Aerodynamics of the Airplane

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

Author: Hermann T. Schlichting | Erich A. Truckenbrodt (transl. by Heinrich J. Ramm)

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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

Diisseldorf Johannesburg Lisbon London Lucerne Madrid Mexico Montreal New Delhi Panama Paris San Juan Sa"o Paulo Singapore Sydney Tokyo Toronto

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 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:


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


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.


flow direction. Then, all flow processes are equal in every cross section normal to

the generatrix of the cylinder; that is, flow about an airfoil of infinite length is two-dimensional. The theory for the calculation of the lift of such an airfoil of infinite span is also termed profile theory (Chap. 2). Particular flow processes that have a marked effect on both lift and drag take place at the wing tips of finite-span wings. These processes are described by the theory of the wing of finite span (Chaps. 3 and 4).

Lift production on an airfoil is closely related to the circulation of its velocity near-field. Let us explain this interrelationship qualitatively. The flow about an airfoil profile with lift is shown in Fig. 24. The lift L is the resultant of the

pressure forces on the lower and upper surfaces of the contour. Relative to the pressure at large distance from the profile, there is higher pressure on the lower surface, lower pressure on the upper surface. It follows, then, from the Bernoulli equation, that the velocities on the lower and upper surfaces are lower or higher, respectively, than the velocity w. of the incident flow. With these facts in mind, it is easily seen from Fig. 2-4 that the circulation, taken as the line integral of the velocity along the closed curve K, differs from zero. But also for a curve lying very close to the profile, the circulation is unequal to zero if lift is produced. The

velocity field ambient to the profile can be thought to have been produced by a clockwise-turning vortex T that is located in the airfoil. This vortex, which apparently is of basic importance for the creation of lift, is called the bound vortex of the wing. In plane flow, the quantitative interrelation of lift L, incident flow velocity w,,, and circulation T is given. by the Kutta-Joukowsky equation. Its simplified derivation, which will now be given, is not quite correct but has the virtue of being

particularly plain. Let us cut out of the infinitely long airfoil a section of width b (Fig. 2-5), and of this a strip of depth dx parallel to the leading edge. This strip of

planform area dA = b dx is subject to a lift dL = (pl - pu) dA because of the pressure difference between the lower and upper surfaces of the airfoil. The vector

dL can be assumed to be normal to the direction of incident flow if the small angles are neglected that are formed between the surface elements and the incident flow direction. The pressure difference between the lower and upper surfaces of the airfoil can be expressed through the velocities on the lower and upper surfaces by applying the

wo,

Figure 24 Flow around an airfoil profile with lift L. 1' = circulation of the airfoil.


4dL Pu wo,

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


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


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


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


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)


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


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)]


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.


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].


a

Figure 2-11 Photographs of the flow about airfoil, after Prandtl and Wieselsberger [471. (a) Attached flow. (b) Separated flow.

inclined flat plate is the fact that the resultant L of the forces is not perpendicular to the plate, but perpendicular to the direction of the incident flow w.. (Fig. 2-9a). Since only normal forces (pressures) are present on the plate surface in a frictionless flow, it could appear to be likely that the resultant of the forces acts normal to the plate, too. Besides the normal component Py = L cos a, however, there is a tangential component P, = -L sin a that impinges on the plate leading edge. Together with the normal component Py, the resultant force L acts normal to the direction

of the incident flow. For the explanation of the existence of a tangential component P, in an inviscid flow-we shall call it suction force-a closer look at the

flow process is required. The suction force has to do with the flow at the plate nose, which has an infinitely high velocity. Consequently, an infinitely high


underpressure is produced. This condition is easier to see in the case of a plate of finite but small thickness and rounded nose (Fig. 2-12a). Here the underpressure at the nose of the plate is finite and adds up to a suction force acting parallel to the plate in the forward direction. The detailed computation shows that the magnitude

of this suction force is independent of plate thickness and nose rounding. It remains, therefore, the value of S = L sin a in the limiting case of an infinitely thin plate.

In real flow (with friction) around very sharp-nosed plates, an infinitely high underpressure does not exist. Instead, a slight separation of the flow (separation

bubble) forms near the nose (Fig. 2-12b). For small angles of attack, the flow reattaches itself farther downstream and, therefore, on the whole is equal to the frictionless flow. The suction force is missing, however, and the real flow around an inclined sharp-edged plate produces drag acting in the direction of the incident flow.

Also, this analysis shows that it is very important for keeping the drag small that the leading edge of wing profiles is well rounded. Figure 2-13 shows (a) the polar curves (CL vs. CD) and (b) the glide angles E = CD/CL of a thin sharp-edged flat plate

and of a thin symmetric profile. In the range of small to moderate angles of attack, the thin profile with rounded nose has a markedly smaller drag than the sharp-edged flat plate. Within a certain range of angles of attack, a is smaller than a (c < a) for

Px = 0

Figure 2-12 Development of the suction force S on the leading edge of a profile. (a) Thin, symmetric profile with rounded nose, suction force present. (b) Flat plate with sharp nose, suction force missing.

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


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


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.


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.


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


\ 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


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].


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.


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)


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


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)


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)


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)


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.


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


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


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.


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)


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


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.


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

_ XÌ

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)].


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)


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 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.


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,


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).


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


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-


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.


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


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.


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.


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


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.


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.


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.


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.


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


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.


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


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.


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 Ì-

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).


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.


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.


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)


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 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


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.


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.


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


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.


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.


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


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.


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 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.


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

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 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


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 .


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:


-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).


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


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


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


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


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.


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


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.


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


Supersonic flow

Subsonic flow Maoo1

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.


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.


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.


.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


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,.


Test

i' 04

/

------- Theory X11

8°

i 0.2

1

o t/c=0,0885

I

-òt/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


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.


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.


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.


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


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.


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


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].


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.


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.


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


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 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)


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).


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.


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).


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.


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


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


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


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


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].


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.


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)


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)


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.


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.


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.


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 a

p1

(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).


, 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


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


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


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.


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.


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<m1

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


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


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.


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)


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


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.


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)


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.


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 ')
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.


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.


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).


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.


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


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: 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 1.

WINGS IN COMPRESSIBLE FLOW 305 Z0

1.8

Y\

\

1.6

7n-l

\

7Th-1.5

\

14

0 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: supersonic leading edge. 306


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


=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.


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


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


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


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.


,\

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 ] .


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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53. Linnel, R. D.: Two-Dimensional Airfoils in Hypersonic Flows, J. Aer. Sci., 16:22-30, 1949.

54. Lock, R. C. and J. Bridgewater: Theory of Aerodynamic Design for Swept Winged Aircraft at Transonic and Supersonic Speeds, Prog. Aer. Sci., 8:139-228, 1967. Lock, R. C.: Aer.

Quart., 12:65-93, 1961; J. Roy. Aer. Soc., 67:325-337, 1963; in K. Oswatitsch and D. Rues (eds.), "Symposium Transsonicum II," pp. 457-486, Springer, Berlin, 1976. Lock, R. C. and E. W. E. Rogers: Adv. Aer. Sci., 3:253-275, 1962. 55. Lock, R. C. and J. L. Fulker: Design of Supercritical Aerofoils, Aer. Quart., 25:245-265, 1974. Lock, R. C., P. G. Wilby, and B. J. Powell: Aer. Quart., 21:291-302, 1970. 56. Love, E. S.: Investigations at Supersonic Speeds of 22 Triangular Wings Representing Two Airfoil Sections for Each of 11 Apex Angles, NACA Rept. 1238, 1955. Lampert, S.: J. Aer. Sci., 24:667-674, 682, 1957. 57. Ludwieg, H.: Pfeilfliigel bei hohen Geschwindigkeiten, Lil.-Ber., 127, 1940. 58. Mangler, K. W.: Calculation of the Pressure Distribution over a Wing at Sonic Speeds, ARC RM 2888, 1951/1955. Mangler, K. W. and D. G. Randall: ARC RM 3102, 1955/1959. 59. Martin, J. C. and I. Jeffreys: Span Load Distributions Resulting from Angle of Attack, Rolling and Pitching for Tapered Sweptback Wings with Streamwise Tips, Supersonic Leading and Trailing Edges, NACA TN 2643, 1952. Harmon, S. M. and I. Jeffreys: NACA TN 2114, 1950. Martin, J. C. and N. Gerber: J. Aer. Sci., 20:699-704, 1953. 60. Maskell, E. C. and J. Weber: On the Aerodynamic Design of Slender Wings, J. Roy. Aer. Soc., 63:709-721, 1959. Kawasaki, T.: AIAA J., 5:439-445, 1967. Lord, W. T. and G. G. Brebner: Aer. Quart., 10:79-102, 1959. 61. Miele, A. and R. E. Pritchard: Two-Dimensional Wings of Minimum Total Drag, in A.

Miele (ed.), "Theory of Optimum Aerodynamic Shapes," pp. 87-101, Academic, New York, 1965. Drougge, G.: in A. Miele (ed.), "Theory of Optimum Aerodynamic Shapes," pp. 79-86, Academic, New York, 1965. Miele, A. and A. H. Lusty, Jr.: in A. Miele (ed.), "Theory of Optimum Aerodynamic Shapes," pp. 137-150, Academic, New York, 1965. 62. Mirels, H.: A Lift-Cancellation Technique in Linearized Supersonic-Wing Theory, NACA

Rept. 1004, 1951. Behrbohm, H.: Z. Flugw., 4:263-268, 1956. Leiter, E.: Z. Flugvv., 23:69-76, 1975. 63. Murman, E. M. and J. D. Cole: Calculation of Plane Steady Transonic Flows, AIAA J., 9:114-121, 1971. Krupp, J. A. and E. M. Murman: AIAA J., 10:880-886, 1972. Magnus,


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.


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.

94. Taylor, G. I.: Recent Work on the Flow of Compressible Fluids, J. Lond. Math. Soc., 5:224240,1930; "Scientific Papers," vol. III, pp. 157-171, Cambridge University Press, Cambridge, 1963. Taylor, G. 1. and J. W. Maccoll: in W. F. Durand (ed.), "Aerodynamic Theory-A General Review of Progress," div. H, Springer, Berlin, 1935, Dover, New York, 1963. 95. Truckenbrodt, E.: Ein Verfahren zur Berechnung der Auftriebsverteilung an Tragflugeln bei Schallanstromung, Jb. WGL, 113-130, 1956.


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.


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,


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 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


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


(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.


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.


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
R)

(6-16a)

where R is the radius of the circular cylinder. For the range -R 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)


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.


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


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.


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.


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


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 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


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.


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.


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.


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


[R() [(' R

,

J cp = 4Nc, tany

s(x)

1

yy

1+

4

ls(z)/ J -

y

s(X)'+Y

s

z

for R2 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


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.


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


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


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.


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.


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,,,


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) 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


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


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 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.


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


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.


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


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.


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.


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


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-
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


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].

REFERENCES 1. Abbott, I. H. and A. E. von Doenhoff: "Theory of Wing Sections, Including a Summary of Airfoil Data," Dover, New York, 1959. 2. Allen, H. J.: Calculation of the Chordwise Load Distribution over Airfoil Sections with Plain, Split, or Serially Hinged Trailing-Edge Flaps, NACA Rept. 634, 1938. 3. Arnold, K. 0.: Aerodynamische Untersuchungen an Flugeln mit Bremsklappen, Z. Flugw.,

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.


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.


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

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