Aerodynamicsof the

AirplaneHermciui Sch!ichthg and Erie lw c ro t

Translated by Heinrich J. Ramm

AERODYNAMICSOF THE AIRPLANE

Hermann SchlichtingProfessor, Technical University of Braunschweig

and Aerodynamic Research Institute (A VA), Gottingen

Erich TruckenbrodtProfessor, Technical University of Munich

Translated byHeinrich J. Ramm

Associate Professor, University of Tennessee Space Institute

McGraw-Hill International Book CompanyNew York St. Louis San Francisco Auckland Beirut Bogota

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Singapore Sydney Tokyo Toronto

This book was set in Press Roman by Hemisphere Publishing Corporation. The editorswere Lynne Lackenbach and Judith B. Gandy; the production supervisor was RebekahMcKinney; 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 Statesof America. No part of this publication may be reproduced, stored in a. retrieval system,or transmitted, in any form or by any means, electronic, mechanical, photocopying,recording, or otherwise, without the prior written permission of the publisher.

1234567890 MPMP 7832109

Library of Congress Cataloging in Publication Data

Schlichting, Hermann, date.Aerodynamics of the airplane.

Translation of Aerodynamik des Flugzeuges.Bibliography: p.Includes index.1. Aerodynamics. I. Truckenbrodt, Erich,

date, joint author. H. Title.TL570.S283313 629.132'3 79-60ISBN 0-07-055341-6

CONTENTS

PrefaceNomenclature

1 Introduction1-1 Problems of Airplane Aerodynamics1-2 Physical Properties of Air1-3 Aerodynamic Behavior of Airplanes

References

Part 1 Aerodynamics of the Wing

viiix

1

1

2

822

23

2 Airfoil of Infinite Spanin Incompressible Flow (Profile Theory) 25

2-1 Introduction 252-2 Fundamentals of Lift Theory 302-3 Profile Theory by the Method of Conformal Mapping 362-4 Profile Theory by the Method of Singularities 522-5 Influence of Viscosity and Boundary-Layer Control

on Profile Characteristics 81References 101

3 Wings of Finite Span in Incompressible Flow 1053-1 Introduction 1053-2 Wing Theory by the Method of ` ortex Distribution 1123-3 Lift of Wings in Incompressible Flow 1313-4 Induced Drag of Wings 1733-5 Flight Mechanical Coefficients of the Wing 1813-6 Wing of Finite Thickness at Zero Lift 197

References 206

Vi CONTENTS

4 Wings in Compressible Flow4-1 Introduction4-2 Basic Concept of the Wing in Compressible Flow4-3 Airfoil of Infinite Span in Compressible Flow

(Profile Theory)4-4 Wing of Finite Span in Subsonic and Transonic Flow4-5 Wing of Finite Span at Supersonic Incident Flow

References

Part 2 Aerodynamics of the Fuselage andthe Wing-Fuselage System

5 Aerodynamics of the Fuselage5-1 Introduction5-2 The Fuselage in Incompressible Flow5-3 The Fuselage in Compressible Flow

References

6 Aerodynamics of the Wing-Fuselage System6-1 Introduction6-2 The Wing-Fuselage System in Incompressible Flow6-3 The Wing-Fuselage System in Compressible Flow6-4 Slender Bodies

References

Part 3 Aerodynamics of the Stabilizersand Control Surfaces

7 Aerodynamics of the Stabilizers7-1 Introduction7-2 Aerodynamics of the Horizontal Tail7-3 Aerodynamics of the Vertical Tail

References

8 Aerodynamics of the Flaps and Control Surfaces8-1 Introduction8-2 The Flap Wing of Infinite Span (Profile Theory)8-3 Flaps on the Wing of Finite Span and on the Tail Unit

References

BibliographyAuthor IndexSubject Index

213

213214

227261276317

325

327

327331348367

371

371376401416425

429

431

431435466477

481

481486506517

521

527

537

PREFACE

Only a very few comprehensive presentations of the scientific fundamentals of theaerodynamics of the airplane have ever been published. The present book is an Englishtranslation of the two-volume work "Aerodynamik des Flugzeuges," which has alreadyappeared in a second edition in the original German. In this book we treat exclusivelythe aerodynamic forces that act on airplane components-and thus on the wholeairplane-during its motion through the earth's atmosphere (aerodynamics of theairframe). These aerodynamic forces depend in a quite complex manner on thegeometry, speed, and motion of the airplane and on the properties of air. Thedetermination of these relationships is the object of the study of the aerodynamics ofthe airplane. Moreover, these relationships provide the absolutely necessary basis fordetermining the flight mechanics and many questions of the structural requirements ofthe airplane, and thus for airplane design. The aerodynamic problems related toairplane propulsion (power plant aerodynamics) and the theory of the modes ofmotion of the airplane (flight mechanics) are not treated in this book.

The study of the aerodynamics of the airplane requires a thorough knowledge ofaerodynamic theory. Therefore, it was necessary to include in the German edition arather comprehensive outline of fluid mechanic theory. In the English edition thissection has been eliminated because there exist a sufficient number of pertinent worksin English on the fundamentals of fluid mechanic theory.

Chapter 1 serves as an introduction. It describes the physical properties of air andof the atmosphere, and outlines the basic aerodynamic behavior of the airplane. Themain portion of the book consists of three major divisions. In the first division (Part1), 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-fuselagesystem. Finally, in the third division (Part 3), Chaps. 7 and 8 are devoted to theproblems 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 ofthe airplane, that is, the aerodynamic interference, are given special attention.

Specifically, the following brief outline describes the chapters that deal with theintrinsic problems of the aerodynamics of the airplane: Part 1 contains, in Chap. 2, theprofile theory of incompressible flow, including the influence of friction on the profile

viii PREFACE

characteristics. Chapter 3 gives a comprehensive account of three-dimensional wingtheory for incompressible flow (lifting-line and lifting-surface theory). In addition tolinear airfoil theory, nonlinear wing theory is treated because it is of particularimportance for modern airplanes (slender wings). The theory for incompressible flowis important not only in the range of moderate flight velocities, at which thecompressibility of the air may be disregarded, but even at higher velocities, up to thespeed of sound-that is, at all Mach numbers lower than unity-the pressuredistribution of the wings can be related to that for incompressible flow by means ofthe Prandtl-Glauert transformation. In Chap. 4, the wing in compressible flow istreated. Here, in addition to profile theory, the theory of the wing of finite span isdiscussed at some length. The chapter is subdivided into the aerodynamics of the wingat subsonic and supersonic, and at transonic and hypersonic incident flow. The lattertwo cases are treated only briefly. Results of systematic experimental studies onsimple wing forms in the subsonic, transonic, and supersonic ranges are given for thequalification of the theoretical results. Part 2 begins in Chap. 5 with the aerodynamicsof the fuselage without interference at subsonic and supersonic speeds. In Chap. 6, arather comprehensive account is given of the quite complex, but for practical casesvery important, aerodynamic interference of wing and fuselage (wing-fuselage system).It should be noted that the difficult and complex theory of supersonic flow could betreated only superficially. In this chapter, a special section is devoted to slender flightarticles. Here, some recent experimental results, particularly for slender wing-fuselagesystems, are reported. In Part 3, Chaps. 7 and 8, the aerodynamic questions ofimportance to airplane stability and control are treated. Here, the aerodynamicinterferences 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 againto subsonic and supersonic incident flow.

A comprehensive list of references complements each chapter. These lists, as wellas the bibliography at the end of the book, have been updated from the Germanedition to include the most recent publications.

Although the book is addressed primarily to students of aeronautics, it has beenwritten as well with the engineers and scientists in mind who work in the aircraftindustry and who do research in this field. We have endeavored to emphasize thetheoretical approach to the problems, but we have tried to do this in a manner easilyunderstandable to the engineer. Actually, through proper application of the laws ofmodern aerodynamics it is possible today to derive a major portion of theaerodynamics of the airplane from purely theoretical considerations. The verycomprehensive experimental material, available in the literature, has been includedonly 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 throughaccumulation of large numbers of experimental results, but rather through synthesis oftheoretical considerations with a few basic experimental results. Through numerousdetailed examples, we have endeavored to enhance the reader's comprehension of thetheory.

Hermann SchlichtingErich Truckenbrodt

NOMENCLATURE

MATERIAL CONSTANTS

0 density of air (mass of unit volume)g gravitational accelerationcP, cv specific heats at constant pressure and constant volume,

respectivelyy = cP/ci1 isentropic exponenta = yp/,o speed of sound coefficient of dynamic viscosityv = /9 coefficient of kinematic viscosityR gas constantT absolute temperature (K)t temperature (C)

FLOW QUANTITIES

p pressure (normal force per unit area)T shear stress (tangential force per unit area)u, v, w velocity components in Cartesian (rectangular) coordinatesu, Wr, w.3 velocity components in cylindrical coordinatesV, U. velocity of incident flowWe velocity on profile contourwt induced downwash velocity, positive in the direction of the

negative z axis

Lx

X NOMENCLATURE

q = (p/2)V2q00 = (,o./2)U!Re = VI/vMa=V/aMay, = U./ate,Ma. cr

dynamic (impact) pressuredynamic (impact) pressure of undisturbed flowReynolds numberMach numberMach number of undisturbed flowdrag-critical Mach numberMach angledisplacement thickness of boundary layercirculationdimensionless circulationvortex densitysource strengthdipole strengthvelocity potential

GEOMETRIC QUANTITIES

x,Y,z

=x/s,n=y/s,z/s

Xf, Xr

xl, xp

AAFAHAyb = 2sbFbHA =b2/A`4H, Ay

C

Cr, Ctc11 =(2/A)foc2(y)dYX = Ct/CrIFcfXf=Cf/cTif

Cartesian (rectangular) coordinates: x = longitudinal axis,y = lateral axis, z = vertical axis

dimensionless rectangular coordinatestrigonometric coordinate; cos $ = qcoordinates. of wing leading (front) and trailing (rear) edges,xo, x1oo, respectivelycoordinates of quarter-point and three-quarter-point lines,x25 , X75, respectivelywing areafuselage cross-sectional areaarea of horizontal tail (surface)area of vertical tail (surface)wing spanfuselage width

span of horizontal tail (surface)aspect ratio of wingaspect ratios of horizontal and vertical tails (surface),respectivelywing chordchord at wing root and wing tip, respectivelywing reference chordwing taperfuselage lengthflap (control-surface) chordflap (control-surface) chord ratioflap deflection

NOMENCLATURE Xi

7m = tan y/ tan

E

V

N25tS = t/c

hxtXhZ(S)Z(t)dFmaxSF = dFinaxliF17F=bFIbD=2RZo

rH

EHrv

sweepback angle of wingleading edge semiangle of delta wing (Fig. 4-59)parameter (Fig. 4-59); m < 1: subsonic flow edge, m > 1:supersonic flow edgetwist angleangle of wing dihedralgeometric neutral pointprofile thicknessthickness ratio of wingprofile camber(maximum) thickness position(maximum) camber (height) positionskeleton (mean camber) line coordinateteardrop profile coordinatemaximum fuselage diameterfuselage thickness ratiorelative fuselage widthdiameter of axisymmetric fuselagewing vertical positionlever arm of horizontal tail (= distance between geometricneutral points of the wing and the horizontal tail)setting angle of horizontal stabilizer (tail)lever arm of vertical tail (= distance between geometricneutral points of the wing and the vertical tail)

AERODYNAMIC QUANTITIES (see Fig. 1-6)

WX, Wy, WZ

"`LX = WX S/V,any = W yCM/ V,Z WZS/VLDYMxM, MyMZDiCL

CD

CMX

angle of attack (incidence)angle of sideslip (yaw)components of angular velocities in rectangular coordinatesduring rotary motion of the airplane

components of the dimensionless angular velocitiesliftdragside forcerolling momentpitching momentyawing momentinduced draglift coefficientdrag coefficientrolling-moment coefficient

Xii NOMENCLATURE

CM,CMyCMZ

Cl

Cm

CmfCifCDiCDp (dcL/da)cp =(p-pc,)/Q.Cp plCP Crd Cp = (pi - pu)qf = 2b/CL,ok = 7r11/cLwae

ag = aai = wi/U,0ao

OW =a+EH+awaw=w/UUNXNId XN

pitching-moment coefficientyawing-moment coefficientlocal lift coefficientlocal pitching-moment coefficientcontrol-surface (hinge) moment coefficientflap (control-surface) load coefficientcoefficient of induced dragcoefficient of profile draglift slope of wing of infinite spanpressure coefficientpressure coefficient of plane (two-dimensional) flowcritical pressure coefficientcoefficient of load distributionplanform functioncoefficient of elliptic wingeffective angle of attackgeometric angle of attackinduced angle of attackzero-lift angle of attackangle of attack of the horizontal taildownwash angle at the horizontal tail locationaerodynamic neutral pointposition of aerodynamic neutral pointdistance between aerodynamic and geometric neutral pointsangle of flow incident on the vertical tailangle of sidewash at the station of the vertical tail

DIMENSIONLESS STABILITY COEFFICIENTS

Coefficients of Yawed Flightacy/aoacMX/a1aCMZ/a 3

side force due to sidesliprolling moment due to sideslipyawing moment due to sideslip

Coefficients due to Angular VelocityacylaQZacMXla QXacMX/aQZacMZ/af?ZacMZ l a X

aCL/a!?y

acJ/aQy

side force due to yaw raterolling moment due to roll raterolling moment due to yaw rateyawing moment due to yaw rateyawing moment due to roll ratelift due to pitch ratepitching moment due to pitch rate

NOMENCLATURE Xiii

INDICES

W wing dataF fuselage data(W + F) data of wing-fuselage systemH data of horizontal stabilizerV data of vertical stabilizerf data of flaps (control surfaces)

CHAPTER

ONEINTRODUCTION

1-1 PROBLEMS OF AIRPLANE AERODYNAMICS

An airplane moves in the earth's atmosphere. The state of motion of an airplane isdetermined by its weight, by the thrust of the power plant, and by the aerodynamicforces (or loads) that act on the airplane parts during their motion. For every stateof motion at uniform velocity, the resultant of weight and thrust forces must be inequilibrium with the resultant of the aerodynamic forces. For the particularlysimple state of motion of horizontal flight, the forces acting on the airplane areshown in Fig. 1-1. In this case, the equilibrium condition is reduced to therequirement 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 R1normal and parallel, respectively, to the flight velocity vector V. For nonuniformmotion 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 theindividual parts, and thus on the whole aircraft, during motion. The most importantparts of the airplane that contribute to the aerodynamic forces are wing, fuselage,control and stabilizing surfaces (tail unit or empennage, ailerons, and canardsurfaces), and power plant. The aerodynamic forces depend in a quite complicatedmanner on the geometry of these parts, the flight speed, and the physical propertiesof the air (e.g., density, viscosity). It is the object of the study of the aerodynamicsof the airplane to furnish information about these interrelations. Here, the followingtwo problem areas have to be considered:

1. Determination of aerodynamic forces for a given geometry of the aircraft(direct problem)

2. Determination of the geometry of the aircraft for desired flow patterns(indirect problem)

I

2 INTRODUCTION

Th Figure 1-1 Forces (loads) on an air-plane in horizontal flight. L, lift; D,drag; W, weight; Th, thrust; R,, re-sultant of aerodynamic forces (result-ant of L and D); Rz , resultant of Wand Th.

The object of flight mechanics is the determination of aircraft motion for givenaerodynamic forces, known weight of the aircraft, and given thrust. This includesquestions of both flight performance and flight conditions, such as control andstability of the aircraft. Flight mechanics is not a part of the problem area of thisbook. Also, the field of aeroelasticity, that is, the interactions of aerodynamicforces with elastic forces during deformation of aircraft parts, will not be treated.

The science of the aerodynamic forces of airplanes, to be treated here, formsthe foundation for both flight mechanics and many questions of aircraft design andconstruction.

1-2 PHYSICAL PROPERTIES OF AIR

1-2-1 Basic FactsIn fluid mechanics, some physical properties of the fluid are important, forexample, density and viscosity. With regard to aircraft operation in the atmosphere,changes of these properties with altitude are of particular importance. Thesephysical properties of the earth's atmosphere directly influence aircraft aero-dynamics and consequently, indirectly, the flight mechanics. In the followingdiscussions the fluid will be considered to be a continuum.

The density o is defined as the mass of the unit volume. It depends on bothpressure and temperature. Compressibility is a measure of the degree to which afluid can be compressed under the influence of external pressure forces. Thecompressibility 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 aparticular flow process lead to noticeable changes in density.

Viscosity is related to the friction forces within a streaming fluid, that is, to thetangential forces transmitted between ambient volume elements. The viscositycoefficient of fluids changes rather drastically with temperature.

In many technical applications, viscous forces can be neglected in order tosimplify the laws of fluid dynamics (inviscid flow). This is done in the theory of liftof airfoils (potential flow). To determine the drag of bodies, however, the viscosityhas to be considered (boundary-layer theory). The considerable increase in flightspeed during the past decades has led to problems in aircraft aerodynamics thatrequire inclusion of the compressibility of the air and often, simultaneously, theviscosity. This is the case when the flight speed becomes comparable to the speed ofsound (gas dynamics). Furthermore, the dependence of the physical properties of airon the altitude must be known. Some quantitative data will now be given fordensity, compressibility, and viscosity of air.

1-2-2 Material PropertiesDensity 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, pressurep, and absolute temperature T is given by the thermal equation of state for idealgases

p =QRT (1-la)

1 1 bR = 287 kgK (air) - )(

where R is the gas constant. Of the various possible changes of state of a gas, ofparticular importance is the adiabatic-reversible (isentropic) change in which pressureand density are related by

p= const

Qy

Here y is the isentropic exponent, with

(1-2)

CP

y - cU(1-3a)

= 1.405 (air) (1-3b)cP and c are the specific heats at constant pressure and constant volume,respectively.

Very fast changes of state are adiabatic processes in very good approximation,because heat exchange with the ambient fluid elements is relatively slow and,therefore, of negligible influence on the process. In this sense, flow processes at highspeeds can usually be considered to be fast changes of state. If such flows aresteady, 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 notfollow Eq. (1-2).

Across a normal compression shock, pressure and density are related by

of-1)+(7+1)PZ

Pie21 4= ( - a)

el (7+1)+(7-1)Pi7+1

= 5.93 (air) (1-4b)7-1where the indices 1 and 2 indicate conditions before and behind the shock,respectively.

Speed of sound Since the pressure changes of acoustic vibrations in air are of sucha 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 compressibilitylaw 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 rootof the absolute temperature. The value given in Eq. (1-5b) is valid for air oftemperature t = 15C or T = 288 K.

Viscosity In flows of an inviscid fluid, no tangential forces (shear stresses) existbetween ambient layers. Only normal forces (pressures) act on the flow. The theoryof inviscid, incompressible flow has been developed mathematically in detail, giving,in many cases, a satisfactory, description of the actual flow, for example, in computingairfoil lift at moderate flight velocities. On the other hand, this theory fails completelyfor the computation of body drag. This unacceptable result of the theory of inviscidflow is caused by the fact that both between the layers within the fluid and betweenthe fluid and its solid boundary, tangential forces are transmitted that affect the flow inaddition to the normal forces. These tangential or friction forces of a real fluid arethe result of a fluid property, called the viscosity of the fluid. Viscosity is definedby Newton's elementary friction law of fluids as

(1-6)

Here T is the shearing stress between adjacent layers, du/dy is the velocity gradient nor-mal to the stream, and u is the dynamic viscosity of the fluid, having the dimensionsNs/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 andinertia forces simultaneously, the quotient of viscosity i and density Q plays animportant 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, dynamicviscosity p, and kinematic viscosity v of air are given versus temperature at constantpressure.

1-2-3 Physical Properties of the AtmosphereChanges of pressure, density, and viscosity of the air with altitude z of thestationary atmosphere are important for aeronautics. These quantities depend on thevertical temperature distribution T(z) in the atmosphere. At moderate altitude (upto about 10 km), the temperature decreases with increasing altitude, thetemperature gradient dT/dz varying between approximately -0.5 and -1 K per 100m, depending on the weather conditions. At the higher altitudes, the temperaturegradient varies strongly with altitude, with both positive and negative valuesoccurring.

The data for the atmosphere given below are valid up to the boundary of thehomosphere at an altitude of about 90 km. Here the gravitational acceleration isalready markedly smaller than at sea level.

The pressure change for a step of vertical height dz is, after the basichydrostatic equation,

dp = - Qg dz_ -ego dH

where H is called scale height.

Table 1-1 Density e, dynamic viscosity , andkinematic viscosity v of air versus temperature tat constant pressure p 1 atmosphere

KinematicTemperature Density Viscosity viscosity

t Q[C] [kg/m3 ] [kg/ms] [m2 /s]

-20 1.39 15.6 11.3-10 1.34 16.2 12.1

0 1.29 16.8 13.010 1.25 17.4 13.920 1.21 17.9 14.940 1.12 19.1 17.060 1.06 20.3 19.280 0.99 21.5 21.7

100 0.94 22.9 24.5

(1-8a)

(1-8b)

6 INTRODUCTION

The decrease in the gravitational acceleration g(z) with increasing height z is

g(z) = r, 2 go (1-9)(ro + z)with ro = 6370 km as the radius of the earth, and go = 9.807 m/s', the standardgravitational acceleration at sea level. With Eq. (1-8) we obtain by integration

H = f g(z) dz = za

(1-10)go

0

+r0

For the homosphere (z < 90 km), the scale height is insignificantly different fromthe geometric height (see Table 1-2).

The variables of state of the atmosphere can be represented by the thermal andpolytropic equations of state,

p = Q RT (1-11a)P c nst (1-llb)= o9 ?6

with n as the polytropic exponent (n

INTRODUCTION 7

which shows that each polytropic exponent n belongs to a specific temperaturegradient dT/dH. Note that the gas constant* in the homosphere, up to an altitudeof H = 90 km, can be taken as a constant.

From Eq. (1-13) follows by integration:

T=Tb7Ln1 R (H - Hb) (1-14)Here it has been assumed that the polytropic exponent and, therefore, thetemperature gradient are constant within a layer. The index b designates the valuesat the lower boundary of the layer. In Table 1-2 the values of Hb, Zb, Tb, anddT/dH are listed according to the "U.S. Standard Atmosphere" [2].

The pressure distribution with altitude of the atmosphere is obtained throughintegration of Eq. (1-12a) with the help of Eq. (1.14). We have

11

n-1Tb

- 1- nnl Ro (H H,)]For the special case n = 1 (isothermal atmosphere), Eq. (1.15a) reduces to

r

P =expL- RTb (H - Hb)

(1-15a)

(1-15b)

In the older literature this relationship is called the barometric height equation.Finally, the density distribution is easily found from the polytropic relation Eq.

Also given in Table 1-2 are the reference values Pb and eb at the layerboundaries. For the bottom layer, which reaches from sea level to H= 11 km,Hb = Ho has to be set equal to zero in Eqs. (1-15a) and (1-15b). The other sea levelvalues (index 0), inclusive of thoseviscosity, are, after [2] ,

go = 9.8067 rn/s2

po = 1.0 atm

o = 1.2250 kg/m3To =288.15K

for the speed of sound and the

to=15Cao = 340.29 m/s

vo = 1.4607 - 10-5 m2 /s

(dT/dH)o = -6.5 K/km

kinematic

*The temperature gradient dT/dH determines the stability of the stratification in thestationary atmosphere. The stratification is more stable when the temperature decrease withincreasing height becomes smaller. For dT/dH= 0 when n = 1, Eq. (1-13), the atmosphere isisothermal and has a very stable stratification. For n = y = 1.405, the stratification is adiabatic(isentropic) with dT/dH = -0.98 K per 100 in. This stratification is indifferent, because an airvolume moving upward for a certain distance cools off through expansion at just the same rateas the temperature drops with height. The air volume maintains the temperature of the ambientair and is, therefore, in an indifferent equilibrium at every altitude. Negative temperaturegradients of a larger magnitude than 0.98 K/100 m result in unstable stratification.

8 INTRODUCTION

Table 1-3 Barometric pressure p, air density o, temperature T, speedof sound a, and kinematic viscosity v versus height z*

z [km] T/To p/po Q/Po I a/ao V/1'0

0 1.0 1.0 1.0 1.0 1.02 0.9549 7.846 - 10-1 8.217-10-1 0.9772 1.1744 0.9097 6.085 - 10-1 6.688-10-1 0.9538 1.3886 0.8647 4.660 - 10-1 5.389-10-1 0.9299 1.6548 0.8197 3.518-10-1 4.292 10-1 0.9054 1.988

10 0.7747 2.615-10-1 3.376-10-1 0.8802 2.41311.019 0.7519 2.234-10-1 2,971-10-1 0,8671 2.67412 0.7519 1.915-10-1 2,546-10-1 0.8671 3.12014 0.7519 1.399-10-1 1.860-10-1 0.8671 4.27116 0.7519 1.022-10-1 1.359. 10-1 0.8671 5.84618 0.7519 7.466 10-2 9.930. 10-2 0.8671 8.00020 0.7519 5.457 - 10-2 7.258- 10-2 0.8671 1.095-10120.063 0.7519- 5.403.10-2 7.186- 10-2 0.8671 1.106- 10'-25 0.7689 2.516-10-2 3.272- 10-2 0.8769 2.474- 10130 0.7861 1.181 10-2 1.503-10-2 0.8866 5.486-10132.162 0.7935 8.567 10-3 1.080-10-2 0.8908 7.696 - 10135 0.8208 5.671-10-3 6.909- 10-3 0.9060 1.236- 10240 0.8688 2.834 - 10-3 3.262-10-3 0,9321 2.743- 10245 0.9168 1.472 10-3 1.605 - 10-3 0.9575 5.819 - 10247.350 0.9393 1.095 .10-3 1.165. 10-3 0.9692 8.170 - 10250 0.9393 7.874 . 10-4 8.383 - 10-4 0.9692 1.136-10352.429 0.9393 5.823-10-4 6.199- 10-4 0.9692 1.536 - 10355 0.9218 4.219.10-4 4.578.10-4 0.9601 2.049.10360 0.8876 2.217-10-4 2.497- 10-4 0.9421 3.645-10361.591 0.8768 1.797 10-4 2.050- 10-4 0.9364 4,397- 10365 0.8305 1.130 10'4 1.360-10-4 0.9113 6.340- 10370 0.7625 5.448.10-5 7.146-10-5 0.8732 1.125 10475 0.6946 2.458-10-5 3.538- 10-5 0.8334 2.100-10479.994 0.6269 1.024-10-5 1.634-10-5 0.7918 4.161- 10480 0.6269 1.023 - 10-5 1.632-10-5 0.7918 4.166-10485 0.6269 4.071-10-6 6.494. 10-6 0.7918 1.047-10590 0.6269 1.622 - 10-6 2.588 - 10-6 0.7918 2,627-105

*After "U.S. Standard Atmosphere" [2].

The numerical values of pressure and density distribution are listed in Table 1-3, towhich the values for the speed of sound and the kinematic viscosity have been added.More detailed and more accurate values are found in the comprehensive tables [2].

Finally, in Fig. 1-2, a graphic representation is given of the distributions ofpressure, density, temperature, speed of sound, and kinematic viscosity versusaltitude. Whereas pressure and density decrease strongly with height, kinematicviscosity increases markedly.

1-3 AERODYNAMIC BEHAVIOR OF AIRPLANES

1-3-1 Similarity LawsThe question of the mechanical similarity of two flows plays an important role inboth the theory of fluid flows and the extensive testing procedures of fluid

INTRODUCTION 9

mechanics. That is, given are two fluids of different physical properties, in each ofwhich one of two geometrically similar bodies is located. Under what conditions arethe two flow fields about the two bodies similar-in other words, under whatconditions do they have a similar set of streamlines? Only in the case ofmechanically similar flow fields is it possible to draw conclusions from theknowledge-which may have been obtained theoretically or experimentally-of theflow field about one body on the flow field about another geometrically similarbody. To ensure mechanical similarity of flow fields about two geometricallysimilar, but not necessarily identical, bodies (e.g., two airfoils) in different fluids ofdifferent velocities, the condition must be satisfied that in each pair of points ofsimilar position, the forces acting on two fluid elements must be similar in directionand magnitude. For the aerodynamics of aircraft, gravitation is of negligibleinfluence and will not be considered for the establishment of similarity laws.

Mach similarity law First, let us consider the case of a compressible, inviscid flow.Here, except for inertia forces, only the elastic forces act on the fluid elements of ahomogeneous fluid. For mechanically similar flows, obviously the relative densitychange caused by the elastic forces must be equal in the two flows. This leads to therequirement that the Mach numbers of both flows, that is, the ratios of flow velocityand sonic speed, should be equal. This is the Mach similarity law. The Mach number

Ma = V (1-16)a

Figure 1-2 Atmospheric pressurep, air density o, temperature T,speed of sound a, and kinematicviscosity v, vs. height z. From"U.S. Standard Atmosphere" [2].

10 INTRODUCTION

is, therefore, a first important dimensionless characteristic number of flow processes.Since the effects of compressibility become noticeable for Ma > 0.3, as pointed outabove, the Mach similarity law needs to be considered only above this limitingvalue. The fluid dynamic laws of an incompressible fluid can, therefore, be taken asthe laws for very small Mach numbers with the limiting case Ma -+ 0.

Reynolds similarity law Let us now consider the case of an incompressible, viscousflow. Here, only inertia and viscous forces act on the fluid element. These twoforces are functions of the following physical quantities: approach velocity V,characteristic body dimension 1, density o, and dynamic viscosity of the fluid. Theonly possible dimensionless combination of these quantities is the quotient

Re - V i V1 (1-17)

where Re is called the Reynolds number. The ratio p/Q = v has been introducedabove in Eq. (1-7) as the kinematic viscosity. This law was found by Reynolds in1883 during investigations on the flow in pipes and is called the Reynolds similaritylaw.

If velocity and body dimensions are not too small, as in aeronautics, theReynolds number is very large because of the very small values of v. This meansphysically that the friction forces are much smaller than the inertia forces in suchcases. Inviscid flow (v -+ 0) corresponds to the limiting case Re --+ -0. The laws offlow with small viscosity often correspond quite well to those without viscosity. Onthe other hand, in many cases even a very small viscosity should not be neglected inthe theory (boundary-layer theory).

For compressible flow with friction, mechanical similarity requires that theMach and Reynolds similarity laws be satisfied simultaneously, which is verydifficult to accomplish in experimental investigations. The Mach similarity law andthe Reynolds similarity law govern decisively the whole realm of theoretical andexperimental fluid mechanics and particularly the laws of aeronautics.

To give a convenient survey of the Mach and Reynolds numbers occurring inthe aerodynamics of aircraft, the diagrams Fig. 1-3 and Fig. 1-4 have been drawn.They show these two dimensionless characteristic quantities versus flight velocityand flight altitude up to z = 20 km. Figure 1-3 shows that, at constant flightvelocity, the Mach number increases with altitude because the sonic speed decreases,as was shown in Table 1-3. At an altitude of 10 krn, the speed of sound hasdropped to 300 m/s. At the same flight velocity, the Mach number at 10 km ofaltitude is about 10% larger than at sea level. This fact is important for theestimation of the aerodynamic properties of an airplane flying near the speed ofsound.

The Reynolds numbers in Fig. 1-4 are those for a reference length of l = 1 in,where 1 may be the wing chord, fuselage length, or control surface chord. TheReynolds numbers of the diagram must be multiplied by a factor that correspondsto the reference length l in meters. Since the kinematic viscosity increasesconsiderably with increasing height (see Table 1-3), the Reynolds number decreases

INTRODUCTION 11

2,2

2.0

18

1.6

14

08

06

0.4

02

11

12 INTRODUCTION

sharply with increasing height for a constant flight velocity, making airplane drag aparticularly strong function of the height.

1-3-2 Aerodynamic Forces and Moments on AircraftLift, drag, and lift-drag ratio Airplanes moving with constant velocity are subject toan aerodynamic force R (Fig. 1-5). The component of this force in direction of theincident flow is the drag D, the component normal to it the lift L.

Lift is produced almost exclusively by the wing, drag by all parts of the aircraft(wing, fuselage, empennage). Drag will be discussed in detail in the followingchapters. It has several fluid mechanical causes: By friction (viscosity, turbulence)on the surfaces, friction drag is produced, which is composed of shear-stress dragand a friction-effected pressure drag. This kind of drag depends essentially on theaircraft geometry and determines mainly the drag at zero lift. It is called form dragor also profile drag. As a result of the generation of lift on the wing, a so-calledinduced drag is created in addition (eddy drag), which depends strongly on theaspect ratio (wing span/mean wing chord). An aircraft flying at supersonic velocityis subject to a so-called wave drag, in addition to the kinds of drag mentionedabove. Wave drag is composed of a component for zero lift (form wave drag) and acomponent caused by the lift (lift-induced wave drag).

The inclination of the resultant R to the incident flow direction andconsequently the ratio of lift to drag depend mainly on wing geometry and incidentflow direction. A large value of this ratio LID is desirable, because it can beconsidered to be an aerodynamic efficiency factor of the airplane. This efficiencyfactor has a distinct meaning in unpowered flight (glider flight) as can be seen fromFig. 1-5. For the straight, steady, gliding flight of an unpowered aircraft, theresultant R of the aerodynamic forces must be equal in magnitude to the weight Wbut with the sign reversed. The lift-drag ratio is given, therefore, after Fig. 1-5, bythe relationship

tall E=D

where a is the angle between flight path and horizontal line.

Horizontal direction

Flight path

(1-18)

Figure 1-5 Demonstration of glide angle E.

INTRODUCTION 13

The minimum glide angle EI,, is a very important quantity of flightperformance, particularly for glider planes. It is given by (L/D)max after Eq. (1-18).The outstanding characteristic of the wing, in comparison to the other parts of theaircraft, is its quite large lift-drag ratio. Here are a few data on LID for incompressibleflow: A rectangular plate of an aspect ratio A = b/c = 6 has a value of (L/D)max of6-8. Considerably greater lifts for about the same drag are obtained when the plateis somewhat arched. In this case (L/D)max reaches 10-12. Even more favorablevalues of (L/D)max are obtained with wings that are streamlined. Particularly, theleading edge should be well rounded, whereas the profile should taper out into asharp trailing edge. Such a wing may have an (L/D)m of 25 and higher.

Further forces and moments, systems of axes We saw that, for symmetric incidentflow, the resultant of aerodynamic forces is composed of lift and drag only. In thegeneral case of asymmetric flow, the resultant of the aerodynamic forces may becomposed of three forces and three moments. These six components correspond tosix degrees of freedom of the aircraft motion. We introduce two systems of axes,depending on the flight mechanical requirements, to describe these forces andmoments (Fig. 1-6).

1. Airplane-fixed system: Xf, Y f, Zf2. Experimental system: Xe, Ye, Ze

The origin of the coordinates is the same in the two systems and is located in thesymmetry plane of the aircraft. Its location in this plane is chosen to suit the specificproblem. For flight mechanical studies, the origin is usually put into the aircraftcenter of gravity. For aerodynamic computations, however, it is preferable to putthe origin at a point marked by the aircraft geometry. In wing aerodynamics it isadvantageous to choose the geometric neutral point of the aircraft, as defined inSec. 3-1.

The lateral axes of the experimental system of axes xe, ye, ze and of thesystem fixed in the airplane xf, yf, z f coincide so that ye = y f. The experimentalsystem is obtained from the airplane-fixed system by rotation about the lateral axisby the angle a (angle of attack) (Fig. 1-6).

For symmetric incident flow, the aerodynamic state of the aircraft is definedby the angle of attack a and the magnitude of the velocity vector. For asymmetricincidence, the angle of sideslip 0* is also needed. It is defined as the angle betweenthe direction of the incident flow and the symmetry plane of the aircraft (Fig.1-6).

Translator's note: According to the definition given by NASA, the angle of sideslip is theangle between the direction of the incident flow and the symmetry plane of the airplane. Theangle of yaw is referred to a chosen direction, which may sometimes be the direction of theairflow past the body, making the angle of yaw equal to the angle of sideslip. Under someconditions, however, as in turning, a different reference direction may be used.

14 INTRODUCTION

Mze C)Plane of irI

low directionIncident f

wz

Reference plane

Z f3e

1-7t z

Figure 1-6 Systems of flight mechanical axes: airplane-fixed system, xf, yf, zf; experimentalsystem, xe, ye, ze; angle of attack, a; sideslip angle, R; angular velocities, wX, wy, wz

Forces and moments in the two coordinate systems are defined as follows:

1. Aircraft-fixed system:x f axis: tangential force Xf, rolling moment Mx fyf axis: lateral force Yf, pitching mdment Mf (or Myf)zf axis: normal force Zf, yawing moment Mzf

2. Experimental system:Xe axis: tangential force Xe, rolling moment MxeYe axis: lateral force Ye, pitching moment Me (or Mye)ze axis: normal force Ze, yawing moment Mze

The signs of forces and moments are shown in Fig. 1-6.It is customary to use lift L and drag D in addition to the forces and moments.

They are interrelated as follows:

L = -Z,, D = -X,? (for 1i = 0) (1-19)Further, because of the coincidence of the lateral axes yf = y,

Yf= Ye Mf=Me =M (1-20)

Dimensionless coefficients of forces and moments For the representation ofexperimental results and also for theoretical calculations, it is expedient tointroduce dimensionless coefficients for the moments and forces defined in thepreceding paragraph. These coefficients are called aerodynamic coefficients of theaircraft. They are related to the wing area AW, the semispan s, the reference wing

INTRODUCTION 15

chord c (Eq. 3-5b), and to the dynamic pressure q = O V'/2, where V is the flightvelocity (velocity of incident flow). Specifically, they are defined as follows.

Lift:

Drag:

Tangential force:

Lateral force:

Normal force:

Rolling moment:

Pitching moment:

Yawing moment:

L = cLA Wq

D = cDA wq

X=cxAwqY=cyAx,qZ=czAwqMx = cmxA W sq

M= cMAwcuq

Mz = c Awsq

(1-21)

A measurement that determines the three coefficients CL, cD, and cm as afunction of the angle of attack a is called a three-component measurement. Thediagram CL(CD) with a as the parameter was introduced by Lilienthal [1]. It iscalled the polar curve or the drag polar. If all six components are measured, forexample, of a yawed airplane, such a test is called a six-component measurement.Normally, the coefficients of forces and moments of aircraft depend considerablyon the Reynolds number Re and the Mach number Ma; in addition to the geometricdata. At low flight velocities, however, the influence of the Mach number on forceand moment coefficients is negligible.

1-3-3 Interrelation between the Aerodynamic Forcesand the Modes of Motion of the Airplane

Motion modes of the airplane After having discussed the aerodynamic forces andthe moments acting on the aircraft, its modes of motion may now be describedbriefly. An airplane has six degrees of freedom, namely, three components oftranslational velocity V, Vy, V, and three components of rotational velocity wx,wy, wZ. They can be expressed, for instance, relative to the aircraft-fixed system ofaxes x, y, z as in Fig. 1-6. The components of the aerodynamic forces, asintroduced in Sec. 1-3-2, and their dimensionless aerodynamic coefficients arefunctions of these six degrees of freedom of motion.

The steady motion of an aircraft can be split up into a longitudinal and alateral motion. During longitudinal motion, the position of the aircraft plane ofsymmetry remains unchanged. It is characterized by the three components ofmotion

Vx, VZ, wy (longitudinal motion)The remaining three components determine the lateral motion

Vy, wx, wZ (lateral motion)

16 INTRODUCTION

It is expedient for the analysis of the interrelation of aerodynamic coefficientsand components of motion to break down the general motion into straight flight, asdescribed by Vx and VV; yawed flight, described by Vy; and rotary motion aboutthe three axes. These rotary motions are, specifically, the rolling motion wx, thepitching motion coy, and the yawing motion wZ. The quantities of angle of attack aand angle of yaw !3,* which were introduced earlier (see Fig. 1-6), are then given by

tan a = Zf and tan Vyf (1-22)Vxfxf

The signs of a, a, o. , wy, and wZ can be seen in Fig. 1-6. At unsteady states offlight, the aerodynamic forces also depend on the acceleration components of themotion.

Forces and moments during straight flight The incident flow direction of anairplane in steady straight flight is given by the angle of attack a (Fig. 1-6). -Theresultant aerodynamic force is fixed in magnitude, direction, and line of applicationby lift L, drag D, and pitching moment M (Fig. 1.6). Let us now give some detailson the dimensionless aerodynamic coefficients introduced in Sec. 1-3-2. Formoderate angles of attack, the lift coefficient CL is a linear function of the angle ofattack a:

CL = (a - ao) deLd (1-23)

where as is the zero-lift angle of attack and dcLlda is the lift slope. A furthercharacteristic quantity for the lift is the maximum lift coefficient CLmax, which isreached at an angle of attack that depends on the airplane characteristics.

For moderate angles of attack and lift coefficients, the drag coefficient CD isgiven by

CD = CDO + k, CL + k2cL (1-24)where CDO is the drag coefficient at zero lift (form drag). The constants kl and k2depend mainly on the wing geometry.

For wings of symmetric profile without twist we have kl = 0, and thus

CD = CDO + k2 CL (1-25)This is the representation of the drag polar.

The pitching-moment coefficient cm is a linear function of the angle of attacka and the lift coefficient cL, respectively:

C M C M O + dCM CL (1-26)L

where cMo is the zero-moment coefficient and dcM/dcL is the pitching-momentslope. The value of cMo is independent of the choice of the moment reference

*The angle R has been designated here as the angle of yaw. For the difference betweenangle of yaw and angle of sideslip see the footnote on page 13.

INTRODUCTION 17

station, whereas dcM/dcL depends strongly on it. The quantity dcM/dcL is alsocalled the "degree of stability of longitudinal motion" (rotation about lateral axis).The resultant of the aerodynamic forces of the airplane is completely determinedonly when its magnitude, direction, and the position of its line of application areknown. These three data are obtained, for instance, from lift, drag, and pitchingmoment. The position of the line of application of the resultant R, for example, onthe wing, can be defined as the intersection of the line of application with theprofile chord (Fig. 1-7a). This point is called the center of pressure or aerodynamiccenter of the wing. With XA, the distance of the center of pressure from themoment reference axis, we have

M=.AZFor small angles of attack, the normal force with the negative sign is, in firstapproximation, equal to the lift:

Z= -Land by introducing the nondimensional coefficients,

xL CM1 27( - a)

C CZ

CM _ dcM CMO1 27b( - )

CL dCL CL

Figure 1-7 Demonstration of location ofaerodynamic center (center of pressure). (a)Aerodynamic center C. (b) Neutral point N.In general, the reference wing chord is c = c.,.

18 INTRODUCTION

This relationship means that the position of the center of pressure generallyvaries with the lift coefficient. The shift of the center-of-pressure position is givenby the term -CMO /CL

.

In agreement of theory with experiment, the pitching moment can generally bedescribed as the sum of a force couple independent of lift (zero moment) and aterm proportional to the lift:

M=M0 -xNLIn words, the pitching moment is the sum of the zero moment and of the

moment formed by the lift force and the distance XN between the neutralpoint and the moment reference line (Fig. 1-7b). Again introducing the non-dimensional coefficients for lift and pitching moment:

XNCM = CMO - CL (1-28)CA

Comparison with Eq. (1-26) yields, for the position of the neutral pointxN dcMcA. dcL

(1-29)

which shows that the pitching-moment slope dcMldcL determines the position ofthe neutral point. The terms dcL/da and dcM/da are designated as derivatives

,--of longitudinal motion.

Forces and moments in yawed flight When an aircraft is in stationary yawed flight,the direction of the incident flow of the wing is determined by both the angle ofattack a and the angle of sideslip 1 (Fig. 1-6). Because of the asymmetric flowincidence, additional forces and moments are produced besides lift, drag, andpitching moment as discussed in the last section. The force in direction of thelateral axis y is the side force due to sideslip; the moment about the longitudinal axis,the rolling moment due to sideslip; and the moment about the vertical axis, the yawingmoment due to sideslip. The derivatives for 0 = 0,

(8C Y) 0=oap

aCMZI

as Q=0are called stability coefficients of sideslip; in particular, acMZ/aa is called directionalstability. All three of these coefficients are strongly dependent on the wingsweepback, besides other influences.

Forces and moments in rotary motion An airplane in rotary motion about the axesx, y, z, as specified by the modes of motion of Sec. 1-3-3, is subject to additionalvelocity components that are produced, for example, locally on the wing and thatchange linearly with distance from the axis of rotation. The aerodynamic forces andmoments that are the result of the angular velocities wX, wy, wZ will now bediscussed briefly.

During rotary motion of the airplane about the longitudinal axis (roll) with

INTRODUCTION 19

angular velocity co, the lift distribution on the wing, for instance, becomesantisymmetric along the wing span. The resulting moment about the x axis canbe called a rolling moment due to roll rate. It always counteracts the rotary motionand is, therefore, also called roll damping. The asymmetric force distribution alongthe span produces also a yawing moment, the so-called yawing moment due to rollrate. Introducing the dimensionless coefficients according to Eq. (1-21), the stabilitycoefficients of sideslip

acMx acmz

aS?and

asp

are obtained.The quantity .Q is the dimensionless angular velocity cw,. It is obtained from wX,

the half-span s, and the flight velocity V:

5Q,; = E. -I,V (1-30)

The rotary motion of an airplane about the vertical axis (yaw) producesadditional longitudinal air velocities on the wing that have reversed signs on the twowing halves and that result in an asymmetric normal and tangential forcedistribution along the wing span, which in turn produces a rolling and a yawingmoment. The yawing moment created in this way counteracts the rotary motionand is called yawing or turning damping. The rolling moment is called rollingmoment due to yaw rate. Again by introducing nondimensional coefficients after Eq.(1-21), further stability coefficients of yawing motion are formed:

acMx

aQZ andacmz

aQZ

Here the nondimensional yawing angular velocity is

(1-31)

The rotary motion of the aircraft about the lateral axis (pitch), Fig. 1-6,produces on the wing an additional component of the incident velocity in the zdirection that is linearly distributed over the wing chord. This results in anadditional lift due to pitch rate and an additional pitching moment that counteractsthe rotary motion about the lateral axis. Therefore, it is also called pitch dampingof the wing. The magnitude of the pitch damping is strongly dependent on theposition of the axis of rotation (y axis). By using lift and pitching-momentcoefficients after Eq. (1-21), the following additional stability coefficients oflongitudinal motion are obtained:

aCL acMasp,, and asp,,

20 INTRODUCTION

The nondimensional pitching angular velocityy

Dy V (1.32)

is made dimensionless with wing reference chord after Eq. (3-5b) contrary tothe rolling and yawing angular velocities Q,, and Qy , respectively, which were madedimensionless with reference to the wing half-span.

Only the most important aerodynamic forces and moments produced by therotary motion of the aircraft have been discussed above. Omitted, for instance, weredetailed discussions of the side forces due to roll rate and yaw rate.

Forces and moments in nonsteady motion Besides the steady aerodynamiccoefficients discussed above, the nonsteady coefficients applicable to accelerated flighthave increasingly gained importance, particularly for flight mechanical stabilityconsiderations. Nonsteady motions are more or less sudden transitions from onesteady state to another or time-periodic motions superimposed on a steady motion.If the periodic motion is very slow (e.g., changes of angle of attack), theaerodynamic forces can be computed from quasi-stationary theory; this means that,for instance, the momentary angle of attack determines the forces. With periodicmotions of higher frequency, however, the aerodynamic forces are phase-shifted(leading or lagging) from the motion. These conditions are demonstrated schemati-cally in Fig. 1-8 for an airplane undergoing a periodic translational motion normalto its flight path.

At nonsteady longitudinal motion, new aerodynamic force coefficients must beused, for example, the derivatives

aCL

ac

aCMand

Angle of attack

W CM

a

Figure 1-8 Schematic presentation ofquasi-stationary and nonsteady aerody-namic forces.

INTRODUCTION 21

W=O w-i a

w

w=a

Figure 1-9 Propagation of sound waves from a sound source moving at the velocity w through afluid at rest. (a) Sound source at rest, w = 0. (b) Sound source moving at subsonic velocity,w = a12. (c) Sound source moving at sonic velocity, w = a. (d) Sound source moving atsupersonic velocity, w = 2a; the sound waves propagate within the Mach cone of apex semiangle g.

where a= daldt is the timewise change of the angle of attack. The nonsteadycoefficients are important both for flight mechanics of the aircraft, assumed to beinflexible, and for questions concerning the elastically deformable airplane (aero-elasticity).

Forces and moments in supersonic flight During the transition from subsonic tosupersonic flight, the aerodynamic behavior of an airplane undergoes a basic change.This becomes obvious when the airplane is taken as the source of a disturbance thatmoves through still air at a velocity V= w. Relative to this moving centerof disturbance, pressure waves emanate with the speed of sound a. A closerinvestigation of this process shows the importance of the speed of sound-especiallythe ratio of flight velocity to sonic speed, that is, the Mach number from Eq.(1-16). In terms of fluid mechanics, the airplane can be considered as a soundsource. Figure 1-9a shows the propagation of sound waves from a sound source atrest on concentric spherical surfaces. In Fig. 1-9b the sound waves, emitted at equaltime intervals, can be seen for a source that moves with one-half the speed ofsound, w = a/2. Figure 1-9c is the corresponding picture for w = a and finally, Fig.1-9d is for w = 2a. In this last case, in which the sound source moves at supersonicvelocity, the effect of the source is felt only within a cone with the apex semiangle, which is given by

22 INTRODUCTION

at a 1smLL =-=-=-WT to Ma (1-33)

This cone is called the Mach cone. No signals can be sent from the source to pointsoutside of the Mach cone, a zone called the zone of silence. No sound is heard,therefore, by an observer who is being approached by a body flying at supersonicspeed. Physically, the process described is obviously identical to a sound source atrest in a fluid approaching from the right with velocity w. We have to keep in mind,therefore, the following characteristic difference: When the fluid velocity is smallerthan the speed of sound (w a, supersonic flow), pressure disturbances can propagateonly within the Mach cone situated downstream of the sound source (Fig. 1-9d).

Now, every point of the airplane surface can be considered as the source of adisturbance (sound source) as in Fig. 1-9, in analogy to the previous discussionwhere the whole airplane was taken as the sound source. It can be concluded,therefore, that because of the different kinds of propagation of the individualpressure disturbances as in Fig. 1-9b and d, the pressure distribution andconsequently the forces and moments on the various parts of the airplane (wing,fuselage, control surfaces) depend decisively on the airplane Mach number, whetherthe airplane flies at subsonic or supersonic velocities.

The above considerations show that subsonic flow has the characteristicproperties of incompressible flow, whereas supersonic flow is basically different. Inmost cases, therefore, it will be expedient to treat subsonic and supersonic flowsseparately.

REFERENCES

1. Lilienthal, 0.: "Der Vogelflug als Grundlage der Fliegekunst," 1889; 4th ed., Sandig,Wiesbaden, 1965.

2. "U.S. Standard Atmosphere," National Oceanic and Atmospheric Administration and NationalAeronautics and Space Administration, Washington, D.C., 1962.

PART

ONEAERODYNAMICS OF THE WING

CHAPTER

TWOAIRFOIL OF INFINITE SPANIN INCOMPRESSIBLE FLOW

(PROFILE THEORY)

2-1 INTRODUCTION

In this chapter the airfoil of infinite span in incompressible flow will be discussed.The wing of finite span in incompressible flow will be the subject of Chap. 3, andthe wing in compressible flow that of Chap. 4. More recent results andunderstanding of the aerodynamics of the wing profile are communicated inprogress reports by, among others, Goldstein [19], Schlichting [56], and Hummel[26].

Wing profile The wing profile is understood to be the cross section of the wingperpendicular to the y axis. Accordingly, the profile lies in the xz plane anddepends, in the general case, on the spanwise coordinate y. The geometry of a wingprofile may be described, as in Fig. 2-la, by introducing the connecting line of thecenters of the inscribed circles as the mean camber (or skeleton) line, and the lineconnecting the leading and trailing edges of the mean camber line as the chord. Thegreatest distance, measured along the chord, is called the wing or profile chord c.The largest diameter of the inscribed circles is designated as the profile thickness t(Fig. 2-1b), and the greatest height of the mean camber line above the chord as themaximum camber h (Fig. 2-1c). The positions of the maximum thickness and themaximum camber are given by the distances xt (thickness position) and xh (camberposition). The radius of the circle inscribed at the profile leading edge is taken asthe nose radius rN; it is usually related to the thickness. The trailing ede angle 27-4)

25

26 AERODYNAMICS OF THE WING

C

Chord

Figure 2-1 Geometric terminology of lift-ing wing profiles. (a) Total profile. (b)Profile teardrop (thickness distribution).(c) Mean camber (skeleton) line (camberheight distribution).

(Fig. 2-1b) is another important quantity. From these designated quantities thefollowing six geometric profile parameters may be formed:

t/c relative thickness (thickness ratio)*hlc relative camber (camber ratio)*xtlc relative thickness positionxh /c relative camber positionrN/c relative nose radius2r trailing edge angle

For the complete description of a profile, the profile coordinates of the upperand lower surfaces, zu(x) and zl(x), must also be known. A profile can beconsidered as originating from a mean camber line z(s)(x) on which is superimposeda thickness distribution (profile teardrop shape) z(t)(x) > 0. For moderate thicknessand 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 signto the lower surface.

*These quantities may be called in the text simply "thickness" and "camber" when amisunderstanding is impossible.

AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 27

For the following considerations, the dimensionless coordinates

X= x and Z= zc C

are introduced. The origin of coordinates, x = 0, is thus found at the profile leadingedge.

Of the large number of profiles heretofore developed, it is possible to discussonly a small selection in what follows. Further information is given by Riegels [501.

The first systematic investigation of profiles was undertaken at the Aero-dynamic Research Institute of Gottingen from 1923 to 1927 on some 40Joukowsky profiles [47]. The Joukowsky profiles are a two-parameter family ofprofiles that are designated by the thickness ratio t/c and the camber ratio h/c (seeSec. 2-2-3). The skeleton line is a circular arc and the trailing edge angle is zero (theprofiles accordingly have a very sharp trailing edge).

The most significant and extensive profile systems were developed, beginning in1933, at the NACA Research Laboratories in the United States.* Over the years theoriginal NACA system was further developed [ 1 ] .

For the description of the four-digit NACA profiles (see Fig. 2-2a), a newparameter, the maximum camber position xh/c was introduced in addition to thethickness t/c and the camber h/c. The maximum thickness position is the same for all

*NACA = National Advisory Committee for Aeronautics.

Teardrop

63-

Z(s)

Mean camberor skeleton

Z (0

a

b

C

69-

65-

66-

a a0h

-0063C

-0.068

h=0,095

C

a-0.2

a=05

a=20h- = 0.055c

Figure 2-2 Geometry of the most important NACA profiles. (a) Four-digit profiles. (b)Five-digit profiles. (c) 6-series profiles.

28 AERODYNAMICS OF THE WING

profiles xt/c = 0.30. With the exception of the mean camber (skeleton) line forXh = XhIC = 0.5, all skeleton lines undergo a curvature discontinuity at the locationof maximum camber height. The mean camber line is represented by two connectedparabolic 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 isequal to that of the four-digit NACA profiles. The relative camber position,however, is considerably smaller. A distinction is made between mean camber lineswith and without inflection points. The mean camber lines without inflection pointsare composed of a parabola of the third degree in the forward section and a straightline in the rear section, connected at the station X= m without a curvaturediscontinuity.

In the NACA 6-profiles (see Fig. 2-2c), the profile teardrop shapes and themean camber lines have been developed from purely aerodynamic considerations.The velocity distributions on the upper and lower surfaces were given in advancewith a wide variation of the position of the velocity maximums. The maximumthickness position xtlc lies between 0.35 and 0.45. The standard mean camber line iscalculated to possess a constant velocity distribution at both the upper and lowersurfaces. 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 expressionsfor the parabolic biconvex profile and the parabolic mean camber line are

Z(t) = 2 t X(1 - X)C

Z(s) = 4 hX(1 - X)

(2.4a)

(24b)Here, t is the maximum thickness and h is the maximum camber height located atstation X = 2

The so-called extended parabolic profile is obtained by multiplication of theabove equation with (1 + bX) in the numerator or denominator. According toGlauert [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 teardropshape 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 inflectionpoints that have various values of the maximum thickness position and maximumcamber position, respectively. The constants a and b are determined as follows:

AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 29

1t 1-2XtTeardrop: 2Xr c b Xt (2-7a)

1 h 1-2XhSkeleton: a=

xh2 cb=

x2jt (2-7b)

Of the profiles discussed above, the drop-shaped ones shown in Fig. 2-2 have arounded nose, whereas those given mathematically by Eq. (2-6) in connection withEq. (2-7a) have a pointed nose. The former profiles are therefore suited mainly forthe subsonic speed range, and the latter profiles for the supersonic range.

Pressure distribution In addition to the total forces and moments, the distributionof local forces on the surface of the wing is frequently needed. As an example, inFig. 2-3 the pressure distribution over the chord of an airfoil of infinite span ispresented for various angles of attack. Shown is the dimensionless pressurecoefficient

Cp =P -P.

q00

versus the dimensionless abscissa x/c. Here (p - p0,) is the positive or negativepressure difference to the pressure po, of the undisturbed flow and q., the dynamicpressure 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 ratiowith the profile NACA 2412 [12]. Reynolds number Re = 2.7 . 106. Mach number Ma = 0.15.Normal force coefficients according to the following table:

a - 1.70 2.8' 7.4 13.9 17.8'

-CZ 0.024 0.433 0.862 1..0,56 0.950

30 AERODYNAMICS OF THE WING

from the profile upper surface as indicated by the constant pressure over a widerange of the profile chord.

The pressures on the upper and lower surfaces of the profile are designated aspu and pl, respectively (see Fig. 2-3), and the difference d p = (p1- pu) is ameasure for the normal force dZ = A pb dx acting on the surface element dA = b dx(see Fig. 2-5). By integration over the airfoil chord, the total normal force becomes

c

Z= -b fd p(x) dx (2-9a)0= c2q.bc (2-9b)

where cZ is the normal force coefficient from Eq. (1-21) (see Fig. 2-3). For smallangles of attack a, the negative value of the normal force coefficient can be setequal to the lift coefficient cL :

JAcp(x)CL = dx0

The pitching moment about the profile leading edge is

(2-10)

M= -b f Ap(x) dx (2-11a)0

cMq.bc2 (2-11 b)where nose-up moments are considered as positive. The pitching-moment coefficientis, accordingly,

1 cCM=--fdcp(x)dx0

2-2 FUNDAMENTALS OF LIFT THEORY

(2-12)

2-2-1 Kutta-Joukowsky Lift TheoremTreatment of the theory of lift of a body in a fluid flow is considerably lessdifficult than that of drag because the theory of drag requires incorporation of theviscosity of the fluid. The lift, however, can be obtained in very goodapproximation from the theory of inviscid flow. The following discussions may bebased, therefore, on inviscid, incompressible flow.* For treatment of the problem ofplane (two-dimensional) flow about an airfoil, it is assumed that the lift-producingbody is a very long cylinder (theoretically of infinite length) that lies normal to the

*The influence of friction on lift will be considered in Sec. 2-6.

AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 31

flow direction. Then, all flow processes are equal in every cross section normal tothe generatrix of the cylinder; that is, flow about an airfoil of infinite length istwo-dimensional. The theory for the calculation of the lift of such an airfoil ofinfinite span is also termed profile theory (Chap. 2). Particular flow processes thathave a marked effect on both lift and drag take place at the wing tips of finite-spanwings. These processes are described by the theory of the wing of finite span (Chaps. 3and 4).

Lift production on an airfoil is closely related to the circulation of its velocitynear-field. Let us explain this interrelationship qualitatively. The flow about anairfoil profile with lift is shown in Fig. 24. The lift L is the resultant of thepressure forces on the lower and upper surfaces of the contour. Relative to thepressure at large distance from the profile, there is higher pressure on the lowersurface, lower pressure on the upper surface. It follows, then, from the Bernoulliequation, 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, itis easily seen from Fig. 2-4 that the circulation, taken as the line integral of thevelocity along the closed curve K, differs from zero. But also for a curve lying veryclose to the profile, the circulation is unequal to zero if lift is produced. Thevelocity field ambient to the profile can be thought to have been produced by aclockwise-turning vortex T that is located in the airfoil. This vortex, whichapparently is of basic importance for the creation of lift, is called the bound vortexof 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 simplifiedderivation, which will now be given, is not quite correct but has the virtue of beingparticularly 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 ofplanform area dA = b dx is subject to a lift dL = (pl - pu) dA because of thepressure difference between the lower and upper surfaces of the airfoil. The vectordL can be assumed to be normal to the direction of incident flow if the smallangles are neglected that are formed between the surface elements and the incidentflow direction.

The pressure difference between the lower and upper surfaces of the airfoil canbe expressed through the velocities on the lower and upper surfaces by applying the

wo,

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

32 AERODYNAMICS OF THE WING

4dL

Puwo,

00-P00

Figure 2-5 Notations for the computation of liftfrom the pressure distribution on the airfoil.

Bernoulli equation. From Fig. 2-4, the velocities on the upper and lower surfaces ofthe airfoil are (w + J w) and (w - J w), respectively. The Bernoulli equationthen furnishes for the pressure difference

1 P=pt - pu = 2 (wo,, + d w)2 - (w - A u')2 - 2Q u J wwhere the assumption has been made that the magnitudes of the circulatoryvelocities on the lower and upper surfaces are equal, I d wji = JA wju = 1Aw1.

By integration, the total lift of the airfoil is consequently obtained as

L=C

f.JpdA=b -1 p dxJ (2-13a)-(A)/4w= 2 obwoo dx (2-13b)

The integration has been carried from the leading to the trailing edge (length ofairfoil chord c).

The circulation along any line 1 around the wing surface is

.17= w d l (2-14a)(1)

AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 33

C' B CI'= fdzvdx- fdzvdx=2 fdwdx (2-14b)

B,u C,[ B

The first integral in the first equation is to be taken along the upper surface,the second along the lower surface of the wing. From Eq. (2-13b) the lift is thengiven by

L = o b zv, l' (2-15)This equation was found first by Kutta [35] in 1902 and independently byJoukowsky [31] in 1906 and is the exact relation, as can be shown, between liftand circulation. Furthermore, it can be shown that the lift acts normal to thedirection of the incident flow.

2-2-2 Magnitude and Formation of CirculationIf 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 clarifiedas to what way the circulation is related to the geometry of the wing profile, to thevelocity of the incident flow, and to the angle of attack. This interrelation cannotbe determined uniquely from theoretical considerations, so it is necessary to lookfor empirical results.

The technically most important wing profiles have, in general, a more or lesssharp trailing edge. Then the magnitude of the circulation can be derived fromexperience, namely, that there is no flow around the trailing edge, but that the fluidflows 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 rearstagnation point, that is, the point at which the streamlines from the upper andlower sides recoalesce, would lie on the upper surface. Such a flow pattern wouldbe possible only if there were flow around the trailing edge from the lower to theupper surface and, therefore, theoretically (in inviscid flow) an infinitely highvelocity at the trailing edge with an infinitely high negative pressure. On the otherhand, in the case of a very large circulation (see Fig. 2-6b) the rear stagnation pointwould be on the lower surface of the wing with flow around the trailing edge fromabove. 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 rearstagnation point exactly on the sharp trailing edge. Therefore, no flow around thetrailing edge occurs, either from above or from below, and smooth flow-off isestablished. The condition of smooth flow-off allows unique determination of themagnitude of the circulation for bodies with a sharp trailing edge from the bodyshape and the inclination of the body relative to the incident flow direction. Thisstatement is valid for the inviscid potential flow. In flow with friction, a certainreduction of the circulation from the value determined for frictionless flow isobserved as a result of viscosity effects.

For the formation of circulation around a wing, information is obtained from

34 AERODYNAMICS OF THE WING

a

b

c

Figure 2-6 Flow around an airfoil for variousvalues of circulation. (a) Circulation l = 0: rearstagnation point on upper surface. (b) Very largecirculation: rear stagnation point on lower sur-face. (c) Circulation just sufficient to put rearstagnation point on trailing edge. Smooth flow-off: Kutta condition satisfied.

the conservation law of circulation in frictionless flow (Thomson theorem). Thisstates that the circulation of a fluid-bound line is constant with time. This behaviorwill be demonstrated on a wing set in motion from rest, Fig. 2-7. Each fluid-boundline 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 offriction, a left-turning vortex is formed with a certain circulation -F. This vortexquickly drifts away -from the wing and represents the -so-called starting or initialvortex -T (Fig. 2-7c).

For the originally observed fluid-bound line, the circulation remains zero, eventhough 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 thisfluid-bound line remains zero for all times according to the Thomson theorem,somewhere within this fluid-bound line a circulation must exist equal in magnitudeto 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 wingand is, therefore, some time after the beginning of the motion sufficiently far awayfrom the wing to be of negligible influence on the further development of the flowfield. The circulation established around the wing, which produces the lift, can be

AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 35

replaced by one or several vortices within the wing of total circulation +1' as far asthe influence on the ambient flow field is concerned. They are called the boundvortices.* From the above discussions it is seen that the viscosity of the fluid, afterall, causes the formation of circulation and, therefore, the establishment of lift. Inan inviscid fluid, the original flow without circulation and, therefore, with flowaround the trailing edge, would continue indefinitely. No starting vortex wouldform and, consequently, there would be no circulation about the wing and nolift

Viscosity of the fluid must therefore be taken into consideration temporarily toexplain the evolution of lift, that is, the formation of the starting vortex. Afterestablishment of the starting vortex and the circulation about the wing, thecalculation of lift can be done from the laws of frictionless flow using theKutta-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 thebound vortices of finite length that stay with the wing and the starting vortex that drifts offwith the flow. In the case of an airfoil of infinite span, which has been discussed so far, the freevortices are too far apart to play a role for the flow conditions at a cross section of atwo-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 stagnantfluid. (b) Wing shortly after beginning of motion;for the liquid line chosen in (a), the circulation1' 0; because of flow around the trailing edge, avortex forms at this station. (c) This vortex formedby flow around the trailing edge is the so-calledstarting vortex -r; a circulation +1' developsconsequently around the wing.

36 AERODYNAMICS OF THE WING

2-2-3 Methods of Profile TheorySince the Kutta-Joukowsky equation (Eq. 2-15), which forms the basis of lifttheory, has been introduced, the computation of lift can now be discussed in moredetail. First, the two-dimensional problem will be treated exclusively, that is, theairfoil of infinite span in incompressible flow. The theory of the airfoil of infinitespan is also called profile theory. Comprehensive discussions of incompressibleprofile theory, taking into account nonlinear effects and friction, are given by Betz[5], von Karman and Burgers [70], Sears [59], Hess and Smith [23], Robinsonand Laurmann [51], Woods [74], and Thwaites [67]. A comparison of results ofprofile 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, bythe method of conformal mapping, and second, by the so-called method ofsingularities. The first method is limited to two-dimensional problems. The flowabout a given body is obtained by using conformal mapping to transform it into aknown flow about another body (usually circular cylinder). In the method ofsingularities, 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. Themethod of singularities produces, in general, only approximate solutions, whereasconformal mapping leads to exact solutions, although these often require consider-able effort.

2-3 PROFILE THEORY BY THE METHODOF CONFORMAL MAPPING

2-3-1 Complex PresentationComplex stream function Computation of a plane inviscid flow requires much lesseffort than that of three-dimensional flow. The reason lies not so much in the factthat the plane flow has only two, instead of three, local coordinates as that it canbe treated by means of analytical functions. This is a mathematical discipline,developed in great detail, in which the two local coordinates (x, y) oftwo-dimensional flow can be combined to a complex argument. A plane,frictionless, and incompressible flow can, therefore, be represented as an analyticalfunction 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 twofamilies of orthogonal curves in the xy plane. By taking a suitable streamline as asolid wall, the other streamlines then form the flow field above this wall. Thevelocity components in the x and y directions, that is, u and v, are given by

AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 37

a 0 d IF c70 0'l-1u

c9x 7y V Jy Jx

The function F(z) is called a complex stream function. From this function, thevelocity field is obtained immediately by differentiation in the complex plane,where

dFd z

= it - i V = w(z) (2-17)Here, w = u - iv is the conjugate complex number to w = u + iv, which isobtained by reflection of w on the real axis. In words, Eq. (2-17) says that thederivative of the complex stream function with respect to the argument is equal tothe velocity vector reflected on the real axis.

The superposition principle is valid for the complex stream function precisely asfor the potential and stream functions, because F(z) = c, F, (z) + c2 F2(z) can beconsidered 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 theundisturbed flow velocity u,,., the complex stream function is

F (z) = u (z + a-) (2-18)For an irrotational flow around the coordinate origin, that is, for a plane

potential vortex, the stream function is

F(z) = irlnz2irwhere r is a clockwise-turning circulation.

(2-19)

Conformal mapping First, the term conformal mapping shall be explained (see [6] ).Consider an analytical function of complex variables and split it into real andimaginary components:

f (z) = f (x + y) (z, y) + i n (x, y) (2-20)The relationship between the complex numbers z =.x + iy and _ + iri in Eq.(2-20) can be interpreted purely geometrically. To each point of the complex zplane a point is coordinated in the plane that can be designated as the mirrorimage of the point in the z plane. Specifically, when the point in the z plane movesalong 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 technicalexpression of this process is that, through Eq. (2-20), the z plane is conformallymapped on the S plane. The best known and simplest mapping function is theJoukowsky mapping function,

= zca

-21)(2-21)

38 AERODYNAMICS OF THE WING

It maps a circle of radius a about the origin of the z plane into the twice-passedstraight line (slit) from -2a to +2a in the plane.

For the computation of flows, this purely geometrical process of conformalmapping of two planes on each other can also be interpreted as transforming acertain system of potential lines and streamlines of one plane into a system of those inanother plane. The problem of computing the flow about a given body can then besolved as follows. Let the flow be known about a body with a contour A in the zplane and its stream function F(z), for which, usually, flow about a circular cylinderis assumed [see Eq. (2-18)]. Then, for the body with contour B in the plane, theflow 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 theplane. At the same time, the known system of potential lines and streamlines aboutthe body A in the z plane is being transformed into the sought system of potentiallines and streamlines about the body B in the plane. The velocity field of the body Bto be determined in the plane is found from the formula

a (2-23)az d = w(z) d

F(z) and w(z) are known from the stream function of the body A in the z plane(e.g., circular cylinder). Here dz/d = 1 If '(z) is the reciprocal differential quotient ofthe mapping function = f(z). The sought velocity distribution i about body Bcan be computed from Eq. (2-23) after the mapping function f(z) that maps bodyA into body B has been found. The computation of examples shows that the majortask 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 integralformulas for the computation of the force and moment resultants on wing profiles infrictionless flow. They are based on the work of Blasius [71 J.

2-3-2 Inclined Flat PlateThe simplest case of a lifting-airfoil profile is the inclined flat plate. The anglebetween the direction of the incident flow and the direction of the plate is calledangle of attack a of the plate.

The flow about the inclined flat plate is obtained as shown in Fig. 2-8, bysuperposition of the plate in parallel flow (a) and the plate in normal flow (b). Theresulting flow

(c) = (a) + (b)

does not yet produce lift on the plate because identical flow conditions exist at theleading and trailing edges. The front stagnation point is located on the lower surfaceand the rear stagnation point on the upper surface of the plate.

U"a

b

v00

z plane

4a-C

plane

Figure 2-8 Flow about an inclined flat plate. (a) Flat plate in parallel flow. (b) Flat plate innormal (stagnation) flow. (c) Inclined flat plate without lift, (c) = (a) + (b). (d) Pure circulationflow. (e) Inclined flat plate with lift (Kutta condition), (e) = (c) + (d).

39

40 AERODYNAMICS OF THE WING

To establish a plate flow with lift, a circulation P according to Fig. 2-8d mustbe 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 thecondition of smooth flow-off at the plate trailing edge; for example, the rearstagnation point lies on the plate trailing edge (Kutta condition). By superpositionof the three flow fields, a flow is obtained around the circle of radius a with itscenter at z = 0. It is approached by the flow under the angle a with the x axis, abeing 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 plateof length c = 4a in the plane. The velocity distribution about the plate is obtainedwith the help of Eq. (2-23) after some auxiliary calculations as

vccs-W) = uC' T i

vt 2 - 4cc2(2-25)

The magnitude of the circulation T is now to be determined from the Kuttacondition. Smooth flow-off at the trailing e