Wing Theory [Course Book ed.] 9781400860777

Table of contents :
CONTENTS
Preface
CHAPTER ONE Fundamental Considerations
CHAPTER TWO. Potential Flow over Ellipsoids
CHAPTER THREE. Two-Dimensional Flow: Wing Section Theory
CHAPTER FOUR. Thin Airfoil Theory
CHAPTER FIVE. Influence of Compressibility
CHAPTER SIX. Effects of Sweep
CHAPTER SEVEN. Wings of High Aspect Ratio
CHAPTER EIGHT. Lifting Surface Theory
CHAPTER NINE. The Minimum Drag of Thin Wings
CHAPTER TEN. Drag of Wings and Bodies in Combination
Bibliography
Index

Citation preview

Wing Theory

WING THEORY Robert T. Jones

PRINCETON UNIVERSITY PRESS / Princeton, New Jersey

Copyright © 1990 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, Oxford All Rights Reserved Library of Congress Cataloging-in-Publication Data Jones, Robert T. (RobertThomas), 1910Wing theory / Robert T. Jones. p. cm. Bibliography: p. Includes index. ISBN 0-691-08536-6 (alk. paper) 1. Airplanes—Wings. 2. Aerofoils. 3. Aerodynamics. I. Title. TL574.A4J62 1990 629.134'32—dc20

89-36136 CIP

This book has been composed in Sabon. Princeton University Press books are printed on acid-free paper, and meet the guidelines for permanence and durability of the Committee on Production Guidelines for Book Longevity of the Council on Library Resources Printed in the United States of America by Princeton University Press, Princeton, New Jersey 10987654321

C O N T E N T S

Preface

ix

CHAPTER ONE Fundamental Considerations

3

Early Theories of Lift / 3 Multiplicity of Solutions / 4 Kutta-Joukowski Flow / 5 Incompressible Potential Flow / 7 Influence of Viscosity: Reynolds Number / 9

CHAPTER TWO Potential Flow over Ellipsoids

12

Ellipsoidal Coordinates: Lami Functions /12 Translation of Ellipsoids /13 Apparent M a s s a n d Velocity Distribution /15 Moment /17 Pressure Distribution /18

CHAPTER THREE Two-Dimensional Flow:

Wing Section Theory Conformal Mapping of Two-Dimensional Flows / 20 Joukowski Airfoils / 22 Extensions of the Joukowski Theory / 26

20

Vl I CONTENTS

Computer Programs for Airfoil Shapes and Pressures / 28 Laminar Flow Airfoils /31 Boundary Layer Measurements in Flight / 35 Special Purpose Airfoils / 38 Influence of Viscosity on Lift / 41 Behavior of Airfoils at Low Reynolds Numbers / 43 Table 3.2/48 Table 3.2 / 50 Appendix 152

CHAPTER FOUR Thin Airfoil Theory Approximations of Thin Airfoil Theory / 53 Table of Thin Airfoil Functions / 54 Effect of Control Flaps /56 Lift Curve Slope / 56 Aerodynamic Center I 57 Development of Lift Starting from Rest / 59 Table 4.1. Thin Airfoil Functions / 66

CHAPTER FIVE Influence of Compressibility Lateral Expansion of Streamlines / 74 One-Dimensional Flow Equation / 75 Prandtl-Glauert Transformation / 77 Transonic Flows: Shock Waves / 80 Supercritical Airfoils / 83 Thin Airfoil Theory for Supersonic Speed / 85

CHAPTER six Effects of Sweep History of the Sweep Effect / 90 Effect of Sweep: The Independence Principle / 91 Wings Ahead of and Behind the Mach Cone / 93 Oblique Wing at Sonic Speed / 94 Lift and Separation / 98 Oblique Flows with Viscosity /100 Pressures Near the Root Section /103 Spanwise Distribution of Drag /103

CONTENTS CHAPTER SEVEN Wings of High Aspect Ratio

VU

105

Prandtl-Munk Theory /105 Span Load Distribution and Minimum Drag /107 Wings Having Upturned Tips / 111 Minimum Drag with Structural Limitation /112 Drag of Wings with "Winglets" /113 Wing Flapping with Minimum Energy /114 Lift Curve Slope: Spanwise Loading /119 Weissinger Method /123 CHAPTER EIGHT Lifting Surface Theory

129

The Elliptic Wing at Subsonic Speed /129 Cambered Surface to Support a Prescribed Lift Distribution /136 Wings of Low Aspect Ratio /140 Supersonic Conical Flow /147 Reciprocal Relations /149 Table 8.1. Conical Flow Functions /150 CHAPTER NINE The Minimum Drag of Thin Wings

159

Conditions for Minimum Drag of Lifting Surfaces /159 Kogan's Theory for the Minimum Drag of a Lifting Surface /161 Minimum Drag Associated with Volume /166 Lifting Surfaces Having Minimum Drag /170 The Minimum Drag of Elliptic Wings as Determined by Kogan's Theory /180 Minimum Drag of Elliptic Wings with Thickness /191 Estimated Lift!Drag Ratios of Oblique Wings /195 Experiments with Oblique Wings /196 CHAPTER TEN Drag of Wings and Bodies in Combination Interference Drag: Busemann Biplane / 200 Wave Drag of Simple Volume Distributions / 202 Supersonic Area Rule / 205 Bibliography / 209 Index / 211

200

P R E F A C E

In this book I have tried to bring out important elements of what might be termed "classical wing theory" and to make more explicit the physical basis of its various components. It is, after all, these earlier theories and the physical insights they provided that formed the basis for the highly successful development of the modern air­ plane. Elaborations of the analysis, useful when methods of calcu­ lation were more primitive, have been omitted since they can now be accomplished better by the computer. Much of this material has been adapted from the Princeton Aeronautical Paperbacks publication, High Speed Wing Theory by Robert T. Jones and Doris Cohen. The present book is, however, wider in scope, giving equal coverage to both low and high speeds, including the behavior of wings at very low Reynolds numbers. A notable advance since that publication has been Μ. N. Kogan's theory of the minimum drag of wings at supersonic speed. Kogan's analysis provides a direct and intuitive connection between the older theory of "induced drag" and clearly shows the loss of effi­ ciency as the speed increases in the supersonic range. Aeronautical calculations today rely increasingly on the awe­ some power of the computer. However, as has been observed, power can corrupt. Equipped with an appropriate address book, giving the location and availability of various programs, the aero­ nautical engineer can now command the solution of a great variety

X I PREFACE

of aerodynamic problems. Moreover, the capacity of the computer has made possible the inclusion of many small physical influences that until now had to be neglected but sometimes create a false impression of high accuracy. However, the basic physical assump­ tions of calculations, if they are discussed at all, are often not given adequate treatment. If "computer aerodynamics" is to realize its full potential, then more attention must be devoted to these under­ lying principles. In his preface to Aerofoil and Airscrew Theory, first published in 1926, H. Glauert observes that the theory is incomplete in the sense that airfoil behavior can be predicted successfully only for a limited range of angles of attack. Now, sixty years later, there is still no satisfactory theory for the maximum lift of an airfoil, nor for the influence of widely used "high lift devices." In this area the engineer must rely on the large body of accumulated experience. The solution of this problem will pose a difficult challenge to the emerging science of computational fluid dynamics. I would like to express my indebtedness to my teachers Max Munk and Albert F. Zahm, now deceased, and to my wife Barbara for her enthusiastic help in organizing and editing the book. Los Altos Hills, California May 1989

Wing Theory

Fundamental Considerations

Early Theories of Lift The modern theory of fluid motion originated in the works of Euler, Lagrange, and other great mathematicians of the eighteenth and nineteenth centuries. Being based on idealized frictionless fluids, these early theories were unable to account for the most commonly observed physical phenomena, and they found little ap­ plication to practical problems. The later development of equa­ tions for the motion of viscous fluids did little to help this situation since these equations, known as the Navier-Stokes equations, are extremely difficult to treat mathematically. It was not until the de­ velopment of the airplane and its requirement for streamlined shapes, low drag, and efficient lift that classical fluid dynamics found a real application. Even so, aerodynamic theory was not prepared to give assist-

4 I CHAPTER ONE

ance in the early development of the airplane. Newton's theory of airflow and, later, the more sophisticated wing theory of Helmholtz and Kirchhoff, gave incorrect predictions and were often used to discount the possibility of flight by humans. Thus it is re­ corded that Lord Rayleigh expressed "not the smallest molecule of faith in aerial navigation except by balloon." The first correct physical description of the flow over a wing seems to have been given by F. W. Lanchester in England in 1894. Lanchester envisioned a flow over a curved thin plate having smooth streamlines with fore and aft symmetry—a flow giving lift without drag. Unfortunately, Lanchester's paper was rejected by the Royal Society and did not appear in print until quite a few years later. In the meantime, W. Kutta in Germany and N. Joukowski in Russia developed a concise mathematical theory for the lifting wing in two-dimensional flow. Experiments by Lilienthal, Langley, and the Wright brothers had of course already demon­ strated that human flight is possible. Multiplicity of Solutions The difficulty of the early theories may be attributed in part to the lack of uniqueness of solutions for the flow over a given shape. Figure 1.1 shows several possible solutions for the flow of a frictionless fluid over a thin flat plate. In the uppermost example, the flow clings to both the leading edge and trailing edges, leaving the airfoil at the rear upper surface. Such a flow develops a moment, but it does not develop lift or drag. In the second example, the flow separates from both leading and trailing edges, forming surfaces on discontinuity and a considerable "dead space" above and be­ hind the airfoil. This is the type of flow considered by Helmholtz and Kirchhoff. Examples (c) and (d) in Figure 1.1 are the more nearly correct physical solutions, although in the case of the perfectly thin plate they would require infinite velocities around the leading edge. In practice, of course, we round the leading edge, permitting the flow to remain attached.

FUNDAMENTAL CONSIDERATIONS | 5 Frictionless Flow (a) Flow Clings to Both Edges Zero Lift Zero Drag (b) Flow Separates at Both Edges Small Lift Large Drag (Helmholtz-Kirchhoff Flow) (c) Flow Clings to Leading Edge Separates at Trailing Edge Large Lift Zero Drag (Kutta-Joukowski Flow) (d) Flow Separates from Upper Surface Reduced Lift Small Drag

1.1. Variety of flow patterns given by hydrodynamic theory.

Kutta-Joukowski Flow In example (c), the Kutta-Joukowski flow, the airfoil develops lift without drag, a characteristic of frictionless potential flow that has come to be known as "d'Alembert's paradox." Wind tunnel experiments with smooth airfoil shapes tend to confirm d'Alem­ bert's paradox with surprising accuracy. Figure 1.2 illustrates the result of an experiment in which the airfoil extended completely across the wind tunnel so as to insure two-dimensional flow. Here the drag of the airfoil is compared to the drag of a circular wire at the same speed. The diameter of the wire having the same drag is only about .006 the width of the airfoil. Such an airfoil may easily develop a lift of one hundred pounds for a drag of one pound; lift to drag ratios approaching three hundred have been measured in these circumstances. Although the airfoil develops pressures of the order of P/2V2 at the nose, these are balanced by forward-acting pressures at the rear, with the result that the drag of the airfoil is almost entirely skin friction. The blunt shape of the wire, however, causes the flow to separate from the rear and negative pressures appear, resulting in a drag about equal to p/lV2 times its frontal

6

I CHAPTER ONE

area. The extraordinary efficiency of a well-shaped airfoil poses a difficult challenge to inventors who would like to devise a better method of producing lift. As mentioned earlier, the boundary condition corresponding to impermeability of the surface is not sufficient to establish the flow uniquely. Of the many flow configurations possible, that which re­ mains attached to the nose but separates smoothly from the trail­ ing edge agrees best with experiments at moderate angles of attack in the normal flight range. The condition that there can be no flow around the trailing edge, known as the Kutta condition, depends physically on the action of the boundary layer, which prevents the flow from turning the sharp corner at the trailing edge. Hence, while the pressures on the airfoil and the lift can usually be pre­ dicted with good accuracy by frictionless flow theory, the develop­ ment of lift depends ultimately on the existence of friction in the fluid. The physical nature of the Kutta condition can be clarified by examining two possible solutions for the flow through a pipe, as illustrated in Figure 1.3. Here we assume that some mechanism inside the pipe (perhaps a jet engine) creates the flow. In the first solution the flow enters the mouth of the pipe from all directions and leaves the same way. Such flow would create no thrust, and the fluid merely recirculates through the pipe. In the second solution, real flow, the boundary layer formed inside the pipe prevents the fluid from turning the corner at the exit, a jet is formed, and the mechanism in the pipe creates a thrust.

·-


I

ZT

1.3. Flows through a pipe.

An important feature of Kutta-Joukowski flow not found in earlier models such as that of Helmholtz and Kirchhoff is the oc­ currence of high velocities and large suction forces around the leading edge when the airfoil is at an angle of attack. It is this suc­ tion force that balances the rearwardly inclined lift on the major portion of the surface, leading to the production of lift without drag in frictionless flow. Figure 1.4 shows the pressures measured on an airfoil at 16° angle of attack and plotted as vectors perpen­ dicular to the surface. The pressures are compared with those cal­ culated by frictionless flow theory.1 At this angle of attack, veloci­ ties amounting to several times the stream velocity occur around the nose of the airfoil. In his Collected Works (1910), C. A. Chaplygin noted that these high velocities around the nose might in some cases exceed the velocity of sound and that continuous flow would then no longer be possible. Incompressible Potential Flow In general, specification of the flow around a body requires the determination of three component velocities at each point in space. 1R. M. Pinkerton, Calculated and Measured Pressure Distributions over the Midspan Section of the NACA 4412 Airfoil, NACA TR 563 (1936).

8

CHAPTER ONE Experimental Pressure Vectors

α = 16°

1.4. Pressure-vector diagrams for the NACA 4412 airfoil at 16° angle of attack.

For flows having a velocity potential the calculation is greatly sim­ plified, being reduced to the determination of a single scalar func­ tion of position. The velocity components are then the derivatives of this function in the different directions. The conditions for the existence of a velocity potential are (1) that no vorticity exists in the region of fluid being considered, and (2) in the case of a com­ pressible fluid, that the density be a single-valued function of the pressure. Any function satisfying the partial differential equation (Laplace's equation), φ

τ XX

+ Φ

^yy

+ Φ = 0:(φ ^ ZZ

'

VTrXX

θ2φ = — , etc.), Λγ2'

can represent the flow of fluid. Laplace's equation and the Ber­ noulli relation for pressure serve as the equation for irrotational motion of a fluid. It is important to note that flows satisfying La­ place's equation also satisfy the equations of motion for a viscous fluid (i.e., the Navier-Stokes equations), since the viscous stresses within the fluid contribute nothing to the acceleration of the fluid elements.2 The primary effect of viscosity originates in the contact of the fluid with a solid surface and results from the fact that the fluid sticks to the surface. The shearing motion of layers of fluid near the surface generates vorticity, and if the flow does not sepa2L. Prandtl and O. G. Tietjens, Fundamentals of Hydro and Aero Mechanics (New York: Dover, 1957).

FUNDAMENTAL CONSIDERATIONS | 9 rate, this vorticity will be confined to a thin layer near the sur­ face—the "boundary layer." In frictionless, potential flow the stress in the fluid is simply a scalar pressure, and the fluid elements move under the influence of the gradient of this pressure. If the flow is steady, the disturbance of pressure is strictly proportional to the square of the velocity, that is, the Bernoulli equation: P. „ p = const. — -V2.

Influence of Viscosity: Reynolds Number In a frictionless fluid the flow configuration and the pressures are independent of the scale of the motion. Bodies of different size experience geometrically similar distributions of pressure propor­ tional to ^V2. Introducing viscosity, we find an additional stress proportional to the first power of the velocity divided by some characteristic length in the flow, a velocity gradient. The additional stress will be μν/λ, where μ is the coefficient of viscosity, suppos­ edly a property of the fluid, independent of the state of motion, and λ is some characteristic length in the flow. Taking the ratio of the stress due to inertia of the fluid and that due to viscosity, we find R = pVA/μ. R is of course the Reynolds number. On the scale of human flight, Reynolds numbers tend to be quite large and the viscous stresses correspondingly small. Thus for a wing having a chord of one foot at a speed of 100 mph, the Reynolds number is about one million. Assuming a laminar boundary layer, the tangential stress due to viscosity will be ap­ proximately .0015 that of the scalar pressure. If flows on different scales of dimensions are to be geometrically and dynamically similar, the Reynolds number must be the same. However, at the Reynolds numbers of airplanes, the variation of the usual coefficients based on area and the square of the velocity is rather slow.

IO I

CHAPTER ONE

101—

Turbu lent

Laminar

107 R

1.5. Laminar and turbulent skin friction.

According to H. Bateman, the earliest recorded experiments to measure the force on a body in a moving stream were made by Edme Mariotte.3 Mariotte's Τταΰέ du Mouvement des Eaux ap­ peared in 1686, a year before the publication of Newton's Principia. Mariotte measured the skin friction on a flat plate. Subsequent experiments of this kind have extended their range enormously and have included various fluids such as water, oil, and air. Figure 1.5 shows a plot of skin-friction coefficient as a function of Reyn­ olds number based on the length of the plate. The measured values lie close to one of two curves, one corresponding to laminar mo­ tion in the boundary layer and the other to turbulent motion. The latter curve is more nearly horizontal, indicating a dependence closer to the square of the velocity. A small degree of surface roughness will render this curve absolutely horizontal, indicating that the viscous stress varies with the square of the velocity the same as the inertial stress, making the flow, at least on a macro­ scopic scale, independent of the Reynolds number. One of the most successful theories of turbulent motion is Prandtl's "mixing length" theory in which the results are independent of the Reyn­ olds number. As mentioned earlier, potential flows are not uniquely deter3 See H. L. Dryden, F. D. Murnahan, and H. Bateman, Hydrodynamics (New York: Dover, 1956).

FUNDAMENTAL CONSIDERATIONS | II

mined by the differential equation together with the condition of no flow through the surface. For any given body, a multiplicity of solutions having surfaces of discontinuity, or external vortices, ex­ ists.4 One might suppose that the introduction of viscosity and the additional boundary condition of zero velocity at the surface re­ quired by the Navier-Stokes equations would lead to a unique so­ lution. Evidently this is not the case. Attempts to prove uniqueness, or to determine the conditions for it, have not been successful ex­ cept at very low Reynolds numbers.5 The simplest exact solution of the Navier-Stokes equation—the laminar flow through a smooth pipe with a parabolic velocity profile—breaks down within a short distance into a seemingly random turbulent motion.

4M. Dupont, "Sur la Multiplicite des solutions en aerodynamique," parts 1 and 2, La Technique Aeronautique (December 15, 1926, and January 15, 1927); see also NACA TM 413 (1927). 5M. O. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow (London: Gordon and Breach Science Pub., 1969).

Potential Flow over Ellipsoids

Ellipsoidal Coordinates: Lame Functions The potential flow over ellipsoids is of interest in aeronautics: For example, an elongated prolate spheroid is useful as a model for the flow around an airship or a fuselage, while a thin flat ellipsoid can be used to demonstrate certain features of the flow over a wing. Moreover, ellipsoids provide simple exact solutions that can be used to assess the validity of certain approximations. Potential flows produced by motion of an ellipsoidal boundary are studied with the aid of ellipsoidal coordinates λ, μ, ν, where χ2 a2 + λ

y2 b2 + λ

ζ1 _ c2 + λ

and a, b, c are the semiaxes of the ellipsoid, λ = 0 is the equation of the base ellipsoid, with confocal surfaces given by other positive values of λ. Solving for x, y, z, we find that the equation has two

POTENTIAL

FLOW

OVER

ELLIPSOIDS

| 13

additional roots, These define a family of intersecting surfaces (hyperboloids of one and two sheets), which, at a great distance, approach ordinary spherical polar coordinate surfaces. Solutions of Laplace's equation are separable in ellipsoidal coordinates as products of Lame functions, thus

Here E and F are Lame functions of the first and second kind. Examples of the first few Lame functions are the following:

Lame products of lower degree bear simple relations to Cartesian coordinates, thus

If the flow is to vanish at infinity, functions of the second kind are required. These are determined by integration:

For a complete discussion of ellipsoidal coordinates and associated functions, the reader should consult Whittaker and Watson or Hobson (see Bibliography). Translation of Ellipsoids For translation of the ellipsoid in the negative direction along the three principal axes, the velocity potentials are given by

Thus for motion in x-direction, the potential on the surface of the base ellipsoid is simply The constants depend in general on elliptic functions, but their values have been tabulated for a number of interesting cases. The flow produced by a moving sphere provides a simple ex-

14 I CHAPTER TWO

2.1. Incompressible flow produced by a moving sphere.

ample. Figure 2.1, adapted from Lamb's Hydrodynamics,1 shows the instantaneous streamlines. Here the potential is φ = V2 Ux~. r The equipotential lines are circles (meridians) with values propor­ tional to rc-coordinate on any concentric sphere, including the solid sphere of radius a. The velocity at the crest is Vi U, opposite in direction to the motion of the sphere. The motion is unsteady in the sense that the flow pattern moves with the sphere and remains attached to it. If we start the sphere from rest, or accelerate it, the whole flow field will be set into motion, and, since the fluid is as­ sumed incompressible, the velocities everywhere will be linked in­ stantly to that of the sphere. Accelerating the flow requires a force that depends in part on the mass of the body itself and also on the 1H.

Lamb, Hydrodynamics (New York: Dover, 1945).

POTENTIAL FLOW OVER ELLIPSOIDS | 15

momentum created in the fluid. The latter can be computed by integrating the momentum over the field or from the unsteady term in the "extended Bernoulli equation" for the surface pressures, 3 c P . .is, Afr . = — ρ—. that H K ' dt

Apparent Mass and Velocity Distribution If the velocity is constant, the fore and aft pressures balance and the fluid offers no resistance to the motion, in accordance with "d'Alembert's paradox." The term proportional to the rate of change of velocity, however, gives positive pressures in front and negative pressures at the rear, resulting in a force that resists the acceleration. In accelerated motion the sphere thus acts as if it had an additional mass equal to the mass of a certain volume of fluid. It is not difficult to show that this additional mass is proportional to the maximum velocity at the surface of the sphere in steady mo­ tion. Since the velocity potential on the surface of the sphere is proportional to the x-coordinate of the surface, the pressure -

plotted along Λ; is a straight line and corresponds to a con­

stant rearward buoyancy in accelerated motion. This gradient of the potential also determines the maximum velocity ViO at the crest of the sphere in steady translation. The factor V2 is called the "inertia coefficient" of the sphere. Hence the inertia coefficient is also the "maximum velocity coefficient." This equality holds for any ellipsoid moving parallel to a principal axis. For a circular cylinder moving at right angles to its long axis, the maximum velocity coefficient is 1.0 and the area of the appar­ ent mass is equal to the area of the cylinder. To obtain the flow over the cylinder at rest in a moving stream, we add the potential Ux. The velocity over the crest of the cylinder then becomes 2 U. Differentiating the potential function in the direction of the sur­ face, we find that the velocity at any point on the surface is simply the projection of the maximum velocity vector in the direction of

16

CHAPTER TWO

Tangent Plane

2.2. Distribution of velocity over an ellipsoid: Munk's Rule.

the surface at that point. This statement holds for any ellipsoid in flow along a principal axis (see Fig. 2.2). Consider a thin, flattened elliptic cylinder resembling the sec­ tion of a wing having a thickness/chord {tic) ratio of 20% (Fig. 2.3). The maximum velocity coefficient in this case is simply tic. At zero angle of attack the velocity at the middle is 1.2U, and the area of apparent mass is 0.2 times the area of the cylinder. Turning the flow direction 90°, the tic ratio of the cylinder becomes 5 instead of Vs. The velocity around the small radii at the ends of the cylinder is now (1+5) times the velocity U. Again, the velocity at any other point on the cylinder is simply the projection of this maxi­ mum velocity. The velocity distribution at any other angle of attack may be obtained by superimposing components due to U and W. When comparing this flow to that around a wing section, we should remember to add a velocity of circulation sufficient to can­ cel the flow around the trailing edge, thereby doubling the velocity around the leading edge. In the limit, as the elliptic cylinder approaches a thin flat strip, the area of apparent mass approaches the area of a circular cylin­ der with the width of the plate as a diameter. In the case of the elliptic disc of finite length, it is found that the area is reduced by the factor 1IE, where E is the complete elliptic integral of the sec­ ond kind, equal to the ratio of the semiperimeter of the ellipse to its span. The edge velocity in perpendicular motion is of course equally reduced. As a consequence, the circulation needed to sat­ isfy the Kutta condition will also be less, and this leads to a reduc­ tion in the lift of a wing of finite aspect ratio.2 2R.

T. Jones, "Theoretical Correction for the Lift of Elliptic Wings," J. Aeronaut.

Set. (November 1941).

POTENTIAL FLOW OVER ELLIPSOIDS | 17

^max ~1·2 xVstream

t

t/c = 0.2

^ Vmax = 6 xVgtream t/c = 5

2.3. Maximum velocities for an elliptic cylinder.

Tabulated values of the inertia factors (or maximum velocity factors) are given in Lamb3 and in Zahm.4 Figure 2.4 shows the maximum velocity for elliptic cylinders and prolate spheroids plot­ ted against the fineness ratio. The relatively small disturbance pro­ duced by the airship shape is noteworthy.

Moment Generally the additional apparent mass of a body is different for different directions of motion. Thus for an elongated spheroid moving crosswise, the flow will resemble locally the flow around a circular cylinder that has an inertia coefficient of 1.0. Hence the volume of the apparent mass will be approximately equal to the volume of the spheroid. For motion in the direction of the long axis, however, the additional "volume" will be only a fraction of the volume of the body. As shown by Munk,5 if the spheroid is placed at an angle α to the flow, it will develop a moment M given by M = p/2 V2 (k 2 — ^1) x volume x sin 2a. 3

Lamb, Hydrodynamics. F. Zahm, Flow and Force Equations for a Body Revolving in a Fluid , NACA TR 323 (1928); see also NASA RP 1050 (1979). 5 Max M. Munk, The Aerodynamic Forces on Airship Hulls, NACA TR 184 (1922). 4 A.

l8 I CHAPTER TWO

Here and k 2 are the inertia coefficients for lengthwise and cross­ wise motion. This moment is unstable in the sense that it tends to increase the angle a. Pressure Distribution

To determine the pressure distribution over an ellipsoid at an angle of attack, one must first of all determine the magnitude and direction of the maximum velocity vector Vm. This is done by re­ solving the stream velocity into components parallel to the princi­ pal axes and multiplying these components by the appropriate ve­ locity factors. Denoting by η the unit (outward) vector normal to the surface, the velocity at a point will be given by the vector product, (η X V m ) X n. Figure 2.5 shows pressure vectors around the forward middle line of a prolate spheroid having an axis ratio of 3 to 1 at an angle of attack of 20°. Theoretically, pressures over the rear of the body are symmetric around the origin, so that the body develops a pure .24

b=

(elliptic cylinders)

b = a (oblate spheroids)

b = c (prolate spheroids)

max

Values Given by Thin Airfoil Theory

O

O

2

4

6

8

10

12

14

Fineness Ratio, a/c

2.4. Maximum velocity increments for ellipsoids.

16

18

POTENTIAL FLOW OVER ELLIPSOIDS | 19

2.5. Pressure vectors in XZ plane. 3/1 ellipsoidal hull at 20° angle of attack.

couple with no lift force. In reality, however, flow separation mod­ ifies the pressure distribution considerably over the rear portions, and some lift is developed. In typical experiments on airship shapes, the measured pitching moments are about 75% of those predicted by frictionless flow theory.

C H A P T E R

r

T H R E E

Bvo-Dimensional Flow: Wing Section Theory

Conformal Mapping of Two-Dimensional Flows If the wing or wing panel is sufficiently long and narrow, the local flow in planes perpendicular to the long axis may be consid­ ered as approximately two dimensional. Such two-dimensional flows are extremely simple since any differentiable function of a complex variable can represent such a flow. As explained by Prandtl and Tietjens,1 every analytic function of such a complex variable can be separated into a pair of distinct functions, one real and one imaginary, thus F(x + iy) = λ

(see Eq. 8.8). For the function of first degree the integration yields = I s,@ - m f.

(8.12)

Here E(k) is the complete elliptic integral of the second kind, equal to the ratio o f the semiperimeter o f the elliptic disc t o its span 2b. The term E(k)x/a yields a parabolic camber shape for sections taken parallel to the flight direction, while the remaining term shows a progressive spanwise twist of the sections along the span. These relations are illustrated in Figure 8.1. As the aspect ratio approaches infinity, the flow approaches that given by function 5 of Table 4.1. Substituting ν for u in formula 9 of Table 4.1 shows that the velocity field in the wake is given by

ν

iw — 2i I Qi

+

.

(8.13)

The downwash along the centerline of the wing (Eq. 8.12, χ = 0) is found to be equal to one-half the final value induced by the trail­ ing vortices in the wake. It may be shown by considerations of symmetry that for any distribution of lift symmetrical about a line at right angles to the flight direction, the lifting line theory gives the correct value of downwash at the center line.

LIFTING SURFACE THEORY | 133 χ

Planform

Spanwise Loading

Downwash x =0

Center

-a

V Distribution of Velocities y=0

Form of Stream Surface y=0

8.1. Velocity distribution given by first Lame function.

Note that the foregoing relations for the ellipsoidal distribution of lift agree qualitatively with the Prandtl theory for wings of high aspect ratio ("lifting line" theory). The more complete calculation shows, however, a reduction of lift in the ratio 1IE, where E is the ratio of the semiperimeter of the ellipse to its span. Lift and downwash distributions of more general form may be obtained by a series of ellipsoidal harmonics, making use of the orthogonal properties of Lame functions. All such distributions

134 I CHAPTER EIGHT fall to zero around the edges, with tangential flow at both the lead­ ing and trailing edge. Lift distributions of the Lame series give span load distributions in terms of Legendre functions of the first kind with velocities in the wake given by functions of the second kind through the Neu­ mann relation, 1 f+1 p , ΓΠ Λ ·

S-® = 2l,

m

In order to show the effect of angle of attack, it is necessary to obtain solutions with flow around the leading edge, and it is to be expected that such solutions will show infinite velocities around the edge. Solutions having the desired properties can be obtained by ap­ plying certain differential operators to the functions uDifferen­ tiating Eq. (8.4) with respect to χ gives x/a

(8.14)

ί - (Γ and w = —E{k).

(8.15)

The downwash is now constant over the disk and Eqs. (8.14) and (8.15) are the solution for a flat elliptic plate in downward motion of the type considered in Chapter 2. The solution corresponds to function 2 of Table 4.1 except that the angle of attack w/V for a given lift distribution is increased by the factor E. Alternatively, for a given angle of attack, the edge velocity around the disc is reduced by the factor 1IE, and as explained in Chapter 7, this reduces the circulation needed to satisfy the Kutta condition. To obtain a circulatory flow around the elliptic disk, we apply the operator d a— da

LIFTING SURFACE THEORY | 135

to Eq. (8.5) while at the same time keeping the ratio bla constant. Geometrically this amounts to an infinitesimal expansion of the scale followed by a subtraction of the original distribution. The result is 1

The chordwise velocity distribution given by this equation agrees with formula 1 of the Table 4.1. The span load distribution, how­ ever, is rectangular, with two finite vortices springing from the tips, and the combination of the circulatory flow with the normal flow due to angle of attack can satisfy the Kutta condition at only two points along the span. By using a series of such functions, the Kutta condition can be extended to more points. Such a series permits the lift curve slope of a flat wing to be determined, and Krienes' result is shown in Figure 8.2. The result differs little from that given by the simple "edge velocity" correction applied to lifting line theory, that is,

da

EA + 2

10 /8/R/ 0/R + 2) V

Φ O

Lif :ing L ine Tl·

8

PCL0 2 7Γ

.6 4

.4

/

/

.2

0

0

L ifting Surface Th eory

1

2

ί = V i -M2

3

4

5

6

7

8

9

/8/R

8.2. Theoretical variations of lift curve slope with aspect ratio for elliptic wings.

136 I CHAPTER EIGHT

where E is the ratio of the semiperimeter of the ellipse to its span, and A is the aspect ratio (see note 2, Chapter 2). Cambered Surface to Support a Prescribed Lift Distribution

The camber and twist of a surface required to support a given distribution of lift can be determined by calculating the downwash at each point produced by the lift. In Prandtl's theory for wings of high aspect ratio, sometimes called "lifting line theory," this downwash is approximated by superimposing two flows, one of them the wing section flow giving the camber of the wing sections and calculated by two-dimensional flow theory. Thus if Γ is the circu­ lation distribution along the chord, the downwash at each point is given by dr/dx

Table 4.1 gives numerous examples of such a calculation; thus if a constant lifting pressure along the chord is desired, the slope of the camber line will be given by example 8 of the table. Following this, we turn the axis 90° and calculate the downwash induced by the vortex wake, using again the two-dimensional flow formula,

This second component of the downwash is reduced by half since the wake vortices extend only behind the wing. Moreover, the downwash induced by the wake is assumed to vary only slowly in the x-direction and is assumed constant over the extent of the wing sections. Thus the wake component simply reduces the effec­ tive angle of attack of the wing. These ideas of the Prandtl lifting line theory may be extended to give the complete flow and downwash of a wing of any aspect ratio by means of Whittaker's solution of Laplace's equation. Whittaker's solution, as mentioned earlier, describes the threedimensional flow as the superposition of two-dimensional flows at varying angles around one of the axes; thus

LIFTING

SURFACE

THEORY

| 137

8.3. Downwash on lifting surface obtained from a series of "lifting lines."

In this light we can see Prandtl's theory as an approximation involving the two angles 0 and To carry out the calculation, we integrate the perturbation velocity u, proportional to the lifting pressure, along the upper surface of the wing to obtain the distribution of I corresponding to the prescribed lift. Since