1 - 4a)
(1-4b)
and 2 indicate conditions before and behind the shock,
respectively.
Speed of sound Since the pressure changes of acoustic vibrations in air are of such a high frequency that heat exchange with the ambient fluid elements is negligible, an isentropic change of state after Eq. (1.2) can be assumed for the compressibility law of air: p(e). Then, with Laplace's formula, the speed of sound becomes (1-5a)
ao = 340 m/s (air)
(1-5b)
where for p/p the value given by the, equation of state for ideal gases, Eq. (1-la), was taken. Note that the speed of sound is simply proportional to the square root
of the absolute temperature. The value given in Eq. (1-5b) is valid for air of temperature t = 15°C or T = 288 K.
Viscosity In flows of an inviscid fluid, no tangential forces (shear stresses) exist between ambient layers. Only normal forces (pressures) act on the flow. The theory of inviscid, incompressible flow has been developed mathematically in detail, giving, in many cases, a satisfactory, description of the actual flow, for example, in computing airfoil lift at moderate flight velocities. On the other hand, this theory fails completely for the computation of body drag. This unacceptable result of the theory of inviscid flow is caused by the fact that both between the layers within the fluid and between the fluid and its solid boundary, tangential forces are transmitted that affect the flow in
addition to the normal forces. These tangential or friction forces of a real fluid are the result of a fluid property, called the viscosity of the fluid. Viscosity is defined by Newton's elementary friction law of fluids as (1-6)
Here T is the shearing stress between adjacent layers, du/dy is the velocity gradient normal to the stream, and u is the dynamic viscosity of the fluid, having the dimensions
Ns/m2. It is a material constant that is almost independent of pressure but, in gases,
INTRODUCTION 5
increases strongly with increasing temperature. In all flows governed by friction and inertia forces simultaneously, the quotient of viscosity i and density Q plays an important role. It is called the kinematic viscosity v, (1-7)
and has the dimensions m2/s. In Table 1-1 a few values for density o, dynamic viscosity p, and kinematic viscosity v of air are given versus temperature at constant pressure.
1-2-3 Physical Properties of the Atmosphere Changes of pressure, density, and viscosity of the air with altitude z of the stationary atmosphere are important for aeronautics. These quantities depend on the vertical temperature distribution T(z) in the atmosphere. At moderate altitude (up to about 10 km), the temperature decreases with increasing altitude, the temperature gradient dT/dz varying between approximately -0.5 and -1 K per 100 m, depending on the weather conditions. At the higher altitudes, the temperature gradient varies strongly with altitude, with both positive and negative values occurring.
The data for the atmosphere given below are valid up to the boundary of the homosphere at an altitude of about 90 km. Here the gravitational acceleration is already markedly smaller than at sea level.
The pressure change for a step of vertical height dz
is,
after the basic
hydrostatic equation,
dp = - Qg dz
(1-8a)
_ -ego dH where H is called scale height. Table 1-1 Density e, dynamic viscosity µ, and kinematic viscosity v of air versus temperature t at constant pressure p 1 atmosphere Kinematic Temperature
Density
t
Q
[°C]
-20 -10 0 10 20
40 60 80 100
[kg/m3
]
1.39 1.34 1.29 1.25 1.21
1.12 1.06 0.99 0.94
Viscosity [kg/ms] 15.6 16.2 16.8 17.4 17.9 19.1 20.3 21.5 22.9
viscosity [m2
/s]
11.3 12.1 13.0 13.9 14.9 17.0 19.2 21.7 24.5
(1-8b)
6 INTRODUCTION
The decrease in the gravitational acceleration g(z) with increasing height z is r,
g(z) =
(ro + z) 2
(1-9)
go
with ro = 6370 km as the radius of the earth, and go = 9.807 m/s', the standard gravitational acceleration at sea level. With Eq. (1-8) we obtain by integration
H = f g(z) dz =
+z
go
(1-10)
a
r0
0
For the homosphere (z < 90 km), the scale height is insignificantly different from the geometric height (see Table 1-2). The variables of state of the atmosphere can be represented by the thermal and polytropic equations of state,
p = Q RT
(1-11a)
P
(1-llb)
9
with n
= c onst
?6
the polytropic exponent (n <,y). From Eq. (1-11) we obtain by
as
differentiation and elimination of do/e, dp
n
dT
T
n-- 1
T
(1.12a)
BT dH
(1 . 12b)
The second relation follows from Eq. (1-8b). Finally, we have
_ _ n-1 9o
dT dH
n
(1-13)
R
Table 1-2 Reference values at the atmosphere layer boundaries, t Hb [km] 0 11
zb [km] 0
20 32 47 52 61
79
88.743
11.019 20.063 32.162 47.350 52.429 61.591 79.994 90
Tb [K] 288.15 216.65 216.65 228.65 270.65 270.65 252.65 180.65 180.65
Pb
[atm] 1
2.234 5.403 8.567
10' 10'2
10-3 10-3 5.823 10-4 1.797 10-4 1.024 10-5 1.622 - 10-6 1.095
°b
dT/dH
n
[kg/rn3]
[K/km]
[-J
1.225 3.639 8.803
10'
10'2 1,322- 10-2 1.427 7.594 2.511 2.001
10-3 10-4 10-4
10'5 3.170 - 10'6
-6.5 0
+1 +2.8 0
-2 -4
0
1.235 1
0.9716 0.9242 1
1.062 1.133 1
`After "U.S. Standard Atmosphere" [2]. tHb, z b, Tb values at the lower boundary of the layer height; dTldH, n values in the layers.
INTRODUCTION 7
which shows that each polytropic exponent n belongs to a specific temperature gradient dT/dH. Note that the gas constant* in the homosphere, up to an altitude of H = 90 km, can be taken as a constant. From Eq. (1-13) follows by integration:
T=Tb7Ln1 Here it
R (H - Hb)
(1-14)
has been assumed that the polytropic exponent and, therefore, the
temperature gradient are constant within a layer. The index b designates the values
at the lower boundary of the layer. In Table 1-2 the values of Hb, Zb, Tb, and dT/dH are listed according to the "U.S. Standard Atmosphere" [2]. The pressure distribution with altitude of the atmosphere is obtained through integration of Eq. (1-12a) with the help of Eq. (1.14). We have
Tb
-
11
1- nnl Ro (H
H,)]
n-1
(1-15a)
For the special case n = 1 (isothermal atmosphere), Eq. (1.15a) reduces to P
r
=expL-
RTb
(H - Hb)
(1-15b)
In the older literature this relationship is called the barometric height equation. Finally, the density distribution is easily found from the polytropic relation Eq.
Also given in Table 1-2 are the reference values Pb and eb at the layer boundaries. For the bottom layer, which reaches from sea level to H= 11 km, Hb = Ho has to be set equal to zero in Eqs. (1-15a) and (1-15b). The other sea level
values (index 0), inclusive of those for the speed of sound and the kinematic viscosity, are, after [2]
,
go = 9.8067 rn/s2
to=15°C
po = 1.0 atm
ao = 340.29 m/s
°o = 1.2250 kg/m3
vo = 1.4607 - 10-5 m2 /s
To =288.15K
(dT/dH)o = -6.5 K/km
*The temperature gradient dT/dH determines the stability of the stratification in the stationary atmosphere. The stratification is more stable when the temperature decrease with increasing height becomes smaller. For dT/dH= 0 when n = 1, Eq. (1-13), the atmosphere is isothermal and has a very stable stratification. For n = y = 1.405, the stratification is adiabatic (isentropic) with dT/dH = -0.98 K per 100 in. This stratification is indifferent, because an air volume moving upward for a certain distance cools off through expansion at just the same rate as the temperature drops with height. The air volume maintains the temperature of the ambient air and is, therefore, in an indifferent equilibrium at every altitude. Negative temperature gradients of a larger magnitude than 0.98 K/100 m result in unstable stratification.
8 INTRODUCTION
Table 1-3 Barometric pressure p, air density o, temperature T, speed of sound a, and kinematic viscosity v versus height z* z [km]
T/To
p/po
Q/Po
I
a/ao
V/1'0
0
1.0
1.0
1.0
1.0
1.0
2
0.9549 0.9097 0.8647 0.8197 0.7747 0.7519 0.7519 0.7519 0.7519 0.7519 0.7519 0.75190.7689 0.7861 0.7935 0.8208 0.8688 0.9168 0.9393 0.9393 0.9393 0.9218 0.8876 0.8768 0.8305 0.7625 0.6946 0.6269 0.6269 0.6269 0.6269
7.846 - 10-1 6.085 - 10-1 4.660 - 10-1
8.217-10-1 6.688-10-1 5.389-10-1
3.518-10-1 2.615-10-1 2.234-10-1 1.915-10-1 1.399-10-1 1.022-10-1
4.292
0.9772 0.9538 0.9299 0.9054 0.8802 0,8671 0.8671 0.8671 0.8671 0.8671 0.8671 0.8671 0.8769 0.8866 0.8908 0.9060 0,9321 0.9575 0.9692 0.9692 0.9692 0.9601 0.9421 0.9364 0.9113 0.8732 0.8334 0.7918 0.7918 0.7918 0.7918
1.174 1.388 1.654 1.988 2.413 2.674 3.120 4.271 5.846 8.000
4 6 8 10 11.019 12 14 16
18
20 20.063 25 30 32.162 35 40 45 47.350 50 52.429 55 60 61.591 65 70 75 79.994 80 85 90
7.466 5.457 -
10-2 10-2
5.403.10-2
2.516-10-2 1.181 8.567
10-2 10-3 5.671-10-3
2.834 - 10-3 1.472 10-3
1.095 .10-3 7.874 . 10-4
5.823-10-4 4.219.10-4 2.217-10-4 10-4
1.797 1.130
3.376-10-1 2,971-10-1 2,546-10-1 1.860-10-1 1.359. 10-1 9.930. 10-2 7.258- 10-2 7.186- 10-2 3.272- 10-2 1.503-10-2 1.080-10-2
6.909- 10-3 3.262-10-3
1.605 - 10-3
1.165. 10-3 8.383 - 10-4 6.199- 10-4
4.578.10-4 2.497- 10-4 2.050- 10-4
10'4
5.448.10-5 2.458-10-5
1.360-10-4
7.146-10-5 3.538- 10-5
1.024-10-5
1.634-10-5
1.023 - 10-5 4.071-10-6
1.622 -
10-6
10-1
1.632-10-5
6.494. 10-6 2.588 - 10-6
1.095-101 1.106- 10'2.474- 101
5.486-101 7.696 - 101
1.236- 102 2.743- 102 5.819 - 102 8.170 - 102
1.136-103 1.536 - 103
2.049.103 3.645-103 4,397- 103 6.340- 103 1.125
104
2.100-104 4.161- 104
4.166-104 1.047-105 2,627-105
*After "U.S. Standard Atmosphere" [2].
The numerical values of pressure and density distribution are listed in Table 1-3, to which the values for the speed of sound and the kinematic viscosity have been added. More detailed and more accurate values are found in the comprehensive tables [2].
Finally, in Fig. 1-2, a graphic representation is given of the distributions of pressure, density, temperature, speed of sound, and kinematic viscosity versus altitude. Whereas pressure and density decrease strongly with height, kinematic viscosity increases markedly.
1-3 AERODYNAMIC BEHAVIOR OF AIRPLANES
1-3-1 Similarity Laws The question of the mechanical similarity of two flows plays an important role in
both the theory of fluid flows and the extensive testing procedures of fluid
INTRODUCTION 9
mechanics. That is, given are two fluids of different physical properties, in each of which one of two geometrically similar bodies is located. Under what conditions are
the two flow fields about the two bodies similar-in other words, under what conditions do they have a similar set of streamlines? Only in the case of mechanically similar flow fields is it possible to draw conclusions from the knowledge-which may have been obtained theoretically or experimentally-of the flow field about one body on the flow field about another geometrically similar body. To ensure mechanical similarity of flow fields about two geometrically similar, but not necessarily identical, bodies (e.g., two airfoils) in different fluids of different velocities, the condition must be satisfied that in each pair of points of similar position, the forces acting on two fluid elements must be similar in direction and magnitude. For the aerodynamics of aircraft, gravitation is of negligible influence and will not be considered for the establishment of similarity laws. Mach similarity law First, let us consider the case of a compressible, inviscid flow. Here, except for inertia forces, only the elastic forces act on the fluid elements of a homogeneous fluid. For mechanically similar flows, obviously the relative density change caused by the elastic forces must be equal in the two flows. This leads to the requirement that the Mach numbers of both flows, that is, the ratios of flow velocity and sonic speed, should be equal. This is the Mach similarity law. The Mach number V
Ma = a
(1-16)
Figure 1-2 Atmospheric pressure p, air density o, temperature T,
speed of sound a, and kinematic viscosity v, vs. height z. From
"U.S. Standard Atmosphere" [2].
10 INTRODUCTION
is, therefore, a first important dimensionless characteristic number of flow processes. Since the effects of compressibility become noticeable for Ma > 0.3, as pointed out
above, the Mach similarity law needs to be considered only above this limiting value. The fluid dynamic laws of an incompressible fluid can, therefore, be taken as the laws for very small Mach numbers with the limiting case Ma -+ 0.
Reynolds similarity law Let us now consider the case of an incompressible, viscous
flow. Here, only inertia and viscous forces act on the fluid element. These two forces are functions of the following physical quantities: approach velocity V, characteristic body dimension 1, density o, and dynamic viscosity µ of the fluid. The only possible dimensionless combination of these quantities is the quotient
Re -
°V i
V1
(1-17)
where Re is called the Reynolds number. The ratio p/Q = v has been introduced above in Eq. (1-7) as the kinematic viscosity. This law was found by Reynolds in 1883 during investigations on the flow in pipes and is called the Reynolds similarity law.
If velocity and body dimensions are not too small, as in aeronautics, the Reynolds number is very large because of the very small values of v. This means physically that the friction forces are much smaller than the inertia forces in such cases. Inviscid flow (v -+ 0) corresponds to the limiting case Re --+ -0. The laws of flow with small viscosity often correspond quite well to those without viscosity. On the other hand, in many cases even a very small viscosity should not be neglected in the theory (boundary-layer theory). For compressible flow with friction, mechanical similarity requires that the Mach and Reynolds similarity laws be satisfied simultaneously, which is very difficult to accomplish in experimental investigations. The Mach similarity law and the Reynolds similarity law govern decisively the whole realm of theoretical and experimental fluid mechanics and particularly the laws of aeronautics. To give a convenient survey of the Mach and Reynolds numbers occurring in the aerodynamics of aircraft, the diagrams Fig. 1-3 and Fig. 1-4 have been drawn. They show these two dimensionless characteristic quantities versus flight velocity
and flight altitude up to z = 20 km. Figure 1-3 shows that, at constant flight velocity, the Mach number increases with altitude because the sonic speed decreases,
as was shown in Table 1-3. At an altitude of 10 krn, the speed of sound has dropped to 300 m/s. At the same flight velocity, the Mach number at 10 km of altitude is about 10% larger than at sea level. This fact is important for the estimation of the aerodynamic properties of an airplane flying near the speed of sound.
The Reynolds numbers in Fig. 1-4 are those for a reference length of l = 1 in, where 1 may be the wing chord, fuselage length, or control surface chord. The Reynolds numbers of the diagram must be multiplied by a factor that corresponds to the reference length l in meters. Since the kinematic viscosity increases considerably with increasing height (see Table 1-3), the Reynolds number decreases
INTRODUCTION 11 2,2
11
2.0
10
8 6 18
2 z=Okm
1.6
14
1J
08 06 0.4
02
0
400
200
600
1000
800
1200
1400
V [km/h] --
1600
1800
2000 km /h 2400
Figure 1-3 Mach number Ma vs. flight velocity V and flight altitude z.
z=Okm 1 40 .106
3 36
4 32
5
1 _ Reference
length [m]
6 I
I
7
8 i
.9
10 11 I
16
i
i
12 13
i
12
i
14
15 16
8
17 18 19
20
4
0
400
800
1200
V [km/h] -
1600
2000 km/h 2400'
Figure
1-4 Reynolds number Re vs. flight
velocity V and flight altitude z.
12 INTRODUCTION
sharply with increasing height for a constant flight velocity, making airplane drag a particularly strong function of the height.
1-3-2 Aerodynamic Forces and Moments on Aircraft Lift, drag, and lift-drag ratio Airplanes moving with constant velocity are subject to an aerodynamic force R (Fig. 1-5). The component of this force in direction of the incident flow is the drag D, the component normal to it the lift L. Lift is produced almost exclusively by the wing, drag by all parts of the aircraft
(wing, fuselage, empennage). Drag will be discussed in detail in the following chapters. It has several fluid mechanical causes: By friction (viscosity, turbulence) on the surfaces, friction drag is produced, which is composed of shear-stress drag and a friction-effected pressure drag. This kind of drag depends essentially on the aircraft geometry and determines mainly the drag at zero lift. It is called form drag or also profile drag. As a result of the generation of lift on the wing, a so-called
induced drag is created in addition (eddy drag), which depends strongly on the aspect ratio (wing span/mean wing chord). An aircraft flying at supersonic velocity
is subject to a so-called wave drag, in addition to the kinds of drag mentioned above. Wave drag is composed of a component for zero lift (form wave drag) and a component caused by the lift (lift-induced wave drag). The
inclination of the resultant R to the incident flow direction and
consequently the ratio of lift to drag depend mainly on wing geometry and incident
flow direction. A large value of this ratio LID is desirable, because it can be considered to be an aerodynamic efficiency factor of the airplane. This efficiency factor has a distinct meaning in unpowered flight (glider flight) as can be seen from Fig.
1-5. For the straight, steady, gliding flight of an unpowered aircraft, the
resultant R of the aerodynamic forces must be equal in magnitude to the weight W but with the sign reversed. The lift-drag ratio is given, therefore, after Fig. 1-5, by the relationship tall
E=D
(1-18)
where a is the angle between flight path and horizontal line. Horizontal direction
Flight path
Figure 1-5 Demonstration of glide angle E.
INTRODUCTION 13
The minimum glide angle
EI,,
is
a very important quantity of flight
performance, particularly for glider planes. It is given by (L/D)max after Eq. (1-18). The outstanding characteristic of the wing, in comparison to the other parts of the aircraft, is its quite large lift-drag ratio. Here are a few data on LID for incompressible flow: A rectangular plate of an aspect ratio A = b/c = 6 has a value of (L/D)max of 6-8. Considerably greater lifts for about the same drag are obtained when the plate
is somewhat arched. In this case (L/D)max reaches 10-12. Even more favorable values of (L/D)max are obtained with wings that are streamlined. Particularly, the leading edge should be well rounded, whereas the profile should taper out into a sharp trailing edge. Such a wing may have an (L/D)m of 25 and higher. Further forces and moments, systems of axes We saw that, for symmetric incident flow, the resultant of aerodynamic forces is composed of lift and drag only. In the general case of asymmetric flow, the resultant of the aerodynamic forces may be composed of three forces and three moments. These six components correspond to six degrees of freedom of the aircraft motion. We introduce two systems of axes, depending on the flight mechanical requirements, to describe these forces and moments (Fig. 1-6). 1. Airplane-fixed system: Xf, Y f, Zf 2. Experimental system: Xe, Ye, Ze
The origin of the coordinates is the same in the two systems and is located in the symmetry plane of the aircraft. Its location in this plane is chosen to suit the specific problem. For flight mechanical studies, the origin is usually put into the aircraft center of gravity. For aerodynamic computations, however, it is preferable to put the origin at a point marked by the aircraft geometry. In wing aerodynamics it is advantageous to choose the geometric neutral point of the aircraft, as defined in Sec. 3-1.
The lateral axes of the experimental system of axes xe, ye, ze and of the system fixed in the airplane xf, yf, z f coincide so that ye = y f. The experimental system is obtained from the airplane-fixed system by rotation about the lateral axis by the angle a (angle of attack) (Fig. 1-6). For symmetric incident flow, the aerodynamic state of the aircraft is defined by the angle of attack a and the magnitude of the velocity vector. For asymmetric incidence, the angle of sideslip 0* is also needed. It is defined as the angle between the direction of the incident flow and the symmetry plane of the aircraft (Fig. 1-6).
Translator's note: According to the definition given by NASA, the angle of sideslip is the angle between the direction of the incident flow and the symmetry plane of the airplane. The
angle of yaw is referred to a chosen direction, which may sometimes be the direction of the airflow past the body, making the angle of yaw equal to the angle of sideslip. Under some conditions, however, as in turning, a different reference direction may be used.
14 INTRODUCTION
Mze C) Plane of
wz irI
Reference plane
Incident f low direction
Zf 3e
z Figure 1-6 Systems of flight mechanical axes: airplane-fixed system, xf, yf, zf; experimental 1-7t
system, xe, ye, ze; angle of attack, a; sideslip angle, R; angular velocities, wX, wy, wz
Forces and moments in the two coordinate systems are defined as follows: 1. Aircraft-fixed system: x f axis: tangential force Xf, rolling moment Mx f yf axis: lateral force Yf, pitching mdment Mf (or Myf) zf axis: normal force Zf, yawing moment Mzf 2. Experimental system: Xe axis: tangential force Xe, rolling moment Mxe Ye axis: lateral force Ye, pitching moment Me (or Mye) ze axis: normal force Ze, yawing moment Mze
The signs of forces and moments are shown in Fig. 1-6. It is customary to use lift L and drag D in addition to the forces and moments. They are interrelated as follows:
L = -Z,,
D = -X,?
(for 1i = 0)
(1-19)
Further, because of the coincidence of the lateral axes yf = y, Yf= Ye
Mf=Me =M
(1-20)
Dimensionless coefficients of forces and moments For the representation of experimental results and also for theoretical calculations, it is expedient to
introduce dimensionless coefficients for the moments and forces defined in the preceding paragraph. These coefficients are called aerodynamic coefficients of the aircraft. They are related to the wing area AW, the semispan s, the reference wing
INTRODUCTION 15
chord cµ (Eq. 3-5b), and to the dynamic pressure q = O V'/2, where V is the flight velocity (velocity of incident flow). Specifically, they are defined as follows. Lift:
L = cLA Wq
Drag:
D = cDA wq
Tangential force:
X=cxAwq
Lateral force:
Y=cyAx,q
Normal force:
Z=czAwq
Rolling moment:
Mx = cmxA W sq
Pitching moment:
M= cMAwcuq
Yawing moment:
Mz = c
(1-21)
Awsq
A measurement that determines the three coefficients CL, cD, and cm as a function of the angle of attack a is called a three-component measurement. The diagram CL(CD) with a as the parameter was introduced by Lilienthal [1]. It is called the polar curve or the drag polar. If all six components are measured, for example, of a yawed airplane, such a test is called a six-component measurement. Normally, the coefficients of forces and moments of aircraft depend considerably on the Reynolds number Re and the Mach number Ma; in addition to the geometric data. At low flight velocities, however, the influence of the Mach number on force and moment coefficients is negligible.
1-3-3 Interrelation between the Aerodynamic Forces and the Modes of Motion of the Airplane Motion modes of the airplane After having discussed the aerodynamic forces and the moments acting on the aircraft, its modes of motion may now be described briefly. An airplane has six degrees of freedom, namely, three components of translational velocity V, Vy, V, and three components of rotational velocity wx, wy, wZ. They can be expressed, for instance, relative to the aircraft-fixed system of axes x, y, z as in Fig. 1-6. The components of the aerodynamic forces, as introduced in Sec. 1-3-2, and their dimensionless aerodynamic coefficients are functions of these six degrees of freedom of motion.
The steady motion of an aircraft can be split up into a longitudinal and a lateral motion. During longitudinal motion, the position of the aircraft plane of symmetry remains unchanged. It is characterized by the three components of motion Vx, VZ, wy
(longitudinal motion)
The remaining three components determine the lateral motion Vy, wx, wZ
(lateral motion)
16 INTRODUCTION
It is expedient for the analysis of the interrelation of aerodynamic coefficients and components of motion to break down the general motion into straight flight, as described by Vx and VV; yawed flight, described by Vy; and rotary motion about the three axes. These rotary motions are, specifically, the rolling motion wx, the pitching motion coy, and the yawing motion wZ. The quantities of angle of attack a and angle of yaw !3,* which were introduced earlier (see Fig. 1-6), are then given by
tan a = Vxf Zf and tan
Vyf
(1-22)
xf The signs of a, a, o. , wy, and wZ can be seen in Fig. 1-6. At unsteady states of flight, the aerodynamic forces also depend on the acceleration components of the motion.
Forces and moments during straight flight The incident flow direction of an airplane in steady straight flight is given by the angle of attack a (Fig. 1-6). -The resultant aerodynamic force is fixed in magnitude, direction, and line of application by lift L, drag D, and pitching moment M (Fig. 1.6). Let us now give some details on the dimensionless aerodynamic coefficients introduced in Sec. 1-3-2. For moderate angles of attack, the lift coefficient CL is a linear function of the angle of attack a: deL
CL = (a - ao) d«
(1-23)
where as is the zero-lift angle of attack and dcLlda is the lift slope. A further characteristic quantity for the lift is the maximum lift coefficient CLmax, which is reached at an angle of attack that depends on the airplane characteristics. For moderate angles of attack and lift coefficients, the drag coefficient CD is given by CD = CDO + k, CL + k2cL
(1-24)
where CDO is the drag coefficient at zero lift (form drag). The constants kl and k2 depend mainly on the wing geometry. For wings of symmetric profile without twist we have kl = 0, and thus CD = CDO + k2 CL
(1-25)
This is the representation of the drag polar. The pitching-moment coefficient cm is a linear function of the angle of attack a and the lift coefficient cL, respectively: CM
C M O + dCM CL L
(1-26)
where cMo is the zero-moment coefficient and dcM/dcL is the pitching-moment slope. The value of cMo is independent of the choice of the moment reference *The angle R has been designated here as the angle of yaw. For the difference between angle of yaw and angle of sideslip see the footnote on page 13.
INTRODUCTION 17
station, whereas dcM/dcL depends strongly on it. The quantity dcM/dcL is also called the "degree of stability of longitudinal motion" (rotation about lateral axis). The resultant of the aerodynamic forces of the airplane is completely determined only when its magnitude, direction, and the position of its line of application are known. These three data are obtained, for instance, from lift, drag, and pitching moment. The position of the line of application of the resultant R, for example, on the wing, can be defined as the intersection of the line of application with the profile chord (Fig. 1-7a). This point is called the center of pressure or aerodynamic
center of the wing. With XA, the distance of the center of pressure from the moment reference axis, we have
M=.AZ For small angles of attack, the normal force with the negative sign is, in first approximation, equal to the lift:
Z= -L and by introducing the nondimensional coefficients, xL
CM
Cµ
CZ
( 1 -27 a)
CM _
dcM
CMO
CL
dCL
CL
Figure
1-7 Demonstration of location of
( 1 -27b )
aerodynamic center (center of pressure). (a) Aerodynamic center C. (b) Neutral point N. In general, the reference wing chord is c = c.,.
18 INTRODUCTION
This relationship means that the position of the center of pressure generally varies with the lift coefficient. The shift of the center-of-pressure position is given by the term -CMO /CL
.
In agreement of theory with experiment, the pitching moment can generally be
described as the sum of a force couple independent of lift (zero moment) and a term proportional to the lift:
M=M0 -xNL In words, the pitching moment is the sum of the zero moment and of the moment formed by the lift force and the distance XN between the neutral point and the moment reference line (Fig. 1-7b). Again introducing the nondimensional coefficients for lift and pitching moment: CM = CMO -
XN
CL
(1-28)
CA
Comparison with Eq. (1-26) yields, for the position of the neutral point xN
dcM
cA.
dcL
(1-29)
which shows that the pitching-moment slope dcMldcL determines the position of
the neutral point. The terms dcL/da and dcM/da are designated as derivatives ,--of longitudinal motion. Forces and moments in yawed flight When an aircraft is in stationary yawed flight, the direction of the incident flow of the wing is determined by both the angle of attack a and the angle of sideslip 1 (Fig. 1-6). Because of the asymmetric flow
incidence, additional forces and moments are produced besides lift, drag, and pitching moment as discussed in the last section. The force in direction of the lateral axis y is the side force due to sideslip; the moment about the longitudinal axis, the rolling moment due to sideslip; and the moment about the vertical axis, the yawing moment due to sideslip. The derivatives for 0 = 0, (8C Y) 0= ap o
aCMZI as
Q=0
are called stability coefficients of sideslip; in particular, acMZ/aa is called directional stability. All three of these coefficients are strongly dependent on the wing sweepback, besides other influences.
Forces and moments in rotary motion An airplane in rotary motion about the axes x, y, z, as specified by the modes of motion of Sec. 1-3-3, is subject to additional velocity components that are produced, for example, locally on the wing and that change linearly with distance from the axis of rotation. The aerodynamic forces and
moments that are the result of the angular velocities wX, wy, wZ will now be discussed briefly.
During rotary motion of the airplane about the longitudinal axis (roll) with
INTRODUCTION 19
angular velocity co, the lift distribution on the wing, for instance, becomes antisymmetric along the wing span. The resulting moment about the x axis can be called a rolling moment due to roll rate. It always counteracts the rotary motion and is, therefore, also called roll damping. The asymmetric force distribution along the span produces also a yawing moment, the so-called yawing moment due to roll rate. Introducing the dimensionless coefficients according to Eq. (1-21), the stability coefficients of sideslip acmz
acMx
and
aS?
asp
are obtained. The quantity .Q is the dimensionless angular velocity cw,. It is obtained from wX, the half-span s, and the flight velocity V: 5Q,; = E. -I,
(1-30)
V
The rotary motion of an airplane about the vertical axis (yaw) produces additional longitudinal air velocities on the wing that have reversed signs on the two wing halves and that result in an asymmetric normal and tangential force
distribution along the wing span, which in turn produces a rolling and a yawing moment. The yawing moment created in this way counteracts the rotary motion and is called yawing or turning damping. The rolling moment is called rolling moment due to yaw rate. Again by introducing nondimensional coefficients after Eq. (1-21), further stability coefficients of yawing motion are formed: acmz
acMx and
aQZ
aQZ
Here the nondimensional yawing angular velocity is (1-31)
The rotary motion of the aircraft about the lateral axis (pitch), Fig. 1-6, produces on the wing an additional component of the incident velocity in the z direction that is linearly distributed over the wing chord. This results in an additional lift due to pitch rate and an additional pitching moment that counteracts the rotary motion about the lateral axis. Therefore, it is also called pitch damping
of the wing. The magnitude of the pitch damping is strongly dependent on the position of the axis of rotation (y axis). By using lift and pitching-moment coefficients after Eq. (1-21), the following additional stability coefficients of longitudinal motion are obtained: aCL
asp,,
and
acM asp,,
20 INTRODUCTION
The nondimensional pitching angular velocity W y CM
Dy
(1.32)
V
is made dimensionless with wing reference chord after Eq. (3-5b) contrary to the rolling and yawing angular velocities Q,, and Qy , respectively, which were made dimensionless with reference to the wing half-span. Only the most important aerodynamic forces and moments produced by the rotary motion of the aircraft have been discussed above. Omitted, for instance, were detailed discussions of the side forces due to roll rate and yaw rate.
Forces and moments in nonsteady motion Besides the steady aerodynamic coefficients discussed above, the nonsteady coefficients applicable to accelerated flight
have increasingly gained importance, particularly for flight mechanical stability considerations. Nonsteady motions are more or less sudden transitions from one steady state to another or time-periodic motions superimposed on a steady motion. If the periodic motion is very slow (e.g., changes of angle of attack), the aerodynamic forces can be computed from quasi-stationary theory; this means that, for instance, the momentary angle of attack determines the forces. With periodic motions of higher frequency, however, the aerodynamic forces are phase-shifted (leading or lagging) from the motion. These conditions are demonstrated schematically in Fig. 1-8 for an airplane undergoing a periodic translational motion normal to its flight path. At nonsteady longitudinal motion, new aerodynamic force coefficients must be used, for example, the derivatives aCL
ac
and
aCM a«
Angle of attack
Figure
1-8 Schematic
presentation of
quasi-stationary and nonsteady aerodynamic forces.
INTRODUCTION 21 W=O
w-i a
w
w=a
Figure 1-9 Propagation of sound waves from a sound source moving at the velocity w through a
fluid at rest. (a) Sound source at rest, w = 0. (b) Sound source moving at subsonic velocity, w = a12. (c) Sound source moving at sonic velocity, w = a. (d) Sound source moving at supersonic velocity, w = 2a; the sound waves propagate within the Mach cone of apex semiangle g.
where a= daldt is the timewise change of the angle of attack. The nonsteady coefficients are important both for flight mechanics of the aircraft, assumed to be inflexible, and for questions concerning the elastically deformable airplane (aeroelasticity).
Forces and moments in supersonic flight During the transition from subsonic to supersonic flight, the aerodynamic behavior of an airplane undergoes a basic change. This becomes obvious when the airplane is taken as the source of a disturbance that moves through still air at a velocity V= w. Relative to this moving center
of disturbance, pressure waves emanate with the speed of sound a. A closer investigation of this process shows the importance of the speed of sound-especially
the ratio of flight velocity to sonic speed, that is, the Mach number from Eq. (1-16). In terms of fluid mechanics, the airplane can be considered as a sound source. Figure 1-9a shows the propagation of sound waves from a sound source at rest on concentric spherical surfaces. In Fig. 1-9b the sound waves, emitted at equal
time intervals, can be seen for a source that moves with one-half the speed of sound, w = a/2. Figure 1-9c is the corresponding picture for w = a and finally, Fig. 1-9d is for w = 2a. In this last case, in which the sound source moves at supersonic velocity, the effect of the source is felt only within a cone with the apex semiangle µ, which is given by
22 INTRODUCTION
smLL =-=-=Ma at
a
WT
to
1
(1-33)
This cone is called the Mach cone. No signals can be sent from the source to points
outside of the Mach cone, a zone called the zone of silence. No sound is heard, therefore, by an observer who is being approached by a body flying at supersonic speed. Physically, the process described is obviously identical to a sound source at rest in a fluid approaching from the right with velocity w. We have to keep in mind, therefore, the following characteristic difference: When the fluid velocity is smaller than the speed of sound (w
a, supersonic flow), pressure disturbances can propagate only within the Mach cone situated downstream of the sound source (Fig. 1-9d). Now, every point of the airplane surface can be considered as the source of a
disturbance (sound source) as in Fig. 1-9, in analogy to the previous discussion where the whole airplane was taken as the sound source. It can be concluded, therefore, that because of the different kinds of propagation of the individual pressure disturbances as in Fig. 1-9b and d, the pressure distribution and consequently the forces and moments on the various parts of the airplane (wing, fuselage, control surfaces) depend decisively on the airplane Mach number, whether the airplane flies at subsonic or supersonic velocities.
The above considerations show that subsonic flow has the characteristic properties of incompressible flow, whereas supersonic flow is basically different. In most cases, therefore, it will be expedient to treat subsonic and supersonic flows separately.
REFERENCES 1. Lilienthal, 0.: "Der Vogelflug als Grundlage der Fliegekunst," 1889; 4th ed., Sandig, Wiesbaden, 1965. 2. "U.S. Standard Atmosphere," National Oceanic and Atmospheric Administration and National Aeronautics and Space Administration, Washington, D.C., 1962.
PART
ONE AERODYNAMICS OF THE WING
CHAPTER
TWO AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY)
2-1 INTRODUCTION In this chapter the airfoil of infinite span in incompressible flow will be discussed. The wing of finite span in incompressible flow will be the subject of Chap. 3, and the wing in compressible flow that of Chap. 4. More recent results and understanding of the aerodynamics of the wing profile are communicated in progress reports by, among others, Goldstein [19], Schlichting [56], and Hummel [26].
Wing profile The wing profile is understood to be the cross section of the wing perpendicular to the y axis. Accordingly, the profile lies in the xz plane and depends, in the general case, on the spanwise coordinate y. The geometry of a wing profile may be described, as in Fig. 2-la, by introducing the connecting line of the centers of the inscribed circles as the mean camber (or skeleton) line, and the line connecting the leading and trailing edges of the mean camber line as the chord. The greatest distance, measured along the chord, is called the wing or profile chord c. The largest diameter of the inscribed circles is designated as the profile thickness t (Fig. 2-1b), and the greatest height of the mean camber line above the chord as the maximum camber h (Fig. 2-1c). The positions of the maximum thickness and the maximum camber are given by the distances xt (thickness position) and xh (camber position). The radius of the circle inscribed at the profile leading edge is taken as the nose radius rN; it is usually related to the thickness. The trailing ede4) angle 2725
26 AERODYNAMICS OF THE WING
Chord
Figure 2-1 Geometric terminology of lift-
ing wing profiles. (a) Total profile. (b) Profile teardrop (thickness distribution). (c) Mean camber (skeleton) line (camber height distribution).
C
(Fig. 2-1b) is another important quantity. From these designated quantities the following six geometric profile parameters may be formed:
t/c hlc
xtlc xh /c
rN/c
2r
relative thickness (thickness ratio)* relative camber (camber ratio)* relative thickness position relative camber position relative nose radius trailing edge angle
For the complete description of a profile, the profile coordinates of the upper
and lower surfaces, zu(x) and zl(x), must also be known. A profile can be considered as originating from a mean camber line z(s)(x) on which is superimposed a thickness distribution (profile teardrop shape) z(t)(x) > 0. For moderate thickness and moderate camber profiles, there results zu,t(x) = z(s)(x) ± z(t)(x)
(2-1)
The upper sign corresponds to the upper surface of the profile, and the lower sign to the lower surface. *These quantities may be called in the text simply "thickness" and "camber" when a misunderstanding is impossible.
AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 27
For the following considerations, the dimensionless coordinates
X=
x c
and
Z=
z C
are introduced. The origin of coordinates, x = 0, is thus found at the profile leading edge.
Of the large number of profiles heretofore developed, it is possible to discuss only a small selection in what follows. Further information is given by Riegels [501. The first systematic investigation of profiles was undertaken at the Aerodynamic Research Institute of Gottingen from 1923 to 1927 on some 40 Joukowsky profiles [47]. The Joukowsky profiles are a two-parameter family of profiles that are designated by the thickness ratio t/c and the camber ratio h/c (see Sec. 2-2-3). The skeleton line is a circular arc and the trailing edge angle is zero (the profiles accordingly have a very sharp trailing edge). The most significant and extensive profile systems were developed, beginning in 1933, at the NACA Research Laboratories in the United States.* Over the years the original NACA system was further developed [ 1 ] .
For the description of the four-digit NACA profiles (see Fig. 2-2a), a new parameter, the maximum camber position xh/c was introduced in addition to the thickness t/c and the camber h/c. The maximum thickness position is the same for all *NACA = National Advisory Committee for Aeronautics. Mean camber or skeleton
Teardrop Z (0
Z(s)
a
b
63-
a a0
h C
-0063
69-
a-0.2
-0.068
C
65-
a=05
h
=0,095
C
66-
a=20
h
- = 0.055
c
Figure 2-2 Geometry of the most important Five-digit profiles. (c) 6-series profiles.
NACA profiles. (a) Four-digit profiles. (b)
28 AERODYNAMICS OF THE WING
profiles xt/c = 0.30. With the exception of the mean camber (skeleton) line for Xh = XhIC = 0.5, all skeleton lines undergo a curvature discontinuity at the location of maximum camber height. The mean camber line is represented by two connected parabolic arcs joined without a break at the position of the maximum camber. For the five-digit NACA profiles (see Fig. 2-2b), the profile teardrop shape is
equal to that of the four-digit NACA profiles. The relative camber position, however, is considerably smaller. A distinction is made between mean camber lines with and without inflection points. The mean camber lines without inflection points are composed of a parabola of the third degree in the forward section and a straight
line in the rear section, connected at the station X= m without a curvature discontinuity.
In the NACA 6-profiles (see Fig. 2-2c), the profile teardrop shapes and the mean camber lines have been developed from purely aerodynamic considerations. The velocity distributions on the upper and lower surfaces were given in advance with a wide variation of the position of the velocity maximums. The maximum thickness position xtlc lies between 0.35 and 0.45. The standard mean camber line is
calculated to possess a constant velocity distribution at both the upper and lower surfaces. Its shape is given by Z(s)
=-
In 2[(l -X) In (1 -X) + X In X]
(2-3)
A particularly simple analytical expression for a profile thickness distribution, or a skeleton line, is given by the parabola Z = aX(l - X). Specifically, the expressions for the parabolic biconvex profile and the parabolic mean camber line are Z(t) = 2 t X(1 - X)
(2.4a)
Z(s) = 4 h X(1 - X)
(24b)
C
Here, t is the maximum thickness and h is the maximum camber height located at station X = 2 The so-called extended parabolic profile is obtained by multiplication of the
above equation with (1 + bX) in the numerator or denominator. According to Glauert [17], such a skeleton line has the form
r
z(S) = aX(1- X)(l + bX)
(2-5)
Usually these are profiles with inflection points.
According to Truckenbrodt [49], the geometry of both the profile teardrop shape and the mean camber line can be given by ,/-,) s-"
Z(X)
-a
X(1 - X)
1+bX
For the various values of b, this formula produces profiles without inflection points that have various values of the maximum thickness position and maximum camber position, respectively. The constants a and b are determined as follows:
AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 29
t 2Xr c
Teardrop:
1
1
a= xh2
Skeleton:
h c
b
b=
1-2Xt Xt
(2-7a)
1-2X x2jt
(2-7b)
h
Of the profiles discussed above, the drop-shaped ones shown in Fig. 2-2 have a rounded nose, whereas those given mathematically by Eq. (2-6) in connection with Eq. (2-7a) have a pointed nose. The former profiles are therefore suited mainly for the subsonic speed range, and the latter profiles for the supersonic range.
Pressure distribution In addition to the total forces and moments, the distribution of local forces on the surface of the wing is frequently needed. As an example, in Fig. 2-3 the pressure distribution over the chord of an airfoil of infinite span is presented for various angles of attack. Shown is the dimensionless pressure coefficient Cp =
P -P. q00
versus the dimensionless abscissa x/c. Here (p - p0,) is the positive or negative pressure difference to the pressure po, of the undisturbed flow and q., the dynamic pressure of the incident flow. At an angle of attack a = 17.9°, the flow is separated
Figure 2-3 Pressure distribution at various angles of attack a of an airfoil of infinite aspect ratio with the profile NACA 2412 [12]. Reynolds number Re = 2.7 . 106. Mach number Ma = 0.15. Normal force coefficients according to the following table: a
- 1.70
2.8'
7.4°
13.9°
17.8'
-CZ
0.024
0.433
0.862
1..0,56
0.950
30 AERODYNAMICS OF THE WING
from the profile upper surface as indicated by the constant pressure over a wide range of the profile chord. The pressures on the upper and lower surfaces of the profile are designated as pu and pl, respectively (see Fig. 2-3), and the difference d p = (p1- pu) is a measure for the normal force dZ = A pb dx acting on the surface element dA = b dx (see Fig. 2-5). By integration over the airfoil chord, the total normal force becomes c
Z= -b
f
d p(x) dx
(2-9a)
0
= c2q.bc
(2-9b)
where cZ is the normal force coefficient from Eq. (1-21) (see Fig. 2-3). For small angles of attack a, the negative value of the normal force coefficient can be set equal to the lift coefficient cL : CL =
JAcp(x) dx
(2-10)
0
The pitching moment about the profile leading edge is
M= -b f Ap(x) dx
(2-11a)
0
cMq.bc2
(2-11 b)
where nose-up moments are considered as positive. The pitching-moment coefficient is, accordingly,
CM=-1
f c
dcp(x)dx
(2-12)
0
2-2 FUNDAMENTALS OF LIFT THEORY
2-2-1 Kutta-Joukowsky Lift Theorem Treatment of the theory of lift of a body in a fluid flow is considerably less difficult than that of drag because the theory of drag requires incorporation of the viscosity of the fluid. The lift, however, can be obtained in very good approximation from the theory of inviscid flow. The following discussions may be based, therefore, on inviscid, incompressible flow.* For treatment of the problem of plane (two-dimensional) flow about an airfoil, it is assumed that the lift-producing body is a very long cylinder (theoretically of infinite length) that lies normal to the *The influence of friction on lift will be considered in Sec. 2-6.
AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 31
flow direction. Then, all flow processes are equal in every cross section normal to
the generatrix of the cylinder; that is, flow about an airfoil of infinite length is two-dimensional. The theory for the calculation of the lift of such an airfoil of infinite span is also termed profile theory (Chap. 2). Particular flow processes that have a marked effect on both lift and drag take place at the wing tips of finite-span wings. These processes are described by the theory of the wing of finite span (Chaps. 3 and 4).
Lift production on an airfoil is closely related to the circulation of its velocity near-field. Let us explain this interrelationship qualitatively. The flow about an airfoil profile with lift is shown in Fig. 24. The lift L is the resultant of the
pressure forces on the lower and upper surfaces of the contour. Relative to the pressure at large distance from the profile, there is higher pressure on the lower surface, lower pressure on the upper surface. It follows, then, from the Bernoulli equation, that the velocities on the lower and upper surfaces are lower or higher, respectively, than the velocity w. of the incident flow. With these facts in mind, it is easily seen from Fig. 2-4 that the circulation, taken as the line integral of the velocity along the closed curve K, differs from zero. But also for a curve lying very close to the profile, the circulation is unequal to zero if lift is produced. The
velocity field ambient to the profile can be thought to have been produced by a clockwise-turning vortex T that is located in the airfoil. This vortex, which apparently is of basic importance for the creation of lift, is called the bound vortex of the wing. In plane flow, the quantitative interrelation of lift L, incident flow velocity w,,, and circulation T is given. by the Kutta-Joukowsky equation. Its simplified derivation, which will now be given, is not quite correct but has the virtue of being
particularly plain. Let us cut out of the infinitely long airfoil a section of width b (Fig. 2-5), and of this a strip of depth dx parallel to the leading edge. This strip of
planform area dA = b dx is subject to a lift dL = (pl - pu) dA because of the pressure difference between the lower and upper surfaces of the airfoil. The vector
dL can be assumed to be normal to the direction of incident flow if the small angles are neglected that are formed between the surface elements and the incident flow direction. The pressure difference between the lower and upper surfaces of the airfoil can be expressed through the velocities on the lower and upper surfaces by applying the
wo,
Figure 24 Flow around an airfoil profile with lift L. 1' = circulation of the airfoil.
32 AERODYNAMICS OF THE WING
4dL Pu wo,
00P00
Figure 2-5 Notations for the computation of lift from the pressure distribution on the airfoil.
Bernoulli equation. From Fig. 2-4, the velocities on the upper and lower surfaces of
the airfoil are (w + J w) and (w - J w), respectively. The Bernoulli equation then furnishes for the pressure difference
1 P=pt - pu = 2 (wo,, + d w)2
- ° (w - A u')2 - 2Q u
Jw
where the assumption has been made that the magnitudes of the circulatory velocities on the lower and upper surfaces are equal, I d wji = JA wju = 1Aw1.
By integration, the total lift of the airfoil is consequently obtained as C
L= f.JpdA=b (A)
= 2 obwoo
-1 J- p dx
/4w dx
(2-13a)
(2-13b)
The integration has been carried from the leading to the trailing edge (length of airfoil chord c). The circulation along any line 1 around the wing surface is
wdl
.17= (1)
(2-14a)
AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 33 B
C'
C
I'= fdzvdx- fdzvdx=2 fdwdx B,u
C,[
(2-14b)
B
The first integral in the first equation is to be taken along the upper surface, the second along the lower surface of the wing. From Eq. (2-13b) the lift is then given by
L = o b zv, l'
(2-15)
This equation was found first by Kutta [35] in 1902 and independently by Joukowsky [31] in 1906 and is the exact relation, as can be shown, between lift and circulation. Furthermore, it can be shown that the lift acts normal to the direction of the incident flow.
2-2-2 Magnitude and Formation of Circulation If the magnitude of the circulation is known, the Kutta-Joukowsky formula, Eq. (2-15), is of practical value for the calculation of lift. However, it must be clarified as to what way the circulation is related to the geometry of the wing profile, to the velocity of the incident flow, and to the angle of attack. This interrelation cannot be determined uniquely from theoretical considerations, so it is necessary to look for empirical results.
The technically most important wing profiles have, in general, a more or less sharp trailing edge. Then the magnitude of the circulation can be derived from experience, namely, that there is no flow around the trailing edge, but that the fluid flows off the trailing edge smoothly. This is the important Kutta flow-off condition, often just called the Kutta condition. For a wing with angle of attack, yet without circulation (see Fig. 2-6a), the rear
stagnation point, that is, the point at which the streamlines from the upper and lower sides recoalesce, would lie on the upper surface. Such a flow pattern would be possible only if there were flow around the trailing edge from the lower to the upper surface and, therefore, theoretically (in inviscid flow) an infinitely high velocity at the trailing edge with an infinitely high negative pressure. On the other hand, in the case of a very large circulation (see Fig. 2-6b) the rear stagnation point would be on the lower surface of the wing with flow around the trailing edge from above. Again velocity and negative pressure would be infinitely high. Experience shows that neither case can be realized; rather, as shown in Fig.
2-6c, a circulation forms of the magnitude that is necessary to place the rear stagnation point exactly on the sharp trailing edge. Therefore, no flow around the trailing edge occurs, either from above or from below, and smooth flow-off is established. The condition of smooth flow-off allows unique determination of the magnitude of the circulation for bodies with a sharp trailing edge from the body shape and the inclination of the body relative to the incident flow direction. This statement is valid for the inviscid potential flow. In flow with friction, a certain
reduction of the circulation from the value determined for frictionless flow is observed as a result of viscosity effects.
For the formation of circulation around a wing, information is obtained from
34 AERODYNAMICS OF THE WING
a
b
Figure 2-6 Flow around an airfoil for various values of circulation. (a) Circulation l = 0: rear stagnation point on upper surface. (b) Very large
circulation: rear stagnation point on lower sur(c) Circulation just sufficient to put rear stagnation point on trailing edge. Smooth flowface.
c
off: Kutta condition satisfied.
the conservation law of circulation in frictionless flow (Thomson theorem). This states that the circulation of a fluid-bound line is constant with time. This behavior will be demonstrated on a wing set in motion from rest, Fig. 2-7. Each fluid-bound
line enclosing the wing at rest (Fig. 2-7a) has a circulation r = 0 and retains, therefore, T = 0 at all later times. Immediately after the beginning of motion, frictionless flow without circulation is established on the wing (as shown in Fig. 2-6a), which passes the sharp trailing edge from below (Fig. 2-7b). Now, because of friction, a left-turning vortex is formed with a certain circulation -F. This vortex quickly drifts away -from the wing and represents the -so-called starting or initial vortex -T (Fig. 2-7c). For the originally observed fluid-bound line, the circulation remains zero, even though the line may become longer with the subsequent fluid motion. It continues, however, to encircle the wing and starting vortex. Since the total circulation of this fluid-bound line remains zero for all times according to the Thomson theorem, somewhere within this fluid-bound line a circulation must exist equal in magnitude to the circulation of the starting vortex but of reversed sign. This is the circulation +T of the wing. The starting vortex remains at the starting location of the wing and is, therefore, some time after the beginning of the motion sufficiently far away from the wing to be of negligible influence on the further development of the flow field. The circulation established around the wing, which produces the lift, can be
AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 35
replaced by one or several vortices within the wing of total circulation +1' as far as the influence on the ambient flow field is concerned. They are called the bound vortices.* From the above discussions it is seen that the viscosity of the fluid, after all, causes the formation of circulation and, therefore, the establishment of lift. In an inviscid fluid, the original flow without circulation and, therefore, with flow
around the trailing edge, would continue indefinitely. No starting vortex would form and, consequently, there would be no circulation about the wing and no lift
Viscosity of the fluid must therefore be taken into consideration temporarily to
explain the evolution of lift, that is, the formation of the starting vortex. After establishment of the starting vortex and the circulation about the wing, the calculation of lift can be done from the laws of frictionless flow using the Kutta-Joukowsky equation and observing the Kutta condition. *In three-dimensional wing theory (Chaps. 3 and 4) so-called free vortices are introduced. These vortices form the connection, required by the Helmholtz vortex theorem, between the bound vortices of finite length that stay with the wing and the starting vortex that drifts off with the flow. In the case of an airfoil of infinite span, which has been discussed so far, the free
vortices are too far apart to play a role for the flow conditions at a cross section of a two-dimensional wing. Therefore only the bound vortices need to be considered.
- --er-o
a
b
Figure 2-7 Development of circulation during set-
ting in motion of a wing. (a) Wing in stagnant fluid. (b) Wing shortly after beginning of motion;
for the liquid line chosen in (a), the circulation 0; because of flow around the trailing edge, a vortex forms at this station. (c) This vortex formed by flow around the trailing edge is the so-called 1'
starting
vortex -r; a circulation +1'
consequently around the wing.
develops
36 AERODYNAMICS OF THE WING
2-2-3 Methods of Profile Theory Since the Kutta-Joukowsky equation (Eq. 2-15), which forms the basis of lift theory, has been introduced, the computation of lift can now be discussed in more detail. First, the two-dimensional problem will be treated exclusively, that is, the airfoil of infinite span in incompressible flow. The theory of the airfoil of infinite span is also called profile theory. Comprehensive discussions of incompressible profile theory, taking into account nonlinear effects and friction, are given by Betz [5], von Karman and Burgers [70], Sears [59], Hess and Smith [23], Robinson and Laurmann [51], Woods [74], and Thwaites [67]. A comparison of results of profile theory with measurements was made by Hoerner and Borst [251, Riegels [50], and Abbott and von Doenhoff [1]. Profile theory can be treated in two different ways (compare [73] ): first, by
the method of conformal mapping, and second, by the so-called method of singularities. The first method is limited to two-dimensional problems. The flow about a given body is obtained by using conformal mapping to transform it into a
known flow about another body (usually circular cylinder). In the method of singularities, the body in the flow field is substituted by sources, sinks, and vortices, the so-called singularities. The latter method can
also be applied to three-
dimensional flows, such as wings of finite span and fuselages. For practical purposes,
the method of singularities is considerably simpler than conformal mapping. The method of singularities produces, in general, only approximate solutions, whereas conformal mapping leads to exact solutions, although these often require considerable effort.
2-3 PROFILE THEORY BY THE METHOD OF CONFORMAL MAPPING
2-3-1 Complex Presentation Complex stream function Computation of a plane inviscid flow requires much less effort than that of three-dimensional flow. The reason lies not so much in the fact that the plane flow has only two, instead of three, local coordinates as that it can be treated by means of analytical functions. This is a mathematical discipline, developed in great detail, in which the two local coordinates (x, y) of two-dimensional flow can
be combined to
a
complex argument. A plane,
frictionless, and incompressible flow can, therefore, be represented as an analytical function of the complex argument z = x + iy :
F (z) = F (x + i y)
= 0 (x, y) + i'(x, y)
(2-16)
where 0 and q, the potential and stream functions, are real functions of x and y. The curves 0 = const (potential lines) and qI = const (streamlines) form two families of orthogonal curves in the xy plane. By taking a suitable streamline as a solid wall, the other streamlines then form the flow field above this wall. The velocity components in the x and y directions, that is, u and v, are given by
AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 37
u
a0
d IF
c9x
7y
V
c70
0'l-1
Jy
Jx
The function F(z) is called a complex stream function. From this function, the velocity field is obtained immediately by differentiation in the complex plane, where
dF dz
= it - i V = w(z)
(2-17)
Here, w = u - iv is the conjugate complex number to w = u + iv, which is obtained by reflection of w on the real axis. In words, Eq. (2-17) says that the derivative of the complex stream function with respect to the argument is equal to the velocity vector reflected on the real axis. The superposition principle is valid for the complex stream function precisely as for the potential and stream functions, because F(z) = c, F, (z) + c2 F2(z) can be considered to be a complex stream function as well as Fl (z) and F2(z).
For a circular cylinder of radius a, approached in the x direction by the undisturbed flow velocity u,,., the complex stream function is
F (z) = u (z +
a-)
(2-18)
For an irrotational flow around the coordinate origin, that is, for a plane potential vortex, the stream function is
irlnz
F(z) =
(2-19)
2ir
where r is a clockwise-turning circulation. Conformal mapping First, the term conformal mapping shall be explained (see [6] ).
Consider an analytical function of complex variables and split it into real and imaginary components: (2-20) (z, y) + i n (x, y) f (z) = f (x + y) The relationship between the complex numbers z =.x + iy and _ + iri in Eq. (2-20) can be interpreted purely geometrically. To each point of the complex z plane a point is coordinated in the plane that can be designated as the mirror
image of the point in the z plane. Specifically, when the point in the z plane moves along a curve, the corresponding mirror image moves along a curve in the plane. This curve is called the image curve to the curve in the z plane. The technical expression of this process is that, through Eq. (2-20), the z plane is conformally
mapped on the S plane. The best known and simplest mapping function is the Joukowsky mapping function,
=z
ca
-21) (2-21)
38 AERODYNAMICS OF THE WING
It maps a circle of radius a about the origin of the z plane into the twice-passed straight line (slit) from -2a to +2a in the plane. For the computation of flows, this purely geometrical process of conformal mapping of two planes on each other can also be interpreted as transforming a certain system of potential lines and streamlines of one plane into a system of those in another plane. The problem of computing the flow about a given body can then be
solved as follows. Let the flow be known about a body with a contour A in the z plane and its stream function F(z), for which, usually, flow about a circular cylinder is assumed [see Eq. (2-18)]. Then, for the body with contour B in the plane, the flow field is to be determined. For this purpose, a mapping function = f (z)
(2-22)
must be found that maps the contour A of the z plane into the contour B in the plane. At the same time, the known system of potential lines and streamlines about the body A in the z plane is being transformed into the sought system of potential lines and streamlines about the body B in the plane. The velocity field of the body B to be determined in the plane is found from the formula
a
az d
= w(z)
d
(2-23)
F(z) and w(z) are known from the stream function of the body A in the z plane (e.g., circular cylinder). Here dz/d = 1 If '(z) is the reciprocal differential quotient of the mapping function = f(z). The sought velocity distribution i about body B can be computed from Eq. (2-23) after the mapping function f(z) that maps body A into body B has been found. The computation of examples shows that the major
task of this method lies in the determination of the mapping function = f (z), which maps the given body into another one, the flow of which is known (e.g., circular cylinder).
Applying the method of complex functions, von Mises [71] presents integral formulas for the computation of the force and moment resultants on wing profiles in frictionless flow. They are based on the work of Blasius [71 J.
2-3-2 Inclined Flat Plate The simplest case of a lifting-airfoil profile is the inclined flat plate. The angle between the direction of the incident flow and the direction of the plate is called angle of attack a of the plate. The flow about the inclined flat plate is obtained as shown in Fig. 2-8, by superposition of the plate in parallel flow (a) and the plate in normal flow (b). The resulting flow (c) = (a) + (b)
does not yet produce lift on the plate because identical flow conditions exist at the leading and trailing edges. The front stagnation point is located on the lower surface and the rear stagnation point on the upper surface of the plate.
U"
a
4a-C
b
v00
z plane
plane
Figure 2-8 Flow about an inclined flat plate. (a) Flat plate in parallel flow. (b) Flat plate in normal (stagnation) flow. (c) Inclined flat plate without lift, (c) = (a) + (b). (d) Pure circulation flow. (e) Inclined flat plate with lift (Kutta condition), (e) = (c) + (d). 39
40 AERODYNAMICS OF THE WING
To establish a plate flow with lift, a circulation P according to Fig. 2-8d must be superimposed on (c). The resulting flow (e) = (c) + (d) = (a) + (b) + (d)
is the plate flow with lift. The magnitude of the circulation is determined by the condition of smooth flow-off at the plate trailing edge; for example, the rear stagnation point lies on the plate trailing edge (Kutta condition). By superposition of the three flow fields, a flow is obtained around the circle of radius a with its center at z = 0. It is approached by the flow under the angle a with the x axis, a being arctan The complex stream function of this flow, taking Eqs. (2-18) and (2-19) into account, becomes
F (z) = (u". - i v") z + (u"" + i v".) z + i
In z
(2-24)
For the mapping, the Joukowsky transformation function from Eq. (2-21) was chosen. This function transforms the circle of radius a in the z plane into the plate of length c = 4a in the plane. The velocity distribution about the plate is obtained with the help of Eq. (2-23) after some auxiliary calculations as
vccsW) = uC' T i
r
2n
(2-25)
vt 2 - 4cc2
The magnitude of the circulation T is now to be determined from the Kutta condition. Smooth flow-off at the trailing edge requires that there-that is, at = +2a-the velocity remains finite. Therefore, the nominator of the fraction in Eq. (2-25) must vanish for = 2a. Hence, because of 4a = c, T = 4rravc,
(2-26a) (2-26b)
= ITCV00
and the velocity distribution on the plate itself becomes, with
u = w" cosy ± sing V c +
fl
and jtj < c/2, (2-27)
The + sign applies to the upper surface, the - sign to the lower surface. With w,, the resultant of the incident flow, and a, the angle of attack between plate and incident flow resultant, the flow components are given by um = w. cos a and v., = w. sin a. At the plate leading edge, t = -c/2, the velocity is infinitely high. The flow around the plate comes from below, as seen from Fig. 2-8e. On the plate trailing edge, t = +c/2, the tangential velocity has the value u = v cos a. At an arbitrary station of the plate, the tangential velocities on the lower and upper surfaces have a difference in magnitude zi u = uu - ul. At the trailing edge, v u = 0 (smooth flow-off). The nondimensional pressure difference between the lower and upper surfaces, related to the dynamic pressure of the incident flow qr, = (o/2)w',, is [see Eq. (2-8)]
AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 41
ACP c - Pr - Pu = uu -2 ui = 2 sin 2a woo
q00
2_
c+2
(2-28)
where uu and ul stand for the velocities on the upper and lower surfaces of the plate, respectively. This load distribution on the plate chord is demonstrated in Fig. 2-9c. The loading at the plate leading edge is infinitely large, whereas it is zero at the trailing edge. By integration, the force resulting from the pressure distribution
on the surface can be computed in principle [see Eq. (2-9)]. In the present case, the result is obtained more simply by introducing Eq. (2-26b) into Eq. (2-15). With L = prrbcw;, sin a
(2-29)
cL = bcq. = 21r sin a
(2-30)
the lift coefficient becomes
This equation establishes the basic relationship between the lift coefficient and the angle of attack of a flat plate in two-dimensional flow. The so-called lift slope for small a is dCL
da
- 2rr
(2-31)
Py
I 11 Li G
-050
b
-025
x
0 C
sx Ic
0.5
C
C
Figure 2-9 Flow around an inclined flat plate. (a) Streamline pattern. (h) Pressure distribution for angle of attack a = 10°. (c) Load distribution.
42 AERODYNAMICS OF THE WING
Figure 2-10 gives a comparison, based on Eq. (2-30), between theory and experimental measurements for a flat plate and a very thin symmetric profile. Up to
about a = 6°, the agreement is quite good, although it is somewhat better for the plate than for the profile. At angles of attack in excess of 8°, the experimental curves lie considerably below the theoretical curve, a deviation due to the effect of viscosity. When the angle of attack exceeds 12°, flow separation sets in. Flows around profiles with and without separation are shown in Fig. 2-11. Naumann [42] reports measurements on a profile over the total possible range of angle of attack, that is, for 0° < a < 360°. Without derivation, the pitching moment coefficient about the plate leading edge (tail-heavy taken to be positive) is given by
-
C.u
M bc2 q.
_ -
-
4
sin2a
(2-32)
From Eqs. (2.30) and (2-32), the distance of the lift center of application from the leading edge at small angles of attack is obtained (see Fig. 2-9) as
XLCM_cL_4 1
(2-33)
C
Since lift and moment depend exclusively on the angle of attack, the center of lift (= center of application of the load distribution in Fig. 2-9c) is identical to the neutral point (see Sec. 1-3-3). An astounding result of the just computed inviscid flow about an infinitely thin I 0.
Theory
cL=2aa% 0.
4
0
1
P
1
t
rofile Go 445-
Flat plate
cai0.
0.4
Plate 03
J
02
Figure 2-10 Lift coefficient cL vs. angle of attack a for a flat plate and a thin symmetric profile. Comparison of theory,
Go 445 t
01
0 0°
2°
40
6°
a ---
8°
10°
12 °
14°
Eq. (2-30), and experimental measurements, after Prandtl and Wieselsberger
[47].
AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 43
a
Figure 2-11 Photographs of the flow about airfoil, after Prandtl and Wieselsberger [471. (a) Attached flow. (b) Separated flow.
inclined flat plate is the fact that the resultant L of the forces is not perpendicular to the plate, but perpendicular to the direction of the incident flow w.. (Fig. 2-9a). Since only normal forces (pressures) are present on the plate surface in a frictionless flow, it could appear to be likely that the resultant of the forces acts normal to the plate, too. Besides the normal component Py = L cos a, however, there is a tangential component P, = -L sin a that impinges on the plate leading edge. Together with the normal component Py, the resultant force L acts normal to the direction
of the incident flow. For the explanation of the existence of a tangential component P, in an inviscid flow-we shall call it suction force-a closer look at the
flow process is required. The suction force has to do with the flow at the plate nose, which has an infinitely high velocity. Consequently, an infinitely high
44 AERODYNAMICS OF THE WING
underpressure is produced. This condition is easier to see in the case of a plate of finite but small thickness and rounded nose (Fig. 2-12a). Here the underpressure at the nose of the plate is finite and adds up to a suction force acting parallel to the plate in the forward direction. The detailed computation shows that the magnitude
of this suction force is independent of plate thickness and nose rounding. It remains, therefore, the value of S = L sin a in the limiting case of an infinitely thin plate.
In real flow (with friction) around very sharp-nosed plates, an infinitely high underpressure does not exist. Instead, a slight separation of the flow (separation
bubble) forms near the nose (Fig. 2-12b). For small angles of attack, the flow reattaches itself farther downstream and, therefore, on the whole is equal to the frictionless flow. The suction force is missing, however, and the real flow around an inclined sharp-edged plate produces drag acting in the direction of the incident flow.
Also, this analysis shows that it is very important for keeping the drag small that the leading edge of wing profiles is well rounded. Figure 2-13 shows (a) the polar curves (CL vs. CD) and (b) the glide angles E = CD/CL of a thin sharp-edged flat plate
and of a thin symmetric profile. In the range of small to moderate angles of attack, the thin profile with rounded nose has a markedly smaller drag than the sharp-edged flat plate. Within a certain range of angles of attack, a is smaller than a (c < a) for
Px = 0
Figure 2-12 Development of the suction force S on the leading edge of a profile. (a) Thin, symmetric profile with rounded nose, suction force present. (b) Flat plate with sharp nose, suction force missing.
AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 45 20
IO
J
49
18
I
i
I
16
Thin profile..
Flat plate 71°
0.7
1,4
26
12
Flat plate Q5
1,0
Q4
08 fl
2
06
a-Z1 ° 04
t0° 021
0.1
01
0
0,02 004
005
008
010
CD -
072
074
0°
016
1
2°
4°
6°
B°
10°
12°
CC -Figure 2-13 Aerodynamic coefficients of a sharp-edged flat plate and a thin symmetric profile for Re = 4 105, A = -, from Prandtl and Wieselsberger [47]. (a) Polar curves, CL vs. CD. (b) a
b
Glide angle, E = CD/CL-
thin profiles; the resultant of the aerodynamic forces is inclined upstream relative to
the direction normal to the profile chord. This must be attributed to the effect of the suction force.
2-3-3 Joukowsky Profiles The Joukowsky transformation (mapping) function Eq. (2-21) is also particularly suitable for the generation of thick and cambered profiles. In Sec. 2-3-1 it was shown that this transformation function maps the circle z = a about the origin in the z plane into the straight line = -2a to = +2a of the plane (Fig. 2-8a).
The same transformation function also allows generation of body shapes resembling airfoils by choosing different circles in the z plane. These shapes may have rounded noses and sharp trailing edges (Fig. 2-14). They are called Joukowsky profiles, after which the transformation function is named. By choosing a circle in the z plane as in Fig. 2-14a, the center of which is shifted by x0 on the negative axis from that of the unit circle and which passes through the point z = a, a profile is produced that resembles a symmetric airfoil shape. It encircles the slit from -2a
to +2a. This
a symmetric Joukowsky profile, the thickness t of which is determined by the location xo of the center of the mapping circle. The profile is
tapers toward the trailing edge with an edge angle of zero. Circular-arc profiles are obtained when the center of the mapping circle lies on the imaginary axis (Fig. 2-14b). When the center is set on +iyo and the circumference passes through z = +a, the same mapping function produces a
46 AERODYNAMICS OF THE WING
Figure 2-14 Generation of Joukowsky profiles through conformal mapping with the Joukowsky mapping function, Eq. (2-21). (a) Symmetric Joukowsky profile. (b) Circular-arc, profile. (c) Cambered Joukowsky profile.
twice-passed circular arc in the plane. It lies between = -2a and = +2a. The height h of this circular arc depends on yo. Finally, by choosing a mapping circle
the center of which is shifted both in the real and the imaginary directions (Fig. 2-14c), a cambered Joukowsky profile is mapped, the thickness and camber of which are determined by the parameters x0 and yo, respectively. As a special case of the Joukowsky profiles, the very thin circular-arc profile (circular-arc mean camber) will be discussed.
Circular-arc profile In the circular-arc profile the mapping circle in the z plane is a circle, as in Fig. 2-14b, passing through the points z = +a and z = -a with its center at a distance yo from the origin on tie imaginary axis. The radius of the mapping circle is R = a-4 + i with a 1 = yo /a. The circle K is mapped into a twice-passed
AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 47
profile in the plane, extending from = -2a to length c = 4a and a camber height h/a = 2 E1 , or
+2a. This profile has a chord
(2-34)
It is easily shown that the profile in the plane is a portion of a circle for any E1 . For small camber (E' < 1), the profile contour is given by
=
2 [1
-4
[1
- 4 (C )2]
(2-35)
C
This profile is also called a parabola skeleton. For small angles of attack, a G< 1, and small camber, the lift coefficient becomes cL = 27r (Cl +2 C
(2-36)
The lift slope dcL/da is again equal to 27r for small angles of attack, as in the case of the inclined flat plate according to Eq. (2-31). For the zero-lift angle of attack this equation yields ao = -2(h/c). The pitching-moment coefficient about the profile leading edge becomes
CM = - 2
(a+4 h)
(2-37)
resulting in cMo = -ir(h/c) for the zero-moment coefficient when ao = -2(h/c). The velocity distribution on the circular-arc profile is given for small camber and small angles of attack by WC-u'c,
1t4C
Y1-4(x)2±
(2-38)
The + sign applies to the upper profile surface, the - sign to the lower profile surface. The second term, which is dependent on the camber, represents an elliptic
distribution over . The third term, which depends on the angle of attack a, corresponds to the expression found for the inclined flat plate [Eq. (2-27)]. At the trailing edge, i = c/2, the velocity on the circular-arc profile is finite, whereas in general its value becomes infinitely large at the leading edge, i _ -c/2. Only for the angle of incidence a = 0 does the velocity remain finite at the leading edge. This is the angle of smooth leading-edge flow (no flow around the leading edge).* Velocity distributions, computed for this case, are shown in Fig. 2-15 for *Translator's note: When the angle of attack of a thin profile (skeleton) is changed from positive to negative values, the stagnation point moves from the lower surface to the upper surface. Only at one angle of attack is the stagnation point exactly on the leading edge. This angle is called the angle of smooth leading-edge flow (S.L.E.F.). Obviously, here, no flow rounds
the leading edge, which-in inviscid flow-would cause infinitely high velocities. Rather, the S.L.E.F. is a smooth flow past the leading edge. Only for a flat plate is the angle of S.L.E.F. equal to the angle of attack a = 0.
48 AERODYNAMICS OF THE WING
Y
015
X h -005
!
\
C
0W
\
1
Figure 2-15 Velocity distribution of circular-arc profile with
025
Exact
--- Approximation 0 -100
-a75
-Q50
-025
0 S
0,25
too
0,75
0.50
/2
camber ratios h/c = 0.05 and 0.15 for smooth leading-edge flow,
two circular-arc profiles of camber h/c = 0.05 and 0.15. For comparison, the exactly computed distributions are also given. The agreement is very good for small camber. For larger camber, some deviations can be seen.
Of particular interest is the largest velocity on the profile at a = 0. It occurs at the profile center t = 0 and is obtained from Eq. (2-38) as wCmax=wo,1 1 +4 k
(2-39a)
+ EL L
(2-39b)
=Woo
1
7r
)
These equations allow a very simple estimation of the maximum velocity on a very thin circular-arc profile with smooth leading-edge flow.
Inclined symmetric Joukowsky profile The symmetric Joukowsky profile may serve as a further example. This profile is obtained from Fig. 2-14a when the mapping circle passes through the point z = +a and is placed with its center on the negative real axis at a distance x0 from the origin. The radius of the circle is
R=a+xo=a(l+E2)
with C2 =
-
xo a
(2.40)
The unit circle and the mapping circle are tangent in z = a; that is, the tangents of the two circles intersect under the angle zero. Since the angles remain unchanged in conformal mapping, the trailing-edge angle of the Joukowsky profile is zero.* For a The Joukowsky mapping function, Eq. (2-21), can be given in more general form in various ways, leading to additional profile shapes that are obtained from mapping circles. For example, when in Fig. 2-14a the mapping circle does not pass through the point +a on the real axis but rather through a point located somewhat farther outside, the sharp trailing edge of the normal Joukowsky profile is replaced by a rounded edge.
AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 49
very small thickness (E2 < 1), the profile chord length is c = 4a and the thickness
t _ C
3 4
/E2 = 1.299c2
(2.41)
The maximum thickness occurs at p = 1200, that is, at point c/4 from the leading edge. The profile contour is given by
= 5E2(1-2C)V'-4(x)2
(2-42)
This profile shape is called the Joukowsky teardrop. The zero-lift direction of this profile coincides with the profile chord (the i; axis). The lift coefficient is CL = 27r(1 + e2) sin a
=21r 1 +0.77
t c
(2-43a) a
(2-43b)
where the second expression is valid for small angles of attack. Accordingly, the lift slope dcL /da increases somewhat with profile thickness. The pitching-moment coefficient about the profile leading edge becomes cm = -(rr/2)(1 + E2 )a, indicating that the lift force center of attack (neutral point) lies at a distance c/4 from the profile nose. The velocity distribution on the contour
of the symmetric Joukowsky profile is obtained in a way similar to that for the circular-arc profile. Presentation of the corresponding expression is omitted. In Fig. 2-16, pressure distributions on a symmetric Joukowsky profile of 15% thickness ratio are presented for various lift coefficients. At an angle of attack a = 0 (CL = 0), the pressure minimum occurs at approximately 15% chord behind the nose. When the angle of attack increases, the minimum moves forward on the suction side and farther back on the pressure side.
Cambered Joukowsky profiles The Joukowsky profile with a mean camber line shaped like a circular arc is obtained by mapping an excentrically located circle with its center at zo = x0 + iyo (see Fig. 2-14c). Further generalizations of the
Joukowsky mapping functions are given by von Karman and Trefftz [7], with profile thickness, camber height, and trailing-edge angle as the parameters. The mean
camber line has the shape of a
circular
arc, however, as in the case of the
Joukowsky profiles, resulting in a considerable shift of the aerodynamic center. For the elimination of this problem, Betz and Keune [7] developed suitable mapping functions.
Experimental results Comprehensive three-component measurements on numerous Joukowsky profiles have been reported in [47]. Figure 2-17 shows a comparison of lift coefficients versus the angle of attack as obtained from theory and tests by Betz [31 ] . The agreement is quite good in the angle-of-attack range from a = -10° to
50 AERODYNAMICS OF THE WING
\ cL°t00 015
Pressure side
c 0
1
50
-05
a qg
a
Suction side
too -1,0
_15
-20 i
-R5
0
01
05
Of
07
ad
09
10
X
Figure 2-16 Pressure distribution of an inclined symmetric Joukowsky profile, t/c = 0.15, for various lift coefficients CL.
a= +10°; the small differences are caused by viscous effects. The moment curves CM(CL) are in agreement with theory up to large thickness ratios in the case of symmetric profiles; in the case of cambered profiles, however, the agreement is good only for small thickness ratios. The theoretical and experimental pressure distributions are also in good agreement, as can be seen from Fig. 2-18.
Concluding remarks The disadvantage of using the method of conformal mapping to determine aerodynamic properties of profiles lies in the necessity of first fording a
mapping function. The resulting profile shape must then be compared with the desired shape. In general, it is not possible to know beforehand the proper mapping
function that is mapping the desired profile shape on the circle. To a first approximation, this problem can be solved as shown by Theodorsen and Garrick [66] ; see also Ringleb [32]. The methods for the treatment of profile theory by means of conformal mapping will not be discussed further, because the method of singularities, which will be discussed next, has proved to be more suitable and allows simpler computation of velocity distributions over a given profile. Furthermore, the method of singularities has the marked advantage over the method of
AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 51 6
Lift
I
1.
Re=105
Theory/
Expe riment
i
122
Drag 0
-02 I
f -04
!
, -8°
-12°
0°
12°
B°
4°
°
Figure 2-17 Lift and drag for plane flow around a cambered Joukowsky profile, after Betz [311. Profile after Fig. 2-18.
conformal mapping that it can be applied to three-dimensional problems (wings of finite span) whereas conformal mapping is strictly limited to two-dimensional problems. The great value of the method of conformal mapping remains nevertheless, because this method allows one to establish exact solutions for the velocity
distribution on certain profiles that then can be compared with approximate solutions as obtained, for instance, by the method of singularities. For the design
zo
Lower surface
05
I
0
a=s° Experiments
25
E
0
ll
07
U ppe r
--- Theory
surfaceRe-;0
Figure 2-18 Comparison of theoretical and experimental pressure distribu-
I
0.2
tions of an inclined cambered Jou-
!
0.9
00
05
0.6
07
0B
02
t0
kowsky profile resulting in the same lift, after Betz [31].
52 AERODYNAMICS OF THE WING
problem, that is, the problem of determining the profile shape for a given pressure distribution, Eppler [13] has developed a procedure that uses conformal mapping.
2-4 PROFILE THEORY BY THE METHOD OF SINGULARITIES 2-4-1 Singularities The method of conformal mapping was applied in Sec. 2-3 to the computation of velocity distributions about a given wing profile. Another means of computing the aerodynamic properties of wing profiles is the method of singularities (see Keune and Burg [33]). This consists of arranging sources, sinks, and vortices within the
profile. Through superposition of their flow fields with a translational flow, a suitable body contour (profile) is produced. The flow field within the contour has
no physical meaning. For the creation of a symmetric profile in a symmetric incident flow field (teardrop profile), only sources and sinks are required, whereas
for the creation of camber, vortices must be added within the profile. This procedure is shown schematically in Fig. 2-19. These sources, sinks, and vortices are termed singularities of the flow. In most cases it is necessary to distribute the singularities continuously over the profile chord rather than discretely.
It is expedient to treat the very thin profile (skeleton profile) first. For such profiles the skeleton theory (Sec. 24-2) produces all essential results for their lift. For representation of the skeleton profile, only a vortex distribution is needed. The symmetric profile of finite thickness (teardrop profile) in symmetric flow. (angle of attack zero) is produced by source-sink distributions (teardrop theory). In this case
the displacement flow about the profile is obtained (Sec. 2-4-3). The cambered
2-19 The singularities method. (a) Cambered profile of finite thickness with angle of attack a. (b) Symmetric profile of Figure
finite thickness in symmetric flow, a = 0. (c) Very thin profile with angle of attack.
AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 53
C
Figure 2-20 The skeleton theory. (a) Arrangement of the vortex distribution on the skeleton line. (b) Arrangement of the vortex distribution on the chord (slightly cambered profile). (c) Circulation distribution along the chord (schematic).
profile of finite thickness is essentially the product of superposition of a mean camber line (skeleton line) with a teardrop profile (Sec. 244).
2-4-2 Very Thin Profiles (Skeleton Theory) Fundamentals of skeleton theory As was stated above, the very thin profile (skeleton profile) is obtained by superposition of a translational flow with that of a distribution of plane potential vortices. This theory has therefore been termed the theory of the lifting vortex sheet. It was first developed by Birnbaum and Ackermann [8] and by Glauert [171, and later expanded in several treatises,
particularly by Helmbold and Keune [22, 32], Allen [3], and Riegels [49]. For the following discussion a coordinate system as shown in Fig. 2-20a is used. Accordingly, the profile chord coincides with the x axis. The coordinate system origin lies on the profile leading edge. The mean camber line is given by z(s')(x). From Fig. 2-20a, the mean camber line is seen to be covered with a continuous vortex distribution. With the assumption that the skeleton profile has only a slight camber and, therefore, rises only a little above the profile chord (x axis), the vortex distribution can be arranged on the chord instead of the mean camber line (Fig. 2-20b). The mathematical treatment of the problem is considerably simplified in this way.
The vortex strength of a strip of width dx of the vortex sheet is, from Fig. 2-20b,
dr = k (x) d x
(244)
54 AERODYNAMICS OF THE WING
Here, k is the vortex density (vortex strength per unit length) or the circulation distribution. By applying the law of Biot-Savart, the velocity components in the x and z directions, respectively, that are induced by the vortex distribution at station x, z are C
U
1
(x, z) =
fk(x')
-
z
(x - x')2
0
+ -`
d x'
(2-45a)
C
w(x z) _ -
1
x-x
fl- (x')
dx'
(245b)
0
For slightly cambered profiles, the velocity components on the skeleton line are
approximately equal to the values on the profile chord (z = 0). The velocity components on the chord are obtained through limit operations as z -> 0 of Eqs. (2-45a) and (2-45b)
U (X) _
k (X)
(2-46a)
1
2n
W (X)
fk(X')
dX1
(2-46b)
0
The dimensionless quantities
X= X C
an d
Z(s) = z (s)
(2-4 7)
C
were introduced in Sec. 2-1, with c being the chord length. The velocity component u is proportional to the vortex density. The upper sign
is valid for the profile upper surface, the lower sign for the lower surface. When crossing the vortex sheet, the velocity component u changes abruptly by an amount
du=uu - ul=k
(248)
The integral for the velocity component w has a singularity at X= X.*
The distribution of the vortex density on the chord is determined by the kinematic flow condition, which requires that the skeleton line is a streamline. Specifically, a translational velocity U. is superimposed on the vortex distribution that forms the angle of attack a with the chord (Fig. 2-20). The kinematic flow condition can also be formulated by the requirement that the velocity components normal to the mean camber line must disappear. Within the framework of the above approximation, it is sufficient to satisfy this condition on the chord instead of the mean camber line, resulting in
*It is necessary to take the Cauchy principal value
f
(Y-e lir n { 111
1
.. d X' ; j.... d Y' j +e
AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 55
U00
ra
-
d7I' (X) l1 1 dX
+w(X) = 0
(249)
This equation relates the angle of attack a and the ordinates of the camber Zisi to the induced normal velocities w.
The velocity distribution on the profile surface and the vortex density are related by
U(X) = U,,,, + 26(X) = Uc,, _ J- k(X)
(2-50)
This relationship is valid for small angles of attack according to Eq. (246a).
The Kutta condition, Sec. 2-2-2, requires that the velocities on the profile upper and lower surfaces be equal at the trailing edge. It is required, therefore, that in Eq. (2-50),
for X= 1
k=0
(2-51)
The total circulation around the profile is determined from the distribution of the vortex density as
T = fk(x)dx=cjk(x)dx 0
(2-52)
0
The pressure difference between the lower and upper surface is obtained by means of the Bernoulli equation:
Pi-PuU.Au=oUUk With Eq. (2-48), the dimensionless pressure coefficient takes the form
dcP(X) = Pi -Pu = 2 k(X)
q
U.
(2-53)
with q. _ U,2o/2 being the dynamic pressure of the incident flow. Consequently, the distribution of the vortex density produces directly the load distribution over the profile chord. From Eq. (2-10), the lift coefficient CL = L/q..bc is expressed by (I
CL
= AJ cp (X) LAX 0
(2-54a)
i 2
UC'
.f k(X) dX
(2-54b)
0
The latter relationship may also be found from the interrelation of lift and circulation after the Kutta-Joukowsky equation (2-15) for w = U.. Equation (2-12) yields the pitching-moment coefficient relative to the profile leading edge, cm = M/q.bc2 (tail-heavy = positive):
c,,r = - f dc1,(X) X dX 0
(2-55a)
56 AERODYNAMICS OF THE WING
C M = - U fk(x)xdx
(2-55b)
0
Computation of the mean camber line from the distribution of circulation Determining the shape of the mean camber line and the angle of attack from a given distribution of circulation k(X) requires two steps. First, from Eq. (2-46b), the distribution of the induced downwash velocity w(X) is obtained along the profile chord. Then, this distribution is introduced into the kinematic flow condition, Eq. (2-49), and the following expression for the shape of the mean camber line is obtained by integration over X: x Z() (X) = a X -}- f w (X) d X + C (2-56) 0
These two steps may be combined into one equation by introducing Eq. (246b) into Eq. (2-56) and integrating over X. The angle of attack and the integration constant C are determined in such a way that the ordinates of the mean camber line disappear on the leading and trailing edges, resulting in
i
Z(.4) (X)
=aX-
(' k(X) in
1
2nJ
Uoo
X_ X, dX' X'
(2-57)
i
0
for the mean camber line and
X=
27C
f 0
U 00
in 1 g,
d X'
(2-58)
for the angle of attack as measured from the chord.
In the case of a constant distribution of circulation along the profile chord, k = 2UOOC, Eqs. (2-57) and (2-58) yield, for the mean camber line and the angle of attack, Z(s)(X)
C [(1 -X)ln(1 -X) +X1nX] with a=0
(2-59)
The maximum camber height is h/c = (In 2/7r)C = 0.221 C and lies at 50% chord. This mean camber line is found in NACA profiles of the 6-series (see Fig. 2-2c; a = 1.0). The lift coefficient is obtained from Eq. (2-54b) as CL = 4C =
In 2 c
(2-60)
Following up on the investigations of Birnbaum and Ackermann, Glauert [171 proposed the following Fourier series expansion for the circulation distribution in the two-dimensional airfoil problem: k (r) = 2 U,,,, (A0 tan `
' A.,, sinn
(2-61)
AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 57
Here
X =j-(1 + cos cp)
(2-62)
so that on the leading edge X = 0 and cp = ir, and on the trailing edge X = 1 and cp = 0. Each term in Eq. (2-61) satisfies the Kutta condition, Eq. (2-51).
By introducing the expression for the distribution of circulation, Eq. (2-61), into the equation for the induced downwash velocity, Eq. (2-46b), the simple relationship * ?1' (1p)
UI
- - (A0 + - i N
1
(2-63)
A!, COS n cp J
is found after integration.
The interrelation of the Fourier coefficients of Eq. (2-63), the shape of the mean camber line, and the angle of attack are obtained with the help of Eq. (2-49) as N
A 0 --r- ,4 A cos n 92 = a n-i
dZ(s)(X) (2-64)
dX
With a given distribution of the circulation, this is a differential equation for the mean camber line Z(s)(X).
The first two terms in Eq. (2-61) represent particularly simple mean camber lines: The distribution of circulation of the first standard distribution becomes k = A0 kl =. 2 Uoo A0 tan
E
= 2 U,,. A0 V
1
19
X
X
(2-65)
The distribution k is shown in Fig. 2-21a. The induced downwash velocity is determined from Eq. (2-63) to be w/U,,, = -A0, leading to
Further, from the kinematic flow condition, Eq. (2-64), it follows that the profile inclination dZ(s)/dX must be constant. This is possible only when Z(s) = 0, and, therefore,
A0 =a
(2-66)
It has thus been shown that the first normal distribution represents flow about the inclined flat plate. The second normal distribution is given by
lc= A1krf=2U... A1sin cp=4U, A1VX(1 -X) *Note that the following relation is valid according to Glauert [ 17 1: :z
1r
z
0
cosncp' cosrp - cosrp
,
sing. rp sin (P
(2-67)
58 AERODYNAMICS OF THE WING
6
5 41
3
a
2
b
2
X 0
02
0V
06
0.8
02
"0
ZO
04'
06
08
10
Figure 2-21 The first and the second normal distributions; circulation distribution by Eq. (2-61). (a) The inclined flat plate. (b) The parabolic skeleton at zero angle of incidence.
This is an elliptic distribution (Fig. 2-21 b). The induced downwash velocity is obtained from Eq. (2-63) as
I = - cosgq =-(2X - 1) and with Eq. (2-56), the shape of the mean camber line is given by Z(') =A 1.X(1 - X) = 4 c X(1 - X)
with a = 0
(2-68)
This is a parabolic mean camber line with camber height h/c = Al /A0 . The results
obtained for the inclined flat plate and the parabolic camber without angle of attack agree with the exact solutions found by the method of conformal mapping for small angles of attack, Secs. 2-3-2 and 2-3-3, respectively.* In particular, the relationships for lift and pitching-moment coefficients are also valid.
Computation of the aerodynamic coefficients Equations will now be presented that
allow one to compute the aerodynamic coefficients directly from a given mean camber line. The lift coefficient is obtained from Eq. (2-54b) after lintegration# with the help of Eqs. (2-61) and (2-62) for the distribution of circulation as CL = 7r(2Ao +A,)
(2-69)
In the same way, the pitching-moment coefficient relative to the leading edge is obtained from Eq. (2-55b) as c111
=-
(2A0+2A1+A2)
(2-70)
4
This equation was first presented by Munk [41]. *Note that /c = X - z T In this process, most of the integrals over cp disappear as a result of the orthogonality conditions of the trigonometric functions.
AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 59
The angle of attack for zero lift (CL = 0) is obtained by setting 2A0 = -A1, and the zero-lift moment coefficient becomes cm,) = -(7r/4)(A1 +A2). Consequently, the pitching-moment coefficient can also be written as 1
CM = CMo - 4 CL
From Eq. (1-29), the neutral-point location
is
(2-71)
given by -dcM/dcL = XN/c.
Consequently, the distance of the neutral point from the leading edge becomes XN C
(2-72)
4
which is independent of the shape of the mean camber line. The Fourier coefficients are found through Fourier analysis: Z
A.0
1 f d'P)
dg
:Z
d92
rdZ'a)
All
J0 dX
0
cosn q7 dq7
(n > 1)
(2-73)
The integrals can be transformed through integration by parts into terms in which the camber line coordinates Z(S) replace the camber line inclination dZ(S)/dX. By introducing the coefficients A0 and Al into Eq. (2.69), the relation
da =
21r
(2-74)
is obtained for the lift slope, independent of the camber line shape, and the lift coefficient from Eq. (1-23) is
CL = 21r(a -a0)
(2-75)
The equations for ao and cMo are given in Table 2-1. On the profile leading edge, X = 0, that is, cp = 7r, in general the vortex density and consequently the velocity are infinitely large (Eq. 2-61). There is an angle of attack, however, for which the velocity remains finite on the leading edge. In Sec. 2-3-3, the designation of angle of smooth leading-edge flow was introduced for this angle of attack. This angle as can be determined from Eq. (2-61) by setting A0 = 0. The expressions for as and for the lift coefficient for smooth leading-edge flow are also presented in Table 2-1. If there is flow around the leading edge, the velocity is infinitely high, streaming either from below to above, or vice versa. The strong underpressures near the leading edge produce a force acting upstream on the leading edge, called suction force in Sec. 2-3-2. The suction force coefficient c8 = S/q.bc can be expressed by 1
7
rk(X) w(X) dX 0
Introducing Eqs. (2-61) and (2-63) into this equation yields c,s.= 2.-;r A2
(2-76)
a
-
P
K N N'
N
O
I
0
N
8
v0 I
GN
K
oA
2
K
K
L
cl
+ 4-
o
0
I..I Wyy
w O V b
O o
a. O y ..i
60
fl.
r tv
z
o
w
O
.yr O
N
¢
0
Wrr
p b
O
a Gn
AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 61
and with CL of Eq. (2-69) and CLS = 7rA1, 1
Cs = 21r (CL - BLS)
z
(2-77)
Consequently, the suction force is zero for smooth leadinb edge flow, but grows with the square of (CL - cLS). For a given distribution of circulation k(X), the coefficient AID in Eq. (2-61) is obtained by the limit operation A o = 2U--
lmo[k(X)V ]
In the integral formulas of Table 2-1 for the computation of the various coefficients, only the. distribution of the mean camber coordinates Z(s')(p) appear besides certain trigonometric functions of gyp. In addition, simple quadrature formulas
are given for the numerical evaluation of the integrals. Accordingly, the profile coordinates Zm = Z(Xm), at the stations Xm are multiplied with once-for-all-computed coefficients Am, ... , F,,,, and the sums are then formed of these products (see Table 2-2).
In Table 2-3 a few results are presented that can easily be verified. Case (a) refers to a uniformly cambered skeleton line from Eq. (2-6); case (b) refers to an asymmetrically cambered line from Eq. (2-5). For the case of a simple parabolic mean camber (Xh = Z), the numerical values are
ao=-2-
CMo=-irh
cLS = 4tr h
US = 0
c
(2-78)
C
The profiles with fixed aerodynamic centers according to the discussion in Sec. 1-3-2 are obtained from the above skeleton family by setting cMo = 0. From Table 2.3, case (b), it follows immediately that b = - s . This camber line has an inflection point (S shape). The case b = 0 is again the simple parabola skeleton. Table 2-2 Coefficients A, B, C, D, E, F for the computation of the aerodynamic coefficients of Table 2-1 for N = 12 (after Riegels f49, 50] ) m
Xm
A.
B.
1
0.9830 0.9330 0.8536 0.7500 0.6294 0.5000 0.3706 0.2500 0.1465 0.0670 0.0170
0.6440 0 0.2357 0 0.1726 0 0.1726 0 0.2357 0 0.6439
-4.8919
2 3
4 .3
6 7
8 9
10 11
0
-0.5690 0
-0.2249 0
-0.1324 0
-0.0976 0
-0.0848
C. 0.6864 0.1667 0.3333 0.2887 0.2387 0.3333 0.0601 0.2887
-0.3333 0.1667
-1.8017
D.
I
-7.9370 -0.2267 -1.0790 -0.1309 -0.4210 0
-0.1402 0.1309 0.0318 0.2267 0.1197
'
E.
Fm
-2.4032
15.6333
0
-0.2357 0
-0.0462
0
2.0944 0
1.1224
0
0 1.12'24 0
0.2357
2.0944
0
0.0462
0
2.4032
0
15.6333
62 AERODYNAMICS OF THE WING
Table 2-3 Aerodynamic coefficients of uniformly and asymmetrically cambered skeleton lines (a) Skeleton from Eq. (2-6)
Coefficient Zero-lift angle
a0
Zero-moment coefficient
cM0
Angle of incidence for S.L.E.F.*
as
h
it h Xh(3 - 2Xh) 2c 1-Xh 1 h 1 - 2Xh 2 c Xh(1 -Xh) h
CLS
*
- 8 (4+3b)
1
C 1-XI,
Lift coefficient for S.L.E.F.
(b) Skeleton from Eq. (2-5)
'r
-
32a
(8+7b)
1 - -gab
2a(2+b)
1
cXh(1-Xh)
*Smooth leading-edge flow.
In the NACA systematic listing, various skeleton line shapes are used (see Sec. 2-1).
Four-digit NACA profiles In Fig. 2-22, zero-lift angles of attack and zero moments are plotted versus the maximum camber height (crest) location. Test results [1] are
also shown for comparison with theory. Because of the slight effect of profile thickness in the range of thickness ratios 0.06 < t/c < 0.15, a mean curve of experimental data is shown. The plotted bars represent three data points each for cambers h/c = 0.02, 0.04, and 0.06. The agreement of theory and experiment is
4
4
t
Z
'
C
3
2
`,,yam
3
I JL
a 0
02
11T a4
Xh--
0.6
08
02
04'
Xh---
a6
0,8
Figure 2-22 Zero-lift angle of attack as and zero-moment coefficient cMo of NACA skeleton lines. Comparison of theory and experiment from NACA Repts. 460, 537, and 610. Curve 1, four-digit skeleton lines. Curve 2, five-digit skeleton lines.
AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 63
satisfactory. As a result of friction, the deviations increase somewhat with a downstream shift of the camber crest.
Five-digit NACA profiles In Fig. 2-22, results for zero-lift angle and zero moment are presented for the skeleton lines without inflection points. Test results from [1] are also shown. The influence of the profile thickness is again negligibly small. The
plotted test data are the results for values of CLS = 0.3, 0.45, 0.6, and 0.9. Agreement between theory and experiment is better than in the case of the four-digit NACA profiles.
NACA 6-profiles The skeleton lines of the NACA 6-series have been established from purely aerodynamic considerations. Preestablished are the resultants of the pressure distributions on the lower and on the upper profile surfaces (Fig. 2-23a). The corresponding skeleton lines are presented in Fig. 2-2c. For the aerodynamic coefficients, the following expressions are established: as =
4n
1
{a
[1
-(1 - a) ln(1 - a) + 1 a- lnaJ
1
as = CIS - 2 CLS 0110 = -
1 !E q-
(2-79a)
(2-79b)
+ 4a2 +a)
(2-79c)
1
Zero-lift angles of attack and zero-moment coefficients for CLS = 1 are given in Fig. 2-23b and c versus the quantity a. These results are compared with test results of NACA Rept. 824 and show satisfactory agreement. Bent plate (flap, wing, control surface) Another valuable application of the skeleton theory is found in the calculation of the aerodynamic coefficients of the flap wing. By replacing the flap wing by a skeleton line, the bent plate, Fig. 2-24, is obtained. This problem was attacked first by Glauert [18] .
With the assumption of a small deflection angle
rj
the ordinates of the
skeleton line Z(s) = Zf, relative to an imaginary chord connecting the leading edge with the trailing edge of the deflected flap, are
(0<X<Xf)
Zf=AfX,?f
Zf=(1 -Xf)(1 -
7f
(Xf<X< 1)
(2-80)
where Af= cf/c is the flap chord ratio (see Sec. 3-1-1).
Since the profile inclinations are constant within the ranges of Eq. (2-80), integrations for the determination of aerodynamic coefficients can easily be performed. It is expedient to introduce in addition the following relationship for the position of the station:
Xf= 1 -Xf=
(1 +cos cpf) 2
(2-81)
64 AERODYNAMICS OF THE WING
The change of the zero-lift angle with flap deflection is measured relative to the fixed portion of the profile (wing, stabilizer) rather than relative to the imaginary chord. The change of angle of attack (= change of the zero-lift angle with flap deflection) is then given by a«° = - I (sin cpf + p f) rlf
2(
Xf(l - Xf) + arcsin
S
)
(2-82a)
a ,05
i1
V0.
CLS=1
a
0` 0
0.2
0.4' X_ 06
08
.10
0.8
%0
10i
O
a
0
63-series o 66-series
I
65-series 0
0 66-series
0.2
0,4'
a__
06
0,3
I
0
C
02
0.4'
a--
06
0.8
to
Figure 2-23 Aerodynamic coefficients for skeleton lines of the NACA 6-profiles at the lift coefficient of the smooth leading-edge flow CLS = 1.0. Comparison of theory [Eqs. (2-79a)(2-79c)] and experiment, after NACA Rept. 824. (a) Pressure distribution d cp; (b) Zero-lift angle ao . (c) Zero pitching-moment coefficient cMo
AIRFOIL OF INFINITE SPAN 1N INCOMPRESSIBLE FLOW (PROFILE THEORY) 65
Figure 2-24 Coordinates used in the skeleton theory of airfoils with flaps.
The term aao l arjf is frequently called the flap or control-surface efficiency, because
it is a direct measure of the lift change caused by the flap (control surface). The flap efficiency vanishes for X f = 0 and amounts to -1 for X f = 1, that is, when the whole profile is being deflected as a flap.
The change of the zero-moment coefficient (moment change at constant lift) becomes
a ° _ - 2 sin pf{1 + cos pf) _ -2 af(1 - Af)3
(2-82b)
f
The results of the above formulas are shown in Fig. 2-25. The theoretical relationship between the aerodynamic flap coefficients and the flap chord ratios is well supported by measurements. In Fig. 2-25, the test results of simple cambered flaps by Gothert [21 ] are added. The deviations are again due to friction effects.
1.0
1.0
i
08
0.8
0.8
I
06
}
e1 p-ac
ro
I
02
09
a 0I 0
0,Z
12
f
Xf= C
i
08
0.6
1D
b
O1
0
1
OZ
1
1
09
Cf
06
(18
ZO
f= C -+
Figure 2-25 Aerodynamic coefficients of a flap wing. (a) Angle of attack derivative. (b) Pitching-moment derivative. cambered flap.
(
) Theory from Eq. (2-82a, b). (---) Tests on a simple
66 AERODYNAMICS OF THE WING
The aerodynamics of flaps and control surfaces will be discussed in more detail in Sec. 8-2-1.
Computation of the velocity distribution on the skeleton line The problem of computing the distribution of circulation and consequently the velocity distribution will now be treated for a given skeleton line shape at a given angle of attack. By
introducing Eq. (2.49) into Eq. (2-46b), the equation defining the circulation distribution becomes 1
dX'
U°°dX This is an integral equation for
a - dZ(s)/dX. It
X - X,
(2-83)
0
the vortex density k with given values of
was first solved by Betz [4]. By taking into account the Kutta condition Eq. (2-51) and Eq. (2-50), the velocity distribution about the skeleton
profile (see also Fuchs [16]) is given by 1
= 1 + Ir- % (a +
U(
±'X' d$' l 1Y-X
('
1 0
(2-84)
For the case of the uncambered profile, Z(s) = 0, the already known result for the inclined flat plate is valid; see Eq. (2-65). To evaluate the quadrature formula for the velocity distribution, Riegels [49] makes the Fourier substitution n
Z(s)
_
I a cosv 9)
(2-85a)
V-1
X = f (1 + cos (P)
.(2-85b)
Introducing these expressions into Eq. (2-84) makes elementary evaluation of the integrals possible.* The velocity distribution of the skeleton profile is then U
c.
U00
an
L+, 2
cos v 99 - 1
va
sin 92
(2-86)
where the upper sign is valid for the upper side, the lower sign for the lower side of
the skeleton profile. Numerical evaluation of this equation by. means of simple quadrature formulas is treated in [49] (see also [28] ). The first term of Eq. (2-86) represents the velocity distribution of the inclined flat plate. For the parabolic skeleton Z(s)
_h
sine cp =
2 c (1 - cos 2cp)
[see Eq. (2-68)] , a1 = h/c, a2 = -h/c, a3 = a4 =
= 0. Therefore, Eq. (2-86) yields,
for a = 0, U(T)
U0
-I =1 =1-!-2u cos2q sin?
`See the footnote on page 57.
4 h-sine C
AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 67 a
result already obtained in Eq. (2-67),
taking into account Eq.
(2-50).
Velocity and pressure distributions for the skeleton lines of the four- and five-digit NACA profiles, Figs. 2-2a and 2-2b, are given in [1 ] . In Fig. 2-26 some pressure distributions are presented for the angle of smooth leading-edge flow. For the skeleton lines of the NACA 6-series, pressure distributions have been presented in Fig. 2-23a.
Pressure distribution for given lift coefficient and moment coefficient The problem of approximating a given skeleton line by superposition of an inclined flat plate and a parabolic skeleton in such a way that lift and zero-moment coefficients of
approximation and given skeleton are equal can be solved with the help of the above-introduced Fourier series expansion. In this case the Fourier coefficients from Eqs. (2-69) and (2-70) become
Ao =
102
2 CL + 22 emo 1
0..4'
0.6
and
03
Al
=-44 cMo
>.0
Figure 2-26 Theoretical pressure distribution of NACA skeleton lines from NACA Rept. 824 at smooth leading-edge flow. (a) Four-digit NACA profile with h/c = 1.0. (b) Five-digit NACA profile with CLS = 1.0.
68 AERODYNAMICS OF THE WING
coefficients
These
are introduced into Eq. (2-61), and the resultant pressure
distribution, taking into account Eq. (2-53), is obtained as dcp(X) = cLho(X) + cMo4h1 (X) with
ho(X) = z
1/i XX and hl(X) =-(l - 4X) 7r
_ X`I
U 11
(2-87) (2-88)
The distributions ho(X) and h1 (X) are shown in Fig. 2-27.
Bent plate (flap wing) Finally, the pressure distribution of the bent flat plate (flap wing) will be mentioned. For the zero-lift angle of attack, CL = 0, the result is
(8c
l =
2
77fp 10
l
In
1-
± cos('Pf
T) 97)
- 2sincpftan
(2-89) 2
J
In Fig. 2-28, the pressure distribution according to this equation is presented for the flap chord ratio Xf = 4 . The cross-hatched area represents the load on the flap, the determination of which as well as that of the flap moment (control-surface moment) will be discussed in Sec. 8-2-1.
2-4-3 Symmetric Profiles of Finite Thickness in Chord-Parallel Incident Flow (Teardrop Theory) Fundamentals of teardrop theory The term teardrop profile means a symmetric profile of finite thickness. With the method of singularities, a teardrop profile is obtained through superposition of a source-sink distribution along the profile chord with a translational flow (Fig. 2-29). Let a continuous source-sink distribution be given along the profile chord, the source strength per unit length of which is q(x). This source distribution induces the velocity component u(x) in the x direction and produces the velocity component w(x) in the z direction (Fig. 2-29). Let z(t)(x) be the equation of the upper surface of the teardrop with the coordinate origin on the
4
3
Z
I
0 ,-,'I
-1 0`
I
0.Z
04 X
08
08
9.0
Figure 2-27 The functions ha and h, for the pressure distribution on the chord at given lift and moment coefficients [Eqs. (2-87) and (2-88)].
AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 69
6
2 0
f A x i s o f t h e fl ap
2
-
t
Figure 2-28 Theoretical pressure distribution of the folded plate (flap wing) of Fig. 2-24 at zero lift [Eq. 0
all
0.6'
X --r
08
to
(2-89)] 0.25.
.
Chord ratio of flap and wing af= cf/c =
leading edge. Then, the relation between source distribution and teardrop shape is obtained easily by applying the continuity equation to the area element ABCD in Fig. 2-29, with the result
(U + u) zW + Z q dx =
(U00
+ u + du dx) (ZM +
d
(r)
dx dx)
From this the source distribution in linear approximation is obtained as {(U"
q (x) = 2 x
=2U
}d
,U)
z('))
(x)
Figure 2-29 Basic elements of teardrop theory. q(x) = source-sink distribution.
(2-90a)
(2-90b)
70 AERODYNAMICS OF THE WING
For teardrops of moderate thickness, the induced velocities u can be disregarded as compared with U., with the exception of the vicinity of the stagnation point. In the case of thin profiles it can be assumed that the velocity components on the teardrop contour are approximately equal to the values on the profile chord. In
analogy to Eq. (2-46), the components of the induced velocity on the chord are obtained as
f q (X')
it X )
dX'
(2-9 la)
0
w(X) = j 1 q(X)
(2-91b)
To obtain a closed profile contour, the total strength of the source-sink distribution must be zero (closure condition): C
f q(x)dx=0
(2-92)
xm0
In computing the source-sink distribution for a given teardrop shape z(t)(x), the closure condition is automatically fulfilled because of Eq. (2-90). With the profile chord length c, the dimensionless quantities
X=x
Z(t) =
and
z(t)
(2-93)
C
will now be introduced. The kinematic flow condition, namely, that the profile contour is a streamline, is
_
d z<<'
uw (x)
TX-
U
It can be verified immediately that this condition is identical to Eq. (2-90b).
Computation of the velocity distribution on the teardrop profile The problem of determining the velocity distribution on the profile contour may be treated first. For thin profiles, the velocity distribution on the contour is little different from
that on the chord, except for a small range on the profile nose. The velocity distribution on the chord is obtained by introducing Eq. (2-90b) into Eq. (2-91a):
U (X) = Uc 1
L
dX'
dZit)
1
1
dX
;
(2-94)
0
According to Riegels [49], the velocity distribution on the contour is then established by division with
x(X) _
l
/ dTO 2
dX
)
(2-95)
AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 71
Here, l/;-.(x) is called the Riegels factor. Since l/u(x) is zero at the leading edge, it
is assured that the velocity goes to zero at the front stagnation point, and the velocity distribution on the contour is finally TWc(X)
U"
- x(X) 1
i
dZ"' dX' 1 + rc j dX' X - X' 1
(2-96)
0
In this way, the computation of the velocity distribution for a given teardrop profile has been reduced to a quadrature formula. By disregarding the Riegels factor, the velocity distribution on a simple biconvex parabola profile becomes W
C -1+ b[i 7-
U co
; (1 -2X)In
(2-97)
Here, 6 is the thickness ratio t/c. The highest velocity occurs at X =1 with the value u.../U. = (4/rr)& Likewise, the velocity distributions for the extended parabola profiles from Eq. (2-6) can be computed (see Truckenbrodt [49] ). For the evaluation of the quadrature formula, Riegels [49] introduces the Fourier series ,Z
X = j(1 + cosrp)
Z(i) = I- 16, sine 99 V-1
(2-98)
in analogy to Eq. (2-85). When this expression for Z(t) is inserted into Eq. (2-96), the velocity distribution on the contour assumes the simple form WC (m) U00
_
1
1
Y. ((P)
,Z
+v-1 v
by sinv sin
(2-99) J
The numerical evaluation of the equation by means of simple quadrature formulas is
treated in [49]. From Eq. (242), the contour of the thin symmetric Joukowsky profile* is given by
Z(t) =
2
= 2e
sin p(1 -cos gyp)
(2-100a)
X(1 - X)3
(2-100b)
where
3 4
t
- 0. 77
5
an d
xt =
1
4
The Fourier coefficients are bl = E, bz = -c/2, and b3 = b4 = the velocity distribution on the contour is given by Note that /c = X - s = z cos p and rl =Z().
= 0. Consequently,
72 AERODYNAMICS OF THE WING We Uoo
1+s(1-2cosq,)
+
V'-
cos 92 - cos 2 T
(2-101)
sing,
(
Figure 2-30 shows the velocity distributions computed by Eq. (2-101) for various
thickness ratios. Within the accuracy of the plot, complete agreement of the approximate and the exact solutions by conformal mapping is obtained. See Fig. 2-16 for CL = 0. Because the trailing-edge angle of the Joukowsky profiles is zero, the velocity distribution has no rear stagnation point on the trailing edge. For the four- and five-digit NACA profiles from Fig. 2-2a, the theoretical velocity distributions may be found in [1] . A few distributions at various thickness ratios S = t/c are shown in Fig. 2-31. They were computed by the procedure of Theodorsen and Garrick [66]. Values computed by the singularities method deviate only slightly from them. The teardrop shapes of the NACA 6-series, Fig. 2-2c, were established from given velocity distributions, which were determined mainly by the location of the maximum velocity. In Fig. 2-32, the velocity distributions are shown for the four profiles of Fig. 2-33 with a thickness ratio 6 = 0.12. Figure 2-33 gives a comparison between theoretical and experimental pressure distributions on the NACA profile NACA 0010 and shows good agreement.
The maximum velocity on a profile is of considerable importance for the D=r-OZO C 0.95
0>0
U
005
08
0.6
Figure 2-30 Velocity distribution on the profile contour for symmetric Joukowsky profiles of various thickness ratios tic in chord-parallel flow.
Figure 2-31 Velocity distribution on the profile contour for four- and five-digit symmetric NACA profiles in smooth leading-edge flow. (NACA Rept. 824.)
AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 73
T-
12
63 -012
1.1
65-012
0.9
0,60
02
0.
X---
as
O.5
20
Figure 2-32 Velocity distribution on the profile contour for the NACA 6-profiles in chord- parallel flow. Profile contours from Fig. 2-33. Thickness ratio tic = 0.12.
behavior of the profile at high subsonic velocities (critical Mach number, Sec. 4-3-2).
In Fig. 2-34, the ratio of the maximum velocity difference to the constant translational velocity is given against the thickness ratio of most of the teardrop profiles discussed above. Accordingly, the maximum velocity depends heavily on the
thickness distribution for an otherwise unchanged thickness ratio. The elliptic profile produces the smallest velocity difference, the Joukowsky profile the largest. 1,0
i
o Upper surfac e u
0,6
Lower surfac e I
-0.2
Theory
-as
cc=00 -10
I
'0
U.
ti
0,2
041
X
06'
C _u
Profile NACA 0010
as
1,0
Figure 2.33 Comparison of theoretical and experimental pressure distributions for the symmetric profile NACA 0010 in chordparallel flow.
74 AERODYNAMICS OF THE WING
I
03 J
Joukow sky profile
NACA profiles 00-
63-
65-
64-
65
of
E lliptic p rofile
0
0.05
Qf t
015
02
C
Figure 2-34 Maximum perturbation velocity Umax of teardrop profiles in chord-parallel flow vs. thickness ratio S = t/c.
Computation of the teardrop profile from a given velocity distribution In analogy to the Birnbaum-Glauert series expansion for the distribution of circulation in the case of skeleton theory [Eq. (2-61)], the source distribution will now be represented in the form of a trigonometric series (see, e.g., Allen [3] ):
± r B sinn
q(,p) = 2 U. (B0 tan
n-1
(2-102)
The relation between x and co is given in Eq. (2-62). The closure condition for the profile contour, Eq. (2-92), is satisfied when
2Bo + B1 =0
(2-103)
By introducing Eq. (2-102) into Eq. (2-91a), the induced velocity in the x direction is obtained as u(9) co
N
= B0± ,b-1
(2-104)
The profile contour is determined by introducing Eq. (2-102) into Eq. (2-90b) and integrating along X: Z
I
Bo sin (p(1 - cosrp) -
- 6 B3(2sin2g9 -
1
12
sin4q,)
B`(3sinT - sin39 ) (2-105)
The first term represents the Joukowsky profile, as can be verified by comparison with Eq. (2-100). Profile shape and velocity distribution are interrelated
by Eq. (2-96), which must now be interpreted as the integral equation for the profile inclination dZ(t) fdX. Following Betz [4] and Fuchs (16], the solution is
AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 75
f
1
___ d0t)
1
dX
it, (X')
U0 Co
X'(1-X')
d11'
X(1-X) X- 1'
(2-106)
where
. =x
U".
We Uro
-1
denotes the induced velocity distribution on the chord, and r is defined in Eq. (2-95). Since the needed profile inclination is a term of this equation, Eq. (2-106) can be solved only through iterations (see Truckenbrodt [68] ). The publications by Eppler [13] and Strand [63] on this subject should also be mentioned. The simplest case of a constant induced velocity u on the chord leads to an elliptic teardrop profile. For a linear distribution of the induced velocity u/U., u
Z') =
(2-107a)
U=c(1-bXU00
[4 - b(1 + 2X)] X(1 - X)
(2-107b)
For b = 0 the profile is elliptic; for b = 3 , the Joukowsky profile [Eq. (2-100)] is obtained.
2-4-4 Inclined Profile of Finite Thickness Computation of the velocity distribution on the profile contour The general case of a cambered profile of finite thickness will now be treated, after the case of the very thin profile (skeleton) discussed in Sec. 2-4-2 and the case of the symmetric profile of finite thickness in chord-parallel flow discussed in Sec. 2-4-3. The general case is obtained essentially by superposition of these two previously discussed cases.
A cambered profile of finite thickness can be composed of a skeleton line Z(S) = z(S)/c and a teardrop profile Z(t) = z(t)lc (Fig. 2-1). In Eq. (2-1), the profile ordinates are given as Zu,1= Z(S) ± Z(t)
(2-108)
where the upper sign is valid on the upper surface, the lower sign on the lower surface. With Z, the ordinates on the upper surface, and Z1, those on the lower surface, Eq. (2-108) can be broken down into Z(S)
= (Zu + Z1)
and
Z(t) = z (Zu - Z1)
(2-109)
z
The velocity distribution of the general profile is the sum of those of skeleton and teardrop. A third distribution has to be added, however, which is produced through the inclination of the teardrop profile. Riegels [49] computed this contribution. The velocity contribution caused by the inclination of the teardrop
profile may be interpreted as the influence of an additional camber and an
76 AERODYNAMICS OF THE WING
additional angle of attack, as was shown in [68]. Accordingly, the following contributions have to be added to the geometric skeleton (mean camber) line Z(s): dZ(s)
= a [1/_ X x Z(1) +
da
=- 2 a `r d0i) d9;
2(1
- X) (
c')
d92 )
-v-o ]
(2 - 11O a)
(2 - 11 Ob )
1_0
The equations for the computation of velocity distribution and aerodynamic coefficients that were derived by the method of singularities in the cases of skeleton and teardrop profiles can be evaluated conveniently through numerical quadrature formulas. Details for the computations are found in [49]. The result of a sample computation from the above outlined method for the computation of velocity distributions on profiles is presented in Fig. 2-35 for the profile NACA 66 (215)-216, a = 0.6. A theoretical velocity distribution from Theodorsen and Garrick [66] (conformal mapping) is also shown, and for comparison with the theory, a measurement from [1] is added. The agreement of the two theoretical curves with the test data is good. Note the appreciable agreement of these pressure distributions for the inclined symmetric Joukowsky profile, obtained by the method of singularities, with those of Fig. 2-16, computed by the method of conformal mapping. Lift and pitching moment The computation of the aerodynamic coefficients for the skeleton profile was discussed in Sec. 24-2. The results in that section are also valid for the inclined profile of finite thickness if the influence that is introduced by the
inclination of the teardrop profile in Eq. (2-110) is taken into account. The resulting relationships for lift slope and neutral-point location are given in Table 2-1.
The other aerodynamic coefficients (zero-lift angle of attack, zero-moment coefficient, angle of attack, and lift coefficient of the smooth leading-edge flow) are equal for the profile of finite thickness and the skeleton profile. For the Joukowsky profile in Eq. (2-100) the lift slope is dCL
da
gN
2_x
dcL L
I a) = ,.on (i -...0. j f _V) i
=4
.
(2-111a)
(2 - 111 b)
in agreement with the solution from the method of conformal mapping, Eq. (2-33).
For the NACA profiles of Fig. 2-2, these formulas show that the lift slope always increases with thickness, whereas the neutral point lies behind the c14 station of the profile. The test data of the lift slope from [1 ] lie in all cases below the theoretical values. When the profile thickness increases, the lift slope of the older
NACA 00-series with a relatively large trailing-edge angle decreases (Fig. 2-2b), whereas it shows the theoretical trend in the newer NACA 6-series with a smaller trailing-edge angle (Fig. 2-2c). In all cases, the lift slope is smaller when the surface
AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 77 1.
o O
CL-O.23
°
O
a - toy o
U pper surface I
1.
o/
1
Lower surface
\O 0
Theory of Theodorsen
---Theory of Riegels C
Measurements at Re = 6
10 6
a
0.2
0.41
as
08
Lower surface
Figure 2-35 Velocity distribution on the profile contour of profile NACA 66 (215)-216, a = 0.6. Lift coefficient cL = 0.23. Comparison of theory and experiment.
is rough than when it is smooth. This behavior shall be taken up again in Sec. 2-5.
A similar comparison shows disagreement of measurement and theory for the neutral-point location of the older NACA series.
Using a procedure by Martensen [28], Jacob [28] extends the singularities method by arranging vortices on the profile contour instead of the profile chord. The investigation of Lan [37] and the comments by Maskew and Woodward [40] should be mentioned. Furthermore, profile theories of higher order for incompressible flow are found in, among others, Keune [22, 32] and Lighthill [39]. An essential contribution to the theoretical and experimental investigations of fluid mechanical behavior behind blunt profiles has been made by Tanner [65]. Base pressure and base drag play an important role in this case.
78 AERODYNAMICS OF THE WING
2-4-5 Special Problems of Profile Theory Airfoil in curved flow So far, it had been assumed in all considerations of profile
theory that the wing moves in straight incident flow. When investigating the interference of the airplane parts with each other, the case is encountered, however, where the wing lies in a curved incident flow field. The aerodynamic problems of such a wing can also be treated with the skeleton theory. A flat plate in curved flow behaves approximately as a cambered skeleton in parallel flow. The variable angles of attack a '(X) along a profile in curved flow can
be replaced by changed equivalent skeleton line inclinations (a-dZ(S)/dX), as shown in Fig. 2-36. With this procedure in mind, dZ(S>
a - a(X) dX =
must be substituted in the formulas of the skeleton theory (Sec. 24-2). By using the expressions of Table 2-1, the mean angle of attack a is obtained as n
-fa'(g7)(1+cosg7)dq7
(2-112a)
0
This is, to express it again in a somewhat different way, the angle of attack that produces in straight flow the same lift CL = 27ra as the variable angle-of-attack distribution. The zero-moment coefficient becomes then n
cJI0 = -' f a' ((p) (cosg7 + cos2g7) d 97
(2-112b)
0
At a constant angle-of-attack distribution a'(X) = a, the above equations produce the relationships for the inclined plate: cz = a, cM0 = 0. Approximate expressions for a and CMo have been derived by Pistolesi [45] and Multhopp (see Chap. 5 [40]), respectively. Using only local a' values on a few characteristic stations of the chord, they arrived at the expressions a = a;;
--- ai
a
(2-113a)
,Plate
a'j
b
o
X
Figure 2-36 Airfoil in curved flow-sche-
matic explanation. (a) Variation of a' on c
plate. (b) Angle of incidence distribution
a'(x). (c) Skeleton profile Z(S).
AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 79
Figure 2-37 Distribution of the induced downwash angle on the extended wing chord for the inclined flat plate.
z
C:ft o = 16 (o
+ 2 c 0 - 3 ai ors )
(2-113b)
where 4o, a'so, a'7s, and ale mean the local angles of attack a' at the stations X= 0, 0.5, 0.75, and 1.0, respectively. Hence, the value of the angle of incidence at the a -chord station should determine the lift when the angle-of-incidence distribution is variable. Compare the formulas of Weissinger [73],
Theorem of Pistolesi The two approximation formulas of Pistolesi and Multhopp are of particular importance when the continuously distributed circulation on the wing chord is replaced by a single vortex at the 4 -chord station, as is customary for simplified treatment of the wing of finite span (lifting-line theory, Sec. 3-34). The Pistolesi approximation for the mean angle of incidence is in agreement with the exact solution, as seen in Fig. 2-37. The Multhopp approximation for the zero
moment produces the right value cMo = 0 for the flat plate, because a' = -a' and aso = 3aiDo when the plate is replaced by a single vortex at the i -chord station.
The velocity near-field of the profile So far, the velocity distributions have always been determined on the profile surface. For certain aerodynamic problems of
airplanes, for example, for investigations into the influence of the wing on the incident flow of the fuselage or horizontal stabilizer, knowledge of the velocity field off the wing is required. This matter will now be discussed to some extent. First, let the wing be replaced by its skeleton profile; that is, it is represented
by its vortex distribution, Eq. (2-44). Of main interest are the z components w of the velocities because they produce a change of the effective angles of incidence on the airplane components that lie before and behind the wing. The induced velocity component w will be considered only in the wing plane, that is, on the x axis.
80 AERODYNAMICS OF THE WING
The distributions of the induced upwash and downwash velocities along the x axis are given by Eq. (2-46b):
f
1
1
TV (X) _ - 2n
X-(XX,
dX'
(2-114)
0
Here k(x) is the circulation distribution along the profile chord and X = x/c, the dimensionless coordinate in direction of the profile chord. The total circulation of the wing 1' is found from the circulation distribution by integration [see Eq. (2-52)]. Evaluation of Eq. (2-114) over the profile chord, 0 < X< 1, was described earlier using the Birnbaum-Glauert substitution for the circulation distribution [Eqs. (2-61) and (2-63)]. Equation (2-114) can also be used to compute the induced downwash velocity before and behind the profile. A singular point of the integrand no longer exists in this case, however. The previous substitution of Eq. (2-62) now must be replaced by
X = (1 ± cosh 99)
(2-115a)
X' _ (1 + Cos (p')
(2-115b)
The lower sign applies to points before the wing and the upper sign to those behind the wing. By introducing Eqs. (2-115) and (2-61) into Eq. (2-114) and integrating, the downwash angle distribution aw = w/UU is obtained as
aw(X) = -A0 (1 - V 1) +r2' An [1 - 2X (1 X
-
gX
(2-116)
forX>1 andX<0.* In the circulation distribution Eq. (2-61), the term with A0 represents the flat plate inclined by the angle a, with A0 = a. The term with A 1 represents the parabola skeleton with camber h/c in a flow parallel to the chord with A 1 = 4h/c. For the first case, the distribution of the downwash angles along the profile chord is
plotted in Fig. 2-37. For comparison is shown the induced downwash angle distribution that is obtained when the wing is substituted by a single vortex of total circulation 1', located in the center of action XA of the distributed circulation, that is, at XA = a of the inclined plate. At large distances before and behind the wing, the distributions of the induced downwash angles of the continuous vortex distribution (lifting wing) and the single vortex (lifting line) are in agreement. It is noteworthy that in the case of the inclined plate, the induced downwash velocities at station X = a are equal for the lifting surface and for the lifting line (Pistolesi theorem).
In the case of flow around thick profiles, the main interest is directed toward the induced velocity u in the x direction before and behind the profile. From Eq. *Note that, according to Jaeckel [301, the following relation applies: n 1
f 0
cosn cp'
± cosh 97 - cos T'
d
,
-(
1)
n+i coshn cp - sinhn rp sink T
AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 81
U.
2-38 Distribution of induced longitudinal velocity on the Figure
extended wing chord for a symmetric Joukowsky profile (6 = tic = thickness ratio).
(2-9la), this component is obtained by introducing the source distribution from Eq. (2-102). The evaluation is analogous to that of w(x) and produces for u(x)/U00 an equation analogous to Eq. (2-116), where the coefficients A7z of the circulation distribution must be replaced by -Bn of the source distribution. The latter satisfy the closure condition Eq. (2-103). In Fig. 2-38, evaluation of such a computation is shown for a symmetric Joukowsky profile. Although it is not possible in this book to cover the problems of unsteady flow that are of importance to airplane aerodynamics, the fundamental study of Wagner [72] on unsteady wing lift in plane flow should not be overlooked.
2-5 INFLUENCE OF VISCOSITY AND BOUNDARY-LAYER CONTROL ON PROFILE CHARACTERISTICS* So far, all the discussions of this chapter have been based on the assumption of inviscid flow of an incompressible fluid. Now, a few data will be given on the effect
of viscosity and the control of the boundary layer close to the wall. The effect of compressibility on the aerodynamic coefficients of a wing profile will be treated in detail in Chapter 4.
2-5-1 Effect of Reynolds Number on Lift The most important quantity characterizing viscosity effects is the Reynolds number [Eq. (1-17)]. For a given profile geometry, this nondimensional quantity determines decisively the aerodynamic coefficients of a wing. The great importance of the Reynolds number as well as of turbulence on the
profile performance is demonstrated in the summary report of Schlichting [52]. *The authors are indebted to K. O. Arnold, who contributed considerably to this section in the original German version of the book.
82 AERODYNAMICS OF THE WING
Investigations on wing profiles in the critical Reynolds number range are reported by Kraemer [34]. First, the influence of the Reynolds number on the lift and its interplay with the geometric profile parameters will be discussed. Then, some information on the profile drag will be given that, as was pointed out earlier, cannot be determined with the theory of inviscid fluids. The Reynolds numbers of the wings that are of interest in modern aeronautics are of the order of Re = U.c/v = 106 to 5 - 107, except for model airplanes and
certain glider planes for which they lie between 105 106 , but even down to Re = 105 , the lift as computed from potential theory is in satisfactory agreement with experimental results when the angle of attack is small to moderately large. This fact can be seen, for example, in Fig. 2-10 for the inclined flat plate and for the profile Go 445 at
Re = 4 - 105, and in Fig. 2-17 for the Joukowsky profile at Re = 105. In these cases the flow is attached to the wing; that is, no boundary-layer separation occurs. Likewise, the pressure distributions on the profile, determined from potential theory, agree well with experiments in this range of angles of attack and Reynolds numbers; see Fig. 2-18 for a Joukowsky profile, Fig. 2-33 for a symmetric NACA profile at
zero angle of attack, and Fig. 2-35 for a cambered NACA profile with angle of attack.
Lift slope For the profile NACA 2412, Fig. 2-39 gives the lift coefficient CL against
the angle of attack a from Jacobs and Sherman [29]. Figure 2-39a shows that for the range from Re = 8 - 104 to 3 - 106 , no important effect of Reynolds number on the lift can be expected as long as the profile is not too much inclined (a< 80). Since the cL(a) curve is linear in this a range, the Reynolds number influence can be described simply by the lift slope dcL/da. This kind of presentation is used in Fig. 2-40 for a few four- and five-digit NACA profiles. These measurements show a slight increase of the lift slope dcL/da with the Reynolds number for Re < 3 - 106; beyond this Reynolds number, up to Re = 10', practically no change occurs.
In addition, the lift slope depends on both the profile thickness and the trailing-edge angle. It decreases with increasing thickness ratio t/c in the four- and five-digit NACA profiles, whereas the opposite behavior is found in the NACA 6series, namely, an increase of dcL/da with increasing thickness ratio tic. Conversely, increasing the trailing-edge angle always results in a reduction of the lift slope. The quotient x = (dcL /da)exp /(dcL /da)theor is plotted in Fig. 241 as a function of the trailing-edge serniangle T (see Fig. 2-1 b). The quotient % goes to 1 when the trailing-edge angle approaches zero (r = 0). When r increases, the quotient x declines to about 0.8 for smooth surfaces, and to 0.7 for rough surfaces (see also Hoerner and Borst [251). The deviations of the measured lift slopes from the theoretical values are caused by the boundary layer and the wake near the trailing edge.
The difference in boundary-layer thickness on the upper and the lower profile
surfaces-thicker above, thinner below-is equivalent to an additional negative
AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 83
18
1.6
Profile NACA 2412
ZA
Re-8.3.10`` -3.3.10 5 a -6.6.105 o -9.3.106 v -3.1 105
0.4
o
0.2
0
-02 -04
F
-8° a
IT
-4°
00
40
15°
12°
1q0
2400
-RO°
a
001
0.02
0.03
0.04
0.05 0.06
cpp
b
Figure 2-39 Lift and drag measurements on the NACA 2412 profile at various Reynolds numbers. (a) Lift coefficient cL vs. angle of attack a. (b) Polar curves CL vs. CDp.
camber; compare also Pinkerton [44]. The boundary layers change the Kutta condition, too, in that the rear stagnation point shifts from the trailing edge to the profile upper surface.
At extremely small profile Reynolds numbers, Re < 10$ , such as occur in free-flight airplane models (see Schmitz [57] ), often no linear relationship exists between lift coefficient CL and angle of attack a, even for very small angles of attack. In this case the measured CL values deviate strongly from theory over the whole angle-of-attack range, because the flow is widely separated from the profile. Conversely, at larger Reynolds numbers, Re > 10', the separation that is caused by 8 NACA 0072 0
n
4412
o
n
4415
o
n
I
f
I
23012
23015
Y
I
.5
Figure 2-40 Reynolds number in-
fluence on lift slope dcL/da for
4 5
5
8
10 6
2
3
Re ---
4
S
6
107
four- and five-digit NACA profiles with smooth surfaces.
84 AERODYNAMICS OF THE WING 1.
. X4 C4 00
a4
0
63
n
,,
o
"
0.04
64 65 66
008
012
Q16
020
024
tan r -,
W
10
Figure 2-41 Comparison of the lift b 04
t
0
004
008
j
a1z
ale
tan ,r -
E
020
024
slope from theory and experiment for NACA profiles of various trailing-edge
where x = (dCL/da)exp/ (dcL1da)theor.'(a) Smooth surface. (b) angles
028
27-,
Rough surface.
a steep pressure rise on the suction side of the profile occurs only at larger angles of incidence, a = 5-200, depending on the profile shape. The lower value of a is valid
for thin profiles. As soon as local separation occurs on the wing, the lift slope decreases. The deviation from the linear characteristic of the theory grows larger with an extention of the range of separated flow on the profile until finally, at large a, the flow is almost entirely separated on the suction side, and the lift drops, as demonstrated in Fig. 2-1 lb. The phenomenon of separation from the wing, which is to be discussed later in detail, has a decisive effect on the maximum lift coefficient CL max- This coefficient is of great aeronautical importance (in take-off and la nding).
Maximum lift The aerodynamic problems of maximum lift are summarized by, among others, Nonweiler [43], Schlichting [54], and Smith [60]. The maximum lift of a profile depends decisively on the flow conditions in the boundary layer on the suction side. At very small Reynolds numbers, the boundary layer is completely laminar and separation occurs near the profile nose (leading-edge stall) because of the strong pressure rise on the suction side immediately downstream of the leading edge. The location of the separation point is almost independent of the Reynolds number. The maximum lift is, therefore, independent of the Reynolds number in this range. Only at a certain larger Reynolds number, the value of which depends on the profile geometry, do the flow characteristics change. The laminar boundary layer still separates; transition to turbulent flow now takes place in the separated flow, however, leading, in general, to reattachment of the turbulent boundary layer farther downstream. In this way, a laminar separation bubble forms between the points of laminar separation and turbulent reattachment. The reattachment point moves upstream with increasing Reynolds number until it finally reaches the
separation point, that is, until the length of the separation bubble becomes zero. The maximum lift increases strongly with Reynolds number as a result of the
AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 85
superposition of two effects: In the first place, lift is gained at a fixed angle of attack because of the reduction of the separation length, and then the wing can be set at a higher angle of attack before the flow ultimately separates. At very high Reynolds numbers, a natural transition from the laminar to the
turbulent boundary layer occurs before the point of laminar separation. This transition point travels upstream with increasing Reynolds number and the length of the turbulent boundary layer, and consequently the boundary-layer thickness increases. As a result of this process, the circulation around the profile is diminished; that is, the maximum lift may again decrease somewhat at high Reynolds numbers. As an example for the CLmax behavior, the maximum lift of profiles of the NACA 6-series is plotted in Fig. 2-42 against Reynolds number for various thickness ratios t/c and camber heights h/c according to Loftin and Smith [29] ; see also Fig. 2-39. In the range Re > 106 of interest to aeronautics, profiles of moderate thickness (t/c - 0.12) produce the largest lift. The influence of camber is reflected in an increase of CLmax with h/c because the critical, separation promoting pressure rise on the profile suction side is occurring at larger angles of attack for increased h/c. The most important geometric parameter affecting
separation at large angles of attack, and thus affecting the maximum lift, is the shape of the profile nose, because this shape determines decisively the pressure distribution in the vicinity of the leading edge. The measurements by Nonweiler [43] of Fig. 2-43 convey some insight into these relationships through curves that show CLmax values for a fixed Reynolds number (Re = 6
106) as a function of the
thickness ratio t/c. The nose radius is characterized by the profile ordinate z1 at x = 0.05c. Accordingly, the nose radius has no effect on CLmax for very thin profiles, whereas for profiles of moderate thickness, CLmax increases considerably with zi It.
A similar parameter, namely, the ordinate zo Ic of the profile suction side at station x/c = 0.0125, has been used by Gault [77]. It allows delineation of ranges of the various separation processes as a function of Reynolds number in a universal diagram. This presentation, Fig. 2-44, is based on measurements on about 150 1.B
1.6
'
AA CA 64 -409 64,-41Z o 64z 415
Ii
o
I
(
1
#A CA 64r - 012 I
I
n
j
o
64, A 272
o
64,-412
(
i
Re--
6 1075 6
9 106
2
3
4 55 810'
Re b Figure 2-42 Maximum lift coefficient of profiles of the NACA 6-series vs. Reynolds number. (a)
a
Effect of thickness ratio. (b) Effect of camber ratio.
86 AERODYNAMICS OF THE WING 18
_Z t
! l
- 06S
r
0,6
1.6
QSS 0.50 Qcf3
0,35
1.0
030
t 0.8
020
0.10
06
009
0
008
t 212
016
0.20
Figure 2-43 Maximum lift coefficient at Reynolds number Re = 6 - 106 vs. thickness ratio tic and nose radius in terms of z, It. z, = z (x/c = 0.05). After [43].
02'9
S
profiles with smooth surfaces at low wind-tunnel turbulence. It shows that profiles with sharp leading edges, or with very small nose radii (zo/c < 0.009), have, at all Reynolds numbers, a specific separation characteristic that is termed thin-airfoil stall. Even at small angles of attack a, separation of the flow over the thin leading edge occurs directly at the profile nose, followed by reattachment. The velocity
profile of the boundary layer at the point of reattachment is neither typically laminar nor typically turbulent. Not before the boundary layer approaches the trailing edge is a fully turbulent flow pattern established (see McCullough and Gault [77] ). Reattachment occurs more and more downstream when the angle of attack
increases, leading to a growing separation range and consequently a gradually diminishing lift slope. As soon as the flow is detached on the whole suction side, CL decreases continuously with increasing a (see also Young and Squire [77] ). 0.036 0.032 2
0.028
0.024
0.020
0.076
Figure 2-44 Separation from profiles vs. Reynolds number and nose radius [in terms of zo /c, with z0 = z (x/c =
0.072
0.0125)], after [77]. (1) Separation from a thin profile. (2) Laminar separation from profile nose. (3) Combination of laminar and turbu-
0.008
0.004 I
I
0 4
6
8 70 6
2
4 Re
6
8 707
Z
4
lent separation. (4) Turbulent separation.
AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 87
A basically different separation characteristic is found on wings of moderate thickness having a moderately large nose radius, the leading-edge curvature of which
is, however, still relatively large.* The steep pressure rise behind the profile nose then leads to separation of the laminar boundary layer at larger angles of attack. The transition to turbulent flow takes place, however, in the separated flow that results in reattachment- farther downstream. A laminar separation bubble is formed, the extent of which decreases with increasing angle of attack because the transition point, and with it the turbulent reattachment point, move closer to the separation point, which moves likewise toward the leading edge. Eventually, the laminar boundary layer separates on the very nose where the contour curvature is too large for the transition to cause reattachment. This process, known as the leading-edge stall, is characterized by a sudden sharp lift drop (see Crabtree (11] and Tani [64] ).
On the other hand, on most thick profiles (tlc > 0.15), that is, at large nose radii, flow reattachment occurs behind the laminar separation point, even at large angles of attack. In this case the maximum lift is determined by two processes that influence each other. These are the expansion of the laminar separation bubble from
the nose, and the turbulent separation that starts at the trailing edge and moves upstream with increasing angle of attack (combined leading-edge and trailing-edge stall). The variation of the lift cL(a) depends on the predominance of one or the
other of these two separation processes. The separation bubble may disappear entirely on very thick, strongly cambered profiles and at very high Reynolds numbers. The reason for this is that the Reynolds number is then large enough for a natural transition to turbulent flow upstream of the station of strong pressure rise.
The turbulent boundary layer separates only a short distance upstream of the trailing edge (trailing-edge stall). This separation point moves upstream continuously
with growing angle of attack, and the lift does not drop abruptly after passing CLmax but very gradually, similarly to the case of the thin profile. The profile shape of optimum lift coefficient at flow without separation can be computed following a procedure of Liebeck [381.
Pressure distribution In Fig. 2-45, pressure distributions on profiles of the NACA 6-
series are presented in the range of the maximum lift at a Reynolds number Re = 5.8 106 according to McCullough and Gault [77]. Separation from thin profiles (NACA 64A006) is characterized by a very slight underpressure near the leading edge. This underpressure is even reduced with an a increase, whereas the separation range (cP = const) grows from the profile nose downstream. Conversely,
very strong suction peak exists on profiles of larger thickness ratio for a< acL max The laminar separation bubble is too short to be noticeable in the a
pressure distribution, if it exists at all. The NACA 631-012 profile causes laminar separation at the nose, resulting in an abrupt collapse of the high underpressure on
*Translator's note: Remember that the term "nose radius" does not necessarily imply a circular nose. The definition of nose radius is of the kind found in Figs. 2-43 and 244. The curvature can, therefore, be relatively large locally on the nose, even if the radius in the above sense is not small.
88 AERODYNAMICS OF THE WING
245 Measured pressure distribution at Reynolds number Re = 5.8 106 on profiles of NACA 6-series with various separation characteristics in the range of maximum lift. M. Separation from thin profile. (2) Laminar separation from Figure
0.2
0.6
0.4.
x/c
0.8
10
profile nose. (3) Turbulent separation.
the leading edge and an immediate flow separation over the entire suction side. This
in turn results in the steep lift drop when the angle of attack for CLmax is exceeded. As soon as turbulent separation has been established, as is the case on the NACA 633-018 profile, the suction peak at the leading edge remains, even when a is larger than acLmax The separated range expands from the trailing edge farther and farther upstream, causing the lift to decrease continuously. The separation characteristics of a given profile may be different for the various Reynolds numbers, as shown in Fig. 2-46 for the example of the pressure
distribution on the profile NACA 4412 at the angle of attack c= 16°
(see
Pinkerton [29] ). For Re = 1 - 105 and 4.5 - 105 , the pressure distribution is similar to that of the profile NACA 64A006 (Fig. 245); that is, separation has the same character on thin profiles, although only at larger angles of attack. The separated range decreases with increasing Reynolds number in this case. According to Fig.
2-44, for the thin profile at Re < 106, there are only two possibilities, namely, laminar separation or turbulent separation near the trailing edge. Transition from one behavior to the other requires that the profile is made thicker when the Reynolds number is reduced. When the Reynolds number is raised to 1.8 - 106, a laminar separation bubble 0.005c long forms on the NACA 4412 profile, and at
x/c = 0.40 turbulent separation sets in. Finally, at Re = 8.2 106 , the flow is attached over the whole profile. A further increase in Reynolds number has practically no influence on the pressure distribution, which agrees quite well with
theory as long as no separation occurs (see Cooke and Brebner [10] ). Note,
AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 89
Profile NACA 4412
+I
Re - 1 105;cL-0.98 o o
-4
o
---
-4.5.105;
- 1.15
6.106 ;
- 136
- 8.2.106;
- 1.67
-
1.
Theory
-2
.
-1
0
1
0
0,2
0.4
0.5
x/c --
0. B
1.0
Figure 2-46 Effect of Reynolds number on pressure distribution on profile NACA 4412 at a large angle of attack (a= 16°).
however, that the theoretical curve of Fig. 2-46 is obtained from a modified theory from Pinkerton [44] and not from pure potential theory. The influence of the boundary layer on the pressure distribution on a profile as
a function of angle of attack is presented in Fig. 2-47. Figure 2-47a gives the distribution at moderate angles, which is susceptible of computation after the
a
b
c
Figure 2-47 Change of pressure distribution on a wing profile with angle of attack [671. (a) Attached flow, medium angle of attack. (b) Beginning of separation from trailing edge CL = CL max- (c) Separation from leading edge with enclosed vortex (bubble).
90 AERODYNAMICS OF THE WING
methods of potential theory. At larger angles of attack, separation sets in first on the upper surface of the profile near the trailing edge (Fig. 2-47b). From there, it travels upstream with increasing angle of attack. At the same time a wake forms in which a vortex (bubble) is embedded. At very large angles of attack, beyond the maximum of the lift coefficient, the wake shifts upstream to the wing nose (Fig. 247c). The flow reattaches again further downstream.
A comprehensive listing of experimental data on the lift problem is found in Hoerner and Borst [25]. Based on studies of Preston [61], Spence [61] makes some recommendations about the theoretical inclusion of the friction effect into the aerodynamics of the wing profile. Theoretical determination of the pressure distribution for separated incompressible flow about profiles of almost any shape is possible using a computational method of Jacob [271, but the abrupt leading edge separation and reattachment of the flow cannot be obtained directly by this method.
2-5-2 Effect of Reynolds Number on Drag When the lift coefficient is small, the profile drag is caused essentially by friction. Its value depends on the position of the transition point and hence the lengths of laminar and turbulent stretches. The local velocities increase with angle of attack, leading to a slight rise of the profile-drag coefficient CDp. A further contributing factor is the increasing length of the turbulent boundary layer with a simultaneous shrinking of the length of the laminar layer. In the CLmax range, the profile drag rises
steeply because of the strong increase in pressure drag caused by local
separation. The Reynolds number has a very strong influence on the magnitude of the profile drag because both the pressure drag and the friction drag decrease with increasing Reynolds number (see Fig. 2-39b). The dependence of the minimum drag coefficient CDmin on the Reynolds number [29] is plotted in Fig. 249 for several four-digit NACA profiles. Laminar separation causes quite high values of the minimum profile drag CDmin for small Reynolds numbers (Re < 5 - 105 ). Symmetric profiles produce minimum drag at CL = 0, cambered profiles at the angle of smooth leading-edge flow. The value of CDmin decreases strongly when the Reynolds number grows. As soon as fully attached flow is established, the trend of the CDmin curve is similar to that of the friction drag of the flat plate (see Fig. 2-48). In this range of Reynolds numbers (Re > 8 - 105), the minimum drag coefficient is raised more and more above the value of friction drag when the profile thickness grows (Fig. 2-49a). The same behavior is found for the camber (Fig. 2-49b). Peculiarities of the drag appear at laminar profiles (see Wortmann [75] ). As an example, three-component measurements on the NACA 662-415 profile are plotted in Fig. 2-50 for various Reynolds numbers (after [29] ). Over a limited range of
small lift coefficients, the profile drag is constant, independent of the angle of attack. It is lower than that of a normal profile if the Reynolds number is large enough to prevent laminar separation. When the Reynolds number grows, CDp decreases; at the same time the dip in the drag curve, that is, the lift range for
91
92 AERODYNAMICS OF THE WING
e NACA 0009 s
a
a
NA CA 0012 " 2412 o 4412 0 0 6412 o
0012 0015
°
0019
--Flat plate
---Flat plate
I
2
105
34
6 810
3
2
6 810710
4
1
3
5 8106
4
2
3
4
6 8l07
Re---b Re -Figure 249 Minimum drag of four-digit NACA profiles vs. Reynolds number. (a) Effect of a
thickness ratio. (b) Effect of camber ratio.
minimum drag, becomes narrower. When the angle of attack is increased, the pressure minimum shifts toward the nose and, in general, the transition point jumps upstream abruptly, causing a very strong increase in profile drag. This process is
observed at reduced a when Re increases and at last, at very large Reynolds numbers, the dip in the drag curve disappears completely. A normal polar curve with an elevated cD min takes over (see [50] ).
Computational determination of profile drag The profile drag of lifting wings can be determined theoretically by means of boundary-layer theory as long as the flow 1.8
1.6
Re-1.106
-2.90 6 1
-
°
3.106
o
-6 10 6 -9 906
o
011
Profile NACA 5o-415 0.4
Tyr-
c
4
0.2
I
I
j
;
0
-0.2 -04
-8°
-4°
0°
40
8°
12°
a--
16°
200
24°
-0.008
0
0.008
0,016
0.1CM ,CDp
Figure 2-50 Three-component measurements on the laminar profile NACA 662 -415 at various Reynolds numbers.
AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 93 is
fully attached. Pretsch
[48]
and Squire and Young [62] were the first
investigators to publish such methods, which were later improved by Cebeci and Smith [9]. The profile drag (= pressure drag plus friction drag) is obtained from the velocity distribution in the wake at large distance from the body in the form +ro
Dp=nb fu(Uu)dy
(2-117)
Here, b is the span of the wing profile, y is the coordinate normal to the incident flow direction, and u(y) is the velocity distribution in the wake. By defining the profile drag coefficient CDp by Dp = cDpbc(Q/2)UU and introducing the momentum thickness 62,x, the drag of both sides of the profile with a fully turbulent boundary layer is given as CD p = 2
6200
(2-118a)
c
0.148
SRe
1
U (35d()108 c
(2-118b)
Here Re = is the Reynolds number and U(x) is the velocity distribution over the profile as obtained for potential flow. The second relationship [Eq. (2-118b)] is derived from the findings of boundary-layer theory (see Schlichting [55] ). For a plate in parallel flow, there is U(x) = U. = const. For some symmetric wing profiles in chord-parallel flow, the coefficients for the profile drag from [62] are summarized in Fig. 2-51. The profile thickness varies
from t/c = 0 (flat plate) to t/c = 0.25 and the Reynolds number ranges from Re = 106 to 108. The profile drag is strongly dependent on the location of the laminar-turbulent transition point xtr, which varies from xtr./c = 0 to 0.4. The increase in profile drag with thickness must be attributed essentially to a rising pressure drag. Truckenbrodt
[48]
extended the drag formula, Eq. (2-118b), to contain
explicitly the profile shape instead of the velocity distribution of potential flow. Application of this method to a large number of NACA profiles produces the simple relationship between the profile-drag coefficient and the thickness ratio t/c, CDp = 2Cft 1 + C c
(2-119)
Here c ft is the drag coefficient of the flat plate with a fully turbulent boundary layer. The constant C lies between C = 2 and 2.5 (see also Scholz [48] ).
The above statements apply to the profile drag at zero lift. The CD values determined in this way agree, in general, satisfactorily with experiments.
A comprehensive presentation of experimental data on the drag problem is found in Hoerner [24]. Truckenbrodt [69] summarized the decisive findings on drag of wing profiles. Progress in the development of profiles of low drag has been reported by Wortmann [76].
y
N
h
/,V/F
/ZZ /l/
U dU3000L
tb
N
N
-
a n
C O
d
11
-Y
dpi 0001.
H
QO
N.Z
N -- CiO_OO b OO
ft
> 4-
AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 95
2-5-3 Boundary-Layer Control on the Wing A change of the flow in the very thin wall boundary layer may, under certain conditions, alter considerably the entire flow pattern around the body. A number of methods have been developed for boundary-layer control that, in some instances, have obtained importance for the aerodynamics of the airplane. The basic principles of boundary-layer control will be explained briefly in this section. In most cases, boundary-layer control is considered in the following contexts: elimination of
separation for drag reduction or lift increase, or only change of the flow from laminar to turbulent, or maintaining of laminar flow. The various methods that have been investigated mainly experimentally, but also theoretically in some instances, can be highlighted as follows: boundary-layer acceleration (blowing into the
boundary layer), boundary-layer suction, maintaining of laminar flow through proper profile shaping (laminar profile). A comprehensive survey of this field is given by Lachmann [36].
Boundary-layer acceleration A first possibility of avoiding separation is given by introducing new energy into the slowed-down fluid of the friction layer. This can be done either by discharging fluid from the body interior (Fig. 2-52a) or, in a simpler way, by taking the energy directly from the main flow. This method consists of injecting fluid of high pressure into the decelerated boundary layer through a slot (slotted wing, Fig. 2-52b). In either case, the velocity in the wall layer increases through energy addition and thus the danger of separation is removed. For practical applications of the method of fluid ejection as in Fig. 2-52a, particular care is required in designing the slot. Otherwise, the jet may disintegrate into vortices shortly after its discharge. More recently, extensive tests [46] have led to the method of discharging a jet at the trailing edge of the wing, which has proved to be
b
c
Figure 2-52 Various arrangements for boundary-layer control. (a) Blowing. (b) Slotted wing. (c) Suction.
96 AERODYNAMICS OF THE WING
very successful in raising the maximum lift (jet flap). The same benefit has been gained from blowing into the slot of a slotted wing. A slotted wing (see Fig. 2-52b) functions in the following way: On the front wing (slat) A-B, a boundary layer forms. The flow through the slot carries this layer out in the free stream before it separates. At large angles of attack, the steepest pressure rise and hence the greatest danger of separation occurs on the suction side
of the slat. Starting at C, a new boundary layer is formed that may reach the trailing edge without separation. Hence, by means of wing slats, separation can be prevented up to much larger angles of attack, so that much larger lift coefficients can be obtained. In Fig. 2-53, polar diagrams (lift coefficient CL vs. drag coefficient
cD) are given of a wing without and with a wing slat and with a rear flap. In the slot between main wing and rear flap (Fig. 2-52b), the processes are the same, in principle, as those in the front slot. The lift gain from a front slat and a rear flap is considerable. Further information on this item will be given in Chap. 8.
Boundary-layer suction Boundary-layer suction is applied for two purposes: to avoid separation and to maintain laminar flow (see Schlichting [53] and Eppler [15] ). In the first case, the slowed-down portions of the boundary layer in a region of rising pressure are removed by suction through a slot (Fig. 2-52c) before they can cause flow separation. Behind the suction slot, a new boundary layer is formed that, again, can overcome a certain pressure rise. Separation may never take place if the slots are suitably arranged. This principle of boundary-layer removal by suction
JA 20
o
z 2s°
2 2°
1.6
r
ac - 9
5.s °
27 °
ae
-1.7°
I
a.u
I
-7.5 ° 0
9°
I
-12° -02
0
I
I
01
02
CD
I
1
03
'
Fig ure 2-53 Po l ar cu rv es foil with slat and flap.
of
an
AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 97 5
103cQ10 2 5
3_
10-2
2
5
2
t. 2
05 90-3
a2l
S
2 10-
10°
a
5
105
2
S
106
2
Rep U.C
s
107
2
s
108
V
Figure 2-54 Drag (friction) coefficients of flat plate in parallel flow with homogeneous suction;
cQ = (-v0)/U.. = suction coefficient; -u,, = constant suction velocity. Curves
1,
2, and 3
without suction. 1, Laminar; 2, transition laminar-turbulent; 3, fully turbulent; 4, most effective suction.
was checked out for a circular cylinder by Prandtl as early as 1904 and has been investigated by Schrenk [58] for wing profiles. In the second case, suction is applied for the reduction of friction drag of wings (see Goldstein [20] ). This is. accomplished if suction causes a downstream shift of
the laminar-turbulent transition point. For this purpose, it turned out to be more favorable to apply areawise-distributed (continuous) suction, for example, through porous walls rather than through slots. In this way the disturbances by the slots were avoided, which could have led to premature transition. That the flow can be kept laminar through suction may be seen from the fact that the friction layer becomes thinner when suction is applied and, therefore, has less of a tendency to turn turbulent. Also, the velocity profile of a laminar boundary layer with suction has a shape, compared with that of a layer without suction, that makes transition to turbulence less likely even when the boundary-layer thickness is equal in both cases. Of particular interest is the drag law of the plate with homogeneous suction, as given in Fig. 2-54, because it is characteristic for the drag savings gained through
suction-maintained laminar flow. In comparison, the drag law of the plate with a turbulent boundary layer (without suction) is added as curve (3). The drag savings that may actually be achieved cannot yet be derived. First, the limiting suction coefficient must be known, which is necessary to keep the boundary layer laminar-even for large Reynolds numbers. This minimum suction coefficient was determined as CQcr = 1.2 - 10-4
up to the highest Reynolds numbers. This remarkably small value is included in Fig. 2-54
as "most favorable suction" (curve 4). The difference between curves 3 "turbulent" and 4 "most favorable" suction represents the optimum drag savings. In the Reynolds number range Re = 106 to 108, they amount to about 70-80% of the fully turbulent drag.
98 AERODYNAMICS OF THE WING
It should be understood, however, that this saving does not take into account the power needed for the suction. Even when taking this power into account, however, the drag savings are still considerable.
Ackeret et al. [2] were the first investigators to prove experimentally that it is possible to hold the boundary layer laminar by suction. Some of their test results on a wing profile are given in Fig. 2-55. This wing profile was provided with a large number of slots. The considerable savings in drag, even including the blower power needed for the suction, is obvious. The favorable theoretical results about drag savings by maintaining laminar flow have been confirmed completely through investigations of Jones and Head [20] on wings with porous surface.
Boundary layer with blowing Another very efficient means of influencing the boundary layer is the tangential ejection of a thin jet at a separation point. This method has been applied very successfully to wings with trailing-edge flaps. By ejecting a thin jet at high speed at the nose of the deflected flap, flow separation from the flap can be avoided and hence lift can be increased considerably. The underlying physical principles are demonstrated in Fig. 2-56. At large deflections, the effectiveness of the flap as a lift-producing element is markedly reduced by flow separation (Fig. 2-56a). The lift of a wing with deflected flap does not reach at all the value that is predicted by the theory of inviscid flow. Flow separation from the flap and a resulting loss in lift may be avoided, however, by supplying the boundary layer with sufficient momentum. This is accomplished by a thin jet of high speed,
tangential to the flap, introduced near the flap nose into the boundary layer (Fig. 2-56b). The lift gain that can be realized through blowing is shown in Fig. 2-56c as T-MTC1 ?'ZT Suction slots
0,8
1 J,
Zy-
6
Re = >2
With out suction
cL= 0 9 With suction 1
CL=0.16
03
cL =
V 'q
02
cL=0.Z3I 01 i
15 1.5
2
Re ------ +
3
S- 10
0
2 COp
.?
S
7 03
Figure 2-55 Reduction of drag coefficient of wing profiles by suction through slots, after Pfenninger [2]. (a) Optimum drag coefficient of wing with suction vs. Reynolds number. (b) Profile-drag polar.
AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 99
Potential theoretical pressure
distribution
Lift-gain through blowing Pressure distribution for separated flow
Figuue 2.56 Flap wing with blowing at the flap nose for increased maximum lift. (a) Flap airfoil without blowing, separated flow. (b) Flap airfoil with
blowing, attached flow.
(c) Pressure
distribution.
c
the difference between the two pressure distributions. The effect of blow jets and jet flaps is discussed in more detail in Sec. 8-2-3. A synopsis of the increase of maximum lift of wings through boundary-layer control has been written by Schlichting [54]
.
Maintaining laminar flow through shaping Closely related to maintaining laminar flow through suction is maintaining a laminar boundary layer through proper shaping of the body. The goal is the same, namely, to reduce the friction drag by shifting the transition point downstream. Doetsch [12] was the first to demonstrate experimentally that considerable drag reductions can be obtained in the case of a wing profile whose maximum thickness is sufficiently far downstream (laminar profile). By shifting the maximum thickness downstream, the pressure minimum, and thus the laminar-turbulent transition point of the boundary layer, is also shifted downstream because, in general, the boundary layer remains laminar in the range of decreasing pressure. Only after the pressure rises does the flow turn turbulent. These conditions are shown in Fig. 2-57 by comparing a "normal wing" of a maximum thickness position of 0.3c and a laminar profile with a maximum thickness position of 0.45c. In the former case the pressure minimum lies at 0.1c, in the latter case at 0.65c. The drag diagram indicates that, in the Reynolds number range from 3 - 106
100 AERODYNAMICS OF THE WING
3
Z5 I
a
2`
9
-1
9
_L
55
10 6
8
2
Re-
u
c
3
4
5
6
8
101
-
v
t NA CA 0009
0.3 C
t NACA 66-009
0.45 c 12
Velocity maximum
H
NACA 66-009
NACA 0009
V
X
as
Q8
Figure 2-57 Drag coefficients and velocity distribution of laminar profile, after [1]. (a) Drag coefficients: 1, laminar; 2, fully turbulent; 3, transito tion laminar-turbulent. (b) Velocity (pressure) distributions.
to 107, the drag of the laminar profile is only about one-half that of the normal profile. The aerodynamic properties of such laminar profiles have been investigated
in much detail in the United States [1]. Practical application of laminar profiles is impeded particularly by the extraordinarily high demand on surface smoothness necessary to ensure that the conditions for maintaining laminar flow are not lost with surface roughness. The studies of Wortmann [75] and Eppler [14, 15] on the development of laminar profiles for glider planes should be mentioned.
AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 101
REFERENCES 1. Abbott, I. H. and A. E. von Doenhoff: "Theory of Wing Sections, Including a Summary of Airfoil Data," McGraw-Hill, New York. 1949: Dover, New York, 1959. Abbott, I., A. E. von Doenhoff, and L. S. Stivers, Jr.: NACA Rept. 824, 1945. 2. Ackeret, J., M. Ras, and W. Pfenninger: Verhinderung des Turbulentwerdens einer Grenzschicht durch Absaugung, Naturw., 29:622-623, 1941; Helm'. Phys. Acta, 14:323, 1941. Pfenninger, W.: J. Aer. Sci., 16:227-236, 1949. 3. Allen, H. J.: General Theory of Airfoil Sections Having Arbitrary Shape or Pressure Distribution, NACA Rept. 833, 1945. 4. Betz, A.: "Beitrage zur Tragfliigeltheorie mit besonderer Beri cksichtigung des einfachen rechteckigen Fli gels," dissertation, Gottingen, 1919; Ber. Ablr. WGL, 1(2):l-tl, 1920. Nickel, K.: Ing.-Arch., 20:363-376, 1952. 5. Betz, A.: Applied Airfoil Theory, in W. F. Durand (ed.), "Aerodynamic Theory-A General Review of Progress," div. J, Springer, Berlin, 1935, Dover, New York, 1963.
6. Betz, A.: "Konforme Abbildung," 2nd ed., Springer, Berlin, 1964. 7. Betz, A. and F. Keune: Verallgemeinerte Karman-Trefftz-Profile, Jb. Lufo., 1:38-47, 1937; Lufo, 13:336-345, 1936. von KarmSn, T. and E. Trefftz: Z. Flug.. Mot., 9:111-116, 1918;
"Collected Works," vol. II, pp. 36-51, Butterworths, London, 1956. Keune, F. and I. Fliigge-Lotz: Jb. Lufo., 1:39-45, 1938. Piercy, N. A. V., E. R. W. Piper, and J. H. Preston: Phil. Mag. and J. Sci., 24, ser. 7:425-444, 1114-1126, 1937. Schrenk, 0. and A. Walz: Jb. Lufo., 1:29-49, 1939. 8. Birnbaum, W. and W. Ackermann: Die tragende Wirbelflache als Hilfsmittel zur Behandlung des ebenen Problems der Tragfliigeltheorie, Z. Angew. Math. Mech., 3:290-297, 1923. Gebelein, H.: Jb. Lufo., 1:27-34, 1938. Jaeckel, K.: Z. Angew. Math. Mech., 33:213-215, 1953; Z. Flugw., 3:46-48, 1955. Kaufmann, W.: Z. Flugw., 3:373-376, 1955; 4:280-281, 1956. Nickel, K.: Z. Angew. Math. Mech., 31:297-298, 1951; Ing.-Arch., 20:363-376, 1952.
9. Cebeci, T. and A. M. 0. Smith: Calculation of Profile Drag of Airfoils at Low Mach Numbers, I. Aircr., 5:535-542, 1968. Cebeci, T., G. J. Mosinskis, and A. M. 0. Smith:.1. Aircr., 9:691-692, 1972. 10. Cooke, J. C. and G. G. Brebner: The Nature of Separation and Its Prevention by Geometric Design in a Wholly Subsonic Flow, in G. V. Lachmann (ed.), "Boundary Layer and Flow Control-Its Principles and Application," pp. 144-185, Pergamon, Oxford, 1961. 11. Crabtree, L. F.: Effects of Leading-Edge Separation on Thin Wings in Two-Dimensional Incompressible Flow, J. Aer. Sci., 24:597-604, 1957; ARC RM 3122, 1957/1959. Moore, T. W. F.: J. Roy. Aer. Soc., 63:724-730, 1959. 12. Doetsch, H. and M. Kramer: Profilwiderstandsmessungen, Jb. Lufo., 1:59-74, 1937; 1:88-97, 1939; 1:54-57, 1940; Lufo., 14:173-178, 367, 371, 480-485, 1937; ZWB Lufo. FB 548, 1936. Doetsch, H.: ZWB Lufo. FR 782, 1937.
13. Eppler, R.: Die Berechnung von Tragfliigelprofilen aus der Druckverteilung, Ing.-Arch., 23:436-452, 1955; 25:32-57, 1957. Betz, A.: Lufo., 11:158-164, 1934. Mangler, W.: Jb. Lufo., 1:46-53, 1938. 14. Eppler, R.: Laminarprofile fur Segelflugzeuge, Z. Flugw., 3:345.-353, 1955; Ing-Arch., 38:232-240, 1969. Raspet, A. and D. Gyorgyfalvy: Z. Flugw., 8:260-266, 1960. 15. Eppler, R.: Ergebnisse gemeinsamer Anwendung von Grenzschicht- and Profiltheorie, Jb. WGL, 109-111, 1959; Z. Flugw., 8:247-260, 1960; Jb. WGLR, 140-149, 1962. 16. Fuchs, R.: Das Stromungsfeld einer ebenen Wirbelschicht, in R. Fuchs, L. Hopf, and F. Seewald, "Aerodynamik, II. Theorie der Luftkrafte," 2nd ed., pp. 54-60, Springer, Berlin, 1935.
17. Glauert, H.: "The Elements of Aerofoil and Airscrew Theory," Cambridge University Press, Cambridge, 1926/1947; "Die Grundlagen der Tragfliigel- and Luftschraubentheorie," (German translation by H. Holl), Springer, Berlin, 1929;ARCRM 910, 1924.
102 AERODYNAMICS OF THE WING
18. Glauert, H.: Theoretical Relationships for an Aerofoil with Hinged Flap, ARC RM 1095, 1927/1928. 19. Goldstein, S.: Flow Past Asymmetrical Cylinders, Aerofoils, Lift, in "Modern Developments in Fluid Dynamics-An Account of Theory and Experiment Relating to Boundary Layers, Turbulent Motion and Wakes," vols. I-II, pp. 441-490, Dover, New York, 1965. 20. Goldstein, S.: Low-Drag and Suction Airfoils, J. Aer. Sci., 15:189-220, 1948. Jones, M. and M. R. Head: Anglo.-Amer. Aer. Conf, III, Brighton, pp. 199-230, 1951. 21. Gothert, R.: Systematische Untersuchungen an Fliigeln mit Klappen and Hilfsklappen, Jb. Lufo., 1:278-307, 1940.
22. Helmbold, H. B. and F. Keune: Beitrage zur Profilforschung, Lufo., 20:77-96, 152-170, 192-206, 1943. 23. Hess, J. L. and A. M. 0. Smith: Calculation of Potential Flow About Arbitrary Bodies, Prog. Aer. Sci., 8:1-138, 1967. James, R. M.: J. Aircr., 9:574-580, 1972. 24. Hoerner, S. F.: "Fluid-Dynamic Drag-Practical Information on Aerodynamic Drag and Hydrodynamic Resistance," 3rd ed., Hoerner, Midland Park, N.J., 1965; AGARD CP 124, 1973;AR 58, 1973. 25. Hoerner, S. F. and H. V. Borst: "Fluid-Dynamic Lift-Practical Information on Aerodynamic and Hydrodynamic Lift, Hoemer, Brick Town, N.J., 1975.
26. Hummel, D.: Neuere Beitrage der deutschen Luftfahrtforschung auf dem Gebiet der Flugzeugaerodynamik, Jb. DGLR, 18:1-40, 1975. 27. Jacob, K.: Berechnung der abgelosten inkompressiblen Stromung um Tragfliigelprofile and Bestimmung des maximalen Auftriebs, Z. Flugw., 17:221-230, 1969. Jungclaus, G.: Z. Flugw., 5:172-177, 1957. Riegels, F. W.: Z. Flugw., 13:433-437, 1965. 28. Jacob, K. and F. W. Riegels: Berechnung der Druckverteilung endlich dicker Profile ohne
and mit Klappen and Vorfliigeln, Z. Flugw., 11:357-367, 1963. Jacob, K.: Z. Flugw., 15:341-346, 1967; Ing.-Arch., 32:51-65, 1963. Martensen, E.: Arch. Rat. Mech. Anal., 3:235-270, 1959. 29. Jacobs, E. N. and A. Sherman: Airfoil Section Characteristics as Affected by Variations of the Reynolds Number, NACA Rept. 586, 1937. Loftin, L. K., Jr. and H. A. Smith: NACA TN 1945, 1949. Pinkerton, R. M.: NACA Rept. 613, 1938. 30. Jaeckel, K.: Eine Formel fur die von einem diinnen Tragfliigelprofil induzierte Geschwindigkeit in Punkten, die auf der verlangerten Sehne liegen, Lufo., 16:53, 209-211, 1939. 31. Joukowsky, N.: Uber die Konturen der Tragflachen der Drachenflieger, Z. Flug. Mot., 1:281-284, 1910; 3:81-86, 1912. Betz, A.: Z. Flug. Mot., 6:173-179, 1915; 15:100, 1924. Muttray, H.: Lufo., 11:165-173, 1934. Schrenk, 0.: Z. F7ug. Mot., 18:225-230, 276-284, 1927. Prandtl, L., C. Wieselsberger, and A. Betz (eds.): "Ergebnisse der Aerodynamischen Versuchsanstalt zu Gottingen," vol. III, pp. 13-16, 59-77; vol. IV, pp. 67-74, Oldenbourg, Munich, 1935.
32. Keune, F.: Aerodynamische Berechnung systematischer Flugelprofile, ZWB Lufo., TB 11,
no. 1, 1944; Jb. Lufo., 1:3-26, 1938; I:36-50, 1940. Ringleb, F.: Jb. Lufo., 1:133-140, 1942. Rossner, G.: Jb. Lufo., 1:141-159, 1942. 33. Keune, F. and K. Burg: "Singularitatenverfahren der Stromungslehre," Braun, Karlsruhe, 1975. Feindt, E.-G.: Z. Flugw., 10:446-456, 1962. 34. Kraemer, K.: Fligelproflle im kritischen Reynoldszahl-Bereich, Forsch. Ing.-Wes., 27:33-46, 1961.
35. Kutta, W.: Auftriebskrifte in stromenden Fliissigkeiten, Illustr. Aeron. Mitt., 6:131-135, 1902.
36. Lachmann, G. V. (ed.): "Boundary Layer and Flow Control-Its Principles and Application," vols. I-II, Pergamon, Oxford, 1961. 37. Lan, C. E.: A Quasi-Vortex-Lattice Method in Thin Wing Theory, J. Aircr., 11:518-527, 1974.
38. Liebeck, R. H.: A Class of Airfoils Designed for High Lift in Incompressible Flow, J. Aircr.,
10:610-617, 1973. Liebeck, R. H. and A.
I.
Ormsbee: J. Aircr., 7:409-415, 1970.
Ormsbee, A. I. and A. W. Chen: AIAA J., 10:1620-1624, 1972.
AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 103
39. Lighthill, M. J.: A New Approach to Thin Airfoil Theory, Aer. Quart., 3:193-210, 1951; ARC RM 2112, 1945. 40. Maskew, B. and F. A. Woodward: Symmetrical Singularity Model for Lifting Potential Flow Analysis, J. Aircr., 13:733-734, 1976. 41. Munk, M. M.: General Theory of Thin Wing Sections, NACA Rept. 142, 1922; 191, 1924.
42. Naumann, A.: Messung eines Profils im Anstellwinkelbereich 0° bis 360°, Jb. Lufo., 1:51-53, 1940; 1:90-100, 1938. Naumann, A. and B. Sann: Jb. Lufo., 1:406-415, 1940. 43. Nonweiler, T.: The Design of Wing Sections-A Survey of Existing Knowledge on Aerofoil Design for Different Conditions, Aircr. Eng., 28:216-227, 1956; 27:2-8, 1955. 44. Pinkerton, R. M.: Calculated and Measured Pressure Distributions over the Midspan Section of the NACA 4412 Airfoil, NACA Rept. 563, 1936. 45. Pistolesi, E.: Betrachtungen fiber die gegenseitige Beeinflussung von Tragfliigelsystemen, Ges. Vor. Lil.-Ges. Lufo., 214-219, 1937. 46. Poisson-Quinton, P.: Quelques Aspects Physiques du Soufflage sur les Ailes d'Avion, Tech. Sci. Aer., 163-195, 1956; Jb. WGL, 29-51, 1956. 47. Prandtl, L. and C. Wieselsberger, Profiluntersuchungen, in L. Prandtl, C. Wieselsberger, and A. Betz (eds.), "Ergebnisse der Aerodynamischen Versuchsanstalt zu Gottingen," vol. 1, pp. 71-112, Oldenbourg, Munich, 1935. Ackeret, J. and R. Seiferth: ibid., vol. III, pp. 26-91. Seiferth, R. and M. Kohler: ibid., vol. IV, pp. 30-66. 48. Pretsch, J.: Zur theoretischen Berechnung des Profilwiderstandes, Jb. Lufo., 1:60-81, 1938; NACA TM 1009, 1942. Helmbold, H. B.: Ing.-Arch., 17:273-279, 1949. Scholz, N.: Jb. Schiffb., 45:244-263, 1951. Truckenbrodt, E.: Ing.-Arch., 21:176-186, 1953. 49. Riegels, F.: Das Umstromungsproblem bei inkompressiblen Potentialstromungen, Ing.-Arch., 16:373-376, 1948; 17:94-106, 1949; 18:329, 1950. Jungclaus, G.: Z. Flugw., 5:106-114, 1957. Riegels, F. W.: Z. F7ugw., 4:57-63, 1956. Riegels, F. W.: Jb. Lufo., 1:10-15, 1940. Riegels, F. W. and H. Wittich: Jb. Lufo., 1:120-132, 1942. Truckenbrodt, E.: Ing.-Arch., 18:324-328,1950. 50. Riegels, F. W.: "Aerodynamische Profile-Windkanal-Messergebnisse, theoretische Unterlagen," Oldenbourg, Munich, 1958, D. G.. Randall (transl.), "Aerofoil Sections," Butterworths, London, 1961. 51. Robinson, A. and J. A. Laurmann: "Wing Theory," (Cambridge Aeronautics Series, II), pp. 80-168, Cambridge University Press, Cambridge, 1956. 52. Schlichting, H.: Einfluss der Turbulenz and der Reynoldsschen Zahl auf die Tragfliigeleigenschaften, Ringb. Luftfahrt., I(A1):1-14, 1937. 53. Schlichting, H.: Absaugung in der Aerodynamik, Jb. WGL, 19-29, 1956. Regenscheit, B.: Jb. WGL, 55-64, 1952. 54. Schlichting, H.: Aerodynamische Probleme des Hochstauftriebes, Z. Flugw., 13:1-14, 1965.
Schlichting, H. and W. Pechau: Z. Flugw., 7:113-119, 1959. Schrenk, 0.: Jb. Lufo., 1:77-83, 1939. 55. Schlichting, H.: "Grenzschicht-Theorie," Sth ed., Braun, Karlsruhe, 1965, J. Kestin (transl.), "Boundary-Layer Theory," 7th ed., McGraw-Hill, New York, 1979. 56. Schlichting, H.: Einige neuere Ergebnisse aus der Aerodynamik des Tragfliigels, Jb. WGLR, 11-32, 1966.
57. Schmitz, F. W.: "Aerodynamik des Flugmodells, Tragfligelmessungen," 4th ed., Lang, Duisberg, 1960; Jb. WGL, 149-166, 1953.
58. Schrenk, 0.: Tragfliigel mit Grenzschichtabsaugung, Lufo., 2:49-62, 1928, 12:10-27, 1935; Z. Flug. Mot., 22:259-264, 1931;Luftw., 7:409-414, 1940. 59. Sears, W. R.: Some Recent Developments in Airfoil Theory, J. Aer. Sci., 23:490-499, 1956. 60. Smith, A. M. 0.: High-Lift Aerodynamics, J. Aircr., 12:501-530, 1975. 61. Spence, D. A.: Prediction of the Characteristics of Two-Dimensional Airfoils, J. Aer. Sci., 21:577-587, 620, 1954. Preston, J. H.: ARC RM 1996, 1943; 2107, 1945; 2725, 1949/1953. 62. Squire, H. B. and A. D. Young: The Calculation of the Profile Drag of Aerofoils, ARC RM 1838, 1937.
104 AERODYNAMICS OF THE WING
63. Strand, T.: Exact Method of Designing Airfoils with Given Velocity Distribution in Incompressible Flow, J. Aircr., 10:651-659, 1973, 12:127-128, 1975. 64. Tani, I.: Low Speed Flows Involving Bubble Separation, Prog. Aer. Sci., 5:70-103, 1964. 65. Tanner, M.: Theoretical Prediction of Base Pressure for Steady Base Flow, Prog. Aer. Sci., 14:177-225, 1973, 16:369-384, 1975. Nash, J. F.: ARC RM 3468, 1965/1967. 66. Theodorsen, T. and I. E. Garrick: General Potential Theory of Arbitrary Wing Sections,
NACA Rept. 452, 1933; 411, 1931. Gebelein, H.: Ing.-Arch., 9:214-240, 1938. Kochanowsky, W.: Jb. Lufo., 1:52-58, 1937, 1:82-89, 1938, 1:72-80, 1940. Mangler, W. and A. Walz: Z. Angew. Math. Mech., 18:309-311, 1938. Wittich, H.: Jb. Lufo., 1:52-57, 1941. 67. Thwaites, B. (ed.): Uniform Inviscid and Viscous Flow Past Aerofoils, in "Incompressible
Aerodynamics-An Account of the Theory and Observation of the Steady Flow of Incompressible Fluid Past Airfoils, Wings, and Other Bodies," pp. 112-205, Clarendon Press, Oxford, 1960. 68. Truckenbrodt, E.: Die Berechnung der Profilforrn bei vorgegebener Geschwindigkeitsverteilung, Ing.-Arch., 19:365-377, 1951. Riegels, F.: Z. Angew. Math. Mech., 24:273-276, 1944. 69. Truckenbrodt, E.: Die entscheidenden Erkenntnisse uber den Widerstand von Tragfliigeln, Jb. WGLR, 54-66, 1966; Tech. Sci. Aer. Spat., 97-111, 1967. Riegels, F.: Jb. WGL, 44-55, 1952.
70. von Karman, T. and J. M. Burgers: General Aerodynamic Theory-Perfect Fluids, in W. F. Durand (ed.), "Aerodynamic Theory-A General Review of Progress," div. E, Springer, Berlin, 1935, Dover, New York, 1963.
71. von Mises, R.: Zur Theorie des Tragflachenauftriebes, Z. Flug. Mot., 8:157-163, 1917; 11:68-73, 87-89, 1920. Blasius, H.: Z. Math. Phys., 58:90-110, 1910. 72. Wagner, H.: ITber die Entstehung des dynamischen Auftriebs von Tragfliigeln, Z. Angew. Math. Mech., 5:17-35, 1925. Forsching, H. W.: "Grundlagen der Aeroelastik," pp. 149-373, Springer, Berlin, 1974. Kiissner, H. G.: Lufo., 13:410-424, 1936. 73. Weissinger, J.: Theorie des Tragfliigels bei stationarer Bewegung in reibungslosen, inkompressiblen Medien, in S. Fliigge (ed.), "Handbuch der Physik, vol. VIII/2, Stromungsmechanik II," pp. 385-437, Springer, Berlin, 1963. 74. Woods, L. C.: "The Theory of Subsonic Plane Flow," (Cambridge Aeronautics Series, III), pp. 301-425, Cambridge University Press, Cambridge, 1961.
75. Wortmann, F. X.: Ein Beitrag zum Entwurf von Laminarprofilen fur Segelflugzeuge and Hubschrauber, Z. Flugw., 3:333-345, 1955, 5:228-243, 1957. Speidel, L.: Z. Flugw., 3:353-359, 1955. Stender, W.: Luftfahrt., 2:218-227, 1956. 76. Wortmann, F. X.: Progress in the Design of Low Drag Aerofoils, in G. V. Lachmann (ed.), "Boundary Layer and Flow Control-Its Principles and Application," pp. 748-770, Pergamon, Oxford, 1961.
77. Young, A. D. and H. B. Squire: A Review of Some Stalling Research-Appendix: Wing Sections and Their Stalling Characteristics, ARC RM 2609, 1942/1951. Gault, D. E.: NACA TN 3963, 1957. Goradia, S. H. and V. Lyman: J. Aircr., 11:528-536, 1974. Kao, H. C.: J. Aircr., 11:177-180, 1974. McCullough, G. B. and D. E. Gault: NACA TN 2502, 1951.
CHAPTER
THREE WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW
3-1 INTRODUCTION For an airfoil of infinite span, the flow field is equal in all sections normal to the airfoil lateral axis. This two-dimensional flow has been treated in detail by profile theory in Chap. 2. For an airfoil of finite span as in Fig. 3-1, however, the flow is three-dimensional. As in Chap. 2, incompressible flow is presupposed.
3-1-1 Wing Geometry The wing of an aircraft can be described as a flat body of which one dimension (thickness) is very small in relation to the other dimensions (span and chord). In
general, the wing has a plane of symmetry that coincides with the plane of symmetry of the aircraft. The, geometric form of the wing is essentially determined
by the wing planform (taper and sweepback), the wing profile (thickness and camber), the twist, and the inclination or dihedral of the left and right halves of the
wing with respect to each other (V form) (see Fig. 3-1). In what follows, the geometric parameters that are significant in connection with the aerodynamic characteristics of a lifting wing will be discussed.
For the description of wing geometry, a coordinate system in accordance with Fig. 3-1 that is fixed in the wing will be established with axes as follows: x axis, wing longitudinal axis, positive to the rear y axis, wing lateral axis, positive to the right when viewed in flight direction, and perpendicular to the plane of symmetry of the wing z axis, wing vertical axis, positive in the upward direction, perpendicular to the xy plane 105
106 AERODYNAMICS OF THE WING
b
Figure 3-1 Illustration of wing geometry. (a)
c
X
Planform, xy plane. (b) Dihedral (V form), yz plane. (c) Profile, twist, xz plane.
It is expedient to select the position of the origin of the coordinates as suitable for each case. Frequently it is advisable to place the origin at the intersection of the
leading edge with the inner or root section of the wing (Fig. 3-1), or at the geometric neutral point [Eq. (3-7)]. The wing planform is given in the xy plane; the twist, as well as the profile, in the xz plane; and the dihedral in the yz plane. The largest dimension in the direction of the lateral axis (y axis) is called the. span, which will be designated by b = 2s, where s represents the half span. Frequently the coordinates will be made dimensionless by reference to the half-span s, and abbreviated notations (3-la)
(3-1 b)
(3-1c)
are here introduced.
The dimension in the direction of the -longitudinal axis (x axis) will be designated as the wing chord c(y), dependent on the lateral coordinate y. The wing chord of the root or inner section of the wing (y = 0) will be designated by Cr, and
the corresponding dimension for the tip or outer section by ct. In Fig. 3-2, the geometric dimensions are illustrated for a trapezoidal, a triangular, and an elliptic planform. For a wing of trapezoidal planforr (Fig. 3-2a), an important geometric
parameter is the wing taper, which is given by the ratio of the tip chord to the root chord:
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 107
A special case of the trapezoidal wing is the triangular wing with a straight trailing edge, also designated as a delta wing (Fig. 3-2b). The wing area A (reference area) is understood to be the projection of the wing on the xy plane. For a variable wing chord, the area is obtained by integration of the wing chord distribution c(y) over the span b = 2s; that is, 3
A
fc(y)dy -3
Quarter-point line NZ3
.\
V4 CU
a
b
b =2S
Quarter-point line
C
b=2s
Figure 3-2 Geometric designations of wings of various planforms. (a) Swept-back wing. (b) Delta wing. (c) Elliptic wing.
108 AERODYNAMICS OF THE WING
From the wing span b and the wing area A, there is obtained, as a measure for the wing fineness (slenderness) in span direction, the aspect ratio
!1=
b2
(3-a)
b Cm
(3 4b)
As mean chord and reference wing chord, especially for the introduction of dimensionless aerodynamic coefficients, the quantities A
Cm
(3-Sa)
b
s C
fc2(y)ciy "`
(3-5b)
A fS
are used, where the ratio 1. For the trapezoidal planform, it may be easily demonstrated that the reference chord c. is equal to the local chord at the position of the center of gravity of the half wing; that is, cP, = c(yc) (Fig. 3-2a and b). The sweepback of a wing is understood to be the displacement of individual wing cross sections in the longitudinal direction (x direction). Representing the position of a wing planform reference line by x(y), the local sweepback angle of this line is tan92(y) =
(3-6)
If x(y) represents the connecting line of points of equal percentage rearward position, measured from the leading edge at the y section under consideration, then this fact is designated by giving the percentage number as an index of the value x. Accordingly, the position of the quarter-chord line is designated by x25(y). For the sake of simplicity, the index will be omitted in the case of the sweepback angle of the quarter-chord-point line. For aerodynamic considerations, furthermore, the geometric neutral point plays a special role. Its coordinates are given by 8
XV95 = A
,lc(y)
x25 (y) dy
y:V25 ` 0
(3-7)
For a symmetric wing planform, the geometric neutral point may be demonstrated to be the center of gravity of the entire wing area, whose quarter-chord-point line is overlaid by a weight distribution that is proportional to the local wing chord. The rearward distance of the geometric neutral point of a wing with a swept straight quarter-chord-point line is equal to the rearward distance of the quarter-chord point of the wing section at the planform center of gravity of the half-wing. Since, for a trapezoidal wing, the wing chord at the center of gravity of the half-wing is equal to the reference chord c1,, the geometric neutral point for this wing lies at the cu/4 point (see Fig. 3-2a and b). Of particular importance is the delta wing, a triangular wing with a straight
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 109
trailing edge (Fig. 3-2b). For the geometric magnitudes of this wing, especially simple formulas are obtained:
_
b
b
Crrt-2 Cr
3
Cu
4
tancp
Cm
3
Cr
-N25 - 2
(3-8)
For a wing of elliptic planform as in Fig. 3-2c, the geometric quantities become
A=
7r
bcr
!1 =
4 b r
C'm Cr
7r
=
= 0.785
c Cr
-
3 7r
= 0.848
(3-9)
4 4 A further geometric magnitude related to the wing planform is the flap (control-surface) chord cf(y). The flap-chord ratio is defined as the ratio of flap chord (control-surface chord) to wing chord: Xf =
Cf(Y)
(3-10)
(Y)
For the description of the whole wing, data on the relative positions of the profile sections are required at various stations in span direction. They are required in
addition to the knowledge of wing planforms and wing profiles. The relative displacement in longitudinal direction is specified by the sweepback, the displacement in the direction of the vertical axis by the dihedral, and the rotation of the profiles against each other by the twist. In what follows, the geometric twist e(y) is defined as the angle of the profile chord with the wing-fixed xy plane (Fig. 3-3).* For aerodynamic reasons, in most
cases the twist angle is larger on the outside than on the inside. The dihedral determines the inclination of the left and the right wing-halves with respect to the *In addition to the geometric twist, there is an aerodynamic twist, characterized by a twist angle measured against the profile zero-lift direction instead of the profile chord.
2
X
Figure 3-3 Illustration of geometric V;gist.
110 AERODYNAMICS OF THE WING
xy plane. Let z(s)(x, y) be the coordinates of the wing skeleton surface. Then the local V form at station x, y is given by
tanv (x, j) =
8z(a)(x, y)
ay
(3-11)
The partial differentiation is done by holding x constant. If the wing is twisted, it must be specified in addition at which station xp(y) the angle v is to be measured. According to Multhopp [61], the aerodynamically effective dihedral has to be taken approximately at the three-quarter point xp = x75 .
3-1-2 Shapes of Actual Wings To convey a concept of the various wing shapes that have actually been used in airplanes, the profile thickness ratio 5 = t/c, the aspect ratio A = b2 fA, and the sweepback angle of the leading edge pf of some airplanes are plotted in Fig. 3-4 against the flight Mach number. The plots show a clear trend of profile thickness and aspect ratio in the transition from subsonic to supersonic airplanes.
The profile thickness ratio decreases sharply with increasing Mach number, reaching values of tic = 0.04 for supersonic airplanes. The aspect ratios are particularly large in the subsonic range for long-distance airplanes but considerably smaller for maneuverable fighter planes. In the supersonic range, the implementation of larger aspect ratios is no longer required for aerodynamic reasons. In this range,
therefore, design considerations have led to aspect ratios as small as A = 2. The sweepback angle is close to zero at low Mach numbers but increases to pf -- 45,0 at high subsonic speeds. In the supersonic range, airplanes with both relatively large sweepback 6pf 60°) and small sweepback ('pf ~ 30°) are found. Truckenbrodt [86] has shown to what extent the geometric wing data of Fig. 34 have been determined by a decisive understanding of the drag of wings.
3-1-3 Lift Distribution The lift distribution over the span is defined in analogy to Eq. (2-9b) as dL = cl(y)c(y)q dy
(3-12)
Here the local lift coefficient has been introduced in analogy to Eq. (2-10) as cl(y) ~ -cn(y).* The lift distribution of a wing in symmetric incident flow is shown in Fig. 3-5b. Finally, in Fig. 3-6 there is also shown the distribution of measured local lift coefficients cl over the span of a rectangular wing at various angles of attack.
By integrating Eq. (3-12) over the span, the total lift L and further, with Eq. (1-21), the lift coefficient are determined as *To distinguish between the coefficients of the total forces and moments, the indices of which are always expressed in capital letters, lowercase letters will'be used for the indices of the coefficients of local forces and moments.
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 11 1
s
L
1
CL = Aq = A
ct(Y)c(y) dy
(3.13)
-s
Only single wings will be treated in this book. Wing systems such as, for example, biplane and tandem arrangements or ring wings (tube-shaped cylindrical surfaces) will not be considered. reports, more recent results and the understanding of the
In progress
aerodynamics of the wing are presented for certain time periods, among others, by Schlichting [72, 74], Sears [781, Weissinger [97], Gersten [20], Blenk [7], Ashley et al. [21, Kuchemann [491, and Hummel [35]. The very comprehensive compilation of experimental data on the aerodynamics of lift of wings of Hoerner and Borst [311 must also be mentioned. (12
.40 `I-
0.1
\
I
i I
i I
I
I
I
\
7
.
I
11
,
4
1
'
1
1
Z5
b)
.
I
'I
+
0
750
t/c = 0.15
}
1
c)
L
C.
05
L
Macr F
1.0
1.5
2.0
25
Ma
3.0
Figure 3-4 Most important geometric wing data of actual airplanes vs. Mach number. Evolution from subsonic to supersonic airplanes. (a) Profile thickness ratio 6 = t1 c. (b) Aspect ratio A. (c) Sweepback angle of wing leading edge of Macr = drag-critical Mach number (see Sec. 4-3-4).
112 AERODYNAMICS OF THE WING
Figure 3-5 Illustration of lift distribution of wings. (a) Geometric designations. (b) Lift distribution over span.
3-2 WING THEORY BY THE METHOD OF VORTEX DISTRIBUTION
3-2-1 Fundamentals of Prandtl Wing Theory The creation of lift of a wing is tied to the existence of a lifting (bound) vortex within the wing (Fig. 3-7). This fact has been explained in Sec. 2-2 by means of Fig. 2-4. The position of the bound vortex on the wing planform is described in Sec. 2-3-2 for the inclined flat plate. Accordingly, it is expedient to position the vortex on the one-quarter point of the local wing chord. An unswept wing in symmetric incident flow is therefore represented by a bound vortex line normal to the incident flow direction. Profile 60 420
C
'4
° -0.4°
0.60
5.4°
O.B9
17.1 °
9.21
°
0.2
--
1
0.4
0.6
0.9
N 1.0
Figure 3-6 Distribution of local lift coefficients for a rectangular wing of aspect ratio A = 5 and
profile Go 420. Reynolds number Re = 4.2 101; Mach number Ma = 0.12.
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 113
Figure 3.7 Vortex system of a wing of finite span.
Since the pressure differences between lower- and upper-wing surfaces decrease
to zero toward the wing tips, producing a circulation around the wing, the flow field of a wing of finite span is three-dimensional. This pressure equalization at the wing tips, shown schematically in Fig. 3-8b, causes an inward deflection of the streamlines above the wing and an outward deflection below the wing (Fig. 3-8a). In this way, streamlines that converge behind the wing have different directions. They
form a so-called surface of discontinuity with inward flow on the upper surface, outward flow on the lower surface (Fig. 3-8c). The discontinuity surface tends to roll up farther downstream (Fig. 3-8d), forming two distinct vortices of opposite
C
d
e
f
r F
°) r
Figure 3-8 Evolution of the free vortices behind a wing of finite span.
114 AERODYNAMICS OF THE WING
sense of rotation. Their axes coincide approximately with the direction of the incident flow (Fig. 3-8e and f). These two vortices have a circulation strength P. Thus, behind the wing there are two so-called free vortices that originate at the wing tips (Fig. 3-7). Far downstream, these two vortices are connected by the starting vortex, the evolution of which was explained in Sec. 2-2-2. The bound vortex in the wing, the two free vortices, originating at the wing tips, and the starting vortex together form a closed vortex line in agreement with the Helmholtz vortex theorem. The vortices produce additional velocities in the vicinity of the wing, the so-called induced velocities. They are, as a result of the sense of rotation of the vortices, directed downward behind the wing. They play an important role in the theory of lift. The
starting vortex need not be taken into account in steady flow for
treatment of the flow field in the vicinity of the wing. This is understandable when it is realized that the wing has already moved over a long distance from its start from rest. In this case the vortex system consists only of the bound vortex in the wing and the two infinitely long, free vortices. These form again an infinitely long vortex line shaped like a horseshoe, open in the downstream direction. This vortex is called a horseshoe vortex.
The very simplified vortex model of Fig. 3-7, having one bound vortex of constant circulation, is still insufficient for quantitative determination of the aerodynamics of the wing of finite span. A further refinement of the two simple free vortices originating at the wing is necessary. The above-mentioned pressure equalization at the wing tips causes the lift, and consequently the circulation, to be reduced more near the wing tips than in the center section of the wing. At the very wing tips even complete pressure equalization occurs between upper and lower surfaces. The circulation drops to zero. The actual circulation distribution is. similar to that shown in Fig. 3-9; it varies with the span coordinate, T =r(y). The variable circulation distribution T (y) in Fig. 3-9 can be thought of as being replaced by a step
distribution. At each step a free vortex of strength d T is shed in the
downstream direction. In the limiting case of refining the steps to a continuous circulation distribution, the free vortices assume an areal distribution (vortex sheet).
A strip of this vortex sheet of width dy has the circulation strength d .P =
(dr/dy) dy. Thus the slope of the circulation distribution T(y) of the bound vortices determines the distribution of the vortex strength in the free vortex sheet.
It was Prandtl [69] who for the first time gave quantitative information on the three-dimensional flow processes about lifting wings based on the above discussed mental picture. Earlier, Lanchester had investigated this problem qualitatively (see von Karman [90] ).
Lift and induced drag From the Kutta-Joukowsky theorem [see Eq. (2-15)], the lift dL of a wing section of width dy and its circulation T (y) are related by
dL = oVT(y) dy
(3-14)
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 115
tiT
y
Figure 3-9 Wing with variable circulation distribution over the span.
The total lift is obtained by integration as s
L = , V F (y) dy
(3-15)
fs As the most important consequence of the formation of free vortices, the airfoil of finite span undergoes a drag even in frictionless flow (induced drag), contrary to the airfoil of infinite span. Physically, the induced drag can be explained by the roll-up of the discontinuity sheet into the two free vortices: During every time increment a portion of the two free vortices has to be newly formed. To this end, work must be done continually; this work appears as the kinetic energy of the vortex plaits. The equivalent of this work is expended in overcoming the drag during forward motion of the wing. On the other hand, the formation of induced drag may also be understood by means of the Kutta-Joukowsky theorem as follows: The downstream-drifting free vortices produce a downwash velocity wi behind and at the wing, after Biot-Savart. At the wing the incident flow velocity of the wing profile is therefore the resultant of the incident flow velocity V and this induced downwash velocity wj. Accordingly, the resultant incident flow direction at the wing is inclined downward by the angle al against the undisturbed incident flow direction, with (3-16)
In general, wi << V and hence aj
sin aZ ~ tan ai.
116 AERODYNAMICS OF THE WING
From the Kutta-Joukowsky theorem (Sec. 2-2-1), the resultant dR of the aerodynamic forces at the wing cross section y (Fig. 3-10) stands normal to the resultant incident flow direction. Hence, normal to the undisturbed flow direction there is a lift component dL = dR cos a1 -- dR and parallel to the undisturbed flow direction a drag component dD1= dR sin ai Maj. The latter is the induced drag of the wing cross section y, which, with Eq. (3-16), becomes dDi = a1dL = dL V
(3-17)
Hence, the total induced drag is obtained through integration over the wing span from y = -s toy = + s, and by noting Eq. (3-14), as
Di = fr)wi)dY
(3-18)
where wr(y) is the distribution of the induced downwash velocity that is variable in the general case.
The distribution of the induced downwash velocity along the span is obtained by applying the Biot-Savart law to the semi-infinitely long free vortex behind the
wing. The contribution of the vortex strip dy' at station y' to the downwash velocity at the location of the lifting line y (Fig. 3-9) is
dwi (y, y') =
1
dl'(y')
47r y - y'
dl' dy' 4n dy' y - y'
=
1
with (y - y') being the distance of the point under consideration (control point) y from the location y' of the free vortex line. From this, the induced velocity at the wing is found by integration over the area of the free vortices as* wt (y) =
4 or
y J dyz' - .!
(3-19)
z
From this equation, the induced downwash velocity wi at the location of the lifting line can be computed when the circulation distribution F(y) is known. Finally, the induced drag can be determined from Eq. (3-18). It should be mentioned here that the induced downwash velocity wm very far behind the wing has twice the value of the downwash velocity wi at the wing from *At station y' = y, the integrand has a singularity. The analysis shows that the integral has to be evaluated through the Cauchy principal value. Hence, the range y - e < y' < y + e must be excluded during integration and the limit operation y-E lim
must be conducted.
I
s
f ... d y" -r f ... d y'
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 117
- - -Zero-lift direction v. Direction of undisturbed incident flow
---.-Effective incident flow direction
Figure 3-10 Evolution of the induced drag.
Eq. (3-19). This is obvious from the fact that far downstream the free vortices can be taken as being infinitely long vortex lines, leading to
w. (y) = -2 iv; (y)
(3-20)
The velocity w. is taken as positive in the direction of the positive z axis (see Fig. 3-21).
Prandtl's integral equation of the circulation distribution The above considerations will now be applied to the derivation of an equation for the determination of the spanwise circulation distribution for a given wing of finite span.
The change of the incident flow direction that results from the downwash velocity induced by the free vortices was explained in Fig. 3-10. This change of flow incidence, at equal geometric angles of attack a, is responsible for the reduced
lift at the cross section y of a finite-span wing in comparison with the lift at the same cross section of an infinitely long wing. For a span element dy of a finite-span wing, Eq. (3-12) yields for the lift:
dL = cl(y)
V 2 c(y) dy
(3-21a)
V2
(3-21b)
2
=
czoo ae(Y)
2
c(Y) dy
Here c(y) is the wing chord at station y (Fig. 3-9) and cz(y) = cl.cxe(y) is the local lift coefficient of the area element dA = c(y) dy; cxe(y) is termed the effective angle
of attack (Fig. 3-10) and cl. = (dcz/da). is called the lift slope for the airfoil of infinite span. The latter value is close to 21r, from the theory of thin profiles (see Chap. 2). For the inclined flat plate, cl. is exactly equal to 27T. Equation (3-21) is based on the concept that a profile cross section of a wing of finite span behaves like that of a wing of infinite span (plane flow) at an angle of incidence ae The geometric angle of attack a(y), measured from the zero-lift position, the effective angle ae(y), and the induced angle ai(y) [Eq. (3-16)] are related by .
118 AERODYNAMICS OF THE WING
(3-22)
a(y)=cz (y)+a1( )
as shown in Fig. 3-10. The effective angle of attack ae is obtained from Eq. (3-21b) with the help of Eq. (3-14), and the induced angle of attack from Eq. (3.19) with ai = wt/ V as Cie =
2T(y)
(3-23a)
Vc(y)ciC* S
1
PLC (y) -
(' d F
d y'
(3-23b)
4nV J dy' y - y -s
Introducing Eq. (3-23) into Eq. (3-22) yields the following basic equation for the determination of the circulation distribution: «(1')
-
2T (y)
Vc(y)c
S
+ i 4n V
dr
d y'
dy
y -y
-s
(3-24)
This is Prandtl's integral equation for the circulation distribution of a wing of finite span as first published by Prandtl in 1918 [691. It is a linear integral equation for the circulation distribution P(y), where I' depends linearly on the angle of attack a. The profile coefficient cl. is known from profile theory (Chap. 2).* With given wing geometry [chord distribution c(y) and angle-of-attack distribution a(y)], the circulation distribution can be determined from Eq. (3-24). This is the so-called direct problem of wing theory. Conversely, if the circulation distribution '(y) is known, either the angle-of-attack distribution (twist angle) a(y)
can be computed from Eq. (3-24) when the chord distribution c(y) is given, or the chord distribution c(y) when the angle-of-attack distribution a(y) is given. This is the so-called indirect problem of wing theory. In either case, from the circulation distribution I'(y) the lift is obtained from Eq. (3-15) and the induced drag from Eq. (3-18).
From a mathematical viewpoint, the direct problem is considerably more difficult than the indirect problem, because in the former case an integral equation has to be solved while in the latter case only a quadrature has to be performed.
Elliptic circulation distribution A particularly simple solution of Eq. (3-24) that is of great practical importance is found for the elliptic circulation distribution along the span. In this case the circulation becomes T (Y) = ro
1 - (S )2
(3-25)
where ro is the circulation at the wing center y = 0 (Fig. 3-11). From Eq. (3-15), the lift becomes *If the profile coefficient
cl. = cLa,
27r.
cj. is known over
the span,
it may be replaced by
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 119
L = 4 obVP0
(3-26)
The induced downwash-velocity is obtained from Eq. (3-19). Execution of the integral yields for points within the span, lyl < b/2, Wi(J)
To
(3 -27a)
=2b = const
ai (y) = L
;rgb2
(3-27b)
This remarkable result shows that, for elliptic circulation distribution, the induced
downwash velocity w1, and consequently the induced angle of attack aj, are constant over the span (Fig. 3-11).
By introducing Eqs. (3-25) and (3-27a) into Eq. (3-18), the induced drag is obtained with To from Eq. (3-26) as DZ
=
(3 -28a) LZ
_ irgb2
(3-28b)
Here, q = (p/2) V2 is the dynamic pressure resulting from the velocity V. The induced drag is proportional to the square of the lift and inversely proportional to the dynamic pressure and the square of the span. Comparison of Eqs. (3-28b) and (3-27b) confirms the relationship DZ = a1L, given in Eq. (3-17).
The geometry of the corresponding wing is obtained in a particularly simple way when starting from the wing without twist, a(y) = a = const. Since, from Eq.
(3-27b), the induced angle of attack an(y) = const, Eq. (3-22) shows that the effective angle of attack along the span must also be constant: ae(y) = const.
Figure 3-11 Elliptic circulation distribution with the corresponding elliptic wing planform and the constant induced downwash WI = const
velocity over the wing span.
120 AERODYNAMICS OF THE WING
From Eqs. (3-23a) and (3-25), it follows that the chord is distributed elliptically over the span: 2
c(y) = c, ri
s
(3-29)
The elliptic wing planform is shown in Fig. 3-11.x` Thus it has been demonstrated that an elliptic wing without twist has an elliptic circulation distribution. From Eq. (3-21), it also has a constant local lift coefficient c1(y) over the span.
Coefficients Finally, the most important results for the induced angle of attack [Eq. (3-27b)] and for the induced drag [Eq. (3-28b)] will also be expressed through the dimensionless coefficients of lift and induced drag. They are defined as follows :
L = cLgA
(3-30a)
DI = cDigA
(3-30b)
with A being the wing planform area. Consequently, Eqs. (3-27b) and (3-28b) yield (3-3 la)
=
C
(3-31 b)
Here A = b2 /A is the aspect ratio of the wing from Eq. (3.9). The important result for the coefficient of the induced drag of Eq. (3-31b) is compared in Fig. 3-12 with test results for a wing of aspect ratio A. = 5. The theoretical curve for the induced drag agrees quite well over the whole CL range with the polar curve of the measured data. The difference between the two curves is about constant over the whole CL range. It is caused by the effect of friction that has been neglected in the above theory. Figure 3-12 suggests splitting up the drag coefficient into a component that is nearly independent of the lift coefficient and a component that is dependent on the lift coefficient. The former is called the coefficient of profile drag CDp, the latter the coefficient of induced drag CD1. They are related by CD = CDp + CDi
(3-32a)
2
= CDp + 7111
(3-32b)
For the geometric angle of attack, Eqs. (3-22) and (3-3 la) yield a= CYe
+
c 7tl1
CL
CL
+7r1
(3-33a)
(3.33 b)
'The elliptic wing consists of two ellipse halves, the large axis of which is the c/4 line.
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 121 1#
1,2
cps
=20°
4=5
CD
22.x°
1,0
Re=2. 7. 106 0.8
87°
N,4C4 24'12 02
0
-0.2 -041
004
0,08
0.19
CD -
016
a20
024
Figure 3-12 Measured polar curve of a wing of aspect ratio A = 5 and theoretical curve for the induced drag, CDi = cL/rrd .
From Eq. (3-21), ae = CL /CL 00 because the constant local lift coefficient c1(y) and
the total lift coefficient CL are equal in this case. The latter equation allows one to determine the lift slope of the wing of finite span as a function of the aspect ratio. From Eq. (3-33b) it follows: dCL
_
CL
(3-34a)
1+CL-
da
7r A
CL
(3-34b)
A +2
CL-
with cL = dcL/da and cL00 = 21r. Equation (3-34b) expresses the degree of reduction of lift slope and consequently also of lift because of the finite aspect ratio when the angles of attack are equal. In Fig. 3-13 this ratio of lift slopes is presented as a function of the aspect ratio. As will be shown later in more detail, the formulas for induced drag and lift slope found here for the elliptic wing are valid for other wing shapes in good approximation. This is true particularly for the rectangular wing, as shown by Betz [5] ; see Figs. 3-32 and 3-57. .
Prandtl's transformation formulas The above-derived results on the effect of aspect ratio on lift and drag have been checked experimentally by Betz and Wieselsberger [99]. For comparison of the polar curves of two wings of aspect ratios Al and :'12 at equal angles of attack, Eq. (3-32b) with CDP2 = CDpI yields CD2 = CD1
+
d IT
1
!3.2
-
1
<11)
(3-35)
122 AERODYNAMICS OF THE WING
Re
t
a
r 1
8
6'
A
10
12
Figure 3-13 Ratio of the lift slope of wings of finite and infinite aspect ratios vs. aspect ratio, cL°, = 27r.
N.
In Fig. 3-14a the measured polar curves are plotted for a number of rectangular wings with aspect ratios :11 = 1, 2, ... , 7. Figure 3-14b shows the result of the transformation of these polars to the aspect ratio A2 = 5 from Eq. (3-35). The transformed curves fall well on one curve, confirming experimentally the validity of Eq. (3-35). In Fig. 3-14b the theoretical polar curve of the reduced drag for A. = 5
is also included. On the other hand, comparison of the lift curves CL (a) of two wings of aspect ratios Al and r12 of equal lift coefficient yields, with Eq. (3-33b), OZ-2 = at
+ L (1 A2
- 1)i
(3-36)
1
11
1
1 L44 .1 1 .
A=S
°
o
+CP
110
0
i
---l1=9 H
!
°
2
_ ace
212
0 g6
CD - -
a20
Figure 3-14 Demonstration of the experimental verification of the transformation formula for the drag, from [991. (a) Measured polars for rectangular wings of aspect ratios A = 1-7. (b) Polar curves transformed to A1= 5 and comparison with the theory of induced drag. Eq. (3-35).
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 123
For the wings of Fig. 3-14, the lift curves were converted to the aspect ratio A,2= 5. Again, the converted curves fall together, confirming experimentally the validity of Eq. (3-36).
The two equations (3-35) and (3-36) can therefore
for the transformation of measured drag polars CD(CL) and lift curves CL (a) at aspect ratio be
used
!11 to those of a wing with a different aspect ratio 112 if both wings have the same profile. These equations are called, therefore, transformation formulas of the wing of finite span.
3-2-2 Integral Equation for Circulation Distribution from Wing Theory Vortex system of the lifting surface To simplify the problem, it was assumed in Sec. 3-2-1 that the circulation representing the wing was concentrated on one line (lifting-line theory); see Fig. 3-7. This concept is a fairly good approximation for a real wing only when its chord is much smaller than its span (wing of large aspect ratio). When the chord is no longer much smaller than the span, it is necessary to replace the concept of a lifting line by that of a distribution of lifting vortices over the wing chord. Such a continuous vortex distribution over the wing chord was the basis for the skeleton theory (Sec. 2-4-2). In the preceding section, the free vortices were assumed to be distributed on the surface. By applying this concept of a continuous circulation distribution logically to the wing of finite span, a vortex distribution on the surface results that varies in chord and span direction (lifting surface). An outline of this lifting-surface theory will now be derived. This theory is of practical importance particularly for wings of small aspect ratio, for swept-back and delta wings, and for yawed wings. This vortex distribution on the surface can be taken to be a distribution of singularities in the sense of Sec. 2-4-2. During the further development of wing theory, instead of vortex distributions, dipole distributions will be used occasionally; see, for example, Prandtl [69 (1936)].
After the fundamental publication of Prandtl on wing theory using vortex distributions, Blenk [69] further developed this theory by extending the twodimensional Birnbaum-Ackermann theory, Chap. 2 (8], to three dimensions. The distribution of vortex strength over a given surface can be accomplished in
various ways. Let the wing surface have an arbitrary shape, and let a rectangular wing-fixed coordinate system be chosen whose y axis is normal to the incident flow direction.
A first possible approach to the replacement of the wing by a vortex distribution is to cover this surface with two areal vortex distributions k,(x, y) and ky(x, y), as in Fig. 3-15. The former distribution consists of vortex lines parallel to the x axis, the latter of those parallel to the y axis. The ky vortices are of the kind that was previously applied to the two-dimensional wing theory (see Fig. 2-20); the kX vortices, however, resemble the free vortices in the vortex sheet behind the wing (see Fig.. 3-9). Only the ky vortices contribute to the lift of the wing when the incident flow is in the x direction. The vortex distributions kx(x, y) and ky(x, y)
124 AERODYNAMICS OF THE WING
Direction of incident flow
Figure 3-15 Wing with areal vortex distribution.
kx = vortex density of vortex lines in the x direction, ky = vortex density of vortex lines in the y direction.
cannot be chosen arbitrarily; rather, they must produce velocities induced by the vortex sheet that satisfy the condition of irrotationality au/ay - av/ax = 0.
According to Eq. (246a), in the vicinity of the vortex sheet (z - 0) the perturbation velocities are
u=+-k. v=+I,kx
(3-37)
where the upper sign is valid above, the lower sign below the vortex sheet. Hence
akx+aky ax
ay
This relationship is called the condition of source-free vortex distribution.
The connection between circulation and rotation (Stokes's theorem) yields - wy, which is another formulation of the spatial vortex
kx wx and ky conservation law.
A second possible way to represent a wing by a vortex distribution consists, as suggested by Glauert [231, of replacing the wing by so-called elementary wings of infinitesimal span dy and of chord c(y) (Fig. 3-16). Each elementary wing occupies its special location within the wing boundaries as defined by the wing geometry. The vortex system of each elementary wing consists of a number of vortex lines,
one behind the other, parallel to the y axis, which is equivalent to a series arrangement of horseshoe vortices as introduced in Sec. 3-2-1. This representation was given for arbitrary wing planforms by Truckenbrodt [841, among others. In Fig. 3-17, this concept is again demonstrated by the example of a yawed swept-back wing. Note that the free vortices of the individual horseshoe vortices have been
drawn separately in this picture, but only for clarity; actually, all of them are located on two parallel lines of distance dy. The circulation distribution density of
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 125 Direction of incident flow I
dy
0
Figure 3-16 Substitution of a lifting wing by
X
an elementary wing of span dy and chord
b
c(y).
the elementary wing in direction of the chord (x direction) is k(x) per unit length.
In the terminology of Fig. 3-15, k corresponds to ky of this figure. From Eq. (2-44), it follows that the circulation of a surface element of the elementary wing with span dy and chord dx is clI'(x, y) = Ic (x, y) cl x
(3-38)
and the total circulation of the bound vortex of the elementary wing at the wing section y becomes Xr
I'(y) = fkJ x
(3-39)
Xf
where xf(y) and x,.(y) designate the x coordinates of the front and rear edges of the section, respectively. The same circulation is found in the two free vortices originating at the trailing edge of the elementary wing. 1
Elementary wing
d
Figure 3.17 Vortex system of a b=2s
l!
yawed wing, from 184].
126 AERODYNAMICS OF THE WING
Within the framework of linear wing theory, that is, limitation to small profile
camber of the individual wing sections and to small angles of attack, it can be assumed that bound and free vortices of all elementary wings lie in the same plane (xy plane). This assumption was also made for the profile theory in Sec. 2-4-2.
Equation for the determination of the circulation distribution To establish an equation for the computation of the circulation distribution, an expression first must be developed for the requirement that the lifting surface carrying the vortices is a stream surface, that is, that the normal component of the resultant velocity is equal to zero on this surface. This is the so-called kinematic flow condition. In Fig. 3-18 a wing cross section y of the lifting surface (skeleton surface) z(S)(x, y) = z(x, y) is sketched. It is located in a flow field of incident flow velocity U. that forms
the geometric angle of attack ag(y) = aF + £(y) with the chord.* Here aF is the angle of attack, measured from the x axis, and c(y) is the twist angle. The kinematic flow condition becomes, in analogy to Eq. (249),
_
u I-XF
1
az(x, y) ax
+ u' (x, y) = 0
(3-40)
J
where w(x, y) is the velocity in the z direction induced by the total vortex system at the point x, y of the xy plane (w > 0 in the direction of the positive z axis). The brackets contain the term describing the angle between the incident flow direction and the skeleton tangent. Equation (3-40) must be satisfied in all points x, y of the lifting surface.
Furthermore, as a next step, the induced velocity w(x, y) on the lifting surface
must be determined from the given vortex distribution k(x, y). To simplify the problem, the induced velocity is computed, however, at the projection of the lifting surface on the xy plane that is identical with the vortex sheet. The induced velocity
w(x, y) at an arbitrary point of the xy plane is obtained by first determining the contribution of one horseshoe vortex of one elementary wing (Fig. 3-19). The total induced velocity w(x, y) is then the result of integrating first over one elementary wing in the x direction and consecutively in the y direction over the total number *Contrary to Sec. 3-2-1, the incident flow velocity is designated now by U,o instead of V.
Section y
Zero-lift direction aF ,Bx Skeleton line z(s) = z(x)
Figure 3-18 Illustration of the kinematic flow condition of wing theory.
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 127
H
Bound vortex element dI'-k (z; y)d.s'
Figure 3-19 Explanation of the determination of
the induced velocity w(x, y) of the horseshoe vortex of an elementary wing.
of elementary wings. Execution of this integration yields the following result, as shown in detail in [84] : 8
w(x, y)
=-
2
Em
47t E-+o
G(x,
6
y; y) -
r G(x' Y,
j
J,) dy' (y - y 'l)2
(3-41)*
-s
with the kernel function
xrV) G(x, y; y') = f.k(xi, y') (1
x - x'
-}-
,J
}1 (x
xf(y)
dx'
(3-42a)
- X')2 - (y - y')2
X
G (x, y; y) = 2 f k (x', y) d x'
(3-42b)
Xf(y)
For the derivation of Eqs. (3-41) and (3-42), th-. Biot-Savart theorem must be applied in such a way that the point in which the induced velocity w(x, y) is to be computed is first positioned outside of the vortex sheet (z * 0). It is then shifted into the vortex sheet (z -+ 0). On the right-hand side of Eq. (3-4 1), the first term represents the self-induction of the elementary wing in the section y' = y (downwash), whereas the second term represents the external induction of all other elementary wings in the sections y' = y (upwash). Finally, introducing Eq. (3-41) into the kinematic flow condition Eq. (3-40) yields U'° I aF
2x
47c
1--G(x,y;y) o a li
-s
(y - y')'
dy
(3-43)
G(x, y; y') is related to k(x, y) through Eq. (3-42). Equation (3-43) is an integral equation for the circulation distribution k(x, y) of the lifting surface in which the angle of attack aF and the wing shape z(x, y) are given quantities. To satisfy the
dy'
f ...dy'±
r...dy'
128 AERODYNAMICS OF THE WING
Kutta condition for the wing, the vortex density k(x, y) at the trailing edge x = xr(y) must disappear [see Eq. (2-51)] . After having determined the vortex density k(x, y) from Eq. (3-43), the resultant of the pressure distribution of lower and upper surface at the point x, y, from Eq. (2-53), takes the form
-
d cP (x, y) _ P1-Pu = 2 k(x,y) q00
U03
(3-44)
Here, q,c = o U. /2 is the dynamic pressure of the incident flow. As in the case of the Prandtl wing theory (Sec. 3-2-1), the wing geometry (twist
and camber) can be established with Eq. (3-43) when the wing area and vortex distribution k(x, y) are given quantities. The indirect problem requires quadratures as in Eqs. (3.42) and (3-43). When the wing geometry (planform and angle of attack) is given, Eq. (3-43) produces the vortex distribution on the wing surface. This direct
problem leads to an integral equation for the vortex distribution k(x, y), the solution of which poses considerable mathematical difficulties. Approximation methods need to be applied, therefore, which can be laid out in various ways.
A first possibility for obtaining an approximate solution is given by imposing
beforehand the vortex distribution k(x, y) in the direction of the wing span y. By selecting for k(y) an expression of m terms, the first of which may, for instance, represent the elliptic distribution, the integral equation Eq. (3-43) can no longer be
satisfied on the whole lifting surface, but only on m sections in the chord direction.
A second possibility for the establishment of approximate solutions consists of imposing beforehand the vortex distribution k(x, y) in the direction of the wing chord x, for example, using the Birnbaum normal distribution of Eq. (2-61). If one selects for k(x) an expression of n terms, then the integral equation can be satisfied only on n lines along the span. Such procedures have been established for n = 1 (first normal distribution) by Weissinger [95], for n = 2 (first and second normal distributions) by Multhopp [62] and Truckenbrodt (84], and for n = 5 by Wagner [91] and also by Kulakowski and Haskell [12]. A third possibility consists of imposing beforehand distributions with m terms over the span and simultaneously distributions with n terms over the chord. In this case the integral equation can be satisfied at (m - n) points suitably distributed over span and chord. Such a procedure was applied by Blenk [69]. More recently, the so-called panel procedure was developed [46] (see Sec. 6-3-1). Previously, Falkner [14] presented a procedure in which discrete vortices were arranged in both the chord and span directions. Also, the work of Jones [39] and Lan [511 must be mentioned. Velocity potential The induced velocity field of the vortex system of a wing can also
be defined by means of a spatial velocity potential 0 (x, y, z). Here the velocity components induced by the vortex system are
co
00
0x
0y
IV =
co cz
(3-45)
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 129
According to Truckenbrodt [84], the potential is S
O(x, Y, Z) =
with
1
47r -8
z G(x,y,z;y')dy' (y- y')',+z'
xr(Y) G (x, y, z; y') = k (x', y') 11 +
x-x
(x-x')2+(y-y')2± z2
(3-46)
d x'
(3.47)
xf(Y)
This expression for the velocity potentials of a lifting surface had been presented earlier in similar form by von Karman [89] and Burgers [69] ; see also [84].
The potential is discontinuous at the lifting vortex sheet and in the free vortex sheet behind the wing. Closer investigation shows that it changes abruptly when crossing the. vortex sheet from the upper to the lower surface. This step of the potential above (index u) and below (index 1) the vortex sheet is given at the lifting surface [xf(y) <x <xr(y)] by z
Ou (x, y)
f k (x', y) dx' - 0r (x, y) =xf(y)
(3-48a)
and the free vortex sheet [x > xr(y)] by x rr(Y)
0u(x, y) - 0(x, y) = f k(x', y) dx' = T(y)
(3-48b)
xf(Y)
Very far upstream and very far downstream of the wing, the function 0, in terms of I' from Eq. (3-39) becomes (3-49a)
0 (- oo, y, z) = 0
zf
8
0 (+ oc, y, z) =
2iz
ry) (y - y')'2 + z2
d y'
(349b)
-s
Equation (3-49b) represents the two-dimensional potential of the induced velocity field in the yz plane far behind the wing (potential in the Trefftz plane [69] ).
Acceleration potential For the treatment of the problem of the lifting surface by means of the Laplace potential equation there is available, besides the method of the velocity potential just discussed, the method of the acceleration potential. This was first published by Prandtl [69 (1936)]. The method of the acceleration potential has been applied to the circular plate by Kinner [44] and to the elliptic plate by Krienes [47] .
3-2-3 Integral Equation for the Circulation Distribution from the Extended Lifting Line Theory The lifting-surface theory of Sec. 3-2-2 can be transformed into a simpler theory of the kind given in Sec. 3-2-1 by replacing the continuously distributed circulation along the
130 AERODYNAMICS OF THE WING
chord by a vortex line, arranged at a suitably chosen station on the local chord
theory). Let x' = xc(y') be the location of this lifting-vortex line which, from the results of Sec. 2-3-2 for the inclined flat plate, is expediently (lifting-line
placed on the quarter-point line (Fig. 2-37). Then the function G(x, y; y') of Eq. (3.42a) becomes
i+
G(x. y; y') = I'(y')
x -x' c (x -x') zT (y - 02
(3-50a)
Here I'(y') is the total circulation around the wing section y'. Furthermore, for y' = y and x > xc this function becomes G(x, y; y) = 21r(y)
(3-50b)
The kinematic flow condition [Eq. (3-40)] can be satisfied in this case at one point of the chord only. This control point has the coordinate xp(y). Expediently, it is placed on the three-quarter-chord station, measured from the leading edge (three-quarter point, theorem of Pistolesi), see Sec. 2-4-5. Hence, the expression in parentheses on the left-hand side of Eq. (3-43) becomes
az(' y) ex
_ a (y)
(3-51)
where a(y) is the measured angle of attack relative to the zero-lift direction (Fig. 3-18).
By introducing Eqs. (3-51) and (3-50) into Eq. (3-43), the integral equation for the circulation distribution from the extended lifting-line theory i§ obtained as
U" (y)
4n
lim
s
r(y)
J
1
(y
(Y')
- YT
+
_XP - xC
dy,
(x -xc )2 ; (y - y')2 (3-52)
Compared with the simple lifting-line theory discussed in Sec. 3-2-1, Eq. (3-52) has the great advantage that it is also applicable to yawed and swept-back wings. This extended lifting-line theory is also called the three-quarter-point method. It was developed in detail and applied particularly by Weissinger [95]. Also Reissner [95] was engaged in the establishment of a solid foundation for this lifting-line theory. For the swept-back wing a vortex arrangement as in Fig. 3-20 is obtained. In Fig. 3-20a the replacement of the wing by a system of elementary wings and in Fig.
3-20b the equivalent vortex system according to Prandtl's concept (Fig. 3-9) are demonstrated.
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 131
a
b
Figure 3-20 Vortex system of a swept-back wing (lifting-line theory). (a) Substitution of the wing by elementary wings. (b) Bound and free vortices according to Prandtl (see Fig. 3-9).
In Prandtl's lifting-line theory and in the three-quarter-point method described above, the wing is replaced by just one lifting line. Wieghardt [101 ] proposed the arrangement of several lifting lines in series. This method can be designated as a
multiple points method. Scholz [77] developed this method in more detail and applied it especially to the cambered rectangular wing.
3-3 LIFT OF WINGS IN INCOMPRESSIBLE FLOW
3-3-1 Methods of Wing Theory The theoretical basis for this section was laid in Sec. 3-2. For practical applications, the computational methods discussed below (simple and extended lifting-line theories, lifting-surface theory) proved to be particularly convenient and may be characterized as follows: The simple lifting-line theory applies only to wings with straight c14 lines in symmetric flow, that is, to unswept wings. It gives good results
for larger aspect ratios (A > 3) and allows the determination of lift distributions over the span from which total lift, rolling moment, and induced drag, but not pitching moment, may be computed. The extended lifting-line theory (three-quarterpoint method) applies to wings of any planform and aspect ratio. Thus, it applies to swept-back and yawed wings. It gives the lift distribution over the span from which total lift, rolling moment, induced drag, and, approximately, pitching moment are obtained. The lifting-surface theory, like the extended lifting-line theory, applies to
any wing and aspect ratio, but gives lift distributions over the span and over the chord from which total lift, rolling moment, induced drag, and also pitching moment, and thus the neutral-point position of the wing, are found. Accurate knowledge of the neutral-point position is particularly important for swept-back wings.
132 AERODYNAMICS OF THE WING
Summaries and detailed presentations on the methods of wing theory in incompressible flow are given by Betz (61, von Karman and Burgers [88], Robinson and Laurmann [701, Thwaites (82], Weissinger [96], von Karrnan [89], Flax [15],
Hess and Smith [28], and Landahl and Stark [52]. The development of the lifting-line theory as a "singular perturbation problem" is due to van Dyke [87] ; see
also the references on page 111. Extensions of wing theory to include nonlinear angle-of-attack effects and the behavior of wings near the ground (ground effects) are found, for example, in [8, 19, 21, 40] and [2, 81, 100], respectively. Although it is not possible in this book to treat the questions of nonsteady flow that are important for airplane aerodynamics, the references [2, 50, 52, 53] shall be mentioned in this connection. Problems of flexible wings are discussed in [221.
Studies on design aerodynamics have been prompted by Kuchemann and accomplished for swept-back wings in particular [3].
3-3-2 Computation of Total Lift Basic formulas The local lift coefficient c1(y) of a wing section y is obtained through integration of the pressure distribution over the wing chord in analogy to Eq. (2-54) as xrr(Y)
ci(y)
c(Y)J v cp
(x,
y) dx
(3-53)
xf(Y)
The total lift coefficient
CL = L/Aq
of the
wing is
thus obtained with
as CL
=
Aff dc, dx dy
(3-54a)
(A)
= A1
cr(y)c(Y) dy
(3-54b)
-s
Compare also with Eq. (3-13). By using Eq. (3-54a), the total lift is obtained through integration of the pressure distribution over the wing chord. With Eq. (3-15), it may also be obtained from the Kutta-Joukowsky theorem. Here the circulation distribution has to be taken from Eq. (3-39), the distribution of vortex
density from Eq. (3-44). Below, a further expression for the total lift will be derived by applying the momentum law. As in Fig. 3-21, a cylindrical control surface is arranged about the wing. The axis of the cylinder runs in the direction of
the incident flow velocity U.,. The two base surfaces I and II of the cylindrical control surface are assumed to be very far upstream and downstream of the wing, respectively. The diameter of the control cylinder is chosen large enough to make pressure and velocity on the cylindrical surface equal to the values per, and U. of the undisturbed flow on surface 1, respectively. In computing the lift from the
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 133
I
Jr
r-- Ii
Z
a Vortex sheet
Fig re 3-21 Computation of lift by means of the momentum law and of the induced drag by the energy law.
momentum law, it can be assumed that the free vortex sheet is parallel to the incident flow direction far downstream of the wing.* The fluid mass permeating an area element dy dz of surface II per unit time is
o U. dy dz. It produces, together with the velocity w induced by the wing, a momentum component in the z direction of magnitude o U.w. dy dz. Since the induced velocity on surface I is zero, the integral of the momentum over the surface
II represents the force exerted normal to the incident flow direction to the wing, that is, the lift
L = -.Q U, f f w, dy dz
(3-55)
(II)
Now, the identity of Eqs. (3-55) and (3-15) will be shown for the not-rolled-up vortex sheet. The field of the induced velocities very far downstream of the wing can be described by means of a two-dimensional velocity potential P(y, z) [see Eq.
(3-49b)], where w. = ao/az. By introducing this expression into Eq. (3-55), integration over z yields +00
L
= -2 Uo-Y=-f 00 [O]Z 00 N dy
(3-56a)
8
_ e U. f I'(y) dy
(3 -56b)
s
On the boundaries y = ±0o and z = ±-, the values 0 vanish, whereas in the vortex sheet, at z = ±0, the potential in the z direction from Eq. (348b) changes abruptly
by the amount d O(y, 0) = 0,,(v, 0) - 01(y, 0) = T (y). The integration limits y = ±o may be replaced by y = ±s = ±b/2 because d0 (y, 0) = 0 outside of the wing span. Introduction into Eq. (3-56a) yields Eq. (3-56b), in agreement with Eq. (3.15). The total lift thus depends only on the circulation distribution over the wing span. It is thus immaterial whether the circulation distribution is created by wing *Kraemer [791 points out the decisive significance of the inclination of the free vortex sheet for computation of the induced drag by means of the momentum law.
134 AERODYNAMICS OF THE WING
planform (aspect ratio, sweepback, taper), wing twist, or camber of the wing surface.
Certainly, Eq. (3-55) is also valid for the rolled-up vortex sheet as in Fig. 3-8.
Now let bo = 2so be the distance between the two free vortices of circulation strength To, whereby the circulation distribution along the span is symmetric (Fig. 3-22). The induced velocity w. at a point of the yz lateral plane very far behind the wing (x - °°) becomes, from the Biot-Savart law, w (X, Y)
so -y 2n [(so + y)2 + z2 + (So -A' + z2 so + y
To
Introducing this expression into Eq. (3-55) and integrating twice yield L = n Uro I'ob0
(3-57)
By taking into account the Kutta-Joukowsky lift theorem, this formula states that the lift of a wing of span b = 2s and of variable circulation distribution T (y) is
equal to the lift of a wing of span bo and over the span constant circulation distribution To. Comparison of Eqs. (3-56b) and (3-57) yields the distance between the two free vortices: S
ho = r
0
f I'(y) dy
(3-58)
0
This relationship can also be interpreted as a statement that the vortex moment about the longitudinal axis (x axis) remains constant during roll-up. For the right
Figure 3-22 Wing with rolled-up vortex sheet.
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 135
wing-half, the moment of the not-rolled-up vortex sheet about the x axis is equal to - foS (d I'/dy) v dy and that of the rolled-up vortex sheet is equal to To bo /2. By equating these two terms and integrating by parts, Eq. (3-58) is obtained directly. Numerical data for bo/b are given in Fig. 7-17.
Introduction of the dimensionless lift distribution For the following computations it is advisable to use the dimensionless quantities Y(77) =
r bU,,. CI(77)C(77)
2b
(3-59a)
(3-59b)
with 77 = y/s [Eq. (3-1b)]. The relationship Eq. (3-59b) is obtained from Eqs. (3-12) and (3-14). The significance of the linear wing theory of Sec. 3-2 is expressed by the fact that the circulation distributions 71 (77) and y2 (77), resulting from two given angle-of-attack distributions al (77) and a2 (77), can be superimposed linearly: a (77) = al (17) + °L2 (77)
(3-60a)
Y(W = YIN) + Y2(77)
(3-60b)
The total lift coefficient is obtained from Eq. (3-56b) or Eq. (3-54b) as i
cLq
=11 rY(?7)d77
(3-61a)
-1 1
dcL
dx
=.11fYu(77)d
(3-61b)
-1
The lift slope is obtained by computing the circulation distribution of the wing without twist yu for a = 1. The aspect ratio of the wing A is given by Eq. (3-4a). The zero-lift angle ao of a symmetric wing without twist is understood, according to Sec. 1-3-3, to be the angle of attack that produces the total lift zero.
It can be determined as follows: For a given angle-of-attack distribution ag(y) (measured from a wing-fixed reference plane), a circulation distribution yg(77), and, with Eq. (3-61a), a total lift coefficient CLg are computed, from which ao and CLO are obtained as
ao = - d a
cLg
(3-62a)
L 1
CLo= i'l f yo (ii) d77 = 0
(3-62b)
1
It is expedient to represent the circulation distribution of the twisted wing for an arbitrary angle of attack by superposition of the distribution of the wing without twist -yz, and a zero-distribution yo of the twisted wing for which the total lift is
136 AERODYNAMICS OF THE WING
zero. Consequently, the circulation distribution of the twisted wing at given angle of attack a = const is given by (3-63)
Y(77) = aYu(y1) + Yo (77)
The zero distribution yo (77) is obtained from (3-64)
70 (77) _ 'Yg(77) + ao 7u (7?)
Through procedures similar to those applied for the lift, integration over 77 for a known circulation distribution y(rl) produces other simple relationships for the lateral distance of the lift center of a wing-half, for the lift force of a wing-half, and also for the rolling moment about the x axis. They are summarized in Table 3-1. Introduction of a Fourier series Computation of the integrals for the coefficients of lift and rolling moment turns out to be. particularly simple when the circulation distribution is expressed as a Fourier polynomial of the form Al
y
2 S' a. sin,uzg
(3 -65a)
Et=1
Table 3-1 Compilation of the formulas for the aerodynamic coefficients of wings of finite span* Symmetric lift distribution
M+ 1 L
CL = A R.
iT/i
M
2
Z yn M + 1 n=1
I ! y(n) d -n
sun an
11 I Av yv v=1
M+ I 2
f y(n)n dry 3'L
_
0
S
f y(n) do
I Bvyv
V=1 M+1
} Gr A v'Yv 2
o
v=1
Antimetrict lift distribution
M-1
Cl =
(A2)12)q.
2 !1 / Y(77) dr
-
/If 2
V=1
Cvyv
M-1 M
cMX =
M
A
sq .
-A f -y(i7)i7 do
1 yn sin - 2(M + 1) n=1
2
I
v=1
Dvyv
*Lift coefficient cL, lateral distance of the lift center of a wing-half rlL, lift coefficient of a wing-half cL, rolling-moment coefficient cMX (sign convention from Fig. 1-6). Coefficients are given in Table 3-2. tFor an explanation of antimetric, see p. 190.
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 137
cos 0 _
with
(3-65b)
This procedure was first introduced by Trefftz [69] and Glauert [23]. The first term in Eq. (3-65a) represents the elliptic circulation distribution y = 2a1 sin L _ 2a,-\I1 --r7 as treated previously in Sec. 3-2-1.
After execution of the integrations over -1 < rl < I and 0 < 6
(3-66a)
cMx=- 2Aa2 7Z
(3-66b)
The coefficients aµ result from Fourier analysis: a,2 =
1 ry
sing 0di
(3-67a)
0
M
1
sin,cl0,,
where the integral
is
(3-67b)
evaluated by a summation formula. Here the symbols
yn = y(in) = y(rln) signify the circulation values at the stations rln = Cos $n
7rn
with On = M+ 1
(3-68)
By introduction into Eqs. (3-66a) and (3-66b), simple quadrature formulas are obtained with p = 1 and p = 2 for the lift coefficient CL and the rolling-moment coefficient cMx (Table 3-1). In addition,-quadrature formulas are also given for the lateral distance of the lift center of a wing-half r7L = YL is and for the lift coefficient of a wing-half cL . Table 3-2 contains the coefficients for the formulas for practical application of the last column of Table 3-1.
Basic equations The starting equation for the Prandtl lifting-line theory has been given as Eq. (3-22),
a (y) = ae (y) -i- ai (y)
(3-69)
where a(y) is the angle of attack relative to the zero-lift direction as in Fig. 3-23. By introducing the dimensionless circulation distribution y(rr) from Eq. (3-59a) with r7 = y/s, and further the dimensionless planform function f(r7) =
2b C1oC 1
ir
b
c(17)
(3-70a)
(3 -70b)
138 AERODYNAMICS OF THE WING
Table 3-2 Coefficients A, B, C, D for the computation of the aerodynamic coefficients of a wing of finite span of Table 3-1, M = 7 and M = 15
B 1
2
i
3
4 1
2 3 15
4 6 7 8
!
0.9239 0.7071 0.3827 0.0000
0.3006 0.5555 0.7256
0.9808 0.9239 0.8315 0 . 7071 0.5556 0.3827 0.1951 0.0000
+
D.
C,,
0.3260 0.4952 0.8593
0.2777
0.:3927
0.2769 0.3951 0.2702 0.0317
0.0766 0.1503 0.2182 0 . 2777 0.3265 0.3628 0.3852 0.1964
0.0751 0.1389 0.1813 0 . 1965 0.1811 0.1395 0.0733 0.0078
0.0797 0.1438 0.2285 0 . 2622 0.3495 0.3266 0.4546
0.0751 0.1388 0.1814
0.3927 0.2777
0.1964
0.1814 0.1388 0-0751
the effective or, respectively, the induced angle of attack becomes* ae(7)) = f(0 Y(n)
(3-71 a)
1
f dy
1
2n
-1
dn'
(3-71 b)
d'1' 77 - 91'
The formula for ai(n) can also be written, through integration by parts, as cc i(77) _
lim
2
2n E--o 1
Y(r1) -
r
d,,"
(3-71c)
-1
By introducing Eqs. (3-71a) and (3-71b) into Eq. (3-69), the Prandtl integral equation for the dimensionless circulation distribution 'y(n) is obtained in the form (77) = f (@)) Y (11) T
i 2n
1
f,
d7j
ds7 n-17
(3 -72a)
*See footnote on page 139. Section y
Zero-lift direction
Chord
Figure 3-23 Wing section y: a(y), angle of attack against zero-lift direction; ag(y), geometric angle of attack against wing chord; a,, (y), zero-lift angle,
a=ag-a0
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 139
In abbreviated form, the integral equation of the simple lifting-line theory can be written: x (l) = ai (')) H -
f(?1) Y (1))
(3-72b)
Schmidt et al. [76] deal with the mathematical formulation of the simple lifting-line theory and present comprehensive results.
Solution with Fourier polynomials A convenient method of solving Eq. (3-72) for the circulation distribution consists of expressing the circulation distribution as a Fourier polynomial such as Eq. (3-65) (M= n). By introducing Eq. (3-65a) into Eq. (3-71b), first the induced angle of attack is obtained*: Al
I n an
sinnfi1
(3-73)
sing
n=1
After introduction of Eqs. (3-65a) and (3-73) into Eq. (3-72b), the following equation is obtained, defining the Fourier coefficients an: M
a (i) sin?5 = Z a [2 / (6) sin 6 H- n] sin n t
(3-74)
n=1
Here the distribution of the angle of attack a(6) and the wing planform f(6) are given beforehand. The coefficients al, a2i ... , am are determined by satisfying Eq. (3-74) at M points t51 , 62, ... , 6M along the span. This results in a system of M linear equations for al to am. Lotz [56] simplified this procedure by introducing Fourier polynomials for the functions a(6) sin 6 and f(i3) sin 6. After the Fourier coefficients an have been determined, the circulation distribution is obtained from Eq. (3-65a) and the distribution of the local lift coefficients from Eq. (3-59b). Weinig [93] suggests that the theory of the lifting line be solved by comparison with the corresponding grid flow.
Wing of elliptic planform The elliptic wing has been treated in Sec. 3-2-1. There it was shown that an elliptic wing without twist has an elliptic circulation distribution
over the span. The elliptic wing with twist may be computed from the above formulas very easily as, among others, Schmidt [69] has shown. For the elliptic wing c = c, 1 - 712 = Cr sin 6 and A = 4b/?TCr [Eq. (3-9)], and thus from Eq. (3-70),
2sin0/(0) _ 'A
(3-75a)
CL cc
k
*Note that, according to [23 ]
,
z
r cos no,
1
7t j Cos'0 - cos 0' 0
(3 -7 5b)
d6'
sin?09 sin
140 AERODYNAMICS OF THE WING
Hence, Eq. (3-74) for the wing with elliptic planforrn becomes M
a(O) sine n=1
(k+n)aasinnO
(3-76)
and the coefficients a, can be computed directly through a Fourier analysis as an
_
2f a 79
1
k+
sink sinnO d6
7r
(3-77)
0
This solution will now be discussed for a few particularly simple angle-of-attack distributions. Setting sin m&
(3-78)
sin,&
the corresponding circulation distribution is obtained with a,, = 0 for n
m and
with am = rm J(k + m) for n = m as
Y(6)=2k
75
(3-79)
gin,
For a wing with the aspect ratio A = 6, that is, k = 3, the results for m = 1, 2, and 3 are presented in Fig. 3-24. Here m = 1 gives the constant angle-of-attack
a
b
-oz
0.2
Figure 3-24 Lift distribution according to the simple lifting-line theory of an elliptic wing at various twists (Eq. (3-78)]; aspect ratio A = 6. (a) Wing planform. (b) m = 1: wing without twist. (c) m = 2: linear angle-of-attack distribution. (d) m = 3: parabolic angle-of-attack distribution.
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 141
7
2
',
C 2
I
I
A3
4(
t
If
7
6'
Figure 3-25 Lift slope of elliptic wings vs. aspect ratio; cL = 2n. (1) Simple lifting-line theory, Eq. (3-80b). (2) Extended lifting -line theory, Eq. (3-98). (- o -) Exact solutions according to Kinner [44] and Krienes [47].
distribution (wing without twist), m = 2 gives the linear angle-of-attack distribution
such as, for instance, is encountered in a rolling motion, and m = 3 gives the parabolic angle-of-attack distribution (symmetric twist, CL = 0).
Circulation distribution and coefficients of lift and rolling moment of the elliptic wing with twist are obtained from Eq. (3-66a) and also from Eq. (3-66b) by introducing the corresponding coefficients a7z according to Eq. (3-77). The lift coefficient thus becomes CL
k+ i
. IT
r()S1fl2CZz
(3-80a)
0
dcL
,cal.
da
k+1
(3-80b)
For the wing without twist, a = const, the coefficient of lift slope is obtained in agreement with Eq. (3-34a). It is presented in Fig. 3-25 as a function of the aspect ratio A l . Also shown are the results based on the extended lifting-line theory that will be treated further in Sec. 3-3-4, and the exact solution for the elliptic wing.
From Eqs. (3-80a) and (3-80b) the zero-lift angle is obtained with Eq. (1-23) as +1
2 f a("q) Y1 -?12d.1 a0=--
(3-81)
For approximate computations, the relationships of the elliptic wing can be applied to other wing shapes.
Quadrature method of Multhopp The simplest and most used method for the computation of the lift distribution of unswept wings according to the simple Here, the wing planform is an exact ellipse, which, e.g., becomes a circular disk for A = 4/ir.
142 AERODYNAMICS OF THE WING
lifting-line theory is that of Multhopp [60]. This method will be briefly sketched now: Starting from the expressions for the circulation distribution [Eq. (3-65)] and for the Fourier coefficients [Eq. (3-67)] in connection with Eq. (3-68), the summation expression, Eq. (3-67b), is introduced into Eq. (3-73). The induced angle of attack at the discrete stations
nv=costg
+1
with t%, =
(v= 1,2,...,M)
(3-82)
... , X)
(3-83)*
is then obtained in the form 1I
ai(77v) = a:, = b,q y, - 2:' b,,t y,,
(v = 1, 2,
n=1
with the universal coefficients __
b"v bvn
=
M+ 1 4 sin tv
(3-84a)
1 -(-1)v-n
sin 6 n
2(M + 1)
(cos 6v - Cos $n) 2
(v
n)
(3-84b)
By introducing expression (3-83) for the induced angle of attack into the integral equation for the circulation distribution Eq. (3-72b), the following system of equations is obtained for the values of -y,,: (bvv +.fv)7v
b, y.
aY +
(v = 1, 2,
... , M)
(3-85)
9t=1
This is a system of M linear equations for the M circulation values 7v = -y(rv) with v = 1, 2, ... , M. In Eq. (3-85), the following relationships apply: av = a(rly)
fv =
2b C100 0v
with cv = c(nv)
(3-86)
For M= 7 and M= 15, the universal coefficients are compiled in Tables 3-3 and 3-4. The values for (v -nf = 2, 4, ... are equal to zero. For the numerical solution of the system of equations, it is significant that the system of M equations can be split up into two systems of (M + 1)12 or (M - 1)/2 equations, respectively, which can be solved conveniently by iteration. By splitting an arbitrary angle-ofattack distribution into its symmetric and antisymmetric contributions, the
procedure of the numerical solution can be further simplified. For a continuous behavior of wing chord c(r1) and angle of attack a(77), usually M = 15 points along the
span are sufficient for all practical purposes. For discontinuous angle-of-attack distributions, as found for flap deflections, Multhopp recommends that one split off *Here, the prime () on the summation sign indicates that the term n = v is to be omitted in the summation.
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 143
and b,,,, for the computation Table 3-3 Universal coefficients of circulation distributions, for M = 7, according to Eq. (3-84)2 4
1
3
2
(7)
(5)
(6)
771,
0.9239
0.3827
0.7071
0.0000
b,.,,
5.2262
2.1648
2.8284
2.0000
1.0180 1.0972 0.0973 0.0180
0.0560 0.7887 0.7887 0.0560
V
n 2(6) 4(4) 6(2)
bm j
-
ft 1.8810 0.1464 0.0332
-
0.8398 0.8536 0.0744
-
1(7) 3(5) 5(3) 7(1 )
,
aAfter Multhopp [60].
the discontinuity stations before applying the above computational procedure; see Chap. 8.
Equations (3-85) are valid for unswept but otherwise arbitrary wings of sufficiently large aspect ratio (Ai > 3) and also for arbitrary angle-of-attack distributions.
Further results of the simple lifting-line theory In Fig. 3-25 the dependence of the lift slope on the aspect ratio is shown for a wing of elliptic planform. This result is approximately valid also for wings of different-for instance, trapezoidal-planforms.
To demonstrate the effect of the aspect ratio on the lift distribution, the circulation distributions over the span were computed for three rectangular wings with c = const and aspect ratios A = 6, 9, and 12. When A increases, the circulation distribution approaches more and more a rectangular distribution. Figure 3-26 demonstrates this fact. Illustrated are the local lift coefficients cl with reference to the total lift coefficient cL along the span. For A °° (plane problem), cl/CL = 1, and for very small aspect ratios (A -+ 0) the lift distribution is elliptic. This can easily be seen from Eq. (3-72b), which for ii --> 0 goes to «(r7) = al(r7) because f(ry) = 0. Hence, for a = const, aZ = const, meaning, from Sec. 3-2-1, that the circulation distribution is elliptic. To show the effect of wing taper on the lift distribution, Fig. 3-27 illustrates the circulation distribution for four trapezoidal wings without twist of aspect ratio A.= 6 and tapers X = ct/c,. = 0, a , a , and 1. The taper has a strong effect on the distribution of the local lift coefficients along the span. This can be seen in Fig. 3-28 in which the curves Cl/CL are shown. The strongly tapered wings have, near the wing tip, local lift coefficients that are considerably larger than the total lift coefficient CL. This fact is significant for the flow-separation characteristics of such
rU O
v
O Q O
O O
Oo
O Gli
G C
.--i CC
*
r
N
0
00 0 000*-i_ 000 O0 O O O0 1-4 X0 Cfl - *1 =
Cq
.-1
0O
O rNMN M C Cl CO 0 OO
M
0 0 .-i , C C c;
eM
M -4 GV 00 M cp [ M N c
00
dJ
OS L
l
CC'
NM O
In
ocici06666
Cl
M O O M 1 CO CO 00 -1 dl Cl - 0
+ GV CV Cl 0 0 0 0
CR
0
OeO.-+000
L^
of
O
v
t M rd ----M ,-,i
.p
0
N
..
E-
C
O CC O CV O M ++
N O
LeD CO N ZC O .- cfl c .-+ O O
(30
00_+-1000
eM
O CO
00
O
eM
-
ay XO N0
d
G
OV O O O
,
G
S.
O
CO
00 ODCQ0000 GV CJ 00000 00
_
G
c-
OC
0 00
O
O
O
I
O
1
Mto 1-40000
x 0 0 0 0 0 0
'4CplO
4
-
C
OO O eH 0 -- LO LID -4 10 000CO CO M 1-4O
I
CI
O
--
4CO00 00ad+
Q
C
144
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 145
1//
-1
12
12 10
0
A-6 9
03
12
02
0,1
05
0.3
97
0.6
00
09
to
Figure 3-26 Lift distribution ci/cL of rectangular wings without twist of aspect ratios e1= 6, 9, 12; also limiting curves for A. -- 0 and A -* -; cL,,, = 21r.
wing shapes at high lift coefficients. With increasing angle of attack, separation begins approximately at the station of maximum local lift coefficient, hence on strongly tapered wings close to the wing tips, but on rectangular wings in the middle of the wing.
3-3-4 Extended Lifting-Line Theory Method of Weissinger The method of the extended lifting-line theory, as explained in its basic aspects in Sec. 3-2-3, has been developed into computational procedures for practical applications by Weissinger [951. The basic equation for the a
=0
oas a 0.50
70
02
0,1
of
0.2
0.3
0.4
05
06
07
OF
09
10
Figure 3-27 Circulation distribution y of a trapezoidal wing without twist of taper ? = 0, a, ;, 1; aspect ratio a = 6; cL. = 27r.
146 AERODYNAMICS OF THE WING
Z=O 12
0.25
0.5 1.D
I
06
05
02
0
01
0,2
0,3
0//
0.5
07
D.6
1J-
0.6
0.9
119
Figure 3-28 Distribution of the local lift coefficients cl/cL over the span for trapezoidal wings without twist of tapers X = 0, 1, 1, 1; aspect ratio ,1 = 6; cj,,, = 27r. 4 2
determination of the circulation distribution using this procedure is Eq. (3-52). With
the dimensionless space coordinates , ri from Eqs. (3-la) and (3-1b) and the dimensionless circulation distribution y from Eq. (3-59), Eq. (3-52) takes the form
a(r!) = i2Zlim e0
4 E
Y(77) -
I
ti (5 P!
1'lY(i1') dl]'
(3-87)
-
(3-88a)
(?1-71)
J
with
' - tc (gyp _ ' )2+ ( -
(3-88b)
S ( , 7p 17) = 2
As shown in Fig. 3-29,
711)2
t,(77') is the position of the lifting line at a distance c14
from the leading edge and tp = tp(rj) the position of the control points. As explained in Sec. 3-2-3, the control points are arranged at three quarters of the local wing chord; thus xp = xC + c12. This choice of the position of the control points (three-quarter points) results from two-dimensional skeleton theory for which the position of the control point cL m = 21r. To introduce another value for cL can be changed by setting (see [831 ): )+01.C(') 27r
(3-89a)
2 P(77)
5(77)
= E p(77) -
(Tn) =
2n
(3-89b)
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 147
By introducing, according to [95] , the function K (71' i1,) = K ($p )1
_
?7')
71
(3-90)
Eq. (3-87) becomes
f
1
cc (27) = 2x:(71) +
2c
K(rl, -77')
dr1'
(3-91)
-1
where a1(77) is taken from Eq. (3-71 c).* The kernel function K(r7, T?') in Eq. (3-90) has been selected to be regular at r7' = r7, whereas the integrand in Eq. (3-87) is singular at this point. By a simple computation it can be shown that 1
K(r1, 1) _
_
2(sep
1
262(17)
(3-92)
The integral equation of the extended lifting-line theory now takes the following form, in analogy to Eq. (3-72b) of the simple lifting-line theory: a (n) = 2 [at (77) + z (ti7)l
(3-93)
1
(3-94)
where
The kernel function K(77, 77') depends exclusively on the geometry of the wing planform [83].
Wing with elliptic planform In Sec. 3-3-3 the wing with elliptic planform was treated by using the simple lifting-line theory. Now this wing shape will be computed using the extended lifting-line theory. A result of the simple lifting-line *2ai is the induced downwash angle far behind the wing, it - -.
Lifting line (quarter-point line) c(r,) Figure 3-29 Sketch for the extended liftingline theory.
----- Line of the control points tp (77)
148 AERODYNAMICS OF THE WING
theory, namely, that the elliptic wing without twist has an elliptic circulation distribution along the span, will be taken to apply here, too. Then the following study shows the difference in the total lift as determined from the simple and from
the extended lifting-line theories, respectively. Since in the present case of an elliptic wing the circulation distribution along the span after the simple theory is assumed to apply, the kinematic flow condition can be satisfied only at one point of the three-quarter-point line. Following Helmbold (27], the three-quarter point of
the wing half-span section will be chosen. The kinematic flow condition thus becomes, from Eq. (3-40),
c +'XE'(4, 0) = 0
(3-95)
Here tp = xp/s is the dimensionless distance of the control point from the c/4 line. We shall not perform the calculation in detail, but the induced downwash angle at the wing. middle section becomes, according to Glauert [23], for elliptic circulation distribution,
a.u(p,0)_-1 1-{-2
E ai
SAP+1 SE
?L
(3-96)
Here E is the complete elliptic integral of the second kind with module 1/ /p + 1, and cx = CL/7rA is the induced angle of attack introduced earlier. To simplify the computation, an approximate expression can be given for Eq. (3-96) that no longer contains the elliptic integral (see [27] ). With Eq. (3-95), this expression becomes 2
a = .1
1 -{-
z sep
z
2
CL 7r "l
(3-97)
The position of the three-quarter points is obtained from Eq. (3-89b) with 0, and further with A = 4b/7rc,. from Eq. (3-9) and k = 7rA/cL- from Eq. (3-75b)* as P
_
CLao
27r
Cr_ 2 b
Irk
By introducing this expression into Eq. (3-97), the lift slope is found to be
da
k-+1 +1
(3-98)
In Fig. 3-25 the lift slope after this formula is presented for ci = 21r, that is, k = /1/2, versus the aspect ratio. For comparison, the curve according to the simple lifting-line theory [Eq. (3-80b)] is also shown. The difference between the two theories is similar to that for the rectangular wing of Fig. 3-32. Equation (3-98) for the extended lifting-line theory evolves from Eq. (3.80b) for the simple theory by formally replacing k by k2 + 1. In an analogous way, the
Fourier coefficients for the circulation distribution of the twisted wing can be modified to comply with the extended lifting-line theory. Thus Eq. (3-77) takes the form `See footnote on page 118.
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 149 z
an =
1
yk2+10, ; n
c.. (t$) sin to sin -10 d 0
2
(3-99)
0
The usefulness of this formula has been confirmed by numerous examples. The rolling-moment coefficient cMX is obtained in closed form by introducing into Eq. (3-66b) the value for a2 from Eq. (3-99) and observing that 77 = cos 6, Eq. (3-65b), as =1
cMX
This is
k'
Tc
J
a (,1) i1 y 1 - 1]2 d )j
(3-100)
-1
a quite simple equation for the determination of the rolling -moment
coefficient.
Quadrature methods For the numerical evaluation of Eq. (3-93), Weissinger [95] presented a refined quadrature method analogous to that of the simple lifting-line theory (method of Multhopp). This method will not be presented here; instead, reference is made to [95]. Comprehensive sample computations using the Weissinger method have been conducted by de Young and Harper [103].
Further results of the extended lifting-line theory In Fig. 3-30 the circulation distribution versus the span at a= 1 is demonstrated for the rectangular wing without twist of aspect ratio 1= 6. For comparison, the curve using the simple lifting-line theory is also given. This figure shows that the extended theory produces a smaller lift than the simple theory for the same angle of attack. Furthermore, Fig. 3-31 illustrates the lift distribution c1/CL of the same wing. The extended lifting-line theory produces a somewhat less full distribution curve than the simple lifting-line
theory. Both of these statements are typical for the extended lifting-line theory. The lift slope of rectangular wings after the extended and after the simple lifting-line theory are compared in Fig. 3-32. The difference between the curves is
Figure 3-30 Circulation distribution of the rectangular wing without twist of aspect ratio A = 6 for a = 1; cL. = 27r. (1) Simple lifting-line theory. (2) Extended lifting-line theory.
150 AERODYNAMICS OF THE WING
000
0,2
02
0,1
03
0, 41
0.7
0.5
0, 5
88
0.9
10
1
Figure 3-31 Lift distribution cllcL of the rectangular wing without twist of aspect ratio A1= 6; c',. = 2ir. (1) Simple lifting-line theory. (2) Extended lifting-line theory:
rather small for large aspect ratios A. It is considerable, however, for small values of .1. The limiting values of dcL/da for :1 --> 0 of the simple [Eq. (3-101a)] and the extended [Eq. (3-101b)] lifting-line theory* are dcL
zA
doL
z
da
2
(A ->-
0)
( 3 - 101 a)
(A -- . 0)
1
(3-101b)
The two limiting values are also indicated in Fig. 3-32; see also Fig. 3-25. ' For A -> 0, a(r1) = ai(r1) in the simple lifting-line theory; for the extended theory, however, a(77) = 2ai(n) because K(n, ra') = 0.
2
2
3
1
5
6
7
d
9
10
17
72
Figure 3-32 Lift slope dcLldca of rectangular wings vs. aspect ratio A; cL = 27r. (1) Simple lifting-line theory. (2) Extended lifting-line theory.
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 151
In Fig. 3-33, results for a trapezoidal wing, a swept-back wing, and a delta wing with aspect ratios between A = 2 and 3 are presented. The geometric data for these three wings are compiled in Table 3-5. Figure 3-33 gives the circulation distribution
for the wing without twist at a = 1. For the trapezoidal wing, the curve using the simple lifting-line theory has been added. In this case, too, it lies above the curve for the extended lifting -line theory. For all three wings, results are shown of the lifting-surface theory, which will be discussed in Sec. 3-3-5. Agreement between the extended lifting-line theory and the lifting-surface theory is good. The values for the lift slope and the neutral-point displacement, together with additional aerodynamic coefficients yet to be discussed, are compiled in Table 3-5.
Transition from extended to simple lifting-line theory It should be shown that the extended lifting-line theory may be transformed into the simple lifting-line theory for large aspect ratio. In performing this limit operation, according to Truckenbrodt [83], the control-point line p(r?) for the kinematic flow condition of the extended lifting-line theory must be shifted toward the lifting line t1(r7), tp -- 1, or 5. -* 0 (Fig. 3-29). Thus the kinematic flow condition becomes « , (a > 0, n) + oc (n) = 0
(A = large)
(3-102)
where S(ri) is defined by Eq. (3-89b). The dimensionless induced downwash velocity according to Biot Savart of a lifting line normal to the incident flow becomes, for a
control point p = xp/s = 8 that lies very close to the lifting line,
-a,,(a - 0, n)
= ai (77) + i 2 (77)
(3-103)
The first term of the right-hand side signifies the contribution of the free vortex, the second term that of the bound vortex. Since, from Eqs. (3-89b) and (3-70a), r5 (?7) = 1/f(r7), it follows from Eq. (3-102) that C(n) = ai(n) + f('r1)Y(17)
(3-104)
10
08
I aTrapezoidal wing L _11
bSwept-back wing
3 3 i
02
0
92
0.#
06'
09
10 0
0.2
09
,o0
71
Figure 3-33 Circulation distribution of three wings without twist of Table 3-5; a = 1; cL = 27r. (a) Trapezoidal wing; cp = 0; A = 2.75; x = 0.5. (b) Swept-back wing; yp = 50°; A = 2.75; = 52.4°; A = 2.31; a = 0. Curve 1, simple lifting-line theory of X = 0.5. (c) Delta wing; Multhopp. Curve 2, extended lifting-line theory of Weissinger. Curve 3, lifting-surface theory of Truckenbrodt.
6
OA
f
4
O
,4n
110 r. 4 Ci
cq
M
CN GV
C
e^
O
ci
C
O C
O
Z
xU
8
Cc
Cpl v G
co cc C O ri
{
-
cd
H
C
V-4
LO
Ln 00
O O
t
Z
I
oN
01 G
(
H H
-
~
N
H
0
M
O
1
L^.
N !:7
C L^
O
O
00
c7
N N
tr.
GU
H
C1
N C?
C
O O
O
CO
N
to
00
00 oc
M CJ O C I
I
O 00
c
Or
N
O
O
GV
C
w O O .C
V N
Cs II
a' , U
.
%V
y
N
v O 6)
152
Ow
G)
Cl-
Q
ry
C] e,..
~
; C)
cn
y
=
`
"..
z
FNr
U d
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 153
Thus it has been shown that the extended lifting-line theory is reduced to the simple lifting-line theory for very large aspect ratios. This limit operation can be applied to the yawed wing according to [831 and leads to the earlier-stated theory of the yawed wing of Weissinger [94]. The limit transition from the extended to the simple lifting-line theory is also treated in [83]. Based on this, Laschka and Wegener [83] developed a quadrature procedure for computing the lift distribution of swept-back wings of large aspect ratio.
3-3-5 Lifting-Surface Theory General formulation of the procedure The integral equation for the computation of
the circulation distribution according to the wing theory was presented as Eq. (3-43). With the dimensionless areal coordinates l;, r1 of Eqs. (3-la) and (3-1b), the dimensionless kernel function, and the local angle of attack of the cambered lifting surface, that is, with G 9
(3 -105a)
b U00
a (5, 7) = aF - a
(3-105b)
Eq. (3-43) becomes 1
2;r
lim
9($''7; ?1')
2 g
d.,
(3-106)
'
6
For the areal distribution of the vortex density k(x, y), the following product expression according to Truckenbrodt [84] is introduced: k (:2', /)
=
1 c
S
U.
(3-107)
c., (I) h,, (x) )r=(J
Here the cn(y) represent the spanwise distribution and the hn(x) the chordwise distribution of the lift. Introducing Eq. (3-107) into Eq. (3-42a) yields
Xr(Y ')
with
H" (x.11; y')
(3-108)
S c,, (y') c(y') H,, (.r.
G (x,
C h,,(x') (1 +
CV) J
1 (:L
xf(v')
-
x- ) (
.i'')=
'/
tl:a
(3-109)
i/')=
According to the skeleton theory of Sec. 2-4-2, the distribution function over the wing chord is expressed as
z
sing
(n = 0, 1...)
(3-110)
154 AERODYNAMICS OF THE WING
where, from Eq. (2-62), X- 1( 1 + cos P)
c(y)
Here c(y) is the chord and x f the position of the wing leading edge at section y. For the values n = 0, 1, and 2, the distributions are given in Fig. 3-34; see Fig. 2-27. The functions ho and h 1 of Eq. (3-110) have been normalized to produce the local lift and moment coefficients relative to the c/4 point through integration over
the chord after introduction into Eq. (3-107) [see Eqs. (2-54) and (2-55)]. The result is cr(y) = co (Y)
(3-112a)
Cm (y) = 14C, (Y)
(3-112b)
The explicit expression for the functions H (x, y; y'), which are dependent only on the wing planform, is obtained by introducing the distribution functions hn of Eq. (3-110) into Eq. (3-109). Writing the function G of Eq. (3-108) in dimensionless form and considering Eq. (3-105a) leads to
977') Zn=0H. .4
with
(3-113a)
77; 77') fn (q')
(3-113b)
2b
Introducing this function g(, n; rl') into Eq. (3-106) finally yields
1 I lim
y7) =
x(
11=0 e-+0
?E H. (, 7J
Hn , r1) f>a (77)
/)'
d f7'
(3-114)
-1
41
3
m=0
2
1
0
_2 02
X-
Of
06'
0
10
Figure 3-34 The functions h 0 , h 1, and 11 2 for the lift
distribution vs. wing chord, from [84); see Eq. (2-88).
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 155
This is a system of integral equations for the (N+ 1) functions f,(n), (n = 0, 1,
... ,
N). Choosing (N+ 1) distribution functions by satisfying the kinematic flow conditions on (N+ 1) control-point lines along the span, (N+ 1) distribution functions can be determined. After having determined the functions fo(ri),fl (r7), and so on, from this system of equations, the lift distribution is obtained from* cr(17)c(77)
(3-115a)
= fo (77) = Y('?)
2b
and the moment distribution (moment coefficient relative to the c/4 point) from C.(2bc(?7)
= 4 /.(n)
(3-115b)
The resultant of the pressure distribution of the lower and upper surfaces (load distribution) follows in analogy to the expression Eq. (3-107) and the relationship Eq. (3-44) as A cp =
=
P1 qC0
N
2b
h" (
C07) n
(3-116)
) A 07)
In the following section this procedure will be explained through numerical execution.
Method of Multhopp and Truckenbrodt Multhopp [62] and Truckenbrodt [84] independently developed methods for the numerical evaluation of the method outlined above. In either publication the two distribution functions ho and h1 mark the basic approach. Multhopp puts the two control-point lines at 34.5 and 90.5% of the local chord. Truckenbrodt prefers positions of the control-point lines on the
trailing edge and the c/4 line of the wing. A comparison of the best known lifting-surface theories is given in [18].
The explanation of the computational procedure now to be given follows closely [84]. As already stated, only two distribution' functions over the chord are chosen, limiting the correlation functions of the method to Ho. and H1. With new designations, 97;97')
(3-117a)
77') =70,97;97')
(3-117b)
Ho( ,97;97')
4H, Eq. (3-114) becomes
1
1
77) = 27 lim e-;0 2 s
2 i(
p, 77;
77)Y(/) -
-J
1
7(fir,,9) ;97)it(97)
97
(97 - 97 )"
Y(1')
d77'
,
' )2
(3-118)
-1
*fa (n)
(3-59).
is identical to the dimensionless circulation distribution y(r7) = T /b U. from Eq.
156 AERODYNAMICS OF THE WING
where fo and fl are replaced by y and µ as in Eqs. (3-115a) and (3-115b). This equation must be satisfied for two values of gyp, namely, for
p = t25 = I
and
tp = tioo = to
(3-119)
Here 25(ri) stands for the c/4 line and tioo(?7) for the trailing edge. The two functions y(77) and 4(r1) of Eq. (3-118) are now to be determined. The angle-of-attack distribution a(p, rl) is given directly by the wing geometry, and the kernel functions i and j are given indirectly as functions of the wing planform. Only the angle-of-attack distribution values on the c/4 line and on the trailing edge are required in Eq. (3-118). Wagner [91] expanded the described lifting-surface method to more than the
two distribution functions over the span ho and hl. Accordingly, the number of control-point lines must be increased. In selecting five distributions ho through h4, in [91 ] the control-point lines are laid on the leading edge, the one-quarter, one-half, and three-quarter point lines, and on the trailing edge. Quadrature methods The numerical solution of Eq. (3-118) is accomplished through an extended quadrature method, following Multhopp's procedure for the lifting-line theory. Because of the considerable extent of the computations, use of an electronic
computer is necessary. Further possible solution procedures for the equations of lifting-surface theory are reported by, among others, Kulakowski and Haskell [12], Cunningham [12], and Borja and Brakhage [9]. The panel method of Kraus and Sacher [46] should also be mentioned.
Lift distribution After having obtained the values y(rl) and µ(r1) by solving the system of equations, the lift distribution along the span follows from Eq. (3-.115a). The load distribution over the wing chord is consequently derived from Eq. (3-116) [compare also Eqs. (2-87) and (2-88)] as zi e22 (X,11)
= C(1b7) [ho(X)
y (?7)j
(3-120)
where the functions ho and h1 are taken from Fig. 3-34.
Lift, rolling moment The total lift coefficient, the lateral distance of the lift center of a wing-half, the lift coefficient of a wing-half, and the rolling-moment coefficient may be determined from the formulas of Table 3-1. Pitching moment The local pitching moment about the local c/4 point is given by Eq. (3-115b). In Fig. 3-35, x25(y) designates the c/4 line and x1(y) the line of the local aerodynamic center. It follows, then, that the moment of a wing section y
about the c/4 point is dM= -dx1 dL. By setting dM = c,,go,c? dy and dL = c1gc.,c dy, the distance between the local aerodynamic force and the local c/4 point becomes, with the help of Eqs. (3-115a) and (3-115b), Jx1(17)
cm(17)
_ x1(rl)-x25(?7) _
4(77)
c(11)
c1(17)
c(17)
7(77)
(3-121)
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 157
Section y
Figure 3-35 Computation of the pitching mo-
ment. xl(y) = position of the local aerody-
X25
namic centers. x,, (y) = local c/4 line, N = neutral point of total wing.
- XI
The pitching moment of the whole wing is obtained from the contributions of the individual sections to the moment about the y axis dM = xl dL, resulting in
- f xl(y) dL = -
8
4
131 =
-S
f [x25 (Y) + J x1(v) ] dL -S
Hence the pitching-moment coefficient cM =117/q.,Acu, with CA as the reference wing chord [Eq. (3-5b)], becomes 1
r
cal = __ i f Cy(71)
-t
x250) C
(77)
Cu
dC/1
(3-122)
Finally, the neutral-point position of the whole wing is obtained from Eq. (1-29).
Results of wing theory and comparison with tests The examples computed in this section include rectangular, trapezoidal, swept-back, and delta wings. Earlier, in Fig. 3-33, circulation distributions of a trapezoidal wing, a swept-back wing, and a delta wing, all without twist, were presented for several computational methods.* The geometric data of these three wings are compiled in *Computation of the lift distribution of delta wings has also been treated by, among others, Garner [17].
158 AERODYNAMICS OF THE WING
Table 3-5. From Fig. 3-33 it was concluded that the difference between the extended lifting-line theory and the lifting-surface theory is quite small. In Fig. 3-36, the lift distribution of three wings without twist is illustrated in the form clc/CLCm versus the span coordinate. In this kind of presentation, the computational results are practically identical. The lift slopes dcL /da of these three wings, based on various theories, are compiled in Table 3-5. Neither the simple nor the extended lifting-line theory allows determination of the local neutral-point position because these methods require that the local neutral point be fixed on the lifting line (c/4 line). Application of wing theory according to Eq. (3-121) is required for local neutral-point determination. In Fig. 3-37, the local
neutral-point positions over the span are plotted for the three wings of Fig. 3-36; see also Table 3-5. The local neutral points of the unswept wing lie before the c/4 line over the whole span. On the other hand, the local neutral points of both of the swept-back wings lie behind the c/4 line near the wing root and before the c/4 line
in the range of the wing tips. The resulting total wing neutral points and the geometric neutral points according to Eq. (3-7) are also shown in Fig. 3-37. The distance between aerodynamic and geometric neutral points is very large, particularly on the delta wing. The numerical data for this displacement are compiled in Table 3-5. Comparisons between theoretically and experimentally determined local neutral points of swept-back wings have been published by Hickey [29]. Additional test results on a series of delta wings from [85] are shown in Fig. 3-38. They have aspect ratios from 1 to 4. Lift slope dcL/da and neutral-point displacement J xN/cu are plotted against the aspect ratio. Here, too, agreement between theory and experiment is good. In Fig. 3-39, the theoretical lift distribution over the span of a delta wing is compared with measurements of Kraemer [85]. Agreement is very good for angles of attack up to about a = 5°. Flow separation from the outer parts of the wings b Swept-back wing 72
70
0.8
10 0.2
02
0.
00
0.8
7.00
02
0#
49S
0.8
100
02
00
00
t8
10
Figure 3-36 Lift distribution clc/cLcm of three wings without twist of Table 3-5 and Fig. 3-33, cL, = 2n; cm = Alb = mean wing chord. Curve 1, simple lifting-line theory of Multhopp. Curve 2, extended lifting-line theory of Weissinger. Curve 3, lifting-surface theory of Truckenbrodt. Curve 3a, lifting-surface theory of Wagner (five-chord distributions).
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 159 c Delta wing
b Swept-back wing
a Trapezoidal wing
Local neutral-point position
Figure 3-37 Local neutral-point positions of three wings without twist of Table 3-5; cl = 27r; lifting-surface theory. Curve 1 according to Truckenbrodt, curve 2 according to Wagner. N25 = geometric neutral point of the whole wing; N = aerodynamic neutral point of the whole wing.
Figure 3-38 Lift and neutral-point positions of delta wings of various aspect ratios with taper
2
3
X = e;
cp = 0.68cy.
Comparison
of
theory and experiment from Truckenbrodt. Profile NACA 0012. (a) Lift slope. (b) Neutral-point displacement; J x N = distance of aerodynamic neutral point N from geometric neutral point N2$.
160 AERODYNAMICS OF THE WING 12
Measurements 10
Azt L N 0S
-
' T
I
Theory
0.0
0.2
0
02
a
2y/
oe --
0.8
10
Figure 3-39 Lift distribution c1c/2bcti of a delta wing of aspect ratio ,i= 2.3; profile NACA 65A005 according to measurements of Kraemer; comparison with lifting-surface theory of Truckenbrodt [84].
causes strong deviations of the measured lift distribution from theory for the large lift coefficients. The local neutral-point positions are compared with theory in Fig. 3-40. Here again, satisfactory agreement is found. For the same wing, the measured pressure
distributions for a few sections along the span are compared in Fig. 3-41 with theory according to Eq. (3-120). In general, the agreement is satisfactory. The
340 Local neutral-point positions of a delta wing of aspect = 2; comparison of theory ratio [84] and measurements [NACA TN 1650]. Profile NACA 0012. Figure
V
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 161 =0396
U0.
Itl
NA CA 0012
0
0,2
a#
x
01
as
10
C
rj=060//
7f=0,0 1
Theory
IL
T_ 92
1-
Of
-x
I
I
06
09
2
10
04'
x
as
09
70
C
0
02
04t
x
06'
OR
10
0
02
04f
x
0.e
08
C
1.0
Figure 3-41 Pressure distribution over wing chord for the delta wing of Fig. 3-40; theory [84] and measurements [NACA TN 1650]. cL = 0.585; profile NACA 0012.
deviations between theory and experiment can be partially explained by the fact that theory is valid for infuiitely thin profiles and, therefore, does not account for the profile thickness.
Now, comparisons of experiment and theory will be made for unswept wings (rectangular wings). Figure 3-42 illustrates lift slope versus aspect ratio. The theoretical curve has been computed according to the multipoint method of Scholz [77] ; it is in agreement with the curve for the extended lifting-line theory in Fig. 3-32. The test points from several sources follow the theoretical curve well. In Fig. 3-43, the neutral-point positions for the same series of rectangular wings are plotted against the aspect ratio. The neutral-point shifts considerably upstream of the c/4 line when the aspect ratio A is reduced. Also included are measurements on rectangular plates that are in good agreement with theory. Results for a series of swept-back wings of constant chord are presented in Fig. 3-44. For both lift slope and neutral-point position, the measurements are in good agreement with theory. Note particularly that the lift slope of the swept-back wing, especially with a large aspect ratio, is considerably smaller than that of the unswept wing, p = 0. This reduction of the lift slope through sweepback can be assessed particularly well by considering the swept-back wing of infinite aspect ratio. Figure 3-45 depicts a span section b of an unswept and of a swept-back wing of infinite span. The section of the unswept wing produces the lift
L = I U!bccL--a Let the swept-back wing with sweepback angle
be inclined to make, in the plane
of the incident-flow direction U., the angle of attack a equal to that of the
162 AERODYNAMICS OF THE WING
Gottingen measurements 0
Profile Clark Y after Zimmermann Flat plate
°
I aft
Profile Go 409'f Wi:
r
ter
Flat plate
Figure 3-42 Lift slope of rectangular wings of various aspect ratios A ; comparison of theory and experiment. Theory from Scholz (multiple-points theory) [77]. Measurements from Wieghardt, Scholz, and NACA Rept.
Profile NACA 0015 after Scholz
1
V
I
Z
7
3
7
A
431.
unswept wing. Then, in the plane normal to the leading edge, the angle of attack is
a* = a/cos cp. For the lift of the swept-back wing, only the velocity component normal to the leading edge, U. cos gyp, is effective. Thus, the cross-hatched surface portion of the swept-back wing has a lift
L* =
2
2 ' (U cos cp) bCCLoo
°L
cos
99
2
U2 bcc'L a cos O M
Hence, the lift coefficient of the swept-back wing is CL = L*/bc(p/2)U! _ CL a cos gyp, whereas that of the unswept wing is (CL ),,= O = CL oo a. The two lift slopes are thus related by C11CL
dC
da
(da
T-0
COST
(3-123)
0.25
0.20
0.05
b
1
2
3
A ---
5
5
Figure 3-43 Neutral-point position of rectangular plates; comparison of measurements and theory, from Scholz [771.
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 163 q
Theory
/ SP=O
xA Measure ments
N
SP
to
Nzs
S
Figure 3-44 Lift and neutral-point position of swept-back wings of con-
stant chord and various aspect ratios; sweep-back angle p = 45°; comparison of theory and measure-
ments from Truckenbrodt. Profile NACA 0012. (a) Lift slope. (b) Neutral-point displacement.
cas9'
Figure 3-45 Geometry and velocity components explaining the lift of swept-back wings of infinite span.
164 AERODYNAMICS OF THE WING
This relationship has been confirmed experimentally by Jacobs [37]. It is also valid, to good approximation, for the pressure distribution along the chord.
To show the effect of the sweepback angle on the lift slope, Fig. 3-46 illustrates, for swept-back wings of constant chord, the lift slope dcL /da versus sweepback angle and aspect ratio according to de Young and Harper [1031. For large aspect ratios 11, the decrease in lift slope with increasing sweepback angle is considerably stronger than for small aspect ratios. For r1 00, the cos .p law of Eq. (3-123) is also shown for comparison. The sweepback angle also strongly affects the circulation distribution over the span. This is apparent in Fig. 3-47, which demonstrates the circulation distribution along the span of a rectangular wing (cp = 0) and a swept-back wing (gyp = 45°). The maximum value of the lift distribution of the swept-back wing is found at the outer
wing portion. Sweepback causes a shift of the station of maximum local lift from the middle toward the outer end. Hence, the separation tendency of the swept-back wing is increased at large angles of attack compared with the. unswept wing. In this respect, sweepback produces unfavorable effects similar to a strong taper of an unswept wing (see Fig. 3-28). Let us deviate from the wing theory discussed here. A swept-back wing theory has been developed by Kuchemann [48] that is not based on the Bimbaum normal distribution over the chord. This method, partially empirical, takes into account the wing thickness and the boundary layer, and also certain nonlinear effects. It therefore agrees very well with test results.
A cylindrical body in a flow that is inclined against its generatrix (yawed cylinder) may be subject to- complex three-dimensional flow processes in the boundary layer. These are of considerable importance to the aerodynamic properties of swept-back wings. At larger lift coefficients, both yawed and swept-back wings
undergo a strong pressure drop toward the rearward wing tip on the suction side 2Z
A cc
0
t l
a
8 6
3
2
I
1
-50°
-900
-30°
200
-10°
0°
10
20°
300
M°
30°
Figure 3-46 Lift slope of swept-back wings of constant chord vs. sweepback angle p and aspect ratio, from [103] ; extended lifting-line theory. Curve forA = oo: cbs cp law from Eq. (3-123).
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 165
Figure 347 Circulation distribution and distribution of the local lift coefficients over the span for two wings of constant chord; aspect ratio it = 5, sweepback V = 0 and p = 45°; cL, = 27r; 1; lifting-surface theory of Truckenbrodt [84]
.
near the wing nose, as shown in Fig. 3-48. In this figure, the isobars for the suction
side of a yawed, inclined wing are seen. The fluid, decelerated in the boundary layer, follows this pressure gradient and consequently a strong cross flow in direction of the rearward wing sets in. Measurements of Jones [37] and Jacobs [37] have shown that, therefore, a marked thickening of the boundary layer is caused on the rearward wing tip and, as a consequence, a premature flow separation results. In airplanes with swept-back wings, this departure of the boundary layer toward the
outside causes separation to occur first at the outer portion of the wing, in the
Figure 3-48 Evolution of cross flow in the boundary layer of a yawed wing (swept-
back wing). Curves of constant pressure (isobars) on suction side of the wing.
166 AERODYNAMICS OF THE WING
range of the aileron. This in turn causes the feared "roll-off" toward the stalled wing. This initiation of separation at the outer portion of the wing, and thus the undesirable "roll-off," can be avoided by providing the wing with a boundary-layer fence (stall fence). This is a thin sheet-metal wall on the suction side of the front wing portion that prevents cross flows in the boundary layer. Liebe [13] describes the improvements in stall behavior by this provision. The work of Queijo et al. [13] includes results of comprehensive measurements on the improvement of the
aerodynamic properties of a wing by means of boundary-layer fences. Compare also the basic studies of Das [13]. Poisson-Quinton [68] makes a contribution to the theoretical and experimental
investigations on the problem of the aerodynamics of folding wings (wings with adjustable sweepback angle).
For wings of small aspect ratio, an essential simplification of wing theory is feasible, according to a proposal first made by Jones [36]. The basic concept of this theory is that the perturbation velocities in the x direction in the flow field about a slender wing are small compared to those in the transverse directions (y and z directions). The potential equation is then reduced to that of a two-dimensional flow in the yz plane (slender-body theory). In connection with this theory, the method of Lawrence [54] for the computation of the lift distribution of wings of small aspect ratio and the treatment of very strongly swept-back wings according to [59] should be mentioned. The application of slender-body theory to wings of extremely large thickness (covering of the wing contour with singularities) has been attacked by Hummel [341.
3-3-6 Nonlinear Wing Theory The wing theory treated
so
far establishes
a linear correlation between lift
coefficient and angle of attack. It is designated, therefore, linear wing theory. It is known from experimental investigation that for wings of very small aspect ratio, A < 1, lift coefficients cL are considerably larger than those obtained from linear theory when plotted against the angle of attack. Figure 3-49 illustrates this behavior for rectangular wings of aspect ratios A = 0.2, 0.5, 1.0, and 5.0 as compiled by Gersten (21]. The dashed theoretical curves represent linear theory as discussed earlier. Although linear theory produces the right lift slope (dcL/da)a=o even for small aspect ratios, strong deviations of the measurements from linear behavior are already obvious for small angles of attack. All wing theories discussed so far are based on the concept that bound and free
vortices lie in the same plane. A linear relation between lift and angle of attack is the necessary consequence.
This much simplified vortex model must be abandoned for a theoretical explanation of the nonlinear relation between lift and angle of attack. A first trial in this direction was made by Bollay [8]. He used a vortex model similar to Fig. 3-50a in which the free vortices no longer lie in one plane but rather are shed in the downstream direction from the wing tips under the angle a/2 with the wing plane. Bollay assumes that the bound vortices are constant over the span. Gersten [21]
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 167 0.8
0.6
0.2
a0 0.8
A= 05 o_ .
06
2 0 0.00
o
I
_j
r
.
0
00,
1 0.6 2
/000,
e 0.2
cj
I
A=10
cow
2
0.8
4
47
Figure 3-49 Measured lift coefficients CL vs.
11=5.0 0.2
a 0 8°
72 °
16°
20°
24°
angle of attack a for rectangular wings of aspect ratios A = 0.2, 0.5, 1.0, and 5.0. Curve 1, linear theory of Scholz. Curve 2, nonlinear theory of Gersten.
refined this vortex model by prescribing a variable circulation distribution over the span (Fig. 3-50b). The CL(a) curves based on this theory are given in Fig. 3-49 as solid lines. They are in very good agreement with this theory (see Winter [102] ). By the same theory, pitching moment, induced drag, and lift distribution along the span have also been determined. Agreement between tests and theory is good in these cases, too. Furthermore, the nonlinear theory has been extended by Gersten to arbitrary wing shapes. It represents an extension of the lifting-surface theory of Sec. 3-3-5 to the nonlinear angle-of-attack range. The cL(a) curves as determined
168 AERODYNAMICS OF THE WING
a
Ya
b
C'
z Figure 3-50 Vortex model of nonlinear wing theory. (a) Vortex model of Bollay. (b) Vortex model of Gersten.
from this theory and the comparison with test data are shown in Fig. 3-51 for a swept-back and a delta wing. It is known from test results that the aerodynamic coefficients of wings of small aspect ratio are strong functions of the wing leading-edge design. This is true particularly for swept-back and delta wings with sharp leading edges which, even at very small angles of attack (a = 3°), promote flow separation from the leading edge of the kind shown in Fig. 3-52. Starting at the wing tips, two vortex sheets form on the two leading edges that roll up into free vortices when floating downstream.
This process was first discussed by Legendre [55] and has been treated in
°
Nonlinear theory o
I
i Linear theory
IL
0 1.0r
Figure 3-51 Lift coefficient of sweptback wings with sharp leading edge and small aspect ratio vs, angle of attack. (- - -) Linear theory from Eq.
0,2
12'
16
090
24
) Nonlinear theory of (3-101b). ( (o) Measurements. (a) SweptGersten. back wing .1 = 1, X = 1, tp = 45°. (b) Delta wing .4 = 0.78, X = 8.
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 169
Figure 3-52 Bursting of the free vortices of a delta wing according to Hummel. Aspect ratio r = 0.78, taper x = 0.125. (a) Vortex formation shown schematically. (b) a = 20°, (3 = 0°, no bursting. (c) a = 30°, g = 0°, bursting of the vortices at large angles of attack. (d) a = 20°, 0 = -10°, bursting of one vortex of yawed wing. (e) a = 20°, p = 0°, bursting of one vortex through an artificial pressure rise.
numerous other publications [4, 10, 33, 55, 98]. The roll-up of vortex sheets has been studied theoretically by Roy [71] and by Mangler and Smith [58, 80]. Roy established details through numerous flow-pattern photographs. Under certain circumstances, a striking change in the structure of the rolled-up vortex sheets can be observed that can be termed bursting of the vortices. Figure 3-52b-e shows smoke pictures of this phenomenon from Hummel [33]. The bursting of vortices occurs (1) at large angles of attack in symmetric incident flow (Fig. 3-52c), (2) at
170 AERODYNAMICS OF THE WING
the vortex of the upstream-turned side of yawed wings (Fig. 3-52d), and (3) when an obstruction is placed into the vortex flow (Fig. 3-52e). Naturally, the bursting of
vortices has a strong effect on the aerodynamic properties of the delta wing; compare [4, 33]. These processes affect lift and pitching moment as well as drag.
Further investigations of nonlinear effects on wings of small aspect ratios, especially on delta wings, are reported in [19, 32, 57, 67]. A very recent survey of the aerodynamic properties of slender wings with a sharp leading edge has been given by Parker [661.
3-3-7 Maximum Lift of Wings of this chapter, the fluid was considered to be incompressible and inviscid when establishing the theory of lift. The wing theory based on this concept is in good agreement with measurements as long as the angle of attack is small to moderate; see, for example, Figs. 3-38, 3-39, 3-42-3-44, and 3-49. Only in the range of large angles of attack does the effect of friction have In the previous sections
significance for the lift. In particular, the maximum lift of a wing is not only determined by its geometry, but it is also considerably affected by friction. Determination of the maximum lift of a wing by strictly theoretical methods is not yet possible. Cooke and Brebner [11] report on flow separation from wings in general terms. Schlichting [73] presents the aerodynamic problems of maximum lift of wings in comprehensive form.
From measurements it is known that the maximum lift coefficient is strongly dependent on the geometric profile parameters (thickness, camber, nose radius) and on the Reynolds number. In Sec. 2-5-1 this relationship was discussed briefly; see, for example, Figs. 2-39 and 242-2-44: These previously reported results should be
supplemented by the statement that the maximum lift of an unswept wing is essentially a problem of two-dimensional flow. A large aspect ratio of unswept wings of finite span cannot have an important effect on flow separation and consequently on the maximum lift because in this case the flow over the major portion of the wing deviates only a little from plane flow. Quite different are the conditions for wings of small aspect ratio. Here the flow around the wing tips reaches to the middle of the wing. For strongly swept-back wings, which includes delta wings, the flow conditions are particularly complex because the leading edge acts in a similar way as the tips of an unswept wing. For these kinds of wings, even the attached flow is much harder to assess than that for unswept wings, because the flow directions in the boundary layer may deviate from that of the outside flow (departure of the boundary layer to the wing tips, boundary-layer fence). Contrary to unswept wings, the flow over strongly swept-back wings without twist separates locally first at the wing tips because the lift load has its maximum there (see Fig. 347). When the angle of attack increases, the separated region expands inward in span direction. This behavior is discussed in more detail in [26]. A very comprehensive
compilation of material on this behavior of swept-back wings at large angles of attack and high Reynolds numbers has been given by Furlong and McHugh [161.
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 171
The effect of the aspect ratio and the sweepback angle on the maximum lift coefficient will now be examined using some test results. In Fig. 3-53, results are plotted for the maximum lift coefficient of rectangular wings and swept-back wings of constant chord (p = 45°). The Reynolds numbers of
these measurements are Re 106. Figure 3-53a confirms that the maximum lift coefficient for A > 2 is almost independent of the angle of attack. For very small aspect ratios, CL max is somewhat larger than for large aspect ratios. Particularly noteworthy in Fig. 3-53b is, for aspect ratios A < 2, the strong increase to values of a - 30° in the angle of attack for which the maximum lift coefficient is obtained. In Fig. 3-54 curves are given for the lift coefficients of a series of delta wings plotted against the angle of attack. When the aspect ratio
7.2
10
1
0s 7
3
X
J 0.6 0.4
02
a 0
35°
30°
25°
Figure 3-53 Maximum lift coefficients of rectangular wings (gyp = 0) and swept-back wings of constant chord (gyp * 0), Reynolds number Re 106. (a) Maximum lift coefficient CLmax Vs-
aspect ratio A. (b) angle of attack a for CL max vs. aspect ratio .i. Curve 1, p = 0°; profile NACA
70°
0015, from Bussmann and Kopfermann [25]. 2, p = 45°; profile NACA 0012, from Truckenbrodt [85 ]. Curve 3, tp = 0°; 6 - 0.10,
'.
Curve f
13
9°
0
Z
3
J1-=
4
6
mean values of various measurements. Curve 4, p = 35°; 6 - 0.10, mean values of various measurements.
172 AERODYNAMICS OF THE WING 1.2
1.0
0O
0.5
x
-A-0 .
'I
i
83
1,61
i
- 02
'
1. 38
3.S4
-0.6 0°
10°
a
20°
30'
00
3. if
Figure 3-54 Lift coefficients CL vs. angle of attack « for delta wings of various aspect ratios .1; taper X = s, thickness ratio 5 = 0.12, Reynolds number Re - 7 101, from Truckenbrodt [85 1.
with Fig. 3-53a shows that the increase in CLmax for small aspect ratios
is
considerably larger than for rectangular and swept-back wings. Also, Fig. 3-55b shows a strong increase of CYCLmax at small aspect ratios in agreement with Fig. 3-53b. Experimental studies on the separation characteristic of delta wings have been carried out by Truckenbrodt and Feindt [85] by means of simple wake measurements.
Figure 3-55 Maximum lift coefficients of delta wings, Reynolds number Re 106. (a) Maximum lift coefficient cLmax vs. aspect ratio .4. (b) Angle of attack a
for CL,max vs aspect ratio A. Curve 1, delta wing; A = 0; pro-
file NACA 0012, from Lange and Wacke [25]. Curve 2, delta wing; A = 8; profile NACA 0012,
from Truckenbiodt [85 1. Curve 3, mean values of various measurements.
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 173
3-4 INDUCED DRAG OF WINGS
3-4-1 Drag of Wings of Finite Span The total drag of a wing of finite span, Sec. 1-3-2, is composed of profile drag and induced drag: D = Dp + Di
(3-124)
The profile drag Dp is created by friction effects. It is almost independent of the wing aspect ratio. A procedure for the theoretical determination of the profile drag has been developed in Sec. 2-5-2. Experimentally, the profile drag can be determined through wake measurements (momentum-loss measurements). The induced drag D= exists only at a wing of finite span. It is created by the flow processes at the wing tips and can be determined from the laws of inviscid flow. In Sec. 3-2-1 it has been demonstrated that the induced drag is proportional to the square of the lift. The drag coefficient cD = D/Aq. of a wing with elliptic lift distribution is, from Eq. (3-32b): CD = CDp + CDi
(3-125a)
2
= CDp + _L
(3-125b)
where A, from Eq. (3-4a), is the wing aspect ratio.
There are two methods available for the determination of the induced drag. They differ in their physical concepts. In the first method, the induced drag is found from the pressure forces that act on the wing itself. In the second method, the induced drag is obtained from energy considerations. The latter approach allows the determination of the induced drag of only the whole wing. Conversely, the first
method produces, within the framework of simple lifting-line theory, the local distribution of the induced drag. Truckenbrodt [86] summarizes the state of the art of the drag of wings. Basic considerations to the drag problem stem from Jones [38]. Also, the comprehensive compilation of experimental data of the wing aerodynamics by Hoerner [30] must be mentioned.
3-4-2 Computation of Induced Drag Application of the Kutta-Joukowsky theorem The induced drag of an unswept wing of finite span from the Prandtl lifting-line theory, Eq. (3-18), is s
Di = 2 f I (y) u- (y) d y
(3-126)
s
Here F(y) is the circulation distribution and wi(y) is the distribution of the induced downwash velocity over the span -s
174 AERODYNAMICS OF THE WING
planforms. Following Fig. 3-16, let the wing be replaced by so-called elementary wings of the infinitesimal span dy and wing chord c(y). The vortex system of an elementary wing (see Fig. 3-17) consists of a number of horseshoe vortices of width dy arranged in series, one behind the other. In Fig. 3-56, two horseshoe vortices are drawn that originate at the stations x1, y1 and X2, Y2 of the wing. Their respective widths and circulation strengths are dy 1i dye and dr1, dr2 . Horseshoe vortex dl'1 induces at station X2, y2 the upward velocity d2 w21i whereas horseshoe vortex d r2 induced at station x1, yl the upward velocity d2w12. In analogy to the Kutta-Joukowsky theorem [see Eq. (3-14)], the lifting-circulation elements dr'1 and d r2 produce forces normal to the upward flow that are the result of the
upward-flow velocities d2w12 and d2w21, respectively. These forces represent contributions to the induced drag. The vortex system of Fig. 3-56 produces the partial induced drag
d4Di = -odT1 d2w12 dy1
-
dl'2 d2w21 dye
(3-127)
where the sense of rotation of the circulation elements has been taken into account.
Because dr = k dx, the induced upward velocities of the exterior induction (y1 =Y2) are found from Eq. (341) with Eq. (3-50a), as long as yl Oy2, as d 2w12 = d2w21
4
_ i 4
x1 - x2
d r2
1 (y1
-- 2
2
i
dr1 Ji + (y2 - y1)-
,/(x1 V (XI
- x2)2 + (Y1 - y2) 2 x2 - x1
dye
(3-128a)
1 dy1
(3-128b)
r (x2 - x1)2 + (y2 - y1)2
44
Figure 3-56 Explanatory sketch for computation of induced drag.
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 17/5
It can be seen that the second terms in the square brackets differ by their signs only. When introducing Eq. (3-128) into Eq. (3-127), these terms do not contribute to the induced drag, leading to: d4D;
=
-
ar dr 2
(y 2 1 T'2
dyi dye
(3-129)
From these considerations there follows immediately that the location of the lifting-circulation elements in the x direction does not affect the drag. Hence, the relationship found earlier [Eq. (3-126)] for the unswept wing is valid for the total induced drag of a wing of arbitrary planform. Since the induced downwash velocity wj(y) from Eq. (3-19) depends only on the circulation distribution over the span, the total value of the induced drag also depends only on the circulation distribution over the span. It is independent of the arrangement of the elementary horseshoe vortices in the. chord direction (flight direction). This result was realized very early by Munk [63, 64] and is known as the Munk displacement theorem. Thus, it is immaterial for the magnitude of the induced drag whether the circulation
distribution is caused by the wing planform (aspect ratio, sweepback, taper), by a wing twist, or by camber of the wing surface.
Application of the energy law Although the total lift of a wing can easily be computed by using the momentum law (Sec. 3-3-2), computation of the induced drag by means of the momentum law is considerably more difficult because the inclination of the vortex sheet has to be considered.* However, when using the energy law, the inclination of the free vortex sheet (Fig. 3-21), relative to the incident flow direction, can be disregarded. Since the induced velocities on surface I (Fig. 3-21) are zero, the mass o U. dy dz permeating the area element dy dz of surface II
per unit time undergoes an energy increase dEu =
wi,) dy dz. Here v. and w. are the induced velocities in the y and z directions, respectively, and the area integral over dEn is the work done by the induced drag U0D1 per unit time. Hence, after division by U., DZ
2
r f (vim
+
dy dz
(3-130)
(.III)
This relationship is valid for both not-rolled-up and rolled-up vortex sheets behind the wing.
The equivalence of Eqs. (3-130) and (3-126) will now be shown for the notrolled-up vortex sheet. The induced velocity field very far behind the wing with components v.(y, z) and z) can be expressed through the two-dimensional velocity potential 0(y, z) as *Kraemer [791 conducted a more detailed study into the application of the momentum law to the computation of the induced drag; see Sears [79].
176 AERODYNAMICS OF THE WING
ao
(3-13 la)
v00
ay
1VC0 = 7a
and
(3-131b)
Here 0(y, z) satisfies the potential equation a-0
32
ay=+
ati2
(3-132)
=
Introduction of Eq. (3-131) into Eq. (3-130) and integration by parts, the first integral with respect to y, the second with respect to z, yield, with Eq. (3-132), e
2
['p
f [(P
];°c0 dz + f
=-oo
az
v=-00
I
rods
where surface II has been extended to infinity. Since the values of 0, a 013y, and aO/az vanish at the boundaries y = ± oo and z = ± oo, whereas the potential according to Eq. (3-48b) changes abruptly in the z direction for z = ±0 by the amount Pu(y, 0) - 01(y, 0) = F(y), with Eq. (3-13lb) the induced drag becomes 4
Di = -
f J r(y) woo (y) dJ
2
(3-133)
-S
The integration limits y = ± -o can be replaced by y = ±s, because 0(y, 0) = 0 beyond the wing span. Now, by introducing Eq. (3-20) with w. (y) = -2wi(y) into Eq. (3-133), Eq. (3-126) is finally obtained, as was to be proved. Equation (3-130). is valid for the not-rolled-up vortex sheet. Kaufmann [411 showed
that the same induced drag is obtained for the rolled-up vortex sheet, where it must be assumed, however, that the cores of the two free vortices have finite velocities.
Practical computation of the induced drag From Eq. (3-126) the formula for the coefficient of induced drag is obtained with y = T/bU.. and ai = wi/U. from Eq. (3-71b), and with r7 = y/s as I
CDi = q _
Z1 f Y(71)ai(77) d11
(3-134)
-1
the aspect ratio of the wing from Eq. (34). By expressing the circulation distribution y by a Fourier polynomial as in Eq. (3-65a), the result of where
;1
is
the integration becomes, with ai from Eqs. (3-73) and (3-65b),
CD i = :z /l' n a;j
(3-135a)
9Z=1
=
2 CL
?lT
- .rul na , zll ya=3
(3-135b)
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 177
coefficient CL was determined from the
In the second relationship, the lift
coefficient al of Eq. (3-66a). In Eq. (3-135b) the first term represents the value for the elliptic circulation distribution [see Eq. (3-31b)] . Since the second term is always positive, the important theorem follows that the induced-drag coefficient for elliptic circulation distribution is a minimum. This theorem is true for fixed aspect ratio and for fixed lift coefficient CL. It was proved first by Munk [63]. A summation formula for the coefficient of induced drag can be derived in a
way similar to that which led to the summation formulas for the lift-related coefficients of Table 3-1. It has the form
Dt =
3f
-rA
(3-136)
S' yn Biz: sin 0
where the values for c have to be computed from Eq. (3-83). Equation .(3-134) for the induced drag will now be applied to trapezoidal wings
with symmetric twist. This example explains the relationship between twist and induced drag. In Sec. 3-3-2 it was shown that the circulation distribution of a symmetrically twisted wing can be put together from that of a wing without twist and a zero distribution. In the same way as the circulation distribution was split up in Eq. (3-63), the induced angle of attack of the twisted wing can be split up: '%i(0 = ix xi'(71) + aio(71)
(3-137)
of the
twisted wing leads, with
Thus Eq. (3-134) for the induced drag a = (da/dcL )CL , to 2
L
CDt=C2
7t11+C1CL+Co
(3-138)
1
with
C2 = 1,12 (L)2 r yu iu c'1
(3-139a)
1
1
C1 =
da j'
dcL J (Yunio
Yoaiu) tdi 7
(3-139b)
1
Co =CDio =1`i f Yomio d?1
(3-139c)
When the wing has no twist, the induced drag is determined just by the term C2. The wing with twist requires, in addition to this term, a term C, that is proportional to CL, and a term Co that is independent of cL. Here, the first term represents a linear twist, the latter a quadratic twist. As can easily be seen by comparison with Eq. (3-31b), the constant C2 is unity for an elliptic circulation distribution. For the wing without twist the coefficient C2 signifies physically, therefore, the ratio of the induced drag to its minimum value for elliptic circulation distribution.
As an example, the induced drag of a trapezoidal wing with twist is given in
178 AERODYNAMICS OF THE WING 01
11-9 '6
I
V
b___V
001
a
7
0
02
0.o
06,
08
70
04C
06
08
10
C
-0.02
100
0
I
-0.0110 0.2
02
10
08
d
Figure 3-57 Induced drag of symmetrically twisted tapered wing of various aspect ratios i and various tapers A from Eq. (3-138) (lifting-surface theory). (a) Wing planform with linear twist. (b) Induced drag of wing without twist, from Eq. (3-139a). (c), (d) Twist contribution to the induced drag from Eqs. (3-139b) and (3-139c). Fig.
3-57. It is based on
symmetric linear twist with as(p) _ 1iilai. The
corresponding circulation distribution has been computed from the lifting surface.
Figure 3-57b indicates that the induced drag of the wing without twist has a minimum for a taper X ~ 0.45 at all aspect ratios A. The value of this minimum is
only a little different from that of the elliptic wing (C2 = 1). For delta wings (X = 0) and rectangular wings (), = 1), cDi is in many instances considerably larger than for elliptic wings. The contribution that is independent of the lift, Fig. 3-57c, is always positive. The sign of the contribution that is linearly dependent on lift,
Fig. 3-57d, depends on the value of the taper. When the taper X -- 0.45, this contribution is zero for all aspect ratios. The reason is found in the nearly elliptic circulation distribution over wings of this taper without twist. Furthermore, by means of Eq. (3-135b), it can be shown that C1 = 0 for elliptic wings with arbitrary twist and that M
CDio = Co = ?s!1 f na,,
n-2
(3-140)
is the contribution of the induced drag caused by the twist for zero lift. Investigations related to the establishment of the local drag distribution along the span are compiled in [1].
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 179
3-4-3 Tangential Force and Suction Force Tangential force Earlier, in Sec. 1-3-2, the wing-fixed components of the aerodynamic force, normal force, and tangential force were introduced together with the flow-field-fixed components of lift and drag. For small angles of attack, the normal force is almost equal to the lift, whereas the tangential force deviates considerably from the drag, even for small angles of attack. The tangential force T is taken as positive in the direction from the wing leading edge to the trailing edge. When limiting the angle of attack to small or moderately large values (see Fig. 1-7a), the tangential force coefficient CT = T/Aq becomes, with T= -X, CT = CD -CLa
(3-141)
where CD is the coefficient of total drag as composed from profile drag and induced drag from Eq. (3-125). By introducing CD from Eq. (3-125b) and a= (aa/aCL)CL into Eq. (3-141), the tangential force coefficient for elliptic circulation distribution becomes
CT = CDP -
(ddCccL
-
i) CL
(3-142)
With dcL/da from Eq. (3-80b) for the simple lifting-line theory and from Eq. (3-98) for the extended lifting-line theory, Eq. (3-142) yields
CT=CDpCT = CDP -
(3-143a)
k2 _+1 2 n-A
CL
(3-143b)
where k = nA/cL . In Fig. 3-58, the difference (CDP - CT)/CL from Eq. (3-143) is shown against the aspect ratio. Accordingly, for large aspect ratios this difference, and thus the tangential force, are independent of the aspect ratio. Figure '3-59 illustrates the dependence of the tangential force coefficient on the lift coefficient for wings of various aspect ratios A. The profile drag coefficient had been taken to be CDP 0.05. It is remarkable that the coefficient of the tangential force assumes negative values when the lift coefficient CL > 0.5. In this case the 0,5
N
Q0.2
%1
-1
0
L
j 1
3
7
Figure 3-58 Tangential force coefficient CT vs. aspect ratio .1. (1) Based on the simple lifting-line theory, Eq. (3-143a). (2) Based on the extended lifting-line theory, Eq. (3-143b). cDp = coefficient of profile drag; cLoo = 2a.
180 AERODYNAMICS OF THE WING
A-12
11
6
I12
CDi
CD
3 0.8
CT
Figure 3-59 Lift coefficient cL vs. tangential force coefficient CT for wings of various aspect ratios A, from Eq. (3-143b). For comparison,
0.2
V
-0.15
-0.10
-0.05
I
0.05
0.70
0.15
0..2
kCD'6+
the drag polars cD(cL) are also shown.
CT; CD --+
tangential component of the resultant of the aerodynamic forces
is directed
upstream along the wing chord. The drag polar curves CD(CL) are also included in Fig. 3-59.
Suction force The discussions about the drag of wings of infinite span of Sec. 24-2 have shown that the flow around the leading edge of an inclined profile produces a
suction force in an inviscid fluid (Fig. 2-12a). This is the result of the strong underpressure in the vicinity of the leading edge. Now, the suction force on wings of finite span will be examined. The suction force is a part of the induced drag [Eq.
(3-124)], with the total drag being split into profile drag and induced drag. Equation (3-125) is therefore the expression for the drag coefficient with suction force. It has been pointed out in Sec. 24-2 that no suction force exists for very sharp leading edges. In this case, the flow around the leading edge causes local separation, eliminating the strong underpressure that results in a suction force. Rather, the resultant force of the pressure distribution over sharp-edged noses acts normal to the wing surface and, therefore, has the component La in the incident flow direction. Thus the drag coefficients with and without suction force are, respectively, CD = CDp + CDi
(3-144a)
CDp +CLa
(3-144b)
CD
The difference of the drag coefficients of Eqs. (3-144a) and (3-144b) yields the suction force coefficient cS = S/Aq., *: CS = CLOY
da dCL
(3-145a)
CDi 1
2
TA) CL
*The suction force is considered positive when acting upstream.
(3-145b)
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 181
Comparison with Eq. (3-142) yields CS = CDp -CT
(3-146)
Consequently, the quantity of Fig. 3-58 is a direct measure of the suction force. In particular, cS = cL /27r for wings of very large span. This result is in agreement with Eq. (2-77), remembering that the lift coefficient for smooth leading-edge flow cLS is zero for symmetric profiles.
In conclusion, a few experimental results on wings of small aspect ratio according to Hansen [44] will be presented, confirming the above considerations. Figure 3-60 shows polar curves for a slender circular disk. To show the effect of the suction force, the leading edge of the disk was formed in several ways, as can be seen from Fig. 3-60. The suction force increases with the leading-edge nose radius. The theoretical curves for the drag coefficient with and without suction force from Eqs. (3-144a). and (3-144b) are added in this figure. The tests show the expected result, namely, that the measured drag coincides with the theoretical curve, including suction forces, when the leading edge is well rounded. When the leading edge is very sharp, however, the measured drag lies close to the theoretical curve without suction force. All measurements with differently formed leading edges lie between the two theoretical curves.
3-5 FLIGHT MECHANICAL COEFFICIENTS OF THE WING
3-5-1 Contributions of the Wing to Stability The methods for the computation of the aerodynamic forces on a wing have been discussed in detail in Secs. 3-2-3-4. This section will show how these methods can be applied to the determination of the flight mechanical coefficients of the wing. A survey on these coefficients has been given previously in- Sec. 1-3-3.
The flight mechanical coefficients are determined by the motion of the wing 0.5
0.4
0.3
i
Figure 3-60 Drag polars of circular wings of Diskl
various degrees of leading-edge rounding, accord-
ing to measurements of Hansen [44]. Theory according to Kinner [44]. The drag at CL = 0 has been subtracted from the measured values. Disks
j
-Q1
0
I and II: cDp = 0.012; disk III: cDp = 0.008. Curve 1, with suction force from Eq. (3-125); OR
0.09
0.06
CD-.-
0.08
0.70
0.72
CD = CDp + 0.274cL. Curve 2, without suction force from Eq. (3-144b); cD = cDp + 0.55cL.
182 AERODYNAMICS OF THE WING
and the wing geometry. In the following discussions, only those coefficients will be considered that are significant for airplane stability. The coefficients that determine maneuverability will be treated later in Chap. 8. In addition to the wing, the other parts of the airplane (fuselage, empennage) contribute, sometimes considerably, to these flight mechanical coefficients. These contributions will be discussed later, too. In the present section, only the contributions of the wing will be discussed. The flight mechanical coefficients of the wing depend on numerous geometric
parameters of the wing, such as wing planform (aspect ratio, taper, sweepback), twist, and dihedral (see Sec. 3-1-1). The dependence of the flight mechanical coefficients on wing geometry is too varied to attempt a complete description of all these interrelations. In some cases the contribution of the wing to the stability coefficients of the whole airplane is small. Further investigations will be restricted to the cases in which the wing makes an essential contribution. Reference will be made to the summary reports of Betz [61, Schlichting [72], and Multhopp [61]. Of the two axis systems of Fig. 1-6, the experimental system will be used.* The coefficients are defined in Eq. (1-21).
The motion of the airplane can be divided into a longitudinal motion and a lateral motion, as has been explained in Sec. 1-3-3. During longitudinal motion, the position of the plane of symmetry of the airplane does not change. This motion is characterized by three parameters: flight velocity IT, angle of attack a, and pitching angular velocity w,, (Fig. 3-61). The lateral motion is defined by sideslip angle
rolling angular velocity wX, and yawing angular velocity wZ (Fig. 3-61). The stability coefficients are understood to be the changes of force and momentum coefficients with the above motion parameters.
3-5-2 Stability Coefficients of Longitudinal Motion Straight flight For longitudinal motion, the resultant aerodynamic force may be represented by lift, drag, and pitching moment. Their dependence on the angle of attack (see Fig. 3-61a) has been discussed in the previous section. The two most important coefficients are lift slope dcLIda and pitching-moment slope dcMfdcL. The
latter determines the position of the aerodynamic neutral point of the wing by Eq. (1-29). The lift slope dcLlda is presented for various wing. shapes in Figs. 3-25, 3-38, 3.42, 3-44, 3-46, and Table 3-5. The neutral-point positions of various wing forms
can be obtained from Figs. 3-37, 3-38, 3-43, 3-44, and Table 3-5. The flight mechanical computations for the neutral-point position require great accuracy. The
neutral-point position depends strongly on the individual planform. In general, therefore, it is required that for its determination the lift distribution should be computed by using the lifting-surface theory (see Sec. 3-3-5). Pitching motion Pitching motion is actually a nonsteady motion. In general it proceeds
slowly enough, however, that it can be treated as "quasi-steady." When the wing *In what follows, the index e of forces and moments and their coefficients, which indicates the axis system used, will be omitted.
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 183 Straight flight
\ cc
x
Figure 3-61 The motion modes of the wing. (a) Straight flight. (b) Yawed flight. (c)-(e) Rotary motions: rolling, pitching, yawing.
(Fig. 3-62) performs a rotary motion with angular velocity coy about the lateral axis through xS (Fig. 3-61 d), a vertical additive velocity VZ = wy(x - xs) is produced that varies linearly over the wing chord. Together with the incident flow velocity V, the rotary motion in chord direction produces an additive angle-of-attack distribution a(x) = VZ/V of magnitude n: (x) = °y (x
xs)
(3-147)
This angle-of-attack distribution produces an additive lift distribution, the integra-
tion of which leads to an additive lift and an additive pitching moment. These quantities are designated lift due to pitch rate and pitch damping. Both depend linearly on wv. It is expedient, therefore, to introduce the coefficients acL /a 2 y as lift due to pitch rate and acM/aQy as pitch damping, where Qy = wycjV is the dimensionless pitching angular velocity and c,1 is the wing reference chord,
introduced earlier by Eq. (3-5b). These coefficients depend only on the wing geometry and the position of the axis of rotation. Now it will be explained how these two quantities can be determined and, in
184 AERODYNAMICS OF THE WING
Figure 3-62 Explanatory sketch for aerodynamic coefficients of the pitching wing.
particular, how their values change with the position of the axis of rotation xs.* It is evident that there is an axis of rotation x0 for which the lift due to
pitch rate is zero. For a rectangular wing, this axis of rotation lies at a distance 4c from the leading edge, according to the Pistolesi's theorem (see Sec. 2-4-5). The pitch damping, however, cannot be zero for any position of the axis of rotation. For the computation of the lift due to pitch rate, the angle-of-attack distribution of Eq. (3-147) is rewritten in the form
a (x) = V (x - xo) + y (xo - XS)
(3-148)
By setting xS = x0 in this equation, the second term becomes zero, whereas, by definition, the first term produces zero lift due to pitch rate, (aCL /a Shy) = 0. The contribution of the first term to the pitch damping will be expressed by (aCMlaQy). The second term in Eq. (3-148) represents a constant angle of attack and gives the total lift due to pitch rate as acL
dcL xo - xs
a Sty s
du
c.
(3-149)
Here dcL/da is the lift slope of the wing. Equation (3-149) shows that the lift due to pitch rate is a linear function of the position of the axis of rotation (see Fig. 3-63a). The moment of the pitching motion is obtained from Eq. (1-28) as CM = CMO -
XN -XS C
CL
JU
which leads with Eq. (3-149) to the pitch damping: GcM
(aQJ S
-
dclyi
(0 2,)O
xy - XSxo - xs dcL da cry Cu,
(3-150)
*For flight mechanical computations, the axis of rotation coincides with the lateral axis through the airplane's center of gravity.
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 185
This equation shows that the pitch damping depends parabolically on xs. In particular, it is immediately obvious that for x s = xN and for xS = xo the pitch damping has the same value, namely, (acM/a Qy )o , as in Fig. 3-63h.
To be able to compute the pitch damping from Eq. (3-150) for an arbitrary position of the axis of rotation x5, the determination of (acM/a-2y)o and of xo/cN, is required, whereas the coefficients dcL/da and xN/cL are known from earlier discussions.
For xs = xN, Eqs. (3-149) and (3-150) yield X0
xN
cA
cu
a rnz
as o
-
do acL '
dcL (BSZyiN
(a cat
(3-15la) (3-151b)
r
Thus the problem of determining the lift due to pitch rate and the pitch damping for an arbitrary
position of the
axis
of rotation has been reduced to the
computation of the two coefficients (aCL/a.Qy)N and (acM/aQy)N for the position
of the axis of rotation in the neutral point. These latter two coefficients are obtained from the lift and pitching moments as determined from lifting-surface wing theory for the angle-of-attack distribution corresponding to Eq. (3-147):
a (x) =
x
xN CA
DY
(3-152)
In Table 3-6, numerical data on the positions of the axis of rotation for zero lift
due to pitch rate and of the corresponding pitch damping are compiled for a trapezoidal wing, a swept-back wing, and a delta wing (Table 3-5). Compare also Garner [62] and Gothert and Otto [24] . In the case of airplanes with a separate horizontal tail, the contribution of the wing to the lift due to pitch rate is small compared with that of the tail surface. An act
aay
a
acM
i9ay
b
Figure 3-63 Lift due to pitch rate (a) and pitch damping (b) vs. position of axis of rotation xs. xN = neutral-point position; x,, = axis of rotation for vanishing lift due to pitch.
186 AERODYNAMICS OF THE WING
Table 3-6 Position of the axis of rotation for zero-lift due to pitch rate and corresponding pitch damping for a trapezoidal wing, a swept-back wing, and a delta wing* irapezoiaai wing
y
2.°
CAI
a r_ BS?,) o
swept-oacK wing
Delta wing
0.533
0.485
0.604
-0.358
- 0.498
- 0.285
*The distance d x0 is measured relative to the geometric neutral point N25 , the position of which is given in Table 3-5. Table 3-6 is based on data from Table 3-5.
accurate computation is therefore not required. On the other hand, in the case of all-wing airplanes, whose total pitch damping is almost completely produced by the wing, a more accurate computation may be required, depending on the specific case.
3-5-3 Stability Coefficients of Lateral Motion Yawed flight During steady yawed flight, the incident flow condition is determined by the sideslip angle a (Fig..3-61b) in addition to the angle of attack a. Because of
the asymmetric incident flow, in addition to lift, drag, and pitching moment, additive forces and moments are created, namely, the side force due to sideslip Y, the rolling moment due to sideslip M, and the yawing moment due to sideslip MZ (see Fig. 1-6). They vary linearly with R for small angles of sideslip. The derivatives of the dimensionless coefficients with respect to the sideslip angle are, therefore, independent of the sideslip angle. They are termed stability coefficients of lateral motion. All three coefficients for a wing are strongly dependent on the sweepback angle and the dihedral angle. First, the wing without dihedral will be treated, followed by a discussion of the effect of the dihedral angle. A fundamental treatment of the yawed wing was first given by Weissinger [94]. The resulting theory can be designated as simple lifting-line theory in the sense of Sec. 3-3-3. In this theory it is assumed that free vortices are shed only from the
edge and that these are parallel to the incident-flow direction. The inclination of the free vortex strips against the wing axis of symmetry is of trailing
secondary effect on the results of the Weissinger theory. In this theory, Weissinger [94] introduced a correction factor taking into account the effect of the wing end
flaps on the rolling moment due to sideslip. Later Gronau [25] made comprehensive computations of the rolling moment due to sideslip and the yawing moment due to sideslip, mainly for swept-back and delta wings, using the method of the extended lifting-line theory (Sec. 3-3-4). Here, too, the effect of the free vortex strip inclination has only approximately been taken into account.
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 187
Test results from various sources for the rolling moment due to sideslip of rectangular, swept-back, and delta wings against the aspect ratio are shown in Fig. 3-64. For comparison, theoretical curves from [25] are added. Agreement between measurements and theory is good. With decreasing aspect ratio, the rolling moment due to sideslip decreases strongly. This presentation reveals further that sweepback causes a strong increase in the rolling moment due to sideslip. This means that the rolling moments due to sideslip of swept-back and delta wings, in particular, are strongly dependent on the lift coefficient (see also Kohlman [45] ). Figure 3-65 gives the corresponding plots for the yawing moment due to sideslip. A wing in asymmetric flow (yawed wing) corresponds aerodynamically to a wing of asymmetric planform (see Fig. 3-17). Based on this concept, its circulation distribution can be computed from the extended lifting-line theory (Sec. 3-3-4), or the lifting-surfaces theory (Sec. 3-3-5). However, the required computation effort is considerably greater than for symmetric incident flow because of the asymmetry of the wing planform.* The result of such a computation is the circulation distribution over the span, measured normal to the incident flow direction. With it, the total lift and the neutral-point positions are obtained .from the formulas of Sec. 3-3-2. In this way the rolling moment about the experimental x axis is also determined. The rolling moment due to sideslip, accordingly, is proportional to the total lift. The circulation distribution for the three wings without twist examined earlier has been computed by this method at three different angles of sideslip. In Fig. 3-66, the circulation distributions for 0° and a = 10° have been presented over the
span coordinate, measured normal to the direction of the incident flow. For all three wings, the circulation distribution changes very little with the angle of sideslip. *Only after electronic computers became available has this procedure gained practical value.
7,4
7.2
'
1.0
I
2 0.8
I
I
i
I
4
5
6
A
Figure 3.64 Rolling moment due to sideslip of rectangular wings, swept-back wings, and delta wings vs. aspect ratio A; theory of Gronau. Measurements: curve 1, rectangular wing (gyp = 0°), (o) from Bussmann and Kopfermann, (v) from NACA Rept. 1091. Curve 2, swept-back wing of constant chord (gyp = 45°), (.) from Gronau, (o) from NACA TN 1669, (v) from Jacobs. Curve 3, delta wing (X = 8), (s) from Gronau, (0) from Lange and Wacke.
188 AERODYNAMICS OF THE WING 7.
I
3
0.2 7
2
1
4
5
8
Figure 3-65 Yawing moment due to sideslip of rectangular wings, swept-back wings, and delta wings vs. aspect ratio e; theory of Gronau. Measurements: curve 1, rectangular wing (gyp = 0°), from Bussmann and Kopferrnann. Curve 2, swept-back wing of constant chord ('p = 45°), from Gronau. Curve 3, delta wing (A = 8), from Gronau.
It should be mentioned that this behavior is typical for wings without dihedral. The
locations of the neutral points for three angles of sideslip are inscribed into the wing planforrns. The coefficients of the rolling moment due to sideslip and the coordinates of the neutral points are compiled in Table 3-7. The yawing moment due to sideslip is caused by the difference in drag of the two wing-halves. It consists of a contribution from the profile drag and one from the induced drag. The latter
a
b
c
08
as
fl-10° I
j
-o
04 02
0.2
0
-1.0
-0.5
0
0
0.5
1.0
-10
-0,5
0
0.5
10
?7
Figure 3-66 Circulation distribution of three yawed wings without twist in sideslipping, based on the lifting-surface theory of Truckenbrodt. Angle of attack a = 1, measured in the section parallel to the incident flow direction, y = r/Vb. Geometric data of the wings from Table 3-5. (a) Trapezoidal wing: V = 0'; A = 2.75; A = 0.5. (b) Swept-back wing: p = 50°; A = 2.75; X = 0.5. (c) Delta wing: , = 5 2.4°; = 2.31; 1\ = 0.
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 189
Table 3-7 Coefficients of the rolling moment due-to sideslip and position of the neutral point for a = 0°, 50, and 100 for a trapezoidal, a swept-back, and a delta wing* Trapezoidal wing
1 acMX CL
ap
0° 50
s
100
yi9 s
Swept-back wing
Delta wing
0.111
0.717
0.580
0.219 0.221 0.223
0.781 0.794 0.814
1.027 1.024 1.018
00
0
0
0
5°
-0.010 -0.020
-0.060 -0.123
-0.050 -0.102
100
*Distances are measured in the wing-fixed coordinate system from the leading-edge station of the wing middle (root) section. Table 3-7 is based on Table 3-5 (see Fig. 3-66).
contribution is proportional to the square of the lift, precisely like the induced drag. The side force due to sideslip of a wing without dihedral can be determined approximately by considering that the profile drag of a yawed wing acts parallel to the direction of the incident flow, but the induced drag acts in the direction of the
wing axis of symmetry. Consequently, in asymmetric incident flow, only the component of the profile drag cy = CDP sin j3 acts in the direction of the wing-fixed lateral axis. Hence the side force slope is acY ao
= CDP
(3-153)
Wing with dihedral The dihedral of a wing is understood to be the inclination of the left and the right wing-halves relative to the xy plane (Fig. 3-61b). The dihedral angle is designated as v; in the general case v may vary along the span. The stability coefficients of yawed flight acy/a ji, acM/ao, and acMZ/a j3 of the wing are strong
functions of the dihedral. For the total airplane, the contributions of the wing to the side force due to sideslip acy/a(3 and to the yawing moment due to sideslip are relatively small. Conversely, the contribution of the wing to the rolling moment due
to sideslip of the total airplane is of decisive significance. Selection of the wing dihedral is governed exclusively by the requirement of a flight mechanically favorable value of the rolling moment due to sideslip.*
The aerodynamic effect of the dihedral in yawed flight is due to the angle of attack, increased by the amount .da of the leading half-wing, and the angle of attack decreased by d a of the trailing half wing. This angle d a can be determined The value of the rolling moment due to sideslip of the total airplane depends on the vertical position of the wing relative to the fuselage in addition to the dihedral.
190 AERODYNAMICS OF THE WING
as follows: From Fig. 3-67a and b, the lateral component of the incident flow V, = V sin a produces on either half-wing a component normal to the wing of amount
V,, = ± V. sine
Together with the component VX = V cos (3 of the incident flow, the additive angle-of-attack change becomes
Ja=
y"` z
= ± tan fi sin'v
(3-154a)
= +(3v
(3-154b)
The second relationship is valid for small angles of sideslip and small dihedral angles. The exact establishment of the dihedral angle from a given wing geometry must be based on Eq. (3-11). The lift distribution of a wing with dihedral during yawed motion may thus be determined by adding the geometric angle-of-attack distribution of the wing without
dihedral to the antimetric* twist from Eq. (3.154) (see also Fig. 3-67). As in Fig. 3-67, the lift (L/2 +A L/2) acts on the leading wing-half, the lift (L/2 -J L/2) on the trailing wing-half. L is the lift for symmetric incident flow and AJ L/2 is the additive lift of one wing-half in yawed motion. For the determination of the aerodynamic forces of the two wing-halves, it has to be realized that, as Fig. 3-67 demonstrates, the resultant incident flow direction is deflected up by the angle J a on the leading wing-half but deflected down by the same angle Aa on the trailing wing-half. These angle-of-attack changes are relative to the angles of symmetric incident flow. The resultant aerodynamic forces on the two wino halves undergo the same direction changes. The exact determination of the side
force due to sideslip, of the rolling moment due to sideslip, and of the yawing moment due to sideslip requires computation of the lift distribution on the given wing for the antimetric angle-of-attack distribution in Eq. (3-154).
Approximate expressions for the aerodynamic quantities of the yawed wing with dihedral giving an explicit account of their dependence on the dihedral angle and the total lift coefficient can be gained, however, through the following estimations: The side force due to sideslip resulting from the dihedral is, from Fig. (3-67b),
Y=2
2
sin v= dLv
with 2 = 2 V2 2 da
da v
being the additive lift of one wing-half, where, from Eq. (3-154b), d a = v(i. Consequently, the coefficient of the side force due to sideslip becomes dap
=
(dCa)y1,2
(3-155)
The coefficient (dcL /da) of this equation can be determined exactly only by computation of the lift distribution on a wing with antimetric twist as in Fig. 3-67c. *Translator's note: The word antimetric, found repeatedly in the text, has been coined by the authors to avoid an inconvenient expression like "acting or pointing in opposite directions but being of equal magnitude."
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 191
C
da
-7
7 t
57 i
da=-v.
r
Figure 3-67 Aerodynamics of a wing with dihedral in sideslipping. (a) Wing planform, xy plane. (b) Dihedral yz plane. (c) Additive antimetric angle-ofattack distribution due to the dihedral ]a = ± vg. (d) Incident flow resultant and aerodynamic-forces resultant of the two wing halves.
As an approximation, however, it may be assumed that this coefficient is equal to that of a wing without twist of aspect ratio A/2. Equation (3-155) reveals, then, that the coefficient of the side force is proportional to the square of the dihedral angle and independent of the total lift coefficient. Introduction into Eq. (3-155) of the lift-slope value for an aspect ratio A/2 from the extended lifting-line theory of Eq. (3-98) yields, for the unswept wing,
192 AERODYNAMICS OF THE WING
n!1
a cy
Vk2+4+2 where k = 7rA/ct
v4
(3-156)
..
Measurements that confirm the above formula are reported in the summary account [721. The rolling moment due to the dihedral is (see Fig. 3-67b)
Mx=2d2 LYL where YL designates the distance of the center of the additive lift of the half-wing, :dL/2, from the wing root. For the rolling-moment coefficient cMx = Mx/qAs results, corresponding to the above discussion, acMx
_
(dCL)()v
( 3-157)
Here, (nL )v = YL IS is the dimensionless distance of the center of the additive lift of the half-wing from the wing root. This equation demonstrates that the coefficient of
the rolling moment due to yaw as a result of the dihedral is proportional to the dihedral angle and independent of the total lift coefficient. To a good approximaWith this value, the following 31r=0.424. tion, (iL)v can be set equal to approximate relationship for the unswept wing is obtained. Here (dcL /da)v from Eq. (3-98) for one-half of the aspect ratio A 12 is again introduced.* acMx
4
8f3
3n
n!1 1/k2
+ 4 +2
V
(3-158)
The additive yawing moment due to sideslip resulting from the dihedral is very small in general. Its sign is such that it tends to turn the leading half-wing further
upstream. This comes about because, as shown in Fig. 3-67d, the resultant aerodynamic force at the leading half-wing is being turned toward the front and at the trailing half-wing toward the rear. Measurements are given in [72]. Rolling motion A linearly variable vertical velocity VZ = Wxy is obtained when the wing executes a rotary motion about the longitudinal axis as in Fig. 3-68a (see also Fig. 3-61c). Superposition with the incident flow velocity V results, from Fig. 3-68b and c, in an additive antimetric angle-of-attack distribution ._l a (?7)
= 77-x
(3-159)
where Qx = coxs/V is the dimensionless angular rolling velocity. This angle-of-attack distribution produces an antimetric lift distribution along the span and consequently
a moment about the x axis that always tends to inhibit the rotary motion. This moment is designated rolling moment due to roll rate or roll damping. The *Through evaluation of Eq. (3-100) with Eq. (3-154), Eq. (3-158) may be established as solution for the elliptic wing.
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 193
b
V
Figure 3-68 Aerodynamics of the rolling wing. (a) Wing planform. (b) Additive antimetric angle-of-attack distribution A s = rDX. (c) Resultant incident flow direction and resultant aerodynamic force of the two wing-halves.
asymmetric force distribution along the span furthermore produces a yawing moment, the so-called yawing moment due to roll rate. These two moments are proportional to the dimensionless rolling angular velocity QX , making their In determining the coefficients acMX/aQX and acMZIaQX independent of aerodynamic force of the two wing-halves from Fig. 3-68c, it should be noted that
.,
relative
to the symmetric incident flow direction, the resultant incident flow
direction of the downward-turned wing-half is deflected upward by the angle A a and that of the upward-turned wing-half deflected by the same angle Aa downward. Consequently, the local aerodynamic forces on the two wind halves undergo the same directional changes.
For the determination of the roll damping of a given wing, the antimetric circulation distribution ya(r1) over the span has to be established following a procedure for the computation of the lift distribution of Sec. 3-3. Hence the roll damping is, from Table 3-1, i
OCM.X
8Q
=
-:li yar/ all
(3-1oua)
with
(3-160b)
is independent of the total lift coefficient of the wing. The roll-damping coefficients of the three wings (trapezoidal, swept-back, delta) examined earlier are found in Table 3-5. Accordingly, the roll-damping coefficient
aCMX/8Q.(
A simple approximate formula for the roll damping of unswept wings is obtained by setting a =,n in Eq. (3-100): acMX
t
QX
'
7cA
V1c2 ; 4
(3-161) 2
194 AERODYNAMICS OF THE WING
of this approximate formula
not recommended for wings of strong sweepback. A more accurate computation should be made. Schlottmann [75] Use
is
demonstrates the theoretical determination of the roll damping of slender wings by a nonlinear theory and experimental confirmation of the computed results.
The yawing moment due to roll rate tends to turn the downward-turning wing-half forward. This behavior can be understood as follows: On the downward-
moving wing-half, the resultant incident flow direction
is
turned upward and
consequently the resultant aerodynamic force turns forward. On the upward-moving
wing-half, the resultant aerodynamic force consequently turns rearward. On a section y of an unswept wing, the force dD' = dD1 - dL d a = A(al -.i a) is thus obtained in direction of the undisturbed incident flow.* Here dDi = dLat from Eq. (3-17). Integration produces the induced yawing moment due to roll rate: +8
MZ= f (a1-da)ydL -8
With dL = e V T dy from Eq. (3-14) and 'y = T /b V and Al a from Eq. (3-159), the coefficient of the yawing moment is determined as cMZ = A f y (a4 - S2x-l) ?7 d
The total circulation y is composed of the contribution of the wing in symmetric incident flow ys and of the contribution ya created by the rotary motion for as = rl that is, y = ys + SQ X`ya . Correspondingly, the induced angle of attack becomes ai = ais + Qxaia . Introduction of these relationships into the above equation yields 1
ac1Z
aslx = 11J L(aia - ?7) Ys -t- atsYai 17 d r1
(3-162)
For wings without twist of elliptic circulation distribution, a simpler evaluation of the integral is possible. The circulation distribution is obtained from Eq. (3-65), specifically, ys with p = 1 and ya with p = 2, 3, ... , M. Correspondingly, the
induced angles of attack are found from Eq. (3-73), at3 with n = 1 and at,, with n = 2, 3, ... , M. By taking into account Eq. (3-65b) with 'r = cos 6 and drl = - sin t$ dzg, the integration over 0 c 6 c n yields the relationship a em' as2x
= 4 Aa1(6a2 - 1)
(3-163)
where al = cL/7rA and a2 = -(2/7rA)(acMX/aQX) from Eqs. (3-66a) and (3-66b). By introducing Eq. (3-161), the following approximate formula for the coefficient of the yawing moment due to Toll rate is finally found:
*The profile drag is not taken into account.
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 195 acMZ
i
a QX
4
yk'2 ± 4 - 1 }"kz _+4 + 2
L
(3-164)
Thus the coefficient of the yawing moment. due to roll rate is proportional to the total lift coefficient.
Yawing motion Motion of the airplane about the vertical axis produces additive longitudinal velocities, of reversed signs on the two wing-halves (Fig. 3-69; see also Fig. 3-61e). An asymmetric lift distribution over the span results, creating a yawing moment and a rolling moment. This yawing. moment counteracts the rotary motion and is termed, therefore, yaw-damping or turn-damping of the wing. It is very small compared with that of the whole airplane, and therefore its computation is omitted.
The rolling moment created by the yawing motion is termed rolling moment due to yaw rate or turning rolling moment. The turning rolling moment tends to turn the forward-moving wing-half upward.
The turning rolling moment can be computed in the following way: Through the rotary motion with angular velocity wZ from Fig. 3-69b, a linear distribution of the longitudinal velocity is generated along the span:
V. (y) = V - a.zJ
(3-165)
To ensure that the wing is a strearnlayer of this inhomogeneous flow field, the kinematic flow condition
V. (x, y) + w(x, v) = 0
(3-166a)
.y
c
a; ,y
Figure 3-69 Yawed wino. (a) Wing planform. (b) Velocity distribution. (c) Resultant incident flow
z yx
direction of the two wing-halves.
196 AERODYNAMICS OF THE WING
(3-166b)
must be satisfied at each point of the wing surface. Here V, is the component of the longitudinal velocity Vx normal to the wing chord; thus, V, = a Vx (Fig. 3-69c). In a homogeneous flow field, Vx = V, the kinematic flow condition becomes w/V + a = 0. Comparison with Eq. (3-166b) demonstrates that the inhomogeneous flow is equivalent to a homogeneous flow with the mathematical angle of attack
ab = a
y =all - Qj)
(3-167)
where nZ = w2s/V. Consequently, the circulation distribution for inhomogeneous flow can be computed by using the computation procedures of Sec. 3-3, but by applying an angle-of-attack distribution as in Eq. (3-167). The resulting circulation distribution
is
lb = b Vyb. A wing strip of width dy thus produces a lift
dL = 0 Vx I'b dy = 0V(1 -0zrt)Tb dy, and the rolling moment becomes 8
S
Mx = - f ydL = -of V-,rby dy -s
-8
And further, the coefficient of the rolling moment cMx = Mx/qAs is found as 1
cMx = -A f (1
- Ox 77) Yb77 d?7
-1
The circulation distribution yb at the angle of attack ab, from Eq. (3-167), may be composed as follows: Yb =
(3-168)
a -2zYa
Here yu is the circulation distribution for a = 1, and ya that for as = rl [see Eq. (3-160)]. For the sake of simplicity, let a wing without twist a = const, be considered. Introduction of Eq. (3-168) into the equation for the rolling-moment coefficient yields
f
1 1 acm, 8S2z - A f yu?72 d + A yak d -1 -1
do L
cz
(3-169)
This equation demonstrates that the coefficient of the rolling moment due to yaw rate is proportional to the angular velocity Q, and the total lift coefficient CL . For a wing without twist of elliptic circulation distribution, the following
approximation formula is obtained with Eq. (3-98) and after evaluation of the integrals similar to those of Eqs. (3-162) and (3-163): cMx
8z
t 4
1 + k2
t
Vk2 + 44
T1
CL
(3-170)
2
This expression is nearly independent of the aspect ratio. For the three wings that have been examined (trapezoidal, swept-back, and delta, Table 3-5), the coefficients of the rolling moment due to sideslip are listed in Table 3-8.
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 197
Table 3-8 Coefficients of the rolling moment due to yaw rate for a trapezoidal, a swept-back, and a delta wing based on Table 3-5
1 acM"
CL &Q
Trapezoidal wing
Swept-back wing
Delta wing
0.410
0.443
0.378
For an accurate computation of the rolling moment due to yaw rate, it must be realized that the rotary motion of the wing causes the free vortices to be shed into a lateral flow. Hence the portions of the free vortices that lie on the wing produce an addition to the lift and thus to the rolling moment due to yaw rate. A detailed computation reveals that the coefficient of the rolling moment due to yaw rate for wings of small aspect ratio (ii < 3) depends considerably on the position of the axis of rotation.
3-6 WING OF FINITE THICKNESS AT ZERO LIFT
3-6-1 Displacement Problem of the Wing The theory of the wing of infinite span as discussed in the previous sections of this
chapter was based on the assumption of a very thin profile (skeleton). For the theory of the wing of finite span, the extension from the skeleton theory to the theory of the inclined wing of finite thickness (profile teardrop) has long been available (Sec. 2-4-4). A similar extension for the wing of finite span and finite thickness is still lacking. However, for the wing of finite span and finite thickness of the wing profiles (symmetric profiles), there does exist a computational method that
allows the determination of the displacement effect of the wing and thus of the pressure distribution on the surface of such wings, provided that the lift is zero. It represents, therefore, a teardrop theory for wings of finite span; note the publications of Keune [42] and Neumark [65] and compare also [43]. The method of singularities is used in which the body within the flow field is replaced by a system of sources and sinks. The fundamentals of this method have been established in Sec. 2-4-3 for two-dimensional flow and applied to the problem of an airfoil in plane flow. For the assessment of the effect of compressibility in both two-dimensional and, in particular, three-dimensional flow, it is important to know the maximum
perturbation velocity on the wing. The computational procedures, treated in the following sections, for the velocity disturbance on wings of finite span and finite thickness are of significance, therefore, for the aerodynamics of the wing of high subsonic velocities.
198 AERODYNAMICS OF THE WING
3-6-2 Method of Source-Sink Distribution Source system of the wing of finite span For the computation of the threedimensional flow field about a slender body resembling a wing of finite span and finite thickness, a distribution of three-dimensional sources and sinks is established in the plane of the surface A (wing planform plane). An area element dx dy carries the source strength d2Q (x, y) = q (x, y) d x d y
(3-171)
when q(x, y) designates the source strength per unit area. The source strength q(x, y) must satisfy the so-called closure condition
f f q (x, y) d.x d y= 0
(3-172)
(A)
in order to form a closed body shape. Compare also the corresponding expressions for the plane case, Eq. (2-92).
Velocity distribution on the wing contour Superposition of the velocity field, produced by the source distribution, with a translational flow of velocity U., the direction of which lies in the source plane (Fig. 3-70), produces a closed stream surface that can be interpreted as the contour of the wing of finite thickness; compare again Sec. 2-4-3. Let u, v, and w be the velocities induced by the source distribution (perturbation velocities) and let z(t)(x, y) = z(x, y) be the shape of the wing contour, symmetric to the xy plane. Then the condition for tangency of the velocity resultant on the entire contour is az w = ( U + u) ax + v y az =U00 ax
( 3 - 173a) (3-173b)
This is the kinematic flow condition. Since, for slender bodies, the velocities u and v are small compared to the incident flow velocity U., except for the immediate
Figure 3-70 Flow around a wing of 1
-b-2S-
finite span and finite thickness at zero lift.
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 199
vicinity of the leading edge and the wing tips, it is sufficient to work with the simplified form, Eq. (3-173b). This is, formally, the same relationship as in the plane case. Both for the kinematic flow condition and for the computation of the pressure distribution on the contour, the velocity components u, v, w are required on the contour. For slender bodies, it is sufficient, however, to compute the velocity components in the wing plane, as in the teardrop theory of plane flow. This simplifies the problem considerably.
The source distribution of Eq. (3-171) constitutes a three-dimensional source. Thus, the velocity potential of the source distribution q(x', y') at a point x, y, z is obtained as 1
4d
ff
q(x'.y')dx'dy'
N
(3-174)
(x - x')2 -± (y - y')s -L
(A)
where the integration is performed over the wing area A covered by sources. The corresponding velocity components are found from Eq. (3-45). At a point x, y of the wing plane z = 0, they become* it (x, y, 0) = 4
1
ff q (x',
y')
(x - x') dx' d y'
-
(x - x')2 + (y -y')
23
(3-175a)
(A) V (x,
y, 0)
4-
['f
(y-y")dy'dx' q (x',
(3-175b)
y'(x - x')2 _L (y - y')23
.(A)
(3-175c)
w(x, y, 0) = ± 1 q(x, y)
The upper sign is valid for z > 0, the lower for z < 0. Hence, the induced velocities normal to the xy plane are discontinuous across the source layer (wing plane). Introduction of Eq. (3-175c) into the kinematic flow condition Eq. (3-173b) yields q (x, y) = 2 U,n
az
-x
(3-176)
Consequently, the source strength is proportional to the slope of the contour in the zx plane [see also Eq. (2-90h)]. The formulas obtained by properly introducing Eq. (3-176) into Eqs. (3-175a) and (3-175b) describe the velocities added at the location of the wing by flow displacement (profile teardrop) of the wing. Presentation of these formulas is omitted. For the wing of infinite span (plane problem), Eq. (2-94) yields *Because of the singular points of the integrands in Eqs. (3-175a)-(3-175c), integration of Eq. (3-175a) must be conducted first over x' and then over y'. For Eq. (3-175b) the reverse order of integration is necessary. If the integration is to be performed in a different order, however, the Cauchy principal value must be taken for the second integration in either equation.
200 AERODYNAMICS OF THE WING Z
U P1
1
Uro
is
(' az dx' J ox, x - x'
(3-177)
0
Also, in this case vpl = 0.
It should be stated here that the velocity differentials (perturbation velocities) of wings of finite span in Eqs. (3-175a), (3-175b), and (3-176) are proportional to the profile thickness ratio S = tic, in analogy to the two-dimensional profile theory (Eq. (3-177)]. The above linear theory is sufficiently accurate for all practical purposes up to about S = 4. The resultant velocity on the contour is
We=j(U"-f-v2 U.+It
(3-178)
when quadratic terms in u and v are neglected.
3-6-3 Results of the Teardrop Theory for Wings of Finite Span Rectangular wing of finite span In the simple case of a rectangular wing c(y) = c of
constant profile over the span, the contour is represented by z(x, y) = z(x) for -s
UPI
Uco
Uo
,
1
du
(3-179)
Uro
with
du U11
_
1
1
2x
f
az f 2 _ s -Fy ax' (x - x')2 + (s ± y)=
dx'
(x -. x')1 ± (s - y)s x - x
I
0
It is easily verified that the quantity J u is negative in general. This means that the perturbation velocities on the contour of a finite wing are. reduced in
comparison with those on the infinitely long wing. For the wing middle (root) section, y = 0, the perturbation velocity becomes
du
__
U"
t I 7L
f
0
az
ax' ( _-_
s
1
dx'
Y(x-x')2-82 x - X'
,r
(3-180)
When the aspect ratio is small, it follows with Z = z/c and X = x/c that u TIT 11
i azZAInA (A.=small) .z
a Y=
(3-181)
For A = 0 this reduces to u = 0.
On the parabola profile Z = 26X(1 -X) of thickness ratio 5 = t/c, the
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 201
maximum (perturbation) velocity at station x = c/2 of the middle section y = 0 is, from Eqs. (3-179) and (3-180), U ax = 7T 5A sinh-1
Thus, for the plane case,
(3-182
1
it follows that Umax pi/U., = 46/7r, in agreement
A1.
with Eq. (2-97).
Results for the rectangular and the parabolic profiles are given in Fig. 3-71. Figure 3-71a shows the maximum perturbation velocity, which lies at X= 0.5, as a function of the aspect ratio for the two sections r7 = 0 and 77 = 0.5. In Fig. 3-71b the maximum perturbation velocities are depicted for various aspect ratios over the span coordinate. In conclusion, it should be stated that the maximum perturbation velocity on the wing of finite span becomes noticeably smaller than on the wing of infinite span only for aspect ratios A < 2.
Elliptic wing The theory of Sec. 3-6-2 for the computation of the perturbation velocities and the above example for the rectangular wing were based on approximate methods valid for small thickness ratios 8. One example of an exact solution will now be given.
A wing of elliptic planform and elliptic profile as in Fig. 3-72 is a general ellipsoid of which the two axial ratios are very different. Let a 1, b, , c1 be the three semiaxes of the ellipsoid. Then (3.183a)
(3.183b) *The trigonometric functions will be given as sinh-' rather than arcsinh.
0.2
a
i
0.5
1.5.
1.0
20
2.5
3.0
0.2
0.4
0.5
0,8
1,0
/1
Figure 3-71 Maximum perturbation velocities on rectangular wings with parabolic profile at zero lift. For infinite span Umax p1 = (4/71)g U. at station x/c = X = 0.5. (a) Dependency on aspect ratio A. (b) Dependency on span coordinate n = y/s.
202 AERODYNAMICS OF THE WING
tr -2c,
For
Figure 3.72 The geometry of the elliptic wing with elliptic profile (general ellipsoid).
general ellipsoid, the velocity
a
distribution on the contour can be
determined in closed form. The pressure distribution on the surface of the ellipsoid in a flow parallel to the x axis is, from Maruhn, Chap. 5 [40], Ic ) 2
=1
A2
1
a
((x
ai)
2
z
a,)'
2
\
z
\ci\ci/
+\bi! \bi)
2
(3-184)
where c p = (p -p.)1(Q12)U.' is the dimensionless pressure coefficient and A = A(bl /a1 , cl /bl ), a quantity that depends on the two axis ratios of the ellipsoid. Equation (3-184) demonstrates that the pressure minimum and thus the velocity maximum lie at x = 0. This velocity maximum is constant along the y axis. From Eq. (3-184), cp min = 1 -A 2 = 1 - (Umax/Uco)2, with Umax = UC, + Umax being the maximum velocity on the contour. Hence, the maximum perturbation velocity becomes Q.1n11X iTCr
_ A (), i1) - 1
(3-185)
In Fig. 3-73, the ratio Umax/Umax pl is plotted against the aspect ratio A for various thickness ratios 5, where Umax p1 = 5U... The curve S ->0 represents linear 1,0
0.5
X co
E
0.2
0.5
1.0
7.5
2,0
0.5
3,0
3.5
V0
4.5
5.0
A
Figure 3-73 Maximum perturbation velocity on an elliptic wing with elliptic profile as in Fig. 3-72 for zero lift, at x = 0, -s
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 203
theory. The curves for the other values of 5 show the deviations of the exact solutions from linear theory.
Swept-back wing Another example is the swept-back wing of constant chord. The wing of infinite span (see Fig. 3-74) is considered first. Its sweep-back angle is cp and the profile z(x, y) = z(xr) is constant over the span, where xr is the x coordinate of the middle (root) section. The wing sections at large distance from the plane of symmetry are always in quasi-two-dimensional flow. Its velocity distribution can be
determined by assuming an incident flow normal to the leading edge of the magnitude U. cos gyp. The result is a perturbation velocity in the x direction that, for the swept-back wing, is smaller by the factor cos p than that for the unswept wing (plane problem): (3-186)
oo) = up, cos p
Now the velocity distribution on the middle section is to be computed. From Fig. 3-74, x - x' = x - x' - ly' I tan gyp. Introduction of this relationship into Eq. (3-175a), together with Eq. (3-176), yields u(Xr)
U
_
1
C
0
az 8x'
f
Xr- Xr - y' taal qi
(xr - Xr- j tan 92)2 + y'2'
tly' dxr
U
The integration requires special caution [see footnote to Eq. (3-175)]. The result for the middle section is
u xr = cosrp U0.
UPI
z 2x tanh-1 (sinT)
This relationship was first published by Neumark [65]. The first term represents the velocity distribution on a wing section far away from the wing plane of symmetry
Figure 3-74 Geometry of the swept-back wing of infinite span.
204 AERODYNAMICS OF THE WING
50
-.N,
section of swept-back wings of infinite span with
o ti..
00,
0.2
Figure 3-75 Velocity distribution at the middle (root)
INN
N
Xr-
O..Y
0..5
0.8
10
generalized parabolic profiles from Eq. (2-6) at zero lift. Sweepback angle p = -45°, 0°, + 45°. (a) Relative thickness position Xt = 0.2. (b) Relative thickness position Xt = 0.3. (c) Relative thickness position Xt = 0.5.
as in Eq. (3-186). The second term represents the change in velocity distribution caused by wing folding. In the case of backward sweepback (gyp >0), the perturbation velocity in the front part of the middle section is reduced, and in the rear part of the middle section it is increased. The above equation has been extended to generalized parabolic profiles. The result is presented in Fig. 3-75 for profiles with relative thickness positions Xt = xt/c = 0.2, 0.3, and 0.5. The curves for the sweepback angles cp = -45°, 0°,
and +45° show a very considerable influence of sweepback on the velocity distribution over the middle section. The maximum perturbation velocities are shown once more separately in Fig. 3-76 over the sweepback angle.
Figure 3-76 Maximum perturbation velocity at middle (root) section of swept-back wings of constant chord and infinite span vs. sweepback angle gyp; see Fig. 3-75.
WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 205 if
-0 0. 70
1.2
023 1.0
-
/
'
71
050 '
1
I
1
075'i 0.90
0.2
ti
1.00
0
-0.2
0,2
0X -
Figure 3-77 Velocity distribution of swept-back wing of constant chord, with aspect ratio .1 = 2.0 and sweepback angle V= 53° at zero lift for several sections along the span, according to Neumark. Wing profile: parabolic profile Xt = 0.5. rnaxO) (4/ir)(8 cos VU°,) = maximum perturbation velocity of swept-back wing of infinite span at section y = -.
For a swept-back wing of constant chord and finite span, corresponding computations have been made by Neumark [65]. The velocity distribution u of a wing of aspect ratio A1= 2 and sweepback angle cp = 53° is illustrated in Fig. 3-77 for various sections along the span. It is related to the maximum perturbation velocity of the swept-back wing of infinite span at a section far away from the wing root [Eq. (3-186)]. For the same wing, the lines of constant velocity (isobars) are drawn on the wing planform in Fig. 3-78. This figure demonstrates particularly well that, as a result of the sweepback, the maximum perturbation velocity increases
Figure 3-78 Isobars of a swept-back wing of aspect ratio .1= 2, with sweepback angle p = 53° at zero lift, from [651. Curves u (x, y) lu p, max(Y _ oo) = const; see Fig. 3-7 7 .
206 AERODYNAMICS OF THE WING
considerably in the vicinity of the wing middle (root) section, and that the velocity maximum at this middle section has shifted far back. Investigations on the pressure distribution over the middle section of a lifting swept-back wing of infinite span have been conducted by Kuchemann and Weber [48], and those on a swept-back wing with an arbitrary symmetric profile by Weber [92].
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208 AERODYNAMICS OF THE WING
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WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 211
92. Weber, J.: The Calculation of the Pressure Distribution over the Surface of Two-
Dimensional and Swept Wings with Symmetrical Aerofoil Sections, ARC RM 2918, 1953/1956; 2993, 1954/1957; 3026, 1955/1957; 3098, 1957/1959.
93. Weinig, F.: Beitrag zur Theorie des Tragfliigels endlicher insbesondere kleiner Spannweite, Lufo., 13:405-409, 1936; 14:434-437, 1937; NACA TM 1151, 1947. 94. Weissinger, J.: Der schiebende Tragflugel bei gesunder Stromung, Jb. Lufo., 1:138-181, 1940; Lil.-Ges. Lufo., S2:5-51, 1938/1939; ZWB Lufo., TB 10, no. 7, 1943. Fuchs, R.:
Lil.-Ges. Lufo., S2:65-82, 1938/1939. Jaeckel, K.: Z. Angew. Math. Mech., 33:65-66, 1953. Weinig, F.: Lufo., 13:45-54, 1937. 95. Weissinger, J.: Uber eine Erweiterung der Prandtlschen Theorie der tragenden Linie, Math.
Nachr., 2:45-106, 1949; NACA TM 1120, 1947. Reissner, E.: Proc. Nat. Acad. Sci. U.S.A., 35:208-215, 1949. 96. Weissinger, J.: Der Tragflugel endlicher Spannweite, in S. Fligge (ed.), "Handbuch der Physik, vol. VIII, Stromungsmechanik II," pp. 402-431, 434-437, Springer, Berlin, 1963. 97. Weissinger, J.: Neuere Entwicklungen in der Tragflugeltheorie bei inkompressibler
Stromung, Z. Flugw., 4:225-236, 1956. 98. Werle, H.: Sur 1'Eclatement des Tourbillons d'Apex d'une Aile Delta aux Faibles Vitesses, Rech. Aer., 74:23-30, 1960. 99. Wieselsberger, C.: Experimentelle Priifung der Umrechnungsformeln, in L. Prandtl, C. Wieselsberger, and A. Betz, "Ergebnisse der Aerodynamischen Versuchsanstalt zu Gottingen," vol. 1, pp. 50-53, Oldenbourg, Munich, 1921. Betz, A.: in L. Prandtl, "Gesammelte Abhandlungen zur angewandten Mechanik, Hydra- and Aerodynamik," p. 389 (footnote), Springer, Berlin, 1961. 100. Widnall, S. E. and T. M. Barrows: An Analytical Solution for Two- and Three-Dimensional Wings in Ground Effect, J. Fluid Mech., 41:769-792, 1970. Kida, T. and Y. Miyai: Aer. Quart., 27:292-308, 1976. 101. Wieghardt, K.: Uber die Auftriebsverteilung des einfachen Rechteckfliigels fiber die Tiefe, Z. Angew. Math. Mech., 19:257-270, 1939; NACA TM 963, 1940. Ginzel, I.: Jb. Lufo., 1:238-244, 1940. 102. Winter, H.: Stromungsvorgange an Platten and profilierten Korpern bei kleinen Spannweiten, Forsch. Ing.-Wes., 6:67-71, 1935; NACA TM 798, 1936. Flachsbart, 0.:. in L. Prandtl, C. Wieselsberger, and A. Betz, "Ergebnisse der Aerodynamischen Versuchsanstalt zu Gottingen," vol. IV, pp. 96-100, Oldenbourg, Munich, 1932. 103. Young, J. De and C. W. Harper: Theoretical Symmetric Span Loading at Subsonic Speeds for Wings Having Arbitrary Plan Form, NACA Rept. 921,. 1948. Young, J. De: J. Aer. Sci., 22:208-210, 1955.
CHAPTER
FOUR WINGS IN COMPRESSIBLE FLOW
4-1 INTRODUCTION The theory of the wing in incompressible flow was discussed in Chap. 2 for the two-dimensional problem (infinite span) and in Chap. 3 for the three-dimensional problem (finite span). In this chapter the wing in compressible flow will be treated. Both subsonic and supersonic flows will be considered, depending on whether the flow velocities are lower or higher than the speed of sound, respectively. The connection between these two kinds of flow is formed by the transonic flow. Flows with very high supersonic velocities, so-called hypersonic velocities, are designated as hypersonic flows.
The influence of compressibility must be taken into account at Mach numbers higher than Ma 0.3. Compressible flow is of great significance for flow about wings, because the Mach numbers in aeronautics are, in general, considerably higher than 0.3. The discussions of this chapter will be organized similarly to those of Chaps. 2 and 3, in that first the airfoil of infinite span in compressible flow (profile theory) and then the wing of finite span in compressible flow will be treated. From gas dynamics it is known that the compressible flows of subsonic and supersonic velocities are basically different; the same is true, naturally, for wing flows (see Fig. 1-9). For the theoretical treatment of compressible wing flow, the so-called linear
theories will be applied predominantly, because they yield results that can be interpreted easily and thus allow the establishment of general validity and of practical conclusions. For the theoretical considerations, mainly inviscid flow will be assumed as in Chaps. 2 and 3. Besides the textbooks on gas dynamics listed in Section II of the Bibliography, 213
214 AERODYNAMICS OF THE WING
basic questions of compressible wing flow are treated by Taylor [94], Prandtl [73, 74], von Karman [1011, Howarth [34], Robinson and Laurmann [78], Sears [83], Kuo and Sears (47], Heaslet and Lomax [30], Jones and Cohen [39] , and Garrick
[26]. Furthermore, more recent results and understanding of the theory of the aerodynamics of wings in compressible flow are presented in progress reports for certain time intervals by, among others, von Karman [102, 104], Ashley et al. [31, Kuchemann [45], Schlichting [81], Landahl and Stark [48], and Hummel [351. Problems of experimental wing aerodynamics are treated by Frick [24]. In this connection, the comprehensive compilations of experimental data on the aerodynamics of lift and drag by Hoerner and Borst [32], and Hoerner [31 ] should be mentioned.
4-2 BASIC CONCEPT OF THE WING IN COMPRESSIBLE FLOW
4-2-1 Temperature Effects in Compressible Flow It is a peculiarity of all compressible flows that their aerodynamic processes are always coupled with thermodynamic processes. The pressure changes in the flow, in general, are connected to temperature changes that may be determined from the equation of state [Eq. (1-la)] . Stagnation flow An outstanding station in the flow about a body is, according to Fig. 4-1, the front (upstream) stagnation point, at which the velocity is zero. The flow quantities at the stagnation point will be designated by the index 0. The
pressure in the undisturbed flow of velocity w is p., the density p., and the temperature T.. At the stagnation point, the velocity is wo = 0, and the pressure, density, and temperature are po, °o, and To, respectively. A pressure increase dp = po --p,c, takes place on the streamline incident on the stagnation point, which causes
a temperature increase 4T = To - T.. The pressure coefficient at the
stagnation point is obtained with steady, that is, isentropic compression as cpo
P-0
1}
Poo W
2
_
y
.,..
y Maw
{(1+7;'M)_ -1
1
-(4=1-)
O°
The dependence of the pressure coefficient at the stagnation point cpo on Ma is shown in Fig. 4-2a. For moderately high Mach numbers of the incident flow,
woo
P., T. Figure 4-1 Temperature rise through compression.
WINGS IN COMPRESSIBLE FLOW 215
3.0
3.0
2.5
Isentropic (continuo
Isentropic
(continuous)ir '0.
184 o
Z.0
1.5
15
1/
With shock wave (discontinuous)
With shock wave (discontinuous)
Approximation 1.0
0
I
I
10L 0
J
Z
1
3
a
Z
1
3
Ma Figure 4-2 Compressible stagnation flow of air with y = 1.405. (a) Pressure coefficient at b
stagnation point. (b) Temperature ratio.
particularly in the subsonic range, Eq. (4-1) is reduced by binomial expansion to 1 + 4Ma,., as also shown in Fig. 4-2. Agreement of the approximation with cP o the exact formula, Eq. (4-1), is quite good up to Ma. = 1. For Mao, -p 0, Eq. (4-1) becomes the well-known formula for the stagnation pressure of incompressible flow, po - p. = (o ./2)w2 , which is -the basis for velocity measurements with the Prandtl
impact-pressure tube (pitot tube). Such a tube measures the pressure difference (Po -p.) in compressible flow as well. At an incident flow of supersonic velocity, the pressure changes from p, to po discontinuously through a shock wave located somewhat upstream of the stagnation
point (Fig. 4-2a). The pressure change can be determined in this case by first computing the pressure jump across the shock wave from the equations of the normal shock. In the subsonic flow behind the shock wave, the pressure change is isentropic. The result of this computation is
c+1 PO
y
(y-k1)2Ma00-
17-1
2[2yMa-(y-1)J
2 yMa;0
()
At very high Mach numbers, May, -* -, the pressure coefficient approaches a finite value, which for air of y = 1.405 is CP0 max = 1.84. It can be seen from Fig. 4-2a that for supersonic incident flow, the pressure increase at the stagnation point with unsteady compression (which describes the physical reality) is considerably smaller than that obtained from the computation of steady compression.
Temperature The pressure increase of the stagnation point is always tied to a temperature increase. It is obtained from the energy equation as
216 AERODYNAMICS OF THE WING
T =T 00 T 2c
woo-
(4-3a)
0
TTo
1+y
,, 1
Ma.2o
(4-3b)
00
This temperature increase at the stagnation point is shown in Fig. 4-2b. It is equally valid for steady and for unsteady compression. It increases with the square of the velocity, and therefore, reaches appreciable values in the supersonic incident flow range. Note that a temperature increase according to Eq. (4-3) occurs not only at the stagnation point and its vicinity but, approximately, everywhere along a solid
wall. In a thin layer (friction or boundary layer) close to the wall, the kinetic energy of the moving gas is transformed into heat through viscous effects (see Fig. 4-3). This results in heating of the wall by an amount d T = T - T,,, which can be represented approximately by a relationship similar to Eq. (4-3). It can be realized,
therefore, that a "heat cushion"
is
found over the entire surface of a body
immersed in a flow of high velocity. In the immediate vicinity of the stagnation point this heating is produced by compression, and on the remaining portion of the surface by friction.
4-2-2 Friction Drag on a Flat Plate in Compressible Flow In Sec. 2-5-2 the friction drag of wing profiles in incompressible flow was discussed. Particularly, in Fig. 248 the influence of the Reynolds number on the drag of a flat plate in chord-parallel incident flow was demonstrated. The insight gained then will now be extended to the case of compressible flow.
Wall flow The compressible boundary layer is decisively affected by the heat transfer between the wall and the streaming fluid. Here, the case of the wall without heat transfer (adiabatic wall) is of particular importance. The laminar boundary layer of compressible flow can be treated theoretically, but theoretical studies dealing with the turbulent compressible boundary layers are
still limited to semiempirical theories of the type of the Prandtl mixing-length hypothesis, in which, however, additional assumptions must be made. Drag coefficients of the flat plate at zero incidence over Reynolds and Mach numbers are given in Fig. 4-4 in comparison with measurements. Agreement between computation and measurement is not satisfactory in all cases. However, some uncertainty of measurement at high Mach numbers should be taken into account. Also, in Fig. 4-5, W.
TW W.
Figure 4-3 Heating of a solid wall through friction; W = velocity boundary
layer: T = temperature boundary layer.
WINGS IN COMPRESSIBLE FLOW 217 100 80 i
60
I
Turbulent
Laminar
V0 1
i
I
zo
10
1328
i ff
e
I
f
da, .
6
/
J
I
!
i
I
B/ps/GS
Lld
V
OO
I
`
-
1. 69
Q6
173 290
Ow
V 2.00 s 12 oZ.19
02
2
9
8 10 I
2
6
'hd
T 6 a 105
V
I
1 48
02
Ada m
le
2
Q10 y
_
4,
6 8106
Y
1
Umc Re -
T -12 71!
6 8107
L
-7
6 8108
M
71
E
4'
6
8 109
vo,
Figure 4-4 Skin-friction coefficient of the flat plate at zero incidence vs. Reynolds number and Mach number for laminar and turbulent flow in the boundary layer and for adiabatic walls, from van Driest.
the ratio of the drag coefficients at compressible and incompressible flow are presented against the Mach number up to very high Mach numbers. The decrease of friction drag is very pronounced at high Mach numbers. Curve 1 of the two
theoretical curves is valid for the adiabatic wall, curve 2 for the wall with heat transfer. Measurements of several authors are in good agreement with theory. For completeness, the friction coefficients of the flat plate at zero incidence are also given for compressible laminar flow.
4-2-3 Similarity Rules for Wing Theory at Compressible Flow Velocity potential (linearization) For slender body shapes (wings) in an incident flow of velocity U.. in the direction of the x axis (longitudinal axis), the local velocities differ only a little from U. in direction and magnitude. The total flow can then be separated into a basic flow and a superimposed perturbation flow by setting Z%=Ucc
2G
tr
TV=v'.
(4-4)
where it, v, w are the perturbation velocities caused by the wing, for which it is required that
c
V
U,
it,
U
By retaining only the largest terms (linearization), the potential equation of compressible flow for such a flow problem takes the form
(1 -
a2
a2 (1)
`20
aX2
r?/-
cZ-
=0
(4-5)
218 AERODYNAMICS OF THE WING
Dhawan
(5
C Coles (plane flow) Brinich, Diaconis (i xial flow) Chapman, Kester xial flow) Seiff (axial flow)
e
°
o
2
e7 Lobb, Winkler, Pen h ac
a
Q.
0
vas c
E
v
(plane flow)
3
o Hill (axial flow)
E
Of 0 03
02
Theory of Wilson, without -- Theory of van Driest, with
heat transfer
0.1
0 1
2
3
6
S
9
10
ma".
Figure 4-5 Ratio of the skin-friction coefficients of the flat plate at zero incidence for compressible and incompressible flow vs. Mach number; Reynolds number Be - 10'. Comparison of theory and experiment.
where Ma = U/a is the local Mach number. This equation is valid for subsonic, transonic, and supersonic flow.* It is nonlinear in the velocity potential. The components of the perturbation velocities are obtained from Eq. (4-5) as U
ao ax
00
-
v
ay
u'
(4-6)
bz
In analyzing the linearized Eq. (4-5) further, the first term requires particular attention because it changes sign when passing the speed of sound (Ma = 1) and thus changes the mathematical character of the differential equation. By retaining
only the linear terms in u/U., the local Mach number Ma in Eq. (4-5) may be expressed by the Mach number of the incident flow Ma. = U../a as follows:
Mat=[1+2(1-}-72 1Ma)
i-]
MaM
(4-7)
For pure subsonic and pure supersonic flow, the simplified potential equation
(1 - Ma")
a2ax'
+
a2 aY2
+
a2o azs
=0
(111a.. > 1)
(4-8)
is obtained by replacing Ma by Ma. in first approximation. This differential equation for 0 is now linear. For pure subsonic flow, it is of *For Ma = 0, Eq. (4-5) reduces to the well-known relationship of incompressible flow. It is not valid for hypersonic flow; see Sec. 4-3-5.
WINGS IN COMPRESSIBLE FLOW 219
the elliptic type, as is the equation for incompressible flow. For pure supersonic flow, however, it is of the hyperbolic type.
When the undisturbed flow velocity is equal to the speed of sound (May, = 1), a transonic flow results whose velocity field may include stations of Ma = 1. In this case, from Eqs. (4-5) and (4-7), it follows that
- 7± i ao aw0 + u ax ax,
This differential
+ "" = 0
a20 ay-
az21
(1Y1a,,
(4-9)
equation for 0 is nonlinear. Therefore, the computation of
transonic flow fields is considerably more difficult than computations of pure subsonic and pure supersonic fields. The potential equations derived above [(4-8) and (4-9)] will now be used to derive similarity rules for three-dimensional wing theory at subsonic, supersonic, and transonic flows. Similarity rules for subsonic and supersonic flow For subsonic flows, similarity rules
can be derived from Eq. (4-8) according to Prandtl [73], Glauert [27], and Gothert [28], the application of which greatly simplifies the computation of compressible potential flows. These procedures can be applied similarly to
supersonic flows; see Ackeret [1]
.
Of the various possible derivations of these
similarity rules, the so-called streamline analogy will be applied. The similarity rules for a wing in subsonic or supersonic incident flow
(Ma.. > 1) are obtained through a transformation of the potential equation, Eq. (4-8). This transformation is such that the Mach number of the undisturbed flow no
longer appears explicitly in the transformed- potential equation. To this end, a transformed reference flow is established for the given flow in a suitable way. The variables of the reference flow are designated by a prime: X, = X
z' =
y' = cl y
0 = C20'
C1Z
UO'C = U11-
(4-10)
By introducing these terms into Eq. (4-8), the factor cl > 0 is determined in such a way that the Mach number is eliminated, resulting in
cl = 1 - 1V1a200 (Maw < 1)
(4-1 la)
cl = 111a 00 -
(4-11b)
1
(?V1a,,. > 1)
These cases can be combined to
cl =
j1
- Mall
(4-12)
With Eq. (4-11) for the transformed reference flow, the following differential equations are obtained for the velocity potential: a201 a x'3
°20' a x'2
+
C20'
020'
oy y
a z,2
a2 0' ay,2
a-45' az12
=0
(Ma. <1)
(4-13)
=0
(Max > 1)
(4-14)
220 AERODYNAMICS OF THE WING
The transformed equation for subsonic flow is identical to the potential equation for incompressible flow, and the transformed equation for supersonic flow
is identical to the linear potential equation, Eq. (4-8), at Mach number May, = f. This transformation shows that computations of subsonic flows of any Mach number
can be reduced to computations of flow at Ma,, = 0, and computations of supersonic flows to those at Ma. = \[2-. The transformation factor c2 in Eq. (4-10) remains undetermined for the time being. Its value will be given later. Application of the transformation formulas, Eq. (4-10), to wings of finite span will now be treated. The coordinate system x, y, z of Fig. 4-6 will be used with its
x axis parallel to the incident (undisturbed) flow. Equation (4-10) describes the procedure for determining the transformed wing from a given wing of a given Mach
number where the flow field about the transformed wing is to be computed, according to the above rules, for subsonic flow at Maw = 0 and for supersonic flow at Ma., _ Nf2-. The transformed wing according to Eq. (4-10) is then obtained from
the given wing by decreasing or increasing, respectively, the dimensions in the directions normal to the incident flow direction (y and z directions) by the factor cl of Eq. (4-12). .For the wing planform, the following relationships between the transformed (primed symbols) and the given data are thus obtained: (4-15a)
7.' = 1
Taper:
Aspect ratio: Sweepback angle:
A
;z
A' = A 1 - Ma.N cot 92' = cotg2
1 - MaLI
(4-15b)
(4-15c)
Figure 4-6 Wing geometry. (a) Wing plan-
form; x = ct/c,, taper; ,1 = b2 /A, aspect ratio; A = wing planform area; 0 = sweepback angle. (b) Profile section y = const; zC(x) = profile contour; h/c = relative camber; t/c = relative thickness; cu = angle of attack.
WINGS IN COMPRESSIBLE FLOW 221 a,
Given wing lU.
b Transformed wing
tTa
i, i-j'2'
Figuze 4-7 Application of subsonic and
M¢ Z.O-ff
supersonic similarity rules to the example of a tapered, swept-back wing. (a) Given wing, to be computed for Mach numbers Ma,, = 0.7, 0.9, 1.1, and 2. (b) Transformed wing for these Mach numbers.
From Eqs. (4-15b) and (4-15c), the remarkable relationship <1' tan rp' _ A. tan T
(4-15d)
is obtained, where it is immaterial to which -of the planform contour lines -the sweepback angle is referred, for example, the leading edge or the trailing edge.
In Fig. 4-7 the transformation of the wing planform is explained through the example of a swept-back wing. The crosshatched wing planform in Fig. 4-7a is the shape of the given wing, the flow field of which is to be determined for the various Mach numbers Ma. = 0.7, 0.9, 1.1, and 2.0. The corresponding transformed wing planforms are shown in Fig. 4-7b, where at Maw < 1 the transformed wings are to . be computed for incompressible flow (Ma.. = 0), and at Maw, > I for Ma. _ In Fig. 4-8, the wing planform transformation as given by Eqs. (4-15a)-(4-15c) is explained in more detail. Here, A'/i1 and cot cc'/cot p are plotted versus Ma... Again, the given wing planform, which is to be computed for the various Mach numbers, has been crosshatched. The open wing planforms represent the trans-
222 AERODYNAMICS OF THE WING
Figure 4-8 Illustration of the application of subsonic and supersonic similarity rules; aspect ratio A' and sweepback angle gyp' of the transformed wing vs. Mach number-
formed wings for the corresponding Mach numbers. When the given Mach numbers respectively, the transformed and the given wings are are Ma = 0 and Ma. = identical. Figure 4-8 shows that, in the subsonic range, an increase of Ma results in a decrease of the aspect ratio whereas the sweepback angle increases. For Ma. -* 1,
the aspect ratio of the transformed wing approaches !1' - 0 and the sweepback angle gyp' -+ 900. In the supersonic range Ma. >,,/2-, the aspect ratio of the transformed wing increases with Ma,. whereas the sweepback angle decreases. In the
limit of very large Ma., the aspect ratio of the transformed wing A1' -)- - and the sweepback angle cp' - 0. The remarkable result is found that for large Mach numbers the three-dimensional wing flow field is converted into a two-dimensional field. The Prandtl-Glauert-Gothert-Ackeret rule is also applicable to asymmetric
incident flow (yawed wings); see Truckenbrodt [281. For the profile cross section
and angle of attack of Fig. 4-6b, Eqs. (4-10) and (4-12) lead to the following expressions: Camber :
Thickness ratio:
Angle of attack:
This shows that for Ma,. <
h'
h
C'
C
t
t
C
C
11 MaL I
a' = a V 11 -MaLI
(4-16a)
(4-16b)
(4-17)
the transformed wing has less camber, is thinner, it and has a smaller angle of attack than the given wing; conversely, for May, > has more camber, is thicker, and has a larger angle of attack. After the effect of the transformation, Eq. (4-10), on the wing geometry has been discussed, the relationship between the pressure distributions of the given and the transformed wing must be studied.
WINGS IN COMPRESSIBLE FLOW 223
The dimensionless pressure coefficients cp = (p - p4/(p U42) assume, within the framework of linearization, the approximate form cP
u = -2 U=
2
Um ax
2"
c'2
a-
2
aO-
U ax,
U11
(4-18a)
(4-18b)
where the velocities of the incident flow U. of the given and transformed flow must be equal. This leads with Eq. (4-10) directly to (4-19)
cP = c2 cp
The still-unknown transformation factor c2 is determined from the kinematic
flow conditions for the two wings (streamline analogy). These are, within the framework of linearized theory, W = UCC aZx
az' w = U00 arc
(4-20)
where w and w' are the z components of the perturbation velocity on the profile contour zC and zc, respectively (Fig. 4-6b). Because w = aO/az and w' = aO'/az', we find with Eq. (4-10): 1 C2
(4-21)
11 - Ma_ 1
The meaning of the subsonic and supersonic similarity rules can now be summarized as follows: From the given wing and the incident flow Mach number, the transformed wing is found by multiplying the dimensions of the given wing in the y and z directions and its angle of attack by the factor cl = I(1 -Mam)I, whereas the dimensions in the x direction remain unchanged. For subsonic
velocities, the flow about the transformed wing is computed from the incompressible equations; for supersonic velocity, however, it is computed from the compressible equations for Ma,, = V2-. If the incident flow velocities are equal for both wings, the pressure coefficients are related by C
P
= P - Pm
el"
(version I)
(4-22)
11- Mal l With regard to practical applications, it is expedient to choose a transformation in which only the dimensions in the y direction (wing planform) are distorted, whereas the dimensions in the z direction (profile and angle of attack) remain unchanged. Such a transformation is obtained from the above version I by removing the distortion in the z direction according to Eqs. (4-16a), (4-16b), and (4-17). Thus, q00
from Eq. (4-22), the pressure coefficient is changed, within the limits of the linearized theory, by the factor pressure coefficient becomes
11 - Mam 1, that is, cP = cp
I 1 --Ma'. 1, and the
224 AERODYNAMICS OF THE WING
Cp =
P-P°°
Cp
(version II)
I1 -Ma.
q°°
(4-23)
This relationship is shown in Fig. 4-9. Thus, the following version is obtained for the subsonic and the supersonic similarity rule. From the given wing and Mach number, a transformed wing is formed by multiplyin the dimensions of the given wing in the y direction with the factor Cl = I(1 -Ma ,',)I, whereas the dimensions in the x and z directions remain unchanged. For the transformed wing thus obtained, the incompressible flow field is computed when the given incident flow Mach number lies in the subsonic range. When the Mach number lies in the supersonic range, however, the flow field about the transformed wing is computed from compressible equations at Ma. = N f2-. For equal incident flow velocities U. of given and transformed wings, the pressure coefficients are interrelated through Eq. (4-23). From the subsonic and supersonic similarity rules, the following generally valid relationships for the aerodynamic coefficients are obtained: Let the function
cr = S' fl
(A' ;
x' A'; cot q,' ;-; C
'
y,
(4-24)
s
describe the dependency of the pressure coefficient on the geometric wing data at Ma,, = 0 or Ma = f . Then the corresponding dependency of the geometric wing data at an arbitrary Mach number is obtained, because of Eqs. (4-15) and (4-22), in the form:
cP=
a
1-Ma"I
f2(A;AVI1-Mat1;cotgpVj1-Ma`,I;x;y (4-2 5a) C
00
S
Here S stands for the relative thickness t/c, the relative camber height h/c, or the angle of attack. This equation can be written in a simpler form: 6
cp =
/3 (A; A tancp; A
Ji
V
I 1 - .Ma2 00 I; x; '-) C
I
s
(4-25b)
From this formula for the pressure distribution, the lift coefficient is obtained in corresponding form by integration over the wing surface: CL
S
Fi(A; A tanfp; A
1- Mat00I)
(4-26)
111 -Mu0l Here 5 stands for the angle of attack or for the relative camber height. By going to the limiting case of the airfoil of infinite span (X = 1, p = 0, A -* 00), the subsonic similarity rule transforms into the well-known Prandtl-Glauert rule of plane flow. A formula analogous to Eq. (4-26) for the drag coefficient (wave drag) that is valid, however, only for supersonic flow (see the discussions of Sec. 4-5-5) is given as CD =
-
Ma7-1
F2 (A, A tan q9, A I Ma - 1)
(4-27)
For wings with zero angle of incidence, S is the wing thickness ratio t/c. In this case, the drag coefficient at zero lift CD = C- DO is proportional to 5 2 .
WINGS IN COMPRESSIBLE FLOW 225
Figure 4-9 Illustration of the applicaV2
c
3
tion of subsonic and supersonic similarity rules (version II): transformation of the pressure coefficients.
The outstanding value of the above formulas lies in their describing the Mach number effect in a simple way. They can, however, also be used to great advantage for the classification of test results. Transonic similarity rule For flows of velocities near the speed of sound (transonic flows), a similarity rule can be derived after von Karman [103] that is related to those for subsonic and supersonic flows. For wings in a flow field of sonic incident velocity (Ma.. = 1), it is obtained from the potential equation, Eq. (4-9). Contrary to the similarity rules for subsonic and supersonic flows, for which the dependency of aerodynamic coefficients from the geometric wing parameters and the Mach number was investigated, only the dependency of the aerodynamic
coefficients on the geometric parameters must now be studied, because Ma. = const = 1.
The problem can be posed in the following way: Given is a wing with all geometric data (planform and profile) at an angle of attack zero. What, then, is the geometry of a reference wing, also in an incident flow field of Maw, = 1, that has an
affine pressure distribution equal to that of the given wing? To answer this question, the following transformation is introduced into Eq. (4-9) [see Eq. (4-10)] :
X, = x
y' = C3 f
z' = C3 Z
.0 = C40'
Uc'o = Uc-
(4-28)
where the quantities without primes refer to the given wing, those with primes to the reference wing. Introducing Eq. (4-28) into Eq. (4-9) yields, with C3 = C4
(4-29)
the following nonlinear differential equation for the velocity potential of the transformed flow:
226 AERODYNAMICS OF THE WING
-y+ i a 0l a201 U,, ax' ax'
+
(a2 0' ay'2
ay 0')
+
=0
az'
(Ma. = Ma' = 1)
(4-30)
For an additional relationship between the constants c3 and c4, the kinematic flow conditions, Eq. (4-20), for both wings have to be established. For chord-parallel incident flow, this relationship is azC aZC =
C3C
axl ax'
S
(4-31)
$'
where 6 = tic is the thickness ratio of the wing profile, which has been assumed to be symmetric. Hence, with Eq. (4-29): (6
C4 = (a,)y
C3 -
13
(4-32)
The distortion of the geometric data of the wing planform is given by the factor c3 in Eq. (4-28). Hence, the following transformations are valid:
2' = 7.
Taper:
Aspect ratio: Angle of sweepback:
Al
=
cot cp'
(4-33a)
s
113
6,
a 1/3
A
(4-33b)
cot cp
(4-33c)
As an example for the transonic similarity rule, the transformation for a swept-back wing is presented in Fig. 4-10.
Transformation of the pressure distribution is obtained in analogy to Eqs. (4-18) and (4-19) merely by replacing c2 by c4i that is, cp =C4c,. With C4 according to Eq. (4-23), it follows that 2/3 CP
(4-34)
C
If the pressure distribution is to be related to the geometric parameters, Eq. (4-34), considering Eqs. (4-33a)-(4-33c) leads to CP = 62/3 f
/
z,
;1 tan!p, A61/3..C
i
(4-35)
Hence it is shown that the pressure coefficient from the transonic similarity rule is proportional to 5 213 , whereas it is proportional to 6 according to the subsonic and supersonic similarity rules of Eq. (4-25). From Eq. (4-35) the following expression is found for the drag coefficient, CD = 55I3 F (, ;1 tang,
-16113)
(4-36)
showing that the drag coefficient is proportional to 51", whereas it is proportional to 62 according to Eq. (4-27).
WINGS IN COMPRESSIBLE FLOW 227
b
Figure 4-10 Application of the transonic similarity rule for sonic incident flow to the example of a trapezoidal swept-back wing. (a) Thickness ratio S = tic = 0.05. (b) Thickness ratio b' = t'/c' = 0.10.
The formulas for the airfoil of infinite span (X = 1, cp = 0, A - 00) will be given in Sec. 4-3-4 in extended form (Mao 1 instead of Ma. = 1).
4-3 AIRFOIL OF INFINITE SPAN IN COMPRESSIBLE FLOW (PROFILE THEORY)
4-3-1 Survey Now that a basic understanding of the compressible flow over wings (slender bodies) has been established in Sec. 4-2, the airfoil of infinite span will be discussed. On the basis of the similarity rules of Sec. 4-2-3, it turns out to be expedient to study pure subsonic and supersonic flows (linear theory) first, that is, flows with subsonic and supersonic approach velocities (Ma.. 1), Secs. 4-3-2 and 4-3-3. The validity range
of linear theory for Ma < 1 is limited by the critical Mach number Mar', for the drag of Sec. 4-3-4. Later, transonic flow (nonlinear theory) will be discussed, at which the incident flow of the wing profile has sonic velocity (Ma.. ~ 1). Lastly, in
Sec. 4-3-5, a brief account of hypersonic flow will be given, characterized by incident flow velocities much higher than the speed of sound (Ma. > 1).
4-3-2 Profile Theory of Subsonic Flow Linear theory (Prandtl, Glauert) The exact theory of inviscid compressible flow leads to a nonlinear differential equation for the velocity potential for which it is
228 AERODYNAMICS OF THE WING
quite difficult to establish numerical solutions in the case of arbitrary body shapes. For slender bodies, however, particularly for wing profiles, this equation can be linearized in good approximation, Eq. (4-8). For such body shapes, explicit
solutions are therefore feasible. In these cases, the physical condition has to be satisfied that the perturbation velocities caused by the body are small compared with the incident flow velocity. This condition is satisfied for wing profiles at small
and moderate angles of attack. Linear theory of compressible flow at subsonic velocities leads to the Prandtl-Glauert rule. It allows the determination of
compressible flows through computation of a subsonic reference flow. As discussed
in Sec. 4-2-3, this subsonic similarity rule (version II) consists essentially of the following.
For equal body shapes and equal incident flow conditions, the pressure differences in the compressible flow are greater by the ratio 1 J 1 -Ma;, than those in the incompressible reference flow. Here, Ma. = U.Jam, is the Mach number, with
U. the incident flow velocity and a the speed of sound. Hence, the pressure distribution over the body contour from Eq. (4-23) becomes 1
f,,
P (x) - p. = y 1 - Mci U'inc(x) - P- ]
(4-37)
Here the quantities of compressible flow are left without index, those of the incompressible reference flow have the index "inc." For the dimensionless pressure coefficient, the formula of the translation from incompressible to subsonic flow is obtained as eP
P
-
Poo
q00
1 V1-Maz 00
Cpinc
(version II)
(4-38)
Here it has been assumed that profile contours and angles of attack of compressible flow and of the incompressible reference flow are equal; that is, Zinc(X) = Z(X)
(4-39a)
ainc = a
(4-39b)
where X = xlc and Z = zlc are the dimensionless profile coordinates according to Eq. (2-2). An experimental check of Eq. (4-38) is given in Fig. 4-11 for the simple case of
a symmetric profile of 12% thickness in chord-parallel flow. Agreement between theory and experiment is very good in the lower Mach number range. At higher Mach numbers some differences are found. In Fig. 4-11, the values of the local sonic speed (Ma = 1) are included, showing that sonic speed is first reached locally at Ma. = 0.73. The lift, obtained by integration of the pressure distribution over the profile chord, increases with the transition from incompressible to compressible flow as 1/-../l -Ma, because of Eq. (4-38). The expression for the lift coefficient is given in Table 4-1, which also contains the transformation formulas for the other lift-related aerodynamic coefficients. For
WINGS IN COMPRESSIBLE FLOW 229
0.6 0.4
vL
02
0' 081
0.6 0.4
0.2
0
0.2
0.6 Od 04 'Y /C ----
1.0
Theory ----Measurement
0.2
0,4
x/c
0.6
0.8
1.0
Figure 4-11 Pressure distributions of the profile NACA 0012 at chord-parallel incident flow for several subsonic Mach numbers May,. Theory according to the subsonic similarity rule, Eq. (4-38); measurements from Amic [88] ; Ma = 1 (wc= a) signifies points where the speed of sound is reached locally.
incompressible flow, the determination of neutral-point position, zero-lift angle, zero-moment coefficient, and angle of attack and angle of smooth leading-edge flow
has been discussed in Sec. 2-4-2. For lift slope and neutral-point position of the skeleton profile, the values found for the inclined flat plate are valid, namely, (dcL/sla}inc- 2rr and ( N/c)inc = lift , respectively. In Fig. 4-12, the theoretical slopes are plotted against the incident flow Mach number. Since, according to Eq. (4-37), the pressure distributions over a body at various Mach numbers are affine to the incompressible pressure distribution, it follows
immediately that the position of the resultant aerodynamic force in the subsonic range (as long as no shock waves are formed) is equal to that in incompressible flow. Also, the drag in the subsonic range is determined by the same processes as in incompressible inviscid flow; that is, it is equal to zero. Comparison with test results In Fig. 4-13, the most important results of the subsonic similarity rule are compared with measurements of Gothert [88]. For 5 symmetric
230 AERODYNAMICS OF THE WING
Table 4-1 Aerodynamic coefficients of a profile in subsonic incident flow based on the subsonic similarity rule (version II)* Pressure distribution
cp
Lift
CL
- cpinc
1
yl - 1ti7aN l
dCL
d
Lift slope Zero-lift angle
ao
Pitching moment
cM
Ma;
dcL
1
y'1
cL in c
- lilaN \
2r
` inc
j/1
-
M(12
o inc 1
CMinc 1
1
1
Angle of smooth leading-edge flow
- Maro
CMO inc
'inc s
«s
Lift coefficient of smooth
1
CLs
leading-edge flow
1 -Ma;, cLsinc
*« _ «inc, It/c = (h/c)inc For aerodynamic coefficients for incompressible flow, see Table 2-1.
14
0.2
0.4
0.6
Ma co
o.d
j
Figure 4-12 Theoretical lift slope at subsonic incident flow according to the subsonic similarity rule.
WINGS IN COMPRESSIBLE FLOW 231 0.14
20
0.18
0,06
P1001- 6101tert
0.15
0.12
0,09 0.04 0 12
0.02 0
41 CJ 0
0
1-0.02
-0--
t C
-0-X -0.Aq
0.15
-0.08 -0.10
1
0 0.3
0.5 0.6
0.7
a
0.8
0.9
-0.120
0..3
0.5 0.6
0.7
0.6
0.85
0.9
b
Figure 4-13 Lift slope (a) and neutral-point position (b) of NACA profiles of various thickness tic vs. Mach number, for subsonic incident flow, from Multhopp; measurements from Gothert; neutral-point position as distance from the c/4 point.
wing profiles of thickness ratios t/c = 0.06, 0.09, 0.12, 0.15, and 0.18, lift slopes are plotted in Fig. 4-13a and neutral-point positions in Fig. 4-13b, both against the Mach number of the incident flow. For comparison, the theory with (dcL/da)lnc = 5.71 is drawn as a straight line in Fig. 4-13a.* In the lower Mach number range, agreement between theory and measurement
is very good, with the exception of the profile of 18% thickness. The theoretical curve follows the experimental data up to a certain Mach number, which shifts toward Ma. = 1 with decreasing profile thickness. The differences between theory and experiment beyond this Mach number are caused by strong flow separation. This fact can also be seen in the presentation of the drag coefficients of the same profiles in Fig. 4-14a.
According to the present linear theory for very thin profiles, the neutral-point position should be independent of Mach number. The experimental results of the profiles of Fig. 4-13b show, however, a considerable dependence of the neutralpoint position on the Mach number when the profile thickness increases. For the same symmetric profiles that have just been discussed with regard to lift slope and neutral-point position, the dependence of the drag (= profile drag) on
the angle of attack a and on the Mach number of the incident flow Ala. is demonstrated in Fig. 4-14. The behavior of the curves for the drag coefficient cDp(Maa,), with t/c as the parameter, is characterized by the near independence of CDp from the Mach number in the lower Mach number range, whereas a very steep `Presented in double-logarithmic scale is dcL/d« vs. (1 -Ma;0).
232 AERODYNAMICS OF THE WING 0.05 0.04 I
0.03
cc 0° A
0,02
r- 01800
I
__j
C
Q
009 0
O tj 0.01 0.009 0.008
0.007 0.006
0.005
a I
o
0.5,
L
0.6
0.7
0.8
May, -
0.85
0.9 00.3
0.5 0.6
0.7
00
085
09
Ma. a b Figure 4-14 Profile drag of NACA profiles of various thickness vs. Mach number, for subsonic
incident flow, from measurements of Gothert. (a) Symmetric incident flow, a= 0°. (b) Asymmetric incident flow, a = 4°.
drag rise occurs when approaching Ma = 1. This drag rise results from flow separation, caused by a shock wave that originates at the profile station at which the speed of sound is locally exceeded. The associated incident flow Mach number In the case of chord-parallel is designated as drag-critical Mach number
incident flow (a = 0) the drag rise and, therefore, Ma.,,, occur closer to Ma = 1 for thin profiles than for thick ones (Fig. 4-14a). For a profile with angle of attack (a * 0), the profile thickness has a negligible influence on the drag rise, as seen in Fig. 4-14b. As would be expected, the drag rise shifts to smaller Mach numbers with
increasing angle of attack of the profile. The effect of the geometric profile parameters of relative thickness ratio, nose radius, and camber on the trend of the curves cDP(Ma,o) is shown in Fig. 4-15.
Attention should be called to the test results reported by Abbott and von Doenhoff, Chap. 2 [1 J , and by Riegels, Chap. 2 [50]. In summary, it can be concluded from the comparison of theory and
experiment that the subsonic similarity rule (Prandtl-Glauert rule) is always in good agreement with measurements before sound velocity has been reached locally on the profile, that is, when no shock waves and corresponding separation of the flow can
occur. Since these two effects are not covered by linear theory, the drag-critical Mach number is at the same time the validity limit of linear profile theory. Determination and significance of the critical Mach number Ma., will be discussed in detail in Sec. 4-3-4.
Higher-order approximations (von Karman-Tsien, Krahn) From the derivation of the
linear theory (Prandtl, Glauert), it can be concluded that the deviations of this approximate solution from the exact solution are increasing when the Mach number approaches Ma = 1. The same is shown in the pressure-distribution measurements of
WINGS IN COMPRESSIBLE FLOW 233
Fig. 4-11. Several efforts have been made, therefore, to improve the Prandtl-Glauert approximation. Steps in this direction have been reported by von Karman and Tsien [96], Betz and Krahn [7], van Dyke [99], and Gretler [29]. By the von
Karman-Tsien formula, the computation of a compressible flow about a given profile is reduced to the determination of an incompressible flow about the same profile. The result is given here without derivation: Cpinc
cp _
(4-40)
/1 -Maro T 2 (1 - Y'1- Ma's, )cpinc It can be seen immediately that this equation becomes the Prandtl-Glauert formula
for small values of cpinc According to von Karman-Tsien, the underpressures assume larger values and the overpressures smaller values than according to Prandtl-Glauert. In Fig. 4-16, the von Karman-Tsien rule and the [Eq. (4-38)]
Prandtl-Glauert rule are compared with measurements on the profile NACA 4412. Obviously, for the higher Mach numbers the von Karman-Tsien rule is in markedly better agreement with experiment than the Prandtl-Glauert rule. At the stagnation point of a profile, both theories give the pressure coefficients too high, whereas the Krahn theory, which will not.be discussed here, describes the behavior at this point accurately. Also, for Maw - 1, Eqs. (4-38) and (4.40) lose validity, as would be expected from the assumptions made in their derivation. The relationship for the critical pressure coefficient cpcr (Ma.) is shown in Fig. 4-16 as a limiting curve (see Sec. 4-3-4, Fig. 4-28).
0.04 0.03
L? 02
c. 0.01
--4
0.009 0.005 0.007 0.4
0.005
0005 'G
0.004
003! 0 e3 1
I
i
I
c5 06
i 0.7
0
i
:%3500.3
05
.s
0.7
0
1
I
0.d 003 0.5 co
I
i
/7 7
b
Figure 4-15 Profile drag of NACA profiles vs. Mach number for subsonic incident flow, from measurements of Gothert. Profile thickness t/c = 0.12; cL = 0. (a) Effect of relative thickness position xt/c. (h) Effect of nose radius rN/c. (c) Effect of camber h/c; relative camber position x,,,/c = 0.35.
234 AERODYNAMICS OF THE WING 2,0
c. =0° X = 0,275
X-_20 - = 0,30
C
C
1.5
i
T cp cr
Cpcr
t 1.2
1.2
i
R
o °
q
1
0
0.8
0.8
° 0.4
0
0
0,4
0.2
0.4
0.6
Aboo -
1.0
0L 0
0.2
0.4
016
Mao---0.
0.8
1.0
b
a,
Figure 4-16 Comparison of measured pressure coefficients in subsonic flow with theory. (1) von Karman-Tsien, Eq. (4-40); (2) Prandtl-Glauert, Eq. (4-38), measurements from [89].
4-3-3 Profile Theory of Supersonic Flow When a slender body with a sharp leading edge is placed into a supersonic flow field
streaming in the direction of the body's longitudinal axis (Fig. 4-17), the leading edge of this body assumes the role of a sound source in the sense of Fig. 1-9d. As a consequence, Mach lines originate at the sharp leading edge, upstream of which the incident parallel flow remains undisturbed. Only downstream of these Mach lines is the flow disturbed by the body. As an example of this behavior, the flow pattern about a convex profile in supersonic incident flow is shown in Fig. 4-18. The Mach lines, at which the pressure changes abruptly, have been made visible by the Schlieren method. The incident flow velocity can be determined quite accurately, with Eq. (1-33), from the angle of the Mach lines that originate at the profile leading edge.
Linear theory (Ackeret) In analogy to the case of subsonic incident flow of Sec. 4-3-2, inviscid compressible flow about slender bodies (wing profiles) can be
Figure 4-17 Supersonic flow over a sharp-edged wedge.
WINGS IN COMPRESSIBLE FLOW 235
Figure 4-18 Supersonic flow over a biconvex profile, Schlieren picture. Mach waves originate at the leading and trailing edges.
computed by a linear approximation theory in the case of supersonic incident flow as well. The linearized potential equation, Eq. (4-8), is valid both for subsonic and supersonic flows. It was Ackeret [1 ] who laid the foundation for this linear theory of supersonic flow. The essential concept of this linear theory is expressed by the requirement that the perturbation velocity u in the x direction is a function only of the inclination of the profile contour area elements with respect to the incident flow direction, of the velocity U, and of the Mach number Maw :
u(x) = -
L(X)
Maw -
1
U.
with w(x) = 3(x)UU
(4-41)
according to the kinematic flow condition (3 > 0: concave; 0 < 0: convex).
The inclinations of the contour on the upper and lower surfaces against the incident flow direction, 6u and zg1, respectively, are given for slender profiles of finite thickness and pointed nose (see Fig. 4-19) as t9
u, t = + a -
dxz
(4-42)
where a is the angle of attack of the chord and z(x) is the profile contour. In linear approximation, the dimensionless pressure coefficient becomes cp = -2u/U,.,, [see Eq. (4-18)], leading with Eq. (4-41) to (x) p - pec = 26(x) (443a) cP Q Maw - 1 2
_
2
- + 1Vla -1
a-
dz(x)
(4-43 b)
dx Here the upper sign applies to the upper surface of the profile, the lower sign to the lower surface. Equation (4-43) confirms the supersonic similarity rule (version II) as
236 AERODYNAMICS OF THE WING
Figure 4-19 Geometry and incident flow vector used in the profile theory at supersonic velocities.
derived in Sec. 4.2-3 [see Eq. (4-25)]. For the further evaluation of Eq. (443), it is expedient to separate the profile contours again, as in the case of the incompressible flow in Chap. 2, into the profile teardrop and the mean camber (skeleton) line [see Eq. (2-1)]
.
Z=
C
= Z(s) ± Z(t)
X= X
and
(444)
Here, as previously in Eq. (2-2), the coordinates have been made dimensionless with
the profile chord c. Again, the upper sign applies to the upper surface of the profile, the lower sign to the lower surface. For the pressure difference between the lower and upper surfaces of the profile (load distribution), Eq. (4.43b) yields with Eq. (4-44): ZJcP(`Y)
= Pt-Pu = 40o
4
a00-1 \
___ dX)
a
(4.45)
The aerodynamic coefficients are easily obtained from the pressure distribution through integration. The lift coefficient is, from Eq. (2-54a), i
cL=
4
dc,(X)dX =
x
(4-46)
JJ
0
It is a remarkable result that the lift coefficient depends only on the angle of attack a and not at all on the profile shape; that is, the zero-lift direction coincides with the profile chord (x axis). The moment coefficient, referred to the profile leading edge (nose up = positive), becomes, from Eq. (2-55a)*: 1
c=-
J c,(X) X dX = o
4
a
M1 2
PO dX
-{-
(4-47)
.1
The lift-related aerodynamic coefficients are compiled in Table 4-2. They include the lift slope dcLlda and the neutral-point position xNlc = -dcMldcL, of which the dependence on the incident Mach number Ma,c > 1 is demonstrated in Fig. 4-20a
and b. For comparison, the dependencies for the skeleton profile in subsonic incident flow, Mam < 1, are also shown (see Table 4-1). These results are identical to those of the inclined flat plate. For Ma. - 1, both linear theories presented here fail, because the assumptions made are no longer valid. This is true particularly for ' The integral of the second equation is obtained through integration by parts.
WINGS IN COMPRESSIBLE FLOW 237
the lift slope, as can be seen from Fig. 4-33. The location of the neutral point is at xN/c = a for subsonic flow and at xN/c = a for supersonic flow. This marked shift toward the rear when the flow changes from subsonic to supersonic velocities should be emphasized.
In addition to lift, drag is produced in supersonic frictionless flow. It is called wave drag. The two forces are expressed by c
C
L=b
D=b
(JPr - JPu)dx
(J PA +JpuJu)dx 0
0
where JP/(X) = pl(x) -p. and d pu(x) = pu(x) -p. are the pressures on the lower and upper surfaces of the profile, respectively, and and zit are the profile inclinations from Eq. (4-42). By using the pressure coefficients from Eq. (4-43b) and
evaluating the integrals under the, assumption that the profiles are closed in front and in the rear, the lift coefficient CL is obtained as in Eq. (4.46), and the drag coefficient CD becomes* 2
May
CD
2 az +
f(--)y
>/
dX
(4-48a)
2d
+J1dX
*Note that, also in subsonic flow, the wing of finite span has a drag that is proportional to the square of the lift (induced drag, see Sec. 3-4-2).
Table 4-2 Aerodynamic coefficients of a profile in supersonic incident flow based on the linear theory (Ackeret) Pressure distribution
c,
_ -Tr-
2 00
Lift slope
Neutral-point position Zero-lift angle
dCL
da
llla - 1 CIO
-
zN C
NO
dZ)
-
4
=
a
i 2
=0
`
1
Zero moment
4
Z131 d X
citito
dcD dcL
i
Wave drag 4 CDo
VMa
- i.
dZ:1, 2
L\dX)
/dZ'c,
2]
'td
) j
238 AERODYNAMICS OF THE WING
Supersonic flow
Subsonic flow Maoo<1
Ma00>1
r
Pr andt/-Gloue rt
2n
Acke ret
Incompressib le 4
2
a 0
02
04
0,6
0.8
70
12
74
16
18
2.0
Mao,
Profile leading edge
b 0
0,2
0,4
0,6
0,8
10
72
74
76
78
20
Mao, ----- 00,6
Figure 4-20 Aerodynamic forces of
the inclined flat plate at subsonic I
Co
02
0,4
0,6
08
10
72
14
16
78
Mao,
2,0
and supersonic flows. (a) Lift slope dcL/da. (b) Position of the resultant of the aerodynamic forces xN. (c) Drag coefficient CD.
Replacing a by CL as in Eq. (446), and Zu, I by Z (s) and Z(') as in Eq. (4-44), results in
cD =
Mad
- 1 c+
i
Ma-1
i
Z 2 dX + r ()y dX (448b) f (iE-)
n
.1
1
0
It should be noted that the total wave drag is composed of three additive
contributions. The first contribution is proportional to CL and independent of the profile geometry. It is plotted in, Fig. 4-20c against the incident flow Mach number.*
The second and third contributions are independent of the lift coefficient and proportional to the square of the relative camber and the relative thickness, respectively. Consequently, it can be seen directly that the flat plate is the so-called best supersonic profile, because the second and third contributions are equal to zero in this case.
The formulas for the drag rise dcDldcL and for the zero drag CD at CL = 0 have been listed once more separately in Table 4-2. A simple explanation of the wave drag will be given for the subsequently discussed case of the inclined flat plate. *See footnote on page 237.
WINGS IN COMPRESSIBLE FLOW 239
Results of linear theory The physical understanding of the last section was applied for the first time by Ackeret [1 ] to a quite simple computation of the flow over a
flat plate in a flow of supersonic velocity U. at a small angle of incidence a. According to Fig. 4-21, the streamline incident on the plate leading edge forms with
the plate a corner of angle a that is concave on the lower side of the plate and convex on the upper side. Consequently, an expansion Mach line originates on the upper side and a compression Mach line on the lower side. At the trailing edge, the compression line is above, the expansion line below the plate. Behind the plate the
velocity is again equal to U. and the pressure equal to p., as it is ahead of the plate. Consequently, there is a constant underpressure pu on the entire upper surface and a constant overpressure pl on the lower surface. The pressure coefficient cp(x) = const follows from Eq. (4-43b) with a* 0 and z(x) = 0. The characteristic difference in the pressure distributions for supersonic and subsonic incident flow is explained in Fig. 4-22. From Fig. 4-22a, at subsonic velocity the pressure distribution produces a force-resultant N normal to the plate, and in addition, the flow around the sharp leading edge produces a suction force S directed upstream along the plate (see Sec. 3-4-3). The resultant of the normal force N and the suction force S is the lift L, which acts normal to the incident flow direction U,,. The resultant aerodynamic force has no component parallel to the incident flow direction; in other words, the drag in the frictionless subsonic flow is equal to zero. For the case of supersonic flow, Fig. 4-22b, the force N resulting from the pressure distribution also acts normal to the plate. However, because there is no flow around the leading edge, no suction force parallel to the plate exists here. The normal force N in inviscid flow therefore represents the total force. Separation into components normal and parallel to the incident flow direction establishes the lift L = N cos a N and the wave drag D = N sin a La. There is another physical explanation for the existence of drag at supersonic incident flow, namely, that for the production of the pressure waves (Mach lines) originating at the body during its motion, energy is expended continuously. As a further example of the pressure distribution on profiles in supersonic flow, a biconvex parabolic profile and an infinitely thin cambered parabolic profile, given by the equations
Z(t)=2 CtX(l -X)
(4-49a)
Expansion 4pu Y
PJ, P/
A
Compression Gyp/
Expansion *'
Figure 4-21 Inclined plate in supersonic incident flow.
240 AERODYNAMICS OF THE WING
Figure 4-22 Pressure distribution and forces on an inclined flat plate in compressible flow. (a) Subsonic incident flow (Ma. < 1). (b) Supersonic incident flow (Mao, > 1).
Z(S)=4kX(l-X)
(4-49b)
are compared in Fig. 4-23. Both profiles are in chord-parallel incident flow, a= 0°. Consequently, from Eq. (4-46), CL = 0 for either profile. The pressure distributions,
as computed from Eq. (4-43), are given in Fig. 4-23. The zero moment of the teardrop profile is equal to zero, whereas that of the skeleton profile is turning the leading edge down (nose-loaded). The lift-independent share of the wave drag is obtained from Eq. (448b) as (4-50a)
CDO =
(4-50b)
These expressions show that the zero-drag coefficients are proportional to the squares of the thickness ratio t/c and the camber h/c, respectively. In Fig. 4-24, the
a
b
Figure 4-23 Pressure distribution at
I
CM) eo
supersonic incident flow for parabolic profiles at chord-parallel incident flow. (a) Biconvex teardrop profile. (b) Skeleton profile.
WINGS IN COMPRESSIBLE FLOW 241
.Expansion line
I
C
d
e
f
Figure 4-24 Pressure distribution on profiles at supersonic incident flow. 1, lower surface; ii, upper surface. (a) Inclined flat plate. (b) Parabolic skeleton at angle of attack at = 0°. (c) Biconvex profile at a = 0°. (d) Circular-arc profile, a = 0° . (e) Biconvex profile, a 0°. (f) Circular-arc profile, a T 0°.
pressure distributions of an inclined flat plate (Fig. 4-24a), a parabolic skeleton (Fig. 4-24b), a symmetric biconvex profile, and a circular-arc profile at angle of attack a = 00 (Fig. 4-24c and d), as well as at a T 0° (Fig. 4-24e and f), are compared. Further, a few data should be given about the dependence of wave drag on the
relative thickness position for double-wedge profiles and parabolic profiles. The
242 AERODYNAMICS OF THE WING
geometry of parabolic profiles was given by Eq. (2-6). In Table 4-3 the results are compiled, and in Fig. 4-25 the contribution to the wave drag that is independent of CL is plotted against the relative thickness position. For a relative thickness position xt = 0.5, the wave drag of the double-wedge profile is
cDo=
t)
4
V Maw
-1
(4-51)
c
Thus, the drag of this double-wedge profile is lower by a factor a than that of the parabolic profile (Xt = 0.5). The double-wedge profile (Xt = 0.5) is the profile of lowest wave drag for a given thickness. Data on additional profile shapes are found in Wegener and Kowalke [21]. Information on the remaining aerodynamic coefficients, namely, zero-lift angle and zero moment, is compiled in Fig. 4-26 for skeleton profiles of all possible relative camber positions. The geometric data of the skeleton line were given in Eq. (2-6). For comparison, the coefficients for subsonic velocities are also shown. The zero-lift angle and the zero moment are plotted against the relative camber position in Fig. 4-26a and b, respectively. In either case the basically different trends at subsonic and supersonic velocities are obvious.
Higher-order approximations (Busemann) The above-stated linear profile theory for supersonic flow, characterized by a local pressure difference (p -p'.) proportional
to the local profile inclination 0 was later extended by Busernann [10] to a
higher-order theory by adding terms of d2 and X93. The pressure coefficient of the extended theory changes Eq. (4-43a) into cp(x) =
Ma;, -1
[1
+K$(x)]
(4-52a)
Table 4-3 Wave drag at supersonic incident flow for double-wedge profiles and parabolic profiles (see Fig. 4-25) Designation
Double-wedge profile
Parabolic profile
Side view
r)
.1
2 Xt
for (I)
Contour C 8
(1-` )for(I1) i - Xt.
- 2 Xr r)
X(1-X) (1 - 2 Xt) X
Wave drag
Vja
-1
ODo 62
1
1
xt(1- X-)
:3X2 (1 - Xt)Z
WINGS IN COMPRESSIBLE FLOW 243 25
2
20
1 N 0
U V
15
r6
3
Figure 4-25 Wave drag at supersonic flow vs. relative thickness position for double-wedge 0
08
0.6
0.4
0.2
Xt.
1.
profile (1) and parabolic profile (2), from [211 (see Table 4-3).
-
with
h' -
1
(31.a. 20
4
- 2)2 + yMal
(4-52b)
(1VIa 0 - 1)3/2
The aerodynamic coefficients can be determined from Eq. (4-52), but no details will be given here. For the lift-independent contribution, an additional term is obtained that is proportional to (t/c)3 for symmetric profiles. Theoretical drag values, computed using this theory of second-order approximation, are compared in Fig. 4-27 with measurements by Busemann and Walchner [10] . Good agreement is obtained.
6
5
,fa,j 1
I
0,
0.2
0.4
0.6
as
10
00,
Xh a
0.2
0..4
Xh
0.6
00
1.0
b
Figure 4-26 Aerodynamic coefficients of cambered skeleton profile at subsonic and supersonic flows. (a) Zero-lift angle a, . (b) Zero-moment coefficient cm,.
244 AERODYNAMICS OF THE WING
Test
i' 04
/
------- Theory X11
8°
i 0.2
1
o t/c=0,0885
I
-`ot/C=0
.t/c=0 1670 .
c
6°
4°
1
8°
1250
,
4 I
0
v
Z
;0 °
d =0
-220
-02
\
-4°
4°
00 7-
-0.4
46\
-8° N.
-8°
-6°
-20
-06
-08
0,05
01
Q75
02
cD --
Q25
03
035
04
Figure 4-27 Drag polars cL(cD) of circular-arc profiles of several thickness ratios t/c at Mach number Ma°o = 1.47, from measurements of Busemann and Walchner; comparison with second-order approximation theory of Busemann.
With greater accuracy than by the above-illustrated theory of second-order approximation, the supersonic flow about thin profiles can be determined by the method of characteristics. Compare, for instance, the publications of Lighthill [51, 52].
4-3-4 Profile Theory of Transonic Flow Both approximation theories for subsonic and supersonic flows discussed in Secs. 4-3-2 and 4-3-3 fail when the incident flow velocity approaches the speed of sound.
In this case the flow becomes of the mixed type; that is, both subsonic and supersonic velocities exist in the flow field. At certain points the flow therefore passes the speed of sound. In transonic flow fields of this kind, shock waves are formed in most cases, and theoretical treatment is made much more difficult. Drag-critical Mach number First, the limiting Mach number should be established up
to which the theory of subsonic flow of Sec. 4-3-2 is still valid. In the case of a wing profile at subsonic incident flow velocity (Ma.. < 1), Fig. 4-13a demonstrated that the lift slope can no longer be described by the linear theory at higher subsonic Mach numbers. The results on the neutral-point position of Fig. 4-13b, and in particular those on the drag coefficients of Figs. 4-14 and 4-15, confirm this fact, which is caused by flow separation on the profile. Depending on the profile shape (thickness ratio, camber ratio, nose ratio) and the angle of attack, a critical Mach number Maw, cr can be established up to which no significant flow separation occurs.
This will be designated as the drag-critical Mach number. It can be defined, for
WINGS IN COMPRESSIBLE FLOW 245
instance, as the Mach number at which the drag coefficient CD is higher by d CD = 0.02 than at May, = 0.6.
The physical reason for flow separation at higher subsonic Mach numbers is that shock waves are formed when sonic velocity is reached locally on the profile and exceeded over a certain range. The critical Mach number Ma.,, is understood, therefore, to be the Mach number of the incident flow at which sonic velocity is reached locally on the profile. The critical pressure coefficient at the critical Mach number Ma., is cpcr. The critical Mach number Ma,ocr is obtained by setting for cpcr the highest underpressure cpmin that occurs at the body. For slender bodies, Cpmin is small and Mao,cr is close to unity. In this case, based on streamline theory of compressible flow, neglecting higher-order terms, cpcr becomes
1 -Ma o, cr
2
c p Cr
=-7+ .
1
Ma;o cr
( 4 - 53 a)
(4-53b)
= Cpmin
From Eq. (4-38), Cpmin is a function of Mach number. Introducing Eq. (4-38) into Eqs. (4-53a) and (4-53b) yields (1 -Ma200cr)3/2 Mat +1 2
(Cpmin )inc
(4-54)
. Cr
In Fig. 4-28, cpcr from Eq. (4-53a) is shown versus Ma, as curve 1. For a given wing profile, Mao, cr is determined by the intersection of curve i according to Eq. (4-53b) with curve 3 according to Eq. (4-38); see also Fig. 4-16. More simply, Ma, can be obtained by starting from Eq. (4-54). This relationship is given as curve 2.
,
The value of cpmin depends strongly on the profile shape and the angle of attack. It is obtained from the velocity distribution of potential flow with Cpmin = -2umax/Uoo. The maximum pressures for various profiles in incompressible flow are plotted in Fig. 2-34 against the thickness ratio. The critical Mach numbers
for chord-parallel flow are shown in Fig. 4-29 for several profiles as functions of 0.
0. '
I
2-A
03 .12
1a
1
02
Cpmin
mm inc
0.1
Fire 4-28 Illustration of determination of drag-critical Mach number Ma-cr of a wing
'
1
O5
0.8
I
0.7
Maoocr
08
-0.
1.0
profile; y = 1.4. Curve 1 from Eq. (4-53), curve 2 from Eq. (4-54), curve 3 from Eq. (4-38).
246 AERODYNAMICS OF THE WING 1,0
0.8
Joukowsky profile
0.2
0
405
0.15
0,10
0,20
0.25
Figure 4-29 Drag-critical Mach number Mao,cr
t
of several profiles at chord-parallel incident
c
flow; see Fig. 2-34.
profile thickness S = t/c and relative thickness position Xt = xtlc. As would be expected, the critical Mach number decreases sharply with increasing thickness ratio for all profiles.
Physical behavior of transonic profile flow When a wing profile is exposed to an incident flow velocity high enough to form areas of local supersonic velocity in its vicinity, shock waves are formed in the ranges where the velocity is reverted from supersonic to subsonic. In these shock waves, pressure, density, and temperature change very strongly. The strong pressure rise in the shock wave frequently leads to flow separation and consequently to a complete change of the flow pattern. This effect causes a strong increase in the drag (pressure drag).
To demonstrate these processes, the pressure distribution on a wing profile is given in Fig. 4-30a for various Mach numbers from measurements in reference [89]. The pressure distribution is steady for Mach numbers at which the maximum velocity on the profile contour is everywhere smaller than the local sound speed,
we 0.7, at which the sonic velocity is exceeded locally, we > a, the pressure rise behind the pressure minimum occurs unsteadily in a shock wave. The height of the pressure jump increases with Mach number. This abrupt pressure rise is very undesirable with respect to the boundary layer, which
tends to separate even at a steady pressure rise. In most cases, the shock wave causes separation of the flow from the wall and thus a strong drag rise, as is obvious from the curve of the drag coefficient versus Mach number of Fig. 4-30b; see also Figs. 4-14 and 4-15. In Fig. 4-31, a Schlieren picture and an interferometer photograph from Holder
[33] are shown of a wing of angle of attack a = 8° in a flow field of Ma.. = 0.9. The formation of the shock wave and a strong separation immediately behind the shock are clearly noticeable.
WINGS IN COMPRESSIBLE FLOW 247
The flow pattern in the transonic velocity range, which is, in general, quite complicated, is displayed schematically in Fig. 4-32 for a biconvex profile in symmetric incident flow. Pressure distributions and streamline patterns are given over a range of increasing Mach number. Figure 4-32a represents the incompressible
case, Fig. 4-32b the subsonic case in which the "sonic limit" has not yet been exceeded anywhere. Figure 4-32c-e demonstrates the formation of the shock wave
after the "sonic limit" of the pressure distribution (critical pressure) has been passed. Figure 4-32f and g represents the typical pressure distribution of supersonic flow that was previously shown in Fig. 4-24. The formation of shock waves in the transonic range also has a strong effect on the lift. This is demonstrated schematically in Fig. 4-33, in which the solid curve represents a typical measurement of the relation between lift coefficient and Mach number, whereas the dashed line corresponds to the linear theory according to Fig.
4-20a. For a better understanding of the measured lift curve, the positions of the shock wave and the velocity distributions on the profile for the points A, B, C, D, and E are shown in Fig. 4-34. At Mach number Ma = 0.75 (point A), a shock wave does not yet form because the velocity of sound has not been exceeded
Q 78
pcr=01527pp I
a 02
04
Qs
x/c---
08
10
006
004
Figure 4-30 Measurements on a wing profile at subsonic incident flow from (891, angle of attack a = 0°. (a) Pressure distribution at
00
A11a°°cr0,7
b 02
04
Mac r
0.6
08
;0
various Mach numbers. (b) Drag coefficient vs. Mach number.
248 AERODYNAMICS OF THE WING
a1
Figure 4-31 Flow about a wing profile at Mach number Ma,. = 0.9. Angle of attack a = 8°, from Holder. (a) Schlieren picture. (b) Interferometer photograph.
WINGS IN COMPRESSIBLE FLOW 249
Sonic limit
b
Shock wave
Local supersonic flow F..
////// //M
U-L"11 ii77.1
e
f c C
g
C)
Q U,
Figure 4-32 Pressure distribution and flow patterns of a biconvex profile in the transonic range (schematic).
G
05
10
Ma.
75
ZO
Figure 4-33 Lift coefficient of wing vs. Mach number. Solid curve: typical trend of measurements. Dashed curve: theory according to Fig. 4-20a.
t
250 AERODYNAMICS OF THE WING
a i=
b
Velocity distribution on profile
Position of shock wave
Z2
u
10
A
Wake flow
U
,75
w°°
08 ,
31x0.6
/
I
O4
-j
I
0
0,5
70
x/c--«
1,6
14
U
12
781
Shock wave
--..
B
f
3d /
10 -',-- -
'
08 0.6
04
0
05
10
x/c --
1,6
14
u
12
7,89
-- -
_
3
C
08, 06 04
0
0.5
10
0.5
70
X/C-.-
1.6
u
14 1.2
298
-,.
u
10 0.8
D
r
06
04 0
-
X/C --
2. 0
U 1.6
1. 4
810 -- 1.2
14
u
08
E 06 04 0
05
x/c---
70
Figure 4-34 Transonic flow over a wing profile at various Mach numbers; angle of attack a = 2°, from Holder. The points A, B. C, D, and E correspond to the lift coefficients of Fig. 4-33. (a) Position of shock wave. (b) Velocity distribution on profile.
WINGS IN COMPRESSIBLE FLOW 251
significantly on either side of the profile. Up to this Mach number, the flow is subsonic and the lift follows the linear subsonic theory (Prandtl, Glauert). At Mac, = 0.81 (point B), the velocity of sound has been exceeded significantly on the front portion of the profile upper surface. A shock wave at the 70% chord is the result. The lower surface is still covered everywhere by subsonic flow. Up to point
B, the lift increases with Mach number. At Mach number 0.89 (point C), the velocity of sound is also exceeded over a large portion of the lower surface. A shock wave therefore forms on the lower surface near the trailing edge. This changes
the velocity distribution over the profile considerably, resulting in a marked lift reduction. At Mach number Ma. = 0.98 (point D), the two shock waves on the upper and lower surfaces are considerably weaker than at Ma. = 0.89 and are located at the trailing edge. The lift, therefore, is again larger than at point C. Finally, at Ma. = 1.4 (point E), pure supersonic flow has been established with a velocity distribution typical for supersonic flow. The magnitude of the lift now corresponds to the linear supersonic theory (Ackeret). All tests indicate that the processes in the shock wave are markedly affected by
the friction layer. This interaction between shock wave and boundary layer is, besides other effects, particularly complicated because the behavior of the boundary
layer changes with Reynolds number, but on the other hand, the shock wave depends strongly on the Mach number. Above a certain shock strength, the pressure rise in the shock causes boundary-layer separation which, in addition to the drag rise
already discussed, leads to strong vibrations as a result of the nonsteady
character of this flow. This phenomenon is also called "buffeting" in aeronautics; see, for example, Wood [109]. Both the Mach numbers of sudden drag rise and of buffeting are influenced by the profile shape and the angle of attack a (see Fig. 4-35). The so-called buffeting limit restricts the Mach number range for safe airplane operation. By increasing the incident flow Mach number to supersonic velocities, the shock moves to the wing trailing edge and the buffeting effects disappear again. For very thin and slightly inclined profiles, this state can be reached without the shock's gaining sufficient strength to excite buffeting while it is moving over the profile.
The individual phases of the flow in Fig. 4-35a are explained by the pressure distributions of Fig. 4-35b. Because of the complicated flow processes above the critical Mach number, a
strictly theoretical determination of the buffeting limit is not possible. However, Thomas and Redeker [109] developed a semiempirical method for the determination of the buffeting limit; see Sinnott [84]. A comprehensive experimental investigation of this problem, which is most important for aeronautics, has been reported in detail by Pearcey [69] and Holder [33]. Similarity rule
for transonic profile flow So far, analytical determinations of
transonic flows with shock waves have succeeded only in a few cases. In some cases, however, a steady transition through the sonic velocity (without shock waves) has
also been observed. In this latter case, transonic flows can be treated theoretically by means of an approximation method. They lead to similarity rules for pressure distribution and drag coefficient (Sec. 4-2-3) that are in quite good agreement with
252 AERODYNAMICS OF THE WING
A... C Attached flow
Flow separated at the shock Shock at the trailing edge
D
£ x/c
Figure 4-35 Behavior of a wing in the transonic velocity range (schematic), from Thomas. (a) Buffeting limit vs. Mach number. (b) Pressure distributions at several Mach numbers.
measurements. It can be shown that the transonic similarity rule remains valid even when the flow includes weak shock waves. Between pressure distribution and drag coefficient of wing profiles of various
thickness ratios t/c and at various transonic Mach numbers of the incident flow (Mi -- 1), the following expressions are valid according to reference [103], and extend Eqs. (4-35) and (4-36): cp
,
x t ,
C
where
Mam
(7+ 1) 1/3
erp
(t/c)5/3
t
CD
x
(t/c)2 /3
,
Mao,
ynoo
r + 1) 1/3 (7
Mat1 - [(7±1)C
moc C
v
(4-55) (4-56) (4-57)
Here, cp is called the reduced pressure coefficient, and ED is the reduced drag coefficient. For the special case Maw = 1 (sonic incident flow), mc, = 0 from Eq. (4-57). From this it follows immediately that the pressure coefficient cp is proportional to (t1c)213 in this case and the drag coefficient proportional to (t/c)s/3 [see Eqs. (4-35) and (4-36), respectively].
WINGS IN COMPRESSIBLE FLOW 253
Malavard [103] checked the similarity rules, Eqs. (4-55)-(4-57), in comprehensive experiments. He clearly verified the transonic similarity rule for pressure distribution and drag coefficient of symmetric biconvex profiles of thickness ratios t/c = 0.06-0.12 at chord-parallel flow of incident Mach numbers of Ma. _ 0.775-1.00. Plotting of the drag coefficient CD against the Mach number in Fig. 4-36a shows the well-known strong drag rise near Ma. = 1 and, moreover, the strong increase of this rise with the thickness ratio t/c.
Theories for the computation of transonic profile flows The transonic profile flow with shock waves can be treated only by nonlinear theory, in contrast to the linear theories of subsonic and supersonic profile flows. There exist numerous trials and
methods for the solution of this task. A survey of the more recent status of understanding of theory and experiment for transonic flow is given by Zierep [111]. So far, the hodograph method, the integral equation method, the parabolic method, and the method of characteristics have been applied to computations. Guderley uses mainly the hodograph method, Oswatitsch generally prefers the integral equation method. The many publications quoted in [63, 66, 79, 84-87, 111 ] show that no generally valid solution has been found for the computation of the pressure distribution of wings on which shock waves form at transonic incident flow. More recent progress has been discussed at the two Symposia Transsonica [67]. Supercritical profiles For wing profiles operating at high subsonic flight velocities, the. drag-critical Mach number Mao, according to Figs. 4-14a and 4-29 can be shifted to higher values by reducing the profile thickness ratio or by lowering
,
010
5
0,08
4
0,10
10,01
008
00
0. 06
0,02
0
a
I
i
0
08
09
70
Mao,--
1.1
72
-12 -10 -08 -06 -04 -02 0
02 04 06 Qd
10
moo
Figure 4-36 Drag measurements on symmetric profiles in the transonic velocity range at chord-parallel incident flow, from Malavard. (a) Drag coefficient CD vs. Mach number Ma.. for
symmetric profiles of various thickness ratios t/c. (b) Reduced drag coefficient cD from Eq. (4-56) vs. reduced Mach number h,,. from Eq. (4-57) for symmetric profiles of various thickness ratios t/c.
254 AERODYNAMICS OF THE WING
the profile lift coefficient.* Profiles at which the critical pressure coefficient cp
Cr
from
Eq. (4-53a) has not yet been exceeded or has just been reached on the suction side (profile upper side) are termed subcritical profiles. On them no shock waves form, and therefore no shock-induced flow separation occurs. Through suitable profile design, local areas of supersonic flow can be created on the profile in which recompression to
subsonic flow occurs steadily or in weak shock waves only. On these profiles the pressure rise in the recompression zone is gradual and therefore does not cause flow separation. Transonic profiles designed according to the stated criterion are termed supercritical profiles.
A few more statements should be made about the evolution from subcritical to supercritical wing profiles. In many designs the product of lift-to-drag ratio and Mach number must be optimized. This request may roughly be transferred to the aim to achieve for a given profile thickness ratio at the design Mach number the highest possible lift at fully attached flow conditions. By starting with the pressure distribution la in Fig. 4-37 found on the suction side of the conventional NACA 64A010 profile a gain in lift first may be obtained by further upstream and downstream extension of the minimum suction pressure just along its critical value *The feasibility of increasing the drag-critical Mach number by sweeping back the wing will be discussed in Sec. 4-4-4.
Figure 4-37 Pressure distributions of various wing profiles. (a) Suction side (upper surface). (b) Pressure side (lower surface). (1) Conventional profile NACA 64A010 at Mao, = 0.76, a = 1.20, measurements of Stivers [651. (2) Roof-top profile. (3) Supercritical profile of thickness ratio t/c = 0.118 with "rear loading," from Kacprzynski [65). Theory: Ma".=0,75, cL = 0.63. Measurements: Ma = 0.77, cL = 0.58.
WINGS IN COMPRESSIBLE FLOW 255
Figure 4-38 Comparison of the contour of a supercritical profile with a conventional profile (NACA 641 A212), thickness ratio t/c = 0.12.
according to curve 2a. Such profiles are called "roof-top profiles." In the range of the profile nose, a strong acceleration of the flow is required, which is accomplished by
increasing the nose radius. The onset of the recompression needed to match the pressure at the profile trailing edge (pressure at the rear stagnation point in inviscid flow) must be chosen to allow establishment of a pressure gradient over the rear portion of the profile that does not cause flow separation. Chordwise linear recompression according to curve 2a has been found to be good in practical applications. A further marked increase in lift is obtained by admitting a local supersonic flow field on the profile suction side, which means choosing pressure distributions exceeding the critical pressure coefficient. That kind of flow implies a further increase in nose radius, and, in addition, a flattening of the upper surface. In this case, an essentially shock-free or weak shock pressure distribution along the profile chord, allowing recompression without separation, curve 3a, is of decisive importance. The pressure distribution over the rear portion of the pressure side of conventional profiles is little different from that on the suction side (curves 1 a and l b). Thus, the rear portion of such profiles contributes little to the lift. A larger difference in the pressure distribution of upper and lower side, curves la and 3b, is obtained through changing the profile lower contour between the range of maximum thickness and the trailing edge such that a reduced local thickness is obtained. This change means, according to Fig. 4-38, the establishment of a corresponding profile camber. Measures of that kind are known as "rear loading." At such profile designs, caution is necessary to avoid flow separation in the recompression region, precisely as it was required on the suction side. A comparison of the geometries of a subcritical and supercritical profile with
"rear loading" and thickness ratios tlc = 0.12 is shown in Fig. 4-38. Systematic investigations on profiles with shock-free recompression from subsonic to supersonic
flow have been made by Pearcy [69]. The first design intended to produce shock-free supercritical profiles, so-called quasi-elliptic profiles, was conducted by Niewland [65] and confirmed in the wind tunnel (Fig. 4-39). Since then, a number of generally applicable design methods for supercritical profiles have been
developed, and profile families have been checked out successfully in the wind tunnel [4, 54, 55].
4-3-5 Airfoil of Infinite Span in Hypersonic Flow By taking into account the similarity rules of Sec. 4-2-3, specific profile theories have been developed for flow about wing profiles (slender bodies) that depend on
256 AERODYNAMICS OF THE WING
Figure 4-39 Pressure distribution of a quasi-elliptic symmetric shock-free supercritical profile in chord-parallel flow, from Niewland, Ma. = 0.786. Measurements: o NPL, 4 NLR.
the values of the incident flow Mach number. For May, < 1 the subsonic flow is described in Sec. 4-3-2, for Maw > 1 the supersonic flow in Sec. 4-3-3, and for Ma., = i the transonic flow in Sec. 4-3-4. For very high Mach numbers of incident flow, that is, Ma. > 1, the theory of supersonic flow does not lead to satisfactory results. For this case of incident flow with hypersonic velocity (Ma., > 4), a few statements on a profile theory of hypersonic flow will be made. First, the following considerations will be based on a slender profile, pointed in front. Theory of small deflections in hypersonic flow Through a concave deflection by the angle > 0, a compression flow is produced that can be computed according to the
theory of the oblique shock. Conversely, an expansion flow is formed behind a convex deflection by the angle < 0 that can be treated as a Prandtl-Meyer corner flow. The fluid mechanical quantities before and behind the deflection will be marked by the indices 1 and 2, respectively. The deflection angle is assumed to be small 161 << 1, which means that the velocities before and behind the deflection differ only by a small perturbation velocity. The range of Mach numbers of the hypersonic flow considered here is Mal > 1 and Mat > 1. The pressure coefficients cp =J p/q i of the pressure change A p = P2 - PI , relative to the dynamic pressure before the deflection q1 = (ol /2)Ul, are obtained as [53] 992 > 0
y(Mal
0)2
1-
] + y-1Ma1 2
lg y
($ > 0)
+92 < 0
(4-58a)
(t < 0)
(4-58b)
WINGS IN COMPRESSIBLE FLOW 257
In either case, the pressure coefficient at small deflections of a hypersonic flow is given as
cp = 62f(Ma1 6)
(4-59)
where Mal t5 is the similarity parameter of hypersonic flow. The parameter will be discussed later in more detail in connection with the hypersonic similarity rule. For large values of Ma l 19 > 1, the expressions
cP = (y + 1)62
(4-60a)
(Ma1 $ -> 00)
2752
7(Ma1 6)2
-Ma1
19 >
2
7-1
(4-60b)
are valid. The latter formula indicates that after deflection, vacuum (p2 = 0) is obtained for values of -Ma1 3 > 2/(y - 1). In Fig. 4-40, the pressure coefficient in relation to the square of the deflection angle cP/02 is plotted as a function of the hypersonic similarity parameter Mal 6 by curves 1 and 2. For comparison, the supersonic approximation of Eq. (443a) for high Mach numbers is 9
a-i 7
(4-61a)
VMi 2
(supersonic approximation) (4-61b)
t52
shown as curve 3. This approximation agrees better with the expansion flow than with the compression flow. The deviations are too large, however, to adopt this approximation as the pressure equation for hypersonic flow with small deflections.
Inclined flat plate in hypersonic flow By setting 6 = ±a in Eqs. (4-58a) and (4-58b), a being the angle of attack, the pressure distributions on the lower and upper surfaces of an inclined flat plate in hypersonic flow can be easily computed. They are constant over the chord. The lift is then obtained from the resultant pressure distribution of the lower and upper surfaces. The lift coefficient is obtained as CL =
cp
a2F(Ma a)
(4-62a)
S
2
3
i
Figure 4-40 Pressure coefficients at hypersonic u, i
2
3
flow (y = 1.4). (1) Expansion: lower sign, from
---- -- S
Eq. (4-58b). (2) Compression: upper sign, from Eq. 5
Ma, 15
(4-58a). (3) Supersonic approximation from Eq. (4-61).
258 AERODYNAMICS OF THE WING
CL = (y + 1)a2
(Ma -+ 00)
(4-62b)
In Fig. 4-41, this result is presented for various Mach numbers of the incident
flow Mat =Ma according to Linnel [53]. It can be seen that the lift coefficient for a fixed angle of attack decreases sharply with increasing Mach number and that the hypersonic theory deviates from the supersonic theory. The curves for Ma = 0 (incompressible flow) and Ma = -- mark the limiting cases. Hypersonic similarity rule Specific similarity rules were established in Sec. 4-2-3 for subsonic, transonic, and supersonic flows. With their help, flows about geometrically similar bodies can be related to each other. Such a similarity rule also exists for hypersonic flow. It was first presented by Tsien [98] and proved to be completely general by Hayes [98]. The relation between pressure coefficient and deflection angle and Mach number is expressed in Eq. (4-59). For symmetric incident flow, the
deflection angle is proportional to the thickness ratio t/c. In this case the Mach number Mal becomes the incident flow Mach number Ma,,. Hence, in analogy to Eqs. (4-35) and (4-36), the following expressions are obtained for the pressure and drag coefficients:
cp = 82f 5 Ma.,
)
(4-63) (4-64)
Hypersonic flow over a blunt profile The flow pattern in the vicinity of the nose of a body in hypersonic incident flow is sketched in Fig. 4-42. Keeping in mind the
Figure 4-41 Lift coefficient of the flat plate vs.
angle of attack « for various Mach numbers (y = 1.4). Hypersonic theory for small angles of I
00
2°
4°
60
8°
a
10°
attack according to Linnell. (-) Hypersonic theory, Eq. (4-62a), Ma -: cL = (y + 1)a2. (-- -) Theory based on Eq. (4-46), Ma -} 0: cL = 2ira.
WINGS IN COMPRESSIBLE FLOW 259
Figure 4-42 Sketch of a hypersonic flow. Zone A: boundary layer with friction and rotation. Zone B: inviscid layer, but with rotation.
important fact that the leading edge of every body is somewhat-even if very little-rounded, it is obvious that a stagnation point always exists on the nose, and therefore a detached shock wave is formed upstream of the stagnation point in which the approaching hypersonic flow is abruptly reduced to subsonic flow. As a result, extremely high temperatures are produced near the stagnation point, which may lead to dissociation and ionization of the gas and thus to deviations from the properties of ideal gases. The thermic equation of state [Eq. (1-1)] is no longer valid, for instance, and the specific heat capacity cp does not stay constant either. The dependence of the temperature rise that occurs near the stagnation point after passage of the shock wave on the Mach. number is presented in Fig. 4-43 for air. The dashed line is valid for the ideal gas (see Fig. 4-2b) and the solid curves for a 24000
/
20000 00
160000
12000°
P. =
12i
10 'Atm
e00o°
10 -2 10_4
4000° i
I
1
I
010
4
B
i
I
12
16
20
24
Figure 4-43 Temperature rise behind normal shock vs. Mach number (temperature before the shock: T. = 222 K). Curve 1: real gas for several values of the static pressure per,. Curve 2: ideal gas (y = 1.4).
260 AERODYNAMICS OF THE WING
real gas at several values of the static pressure p. of the incident flow. Because of dissociation, the temperature rise at high Mach numbers is considerably smaller for real gases than for ideal gases. At larger distances from the stagnation point the shock wave closely approaches
the body contour. It is strongly curved, therefore, particularly near the stagnation point (Fig. 4-42). On the body contour itself, a friction (boundary) layer (range A) forms because of the viscosity, the thickness of which is now of the same order of magnitude, however, as the distance between shock wave and the outer edge of the boundary layer (range B). The formation of the boundary layer is governed by the pressure distribution on the body, which, at hypersonic incident flow, is determined mainly by the shape of the shock wave. This, in turn, depends on the body contour and its boundary layer. There prevails, consequently, a very strong interaction between friction layer and shock wave in hypersonic flow. Another difficulty contributes to the problem. Since the shock wave is curved, the entropy increases in the shock wave are different for each streamline. These increases depend on the shock-wave inclination at the respective stations. Therefore, the flow behind the curved shock is no longer isentropic. This means that the flow behind the shock is no longer irrotational and that the separation into a rotational friction layer and an irrotational outer flow, customary in boundary-layer theory, is no longer possible. On the contrary, the total flow field between shock wave and body contour is now rotational. The friction effects, however, are of significance only in the zone next to the wall, zone A of Fig. 4-42, whereas zone B represents an inviscid, but not irrotational, layer. An important characteristic of hypersonic flow is its small lateral extent. Therefore the flow quantities vary strongly in the lateral direction, whereas they vary only little in direction of the incident flow.* The computations of the flow about a body with a blunt leading edge, and particularly the computation of the shock-wave shape and of the pressure distribution on the body, are very difficult, even when friction is disregarded,
because the flow field contains, side by side, zones of hypersonic, supersonic, and subsonic flow. In the special case (Ma. - co, y -; 1), the incident flow would remain
undisturbed up to the body contour and then be deflected in direction of the contour. Thereby a portion of the horizontal momentum would be transmitted to
the body wall and thus produce the body drag. This special case is termed Newtonian flow because Newton based his theory for the drag of arbitrary bodies on this concept. It leads to the following expression for the pressure coefficient: cP = 2 sine :g
(Newtonian approximation)
(4-65)
with a being the deflection angle.t This relationship serves as a rough approximation for the front portion of the body, whereas the above momentum consideration *The opposite trend is found in transonic flow, in which the changes of the flow quantities are small in the lateral and strong in the longitudinal direction. 1 This formula and its comparison with measurements will be discussed in more detail in Sec. 5-3-3.
WINGS IN COMPRESSIBLE FLOW 261
does not give an answer for the rear body portion. In this context the expression aerodynamic shadow is used.
The methods for the exact computation of hypersonic flows are very lengthy and can be handled only with modern electronic computers. Investigations in this field are still in progress, and many aerodynamic problems-particularly those including the deviations from the properties of ideal gases-are not yet completely solved.
Monographs in book form on hypersonic flow are listed in Section II of the Bibliography. Compare also Schneider [82].
4-4 WING OF FINITE SPAN IN SUBSONIC AND TRANSONIC FLOW
4-4-1 Application of the Subsonic Similarity Rule It has been shown in Sec. 4-2-3 that the computation of flow about a wing of finite span with incident flow Mach number Ma. < 1 can be reduced to the determination of the incompressible flow for a wing of finite span by means of the subsonic similarity rule (Prandtl, Glauert, Gothert). The corresponding problem for the airfoil
of infinite span (profile theory) was discussed in Sec. 4-3-2. Computation of incompressible flows was treated in detail in Chap. 2 for the airfoil of infinite span
and in Chap. 3 for the wing of finite span. The methods of wing theory for incompressible flow therefore have a significance that reaches far beyond the area of incompressible flow.
The second version of the subsonic similarity rule of Sec. 4-2-3 is the starting point for further discussions. In what follows, the reference wing in incompressible flow that is coordinated to the given wing at given Mach number will be designated
by the index "inc." Thus, the transformation formulas for the wing planform according to Eqs. (4-10) and (4-15) are (4-66)
Coordinates:
Xinc = x,yinc
Span:
bins = b
Wing chord:
cinc =C
(4-67b)
Taper:
Ainc = X
(4-68a)
Aspect ratio:
Ainc= A
Sweepback:
cot cpinc
I - Ma
1 -1VIax
1 - Maco
= cot p I
- Mc
(4-67a)
(4-68b) (4-68c)
The geometric transformation for a trapezoidal swept-back wing in straight flight and in yawed flight for Mach number Ma. = 0.8 is presented in Fig. 4-44. For unchanged profile (h/c)inc = h/c, (t/c)inc = t/c, and unchanged angles of attack ainc = a, the pressure coefficient of the given wing cp is obtained according to Eq. (4-23) from that of the transformed wing cpinc as
262
WINGS IN COMPRESSIBLE FLOW 263
CP
_
- y1-Maw Cpinc
(version II)
(4-69)
Compare Figs. 4-8 and 4-9 for the Mach number range 0 <Ma.. < 1. In the case of airfoils of infinite span, the subsonic similarity rule is no longer valid for Maw, = 1 (see Sec. 4-3-2). Approximately, however, it may be applied to May, = 1 in the case of wings of finite span. More details will be given later. Attention should be drawn to the panel method of Kraus and Sacher [44], which includes the influence of compressibility.
4-4-2 Inclined Wing at Subsonic Incident Flow General formulas The local lift coefficient of a wing section is obtained through integration of the pressure distribution over the wing chord according to Eq. (2-10). By taking into account Eq. (4-69), the transformation formulas for the local lift coefficient and, accordingly, for the local pitching-moment coefficient are thus given as C1(y)
Cm
Clinc(Yinc)
=
1 - Ma00
(a -- CYinc)
CM inc(Yinc) () = V 1-Maro
(4-70a)
(4-70b)
For incompressible flow, the wing theory of Sec. 3-3-5 produces the dimensionless lift distribution yinc (?yinc) and the dimensionless pitching-moment distribution from
Eqs. (3-115a) and (3-115b). By introducing Eqs. (4-67a), (4-67b), (4-70a), and (4-70b), the dimensionless distributions for subsonic flow become clc
7=2b=yinc µ
_CmC = 2b
(4-71a) (« _ «inc)
(4-71 b)
1uinc
These equations show that the dimensionless lift and moment distributions remain unchanged during transition from incompressible to compressible flow. It should be noted, however, that the distributions y and yin, and p and pi,,, belong to different planforms (Fig. 4-44). The transformation of the coefficients of total lift and pitching moment, taking into account Eqs. (4-66), (4-67a), (4-67b), and (4-69), results in CL =
CLinc
(4-72a)
M(a
Cal =
-
CMinc
(a = «inc)
(4-72b)
1 1 - M)Ia
coefficient of induced drag in incompressible flow for elliptic lift distribution is, from Eq. (3-31b), CDi inc =CL inc/T-line Introducing CL inc and iliac The
into the above transformation formulas yields the relationship
264 AERODYNAMICS OF THE WING
CDi
_
2 CL
(4-73)
7TA
Hence, the formula for the coefficient of induced drag in relation to the lift coefficient is independent of the Mach number. The transformation formulas for the remaining aerodynamic coefficients are compiled in Table 4-4.
Elliptic wing Simple closed formulas for the lift slope as a function of the Mach number can be established for wings with elliptic planform. For incompressible flows, computations follow Eq. (3-98) of the extended lifting-line theory. Applying the subsonic similarity rule yields dcL da
2rA 1/(1-1YIa2)A-' +4±2 00
(4-74)
Table 4-4 Transformation formulas for the aerodynamic coefficients of an inclined wing of finite span in subsonic flow (Prandtl, Glauert, Gothert),
a - «inc 1 Pressure distribution
cp
Lift
CL
Lift slope
1/1-Ma. 1
- Maw cL inc
dCL
da
1-
Zero-lift angle
co
= aoinc
Pitching moment
CM
Neutral-point position
cpinc
doL
1
Maw da inc
1
- Maw _
xN
r
- cp
c,,
chin c
inc 1
Zero-pitching moment
y 1 - Ma's
CMo inc
1
Rolling moment
CMX
Ma-
cMx inc
i
Induced drag
1
cDi
1-Ma;,
cDi inc
WINGS INCOMPRESSIBLE FLOW 265 3,0
Figure 4-45 Ratio of lift slopes at subsonic and incompressible flow for elliptic wings of various aspect ratios A vs. Mach number of incident flow according to Eq. (4-74).
70
aZ
19
IM
70
48
Mo.
from which the limiting values dcL da
dCL da
2
A
`
(A - 0)
2z j/1
(A - °°)
- Ma.
(4-75a)
(4-75b)
are obtained. Equation (4-75a) is identical to Eq. (3-101b). For very small aspect ratios, the dependence of the lift slope on the Mach number thus disappears.
Equation (4-75b) is identical to the expression of the plane problem from Table 4-1
For the case Maw = 1, the lift slope becomes
dcL_?zl da
2
(May,=1)
(4-75c)
Contrary to the airfoil of infinite span (A = °°), for which (dcL Ida). _ 00, the lift slope of wings of finite span has finite values. The significance of this result will be investigated more closely in Sec. 4-4-4.
The ratio of the lift slopes for Ma,,
0 and Maw = 0 is shown in Fig. 445 for the Mach number. This figure shows that the compressibility influence on the lift slope becomes smaller when the aspect ratio is reduced. This fact was first pointed out by Gothert [28]. several aspect ratios against
Wings without twist The aerodynamic coefficients will be computed for the same wings for which the lift distribution was determined in Sec. 3-3. These were a trapezoidal, a swept-back, and a delta wing, with aspect ratios between A= 2 and A1= 3. These three given wings are depicted in the upper boxes of Fig. 4-46. The geometric data for the wings are compiled in Table 3-4. The second and third rows of boxes show the wings transformed with the subsonic similarity rule for Ma. = 0.4 and 0.8, respectively. The lift distributions of these wings have been
266 AERODYNAMICS OF THE WING
Figure 4-46 Planforms of given and transformed wings for the examples of lift distribution at subsonic incident flow. Given wings: see Table 34. (a) Trapezoidal wing: 0 = 0° , A= 2.75, X = 0.5. (b) Swept back wing: o = 50°, n = 2.75, X = 0.5. (c) Delta wing: p = 52.4°, .i = 2.3 1,
computed according to the wing theory for incompressible flow of Sec. 3-3-5. The results of these computations for the lift distribution of the wing without twist (a = 1) are presented in Fig. 4-47. The lower figures give the dimensionless lift distributions y according to Eq. (4-71a) for Mach numbers Ma. = 0 and Ma, = 0.8. The curves for Ma. = 0 are identical to curve 3 of the distributions in Fig. 3-33. In the upper figures, the local neutral points and the total neutral points N are plotted
on the wing planform. At the upper part of Fig. 4-48, the lift slopes are plotted against the Mach number; at the lower part, the neutral-point displacements with respect to the geometric neutral point. The points for Ma. = 1, shown as open circles, are theoretical values of an approximation method that will be explained in Sec. 4-4-4. They agree with Eq. (4-75c) for trapezoidal and delta wings. In addition, in all six diagrams, measurements by Becker and Wedemeyer [5] are included. The measured lift slopes agree well with theory in all cases. In general, the dependence on Mach number of the neutral-point positions is given satisfactorily by theory.
0)
d
o O
0
0 pO
eo'
LA
c
267
268 AERODYNAMICS OF THE WING
I
a 5
b
C
5 4
4
-'I 't
2 1
0
0.2
0,4
0.6
0.8
1.0
0
62
0.4
0,6
0.6
40
Mao,, 0,
1 0.05
X0.10 ti
015 0.111
0
0.2
0,4
0.6
0,8
Mao.-
025 1.
0.30
0.4
0.6
0.8
Ma.. --
1.0
0
0.2
0.4
0.6
08
1,0
Mcz..-
Figure 4-48 Lift slopes and neutral-point displacements for the three wings of Fig. 4-46 vs. Mach number. ( ) Subsonic similarity rule (wing theory, Sec. 3-3-5); approximation theory for Ma,. = 1; Sec. 4.4-3. (- - -) measurements from Becker and Wedemeyer, profile thickness 6 = 0.05. (a) Trapezoidal wing. (b) Swept-back wing. (c) Delta wing.
Certain discrepancies between theory and experiment of the neutral-point positions can be explained mainly by the effect of the finite .profile thickness disregarded in the theory; compare also Fig. 4-13b. It is noteworthy that the neutral point of the trapezoidal wing shifts considerably upstream under the compressibility influence. However, this theoretical result is only partially confirmed by measurements, because shock waves form when the drag-critical Mach number is exceeded. On the
two other wings, the swept-back and the delta wings, the neutral points are displaced toward the rear. No more detailed statements are needed on the induced drag, since, as shown by Eq. (4-73), the quotient cDi/cL is independent of Mach number and thus equal to that of incompressible flow (see Table 3-4). Further results on the aerodynamic coefficients of delta wings of various aspect
ratios are compiled in Fig. 4-82, together with results for supersonic incident flow.
Data for the compressibility effects on the flight mechanical coefficients at subsonic incident flow, for example, of the rolling, pitching, and yawing wing, are found in Kowalke [5] and Krause [5].
WINGS IN COMPRESSIBLE FLOW 269
4-4-3 Inclined Wing at Transonic Incident Flow It has been shown in Sec. 4-34 that the aerodynamic coefficients of a wing profile undergo strong changes during transition from subsonic to supersonic flow, that is, at transonic flow. The linear approximation methods for incident flows of subsonic and supersonic velocities for the airfoil of infinite span fail when sonic velocity, Ma -- 1, is approached (see Fig. 4-33). For wings of finite span, however, physically plausible results may be obtained at Ma,, = 1. In this case, the same limiting values are obtained for the lift-related coefficients (see, e.g., Fig. 4-82), by approaching Ma. = 1 both from subsonic and from supersonic incident flow. Now, for the lift problem at May, = 1, a few results will be presented that have been obtained according to the method of Truckenbrodt [95] ; compare also the publications of Mangler [58], Mangler and Randall [581, and Spreiter [85]. For tapered swept-back wings, the lift slope and the neutral-point position are shown in Fig. 4-49 as functions of the geometric parameter cr/a for several values of crlao . The wing geometry is seen in Fig. 4-49a, the lift slope in Fig. 4-49b, and the neutral-point position in Fig. 449c. It is noteworthy that for crla > 1 [i.e., when the trailing edge of the inner (root) section lies farther back than the leading edge
of the outer (tip) section], the lift slope is equal to iiA/2 for all wing shapes in agreement with Eq. (4-75c). For cr/a < 1 (i.e., when the trailing edge of the root section lies farther upstream than the leading edge of the tip section), the lift slope is smaller than iiA/2. The neutral point for cr/a > 1 lies at xN/a = 3 (see Fig. 4-49c). For delta wings (do = a = Cr), XN/cr =!.. For cr/a < 1, the neutral point
0.81
I Cr
id0
04!
07
2
06
a4l
a
0
e2-
04
75-77-2-777
b
Figure 4-49 Aerodynamic coefficients of inclined swept-back wings at sonic incident flow Ma. = 1, from Truckenbrodt, (a) Wing geometry. (b) Lift slope. (c) Neutral-point position.
270 AERODYNAMICS OF THE WING
shifts upstream. The linear theory for Maw, = 1 also allows computation of the pressure distribution on the wing surface. Here, for uncambered wings, wing areas of which the local span remains constant in the chord direction (Fig. 4-50a), or decreases (Fig. 4-50b), do not contribute to the lift (d cp = 0). Finally, a few test results [16] are given in Fig. 4-51 for the lift of delta wings
at Mach numbers close to unity. The lift slopes dcL/da are plotted against the parameter A2(Ma', - 1), which results from the similarity transformation of compressible flow [see Eq. (4-26)]. The pronounced peak in the theoretical curve of dcL /da at Ma. = I is not fully confirmed through measurements. In the subsonic and supersonic range, theory is well represented by the measurements. Further experimental results on wings in transonic flow are found in Frick [24].
4-4-4 Wing of Finite Thickness at Subsonic Incident Flow Pressure distribution In this section, the wing of finite span at incident flow of subsonic velocities will be investigated. at zero lift (displacement problem). The pressure distribution of such a wing of finite thickness is of particular interest with
regard to the determination of the drag-critical Mach number at high subsonic incident flow. The concept of critical Mach number has already been explained in Sec. 4-34. The incident flow velocity of the critical Mach number is the lower limit
for the formation of the shock waves, which change the entire flow pattern considerably and, in particular, lead to a strong drag rise (see Figs. 4-14 and 4-15). The pressure distribution Of a three-dimensional wing with symmetric wing profiles at subsonic incident flow is obtained from that of the transformed wing of Eq. (4-69) as Cp inc
e00 U2 2 °°
1-Mao
(6 = 5inc)
(4-76)
where Cp inc is the pressure distribution of the transformed wing for which the
pressure distribution of incompressible flow is to be computed. The computational
Figure 4-50 Pressure distribution on wings without camber at Ma,, = 1. The white areas do not contribute to the lift, _o cp = 0, because for them (a) the span is constant in the chord direction, (b) the span decreases in the chord direction.
WINGS IN COMPRESSIBLE FLOW 271 I
1.
L inear theory
A=3
l ip
Profile NACA 63 X 002 63 A 094
0.4
61AX5 0.2
L
FTI -5
-6
0,6
I
'
l
l
3
7
.7
1.05
1.X1
Figure 4-51 Lift slope of delta wings of various thicknesses; aspect ratio A = 3, from [16]. Comparison with linear theory.
method was given in Sec. 3-6. The transformation of the wing planform follows Eqs. (4-66)-(4-68); the thickness ratio 5 = t/c remains unchanged (version II of the subsonic similarity rule of Sec. 4-3-2).
Drag-critical Mach number On three-dimensional wings, contrary to the plane problem, frequently the wing leading or trailing edges are not perpendicular to the incident flow direction. The simplest cast of that kind is the swept-back wing of constant chord .and infinite span. This case has been treated previously for incompressible flow in Sec. 3-6-3. The sweepback has a significant influence on the
magnitude of the critical Mach number, because only the velocity component normal to the leading edge determines the maximum perturbation velocity on the contour of such wings of finite thickness. From Eq. (4-53a), the critical pressure pcr of the wing in incident flow normal to its leading edge is obtained after multiplication with n U!,,/2 and with Mao cr = U. la. as PCT
2,
x
000 pro
y+1 (
l CL;ro
)
By introducing now, in agreement with the above statement, U= Cr cos .p as the effective velocity instead of U0cr, and again adopting the dimensionless notation, the critical pressure coefficient of the swept-back wing becomes Cp Cr =
Per -Pcc 200
7
r2oo er
2
1 -Ma2mcr cost
y+1
Ilfa2
Cr
(4-77)
272 AERODYNAMICS OF THE WING
Here, as in Eq. (4-76), the pressure coefficient of the swept-back wing is referred to
the dynamic pressure of the incident flow. The relation cpcr(Ma.i.) is shown in Fig. 4-52 for p = 00 (see curve 1 of Fig. 4-28) and for cp = 45°. To determine the critical Mach number of the incident flow Ma.,,, the curve cp min is drawn in Fig. 4-52 up to its intersection with the curve Cpcr (see Fig. 4-28).
Swept-back airfoil of infinite span For the determination of the pressure difference (p - p.) of a swept-back wing, it should be observed that (p is proportional to the dynamic pressure of the effective velocity cost tp. It is also proportional to the thickness ratio or the angle of attack, respectively, determined in the plane of the effective incident flow; that is, it is proportional to (t/c) cos gyp. It follows that in incompressible flow, Pinc -P- = (Pinc -P-)w=o COS cp
.
Referred to the dynamic pressure of the incident flow (p /2)UU, the relation between the pressure coefficients becomes (cpmin)inc = COS Vinc(Cpmin)inc,,p-oWith Eqs. (4-76) and (4-68c) it is s Cpmin
(Cpmin)inc,cp=o
1
with cos Pinc =
1-
cos (p
Q°°
1 '- Mac. COS 2
cp
By substitution, finally, /
cosq7
Cpmin .V1
- Mci' cosaT (Cputin)inc,,p=o
(4-78)
The above-explained procedure has been applied to an example in Fig. 4-52. Chosen were two airfoils of infinite span, one unswept and one with a sweepback angle of 45°. For the unswept airfoil, (Cpmin)inc, V =o = -0.2 has been assumed, resulting in a critical Mach number 0.83. The effect of the sweepback is seen in a shift of the critical Mach number to a considerably larger value of
1.13. This shift is caused by three effects. First, the curve cp cr is shifted to the right because of the sweepback; second, by the sweepback, Cpmin at Ma. = 0 is o.
a
04
I
cpcr,
Y cp;mi n
Figure 4-52 Determination of drag Of I
1
+
Ma,ocr\i Ma00cr 0
0.2
0.4
0.6
40 mam1 0.9
7.2
1.4
1.6
critical Mach number Mao, cr for an unswept and a swept-back airfoil of infinite span. (cp o ,Mac = o
-0.2.
WINGS IN COMPRESSIBLE FLOW 273 2,0
U 1.8
16
1,c
=60°
30 ° 15
0,8
a
0,6
-02
-01
-03
-06
-0.5
-0..q
(cpmin)inc,,p=0 800
Cos f0 /i I
i
.
Um zoo
c
'
Figure 4-53 Drag-critical Mach number of the incident flow of swept-back airfoils of infinite
b
I
span. (a) Effect of pressure coefficient. (b) Effect of thickness ratio (biconvex parabolic
0°
2
Ma,o cr
shape).
reduced; and third, the rise of cprnin with Mach number is much weaker for a swept-back wing than for an unswept wing.
An extension of Fig. 4-52 is given in Fig. 4-53a, where the critical Mach numbers of swept-back airfoils of infinite span are presented relative to (cpmin)inc, =o For a biconvex parabolic profile,(cprnin)inc, =o =-2(urnaxIUU)inc = -(8/ir)(t/c). Corresponding to the example shown in Fig. 4-53a, the sweepback angle
has been evaluated in Fig. 4-53b as a function of the critical Mach number and for several thickness ratios. For S = t/c = 0, this function is
Ma cr =
1
cos
(A -> cc, 5 -. 0)
(4-79)
274 AERODYNAMICS OF THE WING
Thus, sweepback may raise the drag-critical Mach number of very thin profiles considerably above unity.
Middle (root) section of the swept-back wing The discussions about the effect of wing sweepback presented so far are valid only for the straight airfoil of infinite span (see Fig. 4-52). For folded wings (Fig. 3-74), the favorable sweepback effect (raising of the drag-critical Mach number) is not realized fully in the vicinity of the root section. The middle portion of the wing performs somewhat as if it were unswept. For the computation of the critical Mach number of the middle section of
the folded swept-back wing, the following procedure has to be applied: For incompressible flow, the velocity distribution over the root section is given by Eq. (3-187). The maximum velocity over the root section produces the largest underpressure (Cpmin)inc = -2(UmaxlU-)inc. The value Of (urnax/U-)inc of a parabolic profile is plotted in Fig. 3-76 against the sweepback angle ipinc- Conversion of (Cpmin)inc into Cpmin for the various Mach numbers is given by Eq. (4-76), where the sweepback angle also has to be transformed according to Eq. (4-68c). The critical Mach number is then obtained as the intersection of the curves cp min and
cpcr of Fig. 4-52, where for the root section the curve cpcr for p = 0 has to be taken. The result of this computation is presented in Fig. 4-54, for sweepback angles p = 0, 45, and -45° and for several relative thickness positions Xt. The dashed curve for ep = ±45° shows the values for the straight swept-back wing. They
are valid for sections of the folded wing at large distances from the root. It is clearly seen that the swept-back wing (gyp = +45°) has the most favorable critical Mach number of the root section for relative thickness positions of about 30%, 12
p=t45° 10
14
0
i
-45°
+45°
2
1
02
0.2
Xt
0.4
x t=02 0.3
0.5
_. - -
i__9
-
06
o.4
10
Xt
Figure 4-54 Drag-critical Mach numbers for middle (root) and outer (tip) sections of folded swept-back wings of various relative thickness positions; thickness ratio 6 = tic = 0.1. (1) Root section. (2) Tip section.
WINGS IN COMPRESSIBLE FLOW 275
whereas the swept-forward wing (gyp = -45°) is most favorable for relative thickness
ratios of about 70%. These results show that the critical Mach number of the middle section of folded swept-back wings is, in general, considerably lower than that of the tip section. It follows that the favorable sweepback effect of the straight swept-back wing cannot be fully realized by folded wings. Investigations of the drag-critical Mach number of folded swept-back wings were made by Neumark [64]. He also studied the influence of finite aspect ratios on the critical Mach number, but no marked differences with the airfoil of infinite span were found; see Fig. 3-71.
Experimental results Raising of the drag-critical Mach number by sweepback has found practical applications of great importance for airplane design. As has previously been shown in Sec. 4-3-2, increasing the critical Mach number produces a shift of the compressibility-caused drag rise to higher Mach numbers (Fig. 4-14a). It must be expected, therefore, that sweepback causes a shift to higher Mach numbers of the strong rise of the profile-drag coefficients with Mach number, cDp(Ma.). This fact was first realized by Betz in 1939 and has been checked experimentally by Ludwieg [57]. A few of his measurements are plotted in Fig. 4-55. The polars for an unswept and for a swept-back trapezoidal wing (cp = 45°) show the following: The profile drag (CL = 0) of the unswept wing is several times larger at Ma = 0.9 than at Maw = 0.7. Thus the drag-critical Mach number of this wing lies between Maw, = 0.7 and May, = 0.9. For the swept-back wing, however, the profile drag at Maw, = 0.9 is only insignificantly higher than at Ma. = 0.7. In other words, the critical Mach number of this wing lies above Mae, = 0.9. Another example of this important swept-back wing effect is demonstrated in Fig. 4-56. Here, from [71], CDp is shown versus Ma,. for an unswept and a swept-back wing (p = 45°). The sweepback effect is manifested by a shift of the onset of the drag rise from about Maw, = 0.8-0.95. This favorable sweepback effect has been exploited by airplane designers since World War II. The presentation of Fig. 3-4c, namely, sweepback angle versus flight Mach number, shows very clearly that the sweepback angle of airplanes actually built increases markedly when Mach number Ma. = 1 is approached.
Thick wing at sonic incident flow The subsonic similarity rule of Sec. 4-4-3 leads to useful results in computing the lift for incident sonic flow (Maw, = 1). It fails, however, in the computation of the displacement effect of a finitely thick wing at sonic incident flow. The reason is that the pressures on the wing become infinitely high. Compare, for example, [70] for an account of this difference between the lift problem and the thickness problem in the limiting case Mae, -} 1. To obtain useful information on the thickness problem at Ma. = 1, nonlinear approximation
methods have to be applied. The transonic similarity rule (see Sec. 4-3-4) is particularly well suited for classification and systematic presentation of test results on wings of finite span; see Spreiter [103]. Further information on the theory of transonic flow of wings is found in publications by Keune [43] and Pearcey [69] and in reference [68] on the equivalence theorem of wings of small span in transonic flow of zero incidence.
276 AERODYNAMICS OF THE WING
a=12,4°
f
Z/
0or
Maw =0.7 38 °
.4
4 Maw = as
9,B°
"Oo
12,0
174
Maw =0. 9
0.
22 °
= 0°
1 1.80
.4°
0 S
5,6°
-0
-04 '
a
0.1
cD.
VT
430
0,2
0.2
0.1
b
0.
CD
Figure 4-55 Polars, lift coefficient CL, and drag coefficient cD at high subsonic incident flow; Mach number Ma. = 0.7 and 0.9, for a straight and a swept-back wing of profile Go 623, from Ludwieg. (a) Straight wing, b = 80. mm, Cr = 22.5 mm; Re = Uo°c,.1v = 3.0 105 at Ma = 0.7, = 3.5 - 105 at Ma o° = 0.9. (b) Swept-back wing, p = 45°, b' = 57 mm, Cr = 32 mm; Re =
Uocr/v=4.2 105 atMao,=0.7,=5.0 - 10' at Ma.
0.9.
4-5 WING OF FINITE SPAN AT SUPERSONIC INCIDENT FLOW
4-5-1 Fundamentals of Wing Theory at Supersonic Flow Mach cone (influence range) There is an essential physical difference between flows
of subsonic and supersonic velocities, namely, that the disturbances of a sound point source in the former flow propagate in all directions, but in the latter flow only within a cone that lies downstream of the sound source (Fig. 1-9b and d). This so-called Mach cone has the apex semiangle a, which, by Eq. (1-33), is given by
sing =
1 11J. a 00
and
tang =
1
VMaL-1
(4-80)
with Ma. = U. 1a.. The state of affairs just discussed may also be interpreted (see Fig. 4-57) that a given point in a supersonic flow, U. > ate, can influence only the space within the downstream cone, whereas it can itself be influenced only from the space within the upstream cone. Application of this basic fact of supersonic flow on a wing of finite span is demonstrated in Fig. 4-58. The flow conditions at a point x,
WINGS IN COMPRESSIBLE FLOW 277 0.10
Q0.
t
0,06
=45
004 0.0
0
12
Ma
Figure 4-56 Profile drag coefficients vs. Mach number for an unswept and a swept-back wing (gyp = 45°), t/c = 0.12, A = 4.
y, z = 0 on the wing can be influenced only from the crosshatched area A' of the wing that is cut out of the wing by the upstream cone. When the Mach line M.L. lies before the wing leading edge, as in Fig. 4-58, the area between this Mach line
and the leading edge also contributes to the influence on point x, y, z = 0. Downstream, the influence range is bounded by the two Mach lines through the point x, y, z = 0.
Subsonic and supersonic edge The conditions of Fig. 4-57 find an important application in oblique incident flow on a wing edge. If, as in Fig. 4-59a, a Mach line
lies before the wing edge, the component v,, of the incident flow velocity U. normal to the edge is smaller than the speed of sound a.. Such an edge is termed subsonic edge. Conversely, if, as in Fig. 4-59b, the Mach line lies behind the wing edge, then v,, is larger than ate, . In this case, the edge is termed supersonic edge. With p as the Mach angle and y as the angle of the edge with the incident flow direction (Fig. 4-59), the expression m
= tang tan
- tan y Xa., - 1
(4-81)
Figure 4-57 Upstream cone and downstream cone of a point in supersonic flow. µ = Mach angle.
278 AERODYNAMICS OF THE WING
Figure 4-58 Wing in supersonic incident flow. A'= influence range.
allows one to determine whether the edge is subsonic or supersonic. Thus the edges are characterized as follows. Subsonic edge:
vn < a.
p>y
m<1
(4-82a)
Supersonic edge:
vn > a
p
m>1
(4-82b)
The special case y = 00 (m = 1) is a subsonic edge for all supersonic Mach numbers,
and the case y = 900 (m = oc) is a supersonic edge. The concept of subsonic and supersonic edges is of significance not only for the leading edge, but also for the trailing and side edges. This fact is explained in' Fig. 4-60. Here, the subsonic edges are drawn as dashed lines, the supersonic edges as solid lines. For the same wing planform, the Mach lines for three different Mach numbers are drawn. At the lowest U00
Uoo
U00
\\
vn=Uco
a
b
Figure 4-59 Concept of subsonic and supersonic edges. (a) Subsonic edge (0 < m < 1). (b) Supersonic edge (m > 1).
WINGS IN COMPRESSIBLE FLOW 279
, 01
Mach line
11
Figure 4-60 Example for the explanation of subsonic and supersonic edges of swept-back wings. Dashed lines: subsonic edges; solid lines: supersonic edges. (a) Subsonic leading edge and subsonic trailing edge. (b) Subsonic leading edge and supersonic trailing edge. (c) Supersonic leading edge and supersonic trailing edge.
Mach number (Fig. 4-60a), all edges are subsonic, at the highest Mach number (Fig.
4-60c), the leading and trailing edges are supersonic, but the side edges are still subsonic. Distinction between subsonic and supersonic edges is conditioned by the difference in flow patterns in the vicinity of the edges. In Fig. 4-61, the various types of flow patterns are sketched, which are the sections normal to the leading and trailing edges, respectively. In close vicinity to the section plane, the flow may be considered to be approximately two-dimensional. The basically different character of subsonic and supersonic flows over an inclined flat plate was demonstrated in Fig. 4-22. Based on this figure, Fig. 4-61 shows the subsonic leading edge, at which flow around the leading edge is incompressible according to Fig. 2-9a. An .essential characteristic of this flow is the formation of an upstream-directed suction force on the nose (see Fig. 4-22a). Figure 4-61b shows the subsonic trailing edge with smooth flow-off according to the Kutta condition (see Sec. 2-2-2). At such a trailing edge, the pressure difference between the lower and upper surfaces is equal to zero (Fig. 4-22a). Complete pressure equalization between the lower and upper surfaces is achieved. In Fig. 4-61c and d, the supersonic leading edge and the supersonic trailing edge, respectively, are shown.
In both cases, neither flow around the edge nor smooth flow-off is achieved, but Mach lines originate at the edges along which the flow quantities change unsteadily. Between the lower and upper surfaces, a finite pressure difference exists (see Fig. 4-22b).
Finally, the pressure distributions over a wing section are shown schematically for the three different cases of Fig. 4-60. For the section with subsonic leading and
280 AERODYNAMICS OF THE WING
b vn < a,.
Figure 4-61 Typical flow patterns at subsonic and supersonic edges (see Fig. 4-59). (a) Subsonic leading edge, vn < a., flow around edge. (b) Subsonic trailing edge, vn < a-, smooth flow-off (Kutta condition). (c) Supersonic leading edge, un > a-, with Mach lines. (d) Supersonic trailing edge, vn > a,o, with Mach lines.
4-62a,, the pressure distribution is similar to that of incompressible flow, as would be expected. The rear Mach line, however, causes a break in the pressure distribution. In the case of the section with supersonic leading and trailing edges (Fig. 4-62c), the pressures at the leading and trailing edges have finite values. The front Mach line again produces a break in the pressure trailing
edges,
Fig.
distribution.
4-5-2 Method of Cone-Symmetric Supersonic Flow Fundamentals Before the general theory of the three-dimensional wing in supersonic
incident flow is treated in the following sections, a simple special case will be discussed first that has great significance, particularly for wings of finite span. Consider the flow about a triangular plane surface. In Fig. 4-63, two Mach lines originate at the apex A0 of the triangle, where, in this example, the right-hand edge of the triangle is a subsonic edge, the left-hand edge a. supersonic edge. Further, the
flow conditions are studied on a ray originating at the triangle apex. The flow conditions at point A 1 of this ray are determined exclusively by the area that is cut out of the triangle by the upstream cone of A1, supplemented-if applicable-by the area between the Mach line M.L. and the wing leading edge (influence range of A 1). The flow conditions at A2 likewise are determined exclusively by the influence range of A2. The two influence ranges of Al and A2 are geometrically similar, and the flow conditions in Al and A2 must be equal. It follows that the flow properties
WINGS IN COMPRESSIBLE FLOW 281
Cp
b
Cp
Figure 4-62 Pressure distributions over the wing chord (schematic) for a section of an inclined sweptback wing. (a) Subsonic leading and trailing edges. (b) Subsonic
M. L.
leading and supersonic trailing edge. (c) Supersonic leading and trailing edges.
(pressure, density, velocity, and temperature) are constant on the whole ray through A0. This statement is valid for any ray through A0. The flow field thus described is
called a cone-symmetric (conical) flow field, according to Busemann. It is a requirement for the above considerations that the edges of the triangular area be straight lines; they are two special rays of the cone-symmetric flow field.
Figure 4-63 Cone-symmetric flow over triangular flat plate at supersonic flow.
a
282 AERODYNAMICS OF THE WING
a l
i
I
I
j
Figure 4-64 Examples of the application of cone-symmetric flows. (a) Triangular wing of finite thickness at zero lift. (b) Triangular flat plate with angle of attack. (c) Rectangular flat plate with side edges.
A few examples of the application of such cone-symmetric flows are given in Fig. 4-64. Figure 4-64a shows a delta wing with a double-wedge profile in sections
normal to the incident flow direction. This is an example of a wing of finite thickness at zero lift. Figure 4-64b depicts the triangular flat plate with angle of attack (lift problem). The flow over the side edge of an inclined rectangular plate is seen in Fig. 4-64c. In the triangular part of the plate surface, limited by the Mach line M.L., the flow conditions are constant on each of the rays through the corner A0. On the remaining part of, the surface, the flow field is constant because here, in sections normal to the plate leading edge, the flow is two-dimensional and supersonic (see Fig. 4-22b). For the cone-symmetric flow just discussed, the three-dimensional potential equation, Eq. (4-8), assumes a simplified form. By choosing for the cone-symmetric flow the coordinate system according to Fig. 4-65, the perturbation potential 0 (x, y, z) = x f (77, C)
with
7=
y x
and
z S=-
x
(4-83a)
(4-83b)
Figure 4-65 Cone-symmetric flow at supersonic velocity.
WINGS IN COMPRESSIBLE FLOW 283
satisfies the condition that the velocity components from Eq. (4-6) are constant on the rays through the cone apex A. By introducing Eqs. (4-83a) and (4-83b) into Eq. (4-8), the following differential equation of second order for f(77, ) is obtained, 772)
21
02f
- 2 -?7
a>7z
+ (tan 2y
77 as
- ") -ta- = 0 .:.,
(4-84)
where tan p = 1 / Ma. - 1. This equation for the new function f depends only on the two space variables 77 and in the plane normal to the incident flow direction (x direction) (see Fig. 4-65). In the lateral planes (x = const), the v and w components form a quasi-plane flow. Application of the cone-symmetric supersonic
flow was restricted at first to wings with straight edges. Later it was extended to "quasi-cone-symmetric" flows, see [30].
Classification. of ranges The application of this method will be demonstrated for one wing at various Mach numbers by means of Fig. 4-66. The chosen example, a pointed swept-back wing without twist, is shown in Fig. 4-66. In Fig. 4-66a, it has subsonic leading edges only, in Fig. 4-66b only supersonic leading edges. In range I of Fig. 4-66a, the flow is cone-symmetric with the wing apex A as the cone center. In the remaining crosshatched zones, no cone symmetry exists with reference to the centers B and C, since on the Mach lines through B and C the pressure cannot be constant because of range I. In Fig. 4-66b, the pressure is constant over the entire range II, as will be shown later. In range III, there is cone-symmetric flow, the cone Uooj A i /ioho
Figure 4-66 Flow types of inclined wings of finite span at supersonic incident flow; example of a tapered swept-back wing. M.L. = Mach line. (b) (a) Wing with subsonic leading edge, u > Wing with supersonic leading edge, µ < -y. Without hatching = pressure is constant. Single hatching = pressure distribution is cone-symmetric. Cross-hatching = pressure distribution is not conesymmetric.
284 AERODYNAMICS OF THE WING
tip of which is the wing apex Ao, since the pressure is constant on the Mach lines from point A because of range II. Also, range IV is covered by cone-symmetric flow
with reference to point B. In the crosshatched zones, however, the flow is not cone-symmetric. Now, some information will be given on the pressure coefficients in
the various ranges (Table 4-5). The values are referred to the constant pressure coefficient of the inclined flat plate, according to Eq. (443): cPP, _
P_ _ eW U 2
'/Ma
f
-2
(4-85) L%
1
Table 4-5 Basic solutions for the pressure distribution of the inclined flat surface in supersonic incident flow (cone-symmetric flow) for ranges 1, II, III, and IV of Fig. 4-66* CpICPpl
na
U
m E' (m)
A 0<m<1 II
-
m>1
III
A
m>1
IV
B
in > 1
Unwept leading edge (m - oc)
Swept-back leading edge 1
-
t
1 ___V in
1
mz - 1 in
2
m2-1
1
2
arc cos
arc cos
1-t
z
1
m2-t2
1 + 2t m + 1
Vrraz-1
in -
i
are cos (1 + 2t)
0<m--I B
x *cppl from Eq. (4-85); m from Eq. (4-81). Range I, wing with subsonic leading edge, n from Eq. (4-86); 11, wing with supersonic leading edge, range before the Mach line, t from Eq. (4-90); III, wing with supersonic leading edge, range behind the Mach line, t from Eq. (4-90); IV, wing with supersonic leading edge and side edge, t from Eq. (4-92).
tE' (mn) - f V1 - (1 - m''-') sin2,p dip; E'(0) = 1. 0
WINGS IN COMPRESSIBLE FLOW 285
C cp P1
10
Figure 4-67 Inclined wing with subsonic leading edge (0 < m < 1). (a) Wing planform (triangular wing). (b) Pressure distri-
bution on a section normal to the flow direction, m = 0.6.
The index pl designates the plane problem. The upper sign will be used for the upper side, the lower sign for the lower side.
Wing with subsonic leading edge Without going into the details, the computed pressure distributions in sections through the wing, normal to the incident flow direction (0 < f < 1), are tabulated in Table 4-5; see [20, 77] for a wing with subsonic leading edge (range I in Fig. 4-66a). In the present case, m assumes the values 0 < m < 1. From Fig. 4-67a, the following relation applies to Range I:
On the wing,
77 =
y x
cot
runs from -1 to + 1, where
edges. In Fig. 4-67b, the
pressure
tan tan r
-1 and
(4-86)
1 are the leading
distribution is shown. On the two edges,
c, is infinitely large, as would be expected for flow around a sharp subsonic leading edge (see Figs. 4-61a and 4-62a and b). The mean value of the pressure over the width (span) is
286 AERODYNAMICS OF THE WING
Cp
2 E'(na)CPpl
2
c,(0)
(4-87)
Wing with supersonic leading edge The simplest case of a wing with supersonic leading edge is the inclined flat plate in incident flow normal to the leading edge. This problem has been treated before in Sec. 4-3-3 as a plane problem [see Fig. 4-22b and Eq. (4-85)]. The pressure distribution of the swept-back flat plate, the leading edge of which
forms the angle y with the incident flow direction (Fig. 4-68) is obtained by considering that only the component of the incident flow velocity normal to the leading edge, that is, U. sin 'y, is affecting the lift (see Fig. 3-45). In the section normal to the leading edge, the plate angle of attack a* = a/sin y. Here a is the angle of attack in the plane of the velocity U.. Consequently, the pressure distribution of the swept-back inclined flat plate becomes CP _ p - poo
_
2asiny Mao, sine y
Q00 Ua 2
-1
(4-88)
00
The swept-back plate, 'like
the unswept plate, has a constant pressure
distribution over the wing chord. The ratio of the pressure coefficients of swept-back and unswept plates becomes, with Eqs. (4-81) and (4-85),
P_ Cppl
n7,
1/,rn2
-1
(4-89)
where m > 1, according to the assumptions made. It is noteworthy that cP/cpp1 > 1, which signifies that the swept-back plate produces a higher lift per unit area than the unswept plate, presupposing that the angles of attack, measured in the incident
flow direction, are equal. For y = 7r/2, that is, m = oo, cp/cppl = 1, as would be expected. For y = p, that is, m = 1, cp/Cp p1 = o. In this case, the Mach line falls on the leading edge, and thus the incident flow component normal to the leading edge is equal to the speed of sound. Linear supersonic theory therefore fails. These results for two-dimensional flow about a swept-back flat plate can be
applied to the wing of finite span. To that end, an inclined delta wing with
ML
`
\v
Figure 468 Swept-back flat plate with supersonic leading edge.
WINGS IN COMPRESSIBLE FLOW 287
a
Figure 4-69 Inclined wing with supersonic leading edge (in > 1). (a) Wing planform (triangular wing). (The hatched area A' is explained on page 293.) (b) Pressure distri-
bution on a section normal to the flow 1
in
t
direction, in = I.S.
supersonic leading edge (m > 1), according to Fig. 4-69a, may be considered. Here, m is given by Eq. (4-81), and the following relationships apply to Ranges 11 and III:
t
=tan y' _ ?/ cot,u, _ '! Va - 1 00 x tan /c
(4-90)
X
The straight lines t = const are rays through the wing apex, where t runs from 0 to m > 1; t = ±1 represents the Mach line, t = ±m the leading edge. On this wing, the ranges II and III of Fig. 4-66b must be distinguished. The pressure is constant and is given by Eq. (4-89), between the Mach line and the leading edge, that is, in range II (1 < t < m). Details of the computation for range III (0 < t < 1) will not be given here.
In Table 4-5, formulas are listed for the basic solutions in ranges II and III at cone-symmetric supersonic incident flow. Figure 4-69b gives the pressure distribu-
tion in a section normal to the flow direction. Note that the pressures on the portions of the surface that lie before the Mach lines originating at the apex are larger than in the case of a leading edge normal to the incident flow. Conversely, the pressures are considerably smaller behind these Mach lines. The mean value of cp over the span is Cp = Cppl
(4-91)
288 AERODYNAMICS OF THE WING
Wing with a supersonic leading edge and supersonic side edge So far, the wing with a supersonic leading edge has been treated. Now, for a further basic solution, the wing with a supersonic leading edge and a supersonic side edge will be discussed. A side edge is defined as an edge that is parallel to the incident flow in the planform (Fig.
4-70). From point B of the
side edge,
a wedge-shaped range IV of
cone-symmetric flow is formed rearward (see Fig. 4-66b). This range is bounded by
the side edge of the wing and the two Mach lines issuing from A and B. The boundary conditions for the pressure distribution in range IV are cp = 0 on the side edge and cp = crII = const on the Mach line. By using the coordinate system 2, y of Fig. 4-70a, the following relationship applies to
cotu=
Range IV:
VXa'-1
(4-92)
z where t = 0 represents the side edge and T= -1 the Mach line. The relationship for the pressure coefficient is given in Table 4-5. A particularly comprehensive compilation of basic solutions is found in Jones and Cohen [39].
Superposition principle Determination of the lift distributions at supersonic flow over an arbitrary wing shape is not yet possible by means of the basic solutions of
a
i
Leading edge
i'
\0
B
//
I'll Side edge (tip) X T__7
21 t-0
Figure 4-70 Inclined wing with supersonic leading edge and side edge. (a) x
Swept-back wing.
wing.
(b)
Rectangular
WINGS IN COMPRESSIBLE FLOW 289
Figure 4-71 The superposition principle at supersonic velocities. Wing AED : basic; ABCD: given wing.
Table 4-5. In those ranges of the wing that are covered by the Mach cones of several disturbance sources, for example, the crosshatched zones in Fig. 4-66, the basic solutions cannot be immediately applied. For these areas, a solution can be found, however, with the help of a simple superposition procedure, which will be sketched briefly.
Sought is the lift distribution of a tapered swept-back wing without twist, ABCD in Fig. 4-71. To this end, the wing is complemented to a wing with a sharp tip AED for which the basic solution of the lift distribution is known from Table 4-5. To obtain the given wing ABCD from this initial wing AED, a disturbance source is thought to be placed on point B. Two Mach lines under the angle p with
the side edge BC issue from this point. The left-hand Mach line intercepts the trailing edge of the given wing at point F. In the range ABFD of the given wing, no change in lift distribution is caused by the disturbance source B. Now, the following solution has to be added to the solution of the wing AED to obtain the solution for
the given wing ABCD: For the range BEF, a solution is to be found with the following characteristics (so-called compensation wing). In the partial range BEC, the lift of the compensation wing has to be equal but opposite to that of the wing AED so that the total lift disappears in the former after superposition (lift
extinction). In the partial range BCF, the compensation wing must not have a normal velocity component to keep the angle of attack, of this range unchanged after superposition. The details for the computation of such compensation wings cannot be discussed here. A comprehensive listing of the most important compensation wings and their velocity distributions is found, however, in Jones and
Cohen [39]. For the fundamentals of the theory, compare also Mirels [62]. The above method may be applied to a simple example like that given by Fig. 4-72.
4-5-3 Method of Singularities for Supersonic Flow in Sec. 4-5-2, the method of cone-symmetric flow was applied to the computations
of flows about wings in supersonic incident flow. This method is limited to the treatment of special cases, such as wings without twist and with straight edges. Wings of arbitrary planform with twist cannot be treated using this method. For them, the method of singularities is available.
290 AERODYNAMICS OF THE WING
a .11
Cp P1
C 1
1
0
CP2- - CP p I
d.
Figure 4-72 Application of the superposition principle to the inclined rectangular flat plate. (a) Given wing. (b) Basic wing (infinitely wide plate). (c), (d) Compensation wings 1 and 2. (e) Procedure for determination of the
e
pressure distribution.
'-M. L.
A detailed presentation of this method and of its applications is found in Jones and Cohen [39] and Heaslet and Lomax [30] ; see also the basic contribution of Keune and Burg [42]. The basic features of the method of singularities for incompressible flow have been explained in Secs. 3-2 and 3-6. An analogous procedure has been developed for
supersonic flows. The equation for the velocity potential of three-dimensional incompressible flow O(x, y, z) is given for Ma00 > 1 in Eq. (4-8).
Vortex distribution It has been shown in Sec. 3-2-2 that a solution of the potential equation for a wing with lift in incompressible flow can be obtained by means of a vortex distribution in the xy plane. By designating the vortex element at station
x'y' (Fig. 3-17) by k(x', y'), Eqs. (3-46) and (3-47) yield for the contribution of this element to the velocity potential d2 0 (x, y, z ; x', y,) =
with
L;
(x', 4 y')
r=V(x
(y - 01 . z-' (1
I x - x') d x' d y'
-x')s±(y- y')2Tz2
(Ma,. = 0)
WINGS IN COMPRESSIBLE FLOW 291
By applying the supersonic similarity rule Eq. (4-10) with Eq. (4-12), the corresponding solution for supersonic incident flow becomes d20(x, y, z; x', y') = with
2k(x', y')
4n
z
(y - y')2 +
x -xdx'dy' r
(4-93)
r = V (x - x')2 - (Ma2 - 1) [(y - y')2 + z2]
The analogous formula for a source distribution is Eq. (4-101).
For the transition to the potential of supersonic flow, the term in the incompressible equation that is formed by multiplication with the i in the brackets must be eliminated because it is real in the entire space and, therefore, physically impossible in supersonic flow. The term with l 1r in the potential equation of incompressible flow becomes, in the potential equation of the supersonic flow, a term that is real only within the Mach cone. Because a point P is affected by two disturbances in supersonic flow but by only one in subsonic flow, as demonstrated in Fig. 4-73b, the factor before the vortex element k has, for supersonic flow, twice the value of that for incompressible flow.
In order to obtain now the total potential at a point x, y, z, the contributions of the vortex elements have to be integrated in the x y' plane. Here, only the downstream cones of the vortex elements are taken into account; the upstream cones remain unused. Hence, the potential of the vortex distribution, see Eq. (3-46), becomes +8
'(x,y,z)=
r
z
y'), + z2 G(x' y, z; y') dy'
(4-94)
a
l~iguze 4-73 The effect of a sound point source at subsonic and supersonic velocities.
292 AERODYNAMICS OF THE WING
with the kernel function 2:,(y')
G(x, y, z; y') = 2
k(x', y') (x - x') dx'
J
(1Yla2-1)[(y-y')2+z2]
{x-x')9_
(4-95)
In Eq. (4-94), the integration has to be conducted over the width of the upstream cone in span direction (see Fig. 4-74). Integration of Eq. (4-95) has to be conducted over x' in the upstream cone of the point x, y, z from the leading edge to the Mach cone xo (y'), given by
xo(y') = x - f(Ma0 - 1) [(y - y')2 + z2]
(4-96)
Corresponding to Eq. (3-45), the velocity components in the x and z directions at the wing location z = 0 are obtained from Eqs. (4-94) and (4-95) [compare also Eqs. (3-37) and (3-41)] as
u (x, y, 0) = ± , k (x, y)
(4-97) +3
1
4a
lim ,0
2
G(--, y; y)
r 0(x,Y;Y') d
T (y - 01 -
(4-98)
3
with the kernel function z,(y') ,
j
k(x', y') (x - x') dx' (x - x')2 - (Ma's - 1) (y - y')'
(4-99a)
Xf(Y')
(4-99b)
G Xf(Y)
The equation for the vortex density k(x, y) is obtained from the kinematic flow condition, which for the wing without twist with z = 0 and aF = a is given from Eq. (3-40) by
U0 a + w(x, y) = 0
(4-100)
xa ')<
Figure 4-74 The integration range for the velocity
potential of a wing at supersonic incident
flow velocity from Eqs. (4-94) and (4-95).
WINGS IN COMPRESSIBLE FLOW 293
By introducing Eq. (4-98) into Eq. (4-100), the latter equation becomes an integral
equation for the determination of the vortex density k(x, y) in which the wing shape must be given. Solution of this integral equation is quite difficult, as in the incompressible case; see [15, 18, 30]. From the velocity component u, the pressure difference between the lower and upper sides of the wing is obtained in the form of the pressure coefficient from Eq.
(344). A relatively simple solution for the method of singularities was outlined in the early days of aerodynamics in a few examples by Prandtl [75] and Schlichting [80].
Source distribution It has been shown in Sec. 3-6-2 that the potential equation for incompressible flow with Ma. = 0 can be solved through a source distribution on the wing surface. The method of source distributions for supersonic flow has been developed into a computational procedure by Evvard [18] ; see also Puckett [76]. The source element q(x', y') at the station x', y' contributes, from Eq. (3-174), to the perturbation velocity potential the amount d
2O
(x, y, z ; x',
y,)
q( x', y)' dx'dy' 4n r
(Ma = 0)
where, again,
r=
(x - x')2 + (y - y')2 + z2
The corresponding solution at supersonic incident flow becomes, with Eq. (4-12) and the supersonic similarity rule, Eq. (4-10),
d2O(x,y,z;x',y')=-4 1
2q(x', y')
r
dx, dy
(4-101)
where r is given by Eq. (4-93). It can be proved that this expression is a solution of the potential equation, Eq. (4-8). The square root has real values only within the
two Mach cones of the point x', y', z' = 0 (upstream and downstream cones, see Fig. 4-57) with the apex semiangle u, where tang = 1 / Maw, -1. For physical reasons, however, the source element produces a contribution to the potential of only the points x, y, z that lie in the downstream cone of the source element. Equation (4-101) contains an additional factor of 2, however, for reasons that were explained for Eq. (4-93).
The total potential at the point x, y, z is obtained by integration over the contributions of source elements in the x 'y' plane, considering only the downstream cones of the source elements. The upstream cones are not considered. Hence
-,
1
rr
a(x'.v')dx'dy'
(4-102)
- ti [(Y - V')2 ; ='J (A')
Here, A' is the influence range (integration range) of the point x, y, z. It is shown for z = 0 in Fig. 4-58. For z # 0, the influence range is bounded by a hyperbola (see Fig. 4-74).
294 AERODYNAMICS OF THE WING
The velocity components in the x and z directions at the wing location z = 0 are obtained from Eqs. (3-45) and (4-102) as is (x, y)
1
a
2n ax
Jf
q (x', y') dx' d y'
y(x - x')2 - (111ta
(A) w (x, y)
2 q (x, y)
-
1) (y
(4-103) y')2
(4-104)
where the upper sign is valid for z > 0 and the lower for z < 0. The partial differentiation with respect to x in Eq. (4-103) requires particular precautions because the integrand goes to infinity on the boundaries of the integration ranges formed by the Mach lines, and these boundaries depend on x and y. Those integrals are best solved by the method of finite constituents of divergent integrals of Hadamard.*
The pressure coefficient of supersonic flow becomes the same as in incompressible and subsonic flow [Eq. (4-18)] : cp (x, y)
2 u U,)
(4-105)
Equation (4-103) is suitable immediately in the given form for the computation of the velocity distribution on a wing of finite thickness at supersonic flow. (displacement problem) (see Sec. 4-5-5 for a specific discussion). The method of source distribution will now be applied to the inclined wing at supersonic flow (lift problem); the inclined wing with subsonic leading edge cannot be treated by the discussed method of source distribution without complications, because in this case flow around the leading edge is present. Instead of the source distribution, the dipole distribution according to [30] and a vortex distribution of
the kind described above are therefore preferable. A method will be given later, however, by which a wing with subsonic leading edge can be computed after all by the source method. A simple application of the source distribution method is the computation of the inclined wing with supersonic leading edge. Since the incident flow component normal to the leading edge is larger than the speed of sound and, consequently, there is no flow around the leading edge (Fig. 4-61c), the solution for the lower and upper sides of a wedge profile with linearly growing thickness is at the same time the solution for the inclined flat surface (see Fig. 4-64a and b). The starting point for further consideration is the velocity potential of the source distribution of Eq. (4-102). For an inclined wing, source distributions of different signs have to be arranged in the wing plane on the upper and lower wing surfaces. Thus, a pressure discontinuity is produced at the wing that results in lift. Further discussion needs to be conducted for the upper half-space, z > 0, only. The upper source distribution corresponds to the potential P(x, y, z). Then, the velocity components of the perturbation flow are computed with Eq. (3-45). The source strength from Eq. (4-104) is q (x, y) = 2 w (x, y)
(4-106)
`Translator's note: See M. A. Heaslet and H. Lomax in W. R. Sears (ed.), "General Theory of High Speed Aerodynamics," Princeton University Press, Princeton, N.J., 1954, for a discussion of Hadamard's method.
WINGS IN COMPRESSIBLE FLOW 295
For the solution of the problem the following conditions must be satisfied: For the supersonic leading edge, the flow in the range before the wing is undisturbed. For the wing with subsonic leading edge, the flow is undisturbed before the Mach lines. Thus, in these two ranges 0 = 0. On the wing, the kinematic flow condition must be satisfied, namely, U" a (x, y) -{- w (x, y) = 0
(4-107)
where a(x, y) is the angle-of-attack distribution. Thus, from Eq. (4-106), the source distribution of the wing becomes q (x, y) = -2 U. L-4 (x, y)
(4-108)
For the wing with subsonic leading edge, an upwash range with the local streamline inclination X(x, y) lies between the Mach lines and the wing leading edge. In analogy to Eq. (4-108), it follows that q (x, y)
2 U,. (x, y)
(4-109)
In this upwash range, no pressure discontinuity can exist in the z direction, however, requiring that u(x, y) = v(x, y) = 0. Introducing Eqs. (4-108) and (4-109) into Eq. (4-102) yields 0(x' y, z)
= "',
[ff
a (x', y) V (X
dx,dy,
- x')2 - (Ma', - 1) [(y - y')2 + z"]
(R W)
A(x', y') dx- dy'
+ (R u)
V(x - x')2 - (Ma;o - 1) [(y - 02 + Z2]
(4-110)
Here, Rw is the integration range on the wing and Ru that of the upwash zone. These ranges may be explained now through three examples: In Fig. 4-69, a delta wing with two supersonic leading edges is shown. In this case, the range R,, does not exist, whereas-the range R w is identical to the hatched range A'. In Fig. 4-75, a wing with a supersonic and a subsonic leading edge is sketched. As has been shown
Figure 4-75 Application of the singularities method of Eward to the computation of lift distributions of wings at supersonic incident flow. (a) A supersonic and a subsonic leading edge, from Evvard. (b) Two subsonic leading edges, from Etkin and Woodward.
296 AERODYNAMICS OF THE WING
by Evvard [18] , only the integral over the range RW is left for the potential at the point P(x, y, 0), because the integrals over the ranges R,, and R'yy just cancel each other. The wing with two subsonic leading edges is shown in Fig. 4-75b. In this case, the above Evvard theorem, applied twice, leads to the conclusion that, approximately, only the hatched ranges R'W contribute to the integral Eq. (4-110); see Etkin and Woodward [17], Hancock [18], and Zierep [18]. Application of the Evvard procedure is always feasible for wings with supersonic trailing edges. The flows with subsonic trailing edges, however, require consideration of the vortex sheet behind the wing. A contribution to the solution of this problem was made by Friedel [25].
4-5-4 Inclined Wing in Supersonic Flow Before reporting on a general computational procedure for the determination of the lift distribution on wings of finite span in supersonic incident flow, first two particularly simple wing shapes will be treated, namely, the rectangular wing and the triangular wing (delta wing). Fundamentally, these two wings can be computed by the relatively simple method of cone-symmetric flow of Sec. 4-5-2. For arbitrary wing shapes, however, the method of singularities discussed in Sec. 4-5-3 must be used.
Rectangular wing The simple rectangular wing is obtained by setting 7 = rr/2 in Fig. 4-70. Thus, from Eq. (4-81), m = °°. During transition from the swept-back leading
edge of Fig. 4-70a to the unswept leading edge of Fig. 4-70b, the Mach line originating at point A disappears because point A is no longer a center of disturbance. Hence, range II of constant pressure distribution now embraces the entire surface outside of range IV. The solution for the edge zone of the rectangular wing (range IV) is obtained from Table 4-5 for m ->. = as --P
cppl
=
i fl
arccos. (1
2 t)
(4-111)
with t from Eq. (4-92). This pressure distribution is shown in Fig. 4-76. It was first investigated by Schlichting [80]. From Fig. 4-76 it can be seen that the lift of the edge zone is only half as high as that of an area of the same size in plane flow. This solution allows a simple determination of the total lift of a rectangular wing. The lift slope becomes dcL
da
=
4
M2.--1
1
` 2AVMa -1 i
(4-112)
This formula is applicable as long as the two edge zones do not overlap, that is, for
A Afa', - 1 > 2 (Fig. 4-77a). They overlap for 1 < A Ma. - 1 < 2 (Fig. 4-77b). The Mach lines from the upstream corners intersect the wing trailing edge. For A v1_1 -Maw, < 1, they intersect the side edges and are reflected from them as shown in Fig. 4-77c. The pressure distribution in the ranges affected by two Mach cones (simple overlapping) may be gained by superposition (see Sec. 4-5-2).
WINGS IN COMPRESSIBLE FLOW 297
a
1-1
0.5
Figure 4-76 Inclined rectangular plate at supersonic incident flow. (a) Planform. (b) Pressure distribution at the wing edge, from Eq. (4-111).
The lift slope of the rectangular wing is seen in Fig. 4-78a, where Eq. (4-112) is
valid even up to !1 Ma, - 1 = 1. A detailed explanation thereof will be omitted here. In Fig. 4-78b and c, the neutral-point positions and the drag coefficients are also shown. Finally, the pressure distribution over the wing chord and the lift distribution over the span are given in Fig. 4-79 for a rectangular wing of aspect ratio A1= 2.5; in Fig. 4-79a the Mach number Mac. = 1.89, and in Fig. 4-79b
a
b
c
Figure 4-77 Inclined rectangular plate of finite span at supersonic incident flow for several Mach numbers. (a) -I -,a> 2. (h) 1 < J -1 < 2.
(c) .I\AZ -1 < 1.
298 AERODYNAMICS OF THE WING
01/
a.
0
0.5
ZO
20
15
AMaZ-1-
0.5
24
3.0
2.5
30
-
0.4'
N
N
0.5
1.0
1,5
Z0
2.5
f
O
2.0
k
15
0.5
C
0
05
7.0
1.5
A Ma.j-1
2.0
2.5
3,0
Figure 4-78 Aerodynamic forces on inclined rectangular wings of various aspect ratios at supersonic incident flow. (a) Lift slope. (b) Neutral-point position. (c) Drag coefficient.
Ma = 1.13. It can be shown easily that a wing with A Maw - 1 = 1, as at Ma. = 1, has an elliptic circulation distribution. The influence of the profile thickness of an inclined rectangular wing has been investigated, in the sense of a second-order theory, by Bonney [8] ; compare also Leslie [50). Delta wing As a further example, the delta wing will be discussed. This includes wings with subsonic and supersonic leading edges, depending on the Mach number (Figs. 4-67 and 4-69). Wings with subsonic trailing edges are entirely described by range I, as can be
WINGS IN COMPRESSIBLE FLOW 299
concluded from Fig. 4-66a. The corresponding pressure distribution has already been given in Table 4-5 and in Fig. 4-67. In terms of the mean value of the pressure over the span from Eq. (4-87), the total lift is obtained by integration over the wing area as
L=2
U.2,4 J cp p,
where J cp pi = cp pt 1 - cp pi u is the mean pressure difference between the lower and upper surfaces of the unswept plate. With Acppi = 4a/ Ma. - 1, the lift slope of the delta wing with subsonic leading edge becomes dcL d
»z
2z
ro-
E' (art) 1/1t1ci' 2:-r
tiny
(4-113a) 1
(0 < 972. < 1)
(4-113b)
forMa,>1 and0<m<1. ucc
Figure 4-79 Pressure distribution over the chord and lift distribution over the span for the inclined rectangular plate of aspect ratio .1= 2.5 at supersonic incident flow. (a)
t -1=4:Maa,=1.89.(b)
Yla;e-1=3:Ma«,=1.13.
300 AERODYNAMICS OF THE WING
One result of Eq. (4-113b) should be emphasized: For very slender wings (y very small), m approaches zero for any Mach number, and because E'(0) = 1, dcL
= 27r tan y
(4-114a)
dot
(y-+0, 11-0)
=
(4-114b)
Al
2
with tan y =A /4. Thus, the lift slope of very slender wings is independent of the
Mach number when Ma. > 1. The same result was found in Eq. (4-75a) for Ma,o < 1. This is the so-called slender-body theory of Jones [371. For Ma = 1, again m = 0, and in this case Eq. (4-113) is also valid, in agreement with Eq.
(4-75c). Thus it has been shown that the lift slope at Ma = 1 has the value dcL/da = 7rA/2 for arbitrary aspect ratios A, whether Ma. = 1 is reached from the subsonic or from the supersonic range (see Fig. 4-51). The neutral point lies in the surface center of gravity because the pressure differences, averaged in the lateral direction, are constant in the longitudinal direction. Thus, the neutral point lies at xN Cr
=2
(4-115)
3
The drag of a wing with subsonic leading edge is composed of the partial force La, which depends on the pressure distribution on the wing, and the suction force S, which is produced by the flow around the leading edge (see Sec. 3-4-3). Thus, the drag is given by
D=La-S
(4-116)
The contribution La is known from the above discussion. The suction force S can be determined from the vortex density k(x, y) in the vicinity of the leading edge. This relationship for plane incompressible flow is given in Eq. (2-76). Determination of the suction force for compressible flow with subsonic leading edge is treated, for example, in [37] and [77]. For a delta wing with subsonic leading edge (m < 1), the drag coefficient without suction force CD = CLa becomes .
D = CD
= 2 E' (m)
27vaa
nz
E'' (m) )/Ma2c
-1
2
CL re el
(4-117)
Here it has been taken into account that, from Eq. (4-81),
m = tan y Mad - 1 = 4 Ma=w - 1
(4-118)
The coefficient of the suction force is determined from [77] as 2
GS =
V1
- nag
(4-119)
According to Eq. (4-116), this quantity is to be subtracted from the drag coefficient
from Eq. (4-117) to obtain the total drag (reduced drag + wave drag + suction force). Hence
WINGS IN COMPRESSIBLE FLOW 301 2
CD =
{2 E'(7n) -
-
(4-120)
The wing with supersonic leading edge is composed of ranges II and III of Fig. 4-66b only. The corresponding pressure distributions have been given previously in Table 4-5 and in Fig. 4-69b. By taking the mean value of the pressure over the width from Eq. (4-91), the lift slope of a delta wing with supersonic leading edge becomes dcL
-
da
4
Ma - 1
(m > 1)
(4-121)
00
Hence, the lift slope of a delta wing with supersonic leading edge is equal to that of the plane problem (Table 4-2). Likewise, the neutral point of a delta wing with supersonic leading edge lies in the area center of gravity because the pressure difference,. averaged laterally, is constant in the longitudinal direction. Thus the neutral-point position is the same as that of a delta wing with subsonic leading edge, namely, xN/c,. = 3 [Eq. (4-115)] . The total drag (induced + wave drag) is
D=La Since there is no flow around the leading edge, no suction force is created. The drag coefficient becomes, therefore, 4a2 CD
-1 2 = 4 Y Mao -
(4-122a)
VMa
(4-122b)
2
CL
=7m nA
(4-122c)
in agreement with the flat plate of infinite span (see Table 4-2). Equation (4-122c) is obtained with Eq. (4-118). The ratio of the lift slopes of a delta wing from Eqs. (4-113) and (4-121) and that of an inclined flat plate of Table 4-2, with dcL 4 da Ma;, -1 is plotted in Fig. 4-80 against m [Eq. (4-118)]. The lift slope of a delta wing with a subsonic leading edge (0 <m < 1) is considerably smaller than that of a delta wing with a supersonic leading edge (m > 1). The theoretical results for the lift slope of delta wings in the entire Mach number range are compiled in Fig. 4-82a. The values for Ma., < 1 have been established from the linear theory of subsonic incident flow (Sec. 44-2), those for supersonic incident flow from the above formulas, which are also linear. The curve for it = - corresponds to the plane problem in Fig. 4-20a. The neutral-point positions of a delta wing for the entire Mach number range
302 AERODYNAMICS OF THE WING
0.2
Figure 4-80 Lift slope of a delta wing at 0S
2
an y
supersonic incident flow. Subsonic leading edge: 0
tan,u
rn > 1.
1.5
1
are presented in Fig. 4-82b for several aspect ratios A. The curve for t1= corresponds to the plane problem in Fig. 4-20b.
The drag coefficient of delta wings is given in Fig. 4-81, where curve la represents the case Ma < 1 without suction force, Eq. (4-117), and curve lb the case with suction force. Curve 2 is the case m > 1 from Eq. (4-122c).
In incompressible flow it is customary to designate the contribution CD = cL/ rA, caused by the velocity field induced behind the wing, as induced drag. Such a contribution is made to the drag in compressible flow, too, and it is logical to call it induced drag also (Fig. 4-81, curve 3). Subtracting this drag from the total drag at supersonic velocities, the wave drag (Fig. 4-81) is obtained. For practical purposes, separate determination of the induced drag has no particular significance. Only the sum of induced drag and wave drag is required; see Schlichting [80]. The drag coefficient of delta wings without twist at various aspect ratios A is shown in Fig. 4-82c versus the Mach number. The curve for A = 00 corresponds to 6
5
y I'
Suction
force
cs
Wav e drag
lay
JR, 3
1b
2
I
2
__
3
1
Figure 4-81 Drag of delta wings at supersonic incident flow vs. m from Eq. (4-118). in < 1:
I
I
Induced drag i
0,5
tan to
1
15
2
subsonic leading edge. in > 1: supersonic leading edge. Curve la, from Eq. (4-117). Curve lb, from Eq. (4-120). Curve 2, from Eq. (4-122). Curve 3, induced drag from Eq. (3-134).
Z,1t
e -oo
/
r
A-CO
5
Supersonic
-
leading edge
-3
3
=2 2
aI
=1
1
A=0
a
02
0.6
0,4'
oe
to
Ma
12
46
1 4f
18
20
0.25
0
=2
N25
cr
3
3cr
Ao4
0,10
jU
0D5
b
A
b 02
Oaf
0.6
08
10
12
1.6
1.6
2.0
Ma,, 0.5
A-1 0.4
-1
A-2 =3
2 I
I
A =3
l
0.1
c
L_ 02
1
06
aOo
!
Ole
f
10
t I
1
12
1#
i
16
18
20
Mao
Figure 4-82 Aerodynamic forces of inclined delta wings of various aspect ratios at subsonic and supersonic flows. (a) Lift slope. (b) Neutral-point position. (c) Drag coefficient (with suction force). 303
304 AERODYNAMICS OF THE WING
the plane problem in Fig. 4-20c. Note that the aspect ratio has a strong effect on the lift-related drag at subsonic incident flow. Conversely, this effect is negligible for supersonic incident flow. For airplane design, wing forms with large aspect ratio do not offer an advantage at supersonic flight velocities (see Fig. 3-4b).
A survey of the pressure distributions over the wing chord and the lift distributions over the span is found in Fig. 4-83 for delta wings with subsonic and supersonic leading edges. The lift distributions (ctc) are illustrated in Fig. 4-84 for several values of m. It is noteworthy that the lift distributions over the span are elliptic for all wings with subsonic leading edge, 0 < m < 1. For wings with
supersonic leading edge, m > 1, the lift distribution approaches a triangular form at very high Mach numbers (m - co). Systematic measurements to check the three-dimensional wing theory at supersonic incident flow have been published by Love [56] for delta wings with
rounded and sharp-edged noses. In these measurements the aspect ratio A lies
Figure 4-83 Pressure distribution over
the wine chord and lift distribution over the wing span of delta wings at supersonic incident flow. (a) Subsonic leading edge, 0 < m < 1. (b) Supersonic leading edge, in > 1.
WINGS IN COMPRESSIBLE FLOW 305 Z0
1.8
Y\
\
1.6
7n-l
\
7Th-1.5
\
14
0<7T
1
1.2
U 0.8
0.6
O.u
\ 0.2
Figure 4-84 Lift distributions over the span of delta wings at supersonic flow for several values 0
0.2
0,6
0.4
Y
7I ° s
0.8
7.0
of m from Eq. (4-118). 0 < m < 1: subsonic leading edge. m > 1: supersonic leading edge.
between 0.7 and 4, the profile thickness is S = t/c = 0.08, and the relative thickness position Xt = xt/c = 0.18; the Mach numbers are Ma. = 1.62, 1.92, and 2.40.
The results for the lift slope are given in Fig. 4.85. As the abscissa, the parameter in was chosen. The ordinate for the range of subsonic leading edges (rn < 1) is the quantity cot 'y (dcL/da) = (4/i1) do /da ; for the range of supersonic leading edges (m > 1), the quantity (dcL/da) Ma;, -1 is the ordinate. Test results
for the 22 wings at the 3 different Mach numbers lie quite close to one curve, confirming the validity of the supersonic similarity rule of Sec. 4-2-3. The measured curve follows the theoretical curve fairly well. The deviations between theory and measurements at m = 0 and m = I are understandable, because m -- 0 means
transonic flow (Ma
1), and in = 1 signifies transition from a subsonic to a
supersonic leading edge.
The analogous plotting of the drag coefficients is given in Fig. 4-86. Only the values for rounded noses are included. Here also, the measured drag coefficients lie near one single curve, again confirming the supersonic similarity rule. In the range of subsonic leading edges the curve of the measured drag coefficients lies, at the lower values of in, between the theoretical curves with and without suction force. Finally, in Fig. 4-87, the measured neutral-point positions are plotted. Here, too, the supersonic similarity rule finds a satisfactory confirmation. The neutral points of wings with rounded noses lie somewhat more upstream than those with
8 ft.
4
dcL
a
dCL da
M0 1 (m>1)
A da (m<1)
j
`
Zit 6
Xt 0.18
4
a
'fr
°
d
Maw I I 04 1.62
2
°4
1.92
240
Mac, 1 0.5
1.0
2,0
1.5
2.5
7n= 4 Ma00-1 Figure 4-85 Measured lift slopes for delta wings at supersonic incident flow, from Love. 0 < m < 1: subsonic leading edge. m > 1: supersonic leading edge. 0,4
-CD- 1 A cL
' -c-L'
)
i
(m,> 1)
(72<1)
o 0,3
00 °
o ice/
Without suction force
-V
02
.gg
----- -
0--s-
Theory
- ---
/
j
o
°
\
0.1 q>L
c5'G08
M
With suction
Xt- 0'8
forc e
V
M17 00
i
0.5
I'D
15
2.0
2.5
Figure 4-86 Measured drag coefficients due to lift of delta wings in supersonic incident flow, from Love. 0 < in < 1: subsonic leading edge. in > 1: supersonic leading edge. 306
WINGS IN COMPRESSIBLE FLOW 307
0."
Maw
Jr
0 0 4
1. 5Z
1.92
I
2.40
I t
0.3
2°
cl
Theory
6 0 -0-0 00 0
0
ca e
8-0,08
0,1
Xt- 0..18
PMa-1 0
I
I
0.5
1.0
I
I
1,5
2.0
2.5
Figure 4-87 Measured neutral-point positions for delta wings at supersonic incident flow, from Love. 0 < m < 1: subsonic leading edge. m > 1: supersonic leading edge.
sharp-edged noses. The measured neutral-point position moves slightly upstream and increases with Mach number, although, from the linear theory, it should be independent of Mach number.
Swept-back wing Lift slopes of swept-back wings with constant wing chord (taper
X = 1) are given in Fig. 4-88 with A cot y as the parameter (zi = aspect ratio, y = sweepback angle measured from the wing longitudinal axis). The lift slope is referred to that of the plane problem do /da). = 4/ Maw, - 1 and depends on the parameter rn = tan 7/tan p = tan y Ma. - 1 and on the purely geometric quantity . Ai cot y, and may be written as cot
(
The fact that the lift slopes depend only on these three parameters can be realized by setting tan z = cot y in the supersonic similarity rule Eq. (4-26) and observing that A Ma. - 1 /A tan cp = tan y/tan p = m [see Eq. (4-81)] . Under :low conditions rendering the leading edge of the present wing shapes subsonic, the lift slopes-in a way similar to that shown for delta wings (Fig. 4-80)-deviate considerably from those of the plane problem. Conversely, when the leading edge of the present wing shapes is supersonic, the lift slopes are almost equal to those of
308 AERODYNAMICS OF THE WING 1.5
A coty-6 5
3
I
f
0,5 7b
1.5
7.0
ms
tarry tan
y
2.0
2.5
=tang Ma- 1
Figure 4-88 Lift slope of swept-back wings (taper X = 1) at supersonic incident flow, from 155]. 0,< m < 1: subsonic leading edge. m > 1: supersonic leading edge.
the plane problem. For a better illustration, the wing planforms are sketched in Fig. 4-88 for A = 3. However, the diagram applies to other values of A, too. The figure does not include rectangular wings, because the chosen presentation is not applicable to the case of y = ir/2. The lift slopes of the rectangular wing were given earlier in Fig. 4-78a.
Arbitrary wing planforms So far, results have been presented for the linear wing :ieory at supersonic incident flow for the unswept rectangular wing, the delta (triangular) wing, and the swept-back wing. In this section, a few results will be given for a trapezoidal wing, a swept-back wing, and a delta wing; see Fiecke [21 ] . The theoretical lift slopes of these three wings are given in Fig. 4-89 for the Mach number range from Ma. = 0 to, May, = 2.5. For the same Mach number range, the drag coefficients of these three wings are presented in Fig. 4-90. Two curves each apply to the subsonic range and to the supersonic range with subsonic leading edge. The dashed curve applies to the values with suction force, the solid curve to those without. The former are described by the well-known formula for the induced drag CD = C2L/1rA. The drag without suction force is found from CD = CLa = cL(da/dcL),
where the values of dcL/da are taken from Fig. 4-89. It can be expected that the suction force is fully effective on a well-rounded profile nose and that the dashed lines represent the drag coefficients. Conversely, the suction force is negligible on thin profiles with sharp noses, as used in most cases on supersonic airplanes, and thus the upper curve applies. In Fig. 4-91, the neutral-point positions of these three wings are shown schematically against the Mach number. The typical behavior during transition from subsonic to supersonic velocities is seen, namely, that the neutral point moves considerably downstream when a Mach number of unity is
WINGS IN COMPRESSIBLE FLOW 309
=3 5 I
1
1
T
1 A oo
A
2
l
I
0.5
Z5
1.0
2.0
2.5
Figure 4-89 Lift slope vs. Mach number for a trapezoidal, a swept-back, and a delta wing of aspect ratio .s = 3, from [21 ].
0. 50
0.25
;rA 0
7n 0.5
1.0
0.75
1.5
2.0
Maw
2.5
`711, -1
fAL 0
0.5
-25
7.0
2.0
2.5
Maoo 0.7
Figure 4-90 Drag coefficient due to lift vs. 025
Mach number for a trapezoidal, a sweptback, and a delta wing of aspect ratio .1 = 3, from [21 ]. Dashed curve: with suc-
0
0.5
7.0
Ma00
1.5
2.0
2.5
tion force. Solid curve: v'ithout suction force.
310 AERODYNAMICS OF THE WING
Ma
Figure 4-91 Neutral-point position vs. Mach number for a trapezoidal,
a swept-back, and a delta wing, from (21]. (0) Neutral-point position
for Ma < 1. (.)
Neutral-
point position for Ma. > 1.
exceeded. This means an increase in longitudinal stability of the airplane during transition from subsonic to supersonic flight. Finally, a brief account will be given of the experimental confirmation of linear wing theory. In Fig. 4-92, the lift slopes dcL/da are plotted over the Mach number for four different wings (rectangular, trapezoidal, triangular, and swept-back). For the subsonic range, the theoretical curves were determined according to Sec. 4-4-2,
for the supersonic range, from Friedel [251. The measured lift slopes are in good agreement with theory, except for the immediate vicinity of Ma. = 1. Additional details of a three-component measurement in the subsonic and supersonic ranges of the trapezoidal wing of Fig. 4-92b are illustrated in Fig. 4-93. The curves CL(a) of Fig. 4-93a show clearly that the linear range and the coefficient of maximum lift CL are considerably larger in supersonic than in subsonic flow. Also, the pitchingmoment curves CL(cm) in Fig. 4-93c confirm that the linear range is markedly larger
for Ma. > 1 than for Mar < 1. In this connection, the publications [59, 76, 90] are noted; they are concerned with the computation of twisted wings and flight mechanical coefficients of wings at supersonic velocities.
4-5-5 Wing of Finite Thickness in Supersonic Flow General statements In the previous sections, the inclined wing of finite span in supersonic flow was treated (lift problem). Now, the special case of a wing of finite
WINGS IN COMPRESSIBLE FLOW 311
thickness with zero lift (displacement problem) will be discussed in more detail. Of interest here are the pressure distribution over the wing contour and the resulting wave drag. The latter is a strong function of the profile thickness, as was discussed for the plane problem in Sec. 4-3-3. The most general method of determining the pressure distribution of wings of finite thickness at zero lift is the source-sink method of von Karman [100]. The fundamentals of this method for the wing with supersonic incident flow were furnished in Sec. 4-5-3. The basic concept of this method is to cover the planform area of the given wing with a source distribution
q(x, y) in the xy plane. From this, the x component of the velocity on the wing surface u(x, y) is obtained from Eq. (4-103) and the z component w(x, y) from Eq. (4-104). By describing the wing contour by z(t)(x, y) = z(x, y), the kinematic flow condition is expressed by Eq. (3-173b). Introducing this into Eq. (4-104) yields the source distribution Eq. (3-176). Introducing this result into Eq. (4-103) furnishes the pressure, coefficient cp = -2u/U. as az (x ' , y ' ) ax, e:r,
(A')
'7
X
,1
"I /
(4-124) y')2
V(x - x')"- --
Here, A' is the influence range of the point x, y, as indicated in Fig. 4-58 by cross-hatching. The pressure distribution for a given wing contour z(x, y) can thus be determined. Subsonic leading edge
Supersonic leading edge
S
SBT
t
SBT
Theory l1=2,75
0
JO
0
a
Th eory Am .Pw o
C
!Supersonic leading edge Subsonic leading edge Supersonic leading edge
Subsonic leading edge 5
SBT O
SBT
Theory
OO
/1=2.75
=2%S
0
Theory
3 0
1 /V' M cc,)
Figure 4-92 Experimental confirmation of linear wing theory at subsonic and supersonic incident flow. Lift slope vs. Mach number: measurements from Becker and Wedemeyer [51, Stahl and Mackrodt [90] . Theory for supersonic flow from Friedel [25]. SBT = Slender-body theory, Sec. 4-4-3.
I
_3ti
.p --
p u
I
I
0
0
70
a
{
N
l
NNKN,X Al (
h.
p
°0
0
0 0
U f3
312
WINGS IN COMPRESSIBLE FLOW 313
Wave drag The coefficient of wave drag of the wing at zero lift is obtained through integration of the pressure distribution over the wing area A as
f
2
CDO = A
c, (x, y) a2 dx dy
(4-125)
(A)
This formula is applicable to sharp-edged profiles only. The dependence of the drag coefficient on profile thickness ratio, taper, aspect ratio, sweepback angle, and Mach number of the incident flow is given according to the supersonic similarity rule by Eq. (4-27). This relationship is of great value for a systematic presentation of theoretical and experimental results.
Rectangular wing For the wing of rectangular planform and spanwise constant profile z(x, y) = z(x), introducing Eq. (4-124) into Eq. (4-125) and integrating twice yield (see Dorfner [15] ) 1
4
CDO=
YMa'
00
r0,Y) Z - dX
(A'> 1)
- 1 0J
(4-126a)
1
4
VMa.
-1 1
+
2
Wx
3
cl X
0
dZ
nA.' J dX
T-1' dZ dX'
(X - X')2 - A
X - X'
dX' dX
(^1' < 1) (4-126b)
A'
Note that, for A' = A Ma. - 1 > 1, the drag. formula for the rectangular wing of finite span is identical to that of the rectangular wing of infinite span (see Table 4-2).
For a convex parabolic profile Z = z/c = 26X(1 -X) with X = x/c, the integration yields CDO
1
CDo-
=
2n
(4-127a)
(A'_>_1)
L4 arcsin A' - A' ()l1 L
- A'2 - (6 - A'2) cosh-1 A) -L ]
(A' < 1) (4-127b)
where CDO- is given by Eq. (4-50a). The numerical evaluation is given in Fig. 4-94.
Delta wing A few results will be added for delta (triangular) wings. Delta wings with double-wedge profiles have been computed by Puckett [761, those with biconvex parabolic profiles by Beane [76]. Coefficients of the wave drag at zero lift for double-wedge and biconvex parabolic profiles of 50% relative thickness position are shown in Fig. 4-95 as a function of the parameter m = Maw -1-4/4. For the double-wedge profile, CDO
is expressed by Eq. (4-51). For supersonic leading edges
314 AERODYNAMICS OF THE WING
0.6 8
X00.4
}
Figure 4-94 Drag coefficient (wave drag) at zero lift for rectangular wings at super-
02
t0
05
2.0
45
2.5
30
A Ma, 1
sonic incident flow vs. Mach number. Biconvex parabolic profile cDo . from Eq. (4-50a).
(m > 1), cDo /eD o 00 is almost independent of Mach number, whereas it changes
strongly with Mach number for subsonic leading edges (m < 1). Both curves have pronounced breaks at m = 1, that is, when the Mach line coincides with the leading edge. The curve for the double-wedge profile has another break at m = 2 , that is, when the Mach line is parallel to the line of greatest thickness. In Fig. 4-96, a number of measurements on delta wings with double-wedge profiles and 19% relative thickness position are plotted from [56]. Similar to Fig. 4-86, different representations have been chosen for m < 1 and m > 1. At the kind of presentation chosen here, these measurements on 11 wings at Mach numbers Ma. = 1.62, 1.92, and 2.40 fall very well on a single curve- Hence, the supersonic similarity rule of Eq. (4-27) has been confirmed again. The theoretical curve from
I
jr
0.
0.2
00
02
0.4.
0.6
017
1.0 Lan j-,
ia n
1.2
1.4
1.6
1.6
2.0
?.2
4 'M -1
Figure 4-95 Drag coefficient (wave drag) at zero lift for delta wing (triangular wing) vs. Mach number. Profile I: double-wedge profile cDoo,, from Eq. (4-51). Profile II: parabolic profile, cDo,,, from Eq. (4-50a). 0 < in < 1: subsonic leading edge. in > 1: supersonic leading edge.
WINGS IN COMPRESSIBLE FLOW 315
,\
10,
4 CD o
I
(m mil)
Mm
l1
Jo
I
1
7
Theory
fI
a
a
/
5.
i
Theory
2.5
M¢ 1.62
192 2.40
1l < 4
d-0.08 Xt°0,18
4
m-A Figure 4-96 Measured drag coefficients (wave drag) at zero lift for delta wings at supersonic incident flow, from Love, Theory from Puckett. Double-wedge profile of 18% relative thickness position. 0 < m < 1: subsonic leading edge. m > 1: supersonic leading edge.
Puckett [76] for the relative thickness position Xt = 0.18 shows a high peak at m = 1 that is not confirmed by measurements, as would be expected because the incident flow velocity at the leading edge is just sonic. Comparison between theory
and experiment suffers from the uncertainty in the determination of the friction drag, which has to be subtracted from the measured values. The treatment of the thickness problem of a delta wing with sonic leading edge has been compared with transonic flow theory by Sun [93]. Swept-back wing The wave drag coefficients of swept-back wings of constant chord
are illustrated in Fig. 4-97. The corresponding information for the lift slope was given in Fig. 4-88. The wing has a double-wedge profile, of which the drag coefficient in plane flow CDO is obtained from Eq. (4-51). The curves show a pronounced break at m = 1, that is, when the Mach line and leading edge fall together. It should be noted that, according to [15], CDO
CDO,o
in
f -1
for rn > I --
!i cot f
(4-128)
is obtained in the range of the supersonic leading edge if the Mach line originating at the apex (line g) intersects the trailing edge.
316 AERODYNAMICS OF THE WING 2,
1.
I F
1
A cotJ-1
1
05
0
0. .
1.0 ton
12 s ton;
-
1.5
2.0
2.5
Figure 4-97 Drag coefficients (wave drag) at zero lift of swept-back wings (taper X = 1) at supersonic incident flow, from [49]. 0 < m < 1: subsonic leading edge. m > 1: supersonic leading edge. Dashed curve (g) from Eq. (4-128).
Arbitrary wing planforms To conclude this discussion, the total drag coefficient at zero lift (wave drag + friction drag) of the three wings (trapezoidal, swept-back, and
delta) treated earlier (Figs. 4-89-4-91) is plotted in Fig. 4-98 against the Mach number. These three wings have double-wedge profiles with a thickness ratio t/c = 0.05 and an aspect ratio A = 3. Within the Mach number range presented, the 0,03
A=3
4
Wave drag
Friction drag (Re
107)
4-98 Total drag coefficient (wave drag + friction drag) vs. Mach number for a trapezoidal, a swept-back, and a delta wing of aspect ratio e = 3. Double wedge profile tic = 0.05, Yt/c = Figure
0.50, from [21 ] .
WINGS IN COMPRESSIBLE FLOW 317
wave drag is two to three times larger than the friction drag. The latter has been determined from Fig. 4-4 for Reynolds numbers Re 107. Since the wave drag at supersonic incident flow is proportional to (t/c)2, this contribution, and thus the total wing drag at zero lift, can be reduced considerably by keeping t/c small. This fact is taken into account in airplane design by choosing extremely small thickness ratios for supersonic airplanes; compare Fig. 3-4a.
Concluding remarks In addition to the references included in the text, attention should be directed toward summary reports and reports dealing with various theories on the aerodynamics of the wing in supersonic flow [6, 11, 19, 22, 23, 40, 51, 92, 105-107]. The special case of the aerodynamics of the wing of small aspect ratios, first studied by Jones [37], has been investigated comprehensively as the "slender-body theory" for both lift and drag problems [2, 13, 14, 41, 108]. The aerodynamics of slender bodies is treated in Sec. 6-4. The influence of vortex shedding at the lateral wing edges of rectangular wings, and the leading-edge separation on swept-back and delta wings at supersonic flow, are treated in [12, 72, 91 ], based on the understanding of incompressible flow. Based on a suggestion of
Jones, questions concerning the minimum wing drag have been investigated by several authors [36, 61, 97, 1101. In this connection, the investigations on the design aerodynamics of wings at high flight velocities, promoted mainly by Kuchemann, play an important role [9, 38, 46, 60].
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318 AERODYNAMICS OF THE WING
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WINGS IN COMPRESSIBLE FLOW 319
28. Gothert, B.: Ebene and raumliche Stromung bei hohen Unterschallgeschwindigkeiten (Erweiterung der Prandtlschen Regel), Jb. Lufo., 1:156-158, 1941; NACA TM 1105, 1946. Truckenbrodt, E.: Z. Flugw., 5:341-346, 1957. 29. Gretler, W.: in Anwendung direkter and indirekter Methoden in der Theorie der Unterschallstromungen, "Ubersichtsbeitrage zur Gasdynamik," pp. 95-116, Springer, Wien, 1971; Jb. WGL, 171-179, 1961; Act. Mech., 1:109-134, 1965. Sells, C. C. L.: Proc. Roy. Soc. A, 308:377-401, 1968. 30. Heaslet, M. A. and H. Lomax: Supersonic and Transonic Small Perturbation Theory, in W.
R. Sears (ed.), "General Theory of High Speed Aerodynamics," Sec. D, Princeton University Press, Princeton, N.J., 1954. Heaslett, M. A. and J. R. Spreiter: NACA Rept. 1318, 1957. 31. Hoerner, S. F.: "Fluid-Dynamic Drag-Practical Information on Aerodynamic Drag and Hydrodynamic Resistance," 3rd ed., Hoerner, Midland Park, N.J., 1965; AGARD-CP 124, 1973; AR 58, 1973. 32. Hoerner, S. F. and H. V. Borst: "Fluid-Dynamic Lift-Practical Information on Aerodynamic and Hydrodynamic Lift," Hoerner, Brick Town, N.J., 1975. 33. Holder, D. W.: Transsonische Stromung an zweidimensionalen Fliigeln, Z. Flugw., 12:285-303, 1964; J. Roy. Aer. Soc., 68:501-516, 1964. 34. Howarth, L. (ed.): Flow Past Aerofoils and Cylinders, in "Modern Developments in Fluid Dynamics-High Speed Flow," pp. 612-687, Clarendon, Oxford, 1953/1964. 35. Hummel, D.: Neuere Beitrage der deutschen Luftfahrtforschung auf dem Gebiet der Flugzeug-Aerodynamik, Jb. DGLR, 18:1-40, 1975. 36. Jones, R. T.: Theoretical Determination of the Minimum Drag of Airfoils at Supersonic Speeds, J. Aer. Sci., 19:813-822, 1952; 18:75-81, 1951; "Collected Works," NASA TM X-3334, pp. 567-578, 557-565, National Technical Information Service, Springfield, Va., 1976. Ginzel, I. and H. Multhopp: J. Aerosp., 27:13-20, 36, 1960. Graham, E. W.: J. Aer. Sci., 19:823-825, 1951. Jones, R. T.: In A. Miele (ed.), "Theory of Optimum Aerodynamic Shapes," pp. 125-134, Academic, New York, 1965; "Collected Works," NASA TM X-3334, pp. 645-656, National Technical Information Service. Rott, N.: J. Aer. Sci., 20:642-643, 1953. Sedney, R.: J. Aer. Sct, 21:639-640, 1954. Smith, J. H. B.: Aer. Quart., 12:201-218, 1961. 37. Jones, R. T.: Properties of Low-Aspect-Ratio Pointed Wings at Speeds Below and Above the Speed of Sound, NACA Rept. 835, 1946; J. Aer. Sci., 17:307-310, 1950, "Collected Works," NASA TM X-3334, pp. 369-375, 533-538, National Technical Information Service, Springfield, Va., 1976. Bera, R. K.: J. Aircr., 11:245-247, 1974; 12:692, 1975: 11:777-779,1974. 38. Jones, R. T.: Aerodynamic Design for Supersonic Speeds, Adv. Aer. Sci., 1:34-51, 1959; Z. Flugw., 4:257-268, 1956. "Collected Works,"NASA TMX-3334, pp. 625-644, 604-608, National Technical Information Service, Springfield, Va., 1976. 39. Jones, R. T. and D. Cohen: Aerodynamics of Wings at High Speeds, in A. F. Donovan and H. R. Lawrence (eds.), "Aerodynamic Components of Aircraft at High Speeds," Sec. A, Princeton University Press, Princeton, N.J., 1957. 40. Keune, F.: Einfluss von Spannweite, Dicke, Anstellwinkel and Machzahl auf die Stromung urn Fligel kleiner and grosser Spannweite, Z. Flugw., 2:292-298, 1954; 4:276-280, 1956. 41. Keune, F.: Einfluss endlicher Spannweite oder enalicher Flache im Endquerschnitt vorn spitzer Fliigel kleiner Streckung oder schlanker Rotationskorper auf die Stromung am and hinter dem Kdrper, Jb. WGLR, 186-203, 1964. 42. Keune, F. and K. Burg: "Singulaiitatenverfahren der Stromungsl.ehre," Braun, Karlsruhe, 1975. 43. Keune, F. and K. Oswatitsch: Nichtangestellte Korper kleiner Spannweite in Unter- and Z. Flugw., 1:137-145, 1953. Gersten, K.: Z. Angew. IYlath. Mech., 43:T 130-135, 1963. Keune, F.: Jb. WGL, 176-186, 1955; Z. Flugw., 4:47-53, 1956; Jb. WGL, 67-82, 1957; Z. Angetiv. Math. Mech., 37:300-301, 1957; Z. Flugw., 5:121-124, 243-247,1957.
320 AERODYNAMICS OF THE WING
44. Kraus, W. and P. Sacher: Das Panelverfahren zur Berechnung der Druckverteilung von Flugkorpern im Unterschallbereich, Z. Flugw., 21:301-311, 1973. 45. Kuchemann, D.: Entwicklungen in der Tragfli geltheorie, Jb. WGLR, 11-22, 1967; Jb. WGLR, 66-77, 1962. 46. Kuchemann, D.: Aircraft Shapes and Their Aerodynamics for Flight at Supersonic Speeds, Adv. Aer. Sci., 3:221-252, 1962; Jb. DGLR, 41-52, 1968. Gonor, A. L.: Prog. Aerosp.
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48. Landahl, M. T. and V. J. E. Stark: Numerical Lifting-Surface Theory-Problems and Progress, AIAA J., 6:2049-2060, 1968. Tsakonas, S.: AIAA J., 7:1661, 1969. 49. Lawrence, T.: Charts of the Wave Drag of Wings at Zero Lift, ARC CP 116, 1953. 50. Leslie, D. C. M.: An Introduction to Second-Order Wing Theory, J. Aer. Sci., 24:99-106, 1957. Clarkson, M. H.: Quart. J. Mech. App. Math., 7:203-221, 1954. Fell, J. and D. C. M. Leslie: T. Aer. Sci., 21:851-852, 1954. 51. Lighthill, M. J.: The Supersonic Theory of Wings of Finite Span, ARC RM 2001, 1944; 1929, 1944; J. Fluid Mech., 1:337-348, 1956. 52. Lighthill, M. J.: Higher Approximations, in W. R. Sears (ed.), "General Theory of High Speed Aerodynamics," Sec. E, Princeton University Press, Princeton, N.J., 1954.
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WINGS IN COMPRESSIBLE FLOW 321
R. and H. Yoshihara: AIAA J., 8:2157-2162, 1970. Newman, P. A. and E. B. Klunker: AIAA J., 10:971-973, 1972. Nixon, D.: J. Aircr., 11:122-124, 1974. Rohlfs, S. and R. Vanino: Z. Flugw., 23:239-245, 1975. Sato, J.: AIAA J., 11:58-63, 1973. Stanewsky, E. and H. Zimmer: Z. Flugw., 23:246-256, 1975. Steger, J. L. and H. Lomax: AIAA J., 10:49-54, 1972. 64. Neumark, S.: Critical Mach Numbers for Swept-Back Wings, Aer. Quart., 2:85-110, 1950. Scholz, N.: Jb. WGL, 319-324, 1960.
65. Nieuwland, G. Y.: Theoretical Design of Shockfree, Transonic Flow Around Airfoil Sections, Aerosp. Proc., 1:207-239, 1966. Cahn, M. S. and J. R. Garcia: J. Aircr., 8:84-88, 1971. Kacprzynski, J. J.: Nat. Res. Coun. Can., 1972. Nieuwland, G. Y. and B. M. Spee: in M. van Dyke, W. G. Vincenti, and J. V. Wehausen (eds.), "Annual Review of Fluid Mechanics," vol. 5, pp. 119-150, Annual Reviews, Palo Alto, Calif., 1973;AGARD-CP 35, 1968. Stivers, L. S., Jr.: NACA TN 3162, 1954. 66. Oswatitsch, K.: Die Geschwindigkeitsverteilung bei lokalen [Jberschallgebieten an flachen Profilen, Z. Angew. Math. Mech., 30:17-24, 1950; Acta Phys. Aust., 4:228-271, 1950. Crown, J. C.: AIAA J., 6:413-423, 1968. Gullstrand, T. R.: Z. Flugw., 1:38-46, 1953. Hansen, H.:- in K. Oswatitsch and D. Rues (eds.), "Symposium Transsonicurn II," pp. 183-190, Springer, Berlin, 1976; Z. Flugw., 24:340-349, 1976. Norstrud, H.: Z. Flugw., 18:149-157, 1970; Aer. Quart., 24:129-138, 1973. 67. Oswatitsch, K. (ed.): "Symposium Transsonicum I," Springer, Berlin, 1964. Oswatitsch, K. and D. Rues (eds.): "Symposium Transsonicum II," Springer, Berlin, 1976.
68. Oswatitsch, K. and F. Keune: Ein Aquivalenzsatz fur nichtangestellte Fligel kleiner Spannweite in schailnaher Str6mung, Z. Flugw., 3:29-46, 1955; Z. Angew. Math. Phys., 7:40-63, 1956. 69. Pearcey, H. H.: Shock-Induced Separation and Its Prevention by Design and Boundary Layer Control, in G. V. Lachmann (ed.), "Boundary Layer and Flow Control-Its Principles and Application," pp. 1166-1344, Pergamon Press, Oxford; in K. Oswatitsch (ed.), "Symposium Transsonicum I," pp. 264-275, Springer, Berlin, 1964. 70. Pearcey, H. H.: The Aerodynamic Design of Section Shapes for Swept Wings, Adv. Aer. Sci., 3:277-322, 1962. 71. Polhamus, E. C.: Summary of Results Obtained by Transonic-Bump Method on Effects of Plan Form and Thickness on Lift and Drag Characteristics of Wings at Transonic Speeds, NACA TN 3469, 1955. 72. Polhamus, E. C.: Predictions of Vortex-Lift Characteristics by a Leading-Edge Suction Analogy, J. Aircr., 8:193-199, 1971. Lamar, J. E.: J. Aircr.,.13:490-494, 1976. 73. Prandtl, L.: Uber Stromungen, deren Geschwindigkeiten mit der Schaligeschwindigkeit vergleichbar sind, J. Aer. Res. Inst. Tokyo Insp. Uni., 5, no. 65:25-34, 1930; Aerodynamik-Vorlesung Gottingen 1922, "Gesammelte Abhandlungen zur angewandten Mechanik, Hydro- and Aerodynamik," pp. 998-1003, Springer, Berlin, 1961. 74. Prandtl, L.: Allgemeine Betrachtungen fiber die Stromung zusammendruckbarer Fliissigkeiten, Z. Angew. Math. Mech., 16:129-142, 1936; "Gesammelte Abhandlungen zur angewandten Mechanik, Hydro- and Aerodynamik," pp. 1004-1026, Springer, Berlin, 1961; L'Aerotecnica, 16:511-528, 1936; NACA TM 805, 1936. 75. Prandtl, L.: Theorie des Flu- zeugtragfligels im zusammendriickbaren Medium, Lufo., 13:313-319, 1936: "Gesammelte Abhandlungen zur angewandten Mechanik, Hydro- and Aerodynamik," pp. 1027-1045, Springer, Berlin, 1961. 76. Puckett, A. E. and H. J. Stewart: Aerodynamic Performance of Delta Wings at Supersonic Speeds, J. Aer. Sci., 14:567-578, 1947. Beane, B.: J. Aer. Sci., 18:7-20, 1951;
20:138-140, 1953. Brown, C. E.: NACA Rept. 839, 1946. Brown, C. E. and M. S. Adams: NACA Rept. 892, 1948. Haskell, R. N., J. J. Hosek, and W. S. Johnson, Jr.: J. Aer. Sci., 22:274, 278-280, 1955. Kainer, J. H.: J. Aer. Sci., 20:469-476, 749-750, 1953; 22:598-606, 1955. Malvestuto, F. S., K. Margolis, and H. S. Ribner: NACA Rept. 970,
1950. Puckett, A. E.: J. Aer. Sci., 13:475-484, 1946. Roberts, R. C.: J. Math. Phys., 27:49-55, 1948. Zienkiewicz, H. K.:.7. Aer. Sci., 21:421-423, 792, 1954.
322 AERODYNAMICS OF THE WING
77. Robinson, A.: Aerofoil Theory for Swallow Tail Wings of Small Aspect Ratio, Aer. Quart., 4:69-82, 1952; ARC RM 2548, 1946/1952; J. Roy. Aer. Soc., 52:735-752, 1948. 78. Robinson, A. and J. A. Laurmann: Aerofoils in Compressible Flow, in "Wing Theory," pp. 298-480, Cambridge University Press, Cambridge, 1956.
79. Rubbert, P. E. and M. T. Landahl: Solution of the Transonic Airfoil Problem Through Parametric Differentiation, AIAA J., 5:470-479, 1967. 80. Schlichting, H.: Tragfliigeltheorie bei Uberschallgeschwindigkeit, Lufo., 13:320-335, 1936; Z. Angew. Math. Mech., 16:363-365, 1936; NACA TM 897, 1939. 81. Schlichting, H.: Einige neuere Ergebnisse aus der Aerodynamik des Tragflugels, Jb. WGLR, 11-32, 1966. 82. Schneider, W.: Hyperschallstromungen-Entwicklungsrichtungen der Theorie, in "Ubersichtsbeitrage zur Gasdynamik," pp. 163-194, Springer, Wien, 1971.
83. Sears, W. R.: Small. Perturbation Theory, in "General Theory of High. Speed Aerodynamics," Sec. C, Princeton University Press, Princeton, N.J., 1954; in M. van Dyke, "Perturbation Methods in Fluid Mechanics," Academic, New York, 1964. 84. Sinnott, C. S. and J. Osborne: Review and Extension of Transonic Aerofoil Theory, ARC
RM 3156, 1958/1961. Fitzhugh, H. A.: J. Aircr., 7:277-279, 1970. Sinnott, C. S.: J. Aerosp., 26:169-175, 1959; 27:767-778, 1960; 29:275-283, 1962; ARC RM 3045, 1955/1957. Smetana, F. 0. and D. P. Knepper: J. Aircr., 10:124-126, 1973. 85. Spreiter, J. R.: The Local Linearization Method in Transonic Flow Theory, in K. Oswatitsch (ed.), "Symposium Transsonicum I," pp. 152-183, Springer, Berlin, 1964. Hosokawa, I.: J. Aerosp., 28:588-590, 1961; 29:604, 1962; in K. Oswatitsch (ed.), "Symposium Transsonicum I," pp. 184-199, Springer, Berlin, 1964. 86. Spreiter, J. R. and A. Alksne: Theoretical Prediction of Pressure Distributions on Nonlifting Airfoils at High Subsonic Speeds, NACA Rept. 1217, 1955; 1359, 1956. Rotta, J.: Jb. WGL, 102-109, 1959; in K. Oswatitsch (ed.), "Symposium Transsonicum I," pp. 137-151, Springer, Berlin, 1964. 87. Spreiter, J. R. and S. S. Stahara: Developments in Transonic-Flow Theory, Z. Flugw., 18:33-40, 1970; AIAA J., 8:1890-1895, 1970. Spreiter, J. R.: J. Aerosp., 26:465-487, 517, 1959; J. Aer. Sci., 21:70-72, 1954. 88. Stack, J. and A. E. von Doenhoff: Tests of 16 Related Airfoils at High Speeds, NACA Rept. 492, 1934. Amic, J. L.: NACA TN 2174, 1950. Gothert, B.: ZWB Lufo. FB 1490, 1941. Gothert, B. and G. Richter: Jh. Lufo., i:101-110, 1941. 89. Stack, J., W. F. Lindsey, and R. E. Littell: The Compressibility Bubble and the Effect of Compressibility on Pressures and Forces Acting on an Airfoil, NACA Rept. 646, 1938. 90. Stahl, W. and P. A. Mackrodt: Dreikomponenten-Messungen bis zu grossen Anstellwinkeln an fiinf Tragfltigeln mit verschiedenen Umrissformen in Unterschall- und UberschalI-
stromung, Z. Flugw., 11:150-160, 1963; 13:447-453, 1965; Jb. WGLR, 159, 1962. Lipowski, K.: Z. Flugw., 13:453-458, 1965.
91. Stanbrook, A. and L. C. Squire: Possible Types of Flow at Swept Leading Edges, Aer. Quart., 15:72-82, 1964. Kiichemann, D.: J. Roy. Aer. Soc., 57:683-699, 1953. 92. Stewart, H. J.: A Review of Source Superposition and Conical Flow Methods in Supersonic Wing Theory, J. Aer. Sci., 23:507-516, 1956; Quart. App. Math., 4:246-254, 1946.
93. Sun, E. Y.
C.: Vergleich der Behandlung des Dickenproblems eines Deltaflugels mit
Schallvorderkanten mit der Theorie der schallnahen Stromung, Z. Angew. Math. Mech.,
46:T 219-220, 1966; 43:T 172-173, 1963; 48:T 250-251, 1968; J. M c., 3:141-163, 1964.
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WINGS IN COMPRESSIBLE FLOW 323
96. Tsien, H. S.: Two-Dimensional Subsonic Flow of Compressible Fluids, J. Aer. Sci., 6:399-407, 1939. von Karman, T.: "Collected Works," vol. IV, pp. 146-150, Butterworths, London, 1956. Laitone, E. V.: J. Aer. Sci., 18:350, 1951. Norstrud, H.: J. Aircr., 8:123-125, 1971. 97. Tsien, H. S.: The Supersonic Conical Wing of Minimum Drag, J. Aer. Sci., 22:805-817, 843, 1955. Cohen, D.: J. Aer. Sci., 24:67-68, 1957. Germain, P.: Reds. Aer., 7:3-16, 1949. Lance, G. N.: Aer. Quart., 6:149-163, 1955. 98. Tsien, H. S.: Similarity Laws of Hypersonic Flows, J. Math. Ples., 25:247-251, 1946. Hayes, W. D.: Quart. App. Math., 5:105-106, 1947. 99. van Dyke, M. D.: The Second-Order Compressibility Rule for Airfoils, J. Aer. Sci., 21:647-648, 1954; NACA Rept. 1274, 1956. Hayes, W. D.: J. Aer. Sci., 22:284-286, 1955. Imai, I.: J. Aer. Sci., 22:270-271, 1955. 100. von Kirman, T.: The Problem of Resistance in Compressible Fluids, Volta-Kongress Rom, 222-276, 1935; "Collected Works," vol. III, pp. 179-221, Butterworths, London, 1956. 101. von Karman, T.: Compressibility Effects in Aerodynamics, J. Aer. Sci., 8:337-356, 1941; "Collected Works," vol. IV, pp. 127-164, Butterworths, London, 1956. 102. von Karman,.T.: The Supersonic Aerodynamics-Principles and Applications, J. Aer. Sci., 14:373-409, 1947; "Collected Works," vol. IV, pp. 271-326, Butterworths, London, 1956. 103.
von Karman, T.: The Similarity Law of Transonic Flow, J. Math. Phys., 26:182-190, 1947; "Collected Works," vol. IV, pp. 327-335, Butterworths, London, 1956. Guderley,
G.: MOS (A) RT 110, 1946. Malavard, L.: Jb. WGL, 96-103, 1953. Oswatitsch, K.: ARC RM 2715, 1947/1954. Spreiter, J. R.: NACA Rept. 1153, 1953. 104. von Karman, T.: Some Significant Developments in Aerodynamics Since 1946, J. Aerosp.
Sci., 26:129-144, 154, 1959; "Collected Works," vol. V, pp. 235-267, von Karman Institute, Rhode-St. Genese, 1975. 105. Vincenti, W. G.: Comparison Between Theory and Experiment for Wings at Supersonic Speeds, NACA Rept. 1033, 1951. 106. Ward, G. N.: Supersonic Flow Past Thin Wings, Quart. J. Mech. App. Math., 2:136-152, 374-384, 1949. 107. Ward, G. N.: "Linearized Theory of Steady High-Speed Flow," Cambridge University Press, Cambridge, 1955. 108. Weber, J.: Numerical Methods for Calculating the Zero-Lift Wave Drag and the Lift-Dependent Wave Drag of Slender Wings, ARC RM 3221, 1959/1961; 3222, 1959/1961. 109. Wood, C. J.: Transonic Buffeting on Airfoils, J. Roy. Aer. Soc., 64:683-686, 1960.
Redeker, G.: Z. Flugw., 21:345-359, 1973. Thomas, F.: Jb. WGLR, 275, 1965; 126-144, 1966.
110. Yoshihara, H., J. Kainer, and T. Strand: On Optimum Thin Lifting Surfaces at Supersonic Speeds, J. Aerosp. Sci., 25:473-479, 496, 600, 1958. Anliker, M.: Z. Angew. Math. Phys., 10:1-15, 1959. Jones, R. T.: J. Zerosp. Sci., 26:382-383, 1959. Strand, T.: J. Aerosp. Scl, 27:615-619, 1960. 111. Zierep, J.: Theorie and Experiment bei schallnahen Stromungen, in "Ubersichtsbeitrage zur Gasdynamik," pp. 117-162, Springer, Wien, 1971; in K. Oswatitsch (ed.), "Symposium Transsonicum I," pp. 92-109, Springer, Berlin, 1964. Burg, K. and J. Zierep: Act. Mech., 1:93-108,1965.
PART
TWO AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
CHAPTER
FIVE AERODYNAMICS OF THE FUSELAGE
5-1 INTRODUCTION
5-1-1 Geometry of the Fuselage Whereas the main function of the airplane wing is the formation of lift, it is the main function of the fuselage to provide space for the net load (payload). It is required, therefore, that the wing at given lift and the fuselage at given volume have the least possible drag. Consequently, the fuselage has, in general, the geometric shape of a long, spindle-shaped body, of which one dimension (length) is very large in comparison with the other two (height and width). The latter two dimensions are of the same order of magnitude. In Fig. 5-1, a number of idealized fuselage shapes are compared. In general, the plane of symmetry of the fuselage coincides with that
of the airplane. The cross sections of the fuselage in the plane of symmetry and normal to the plane of symmetry (planform) have slender, profilelike shapes. The most important geometric parameters of the fuselage that are of significance for aerodynamic performance will now be discussed.
In analogy to the description of wing geometry, a fuselage-fixed rectangular coordinate system as in Fig. 5-1 will be used, with x axis: fuselage longitudinal axis, positive in rearward direction v axis: fuselage lateral axis, positive toward the right when looking in flight direction z axis: fuselage vertical axis, positive in upward direction 327
328 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
Figure 5-1 Geometric nomenclature for fuselages. (a) General fuselage shape.
Skeleton; angle of attack
Z}
(b), (c)
Fuselage teardrop
with noncircular cross sections. (d)
zF(x)
Fuselage teardrop with circular cross sections (axisymmetric fuselage). (e)
e
Fuselage line.
mean
camber
(skeleton)
In general, it is expedient to place the origin of the coordinates on the fuselage nose. For axisymmetric fuselages, utilization of cylinder coordinates as in Fig. 5-ld is frequently preferable, where r stands for the radius and $ for the polar angle. The main dimensions of the fuselage are the fuselage length 1F, the maximum fuselage width bFinax, and the maximum fuselage height hFmax (Fig. 5-1). The fuselage cross sections in the yz plane are usually oval-shaped (Fig. 5-1 b and c). The simplest case is the fuselage with circular cross sections as in Fig. 5-1d, with bF max - hF max - dF max , where dF max is the maximum fuselage diameter. From these four main dimensions, the following relative quantities can be formed: dFinax
-S
fuse age 1
Finax _ S*
+1,; is kn ess is +'Io
fi1Se age W I'dt h rat 10 1
IF
hFmax
= bF*
fuselage height ratio
_
f use1abe Q cross-sect'ion Idt'io
1F
hFinax bFmax
F
AERODYNAMICS OF THE FUSELAGE 329
The first three quantities are measures of the slenderness or fineness ratio of the fuselage. For the fuselage of circular cross section, 5F = SF = SF* and XF = 1. A more detailed description of fuselage geometry can be given by introducing
the fuselage mean camber line. As shown in Fig. 5-la, this line is defined as the connection of the centers of gravity of the cross-sectional areas AF(x). The line connecting the front and rear endpoints of the skeleton line is designated as the fuselage axis; it should coincide with the x axis. The fuselage skeleton line zF(x) as shown in Fig. 5-le lies in the fuselage symmetry plane. The largest distance of the skeleton line from the fuselage axis is designated as fF.
In analogy to the wing shape, Sec. 2-1, a general fuselage shape as shown in Fig. 5-la can be thought of as being composed of a skeleton line ZF(x) on which cross sections AF(x) are distributed. The body with this cross-section distribution is also termed a fuselage teardrop. In the case of noncircular cross sections of the fuselage, fuselage teardrops are characterized by the distributions hF(x) and bF(x) as in Fig. 5-lb and c. In the case of circular fuselage cross sections, the fuselage teardrop is determined uniquely by the distribution of the radii R(x) (Fig. 5-1d). The geometric parameters of a wing (teardrop and skeleton) can be selected first for the required aerodynamic performance. For fuselages this procedure is possible only to a very limited degree, because the fuselages must satisfy important requirements that may not be compatible with the aerodynamic considerations. For theoretical investigations on the aerodynamic properties of fuselages, the profile teardrops discussed in Sec. 2-1 are well suited.
The ellipsoid of revolution of Fig. 5-2a is a simple fuselage configuration for subsonic velocities. Another simple fuselage of axial symmetry that is used particularly for supersonic flight velocities is the paraboloid of revolution with a pointed nose as shown in Fig. 5-2b.* To accommodate jet engines, fuselage configurations with blunt tails may be chosen. Among the design parameters not only fuselage length and diameter play an important role, but also fuselage volume and surface area. Volume and surface area of axisymmetric fuselages are given by IF
VF =
JR2(x) dx
(5-1a)
0
*The axis of rotation is parallel to the tangent at the vertex.
IF
=i
'Fo
I
Figuae 5-2 Special axisymmetric fuselages. (a) Ellipsoid of revolution. (b) Paraboloid of revolution.
330 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
lF
SF = 27r f R(x) ds
(5-1 b)
0
where s is the path length along the fuselage contour and ll is the associated length of a meridional section measured on the fuselage contour. Finally, a few data are given here for the volume of the ellipsoid of rotation and the paraboloid of rotation (1F = lFo) of Fig. 5-2, respectively: VF =
a3lFAFinax
VF = is1FAFinax
(ellipsoid)
(5-2a)
(paraboloid)
(5-2b)
Here, 1F is the fuselage length and AFinax is the maximum fuselage cross-sectional area, also called the frontal area.
5-1-2 Forces and Moments on the Fuselage The following sections will be devoted to a detailed discussion of fuselage aerodynamics. To give a feeling for the magnitudes of the forces and moments acting on the fuselage, a typical measurement on a fuselage will be presented first. In Fig. 5-3, some results of a three-component measurement on an axisymmetric fuselage by Truckenbrodt and Gersten [50] are plotted. Here, the following dimensionless coefficients have been introduced for the components of the resultant force (lift and drag) and for the pitching moment: Lift:
LF = CLF VF 3 q00
Drag:
DF = CDF VF
Pitching moment:
MF=cmFVFq
11 3
q.
(5-3)*
where q _ (9/2) U! is the dynamic pressure of the incident-flow velocity U. and VF is the fuselage volume. Figure 5-3 shows the lift coefficient cLF, the drag coefficient cDF, and the pitching-moment coefficient cMF plotted against the angle
of attack a. The position of the axis of reference for the pitching moment is indicated in Fig. 5-3. In the range near a = 0, the lift coefficient changes linearly with angle of attack a. At larger angles of attack, CLF grows more than linearly. This lift characteristic CL(a) is very similar to that of a wing of very small aspect ratio (see Fig. 3-49). The drag coefficient CDF is approximately proportional to the square of the angle of attack, similar to that of the wing. In the range of large angles of attack, the pitching-moment coefficient depends almost linearly on the angle of attack. Forces and moments, in addition to those discussed above, act on the fuselage *Fusela?e volume is introduced in this case as a quantity of reference in compliance with the theory of fuselages (see Sec. 5-2-3). The drag coefficient is frequently referred to the surface SF or the frontal area AFinax of the fuselage.
AERODYNAMICS OF THE FUSELAGE 331 0.6
00
0.2
0
-0.2
-0.4
-0.6
-6°
00
6°
f2°
2/f0
19°
-*a
Moment reference point
.90°
Figure
5-3 Three-component
mea-
surements CLF, cDF, and cMF vs. angle of attack on an axisymmetric fuselage. Reynolds number Re = 3 106. Theory (5-34).
for cMF from Eq.
as a result of the turning and sideslip motions of the airplane, as has been discussed for the wing in Sec. 3-5. The summary reports of Munk [41] , Wieselsberger [58], Goldstein [141,
Thwaites [47], Howarth [22], Heaslet and Lornax [17], Brown [5], Ashley and Landahl (4], Hess and Smith [181, and Krasnov [28] deal with the questions of flow over a fuselage in incompressible, and, to some extent also in compressible flow. Also, the survey of Adams and Sears [1 ] must be mentioned. Furthermore, the comprehensive compilations of experimental data on the aerodynamics of drag and lift of fuselages of Hoerner [19] and Hoerner and Borst [20] should be pointed out.
5-2 THE FUSELAGE IN INCOMPRESSIBLE FLOW
5-2-1 General Remarks Now that some experimental results have been given. the theory of flow over fuselages will be presented. Fuselage theory can be established, similar to profile theory, by two different approaches.
The first approach consists of the establishment of exact solutions of the three-dimensional potential equation, which can be done successfully in only a few cases. The second approach is the so-called method of singularities, in which the flow pattern about the fuselage is formed by arranging sources, sinks and, if necessary,
332 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
dipoles on the fuselage axis. This procedure is fairly simple for bodies of revolution (see von Karman [54] and Keune and Burg [26] ). An extention of this method for the computation of the flow over fuselages consists of arranging ring-shaped source distributions on the body surface (see Lotz [34], Riegels [32], and Hess [18] ). By
this method, body shapes can be treated whose cross sections deviate somewhat from circles.
First, the fuselage in axial flow will be discussed, then the fuselage in oblique flow.
5-2-2 The Fuselage in Axial Flow Pressure distribution by the method of source-sink distribution The method of source-sink distribution for bodies of revolution in axial flow was first presented in detail by Fuhrmann [13). The flow over such a body can be represented, as in Fig.
5-4, by a distribution q(x) of three-dimensional sources on the body axis that is superimposed by a translational flow U.. Compare the discussions of the plane problem (profile teardrop) of Sec. 2-4-3. The connection between the source distribution q(x) and the fuselage contour R(x) can be established intuitively through application of the continuity equation to the volume element ABCD of Fig. 5-4: (U""
-{-u)nR2-}-gdx=(U.. +u+dx
dx)v(R-{-RdX2
Hence, it follows the source distribution (5-4a)
dx [(U.+u)R2]
q(x)
_
(R2) 40!L = U`'° dAF 0O dx
U00
dx
Figure 5-4 Fuselage theory at axial flow. q (x) = source-sink distribution.
(54b)
AERODYNAMICS OF THE FUSELAGE 333
Except for the vicinity of the stagnation point, u << U. for slender bodies, to which the second relationship applies. The quantity AF(x) = rrR2 (x) is the local fuselage cross section. The closure condition for a closed fuselage contour [see Eq. (2-92)] is automatically satisfied by the expression for the source distribution if AR = 0 at the nose and at the tail. For the induced velocity components in the axial and radial directions,
u(x , r) =
IF q( x)
1 4:s
(x - x) dx
( x - x')2 ±
(5-5a)
r23
0
(x , r) W,
=
r 4 7v
IF
f
q(x') dx' (x - x')2 4 r2
(5-5b)
3
0
To determine the velocity distribution on the surface of slender fuselages, the values of the induced velocities u and w, for small values of r are needed. Special caution is re-
quired in establishing these values, because on the fuselage axis even the induced velocities are singular. Expansion of u and w, for small values of r leads, under the assumption that dq/dr is steady in the vicinity of the point x' = x, to the following expressions:
it(x,r 0)_-lim 2 (1 -In4 n Ego
2s dq(x) r ) dX
w,.(x,r-a 0 ) =
-{-
x-e q(x') dx'
r (x - z')
,J
0
IF
dx
z+e
q(x)
1
(5-6a)
(x - x')2
(5-6b)*
2n r
lim(rwr )=Uc R dR
(5-6c)
dx
2-4
Equation (5-6c) is obtained by introducing Eq. (54b) into Eq. (5-6b). These equations show that the two components of the induced velocity become infinitely large on the fuselage axis (r = 0). This constitutes a basic difference to the plane case (profile theory); see Eq. (2-91). The radial velocity component is given, from Eq. (5-6b), by the limiting value (boundary condition) on the fuselage axis. Hence,
the induced velocities on the surface of the slender body [r = R(x)] are finally obtained from Eqs. (5-6a) and (5-6b) and by introducing the expression Eq. (5-4b) for the source distribution q(x) as U, (x)
1
Uro
4 Ego
EM 2 1 - In 1
1
(
d2(R2) d x2
x-e
2E
d2
IF1
d(R2)
(R=)
dx2
IF d(R-)
dx'
-i
dx'
J dx' (x - x')2 x- +Je dx' (x - x')2
0
In R (x)
(5-7a)-
IF
`The validity of Eq. (5-6b) is immediately established from the continuity equation.
tAt stations of the fuselage contour at which the curvature d ZR/dx2 is unsteady, it must be set: da (R2)
1
2 C(
d2 (R2)
dx2 )1_0
d2 (R2)
1
( dx2 )x+oJ
334 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
_
IV,(x)
d R (x)
(5-7b)
dx
U00
The latter equation is equivalent to the kinematic flow condition, which states that the direction of the velocity vector on the surface is tangent to the surface. Pressure distribution From the Bernoulli equation, the pressure distribution on the surface of the body is obtained as Cp =
1'
4
(WC)2
°D = i
[2
(.J±_)2+ (wr)2]
U
where WC = (U. + u)2 + w2 is the velocity on the fuselage contour. As in wing profile theory, the quadratic terms of the induced velocities may be disregarded. Thus, the first approximation of the equation becomes
cp(x) = -2 Ux)
(first approximation)
(5-8)
U.
A more accurate formula for the pressure distribution is obtained by retaining the term wY because, by Eq. (5-7b), w,, is proportional to the slope of the contour dR/dx, the influence of which is thus taken into account more effectively. A second approximation is thus obtained as cp(x)
2
u(x) Um
- rdR(x)12 dx
(second approximation)
(5-9)
Equations (5-8) and (5-9), together with Eq. (5-7a), allow the determination of the relationship between the pressure coefficient and the fuselage thickness ratio SF = dFinaxllF. This relationship is found as cp(x) = Lf(x) + g(x) ln5F] 82F w ith
1
g (x)
IF'
R21W,.,
(5-10a)
d2(R2)
(5 - 10b)
dx2
where the functions f(x) and g(x) depend only on the fuselage form but not the thickness ratio.
Examples A few examples of this method of source-sink distribution will now be discussed. The induced velocity u(x) of an ellipsoid of revolution of thickness ratio 5F = dFinax/lF is obtained from Eq. (5-7a) with X = x/lF as I
U
U
14X
+ In
SFJ S2 2
F
(5-11 a)
The pressure distribution of an ellipsoid of revolution of thickness ratio SF = 0.1 is given in Fig. 5-5. Both the first approximation from Eq. (5-8) and the second approximation from Eq. (5-9) are shown. For comparison, the exact solution is given, and will be discussed in the next section. The second approximation agrees
AERODYNAMICS OF THE FUSELAGE 335 -0.0 Poraboioi01111
2
-0.06
\ -0.04
/
3-
\\
E//rp soid (Il
0 I 0.02 ,
{
0.04 0
0.2
0.6
0, 4
0.8
1.0
IF
U. L IM 01
-a
- - -- --- - II
Figure 5-5 Pressure distribution on bodies of r evo luti o n (ellipso id , para b o l o id) at i n c ompres sible axial flow. Fuselage thickness ratio 6F =
0.1. (1) Exact solution from Eq. (5-14) or Lessing. (2) Second approximation, Eq. (5-9). (3) First approximation, Eq. (5-8).
well with the exact solution over the entire contour. The first approximation deviates at the front and rear portions.
For the maximum perturbation velocity at the ellipsoid of revolution that occurs at station X = 2, Eq. (5-1 la) yields
Ua'` _ - (1
-}- In
f) SF
(ellipsoid)
(5-11 b)
This value is plotted against the fuselage thickness ratio in Fig. 5-6. Here, too, the exact solution is shown for comparison. At large thickness ratios the values of the 0z
Ellipse profile
Paraba/oid
/
0.1
02
03
SF
Figure 5-6 Maximum perturbation velocity of bodies of revolution in axial flow vs. thickness ratio SF. (1) Exact solution from Eq. (5-15) or Lessing, respectively. (2) Approximation from Eq. (5-11b) or (5-12b), respectively.
336 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
exact solution are larger than those of the approximation solution of the source-sink
method with the source distribution on the axis. Also included in Fig. 5-6 is the perturbation velocity for the plane problem of the elliptic profile as in Fig. 2-34. In this case umax/U = 5 (= SF). The comparison of the curves for the elliptic profile and the ellipsoid demonstrates by how much the maximum perturbation velocity at the body of revolution is smaller than that at the wing profile of the same thickness ratio.
For the induced velocity of a paraboloid of rotation [see Eqs. (2-6) and (2-7a)] , Eq. (5-7a) yields u(X)
Uro
= 2 [1 - 6X(1 - X)] [3 + -- InX (1 - X) -f- 2ln5F] 62V
(5-12a)
The corresponding pressure distribution (second approximation) for 5F7=0.1 is shown as curve 2 of Fig. 5-5. The maximum perturbation velocity, lying again at X = 1, is obtained from Eq. (5-12a) as umaX
(3 -E- 2 InsF) SF
Uc
(5-12b)
(paraboloid)
This value is represented by curve 2 of Fig. 5-6.
The computations discussed so far are based on source distributions on the fuselage axis. Results of Lessing [32] for distributions of source rings on the body surface are included in Figs. 5-5 and 5-6 as curves 1. These results can be considered
to be "exact." The considerable improvement of the theory based on surface distribution over that based on axial distribution is obvious in Fig. 5-6. The pressure distribution for a body of revolution, composed of a half-ellipsoid
of revolution and a matching infinitely long cylinder, is given in Fig. 5-7. For evaluation of Eq. (5-7a) at the station of the curvature discontinuity, x/lp = 2 , the
-001
0
of
02
0.3
Eliip roid-
04
Os
0.5
0,7
98
0.9
10
1,1
Cylinder
Figure 5-7 Pressure distribution on an axisymmetric half-body (dFinax/lF = 0.1) in axial flow (source distribution on the axis).
AERODYNAMICS OF THE FUSELAGE 337
-0oe1
-a.06
05
X=--i
041
U
r-f/ip,roid
06
to
0,7
-Cylinder
H
0,321 IF
04'O'-IF.
IF
Figure 5-8 Pressure distribution on a body of revolution with cylindrical center section. 6F = dFinax/lF = 0.09 (source distribution on the axis).
footnote to this equation must be observed. In this way, the specifically marked value of cn is obtained.* Finally, Fig. 5-8 shows the pressure distribution of a body of revolution composed of a frontal half-ellipsoid of revolution, a rear half-paraboloid of revolution, and a matching cylindrical center section. For the marked points at the stations of curvature discontinuity, the comment that was made for Fig. 5-7 applies.
A body of revolution of the airship kind has been studied particularly by Fuhrmann [13]. The flow pattern produced by a slender body of revolution (so-called streamlined body) is illustrated in Fig. 5-9. Its generating source-sink distribution is indicated in the upper picture. The theoretical pressure distribution is in excellent agreement with measurements.
Exact solutions A few more data will be given on the exact solutions for fuselages in axial incident flow. Such exact solutions of the spatial potential equation can be found in closed form for a few cases only. The general ellipsoid in axial incident flow was first investigated by Tucker-
mann [36] and Zahm [36] and later, more explicitly, by Maruhn [36]. The pressure distribution on the surface of the ellipsoid, Fig. 5-10, in incident flow parallel to the x axis is given in [36] as
(r)2
]-L
-
[(__)2sin2 t9 - ()'cos2]Jo
`The dashed curves of Fig. 5-7 are the results of Eq. (5-7a) without consideration of the footnote given there.
338 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
a
U.
U_
uo
R4
-o- Measurement
a8 b8
----- Theory
R
J
I
0
i
PaW 12
Re= UHF =1.3.106 V
nn
0
ai
b
U2
0
Of
!25
x/IF-
06
07 M
49
to
Figure 5-9 Streamline pattern and pressure distribution of a body of revolution in axial flow (from Fuhrrnann, body III). (a) Streamline pattern. (b) Pressure distribution on the body surface.
where a, b, and c are the semiaxes of the ellipsoid. The origin of the coordinates
lies at the center of the ellipsoid. The quantity A is a function of the two axis ratios a/c and b/c; it is presented in Fig. 5-11, from (36]. The special case of an ellipsoid of revolution in axial incident flow is obtained from Eq. (5-13) for b = c as
c -I-A2. p
-
1L1
!Z
-2d-
U
)
(5-14) \a
AZ
_2b
Figure 5-10 Geometry of a general ellipsoid.
AERODYNAMICS OF THE FUSELAGE 339
C =Z 1,
125
120
110
105
969
Figure 5-11 Coefficient A for the determination of the pressure distribution on a general
100
0
as
2.5
)0
ellipsoid in
axial incident flow, from Eq.
(5-13), vs. the two axis ratios a/c and b/c.
Here, bla = 5F is the thickness ratio of the body of revolution. The evaluation of Eq. (5-14) is included in Fig. 5-5 as the exact solution. The pressure minimum cpmin = 1 -A2 lies at x = 0. Hence, the maximum perturbation velocity becomes
UmaxA
(5-15)
U
where z
A=
2 2
ao
with ao =
2 SF (tanh-1
y1 - cSF
a
/1
- -1aF
SF)
(5-16)
The relation between umax/Ue and 5F is shown in Fig. 5-6 as the exact solution for the ellipsoid of revolution. For small values of 5F, the three equations above yield the relationship Eq. (5-11b) that was derived by means of the singularities method.
Effect of viscosity So far in this chapter, the fluid has been assumed to be inviscid and incompressible. The effect of compressibility on the aerodynamic properties of a fuselage will be treated in the following sections. First, a few data will be given on the effect of viscosity in incompressible flow (effect of Reynolds number). At
moderately large Reynolds numbers (Re > 105), the effect of viscosity on the pressure distribution on bodies of revolution in axial incident flow is quite small. This can be seen, for instance, from Fig. 5-9, in which the pressure distribution computed for inviscid flow is compared with measurements. The slight deviations of
the pressure distribution as obtained from potential theory from that found in viscous flow is responsible for the pressure drag of the body of revolution. In addition there is a friction drag, which is produced by the wall shear stress. The effect of friction on the flow about fuselages is determined from
340 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
boundary-layer computations, quite similar to the case of wing profiles. In the latter case the boundary layers are two-dimensional, whereas in the case of fuselages with circular cross sections in axial incident flow, the boundary layers are axisymmetric.
The computational procedures for the latter are very similar to those for the two-dimensional boundary layers, both laminar and turbulent. These boundary-layer computations for a given body produce distributions of boundary-layer thicknesses (momentum thickness and displacement thickness) and
of a form parameter of the boundary-layer profiles over the contour. They determine drag and position of the separation point. Young [59] and Scholz [59] conducted comprehensive computations of the drag of bodies of revolution. They
found that the contribution of the wall shear stress to the drag of bodies of revolution is, in general, equal to that of the flat plate in parallel incident flow of equal surface area and equal Reynolds number with reference to the body length.
For fully turbulent flow, the body drag due to friction DFf may be obtained approximately from the flat plate drag Dp from the formula DFf = Dp(l + CSF)
(5-17)
with c 0.5. Here Dp is the drag of the flat plate in parallel incident flow of the same surface area SF and the same length IF as those of the body of revolution. Hence, Dp = CfSFgo,, where the coefficient c f for smooth surfaces can be taken from Fig. 2-48. Further data on fuselage drag are found in Hoerner [19].
5-2-3 The Fuselage in Asymmetric Incident Flow General remarks Now the asymmetric inviscid flow about an inclined fuselage as in Fig. 5-12 will be considered. First, it is important to state that, in inviscid flow, only a moment, not a resultant force, is acting on the -inclined fuselage. This is caused by the
underpressures on the upper side of the body nose and the lower side of the tail and, conversely, the overpressures on the lower side of the nose and the upper side of the tail. This pressure distribution results in a moment MF that attempts to turn the fuselage nose up (unstable moment). At small angles of attack a, this moment is proportional to the angle of attack. The fuselage-wing interaction changes the magnitude of this moment greatly (see Chap. 6). However, the moment of the fuselage alone will be treated here, first in inviscid flow and later with consideration of friction. It should be mentioned that
Figure 5-12 Inviscid flow about an inclined fuselage.
AERODYNAMICS OF THE FUSELAGE 341
01
0.2
03
0.41
0.5
aC
-
0.5
0.7
0.8
0,9
10
Figure 5-13 Coefficient k for the computation of the moment of an inclined general ellipsoid of Eq. (5-18b), from Vandrey.
the effect of friction on the aerodynamic properties of the fuselage is considerable. The moment in inviscid flow can be obtained from simple momentum considerations. Computation of the pressure distribution on the fuselage surface requires application of potential theory. As in the case of the fuselage in axial incident flow, exact solutions and approximate solutions to the singularities method are known. Finally, the effect of friction can be determined with the help of boundary-layer theory.
Fuselage moment by the momentum method of Munk An early account of the computation of the moment of an inclined fuselage was given by Munk [41]. It is based on an application of the momentum law. The momentum far behind a body moving at constant velocity in an inviscid fluid remains unchanged and no resultant
force acts on the body, but this does not exclude the existence of a free force couple. According to the Munk theory, lift and pitching moment (free force couple) of a fuselage at an angle of incidence a and at free stream velocity U. are
LF = 0
(5-18a)
MF = 2kga, VFa
(5-18b)
Here, q,o = (o/2)UU is the dynamic pressure of the incident flow, VF is the body volume, and k is a factor describing the ratio of the volume of the fluid quantity moving with the body to the body volume. Values of k for general ellipsoids have been given by Zahm '[36] and presented graphically by Vandrey [40] The coefficients k for general ellipsoids of volume VF = s 7rabc are given in Fig. 5-13 as a function of the axis ratios c/a and b/c. Accordingly, the coefficient k for slender ellipsoids of revolution (b = c and c/a < 0.2) differs little from unity. Thus, from .
Eq. (5-18b), the moment of slender bodies of revolution is obtained from the simple approximation formula
342 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
Mp=2q. VFa
(5-19)
Note that the unstable moment of the aerodynamic forces acting on an inclined slender fuselage of revolution is proportional to the angle of attack a and the body volume VF [Eq. (5-1a)].
Pressure distribution by the method of dipole distribution The flow field of an inclined body of revolution can be computed by the singularities method. In the simplest approach, a distribution of spatial dipoles as in Fig. 5-14 is arranged on the body axis.* The axes of the dipoles are parallel to the z axis. The potential of the dipole distribution is IF
m(x') dx' i' (x - x')2 ± r23
r cos
4n
J
t/
0
cos 6 m (x) 2n r
(5-20a)
(5-20b)t
where m(x) stands for the dipole strength. The second relationship results from the expansion of the potential for small distances r from the axis, as required for slender bodies. The velocity components in axial, radial, and circumferential directions, respectively, are obtained from Eq. (5-20b) as
a0
cos l drn(x) 2n r dx 1
8x
U',,,=
ao
1
ar
2n
1
a4)
?1
e6
=
cos i r1
sin 6
2n r"
(5-21a) (5 - 21 b)
(x)
()
912 (X)
( 5 - 21 c)
The dipole strength is determined from the kinematic flow condition, which demands that, on the body, the velocity component normal to the surface is zero. *For asymmetric incident flow, the method of source distributions on the surface with nonaxisymmetric distribution has been successful.
Note that this expression for the potential of a very long body of revolution is identical to the potential of the dipole distribution of a circular cylinder.
Figure 5-14 Illustration for the theory of a fuselage of revolution at asymmetric incident flow.
AERODYNAMICS OF THE FUSELAGE 343
z
zf
Teardrop R(r)
Skeleton line zp(x)
a/z) t zF ; fuselage axis .cc(,-)
U" =a(.)-cos 0'
Figure 5-15 Illustration for the theory of a cambered fuselage with angle of attack.
For a body with a cambered skeleton line as in Fig. 5-15, which is a generalization of Fig. 5-14, the kinematic flow condition becomes* a (x) U,,,, cos 3 + wr (X) = 0
for
r=R
(5-22)
Here a(x) is the local angle of attack of the skeleton line referred to the incident flow direction of V. as given by a(x)
=a- dzF(x) dx
(5-23)
where a is the angle of attack of the fuselage axis and zF(x) is the skeleton line of the fuselage. Introduction of wr from Eq. (5-21b) into Eq. (5-22) yields, for the dipole distribution, m (x) = 2n U,,. a (x) R2 (x) = 2 U,,. a (x) AF(x)
(5-24)
where AF(x) is the cross-sectional area of the fuselage.
Pressure distribution The inclination of the fuselage causes a pressure distribution on the body surface that, from Eq. (5-8), is given in first approximation as cp(x, 6) = -2u(x, 6)1U.. Introducing Eq. (5-24) into Eq. (5-21a) yields cP (x, t) = - 2
cos z$
d
B (x) d x [x (x) R2 (x))
(5-25a)
If the angle of attack is constant along the fuselage axis, this equation takes the simpler form cp(x, 6) = -4a cos t5
d dxx)
fu(x) = const]
(5-25b)
Cc
An example of these pressure distributions is given in :gig. 5-16 by means of ellipsoids of revolution of thickness ratios 5F = dFinaxllF = 0.1 and 0.2 and angle "Here, the dipole distribution can be left on the body axis, as in the case of the plane skeleton theory (see Sec. 2-4-2).
344 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
Ex act 1.0
5 F=01 05
.- -1.0
19, 1
1
Exact -1.5
-20 02
OB
1.0
Figure 5-16 Pressure distribution result-
ing from asymmetric incident flow on ellipsoids of revolution of thickness ratios
5F = 0.1 and 0.2 from Eq. (5-26). Exact solution of Eq. (5-33).
of attack a = const. The following expression for the pressure distribution is easily found :
cP_-2acost$ 1-2X
SF
(5-26)
YJ_
Lift distribution The lift distribution is obtained from the pressure distribution by integration. A fuselage portion of length dx is supported by the lift force dLF of magnitude In
dLF
P (x) d x f cP cos 0 d t$
(5-27)
0
Observing Eq. (5-25a) and integrating over t3 yields the lift distribution, dLF
= 2zq. dx [x (x) R2 (x)]
(5-28)
This relationship has been derived by Multhopp [40] from momentum considerations.
AERODYNAMICS OF THE FUSELAGE 345
Equation (5-28) shows directly that the total lift force of a closed body vanishes, because 1
LF =
dx = 27rq. [u(x)R (x)] 1 fF dLF dx Z
(5-29a)
0
if R(x) = 0 at the nose and tail of the body [see Eq. (5-18a)]. Pitching moment The pitching moment of the fuselage at constant angle of attack a(x) = const is obtained from Eq. (5-28) through integration by parts as IF
MF = - f
IF
dd Fx dx = 27rq.a 1 R2(x) dx = 2q., VFa
(5-29b)
0
0
where VF is the fuselage volume from Eq. (5-la). In this way, the Munk approximation formula for the moment of slender bodies of revolution has also been obtained by means of the singularities method. Because LF = 0, the fuselage moment is independent of the location of the reference axis. It is a so-called free moment.
As an example, Fig. 5-17 illustrates, for theory and experiment, the lift distribution from [16] of an inclined ellipsoid of revolution of thickness ratio The theoretical lift distribution is obtained from Eq. (5-28) with 6F=-!.
a(x) = const as dLF
dh = 2nq (1 - 2X) dFinax«
(5-30)
included in Fig. 5-17 as the solid line (line 1). The measurements agree well with theory in the front portion of the fuselage, but some This approximation is
deviations are found for the rear portion. For comparison, see Sec. 5-2-2. The above. discussions apply to bodies of revolution. To determine the moments
of bodies of noncircular cross sections at constant angle of attack, it should be realized that, essentially, only the fuselage width distribution bF(x) determines the
20
Figure 5-17 Lift distribution of an ellipsoid of revolution of thickness ratio 6F = -',. (1) Approximation theory from Eq. (5-30). (2) Exact theory (inviscid). (3) Theory with friction, from Hafer [16].
346 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
moment caused by the inclination. Equation (5-29b) can therefore be applied to fuselages of noncircular cross sections by substituting DF/2 for R and introducing a correction factor k*. This leads to
MF = 2k*q VFa
(5-31a)
IF
with
VF
= 4 r bF(x) dx
(5-31 b)
0
Here VF is the volume of a body of revolution that has the body width for its diameter. The correction factor can be determined by comparing Eq. (5-31a) with the exact equation Eq. (5-18b) for general ellipsoids. Because VF = (b/c)VF, we have k* = kc/b, where k is given in Fig. 5-13. The values of k* thus computed are presented in Fig. 5-18 as functions of fuselage width ratio 5* = bFinax/lF and the cross-section ratio of the fuselage XF = hFinax/bFinax (see Fig. 5-1). It follows, therefore, that the factor k* is almost unity for slender fuselages of all practical cross-section ratios
AF. Thus, the above discussion has shown that for the
computation of the moment of slender fuselages of noncircular cross sections, Eq.
(5-29b) may be used in good approximation if the radius R is replaced by the semiwidth bF/2.
The moment of the fuselage of variable angle of attack a(x) is obtained by using the semiwidth bF/2 in Eq. (5-28) instead of R. Hence, integration over the fuselage length yields for the pitching moment, from Eq. (5-29b), IF
MF = q. 2 f a(x)bF(x) dx
(5-32)
0
This equation is applicable to the fuselage with cambered skeleton line from Eq. (5-23) and to fuselages in curved flow as encountered during rotation about the lateral axis [see Eq. (3-147)]. Furthermore, this relationship is important for the 10
AF-Z
0.8
-rI
3
J
02 I
t
I
0
Q1
0,2
2
02
0,¢
Figure 5-18 Coefficient k* for the computation fuselage of noncircular cross sections, from Eq. (5-31a).
0 of the moment of an inclined
AERODYNAMICS OF THE FUSELAGE 347
computation of the fuselage moment when a wing is attached to the fuselage (see Chap. 6).
The above considerations on the lift distribution and on the moment furnish, accordingly, the side force distribution and the yawing moment due to sideslip for a yawed fuselage.
Exact solutions A few data will now be given on the exact solutions for inclined ellipsoids of revolution. Maruhn [36] determined the pressure distribution of the inclined ellipsoid of revolution at small angles of attack as
eP = (cp)a-o ; 2 B
a
b b a,
1 - I - COs ? N
(5-33)
1 - C1 - (a)-] (a )-
Here, bla = 5F is the fuselage thickness ratio. The quantity B is defined as
B4 _ a2 8
2
where ao is given by Eq. (5-16). The angle-of-attack-dependent pressure distribution
for this exact solution is shown in Fig. 5-16 for 6F = 0.2. In the vicinity of the nose and the tail, the exact solution gives somewhat smaller values of the pressure
coefficient than the approximate solution by the method of singularities. This means that the correction factor k* for the moment in Eq. (5-31a) is somewhat smaller than unity. In Fig. 5-17, too, the exact solution for the lift distribution is included as curve 2. Near the nose and the tail, the exact solution deviates somewhat from the approximation solution. In the vicinity of the nose, the measurements agree quite well with the exact solution. Larger differences remain, however, near the tail. They are caused by viscosity effects to be treated in the next section.
The values of the moment from the exact solution have already been given in Eq. (5-1 gb). From Eq. (5-31a), the theoretical moment coefficient cMF = AfFlg.. VF is obtained as cMF = 2k*a
(5-34)
This theoretical value, with k* = 0.95, is compared in Fig. 5-3 with a measurement. The moment slope dcMFlda from this theory is considerably steeper than that of the measurement. This difference is caused by viscosity effects. Viscosity effects are also responsible for the deviation of the measured lift from zero, as seen in Fig. 5-3.
Viscosity effects Qualitatively, viscosity affects the flow over the inclined fuselage (Fig. 5-12) in such a way that the pressure on the tail section is reduced, because the inviscid outer flow is forced outward by the boundary layer. Consequently, the negative lift of the tail section is somewhat smaller than the positive lift of the nose section. Overall, therefore, viscosity effects cause a positive lift, which is also termed friction lift. This fact may be seen in Fig. 5-17 for the lift distribution. The friction
348 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
lift changes the moment; specifically, it creates an additive nose-down moment with reference to the lateral axis through the fuselage center. Hafer [16] has described a
method for the approximate computation of the viscosity effect by means of boundary-layer theory. Accordingly, the boundary-layer displacement thickness 51 is determined along the fuselage surface at axial incident flow. A rather strong growth
of the boundary-layer thickness near the tail is found, as sketched in Fig. 5-19. Consequently, to compute the pressure distribution, the local fuselage radius R(x) in Eq. (5-25a) must be replaced by the radius [R(x) + 51(x)] , that is, the radius R(x) enlarged by the displacement thickness 51. The lift distribution is obtained from the
pressure distribution corrected for viscosity by integration. In Fig. 5-17, the lift distribution, computed in this way from [16], is also shown. Through the correction for viscosity, better agreement is reached with the measurements, particularly in the vicinity of the tail. Lift, pitching moment, and neutral-point position are determined through further integrations. In Fig. 5-20, the lift slope dcLFlda, the moment slope dcMFlda, and the neutral-point position xNF/IF are plotted against the inverse fuselage thickness ratio 1F/dFinax for several axisymmetric
fuselages. These measurements were taken by Truckenbrodt and Gersten [501. Curves 1 are those from the inviscid theory, curves 2 from the viscous theory of Hafer [161. The latter theory agrees quite well with measurements.
5-3 THE FUSELAGE IN COMPRESSIBLE FLOW
5-3-1 Similarity Rules for Fuselage Theory of Compressible Flow Velocity potential (linearization) For slender fuselages under a small angle of incidence, the magnitude and direction of the local velocities are only a little different from the velocity of the incident flow U,,. It is expedient, therefore, to split up the total flow into a basic, undisturbed flow and a superimposed perturbation flow:
U = U" + it
w o = Wo
W,. = u'r
(5-35)
where u, w,., w6 are the perturbation velocities with it 4 U00
Wr < U00
iao < Um
IF
Figure 5-19 Viscosity effect on the flow about fuselages. S, (x) = displacement thickness of the boundary layer.
AERODYNAMICS OF THE FUSELAGE 349
a5
2,5
04
20
k
.3 3
0
Z
02
X
0-7 a1
05
D
0
6
a
to
a
72
6
1k
F
B
b
dFmax
12
10
14
IF dFinax
c
dFinax
Figure 5-20 Effect of viscosity on the aerodynamic coefficients of inclined axisymmetric fuselages. Position of the moment reference point x0 is different for all fuselages: 0.251F < x,, <0,51,F. (I) Theory without friction. (II) Theory with friction, from Hafer. (a) Lift 2/ 3 slope, cLF = LF/q . VF . (b) Pitching-moment slope cMF = (c) Neutral-point
position xNF
By retaining only the largest terms (linearization), the potential equation of compressible flow becomes, in analogy to Eq. (4-5),
(1 - Mat) a2 + 0X'-
020 ar2
+
1
ao
+
r ar
1 a20 r2
a,&2
=0
(5-36)
Here Ma = U/a is the local Mach number. Equation (5-36) applies to subsonic, transonic, and supersonic flows. The components of the perturbation velocity become
ao
zc= ax
Wr= ar
Iva= ir ao
(5-37)
1?6
The relationship between the local Mach number Ma and the Mach number of the incident flow Maw, = U. /a. is given by Eq. (4-7). For purely subsonic and purely supersonic flows, Ma can be replaced approximately by Ma,. Hence, the following linear differential equation for the potential is obtained in analogy to Eq. (4-8):
(i-1Vda200
)a2
ax-
cr2
r ar
t °-
rz a79-
= 0 (Maw
1)
(5-38)
In analogy to Eq. (4-9), the equation for transonic flow becomes y
1 a0 a2(
a-(
ax ax2
er2
U,-,
1
ao ar
+ 1- a20
a.8=
= 0 (Max = 1)
(5-39)
Contrary to Eq. (5-38), this differential equation for the potential is nonlinear. in analogy to the case of the wing of finite span, the potential equations derived above, Eqs. (5-38) and (5-39), will now be applied to the development of similarity rules for subsonic, transonic, and supersonic flows.
350 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
Subsonic and supersonic similarity rules The similarity rules for subsonic and supersonic flows are obtained through a transformation of the potential equation [Eq. (5-38)]. To this end, the given compressible flow is transformed into a flow, the potential equation of which no longer contains the Mach number. This transformation is accomplished, in analogy to Eq. (4-10), by setting
x' = x
r' = clr
0 = c20'
aV _
U' = U00
(5-40)
where the primes signify the transformed quantities. The factor cl is determined in
such a way that the Mach number Ma no longer appears in the transformed potential equation. The factor c2 is obtained by applying the streamline analogy (kinematic flow condition). These factors cl and c2 are given by the expressions Eqs. (4-12) and (4-21) derived earlier. The transformed potential equations are, in analogy to Eqs. (4-13) and (4-14), a20'
a20'
ax ,"- -l- are, a20' ax'2
a20' ar'2
+
1 a0' r ar ,
r'
a20' + r2 a2, = 0 1
=0
r'2 a6'2
L9 r'
(Macc < 1) (
M a,,,, >
1)
(5-41)
( 5 - 42 )
The transformed potential equation for subsonic flow is identical to the potential equation (Ma = 0). The transformed potential equation for supersonic flow is identical to the linear potential equation Eq. (5-38) for Ma =-\/2-. These transformations show that the computation of subsonic flows of any Mach number can be reduced to the computation of the flow at Ma = 0 and the computation of supersonic flows of any Mach number to that at Ma = s. This is the Prandtl-Glauert-Gothert-Ackeret rule for fuselages. It can be formulated in the following way, corresponding to version I for wings of finite span (Sec. 4-2-3).
From the given fuselage and the given Mach number, a transformed fuselage is obtained by a distortion of its dimensions in the y and z directions and of its angle of attack by the factor ci = [1 -Ma2,1. Its dimensions in the x direction remain unchanged. For the fuselage, transformed in this way, the incompressible flow has to be computed if the given Mach number is subsonic. If the given Mach number is has to be computed. supersonic, however, the compressible flow for Ma = The transformation formulas of the geometric quantities of the fuselage are Thickness ratio:
d=
Camber ratio:
lF =
Angle of attack:
a' =
I
dF F
(5-43a)
I ZF
(5-43b)
Maw, I
F
I 1 -Ma2, j a
(5-44)
Hence, when the velocities of the incident flow of the given and the transformed fuselages are equal, the pressure coefficients are related by
AERODYNAMICS OF THE FUSELAGE 351 P
cP
Po
=
c'P
(5-45)*
Ma
The geometric transformation of Eq. (5-43a) is illustrated in Fig. 5-21, in which the transformed thickness ratio is plotted against the Mach number. The hatched
body is the given body the flow over which is computed for different Mach numbers. The transformed bodies belonging to the given Mach numbers are drawn without hatches. The flow about these transformed bodies has to be computed as
incompressible flow when Ma < 1, and as flow at Maw, =
when Ma. > 1.
Applications of this rule will be discussed in Secs. 5-3-2 and 5-3-3. Transonic similarity rule The similarity rules explained above apply only to
subsonic and supersonic flows. Now, a similarity rule for fuselages at transonic flow (Ma. = 1) of axial incidence will be given. This similarity rule was first formulated by von Karman [56]. A more detailed presentation of this similarity rule was later given by Keune and 0swatitsch [27]. The following simplified derivation should be sufficient.
By starting with the nonlinear potential equation, Eq. (5-39), the problem may be formulated as follows: Given is an axisymrnetric fuselage of revolution at May, = 1. Then, what is the pressure distribution over an affine reference fuselage at
the same incident flow Mach number Mam = 1? In analogy to Eq. (4-28), the following transformation is introduced:
x'=x r'=car 0=c4P' U,= U.
(546)
*The validity of this transformation formula for the pressure distribution reaches beyond
the framework of the first approximation of Eq. (5-8), as has been shown, e.g., by Truckenbrodt [49 ]. It applies to the second approximation of Eq. (5-9) as well.
L U. CIO
I
to
Figure 5-21 The application of the 0
1
IT Maw
2
3
Prandtl-Glauert-Ackeret rule to fuselages. Thickness ratio 5' of the transformed fuselage vs. Mach number.
352 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
Again, the quantities with primes signify the reference fuselage, those without primes the given fuselage. Substituting Eq. (5-46) in Eq. (5-39) yields, in analogy to Eq. (4-29), C3 = C4. To establish another relationship between the constants c3 and c4, the radial velocity component w,. is derived from the boundary condition Eq. (5-6c):
lim (r' iv'.) = U R'
lim (r -w,.) = U B dR dx
r
r->0
d
d a:'
(5-47)
Be cause
of the affinity of the two fuselages, R'= (SF/SF)R , with 5F and SF being the fuselage thickness ratios. With W. = aO/ar and w' = W/ar', C4
(IF
2
_
and
5F
5F
(5-48)
F
Finally, the relationship between the pressure distributions cp and c, of the two fuselages remains to be determined. Because cp = -2u/U,. = -(2/U,.)aO/ax, this relationship is obtained immediately as S
Cp = C4 c 7, _ (SF
2
(5-49)
CV,
This is the well-known von Karman similarity rule for bodies of revolution at transonic incident flow.
As was first shown by Oswatitsch, a correction term to this formula can be determined, leading to
cp = cP(SF ) + 2g(x)5F In SF 2
I
F
(5-50)
Here, g(x) is given by Eq. (5-1Ob).
5-3-2 The Fuselage in Subsonic Incident Flow Computational procedures In Sec. 5-3-1 it was shown that at Mach numbers
Ma < 1, the computation of the flow about a fuselage may be reduced to the determination of the incompressible flow for a fuselage that is suitably transformed. The computation of the incompressible flow over a fuselage was discussed in detail
in Sec. 5-2. The starting point for further consideration is the subsonic similarity rule. By assigning the index "inc" to the reference fuselage that corresponds to the given fuselage at a given Mach number, the transformation formulas for the geometric data of the fuselage become, from Eqs. (5-40), (543), and (5-44), Coordinates:
xinc = x
rinc = r
Fuselage radius:
Rinc = R
1 -Mao,
Fuse lage l engt h :
1
'Finc = 6F
Angle of attack:
Oinc = a
t,inc = d
(5-51) (5-52a)
( 5 - 52b )
Finc = lF
Thickness ratio:
1 -Mao
1 -Mao,
l -Ma o
(5-52c)
(5-52d)
AERODYNAMICS OF THE FUSELAGE 353
The transformation formula for the pressure coefficient is, from Eq. (5-45), 1
Cp
(5-53)
= 1 _Mao, Cpinc
This computation procedure will now be applied to fuselages in axial and inclined incident flow at subsonic velocities. The fuselage at axial incident flow The pressure coefficient for incompressible flow from Eq. (5-10a) can be given in the form Cp inc = [f(x) +g(x) In SFinc] SFinc - (Cp)Ma,o =o
where the functions f (x) and g(x) are independent of the thickness ratio of the fuselage. Introducing Eqs. (5-52c) and (5-53) into the above equation yields Cp = (Cp) Ila
a°O°
i d 2AF n dzZ
In
1 - Ala
(5-54)
when taking into account that, from Eq. (5-l0b), g(x)SF = -(1 /n)(d 2AF/dx2) with AF = nR2 as the fuselage cross section. From Eq. (5-54), it can be seen that the influence of compressibility on the pressure distribution is taken into account by a term additive to the pressure distribution at incompressible flow. It is proportional to the second derivative of the distribution of the fuselage cross section. Since, in general, this derivative is negative, the additive term represents an increase in the negative perturbation pressure. The similarity rule of Sec. 5-3-1 is thus confirmed, namely, that the computation of subsonic flow of arbitrary Mach number, 0 <Ma, < 1 may be reduced to the computation for Ma. = 0. The pressure distributions for the paraboloid of revolution of thickness ratio 5F = 0.1 are shown in Fig. 5-22 for several Mach numbers. Marked changes of the pressure distribution because of the compressibility effect are found only near the fuselage center section (see Krause [29]). Drag-critical Mach number The critical Mach number of the incident flow Ma,, cr at
which the velocity of sound is reached locally on the body is obtained, from Eq. (4-53b), from the lowest pressure on the body cpmin. In Fig. 5-23, determination of the drag-critical Mach number for paraboloids of revolution is demonstrated for several thickness ratios SF. As the figure shows, the intersections of the curves Cpmin versus Ma., of the various paraboloids of revolution from Eq. (5-54) with the curve from Eq. (4-53b) have to be established. For comparison see also Fig. 4-28. The critical Mach number, determined in this way, is plotted in Fig. 5-24 against the fuselage thickness ratio. The critical Mach numbers of ellipsoids of revolution are included in this figure. They are somewhat larger than those of the paraboloids. Comparison of these critical Mach numbers of bodies of revolution with those of wing profiles of Fig. 4-29 shows that, for the same thickness ratio (5F = 6), the critical Mach number for three-dimensional flow is considerably larger than for plane flow.
The drag-critical Mach number is of significance for the drag rise at high subsonic Mach numbers; compare Fig. 4-14 for wing profiles. Finally, the drag
354 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
Mom o8
A /-a06
.0
o02
aoa 1
1
1
1
I
i
aoa
of
X-fF oW
4dFinax
to
Figure 5-22 Pressure distribution on a paraboloid of revolution of thickness ratio 6F = 0.1 in axial flow for several Mach numbers.
X
IF
coefficients at axial incident flow, from Gothert [15] are presented in Fig. 5-25 for a few relatively thick fuselages as a function of the Mach number (see Krauss [30] ). These measurements show that the drag rise for fuselages lies at higher Mach numbers than for wing profiles of the same thickness, as would be expected from theory.
The fuselage in asymmetric flow The pressure distribution due to the angle of attack is given for incompressible flow by Eq. (5-25) when the index inc is added to all quantities. Introducing Eqs. (5-51)-(5-53) into this equation yields the pressure distribution at compressible flow as
cP (x, t3) _ -2
d B (x) d x [x (x) R2 (x)]
cos
(5-55)
Figure 5-23 Determination of the drag-critical Mach number Ma..cr of paraboloids of revolution of
thickness ratio SF at axial incident flow. Curve i from Eq. (5-54) and Fig. 5-6. Curve 2 from Eq. (4-53b).
AERODYNAMICS OF THE FUSELAGE 355 S
0
005
015
010
020
025
i00
Z95
Ellipsoid
!90
Paraboloid
H_ _
Figure 5-24 The drag-critical Mach number Maa,cr
of paraboloids and ellipsoids of revolution
vs.
thickness ratio bF and axial incident flow.
q
By comparison with Eq. (5-25a), it is apparent that the pressure distribution due to
the angle of attack is independent of the Mach number. It follows that the relationships of Sec. 5-2-3 for the lift distribution, the lift, and the moment in incompressible flow apply directly to compressible subsonic flow.* Studies of the computation of the pressure distribution on fuselages of arbitrary cross section shapes, for both subsonic and supersonic flows, have been conducted, for example, by Hummel [23]. A nonlinear second-order theory is given by Revell [421
5-3-3 The Fuselage in Supersonic Flow Fundamentals The essential difference between subsonic and supersonic flows has already been explained by Fig. 1-9. Furthermore, the specific problems of the wing
-
*This is true also for supersonic incident flow, as will be shown in Sec. 5-3-3. 015
0
1
an
2
GIV
3
6
F
"-
-
0q t4L
0..113
» - 0387
6Finax
1
'
Z
0353
IF
co;
002
0 Oa
0.3
04
05
No.
06
07
08
0S
Figure 5-25 Drag coefficients of bodies of revolution in axial incident flow vs. Mach number, from measurements of Gothert. c.DF refers to the frontal area.
356 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
of finite span at supersonic velocities were discussed in Sec. 4-5. The essential physical difference between flows of subsonic and supersonic velocities lies in the fact that, at the latter, a given point can affect only the space enclosed by the downstream cone. This point itself can be affected only by disturbances within the upstream cone. The application of these fundamental facts of supersonic flow to a fuselage is explained in Fig. 5-26. The flow at a station x, r can be influenced only by the crosshatched range cut out of the fuselage by the upstream cone of apex semiangle M. The Mach angle u is related to the approach Mach number Ma,. by Eq. (4-80). The upstream cone to the point (x, r) intersects the fuselage axis (x axis) at the point
xo = x - r cot4u = x - r
am
-1
(5-56)
In the following discussions, the length x0 will be termed "influence length." The Mach cone generated by the fuselage nose is also sketched in Fig. 5-26. The supersonic flow about a circular cone (fuselage nose tip) in axial incident flow represents the simplest case of a cone-symmetric supersonic flow, which has been discussed previously in Sec. 4-5.
Now the. slender body of revolution at flows without (axial) and with a small angle of incidence will be treated. Either case can be computed approximately with the method of singularities (source-sink and dipole distributions, respectively). This method has been presented previously in Sec. 5-2 for incompressible flow. Another possibility is the application of the method of characteristics. Besides the linear theory of supersonic flow over fuselages, which will be presented below in detail, nonlinear theories of higher order have been developed by, for example, van Dyke [51 ] and Lighthill [33]. Comprehensive presentations concerning the fundamentals of the aerodynamics of fuselages in compressible flow are found in the pertinent publications on gas dynamics, listed in, Section II of the Bibliography. The fuselage at axial incident flow The axisymmetric fuselage in axial incident flow of supersonic velocity can be treated by means of the source-sink method in a way similar to that which has been explained for incompressible flow (Sec. 5-2-2). This
Figure 5-26 Fuselage theory at supersonic incident flow.
AERODYNAMICS OF THE FUSELAGE 357
method was developed by von Karman and Moore [55]. The relationship between
the source distribution q(x) and the fuselage contour R(x) can be established through the same considerations as in the case of incompressible flow; that is, here,
too, Eq. (5-4b) is valid. The procedure for translating the source-sink method of incompressible flow into that of supersonic flows has been treated in detail for the wing in Sec. 4-5-3 and can be applied to the fuselage. The potential 0 (x, r) of the flow induced by the linear source distribution q(x) on the x axis is given [see also Eq. (4-102)] as 70
i
(
ir=-2
g(Z) dx'
)
V(x
(5-57)
- x')= - (Ma"
0
Here x0 is the influence length from Eq. (5-56). The velocity components are obtained for the entire space in the well-known way as 2G =
(5-58)
2C,
ax
or
In executing these differentiations it should be noted that the upper limit x = x0 of the integral in Eq. (5-57) depends on x and r, and that. for x = x0, that is, on the Mach cone, the denominator of the integrand vanishes. To determine the velocity distribution on the fuselage surface, the values of the induced velocities are needed for small radial distances r; see Eqs. (5-6a) and (5.6b). Equations (5-57) and (5-58) yield U'-E
2E
q(x') dx'
1 dq(x)
J
(x - x')2 (5-59a)
zv,(x,r-* 0)=
I q(x) 2z r
(5-59b)
The final form of the induced velocity components is obtained by introducing Eq. (5-4b) into Eqs. (5-59a) and (5-59b) as U
1
')
3
Uro
-
1 - In IF
lim EAU 1
d2 (B2) d
w,(x) L' cc
_
dR(x) dx
X-E
2E) d2(R2)
In (R(x) IF
dx +
]iMa
JC d(R2)
-
dx'
dx' (x - x')-
1)
1 d (R2) E
dx (5-60) (5-61)
To determine the pressure distribution from the induced velocities, the formulas
of the incompressible flow are directly applicable, that is, Eq. (5-8) for the first approximation and Eq. (5-9) for the second approximation. In analogy to Eq.
358 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
(5-10a), the dependence of the pressure distribution on the body thickness ratio 8F can be found for fuselages in axial incident flow. By observing Eqs. (5-60) and (5-61), this dependence is obtained as (5-62) cp(x) = [fi(x) + 91(x) ln(SF Maw, - 015F" The functions fl (x) and gl (x) depend on the fuselage geometry. They do not depend, however, on the thickness ratio of the fuselage. Consequently, Eq. (5-62) for the pressure distribution may be written in the following form: Cp =: (Cp)Ma0.=y5 -
1 ddx
1 Ma - 1
(5-63)
This equation is analogous to Eq. (5-54) for subsonic incident flow.
It may be seen from Eq. (5-63) that, at supersonic incident flow, the compressibility effect on the pressure distribution is given by a term additive to the pressure distribution at Ma = N/2-. This confirms the similarity rule of Sec. 5-3-1, stating that the computation of a supersonic flow of arbitrary Mach number can be reduced to the computation at Ma _ The above computational procedure for the pressure distribution of fuselages in supersonic axial flow will be explained now
by means of a few examples. The supersonic flow over the nose tip of a cone-shaped body was treated early by Taylor and Maccoll [46], Tsien [48], and Busemann [6]. Results for a blunt-body nose in supersonic incident flow have been published by Holder and Chinneck [21] and van Dyke [521. In Fig. 5-27, the pressure coefficient for Ma,,, = (second approximation) is presented for the paraboloid of revolution of thickness ratio 6F = 0.1. The .
functions fi and gl of Eq. (5-62) become in this case
fi(X) = -4(22X2 - 16X + 1) - 8(6X2 - 6X + 1)ln(1 -X)
g1(X)=-8(6X2-6X+1)
(5-64)
with X= x/1F. For comparison, the pressure coefficient from the method of characteristics is also shown. Agreement of these two computational methods is very
good. Furthermore, the pressure distribution for incompressible flow (Ma = 0) from Fig. 5-5 is added. It is noteworthy that, in supersonic flow, the pressure minimum lies behind the middle of a body that is symmetric to X = 0.5. Furthermore, it should be noted that for the same shape of the body cross sections, the pressure distribution in the axisymmetric case shows a completely different character than in the plane case, as may be verified by comparison with Fig. 4-23a. As in the case of a wing, the pressure distribution over the total surface of an axisymmetric fuselage in supersonic incident flow results in a force in flow direction that is different from zero. As in the case of the wing, this force is termed wave drag. It is caused by the Mach waves originating at the body. Computation of the
wave drag may be done either with the help of the momentum law or through direct integration of the pressure distribution over the surface. Only the latter computational procedure will be described below.
Integration of the pressure distribution over the surface (component of the
AERODYNAMICS OF THE FUSELAGE 3 59
-Q08
-006
M aw; 0 -004
I
I
I
0.02
1
Figure 5-27 Pressure distribution on a paraboloid of revolution of thickness ratio 6p = 0.1 at Ma . =
004 0
08
0.2
to
f and Ma = 0 in axial flow. Curve 1, singularities method; second approximation from Eqs. .(5-62) and (5-64). Curve 2, linear method of characteristics.
pressure force in the x direction) yields the wave drag of the body of revolution in axial incident flow as IF
DF = 21r f (p -pc)R
dR dx
IF
dx = q f cp
dAF dx
dx
0
0
To establish the effect of the Mach number on the wave drag, substituted for cp in Eq. (5-65). Integration by parts yields
DF = (DF)Ma. =
1
2
2n q-
dAF (IF dx
2
In
2
Maw - 1
For R = 0 or dRldx = 0 at the fuselage tail, the wave drag is independent of the Mach number because dAF f dx = 27rR(dR/dx) = 0. Its value becomes equal to that
for Ma. _. In this case the wave-drag coefficient of the fuselage with reference to the frontal area AF is obtained from Eq. (5-66) as
DF CDF =
q-AFinax
- (CDF)Ma = f = number SF
(5-67)
Here the "number" depends on the fuselage geometry, but not on the thickness ratio. Consequently, the coefficient of the wave drag, referred to the frontal area, is *In this section, the drag of the fuselage as obtained in inviscid flow (wave drag) is designated as DF. Because of viscosity effects (friction), a contribution DF must be added to this drag [see Eq. (5-17)].
360 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
proportional to 52.* Evaluation of the above equations for the paraboloid results in
cDF = s 5F = 10.6752
(paraboloid)
(5-68)
where cp(x) of Eq. (5-62) has been substituted, using the expressions of Eq. (5-64). The coefficients of the wave drag of truncated paraboloids of Wegener and Kowalke [11] are given in Fig. 5-28. For the paraboloid cut off in the middle (IFIlFO = it becomes 2),
CDF = s 52 = 4.6752
(paraboloid tip)
(5-69)
This drag coefficient does not include the contribution made by the suction pressure on the blunt tail surface (so-called base pressure). For paraboloids of thickness ratios 5F = 0.1 and 0.2, the drag coefficients as determined by the method of source distributions are compared in Fig. 5-29 with those from the linear method of characteristics. The deviations of the coefficients from the two methods are very small for 5F = 0.1. They are no longer negligible for 5F = 0.2, however. By substituting in Eq. (5-65) the expression for cp(x) of Eq. (5-62), a formula for the wave drag of a general pointed body of revolution is obtained that depends on the *In this connection it should be remembered that the wave-drag coefficient of the wing of finite span, referred to the planform area, is likewise proportional to the thickness ratio (Sec. 4-3-3). 12
F M;10.1
Z
0.2 10
8
16
Z
0
Q5
06
07
F
Q8
09
1.0
1Fo
Figure 5-28 Coefficients of wave drag for truncated paraboloids of revolution vs. thickness ratio 8F = dFinaxllF and Mach number, from Wegener and Kowalke.
AERODYNAMICS OF THE FUSELAGE 361 05
I
0.
6F-02
03
6F=0.1
0.1
-- 2 Figure 5-29 Coefficients of wave drag for paraboloids of revolution of thickness ratios 5F = 0.1 and 0.2. Comparison of the singularities method (1),
0 2.5
ao
10
from Eq. (5-68), and the method of characteristics (2).
body geometry. von Karman and Moore [55] and Ward [57] established the following equation; see the derivation [5] IF
AF' (1F) f AF in
DF = 2 U!
1 - lF
dx
\ 0
IF IF
-2f 1
0
-
f AF(x')AF11 (x) In 0
[AF(lF)] 2 In
R(lF) 2lF
x -xl lF
z Maw
-
dx' dx
(5-70)
being the fuselage crosswhere AF = dAF/dx and AF = d 2AF/dx2 , with sectional area. With this formula, the wave drag at given body geometry may be determined through relatively simple quadratures (see [9] ).
362 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
Das [8]
discusses some basic questions about the connection between the
various theories for the computation of the wave drag of fuselages, about the ranges
of their applicability and the limitations in their accuracies. Both the various theories (linear, nonlinear) and the test results are compared. The summary report on wave drag of fuselages by Morris [39] and the investigations on the base drag of bodies with blunt tails of Tanner [45] should be mentioned here. The computation of the friction drag of bodies of revolution in supersonic flow has been treated by Young [601. The evaluation of drag measurements for the determination of the wave drag includes considerable uncertainties, because the measured total drag is composed of friction drag and, if the tail is blunt, base drag, besides the wave drag. Measurements in which these three contributions were determined individually have been
conducted by Chapman and Perkins [10] and Evans [101. In Fig. 5-30, the test results of [10] for a truncated paraboloid are plotted as drag coefficients against the Mach number. The comparison of these measurements with theory was accomplished by adding to the measured base drag the theoretical friction drag from
Fig. 4-5 and the wave drag from Fig. 5-28. Agreement of the drag coefficients computed in this way with the measurements is quite good. It should be mentioned, however, that there are cases of larger differences between measurements and 020
Measurement
Re =3
- Theory
10'
0.15
Wave drag
U
Frictio n drag
005
Base drag (measurement) 0 125
175
1,50
200
MGM +
dFinax j
Figure 5-30 Measured drag coefficients of a truncated paraboloid of revolution in axial flow (dFinax/ IF=0.07) at supersonic velocities, from Evans. Comparison with theory. Curve 1, base drag. Curve 2,
base and friction
08-IF IF
total drag.
drag. Curve 3,
AERODYNAMICS OF THE FUSELAGE 363
4
3
x M E LL.
13
U.
IF 0.71
4
3
Theor y 2
0.05
00#
0.03 70
Figure 5-31 Drag coefficients (pressure
drag without base drag) of slender fuselages vs. Mach number Ma,,, from measurements of [3] (body contour shown with increased ordinates). (1) Optimum body, from Haack and Sears, dFinax/IF = of revolution, (3) Cylindrical body, dFinax/IF = 0.091. dFinax/IF = 0.08. (4) Cylindrical body with contraction, dFinax/IF= 0.08. 0.086.
is
2.0
2,S
3,0
Ma,. -
40
(2)
Paraboloid
theory. Additional test results are given in Fig. 5-31, namely, the coefficients of the pressure drag CDF of four slender fuselages in axial incident flow plotted against the Mach number Ma.. These drag coefficients do not include the base drag. Fuselage i is a body of minimum wave drag for a given volume and a given length, from Haack
[43] and Sears [43]. Fuselage 2 is a paraboloid of revolution. Fuselages 3 and 4 have cylindrical tail sections. For fuselages 2 and 3, the theoretical values of Eq. (5-70) are also shown.
Another optimum fuselage configuration with pointed nose and blunt tail was specified by von Karman [55]. Also, Das [7] concerned himself with the determination of optimum shapes of a fuselage with regard to its drag at supersonic flow. A compilation of additional test results and of comparisons with theory is found in Fiecke [11 ] . Miles [38] derived a linear theory for the computation of the wave drag of fuselages at supersonic incident flow. The flow picture of Fig. 5-32 gives a more profound insight into the flow about a fuselage in supersonic flow of axial incidence. In particular, it shows clearly the bow wave and the tail wave at a Mach number of Ma. = 3.5.
364 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
Figure 5-32 Shadowgraph picture of a fuselage at Mach number Mao, = 3.5.
Body of revolution with a blunt nose in hypersonic incident flow In Sec. 4-3-5, the profile with a blunt nose in hypersonic incident flow was treated. For the computation of the pressure distribution on the body surface, Newton's approximation, Eq. (4-65), was furnished as the simplest expression. This relationship, which was established for plane flow, can be applied likewise to axisyrnmetric flow as present in the case of fuselages. The pressure distribution on a half-body consisting of a cylinder with a matching spherical nose pertaining to such a hypersonic flow is plotted in Fig. 5-33. According to Newton's concept of momentum transfer from the flow particles to the body, the pressure distribution would be given by Eq. (4-65). The real flow does not correspond to this concept, and Eq. (4-65) cannot properly represent the pressure distribution. Nevertheless, a very good approxima1.0
as 0.6
a4
Figure 5-33 Pressure distribution of a half-body with spherical nose, from Lees.
1
-01 ZOL
0.4
0.6
1.2
s R
1.6
2.0
2.4
2.B
(o) Mao,, = 5.8, Re = UooR/v. = 1.2 105. (o) Mao, = 3.8, Re = U.RR/vo, _
) Modified Newtonian 1.4 - 105. ( approximation, from Eq. (5-71).
AERODYNAMICS OF THE FUSELAGE 365
tion for the pressure distribution is obtained, at least near the stagnation point, by substituting in Eq. (4-65) the actual value at the stagnation point for the factor 2. Thus, the so-called modified Newton formula is obtained: cp = op mas Sln2 t.
(5-71)
This relationship is also given in Fig. 5-33, showing very good agreement. It should be emphasized, however, that Eq. (5-71) is an empirical relationship. The fuselage in asymmetric incident flow The fuselage in asymmetric incident flow of supersonic velocity can be treated by means of a dipole distribution on the body axis, similar to the method presented in Sec. 5-2-3 for incompressible flows. The
adaptation of the dipole distribution of incompressible flow to supersonic flow follows the rules explained for the axial incident flow. The potential 0 (x, r, t9) of a line
distribution of three-dimensional dipoles m(x) on the x axis becomes, in
analogy to Eq. (5-20a), TO
r (Ma' - 1) cos z
m(x') dx'
2,r
*
(x - x')'2 - (Ma' - 1) r23 0
Here, x0 is the influence length from Eq. (5-56) and Fig. 5-26. The expansion of 0 (x, r, 6) for small radial distances r yields 0 (x, r -* 0, t$) =
cos 27r
m (x)
r
(5-73)
in agreement with Eq. (5-20b) for incompressible flow. Consequently, the velocity components determined from Eq. (5-21) for supersonic flow are identical to those for incompressible flow. Furthermore, the kinematic flow condition of Eq. (5-22), and hence the determining equation for the dipole distribution Eq. (5-24), applies directly to supersonic flows. Finally, it follows that the formula for the pressure distribution at incompressible flow, Eq. (5-25a), is also valid for any supersonic
Mach number of incident flow. Since it has been found that Eq. (5-55) for the pressure distribution at subsonic incident flow is identical to Eq. (5-25a), the remarkable result is obtained that, over the entire Mach number range, the pressure distribution due to the angle of attack of the fuselage, and the lift distribution, the lift, and the moment, can be determined from the formulas for incompressible flow. For instance, the lift of a fuselage, truncated in the rear, at supersonic incident flow is, from Eq. (5-29a),
LF = 2aq.AFt
(5-74)
where AFt is the cross-sectional area of the fuselage tail. The sign f signifies, according to Hadamard, that only the finite part of this integral has to be taken.
366 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM 12
dFinax 10
Um
F
09
ios Measurement,
Figure 5-34 Lift coefficient CLF = LF/ AFrnax4- of a slender body of revolution with blunt tail vs. angle of attack a, from 153]. Body thickness ratio dFmax/
0.#
0.Z
IF=0.10, Mach number Mac, =1.97, a
Reynolds number Re = UO,lF/v - 106 , linear theory from Eq. (5-74).
e
All computational methods for the lift of fuselages treated so far lead to a linear dependence of the fuselage lift on the angle of attack. At larger angles of attack, however, the lift increases more than linearly with angle of attack. As an example, in Fig. 5-34 the lift coefficient CLF of a slender body of revolution with a blunt tail is plotted against the angle of attack for Mach number Ma., -- 2. Compare
also Fig. 5-3 for the case of incompressible flow. This lift characteristic much resembles that of a wing of extremely small aspect ratio (see Sec. 3-3-6). The nonlinearity is caused by viscosity effects. At larger angles of attack, the flow separates on the upper and lower surfaces of the fuselage because of cross flow over
the body. Subsequently, the flow rolls up and, as in the case of the flow over the side edges of a wing of small aspect ratio, free vortices form that are shed from the body under an angle different from zero (see Fig. 3-S0a). The formation of the vortex sheet on slender bodies at large angles of attack is sketched in Fig. 5-35 for a rectangular wing and for a delta wing of small aspect ratio, and for a slender fuselage. Details of the flow about slender bodies at large angles of attack and the theoretical determination of the nonlinear lift characteristic are treated in [2, 24, 35, 37, 51 ]
.
For transonic flow about bodies of revolution, generally valid solutions are not
yet available. However, the investigations of Keune and 0swatitsch [25, 271, Spreiter [441, Fink [12], and Krupp and Murman [311 must be mentioned here.
a,
b
c
Figure 5-35 Vortex formation on wings of small aspect ratio and on slender bodies, leading to nonlinear lift characteristics. (a) Rectangular wing. (b) Delta wing. (c) Fuselage.
AERODYNAMICS OF THE FUSELAGE 367
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13. Fuhrmann, G.: Theoretische and experimentelle Untersuchungen an Ballonmodellen, Jb. Mot. -Luft.-Stud.-Ges., 5:63-123, 1911/1912. Keune, F.: Z. Flugw., 6:336, 1958. Miiller, W.: Ing.-Arch., 19:282-295, 1951; 20:57-66, 1952. 14. Goldstein, S.: Flow Past Solid Bodies of Revolution, in S. Goldstein (ed.), "Modern Developments in Fluid Dynamics-An Account of Theory and Experiment Relating to Boundary Layers, Turbulent Motion and Wakes," vol. 2, pp. 491-549, Dover, New York, 1965. 15. Gothert, B.: Hochgeschwindigkeitsmessungen an rotationssymmetrischen Korpern, ZWB
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368 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
21. Holder, D. W. and A. Chinneck: The Flow Past Elliptic-Nosed Cylinders and Bodies of Revolution in Supersonic Air Streams, Aer. Quart, 4:317-340, 1954. 22. Howarth, L.: Flow Past Bodies of Revolution, in L. Howarth (ed.), "Modem Developments in Fluid Dynamics-High Speed Flow," vol. 2, pp. 688-756, Clarendon, Oxford, 1953.
23. Hummel, D.: Berechnung der Druckverteilung an schlanken Flugkorpem mit beliebiger Grundriss- and Querschnittsform in Unter- and Uberschalllstromung, Jb. DGLR, 158-173, 1968. Rothmann, H.: Z. Flugw., 20:98-105, 1972. 24. Kelly, H. R.: The Estimation of Normal-Force, Drag, and Pitching-Moment Coefficients for Blunt-Based Bodies of Revolution at Large Angles of Attack, J. Aer. Sci., 21:549-555, 565, 1954. Buford, W. E.: J. Aer. Sci., 25:103-108, 1958. 25. Keune, F.: Uber den Kompressibilitatseinfluss bei and nahe Machzahl Eins fdr Korper kleiner Streckung and schlanke Rotationskorper, Z. Flugw., 4:47-53, 1956; Jb. WGL, 176-186, 1955; Jb. WGLR, 186-203, 1964. 26. Keune, F. and K. Burg: "Singularitatenverfahren der Stromungslehre," Braun, Karlsruhe, 1975.
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28. Krasnov, N. F.: "Aerodynamics of Bodies of Revolution" (transl., 2nd Russian ed.), American Elsevier, New York, 1970.
29. Krause, F.: Unterschallstromung um nicht angestellte, dicke Rotationskorper, Ing.-Arch., 32:1-25, 1963. 30. Krauss, E. S.: Effect of Bluntness of Elliptic Nose Shape on the Drag of Bodies of Revolution in Axisymmetric Subsonic Flow, Z. Flugw., 15:171-175, 1967; 16:429-437, 1968; 20:81-90, 1972; 22:15-24, 1974. 31. Krupp, J. A. and E. M. Murman: Computation of Transonic Flows Past Lifting Airfoils and Slender Bodies, AIAA J., 10:880-886. 32. Lessing, F.: Anwendung des Singularitatenverfahrens der Oberflachenbelegung auf rotationssymmetrische Korper, Ing.-Arch., 38:400-406, 1969. Geissler, W.: Z. Flugw., 20:457462, 1972. Riegels, F.: Abh. Braunschw. Wiss. Ges., 4:146-165, 1952. Vandrey, F.: ARC RM 3374, 1951/1964. 33. Lighthill, M. J.: Supersonic Flow Past Bodies of Revolution, ARC RM 2003, 1945; Quart. J. Mech. App. Math., 1:76-89, 90-102, 1948. 34. Lotz, I.: Zur Berechnung der Potentialstromung um quergestellte Luftschiffkorper, Ing.-Arch., 2:507-527, 1931; NACA TM 675, 1932. Kiichemann, D.: Jb. Lufo., 1:547-564, 1940. Weinel, E.: Ing.-Arch., 3:149-151, 1932. 35. Marshall, F. J. and F. D. Deffenbaugh: Separated Flow over a Body of Revolution, J. Aircr., 12:78-85, 1975. Angelucci, S. B.: J. Aircr., 8:959-966, 1971. Schindel, L. H.: J. Aircr., 6:537-543, 1969. 36. Maruhn, K.: Druckverteilungsrechnungen an elliptischen Riimpfen and in ihrem Aussenraum, Jb. Lufb., 1:135-147, 1941; 1:263-279, 1942. Tuckermann, L. B.: NACA Rept. 210, 1925. Zahm, A. F.: NACA Rept. 323, 1929. 37. Mello, J. F.: Investigation of Normal Force Distributions and Wake Vortex Characteristics of Bodies of Revolution at Supersonic Speeds, J. Aerosp. Sci., 26:155-168, 1959. 38. Miles, J. W.: On the Sonic Drag of a Slender Body, J. Aer. Sci., 23:146-154, 1956. 39. Morris, D. N.: A Summary of the Supersonic Pressure Drag of Bodies of Revolution, J. Aerosp. Sci., 28:563-572, 1961. 40. Multhopp, H.: Zur Aerodynamik des Flugzeugrumpfes, Lufo., 18:52-66, 326, 1941; NACA TM 1036, 1942. Vandrey, F.: Jb. Lufo., 1:367-370, 1940. 41. Munk, M. M.: Aerodynamics of Airships, in W. F. Durand (ed.), "Aerodynamic Theory-A General Review of Progress, 1934, 1936," div. C, Q, Springer, Berlin, Dover, New York, 1963;NACA Rept. 184, 1924. 42. Revell, J. D.: Second-Order Theory for Steady or Unsteady Subsonic Flow Past Slender Lifting Bodies of Finite Thickness, AIAA J., 7:1070-1078, 1969.
AERODYNAMICS OF THE FUSELAGE 369
43. Sears, W. R.: On Projectiles of Minimum Wave Drag, Quart. App. Math., 4:361-366, 1947. Adams, M. C.: NACA TN 2550, 1951. Ferrari, C.: in A. Miele (ed.), "Theory of Optimum Aerodynamic Shapes," pp. 103-124, Academic, New York, 1965. Haack, W.: Lil.-Ber.,
139:14-28, 1941. Harder, K. C. and C. Rennemann, Jr.: NACA Rept. 1271, 1956. Ramaswamy, M. A. and S. Viswanathan: J. Airer., 12:1001-1002, 1975. Schmidt, W.: Z. Flugw., 7:194-201, 1959.
44. Spreiter, J. R.: Aerodynamics of Wings and Bodies at Transonic Speeds, J. Aer. Sci., 26:465-486, 517, 1959; in K. Oswatitsch (ed.), "Symposium Transsonicum I," pp. 152-183, Springer, Berlin, 1964. Heaslet, M. A. and J. R. Spreiter: NACA Rept. 1318, 1957. Hosakawa, I.: in K. Oswatitsch (ed.), "Symposium Transsonicum I," pp. 184-199, Springer, Berlin, 1964. Spreiter, J. R. and S. S. Stahara: Z. Flugw., 18:33-40, 1970; AIAA J., 9:1784-1791, 1971. 45. Tanner, M.: Reduction of Base Drag, Prog. Aerosp. Sci., 16:369-384, 1975. 46. Taylor, G. I. and J. W. Maccoll: The Air Pressure on a Cone Moving at High Speeds, Proc. Roy. Soc. A, 139:278-311, 1933;" "Scientific Papers," vol. III, pp. 182-209, Cambridge University Press, Cambridge, 1963.
47. Thwaites, B.: Uniform Flow Past Bodies of Revolution, in B. Thwaites (ed.), "Incompressible Aerodynamics-An Account of the Theory and Observation of the Steady Flow of Incompressible Fluid Past Aerofoils, Wings, and Other Bodies," pp. 369-421, Clarendon, Oxford, 1960. 48. Tsien, H.-S.: Supersonic Flow over an Inclined Body of Revolution, J. Aer. Sci., 5:480-483, 1938. Laitone, E. V.: J. Aer. Sci., 14:631-642, 1947; Quart. App. Math., 5:227-231, 1947. Rakich, J. V. and J. W. Cleary: AIAA J., 8:511-518, 1970. 49. Truckenbrodt, E.: Zur Aerodynamik der Rumpfkorper bei kompressibler Strornung, Z. Flugw., 6:15-20, 1958.
50. Truckenbrodt, E. and K. Gersten: Experimentelle and theoretische Untersuchungen an Deltafliigel-Rumpf-Anordnungen bei symmetrischer Anstromung, Z. Flugw., 5:204-216, 1957. Jacobs, E. N. and K. E. Ward: NACA Rept. 540, 1935. Moller, E. and H. Trienes: Z. Flugw., 1:2-8, 1953. Sherman, A.: NACA Rept. 575, 1936.
51. van Dyke, M. D.: First- and Second-Otder Theory of Supersonic Flow Past Bodies of Revolution, J. Aer. Sci., 18:161-178, 216, 1951; J. Fluid Mech., 1:1-15, 1956. Broderick, J. B.: Quart. J. Mech. App. Math., 2:98-120, 1949. Moore, F. K.: J. Aer. Sci., 17:328-334, 383, 1950. 52. van Dyke, M. D.: The Supersonic Blunt-Body Problem-Review and Extension, J. Aer. Sci., 25:485-496, 1958. Traugott, S. C.: J. Aerosp. Sci., 27:361-370, 1960. 53. Voellmy, H. R.: Experimentelle Untersuchungen an verschieden stark konvergenten, schlanken Rotationskorpern bei massig hohen Uberschallgeschwindigkeiten, Mitt. Inst. Aero. ETH Zurich, Mitt. 24, 1958. 54. von Karman, T.: Berechnung der Druckverteilung an Luftschiffkorpern, Abh. Aer. Inst. TH Aachen, 6:3-17, 1927; "Collected Works," vol. II, pp. 253-273, Butterworths, London, 1956; NACA TM 574, 1930. Moran, J. P.: J. Fluid Mech., 17:285-304, 1963. 55. von Karman, T. and N. B. Moore: Resistance of Slender Bodies Moving with Supersonic Velocities, with Special Reference to Projectiles, Trans. Amer. Soc. Mech. Eng., 54:303310,1932; "Collected Works," vol. 11, pp. 376-393, Butterworths, London, 1956. von Kirmin. T.: Volta-Kongress Rom, 222-276, 1935; "Collected Works," vol. III, pp. 179-221, Butterworths, London, 1956. Stetter, H. J.: Z. Angew. Math. Mech., 37:145-146, 1957. 56. von Karmin, T.: The Similarity Law of Transonic Flow, J. Math. Phys., 24:182-190, 1947; "Collected Works," vol. IV, pp. 327-335, Butterworths, London, 1956. 57. Ward, G. N.: Supersonic Flow Past Slender Pointed Bodies, Quart. J. Mech. App. PMlath., 2:75-97, 1949. Berndt, S. B.: Z. Angew. Math. Mech., 35:362, 1955. Fraenkel, L. E.: ARC RM 2954, 1952/1955. Kahane, A. and A. Solarski: J. Aer. Sci., 20:513-524, 1953. 58. Wieselsberger, C.: Airplane Body (Non Lifting System) Drag and Influence on Lifting System, in W. F. Durand (ed.), "Aerodynamic Theory-A General Review of Progress," 1935, div. K, Springer, Berlin, Dover, New York, 1963.
370 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
59. Young, A. D.: The Calculation of the Total and Skin Friction Drag of Bodies of Revolution at Zero Incidence, ARC RM 1874, 1939. Cebeci, T., G. J. Mosinskis, and A. M. 0. Smith: J. Aircr., 9:691-692, 1972. Granville, P. S.: Dav. Tay. Mod. Bas. Rept. 849, 1953. Scholz, N.: Jb. Schiffb., 45:244-263,.1951. 60. Young, A. D.: The Calculation of the Profile Drag of Aerofoils and Bodies of Revolution at Supersonic Speeds, Jb. WGL, 66-76, 1953; ARC RM 2204, 1945.
CHAPTER
SIX AERODYNAMICS OF THE WINGFUSELAGE SYSTEM
6-1 INTRODUCTION
6-1-1 General Remarks on the Interactions among Parts of the Airplane The aerodynamic coefficients of the major components of the airplane-wing, fuselage, empennage-are quite well established through theory and systematic measurements. The aerodynamics of the wing was treated thoroughly in Chaps. 2-4. The findings established there apply accordingly to the empennage (vertical
stabilizer and rudder, and horizontal stabilizer and elevator; see Chap. 7). The aerodynamics of the fuselage was the subject of Chap. 5. When these individual parts are assembled into a complete airplane, however, their interaction (interference) plays a very important role in the formation of aerodynamic forces. In many cases these interference effects are of the same order of magnitude as the contributions of the individual parts to the aerodynamic forces of the airplane as a whole. For this reason, consideration of these interactions is indispensible to the study of the aerodynamics of the airplane. The physical processes behind the aerodynamics of the interactions, are, of course. much harder to conceive than those of the aerodynamics of the individual parts. Consequently, the theoretical study of the interference problem has been attacked much later and is, even today,
not yet established to the extent of that of the individual parts. The theory of interference aerodynamics is available to a large extent for inviscid flow only. 371
372 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
Most important of the numerous interference effects among the various airplane components are the interactions between the wing and the fuselage and between the wing and the empennage. The interference between the wing and the fuselage is felt mainly in a changed lift distribution over these parts. The effect of the wing on the empennage, on the other hand, lies mainly in a changed incident flow direction of the empennage caused by the induced velocity field of the wing. A further important interference effect is the so-called ground effect, which is created during flight near the ground. Hereby for equal lift, the lift slope is increased and the induced drag is usually reduced. This problem has been treated in detail in theory and experiment; see the references cited in Sec. 3-3-1. In this chapter, only the interaction between wing and fuselage will be investigated. The interference problems related to the empennage will be treated in Chap. 7 together with the aerodynamics of the empennage.
6-1-2 Geometry of the Wing-Fuselage System For a better understanding of the aerodynamics of the wing-fuselage system to be
discussed below, the geometry of such a system will be discussed first. The geometry of the wing has been described in Sec. 3-1 (Figs. 3-1 and 3-2), that of the
fuselage in Sec. 5-1 (Fig. 5-1). The geometry of the wing-fuselage system is illustrated in Figs. 6-1 and 6-2. Figure 6-1 gives the plan view and the side view of a
wing-fuselage system, Fig. 6-2 the rear view of two wing-fuselage systems. The position of the wing relative to the fuselage is defined by the wing rearward position e, the wing high position zo, and the angle of wing setting 60. As shown in Fig. 6-1, the wing rearward position e is the distance between the geometric neutral
\
Wing chord Fuselage axis
o
If
10
ct
FZ,>/o
\.
V
N25p_
41
e
Figure 6-1 Geometry of a wing fuselage system (side view and plan view).
AERODYNAMICS OF THE WING-FUSELAGE SYSTEM 373
z 7
a!
bFo - b -2s
Figure 6-2 Geometry of wingfuselage systems (rear view). (a)
High-wing system without dihedral. (b) Mid-wing system with dihedral.
point of the wing (Sec. 3-1) and the fuselage nose. According to Fig. 6-2a, the wing high position zo is the distance between the wing and the fuselage axis. Its values are
High-wing airplanes:
zo > 0
Mid-wing airplanes:
zo = 0
Low-wing airplanes:
zo < 0
A typical mid-wing airplane with dihedral is sketched in Fig. 6-2b. The angle of wing setting co is, from Fig. 6-1, the angle between the chord of the wing root section and the fuselage axis. When the wing penetrates the fuselage, the portion of the wing shrouded by the fuselage requires special explanation. In the case of a swept-back trapezoidal wing, it is advantageous to replace the portion of the wing shrouded by the fuselage by a rectangular wing section. This rectangle is formed by the length of the root section to and by the mean fuselage width in the range of the wing bFo . For conventional wing-fuselage systems, bFo is almost equal to the maximum fuselage width bFinax, according to Fig. 5-1. The wing thus defined will be termed the "substitute wing," whereas the wing from which it has been derived will be termed the "original wing." Another important geometric parameter of a wing-fuselage system is the ratio of fuselage width bFo and wing span b: rIF
= bFo b
(relative fuselage width)
6-1-3 Aerodynamic Coefficients It is advantageous and generally customary to refer the aerodynamic coefficients of a wing-fuselage system to the geometric quantities of the original wing. A summary of the aerodynamic coefficients of the wing has been given by Eq. (1-21). These
definitions of the aerodynamic coefficients are applicable directly to the wingfuselage system when the forces and moments of the wing-fuselage system are
374 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
substituted. The reference axes and the signs of forces and moments are shown in Fig. 1-6. To
convey a feeling for the magnitude of the interference effects on
wing-fuselage systems, a few test results are given in Figs. 6-3 and 6-4. In Fig. 6-3a
the lift coefficient CL is shown plotted against the angle of attack a for a simple mid-wing system of a rectangular wing and an axisymmetric fuselage, and for the
wing alone. In the range of moderate angles of attack, the fuselage does not noticeably affect the trend of the CL(a) curve. The coefficient of maximum lift CL max, however, is markedly reduced by the fuselage. This can be understood by realizing that the flow about the wing of a mid-wing airplane is strongly disturbed by the fuselage, leading to premature flow separation. The lift coefficient CL versus the pitching moment coefficient cm for the wing alone and the wing-fuselage system is plotted in Fig. 6-3b. Here the fuselage causes a strong increase in the pitching-moment slope dcM/dcL. The inclined fuselage alone has a pitching moment
that tends to turn it into a crosswind position (see Sec. 5-2-3), and this pitching moment obviously is greatly increased by the effect of the wing. In Fig. 6-4, the rolling-moment coefficient cMX is plotted against the angle of
i
1.2
Figure 6-3 Lift and pitching
moment of a mid-wing system
and of the wing alone, from Molier and Trienes. Fuselage: ellipsoid of revolution of axis ratio 1: 7. Wing: rectangle of
0
aspect
-0,2 - 04 .80
ratio
el = 5,
profile
NACA 23012. (a) Lift coefficient cL vs. angle of attack a. 0°
8°
16
PX ° -0.08
0
-0.04
CM
0.04
0.08
Lift coefficient CL vs. pitching-moment coefficient cM. (b)
AERODYNAMICS OF THE WING-FUSELAGE SYSTEM 375
0.
0.0
Wing + fus elage 0.0.
Wing
0.0
ac'B,5° - 0.0
-0.0
0"'"300
Figure 6-4 Rolling moment due to side-20°
-10
0
P_
sideslip
70
.10°
slip of a high-wing system and of the wing alone, from M6ller; fuselage and wing of Fig. 6-3.
for a high-wing system also consisting of a rectangular wing and an
axisymmetric fuselage. The difference between the trends of the curves cMx(3) for the wing alone and for the wing-fuselage system is quite large. The effect of the fuselage of a high-wing airplane consists of a strong increase in the rolling moment due to sideslip acm,/aa. This effect is caused by the cross flow over the fuselage. The interference effects shown in Figs. 6-3 and 6-4 can be treated theoretically. Other interference problems, particularly those of the drag of wing-fuselage systems, are hardly accessible to theoretical determinations. Therefore, in these cases experimental studies are indispensible [11, 15].
Summary reports about the interactions between the wing and fuselage in incompressible, and to some extent in compressible flow, have been published by Wieselsberger [511, Muttray [34], Schlichting [39, 41], Ferrari [6], and Lawrence and Flax [26], as well as Ashley and Rodden [2]. Surveys of the aerodynamics of slender bodies have been given by Adams and Sears [1] and Gersten [7].
376 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
6-2 THE WING-FUSELAGE SYSTEM IN INCOMPRESSIBLE FLOW
6-2-1 Fluid Mechanical Fundamentals of the Wing-Fuselage Interference In the next section the quantitative computation of the interactions between wing and fuselage will be discussed, but a few physical explanations will be given here first. When putting together a wing and a fuselage, a flow about the wing-fuselage system results, with the fuselage lying in the flow field of the wing and the wing in the flow field of the fuselage. Thus, an aerodynamic interference exists between the fuselage and the wing, in that the presence of the fuselage changes the flow about
the wing and the presence of the wing changes the flow about the fuselage. Consequently, the computation of the flow about a wing-fuselage system can be accomplished by first computing the flows about the wing and fuselage separately,
and then adding the interference effects of the wing on the fuselage and the fuselage on the wing. These interference effects are obtained by satisfying the kinematic flow condition (zero normal component of the velocity on the surface of the wing-fuselage system).
The, flow field of a wing-fuselage system at subsonic velocity in symmetric, incident flow (angle of sideslip 13 = 0) is illustrated in Fig. 6-5. Figure 6-5a shows
the flow about the fuselage as affected by the wing. Along the fuselage axis, additive velocities normal to the fuselage axis are induced by the wing, which are
directed upward before the wing and downward behind it. In the range of the wing-fuselage penetration, the flow is parallel to the wing chord, corresponding to a constant downwash velocity along the wing chord. The fuselage is therefore in a curved flow with an angle-of-attack distribution a(x) varying along the fuselage axis as shown in Fig. 6-5a. This angle-of-attack distribution, induced by the wing, shows that the fuselage is subjected to an additive nose-up pitching moment. The effect of the fuselage on the flow about the wing is sketched in Fig. 6-5b. The component of the incident flow velocity normal to the fuselage axis
U. sin a. x
generates additive upwash velocities in the vicinity of the
fuselage. The effect on the wing of these induced velocities normal to the plane of the wing is equivalent to an additive symmetric angle-of-attack distribution over the wing span (twist angle). The flow field of a wing-fuselage system at asymmetric incident flow is shown schematically in Fig. 6-6. The flow about the wing-fuselage system with the angle of
can be thought to be divided into an incident flow parallel to the plane of symmetry of the velocity U. cos a U. and an incident flow normal to the plane of symmetry of the velocity U. sin 03 U43. The latter component of the incident flow generates a cross flow over the fuselage as illustrated in Fig. 6-6b, c, and d for a high-wing, a mid-wing, and a low-wing system, respectively. This cross flow over the fuselage results in an additive antimetric* distribution of the normal velocities along the span that is equivalent to an antimetric angle-of-attack distribution a(y). sideslip
*See footnote on page 190.
AERODYNAMICS OF THE WING-FUSELAGE SYSTEM 377
Upwash
Downwash
1
X
b
y
Figure 6-5 Symmetric flow about a wing-fuselage system (schematic). (a) Flow in the airplane plane of symmetry and angle-of-attack distribution «(x) on the fuselage axis. (b) Flow in a plane normal to the fuselage axis and angle-of-attack distribution cx(y) over the wing span.
The
lift distributions
over
the wing span generated by this angle-of-attack
distribution have reversed signs for high-wing and low-wing airplanes. The rolling moment (rolling moment due to sideslip), as affected by this antimetric lift distribution, is zero for the mid-wing airplane, positive for the high-wing airplane, and negative for the low-wing airplane. These findings are confirmed by the test results of Fig. 6-4, which show that the rolling moment due to sideslip Zc,L IaQ of a high-wing airplane is larger than for the wing alone. The effect of the fuselage on the wing in yawing motion may be interpreted, therefore, as the effect of a positive dihedral of the wing on the high-wing airplane, and as that of a negative dihedral on the low-wing airplane.
378 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
b
Figure d
over
a
6-6 Asymmetric flow wing-fuselage system
(schematic). (a) Wing planform. (b)
High-wing
airplane
with
angle-of-attack distribution a(y). (c) Mid-wing airplane. (d) Lowwing
airplane
with
attack distribution a(y).
angle-of-
AERODYNAMICS OF THE WING-FUSELAGE SYSTEM 379
6-2-2 The Wing-Fuselage System in Symmetric Incident Flow Total lift of a wing-fuselage system The first attempt at a theoretical description of the interference of a wing-fuselage system was made by Lennertz [27]. First, only the lift distribution on the wing and fuselage of such a system will be investigated. For simplicity, let the fuselage be an infinitely long circular cylinder as shown in
Fig. 6-7, whereas the unswept wing has an infinite span. For the portion of the wing not shrouded by the fuselage, let the lift distribution over the span be known and thus the circulation distribution r(y). The vortex system of the wing can be composed, from Fig. 3-20a, of horseshoe vortices of width dy and vortex strength T, as shown in Fig. 6-7b. To determine the lift of this arrangement generated at the fuselage, the kinematic flow condition must be satisfied on the fuselage surface, thus making the fuselage surface a stream surface. In a cross section normal to the fuselage axis far behind the wing, the flow in the yz plane is two-dimensional. The kinematic flow condition can here be satisfied by means of the reflection principle; that is, for every free vortex outside of the fuselage, a vortex reflected with respect to a circle has to be placed into the fuselage that has the same vortex strength but
y
Figure 6-7 Determination of the total lift
of a
wing-fuselage system. (a)
Rear view. (b) Plan view with vortex system. (c) Circulation distribution in span direction.
380 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
the opposite sense of direction of rotation. The reflected vortex belonging to the free vortex at station y is located at a distance YF = R2ly from the fuselage axis, where R is the radius of the fuselage cross section.* Thus, a circulation distribution is obtained on the fuselage as demonstrated in Fig. 6-7c.
The lift of the wing portion not shrouded by the fuselage L!, is obtained through integration of the circulation distribution over the span from Eq. (3-15) as L w= 2 e U... f r(y) d y
(6-2)
y=R
Analogously, the lift of the fuselage becomes, with dyF = -(R2 /y2) dy and r(yF) = r (y) for the bound vortex, S
R
r'' dyF 2Q Ucc f r(Y) LF = 2,o U f r (YF) = yF'-R 1S
y=R
dy y2
(6-3)
The total lift of the wing-fuselage system follows from Eqs. (6-2) and (6-3) as S
L(w+F) =Lw +LF= 2eUo, rr(Y) 1+ Rz) du y
.1
y=R
(6-4)
For numerical evaluation of this equation, an assumption must be made about the circulation distribution r(y). The simplest case is a constant circulation distribution T (y) = To = const. Here, Eqs. (6-2) and (6-3) yield for the ratio of fuselage lift to wing lift and for the ratio of fuselage lift to total lift: LF W
RS
1P,
LF
?7F
L(W+F)
1 + r1F
(6-5)
The latter ratio is presented in Fig. 6-8 versus the relative fuselage width 17F = R/s as curve 1. Lawrence and Flax [26] and Luckert [32] have shown that curve 1 of Fig. 6-8 may also be applied, in very good approximation, to different lift
distributions. Curve 2 of Fig. 6-8, from Spreiter [44], applies to wings of small aspect ratio (cf. Sec. 6-4). The result of Eq. (6-3) for the lift of fuselages may also be obtained from the integral of the pressure over the body surface or by means of the momentum theorem. The above considerations fail to give information about the distribution of the
lift of the fuselage over its length. This problem will be treated in the following section.
Lift distribution of the fuselage To determine the lift distribution over the fuselage length under the influence of the wing, the corresponding considerations for the
fuselage alone of Sec. 5-2-3 may be applied. It was shown there that the lift *It can easily be seen that the flow pattern of the two counter-rotating vortices at y and YF and at (y + dy) and (yF + dyF), respectively, contains, as a streamline, the circle of radius R about the origin.
AERODYNAMICS OF THE WING-FUSELAGE SYSTEM 381
01
--0,3
0.2 77F =
R
0.4
0.5
Figure 6-8 Ratio of the fuselage lift Lp to the total lift of a. wing-fuselage system L(yy+F) vs. relative fuselage width 77F = R/s. Curve 1, theory from Lennertz (r = const). Curve 2, theory from Spreiter (slender-body theory).
distribution over the fuselage length for a fuselage as shown in Fig. 6-9 is given by Eq. (5-28) as clxF
= 4'00 y d [,x(x)bF(x))
(6-6)
Here dLF is the lift force of a fuselage section of length dx, bF(x) is the local fuselage width, a(x) is the local angle of attack of the fuselage axis, and q. = oUU/2 is the dynamic pressure of the incident flow. To compute the lift distribution of the fuselage alone, the angle of attack in this equation has to be
S
Figure 6-9 The lift distribution of an IF
inclined fuselage.
382 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
taken as a(x) = a = const. For the lift distribution over the fuselage under the effect of the wing, the angle of attack has to be expressed as ti(x) = aoo 1i' aw(x)
(6-7)
where ax,(x) represents the upwash and downwash angles induced by the wing at the location of the fuselage (see Fig. 6-5a). For bF(O) = 0 = bF(IF), the total lift of
the fuselage under the influence of the wing is obtained from Eq. (6-6), in agreement with Eq. (5-29a), as LF = 0. As was shown in Sec. 5-2-3, this relationship is valid for inviscid flows. To compute the pitching moment at a variable angle-of-attack distribution a(x), Eq. (5-32) is already available. This pitching moment is independent of the position
of the reference axis because it
is
a free moment. The above method for the
computation of the wing-fuselage interference was developed by Multhopp [32]. The computation of the lift distribution over the fuselage length from Eq. (6-6) and
of the pitching moment from Eq. (5-32) requires the determination of the distribution along the fuselage axis of the angle of attack induced by the wing. This is a problem of wing theory that has already been treated in Sec. 24-5 for the two-dimensional case and basically in Sec. 3-2 for the three-dimensional case. A
comprehensive presentation of the computational procedures for the induced velocity fields of wings will be given in Chap. 7.
The fundamentals of the method for the computation of the lift distribution
and of the pitching moment can be understood from the simple case of a wing-fuselage system with a wing of infinite span, as shown in Fig. 6-10. The induced angle of attack of the inclined flat plate is given by Eq. (2-116) with A0 = a and A = 0 for n > 1 [see also Eq. (2-66)]. Hence, Eq. (6-7) yields, for the local angle of attack,
a(X) = a V XA,-1
for X>1 and X < 0
(6-8a)
where X = x/c is the dimensionless distance from the plate leading edge. This distribution is shown in Fig. 6-l Ob. Within the range of the wing, 0 < X < 1, there
is a,(x) = -a and thus a(X) = 0
for 0 < X < 1
(6-8b)
The local angle of attack a(x) from Eqs. (6-8a) and (6-8b) is discontinuous at the wing leading edge: The quantity a(x) drops abruptly from an infinitely large positive value to zero. At this station, daldx has an infinitely large negative value, requiring special attention when determining the lift distribution from Eq. (6-6). For clearness in the computation of the lift distribution, the discontinuity of the a(x) curve has been drawn in Fig. 6-10b as a steep but finite slope. With the local angle-of-attack change thus established, the lift distribution of Fig. 6-10c is obtained.* It has a large negative contribution in the form of a pronounced peak For a blunt fuselage nose and tail, Eq. (6--6) gives finite values for dLF/dx, contrary to the exact values dLF/dx = 0.
AERODYNAMICS OF THE WING-FUSELAGE SYSTEM 383
Figure 6-10 Computation of the lift distribution on the fuselage of a wing-fuselage system. (a) Geometry of the wing-fuselage system. (b) Angle-
of-attack distribution a(x). (c) Lift distribution dLF/dx.
directly before the wing leading edge. This is caused by the large negative value of da/dx close to the wing nose. The magnitude of this negative contribution is easily found when one realizes that for the fuselage section from the fuselage nose to a station shortly behind the wing leading edge, the lift force must be zero according
to Eq. (5-29a), because bF = 0 at the fuselage nose and a = 0 shortly behind the wing leading edge. Accordingly, the positive contribution LFI and the negative contribution LF2 are equal.
On the other hand, the lift distribution of the wing alone (without fuselage interference) has a strongly pronounced positive peak in the vicinity of the wing leading edge. Actually, this positive lift peak of the wing is reduced by the negative lift peak of the fuselage LF2 mentioned above. Hence, a lift distribution over the fuselage is obtained, including the shrouded wing area, given as the solid curve of Fig. 6-10c. Finally, this analysis shows that the total lift of the fuselage in the wing-fuselage system is approximately equal to the lift of the shrouded wing portion.
An example of this computational procedure and a comparison with measure-
ments is given in Fig. 6-11. The fuselage is an ellipsoid of revolution of axis ratio I : 7 that is combined with a rectangular wing of aspect ratio A = 5 in a mid-wing arrangement. Curve 1 shows the theoretical lift distribution from Eq. (6-6). It is in quite good agreement with the measurements in the ranges before and
384 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
o
r r
2 3
o
0
2 1
-Z
i -1
3
4
Figure 6-11 Lift distribution on the fuselage of a wing-fuselage system (mid-wing airplane). Fuselage: ellipsoid of revolution of axis ratio 1 : 7. Wing: rectangle of aspect ratio A = 5. Measurements from [41]; theory: curve 1 from Multhopp, curve 2 from
Lawrence and Flax, curve 3 from IF
curve 2, from Adams and Sears.
behind the wing. No result is obtained by this computational procedure within the range of the wing. The measured lift distribution shows a pronounced maximum in the vicinity of the wing leading edge. Curve 2 represents' an approximation theory
of Lawrence and Flax [26], which will be discussed later; it is in satisfactory agreement with the measurements in the range of the wing. Curve 3 will also be explained later. The -influence - of the wing - shape -- on- the - wing-11 selage - interference can - be
assessed best by means of the angle-of-attack distribution induced on the fuselage axis. For unswept wings, Fig. 6-12 illustrates the effect of the aspect ratio on the distribution of the angle of attack. All the wings have an elliptic planform. The angle-of-attack distribution has been computed using the lifting-line theory. For an elliptic circulation distribution its value becomes, Eq. (3-97),
= x/s and the coordinate origin x = 0 lies on the c/4 line. Because = 8X/irf with X = x/l, and with the relationship between CL and a. of Eq.
where
(3-98), Eqs. (6-9) and (6-7) yield
AERODYNAMICS OF THE WING-FUSELAGE SYSTEM 385
(A)2 a (x)
a
=1--
4
+ Ya _- X
2 a+
X
2
(6-10)
1+
In Fig. 6-12, a/a. is shown versus X.* Hence, in the range before the wing, the upwash angles become markedly smaller when the aspect ratio A is reduced. In the range behind the wing, however, the downwash angles increase with decreasing aspect ratio. At the 4 c point, all curves have the value a = 0, as should be expected because of the computational method used (extended lifting-line theory = threequarter-point method). The effect of the sweepback angle on the distribution of the angle of attack is shown in Fig. 6-13 for a wing of infinite span, constant chord, and unswept middle section. This latter section represents the shrouding of the wing by the fuselage as shown in Fig. 6-1. The induced angle-of-attack distribution on the x axis is obtained from the lifting-line theory according to Biot-Savart as aw (x) = -
r
with
U.c
r
x-}
xa+gb,sin9
2vU.. x xcosrp+sF.sin T
= 7 acc cos
(6-1 la) (6-11b)
where r is the circulation of the lifting line, cp is the sweepback angle, and SF is the semiwidth of the unswept middle section. The relationship between the circulation *For this illustration, the coordinate origin has been laid on the leading edge. tHere, the coordinate origin lies at the c/4 point of the root section.
9.
16 10, -2
-1
0
1
2
Figure 6-12 Distribution of the local angle of attack on the fuselage axis of wing-fuselage systems with wings of several aspect ratios A.
386 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM 3
I
I
r
r
SF 14
D
00 ooo.ol
Q
Z
-1
X
2
3
C
Figure 6-13 Distribution of the local angle of attack on the fuselage axis of wing-fuselage systems with swept back wings of infinite span and with rectangular middle portion (lifting-line theory). Solid curves, of = sp/c = 0. Dashed curves, ep = 0.5.
r and the angle of attack ate, of a swept-back wing of infinite span is expressed by Eq. (6-1 lb), because
cL = 2r/ U.c and cL = 27ra. cos p from Eq. (3-123).
Consequently, Eq. (6-1 la) maybe written in the form a (X)
cos 97 X + XZ + QF sin 97
a00
2X X cos g)+ aF sin T
(6-12)
with X = x/c and QF = sF/c. The angle-of-attack distributions computed by this equation are plotted in Fig. 6-13 for sweepback angles cp = 0, +45, and -450, and for (Yp = 0 and 0.5.* From Fig. 6-13 it can be seen that the upwash before the wing is reduced in the case of a backward-swept wing and the downwash behind the wing is increased. In the case of a forward-swept wing, the reverse occurs. As would
be expected, introduction of the rectangular middle section reduces the effect of sweepback. The distribution of the induced angle of attack on the fuselage axis for the swept-back wing without a rectangular middle section (sF = 0) is given, from Eq. (6-1 la), as
() _ - 2' U0,rx cos 9
06W x
1
'-x
'
sin
(6-13)
I xi
Since aw = -T/27rU.x for the unswept wing, Eq. (6-13) shows that the effect of the sweepback angle on the induced downwash angle may be expressed by a factor. The procedure discussed so far for the determination of the wing influence on the angle-of attack distribution of the fuselage does not give any information about *Compare the footnote on page 385.
AERODYNAMICS OF THE WING-FUSELAGE SYSTEM 387
the distribution in the range of the wing, as may be seen from Fig. 6-11. Lawrence and Flax [26] developed a method allowing determination of the angle-of-attack distribution over the entire fuselage length, including the shrouded wing section. The basic concept of this method is indicated in Fig. 6-14. Contrary to the previous approaches, which were based on an undivided wing, now the fuselage is taken as being undivided and the wing as divided. Consequently, the effect of the two partial
wings on the fuselage is determined, whereby both the x component and the z component of the induced velocity must be taken into account.
The first contribution to the lift distribution is generated by the longitudinal velocity components u(x) because they determine the pressure distribution on the fuselage surface by cp = -2u/U.. The induced velocities on the surface z = R cos 6 can be expressed by
_ z (z-c) = UUR cos 6 do u = z (a az -u z=0 dx 8x z=o 1
Here it has been taken into consideration that au/az = 8w/ax, because the flow is irrotational, and further that the simple relationship daw/dx = da/dx follows from
Eq. (6-7). The second contribution to the lift distribution is generated by the
b -2s
I
U-
9
Figure 6-14 Computation of the lift distribution on the fuselage of a wing-fuselage system according to the theory of Lawrence and Flax.
388 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
upwash velocities on the fuselage axis resulting from the vortex system of the two wing parts. The corresponding pressure distribution is obtained from Eq. (5-25a). Thus, the resulting pressure distribution on the fuselage is C (x, $) _ -4 cos 6 dx [a(x)R(x)]
Introduction of this expression into Eq. (5-27) and integration over 0 <
(6-14) < 27r
yield the total lift distribution dd F = 4 1rq R (x)
dx [a(x)R(x) ]
(6-15)
Note the difference from Eq. (5-28). For the case a = const (fuselage alone), the equations are identical. Lawrence and Flax [26] have evaluated Eq. (6-15) assuming that the circulation distribution is constant on either wing part. This result is given as curve 2 of Fig. 6-11. For the fuselage portions before the' wing and within the
wing range, agreement of this approximation theory with measurement is quite good. For the fuselage portion behind the wing, the deviations from measurement are considerable. Therefore, a correction for this range has been given by Adams and Sears [1], shown as curve 3. It should be mentioned in this connection that the computational procedure of Multhopp [32] leads to nearly the same results.
Lift distribution of the wing Since the effect of the wing on the fuselage has been discussed, the effect of the fuselage on the lift generation on the wing will now be investigated more closely. A typical test result on this problem is shown in Fig. 6-15. For a mid-wing system consisting of a rectangular wing and an axisymmetric
5.6°
0.
f
1.
CC
71.4'
-7y I
I
0
0.2
0.6
0.4
7-3
---Wing alone
0.8
-Mid-wing airplane
1.0
Figure 6-15 Measured lift distributions on the span for a mid wing system and for the wing alone at several angles of attack, from [41]. Fuselage: ellipsoid of revolution of axis ratio 1:7. Wing: rectangle of aspect ratio A = 5. The curves for the mid-wing airplane include only the fuselage lift within the wing range.
AERODYNAMICS OF THE WING-FUSELAGE SYSTEM 389
Figure 6-16 Induced angle-ofattack distribution of a wing-fuse-
lage system. The fuselage is an infinitely long circular cylinder;
a
R = co /2. Curve 1, angle-of-attack distribution a(x) = a,,. = const
over the entire fuselage length. Curves 2 and 3, angle-of-attack distributions a(x) before and behind
-0
We-
the wing are constant,
a(x) = 0 within the wing range. Curve 2 for the upswept wing, curve 3 for the swept-back wing = 45°. Curves 2 and 3 give the distribution of the induced angle of attack on the a -point line of the wing.
fuselage, and for the wing alone, distributions of the local lift coefficients over the
span are shown. These data have been extracted from comprehensive pressure distribution measurements of Moller [15] on wing-fuselage systems. In the case of the wing-fuselage system, the lift coefficients refer to the wing portion shrouded by the fuselage. The lift distributions on the wing portions outside of the fuselage at three different angles of attack are consistently little affected by the fuselage.
However, within the fuselage range, a considerable drop in the lift coefficient occurs. This reduced wing lift within the fuselage range has previously been discussed in connection with Fig. 6-10c.
For the theoretical determination of the influence of the fuselage on the lift distribution of the wing, the additive angle-of-attack distribution from Fig. 6-5b has to be determined that is the result of the cross flow over the fuselage. Figure 6-16 shows as curve 1 the additive angle-of-attack distribution induced by an infinitely long fuselage of circular cross section. Outside the fuselage, the induced angle of attack J a = w/U for mid-wing systems is given by da(y) a00
=
R2 y
(y
>R)
(6-16a)
where R is the radius of the circular cylinder. For the range -R
390 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
da
a = - (1 -2)
(0
(6-16b)
The angle-of-attack distribution thus determined has a very sharp peak of d a/a = +1 on the fuselage side wall, whereas the value d a/a = -1 is reached on the fuselage axis, that is, the local angle of attack a = a + d a = 0 on the axis. By using this angle-of-attack distribution of the wing according to curve 1 of Fig. 6-16, the fuselage influence on the wing is greatly overrated because it is based on the assumption that the angle of attack of the fuselage is a = a within the wing range, too. Multhopp [32] computed lift distributions with angle-of-attack distributions of this kind. Compare also Liess and Riegels [32] and Vandrey [47].
A better approximation for the fuselage influence on the wing is obtained under the assumption that the wing turns the flow within the wing range parallel to the fuselage axis, that is, that a = 0 in this range. The corresponding distribution of the induced angles of attack over the span can be determined by arranging a dipole distribution on the fuselage axis that is dependent on x. This procedure has been given for the fuselage alone in Sec. 5-2-3. With r cos 6 = z and r2 = y2 + z2 and
with m from Eq. (5-24), Eq. (5-20a) yields for 4 a = w/U = (aq/az)/U. in the wing plane z = 0, IF
a(x') R2(x')
1
da(x, y) =
(x -x')2 -1-y2 a
2
dx'
(6-17)
0
valid for y >R. For an infinitely long fuselage of constant width whose angle of attack is constant before and behind the wing and zero (a = 0) within the wing range, the result is 4a (x, y) MOO
- 1 R2 - l4 - x 2 y2 0u - x)3 + y3 2
x 11x2 +. y3
(6-18)
Here, l0 is the wing chord at the fuselage side wall. The distribution of the induced angle of attack, computed with Eq. (6-18), is shown in Fig. 6-16 as curves 2 and 3 for an unswept wing and for a swept-back wing with p = 450, respectively. The computed values are valid for the a c point of the wing. Comparison of curves 2 and
3 with curve 1 demonstrates that this refined computational method leads to a considerably smaller fuselage influence.
Neutral-point position of wing-fuselage systems Besides the changes of the lift distributions of fuselage and wing, the change of the neutral-point position is of particular importance for flight mechanical applications (see Sec. 1-3-3). The distance of the neutral point from the moment reference axis is generally given by xN = -dM/dL. Hence, for the wing-fuselage system it becomes XN
M(W+F)
(6-19a)
(W +F)
dMw
dLw
dMF dLw
(6-19b)
AERODYNAMICS OF THE WING-FUSELAGE SYSTEM 391
where M(W+F) is the pitching moment and L(W+F) is the total lift of the wing-fuselage system. The pitching moment of the wing-fuselage system may be
composed of the contributions of the fuselage MF and of the wing MW. The fuselage contribution can be computed as described previously. The wing contribu-
tion will be taken to be the moment of a wing with rectangular middle section (substitute wing). Since the fuselage influence on the wing is generally small, it can often be disregarded (see Hafer [11]). The lift of the wing-fuselage system L(w+F)
given approximately by the lift of the wing alone Lw, as was shown earlier. Because M(W +F) = MW + MF and L(w+F) LF, Eq. (6-19b) is obtained. The first term gives the neutral-point position of the wing with rectangular middle section, which can be determined through computation of the lift distribution of such a wing according to the lifting-surface method. The second contribution gives the neutral-point displacement caused by the fuselage including the influence of the is
wing on the fuselage. It is advantageous to refer the neutral-point position of the wing-fuselage system to the position of the neutral point of the wing alone, that is, of the original wing (Fig. 6-1). As reference chord, that of the original wing is chosen likewise. The neutral-point displacement of the wing-fuselage system from the aerodynamic neutral point of the wing alone becomes, from Eq. (6-19b), (A XN)(W+F)
_ (A 4N')W +
(A XN)F
CA
CA,
CA
(6-20)
Here (A xS)W is the neutral-point displacement because of the planform change of the wing (introduction of the rectangular middle section into the range shrouded by the fuselage) and (A xN)F is the neutral-point displacement because of the fuselage. Obviously, the first contribution can be of real importance for only swept-back and
delta wings. By considering, as a first approximation, the displacement of the geometric neutral point only, the neutral-point displacement of the swept-back wing of constant chord becomes (A x125)W
4
C
!1'ij
tan rp
(6-21)
with 'qF as the relative fuselage width from Eq. (6-1).
The second contribution in Eq. (6-20), that is, the neutral-point displacement due to the fuselage, is obtained from the fuselage moment MF by the relationship (AXN)F CA
_
-
1
1 dMr dam
Acu qro daa; dCL
(6-22)
where dcL/dca is the lift slope of the wing (see Sec. 3-5-2). The neutral-point displacement caused by the fuselage of Eq. (6-22) depends mainly on the following geometric parameters, as intuitively plausible: wing rearward
position, fuselage width ratio, and sweepback angle. In Figs. 6-17-6-19, a few computational results from Hafer [11] on the influence of these parameters are presented and compared with measurements.
The neutral-point displacement due to the wing rearward position for an
392 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM -0.16
4
ti 0.08
7
o Mid-wing 0 High-wing A Low-wing
of
02
0-7
airplane
0,5
0.4
0.5
0.7
e
IF
Figure 6-17 Neutral-point shift of wing-fuselage systems due to the fuselage effect vs. the wing rearward position, from Hafer. Fuselage: ellipsoid of revolution of axis ratio 1: 7. Wing: rectangle of aspect ratio A = 5.
unswept wing is given in Fig. 6-17 as a function of the widely varied wing rearward
position. The fuselage causes an upstream displacement of the neutral point (destabilizing fuselage effect) that increases with the rearward wing position. The wing high position, also varied in these measurements, has no marked effect.Agreement between theory and experiments is good.
Figure 6-18 illustrates the effect of the sweepback angle on the neutral-point -0,
-0.16
1-0,08
Theory
k
A=0.2 (
0
0.04 0
V
01i9
_Z L'
-10°
0°
10u
k=0.2 Zllu
.V°
40 q
9
F Figure 6-18 Neutral-point shift of wing-fuselage systems due to the fuselage effect vs. sweepback angle of the wing, from Hafer. Fuselage: ellipsoid of revolution of axis ratio 1: 7. Wing: aspect ratio A = 5; taper a = 1.0 and 0.2.
AERODYNAMICS OF THE WING-FUSELAGE SYSTEM 393
Figure 6-19 Neutral-point shift of wing-fuselage systems due to the fuselage effect vs. wing rearward position, from Hafer. (I) Sweptback wing; A = 2.75; X = 0.5; p = 50°. (II) Delta wing: A = 2.33; X=0.125. Curve 1, fuselage with pointed nose. Curve 2, fuselage 611F
with rounded nose.
position caused by the fuselage. The measurements are for wing-fuselage systems with wings of constant chord (A = 1) and with trapezoidal wings (A = 0.2). The neutral-point displacement becomes markedly smaller when the sweepback angle increases. It is noteworthy that the neutral-point displacement is almost zero for strong sweepback (gypcz:l 45°). Here, too, agreement between theory and measurement
is quite good. The first theoretical studies about the effect of the sweepback angle on the neutral-point displacement
caused by the fuselage was conducted by
Schlichting [401.
Finally, in Fig. 6-19, results are given on the influence of the wing rearward position of a swept-back wing and a delta wing. The swept-back wing has the aspect ratio A = 2.75, the taper A = 0.5, and the sweepback angle of the quarter-point line cp = 50°. The neutral-point position of this wing has been shown in Fig. 3-37b. The delta wing has the aspect ratio A = 2.33 and the taper A = 0.125. The results of Fig. 6-19 are given for two different fuselage shapes, namely, a pointed and a rounded fuselage front portion. For either wing, in agreement with Fig. 6-17, a considerable increase in the neutral-point displacement is caused by the fuselage
when the wing is moved rearward. Here, too, agreement between theory and measurement is good. Important contributions to the interference between a swept-back wing and a fuselage are also due to Kuchemann [24].
Drag and maximum lift of wing-fuselage systems The interference effect of wind fuselage systems on drag and maximum lift lies mainly in the altered separation behavior when wing and fuselage are put together. These effects are hardly accessible to theoretical treatment, however, and their study must be limited to experimental approaches. The first summary report hereof comes from Muttray [34] ; compare also Schlichting [38]. Very comprehensive experimental investiga-
tions on the interaction of wing and fuselage, particularly concerning the drag
394 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
problem, have been conducted by Jacobs and Ward [15] and by Sherman [151. For drag and maximum lift of a wing-fuselage system, the low-wing arrangement is particularly sensitive, because the fuselage lies on the suction side of the wing, strongly influencing the onset of separation at larger lift coefficients. Through
careful shaping of the wing-fuselage interface by means of so-called wing-root fairings, the flow can be favorably affected in this case, that is, the onset of separation can be shifted to larger angles of attack.
The investigations of Jacobs and Ward [15] and of Sherman [15] cover a comprehensive program on two different 'fuselages (circular and rectangular cross sections) and two wings of different profiles (symmetric and cambered). Varied were the wing rearward position, the wing high position, and the wing setting angle. Included in the study was the effect of wing-root fairings. The drag of a wing-fuselage system depends predominantly on the wing high position, and very little on its rearward position and its setting angle. In Fig. 6-20, the lift coefficient CL is plotted against the coefficient of the form drag 2 CL
_
CDe -
CD
-
(6-23)
of several wing-fuselage systems. The coefficient of the form drag is obtained as the difference of the coefficients of total drag and induced drag. These wing-fuselage systems are a mid-wing airplane with round fuselage and low-wing airplanes with round and square fuselages. For comparison, the wing alone is added as curve 1. A strong drag increase above a certain lift coefficient is characteristic for wing-fuselage
systems. It is the result of the onset of separation caused by the fuselage. This 7
1.
2 I
i
--
-
4
I
0.2
0
Figure 6-20 Lift coefficients of wing-fuselage systems vs. drag coefficients, from Jacobs and
j
-0.2 -04 0
Ward. CDe = coefficient of the form drag I
0.02
0.04
I
0.06
cDe
I
0.06
from Eq. (6-23). Fuselages with circular and
i
0.10
0.12
0.14
square cross sections, wing profile NACA 0012.
AERODYNAMICS OF THE WING-FUSELAGE SYSTEM 395
1.e
Wing
L-7-
I
___L - Wing + fuselage
1,2
0,4
1,6
.o Wing 1
_
X M
E
J
1
T
4
Q
ge
Mid-wing airplane
t -Q8
-0.4
0
04
0.8
L
Figure 6-21 Maximum lift coefficients of wing-fuselage systems, from [38). Fuselages with circular cross sections, wing profile NACA 0012. (a) Maximum lift coefficient vs. wing rearward position, zo /10 = 0. (b) Maximum lift coefficient vs. wing high position, e0 /10 = 0.
phenomenon is most pronounced in the low-wing system with round fuselage, curve 3, where separation begins very early at CL = 0.6. Here fuselage side wall and wing upper
surface form an acute angle that particularly promotes boundary-layer separation. Considerably more favorable than the low-wing airplane is the mid-wing airplane, curve 2, because here the wing is attached to the fuselage at a right angle. By going from a round to a square fuselage, the conditions may be further improved, as shown by curve 4 for the low-wing airplane. Theoretical results on the pressure distribution at the wing-fuselage interface are given by Liese and Vandrey [47] for the case of a symmetric wing-fuselage system (mid-wing) in symmetric incident flow (CL = 0).
The maximum lift of wing-fuselage systems depends on both the wing high position and the wing rearward position. A survey of the CLmax values for several high and rearward positions is given in Fig. 6-21. From Fig. 6-21a, the maximum lift coefficient CLmax decreases with increasing rearward position. In the most favorable case, CLmax of a wing-fuselage system is equal to that of the wing alone. With regard to the wing high position, the mid-wing arrangement is least favorable, as shown by Fig. 6-21b (compare also Fig. 6-3). From this value for the mid-wing arrangement, CLmax increases when the wing is shifted to both high- and low-wing positions.
396 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
6-2-3 The Wind Fuselage System in Asymmetric Incident Flow Rolling moment due to sideslip of a wing-fuselage system In asymmetric incident
flow of a wing-fuselage system, the lateral component of the flow about the fuselage creates an additive antimetric distribution of the angle of attack of the wing as discussed in Sec. 6-2-1 and demonstrated in Fig. 6-6. It has reversed signs for high-wing and low-wing airplanes, and it is zero for mid-wing airplanes. This antimetric angle-of-attack distribution generates an antimetric lift distribution at the wing and thus a rolling moment due to sideslip. This additive rolling moment due to sideslip caused by the fuselage also has reversed signs for high-wing and low-wing airplanes.
For a theoretical assessment of the influence of the fuselage on the lift distribution of the wing, the antimetric angle-of-attack distribution as shown in Fig. 6-6 must be determined as caused by the cross flow over the fuselage with velocity U,. sin 0 ~ U43. This angle-of-attack distribution d a = w/U for an infinitely long fuselage with circular cross section (radius R) becomes
= - 2R2
(Y >yo)
Z3)2
(y2
(6-24)
where the fuselage cross section is given as in Fig. 6-22 as yo + zo = R2 . Within the range of the fuselage, that is, for yo < y < +yo , d a has to be taken as being zero, Ace= 0. For the wing without dihedral, following Fig. 6-2a, z has to be replaced in Eq. (6-24) by zo (z = zo). Thus the angle-of-attack distribution may be expressed by the dimensionless coordinates yls = rt and zo/s = o with T1F = R/s as the relative
fuselage width. The angle-of-attack distributions computed by this method are shown in Fig. 6-22 for two values of o . They have a very pronounced maximum near the fuselage axis (at 77 = 0.578"0), which, however, in some cases lies within the fuselage, and thus does not contribute to the lift distribution. To determine the angle-of-attack distribution of a fuselage of finite length, a consideration equivalent to that of Sec. 6-2-2 [see Eq. (6-17)] leads to IF
d a(x, y, z)
2o
f 0
x2 yzR(x')
(
)
+ Z2 5
dx'
(6-25)
Y
As will be shown later, it is sufficient in most cases, however, to assume an infinitely long fuselage.
In Fig. 6-23, the rolling moments due to sideslip acm la¢ of a low-wing, a mid-wing, and a high-wing fuselage system from measurements of M611er [15] are
plotted against the lift coefficient CL. For comparison, the values for a wing without dihedral and for a wing with a dihedral of v = 30 are also shown. The fuselage causes a parallel shift of the curve for the wing alone. Thus the fuselage influence is reflected in a contribution to the rolling moment due to sideslip, independent of the lift coefficient, corresponding to the contribution of the
18
16
14
12
10
0.2
0,4
0,6
17 -W
1.0
1018
Figure 6-22 Additive angle-of-attack distribution of wing-fuselage systems at asymmetric incident flow. Fuselage of circular cross section.
0
Oa
0.1,
OR
.01101
.000
L
I
-0.04
Figure 6-23 Coefficient of the rolling moment due to sideslip acM /a(3 vs. lift coefficient CL of wing-fuselage systems,
i
OHM
'
from Nloiler. Fuselage: ellipsoid of revolu-
-0.12 'L
-0.1
02
tion of axis ratio 1:7. Wing: rectangle
i
0,4
CL
06
1.0
A = 5. L = low-wing airplane, M = midairplane, 11 = high-wing airplane,
wing
W = wing alone (v = angle of dihedral). 397
398 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
dihedral at the wing alone. Figure 6-23 shows that the effect of the fuselage on the rolling moment due to sideslip may be replaced by that of an "effective dihedral"
of the wing. Here the high-wing airplane has a positive effective dihedral, the low-wing airplane a negative effective dihedral. This fact is taken into account in airplane design: In order to obtain approximately the same rolling moment due to sideslip for different wing high positions, the low-wing airplane is given a considerably larger geometric dihedral than the high-wing airplane. Following the above procedure, Jacobs [16] determined theoretically the
fuselage influence on the rolling moment due to sideslip for an infinitely long fuselage. In Fig. 6-24, results are plotted of his computations for the additive rolling moment due to sideslip d (acMX/a p) as a function of the wing high position zo /R Here the coefficient of the rolling moment due to sideslip is defined as Mx = with s being the semispan of the wing. These theoretical results are compared. with measurements by Bamber and House [16] and by Moller [15]. Theory and measurements are carried to large wing high positions at which wing .
and fuselage no longer penetrate each other. Agreement between theory and measurement is very good. A closed formula may be obtained for the rolling moment due to sideslip caused by the fuselage by introducing into Eq. (3-100) the angle-of-attack distribution from Eq. (6-24) with z = zo or yls = rl, zo/s = fl, and R/s = nF, respectively: +1 7rA
4+2
8 co r 1F
7J2
7
1
i - n2
(rl2+ C0)2
d
77
(6-26)
770
all
r
a o
2
i
i
i
2R
i
-0.
2 1
Figure 6-24 Additive rolling moment due to sideslip vs. wing high position, from Bamber and House and from Moller. Theory from
-0.06
-
2 -008
Jacobs [the theoretical curves have been corrected considering the ex-1
I
2
tended lifting-line theory (cL,0 = Relative fuselage width 'nF = 1:7.5; aspect ratio A = 5. 21r) ]
.
AERODYNAMICS OF THE WING-FUSELAGE SYSTEM 399 7
ZO
R
7.0 ±0,8
60
±0.6
2R 2s
5
±0.4
Z30
20
±0.2
10
Figure 6-25 Effective dihedral angle veff for a 0.0
0.04
012
wing-fuselage system of np = R/s and wing high position z0 /R for fuselages of circular
0,20
0,16
cross sections. Theory from Eq. (6-28).
T1F
Here, r70 = 7F - o is the coordinate on the fuselage surface and k = 7r:1/c' A/2. For a simplified integration in Eq. (6-26), Multhopp [32] gave the value of unity to the square root in the integrand and changed the upper integration limit from unity to 2/7r. For o < (2/ir)2, this leads to a
"
"A k z T4
i 2
. (77F
1
Z + arcsin F - :z Co
(6-27)
T?F
which is valid for fuselages with circular cross sections and wing high positions
-R
eff -
2 1_ 3 nF B 0
1-( LO
),
-r-
aresin R - 777F R
(6-28)
In Fig. 6-25, the computed effective dihedral angle is plotted against the relative fuselage width 71F for several wing high positions zo /R. The effective dihedral angle increases strongly with increasing relative fuselage width 77F and increasing wing high position. For instance, for 71F = 0.12 and z0 /R = ± 1, its values are Veff = +3 and
-3°, respectively. Multhopp [32] conducted that kind of computation for fuselages of elliptic cross sections. Computations of the rolling moment due to sideslip for other fuselages have been conducted by Maruhn [16] . Some of his results are presented in Fig. 6-26. Fuselages with angular cross sections produce a particularly large rolling moment due to sideslip. All the theoretical results discussed so far are valid for
400 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
io
bF F 0.18
0.16
0.14
0,08
0.05
Figure 6-26 Additive rolling moment due to sideslip of wing-fuselage systems vs. wing high position for several
/A17
0.04
0,02
0 7
22
114
OB
bF
98
to
1.2
1.,
shapes of the fuselage cross section, from Maruhn [the theoretical curves have been corrected considering the extended lifting-line theory (c' , = 2ir)]. Wing: ellipse A = 3.8. Relative fuselage width 'RF = 6. Fuselage crosssection ratios hF/bF = 1.0 and 1.5.
infinitely long fuselages. Braun and Scharn [16] computed the effect of fuselages of finite lengths.
Yawing moment due to sideslip and side force due to sideslip of a wing-fuselage system The wing-fuselage arrangement has a quite small effect on the yawing moment due to sideslip. Essentially, the right value for the yawing moment due to
of a wing-fuselage system can be obtained by adding the stabilizing contribution of the wing (Sec. 3-5-3) to the destabilizing contribution of the
sideslip
fuselage (Sec. 5-2-3). Figure 6-27 shows the yawing moment due to sideslip of three different wing-fuselage systems (low-wing, mid-wing, and high-wing arrangements)
from measurements of Moller [15]. For comparison, the wing alone and the fuselage alone are also shown. Obviously, no substantial interference exists. Furthermore, it should be noted that, for the yawing moment of the entire airplane, the usually destabilizing contribution of wing and fuselage is much smaller than the stabilizing contribution of the vertical tail assembly (see Chap. 7). The interference
of wing and fuselage is more pronounced, however, for the side force due to sideslip. Figure 6-28 shows the side force due to sideslip, again from measurements
of Moller [15], for the three wing-fuselage systems of Fig. 6-27. Note that at CL = 0.2, the side force due to sideslip for the high-wing and the low-wing airplanes is about twice as large as for the mid-wing airplane. Also, the coefficient of the side
AERODYNAMICS OF THE WING-FUSELAGE SYSTEM 401 0.1
--
F1
0.12
Figure 6-27 Yawing moment due
0
to sideslip of wing-fuselage systems vs. lift coefficient. Measurements from Moller. Fuselage: ellipsoid of revolution 1:7. Wing: rectangle A = 5. L = low-wing airplane, M =
1-7 1 W
0
02 CL
0.6
0.4
0.8
mid-wing airplane, H = high-wing airplane, W = wing alone, F = fuse-
1.0
lage alone.
force due to sideslip of the high-wing and the low-wing airplanes depends strongly on the lift coefficients. The larger values of acy/aa and their dependence on the lift coefficient for low-wing and high-wing planes find their explanation in the induced sidewash. Puffert [16] and Gersten and Hummel [8] studied these phenomena theoretically.
6-3 THE WING-FUSELAGE SYSTEM IN COMPRESSIBLE FLOW
6-3-1 The Wing-Fuselage System in Subsonic Incident Flow Fundamentals The following discussions on the flow about a wing-fuselage system
at subsonic velocities will be limited to the case of straight flight. The effect of 0.2,
0.
I, y
0.05
W 0.2
CL
0.4
-
0.5
0.8
1.0
Figure 6-28 Side force due to sideslip vs. lift coefficient. Measurements from Moller (system as in Fig. 6-27).
402 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
compressibility on the flow about a wing has been explained by means of the Prandtl-Glauert-Gothert rule in Sec. 4-4-2 for wings, and for fuselages in Sec. 5-3-2.
This rule renders feasible the determination of a subsonic flow (Ma., < 1) about wings and fuselages by means of a transformation to incompressible flow. By this means the incompressible flow will be computed for a transformed wing and a transformed fuselage. The transformation of the geometric quantities for wing and fuselage is given by Eqs. (4-66), (4-67a), (4-67b), (4-68a)-(4-68c), (5-51), (5-52a), and (5-52b), where the quantities for incompressible flow are marked by the index "inc" and those for the compressible flow are given without the index. These quantities are as follows: Coordinates:
1 -Ma.
xinc = x, yinc =Y
(6-29)
Zinc = Z
1 _-Ma".
Span:
binc = b
1 -Ma;,
Wing chord:
Cinc = C
(6-30b)
Taper:
zinc = a
(6-30c)
(6-30a)
A inc=A 1-Mal
Aspect ratio: Sweepback:
cot Pinc = cot P
Fuselage width:
bF inc = bF
Fuselage length:
IF inc =1F
(6-30d)
1 -Mat
(6-30e)
l -Mat
(6-3 la)
(6-31b)
By computing the incompressible flow for the transformed wing-fuselage system at
the angle of attack of the compressible flow, the transformation of the pressure coefficient from Eq. (4-69) becomes 1 -MaCpinc
CP
1
(«inc = a)
(6-32)
Through an analogous transformation, the lift coefficients and the pitching-moment coefficients of wino fuselage systems are obtained. The discussions about the incompressible flow over wing-fuselage systems of Sec. 6-2-2 led to the conclusion that the lift slope of the wing-fuselage system is
little different from that of the wing alone if the relative width of the fuselage is small to moderately large. Consequently, the relationship Eq. (4-74) for the wing alone applies directly to the wing-fuselage system, or dC
2-rA
,
(da)(W+F)
V(1
- Ma") Ac
4- 2
(6-33)
*For the fuselage, a different transformation formula of the pressure coefficient was given by Eq. (5-53), where the angle of attack was transformed according to Eq. (5-52d). However,
within the framework of the linear lift theory, the Eq. (6-32) for the fuselage is equivalent to Eqs. (5-52d) and (5-53).
AERODYNAMICS OF THE WING-FUSELAGE SYSTEM 403
The dependence of the neutral-point position on the Mach number of a wing-fuselage system follows immediately from the relationships just stated, because XN/C = -dCM/dCL, as XN(W+F)
XN(W +F) inc
CA
CA
(6-34)
Here XN(W +F) inc is the neutral-point position of the wing-fuselage system at
incompressible flow as transformed according to Eqs. (6-29)-(6-31). As an example, the lift slopes and neutral-point positions are presented in Fig. 6-29 against the Mach number. These measurements of Schneider [42] are compared with
theoretical results. The lift slope is little affected by the fuselage; the neutral-point displacement, however, shows a considerable fuselage influence. Results for wing-fuselage systems with a rectangular and a delta wing are also available in [42]. Investigation of the wing-fuselage system by means of the panel method As a result of
utilizing efficient computers, methods have become more useful that are based on singularities distributed on the body surface, thus satisfying exactly the kinematic boundary conditions. Such generalizations have the advantage that geometric restrictions in the body shape are essentially eliminated. Based on the computational procedure for the displacement flow of Smith and Hess [13], the simultaneous treatment of the displacement flow and the lift flow of wing-fuselage systems has been presented independently by Kraus and Sacher [22] and Labrujere et al. [25]. In the method of Kraus and Sacher, the displacement flow is generated through a s
0
Nas
01 1
i
0
-02
-
o
-0.3
±
OS
a
I
_041
S
0.6
07
0.8
Mc -
99
70
OS
b
0.5
07
0.3
0.9
70
He
Figure 6-29 Lift slope (a) and neutral-point shift (b) due to the fuselage effect vs. Mach number for axisymmetric fuselage with swept-back wing. Measurements from Schneider; theory for o) Wing alone, A = 2.75, A = 0.5, p = 52.4°. incompressible flow from Sec. 6-2-2. (o
(v - -- n) Wing and fuselage, 1F/d = 12.5, e/d = 7.25, b/d = 5.0. (c - - - - - c) Wing and fuselage, IFId = 10, e/d = 6.5, b/d = 3.33.
404 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
source-sink distribution on the surface of the body, the abrupt change of the potential of the circulation flow through a vortex distribution within the body. The total potential results from the superposition of the individual contributions. The defining equations for the as yet unknown singularities are established by means of the kinematic boundary condition, to be satisfied on the surface. The flow conditions for the displacement problem are expressed by the requirement that no flow is to penetrate the body surface, that is, that the velocity normal to the body surface is zero. For the lift problem (circulation flow), this condition requires
smooth flow-off at the trailing edges of the lift-producing surfaces. The distribution of the singularities and the perturbation potentials are obtained from the solutions of the equations defining the singularities. Thus the total potential and the velocities
and pressure coefficients are obtained in the entire flow field and on the body surface. The pressure coefficients are computed with all three of the components of the perturbation velocity. A suitable approach to the solution of the defining equations is found in the panel method. The singularities, first assumed to be distributed continuously, are now assumed to be constant on small flat surfaces (panels) and thus are accessible to analytic integration of their defining equations. For the displacement flow, panels covered with a constant source-sink density as in Fig. 6-30a are distributed on the surface. For the circulation flow, panels on the inner surface are used, and on the edges of these panels a vortex filament is laid of constant vortex strength. This leads, as in Fig. 6-30b, to the well-known picture of lifting surfaces consisting of vortex ladders composed of individual horseshoe vortices forming elementary wings.
On each surface panel, carrying a singularity density assumed to be constant, a control point lies in its center of gravity at which the kinematic flow condition is to
be satisfied. Hence there exist as many control points as surface panels with singularity densities individually assumed to be constant but as yet unknown in magnitude. To each vortex ladder (consisting of several inner panels with a vortex filament of constant circulation strength on their edges) a control point is assigned at the trailing edge of the wing in which the Kutta condition is satisfied. Thus there are as many Kutta control points as vortex ladders with unknown total circulation
strength (the circulation distribution within each of the vortex ladders on the individual panels is assumed a priori to be a Birnbaum distribution).
The requirement of the defining equation that the kinematic boundary condition has to be satisfied at all control points for all perturbation potentials of all panels leads to a system of linear equations of a form similar to that derived for the lifting-surface method (see [221). This, in most cases very extensive,. system of linear equations is solved through iteration by means of a Gauss-Seidel procedure.
By this means the singularity strength and thus the velocity and the pressure distribution at the control points are obtained. A typical result of a computation of a wing-fuselage system is shown in Fig. 6-31 by means of the pressure distributions on a few selected fuselages and wing sections in comparison with measurements by Schneider [42]. The effect of compressibility has been taken into account through the Gothert rule (see (22]). Further methods of general validity for the computation of the aerodynamics of wing-fuselage systems at subsonic flow have been developed by Woodward [521, Giesing et al. [91, and Korner (211.
AERODYNAMICS OF THE WING-FUSELAGE SYSTEM 405
b
Figure 6-30 Application of the panel method of Kraus and Sacher to a wing-fuselage system. (a) Outer surface (source-sink distribution). (b) Inner surface (vortex distribution).
6-3-2 The Wing-Fuselage System in Supersonic Incident Flow General remarks Numerous contributions to the aerodynamics of the wing-fuselage interference at supersonic velocities have been published. However, the establish-
ment of a simple, generally valid method for its computation, like the method already available for incompressible flow (Sec. 6-2), has not yet been devised. Summary presentations have been given by Lawrence and Flax [26], Ferrari [6], Pitts et al. 137], and Ashley and Rodden [2].
406 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
Section A
Section 1 -0.4
cp cr
Swept-back wing
-Q2 1
-e1= 6.0 450 = 30° X = 0.33 Profile RAE 101.9%, mod.
U.S. 0
a2
Relative fuselage width
L.S.
Section C
77F=D/b=0.11
0
02
Q.
06
08
t0
X/C
Section 2 Section E
AA_
;, 1-useiage stations 08
a
b
w
X/C
C
U.S. = upper side L.S. = lower side
Figure 6-31 Wing-fuselage system at subsonic incident flow, c = 8°, Ma. = 0.7. Measurements from Schneider, theory from Kraus. (a) Geometry. (b) Pressure distributions, wing sections. (c) Pressure distributions, fuselage sections.
The earlier theories on the wing-fuselage interferences were based on the assumptions made for the theories of the wing and of the fuselage and were limited to specific wing-fuselage systems. The first work of this kind came from Kirkby and Robinson [5]. Here a wing of large aspect ratio, attached to a conical body, is treated by means of the stripe method. This method does not take into account the lift loss at the wing-fuselage interface and the effect of the wing on the fuselage due to the large ratio of wing span to fuselage diameter. According to Cramer [4], these two contributions cancel each other to a large extent and the total lift is obtained relatively well. Fundamental investigations in the field of wing-fuselage interference at supersonic velocities have been conducted by Ferrari [5]. He was concerned with the problem of a rectangular wing of large aspect ratio on a cylindrical fuselage with a pointed nose. The solution is accomplished through an iteration procedure. After having determined separately the potential functions for the wing and for the fuselage, these functions are combined and corrected step by step in such a way that the boundary conditions are satisfied exactly for one part of the system only, either for the wing or for the fuselage, whereas the boundary conditions of the other part are disregarded. This procedure converges after a few steps. Browne et al. [3] investigated a delta wing with conical fuselage (wing and fuselage apex coincide) by the method of cone-symmetric flows. Results are given for wings with subsonic and supersonic leading edges. Although this method does not offer extensive practical applications, its exact solutions are valuable for comparisons with approximation solutions. The results of Lock [28] illustrate this fact.
A rectangular wing with a cylindrical fuselage has been treated by Morikawa
AERODYNAMICS OF THE WING-FUSELAGE SYSTEM 407
[311 and Nielsen [37]. By applying the Laplace transform, an exact solution is obtained in the form of a series. Morikawa offers only an approximate solution, but Nielsen succeeded in retransforming the solution. The computational procedure of Woodward [52], applicable to subsonic and supersonic flows as well, should be mentioned.
Now it will be shown that the essential relationships may be gained through simple, physically plain considerations without lengthy mathematical derivations; see also Schrenk [43]. In analogy to incompressibly flow, the wing-fuselage interference for supersonic flow will be analyzed by first discussing the effect of the wing on the fuselage and then the effect of the fuselage on the wing. Lift distribution
of the fuselage As has been shown in Sec. 5-3-3, the lift
distribution at supersonic incident flow of the fuselage alone may be determined from the relationship for incompressible flow, by setting a(x) = a. = const in Eq. (5-28), as dLF
= 27ra q d(R2) dx
In Fig.
(6-35)
6-32a, the supersonic flow for the simple wing-fuselage system of an
axisymmetric fuselage and a rectangular wing in mid-wing position is demonstrated,
schematically. The absence of an influence of the wing on the fuselage portion before the wing in supersonic flow marks the important difference between supersonic incident flow and incompressible flow. Hence, the lift distribution for the front portion of the fuselage in a wing-fuselage system is identical to that of the fuselage alone as given by Eq. (6-35). Thus the lift of the fuselage front portion becomes
LFf = 27ra.qRa
(6-36)
For the remaining portion of the fuselage, a simple survey will be given of the lift distribution created in addition to Eq. (6-35) by the wing on the fuselage. To simplify the problem, a wing of infinite span has been assumed in Fig. 6-32. In this case the wing generates perturbation velocities only in the range between the Mach
lines m, and m2 originating at its leading edge and its trailing edge. Under the simplifying assumption that the lift distribution of the wing is unchanged in the fuselage range, no additive lift force, caused by the wing, acts on the rear fuselage portion either. Hence the fuselage feels an additive lift force only within the range between the Mach lines ml and m2. This lift due to the wing influence is caused by the velocities induced by the wing in the x direction, that is, u(x), and in the z direction, that is, w(x). In Fig. 6-32b and c, the distribution of the induced velocities u(x) and w(x) is given. Their variation on the fuselage surface at z = 0 and 6 = 90°, respectively, is marked by the dashed curve, and that at y = 0 and 6 = 00, respectively, by the dash-dotted curve. These two curves are merely displaced from each other in the longitudinal
408 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
m7
U(X) e=go
-
b
C
(dLF)
MF2
dx e
x Figure 6-32 Computation of the inter-
ference of wing-fuselage systems at dLF
supersonic incident flow. (a) Geometry of the wing-fuselage system. (b) Distri-
n
bution of the longitudinal velocity u(x). (c) Distribution of the vertical
i
dx
velocity w(x). (d) Lift distribution due
to the longitudinal velocity. (e) Lift distribution due to the vertical veloc-
f
ity. (f) Resultant lift distribution.
direction. The maximum values of the induced longitudinal and vertical velocities are given in Eqs. (441a) and (4-41b) as 2c =
a°°
VMa5-1
w = - aco UN
Um
(6-37a)
(6-37b)
The solid curves signify the mean values of the induced velocities over the circumference. They are essential for the computation of the lift distribution of the
AERODYNAMICS OF THE WING-FUSELAGE SYSTEM 409
fuselage. The following expressions from Ferrari [5 ] are obtained for 0 < x <xo , if in this range R(x) = Ro = const: Ft (x)
U".
`i-
1-
x0 Y
Y
w(x)
(arcsin
UM
x
(6-38a)
co
-
x
1
x
(6-38b)
Here xo =R0 '/Ma w, -1. Corresponding formulas are valid for c <x< c -F-x0.
From these mean values of the induced velocities, two contributions are obtained to the lift distribution of the fuselage in the wing range, namely, (dLF/dx)1 from the longitudinal velocity u(x), and (dLFldx)2 from the vertical velocity a,,, U + w(x). For the case of a constant fuselage radius R(x) = Ro within the wing range 0 <.x < c + xo , these contributions are
(.F)
u(x)
= 8q.Ro U
dx
dLF dx
= 2nq.Ro 2
(6-39a)
fix) ddx U2
(6-39b)
The qualitative trend of these two contributions is shown in Fig. 6-32d and e. The resulting lift distribution as the sum of these two contributions is presented in Fig.
6-32f. Integration of the contribution of Eq. (6-39b) results in LF2 = 0,
because, according to Fig. 6-32c, w(x) is equal to zero before and behind the wing. From Fig. 6-32b, the lift force is obtained through integration of Eq. (6-39a), because LF1 = LF, as LF = 8 q
,
°L°°
VMa' -1
Roc
(6-40)
This simple relationship for the lift of the fuselage due to the wing applies to the cases where the intersection of the front Mach line ml with the fuselage surface at y = 0 lies before the wing trailing edge (see Fig. 6-32a). There is another interpretation of this result of Eq. (6-40), namely, that the fuselage lift due to wing effects is equal to the lift in plane flow of the wing portion AA = 2Roc shrouded by the fuselage. Also, the case has been investigated by Ferrari [6] where the front Mach line intersects the fuselage upper edge behind the wing trailing edge. In this case the computation of the additive fuselage lift is considerably more difficult than explained above.
In Fig. 6-33 the lift distribution of the fuselage under the influence of the wing is shown for an example of Cramer (4]. The theoretical curve has been computed
by Ferrari [5]. At the Mach number Mac, = 2 used in this study and the chosen geometry of the wino fuselage system, the Mach line from the leading edge of the wing intersects the upper edge of the fuselage behind the wing trailing edge. Therefore, contrary to Fig. 6-32f, the lift distribution reaches far beyond the wing trailing edge. Agreement between theory and measurement is good.
410 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM 0
CL,, a8 0
Theory Measurement
-0
2
j c-7.4Ro
3
4
S
7
6
a
RO
t-- _,
,Z
Figure 6-33 Lift distribution on the fuselage for a wing-fuselage system (mid-wing airplane) at supersonic velocities, from Cramer. Mach number Ma. = 2, angle of attack a = 8°. Theory from Ferrari, wing chord c = 1.4R0 , wing span b = 8R 0 .
As stated earlier, the above theoretical results apply to the case of wings of very large span. The effect of the wing aspect ratio can be seen in Fig. 6-34, where, from [6], the additive lift distributions of the wing are plotted for wing-fuselage systems with wings of several aspect ratios. The Mach number is also Ma. = 2. When the aspect ratio decreases, the additive lift force decreases considerably, as would be expected. For the test series of Fig. 6-34, the ratio of the fuselage lift LF and the lift of
4-4.3
e
'
I
.8
2
1 0S
2
7
Figure 6-34 Effect of wing aspect ratio on the lift distribution on the fuselage for wing-fuselage systems (mid-wing airplane) at supersonic velocities, from Ferrari. Mach number Ma. = 2, angle of
attack a. = 8°.
AERODYNAMICS OF THE WING-FUSELAGE SYSTEM 411 1.0
11-asti 0.8
06
I Figure 6-35 Ratio of fuselage lift LF to wing 4.3
0,
0.2
lift L'W vs. relative fuselage width nF for wing-fuselage systems at supersonic velocities, from Ferrari (system as in Fig. 6-34).
Mac. z OCm °B°
0,2
0,4
0.6
0.8
Mach number Ma = 2, angle of attack a, = 8°. Curve 1, theory from Lennertz.
1.0
'7F --
Curve 2, slender-body theory.
the wing portion not shrouded by the fuselage, L'' , is plotted in Fig. 6-35. The data are compared with theoretical curves: Curve 1 reflects the theory of Lennertz [27] from Eq. (6-5), valid also for supersonic flow. Curve 2 gives the theory of wing-fuselage systems with wings of small aspect ratio of Sec. 6-4. The two theoretical curves are not very different. The measured data agree quite well with the theoretical curve (2). The lift distribution leads to the pitching moment from Eq. (5-32). By
introducing the two contributions of the lift distribution from Eqs. (6-39a) and (6-39b), the following two contributions to the pitching moment, referred to the middle of the wing, are obtained, where the contribution of the fuselage front portion has been disregarded: C+xo
MF1= -8q".Po 1 X -
2G
C1x 2
(6-41a)
U,,
U
2;zq,,. '. p20 C
(6-41b)
C+x
MF2 = 2zq.Po
E(x)
L
tlx
(6-42a)
0
_ -27rq,c Plc
(6-42b)
Here x is measured from the wing leading edge. From Fig. 6-32e, MF2 is a free moment. The total moment (without the fuselage front portion), referred to the middle of the wing, thus becomes
MF=
4,-z q,
R2c
(6-43)
Note that this additive moment is independent of the Mach number. The neutral-point displacement due to the wing influence on the fuselage with reference to the neutral point of the wing alone, XJ.T W = c/2, becomes
412 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
MF - _ L (W+F)
(,XN)(W+F) _
For small relative fuselage widths qF = 2R0 /b, L(w+F) may be approximated by Lw. For the wing of large aspect ratio, from Eq. (4-46), this leads to L W = q0,a,,
4
VM a'
-1
be
Hence with A = b/c > 1, the approximate expression is obtained: (d xN)(W +F) C
=
Ir
4
211
?IF
Ma a,
-
1
(644)
This stabilizing neutral-point displacement due to the effect of the wing on the fuselage counteracts the destabilizing contribution of the fuselage front portion. The above results on the effect of the wing are valid for the unswept wing of large aspect ratio, that is, for wings with supersonic leading edges. For wings with small aspect ratio, the discussions of Sec. 6-4 should be examined.
Lift distribution of the wing The effect of the fuselage on the lift distribution of the wing at supersonic velocities can be determined approximately by the method applied in Sec. 6-2-2 to incompressible flow. The additive angle-of-attack distribution, caused by the cross flow over the fuselage as in Fig. 6-5b, creates additive lift locally on the wing. Under the assumption of an infinitely long fuselage, the additive angle-of-attack distribution for a given fuselage cross-section shape is the same as in incompressible flow, because the velocity of the cross flow of the fuselage is considerably lower than the speed of sound. Equations (6-16a) and (6-16b) give the distribution of the induced angle of attack for a fuselage of circular cross section (radius R) with a wing in mid-wing position. The computation of the approximate lift distribution along the span for the given angle-of-attack distribution may be conducted very easily with the so-called stripe -method.* Hence, the local lift coefficient becomes
CI(y) ` with d a{y) from Eq. (6-16a).
_
4a,,, M a., _1
h
AM(y) )
a
(6-45)
An example for a wing-fuselage system of an axisymmetric fuselage and a rectangular wing is given in Fig. 6-36. It shows the lift distribution plotted against the span for the Mach number Ma. = 2 at an angle of attack a,. = 8°. Curve 1 reflects the theory of the stripe method, Eq. (6-45), and curve 2 a theory of Ferrari [6]. Both theories agree quite well with the measurements, except for the stripe method in the immediate vicinity of the fuselage. For comparison, the theory for *The stripe method is a procedure whereby the local lift coefficient is set proportional to the local angle of attack based on the lift slope of the plane problem, which from Eq. (4-46) is given for supersonic velocities by (dc1/da),,, = 4 f Ma«, -1.
AERODYNAMICS OF THE WING-FUSELAGE SYSTEM 413 0.
0.6
Mmco = 2
i
oGo,=B 05
t
0.4 4
0.2
0,1
L
M I
0.2
0.4
0.5
0.8
1.0
7 y Figure 6-36 Lift distribution on the wing due
to the fuselage effect for a wing-fuselage system
(mid-wing airplane) at supersonic
velocities. Curve 1, theory, stripe method, from Eq. (6-45). Curve 2, theory from Ferrari. Curve 3, measurements from Ferrari. Curve 4, theory, wing alone.
the wing alone is added as curve 4. Obviously, the influence of the fuselage on the lift distribution of the wing is rather large. The above results on the effect of the fuselage on the lift distribution of the wing apply to wings of large aspect ratios. For wings of small aspect ratios, reference should again be made to Sec. 6-4.
Wave drag The problem of the determination of the wave drag of wing-fuselage systems at supersonic velocities has been attacked by Vandrey [48], Lomax and Heaslet [30], Jones [181, and Keune and Schmidt [19]. Also, the experimental investigations of Schneider [42] should be mentioned.
6-3-3 The Wing-Fuselage System in Transonic Incident Flow The following discussions on the interference in wing-fuselage systems in transonic
flow will be restricted mainly to the drag problem. The drag of wing-fuselage systems near Ma = 1 is generally larger than the sum of the drags of the wing
414 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
alone and the fuselage alone. Here the wave drag at zero lift is the major factor. Figure 6-37 shows drag measurements by Whitcomb [50] on wing-fuselage systems in the Mach number range from Ma,0 = 0.85 to Ma00 = 1.1 with CL = 0. The tested models are shown in Fig. 6-37a, their total drag in Fig. 6-37b, and the drag remaining after subtraction of the friction drag in Fig. 6-37c. The curve for the fuselage alone (model 1) shows a strong drag rise near Mao, = 1. The simple combination of wing and fuselage (model 2) produces a particularly large drag in the transonic range. Whitcomb [50] showed that by contracting the fuselage within the wing range, the drag in the transonic range may be greatly reduced (model 3). This contraction of the fuselage has to be chosen such that the wing-fuselage system and the original fuselage (model 1) have approximately equal distributions of the cross-sectional areas normal to the fuselage axis. This rule for the distribution of cross-sectional areas of a wing-fuselage system is called the "area rule." Figure 6-38
shows the application of this rule to an airplane, where Fig. 6-38a gives the plan view of the airplane, Fig. 6-38b the contour of an axisymmetric body of equal cross-sectional area distribution AF(x), and Fig. 6-38c the variation of this cross-sectional area along the fuselage axis dAFldx. In Fig. 6-38c, the case without
a
b
/.169
0.020
2 0. 015 I
i
0. 012
OSM 1
,
C 0.072
Figure 6-37 Drag coefficients of wingfuselage systems and axisymmetric fuse-
0.004E
0.84
ON
0.92
OX Maw
1,90
74)
1.0
1J2
lages in the transonic Mach number range, from Whitcomb. (a) Geometry. (b) Total drag coefficients CD at zero lift. (c) Coefficients of wave drag.
AERODYNAMICS OF THE WING-FUSELAGE SYSTEM 415
A
Figure 6-38 The area rule for transonic flow. (a) Airplane planforrn. (b) Distribu-
tion of the cross sections AF(x) of the equivalent body of revolution. (c) Variation of the cross-sectional area distribution along the fuselage dAFldx.
area contraction is drawn as a solid line, the case with area contraction as a dashed line. The fuselage area contraction has been chosen for as smooth a dAF/dx variation as possible.
For the experimental proof, Whitcomb [50] also tested a fuselage whose cross-sectional area distribution is equal to that of the wing-fuselage system without
contraction (model 4 of Fig. 6-37). This model indeed has the same drag rise as model 2 in the transonic range. The theoretical basis of this phenomenon has been studied by Jones [18] and Oswatitsch [35], as well as by Keune and Schmidt [19]. Finally, it may be seen in Fig. 6-39 that the advantage of the area rule is limited to 0
2
i
f 0.7
0
0.8
7 0.9
1,0
47
1.3
1.4
Ma,,
Figure 6-39 Drag coefficients of wing-fuselage systems at zero lift in the transonic Mach number range; measurements from Jones. Curve 1, without fuselage contraction. Curve 2, with fuselage contraction, computed for Ma. = 1. Curve 3, with fuselage contraction, computed for Ma. = 1.2.
416 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
the transonic Mach number range. This figure gives the drag coefficients of 3 wing-fuselage systems in the Mach number range from Ma = 0.8 to Ma = 1.4. Model 1 is the fuselage without contraction, whereas models 2 and 3 are fuselages
with two different contractions. The contraction of model 2 has been chosen for largest drag reduction at Ma = 1, whereas that of model 3 is for lowest drag at Ma = 1.2. These tests show that contraction according to the area rule yields favorable results only in the transonic range. In the supersonic range, the results are even less favorable than for fuselages without contraction.
In this connection, the comprehensive experimental studies should be mentioned that Schneider [42] conducted on wing-fuselage systems with three different wings (rectangular, swept-back, and delta wings). A computation of the pressure
distribution on wing-fuselage systems at an incident flow of Ma = 1 and a comparison with measurements have been conducted by Spreiter and Stahara [45]. Compare also the computational methods in [201. .
6-4 SLENDER BODIES In the previous sections of this chapter wing-fuselage systems with wings of large to moderately large aspect ratios have been discussed. Now systems with wings of small. aspect ratios will be treated. Here the slender triangular wings (delta wings) with large sweepback play a special role. With flight velocities having increased from subsonic to supersonic speed ranges over the past decades, this kind of slender body
(Fig. 6-40) has become most important. They are characterized by aerodynamic coefficients that are largely independent of the Mach number but depend, to a large extent nonlinearly, on the angle of attack (see Secs. 3-3-6 and 5-3-3). The theory for lift computation developed by Munk [33] for slender fuselages and by Jones [17] for wings of low aspect ratio has been extended by Ward [49] and Spreiter [44] to wing-fuselage systems with wings of low aspect ratio; see also
Jacobs [44]. The basic thought underlying this theory is the fact that changes in the perturbation velocities about slender bodies are small in the x direction (fuselage axis, wing longitudinal axis) compared with those of the perturbation velocities in
dF b i,z
ix
ix
Figure 6-4U Slender bodies: wing, tuselage, wing fuselage system.
AERODYNAMICS OF THE WING-FUSELAGE SYSTEM 417
Z,
V"
Figure 6-41 Theory of wing-fuselage systems with wings of small aspect ratio. (a) Sketch of the wing-fuselage system. (b) Cross section x = const. (c) Conformal mapping of cross section x = const of b.
C
31
the y and z directions normal to the x direction. This causes the potential equation, Eq. (4-8), to be reduced to that of two-dimensional flows in the yz plane: a20
aft
-}-
a2 = 0 az2
(6-46)
where v = a0/ay and w = aOlaz are the induced velocities in the lateral plane. Since Eq. (6-46) is valid for both incompressible and compressible flows, the results given below can be applied to both subsonic and supersonic incident flows.
The potential equation, Eq. (6-46), is to be solved for each cross section x = const (Fig. 6-4la), which can be accomplished by conformal mapping, for instance. The flow about a wing-fuselage system (Fig. 6-41b) can therefore be determined from the flow about a flat plate at normal incidence (Fig. 6-41c). Some results from Spreiter [44] and Ward [49] will now be discussed; see also Ferrari [6] and Haslet and Lomax [12].
Pressure distribution For wino fuselage systems consisting of a delta wing and an infinitely long body of circular cross section, pressure distributions for two sections 1 and 2 normal to the axis are shown in Fig. 6-42. The load distribution on the wing is
418 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
[R() [(' R
,
J cp = 4Nc, tany
s(x)
1
yy
1+
4
ls(z)/ J -
y
s(X)'+Y
s
z
for R2
J
and that on the body is
1 - (s x) /4
Jcp =4.xrotanyAI+
4
for 0
(647b)
(y
For the wing alone, Eq. (6-47a) yields, with R = 0, 4N,,,, tan y
J Cp =
1 - (S(x) )3
(648)
In Eqs. (647a)-(648), y is the leading-edge semiangle of the wing, s(x) = x tan y is the local half span, and R(x) is the body radius. The load-distribution curve in Fig.
Figure 6-42 Load distribution over the span for a wing-fuselage system with a delta wing (slender-body theory). Curves 1 and 2 for the
wing-fuselage system. Curve 1' for the wing -4
-2
0
2
4
6
R
alone.
AERODYNAMICS OF THE WING-FUSELAGE SYSTEM 419 70 i1
H
y 6
y=R
C
6
Wing + fusela
4 I
Wing al one
=0
Wing + fus elage
Figure 6-43 Load distribution in the longitudinal direction on the middle section (y = 0) and on the section at
t arry I
2
.3
R -tarry'
6
the wing root (y = R) for a wingfuselage system with a delta wing (slender-body theory).
6-42 shows that the influence of the fuselage on the pressure distribution is greater at the front portion of the wing than at the rear portion. For comparison, the load distribution of the wing alone is also drawn for cross section 1 (curve 1 ').
In Fig. 6-43 the load distribution in the longitudinal direction along the wing-root section y =R is shown for the wing-fuselage system of Fig. 6-42. The influence of the body is seen in a somewhat smaller load decline in the axial direction than for the wing alone. The load distribution for the middle section (y = 0) is also given.
A procedure for computing the pressure distribution on slender bodies with arbitrary planform and cross-section shape is given by Hummel [14].
Lift distribution In Fig. 6-44, the lift distribution is shown versus the span of the wing-fuselage system of Figs. 6-42 and 6-43. The relative body width is ?7F = 3. The
effect of the body on the lift distribution is considerable. An example of the lift distribution over the body length is shown in Fig. 6-45. Note that the fuselage contributes to the lift only in the range of the wing. Close to the wing nose, the fuselage lift increases strongly; at the wing trailing edge, it drops abruptly to zero. Total lift In Fig. 6-8, curve 2, the ratio of body lift LF and total lift L(W+F) of a delta wing and an infinitely long body of circular cross section, according to this theory, was plotted as a function of the relative body width r?F. Comparison of curves 2 and 1 in Fig. 6-8 shows that the slender-body theory yields almost the same values of LFIL(w+F) as the theory of Lennertz [27], which is valid for arbitrary aspect ratios. We can conclude, therefore, that the values of LF/L(w+F) from the slender-body theory can also be used for wing-body systems with wings of larger aspect ratios. The total lift for wing-body systems from Fig. 6-44 is
L(w+F) =
-R2)2
(6-49)
420 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
2R 2s
IT
Figure 6-44 Lift distribution over the span
y
for a wing fuselage system with a delta wing, 77F = a (slender-body theory). Curve 1, wing + fuselage. Curve 2, wing + flattened fuselage. Curve 3, wing alone.
8
1
4
y
n
0
LT
4
5
R tan7
r
Figure 6-45 Lift distribution of the fuselage for a wing-fuselage system with a delta wing (slender-body theory).
AERODYNAMICS OF THE WING-FUSELAGE SYSTEM 421
Hence, when referring the lift coefficient CL to the wing area A = crs and the dynamic pressure of the incident flow q., setting A = 4s/c, and riF = R/s, the lift slope becomes dCL
da
_
?r
A (1 _ r1F)2
(6-50a)
2
(W +F)
(dcL)rri W
(6-50b)
The second relationship applies to the wing alone (riF = 0); see Eq. (3-101b). In Fig. 6-46, the ratio of the total lift to the lift of the wing alone, that is, L(w+F)/L w, is plotted as curve 1 against the relative fuselage width ?7F- With increasing i7F, the ratio L(w+F)/Lw decreases strongly and becomes zero for 77F = 1.
At a fuselage that is pointed in front, this finite front portion of the fuselage produces a lift additive to that of the infinitely long front portion from Eq. (6-6):
LF f = 2na. q.Ro
(6-51)
This means an increase of the lift slope over the value of Eq. (6-50a), and the lift slope becomes dcL
da (W+F) = 2 A (1 - r1F + 77F)
(6-52)
The ratio L(w+F)/Lw for this case is plotted against the relative fuselage width 17F as curve 2 in Fig. 6-46.
Neutral-point position Finally, in Fig. 6-47, a few results are shown on the shift of the neutral point as caused by the fuselage. For the wing-fuselage system of Fig. 1
0.
2 a, U-
Q
0.
0.2
0,4 77F
0.6
0.3
1.0
Figure 6-46 Ratio of total lift to wing lift for wing-fuselage systems with a delta wing (slender-body theory). Curve 1, infinitely long fuselage. Curve 2, fuselage of finite length (with fuselage nose).
422 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
Figure 6-47 Neutral-point shift of wingfuselage systems with a delta wing (slender-
body theory), from Spreiter. 02
0.4
0.6
0.8
Curve
1,
wing + fuselage. Curve 2, wing + flattened fuselage. Curve 3, substitute wing (with rectangular middle portion).
1.0
'1F-
6-44, the shift of the neutral point caused by the fuselage, relative to the neutral point of the wing alone (dxN/cF,)(W+F), is plotted as curve I of Fig. 6-47 against the relative fuselage width. From the theory for small aspect ratios, the neutral point of the wing alone lies at a distance 3c, from the wing nose. When ?7F increases, the neutral point moves rearward by an amount "XN Cµ
2
_ (W+F)
2+ 1F
(1
(6-53)
)2
For 7?F = 1, the shift of the neutral point becomes (d xN/cµ)(yy+F) = that is, in 2; trailing this case the neutral point of the wing-fuselage system is located at the wing edge, as can easily be understood from inspection of Fig. 6-45. In Fig. 6-47, curve 2, the neutral-point shift is given for a "flat" fuselage (height zero). The difference to curve 1 is relatively small. For the case of a flat fuselage, the lift distribution over the span is
also shown in Fig. 6-44 as curve 2. For comparison, the
neutral-point shift for a wing with rectangular middle section (substitute wing) is given in Fig. 6-47, curve 3.
At last the case of a fuselage with a front portion of finite length will be discussed. The moment of the fuselage front portion, relative to the axis through the wing neutral-point, is given from Eqs. (6-6) and (5-32) as
if (R2 (x) dx + (xNyy
MFf =
- lf)Ro
(6-54)
0
Here, if is the length of the fuselage front portion from Fig. 6-48, and xNW is the distance of the wing neutral point from the fuselage nose. The distance (xNW - l f) is easily determined as 3cr(2 - 371F). Evaluation of Eq. (6-54) for a fuselage with a parabolic nose yields
AERODYNAMICS OF THE WING-FUSELAGE SYSTEM 423
1-
MFf = 2:cc q R0
3
f
I
71F +
Y/
(6-55)*
where cu = 3 Cr.
For the wing-fuselage system with a fuselage front portion of finite length, the neutral-point shift relative to that of the wing alone is M(W +F) °°
lJ X jy
L(W
+F) °o
+MFf I LFf
The index - refers to wing-fuselage systems with infinitely long fuselages, where L(w+F) is computed from Eq. (6-49) and M(w+F)- _ -J/ XNL(W+F)OO With JxN from Eq. (6-53). The values of LFf and MFf are given by Eqs. (6-51) and (6-55), respectively. For a fuselage with elliptic nose section, the factors of if/Cr must be replaced by 1.
0.
-=0 If
.Cr
1s
0.,
L
0
a
'
i
2Ro
-- 0,1
-0.2 7.0
-0,3 -0,4
- 050
0.2
0.4
0.6
'1F
0.8
1,0
Figure 6-48 Neutral-point shift of wingfuselage systems with a delta wing and a fuselage of finite length (slender body theory), from Eq. (6-56).
424 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
By substitution, the final result is 2
JC N
(W +F)
=
2F
1
4
1 - 71F + 'OF 2
2 - 571, + 471, -
8 if 5 Cr
(6-56)
The shift of the neutral point according to Eq. (6-56) is plotted in Fig. 6-48 as a function of the relative body width r1F for various lengths of the fuselage front portion lf/cr. These plots show that the shift of the neutral point d XN is positive (stabilizing) for small values of l flcr, as in the case of an infinitely long fuselage (Fig. 6-47). At larger values of Zflcr, however, the unstable contribution of the fuselage front portion is predominant, making v XN negative.
dFinax
--
---
I
F
..
2
o
3
0
b
I
! '
--E-I
I
I
I
I
'Theory for wing alone
1
1
C
-70'
0°
J-
75°
10,
20°
25'
30'
S'
40°
cc
Figure 649 Lift coefficient vs. angle of attack cL(c) for slender bodies, from measurements of Otto. (a) Wing alone, ,i = 1. (b) Fuselages 1, 2, 3 alone: dFinax/IF = 0.10, 0.10, 0.05. (c) Wing-fuselage systems with fuselage 2, Re = U.lF/v = 7.5
106
.
AERODYNAMICS OF THE WING-FUSELAGE SYSTEM 425
Test results Finally, some test results will be presented that show the nonlinear lift characteristics CL(a) for slender bodies. In Fig. 6-49 the lift coefficients for three wings, three fuselages, and three wing-fuselage systems in incompressible flow are presented from Otto [36]. The lift coefficients of the fuselages (Fig. 6-49b) are
referred to the wing area. For the wings alone, the results of linear theory for slender bodies according to Eq. (6-50b) are also shown. In all three cases (wing, fuselage, wing-fuselage system), the deviation from linear theory is considerable. Corresponding investigations on slender conical wino fuselage systems in supersonic
incident flow have been reported by Stahl 146]. Measurements on the vortex system of inclined wing-fuselage systems have been conducted by Grosche [10] .
The design of slender, integrated airplanes for supersonic flight has been proposed-by, among others, Kuchemann [23]. Design questions for airplanes in the transonic flight mode are discussed by Lock and Bridgewater [29].
REFERENCES 1. Adams, M. C. and W. R. Sears: Slender-Body Theory-Review and Extension, J. Aer. Sci., 20:85-98, 1953. 2. Ashley, H. and W. P. Rodden: Wing-Body Aerodynamic Interaction, Ann. Rev. Fluid Mech., 4:431-472, 1972. 3. Browne, S. H., L. Friedman, and I. Hodes: A Wing-Body Problem in a Supersonic Conical Flow, J. Aer. Sci., 15:443-452, 1948. 4. Cramer, R. H.: Interference between Wing and Body at Supersonic Speeds-Theoretical and Experimental Determination of Pressures on the Body, J. Aer. Sci., 18:629-632, 1951; NAVORD Rept. 3146, 1952. 5. Ferrari, C.: Interference between Wing and Body at Supersonic Speeds-Theory and Numerical Application, J. Aer. Sci., 15:317-336, 1948; 16:542-546, 1949. Kirkby, S. and A. Robinson: Coll. Aer. Cranfield, Rept. 7, 1957. Laitone, E. V.: J. Aer. Sci., 16:510-511, 1949.
6. Ferrari, C.: Interaction Problems, in A. F. Donovan and H. R. Lawrence (eds.), "Aerodynamic Components of Aircraft at High Speeds," Sec. C, Princeton University Press, Princeton, N.J., 1957. 7. Gersten, K.: Neuere deutsche Beitrage zur Aerodynamik schianker Flugkorper, Land. Forsch. Nordrhein-Westfalen, Jb., 229-258, 1969. Humrnel, D.: Jb. DGLR, 18:1-40, 1975. 8. Gersten, K. and D. Hummel: Experimentelle and theoretische Untersuchungen iiber die ,
Interferenzeinfliisse an schiebenden Fliigel-Rumpf-Anordnungen mit Pfeil- and Deltafliigein, DLR FB 66-77, 1966. 9. Giesing, J. P., T. P. Kalman, and W. P. Rodden: Subsonic Steady and Oscillatory Aerodynamics for Multiple Interfering Wings and Bodies, J. Aircr., 9:693-702, 1972. Hua, H. M.: J. Aircr., 10:422-426, 1973; 11:366-368, 1974; 12:916, 1975. Kalman, T. P., W. P. Rodden, and J. P. Giesing: J. Aircr., 8:406-413, 1971. 10. Grosche, F.-R.: Wind Tunnel Investigation of the Vortex System near an Inclined Body of Revolution with and without Wings, AGARD CP 71, 2:1-13, 1971; Z. Flugw., 18:208-217, 1970. 11. Hafer. X.: Untersuchungen zur Aerodynamik der Flugel-Rumpf-Anordnungen, Jb. WGL,
191-207, 1957. Liebe, H.: Jb. Lufo., 1:280-289, 1942. Moller, E. and H. Trienes: Z. Flugw., 1:2-8, 1953. Truckenbrodt, E. and K. Gersten: Z. Flugw., 5:204-216, 1957. 12. Heaslet, M. A. and H. Lomax: Supersonic and Transonic Small Perturbation Theory, in W.
R. Sears (ed.), "General Theory of High Speed Aerodynamics," Sec. D, pp. 249-275, Princeton University Press, Princeton, N.J., 1954; NACA Rept. 1185, 1954.
426 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
13. Hess, J. L. and A. M. O. Smith: Calculation of Potential Flow About Arbitrary Bodies, Prog. Aer. Sci., 8:1-138, 1967. Carmichael, R. L.: AGARD CP 71, 4:1-3, 1971. Smith, A. M. O. and J. L. Hess: Douglas Aircr. Rept. ES 29988, 1958. 14. Hummel, D.: Berechnung der Druckverteilung an schlanken Flugkorpern mit beliebiger Grundriss- and Querschnittsform in Unter- and Uberschallstromung, Jb. DGLR, 158-173, 1968.
15. Jacobs, E. N. and K. E. Ward: Interference of Wing and Fuselage from Tests of 209 Combinations in the NACA Variable-Density Tunnel, NACA Rept. 540, 1935. Moller, E.: Jb. Lufo., 1:336-365, 1942; ZWB Lufo. TB 11, no. 5, 1944. Sherman, A.: NACA Rept. 575, 1936. 16. Jacobs, W.: Berechnung des Schiebe-Rollmomentes fur Flugel-Rumpfanordnungen, Jb. Lufo., 1:165-171, 1941. Bamber, M. J. and R. O. House: NACA TN 703, 1939; 730, 1939. Braun, G. and H. Scharn: Jb. Lufo., 1:246-262, 1942. Maruhn, K.: Jb. Lufo., 1:263-279, 1942. Puffert, H. J.: Z. Flugw., 3:323-331, 1955. 17. Jones, R. T.: Properties of Low-Aspect-Ratio Pointed Wings at Speeds Below and Above the
Speed of Sound, NACA Rept. 835, 1946; "Collected Works," NASA TM X-3334, pp. 369-375, National Technical Information Service, Springfield, Va., 1976. Ashley, H. and M. Landahl: "Aerodynamics of Wings and Bodies," pp. 99-123, Addison-Wesley, Reading, Mass., 1965.
18. Jones, R. T.: Theory of Wino Body Drag at Supersonic Speeds, NACA Rept. 1284, 1956; Adv. Aer. Sci., 1:34-51, 1959;AIAA J., 10:171-176, 1972; "Collected Works," NASA TM X-3334, pp. 609-623, 625-644, 657-664, National Technical Information Service, Springfield, Va., 1976. 19. Keune, F. and K. Oswatitsch: Aquivalenzsatz, Ahnlichkeitssatze fur schallnahe Geschwindigkeiten and Widerstand nicht angestellter Korper kleiner Spannweite, Z. Angew. Math. Phys.,
7:40-63, 1956; Z. Flugw., 1:137-145, 1953. Keune, F. and W. Schmidt: Jb. WGL, 150-155, 1956; Z. Angew. Math. Mech., 36:301-303, 1956. Keune, F.: Jb. WGL, 176-186,
1955; Z. Flugw., 5:121-125, 1957. Keune, F., H. Riedel, and H. Emunds: Z. Flugw., 20-257-261, 1972.
20. Klunker, E. B. and P. A. Newman: Computation of Transonic Flow About Lifting Wing-Cylinder Combinations, J. Aircr., 11:254-256, 1974. Rohlfs, S. and R. Vanino: Z. Flugw., 23:239-245, 1975. 21. Korner, H.: Berechnung der potentialtheoretischen Stromung um Flugel-RumpfKombinationen and Vergleich mit Messungen, Z. Flugw., 20:351-368, 1972. 22. Kraus, W. and P. Sacher: Das Panelverfahren zur Berechnung der Druckverteilung von Flugkorpern im Unterschallbereich, Z. Flugw., 21:301-311, 1973. Kraus, W.: NASA-TT F-14117, 1972; VKILect. Ser. 87, 1976. 23. Kuchemann, D.: Aircraft Shapes and Their Aerodynamics for Flight at Supersonic Speeds, Adv. Aer. Sci., 3:221-252, 1962; Prog. Aer. ScL, 6:271-353, 1965; Jb. WGLR, 85-93, 1964. Baals, D. D., A. W. Robins, and R. V. Harris, Jr.: J. Aircr., 7:385-394, 1970. Kane, E. J. and W. D. Middleton: AGARD CP 71, 3:1-14, 1971. 24. Kuchemann, D.: Some Remarks on the Interference between a Swept Wing and a Fuselage, AGARD CP 71, 1:1-9, 1971. Kuchemann, D. and J. Weber: ARC RM 2908, 1953/1956. 25. Labrujere, T. E., W. Loeve, and J. W. Slooff: An Approximate Method for the Calculation of the Pressure Distribution on Wing-Body Combinations at Subcritical Speeds, AGARD CP 71, 11:1-15, 1971.
26. Lawrence, H. R. and A. H. Flax: Wing-Body Interference at Subsonic and Supersonic Speeds-Survey and New Developments, J. Aer. Sci, 21:289-324, 328, 1954; 22:208, 1955. Flax, A. H.: J. Aer. Sci., 20:483-490, 1953. Lawrence, H. R.: J. Aer. Sci., 20:541-548, 1953.
27. Lennertz, J.: Beitrag zur theoretischen Behandlung des gegenseitigen Einflusses von Tr agflache and Rumpf, Z. Angew. Math. Mech., 7:249-276, 1927; Z. FZug. Mot., 18:11-13, 1927; in W. F. Durand (ed.), "Aerodynamic Theory-A General Review of Progress," div. K. pp. 152-157, Springer, Berlin, 1935, Dover, New York, 1963.
AERODYNAMICS OF THE WING-FUSELAGE SYSTEM 427
28. Lock, R. C.: Theoretical Pressure Distribution at Zero Lift at Supersonic Speeds for Slender Delta Wings Having Fuselages of Circular Cross Section, Aer. Quart., 12:95-130, 1961. Jones, J. G.: Aer. Quart., 11:51-70, 1960. 29. Lock, R. C. and J. Bridgewater: Theory of Aerodynamic Design for Swept Winged Aircraft at Transonic and Supersonic Speeds, Prog. Aer. Sct, 8:139-228, 1967. Gustavsson, A. and R. Vanino: Z. Flugw., 23:257-262, 1975. Lock, R. C.: in K. Oswatitsch (ed.), "Symposium Transsonicum I," pp. 276-287, Springer, Berlin, 1964. Lock, R. C. and E. W. E. Rogers: Adv. Aer. Sci., 3:253-275, 1962. 30. Lomax, H. and M. A. Heaslet: Recent Developments in the Theory of Wing-Body Wave Drag, J. Aer. Sci., 23:1061-1074, 1956; .NACA Rept. 1282, 1956. Bonner, E.: I. Aircr.,
8:347-353, 1971. Ferri, A. and J. H. Clarke: J. Aer. ScL, 24:1-18, 1957. Ferri, A., J. H. Clarke, and L. Ting: J. Aer. Sci., 24:791-804, 1957. Graham, M. E.: J. Aer. ScL, 24:142-144, 1957. Licher, R. M.: T. Aer. Sci., 23:1037-1043, 1956. 31. Morikawa, G. K.: Supersonic Wing-Body Lift, J. Aer. Sci., 18:217-228, 503-504, 1951; Quart. App. Math., 10:129-140, 1952. 32. Multhopp, H.: Zur Aerodynamik des Flugzeugrumpfes, Lufo., 18:52-66, 1941; NACA TM 1036, 1942. Liess, W. and F. Riegels: Jb Lufo., 1:366-373, 1942. Luckert, H. J.: Can. Aer. J, 1:205-217, 1955. Weber, J., D. A. Kirby, and D. J. Kettle: ARC RM 2872, 1951/1956. 33. Munk, M. M.: The Aerodynamic Forces on Airship Hulls, NACA Rept. 184, 1924.
34. Muttray, H.: Die gegenseitige Beeinflussung der Einzelteile am Flugzeug ohne laufende Schraube, Ringb. Luft., I A 4, 1937; Lufo., 2:33-39, 1928; 11:131-139, 1934. 35. Oswatitsch, K.: The Area Rule, App. Mech. Rev., 10:543-545, 1957. Oswatitsch, K. and F. Keune: Z. Flugw., 3:29-46, 1955. 36. Otto, H.: Calculation of Nonlinear Lift and Pitching Moment Coefficients for Slender Wing-Body Combinations, J. Aircr., 11:489-491, 1974; Z. Flugw., 22:187-200, 1974; DLR-FB 73-66, 1973.
37. Pitts, W. C., J. N. Nielsen, and G. E. Kaattari: Lift and Center of Pressure of Wing-Body-Tail Combinations at Subsonic, Transonic, and Supersonic Speeds, NACA Rept.
1307, 1957. Flax, A. H.: J. Aircr., 11:303-304, 1974. Nicolai, L. M. and F. Sanchez: J. Aircr., 10:126-128, 1973. Nielsen, J. N.: Thesis, Cal. Inst. Tech., 1951. 38. Schlichting, H.: Neuere Beitrage der Forschung zur aerodynamischen Fliigelgestaltung (Umriss, Verwindung; Rumpfeinfluss), Jb. Lufo., 1:113-132, 1940. 39. Schlichting, H.: Die Stabilitatsbeiwerte des Flugzeuges unter Beriicksichtigung der Interferenz von Flugel, Rumpf and Leitwerk, Sonderheft: Flugmech. Probleme, Akad. Lufo. 2/43 g, 3-23, 1943. 40. Schlichting, H.: Calculation of the Influence of a Body on the Position of the Aerodynamic Centre of Aircraft with Swept-Back Wings, ARC RM 2582, 1947/1952.
41. Schlichting, H.: Monograph on the Aerodynamics of Mutual Interference between the Components of the Airplane, Nat. Res. Coun. Can., Tech. Transl. TT-92, 1949; Inst. Stro. THBraunschw. 46/5, 1946. 42. Schneider, W.: Experimentelle Bestimmung von Flugel-Rumpf-Interferenzen im Machzahlbereich Ma = 0,5 bis Ma = 2,0, DLR FB 67-93, 1967; AVA 70 A 23, 1970, 70 A 40, 1970; AGARD CP 35, 1968. 43. Schrenk, 0.: Angenaherte Berechnung der gegenseitigen Beeinflussung zwischen Flugel and Rumpf im Uberschallbereich, Z. Angew. Math. Phys., 1:202-209, 1950. 44. Spreiter, J. R.: The Aerodynamic Forces on Slender Plane- and Cruciform-Wing and Body Combinations, NACA Rept. 962, 1950; J. Aer. Sci., 19:571-572, 1952. Cambell, G. S.: J. Aer. Sci., 25:60-62, 1958. Jacobs, W.: Jb. WGL, 168-171, 1955. 45. Spreiter, J. R. and S. S. Stahara: Aerodynamics of Slender Bodies and Wing-Body Combinations at M. = 1, AIAA J., 9:1784-1791, 1971. 46. Stahl, W.: Untersuchungen an schlanken, kegligen Rumpf-Flugel-Kombinationen in Uberschallstromung, insbesondere hinsichtlich Volumenverteilung and Wolbung, Z. Flugw., 18:461-473, 1970. 47. Vandrey, F.: Zur theoretischen Behandlung des gegenseitigen Einflusses von Tragfiiigel and
428 AERODYNAMICS OF THE FUSELAGE AND THE WING-FUSELAGE SYSTEM
Rumpf, Jb. Lufo., 1:158-166, 1938; Lufo., 14:347-355, 1937; Jb. Lufo., 1:367-370, 1940. Liese, J. and F. Vandrey: Jb. Lufo., 1:326-335, 1942. 48. Vandrey, J. F.: Der gegenseitige Einfluss zwischen einem geraden Fliigel and einem Rumpf bei Nullauftrieb im Uberschallgebiet, Z. Flugw., 4:44-46, 1956. 49. Ward, G. N.: Supersonic Flow Past Slender Pointed Bodies, Quart. J. Mech. App. Math.,
2:75-97, 1949. Miles, J. W.: J. Aer. Sci., 19:287, 1952. Stocker, P. M.: Aer. Quart., 3:61-79, 1951. Yang, H. T.: AIAA J., 10:1535-1536, 1972. 50. Whitcomb, R. T.: A Study of the Zero-Lift Drag-Rise Characteristics of Wing-Body Combinations Near the Speed of Sound, NACA Rept. 1273, 1956. 51. WieseLsberger, C.: Airplane Body (Non Lifting System) Drag and Influence on Lifting System, in W. F. Durand (ed.), "Aerodynamic Theory-A General Review of Progress," div. K, Springer, Berlin, 1935, Dover, New York, 1963.
52. Woodward, F. A.: Analysis and Design of Wing-Body Combinations at. Subsonic and Supersonic Speeds, J. Aircr., 5:528-534, 1968; NASA CR 2228, 1973. Bradley, R. G. and B. D. Miller: J. Aircr., 8:400-405, 1971.
PART
THREE AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
CHAPTER
SEVEN AERODYNAMICS OF THE STABILIZERS
1
7-1 INTRODUCTION
7-1-1 Function of the Stabilizers and Control Surfaces The main parts of an airplane are the wing, the fuselage, and the tail unit or empennage. The aerodynamics of the wing has been discussed in detail in Chaps.
2-4, and that of the fuselage alone and the interaction between the wing and fuselage in Chaps. 5 and 6, respectively. Now, in Chaps. 7 and 8, the aerodynamics of the stabilizing and control surfaces will be discussed. Generally, an airplane has (see Fig. 7-1) a horizontal tail consisting of a horizontal stabilizer (tail plane) with an elevator, a vertical tail consisting of a vertical stabilizer (fin) with a rudder, and two ailerons.
A primary purpose of the tail unit is the stabilization of the airplane. This means that the airplane should have the tendency to return to a stationary flight attitude after a small disturbance. This process should take place "by itself"; that is,
the aerodynamic forces should move the airplane back to the original attitude without application of the control surfaces. Another equally important function of the tail unit is control of the airplane.
Whereas the horizontal stabilizer and the elevator control the motion about the lateral axis, the vertical stabilizer with the rudder and the ailerons control that about the vertical and longitudinal axes (Fig. 1-6). The control of the airplane requires establishment of an equilibrium of the moments about the three axes. Here, 431
432 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
rH
-
Elevator
Figure 7-1 The geometry of the tail surfaces (empennage).
in addition to the moments of the aerodynamic forces, those of the inertia forces play a role. As has already been pointed out in Sec. 1-3-3, the motion of the airplane about the lateral axis is termed longitudinal motion, that about the vertical and longitudinal axes lateral motion (side motion). Consequently, the horizontal stabilizer and the elevator stabilize and control the longitudinal motion. The vertical stabilizer and the rudder stabilize and, together with the ailerons, control the lateral motion.
Generally, each of the three control assemblies has the form of a wing with a control surface as shown in Fig. 2-24. It consists of a fixed and a movable part. The fixed part is termed a fin or vertical stabilizer at the vertical tail and a horizontal stabilizer or tail plane at the horizontal tail. The movable part is the control surface. It is termed a rudder at the vertical tail and an elevator at the horizontal tail. In Fig. 7-1, the horizontal stabilizer and the elevator and the vertical fin and the rudder are indicated by hatches. The changes of the moments required for control are effected by deflections of the control surfaces. At the horizontal and vertical tail assemblies, the moments may also be controlled by a stabilizer adjustment (stabilizer trim). The horizontal tail of many airplanes does not have a separate stabilizer and elevator. Here, the change of the moment about the lateral axis is achieved by displacement of the entire horizontal surface. The aerodynamic effect of the horizontal tail is illustrated in Fig. 7-2 for an airplane with and without a horizontal tail. The lift coefficient is plotted against both the angle of attack and the moment coefficient. According to Fig. 7-2a, the contribution of the horizontal tail to the total lift is relatively small. Figure 7-2b gives the moment curves for several setting angles EH at the tail plane. Comparison
AERODYNAMICS OF THE STABILIZERS 433
with the curves for the airplane without a horizontal tail shows that at all setting angles EH of the tail plane, the horizontal tail causes a considerable increase in the stability coefficient aCM/acL, as defined-in Sec. 1-3-3. A change in the tail-plane setting angle EH causes a parallel shift of only the moment curve CM(CL).* When the moment reference axis passes through the center of gravity at a steady flight attitude, the equation cm = 0 describes the moment equilibrium about the lateral axis. Figure 7-2b shows that this condition can always be satisfied by choosing the proper setting angle EH of the tail plane for a given lift coefficient. The results on wing-fuselage-tail systems at subsonic, transonic, and supersonic incident flows reported by Pitts et al. [26] should be pointed out. Schlichting [34] gives a summary of the importance of the interference among wing, fuselage, and tail unit for the stability coefficients of the airplane. The aerodynamics of the tail units will be treated in two parts: The problems concerning the, tail surfaces without deflection of the control surfaces (stabilization)
will be covered in Chap. 7, those concerning the effect of the control surfaces (control) and of the flaps (lift increase) will be discussed in Chap. 8.
7-1-2 Geometry of the Tail Surfaces The geometry of the horizontal and the vertical tails may be described basically like
that of a wing (see Sec. 3-1). In general, the horizontal tail has a symmetric planform and the vertical has an asymmetric side elevation (Fig. 7-1).
The planform of the horizontal tail is defined, in analogy to Sec. 3-1, by the following main quantities:
*The conclusion should not be drawn that the fin setting angle does not affect the longitudinal stability, because the determination of the degree of stability must always be related to an equilibrium state. a
b
10
0.8
02
Without 'r, ccWith I
Me m9
horizon tal tail
0
=0z -'F°
0°
'f°
8°
a
12°
16°
20°-0,3
-02
-0.1
0
03
cM
Figure 7-2 Wind tunnel measurements on an airplane (Messerschmitt model Me 109) with and without empennage. EH= setting angle of the tail plane (see Fig. 7-6a). (a) Lift coefficient vs. angle of attack. (b) Lift coefficient vs. pitching-moment coefficient.
434 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
Span of the horizontal tail bH Area of the horizontal tail AH Aspect ratio of the horizontal tail AH = b%I/AH Setting angle of the tail plane (Fig. 7-6a) £H Deflection of the elevator (Fig. 7-6a) rlx
The position of the horizontal tail relative to the airplane is given by the lever arm rH of the horizontal tail, defined as the distance between the geometric neutral points of the horizontal tail and the wing. The geometric neutral point is defined in Sec. 3-1.
For some airplanes, the high position of the horizontal tail relative to the wing plays some role. For the aerodynamic effects
of the horizontal tail, the following two
dimensionless quantities, which express size and location of the horizontal tail relative to the wing quantities, are particularly important; area ratio AH/A and relative tail-surface distance rHlcµ. Here, A is the wing area and cµ is the reference wing chord according to Eq. (3-5b). For a large number of airplanes, the area ratio lies between AH/A -- 0.15 and 0.25 and the relative tail distance between rH/cµ 2 and 3. The side elevation of the vertical tail is described by the following quantities (Fig. 7-1):
Height of the vertical tail h v Area of the vertical tail A V Deflection of the rudder 71V
The location of the vertical tail relative to the airplane is given by the lever arm ry of the vertical tail, defined as the distance between the geometric neutral points
of the vertical tail and the wing. A general definition of the aspect ratio of the vertical tail is not feasible because of the great variety of tail-surface shapes and the various positions of the vertical tail relative to the fuselage and to the horizontal tail (see Sec. 7-3-2).
For the aerodynamic effect of the vertical tail the following two dimensionless
quantities are important, as for the quantities for the horizontal tail: area ratio AV/A and relative tail-surface distance ry/s, where s = b/2 is the wing semispan. Approximately, A VIA = 0.1-0.2 and r y/s = 0.5 -1.0.
On many newer airplanes, the horizontal tail has been eliminated so that the airplane has only a vertical tail as shown in Fig. 7-3. Such an airplane is termed an all-wing airplane (flying wing). Here the function of the elevator (control about the lateral axis) has been assigned to a control surface (elevator) of width bH. Besides the most commonly used central arrangement of the vertical tail as shown in Figs. 7-1 and 7-3, various other arrangements are also found. For instance, Fig. 7-4a shows two fins (vertical tail surfaces) at the tips of the horizontal tail. Figure 7-4b illustrates a tail surface with large dihedral (V tail surface), combining the functions of both the horizontal and the vertical tails.
AERODYNAMICS OF THE STABILIZERS 43 5
Figure 7-3 The geometry of the empennage of an all-wing airplane.
For the aileron, to be discussed in more detail in Chap. 8, the aileron span SA as shown in Figs. 7-1 and 7-3 is important in addition to the aileron chord ratio.
7-2 AERODYNAMICS OF THE HORIZONTAL TAIL
7-2-1 Contribution of the Horizontal Tail to the Aerodynamics of the Whole Airplane Airplane in straight flight The lift acting on the horizontal tail adds considerably to the pitching moment of the whole airplane because of. its large lever arm compared
bH
a
Figure 7-4 Various forms of horizontal tail. (a) Horizontal tail with two fins. (b) Horizontal tail with large dihedral.
436 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
to the wing chord (see Fig. 7-1). Let LH be the lift of the horizontal tail andr r' the distance of this lift force from the moment reference axis (usually the lateral axis through the wing center of gravity). Then, from Fig. 7-5, the contribution of the horizontal tail to the pitching moment of the whole airplane is
MH = -rHLH
(7-1)
where the nose-up pitching moment is taken as positive. Here the contribution of the tangential force of the horizontal tail to the pitching moment has been
disregarded because of the small high position of the tail surface relative to the fuselage axis. For the contributions of the horizontal tail to the lift LH and to the pitching moment MH, dimensionless coefficients are introduced through LH = CIHAHgH
(7-2a)*
MH = cMHAcAq00
(7-2b)
Here qH is the dynamic pressure at the location of the tail surface. It is, in general, smaller than the dynamic pressure of the undisturbed flow qi, because of the effect of the wing on the tail surface. The moment coefficient of the tail surface referred to the wing quantities is obtained from Eqs. (7-1)-(7-2b) as qH AH rH CMH - -CIH q A C. CIH =
with
dclH daH
(aH - aCH 77H
(7-3a)
(7-3b)
al7f,
The lift coefficient of the horizontal tail CIH depends on, in addition to the geometric data, its angle of attack aH and the elevator deflection i7H (see Fig. 7-6a). The term dclHldcH represents the lift slope of the horizontal tail without interference, and (aaH/a7IH)TIH the change in the direction of the horizontal tail for zero lift caused by the elevator deflection. For the plane problem of the airfoil with control surface (flap), this coefficient has been given as a function of the control-surface chord ratio; for additional information see Chap. 8.
Generally, the incident flow direction of the horizontal tail is considerably different from that of the wing because the tail surface is strongly influenced by the wing and fuselage and lies in the wing downwash (interference). The incident flow
directions of the wing and horizontal tail differ, as shown in Fig. 7-6a, by the downwash angle a, = w/U.., induced by the wing and fuselage at the location of *Note that the index
I
has been chosen for the lift coefficient with reference to the
tail-surface quantities.
Figure 7-5 Contribution of the horizontal tail to the pitching moment (schematic). C.G. = center
of gravity of the airplane. W = weight of the airplane.
AERODYNAMICS OF THE STABILIZERS 437
a Incident flow direction of horizontal tail
rH rH b
-L
LoHLH
0XNH
rH N
Figure 7-6 Aerodynamics of a horizontal tail in straight flight. N25 = geometric neutral point, N = aerodynamic neutral point, (N25 )H = geometric neutral point of horizontal tail. (a) Incident flow direction of the horizontal tail, aH = a + eH + a W. (b) Aerodynamic forces on the wing and horizontal tail.
the tail surface. Here, w < 0 means downwash and w > 0 means upwash. The angle of attack of the horizontal tail thus becomes
aH = a -{
(7-4)
EH + Y-W
where off is the setting angle of the horizontal tail relative to the wing chord and a is the angle of attack of the wing. Hence the contribution of the horizontal tail to the pitching moment at zero elevator deflection becomes CMH = - d«H (a + LH + aw) qH AH rH
(77H = 0)
(7-5)
k
where rH is the distance of the neutral point of the horizontal tail from the moment reference axis of the airplane. The change of the moment with the angle of
attack at fixed setting angle of the horizontal tail (stability coefficient) is then obtained as acMH
( aa
yaap qH AH"H
dd1H
a0t) q,0 A CA
daH (
)eH=const
The quantity
a'%H_1 oa
,
I
afxw G-x
`For simplicity it has been assumed that the ratio of the dynamic pressures gHJq, is independent of the angle of attack a.
438 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
is termed the efficiency factor of the horizontal tail. For the moment change with setting angle of the horizontal tail at constant angle of attack, Eq. (7-5) yields dCIH qH AH r'y
CMH
( raeH )'%=Const
dag q. A CA
Comparison of Eqs. (7-6) and (7-8) shows that the moment change with angle of attack (stability contribution of the horizontal tail) depends on the interference between the wing and the horizontal tail. It is proportional to the efficiency factor of the horizontal tail, aaH/aa = (1 + The efficiency factor of the horizontal tail is generally considerably less than unity, as will be shown more accurately later. The moment change with setting angle of the horizontal tail
(control), however,
is
not affected by the interference if the ratio qH/q is
disregarded.
To establish the contribution of the horizontal tail to the lift of the whole airplane, it is advantageous to define the lift coefficient of the horizontal tail, in analogy to Eq. (7-2b), as (7-9)
LH = CLgAq,o
In analogy to Eq. (7.5), CLH
daH (a + EH + aw) qH AH
(7-10)
Here the comments made in connection with Eq. (7-5) apply also to the derivatives of C1H with respect to a and EH. In the investigations made so far of the contribution of the horizontal tail to the pitching moment and the lift of the whole airplane, the respective coefficients
have been established as functions of the angle of attack of the airplane and the setting angle of the horizontal tail. For some problems it is more favorable, however, to establish the contribution of the tail surface to the angle of attack and to the pitching moment as a function of the lift coefficient of the whole airplane and of the setting angle of the horizontal tail. The lift coefficient of the whole airplane is composed of that of the airplane without the horizontal tail CL OH, and the contribution of the horizontal tail CLH, that is, CL = CLOH + CLH. Hence the lift slope of the whole airplane, without consideration of the effect of the tail surface on the wing at fixed setting angle of the horizontal tail, is obtained from Eq. (7-10) as dCL
dc1H
1+
aaw
4H AH
(7-11)
as ) qr A M),-H=const (doc)OH The sought change of the angle of attack with the lift coefficient of the whole dag (
airplane is given by the reciprocal value of the right-hand side of Eq. (7-11).
The change of the angle of attack with the setting angle of the horizontal tail EH at constant lift coefficient of the whole airplane becomes
AERODYNAMICS OF THE STABILIZERS 439
_
as
8a)
aCLH
(aCL EH=const ( aEg )a=const
(a EH cL=const
(7-12)*
Here the second factor is given by Eq. (7-10). Like the wing alone, the whole airplane has a neutral point, that is, a point on
which that portion of the lift force of the whole airplane acts that is proportional to the angle of attack (compare Sec. 1-3-3). As shown in Fig. 7-6b, let the distance of the neutral point of the whole airplane from the neutral point of the airplane without the tail unit be designated as xNH. This distance is identical to the neutral-point displacement caused by the horizontal tail. This neutral-point displacement XNH, as shown in Fig. 7-6b, can be determined from the moment equilibrium about the neutral point NOH of the airplane without the tail unit XNHL = rHNLH. Here rHN is the distance of the neutral point of the horizontal tail from the neutral point of the airplane without a horizontal tail. The result is (aCLH\
as XNH = aCL
eA-cons t
rHN
( as )eK=const
Introducing Eqs. (7-10) and (7-11) into this equation finally leads to the neutral-point displacement caused by the horizontal tail, XNH Cµ
dCIH
aaw
dag ( i
aa )
dCL
+
( da )oH
+
dCIH
dag (
+
qH AH
q. A aaw
rHN
qH AH cu
( 7 -1 3)
aa) q= A
In this equation, the first fraction on the right-hand side determines the percentage of the tail-surface lever arm by which the airplane neutral point is shifted rearward relative to the neutral point of the airplane without a horizontal tail. In many cases, the second term of the sum of the denominator can be disregarded in comparison with the first term. The neutral-point position of the whole airplane is obtained
from the neutral-point position of the wing alone (Chaps. 3 and 4), from the neutral-point displacement caused by the fuselage (including the wing-fuselage
interference, Chap. 6), and from the neutral-point displacement caused by the tail surface as given above.
The change of the moment coefficient of the horizontal tail as a function of the lift coefficient from Eq. (1-29) is obtained immediately from the neutral-point displacement caused by the horizontal tail in the form *This equation follows from consideration of the total differentials
da = aCL dcL + aH dEH with acL/aEH = BCLH/aEH.
and
dCL = aa« dcti + a H dEH
440 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES 3CMH aCL ej,t=const
XNH
(7-14)
CA
Finally, the change of the moment with the setting angle of the horizontal tail may
be determined for a fixed lift coefficient of the whole airplane. This is a free moment because the total lift remains constant. As shown in Fig. 7-6b, MH = -rHNLH, where LH is the lift of the horizontal tail caused by the change of the setting angle of the horizontal tail, and rHN is the distance between the neutral points of the horizontal tail and of the whole airplane. Thus, observing Eq. (7-10) and with rHN = rHN - XNH, the following relationship is obtained: aCMH
aCH
__ _ dCIH qH AH rHN cL = const
daH q.. A
Cµ
(7-15)
This relationship is also valid for The two coefficients of Eqs. (7-14) and (7-15) can be taken from Fig. 7-2b, the first as the difference of the slopes of the curves CM(CL) with and without tail surface and the second from the curves with different setting angles EH of the horizontal tail. To evaluate the above equations for the contribution of the horizontal tail to
the lift and the moment, attention must be paid to the ratio qH/q,,. Special attention must be paid, however, to the angle of incidence caH of the horizontal tail,
because it depends strongly on the interference between the wing and the tail surface. The angle of incidence of the horizontal tail aH is decisively affected by the induced downwash angle a,,< 0 caused by the wing at the location of the horizontal tail [see Eq. (7-4)]. In Fig. 7-7a, the change of aH with the rearward position of the tail surface is shown under the assumption of a horizontal tail chord parallel to the wing chord (EH = 0). At the wing trailing edge, aH = 0 because here the kinematic flow condition requires that a + a11, = 0. With increasing distance
b
U. -0-
TH
Figure 7-7 Angle of incidence of the horizontal tail. (a) In straight flight at angle of attack a. (b) In pitching motion with angular velocity wy.
AERODYNAMICS OF THE STABILIZERS 441
from the wing trailing edge, aH increases and assumes a constant value at a large distance that is considerably smaller than a. The distribution of a11, and thus of aH behind the wing can be computed with wing theory. This matter will be discussed below.
The lift slope of the horizontal tail dclH/daH for a horizontal tail free of interference may be determined with wing theory.
Airplane in pitching motion So far only the influence of the airplane angle of attack on the aerodynamics of the horizontal tail has been considered. In addition, however, the rotational motion of the airplane about the lateral axis is particularly
important for the aerodynamics of the horizontal tail. During rotation of the airplane about the lateral axis with angular velocity wy, an angle-of-incidence distribution aH of the horizontal tail is created as shown in Fig. 7-7b that increases
linearly with distance from the axis of rotation. This angle of incidence at the location of the horizontal tail (three-quarter point) becomes, with the distance from the axis of rotation rH, aH
with
Sly
=
V VC x
(7-16a)
u
wycu
(7-16b)
V
as the dimensionless pitching angular velocity. By introducing this expression for aH into Eq. (7-3b) and the resulting formula into Eq. (7-3a), the change of the moment coefficient with the pitching angular velocity is obtained, with rH -- rH, as aCMH
any
dc1H qH AH
daH q. A
(7-17)
This coefficient is termed the contribution of the horizontal tail to the pitch damping. Comparison of this formula with Eq. (7-6) shows that the contribution of the horizontal tail to the stability is proportional to (AH/A)(rJlcu), and that to the damping is proportional to (AH/A)(rHlc2)2.
7-2-2 The Horizontal Tail in Incompressible Flow The horizontal tail without interference The further discussions on the aerodynamics of the horizontal tail of this section will deal first with incompressible flow and then with compressible flow at subsonic and supersonic velocities. The horizontal tail without interference from fuselage and wing will be treated first, followed by an account of the effect of the wing on the horizontal tail. For the horizontal tail in incompressible flow without interference, the three-dimensional wing theory of Chap. 3 can largely be applied. Of the aerodynamic coefficients, first the lift slope de1H/daH for small and moderately large aspect ratios AH is required. In Fig. 7-8 a few theoretical curves are given for the lift slope of the horizontal tail as a function of the aspect ratio AH. A
442 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
Figure 7-8 Lift slope of a horizontal tail without interference for incompressible flow vs. aspect ratio of the tail surface dN
AH (lifting-surface theory).
rectangular, a swept-back, and an elliptic wing are described. The elliptic wing follows, from Eq. (3-98), the simple formula dCIH
da$
_
_H
kg +
i+
i
with k$ ==;ig CL
2
Ax
(7-18)
Further information on the lift slope and comparisons with measurements have been given in Sec. 3-3. There, the neutral-point position can also be found, which is required for the determination of the tail-surface lever arm. The above data for the lift slope can be applied to a horizontal tail without a vertical tail surface and also to a horizontal tail with a single vertical tail.
For a horizontal tail with two fins, as shown in Fig. 7-9, the lift slope is considerably larger because of the end-plate effect. Theoretical investigations on wings with end plates have been conducted by Mangler [221. The effect of end plates on the lift slope can be taken into account approximately by introducing, besides the geometric aspect ratio /1H, a so-called effective aspect ratio A*. For a horizontal tail with end plates, these two values Aff and AH are related by the empirical formula
1 = AB 1 -{- 21 bCV Ccv
V H
a
(7-19)
Measurements on the effect of end plates were first published by Prandtl and Betz [27]. In Fig. 7-9, the lift slopes dc1H/doaf, based on those measurements, are given as a function of the effective aspect ratio A * . The solid curve applies to the rectangular wing of Fig. 7-8.
Effect of the fuselage on the horizontal tail The interference of the wing and fuselage with the horizontal tail consists of a reduction of the dynamic pressure at the location of the tail surface and also in an altered incident flow direction of the tail surface. The reduction in dynamic pressure is caused mainly by the boundary layer at the wing-fuselage interface, and the change in incident flow direction of the
AERODYNAMICS OF THE STABILIZERS 443
tail surface by the induced velocity field of the wing-fuselage system. Whereas the induced velocity field can be reasonably well determined theoretically, the dynamic pressure reduction must be found experimentally. It is desirable that the value of the ratio qH/q, be as close to unity as possible and that it be essentially independent of the angle of attack of the airplane. Both requirements can be satisfied through suitable selection of the horizontal tail relative to the wing and the fuselage; compare Hafer [13]. Now, the influence of the fuselage on the horizontal tail will be discussed first.
The arrangement of a horizontal tail on the fuselage corresponds basically to a wing-fuselage system as treated in Sec. 6-2. There is the difference, however, that the fuselage usually does not extend behind the tail surface. It is very difficult to establish a general procedure for the computation of the influence of the fuselage on the tail plane because of the many different arrangements of the horizontal tail
(high, mid, low surface) and the various shapes of the tail of the
fuselage.
Therefore, a review of some test results on this influence must suffice. Koloska [13] reports three-component measurements on fuselage-tail surface systems. The tail surfaces were rectangular of aspect ratio AH = 2 and 1.2, attached
to a partial fuselage. The lift slopes as affected by the fuselage, dc1H/daH, are considerably smaller than those for the horizontal tail without interference as shown in Fig. 7-8. In Fig. 7-10, values of dc1H/daH under the influence of the fuselage are given as a function of the aspect ratio of the horizontal tail AH and the relative fuselage width bF/bH. Accordingly, to give an example, at an aspect ratio AH = 2 and a relative fuselage width bF/bH = 0.3, the fuselage effect reduces the lift slope by about 20%.
Effect of the wing on the horizontal tail The effect of the wing on the tail surface consists essentially of a change of the angle of incidence of the horizontal tail
AH
3 3
AH= Ly = 3
2
S°
i8 3+`
6
°
Theory c
bH
V-1
rcH' 1
!,
!
1
Figure 7-9 Measured lift slope of horizontal tail with end plates, from [27],
Profile 90535
i
0
b
2
3
A*H
vs. effective aspect ratio of the tail
1
6
surface A -, from Eq. (7-19). Theoretical curves from Fig. 7-8 for AH = ' H-
444 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
9
i3
Z2
O
122
Q3 0.4 O'S'
0
1
2
3
AH
9
5
Figure 7-10 Lift slope of the horizontal tail as affected by the fuselage vs. aspect ratio of the horizontal tail fox several relative fuselage widths bF/bH, from Koloska.
because of the induced downwash velocity behind the wing. The relationship between the angle of incidence of the horizontal tail aH and that of the wing a is given by Eq. (7-4); the change of the angle of incidence of the horizontal tail with the angle of attack of the airplane is given by Eq. (7-7). In general, the coefficient 0) and, for a given wing, depends only on the aax,/aa is negative position of the tail surface. The coefficient aaH/aa acts as an efficiency factor of the horizontal tail [Eq. (7-6); see also Eq. (7-13)]. Its value is usually between 0 and 1 and signifies that the downwash reduces the stabilizing effect of the horizontal tail. The aim of the remainder of this section is the determination of this efficiency factor as a function of geometric and aerodynamic, data of the wing and of the position of the horizontal tail relative to the wing. The induced downwash velocity is generated by the vortex system of the wing (bound and free vortices). Figure 7-11 illustrates schematically the vortex system of a given circulation distribution. Figure 7-11a shows the free, not yet rolled-up vortex sheet, whereas in Fig. 7-1 lb the free vortex sheet is rolled up into two single vortices at a certain distance behind the wing. A plane vortex sheet as in Fig. 7-1 la is unstable and tends to roll up into two single vortices (see also Figs. 3-8 and 3-22). From the known vortex system of a wing, the field of the induced downwash velocities is obtained with the Biot-Savart law. The vortex system of a given wing is obtained from the lift distribution as described in Sec. 3-3. In Fig. 7-12, the induced downwash and upwash angles on the longitudinal axis (.x axis) are shown for an elliptic wing without twist. The induced downwash angle a,, is referred to the induced angle of attack at = CL11TA of the wing by Eq. (3-3 la). The ratio a,^,/al is dependent on the angle of attack of the
AERODYNAMICS OF THE STABILIZERS 445
r1
Figure 7-11 The vortex system behind a wing (schematic). (a) Not-rolled-up vortex sheet. (b) Rolled-up vortex sheet.
wing. The relative downwash angle a11,/aj is given as a function of the dimensionless longitudinal coordinate = x/s. The solid curve represents the induced downwash angle of the total vortex system (bound and free vortices) from Eq. (3-96), and the
dashed curve represents the contribution of the free vortices. The difference between the solid and the dashed curves is the contribution of the bound vortices. This latter contribution becomes meaningless for t > 1. For such distances of the tail surface, the induced downwash angle is determined predominantly by the contribution of the free vortices, so that
a,, = -2ai
(7-20)
3
2
I
i
I
i
i
I
I
I
I
-
r
-
I
l
t
I
I
i
Z'
I
I
I
I
I
-2s Free vortices
'
i
I
; Bound vortices
-2
CL
7rA -3
Total vortex system -9
-1,2
-1.0
-08
0.2
at
06
0.8
1,0
1.2
Figure 7-12 Induced downwash angle ai,,, on the x axis of a wing of elliptic planform, from [37]. Contribution of free and bound vortices.
446 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
To obtain the induced downwash angle at the location of the horizontal tail, the position of the horizontal tail relative to the vortex sheet must also be known. Here it must be realized that, in general, the vortex sheet behind the wing lies neither in the wing plane nor in a plane parallel to the incident flow direction. Its shape is curved as shown in Fig. 7-13. Its distance from the wing chord z = 0 is given by zI(x, y). Because of the kinematic flow condition, the vortex sheet at the wing trailing edge xr is tangent to the wing plane; farther downstream it is deflected more and more upward from the wing plane. Its position may be easily determined from the equation x
zl (x, y) = f [a -l
(7-21)*
a,,, (x, y) ] dx
X.
whose validity is obvious from Fig. 7-13. The location of the wing trailing edge is given by x,.(y). Once the position of the vortex sheet is found, the distance of the horizontal tail from the vortex sheet, needed to determine the induced downwash angle at the location of the horizontal tail, is given by (z - zl ). Now, by means of theoretical results and measurements, we shall discuss the influence of the wing shape and of the lift distribution on the distribution of the downwash angle behind the wing. For the not-rolled-up vortex sheet with a given circulation distribution T(y) = bU,7(y), the downwash angle at z = zl is obtained from lifting-line theory by the Biot-Savart law from Eqs. (341), (3-50a), and (3-50b) as
1 lim
aw
2 rv
+1
4 E
y M - jr (,7y-01") 77/)2
1
+
l
d
L
I (S - ei)2 + (71 - 17 /)Z
(7-22)
*To evaluate the integral, the induced downwash aN, < 0 should be considered to be constant.
Um
Figure 7-13 Position of the vortex sheet behind the wing (schematic).
AERODYNAMICS OF THE STABILIZERS 447
T= consf
at d z CL
111
I
b-as 3,5
IrW 1
t
Z
CL
a',1td
l(yf c
2.5
Approximation to 2 3
j
40 0
0.2
a#
0.6
08
10
1.'
1.2
16.
18
20
Figure 7-14 Downwash angle in the vortex sheet ( = 1', ) for 77 = 0 (plane of symmetry of the airplane) behind unswept wings, from Truckenbrodt, computed by lifting-line theory. Curve 1, constant circulation distribution. Curve 2, elliptic circulation distribution. Curve 3, parabolic circulation distribution.
Here t = x/s, rl = y/s, and = z/s are the dimensionless coordinates, and i = t(n') gives the location of the lifting line in the wing from Fig. 3-29. For unswept wings the coordinate origin lies on the lifting line and t' = 0. For the numerical evaluation of this equation, a quadrature procedure has been developed by Multhopp [25]. Other computational methods and results have been published by Glauert [111, Lotz and Fabricius [21 ], and Helmbold [211. The effect of the lift distribution on the downwash distribution in the plane of symmetry of the wing (77 = 0) and in the vortex sheet = si is shown in Fig. 7-14 for the rectangular, elliptic, and parabolic lift distributions. The downwash angle ati is referred to the induced angles of attack ai in the middle of the wing (r1= 0), whose values are also given in the figure. Hence, for all three lift distributions the ratio a,,/cat = 2 far downstream of the wing. This result, which has been given in Eq. (7-20), is obtained by setting t -o in Eq. (7-22) and comparing with Eq. (3-71c). The curves of Fig. 7-14 demonstrate that the kind of lift distribution over the wing span has a considerable influence on the values of the downwash angle at small distances from the wing. For a constant circulation distribution the downwash is expressed by the simple formula
0) = 1
ti
-
1
L
W = const)
(7-23)
where cL/27r%1= aj(0). This formula is obtained from Eq. (7-22), but also directly
from the horseshoe vortex by means of the Biot-Savart law. For the elliptic circulation distribution the downwash angle becomes, according to Glauert [11],
448 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
-aw(,0)1-{-
V
zll
79
(T=To 1-rte)
where E is the complete elliptic integral of the second kind with the module 2 + 1. In the present case, ai(0) = CL/TtJl. The downwash angle at some 1/ distance behind the wing is given by an approximation formula of Truckenbrodt [25] as -a. ($, 77) -- tai (q) rt
ti
(2
+
1
(7-25a)
CL
-r/1
(5 >0)
2) CL
(7-25b)
45 n"1
This last expression applies to elliptic circulation distributions with cap = CL/irA. The result of this formula is added in Fig. 7-14 as an approximation.
The effect of the wing planform on the distribution of the downwash angle over the span at a distance = 1 behind the wing is shown in Fig. 7-15. The three wings have an aspect ratio A = 6 and taper ratios X = 1.0, 0.6, and 0.2. This figure shows that the shape of the wing planform decisively affects the distribution of the downwash angle over the span. Hence the effectiveness of the horizontal tail is much smaller for a highly tapered trapezoidal wing than for a rectangular wing.
The solid curves were determined by a computational procedure of Multhopp [25], whereas the dashed curves were computed using the approximation formula Eq. (7-25a). Figure 7-16 shows the effect of the sweepback angle on the distribution of the downwash angle behind the wing. For simplicity, constant circulation distribution over the span has been assumed for all those sweepback angles. The distribution of the downwash angle over the longitudinal axis shows that the downwash is much
I---- b42s--i v+ "_1K
3
I
,z
i
_'Ellipse
- Ellipse
---r-Eclipse -
X32
I
b 02
0.
0.6
08
100
c
l
0.2
i
!
9'/
0,6
08
10 0
0.2
0
0.6
08
10
Figure 7-15 Downwash-angle distribution over the span in the vortex sheet at distance = x/s = 1
behind the wing, for 3 unswept wings of aspect ratio A = 6, computed by simple lifting-line theory. (a) Rectangular wing. (b) Trapezoidal wing of taper A = 0.6. (c) Trapezoidal wing of taper X = 0.2. Solid curves, exact solution from Multhopp. Dashed curves, approximate solution from Truckenbrodt.
AERODYNAMICS OF THE STABILIZERS 449 8
Z r= const
0°
-4s°
-
b= 2s
0
D..s
10
za
IS'
Figure 7-16 Distribution of the downwash angle aw on the x axis behind swept-back wings of constant circulation distribution.
greater at a backward-swept wing than at a forward-swept wing. The Biot-Savart law leads to the following simple formula for the downwash distribution: 0)
= 11
-{-
tanc)2 -{- 1 +
tangy)] I
1
2nd.
(7-26)
where CL/27rzl = a=(0). Systematic measurements on the downwash of swept-back wings have been conducted by Trienes [40] by the probe surface method. Note also the investigations of Silverstein and Katzoff [38] and of Alford [2]. The results obtained so far were based on the flow with a not-rolled-up vortex sheet. A few data will now be given of the influence of the vortex sheet roll-up on the downwash at the location of the horizontal tail. As has been described by Fig.
7-11b and in more detail in Sec. 3-2-1, the vortex sheet rolls up into two single vortices at some distance behind the wing. They have the circulation To of the root section of the wing, and, from Eq. (3-58), are apart by bo far behind the wing. In Fig. 7-17, the ratio bo/b is plotted against the aspect ratio for a rectangular wing according to Glauert [11]. For an elliptic circulation distribution the ratio is constant: IT
b
4
(elliptic circulation distribution)
(7-27)
For rectangular wings, bo/b increases from this value when the aspect ratio A becomes larger. For very large A, it approaches unity asymptotically, which is the value
of the constant circulation distribution. A simpler computation of the
downwash at a rolled-up vortex sheet is possible by considering a horseshoe vortex as in Fig. 7-17 of strength TO whose free vortices have the distance bo. This quite idealized picture of the roll-up process has not been fully confirmed by measurements of Rohne [16], as seen from Fig. 7-18. Here, the ratio bo /b and the distance ao behind the wing at which the rolling-up process has been completed
450 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
7-----I-
10
r=cnnst
0.8
X0
Rectangular wing
TTF
Elliptic wing
r(y)
r Figure 7-17 Aerodynamics of the rolled-
up vortex sheet behind a wing (schematic). Ratio b0 /b vs. aspect ratio of the wing A. Rectangular wing from [I I ] .
6 A-
have been plotted against the lift coefficient. The measured ratio bo lb is noticeably larger than the theoretical value of Fig. 7-17. A summary report on early downwash measurements is given by Flugge-Lotz and Kuchemann [8]. Studies of the physical explanation of the roll-up process were first made by
Kaden [16] and Betz [161, somewhat later by Kaufmann [16] and Spreiter and Sacks [39]. More recently, additional insight has been gained, to some extent, through the use of efficient computers [3, 4, 12, 30, 421. To convey a feeling for
al
I
o Rectangular wing c
T
Trapezoidal wing
1.0
b0
b=25
1019
b
0.7 0 d
10
0.5
15
CL
Figure 7-18 Measurements of the aerodynamics of the rolled-up vortex sheet behind a wing, from Rohne. (a) Vortex system. (b) Tested wings (profile Go 387). (c) Distance ao at which the
rolling-up process is completed. (d) Distance bo between the two rolled-up vortices. Dashed straight line, theory according to Fig. 7-17.
AERODYNAMICS OF THE STABILIZERS 451
the magnitude of the effect of the wing on the horizontal tail, the efficiency factor of the horizontal tail from Eq. (7-7) is plotted in Fig. 7-1.9 against the aspect ratio. These values apply to very large distances of the tail surface from the wing Q.-* -0) and for wings with elliptic circulation distributions. With the value for the lift slope of Eq. (3-98), the efficiency factor of the horizontal tail for not-rolled-up vortex sheets becomes aaH as
-1+
aaw as
=
1/112 +4 142
-2
+4+2
(` ' °O)
(7-28a)
(y > cc)
(7-28b)
For the rolled-up vortex sheet (horseshoe vortex) it is
YA2+4-2 (bo)
8a a«
=
z
1//1z+4+2
-1
-
with bo lb = it/4 from Eq. (7-27). At small wing aspect ratios, the efficiency factor of the horizontal tail is relatively small; it increases strongly with A.
All the results on downwash obtained so far apply to control points in the vortex sheet. The horizontal tail lies, depending on the angle of attack of the airplane, in, above, or below the vortex sheet. Outside the vortex sheet the downwash is always smaller than in the sheet. This will be shown by the following examples. Before pursuing this matter, however, the position of the vortex sheet (Fig. 7-13) will be discussed. With the help of Eq. (7-21), the position of the vortex sheet is obtained from the distribution of the downwash behind the wing. In Fig. 7-20 the position of the vortex sheet in the root section r7 = 0 behind the wing is shown for an elliptic wing. The distance between vortex sheet and the wing plane is proportional to the angle of attack of the wing. For the downwash angle outside of the vortex sheet, the following equation is obtained for a given circulation distribution by generalization of Eq. (7-22) according to lifting-line theory: +1
w
(77, ) =
1
_ where r =
f () i
)- - (b ( [[I:/;(I::;:]2 \i +
(77
{7 -77')2+( -C1)2
\S - J)2 + ( - 771)2
r3
_i d ?7'
(7-29)
+ (S - J 1 )z
1.0
Figure 7-19 Efficiency factor of the horizontal tail aaH/aa in incompressible flow 02
vs. aspect ratio of the wing for rolled-up and
not-rolled-up vortex sheets. Computation from lifting line theory for elliptic circulation distribution at a very large distance behind the wing
452 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
Figure 7-20 Position of the vortex
0
sheet behind elliptic wings of several aspect ratios A (see Fig. 7-13).
Z
1
r/cr
The quantities used in this equation are defined in connection with Eq. (7-22). Equation (7-29) is converted into Eq. (7-22) by According to Multhopp [25], the change in downwash with distance from the vortex sheet is given by
a Lcx , - I C -- C11 Idly
8a,n
(7-30)
d.?71
Thus the curves of the downwash angle o
against the distance from the vortex
sheet have, in general, a break at the station of the vortex sheet. Experimental results of this kind for unswept and swept-back wings are plotted in Fig. 7-21, from
Trienes [40]. They have been obtained by the probe surface method, which is c-450 300 190 00 -30°
-04
-37 02
b 0.2
0.4
06
08
Oaf
0
-
02
0.4
- aa-W/ as
0,6
----
0e
4
Cr i.
b=2s
Figure 7-21 Downwash distribution outside the vortex sheet; measurements of Trienes by means of the probe surface method. xH = s = rearward position, and H = aH/s the relative high position of the horizontal tail, a,,, = downwash angle as averaged over the probe surface. (a) For an upswept trapezoidal wing. (b) For a swept-back wing of constant chord. Hatched area = probe surface.
AERODYNAMICS OF THE STABILIZERS 453
described in [40] and, therefore, are mean values of the downwash angle a,, over the span of the horizontal tail surface. These experimental results confirm that the downwash angle has a peak value in the vortex sheet. Finally, in Fig. 7-22, theoretical downwash distributions from Glauert [11] are included for the transverse plane far behind the elliptic wing. They show that, for any high position, downwash prevails within the wing span range and upwash outside this range.
To compute the downwash in the vortex sheet, as pointed out above, a quadrature method based on lifting-line theory has been given by Multhopp [25]. An extension of this quadrature method for the computation of the downwash outside the vortex sheet has been developed by Gersten [10] for both the theories of the lifting line and of the lifting surface. The induced downwash velocity according to lifting-surface theory is obtained from the velocity potential of Eq. (3-46), where w = aO/az, as 4-3
IV (x, y, z) =4 z
Gl (x, y. z;
_ y)) 3_
y,)
[(yy
8
-
-
+S
4
G2(x,y,z;y')
+ (zz
(y
_S
-y)2
_ 11) 2
-
)2]'
d y,
-
2 +(z-21)2dy'
(7-31)
a
rry
Upwash
0.
1.6
y o,z
X
z 0.3
y!,
-04
Downwash -0.8
' 1=QS -12
CL
aL
0,1
JCI
f
03
02 -1.6
7-22 Theoretical downwash and upwash angle distributions over the span outside the vortex sheet for an elliptic wing, from Glauert. Figuze
0.2
0,6
ae
1,0
12
16
454 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
Here, G1 is the expression of Eq. (3-47), and G2 (=C, y, 2; Y')
=
zr(y') Z- 1'r'
,
') (, - ')x dx'
V (X - x')2 + (y - y')2 + (z Xf(y')
-
21)23
(7-32)
In analogy to the lifting-surface method of Sec. 3-3-5, Gersten [10] based the evaluation of Eq. (7-31) on two fundamental functions for the vortex density k. In this way he succeeded in developing a relatively simple computational procedure to determine the downwash.
Stabilization by the horizontal tail (neutral-point displacement) This discussion of the downwash will now be concluded with a simple reflection on the displacement of the neutral point of the airplane caused by the horizontal tail xNH (see Fig. 7-6). The analytical expression for this quantity has been given by Eq. (7-13). Let the wing and horizontal tail be of elliptic planform and the distance between the two neutral. points be rffN. The aerodynamic coefficients in Eq. (7-13) have already
been discussed in detail. The lift slope of the airplane without horizontal tail (dcL/da)OH is taken to be equal to that of the wing according to Eq. (3-98). The lift slope of the horizontal tail without interference has been given in Eq. (7-18) and the efficiency factor of the horizontal tail (1 + in Eq. (7-28a). Under the assumption that qH/q = 1, introduction of these expressions into Eq. (7-13) yields, after some intermediate steps, AH
a waH A xNH 1 + awaH
AH rHN
(7-33)
A
Here
aw =
A
d2+4+2 4g ag =
!l$+4±2
(7-34a)
(7-34b)
Equation (7-33) expresses a remarkably simple relationship between the neutralpoint shift caused by the horizontal tail and the four geometric parameters: aspect ratio of the wing A and of the tail surface AH, respectively; ratio of the areas of horizontal tail and wing AHIA; and distance between the neutral points of the tail surface and the wing rHJ1. This relationship is shown in Fig. 7-23. In this diagram is also shown the neutral-point displacement that would be obtained without interference. It is computed, for simplicity, by the stripe method, in which the lift slopes of wing and horizontal tail are set equal to 27r. This case is obtained from Eq. (7-33) with aw = aH = 1 as _ A (7-35) xNH ` A +HAH rHN
AERODYNAMICS OF THE STABILIZERS 455 016
Stripe method
NW
Wing ........ ,.
.
2 xNHI 0.12
T a il
,""i
I
f
s ur ace
/
A-6;Ay=61 00 A=12;11y=6 002
Figure 7-23 Neutral-point displacement caused
by the horizontal tail of wing-horizontal tail 0
02
0.1
AH/A
03
systems vs. the area ratio AHIA, from Eq. (7-33). Stripe method from Eq. (7-35).
The difference between this curve and the others indicates the interference effect
of the wing on the horizontal tail with respect to the neutral-point displacement, including the influences of the finite aspect ratios of wing and tail surface.
Stability at nose-high flight attitude (stall) When an airplane gets into the nose-high flight attitude, safety requires that the pitching-moment curves in this range still be stable (aCMl aCL < 0). For many wing shapes, for example, swept-back wings of large aspect ratio, this condition is not fulfilled. There are a number of measures, such as, for example, boundary-layer fences and slat wings, that lead to a wing stall
behavior ensuring that no nose-up (tail-heavy) pitching moment (pitch up) can occur. Particular attention must be paid to the effect of the downwash as changed by the partial flow separation from the wing on the horizontal tail. Besides the wing planform, the position of the horizontal tail relative to the wing plays an important role, and particularly the high position of the tail surface. Furlong and McHugh [9] give a detailed report on this problem. Severe stability problems can arise, particularly for swept-back-wing airplanes with a tail surface in extreme high position (T fin) at very large angles of attack.
Here the horizontal tail lies in the separated flow of the wing, and its incident flow has a very low velocity. This leads to an unstable action and an almost complete loss of maneuverability. Then the angle of attack increases more and more until, eventually, at a very large angle of attack, a stable flight attitude is again established. Because of the lack of control effectiveness, it is impossible to change this extreme flight attitude, and the airplane is in danger of crashing. This flight attitude is termed "super-stall" or "deep stall." Byrnes et al. [6] have studied this problem in detail.
456 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
7-2-3 The Horizontal Tail in Subsonic Incident Flow The effect of compressibility on the aerodynamic coefficients had been determined by means of the Prandtl-Glauert-Gothert rule for the wing in Sec. 4-4 and for the wing-fuselage system in Sec. 6-3-1. In the same way, this effect can be determined for the horizontal tail. Through a transformation, the subsonic similarity rule allows one to reduce the compressible subsonic flow about the whole airplane to incompressible flow. Here the incompressible flow is computed for a transformed airplane as shown by an example in Fig. 7-24 for Ma = 0.8. The transformation of
the geometric data is given in Eqs. (6-29)-(6-31). For the geometric data on the horizontal tail, Eqs. (6-30a)-(6-30e) apply accordingly. For the transformation of the distance of the tail surface from the wing, the relationship rHinc = rH has to be added, observing Eq. (6-29). The same relationship as for the wing alone applies to the dependence of the lift slope of the horizontal tail without interference on the Mach number Ma.. Hence, with Eq. (4-74), the relationship dc1H
dxH
2nAH V(1
- Ma') A'2 + 4 -F 2
(7-36)
is obtained, which is shown in Fig. 4-45. By computing the incompressible flow for
the transformed airplane at the angle of attack of the subsonic flow, that is, for «inc = a, the induced downwash angle in the vortex sheet becomes aw(S, n) = aw inc(inc, Thnc)
(7.37a)
= - 2ai inc
(7-37b)
( -* °°)
This relationship allows one to determine in a very simple manner the downwash field of compressible flow from that of incompressible flow. A simple approximation formula for the downwash of incompressible flow at some distance behind the wing has been given by Eq. (7-25b). With the above transformation and with Eq. Lla
yi nc
Figure 7-24 The Prandtl-Glauert rule at subsonic incident flow velocities. (a) Given airplane. (b) Transformed airplane.
AERODYNAMICS OF THE STABILIZERS 457 23
1.5
0.6
OZ
08
110
Figure 7-25 Effect of Mach number on the downwash angle at the longitudinal axis behind a wing of elliptic circulation distribution, from Eq. (7-38).
Ma.
(4-72a), this formula can be reduced to subsonic flow. For elliptic lift distribution there results
- aw =
L
J_451 (1 _ Mat00 )J rcll
(7-38)
In Fig. 7-25 the downwash angles so computed for
= 1, 1.5, and 2 have been
12
plotted against the Mach number Maw, .
As a further result, in Fig. 7-26 the efficiency factors of the horizontal tail from Eq. (7-28a) are plotted against the Mach number for several aspect ratios. The analytical expression is caXE
- 1 + 8a, _
8a
'am
A2 (1
- Moo) + 4 - 2
VA2(1-Mat)+4+2
oo)
(7-39)
0.7r
Q6F-
02
0,1
Figure 7-26 Efficiency factor of the horizontal tail vs. Mach number for elliptic wings of various aspect ratios A, from Eq. (7-39) for Ma,,,
458 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
This figure indicates the remarkable result that the efficiency factor decreases
strongly with increasing Mach number at all aspect ratios A. For Ma = 1, the efficiency factor of the horizontal tail becomes zero at all aspect ratios, a result in agreement with slender-body theory (see also Sacks [32] ). Finally, in Fig. 7-27, the
efficiency factor of the horizontal tail acH/aa for a delta wing of aspect ratio A= 2.31 is given for several Mach numbers as a function of the tail surface distance. Accordingly, the efficiency factor changes only a little with Mach number in the range 0 < Ma., <0.8.
7-2-4 The Horizontal Tail in Supersonic Incident Flow Fundamentals The influence on the horizontal tail of the forward airplane components (wing and fuselage) is, at supersonic incident flow, generally greatly
different from that at subsonic incident flow. This difference is a result of the limited influence zones at supersonic incident flow as shown in Fig. 7-28. The flow
at a point of the horizontal tail can be affected only by the parts of the airplane lying within the upstream cone of this point. This cone is, from Fig. 4-58, the Mach
cone of the generating semiangle p, located upstream of the control point under consideration and with axis parallel to the incident flow direction. The relation between incident flow Mach number and Mach angle is given by Eq. (4-80). The upstream cone cuts out of the airplane the influence zone that affects the horizontal tail (see also Fig. 4-58). This influence zone is marked in Fig. 7-28 for two Mach numbers (Mach lines ml and ?n2, respectively). The influence zone shrinks with increasing Mach number; that is, it would be expected that the effect on the horizontal tail, particularly of the upstream-lying wing, decreases with increasing Mach number. Furthermore, Fig. 7-28 demonstrates that the distance between the horizontal tail and the wing is of paramount importance for the magnitude of the interference. At constant Mach number, u = const, the horizontal os
0.1
Figure 7-27 Effect of Mach number on the 1.0
0.8
06
0.4
cr/J
02
0
efficiency factor of the horizontal tail behind a delta wing of aspect ratio A = 2.31.
AERODYNAMICS OF THE STABILIZERS 459
mz
I,-,
\
I I- \ I
mz
u,
Figure 7-28 Effect of wing and fuselage on the horizontal tail at supersonic velocity.
tail is less affected when it is close to the wing than when it is farther away. To establish computational methods for the determination of the downwash at the location of the horizontal tail, those for incompressible flow must be modified to take into account whether, as in Fig. 7-28, the influence zone of the horizontal tail encloses, at the respective Mach number, only a part of the wing (ml) or the whole wing (m2).
As a first step, the physical character of the downwash field generated by a wing in supersonic incident flow will be discussed qualitatively by means of Fig. 7-29. Here a rectangular wing is sketched with its circulation distribution as in Fig. 4-79a. It generates downwash and upwash velocities only within the two Mach cones originating at the two forward corners. In the middle part of the wing of width b* the flow is purely two-dimensional, and according to Fig. 4-21 does not generate a downwash behind the wing. Thus the triangular zone I of Fig. 7-29 remains without downwash (a.,, = 0). From the triangular surface zones at the wing tips in which the circulation drops off, free vortices are shed downstream as in incompressible flow. Thus downwash velocities (ati < 0) are in zone II behind the wing. Conversely, upwash velocities (a,,> 0) prevail in the two zones III that contain the outer halves of the two Mach cones. In the entire range IV before and beside the wing, outside of the Mach cones aw = 0. The horizontal tail without interference in supersonic flow According to Sec. 7-2-1, the contribution of the horizontal tail to the pitching moment and to the lift of the
whole airplane depends on the lift slope of the tail surface dcjH/daH and on the efficiency factor aaH/aa = 1 + aa,/aa. First, a few data will be given on the lift slope dclH/doH of the horizontal tail without interference. They may be taken from Sec. 4-5-4, in which the theory of wings of finite span at supersonic incident flow
460 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
i
y I
W>0
W.0
Figure 7-29 Induced downwash and
upwash fields in the vicinity of a rectangular wing in supersonic incident flow (schematic).
was discussed. For a horizontal tail of rectangular planform as in Eq. (4-112), the lift slope becomes d clH
4
-j
1
1
2 AH Maw --1
(7-40)
if AH Ma;, - 1 > 1. The first factor represents the lift slope in plane flow, the second the correction for the finite aspect ratio of the horizontal tail. This relationship is illustrated in Fig. 4-78a.
Influence of the wing on the horizontal tail in supersonic incident flow For quantitative assessment of the qualitative findings about the downwash at supersonic flow, first the simple case of a wing with constant circulation distribution over the span will be investigated. In this case, for supersonic flow the effect of the wing on
its vicinity can also be described by means of a horseshoe vortex, whose bound vortex lies on the wing half chord. The effect of the two free vortices is restricted, however, to the range within the Mach cones originating at the wing tips. Only the downwash on the x axis will be computed for this arrangement. This can be done by means of the results for the horseshoe vortex at incompressible flow according
to Eq. (7-23), which may be applied to supersonic flow by referring to the corresponding discussion of Sec. 4-5. Thus, the distribution of the downwash angle on the x axis behind the wing becomes
0) = cL
,
(Mad
- 1)
(7-41)
5
where cL121rA = aj(0). The downwash distribution according to this equation is shown in Fig. 7-30 for several Mach numbers. These curves demonstrate that, as has
AERODYNAMICS OF THE STABILIZERS 461
already been discussed in connection with Fig. 7-29, no downwash at all exists on the middle section over a certain stretch closely behind the wing (down to l;a = Ma.. - 1). For large distances, > 0, first the downwash increases strongly and then reaches the asymptotic value ati,, = -2a, = CL /trzl for which is the value for incompressible flow (see Fig. 7-14).
To show more accurately an induced velocity field of a free vortex at supersonic flow, the velocity distribution will now be considered in a Mach cone originating, as shown in Fig. 7-31, at the tip of a semi-infinite wing. This flow was first studied by Schlichting [33]. In Fig. 7-31c the streamline pattern is shown in a lateral plane x = const, normal to the Mach cone axis. Here the cone shell is a singular surface because it is formed completely by Mach lines. The streamline pattern within the Mach cone consists partially of closed streamlines encircling the vortex filament and partially of streamlines entering the cone on one side and leaving it on the other. Near the cone axis, the flow is comparable to that in the
vicinity of a vortex filament in incompressible flow. The distribution of the downwash velocity over the Mach cone diameter for the plane z = 0 is obtained according to [33] as
w=
TO
2ny
1-
xtanu
(7-42) y
This distribution is shown in Fig. 7-31d, where x tang =R is the radius of the Mach cone at the distance x. Because w = TQ/21ry in the potential vortex, it can be concluded from Eq. (7-42) that, at supersonic flow, the distribution of the induced velocity near the axis y = 0 deviates only a little from that at incompressible flow. Both distributions are given in Fig. 7-3 Id. Lagerstrom and Graham [17] gave an exact solution for the downwash field of the inclined plate of semi-infinite span. They obtained it by means of the cone-symmetric flow (Sec. 4-5-2) by first establishing the solution for the laterally cut-off plate of infinite chord, which is `CCW
as
= -1 (t <0)
(743a)
Figure 7-30 Downwash at the longi-
tudinal axis of a wing of constant 0.5
10
15
2.0
2.5
30
circulation distribution (horseshoe vortex) at supersonic velocities of several Mach numbers Ma,o, from Eq. (7-41).
462 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
LT
a y
i f--R
Section R --=cons z I
y
c
Figure 7-31 Velocity distribution within the Mach
cone of a free vortex at supersonic flow. Semi-
d
infinitely long wing of constant circulation distribution, from Schlichting. (a) Circulation distribution. (b) Wing planform and Mach cone. (c) Streamline pattern of the section x = const. (d) Downwash and
-R
upwash velocities in the plane z = 0, Solid curve, from Eq. (7-42). Dashed curve, plane potential vortex.
=t t
i
+
O
arcta n
2
t (i - t) _ I (0 < t <+ 1
1-2t
)
(7-43b)
Here, as in Fig. 7-32, t = y/x tan µ = y/R. This solution leads to that for the downwash field of the laterally cut-off flat plate of finite chord by superposition. In Fig. 7-32, the distribution of the downwash factor aa,,,/aa in the plane of the plate is shown for several distances x/c behind the plate. There are downwash velocities within the inner half of the Mach cone, upwash velocities within the outer half. The curve for x/c = 1 applies on the inner half to points immediately behind the trailing edge, whereas, from Eq. (7-43a), aa,1J/aa = -1 for points on the surface. At a very large distance (x - 00), the following expressions are obtained: 2
ac
It
aa
aaW
as
__
2 1c
(-Ro
1-
(7-44a)
(y > 0 and y < -Ro)
(7-44b)
Here Ro = c tan p is the radius of the Mach cone at the wing trailing edge. The downwash field of the rectangular wing of finite chord and finite span is
AERODYNAMICS OF THE STABILIZERS 463
t
y
\
0
I
-' Upwash
0.
t=-1 / =- Ro
t °9
A--- R
ya
2
f
C
0.4
l5
f
I
02 10
-Q2
5 2
10 I
!
C -06
2
Downwash
at
-110
-0.2
0.2
0C
08
1.0
Figure 7-32 Distribution of the downwash factor behind a semi-infinitely long flat plate of chord c at supersonic velocities for several distances x/c, from [ 17] .
obtained from the above solution by superposition. In Fig. 7-33, the downwash factor aa,,/aa for the middle section according to Laschka [18] is plotted against the distance x/c and with /1 Ma,,, - 1 as the parameter. Here the downwash factor
shows the same trend as seen in Fig. 7-30. For A Ma., - 1 < 2, Mach lines originating at the 2 forward corners intersect each other on the wing. Thus there is t y/7z
I
A d -1 a 1.0
z
:
rT 3
zs
,
-
2.0
(1T7
4
,l'/C - -
Figure 7-33 Distribution of the downwash factor on the longitudinal axis behind rectangular wings at supersonic incident flow for various values of the parameter =1 Mao, -1, from [ 18] . Asymptotic values for x from Eq. (7-45).
464 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
no zone behind the wing in which the downwash is zero. At a very large distance behind the wing (x -), there is, for y = 0,
a=2N
1- 1-
2
A Ma's - 1
(A - , I M - 1 > 2)
For A Ma;. - 1 < 2 the result is
(7-45)
a rectangular
wing of aspect ratio Al = 2, the downwash factor aax,/aa is given in Fig. 7-34 at several distances x/c as a function of the Mach number Ma.. The very strong influence of the Mach number on the efficiency factor of the horizontal tail is obvious.
Experimental studies about the downwash behind the rectangular wing at supersonic velocities have been conducted by Davis [1] and by Adamson and Boatright [ 1 ]
.
The above theoretical results have been obtained with the lifting-surface theory. Mirels and Haefeli [24] developed a lifting-line theory that has been applied to both
rectangular and delta wings. The results of this lifting-line theory agree with the lifting-surface theory at some distance behind the wing, as would be expected. Another computational method for the downwash, applying dipole distributions, has been given by Lomax et al. [20]. By using this method, comprehensive computations of examples have been conducted on delta wings with subsonic leading
edges. Likewise, delta wings with subsonic leading edges have been treated by Robinson and Hunter-Tod [29] and by Ward [411. Some results for delta wings with a supersonic leading edge are found in Lagerstrom and Graham [17].
The results presented so far in Figs. 7-32-7-34 apply to the conditions on the vortex sheet (z = 0). In conclusion, a few data will now be given for the downwash factor outside the vortex sheet. In Fig. 7-35, aaw/aa is plotted against the vertical coordinate for several values of the parameter Ram, - 1. As for incompressible 1.
I
IN r
09
t
77 d
C
0.2
Figure 7-34 Distribution of the downwash
factor on the longitudinal axis behind a rectangular wing of aspect ratio A = 2 at
0 1.0
1,2
14
16
18
Mam
20
2,2
1,4
supersonic incident flow for several Mach
numbers Ma., from [18].
AERODYNAMICS OF THE STABILIZERS 465 13 1.2
1,1
1,
0.9
0.8
A Maw-1=Z.O
30l 0.
0.3
02
y=0 X=W
0.1
III
0
-02
0
0.2
0.
06
Figure 7-35 Downwash factor in the root section 0.8
1.0
1.2
14'
(y = 0) behind rectangular wings vs. the high position at supersonic velocities, from [18 ].
flow (Fig. 7-21), the downwash factor decreases strongly with increasing distance from the vortex sheet. Corresponding results for delta wings are found in [20]. Now a computational method that is analogous to that for incompressible flow will be briefly described. The transformation from incompressible to supersonic flow has been explained in Sec. 4-5. Accordingly, Eq. (7-31) for the downwash velocity is also valid for supersonic flow if the function G1 is replaced by the function G of Eq. (4-95) and the function G2 , corresponding to Eq. (7-32), by G2 (x, y, z; y) = -
2co(a -
M ' 1) lc (x ' , ii) ' (x - x' ) d x ' (x - x')2 - (Maw - 1) [(y - y')2 + zz]3
(7-46)
Xf(y')
Here xo(y') is the location of the Mach line according to Eq. (4-96). Laschka [18]
suggests that one compute G in Eq. (4-95) and G2 in Eq. (7-46) by taking the vortex density k as a constant over the chord x and as a variable over the span y, that is, k(x, y) = k(y). Thus G1 and G2 can be integrated in closed form. For the determination of the downwash velocity w in Eq. (7-31), only an integration over the span coordinate remains to be done. Ferrari [7] gives a summary survey of the downwash in compressible flow. The integral integrals.
must be evaluated after the Hadamard method of finite parts of divergent
466 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
7-3 AERODYNAMICS OF THE VERTICAL TAIL
7-3-1 Contribution of the Vertical Tail to the Aerodynamics of the Whole Airplane The airplane in sideslipping flight The function and the geometry of the vertical tail have already been described in Sec. 7-1. As shown in Fig. 7-36, the vertical tail at asymmetric incident flow of the airplane of sideslip angle 0 is subject to a side force Yv. Because of its large lever arm, this side force generates the predominant portion of the yawing moment due to sideslip of the whole airplane. Moreover, the vertical tail also contributes to the side force due to sideslip and the rolling moment due to sideslip of the airplane. The contribution of the vertical tail to the yawing moment ,due to sideslip of the airplane is MZ v
= -rVYv
(7-47)
where, from Fig. 7-36, r'y is the distance of the side force vector of the vertical tail
from the moment reference axis that generally coincides with the vertical axis through the airplane center of gravity.
In analogy to Eqs. (7-2a) and (7-2b) for the horizontal tail, dimensionless coefficients may be introduced for the side force Yy and the yawing moment MZ y of the vertical tail by
Incident flow direction of the vertical tail
YV = c1VAVgV
(7-48a)
Mz V = CMz VAsq
(7-48b)
Figure 7-36 Incident flow direction of the vertical tail. C.G. = center of gravity of the airplane.
AERODYNAMICS OF THE STABILIZERS 467
Here qV is the dynamic pressure at the location of the vertical tail, which is generally smaller than the dynamic pressure of the undisturbed flow q. because of the interference of wing and fuselage with the vertical tail. The coefficient of the yawing moment of the vertical tail, referred to the wing quantities, is obtained from Eqs. (7-47)-(7-48b) as
gVAVrV CMzV=-C1vq A s with
ctV
dcjv day
(7-49a)
v - aay
(7-49b)
The lift coefficient of the vertical tail ctv depends on the angle of attack ay (angle of sideslip 1iv) and the rudder deflection 77V of the vertical tail, in addition to its geometric data. The term dctvldav stands for the lift slope of the interference-free vertical tail and (aav/arlv)riv stands for the change in the zero-lift direction of the vertical tail caused by the rudder deflection. In some cases the incident flow direction of the fin (3v is considerably different
from that of the airplane 0 because of the interference of wing and fuselage with the vertical tail. The two incident flow angles differ, as shown in Fig. 7-36, by the sidewash angle av = v/U. induced by the wing and fuselage at the location of the vertical tail: (7-50)
3V=0+3v
Hence, for a rudder deflection of zero, the contribution of the vertical tail to the yawing moment is given as rs
dc, V
CMzV =
day (0 + v)
A
(7-51)
(77V = 0)
Q moment with the angle of sideslip It follows, then, that the change in yawing (contribution of the vertical tail to the directional stability, Sec. 1-3-3) becomes aCMZV
ao
dcty daV
aLv
1+
qv Av rv
a3 q A
(7-52)*
s
The quantity
a0V = I +
a
(7-53)
is designated as the efficiency factor of the vertical tail. From Eq. (7-52) it follows that the contribution of the vertical tail to the directional stability is proportional to the efficiency factor. To establish the contribution of the vertical tail to the side force of the whole airplane, the coefficient of side force of the vertical tail is defined, in analogy to Eq. (7-9) for the horizontal tail, as *Here for simplicity it has been assumed that the ratio qv/q. is independent of the sideslip angle g.
468 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
Yy = cyyAq,0
(7-54)
In analogy to Eq. (7-52), the contribution of the vertical tail to the side force due to sideslip becomes
acyy _ dc,V as day
A 1 +a0u 30) qv Ay
q.
(7-55)
Hence, the contribution of the vertical tail to the side force due to sideslip, too, is proportional to the efficiency factor of the fin. Generally, the vertical tail also contributes to the rolling moment due to sideslip because the point of application of the vertical tail side force lies, in most cases, considerably above the airplane's longitudinal axis.
The airplane in yawing motion Besides the sideslipping considered so far, the rotary
motion of the airplane about the vertical axis (yawing motion) is also of great importance to the aerodynamics of the vertical tail. A rotary motion about the vertical axis with angular velocity o 9V =
with
Px =
generates a sideslip angle at the vertical tail w y
= Qz
sr
( 7 - 56a)
CUzS
(7-56b)
V
as the dimensionless angular sideslip velocity. By introducing this expression for py into Eq. (7-51) considering Eq. (7-50), the change in the coefficient of the yawing moment with the angular sideslip velocity becomes acMzv as2z
_ _ dcly qv Ay rv day , qc A
s
2
(7-57)
This coefficient is termed the contribution of the vertical tail to the sideslip or yaw damping. Comparison of this formula with Eq. (7-52) shows that the contribution
of the vertical tail to the directional stability
in terms of the geometric quantities, proportional to (A y/A)(r'y/s) and that to the yaw damping is is,
proportional to (A y/A)(r'y/s)2 . The following discussions will be limited to incompressible flow.
7-3-2 The Vertical Tail without Interference To evaluate the above equations, the lift slope dcl y/day must be known for the interference-free vertical tail. Basically, this can be computed with the methods of three-dimensional wing theory. Since the shapes of the vertical tails are in most cases quite asymmetric, this task is particularly complicated. Hence, wind tunnel measurements are indispensable for the acquisition of these aerodynamic quantities
of the vertical tail. An attempt has been made in Fig. 7-37 to represent the measured lift slopes of single-fin assemblies with partial fuselages as a function of a uniquely defined aspect ratio ply = bv/A V. The meaning of A y and bV is obvious
AERODYNAMICS OF THE STABILIZERS 469
° 9 Circular
cross section of the fuselage
Rectangular 0
Without
. a With 0,5
10
1
horizontal tail 2.0
15
Av
is'
Figure 7-37 Measured lift slopes of an interference-free vertical tail with partial fuselages from DVL measurements and Koloska [ 13 ].
from the sketch in Fig. 7-37. The aspect ratios Ay he between 1 and 2. Fuselages of round and rectangular cross sections and with horizontal and vertical tail edges were investigated as well as systems with and without horizontal tails. The ratio of the fuselage
height hF to the span b y of the vertical tail was limited by
hF/b y = 0.35 and 0.5. Curve 1 of Fig. 7-37 shows the theoretical trend for the lift slope as in Fig. 3-32. It represents approximately the test points for vertical tails with circular fuselages and with horizontal tails. For such vertical tails, the lift slope follows the relationship dcry
day
_
2r11y
4V+ 4 + 2
(7-58)
Curve 2, lying considerably lower than curve 1, represents a vertical tail with fuselages of rectangular cross section and without horizontal tail surfaces. Between
these curves he, as curve 3, the results for systems of circular fuselages without horizontal tails and those of rectangular fuselages with horizontal tails. Additional measurements for vertical tail assemblies with two fins are given in [35]. Theoretical studies of the lift slope of a vertical tail with a horizontal tail have been conducted by Rotta [31]. It is almost impossible to give generally valid data for the aerodynamic coefficients of vertical tails at compressible flow.
7-3-3 Effect of the Wing-Fuselage System on the Vertical Tail Fundamentals As has been shown in Sec. 7-2-2, the effect of the wing on the horizontal tail at symmetric incident flow lies essentially in the downwash of the
470 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
wing. The fuselage and the relative position of wing and fuselage (wing high position) contribute little to the interference. In all cases, however, the effectiveness of the horizontal tail is reduced by the wing and fuselage. Considerably different conditions prevail for the effect of the wing and fuselage on vertical tails at asymmetric incident flow. Schlichting and Frenz [35, 36] showed that vertical tails are markedly affected only by a combination of wing and fuselage. This interference results in an increase or in a reduction of the effectiveness of the vertical tail, depending on the high position of the wing. This
influence on the vertical tail
is
caused
physically by the quite asymmetric
circulation distribution over the span of wing-fuselage systems. This asymmetry, explained by Fig. 6-6, causes a rolling moment due to sideslip. This fact has been discussed in Sec. 6-2-3. In Fig. 7-38, this antirnetric circulation distribution along the span is illustrated for a sideslipping high-wing airplane. The lift increase of the leading wing-half and the lift decrease of the trailing wing-half generate a pressure drop on the upper side of the wing toward the advancing wing-half. This pressure drop leads to an induced flow, as explained in Fig. 7-38, which revolves around the wing. This velocity induced at the wing is effective at the vertical tail as an induced
a.
b
C
-u
Figure 7-38 Evolution of induced sidewash of a wing-fuselage
tv
system
in yawed flight. (a),
(b)
Geometry (high-wing airplane). (c) r(y) = circulation distribution, rg(y) = circulation distribution at symmetric incident flow. (d) Induced velocity field at the location of the vertical tail.
AERODYNAMICS OF THE STABILIZERS 471
lateral velocity of about the same magnitude. Figure 7-38d shows immediately that, for conventional positions of the vertical tail, the incident flow angle of the vertical tail is decreased by the lateral velocity v, that is, that the effectiveness of the
vertical tail is reduced. As in Fig. 6-6d, the sign of the induced lateral velocity is reversed for the low-wing airplane from that of the high-wing airplane. This results, for the same relative positions of fuselage and vertical tail, in an increased effectiveness of the vertical tail. In consequence of its evolution, the lateral velocity induced by the fuselage-wing interference is proportional to the sideslip angle Q and independent of the angle of attack a. Thus the resultant velocity in the y direction at the location of the vertical tail is
Vy = U. +
Vg -IL
3vf
(7-59)
where f3U00 is the lateral velocity due to the sideslip angle, vg is the induced lateral velocity at symmetric incident flow, and 13vR is the additional induced lateral velocity due to sideslip as in Fig. 7-38d. The effective sideslip angle of the vertical tail is (7-60)
Hence the efficiency factor of the vertical tail is
U a = 1 -f-
(7-61)
because vg is independent of 0. Because vR/U = Eq. (7-61) is identical to Eq. (7-53). Equation (7-61) shows that only the lateral velocity due to sideslip is required to determine the efficiency factor of the vertical tail. For an experimental confirmation of the above considerations, a few test results on the efficiency factor are plotted in Fig. 7-39b for the wing-fuselage-vertical tail system of Fig. 7-39a. From measurements of the yawing moment due to sideslip with (wT7) and without (oT7) vertical tail, a mean efficiency factor of the vertical tail has been established by Jacobs [14] in the form acMz
aCMZ
apv
ag
WV
a13
oTf
aCMZ
a13 IV where (acMZ/a1i)y is the contribution of the vertical tail to the yawing moment.* This experimentally determined efficiency factor is given in Fig. 7-39b as a function of the high position of the vertical tail. The result is for the low-wing airplane:
a
>1
(stabilizing)
'This has been determined as the difference of the measurements on fuselage and vertical tail and of the fuselage alone.
472 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES 1,6
airplane Mid-win 9 airplane
1.2
.
a--1° 0
-Q2
b
a.
Low-wing airplane Mid-wing airplane
'Low-wing 0_-r
0
High-wing airplane
High-win 9
airplane
CL .= 0
06-02
0.2
zV
c
S
0
0.2
Oq
zV- -
0.6
Figure 7-39 Efficiency factors of the vertical tail for high-wing, mid-wing, and low-wing airplanes at several high positions of the fin. (a) Geometry. (b) Measured efficiency factors from Jacobs. (c) Theoretical efficiency factors from Jacobs.
and for the high-wing airplane: (destabilizing)
1
Thus the above conclusions have been confirmed.
Theoretical determination of sidewash Computation of the distribution of the induced sidewash velocity for a known circulation distribution can basically be done
like that of the downwash, namely, with the help of the Biot-Savart law. A few qualitative considerations may be noted first. In Fig. 7-40, a symmetric and an asymmetric circulation distribution are compared. Because the circulation distributions have been taken as constant, the symmetric distribution of Fig. 7-40a produces one horseshoe vortex, and the antimetric distribution of Fig. 7-40b two horseshoe vortices turning in opposite directions. It is immediately obvious that in
a
b
r y
y
r
i
r
Figure 7-40 Determination of T v
i2r -i' -w
-v
the induced sidewash. (a) Symmetric circulation distribution. (b) Antimetric circulation distribution.
AERODYNAMICS OF THE STABILIZERS 473
the middle plane, y = 0, a downwash velocity -w is obtained for the symmetric circulation distribution but a sidewash ±v for the antimetric distribution, having reversed signs on the upper and lower sides. The latter results essentially from the counterclockwise-turning "double vortex," shed in the middle. However, this highly idealized vortex model is insufficient to determine the induced sidewash quantitatively.
The computation of the induced sidewash must be based on a variable circulation distribution T (y), for example, like that for the sideslipping wingfuselage system of Fig. 7-38. The sidewash velocity very close to the vortex sheet is obtained in analogy to Eq. (2-46a) as
vu,, =
(7-62)
(z =Z1)
1 dr
where the upper sign applies above the vortex sheet and the lower sign below. The validity of this equation can also be checked by inspecting Fig. 7-38c and d. There,
the slope of the circulation distribution is shown for y = 0, and the sign of the sidewash velocity v is indicated. The induced sidewash angle av = vl U,,, is obtained
from Eq. (7-62) by introducing the dimensionless circulation distribution 7 = T/bUU and the dimensionless coordinate in the span direction 17 =y/s as
y=f
(7-63)
G = J 1)
d_y
By introducing the expression 7(r?) = 7g('7) + R7,8(77) for the circulation distribu-
tion, where yg is the distribution in straight flight and !370 the additive circulation for sideslipping flight, Eq. (7-63) yields, for the efficiency factor of the vertical tail in the vortex sheet,
aP = i
d
( = j1)
(7-64)
The above derivation shows that Eqs. (7-62)-(7-64) are valid for any distance behind the wing for a not-rolled-up vortex sheet.
From Eq. (7-64) it is seen that the efficiency factor changes abruptly in the vortex sheet. The quantity dyp/dr? is obtained from the circulation distribution of the sideslipping wing-fuselage system. The yp distribution for the high-wing airplane is illustrated in Fig. 7-41a. The determination of the induced sidewash outside the
vortex sheet has been studied by Jacobs and Truckenbrodt [141. By applying the Biot-Savart law, the induced sidewash angle for a given circulation distribution 7(77) is obtained from lifting-line theory as -r 1
1f 2
')'n/' (aryl
-1
y
-/51)s]2 (
Y
(Y - 77,) (C - S1)
s
r
pp
7/i
3
(7-65)
with r as in Eq. (7-29). For unswept wings and a very large distance ( Jacobs [14] gave a simple procedure for the evaluation. The solution for arbitrary wing
474 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES c 0.4
0.2
-0.2
1
-10
-0.5
0
0.5
10
15
a13v aj3
b
(rr'
.,I-
Y
-
-02
0
02
06
y
Figure 741 Sidewash factors of a wing-fuselage system from [14], computed with simple lifting-line theory. (a) Additive circulation distribution of the high-wing system, b/2R = 7.5; i = 5. (b) Streamline pattern of the induced velocity field. (c) Distribution of the sidewash factor over the vertical position in the root plane y = 0. (d) Distribution of the
rectangular wing,
sidewash factor over the span for several high positions.
planforms has been studied by Gersten [101. For large distances behind the wing it
suffices to use the values for - o. In conclusion, results of a few sample computations will be reported. In Fig. 741 the induced sidewash field is given for a high-wing system. Figure 7-41a illustrates the geometry and the additive circulation distribution 'yQ due to the sideslipping. Figure 7-41b represents the streamline pattern of the induced velocity field very far behind the wing, and Fig. 7-41c gives the distribution of the sidewash
factor a3,/ap as a function of the distance from the vortex sheet for the middle plane 7 = 0. This figure demonstrates the discontinuity of the sidewash factor at the vortex sheet i and the strong drop with distance from the vortex sheet. Figure 7-41d gives the distribution of the sidewash factor in the span direction for several distances from the vortex sheet. In Fig. 7-42 for a high-wing and for a low-wing airplane the curves of constant local efficiency
factor of the vertical tail a1V/aa = const are shown for the
transverse plane at the location of the vertical tail. The total efficiency factor of the
AERODYNAMICS OF THE STABILIZERS 475
vertical tail is obtained from this through integration over the vertical tail height. The field of the curves as v/ag = const is independent of the angle of attack of the airplane. There is, however, a dependence of the efficiency factor of the vertical tail on the angle of attack because, with a change of the angle of attack, the vortex sheet is displaced relative to the vertical tail (see Fig. 7-20). This influence is quite noticeable, as may be seen by comparing the cases CL = 0 and CL = 1 in Fig. 7-42. For the system of wing, fuselage, and vertical tail of Fig. 7-39a, Jacobs [14] applied this method to determine the efficiency factors theoretically (Fig. 7-39c). The agreement with measurements in Fig. 7-39b is satisfactory. The problem area of the interaction of wing, fuselage, and vertical tail at sideslipping has been investigated by Puffert [28] The concepts established for the induced sidewash have been translated into that for the rolling wing by Michael [23 ] and by B obbitt [5 ] . .
7-3-4 Interaction of the Vertical Tail and the Horizontal Tail The flow conditions at the vertical and horizontal tails are affected not only by the fuselage and wing but also considerably by their mutual interaction. Of special a.
cL=O Vortex sheet
CL=7
105 1.1
0.
0.95
1.05 1.1
1.2
OB
15
0 095
Figure 7-42 Local efficiency factors of the vertical tail. Curves apv/ag = const, from [14], b12R = 7.5. Wing of rectangular planform ii = 5. (a) High-wing airplane. (b) Low-wing airplane.
476 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
a
TV
b
Figure 7-43 Interference between vertical and horizontal tails. Circulation distribution and free vortex sheet of a sideslipping vertical and horizontal tail system, from Laschka [191.
interest here are the conditions at the tail unit at sideslipping and rolling. A tail unit at which the middle section of the horizontal tail lies over the root of the vertical tail will be considered to demonstrate this fact.
On a
vertical tail
in an incident flow of sideslip angle 0, a circulation
distribution is generated that does not drop to zero at the root section but rather has a finite value because of the end-plate effect of the horizontal tail. A circulation discontinuity results now in the shedding of a single vortex that turns in a direction opposite to that of the rest of the free vortices. This vortex in turn induces at the
horizontal tail a downwash exceeding the counteracting induction effect of the continuous free vortex sheet. The resulting circulation distribution at the horizontal tail has, as shown in Fig. 7-43b, a discontinuity in the middle of the horizontal tail; it is antimetric and generates a rolling moment due to sideslip that is reversed from that of the vertical tail (see Fig. 7-43, from Laschka [19]).
To reduce the load induced on the horizontal tail by the sideslipping vertical tail, a positive dihedral may be provided. This increases, however, the total rolling moment due to sideslip. On the other hand, the rolling moment due to sideslip of the tail unit dihedral.
may be reduced by providing the horizontal tail with a negative
By extending and applying a suitable panel method as described in Sec. 6-3-1 for the wing-fuselage system, the pressure distributions, and thus the acting forces
AERODYNAMICS OF THE STABILIZERS 477
and moments, can also be determined for the whole airplane; compare, for example, [15]
.
REFERENCES 1. Adamson, D. and W. B. Boatright: Investigation of Downwash, Sidewash, and Mach Number
Distribution Behind a Rectangular Wing at a Mach Number of 2.41, NACA Rept. 1340, 1957. Davis, T.: J. Aer. Sci., 19:329-332, 340, 1952. 2. Alford, W. J., Jr.: Theoretical and Experimental Investigation of the Subsonic-Flow Fields Beneath Swept and Unswept Wings with Tables of Vortex-Induced Velocities, NACA Rept. 1327, 1957.
3. Bilanin, A. J. and C. duP. Donaldson: Estimation of Velocities and Roll-up in Aircraft Vortex Wakes, J. Aircr., 12:578-585, 1975. Donaldson, C. duP., R. S. Snedeker, and R. D. Sullivan: J. Aircr., 11:547-555, 1974. 4. Bloom, A. M. and H. Jen: Roll-up of Aircraft Trailing Vortices Using Artificial Viscosity, T. Aircr., 11:714-716, 1974.5. Bobbitt, P. J.: Linearized Lifting-Surface and Lifting-Line Evaluations of Sidewash Behind Rolling Triangular Wings at Supersonic Speeds, NACA Rept. 1301, 1957. 6. Byrnes, A. L., W. E. Hensleigh, and L. A. Tolve: Effect of Horizontal Stabilizer Vertical Location on the Design of Large Transport Aircraft, J. Aircr., 3:97-104, 1966. 7. Ferrari, C.: Interaction Problems, in A. F. Donovan and H. R. Lawrence (eds.), "Aerodynamic Components of Aircraft at High Speeds," Sec. C, Princeton University Press, Princeton, N.J., 1957. 8. Fliigge-Lotz, I. and D. Kuchernann: Zusammenfassender Bericht iixber Abwindrnessungen
ohne and mit Schraubenstrahl, Jb. Lufo., 1:172-193, 1938. Fage, A. and L. F. G. Simmons: ARC RM 951, 1925. Muttray, H.: Lufo., 12:28-37, 1935. Petersohn, E.: Z. Flug. Mot., 22:289-300, 1931. 9. Furlong, G. C. and J. G. McHugh: A Summary and Analysis of the Low-Speed Longitudinal Characteristics of Swept Wings at High Reynolds Number, NACA Rept. 1339, 1957. 10. Gersten, K.: Uber die Berechnung des induzierten Geschwindigkeitsfeldes von Tragflugeln, Jb. WGL, 172-190, 1957; 151-161, 1955. 11. Glauert, H.: "The Elements of Airfoil and Airscrew Theory," Cambridge University Press, Cambridge, 1947. "Die Grundlagen der Tragfliigel- and Luftschraubentheorie," (German txansl. by H. Holl), Springer, Berlin, 1929. 12. Hackett, J. E. and M. R. Evans: Vortex Wakes Behind High-Lift Wings, J. Aircr., 8:334-340, 1971. 13. Hafer, X.: Windkanalergebnisse zum Interferenzproblem moderner Hochleistungsflugzeuge, Z. Flugw., 6:20-28, 1958; Jb. WGL, 161, 1955. Koloska, P.: ZWB Lufo. UM 7301, 1944. 14. Jacobs, W. and E. Truckenbrodt: Der induzierte Seitenwind von Flugzeugen, Ing.-Arch., 21:1-22, 1953. Jacobs, W.: Ing.-Arch., 21:23-32, 1953. 15. Kalman, T. P., W. P. Rodden, and J. P. Giesing: Application of the Doublet-Lattice Method to Nonplanar Configurations in Subsonic Flow, T. Aircr., 8:406-413, 1971. 16. Kaufmann, W.: Der zeitliche Verlauf des Aufspulvorganges einer instabilen Unstetigkeitsflache von endlicher Breite, Ing.-Arch., 19:1-11, 1951; Z. 17ugw., 5:327-331, 1957;
Bay. Akad. Wiss., Math. Nat. Abt., 109-130, 1946. Betz, A.: Z. Angew. Math. Mech., 12:164-174, 1932; NACA TM 713, 1933. Jordan, P. F.: J. Aircr., 10:691-693, 1973. Kaden, H.: Ing.-Arch., 2:140-168, 1931, Rohne, E.: Z. Flugw., 5:365-370, 1957. Wurzbach, R.: Z. F7ugw., 5:360-365, 1957. 17. Lagerstrom, P. A. and M. E. Graham: Methods for Calculating the Flow in the Trefftz-Plane Behind Supersonic Wings, J. Aer. Sci., 18:179-190,.1951. 18. Laschka, B.: Uber das Abwindfeld hinter Tragflugeln bei Uberschallgeschwindigkeit, Jb. WGL, 101-102, 1959; Z. F7ugw., 9:33-45, 1961.
478 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
19. Laschka, B.: Interfering Lifting Surfaces in Subsonic Flow, Z. Flugw., 18:359-368, 1970. 20. Lomax, H., L. Sluder, and M. A. Heaslet: The Calculation of Downwash Behind Supersonic Wings with an Application to Triangular Plan Forms, NACA Rept. 957, 1950. 21. Lotz, I. and W. Fabricius: Die Berechnung des Abwindes hinter einem Tragflugel bei Beriicksichtigung des Aufwickelns der Unstetigkeitsflache, Lufo., 14:552-557, 1937; Ringb. Lufo., I A 10, 1937. Helmbold, H. B.: Z. Flug. Mot., 16:291-294, 1925; 18:11, 1927. 22. Mangler, W.: Die Auftriebsverteilung am Tragflugel mit Endscheiben, Jb. Lufo., 1:149-154,
1938; Lufo., 14:564-569, 1937; 16:219-228, 1939. Hubert, J.: Jb. Lufo., 1:129-138, 1937. Schrenk, 0.: Lufo., 14:570-572, 1937. 23. Michael, W. H., Jr.: Analysis of the Effects of Wing Interference on the Tail Contributions to the Rolling Derivatives, NACA Rept. 1086, 1952. 24. Mirels, H. and R. C. Haefeli: Line-Vortex Theory for Calculation of Supersonic Downwash, NACA Rept. 983, 1950; J. Aer. Sci., 17:13-21, 1950. 25. Multhopp, H.: Die Berechnung des Abwindes hinter Tragfliigeln, Jb. Lufo., 1:167-171, 1938; Lufo., 15:463-467, 1938. Richter, W.: Lufo., 20:69-76, 1943. Scharn, H.
and G. Braun: Lufo., 18:179-183, 1941. Truckenbrodt, E.: Ing.-Arch., 18:233-238, 1950. C., J. N. Nielsen, and G. E. Kaattari: Lift and Center of Pressure of Wing-Body-Tail Combinations at Subsonic, Transonic, and Supersonic Speeds, NACA Rept. 1307, 1957.
26. Pitts, W.
27. Prandtl, L. and A. Betz: Untersuchungen an Fliigeln mit Endscheiben, in L. Prandtl, C. Wieselsberger, and A. Betz (eds.), "Ergebnisse der Aerodynamischen Versuchsanstalt zu Gottingen," vol. III,:4th ed., pp. 17-18, 95-99, Oldenbourg, Munich, 1935. 28. Puffert, H. J.: Uber die gegenseitige Beeinflussung von Fliigel, Rumpf and Leitwerken bei Schraganblasung, Z. Flugw., 3:323-331, 1955. 29. Robinson, A. and J. H. Hunter-Tod: Bound and Trailing Vortices in the Linearized Theory
of Supersonic Flow and the Downwash in the Wake of a Delta Wing, ARC RM 2409, 1952/1947.
30. Rossow, V. J.: On the Inviscid Rolled-up Structure of Lift-Generated Vortices, J. Aircr., 10:647-650, 1973. 31. Rotta, J.: Luftkrafte am Tragflugel mit einer seitlichen Scheibe, Ing.-Arch., 13:119-131, 1942.
32. Sacks, A. H.: Vortex Interference Effects on the Aerodynamics of Slender Airplanes and Missiles, J. Aer, Sci., 24:393-402, 412, 1957. von Baranoff, A.: Jb. WGL, 75-79, 1959. Morikawa, G.: J. Aer. Sci., 19:333-340, 1952. 33. Schlichting, H.: Tragfliigeltheorie bei Uberschallgeschwindigkeit, Lufo., 13:320-335, 1936; NACA TM 897, 1939. 34. Schlichting, H.: Die Stabilitatsbeiwerte des Flugzeuges unter Beriicksichtigung der Interferenz von Flilgel, Rumpf and Leitwerk, Sonderheft: Flugmech. Probleme, Akad. Lufo. 2/43 g, pp. 3-23, 1943. 35. Schlichting, H. and W. Frenz: Uber den Einfluss von Fliigel and Rumpf auf das Seitenleitwerk, Jb. Lufo., 1:300-314, 1941. Staufer, F.: Jb. Lufo., I:383-391, 1940; 1:294-299, 1941. 36. Schlichting, H. and W. Frenz: Systematische Sechskomponentenmessungen fiber die gegenseitige Beeinflussung von Fli gel, Rumpf and Leitwerk, ZWB TB 11, no. 6, 1944. 37. Schulz, G.: Der Abwind auf der L'angsachse des Fliigels bei Betzscher Zirkulationsverteilung, Lufo., 19:367-373, 1942.
38. Silverstein, A. and S. Katzoff: Design Charts for Predicting Downwash Angles and Wake Characteristics Behind Plain and Flapped Wings, NACA Rept. 648, 1939. Silverstein, A., S. Katzoff, and W. K. Bullivant: NACA Rept. 651, 1939. 39. Spreiter, J. R. and A. H. Sacks: The Rolling Up of the Trailing Vortex Sheet and Its Effect on the Downwash Behind Wings, J. Aer. Sci., 18:21-32, 72, 1951.
AERODYNAMICS OF THE STABILIZERS 479
40. Trienes, H. and E. Truckenbiodt: Systernatische Abwindmessungen an Pfeilfliigeln, Ing.-Arch., 20:26-36, 1952. 41. Ward, G. N.: Calculation of Downwash Behind a Supersonic Wing, Aer. Quart., 1:35-38, 1950.
42. Williams, G. M.: Viscous Modelling of Wing-Generated Trailing Vortices, Aer. Quart., 25:143-154, 1974.
CHAPTER
EIGHT AERODYNAMICS OF THE FLAPS AND CONTROL SURFACES
8-1 INTRODUCTION
8-1-1 Function of the Flaps and Control Surfaces As has been explained in Sec. 7-1, the tail surfaces of an airplane serve a twofold purpose, namely, to stabilize and to control the airplane. In general, the tail surfaces
consist of a fixed part, the stabilizer, termed a fin on the vertical tail and a (horizontal) tail plane on the 'horizontal tail, and a movable part, the control surface, termed an elevator on the horizontal tail and a rudder on the vertical tail. There is another set of control surfaces attached to the wing, termed ailerons; see Figs. 7-1 and 7-3. The tail surfaces, with the control surfaces fixed, serve to stabilize the airplane. The corresponding aerodynamic problems have been discussed in detail in Chap. 7. The airplane is controlled by deflection of the control surfaces. Control about the lateral axis is accomplished with the elevator, that about the vertical axis and the longitudinal axis with the rudder and the ailerons. The geometry of the tail surfaces and of the ailerons is that of an airfoil with a flap (flap-wing) as shown in Fig. 8-1 (see also Fig. 2-24). The aerodynamic effect of the control surfaces consists of an additive lift produced by their deflection. This lift, acting on the tail surfaces or the wing, respectively, controls the airplane. The aerodynamic forces acting on the control surfaces generate a moment that, referred to the control-surface axis of rotation, is termed control-surface moment or hinge moment. The effect of the control surface should be strong enough to generate an additive lift that, for a given control-surface deflection, is as large as possible. At the 481
482 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
same time, however, the hinge moment should be as small as possible so that the forces needed for the operation of the control surfaces also remain small. A control surface in the form of a simple flap as shown in Fig. 8-1 has relatively large hinge moments. Efforts have therefore been made to reduce the moments required to move the control surfaces. This has been accomplished by means of so-called control-surface balances, as shown in Fig. 8-2. The most important types of aerodynamic control-surface balances are the inner balance (nose balance) as shown
in Fig. 8-2a, the balance tab as shown in Fig. 8-2b, and the outer balance (horn balance) as shown in Fig. 8-2c. In all cases of control-surface balance, it is important that the lift increase caused by the control-surface deflection (controlsurface effectiveness) should, if possible, not be reduced by the control-surface balancing.
The airfoil with control surface of Fig. 8-1 may serve two purposes: first, to control the airplane, and second, to be used as a landing device. In the latter case, its effect is to increase the maximum lift of the airplane, thus holding down the landing speed. This lift increase is usually accompanied by a drag increase. In Fig. 8-3, several designs of such landing flaps are shown. In the arrangements of Fig. 8-3a-e, the flaps are attached to the rear end of the wing, whereas in Fig. 8-3f and g, flaps are shown in front of the wing (slat, nose flap). Some of these arrangements are also employed as take-off assistance to reduce take-off distance. Finally, a few more forms of flaps may be mentioned, namely, the system of a Axis of rotation
Axis of rotation
Balance tab
Axis of rotation Horn
-
Ru dd er
Axis of rotation Horn
Elevator
C
Figure 8-2 Various forms of aerodynamic control-surface balances. (a) Inner balance (nose balance). (b) Balance tab. (c) Outer balance (horn balance).
AERODYNAMICS OF THE FLAPS AND CONTROL SURFACES 483 a-<
d
Figure 8-3 Several control-surfaces and
flaps. (a) Cambered flap. (b) Slot flap. (c) Double-section wing. (d) Fowler flap. (e) Split (spreader) flap. (f) Slat. (g) Nose flap.
brake flap (air brake) on the upper and lower sides of the wing (see Fig. 8-28).
They have the shape of a rectangular plate and are set normal to the flight direction. It is the function of the air brakes in their extended position to increase strongly the drag of the airplane, thus reducing considerably the speed and generating a steeper glide angle (brake effect).
8-1-2 Geometry of the Flaps and Control Surfaces For the aerodynamics of the wing with control surface (flap), the most important geometric parameters as shown in Fig. 8-1 are as follows: Control-surface angle (flap angle): rlf Control-surface chord ratio (flap chord ratio): a f = cf/c
These quantities have already been given for the whole wing with control surface (flap wing) in Sec. 2-4-2 and in Fig. 2-24. If the control surface does not extend over the whole span, as in, for example, the aileron in Fig. 8-4a, the span of the control surface bA = 2SA becomes another important geometric quantity. On the horizontal tail plane and the fin, the control surface usually extends over the
whole span of the horizontal tail bH and the height of the vertical tail h y,
484 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
b
Axis of rotation
by Axis of rotation
Figure 8-4 Geometry of the control surface. (a) Ailerons. (b) Elevator. (c) Rudder.
respectively (Fig. 8-4b and c). In many cases the control-surface chord ratio of is varied along the span. In this case it is preferable to use the control-surface area
ratio A f/A' instead of the control-surface chord ratio cf/c, where Af is the control-surface area and A' is the wing area within the span range of the control surface.
8-1-3 Aerodynamic Coefficients of the Flaps and Control Surfaces The following aerodynamic coefficients are introduced for the wing with control surface:
Lift:
L = cLAgr
(8-1)
Pitching moment:
M = cMAcq.
(8-2)
Control-surface moment:
Mf = cm fA fcfgc,
(8-3)
Here the lift coefficient and the pitching-moment coefficient are referred to the geometric quantities of the wing, as in the case of the wing without control surface [see Eq. (1-21)].. The control-surface moment M' (flap moment, hinge moment) is referred to the axis of rotation of the control surface; its sign can be seen from Fig. 8-1. The coefficient of the control-surface moment c,,. f is referred to the geometric
AERODYNAMICS OF THE FLAPS AND CONTROL SURFACES 485
quantities of the control surface. These three aerodynamic coefficients depend on the angle of attack a and the control-surface angle r1f. As an example of measurements, the lift coefficient CL of a simple flap wing is plotted against the angle of attack in Fig. 8-5a for several flap angles r7f. The flap deflection 17f causes, corresponding to Fig. 2-24, an additive camber and thus, at constant angle of attack, an increase in lift. The curves cL(a) for several angles 17f are parallel to each other. The dependence of the lift coefficient on a and 17f for small angles may be expressed as
cL=
asa+a
f = a« a - a
(8-4a)
r?f
(8-4b)
f 77f
where as/ar? f indicates the change in the zero-lift direction of the wing because of the flap deflection (flap effectiveness) [see Eq. (7-3b)] . The coefficient as/ar?f depends strongly on the control-surface chord ratio. Data on this effect have been given in Fig. 2-25a for a flap wing of infinite span. In Fig. 8-5b, the lift coefficient CL is plotted against the moment coefficient cm for several flap angles 77f. The flap angle causes a parallel shift of the moment curves. The dependence of the moment coefficient cm on CL and 17f for small values of these parameters may be expressed as CM .
acM aCL
CL +
acM
(8-5)
17f
1.
f -20° 16
nf = 20 °
(
1.0
11°
I
50 0°
0.
{
i
{
E Ay
/I X /V
-5- 1
F)
i 10
l
-11;16°
{
i
I
i
7
20° '
I
-1.0
-15
a
=25°
-5° 0° 5°
15°
25° -0.4 b
-0.2
0.2
0.4
CM
Figure 8-5 Aerodynamic coefficients of a simple rectangular flap wing from measurements of Gothert [18]. (a) Lift coefficient CL vs. angle of attack a for several flap angles
rrf. (b) Lift coefficient CL vs. pitching-moment coefficient cMfor several flap angles 77f.
486 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
Here, acM/art f gives the change of the zero moment with the flap deflection. This coefficient depends strongly on the flap chord ratio. Data on the wing of infinite span have been given in Fig. 2-25b. Frequently it is advantageous to specify the location of the aerodynamic center of the additional forces generated by the flap deflection. This point is termed the flap neutral point. The distance of the flap neutral point from the neutral point of the wing without flap deflection (= neutral-point displacement) is obtained from Eqs. (8-5) and (8-4) as (d XN)f
aCM/a7lf
C
aCL/ar?f
(s-6)
where acL/arlf may be taken from Eq. (8-4a). In Fig. 8-6, cL is shown as a function of the control-surface moment coefficient Cm f for several values of r7f. Here, too, a linear relationship applies of the form Cmf
aCmf
aCL CL +
aCmf 77f
(8-7)
a7jf
The dependence of this coefficient on the flap chord ratio will be discussed in Sec. 8-2. The condition cm f = 0 determines a certain coordination of r7f and CL and thus also of r?f and a for self-setting of the free control surface.
8-2 THE FLAP WING OF INFINITE SPAN (PROFILE THEORY) 8-2-1 The Flap Wing in Incompressible Flow The flap wing as a bent plate The fundamentals of the theory of the flap wing of infinite span in incompressible flow have been given in Sec. 2-4-2. In its simplest form, the wing with a deflected flap is replaced by a bent plate as shown in Fig. 2-24, on the chord of which, according to Glauert [16], a vortex distribution is arranged.
For the coefficients of the flap effectiveness, the expressions of Eq. (2-82)* are
a77fa = - 2 ( f(1 - Xf) + aresin ) a 77f
= -21 - Xf)3
(8-8a)
(8-8b)
In Fig. 8-7 these theoretical coefficients have been given against the flap chord ratio Xf. The problem of the single-bent plate has been solved by Keune [24] with the
method of conformal mapping. The most important result of this study is the confirmation of Glauert's approximate solution for small flap angles. For larger flap *The index 0 has been omitted.
AERODYNAMICS OF THE FLAPS AND CONTROL SURFACES 487
1.
77f °GO°
I
170
7!
I
IN
1,
I
II
5° 0° l-IL
11 "Nl
i
5 -11°I
I
I
I
1
I
-0
i
!
I
-7,0 1
i,
I
I
-0.2
02
0.4
Cmf
Figure 8-6 Lift coefficient CL vs. control-surface coefficient cm f of a simple flap wing (aspect ratio A = 3.5; flap chord ratio Xf = 0.5), from measurements of Gothert [18 ].
V.°
j
0.7
os
The ory
Mea surem ent
Th eory
I
Me asurem ent Gt4
03 1
Cambered flap Slot flap
Cambered flap
-- Slot fla p
72
Double-section wing 0.
S p l it
1
fl ap (spreader
I
flap) 7
a
0,2
Af
cf
=C
0.3
0.4
05
0.60
b
0.1
0.2
0.3
0.4
Xf cf = C _
0.6
0.5
Figure 8-7 Flap effectiveness of several designs: theory and measurements. (a) Angle-of-attack change due to flap deflection as/arlfe vs. flap chord ratio Xf. (b) Pitching-moment change due to flap deflection acM/anfe vs. flap chord ratio Xf.
488 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
angles, the deviations are more pronounced. In Fig. 8-7 these results are added to the results of comprehensive test series on wings of various flap shapes. The measured coefficients have been taken from test series for small flap angles. The coefficients thus
obtained have been designated as aoz/arlfe and acM/an fe. Comparison of theory and experiment shows that the measured values are smaller than the theoretical ones
for both the change in the angle of attack and the change in the moment. The curve for the wing with a split flap (spreader flap) shows the largest deviation from the theoretical curve. For larger flap deflections, the flap effectiveness declines. This trend is shown in Fig. 8-8 by assigning an effective flap angle rife to each geometric flap angle r?f. This coordination applies approximately to the moment change as well.
The differences between the theoretical curves and the measurements in Fig. 8-7a and b cannot be fully explained by the influence of the profile thickness. They should essentially be due to friction effects. For theoretical studies of the flap wing, it is advisable to apply empirical corrections to the coefficients of the flap wing as obtained from profile theory. This is accomplished simply by multiplying the effect of the camber on the coefficients as/ar1f and acM/arlf with an empirical factor x. Then the adjusted coefficients assume the form
f= acm
(8-9a)
-
acM
ar f
(8-9b)
17f X-1
Here the terms with the index x = 1 are the theoretical values from Eqs. (8-8a) and
(8-8b). In Fig. 8-9, these coefficients for x = 0.75 are also shown; they agree satisfactorily with the measurements of Fig. 8-7,
In Fig. 8-10, the theoretical values for the position of the flap neutral point from Eq. (8-6) are plotted against the flap chord ratio with aCL/aa = 27r. In this figure, the distance between the
/0
neutral point and the leading edge,
flap
A -/
00
Cambered flap
--Slot flap -10°
-
Double-section wing
-----Split flap (spreader flap)
-Z0°
-70°
00
100
r? f
200
,300
400
°
V[/°
Figure 8-8 Correlation between the effective flap deflection rife and the geometric flap deflection 17f for several flap designs (see Fig. 8-7).
AERODYNAMICS OF THE FLAPS AND CONTROL SURFACES 489 7,
Theory 0.
x =10
..75
/10
0.
Theory
a=1,0
0.75
0,2
Figure 8-9 Reduction of flap effectiveness from Eqs. (8-9a) and (8-9b). (a) Change of angle of attack due to flap deflection. (b)
0 0.2
0.4
0.5
10.6
1.0
Change of pitching moment due to flap deflection.
xNf = c/4 + (d xN)f, is given, where c14 is the position of the wing neutral point. It is noteworthy that, for small flap chords, the flap neutral point lies at c/2. This is in consequence of the fact that the deflection of even a small flap strongly affects the pressure distribution on the front portion of the wing.
Computation of the flap loading (control-surface loading) and of the flap moment (control-surface moment) requires that the pressure distribution on the deflected flap be known. The theoretical pressure distribution on a bent plate is illustrated in Fig. 2-28, whereas Fig. 8-11 gives the experimentally determined pressure distribution on a wing with split flap from Schrenk '[40] (see also Seiferth [41]). The aerodynamic force on the flap (flap loading), the knowledge of which is important for computation of the structural strength of the flap, is obtained from the pressure distribution on the flap as
Lf = bf fl_p)dxcjfbfcfq . (cf)
(8-10)
490 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES 0.5
Lf
0.7
0o
0
0.2
0.6
0,4
08
T0
Figure 8-10 Position of the flap neutral point
vs. the flap chord ratio for incompressible flow.
where cif is the coefficient of flap loading (control-surface loading). The dependence
of the coefficient of flap loading on the lift coefficient and on the flap angle is given, in analogy to Eq. (8-7), as
cif
acifCL
L
+
a f of
(8-11)
f
The theory of the bent plate yields the coefficients acif
_
acL
ac77ff =
2
af g
[aresin
-
- Af) ]
0 - Xf)
(8-12a) (8-12b)
In Fig. 8-12 the two coefficients have been plotted against the flap chord ratio Xf.
The flap moment (control-surface moment) of a wing portion of width bf, referred to the control-surface axis of rotation, is
Mf= -bf f(p,-p)(x-x1) dx
Figure 8-11 Pressure distribution on a wing with a slot flap, from Schrenk.
(8-13)
AERODYNAMICS OF THE FLAPS AND CONTROL SURFACES 491 J.
Lf
2 cf
c
2.
1,
acIf aT?f 1.0
acI f
acL a
0
Xf -
0,2
0.4
0.6
06
1,0
Figure 8-12 Flap loading; theory from Glauert. Curve 1, change of the coefficient of flap loading with lift coefficient. Curve 2, change of the coefficient of flap loading with flap angle.
where xf is the position of the axis of rotation as shown in Fig. 8-1 and c f is the flap chord. The theory of the flap wing (bent flat plate, Sec. 2-4-2) yields the. following relationships for the control-surface moment coefficient cmf: ac m f = - 21rXf 1 2 [( 3 - 2Xf) f(1 - Af) - (3 f) arcsin ac,,
-0
'
ac ,n f
4
1- f
r?f
W
Af
a
[arcsin
-
X_ f(1 - Af) ]
/]
(8-14a)
( 8 -14b)
In Fig. 8-13, these coefficients are plotted against the flap chord ratio Af. Test results for simple cambered flaps are also shown. They lie considerably below the theoretical curves. These differences are caused by the influences of the profile thickness and, particularly, of the friction.
To reduce the control-surface moment Mf, several forms of control-surface balance arrangements have already been shown in Fig. 8-2. Of these, only the inner balance and the balance tab can be considered two-dimensional problems. At the inner balance, the control-surface moment is decreased by moving the axis of rotation rearward. Then, in deflecting the control surface, a control-surface "nose" protrudes from the profile, forming a contour that is hardly accessible to computation. To determine the aerodynamic coefficients of the flap wing with inner balance, mainly experimental studies have to be applied, such as, for example, those
published by Gothert [181. The aerodynamic coefficients of a flap wing with balance tab were first treated by Perring [16] using the theory of the multiple-bend plate. A comparison of his theoretical results with measurements is given by Gothert (18]. In this case the effect of friction is particularly strong. The flap wing as a wing profile Several investigators have studied theoretically not only the flap wing of finite thickness but also the effect of flap arrangement and
492 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES 0. 1
4
Theory eo
Measurement
a 01,0 ae
f MCS 77f
C
O
Theory
Measu rement
Figure 8-13 Coefficient of control-surface moment vs. flap chord ratio Xf; theory from Gothert. (a) Change of the coefficient of control-surface moment with lift coefficient. (b) Change of the coefficient of the control-surface moment with flap deflection.
0.2
b
0.2
0,4 c
0.6
0.e
1,0
Af = c
flap shape and, particularly, that of a slot between the fixed airfoil and the movable
flap. In particular, the publications of Allen [2], Flugge-Lotz and Ginzel [13], Keune [241, and Jacob and Riegels [22] should be pointed out. The results of these studies have been presented systematically by Gothert [18] within the framework of an experimental study. Furthermore, comprehensive test results on flap wings have been reported by Wenzinger [49] and by Keune [24]. Summary accounts of these studies are found in [7] and [45] ; compare also [1, 35]. Theoretical investigations on the behavior of the boundary layer of flap wings and comparisons with measurements have been conducted by Goradia and Colwell [171.
8-2-2 The Flap Wing in Compressible Flow Lift and moment The theory of the flap wing of infinite span in compressible flow may be derived approximately from the profile theory of compressible flow as given in Sec. 4-3. There solutions were obtained for subsonic incident flow using the subsonic similarity rule (Prandtl, Glauert) and for the supersonic incident flow using the supersonic similarity rule (Ackeret). The following formulas apply for fixed flap chord ratios Xf ='Afinc For subsonic incident flow (Maw < 1), as a??f
_
cmm
_ amf
(8-15a) inc
AERODYNAMICS OF THE FLAPS AND CONTROL SURFACES 493 acm
-
1isl
1
(8-15b)
}/1 -MaL (t'aC?7.f Ji nc Here the terms marked by "inc" are those of the incompressible flow from Eqs. (8-8a) and (8-8b) and as shown in Fig. 8-7. For supersonic incident flow (Ma > 1), a77f
as a77f
_ - Af
(8-16a)
acb,
2
a 77f
VMa- -- 1
- Xf( 1
- Tf)
( 8 -1 6 b)
In Fig. 8-14, the changes in zero-lift angle and zero moment caused by the flap deflection are given as a function of the flap chord ratio. By using the above coefficients, the position of the flap neutral point can be
com uted with Eq. (8-6), where it has to be considered that acL/aa= 27r/ _-a" for Maw, < 1 and acL/aa= 4/ Ma;.- 1 for Ma.. > 1. The position of
7the flap neutral point is given in Fig. 8-15 against the flap chord ratio af. Here the 1
relationships xNf = c/4 + (A XN)f applies for Ma,0 < 1 and xNf = c/2 + (A XN)f for
At supersonic velocities the flap neutral point lies much farther back than at, subsonic velocities, as should be expected. The following expressions are obtained for the coefficients of the flap moment (control-surface moment) at subsonic incident flow (Ma,, < 1): aCyyi f
aCL
`
aC,n
f
acL
(8 -17a) inc
f 1.
08
0
/
M¢c
0.2
0.6
0.4
fRaco-,
0.8
,.0
Cf
c
Figure 8-14 Aerodynamic coefficients of a flap wing at subsonic and supersonic incident flows. (a)
Change of the zero-lift angle with flap deflection. (b) Change of the zero moment with flap deflection.
494 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES 1.0
0,8
XNf
- c Md.-1 2
Ma 0,2
Figure 8-15 Position of the flap neutral 01
point vs. the flap chord ratio for compressible flow (subsonic and supersonic veloci-
f
0.2
0.4
0,6
0,8
1,0
ties).
aCmf
al?f
acm f -Ma.
i
aff
(8-17b) inc
Again, the coefficients marked inc are those of incompressible flow from Eq. (8-14) and Fig. 8-13. Corresponding relationships are found for the coefficients of flap loading.
For supersonic velocities (Maw > 1), the Ackeret rule yields (see Sec. 4-3-3) acmf
1
aCL
2
(8-18a)
aCmf
2
8 18b
Ma;. - 1 The coefficients of flap loading are determined immediately as clf = 2Cm f by realizing that the pressure distribution over the flap chord is constant. aI7f
8-2-3 Take-off and Landing Devices* General remarks As has been mentioned in Sec. 8-1, the take-off and landing devices on the wing serve to increase the maximum lift coefficient. A great variety
of arrangements are utilized to increase the maximum lift. The older kinds of take-off and landing devices consist of flaps and balance tabs attached to the wing trailing edge or the wing nose (Fig. 8-3). More recently, devices have frequently
been used that increase -the lift through boundary-layer control by suction or
ejection. A brief account of this method has been given in Sec. 2-5-3. A comprehensive survey of the various methods for the increase in maximum lift is included in Lachmann [28]. The effect of take-off and landing devices on the lift characteristic CL(a) of a *The assistance of K. O. Arnold in preparing this section is gratefully acknowledged.
AERODYNAMICS OF THE FLAPS AND CONTROL SURFACES 495
wing is presented schematically in Fig. 8-16. Curve 1 gives the values without flap deflection. Curve la shows the increase in the coefficient CLmax by boundary-layer control at the wing nose. Curve 2 gives the values with flap deflection, and curve 2a again the increased values of CLmax through boundary-layer control at the nose. Curves 3 and 3a give the corresponding data when, in addition, the boundary layer at the flap nose is controlled as well. The summary report about theoretical and experimental studies on boundary-layer control by Carriere et al. [8] should be mentioned. Earlier, a paper on the properties of flap wings was given by Young [55]
.
Flaps The simplest method of increasing CLmax is the deflection of a cambered flap as shown in Fig. 8-17a. This effect is obtained because the flap deflection increases
the effective camber of the wing, resulting in a lift augmentation that may be considerable. As an example, Fig. 8-17a shows CL against the angle of attack for several flap deflections. The increase in CLmax depends on the flap chord ratio X f; the highest values are usually obtained for Xf = 0.20-0.25 [7]. A quite simple landing device in terms of design is the split flap as shown in Fig. 8-3e. This is a flat plate lying against the lower side of the wing and turning about its forward edge. The lift curves cL(a) of Fig. 8-17b for several flap angles qf are similar to those of the cambered flap (compare Fig. 8-17a). The effectiveness of the split flap is, according to Gruschwitz and Schrenk [19], due not only to an increased camber but also to a reduction of the static pressure on the suction side
of the profile. In Fig. 8-18, the pressure distribution is shown for a wing with
7f
-60°
i
3
?f =40° ZS
10 7
01
'7f 0 S°
10'
7S°
a-
20°
IS'
30°
Figure 8-16 Effect of flap deflection and boundary-layer control on the lift of a flap wing (schematic). Explanations in the text.
496 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES 2.4 0
15°
a = 30° 1,6
o = 45° 60° 0001,
0
00
a
16 °
32°-16°
b
a-
Figure 8-17 Lift coefficients of flap wings vs. angle of attack a for several flap deflections T ?f. Profile NACA 23012, Reynolds number Re = 6 105, from [491. (a) Simple cambered flap, flap chord ratio Xf = 0.2. (b) Split flap, Xf = 0.2.
qr
01.1
-4
-5
Figure 8-18 Pressure distribution on a wing with deflected split flap, from [19). Curve 1, without flap
-6
deflection. Curve 2, with flap deflection.
AERODYNAMICS OF THE FLAPS AND CONTROL SURFACES 497
deflected split flap. Because of the flow around the sharp trailing edge of the deflected plate, a strong low-pressure range is formed in the wake of the flap, having an effect up to the upper side of the wing. Basically, the CLmax value increases with Reynolds number. In Fig. 8-19, the results on the effect of the Reynolds number on the value of CLmax are given, both for a wing without a flap and one with a 600 deflection of a split flap. Young [54] reports on the separation characteristics of flap wings. Flaps extending over only a portion of the wing span will be treated in Sec. 8-3. The effectiveness of the simple cambered flap is limited by the flow separation occurring at large deflection 77f right behind the flap nose. By boundary-layer control at the station of greatest danger of separation, the lift-increasing effect of the cambered flap can be improved, as shown schematically in Fig. 8-16. Boundary-layer control by suction or ejection requires a considerable design and construction effort and will be discussed later in more detail. On the other hand, the slotted flap as shown in Fig. 8-3b, first suggested by Betz [6] and by Lachmann [27] , represents a simple design for natural boundary-layer control. The slotted flap functions in such a way that the air, flowing through the slot from the lower to the
upper side, carries the boundary layer, formed on the wing, into the free flow before separation can occur. Starting at the flap nose, a new boundary layer forms that can again grow over a larger distance before separation. The maximum lift coefficient CLmax depends on the separation processes at the
main wing in front of the flap as discussed in detail in Sec. 2-5-1. The most unfavorable flow conditions occur shortly behind the profile nose of the wing and at large angles of attack, a -_ CLmax). Here, the pressure increase that follows the 3
I
!
!
i
I
!
2.4
0.6
°
NACA64, -412
e
NACA 64,3 -418 NACA 23012
With 04 !
j--- Without
split fla p '
Figure 8-19 Change of maximum lift coeffi-
cient with Reynolds number for a wing Z.0 Re
3.0
4.0 5.0-10
without and with a split flap. Flap chord ratio Xf= 0.20, flap angle r?f= 60°, from [71.
498 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
suction peak usually leads to boundary-layer separation at the wing leading edge (see Fig. 2-44). By boundary-layer control, similar to that of the trailing-edge flap, separation can be shifted to larger angles of attack. The extension of the linear range of the cL(a) curve of Fig. 8-16 leads to a considerable additional lift gain. Another effective arrangement for the increase of the maximum lift is the slat (flap before the wing leading edge) as shown in Fig. 8-3f, whose characteristics have already been discussed in Sec. 2-5-3. A polar curve of it is given in Fig. 2-53. Figure 8-20 shows the lift coefficient plotted against the angle of attack for a wing without
and with a slat. In agreement with profile theory, the slat does not generate a noticeable change of the profile camber, because this would cause a parallel shift of the CL(a) curves without and with slat. Because of natural boundary-layer control, the maximum lift coefficient of a wing with a slat is reached at very large angles of attack. An effect similar to that of the slat is produced by the so-called nose flap, first proposed by Kruger [44]. Here, the increase of a(CLmax) results from a different effect, namely, the shape of the profile nose, responsible for the separation process, which is changed favorably by the flap deflection (see also Fig. 2-44). In addition to the conventional landing devices on the trailing edge discussed so far, the double-section wing as shown in Fig. 8-3c and the Fowler flap as shown in Fig. 8-3d must be mentioned. The former is a simpler design of the slotted flap. The latter consists of a flap that is driven out rearward and deflected. A simultaneous camber and area increase is thus accomplished. Frequently, several landing devices are utilized in combination to establish a maximum lift that is as large as possible. As an example, Fig. 8-21 gives the lift coefficient of the profile Go 819 with a slat and a double-section flap against the angle of attack. The favorable effect on the boundary layer of the flow through the slot between the slat and the main wing is clearly indicated by comparison with the measurement when the nose slot is closed. In this latter case, the cLmax values for
J
Figure 8-20 Lift coefficient CL(a) of a wing with slat,
from (48]. Profile Clark Y, Reynolds number Re = 6 - 105. Curve 1, without slat. Curve 2, with slat.
AERODYNAMICS OF THE FLAPS AND CONTROL SURFACES 499 J
is Z.0
as
Nose slot closed 0
Nose slot open
i
9,06
°
0 819
-05
-10 -S°
0°
5°
70-
7y a
20°
25°
a-
Figure 8-21 Lift coefficient CL(Q) for the profile Go 819 slat and double-slot flap, from Wuest [53].
all measured flap angles are lower by dCLma,X ~ 0.6; also, the flow separation leads to a larger lift drop than for the open nose slot. Comprehensive data on the maximum lift coefficient of wings with and without landing devices are given in [32, 33, 461.
Suction In an effort to increase further the maximum lift of wings, suction was studied quite early (see Betz [4] ). The suction intensity is defined by a dimensionless suction coefficient as
cQ = AQ,,
(8-19)
Here Q is the volume removed per unit time, A is the wing area, and U. is the incident flow velocity. The maximum lift can be increased considerably by slot suction. Comprehensive tests on this method were conducted by Schrenk [4]. The most effective method, particularly for thick profiles, was found to be slot suction with a flap wing. Lift coefficients up to about CL = 4 may be obtained, as shown in Fig. 8-22 for a thick profile with flap and suction. Here the coefficients of suction
are about cQ = 0.01-0.03 and the suction pressures cp = (p - p,,,)/q. = -2 to -4, where Q stands for the total flow volume removed, p for the pressure in the suction slot, and q,. _ (g. /2)UU for the dynamic pressure of the incident flow. The effect of suction lies in its keeping the flow essentially attached to the flap. The greatest danger of separation is near the flap nose. If the decelerated boundary layer at this
500 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
//
30
c U .Joz
/
F
O
/
al
J
IP
77f'95° 17f -,70.
Suction
10
of
c
0
1.0
2,0
Y0
.10
-cp ;
Figure 8-22 Lift coefficients of flap wings with slot suction, from Schrenk.
station is removed strongly enough by suction, the flow over the entire trailing-edge flap may be kept attached. After favorable wind.tunnel. results had been obtained, for flap profiles with suction, the Aerodynamische Versuchsanstalt Gottingen (AVA) conducted the first flight tests of the suction effect in the early 1930s. The possible
gain in lift for fully attached flap flow (CQ = cQL) over the lift of uncontrolled flow (cQ = 0) may be seen in Fig. 8-23. This diagram shows CL as a function of
deflection at several angles of attack of the wing. Note that the lift for potential flow is reached when the suction is just strong enough for complete prevention of separation. Arnold [4] studied the computation of the required amount cQL . More recently, both slot suction and continuous suction through flap
perforated walls have been applied, the latter at the trailing-edge flap as well as at the wing nose. Further developments of suction procedures have been summarized by Regenscheit [36] and Schlichting [36]. The continuously distributed suction has been studied theoretically by Schlichting and Pechau [381. Flight tests by Schwarz [38] and by Schwarz and Wuest [38] confirm the feasibility of nose suction.
Ejection The boundary layer may be controlled by ejection as well as by suction for increased maximum lift. This method has been applied most successfully to the wing with a trailing-edge flap. By tangential ejection of a thin jet of high velocity at the nose of the deflected flap, flow separation from the flap can be prevented and the lift can be increased. Critical for the effectiveness of ejection is, according to Williams [51], the dimensionless momentum coefficient
ci _
e.A
(8-20)
AERODYNAMICS OF THE FLAPS AND CONTROL SURFACES 501
where the index j refers to the conditions in the jet and the index C- to those of the incident flow. Comprehensive studies on the lift increase of flap wings with ejection have been conducted by Thomas [43]. In Fig. 8-24, a typical result of these measurements is given, namely, the gain in the lift coefficient JCL against the momentum coefficient c1 for several flap angles 77f. The curves A CL versus c1 clearly show two ranges: first,
a very steep increase at small momentum coefficients; and second, a considerably smaller increase at large momentum coefficients. The first range is that of boundary-layer control. It extends to the momentum coefficient that just suffices to produce complete flow attachment back to the flap trailing edge, thus completely preventing separation. The second range of considerably smaller lift gain with the momentum coefficient is the range of supercirculation. Here, the "hard jet" (of very high momentum) acts similarly to an extended mechanical flap. In Fig. 8-25, the lift coefficient of a wing at fixed flap deflection is plotted against the angle of attack for several momentum coefficients cy. The ejection has a similar effect as an increased camber (flap deflection). Flow separation sets in at smaller angles of attack, however, than without ejection. Inspecting Fig. 8-16 shows that an additional lift gain can be generated by combination with a boundary-layer
Theory cp -0
Z.
or a=70° 70
j
Zs
7.2
o81
--- -
r Jam
05, m
rfl
0 00
75°
30°
K. ?7f
60°
Figure 8-23 Lift increase due to slot suction at the trailing-edge flap for completely attached flap flow, from Arnold. (- - -) Measurements with) Measurements with out suction. ( suction.
502 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES 6.0
5.0
U
NACADO1o
'
-
77f -900
f
w" I 75 60
a --5°
45° !
2.0 150
I0 0
Figure 8-24 Flap wing with ejection, lift increase AcL vs. momentum coefficient ci for various flap 0.4
0.2
0.8
0.6
cj - Qj vj s/y°, c
angles ref at constant angle of attack a = -5°, from Thomas.
2. A0.131
0,053
2
Q037
2.0
0.018
7.7
1.5
°
1.0
0,75
0.5
0.25
0
-10
-5
If
5°
10°
200
Figure 8-25 Lift coefficient of a wing with ejection over the trailing-edge flap, from Williams, profile tic = 0.08. Flap deflection 77f= 45°, flap chord ratio Af = 0.25.
AERODYNAMICS OF THE FLAPS AND CONTROL SURFACES 503
control at the wing nose, either by suction or by ejection (see Gersten [15] ). Even the flow is completely attached, a further increase in lift may be accomplished by stronger ejection on the flap. This is the result of supercirculation and the jet reaction force. This problem area has been summarized by PoissonQuinton [34] and by Williams [511; see also [28]. Levinsky and Schappelle [29] developed a method aimed at maintaining potential flow through tangential ejection when
on flap wings.
Jet flaps Effects very similar to those generated by a solid trailing-edge flap are obtained by ejecting a high-speed jet under a certain angle nj near the wing trailing
edge. This method, illustrated in Fig. 8-26, is termed a jet flap. The vertical component of the reaction force of the jet is supplemented by an induced lift that may be many times larger than the jet reaction (supercirculation). This effect has been studied by many experiments [34, 521. In Fig. 8-26, the theories of Spence [42] and Jacobs [9] are compared with experiments on a symmetric profile with jet flap. The figure shows the dependence of the lift slopes acL jaa and acL /ar7j on the momentum coefficient c1 as defined by Eq. (8-20). Here the momentum coefficients cj are much larger than in Fig. 8-25. Up to values of about ci = 0.1, the jet acts on the boundary layer; for larger values of cj it essentially causes the circulation to increase (supercirculation). Either lift. slope increases strongly with increasing cp For cj = 4, the lift slope acL/aa has about twice the value of that without ejection (c1 = 0). The agreement of theory
1
dcL
1,
1
J/
Theor : rY
/
e
4
//
+
I
2 6
I
4
5
Figure 8-26 Profile with jet flap, comparison of theory and experiment for lift slopes acL/aa and acL/ary. Theory from Spence and Jacobs. Measurements from
Dimmok [52]. (c) nj = 31°. (o) 58°.
504 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
and experiment is good. Helmbold [20] studied the theory of the wing of finite span with jet flap. A comprehensive wing theory for the wing of finite span with jet flap has been
developed by Das [9]. An example of this theory and a comparison with experiments is given in Fig. 8-27 for a swept-back wing with a jet flap spanning the entire trailing edge. Agreement between theory and experiment is good. Murphy and Malmuth [9] report on the computation of the aerodynamics of the jet flap wing in transonic flow. The jet flap wing. near the ground has been studied by Lohr [30].
The aerodynamic problems of the maximum lift have been summarized by Schlichting [37]. Questions of the practical application of the jet effect to the generation of high lift on wings with and without flap are discussed in the summarizing paper of Korbacher [251,
Air brakes, spoilers The aerodynamic effect of air brakes has been investigated repeatedly (see Arnold [3] ). In particular, various positions of the brakes on the lower and upper sides of the wing have been studied. Figure 8-28 shows the result of three-component measurements for a wing with air brakes over the entire span. The polar curves illustrate the very large drag increase. Compared with the wing alone, the drag coefficient is about 20 times larger. Devices of a similar kind mounted only on the upper side of the wing are also. termed spoilers. By extending them on only one side of the wing, they can be used
U.
7
V. 3.
lo,
C' 2.8
A
cj=2 14
72
/
0.8
/
i
i * X0.2
---
I
Theory Measurerne+
0
-8°
40
00
8°
72°
a ----
Z0°
t
Figure 8-27 Lift coefficient of a swept-back wing with jet flap; comparison of theory and measurement from Das [9 ]. Aspect ratio .4 = 3.5,
sweepback angle p= 45°, jet angle rj = 30°.
AERODYNAMICS OF THE FLAPS AND CONTROL SURFACES 505
_F Z-T
--yr c U.,c/4-I
*11
S+P -0.8
-1G -20'
-70'
70'
20'
30'
26
WO
S+p
P
-0,8
0,7
02
0S
03
CD
8-28 Three-component measurements on a rectangular wing with air brake, from Reller [ 3 ]. Aspect ratio A = 5.1; flaps Figure
-08
extend over the entire span. WO, wing
-76
-0.08
008
0.16'
0.2'
without flap; S, flap on suction side; P, flap on pressure side.
506 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
for control about the vertical and longitudinal axes. The flow separation from the wing caused by the spoiler leads to a strong, one-sided lift loss and thus to a rolling moment. Wing tunnel test results on spoilers and a few computations on the effect of the spoiler are found in [11, 21, 23, 50] .
8-3 FLAPS ON THE WING OF FINITE SPAN AND ON THE TAIL UNIT
8-3-1 Flaps on the Wing in Incompressible Flow Computational methods The aerodynamics of the flap wing of infinite span (plane
problem) has been discussed in the previous section. Now the effect of a flap (control surface) on a wing of finite span will be treated. A further geometric parameter, the span of the flap, is added (see Figs. 7-1, 7-3, and 8-4a). Furthermore,
in many cases the flap chord ratio varies over the flap span (see Fig. 8-1). To determine the lift distribution, a wing with a deflected flap is equivalent to a wing
with an additional angle-of-attack distribution over the span (twist). For a flap covering only a portion of the span, this additional angle-of-attack distribution is discontinuous. The angle-of-attack distribution that is equivalent to a given flap. deflection is obtained from the theory of the flap wing of infinite span as
af(n) _ -
aQ_17f
r?f
(8-21)
where aa/ar f is the local flap effectiveness from Eqs. (8-8a) and (8-9a) and from Figs. 8-7a and 8-9a. If the flap chord ratio Xf varies over the span, it is a function of the span coordinate 77 =y/'s.
According to the procedure for the computation of the lift distribution on wings of Sec. 3-3, the additive circulation distribution caused by the flap deflection can be determined for such an angle-of-attack distribution. Special attention should be paid to the station of discontinuity in the angle of attack. The case of a symmetric angle-of-attack distribution corresponds to a landing
flap at the wing or an elevator at an all-wing airplane as shown in Fig. 7-3. The antimetric angle-of-attack distribution corresponds to the ailerons (Figs. 7-1 and 7-3).
Following simple lifting-line theory (Sec. 3-3-3), Multhopp (Chap. 3, [60] ) developed a method for handling the discontinuity in the angle-of-attack curve. In Fig. 8-29, a result of this method for a trapezoidal wing of aspect ratio A = 2.75 and taper X = 0.5 is shown as curve 1. The station of discontinuity in the angle-of-attack distribution of lies at r70 = 0.5. In Fig. 8-29a it is symmetric, in Fig. 8-29b it is antimetric. According to Fig. 8-29, the symmetric flap deflection at the wing outside generates a considerable lift, even in the wing middle section. The circulation distributions according to extended lifting-line theory (three-quarterpoint method, Sec. 3-34) are also shown in Fig. 8-29 as curves 2. As should be expected, extended lifting-line theory gives a smaller lift than simple lifting-line
AERODYNAMICS OF THE FLAPS AND CONTROL SURFACES 507
21
V
7
- 170
/_"
Of
0.
'0
a0
0.4
of
- , -rIo 770
/IZ11. r"o b 07
n_ 1
22
n. 3
n. 4
as
n_ B
11 7
/10
2R
L
Figure 8-29 Circulation distribution over the span due to a discontinuous angle-of-attack distribution for a trapezoidal wing of aspect ratio A= 2.75; taper X = 0.5. Curve 1, simple lifting line theory. Curve 2, extended lifting-line theory. (a) Symmetric angle-of-attack distribution. (b) Antimetric angle-of-attack distribution.
theory. A computational method for the lift distribution on wings with flaps, based on lifting-surface theory (Sec. 3-3-5), is given in [46]. This method requires the availability of the angle-of-attack distributions caused by the flap deflection on the c/4 line (il) and on the trailing edge (i;,). They are, considering Eq. (8-21), 4 ac3c. a,
elf
«f( r,
a«
2 acrn
anf
2
ae f
of
(8-22a)
)17f
(8-22b)
where the coefficients as/ary and ac,n /ary from Eqs. (8-8a) and (8-8b) and from Eqs. (8-9a) and (8-9b), respectively, are known from the profile theory of the flap wing and depend only on the control-surface chord ratio.* An improved method for describing the effect of the angle-of-attack discontinuity has been given by Hummel [46]. Lift distributions of wings with deflected flaps (angle-of-attack distribution with a break) have been computed by Bausch [5] from simple lifting-line theory for a wing of elliptic planform. For a wing with a trapezoidal planform, corresponding computations have been published by Richter [5]. A large number of computations *Since these equations contain local coefficients, the coefficient CM of Eq. (8-5) has here been written as c,n.
508 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
have been conducted by de Young [10], who applied extended lifting-line theory;
however, he did not exclude the station of discontinuity in his computations. Investigations, applying lifting-surface theory, have been conducted by Truckenbrodt and Gronau [46] on delta wings with deflected flaps.
A summary of American tests on wings of finite span with flaps that extend only over a portion of the span is given in [14]. It includes the separation characteristics of such wings; compare the publications [31, 541. Results of a few sample computations of the lift distribution of wings with flap and control-surface deflections will be given in the following section.
Landing flaps, elevators For the wing of elliptic planform, the change in the mean zero-lift angle caused by the flap deflection is obtained according to Sec. 3-3-3. For a sectionwise-constant, symmetric angle-of-attack distribution, Eq. (3-81) yields, after integration,'
as - -1 +
2
(arccos 77, - 770
1 - rlo)
(8-23)
Here the flap (control surface), having a constant flap chord ratio, extends from -r?o to +rro. The relationship between of and the flap angle rrf is given by the theory of the two-dimensional flap wings of Eq. (8-21). The coefficient as/aaf is. shown in Fig. 8-30 as a function of the flap span. This result is obtained by both simple and extended lifting-line theories.
A further example, in which Truckenbrodt and Gronau [46] applied liftingsurface theory, is shown in Fig. 8-31. It deals with a delta wing of aspect ratio A1* = 2b*/cr = 2 equipped with a flap that is symmetrically deflected. The. flap chord ratio Xf = cf/c, however, varies between Af = s at the wing root and Af = 1 at the wing tips. The local flap effectiveness was obtained by introducing Eqs. (8-9a) and (8-9b) into Eqs. (8-22a) and (8-22b). The changes of the mean zero-lift angle
a«/arlf and of the mean zero-moment coefficient acM/anf were computed first. 10f
0.e
0.6
0.2
0,2
0.4 PO
0A
0.8
1.0
Figure 8-30 Change of the mean zero-lift angle due to flap deflection for an elliptic wing with various forms of the flap, from Bausch.
a
?7f
.
Oe
Theory
.°
Measure ments
0
OZ 0°
0°
4°
8°
12°
16°
20°
24°
Mc
-5°
Theory
;, ,
Measureme
°
I
is
8-31 Measured aerodynamic coefficients of a delta wing with symmetrically deflected flap Figure
nf5°
I
-02 . 0.16 -0.72 -0.09 -0.04 C
H
0
cM
0.04
0.G08
0.12
0.16
extending over the entire trailing edge. Aspect ratio A * = 2, profile NACA 0012; comparison of theory (;, = 0.75) and experiment, from Truckenbrodt and Gronau. (a) Geometry. (b) Lift coefficient vs. angle of attack. (c) Lift coefficient vs. pitching-moment coefficient.
Figure 8-31b and c shows the good agreement of the theoretical results CL(a) and cyl(a) with measurements at small flap deflections. Ailerons In Fig. 8-32, the rolling-moment coefficients are given for a wing of elliptic planform and antimetric control-surface deflection. Figure 8-32a gives the
510 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES 2
2 ,
I
of i
1
I
0
4
2
a
6
10
8
12
A 1.0
0.8
1
1o
Figure 8-32 Rolling-moment coeffi-
cient vs. flap deflection for an elliptic wing, from Bausch. (a) Flap extending over the entire half-span; curve 1,
b
0.2
0,4
0,6
0.8
extended lifting-line theory; curve 2, simple lifting-line theory. (b) Effect of
1.0
the flap span.
r10
rolling moment of the ailerons plotted against the aspect ratio with each aileron extending over the entire half-span. The extended lifting-surface theory of Eq. (3-100) yields acMX
4
orA
3n Vkz + 4 + 2
(8-24a)
aaf where k = irA/cL.. A/2. For comparison, this coefficient according to simple lifting-line theory is added. The rolling moment of the ailerons for the case of an aileron extending over only a part of the wing half-span is shown in Fig. 8-32b. In this case, Eq. (3-100) yields (acMX
f
sax
1 -170 3
(8-24b)
asf ino =0 where (acMxlaaf)no _o is given by Eq. (8-24a) and Fig. 8-32a. For the delta wing of Fig. 8-3la, the theoretical coefficients of the aileron rolling moment of antimetrically deflected ailerons extending over the entire half-span are compared in
Fig. 8-33 with measurements. Agreement between theory and experiment is good for small and moderate angles of attack.
AERODYNAMICS OF THE FLAPS AND CONTROL SURFACES 511
Aileron investigations and comprehensive experimental results are summarized
in [12, 45].
8-3-2 Flaps on the Wing in Compressible Flow The flap wing of finite span in compressible flow may be treated according to the theory of the wing of finite span as discussed in Secs. 4-4 and 4-5. Subsonic incident flow At subsonic velocities, the subsonic similarity rule (Prandtl-
Glauert) of Sec. 4-4-1 applies. It requires the determination of a wing, to be computed for incompressible flow, that is transformed from the given geometry of the wing of finite span at compressible flow. These transformation formulas for the geometries of the wings are found as Eqs. (4-66)-(4-68). The influence of compressibility on the aerodynamic coefficients of the wing is obtained from the transformation formulas Eqs. (4-69)-(4-72). Here, the angle-of-attack distribution due to the flap deflection remains unchanged and is determined with lifting-surface theory from Eq. (8-22). Accordingly, Eqs. (8-15a) and (8-15b) give the changes of the angle of attack and of the momentum coefficient with the flap deflection. However, these equations for the incompressible reference flow now have to be evaluated for the transformed wing planform from Eq. (4-15). In Fig. 8-34, the results of sample computations for wings of finite span with deflected flaps are shown. They are the three wings discussed several times previously, namely, a trapezoidal, a swept-back, and a delta wing; see Table 3-4.
Supersonic incident flow The computation of the aerodynamic effect of a flap on a
wing of finite span at supersonic velocities is in some respect simpler than at subsonic velocities. This becomes obvious from Fig. 8-35, which shows a rectangular
Figuae 8-33 Measured roiling-moment coef-
ficients of a delta wing as shown in Fig. 8-31a, with flaps extending over the entire half-span for several I
l
f 10°
of
15°
±20°
angles of attack a.
Comparison of theory (% = 0.75) and measurements from Truckenbiodt and Gronau.
512 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
a
ZA
b
c
0.8 10.6
f t
0
24
1
0.8
0,4
0
0.2
0.4
0.6
0.8
Ma°,--
7.00
0.2
0.4
0.6
0.6
Mao,
1.00
0.2
0.4
0.6
06
1.0
mat,
I-
Figure 8-34 Change of the zero-lift angle and the zero-moment due to flap deflection for several wings with flaps extending over the entire trailing edge vs. Mach number for subsonic incident flow, from Kowalke [26] ; lifting surface theory, % = 1. (a) Trapezoidal wing: A = 2.75,1 = 0.5, yp = 0°, Af= (b) Swept back wing: .i = 2.75, A = 0.5, P = 50°, xf= s. (c) Delta wing: A = 2.31, X = 0,'P = 52.4°, Xf= Cr/8 = cont,
wing and a delta wing with flaps of constant chord extending over the entire trailing edge. When the flap is being deflected, an additive lift is generated only on this flap
that is equal to the lift of a rectangular wing of span b and of the flap chord cf. The lift of the wing lying before the flap is not changed by the flap deflection. To compute the lift caused by flap deflection, the results for the rectangular wing of Sec. 4-5-4 may be recalled. From Eq. (4-112), the lift coefficient produced by the flap and referred to the total wing area A is given as acL a77f
_ Af
1^
4
A Maw-1
1
Cf
2
b }'Magi - 1
(8-25)
which is valid for cf c b Ma;° - 1, but independent of the wing shape. For the rectangular wing of Fig. 8-35, the change of the zero-lift angle caused
by the flap deflection can easily be determined. Because aa/arrf = -(acL/ark f)1 (acL/aa), Eqs. (8-25) and (4-112) yield
as arjf
_
2A Ma-00
1-Af
2A}IMac-1-1
(8-26)
where 1 f = c f/c = A fIA is the control-surface chord ratio. In this equation, the
fraction on the right-hand side, which is always greater than unity, gives the
AERODYNAMICS OF THE FLAPS AND CONTROL SURFACES 513
J
Figure 8-35 Aerodynamics of the flap wing at supersonic incident flow. (a) Rectangular wing with flap extending
over the entire trailing edge. (b) Delta wing with flap extending over the entire trailing edge.
correction of the value for the two-dimensional flap wing, as can be verified by comparison with Eq. (8-16a). The pressure distribution on the flap of a wing in supersonic incident flow may
also be established quite easily. Figure 8-36 shows a flap design in which the right-hand-side edge of the flap is an "outer edge," the left-hand edge an "inner edge," both of which are parallel to the incident flow direction. When the flap is deflected, Mach lines originate at either upstream edge. In the case of no intersection of these Mach lines on the flap, the pressure distribution in zone 1 is
77
t=1
tr-1
t=1
Figure
8-36 Pressure distribu-
tion due to flap deflection on a
t=-1 0
1
t=-1
0
rectangular flap at supersonic incident flow.
514 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
that for plane flow. The resultant pressure coefficient on the upper and the lower side is, therefore, with Eq. (4-85) and Table 4-5, 4'77 f
Cpl =C pp1 =
,
)/Maw
(8-27a)
-1
The flow in zones 2 and 3 is cone-symmetric. For zone 2, Eq. (4-111) yields cp2 =
1
arccos (1 + 2t)cppl
(8-27b)
7c
For zone 3, Tucker and Nelson [47] found the expression cp3 =
i arccos (-t) Cppl
(8-27c)
In these expressions t = y/x tan p = (y/x) -NIMa, - 1, and y is measured from the upstream corners of the flap. In Fig. 8-36, the pressure distributions are shown for a section x = const. On the side of the inner edge, the flap deflection causes, within the range of the Mach cone, a lift on the undeflected wing that is equal to the lift loss at the adjacent portion of the flap. Furthermore, Fig. 8-37 shows a flap arrangement with a swept-back outer edge of the flap such as, for example, is found in delta wings. In Fig. 8-37a, the outer. edge is a subsonic edge. If the two Mach lines originating at the two, upstream flap corners do not intersect on the flap, zone 1 has again, as in Fig. 8-36, the pressure distribution of plane flow. In the case of the subsonic edge (m < 1) of Fig. 8-37a, the pressure distribution of zone 4 is of the kind given in Fig. 4-67 for a delta wing with a subsonic leading edge. In the case of the supersonic edge (Mac, > 1) of Fig. 8-37b, where the Mach cone from the right-hand upstream corner lies entirely on the flap, the pressure distributions of zones 5 and 6 are of the kind given in Fig.
a
b
Flap
l
I
Figure 8-37 Pressure distribution due to flap deflection on a trapezoidal flap at supersonic incident flow, from Tucker and Nelson. (a) Subsonic outer edge, m = 4. (b) Supersonic outer edge, m = 1 .
AERODYNAMICS OF THE FLAPS AND CONTROL SURFACES 515 1.0
0.8
2 0.6
a 04
bf
3
7
-
0.2
0
3
2
1
bf
cf
7
->
Figure 8-38 Lift due to flap deflection at supersonic incident flow.
Curve 1, inner flap. Curve 2, tip flap. Curve 3, full-span flap.
4-69 for a delta wing with a supersonic leading edge. The pressure in zone 6 is constant, Eq. (4-89): Cps -
Y1L
n
CP p1
(8-28)
ynv -1 with m = tan 7/tan µ. The pressure distributions in zones 4 and 5 have been determined by Tucker and Nelson [47].
Finally, a few data will be given. in the following two figures on the lift produced by the flap deflection and on the position of its center of application. Figure 8-38 gives the total lift of three rectangular flaps. Flap 1 has two inner edges (inner flap), flap 2 an inner and an outer edge (tip flap), and flap 3 two outer edges
bf.
(full-span flap). Shown in this figure is the ratio of the total lift produced by the flap to the lift of the two-dimensional flap wing as a function of the quantity Ma', - 1 /c f. Flap 1 does not cause any lift loss compared with the two-dimensional flap wing; Eq. (8-25) applies to flap 3. The lift of flap 2 is the arithmetic mean of those of flaps 1 and 3. Figure 8-39 shows the position of the lift force of the flap (flap neutral point). Here, xf is the distance of the flap neutral point from the axis of rotation. For flap 1, the flap neutral point lies at the flap half-chord. It shifts forward for flaps 2 and 3. The rolling moment due to aileron deflection can be computed very easily by realizing that the lift force at antimetrically deflected flaps acts, in very good approximation, on the half-span of the flap.
Further information on rectangular flaps is found in Schulz [471. Flaps on rectangular, delta, and swept-back wings have been investigated by Lagerstrom and Graham [47]. Flaps with outer (horn) balances have been studied by Naylor [47].
8-3-3 Control Surfaces on the Tail Unit in this section, a brief discussion will be given of the aerodynamic forces generated by the control-surface deflection of the tail unit and their effect on the force and
516 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
0.
r O'N
2
0,4
3 0.3
f 0.2
bf
C,
t
r
i
i
0.1
3
2
1
bf
Maw -1
cf
Figure 8-39 Position
of the flap
neutral point for flap designs of Fig.
--_-
8-38.
moment equilibrium of the whole airplane. For the case of zero control-surface deflection, the contributions of the horizontal tail and the vertical tail, respectively, to the aerodynamic forces of the whole airplane have been given in Secs. 7-2-1 and 7-3-1.
Elevator For the contribution of the horizontal tail with deflected elevator to the pitching moment of the whole airplane, Eqs. (7-3a) and (7-3b) yield
Ms - -
CMH
do E
H Ca
aax a'7H
77H
4'H AH rH
q. A cg
Here, from Fig. 7-5, rH is the distance of the lift force of the horizontal tail from the moment reference axis of the airplane. The change in the moment caused by the elevator deflection at constant angle of attack is thus obtained as (acMH)
a77H a=const
= dcrH a«H qa Ax rH dcH a77Hgoo A Cµ
(8-29)
Here, the quantity rH of the previous equation has been replaced by the lever arm ,rH, which is the distance of the flap neutral point from the moment reference axis of the horizontal tail. For the two-dimensional flap wing in incompressible flow, the position of the flap neutral point is given in Fig. 8-15. The change in the pitching moment caused by the elevator deflection at constant lift coefficient (zero-moment coefficient) is obtained in analogy to Eq. (7-15) by substituting -(acH/3r1H)77H for EH as
_delHaoH4HAHrHN
aCMg a77H
cL=const
das a?7H q. A
(8-30)
Cu
Here rHN is the distance of the neutral point of the elevator from the neutral point of the whole airplane (see Fig. 7-6b).
AERODYNAMICS OF THE FLAPS AND CONTROL SURFACES 517
Rudder The contribution of the vertical tail with a deflected rudder to the yawing moment of the whole airplane becomes, from Eqs. (7-49a) and (7-49b), cMZv
-`dcty day
aav Qv
arty
(V)
gvAv r'v q0 A s
Here, r'y from Fig. 7-36 is the distance of the side force of the vertical tail from the moment reference axis of the airplane. The change in the yawing moment caused by the rudder deflection is thus given as
acMZV
an v
= dcrv aav qv AV r'v
day anv q. A
s
(8-31)
Here the quantity ry of the previous equation has been replaced by the lever arm r 'y , which is the distance of the flap neutral point from the moment reference axis of the vertical tail. Rudder moments Information on the rudder moments of the airfoil of infinite span for incompressible flow is found in Sec. 8-2. The control-surface moments of the elevator and rudder and also of the ailerons cannot, in general, be computed with sufficient accuracy, because for the control-surface moments the transformation, from the airfoil of infinite span (plane problem) to the wing of finite span is not possible in a reliable way. The control-surface moments for control surfaces with
balance provisions of Fig. 8-2 (inner balance, outer balance, balance tabs) are particularly difficult to determine because they are greatly affected by the boundary
layer as well as by' inviscid flow problems. The control-surface moments must therefore be determined largely through wind tunnel and flight tests (see. Stiess [18]). Some wind tunnel measurements on the control-surface moments of tail surfaces with inner and outer balances were reported by Schlichting and Ulrich [39].
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14:276-281, 1966. Fuchs, D.: Lufo., 15:19-27, 1938. Jacobs, H. and A. Wanner: Jb. Lufo., 1:313-318, 1938. Reller, E.: ZWB Lufo. FB 1689/1, 1942; 1689/2, 1943. Voepel, H.: Jb. Lufo., 1:82-95, 1941. Wanner, A.: Jb. Lufo., 1:308-312, 1940. 4. Arnold, K. 0.: Untersuchungen fiber die Auftriebserhohung eines Klappenflugels durch Schlitzabsaugung, Z. FTugw., 15:37-56, 1967. Betz, A.: Akad. Lufo., H.49, 51-81, 1939. Cook, W. L., S. B. Anderson, and G. E. Cooper: NACA Rept. 1370, 1958. Schrenk, 0.: Lufo., 2:49-62, 1928; 12:10-27, 1935; Z. Flug. Mot., 22:259-264, 1931; Luftw., 7:409-414, 1940. 5. Bausch, K.: Zahlenergebnisse fur elliptische Flugel mit stuckweise konstanter and stuckweise linearer Anstellwinkelverteilung, Lufo., 15:260-274, 1938. Kolscher, M.: Jb. Lufo., 1:129-135, 1938. Richter, W.: Lufo., 20:69-76, 1943.
518 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
6. Betz, A.: Die Wirkungsweise von unterteilten Fliigelprofilen, Ber. Abh. WGL, no. 6, 23-26, 1922. Nickel, K.: Ing.-Arch., 20:363-376, 1952. 7. Cahill, J. F.: Summary of Section Data on Trailing-Edge High-Lift Devices, NACA Rept. 938, 1949. 8. Carriere, P., E. Eichelbrenner, and P. Poisson-Quinton: Contribution theorique et experimentale a 1'etude du controle de la couche limitee par soufflage, Adv. Aer. Sci., 2:620-661, 1959.
9. Das, A.: A Lifting Surface Theory for Jet-Flapped Wings, J. Aerosp. Sci., 29:499-500, 1962; Jb. WGL, 112-133, 1960; Abh. Braunschw. Wiss. Ges., 17:21-50, 1965. Jacobs, W.:
Z. Flugw., 5:253-259, 1957. Mascheck, H: J.: Z. Angew. Math. Mech., 40:T 140-142, 1960. Murphy, W. D. and N. D. Malmuth: AIAA J., 15:46-53, 1977. 10. de Young, J.: Theoretical Symmetric Span Loading Due to Flap Deflection for Wings of Arbitrary Plan Form at Subsonic Speeds, NACA Rept. 1071, 1952; 1056, 1951. 11. Fischel, J. and J. M. Watson: Investigation of Spoiler Ailerons for Use as Speed Brakes or Glide-Path Controls on Two NACA 65-Series Wings Equipped with Full-Span Slotted Flaps, NACA Rept. 1034, 1951. 12. Fischel, J., R. L. Naeseth, J. R. Hagermann, and W. M. O'Hare: Effect of Aspect Ratio on
the Low-Speed Lateral Control Characteristics of Untapered Low-Aspect-Ratio Wings Equipped with Flap and with Retractable Ailerons, NACA Rept. 1091, 1952. 13. Fliigge-Lotz, I. and I. Ginzel: Die ebene Strornung urn ein geknickte Profil mit Spalt, Jb.
Lufo., 1:55-66, 1939; Ing.-Arch., 11:268-292, 1940. Losch, F.: Lufo., 17:1-2, 22-31, 1940. Sohngen, H.: Lufo., 17:17-22, 1940. 14. Furlong, G. C. and J. G. McHugh: A Summary and Analysis of the Low-Speed Longitudinal Characteristics of Swept Wings at High Reynolds Number, NACA Rept. 1339, 1957. Sivells, J. C. and S. H. Spooner: NACA Rept. 942, 1949. 15. Gersten, K. and R. Lohr: Untersuchungen fiber die Auftriebserhohung eines Tragfliigels bei gleichzeitigem Ausblasen an der Hinterkantenklappe and an der Profilnase, DFL-Ber. 189, 1962; Ass. Ital. Aer., 1966. 16. Glauert, H.: Theoretical Relationships for an Aerofoil with Hinged Flap, ARC RM 1095, 1927. Perring, W. G. A.: ARC RM 1171, 1928.
17. Goradia, S. H. and G. T. Colwell: Analysis of High-Lift Wing Systems, Aer. Quart., 26:88-108, 1975. Foster, D. N.: J. Aircr., 9:205-210, 1972. 18. Gothert, R.: Systematische Untersuchungen an Fli geln mit Klappen and Hilfsklappen, Jb. Lufo., I:278-307, 1940; Ringb. Luft. I A 13, 1940. Bausch, K. and H. Doetsch: Jb. Lufo., 1:182-203, 1940. Kupper, A.: Lufo., 20:22-28, 1943. Staufer, F.: Jb. Lufo., 1:245-247, 1940. Stiess, W.: Jb. Lufo., 1:252-277, 1941. 19. Gruschwitz, E. and O. Schrenk: fiber eine einfache Moglichkeit zur Auftriebserhohung von Tragfliigeln, Z. Flug. Mot., 23:597-601, 1932. 20. Helmbold, H. B.: Theory of the Finite-Span Blowing Wing, J. Aer. Sci., 24:339-344, 370,
1957; 22:341-342, 1955; Ing.-Arch., 23:209-211, 1955. Spence, D. A.: J. Aer. Sci. , 23:92-94, 1956.
21. Heyser, A. and F. Maurer: Experirnentelle Untersuchungen an festen Spoilern and Strahlspoilern bei Machschen Zahlen von 0,6 bis 2,8, Jb. WGL, 268, 1961; Z. Flugw., 10:110-130, 1962. Gothert, B.: Ber. Lil.-Ges. Lufo., 156:64-68, 1942. Knoche, H.-G.: Jb. WGL, 262-268, 1961. Naumann, A.: Jb. WGL, 203-204, 1955. Seibold, W.: Jb. WGL, 192-202, 1955. 22. Jacob, K. and F. W. Riegels: Berechnung der Druckverteilung endlich dicker Profile ohne and mit Klappen and Vorflugeln, Z. Flugw., 11:357-367, 1963. 23. Jones, A. L., O. P. Lamb, and A. E. Croak: A Method for Predicting Lift Effectiveness of Spoilers at Subsonic Speeds, J. Aer. Sci., 23:330-334, 376, 1956. 24. Keune, F.: Auftrieb einer geknickten ebenen Platte, Jb. Lufo., 1:48-51, 1937; Lufo., 13:85-87, 1936; NACA TM 1340, 1955; Jb. Lufo., 1:54-59, 1938; Lufo., 14:558-563, 1937. Hay, J. A. and W. J. Egginton: J. Roy. Soc., 60:753-757, 1956. Jungclaus, G.: Z. F7ugw., 5:106-114, 1957. Rossow, V. J.: J. Aircr., 10:60-62, 1973. Walz, A.: Jb. Lufo., 1:265-277, 1940. Weinberger, W.: Lufo., 17:3-11, 1940.
AERODYNAMICS OF THE FLAPS AND CONTROL SURFACES 519
25. Korbacher, G. K.: Aerodynamics of Powered High-Lift Systems, Aniz. Rev. Fluid Mech., 6:319-358, 1974. 26. Kowalke, F.: Die flugmechanischen Beiwerte von Tragflugeln bei Unterschallgeschwindib keit, Jb. WGL, 40-48, 1958. 27. Lachmann, G. V.: Die Stromungsvorgange an einem Profil mit vorgelagertem Hilfsflugel, Z. Flug. Mot., 14:71-79, 1923; 15:109-116, 1924. Petrikat, K.: Jb. Lufo., 1:248-264, 1940. Strassl, H.: Jb. Lufo., 1:67-71, 1939. Weinig, F.: Lufo., 12:149-154, 1935. 28. Lachmann, G. V. (ed.): "Boundary Layer and Flow Control-Its Principles and Application," Pergamon, Oxford, 1961. 29. Levinsky, E. S. and R. H. Schappelle: Analysis of Separation Control by Means of Tangential Blowing, J. Aircr., 12:18-26, 1975. 30. Lohr, R.: Der Strahlklappenfligel in Bodennahe unter besonderer Berdcksichtigung grosser Anstell- and Strahiklappenwinkel, Z. Flugw., 24:187-196, 1976. Kida, T. and Y. Miyai: AIAA J., 10:611-616, 1972. Lissaman, P. B. S.: AIAA J., 6:1356-1362, 1968. 31. McCullough, G. B. and D. E. Gault: Examples of Three Representative Types of Airfoil-Section Stall at Low Speed, NACA TN 2502, 1951. Gault, D. E.: NACA TN 3963, 1957.
32. Nonweiler, T.: Maximum Lift Data for Symmetrical Wings-A Resume of Maximum Lift Data for Symmetrical Wings, Including Various High-Lift Aids, Aircr. Eng., 27:2-8, 1955; 28:216-227, 1956. 33. Pleines, W.: Die Mittel zur Vergrosserung von Hochstauftrieb and Gleitwinkel, Ringb. d. Luftf I A 7, 1936. 34. Poisson-Quinton, P.: Einige physikalische Betrachtungen iiber das Ausblasen an Tragflugeln, Jb. WGL, 29-51, 1956. Poisson-Quinton, P. and H. Jacquignon: Jb. WGL, 149-163, 1960.. 35. Riegels, F. W.: "Aerodynamische Profile, Windkanal-Messergebnisse, theoretische Unterlagen," Oldenbourg, Munich, 1958; "Aerofoil Sections" (English transl. by D. G. Randall), Butterworths, London, 1961. 36. Schlichting, H.: Absaugung in der Aerodynamik, Jb. WGL, 19-29, 1956. Regenscheit, L. B.: Jb. WGL, 55-64, 1952. 37. Schlichting, H.: Aerodynamische Probleme des Hochstauftriebes, Z. Flugw., 13:1-14, 1965. Schrenk, 0.: Jb. Lufo., 1:77-83, 1939. 38. Schlichting, H. and W. Pechau: Auftriebserhohung von Tragflugeln durch kontinuierlich verteilte Absaugung, Z. Flugw., 7:113-119, 1959. Schwarz, F.: Z. Flugw., 11:142-149, 1963. Schwarz, F. and W. Wuest: Z. F7ugw., 12:108-120, 1964. 39. Schlichting, H. and A. Ulrich: Die Seitenstabiliti t eines Flugzeuges mit losgelassenem Seitenruder, Jb. Lufo., 1:172-191, 1941. 40. Schrenk, 0.: Druck- and Geschwindigkeitsverteilung fangs der Fliigeltiefe fur verschiedene Flugzustande, Ringb. Luftf. I A 11, 1938. 41. Seiferth, R.: Kraftmessungen and Druckverteilungsmessungen an zwei Fli geln mit Klappe and Diisenspalt, Jb. Lufo., 1:36-51, 1941; 1:84-87, 1939. Kiel, G.: Lufo., 14:71-84, 1937. Ruden, P.: Jb. Lufo., 1:75-86, 1937.
42. Spence, D. A.: The Lift Coefficient of a Thin Jet-Flapped Wing, Proc. Roy. Soc. A, 238:46-68, 1956; 261:97-118, 1961. Erickson, J. C., Jr.: J. Aerosp. Sci., 29:1489-1490, 1962. Kerney, K. P.: AIAA J., 9:431-435, 1971. Korbacher, G. K.: AJAA J., 2:64-71, 1964. Malmuth, N. D. and W. D. Murphy: AIAA J., 14:1250-1257, 1976. Maskell, E. C.
and D. A. Spence: Proc. Roy. Soc. A, 251:407-425, 1959. Murphy, W. D. and N. D. Malmuth: AIAA J., 15:46-53,1977. 43. Thomas, F.: Untersuchungen fiber die Erhohung die Auftriebes von Tragfligeln mittels Grenzschichtbeeir_flussung durch Ausblasen, Jb. WGL, 243-244, 1961; Z. Flugw., 10:46-65, 1962. Streit, G. and F. Thomas: Jb. WGLR, 119-132, 1962. 44. Thomas, F.: Einige Untersuchungen an Nasenklappenflugeln kleiner Streckung mit and ohne Rumpf, Z. F7ugw., 11:439-446, 1963. Kriiger, W.: AVA 43/W/64, 1943; ZWB Lufo. UM 3049, 1943. 45. Toll, T. A. and Langley Research Staff: Summary of Lateral-Control Research, NACA Rept. 868, 1947.
520 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES
46. Truckenbrodt, E. and K.-H. - Gronau: Theoretische and experimentelle Untersuchungen an Deltafliigeln mit Klappen bei inkompressibler Stromung, Z. Flugw., 4:236-246, 1956. Hummel,D.: Festschrift E. Truckenbrodt, pp. 174-191, 1977. 47. Tucker, W. A. and R. L. Nelson: Theoretical Characteristics in Supersonic Flow of Two Types of Control Surfaces on Triangular Wings, NACA Rept. 939, 1949. Lagerstrom, P. A. and M. E. Graham: J. Aer. Sci., 16:31-34, 1949. Naylor, D.: J. Aer. Sci., 24:574-578, 610, 1957. Schulz, G.: Z. F7ugw., 5:15-22, 1957. 48. Weick, F. E. and J. A. Shortal: The Effect of Multiple Fixed Slots and a Trailing-Edge Flap on the Lift and Drag of a Clark Y Airfoil, NACA Rept. 427, 1932. 49. Wenzinger, C. J.: Wind-Tunnel Investigation of Ordinary and Split Flaps on Airfoils of Different Profile, NACA Rept. 554, 1936. 50. Wenzinger, C. J. and F. M. Rogallo: Wind-Tunnel Investigation of Spoiler, Deflector, and Slot Lateral-Control Devices on Wings with Full-Span Split and Slotted Flaps, NACA Rept. 706, 1941. 51. Williams, J.: British Research on Boundary-Layer Control for High Lift by Blowing, Z. F7ugw., 6:143-150, 1958. 52. Williams, J., S. F. J. Butler, and M. N. Wood: The Aerodynamics of Jet Flaps, Adv. Aer.
Sci., 4:619456, 1962. Butler, S. F. J. and J. Williams: Aer. Quart., 11:285-308, 1960. Davidson, I. M.: J. Roy. Aer. Soc., 60:25-50, 1956. Dirnmock, N. A.: Aer. Quart., 8:331-345, 1957. Hirsch, R.: Aircr. Eng., 29:366-375, 1957; 30:11-19, 1958. Stratford, B.
S.: Aer. Quart., 7:45-59, 85-105, 169-183, 1956. Williams, J.: Z. Flugw., 6:170-176, 1958. Williams, J. and A. J. Alexander: Aer. Quart., 8:21-30, 1957. 53. Wuest, W.: Messungen an einem Fliigelprofil mit Nasenabsaugung im Vergleich zu einem Profil mit Nasenspalt, AVA 62-03, 1962. 54. Young, A. D.: A Review of Some Stalling Research, ARC RM 2609, 1942/1951. 55. Young, A. D.: The Aerodynamic Characteristics of Flaps, ARC RM 2622, 1947/1953.
BIBLIOGRAPHY
1. Books and handbooks-Contributions to the aerodynamics of the airplane
Abbott, I. H. and A. E. von Doenhoff: "Theory of Wing Sections, Including a Summary of Airfoil Data," Dover, New York, 1959. Alexandrow, W. L.: "Luftschrauben" (transl. from the Russian), Verlag Technik, Berlin, 1954. Ashley, H. and M. T. Landahl: "Aerodynamics of Wings and Bodies," Addison-Wesley, Reading, Mass., 1965. Belotserkovskii, S. M.: "The Theory of Thin Wings in Subsonic Flow" (transl. from the
Russian), Plenum, New York, 1967. Bonney, E. A.: "Engineering Supersonic Aerodynamics," McGraw-Hill, New York, 1950. , M. J. Zucrow, and C. W. Besserer: "Aerodynamics, Propulsion, Structures, and Design Practice (Principles of Guided Missile Design)," Van Nostrand, Princeton, N.J., 1956. Carafoli, E.: "Tragfldgeltheorie, inkompressible Fliissigkeiten (transl. from the Romanian), Verlag Technik, Berlin, 1954. ,
D. Mateescu, and A. Nastase: "Wing Theory in Supersonic Flow," Pergamon, Oxford,
1969.
Clancy, L. J.: "Aerodynamics," Pitman, London, 1975.
Dommasch, D. 0., S. S. Sherby, and T. F. Connolly: "Airplane Aerodynamics," 4th ed., Pitman, New York, 1967.
Donovan, A. F. and H. R. Lawrence (eds.): "Aerodynamic Components of Aircraft at High Speeds," vol. VII of T. von Karman, H. L. Dryden, and H. S. Taylor (eds.), "High Speed Aerodynamics and Jet Propulsion," Princeton University Press, Princeton, N.J., 1957. , H. R. Lawrence, F. Goddard, and R. R. Gilruth (eds.): "High Speed Problems of Aircraft
and Experimental Methods," vol. VIII of T. von Karman, H. L. Dryden, and H. S. Taylor (eds.), "High Speed Aerodynamics and Jet Propulsion," Princeton University Press, Princeton, N.J., 1961. Durand, W. F. (ed.) : "Aerodynamic Theory-A General Review of Progress," Springer, Berlin, 1934-1936, Dover, 1963.
Frankl, F. 1. and E. A. Karpovich: "Gas Dynamics of This Bodies" (trans!. from the Russian), Interscience, London, 1953. 521
522 BIBLIOGRAPHY
Fuchs, R., L. Hopf, and F. Seewald: "Aerodynamik," vol. I. "Mechanik des Flugzeuges," 1934; vol. II. "Theorie der Luftkrafte," 2nd ed., 1935, Springer, Berlin. Glauert, H.: "The Elements of Aerofoil and Airscrew Theory," Cambridge University Press, Cambridge, 1947. "Die Grundlagen der Tragfliigel- and Luftschraubentheorie" (German transl. by H. Holl), Springer, Berlin, 1929. Grammel, R.: "Die hydrodynamischen Grundlagen des Fluges," Vieweg, Braunschweig, 1917. Houghton, E. L. and A. E. Brock: "Aerodynamics for Engineering Students (SI Units)," 2nd ed., Arnold, London, 1970. Houghton, E. L. and R. P. Boswell: "Further Aerodynamics for Engineering Students (Metric and Imperial Units)," Arnold, London, 1969. Krasnov, N. F.: "Aerodynamics of Bodies of Revolution" (transl., 2nd Russian ed.), American Elsevier, New York, 1970. Kiichemann, D.: "The Aerodynamic Design of Aircraft," Pergamon, Oxford, 1978.
Lanchester, F. W.; "Aerial Flight," Constable, London, 1907. "Aerodynamik" (German transl. by C. Runge and A. Runge), Teubner, Leipzig, 1911. Martynov, A. K.: "Practical Aerodynamics" (transl. from the Russian), Pergamon, Oxford, 1965. McCormick, B. W., Jr.: "Aerodynamics of V/STOL Flight," Academic, New York, 1967. Miele, A. (ed.): "Theory of Optimum Aerodynamic Shapes," Academic, New York, 1965. Miene Thomson, L. M.: "Theoretical Aerodynamics," 4th ed., Macmillan, London, 1966. Pope, A.: "Basic Wing and Airfoil Theory," McGraw-Hill, New York, 1951. Proll, A.: "Grundlagen der Aeromechanik and Flugmechanik," Springer, Vienna, 1951. Rauscher, M.: "Introduction to Aeronautical Dynamics," Wiley, New York, 1953. Riegels, F. W.: "Aerodynamische Profile, Windkanal-Messergebnisse, theoretische Unterlagen," Oldenbourg, Munich, 1958. "Aerofoil Sections" (English transl. by D. G. Randall), Butterworths, London, 1961. Robinson, A. and J. A. Laurmann: "Wing Theory" (Cambridge Aeronautics Series II), Cambridge University Press, Cambridge, 1956. Schmidt, H.: "Aerodynamik des Fluges, Eine Einfi. hrung in die mathematische Tragflachentheorie," De Gryter, Berlin, 1929. Schmitz, F. W.: "Aerodynamik des Flugmodells, Tragflugelrnessungen," 4th ed., Lange, Duisburg, 1960.
Sears, W. R. (ed.): "General Theory of High Speed Aerodynamics," vol. VI of T. von Kirmin, H. L. Dryden, and H. S. Taylor (eds.), "High Speed Aerodynamics and Jet Propulsion," Princeton University Press, Princeton, N.J., 1954. Theodorsen, T.: "Theory of Propellers," McGraw-Hill, New York, 1948. Thwaites, B. (ed.): "Incompressible Aerodynamics-An Account of the Theory and Observation of the Steady Flow of Incompressible Fluid Past Aerofoils, Wings, and Other Bodies," Clarendon, Oxford, 1960. von Mises, R.: "Theory of Flight," Dover, New York, 1960. "Fluglehre" (German version by K. Hohenemser), 6th ed., Springer, Berlin, 1957. Weinig, F.: "Aerodynamik der Luftschraube," Springer, Berlin, 1940. Weissinger, J.: "Theorie des Tragfliigels bei stationarer Bewegung in reibungslosen, inkom-
pressiblen Medien," in S. Fliigge and C. Truesdell (eds.), "Handbuch der Physik," vol. VIII/2. "Stromungsmechanik II," pp. 385-437, Springer, Berlin, 1963. Woods, L. C.: "The Theory of Subsonic Plane Flow'" (Cambridge Aeronautics Series III), Cambridge University Press, Cambridge, 1961.
II. Books and handbooks-Aerodynamics of fluid mechanics (selection) Albring, W.: "Angewandte Stromungslehre," 4th ed., Steinkopff, Dresden, 1970. Batchelor, G. K.: "An Introduction to Fluid Dynamics," Cambridge University Press, Cambridge, 1967. Becker, E.: "Gasdynamik," Teubner, Stuttgart, 1965. "Gas Dynamics" (English transl by E. L. Chu), Academic, New York, 1968. Betz, A.: "Konforme Abbildung," 2nd ed., Springer, Berlin, 1964. Chang, P. K.: "Separation of Flow," Pergamon, Oxford, 1970.
,
BIBLIOGRAPHY 523
Chernyi, G. G.: "Introduction to Hypersonic Flow" (transl. from the Russian), Academic, New York, 1961. Cox, R. N. and L. F. Crabtree: "Elements of Hypersonic Aerodynamics," Academic, New York, 1965.
Curie, N. and H. J. Davies: "Modern Fluid Dynamics," vol. I. "Incompressible Flow," 1968; vol. II. "Compressible Flow," 1971, Van Nostrand Reinhold, London. Currie, I. G.: "Fundamental Mechanics of Fluids," McGraw-Hill, New York, 1974. Dorfner, K.-R.: "Dreidimensionale Uberschallprobleme der Gasdynamik," Springer, Berlin, 1957. Dryden, H. L., F. D. Murnaghan, and H. Bateman: "Hydrodynamics," Dover, New York, 1956. Duncan, W. J., A. S. Thom, and A. D. Young: "An Elementary Treatise on the Mechanics of Fluids (SI Units)," 2nd ed., Arnold, London, 1970. Eskinazi, S.: "Vector Mechanics of Fluids and Magnetofluids," Academic, New York, 1967. Ferrari, C. and F. G. Tricomi: "Transonic Aerodynamics" (transl. from the Italian), Academic, New York, 1968. Ferri, A.: "Elements of Aerodynamics of Supersonic Flows," Macmillan, New York, 1949. Flugge, S. and C. Truesdell (eds.): "Handbuch der Physik" ("Encyclopedia of Physics"), vols. VIII/1, VIII/2, IX. "Stromungsmechanik I, 11, III" ("Fluid Dynamics I, II, III,"), Springer, Berlin, 1959, 1960, 1963. Forsching, H. W.: "Grundlagen der Aeroelastik," Springer, Berlin, 1974. Goldstein, S. (ed.): "Modern Developments in Fluid Dynamics-An Account of Theory and Experiment Relating to Boundary Layers, Turbulent Motion and Wakes," vols. I and II, Dover, New York, 1965.
Guderley, K. G.: "Theorie schallnaher Stromungen," Springer, Berlin, 1957. "The Theory of Transonic Flow" (English transl. by J. R. Moszynski), Pergamon, New York, 1962. Hayes, W. D. and R. F. Probstein: "Hypersonic Flow Theory," 2nd ed., vol. I. "Inviscid Flows," 1966; vol. II. "Viscous and Rarefied Flows" (in preparation), Academic, New York. Hilton, W. F.: "High-Speed Aerodynamics," Longmans, Green, London, 1952. Hoerner, S. F.: "Fluid-Dynamic Drag-Practical Information on Aerodynamic Drag and Hydrodynamic Resistance," 3rd ed., Hoerner, Midland Park, N.J., 1965.
and H. V. Borst: "Fluid-Dynamic Lift-Practical Information on Aerodynamic and Hydrodynamic Lift," Hoerner, Brick Town, N.J., 1975. Howarth, L. (ed.): "Modern Developments in Fluid Dynamics-High Speed Flow," vols. I and II, Clarendon, Oxford, 1964. Karamcheti, K.: "Principles of Ideal-Fluid Aerodynamics," Wiley, New York, 1966. Keune, F. and K. Burg: "Singularitatenverfahren der Stromungslehre," Braun, Karlsruhe, 1975. Kuethe, A. M. and C.-Y. Chow: "Foundations of Aerodynamics-Bases of Aerodynamic Design," 3rd ed., Wiley, New York, 1976.
Lachmann, G. V. (ed.): "Boundary Layer and Flow Control-Its Principles and Application," vols. I and II, Pergamon, Oxford, 1961. Liepmann, H. W. and A. Roshko: "Elements of Gas Dynamics," Wiley, New York, 1957.
Loitsyanskii, L. G.: "Mechanics of Liquids and Gases" (transl., 2nd Russian ed.), Pergamon, Oxford, 1966.
Miles, E. R. C.: "Supersonic Aerodynamics-A Theoretical Introduction," Dover, New York, 1950.
Miles, J. W.: "The Potential Theory of Unsteady Supersonic Flow," Cambridge University Press, Cambridge, 1959. Milne-Thomson, L. M.: "Theoretical Hydrodynamics," 5th ed., Macmillan, London, 1968. Oswatitsch, K.: "Grundlagen der Gasdynamik," Springer, Vienna, 1977. "Gas Dynamics"
(English transl., 1st ed., by G. Kuerti), Academic, New York, 1956. (ed.): "Symposium Transsonicum I," Springer, Berlin, 1964; Oswatitsch, K. and D. Rues (eds.): "Symposium Transsonicum II," Springer, Berlin, 1976. Pai, S.-I.: "Introduction to the Theory of Compressible Flow," Van Nostrand, Princeton, N.J., 1959.
Prandtl, L. and Tietjens, 0.: "Hydro- and Aeromechanik," vol. I, 1929; vol. 11, 1944, Springer, Berlin.
524 BIBLIOGRAPHY
"Hydro- and Aeromechanics," vols. I and II (English transl. by L. Rosenhead and J. P. den Hartog), Dover, New York, 1957. , K. Oswatitsch, and K. Wieghazdt (eds.): "Fiihrer durch die Stromungslehre," 7th ed., Vieweg, Braunschweig, 1969. "Essentials of Fluid Dynamics" (English transl., 3rd ed.), Blackie, London, 1952.
Sauer, R.: "Einfiihrung in die theoretische Gasdynamik," 3rd ed., Springer, Berlin, 1960. "Nichtstationare Probleme der Gasdynamik," Springer, Berlin, 1966.
Schlichting, H.: "Grenzschicht-Theorie," 5th ed., Braun, Karlsruhe, 1965. "Boundary-Layer Theory" (English transl. by J. Kestin), 7th ed., McGraw-Hill, New York, 1979. Shapiro, A. H.: "The Dynamics and Thermodynamics of Compressible Fluid Flow," vol. 1, 1953, vol. II, 1954, Ronald, New York. Shepherd, D. G.: "Elements of Fluid Mechanics," Harcourt, Brace, World, New York, 1965. Truckenbrodt, E.: "Stromungsmechanik-Grundlagen and technische Anwendungen," Springer, Berlin, 1968. Truitt, R. W.: "Hypersonic Aerodynamics," Ronald, New York, 1959. van Dyke, M.: "Perturbation Methods in Fluid Mechanics," Academic, New York, 1964.
von Karman, T.: "Aerodynamics-Selected Topics in the Light of Their Historical Development," Cornell University Press, Ithaca, N.Y., 1954. "Aerodynamik-Ausgewahlte Therrien im Lichte der historischen Entwicklung" (German transl. by F. Seewald), Interavia, Genf, 1956. , H. L. Dryden, and H. S. Taylor (eds.): "High Speed Aerodynamics and Jet Propulsion," vols. I-XII, Princeton University Press, Princeton, N.J., 1954-1964. von Mises, R. and K. O. Friedrichs: "Fluid Dynamics," Springer, New York, 1971.
Ward, G. N.: "Linearized Theory of Steady High-Speed Flow, Cambridge University Press, Cambridge, 1955. White, F. M.: "Viscous Fluid Flow," McGraw-Hill, New York, 1974. Wieghardt, K.: "Theoretische Stromungslehre, Eine Einfiihrung," Teubner, Stuttgart, 1965. Zierep, J.: "Theoretische Gasdynamik," 3rd ed., Braun, Karlsruhe, 1976.
III. Collected treatises and general survey papers* Betz, A.: Lehren einer fiinfzigjahrigen Stromungsforschung, Z. Flugw., 5:97-105, 1957. Dryden, H. L.: Gegenwartsprobleme der Luftfahrtfoischung, Z. Flugw., 6:217-233, 1958. FIAT Review of German Science: "Naturforschung and Medizin in Deutschland, 1939-1946," vol. 5, pt. 3, A. Walther (ed.), "Mathematische Grundlagen der Stromungsmechanik," 1947; vol. 11, A. Betz (ed.), "Hydro- and Aerodynamik," 1947. Jones, R. T.: "Collected Works," NASA TM X-3334, National Technical Information Service, Springfield, Va., 1976. Kiichemann, D., P. Carriere, B. Etkin, W. Fiszdon, N. Rott, J. Smolderen, I. Tani, and W. Wrist (eds.): "Progress in Aeronautical Sciences," Pergamon, Oxford, 1961. Prandtl, L.: "Gesammelte Abhandlungen zur angewandten Mechanik, Hydro- and Aerodynamik," vols. I-III, Springer, Berlin, 1961. and A. Betz: "Vier Abhandlungen zur Hydrodynamik and Aerodynamik (Fliissigkeit mit kleiner Reibung; Tiagfldgeltheorie, I. and U. Mitteilung; Schraubenpropeller mit geringstem Energieverlust)," Kaiser Wilhelm Institut, Gottingen, 1927. ,
C. Wieselsberger, and A. Betz: "Ergebnisse der Aerodynamischen Versuchsanstalt zu
Gottingen," vol. I, 4th ed., 1935; vol. II,1923;vol. 111, 1935; vol. IV, 1932, Oldenbourg, Munich.
Taylor, G. I.: "Scientific Papers," vol. I, 1958; vol. II, 1960; vol. III, 1963; vol. IV, 1971, Cambridge University Press, Cambridge. van Dyke, M., W. G. Vincenti, and J. V. Wehausen: "Annual Review of Fluid Mechanics," Annual Reviews, Palo Alto, Calif., 1969-1979.
*Note the special survey papers listed in the individual chapters.
BIBLIOGRAPHY 525
von Karman, T.: "Collected Works," vols. I-IV, 1902-1051, Butterworths, London, 1956; vol. V, 1952-1963, von Karman Institute, Rhode-St. Genese,1975. : Supersonic Aerodynamics-Principles and Applications, J. Aer. Sci., 14:373-409, 1947; "Collected Works," vol. IV, pp. 271-326, Butterworths, London, 1956. Some Significant Developments in Aerodynamics Since 1946, J. Aerosp. Sci., 26:129144, 154, 1959; "Collected Works," vol. V, pp. 235-267, von Karrnan Institute, Rhode-St. Genese, 1975. IV. Yearbooks, irregular periodicals, and journals
AFITAE (AFITA), Association Franraise des Ingenieurs et Techniciens de l'Aeronautique et de I'Espace, Paris: Technique et Science Aeronautiques, 1950-1961; Technique et Science Aeronautiques et Spatiales, 1962-1967; l'Aeronautique et 1'Astronautique, 1968-. AGARD, Advisory Group for Aerospace (Aeronautical) Research and Development, Neuilly-surSeine, Paris: Agardographs, Reports, Conference Proceedings, Lecture Series, 1952-. AIAA (IAS), American Institute of Aeronautics and Astronautics (Institute of the Aeronautical Sciences), New York: Journal of the Aeronautical Sciences, 1934-1958; Journal of the Aerospace Sciences, 1958-1962; Aeronautical Engineering Review, 1942-1958; Aerospace Engineering, 1958-1963; Astronautics and Aerospace Engineering, 1963; Astronautics and Aeronautics, 1964-; AIAA Journal, 1963-; Journal of Aircraft, 1964-; Journal of Spacecraft and Rockets 1964-; Journal of Hydronautics, 1967-. AIDA, Associazione Italiana di Aerotecnica, Rome: L'Aerotecnica, 1920-. ARC, Aeronautical Research Council, London: Reports and Memomoranda, 1909-; Current Papers, 1950-. ARL, Aeronautical Research Laboratory, Melbourne: Technical Reports, Notes, Annual Reports, 1939-. DFVLR (AVA/DVL/DFL), Deutsche Forschungs- and Versuchsanstalt fur Luft- and Raumfahrt, Porz-Wahn, K61n: Berichte 1953-1964 (AVA), 1955-1964 (DVL), 1956-1963 (DFL); DLR-Mitteilungen, 1964-1975; DLR-Forschungsberichte, 1964-; Jahresberichte, 1969-. DGLR (WGLR/WGL), Deutsche Gesellschaft fur Luft- and Raumfahrt, K61n: Jahrbiicher,
1912-1936, 1952-1961 (WGL), 1962-1967 (WGLR), 1968- (DGLR); Zeitschrift fur Flugtechnik and Motorluftschiffahrt, 1910-1933; Zeitschrift fur Flugwissenschaften, 19531976, in cooperation with DFVLR; Zeitschrift fur Flugwissenschaften and Weltraumforschung, 1977-, in cooperation with DFVLR. Dt. Akad. Lufo., Deutsche Akademie der Luftfahrtforschung: Schriften, Mitteilungen, Jahrbncher, 1938-1944. FFA, Flygtekniska Forsoksanstalten (The Aeronautical Research Institute of Sweden), Stockholm: Reports, Memoranda, 1945-. LGL, Lilienthal-Gesellschaft fur Luftfahrtforschung: Luftwissen, 1934-1944 (eds.: Reichsluftfahrtministerium). NASA (NACA), National Aeronautics and Space Administration (National Advisory Committee for Aeronautics), Washington, D.C.: NACA Rept., TN, TM, 1915-1958; NASA CR, SP, TM, TN,TT,1959. NLL, National Luchtvaartlaboratorium, Amsterdam: Reports, Technical Notes, 1921-. NRCA, National Research Council of Canada, Ottawa: Canadian Aeronautical Journal, 1955-1961; Canadian Aeronautics and Space Journal, 1962-. ONERA, Office National d'Etudes et des Recherches Aerospatiales, Chatillon-sous-Bagneux, Paris: La Recherche Aeronautique, 1950-1963; La Recherche Aerospatiale, 1963-; Notes Techniques, 1951-. RAE, Royal Aircraft Establishment, Farnbourough: Reports, Technical Notes, 1909. RAeS, The Royal Aeronautical Society, London: Journal of the Royal Aeronautical Society, 1897-1967; The Aeronautical Journal, 1968-; The Aeronautical Quarterly, 1949/1950-; Data Sheets (ESDU), 1965-.
526 BIBLIOGRAPHY
VKI, von Karman Institute for Fluid Dynamics, Rhode-St. Genese, Brussels: Lecture Series, 1962.
ZWB, Zentrale fur wissenschaftliches Berichtswesen der Luftfahrtforschung, Berlin-Adlershof: Forschungsberichte, Untersuchungen and Mitteilungen, 1933-1945; Jahrbiicher der deutschen Luftfahrtforschung, 1937-1942 (Jb. 1943 as preprint); Ringbuch der Luftfahrttechnik, 1937-; Luftfahrtforschung, 1928-1943.
AUTHOR INDEX
Abbott, I. H., 27, 36, 62, 63, 67, 72, 76, 100, 101, 492, 517, 521
Ackeret, J., 27, 42, 43, 45, 49, 98, 101, 103, 219, 232, 239, 317 Ackermann, U., 132, 210 Ackermann, W., 53,101, 123 Adams, M. C., 293, 310, 313, 317, 318, 321, 331, 363, 367, 369, 375, 388, 425
Adamson, D., 464, 477 Albring, W., 521, 522 Alexander, A. J., 503, 520 Alexandrow, W. L., 521 Alford, W. J., Jr., 449,477 Alksne, A., 253, 322 Allen, H. J., 53, 74, 101, 366, 367, 492, 517
Alway, G. G., 128, 155, 185, 209 Arnic, J. L., 229, 322 Anderson, S. B., 499, 500, 517 Angelucci, S. B., 366, 368 Anliker, M., 317, 323 Arnold, K. 0., 363, 367, 499, 500, 504, 505 517
Ashill, P. R., 178, 206 Ashley, H., 111, 132, 206, 214, 331, 367, 375, 405, 416, 425, 426, 521
Baals, D. D., 425, 426 Babaev, D. A., 317, 318 Bagley, J. A., 132, 206 Bamber, M. J., 398, 399, 400, 401, 426 Barrows, T. M., 132, 211 Bartlett, G. E., 169, 170, 206 Batchelor, G. K., 522 Bateman, H., 523
Bauer, F., 255, 317 Bausch, K., 139, 210, 485, 487, 491, 492, 507, 517, 518 Beane, B., 293, 310, 313, 321 Becker, E., 266, 268, 311, 317, 522 Behrbohm, H., 289, 317, 320 Belotserkovskii, S. M., 521 Bera, R. K., 300, 317, 319, 361, 367 Berndt, S. B., 361, 369 Besserer, G. W., 521 Betz, A., 33, 36, 49, 51, 52, 66, 74, 75, 101, 102, 121, 132, 182, 206, 233, 317, 443, 449, 450, 477, 478, 497, 499, 500, 517, 518, 522, 524
Bhateley, I. C., 170, 209 Bilanin, A. J., 450, 477 Birnbaum, W., 53, 101, 123 Black, J., 169, 170, 206 Bland, S. R., 132, 208 Blasius, H., 38, 104 Blenk, H., 111, 114, 118, 123, 128, 129, 132, 206, 209 Bloom, A. M., 450, 477 Boatright, W. B., 464, 477 Bobbitt, P. J., 475, 477 Bollay, W., 132, 166, 206 Bonner, E., 413, 427 Bonney, E. A., 298, 317, 521 Borja, M., 156, 206 Borst, H. V., 36, 90, 102, 111, 207, 214, 319, 331, 367, 523
Bradley, R. G., 170, 209, 404, 407,428 Brakhage, H., 156, 206 Braun, G., 398, 399, 400, 401, 427, 447, 452, 453, 478 527
528 AUTHOR INDEX Braunss, G., 132, 210 Brebner, G. G., 88, 101, 132, 170, 206, 317, 320
Bridgewater, J., 255, 320, 425, 427 Brock, A. E., 522 Broderick, J. B., 356, 366, 369 Brown, C. E., 169, 206, 293, 310, 313, 317, 318, 321, 331, 361, 367
Browne, S. H., 406,425 Bryer, D. W., 169, 170, 206 Buford, W. E., 366, 368 Bullivant, W. K., 449, 47 8
Burg, K., 52, 102, 197, 208, 253, 290, 296, 310, 311, 318, 319, 323, 332, 368, 523 Burgers, J. M., 36, 104,114, 118, 123, 128, 129,132, 209, 210
Busemann, A., 242,243, 318, 358, 367 Bussmann, K., 171, 172, 186, 187, 207 Butler, S. F. J., 503, 520 Byrd, P. F., 131, 161,210 Byrnes, A. L., 455, 477 Cahill, J. F., 492, 495, 497, 518 Cahn, M. S., 254, 255, 321 Campbell, G. S., 380, 416, 417, 427 Carafoli, E., 317, 318, 521 Carmichael, R. L., 403, 426 Carriere, P., 495, 518 Cebeci, T., 93, 101, 340, 370 Chang, P. K., 522 Chapman, D. R., 362, 367 Chaudhuri, S. N., 164, 206, 208 Chen, A. W., 87, 102 Cheng, H. K., 169, 206, 317, 318 Chernyi, G. G., 523 Chinneck, A., 358, 368 Chow, C.-Y., 523 Clancy, L. J., 521 Clarke, J. H., 413, 327 Clarkson, M. H., 298, 320 Cleary, J. W., 358, 369 Cohen, D., 214, 289, 290, 317, 319, 323 Cole, J. D., 253, 320 Colwell, G. T., 492, 518 Conolly, T. F., 521 Cook, W. L., 499, 500, 517 Cooke, J. C., 88, 101, 170, 206 Cooper, G. E., 499, 500, 517 Cox, R. N., 523 Crabtree, L. F., 87, 101, 523 Cramer, R. H., 406, 409, 425 Crowk, A. E., 492, 506, 518 Crown, J. C., 253, 321
Cunningham, A. M., Jr.,128, 128,156, 206 Curie, N., 523 Currie, I. G., 523
Das, A., 166, 170, 206, 209, 317, 318, 362, 363, 367, 504, 518
Davidson, I. M., 503, 520 Davies, H. J., 523 Davis, T., 464, 477 Deffenbaugh, F. D., 366, 368 Diesinger, W. H., 293, 296, 318 Dimmock, N. A., 503, 520 Doetsch, H., 29, 99, 101, 171, 172, 186, 187, 207, 485, 487, 491, 492, 517, 518
Dommasch, D. 0., 521 Donaldson, C. duP., 450,477 Donovan, A. F., 521 Dorfner, K.-R., 293, 313,315, 318, 523 Drougge, G., 317, 320 Dryden, H. L., 521, 523, 524 Duncan, W. J., 523 Durand, W. F., 521
Edwards, R. H., 169, 206 Egginton, W. J., 486, 492, 518 Eichelbrenner, E., 495, 518 Emerson, H. F., 271, 318 Eminton, E., 361, 367 Emunds, H., 413, 415, 426 Eppler, R., 52, 75, 100, 101 Erickson, J. C., Jr., 503, 519 Eskinazi, S., 523 Etkin, B., 296, 318 Evans, A. J., 362,367 Evans, M. R., 450,477 Evvard, J. C., 293, 296, 318
Fabricius, W., 447, 478 Fage, A., 450,477 Falkner, V. M., 128, 155, 185, 207, 209
Feindt, E. G., 52,102, 153, 158,171, 172, 210 Fell, J., 298, 320 Fenain, M., 317, 318 Ferrari, C., 363, 369, 375, 405, 406, 409, 417, 425, 465,477, 523 Ferri, A., 413, 427, 523 Fiecke, D., 242, 243, 308, 309, 310, 316, 318, 363, 367 Filotas, L. T., 139, 210 Fink, M. R., 366, 367 Fink, P. T., 168, 169, 209
AUTHOR INDEX 529 Fischel, J., 506, 511, 518 Fitzhugh, H. A., 251, 253, 322 Flachsbart, 0., 167, 211 Flax, A. H., 132, 207, 317, 318, 375, 380, 384, 387, 388, 405, 407, 426, 427
Graham, E. W., 317, 319, 515, 520 Graham, M. E., 413, 427, 461, 463, 464, 477 Grammel, R., 522 Granville, P. S., 340, 370
Fliigge, S., 523
Gronau, K.-H., 171, 172, 186, 187, 207, 499, 507, 508, 520 Grosche, F.-R., 425 Gruschwitz, E., 495, 496, 518 Guderley, K. G., 225, 253, 275, 323, 523 Gullstrand, T. R., 253 Gustavsson, A., 425, 427 Gyorgyfalvy, D., 100, 101
Fliigge-Lotz, I., 49, 101, 450, 477, 492, 518 Forsching, H. W., 81, 104, 132, 208, 214, 318, 523
Foster, D. N., 492, 518 Fowell, L. R., 317, 318 Fraenkel, L. E., 361, 369 Frankl, F. I., 521 Frenz, W., 470, 478 Frick, C. W., 214, 270, 318 Friedel, H., 296, 310, 311, 318 Friedman, L., 406, 425 Friedrichs, K. 0., 524 Fuchs, D., 504, 505, 517, 522
Fuchs, R., 74, 101, 114, 118, 123, 128, 129, 132, 153, 186, 209, 210, 211 Fuhrmann, G., 332, 337, 367 Fulker, J. L., 255, 320 Furlong, G. C., 170, 207, 455, 477, 508, 518 Garabedian, P., 255, 317 Garcia, J. R., 254, 255, 321 Garner, H. C., 128, 132, 155, 157, 170, 178, 185, 206, 207, 209 Garrick, I. E., 50, 72, 76, 104, 214, 318 Gault, D. E., 85, 86, 87, 104, 508, 519 Gebelein, H., 50, 72, 101, 104, 114, 118, 123, 128, 129, 132, 209 Geissler, W., 332, 336, 368 Gerber, N., 310, 320 Germain, P., 317, 323 Gersten, K., 111, 132, 166, 207, 210, 275, 319, 330, 348, 369, 375, 391, 401, 425, 453, 454, 474, 477, 503, 518
Giesing, J. P., 178, 206, 404, 425, 477 Gilruth, R. R.; 521 Ginzel, I., 131, 211, 317, 319, 492,518 Gispert, H.-G., 233, 317 Glauert, H., 28, 53, 56, 57, 63, 101, 102, 124, 137, 148, 207, 219, 318, 447, 449, 450, 453, 477, 486, 491, 518, 522
Goddard, F., 521 Goldstein, S., 25, 97, 98, 102, 331, 367, 523 Gonor, A. L., 317, 320 Goradia, S. H., 85, 86, 87, 104, 492, 518 Gothert, B., 219, 222, 229, 265, 318, 322, 354, 367, 506, 518 G6thert, R., 65, 102, 185, 207, 485, 487, 491, 492, 517, 518
Gretler, W., 129, 141, 208, 233, 319
Haack, W., 363, 369 Hackett, J. E., 450, 477 Haefeli, R. C., 464, 478 Hafer, X., 345, 348, 367, 375, 391, 425, 443, 469, 477 Hagermann, J. R., 511, 518 Hallstaff, T. H., 293, 296, 318 Hancock, G. J., 178, 206, 293, 296, 318 Hansen, H., 253, 321 Hansen, M., 129, 141, 181, 208 Hantzsche, W., 233, 317 Harder, K. C., 363, 369 Harmon, S. M., 310, 320 Harper, C. W., 149, 164, 170, 207, 211 Harris, R. V., Jr., 425, 426 Haskell, R. N., 128, 156, 206, 293, 310, 313, 321
Hay, J. A., 486, 492, 518 Hayes, W. D., 233, 258, 323, 523 Head, M. R., 97, 98, 102 Heaslet, M. A.,214, 283, 290, 293, 294, 319, 331, 366, 367, 369, 413, 417, 425, 427, 464, 465, 478 Heimbold, H. B., 53, 77, 93, 102, 103, 148, 207, 447, 478, 504, 518 Hensleigh, W. E., 455,477 Hess, J. L., 36, 102, 132, 207, 331, 332, 367,
403,426 Hewitt, B. L., 155, 207 Heyser, A., 506, 518 Hickey, D. P., 158, 207 Hilton, W. F., 523 Hirsch, R., 503, 520 Hodes, I., 406, 425 Hoerner, S. F., 36, 82, 90, 93, 102, 111, 171, 172, 173, 186, 187, 207, 214, 319, 331, 340, 367, 523
Holder, D. W., 251, 319, 358, 368 Hopf, L., 74, 101, 522
530 AUTHOR INDEX Hosakawa, I., 253, 269, 322, 366, 369 Hosek, J. J., 293, 310, 313, 321 Hough, G. R., 128, 208, 232 Houghton, E. L., 522 House, R. 0., 398, 399, 400, 401, 426 Howarth, L., 331, 368, 523 Hua, H. M., 404, 425 Hubert, J., 442, 478 Hucho, W.-H., 132, 207 Hueber, J., 139, 209 Hummel, D., 25, 102,111, 132, 166, 169, 170, 207, 208, 210, 214, 319, 355, 368, 375, 401, 419, 425, 426, 499, 507, 508, 519 Hunter-Tod, J. H., 464, 478 Hiirlimann, R., 170,207 Imai, I., 233, 323 Jacob, K., 66, 77, 90, 102, 495, 518
Jacobs, E. N., 82, 85, 88, 90, 102, 330, 348, 369, 375, 389, 393, 394, 396, 398, 400, 426 Jacobs, H., 504, 505, 517 Jacobs, W., 164, 165, 171, 172, 186, 187, 207, 208, 380, 398, 399, 400, 401, 416, 417, 426, 427, 473, 474, 475, 477, 503, 504, 518
Jacquignon, H., 503, 519 Jaeckel, K., 53, 80, 101, 102, 123, 139, 153, 186, 210, 211 James, R. M., 36, 102, 361, 367 Jameson, A., 255, 317 Jaquet, B. M., 166, 207 Jaszlics, I., 169, 170, 206 Jeffreys, L, 310, 320
Jen, H., 450,477 Johnson, W. S., Jr., 293, 310, 313, 321 Jones, A. L., 492, 506, 518 Jones, 1. G., 317,_320,4063 427 Jones, M., 97, 98, 102 Jones, R. T., 164, 165, 166, 173, 208, 214,
289,290,300,317,319,323,413,415, 416, 426, 524
Jones, W. P., 128, 208 Jordan, P. F., 128, 142, 143, 144, 156, 181, 208, 209, 210, 449, 450, 477 Joukowsky, N., 33, 49, 51,102 Jungclaus, G., 28, 53, 61, 66, 70, 71, 75, 90, 102, 103, 104, 486, 492, 518
Kaatari, G. E., 405, 407, 427, 433, 478 Kacprzynski, J. J., 254, 255, 321
Kaden, H., 449, 450, 477 Kahane, A., 242, 243, 318, 361, 369 Kainer, 1. H., 293, 310, 313, 317, 321, 323 Kalman, T. P., 178, 206, 404, 477 Kandil, 0. A., 132, 208 Kane, E. J., 425, 426 Kao, H. C., 85, 86, 87, 104 Kaplan, C., 233, 317 Karamcheti, K., 523 Karpovich, E. A., 521 Katzoff, S., 449, 478 Kaufmann, W., 53, 101, 123, 176, 208, 317, 320, 449, 450, 477 Kawasaki, T., 317, 320 Kelly, H. R., 366, 368 Kerney, K. P., 132, 210, 503, 518 Kestin, J., 524 Kettle, D. J., 380, 382, 388, 390, 399, 427 Keune, F., 49, 50, 52, 53, 77, 101, 102, 197, 208, 275, 290, 317, 319, 321, 332, 337, 351, 366, 367, 368, 413, 415, 426, 427, 486, 492, 518, 523 Kida, T., 132, 211, 504, 519 Kiel, G., 489, 519 Kinner, W., 181, 208 Kirby, D. A., 380, 382, 388, 390, 399,427 Kirkby, S., 406, 409, 425 Kirkpatrick, D. L. I., 168, 169, 209 Klunker, E. B., 253, 317, 318, 321, 416,426 Knepper, D. P., 251, 253, 322 Knoche, H.-G., 506, 518 Kochanowsky, W., 50, 72, 76, 104 Kohler, M., 27, 42, 43, 45, 49,103 Kohlman, D. L., 169, 170,187, 206, 208 Koloska, P., 443, 469, 477 Kolscher, M., 507, 517 Kopfermann, K., 171, 172, 186, 187, 207 Korbacher, G. K., 503, 504, 519 Korn, D., 255, 317 Korner, H., 404, 426 Koster, H., 358, 3-63,-367 Kowalke, F., 242, 243, 308, 309, 310, 316, 266, 268, 311, 317, 318, 363, 369, 512, 519 Kraemer, K., 82, 102, 133, 153, 158, 171, 172, 210
Krahn, E., 232, 233, 317 Kramer, M., 29, 99, 101, 102 Krasnov, N. F., 331, 368, 522 Kraus, W., 128, 156, 208, 263, 320, 403, 404, 426 Krause, F., 266, 268, 311, 317, 353, 368 Krauss, E. S., 354, 368 Kreuter, W., 142, 143, 144, 209
AUTHOR INDEX 531 Krienes, K., 129, 141, 208 Kriesis, P., 132, 166, 206 Kriiger, W., 498, 519 Krupp, J. A., 253, 320, 366, 368 Krux, P., 317, 318 Kdchemann, D.,111, 132, 164, 166, 206, 208, 214, 317, 320, 322, 332, 368, 393, 425, 426, 450, 477, 524 Kuethe, A. M., 523 Kulakowski, L. J., 128, 156, 206 Kunen, A. E., 170, 209 Kuo, Y. H., 214, 320 Kupper, A., 139, 210, 485, 487, 491, 492, 517, 518 Kiissner, H. G., 81, 104, 132, 208 Kutta, W., 33,102
Labrujere, T. E., 155, 207, 403, 426 Lachmann, G. V., 36, 88, 95, 101, 102, 494, 497, 503, 519, 523
Lagerstrom, P. A., 461, 463, 464, 477, 515, 520 Laidlaw, W. R., 166, 208 Laitone, E. V., 233, 323, 358, 369, 406, 409, 425 Lamar, J. E., 317, 320 Lamar, J. R., 128, 155, 185, 209 Lamb, O. P., 492, 506, 518 Lambourne, N. C., 169,170, 206 Lamla, E., 233, 317 Lampert, S., 304, 314, 320 Lan, C. E., 77, 102, 128, 208, 232 Lance, G. N., 317, 323 Lanchester, F. W., 522 Landahl, M. T., 111, 132, 206, 208, 214, 253, 317, 322, 331, 367, 416, 426, 521 Lange, G., 171, 172, 186, 187, 207 Laschka, B., 132,146,147, 151, 153, 208, 210, 463, 464, 465, 476, 477, 478 Laurmann, J. A., 36, 103, 132, 209, 214, 322, 522 Lawford, J. A., 166, 207
Lawrence, H. R., 166, 208, 375, 380, 384, 387, 388, 405, 426, 521 Lawrence, T., 316, 320 Leelavathi, K., 366, 367 Lees, L., 242, 243, 318 Legendre, R., 168, 169, 209 Lehrian, D. E., 128,132, 170, 207 Leiter, E., 289, 293, 296, 318, 320 Lennertz, J., 379, 411, 419,426 Leslie, D. C. M., 298, 320 Lessing, F., 332, 336, 368
Levinsky, E. S., 503, 519 Licher, R. M., 413, 427 Liebe, H., 375, 391, 425 Liebe, W., 166, 206 Liebeck, R. H., 87, 102 Liepmann, H. W., 523 Liese, J., 390, 395, 428 Liess, W., 380, 382, 388, 390, 399, 427 Lighthill, M. J., 77, 103, 244, 317, 320, 356, 368
Lilienthal, 0., 15, 22 Lincke, W., 132, 210 Lindsey, W. F., 234, 246, 247, 322 Linnel, R. D., 256, 320 Lipowski, K., 310, 311, 322 Lissaman, P. B. S., 504, 519 Littell, R. E., 234, 246, 247, 322 Lock, R. C., 255, 320, 406, 425, 427 Loeve, W., 403, 426' Loftin, L. K., Jr., 82, 85, 88, 90, 102 Lohr, R., 503, 504, 518, 519 Loitsyanskii, L. G., 523 Lomax, H., 214, 25 3, 283, 290, 293, 294, 319, 321, 331, 367, 413, 417, 425, 427, 464, 465, 478 Lord, W. T., 317, 320 Losch, F., 492, 518 Lotz,.I., 139, 209, 332, 368, 447,478 Love, E. S., 304, 314 Luckert, H. J., 380, 382, 388, 390, 399, 427 Ludwieg, H., 170, 209, 275, 320 Lusty, A. H., Jr., 317, 320 Lyman, V., 85, 86, 87, 104 Maccoll, J. W., 214, 322, 358, 369 McCormick, B. W., Jr., 522 McCullough, G. B., 85, 86, 87, 104 McDonald, J. W., 132,206 McHugh, G. C., 170, 207 McHugh, J. G., 455, 477, 508, 518 Mackrodt, P. A., 310, 311, 322 McLean, F. E., 317 Maddox, S. A., 170, 209 Magnus, R., 253, 320 Maki, R. L., 170, 207 Malavard, L., 225, 253, 275, 323 Malrnuth, N. D., 503, 504, 518,519 Malvestuto, F. S., 293, 310, 313, 321 Mangler, K. W., 50, 72, 75, 100, 101, 104, 128,
132, 155,166,169, 185, 206, 209, 269, 317, 318, 320, 442, 478 Margolis, K., 293, 310, 313, 321 Marshall, F. J., 366, 368
532 AUTHOR INDEX Martensen, E., 66, 77, 102 Martin, J. C., 310, 320 Martynov, A. K., 522 Maruhn, K., 337, 338, 341, 347, 368, 398, 399,
400,401,426 Mascheck, H.-J., 503, 504, 518 Maskell, E. C., 317, 320, 503, 518 Maskew, B., 77, 103, 202 Mateescu, D., 521 Mattioli, G. D., 114, 118, 123, 128, 129, 132, 209
Maurer, F., 506, 518 Mello, J. F., 366, 368 Michael, W. H., Jr., 169, 206, 475, 478 Middleton, W. D., 425, 426 Miele, A., 317, 320, 5 22 Miles, E. R. C., 523 Miles, J. W., 363, 368, 416, 417, 428, 523 Miller, B. D., 404, 407, 428 Milne-Thomson, L. M., 522,523 Mirels, H., 166, 209, 289, 320, 464, 478 Miyai, Y., 132, 211, 504, 519
Naylor, D., 515, 520 Nelson, R. L., 515, 520 Neumark, S., 197, 203, 205, 209, 275, 321 Newman, P. A., 253, 321, 416, 426 Nickel, K., 53, 66, 74, 101, 123, 175, 209, 497, 518 Nicolai, L. M., 405, 407, 427 Nielsen, J. N., 405, 407, 427, 433, 478 Niemz, W., 124, 125, 127, 128, 129, 153, 154, 155, 160, 161, 210
Nieuwland, G. Y., 254, 255, 321 Nixon, D., 253, 321 Nonweiler, T., 84, 85, 103, 499, 519 Nostrud, H., 233, 253, 321, 323
O'Hare, W. M., 511, 518 Orrnsbee, A. I., 87, 102 Osbome, J., 251, 253, 322 Oswatitsch, K., 225, 253, 275, 321, 323, 351, 366, 368, 413, 415, 426, 427, 523, 524 Otto, H., 185, 207, 425, 427
Moller, E., 171, 172, 186, 187, 207, 330, 348, 369, 375, 389, 391, 393, 394, 396, 398,
400,425,426 Mook, D. T., 170, 209 Moore, F. K., 356, 366, 369 Moore, K. C., 317, 320
Pai, S.-I., 523 Panico, V. D., 361, 367 Pappas, C. E., 170, 209 Parker, A. G., 170, 209
Moore, N. B., 357, 361, 363, 369 Moore, T. W. F., 87, 101
Pearcey, H. H., 132, 206, 251, 255, 275, 321 Pechau, W., 84, 99, 103, 500, 519
Moran, J. P., 332, 357, 361, 363, 369 Morikawa, G. K., 407, 427, 458, 478 Morris, D. N., 362, 368 Mosinskis, G. J., 93,101, 340, 370 Muter, W., 332, 337, 367 Multhopp, H., 78,128, 142,143, 144, 155, 182, 185, 209, 317, 319, 341, 344, 368, 380, 382, 388, 390, 399, 427, 447, 452, 453, 478 Munk, M. M., 58, 103,175,177, 209, 317, 318, 331, 341, 368, 416, 427 Murman, E. M., 253, 320, 366, 368 Murnaghan, F. D., 523 Murphy, W. D., 503, 504, 518, 519 Muttray, H., 33,49,51, 102, 375, 398, 427,
Peckham, D. H., 169, 170, 206 Perkins, E. W., 362, 366, 367 Perring, W. G. A., 486, 491, 518 Petersohn, E., 450, 477 Petrikat, K., 497, 519 Pfenninger, W., 98, 101 Piercy, N. A. V., 49, 101 Pike, J., 317, 320 Pinkerton, R. M., 82, 83, 85, 88, 89, 90, 102,
450,477
103 Piper, E. R. W., 49, 101 Pistolesi, E., 78, 103 Pitts, W. C., 405, 407, 427, 433, 478 Pleines, W., 499, 518 Pohlhamus, E. C., 170, 209, 275, 317, 321 Poisson-Quinton, P., 95, 103, 166, 209, 495,
503,518,519 Naeseth, R. L., 511, 578 Nagaraja, K. S., 164, 206, 208 Nash, J. F., 77, 104 Nastase, A., 317, 318, 521 Naumann, A., 42, 103, 506, 518 Nayfeh, A. H., 132, 208
Pope, A., 522 Powell, B. J., 255, 320 Prandtl, L., 27, 33, 42, 43, 45, 51, 102, 103, 114, 118, 123, 128, 129, 132, 209, 214, 219, 293, 443, 478, 523, 524 Preston, J. H., 49, 90, 103 Pretsch, J., 93, 103
AUTHOR INDEX 533 Pritchard, R. E., 317, 320 Probstein, R. F., 523 Proll, A., 522 Puckett, A. E., 293, 310, 313, 321 Puffert, H. J., 398, 399, 400, 401, 427, 475, 478
Queijo, M. J., 166, 206
Rakich, J. V., 358, 369 Ramaswamy, M. A., 363, 369 Randall, D. G., 269, 320 Ras, M., 98, 101 Raspet, A., 100, 101 Rauscher, M., 522 Redeker, G., 169, 170, 207, 251, 323 Regenscheit, L. B., 96, 103, 500, 519 Reissner, E., 130,145, 149, 211 Reller, E., 504, 505, 517 Rennemann, C., Jr., 363, 369 Revell, J. D., 355, 368 Ribner, H. S., 293, 310, 313, 321 Richter, G., 229, 322 '
Richter, W., 447, 452, 453, 478, 507,517 Riedel, H., 413, 415, 426 Riegels, F. W., 27, 28, 36, 53, 60, 61, 66, 70, 74, 75, 76, 77, 90, 92, 93, 102, 103, 104, 173, 210, 332, 336, 368, 380, 382, 388, 390, 399, 427, 492, 518, 519, 522 Ringleb, F., 50, 53, 77, 102 Roberts, R. C., 293, 310, 313, 321 Robins, A. W., 425,426 Robinson, A., 36,103, 132, 166, 209, 214, 285, 300, 322, 406, 409, 425, 464, 478, 522
Rodden, W. P., 178, 206, 375, 404, 405, 425, 477 Roe, P. L., 317, 320 Rogallo, F. M., 506, 520 Rogers, E. W. E., 255, 320, 425,427 Rogmann, H., 124, 125, 127, 128, 129, 153, 154, 155, 160, 161,210 Rohlfs, S., 253, 321, 416, 426 Rohne, E., 449, 450, 477 Roshko, A., 523 Rossner, G., 50, 53, 77, 102, 114, 118, 123, 128,129, 132, 209
Rossow, V. J., 450, 478, 486, 492, 518 Rothmann, H., 355, 368 Rott, N., 317, 319 Rotta, J., 253, 322, 469, 478 Roy, M., 87, 101, 163, 169, 209
Rubbert, P. E., 253, 322 Ruden, P., 489, 519 Rues, D., 253, 321 Sacher, P., 128, 156, 208, 263, 320, 403, 404, 426 Sacks, A. H., 450, 458, 478 Sanchez, F., 405, 407, 427 Sann, B., 42, 103 Sato, J., 253, 321 Sauer, R., 524 Schappelle, R. H., 503,519 Scharn, H., 398, 399, 400, 401, 426, 447, 452,
453,478 Schindel, L. H., 366, 368 Schlichting, H., 25, 81, 84, 93, 96, 99, 103, 111, 170, 182, 192, 209, 210, 214, 293, 302, 322, 375, 384, 388, 393, 395, 427,
433,469,470,478,500,504, 517, 519, 524
Schlottmann, F., 194, 210 Schmidt, H., 114, 118, 123, 128, 129,132, 139, 209, 219, 522 Schmidt, W., 363, 369, 413, 415, 426 Schmitz, F. W., 83, 103, 522 Schneider, W., 261, 322, 403, 404, 413, 416, 427 Scholz, N., 93, 103, 131, 161, 210, 275, 321, 340, 370 Schrenk, 0., 33, 49, 51, 84, 99, 101, 102, 103, 403, 404, 413, 416, 427, 442, 478, 489, 495,496,499, 500, 504, 517, 518, 519 Schroeder, H.-H., 317, 318, 363, 367
Schubert, H., 139, 210 Schultze, E., 158, 207 Schulz, G., 445, 478, 515, 520 Schwarz, F., 500, 519 Sears, W. R., 59, 97, 103, 111, 133, 175, 210, 214, 317, 320, 322, 363, 369, 375, 388, 425, 522
Sedney, R., 317, 319 Seewald, F., 74, 101, 522 Seibold, W., 506,518 Seiferth, R., 27, 42, 43, 45, 49, 103, 489, 519 Sells, C. C. L., 233, 319 Shapiro, A. H., 524 Shepherd, D. G., 524 Sherby, S. S., 521 Sherman, A., 82, 85, 88, 90, 102, 330, 342, 369,375,389, 393, 394, 396, 398, 400, 426 Shortal, J. A., 498, 520 Silverstein, A., 449, 478
534 AUTHOR INDEX Simmons, L. F. G., 450, 477 Sinnott, C. S., 251, 253, 322 Sivells, J. C., 508, 518
Slooff, J. W., 403, 426 Sluder, L., 464, 465, 478 Srnetana, F. 0., 251, 253, 322 Smith, A. M. 0., 36, 84, 93, 101, 102, 132, 207, 331, 332, 340, 367, 370, 403, 426 Smith, C. W., 170, 209
Smith, H. A., 82, 85, 88, 90,102 Smith, J. H. B., 169, 209, 210, 317, 318, 319 Snedeker, R. S., 450, 477 Sohngen, H., 139, 210, 492, 518 Solarski, A., 361, 369 Spee, B. M., 254, 255, 321 Speidel, L., 90, 100, 104 Spence, B. F. R., 128, 155, 185, 209, 504, 518 Spence, D. A., 90, 103, 503, 519 Spoonner, S. H., 508, 518 Spreiter, J. R., 214, 225, 253, 269, 275, 283, 290, 293, 294, 319, 322, 323, 366, 369,
380,416,417,427,450,478 Squire, H. B., 85, 86, 87,93,94, 103, 104, 170,207 Squire, L. C., 317, 322 Srinivasan, P. S., 169, 170, 207 Stack, J., 229, 234, 246, 247, 322
Taylor, G. L, 214, 322, 358, 369, 524 Taylor, H. S., 521
Theodorsen, T., 50, 72,104, 522 Thom, A. S., 523 Thomas, F., 132, 210, 251, 323, 501, 519 Thwaites, B., 36, 89, 104,132, 210, 331, 369, 522 Tietjens, 0., 523 Ting, L., 413; 427 Toll, T. A., 492, 511, 519 Tolve, L. A., 455, 477 Traugott, S. C., 358, 369 Trefftz, E., 49, 101, 114,118, 123, 128, 129, 132, 209 Tricomi, F. G., 5 23 Trienes, H., 330, 348, 369, 375, 391, 425, 449, 452, 453, 479
Trilling, L., 169, 170,206 Truckenbrodt, E., 28, 53, 61, 66, 70, 71, 75, 76,93, 103, 104, 124,125,127, 128, 129, 146, 147, 151,154, 154, 155 ,15 8, 160, 161, 171, 172,173, 210, 219, 222, 265, 269, 319, 322, 330, 348, 351, 369, 375, 391, 425, 447, 449, 452, 453, 473, 474,475, 477, 478, 479, 499, 507, 508,
Stahara, S. S., 253, 322, 366, 369, 416, 427 Stahl, W., 310, 311, 322, 425, 427 Stanbrook, A., 317, 322
520,524 Truitt, R. W., 524
Stanewsky, E., 253, 321 Stark, V. J. E., 132, 208, 214, 320
Tsien, H. S., 232, 233, 258, 317, 323, 358, 369 Tucker, W. A., 515, 520 Tuckermann, L. B., 337, 338, 341, 347, 368
Staufer, F., 469, 470, 478, 485, 487, 491, 492, 517, 518
Steger, J. L., 253, 321 Stender, W., 90, 100, 104 Stetter, H. J., 357, 361, 363, 369 Stevens, J. R., 132, 206 Stewart, H. J., 293, 310, 313, 317, 321, 322 Stiess, W., 485, 487, 491, 492, 517,518 Stivers, L. S., Jr., 27, 36, 62, 63, 67, 72, 76, 100, 101, 254, 255, 321 Stocker, P. M., 416, 417, 428 Strand, T., 75, 104, 317, 323 Strassl, H., 497, 519 Stratford, B. S., 503, 520 Streit, G., 501, 519 Subramanian, N. R., 366, 367 Sullivan, R. D., 450,477 Sun, E. Y. C., 315, 322 Szabo, I., 129, 141, 208
Tani, I., 87, 104 Tanner, M., 77, 104, 362, 369
Tsakonas, S., 132, 208, 214, 320
Ulrich, A., 517, 519 Ursell, F., 317, 318
van der Decken, J., 132, 210 Vandrey, J. F., 332, 336, 341, 344, 368, 390, 395, 413, 427, 428 van Dyke, M. D., 132, 210, 233, 323, 356, 358, 366, 369, 524 Vanino, R., 416, 425, 426, 427 Vidal, R.J., 169, 170, 206 Vincenti, W. G., 317, 323, 524 Viswanathan, S., 363, 369 Voellmy, H. R., 366, 369 Voepel, H., 504, 505, 517 von Baranoff, A., 458,478 von Doenhoff, A. E., 27, 36, 62, 63, 67, 72, 76, 100, 101, 229, 322, 492, 517, 521 von Karmar_. T., 36, 49,104,114, 129, 132, 210, 214, 225, 232, 233, 253, 275, 311,
AUTHOR INDEX 535 von Kirman, T. (Cont.), 323, 332, 351, 357, 361, 363, 369, 521, 524, 525 von Mises, R., 38, 104, 522, 524 Wacke, 171, 172, 186, 187, 207 Wagner, H., 81, 104 Wagner, S., 128, 156, 210 Walchner, 0., 242, 243, 318 Walz, A., 49, 50, 72, 101, 104, 486, 492, 518 Wanner, A., 504, 505, 517 Ward, G. N., 317, 318, 323, 361, 369, 416, 417, 428, 464, 479, 524
Ward, K. E., 330, 348, 367, 375, 389, 393, 394, 396, 398, 400, 426 Watson, E. J., 128, 155, 185, 209 Watson, J. M., 506, 518 Weber, J., 164, 166, 206, 207, 208, 211, 317, 320, 323, 380, 382, 388, 390, 393, 399, 426, 427
Wedemeyer, E., 266, 268, 311, 317 Wegener, F., 146, 147, 151, 153, 210, 242, 243, 308, 309, 310, 316, 318, 363, 367 Wehausen, J. V., 524 Weick, F. E., 498, 520 Weinberger, W., 486, 492, 518 Weinel, E., 332, 368 Weinig, F., 139, 153, 186, 211, 497, 519, 522 Weissinger, J., 36, 79,104, 111, 130-,132, 141, 142, 143, 144,149, 153, 186, 209, 211,
Widnall, S. E., 132, 211, 214, 317 Wieghardt, K., 131, 211,524 Wieland, E., 132, 207
Wieselsberger, C., 27, 33,42,43,45,49,51, 102,103, 121, 122, 211, 331, 369, 375, 428, 524 Wilby, P. G., 255, 320 Williams, G. M., 450, 479 Williams, J., 500, 503, 520 Winter, H., 167, 211 Wittich, H., 28, 50, 53, 61, 66, 70, 71, 72, 75,
76,103,104 Wolhart, W. D., 166, 207 Wood, C. J., 251, 323 Wood, M. N., 503, 520 Woods, L. C., 36,104, 522 Woodward, F. A., 77, 103, 202, 296, 318, 404,
407,428 Wortmann, F. X., 90,93,100,104 Wuest, W., 499, 500, 518, 520 Wurzbach, R., 449, 450, 477 Yang, H. T., 416, 417, 428 Yoshihara, H., 253, 317, 321, 323 Young, A. D., 85, 86, 87, 93, 94, 103, 104, 149, 164, 170, 207, 211, 340, 362, 370, 495,49-7, 498, 508, 520, 523 Young, J. de, 149, 164, 211, 508, 518
522
Wellmann, J., 317, 318, 363, 367 Wendt, H., 233, 317 Wentz, W. H., Jr., 169, 170, 206 Wenzinger, C. J., 492, 496, 506, 520 Werle, H., 169, 211 Whitcomb, R. T., 414, 415, 428 White, F. M., 5 24
Zahm, A. F., 337, 338, 341, 347, 368 Zienkiewicz, H. K., 293, 310, 313, 321 Zierep, J., 253, 323, 524 Ziller, F., 114, 118, 123, 128, 129, 132, 209 Zimmer, H., 253, 321 Zucrow, M. J., 521
SUBJECT INDEX
Acceleration potential, 129 Aerodynamic center (center of pressure), 17 Aileron:
geometry of, 431, 484 rolling moment of, 510, 515 Airfoil theory, 123, 131, 153, 263, 269, 280, 288, 290, 453 nonlinear, 166 [See also Wing (airfoil) ]
Angle of attack (incidence), 13, 16 of fuselage, 376, 382, 384, 387 of horizontal stabilizer, 437, 440, 443 of smooth leading-edge flow, 60 of wing, 56, 117, 376, 389, 396,412 Angle of incident flow, 78 Area rule, 414 Atmosphere, 5, 8
Balance tab, 482, 491 Blowing [see Ejection (blowing)] Boundary-layer control, 81, 95 Boundary-layer fence, 166, 455, 494 Brake flap (air brake), 483, 504 Buffeting, 251 Bursting of vortex, 169
Cambered flap, 483, 487, 495 Center of pressure [see Aerodynamic center (center of pressure) ] Characteristics, method of, 244, 358, 360 Circular wing, 181 Circular-arc profile, 46 Circulation, 33
Circulation distribution: over profile, 54, 56 over wing, 114, 117, 123, 126, 129, 136, 140, 298, 379
Circulation (lift) distribution: constant (rectangular), 447, 448, 460 elliptic, 118, 263, 444, 447, 449, 453 parabolic, 447 Closure condition, 70, 198, 333 Coefficients, aerodynamic: definition of, 14, 330, 436, 485 effect of friction (viscosity) on, 81, 170, 347 Compression shock (bow wave), 245, 246, 250, 259 Conical flow, 280, 461 Control surface, balance, 482, 491 [See also Flap (control surface)] Coordinate systems, 13, 105, 327
Delta wing, 106, 108 drag of, 152, 178, 268, 302, 305, 308, 313, 316
lift of, 152, 157, 168, 171, 266, 269, 301, 305, 308
lift distribution of, 151, 158, 266, 304, 419 neutral point of, 152, 158, 266, 269, 301, 307, 308, 393 pressure distribution of, 160, 285, 287, 304, 417 suction force on, 300 Density, 3 Dipol distribution, 123, 342, 365, 390 Direct problem, 1, 118, 128 537
538 SUBJECT INDEX Double-section flap, 498 Double-section wing, 483, 487, 498 Drag, 12, 14 (See also Induced drag; Profile drag (friction
Fuselage (Cont.):
pitching moment of, 330, 340, 345, 346, 348 (See also Ellipsoid; Paraboloid)
drag); Wave drag]
Ejection (blowing), 95, 98, 500 Elementary wing, 124, 126, 130, 174, 379 Elevator, 432, 484, 508, 516 Ellipsoid, 201, 329, 334, 337, 343, 345, 347, 353, 374, 392, 397, 401
Elliptic wing, 109, 119, 120 downwash and upwash of, 383, 384, 444, 453
drag of, 119, 178 lift of, 118, 121, 146, 264 lift distribution of, 141 perturbation velocity of, 202 End plate, 442 Energy law, 175
Fin (see Stabilizer)
Flap (control surface), 63, 109, 481, 491 angle of attack, change by, 64, 96, 486, 492, 493, 508, 512 control-surface moment of, 484, 486, 490,
493,517 geometry of, 481, 483 lift of, 484 loading of, 489, 494 moment change by, 65, 486, 493, 512 neutral point of, 486, 488, 493, 516 pressure distribution on, 489, 496, 513 rolling moment of, 509 Flap, double-section, 498 Flap wing, 96, 482 [See also Lift (lift slope), of flap-wing system] Flap with trailing edge blowing, 98
Fowler flap, 483, 498 Fuselage: in curved flow, 346, 376
drag of, 330, 354, 358, 360, 362 geometry of, 327, 363 lift of, 330, 348, 365, 380 lift distribution of, 344, 380 neutral point of, 348 perturbation velocity on, 335 (See also Induced velocity) pressure distribution on, 332, 334, 343, 347, 352, 353, 354, 358
Glide angle, 12 Gottingen profile system, 27
Ground effect, 132, 371, 504 High-wing airplane, 373, 375, 378, 395, 396, 400, 470, 472, 474 Hinge moment (see Flap, control-surface moment of) Horizontal tail, 432, 433 and vertical tail, control-surface balance of, 482, 491 dynamic pressure ratio of, 437 efficiency (downwash) factor of, 438, 444, 451, 457, 462 geometry of, 434 lift of, 436, 438, 441, 443, 456, 459 neutral-point shift caused by, 439, 454 pitch damping of, 441 pitching moment of, 436, 437, 440 Horn balance, 482
Indirect (design) problem, 1, 118, 128 Induced angle of attack, 115, 117, 119, 138, 139, 142 Induced drag, 114, 119, 152, 173, 175, 176, 264, 301 Induced velocity (source, dipole), 80, 199, 293, 333, 357 Induced velocity (vortex): downwash, 57, 80, 115, 119, 291, 444, 453, 456 sidewash, 472 Influence zone (line), 277, 283, 292, 295, 356, 458 Interference: of fuselage-horizontal tail system, 442 of vertical-horizontal tail system, 475 of wing-fuselage system, 371, 376, 405,413 of wing-fuselage-vertical tail system, 467, 470 of wing-horizontal tail system, 436, 443,
456,458 Jet flap, 503 Joukowsky profile, 45, 46, 48, 72, 246
SUBJECT INDEX 539
Kinematic flow condition, 54, 70, 126, 198, 235, 292, 379 Kutta (flow-off) condition, 33, 40, 66, 128, 279
Kutta-Joukowsky lift theorem, 30, 134 Laminar flow, maintenance of, 96, 97, 99 Laminar profile, 99 Landing device, 482, 494 Landing flap, 482, 508 Lateral motion, 15, 181, 186,432 Lift (lift slope), 12, 14, 16, 110, 135 of flap-wing system, 485, 486, 492, 494 of fuselage, 330, 348, 365, 380, 393, 402 of smooth leading-edge flow, 60, 230 of stabilizer: horizontal, 436, 438, 441, 456, 459 vertical, 469 of wing: compressible, 224, 229, 230, 237, 249, 264, 269 incompressible, 30, 41, 49, 55, 58, 60, 81, 84, 114, 132, 136, 156, 166, 170
of wing-fuselage system, 374, 379, 382,419 Lift distribution (circulation distribution): of fuselage, 330, 344, 380, 387, 407, 409 of wing, 110, 135, 263, 269, 388, 412, 419, 506 Lifting-line theory: simple, 131, 137, 151, 446, 451, 506 extended, 129, 131, 145, 151, 506 Lifting-surface theory, 153, 507 (See also Airfoil theory) Longitudinal motion, 15, 181, 432 Low-wing airplane, 373, 378, 394, 396, 400, 471, 474
Mach cone, 22, 276 Mach number, 9 drag-critical, 227, 232, 244, 271, 274, 353 Maximum lift, 84, 96, 170, 393, 494, 497 Method of characteristics, 52, 244, 358, 360 Mid-wing airplane, 373, 374, 378, 394, 395, 396, 400, 472 Momentum law, 132, 175, 341 Multhopp's quadrature method, 141 Multiple-points method, 131 Munk displacement theorem, 175
NACA profiles, 27, 62, 67, 72, 76, 82, 228, 230, 233, 271
Neutral point: of fuselage, 348 of horizontal tail, 439, 454 of wing: geometric, 108 aerodynamic (general), 18 compressible, 230, 237, 264, 269 incompressible, 42, 59, 60, 157 of wing-fuselage system, 390, 421 Nonlinear lift effects, 166, 330, 366, 4.25 Normal force, 14 Nose balance, 482 Nose flap, 483, 498 Panel method, 403 Parabolic profile (biconvex), 28, 47, 58, 62, 66, 71, 200, 204, 239, 242, 246, 247, 253, 313 Paraboloid, 329, 336, 353, 358, 360, 362 Perturbation velocity, 72, 200, 336 Pistolesi's theorem, 79, 80 Pitch: damping, 19, 183, 441 lift due to, 183
motion, 16, 182,441 Pitching moment: of flap-wing system, 484, 48S of fuselage, 345, 348 of horizontal tail, 436 of wing, 14, 18 compressible, 230, 264 incompressible, 55, 58, 156 of wing-fuselage system, 374, 3 82 Plate, flat: in chord-parallel flow, 90, 97, 216 inclined (with angle of attack): compressible, 229, 238, 239, 257, 286, 461 incompressible, 38, 57, 78 Polar curve (drag), 15, 120, 121, 181, 275, 394 Prandtl wing theory, 112, 117, 138 transformation formulas for, 121 Pressure distribution (pressure coefficient): on flap, 68, 489, 496, 513 on fuselage, 332, 334, 343, 347, 352, 353, 354, 358, 364 on wing: compressible, 214, 223, 224, 226, 228, 230, 235, 237, 241, 246, 257, 258, 260, 261, 270, 285, 294, 311 incompressible, 28, 55, 67, 72, 87, 128, 155
on wing-fuselage system, 402, 406, 417 Pressure equalization, wing, 113 Profile: computation of: skeleton (mean camber) line of, 56
540 SUBJECT INDEX
Profile, computation of (Cont.): teardrop of, 74 with fixed aerodynamic center, 61 friction effect on, 81 geometry of, 26 supercritical, 253 [See also Circular-arc profile; Joukowsky profile; NACA profiles; Parabolic profile (biconvex); Wedge profile] Profile drag (friction drag): of fuselage, 330, 354 of wing, 90, 92, 97, 120, 173, 216, 230, 253, 275
of wing-fuselage system, 394 Profile theory: based on: conformal mapping, 36 singularities method, 52 skeleton theory, 53, 486
teardrop theory, 68 hypersonic, 255, 260 incompressible, 25 subsonic, 227, 230, 232 supersonic, 234, 242 transonic, 244, 253
Rectangular wing: downwash and upwash of, 385, 448, 449,
459,462 drag of, 178, 275, 297, 313 lift of, 149, 161,166, 171, 296, 311, 374 lift distribution of, 143, 149, 297, 412 neutral point of, 161, 297, 392 perturbation velocity on, 201 pressure distribution on, 296 Reference wing chord, 108 Reynolds number, 10, 81, 90 Riegels factor, 70 Roll damping, 19, 192 Roll motion, 16, 192 Rolling moment: of wing, 14, 136, 149, 156, 264, 396 of wing-fuselage system, 374, 396 due to sideslip, 18, 375, 396, 466 due to yaw rate, 19, 192 Roll-up of vortex, 134, 168, 444, 449 Rudder, 432, 484, 517
Separation of flow, 42, 83, 88, 96, 168, 170, 244, 246, 366, 394, 455, 498 Side force, 14, 466 due to roll rate, 20 due to sideslip, 18, 186, 190, 400, 466
Side force (Cont.): due to yaw rate, 20 Sideslip:
angle of, 13, 16, 471 definition of, 13 Sideslipping (yawed) flight, 16, 18, 186, 466 Similarity rule: hypersonic, 258, 364 subsonic, 219, 233, 261, 350, 402, 456, 492, 511
supersonic, 219, 350,492 transonic, 225, 251, 351 Singularities method: for fuselage, 331, 342, 356, 365 for wing, 52, 123, 197, 289 for wing-fuselage system, 403
Slat (flap), 96, 455, 483,498 Slender body, theory of, 265, 300, 311, 416, 458 Slot flap, 483, 487, 490, 497, 498 Slotted wing, 96 Sonic incident flow, 269, 275 Sound, speed of, 4, 332 Source-sink distribution, 198, 293, 311, 356 Split (spreader) flap, 483, 487, 495 Spoiler, 504 Stabilizer, 481 horizontal (tail plane), 431, 432 vertical (fin), 431, 432 Stagnation point, 214, 259, 365 Stall fence (see Boundary-layer fence) Starting vortex, 34 Straight flight, 16, 182, 435 Streamline analogy, 219 Subsonic edge, 277, 514 Subsonic incident flow, 263, 270, 285, 352, 402, 456 Substitute wing, 373, 385 Suction, 96, 499 Suction force, 43, 59, 96, 180, 300, 308 Supercirculation, 503 Superposition principle, 288 Supersonic edge, 277, 286, 288, 514 Supersonic flight, 21 Supersonic incident flow, 276, 296, 310, 355, 405, 458 Super-stall, 455 Swept-back wing, 108 downwash and upwash of, 385, 448, 452 drag of, 152, 270, 275, 308, 315, 316 drag-critical Mach number of, 271, 274, 353
lift of, 152, 161, 164, 168, 171, 266, 269, 307, 308 lift distribution of, 151, 157, 164, 266
SUBJECT INDEX 541 Swept-back wing (Cont.): neutral point of, 152, 158, 161, 308, 392, 393 velocity distribution of, 203
Vortex (wing) (Cont.): free, 113, 115, 131, 166,460 horseshoe, 114, 124, 126, 379 starting, 114 Vortex sheet, 53, 114, 123, 134, 169, 290, 366, 444, 449, 453, 464, 486
Tail plane (see Stabilizer) Tail surface (see Horizontal tail, and vertical tail) Take-off assistance, 482, 494 Tangential force, 14, 179 Temperature increase: through compression, 215, 259 through friction, 216 Three-quarter-point method, 130, 146, 385 Trailing edge: angle, 25, 82 ejection, 98 Transformation, geometric, 220, 261, 350, 352, 402, 456, 511 Transonic (incident) flow, 219, 226, 269, 413 Trapezoidal wing, 106 downwash of, 448, 452 drag of, 152, 308, 316 lift of, 152, 266, 269, 308 lift distribution on, 143, 151, 158, 266 neutral point of, 152, 158, 266, 269, 308
Wave drag:
of fuselage, 358, 360 of wing, 224, 226, 237, 258, 301, 313 of wing-fuselage system, 413 Wedge profile, 242, 313 Wing (airfoil): aspect ratio of, 107, 121 dihedral (V shape) of, 105, 109, 189, 373, 398, 399 pressure equalization on, 113 reference chord of, 108 taper of, 106, 107 twist of, 105, 135, 177 (See also Airfoil theory; Delta wing; Elliptic wing; Rectangular wing; Swept-back wing; Trapezoidal wing) Wing in curved flow, 78 Wing, lifting (with angle of attack), of finite thickness (displacement), 68, 197, 270, 310 Wing-fuselage system:
Unsteady motion, 20
Velocity distribution on contour, 66, 70, 71, 75, 198, 292, 333 Velocity near-field of profile, 79 Velocity potential: of fuselage, 342, 348, 357, 365 of slender bodies, 417 of wing: compressible, 217, 218, 225, 293 incompressible, 128, 199
drag of, 393, 413 geometry of, 371 lift of, 374, 379, 382, 393, 402, 410, 419, 425 neutral point of, 380, 403, 411, 421 pitching moment of, 374, 382,.411 pressure distribution over, 402, 417 rolling moment of, 374, 396 side force on, 400 yawing moment of, 400
Vertical tail, 432, 433
dynamic pressure ratio of, 467 efficiency (sidewash) factor of, 467, 471, 473 geometry of, 434 side force (lift) of, 466 yawing moment of, 466 Viscosity, 4 Vortex density [see Vortex strength (circulation distribution) ] Vortex strength (circulation distribution), 53, 123, 153 Vortex (wing) :
bound (lifting), 31, 35, 80, 113, 131, 166 bursting of, 169
Yaw (turning) damping, 19,468 Yawed (sideslipping) flight, 18, 186, 466 angle of, 467, 471 Yawing moment, 14 due to roll rate, 19, 193 due to sideslip, 18, 186, 400, 466 Yawing motion, 19, 195, 468 Zero moment, 16 compressible, 230, 237, 264 incompressible, 60, 76 Zero-lift angle, 16 compressible, 230, 264 incompressible, 60, 135, 141, 237
