Early Developments of Modern Aerodynamics

Early Developments of Modern Aerodynamics J.A.D. Ackroyd B.P. Axcell A.I. Ruban BUTTERWORTH HEINEMANN Early Developm...

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Early Developments of Modern Aerodynamics

J.A.D. Ackroyd B.P. Axcell A.I. Ruban

BUTTERWORTH HEINEMANN


Early Developments of Modern Aerodynamics Eur. Ing. J. A. D. Ackroyd, Eur. Ing. B. P. Axcell Manchester School of Engineering and

A. I. Ruban Department of Mathematics University of Manchester, Oxford Road, Manchester, UK

OXFORD

AUCKLAND BOSTON

JOHANNESBURG

MELBOURNE NEW DELHI

Butterworth-Heinemann Linacre House, Jordan Hill, Oxford OX2 8DP 225 Wildwood Avenue, Woburn, MA 01801-2041 A division of Reed Educational and Professional Publishing Ltd A member of the Reed Elsevier plc group First published 2001  J. A. D. Ackroyd, B. P. Axcell, A. I. Ruban 2001

All rights reserved. No part of this publication may be reproduced in any material form (including photocopying or storing in any medium by electronic means and whether or not transiently or incidentally to some other use of this publication) without the written permission of the copyright holder except in accordance with the provisions of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London, England W1P OLP. Applications for the copyright holder’s written permission to reproduce any part of this publication should be addressed to the publishers British Library Cataloguing in Publication Data Ackroyd, J. A. D. Early developments of modern aerodynamics 1. Aerodynamics 2. Aerodynamics – History I. Title II. Axcell, B. P. III. Ruban, A. I. 629.10 323 Library of Congress Cataloguing in Publication Data Ackroyd, J. A. D. Early developments of modern aerodynamics / J. A. D. Ackroyd, B. P. Axcell, and A. I. Ruban p. cm. Includes bibliographical references and index. ISBN 0 7506 5133 4 1. Aerodynamics – History. I. Axcell, B. P. II. Ruban, A. I. III. Title. TL570 .A29 2001 2001037428 629.1320 009–dc21 ISBN 0 7506 5133 4 Typeset in 10/12pt Times Roman by Laser Words, Madras, India

Contents

Introduction

1

1

A survey of the basic principles of modern aerodynamics

3

2

From antiquity to the age of Newton

14

3

From Newton to the emergence of the field theory of fluid flow

19

4

The era of the whirling arm and the invention of the aeroplane

27

5

Nineteenth-century developments in the understanding of fluid flow

42

6

Practical developments in aeronautics

51

7

Frederick William Lanchester (1868–1946)

57

8

Lift forces in flowing fluids (Kutta, 1902) Commentary on Kutta (1902)

70 74

9

On the motion of fluids with very little friction (Prandtl, 1904) Commentary on Prandtl (1904)

77 84

10

On annexed vortices (Zhukovskii, 1906a) Commentary on Zhukovskii (1906a)

88 105

11

Boundary layers in fluids with small friction (Blasius, 1908) Introduction I Boundary layer for the steady motion at flat plate immersed parallel to the streamlines II Calculation of the separation position behind a body immersed in a steady flow III Emergence of the boundary layer and the separation position with a sudden onset of motion from rest IV Formation of the separation position with motion uniformly accelerating from rest

107 107 110 118 125 130

vi

Contents

12

13

V Application of the results of the separation problems to the circular cylinder Commentary on Blasius (1908)

137 142

On a two-dimensional flow related to the fundamentals of the problems of flight (Kutta, 1910) 1 Introduction 2 General expression 3 The arched shell of circular form 4 The lift force on the shell 5 A numerical example: the rounding of the leading edge 6 Flat plate and shell under various angles of attack ˇ 7 Final considerations Commentary on Kutta (1910)

145 145 147 149 155 160 171 180 183

On the pressure exerted by a plane-parallel flow on obstructing bodies (aeroplane theory) (Chaplygin, 1910) 1 Introduction 2 Derivation of the general formula for components of the pressure force 3 Pressure force in the presence of standing vortices 4 The pressure force on a segment of a circular cylinder 5 The pressure force acting on a complex wing 6 A wing with one cuspidal point 7 Other wing forms 8 The flow past a plate shaped into a circular arc in the presence of a vortex 9 Calculation of the rotating moment Commentary on Chaplygin (1910)

187 189 191 198 208 211

14

On the contours of the aerofoils of hang gliders (Zhukovskii, 1920) Commentary on Zhukovskii (1910)

223 229

15

Subsequent developments

232

References

234

Index

241

186 186

213 218 220

Introduction

The modern science of aerodynamics began to emerge about one hundred years ago and it is therefore both timely and appropriate that we celebrate its inception. Rather by chance, its emergence almost coincided with the Wright brothers’ demonstration to the world that the dream of powered flight had become reality. Yet the new science of aerodynamics was not only providing the explanation for why flight is possible but was also indicating how it might be accomplished with far greater efficiency in the future. Indeed, the basic principles of the subject, emerging in a mere half-dozen published papers, provided the scientific wherewithal and set the stage for one of the most rapid, and dramatic, technological developments the world has seen. In popular terms, each time we take our holiday or business flight we do so by courtesy of those principles established a century ago, and in the same sense as embarkation on a sea voyage requires the support of the far older buoyancy principle of Archimedes. Clearly, then, there is much to celebrate. Our form of celebration adopted here, however, has been suggested by more cogent reasons than merely that of ritual applause for all this scientific, technological and industrial exuberance. In the first place, those half-dozen or so publications are now not easily obtained, so that a case stands for their re-publication. Secondly, the papers are not in English, and therefore the detail of their content often remains obscure to those unfamiliar with the native language of their authors. While it is both right and proper to see citations to Kutta (1902) and Prandtl (1904) as the origins of modern lift and drag theories, such references have a tendency to become rather more ritual gestures than acknowledgements of known work. So a third reason for our form of celebration is that it provides an opportunity to present English translations of these key papers which ushered in the modern science of flight. A fourth and more general reason is that it is often informative, as well as illuminating, to be able to study the manner in which notable scientists of the past made and described their discoveries. To this point, the German polymath, Gottfried Wilhelm von Leibniz (1646–1716), provides apt justification: It is an extremely useful thing to have knowledge of the true origins of memorable discoveries, especially those that have been found not by accident but by dint of meditation. It is not so much that thereby history may attribute to each man his own discoveries and others should be encouraged to earn like commendation, as that the art of making discoveries should be extended by considering noteworthy examples of it.

2


Thus our subject’s noteworthy historical examples have been collected within a single volume, presented here in English translation, some for the first time, in the hope that both those familiar with the subject and also newcomers to the field may draw both instruction and inspiration from their art of revelation. To set the scene for these seminal papers, our celebration begins with a modern view of the essential features of the subject. This is followed by a historical survey of the key ideas which preceded the papers. The survey is based on material appearing in the Aeronautical Journal (Ackroyd, 1992, 2000 ), its use being by kind permission of the Royal Aeronautical Society. Readers familiar with the subject will realize that one paper has been omitted from the collection presented in translation here. This is Zhukovskii’s paper ‘On the fall through the air of light bodies of elongated shape, animated by a rotary movement’ (Zhukovskii, 1906b). Our reason for this omission is that the paper adds little to the discovery, announced in Zhukovskii (1906a), of what later became known as the Kutta–Zhukovskii theorem. Nonetheless, we are grateful to Dr Laurent Dala and Ms Helen Sharples for providing a translation of Zhukovskii (1906b) from the original French. A brief indication of the paper’s contents is provided in Chapter 10. A note on references is appropriate, however, before we begin. Apart from the original referencing schemes retained in the papers re-published here, the system adopted is that recommended by the Royal Society. This has the advantage, particularly in matters of historical progression, of emphasizing the timing of an idea through the date of publication; thus, for example, Prandtl (1904). However, when review publications are cited, these are indicated by use of italics; thus, for example, Ackroyd (1992).

1 A survey of the basic principles of modern aerodynamics The briefest trawl through the contents of this book will reveal that our subject here, in a broad sense, is fluid flow. The term ‘fluid’ here embraces both gases and liquids which, in simple terms, differ only in the magnitudes of their material properties such as density and viscosity. Our stress on viscosity will no doubt surprise newcomers to the subject. However, it must be emphasized at the outset that, without viscosity, all flight, even in nature, would be impossible since wings would produce no lift. From this it follows that both marine and aircraft propellers – essentially rotating wings – would be useless, while ships driven since antiquity by sails – flexible wings – would have suffered an extremely short history. Moreover, without viscosity we would all be in severe danger of asphyxiating in our own exhalations. Thus an understanding of the role of viscosity emerges as not only vital to any explanation of flight but also vital, quite literally, to life itself. As a material property, viscosity acts so as to transmit shearing effects throughout a fluid. Its action can be understood from the situation shown in Fig. 1.1. This figure is taken from Lanchester (1907), one of the first to grasp viscosity’s role in flight and who, himself, drew the figure from Maxwell (1870) who was the first to measure its properties. Fig. 1.1 shows a fluid sandwiched between two long parallel solid plates. The upper plate moves at constant speed, as shown, whereas the lower plate is stationary. The fluid sticks to the surface of each plate, that is the fluid has the same velocity as that of the solid surface with which it is in contact. This phenomenon is now known as the no-slip condition. It occurs because of interaction on the molecular scale between the fluid and the solid. Because of the imposition of this condition, there is a continuous velocity variation in the fluid between the plates. This variation is depicted by the lines ab in the figure. If we imagine that we can draw a rectangle in the fluid at one instant, and then watch this shape as time progresses, then what we would see is shown in Fig. 1.2. Our rectangular element distorts, its angles change, it suffers increasing angular strain as time progresses. And this continuous straining is caused by the viscous shearing forces shown, which act around the periphery of our rectangle. The action is akin to shearing a thick book lying on a table by pushing sideways across its cover. In the fluid case, however, as the kinetic theory of Maxwell (1860) reveals for gases, these viscous shearing forces are the consequence of the random motion of fluid molecules. Those from a higher speed fluid layer randomly collide with molecules in a layer of lower speed, or vice versa, thereby exchanging momentum between layers. Such momentum changes on the molecular scale thus create at the macroscopic level the viscous forces between layers. These viscous forces not only act upon each and every

4

Early Developments of Modern Aerodynamics C

V

D

a b a b a b a b a b

A

h

B

Fig. 1.1 The viscous flow between two long parallel plates (adapted from Lanchester, 1907).

fluid element but also, as it turns out, are directly related to the continuous straining of the element. And these distortions, in turn, are related directly to the velocity variations within the flow, specifically combinations of the flow’s velocity gradients which have been imposed by the no-slip condition. At this point we need to be rather more specific about the form of these viscous forces, by stating that they turn out to be calculable as the simple multiple of a flow’s continuous straining and the fluid’s coefficient of viscosity. This coefficient, like density, is a property of the fluid itself, not the flow. Depending on the fluid, the numerical value of this coefficient varies enormously. Air and water, for example, possess extremely small viscosity values whereas those for fluids such as oil and treacle are very large. Here, however, we are interested predominantly in airflows. The essential point in such cases is that, even though the air’s viscosity coefficient is extremely small, the viscous forces within its flows are appreciable wherever the flow’s straining becomes intense. Now suppose that our rectangular element of Fig. 1.2 is replaced by one which is circular. If you wish, you can think of the fluid now as being composed of thousands of tiny ball bearings. One of these is shown in Fig. 1.3, again subject to the viscous shearing forces described earlier. Clearly, the action of such shearing forces will cause the balls to spin about their own centres. In other words, a further action of viscosity is to cause rotation, or vorticity, within each tiny element of the fluid. One other force shown acting in Fig. 1.3 is that due to pressure. But such forces, acting always normal to the ball’s surface so that their lines of action therefore pass always through its centre, cannot alone change whatever angular momentum the ball possesses because such pressure forces can create no moments about the ball’s centre. Similarly, the ball’s own weight, also acting through its centre, is equally unproductive in changing angular momentum. Viscous shearing force

Fig. 1.2 The angular distortion as time progresses of a fluid element shown in Fig. 1.1.

A survey of the basic principles of modern aerodynamics Viscous shearing force

Pressure

Weight

Fig. 1.3 The forces on a cylindrical fluid element.

The consequences of viscosity are then twofold. One is that its shearing forces act throughout the fluid but, in particular, act also upon the surfaces of any solid object within the flow. Such surface shearing forces, or surface skin friction, in conjunction with pressure variations around the object, cause its resistance to motion. An obvious manifestation of this resistance is drag, the resistance component defined to be in opposition to the object’s direction of motion. However, viscosity’s second and far more exploitable consequence is the creation of vorticity within a flow about an object. As we shall see, it is this creation of vorticity which is the root cause of lift, the resistance component defined to be perpendicular to the object’s direction of motion. At a superficial level, lift can be seen merely as the consequence of top-to-bottom pressure imbalance acting on a body. Yet such a view cannot explain the origin of such pressure imbalances. It is not entirely surprising, then, that viscosity’s subtle role here took such a long time to be understood, so long, in fact, that aerodynamics is very much a twentieth century science. The genesis of modern aerodynamics sprang from two requirements, seen initially to some extent as disparate but rapidly realized as being intimately connected. The first was a general requirement to understand a fluid’s ability to resist a body’s motion, particularly the phenomenon of drag which had bemused the scientific world since the beginnings of recorded history. As can be judged from the above, little headway could be expected here until an understanding of viscosity’s role in this had been achieved. This understanding began to emerge during the mid-nineteenth century, notably in the work of Stokes (1845, 1851). Nonetheless, the mathematical problems posed here were of such fearsome complexity that little in the way of results useful to the practical scientist and engineer could be drawn from this new viscous flow theory. By good fortune, an approximation to that theory, circumventing at least some of its difficulties, emerged in the work of Prandtl (1904). This introduced the concept of the thin boundary layer (Fig. 1.4) wrapped around the surfaces of a body and within which the action of Velocity gradient

No-slip condition

Boundary Layer

Fig. 1.4 Boundary layer flow about a streamlined body (the boundary layer’s thickness is grossly exaggerated).

5

6


viscosity is largely confined. Because of the imposition of the surface no-slip condition and the thinness of the layer, here the velocity gradients are enormous and the fluid straining is then intense. In a fluid such as air, the multiple of its small viscosity value and such intense straining produces appreciable values for the viscous forces within this layer. While the flowing fluid exterior to this boundary layer also experiences straining, here this is far less severe. Consequently, the viscous forces in this exterior region are so small that the flow here behaves as if it is entirely inviscid and possesses no vorticity. Thus Prandtl’s boundary layer emerges as the source of both vorticity and fluid skin friction drag. Application of the thin boundary layer concept was seen initially as quite general and there is no indication that its subsequent vital role in the technology of the aeroplane was at first envisaged. Here it is important to understand that at this point in history the viability of the still-nascent powered aeroplane remained in considerable doubt. A mere eight years before the announcement of the boundary layer concept, seven years before the first powered flights of the Wrights, in 1896 Lord Kelvin had declined as follows an invitation to join the Aeronautical Society in London (Gibbs-Smith, 1985 ): I have not the smallest molecule of faith in aerial navigation other than ballooning or of expectation of good results from any of the trials we hear of. Others, however, maintained more open minds. Nonetheless, although the Wright brothers had achieved powered flight in the year preceding Prandtl’s paper, the world at large knew next to nothing of their exploits. Indeed, at that time mankind stood transfixed by the recent, repeated and resoundingly public failures of the far more prestigious Samuel Pierpont Langley in his attempts near Washington, DC to achieve powered flight with his ‘Great Aerodrome’, as he called his man-carrying aeroplane. The Wrights did not receive just acclaim, and widespread publicity, for their immense technological achievement until Wilbur Wright demonstrated their Flyer Type A near Le Mans, France, in 1908. Thus it is hardly surprising that, in 1904, the discoverer of the boundary layer concept did not have application to the aeroplane at the forefront of his mind. Although an early application of the concept was to an aeronautical subject, the recipient was a particularly German preoccupation at that time, the airship, but not the aeroplane. An interest in the aeroplane, more specifically the glider, did, however, provide the spur to recognition of the second requirement which led to the birth of modern aerodynamics. This was a need to understand the remarkable lifting effect achieved by arched, or cambered, surfaces. Experimental results for such surfaces, provided by Phillips (1884, 1891) and more notably by Lilienthal (1889), indicated enhanced lifting effects compared to those achieved with the flat surfaces hitherto investigated. Indeed, Lilienthal’s application of his results to his invention of the hang-glider, and his subsequent public demonstrations of examples of this in flight, convinced two scientists that here there was a phenomenon worthy of analytical investigation. According to Seifert (1961), the Russian mechanics professor Nikolai Egorovitch Zhukovskii visited Lilienthal in Berlin in 1895, witnessed a gliding demonstration and purchased an example of a Lilienthal glider. He then returned to Moscow to take up the problem himself. His counterpart in Munich, Sebastian Finsterwalder, in contrast, set the problem for his doctoral student Martin Wilhelm Kutta. The subsequent theoretical investigations


of Kutta (1902, 1910) and Zhukovskii (1906a, 1910) thus provided the genesis of modern lift theory. Translations of these papers, together with that of Prandtl (1904), are included in the present volume. The new lift theory of Kutta and Zhukovskii was based on the idea that a cambered surface produces lift through its ability to generate a vortex about itself. Quite why a cambered surface is capable of doing this was not clear at the time, however. Nonetheless, as we shall see, the vortex creates the top-to-bottom pressure imbalance, and hence lift, mentioned earlier. Now this linking of a side force to a vortex was not entirely a new idea. Indeed, the phenomenon of the sideways force experienced by spinning bodies in a fluid stream had been observed for at least two centuries before this new lift theory emerged. Newton (1672) comments on it in connection with the swerving flight of the sliced tennis ball. The ‘father of modern gunnery’, Benjamin Robins, demonstrated the effect in London in 1746 in connection with the wayward flight of shot (see Wilson, 1761). In 1852 Heinrich Gustav Magnus took the demonstration further in his experiments in Berlin (Magnus, 1852), thereby giving his name to the effect. Later, Lord Rayleigh (1877) took up the spinning tennis ball problem, carrying out an analytical investigation in which he assumed correctly that the spinning object generates a vortex about itself. The crucial difference between these phenomena and the subject of the new Kutta–Zhukovskii theory is that, in the latter case, the body by some means creates the vortex about itself without being itself in a state of rotation. An obvious explanation in the case of the spinning object is that it must drag fluid around its rotating surface so as to create its vortex. From what has been said earlier, it will be seen that this dragging of the fluid by the rotating surface is caused, in fact, by the viscous shearing action consequent upon the imposition of the no-slip condition at the rotating surface. Thus it is not entirely surprising that it was the viscous boundary layer concept which resolved the problem of explaining the means by which the non-rotating cambered surface creates its own so-called ‘bound’ vortex. We have seen already that elements of the viscous fluid within a boundary layer must not only be subject to intense shear but are also caused to spin by the action of these viscous shear forces around their peripheries. Beyond this point, however, explanation of the lift force becomes by no means straightforward and depends on a deeper understanding of fluid flow. One explanation rests on a theorem communicated by William Thomson (later Lord Kelvin; the irony here will no doubt be evident) to George Gabriel Stokes in 1850. Crudely put, this states that the aggregate of all the spins possessed by all such boundary layer fluid elements is equivalent, in effect, to a circulatory vortex around the body in question. A related but more physically appealing explanation begins with the observation that boundary layers cannot tolerate passage around sharp corners. In such circumstances, the boundary layer peels off, or separates, at the corner. Thus modern aerofoil surfaces, otherwise smoothly contoured apart from their sharp trailing edges, impose boundary layer separation at their tails. As we shall see presently, the effect of this on the flow exterior to the boundary layer, behaving as if it is entirely inviscid and having no vorticity, is then as if a ‘bound’ vortex is present about the surface. Surprisingly, these two explanations amount to the same thing. However, the vital conclusion to be drawn from this is that it is the viscous boundary layer’s behaviour which is the cause of lift. Indeed, much of our ability to take the aeroplane’s performance far beyond that achieved by the Wrights has depended on an understanding of the nature of the boundary layer. From this we have learned to control its behaviour

7

8


so as to maximize lift while minimizing drag. Particularly in the latter respect, the recipe has been to adopt a smooth, carefully contoured, or streamlined, shape tapering gently to a sharp tail, at which point alone boundary layer separation is enforced. Careful choice of body shape is crucial, then, in the control of the boundary layer. Moreover, this controlled behaviour on the part of the boundary layer controls the geometry of the inviscid flow exterior to the boundary layer so as to establish within this exterior flow the top-to-bottom pressure imbalance which, in turn, produces lift. Now the details of such exterior inviscid flows are calculable using methods which had first begun to emerge as early as the eighteenth century. In particular, in a paper by Leonhard Euler in 1755 (see Truesdell, 1954 ) it is shown that inviscid flows possessing no vorticity (irrotational flows) obey an equation later known as Laplace’s equation. And Euler demonstrates, however briefly, that this equation has certain simple solutions which represent what we now call the uniform stream, the source, the sink, and the vortex. The flow fields of these simple cases are shown in Fig. 1.5. Furthermore, Euler’s demonstration shows that the Laplace equation has what is called a mathematically linear form. The useful outcome of this property is that all such simple solutions are additive, in the sense that their flow velocities can be added vectorially so as to obtain more complicated flows. Thus, for example, placing a source and a sink close together produces the flow field of Fig. 1.6a. And the addition to that of a uniform stream produces the flow field of Fig. 1.6b. Here it is seen that the flow of the source/sink combination is now enclosed within a single streamline forming a complete loop, and this can be taken to represent the surface of a solid body of oval shape. Hence, the details of the flow field exterior to this body can be calculated. If we bring the source and the sink closer and closer together under certain mathematical conditions, ultimately what we obtain is the source/sink combination called the dipole, or doublet. The addition of a uniform stream now produces the flow about a circular cylinder (Fig. 1.7a). If we add to that a vortex centred at the cylinder’s centre, we then obtain the flow field of Fig. 1.7b, which is that analysed by Lord Rayleigh (1877), as mentioned above. Here we notice the top-to-bottom asymmetry of the flow. This feature is worth further examination, since it is this which is the cause of the top-to-bottom pressure imbalance and hence the sideways, or lift, force which Rayleigh (1877) calculates.

V

(a)

+

(b)

−

(c)

(d)

Fig. 1.5 Irrotational flow fields: (a) the uniform stream, (b) the source, (c) the sink, (d) the vortex.


+

+

−

(a)

−

(b)

Fig. 1.6 (a) The flow field of a source/sink pair. (b) The source/sink pair combined with a uniform stream.

(a)

(b)

Fig. 1.7 (a) The flow about a circular cylinder. (b) The flow about a circular cylinder created by the addition of a vortex centred at the cylinder’s centre.

In Fig. 1.7b it will be seen that the streamlines above the cylinder become tightly bunched, whereas a widening of the streamlines is evident at the lower surface. This behaviour is akin to that seen in a river flow, in which widening banks produce a placid, slow-moving stream whereas faster flow occurs if the river passes rapidly through a narrow gorge. All this is the consequence of the so-called continuity principle, which insists that the same quantity of water per unit time must pass through each section of the river. Between any pair of streamlines shown in Fig. 1.7b, then, the same principle must apply. From this we can conclude that the flow speed has increased over the top of the cylinder, while a slower flow occurs at the cylinder’s lower surface. This is consistent with the effect expected of the clockwise spinning vortex, which must augment the flow speed over the cylinder’s top while detracting from it at the bottom. At this point it is necessary to introduce a further concept encapsulated in the so-called Bernoulli equation. This insists that an increase in flow speed must be accompanied by a corresponding decrease in pressure, and vice versa. Consequently, the cylinder’s upper surface experiences a pressure lower than that on the under surface so that the cylinder is sucked upward by this effect. This provides one explanation of the lift force

9

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obtained. In another view, the effect of the vortex is to induce upward motion in the flow approaching the cylinder, whereas a downward flow occurs aft of the cylinder. Thus the cylinder first destroys the upward momentum of the approaching flow then imparts downward momentum to the departing flow. Both of these processes require the cylinder to be pushing downward on the flow. The equal but oppositely directed reaction on the cylinder is then its upward lift force. All of these features – upper surface suction and upflow/downflow – are also characteristics of the lifting wing. In the case of the cambered wing it will be appreciated that the boundary layer’s insistence on separating at the downwardly inclined tail imparts downflow to the surrounding inviscid flow field, thereby controlling the whole of its geometry. Through this process, an upward direction is induced in the inviscid part of the flow field’s approach to the wing. It is the carefully contoured wing’s ability to produce, not only downflow at its tail, but also upflow towards its nose which gives the wing its high lifting properties. However, in the vortex flow of Fig. 1.7b, and indeed in the other entirely inviscid flow cases illustrated, the fore-and-aft flows are always completely symmetric. Consequently, according to our momentum arguments the flows in this respect suffer no ultimate change in their horizontally-directed momentum so that the body experiences no drag force. This remarkable result, correct for the steady irrotational flow of an inviscid fluid, was discovered at about the same time by both Euler and the French mathematician, d’Alembert. The result, usually referred to as the d’Alembert Drag Paradox, is so much at variance with practical experience that it bedevilled the theory of inviscid fluid flow until the dramatic effects of viscosity came to be understood through the work of Stokes (1851) and Prandtl (1904). From the latter work, for well streamlined bodies it then became clear that the d’Alembert Paradox is almost correct: the drag is the zero value of the Paradox, plus a small correction contributed by the viscous effect of the boundary layer. For blunt, badly streamlined bodies, in contrast, the situation experienced in reality is far more complicated. For such shapes (those, for example, shown in Figs 1.6b, 1.7a, 1.7b) the inviscid part of the flow field accelerates over the forward half of the body and then decelerates over the rear half in an attempt to return to its initial speed condition. In this latter process the boundary layer fluid is further retarded by viscous forces. Indeed, such retardation is often so severe that boundary layer fluid is brought to rest, the consequence being that the boundary layer separates well before the rear of the body and backflow occurs aft of separation. The result of this is a wide wake in which the pressure is low, the consequent fore-and-aft pressure imbalance being then the overwhelming contributor to the body’s large drag force. Body streamlining is thus the panacea for such behaviour. In particular, the long gently tapered rear surface reduces the boundary layer’s rate of retardation so as to prevent its separation at any point other than the sharp tail. Much of the thinking here also emerged in the original paper of Prandtl (1904). Three further points should be made concerning inviscid flow theory and its application. The first stems immediately from the point made in the preceding paragraph: while we may calculate inviscid flow patterns using the ‘building block’ method of the source, sink and vortex indicated earlier, such flow patterns are not entirely achievable in practice if the body shape is badly streamlined. Accepting this restriction, the second point is that the flow patterns illustrated in Figs 1.5–1.7 are similar to those of the flux lines in electric and magnetic field cases; the source and sink of Fig. 1.5 resemble


North and South poles, and so on. In fact the resemblance is no mere chance, since all such problems are governed by Laplace’s equation and therefore exhibit similar mathematical solutions. Indeed, some of the earliest advanced methods for dealing with Laplace’s equation sprang from William Thomson’s interest in the field problems of electricity and magnetism. The third point concerns what is called the mathematically singular behaviour exhibited by three of our ‘building block’ solutions shown in Fig. 1.5. The source and the sink pass fluid at a constant rate between each pair of their streamlines, and to do so the flow speeds at their point-origins must be infinite. However, this mathematically singular behaviour does not trouble us in application since these singular points of infinite flow speed are always inside the body shapes we investigate, whereas it is the flow exterior to such bodies we wish to calculate. Such comments also apply to the vortex shown in Fig. 1.5, yet this flow’s singularity has a further and far more dramatic consequence. This is revealed in Fig. 1.8, which shows that the flow speed, q, on any circular streamline (q being constant along the entire length of that streamline) is inversely proportional to the streamline’s radial distance from the centre, r. That is q D k/r, where k has a constant value throughout the vortex. If we apply the Thomson/Stokes theorem mentioned earlier, we find a surprising and most useful result. The first step in this theorem’s application requires the calculation of what is called circulation. In this case this is simply the multiple of the flow speed, qDk/r, along a particular streamline and the complete length of that streamline, which is 2 r. The result is 2 k, which has a constant and finite value throughout the vortex. According to the theorem, this finite value of the circulation must be equal to the aggregate of the vorticity within the circulation circuit. In other words, there must then be a finite amount of vorticity somewhere within this circuit of our streamline, which happens to enclose the origin of the vortex. However, if we were to carry out a further circulation calculation, but around a circuit which deliberately excludes the origin (for example, around the circuit shown by dashes in Fig. 1.8), the result we would find is zero; there is no vorticity within this circuit. Clearly then, since our dashed circuit could lie anywhere within the vortex, provided that the circuit excludes the origin, this part of the vortex flow is irrotational, like the source and the sink. Yet this is not true if our circuit includes the origin, for which case we obtained the finite circulation 2 k. The answer to this conundrum is that all of the vorticity lies at the vortex centre where it has, like the speed q, an infinite value. However, this infinite vorticity is spread over the zero area of a point, a most singular behaviour even in the mathematical sense. Nonetheless, according to the theorem, the multiple of this q = kr r

Fig. 1.8 The flow field of the irrotational vortex; q / 1/r.

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spin and its area has the finite value 2 k, as already determined by our calculation for the circuit enclosing this singular point. By including a vortex in a flow field we can then add to an inviscid flow any value of vorticity, and circulation, which suits our purpose. It is this feature of inviscid flow theory which both Kutta (1902, 1910) and Zhukovskii (1910) began to exploit with dramatic success. In effect, as was later realized from an understanding of the boundary layer, they were replacing the actual vorticity present in that layer by the mathematically far more tractable vorticity-at-apoint present in the inviscid flow vortex. All that was required, as it turned out, was the erection of a physical principle which would determine the choice of k for the bound vortex to be used. This principle, it was realized, was the condition that the inviscid flow field departs smoothly at the sharp trailing edge of a wing, a condition which in reality is imposed by boundary layer separation at that edge. Thus far the flow fields described have been those about the distinctly hypothetical cases of wings and bodies of infinite span. In the subject of aerodynamics, in contrast, we are interested in wings of finite span. Here the question arises: what happens to the bound vortex, and its circulation, at the wing’s tips? The answer depends entirely on another of those theorems which make aerodynamics such a mathematical subject. In simple terms, this theorem, first stated by von Helmholtz (1858), requires that a vortex cannot end and that its circulation, in some way, must be preserved. The smoke ring provides a good example of this; its vortex structure never ends because it exists as a closed toroidal loop having the same circulation all the way round. In the case of the finite span wing, the bound vortex becomes cast off towards the tips, the released pair of vortices turning themselves roughly through right angles so as to spin away downstream of the wing. And this vortex system also never ends. The so-called trailing vortices can be traced right back to the airfield at which the wing first generated lift. Here the loop is finally closed by the starting vortex (see Fig. 1.9) which is left behind at the airfield. The circulation of this vortex system is preserved all the way round the loop so that the starting vortex has a spin sense opposite to that of the wing’s bound vortex. In fact, this starting vortex is generated by the sudden shedding of boundary layer vorticity at the instant at which the wing begins to move forward so as to generate its own bound lifting vortex. At that instant, the boundary layer adjusts itself so as to continue to insist upon separating at the sharp trailing edge. The trailing vortices behind high-flying aeroplanes are, of course, a familiar sight nowadays. Their distinctive white vapour emanates originally from each engine

Bound vortex Trailing vortex

Trailing vortex Starting vortex

Fig. 1.9 The trailing vortex system behind a wing of finite span.

A survey of the basic principles of modern aerodynamics Wing, one second later

Additional swirling kinetic energy

Thrust

Drag

Wing

Fig. 1.10 The continuous creation of swirling kinetic energy caused by the continual extension of the trailing vortices.

exhaust, but these vapour trails quickly become swept into the trailing vortices so as to form the familiar pair of white trails persisting for miles. Two consequences spring from the presence of the trailing vortex systems of finite wings. First, the position and manner in which the bound vortex is cast off from the wing depend on the wing’s planform shape. This, in turn, leads to a modification of the lift produced by the wing, the lift now being dependent also on wing aspect ratio (the ratio of the wing’s span to its average chord). Secondly, as illustrated in Fig. 1.10, as a wing flies forward it leaves behind two increasing contributions to its trailing vortices. Each additional contribution possesses kinetic energy in its swirling motion. There is a basic theorem of dynamics which requires that, in creating additional kinetic energy, work must be performed – a force must move through some distance. In the case of the wing, a continuing propulsive thrust is required to push the wing so as to create continuously this new swirling kinetic energy. But since the wing moves forward at constant speed, this propulsive force must be opposed by an equal but reactive drag force. This additional drag force is now called induced drag, the drag induced by the presence of the trailing vortices. Many of the ideas here sprang originally from the work of the early motor engineer, Frederick William Lanchester (Lanchester, 1907), who was Britain’s first great, and initially sadly neglected, aerodynamicist. From the foregoing, it will be apparent that the new ideas in aerodynamics which appeared at the beginning of the twentieth century had their roots in the understanding of vortices and other more general fluid flow phenomena developed by that time. Thus, as a prelude to our collection of early papers on aerodynamics, Chapters 2–7 present a historical survey of what went before. In this it will rapidly become apparent that, in the very earliest arguments, the action of a moving fluid was so seriously misunderstood as to provide regrettable support for principles of dynamics at considerable variance to those now accepted. Thus the evolution of our modern principles of dynamics itself is also germane to our survey.

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2 From antiquity to the age of Newton A brief description of antiquity’s thinking is necessary in order that we appreciate the immense intellectual journey which had to be undertaken before a rational understanding of fluid flow, and indeed dynamics, could begin to emerge. As to flight itself, the legends and those more factual accounts of the tower-jumpers and other supposed emulators of bird flight are legion. Their participants’ doomed exploits, however, need not detain us. A dominant influence throughout this period was the work of Aristotle (384–322BC). This his medieval translators greeted almost as holy writ, a bequest from a long lost Golden Age with which some hoped to drag their world from its medieval barbarism. No doubt Aristotle, that great advocate of careful observation and rational thought, would have been horrified had he sensed the two millennia of stultifying influence his thinking on the physical sciences would produce. Nonetheless, his ideas, and extensions of them, reigned supreme until Galileo Galilei (1564–1642) sought in a highly public manner to displace them, an exercise which it is now believed contributed to his eventual brush with the Inquisition. Readers interested in discovering more about this fascinating period of the history of science should consult Cohen and Drabkin (1958) as a useful source on Greek science, Clagett (1959) on mechanics in the Middle Ages and Westfall (1971) on the eventual shift towards the dynamics of Isaac Newton (1642–1727). After Galileo progress was rapid indeed, Newton, it will be seen, being born in the year in which Galileo died. As to the histories of fluid mechanics and flight itself, the book by Rouse and Ince (1957) provides a comprehensive review of the evolution of our understanding of the former subject whereas Gibbs-Smith (1985) remains the authority on the history of aeronautics. Some of Greek science’s earliest valid statements on the balance and the lever came from Aristotle and his school. To the later, more precise contributions of Archimedes (287–212BC) on the balance, lever, centre of gravity and buoyancy, subsequent thinking has needed to add little. While all of this produced significant headway in the mechanics of bodies at rest (Cohen and Drabkin, 1958), the same cannot be said for bodies in motion. How far the journey had to be here is revealed by the following encapsulation of the physical principle underlying antiquity’s arguments on dynamics (Westfall, 1971): Everything that is moved is moved by something else. Thus Aristotle argues (Cohen and Drabkin, 1958) that the spear or arrow in motion must continue to be pushed by the only material substance in contact, which is the


air. Paradoxically, elsewhere he argues that the air can also resist motion. At the very beginning of the arduous history of dynamics, then, the action of the air enters the story in such a manner as to confuse rather than to clarify. And although this view of the air’s action was disputed, sometimes effectively (Clagett, 1959), the basic principle that everything that moves must be pushed remained the accepted rule for two millennia. The usual version of this belief became that in which a force is implanted in the body at the act of projection, this force being eroded, some maintained, by such means as the air’s action. We are thus a long way indeed from the principle of inertia. Another of antiquity’s dominant beliefs was that of natural levels and natural tendencies (Cohen and Drabkin, 1958). Thus earthly objects possess a natural tendency to return to their natural level, which is the Earth’s surface; fire naturally ascends to the supposed fiery regions above, and so on. In his discussions on the animals, Aristotle proposes the extension that creatures exist at their natural levels; fish, being watery creatures, remain in that medium whereas birds, seen as airy creatures, remain largely in the air. Nothing more, it seemed, need be explained. Indeed, so dominant did these beliefs become that recorded thinking provides merely a fantastical embroidery upon them (Hart, 1985). Thus a bird’s feathers are constructed so as to collect more sustaining air; the bird swells its breast to ingest more lightening air; the feather’s hollow quill contains a lightening volatile vapour – this last from the pen of Galileo’s great admirer, Pierre Gassendi (1592–1655), in a posthumous work of 1658. Only two persons appear on record (Hart, 1985) as suspecting that the bird sustains itself through the action of the moving air. The first is Frederick II of Hohenstaufen (1194–1250), King of Germany, Sicily, and Holy Roman Emperor, who reveals this belief around 1250 in his book on falconry. The other, of course, is Leonardo da Vinci (1452–1519). Using the analogy of bouncing on a feather bed, he suggests (c. 1503) that the downward beat of a bird’s wing sustains the bird by compressing the air beneath (Giacomelli, 1930; Hart, 1985). A similar argument based on density increase is used to explain the air’s resistance to forward motion, yet a motion sustained, according to the old thinking, by a force imparted at projection. While Leonardo’s scientific work has been much admired in modern times – the keen observation revealed in his flow sketches, for example – it has to be recognized that the principles he employs run the whole gamut of opinion prevalent at that time (see Truesdell, 1968 for a necessary corrective to more adulatory assessments). Nonetheless, he grasps the principle of relative motion in his study of bird flight. Here he states correctly that the air’s action is the same whether the bird is at rest in a moving airstream – hovering at a cliff edge in a strong breeze is the example cited – or is moving through air at rest. Moreover, he suggests, but through a spurious argument, that air resistance is directly proportional to speed. His aeronautical projects retain their fascination (Gibbs-Smith, 1967), despite the belief nowadays that he never constructed any of his ornithopter designs. Moreover, his work here had little influence, remaining largely unstudied until powered flight had become a reality. Nonetheless, Leonardo appears to be the first on record to state correctly the continuity principle as applied to rivers and the circulation rule for vortices, both based, presumably, on observation. Thus, as to the continuity of river flow (Truesdell, 1968; his suggested correction in square brackets):

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If the water is not added to or taken away from the river, it will pass with equal quantities in every degree of its breadth [length?], with diverse speeds and slownesses, through the various straightnesses and breadths of its length. This fundamental principle, that the quantity of flow (essentially, quantity of matter) must remain unchanged, must be obeyed no matter what dynamic conditions apply to a flow. Thus far, however, Leonardo’s declaration of the principle had already been implied in statements by such Greek scientists as Hero of Alexandria (First Century AD) (Rouse and Ince, 1957). But elsewhere Leonardo then adds (Truesdell, 1968): By so much as you will increase the river in breadth, by so much you will diminish the speed of its course. It is this statement of a direct, and correct, proportionality which is the crucial step. For the vortex, we find in Leonardo’s notes the first statement of the circulation principle that the product of speed and length of circuit has the same value on each circular streamline (Truesdell, 1968): The helical or rather rotary motion of every liquid is so much the swifter as it is nearer to the centre of its revolution. . . . we have the same motion, through speed and length, in each whole revolution of the water, just the same in the circumference of the greatest circle as in the least. . . Some three centuries were to pass before this property of the vortex emerged from the mathematical theory of fluid flow. The property itself follows immediately from the discussion of Fig. 1.8, Chapter 1. As to dynamics, no fundamental change occurred until Galileo Galilei (1564–1642) proposed his inertia principle (Galilei, 1632) in contradiction to all that had gone before. Here, but for its falling motion and resistance, a moving body is seen as requiring no action in order to continue its motion. Yet this motion is construed as being parallel to the Earth’s surface; the principle is then an incorrect ‘circular’ inertia principle. Nonetheless, having accepted the idea he then realizes that some action, or force, is required in order to change motion. At one point he comes close to stating a version of the Second Law of Motion but then makes little further use of this vital step. As to falling motion itself, he recognizes that, while bodies fall at increasing rates, implying that some action takes place, heavy bodies of different weights fall together. This puzzling result raised for him unresolved questions concerning not only the status of weight as a force but also the nature of force itself. Devoid as he was of a mass concept, falling motion thus remained for him a natural motion. Nonetheless, he concludes correctly that, but for resistance, all bodies fall with the same constant acceleration. Completing the programme begun by the Oxford Schoolmen (Clagett, 1959) around 1300, later he published for the first time a correct derivation of the kinematic relations for this unresisted falling motion (Galilei, 1638). Moreover, here he concludes that, again in the absence of resistance, all descending bodies are capable of rising to their initial release positions. This Galilean Principle, as it came to be known when applied later, also provided the heart of the ‘vis viva’ (literally, live


force, or kinetic energy) concept of Gottfried Wilhelm von Leibniz (1646–1716), who consigned weight to the status of ‘dead force’. As to resistances caused by fluid media such as air and water, Galilei (1638) appears to propose that they be considered as two distinct types. The first is buoyancy and, as an astute reader of Archimedes, its practical application he thoroughly understands. The other form of resistance, he recognizes, is that due to motion through a fluid. This, he argues, is caused by fluid interaction with fine irregularities at a body’s surface. Since we cannot credit Galileo with the modern understanding that such interactions take place, in fact, on the molecular scale, his argument can be accounted as no more than a worthy guess. However, from this he, like Leonardo before him, suggests that this resistance to motion is proportional to body speed. Unlike Leonardo, in contrast, the latter conclusion Galileo supports through his observations of the decay in the slow motions of pendulums, their periods having been earlier determined, again by experiment, as being proportional to the square root of their lengths. As we shall see, this resistance result for very low speeds received further support in the magnificently detailed study of pendulum motion conducted during the nineteenth century by Stokes (1851). Although Galileo appears to have had little interest in the problems of fluid flow, one of his former pupils, Antonio (Benedetto) Castelli (1578–1643), achieved fame in the subject, publishing the continuity principle’s detailed proportionalities in 1626 (Rouse and Ince, 1957). Moreover, Castelli’s former secretary and Galileo’s constant companion at the end of his life, Evangelista Torricelli (1608–1647), deduced by observation that the Galilean kinematic relations for falling motion apply to the issue of liquids from vessels, publishing his findings in 1644. Earlier, in 1630, in effect he had invented the mercury barometer. Lacking a pressure concept, however, he was able to give only a partial explanation of its operation. In 1648 Blaise Pascal (1623–1662) used such an instrument to deduce that atmospheric pressure reduces with altitude. Pascal is often credited as the discoverer of the modern pressure principle, the equality of pressures at a point. However, Truesdell (1954) takes the view that this principle may not necessarily be inferred from Pascal’s actual statements. Moreover, the latter go little further than those announced earlier by the Dutchman, Simon Stevin (1548–1620), in 1586. The first person to propose a straight-line inertia principle was Galileo’s contemporary, the French philosopher and geometer René du Perron Descartes (1596–1650) (Descartes, 1644). He then adopted the view that all changes in motion occur through collision. However, his attempts to analyse collision are notable mainly for their lack of success (Westfall, 1971), occasioned largely by his inability to recognise that velocity is a vector quantity which therefore possesses direction. Nonetheless, having grasped the straight-line inertia principle, for circular motion he realized that some action is required although here again he made little headway in analysing the problem. Such was his reputation, however, that these two problems, collision and circular motion, became the key problems to be faced by seventeenth-century science (Westfall, 1971). In both of these his one-time protégé, the Dutchman Christiaan Huygens (1629–1695), was far more successful. Largely avoiding the currently highly confused concept of ‘force’, Huygens produced a successful collision theory in 1669 based on the concept of relative motion coupled to the Galilean Principle mentioned above (Westfall, 1971). This also provided the basis for his successful analysis of unresisted simple pendulum

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motion, from which he was able to confirm the observation of Galilei (1638) that a pendulum’s period is proportional to the square root of its length. By 1673 he had extended his analysis to the compound pendulum. As a result, the accurate measurement of time during motions, a perpetual underlying problem in early experiments in dynamics, became possible. As to circular motion, here a force concept could not be avoided. Despite a correct calculation of the acceleration involved, however, Huygens concluded from his over-complex model that circular motion is caused by a centrifugal (centre-fleeing) force. This, then, was Newton’s inheritance. While his comment that he had seen further because he had stood on the shoulders of giants might have been intended as a cruel gibe at the expense of his arch irritant, the crook-backed Robert Hooke (1635–1703), as some now maintain, the remark still stands as a fair reflection of Newton’s debt to his predecessors. After overcoming an initial inability to accept the inertia principle, Isaac Newton (1642–1727) constructed his own collision theory, tackling at last the difficult concepts of mass and ‘force’ so as to arrive at rules which link ‘force’ to change of motion and to a ‘force’s’ equally reactive consequence (Westfall, 1980). In doing so, he corrected Huygens’s misconception of circular motion through the introduction of centripetal (centre-seeking) force, pointing out that centrifugal force is merely the inevitable, yet oppositely directed, reaction to centripetal force. Application of his ideas on circular motion to the Moon’s revolutions about the Earth suggested an inverse square relation for gravitational attraction which he then found to fit the conclusions of Johannes Kepler (1571–1630) in 1609 and 1619 regarding the elliptic motions of the solar planets. However, it was not until the years of 1680–1681 that a successful application of these ideas to the comets of those years – in fact a single comet, as Newton realized – impelled him to the conclusion that he had discovered a gravitational relation of universal validity. This determined under all circumstances the relationship between a body’s mass and its force of weight. Thus a mechanics of motion, having wide application, at last came into being. Not only could rational sciences of fluid dynamics and of flight itself now begin to emerge but it is also difficult to envisage any serious success in either of these areas prior to this time. As we shall see, the invention of the aeroplane occurred within a century of Newton’s death.

3 From Newton to the emergence of the field theory of fluid flow It is difficult to imagine how the astonishing progress in the mechanical sciences seen over the last three centuries could have been achieved without the dynamical principles set down by Isaac Newton (1642–1727) in the Principia (Newton, 1687). Here he provides the details of his discoveries outlined at the close of the last chapter. That is not to suggest, however, that what we nowadays rather blithely term ‘Newtonian mechanics’ emerged here fully-fledged. The Principia’s stated Laws of Motion have restrictions, their genesis being in Newton’s deliberations on the impulsive collision of bodies treated as point masses. While his stated First and Third Laws (no net force in straight line motion at constant speed, action and reaction are equal but oppositely directed) are recognizable in modern terms, the crucial Second Law is not. This, as stated in Principia, relates change (not rate of change) of momentum to ‘motive force impressed’, the latter therefore being what we now call ‘impulse’. A further evolutionary step was thus required, as we shall see, before the Second Law could emerge in its modern form. Yet Newton (1687) distinguishes between two types of ‘force’: those that ‘be impressed altogether and at once, or gradually and successively’. Close reading of the Principia’s application of these ideas to situations involving such ‘gradual and successive’ (continuing) forces reveals that an element of time is slipped into the analysis at crucial junctures – an example of Newton’s unerring instinct in dynamics – so as to achieve the modern form of result. Newton’s application of these principles to the subject of fluid resistance produced the first, rationally argued relations in an area long fraught with misunderstanding. The resistance R, caused by flow collision with, or by deflection around, a body has the form R / V2 S, 3.1 being the fluid density, V the flow velocity, S being an area characteristic of the body. Prior to this date, the only results of this form had come from the experiments of Edme Mariotte (1620–1684) in which the vertical efflux from a water vessel impinged upon the horizontal arm of a balance (Rouse and Ince, 1957 ). The results indicated the above dependence on V2 but emerged in complete form only at posthumous publication (Mariotte, 1686). Newton’s V2 relation (3.1) for resistance is clearly at variance with Galileo’s experimental findings in which, for the slow oscillations of pendulums, resistance is

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proportional to V. Indeed, Newton (1687) reaches the same conclusion from his own pendulum experiments. He puts this down to a form of low speed ‘attrition’ in fluids, an effect which is dominated by the V2 collision/deflection resistance of relation (3.1) at higher speeds. The resolution of this difficulty did not emerge until the teachings of viscous flow theory had been absorbed around the beginning of the twentieth century. Thus an explanation will be delayed until we reach the appropriate point in the story. Meanwhile, the V2 resistance result will provide the focus of our interest. It is in Book II of Principia that Newton sets down his thinking on resistance. Here he distinguishes between two types of fluid: ‘a rare medium’ of disconnected, non-interacting particles which collide directly with a body, and a ‘continued medium, as water, hot oil, and quicksilver’, in which fluid particles press against each other as well as against the body. His application of the ‘rare medium’ concept to a particular problem – a comparison of the resistances of a sphere and its circumscribing cylinder, flat face normal to the particle stream – provided a calculation method later applied to a flat plate at incidence, ˛, giving the result for inelastic particles, R D V2 S sin2 ˛.

3.2

Its definitive derivation is due to Euler (1745) who nonetheless affirms that the rare medium is (the translation is by Truesdell, 1954 ) . . . contrary to the nature of all fluid matter, and in the whole world no such substance is to be found; nonetheless it serves to lay a basis for the understanding of resistance. Despite such strictures, the sine squared incidence relation was used by the nineteenth century so as to demonstrate the impossibility of human flight. In such demonstrations the relation’s contribution to any lift calculation would be so small as to require a compensatory, and massively over-large, value of wing area S. However, as we shall see, for wings the correct incidence behaviour is a sine, not a sine squared, relation. At first glance it is difficult to believe that the difference between such closely related mathematical functions might have such dramatic consequences. Yet at the small incidences common in flight this is the case; while sin 5° ¾ D 0.09, its square has barely a tenth of that value. Even in the last decade of the nineteenth century the debate as to the sine squared relation’s validity rumbled on, Langley (1891), for example, troubling to note that he can disprove the relation from his own experiments on flat surfaces. Yet during the 1950s the rare medium concept became resurrected so as to deal successfully with the analysis of flight in the low densities at the edge of the Earth’s atmosphere. Moreover, Lanchester (1907) seems to have been the first to suggest that the sine squared incidence relation might be applicable to wings of very low aspect ratio, the reasoning for this not being based, however, on the rare medium concept. Newton’s largely misconceived attempts to apply his continued medium concept emerged through successive editions of Principia. Throughout he employs what became known, notoriously, as the ‘ice cataract’ argument (Fig. 3.1). His application to that problem is investigated experimentally by Torricelli, the efflux from a water vessel. In Fig. 3.1a water is shown in downward flow from a cylindrical vessel, the water emerging at the hole EF. Yet the only water in motion, Newton supposes, falls freely under gravity in the region ABFE. For this flow Newton uses the continuity principle

From Newton to the emergence of the field theory of fluid flow P

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Fig. 3.1 (a) Water efflux from a vessel; the ‘ice cataract’. (b) Circular disk exposed to the efflux and supporting its own ‘ice forebody’ (Newton, 1687).

freely, accepting it as now well known. The remaining water (shown shaded in the adaptation of Newton’s figure) is stationary, as if congealed to ice. A circular disk PGQ (Fig. 3.1b) is then exposed to this efflux, its resistance being taken to be equivalent to the weight of the ice column HPQ (again shown shaded) now supposed to rest upon the disk. The shape of this column, Newton argues, is that of a paraboloid of revolution, from which it follows that for the disk, a sphere, or indeed any other shape of body exposed to the efflux, the constant of proportionality in equation (3.1) is 1/4. The resistance here being entirely the drag force D, equation (3.1) can be recast in the modern form of the drag coefficient CD as CD D

1 2D D . 2 V2 S

Here S is the maximum cross-sectional area of the disk, sphere or other body presented to the flow. To achieve this result, however, one suspects that Newton had devised an argument so as to fit a result already known. His own observations of the kinematics of the falling motion of spheres in water in 1708, and earlier, as well as those in air observed later at St Paul’s Cathedral by Francis Hauksbee (c.1666–1713) in 1710 and John Theophilus Desaguliers (1683–1744) in 1719, all inferred the value CD D 1 2 . Thereby equation (3.1) – the ‘common rule’ as it later became known – gained credence. Book II of Principia touches on one further topic relevant to our survey. This suggests the idea that internal resistance, within a flow itself, can be created by the flow’s own velocity gradients. The idea is introduced via the following hypothesis: The resistance arising from the want of lubricity in the parts of a fluid, is, cæteris paribus, proportional to the velocity with which the parts of the fluid are separated from each other.

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The further idea that the resistance is also inversely proportional to the separation distance between contiguous fluid layers is introduced into the analytical application of the concept. Thus Newton’s ‘want of lubricity’ relation, in practice, emerges as a proportionality between internal resistance and velocity gradient. Newton’s purpose here is to explain how the rotation of a solid cylinder in a bath of water, initially stationary, creates a vortex motion about the revolving cylinder ALF shown in Fig. 3.2. One further hypothesis is absolutely necessary to Newton’s argument and, indeed, is dictated by the nature of the problem attempted. This is that the water in contact with the cylinder’s surface must stick to that surface, otherwise no fluid can be induced to move. This is the first – and correct – use of the no-slip condition, which, we recall, applies at all solid surfaces exposed to fluid motion. The nature of the problem again dictates the requirement that the rest of the water must then be drawn into motion by some means. This, Newton believes correctly, is supplied by the action of ‘want of lubricity’ (viscous action) between the contiguous concentric layers of water shown in Fig. 3.2. For this action he proposes his vague notion of ‘impressions’, such ‘impressions’ being nonetheless taken to be proportional to the velocity gradients so created. His solution for the complete flow is then obtained by integration, performed ‘by the known methods of the quadratures of curves’. The graph abcde shown in Fig. 3.2 represents the variation of the fluid’s angular velocity throughout this flow. It is instructive to study more closely Newton’s method here. Since his flow is subject to continuous radial variation, he divides the fluid into the series of concentric annular ring elements illustrated in Fig. 3.2. This is the first occasion that this fundamental and now time-honoured procedure is used. He then applies physical principle to one arbitrary ring element so as to obtain his relation for the ‘impression’ upon it. In essence, this relation has a differential form, the solution for the complete flow being then obtained by integration, as already noted. However, as stated in Chapter 1, it is a fluid’s shear strain which is fundamentally related to viscous action, a fact which Newton was in no position to appreciate. In such annular flow systems this shear strain a

b c K

d

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Fig. 3.2 The viscous vortex (Newton, 1687).

A B C D E

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has a more complex relationship than that simply of velocity gradient. Moreover, the physical principle which should then be applied is that it is the viscous torques imposed on one ring by its neighbours which govern the ring’s rotation. In Newton’s method it is not clear that his ‘impressions’ have this purpose and certainly his resulting differential relation for velocity variation is incorrect. Thus Newton’s solution for what we now see to be the viscous vortex is in error. Nonetheless, it is regrettable that the two crucial ideas at its root – internal resistance and the no-slip condition – lay in abeyance for rather more than a century before being thoroughly revised by Navier, Stokes and others, as we shall see presently. The idea that a medium possessing continuous variations in its properties might be analysed by applying the appropriate physical principle to an arbitrary, but elementary, part of it was to be arrived at independently by at least one other mathematician within Newton’s lifetime. Since such approaches inevitably generated differential relations for the medium’s properties in question, a grasp of the new differential calculus of Newton and von Leibniz was therefore vital to success. Before leaving Newton, one final item should be mentioned, pre-dating even the first publication of Principia. This is his observation of the curving flight of a sliced tennis ball which is used as an analogy in his early deliberations on the refraction of light (Newton, 1672; our explanatory addition in brackets): For, a circular as well as a progressive motion being communicated to it [the ball] by the stroak, its parts on that side, where the motions conspire, must press and beat the contiguous air the more violently than the other, and there excite a reluctancy and reaction of the air proportionally greater. Here, despite the difficulties of a still developing science, and indeed terminology, we have an early vital clue to the secrets of flight. In truth, Book II of Principia is the curate’s egg of that work and, as far as the development of mathematical physics is concerned, it seems to have left Britain for some time in a state of post-Newtonian torpor. Yet the mistakes of great men can sometimes have a more seminal influence than their more solid achievements. Newton’s misconceptions, notably the ‘ice cataract’ argument, dictated the programme of research for at least the next century. The necessary corrective steps – leading to the field theory of fluid motion, albeit inviscid – were left to the Continental mathematicians Daniel Bernoulli (1700–1782) and his father Johann Bernoulli (1667–1748), Jean Le Rond d’Alembert (1717–1783), most notably Leonhard Euler (1707–1783), and Joseph-Louis, Comte de Lagrange (1736–1813). Initially, the world learned of these developments largely from Lagrange (1788), who adopted a regrettable tendency to laud the work of d’Alembert while giving Euler little credit for his achievements. Modern flow theory begins with the work of the Swiss-born physiologist and mathematician, Daniel Bernoulli (1738). In this he analyses the issue of water from vessels, aptly naming the subject hydrodynamics. In more modern times the term has been taken to apply more specifically to the study of inviscid flows, although in his study Daniel demonstrates some awareness of the effects of fluid friction. In discussing the flow of water in pipes, for example, he remarks that (the translation is by Truesdell, 1954 ) . . . the fluid is moved a little slower on the walls of the vessel, but faster in the middle, which is by reason of attrition. . .

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Early Developments of Modern Aerodynamics A

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G

Fig. 3.3 Water efflux from a vessel (Bernoulli, 1738).

Others were to make similar observations, although none seem to have grasped their full significance until the emergence of the boundary layer concept of Prandtl (1904). In many of his analyses, Daniel Bernoulli (1738) employs the continuity principle in conjunction with von Leibniz’s ‘vis viva’ (kinetic energy) concept as then understood. Tortuously he hammers his way to correct results for flow situations such as that depicted in Fig. 3.3. However, the water flow is treated en masse, so to speak, and Daniel conceives of pressure only in terms of the height of the water column (manometer) shown in Fig. 3.3. This height he is able to calculate correctly, finding experimental confirmation for his result. However, Daniel is also aware of the Newtonian rare medium concept and finds a situation in which its method can be applied correctly to a continuous fluid. This is the case in which a fine water jet strikes a large angled plate. Here all of the jet’s fluid is eventually deflected so as to move parallel to the plate’s surface. Since all of the jet’s momentum component perpendicular to the plate’s surface is destroyed in this process, as is that of the particles in Newton’s ‘rare medium’ case, the plate consequently experiences a force of resistance which Daniel is able to calculate correctly by the Newtonian method. This resistance he shows to be directed perpendicularly to the plate’s surface and to be proportional to the sine of the plate’s incidence angle. This dependence on the sine of the angle, not the Newtonian sine squared dependence, caused some confusion for those subsequently unable to realize that here the plate was not completely immersed in a moving fluid stream. Meanwhile, Daniel’s father Johann Bernoulli, at Newton’s death the recipient of the mantle of Europe’s leading mathematician, was beginning his own assault on the water vessel problem. His book (Bernoulli, 1743) appeared after publication of Daniel’s work, yet he dated it 1732 in an evident attempt to claim priority for his findings. This disgraceful behaviour consigned the work to the category of rank plagiarism until Truesdell (1954) examined the whole matter in detail. From this it emerged that Bernoulli (1743) contains at least two significant advances. The first is that Johann replaces Daniel’s water columns as measures of pressure by forces due to pressure


within the fluid itself. Secondly, and more significantly, Johann, having recognized that flow conditions must vary continuously along pipes of varying area according to the continuity principle, then takes the vital step of analysing conditions at an elementary slice of fluid across the pipe. As Truesdell (1968) points out, this idea Johann had seen his brother, Jakob Bernoulli (1655–1705), use in 1690 in an analysis of the catenary problem, the determination of the curve of a rope suspended between two fixed points. To solve this problem Jakob had divided the rope into elementary slices, then applied the physical principle of static equilibrium to an arbitrary slice. In essence, the idea is related to that used by Newton in the viscous vortex problem. However, the physical principle which Johann now applies to his moving fluid slice – or ‘eddy’ as he confusingly calls it – is no longer his hitherto preferred ‘vis viva’ concept, but Newton’s version of the Second Law from Principia. Upon integration, the result is a relation recognizably close to the modern statement of the so-called Bernoulli equation, which we will come to presently. This relates continuous pressure and velocity variations along an inviscid flow streamtube. However, Johann’s explanations are so wrapped about in circumlocutions that not until Truesdell (1954) had unravelled Johann’s correspondence with his former pupil and Daniel’s close friend, Leonhard Euler, did it become clear that Johann’s vagueness concerning his ‘eddy’ was deliberate, prompted by his fear lest it be confused with Newton’s notorious ‘ice cataract’. As we shall see, it was this combination of a focus on an infinitesimally small material element coupled with the application to it of the powerful Second Law which Euler promptly seized upon, generalized, and exploited with dramatic success. As Euler was embarking on his massive investigation of fluid flow, France’s leading mathematician, d’Alembert, also took up the problem. Recognizing that the flow about a body presents a field of continuous variation in velocity, d’Alembert (1752) was able to obtain the modern differential form of the continuity principle for a constant density fluid. This principle, he understood, must be obeyed whatever the dynamic conditions applicable to the flow. To that end, he introduced the concept of streamlines, the continuity principle then asserting that the flow rate between any pair of streamlines must remain constant. He then attempted to obtain the equations of motion for a fluid flow, showing in the process how complex variable theory can be employed in their solution. Beyond this, however, his attack began to founder. He lacked a complete understanding of pressure and assumed, moreover, the universal validity of certain differential relations, the physical meaning of which asserts that all fluid elements do not spin about their own axes (the zero vorticity condition of irrotationality). His rather unsatisfactory attempt to calculate resistance produced the result, correct as it turned out for the steady flow of an inviscid fluid, that, while resistance obeys the Newtonian equation (3.1), the constant of proportionality in that relation is precisely zero. This is the genesis of the now well-known d’Alembert Zero Drag Paradox. Although he returned to the problem later, introducing in doing so the concept of the stream function which measures flow rate between streamlines, he made little further progress. Earlier, however, Euler (1745) had arrived at the same zero resistance result, and by more rigorous means. And now Euler was beginning to exploit the method, mentioned earlier, which he had learned from Johann Bernoulli. As Truesdell (1954) relates, not only was Euler the first to arrive at the modern conception of the equality of pressures at a point, the first to derive the equations of continuity and motion in general terms

25

26


for an inviscid fluid of variable density, but he was also the first to obtain from them the modern form of Bernoulli’s equation. This states that along any streamline within the steady, inviscid, irrotational flow of a constant density fluid, the pressure, p, and the velocity, q, are related by the equation pC

q2 D constant. 2

3.3

Thus, if the velocity increases along a streamline, this must be accompanied by a corresponding decrease in pressure. The continuity principle then relates velocity changes to changes in the distance between streamlines. Later, Euler went on to show that if certain additional differential relations are satisfied (these representing the zero vorticity condition which d’Alembert had assumed as being always enforced) the flow equations reduce to that usually associated with the later work of Pierre-Simon, Marquis de Laplace (1749–1827) (Laplace’s equation). All this emerged in Euler’s mature papers on fluid flow during the period 1752–1766. Almost in passing, Euler also showed that solutions of the Laplace equation are additive, giving examples which represent the source, the sink and the vortex. This vortex, incidentally, has the circulation property stated earlier by Leonardo and is that described in Chapter 1. In parallel with these major developments, our modern view of the mechanics of motion also began to emerge. Thus the idea proposed by d’Alembert (1743) of imposing opposite accelerations on bodies in accelerated motion, so as to treat them as problems in statics, became refined at the hands of Euler and Lagrange so as to emerge as the ‘d’Alembert Principle’ used nowadays. And Euler’s grasp of Newton’s underlying method in dynamics, revealed in papers published between 1747 and 1750, caused him to assert as ‘the unique foundation of all mechanics’ the principle that force is proportional to the product of body mass and acceleration. Moreover, the latter is expressed as the second derivative with respect to time of the spatial ordinates. It is here, then, that the modern version of the Second Law finally emerges. All that is then required, Euler states, is ‘to apply this fundamental principle adroitly’ to finite solid bodies of distributed mass, as distinct from the mass-points of Newton. When combined with the additional principle of moments, the adroitly obtained result is the system of equations of rotational motion for a finite solid body in which the moments of inertia appear. The interested reader is directed to Truesdell (1954, 1968) for his magisterial surveys of these developments in both the dynamics of solid bodies and the field theory of fluid flow.

4 The era of the whirling arm and the invention of the aeroplane While the field theory of fluid flow began its mighty journey in Continental Europe, attempts to confirm the common rule for resistance began in Britain. Ironically, the first such attempt indicated departures from that rule. These were detected by Benjamin Robins (1707–1751) (Robins, 1742), using his ballistic pendulum (Fig. 4.1) to infer, in effect, CD values from observation of the kinematics of the decaying motion of spherical shot; Principia supplied the kinematic results for motions resisted as V2 . Such departures from the common rule, he concluded, were associated with the fact that the spherical musket balls under test had travelled near to, and beyond, the speed of sound. Rogers (1982) has shown (Fig. 4.2) how close Robins’s results are to the modern data of Hoerner (1958), which show an additional dependence on Mach number (the ratio of the speed of the object to the local speed of sound). As Fig. 4.2 suggests, however, Mach number effects were not to trouble the builders of the early low speed (low Mach number) aeroplanes. In a paper read before the Royal Society in 1747 Robins (see Wilson, 1761) then provided a more direct method for the confirmation of the common rule at low speeds. In this he describes experiments carried out in the previous year using the whirling arm (Fig. 4.3) built for him by John Ellicott (c.1706–1772), also a Fellow of the Royal Society. Not only was a sphere tested, thereby confirming Newton’s result of CD D 12 , but a number of other shapes tested – pyramids and plates in various configurations – established that body shape and attitude to the flow affect the value of CD , contrary to Newton’s belief. Having suspected that other observed wayward motions of shot might be ascribed to a whirling effect caused by passage along the gun barrel, in 1746 Robins demonstrated the consequence of this using a wooden pendulum bob suspended by a double string ‘about eight or nine feet long’ (see Wilson, 1761). By twisting these strings together, so that the bob rotated about its stringwise axis, the pendulum motion was then seen to include a sideways motion, the direction of which he found he could predict correctly as being towards the bob’s side . . . on which the whirl would combine with the progressive motion; for on that side always the deflecting power acted; as the resistance was greater here, than on the side, where the whirl and progressive motion were opposed to each other.


A

F

E

S

R

G K O

H V

L

N

N

28

B

I D

M

W

C

Fig. 4.1 Ballistic pendulum (Robins, 1742).

The deflecting power, however, is not seen here as being due to pressure differences laterally across the body, as Magnus (1852) was later to show, but to some vague notion of differential resistance. As a simple, convenient experimental tool the whirling arm rapidly gained popularity, perhaps by a process of parallel invention. We have it on the authority of Smeaton (1759) that his friend, a certain Mr Rouse of Harborough, independently produced a whirling arm at about the same time as that used by Robins. Rouse’s results gave a value of CD D 1.94 for plates held normal to the direction of motion, yet this value, far too high and often incorrectly attributed to Smeaton (1759), reverberated through history to confuse both Lilienthal and, initially, the Wrights. The modern value (ESDU, 1970) is 1.14. John Smeaton (1724–1792) himself was aware of Robins’s work and adapted the whirling arm for his windmill tests, the model windmill being placed on an arm which

The era of the whirling arm and the invention of the aeroplane 1.5

HOERNER (1958) 1.0 CD 0.5

ROBINS (1742) 0 0.6

0.8

1.0

1.2

1.4

1.6

MACH NUMBER

Fig. 4.2 Ballistic pendulum results (Robins, 1742); drag coefficient, CD ¾ Mach number, M (after Rogers, 1982).

A

F

G

B B

C

H

P L

M

Fig. 4.3 Whirling arm, 1747 (Wilson, 1761).

was here (Fig. 4.4) caused to rotate by the experimenter pulling steadily on the nearhorizontal string shown in the figure. The windmill’s power output was measured by the rate at which the weight was lifted in the scalepan, timing being provided by the pendulum shown. Smeaton carried out his initial tests in 1752 and 1753. In the following year he travelled the Low Countries so as to gain experience of practice there. In his paper Smeaton (1759) states that he has delayed publication until he can confirm his results through full-scale comparisons. Dutch practice had a significant effect on Smeaton’s programme since the Dutch windmill, as Drees (1979) reveals, had by then enjoyed major development. Around

29

30


N

L

Q S

M P T

D

F

O

I

R G

C W V Y

H Z E

X B

A Inches 0 12 6

1

2

3

4

5

6

7

Scale of feet

Fig. 4.4 Whirling arm (Smeaton, 1759).

1550 the blade mainspar had been moved from mid-chord to a position one-third chord from the blade’s leading edge. Blade twist, giving greater blade incidence at the root, had been introduced shortly after 1660. A final move of the single mainspar to the structurally most advantageous quarter chord position occurred around 1650, the leading quarter chord of the blade, a wooden slat, also being drooped slightly so as to produce a camber effect. When it finally emerged, wing theory was to confirm this choice of the quarter chord for the mainspar position, the theory predicting that, at this point, the aerodynamic moment coefficient remains constant with change of incidence. Therefore, a single mainspar placed at this point has less chance of failure in torsion. Some of the Dutch windmill’s evolutionary steps are reflected in Smeaton’s tests but not, regrettably, the introduction of camber. He began with rectangular untwisted blades with mainspars at their leading edges, the triangular leading sections shown in Fig. 4.4 being added later. Having found the blade angle, 15° , which gave maximum power, he then tried a twist distribution calculated by Colin MacLaurin (1698–1746) and based on the sine squared incidence relation. When viewed from behind this produced a blade trailing edge shape ‘convex to the wind’, as Smeaton aptly describes it, but which increased the power by a mere 14%. Twisting the blades according to Dutch practice, however, produced the far more significant power increase of 38%. Here the blade trailing edge shape Smeaton describes as being ‘concave toward the wind’, a phrase which has misled some (see Pritchard, 1957; Ackroyd, 1984 ) into the claim that Smeaton was the first to discover the lift advantages of camber. Smeaton’s final step was to add the triangular leading edge sections shown in Fig. 4.4. This placed about one fifth of the blade area – not quite the one quarter of Dutch practice – ahead


of the mainspar but created a triumphant power increase, compared with the untwisted rectangular blades, of 77%. By 1756 Jean-Charles de Borda (1733–1799) had constructed his own whirling arm at Dunkerque. Borda (1763) makes no mention of Robins’s work and it may be that he independently invented this apparatus. He used a symmetric double arm rotating in the vertical plane for his tests in air, the two arm extensions carrying identical bodies. He later supplemented these with tests in water (Borda, 1767) using a single horizontal arm rotating in air, the single test body being suspended beneath the arm while moving through a circular water bath. Testing a wide variety of body shapes, he, like Robins, found a clear dependence of drag on shape but had difficulty correlating his results in terms of fluid density, probably because of the shallowness of the water bath used. Other pitfalls in whirling arm testing are suggested by the results of Richard Lovell Edgeworth (1744–1817) (Edgeworth, 1782). Perhaps because his arm was large enough to sweep through most of the containing room, causing interference effects, he was unable to correlate his normal plate results in terms of plate area. However, prompted by reports of sailors that bellying sails are superior to sails drawn taut, he bent a plate into progressively tighter arcs to discover that an increasing camber results in greater drag. Rather more carefully conducted whirling arm tests were reported by Charles Hutton (1737–1823) (Hutton, 1790). To confirm the dependence on V2 in equation (3.1), his tests covered eighteen different speeds but only one body shape was used, a solid hemisphere, set curved face then flat face forward. The latter case produced a drag 2.45 times that obtained in the former case. Yet here, as in all early work on resistance, there lingers the belief that the forward faces of bodies are solely the source of resistance, a belief supported by the Newtonian rare medium concept. According to this only the forward surfaces of the body, up to its maximum cross section, experience collision with the approaching ‘beam’ of particles, surfaces aft of that section remaining in the beam’s ‘shadow’, so to speak, so as to experience no collision. After remarking that this concept yields the smaller value of 2.0 for the above factor, Hutton (1790) first doubts the appropriateness of the concept, veers towards the truth, yet then immediately dismisses the outrageous thought (our explanatory addition in brackets): . . . some small part of it [the resistance] may arise from the different figure of the hinder parts of the hemisphere, although I hardly suspect that this may cause any sensible difference. Thus far the effect of incidence variation on plate resistance had not been explored, perhaps because the above misconception as to the source of resistance had coloured thinking here. Borda (1763, 1767), for example, had not tested plates but wedges of various angles, believing, according to the Newtonian concept, that each wedge face would behave as a single plate at incidence. Yet as to his resistance results for wedges Borda (1767) says (our translation), . . . they were very far from proportional to the squares of the sines of the angles of incidence; and even when those angles were very acute, I have found that the resistances did not diminish as much as the simple sines.

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32

Early Developments of Modern Aerodynamics f

g c

b K

d

a

e

n

G

z

h a

W

u

D

E A

F

m

B

C

R

T

Fig. 4.5 Whirling arm (Vince, 1795).

The first to explore plate resistances throughout the incidence range was Samuel Vince (1749–1821). Vince (1795) describes his cruciform double arm (Fig. 4.5) to carry four identical plates, the whole assembly to be placed in a water bath. In his later experiments (Vince, 1798) the plates were set at alternating positive and negative incidences. No doubt the resulting upflow/downflow and general turbulence affected his measurements so that, at small plate incidences, he could detect no dependence on either a sine or a sine squared incidence relation. At the time this, if nothing else, cast welcome doubt on the latter relation although nowadays one doubts the accuracy of Vince’s results. However, an incidence correction of a mere 3° to the small incidence results, in an attempt to allow for the upflow/downflow, produces results close to a sine relation. Vince (1798), however, does not attempt such a correction. In contrasting his whirling arm results with those in which a water jet impinges on a large angled plate (as mentioned earlier, the problem first analysed successfully by Bernoulli, 1738), he realizes an important point. He remarks that in the jet case the water cannot flow to the back face of the plate whereas in his whirling arm experiments flow occurs at the plates’ rear surfaces. This, then, was the limited understanding achieved when the aeroplane came into being. It is generally accepted that this was the invention of Sir George Cayley (1773–1857) in 1799 at Brompton, near Scarborough in Yorkshire. Perhaps wishing permanently to record this event he went to the trouble of engraving his idea, and its date, on one face of a silver disk now in the Science Museum, London (Fig. 4.6a). Here we see Cayley’s vital break with the flapping wing past, the flying machine having a propulsive system, albeit a flapper or paddle arrangement, which is entirely separate from a fixed lifting wing. The reverse of the disk (Fig. 4.6b) shows Cayley thinking

The era of the whirling arm and the invention of the aeroplane (a)

(b)

Fig. 4.6 Cayley’s silver disk, 1799: (a) the aeroplane; (b) air resistance on a wing.

scientifically about the problem of flight, the overall air resistance being decomposed via the triangle of forces into its lift and drag components. No doubt this thinking led to his later prophetic statement (Cayley, 1809) on powered flight: The whole problem is confined within these limits. viz. To make a surface support a given weight by the application of power to the resistance of air. The simple, the obvious, can sometimes only become so having once been stated. Note, however, that the overall resistance shown in Fig. 4.6b lies perpendicularly to the wing’s surface, an oversimplification supported by Newton’s rare medium concept but one which Cayley retained throughout his career. As to the application of power, despite his invention of the hot air engine, and indeed his construction of an ingenious gun-powder engine, here he was balked for want of a suitably light and powerful prime mover. His significant aeroplanes therefore remained gliders. In 1804 Cayley began to provide a practical basis for his invention through tests using the whirling arm shown in Fig. 4.7 (Gibbs-Smith, 1962 ). This he adapted by incorporating a horizontal hinge so that lift forces could be measured by balancing the arm. Because of the rotary motion of the test body, a paper plate 1 ft square (0.093 m2 ), he recognized that the plate’s lifting effect, dependent on V2 , would not be distributed evenly across the plate. The outer tip of the plate moving faster than the inner tip, he argued that the centre of pressure location (the point at which the effective lift force acts) would lie outboard of the midspan position. For his balance calculations he estimated this location by applying the common rule to chordwise strips of the plate followed by numerical integration. His results for the lift force L, covering an incidence range of 3° to 18° in 3° steps, have been set in the modern notation of lift coefficient, CL , where 2L , CL D V2 S by Yates (1954). These are shown in Fig. 4.8 to agree remarkably well with modern data (ESDU, 1970). He also measured the drag of the plate when set normal to the

33

34


Fig. 4.7 Cayley’s whirling arm, 1804 (Gibbs-Smith, 1962). 1.0 0.8 0.6 CL

CAYLEY + 4.57 m/s O 6.65 m/s +

+

ESDU (1970) +

0.4 + 0.2

+

+

0 5

10

15

20

25

α°

Fig. 4.8 Cayley’s whirling arm results, 1804; CL ¾ incidence, ˛ (after Yates, 1954).

motion. His results give a CD value of 1.5, a significant improvement on the Rouse value of 1.9 but still some way from the correct result of 1.14. Later in 1804 Cayley attempted to compare his whirling arm results with the performance of his modification to a simple kite, the world’s first practical glider (Fig. 4.9). The kite itself formed the wing of a markedly low aspect ratio, this being set at 6° to the rod fuselage. Using the weight shown forward he chose a centre of gravity position near the mid-area of the wing, perhaps in the belief that the latter would be the centre of pressure location as predicted by the Newtonian rare medium concept. Be that as it may, as early as 1801 he had already begun to doubt the accuracy of the sine squared incidence relation (3.2), favouring a dependence nearer to the sine of the incidence angle (Hodgson, 1933a,b). This he may have drawn from the work of d’Alembert,


G

Fig. 4.9 Cayley’s glider, 1804 (Gibbs-Smith, 1962).

Condorcet and Bossut (1777) in which towing tests with barge models having wedge shaped bows of various angles, but not angled plates, are reported. Cayley (1809) maintains this view, perhaps by then encouraged by his own results shown in Fig. 4.8. However, with his glider’s centre of gravity positioned as indicated earlier he then found that the glider (Gibbs-Smith, 1962 ) . . . would proceed uniformly in a right line for ever with a velocity of 15 feet per second when its tailplane was set at the positive incidence of 11.5° relative to the rod fuselage. At this point in Cayley’s work there is no mention of trim considerations, the fact that in its straight line, non-rotating motion the glider must experience zero moment about its centre of gravity. Had he done so, due to the tail’s lift, he could have deduced that the wing’s centre of pressure must have been forward of a mid-area location. In flight the glider’s path was directed downward by 18° from the horizontal, from which we deduce the modest L/D ratio of 3. As Yates (1954) calculates, the glider probably set itself in trim at a wing incidence of 20° to that path – not merely the 6° wing setting angle assumed at the time by Cayley – and with a CL value of 0.7. Cayley’s misunderstanding of true incidence caused him to suspect the apparently large discrepancy in his glider’s lift, compared with his whirling arm results, to be due to wing camber effects. The latter he already suspected as enhancing the lift of birds and indeed anticipated further beneficial effects from the use of high aspect ratio wings. A later glider (Fig. 4.10), tested in 1808, incorporated both of these features while retaining the earlier glider’s positive tail incidence (Gibbs-Smith, 1962 ). Here trim considerations were applied, indicating to his surprise a centre of pressure not only forward of the wing mid-area but also a little aft of the quarter chord point. As wing

Fig. 4.10 Cayley’s glider, 1808 (Gibbs-Smith, 1962).

35

36


Fig. 4.11 Cayley’s ‘Solid of Least Resistance’, 1808 (Gibbs-Smith, 1962).

theory was later to predict, the quarter chord point is the asymptote to the forward movement of a wing’s centre of pressure as incidence is increased. In 1809 Cayley obtained his axisymmetric ‘solid of least resistance’ (Fig. 4.11) based on his measurements of the girth distribution of a ‘well fed’ trout. A section of this shape, as von Karman (1954) notes, happens to be close to that of a modern NACA low drag aerofoil section. In this same year a further flying machine was constructed, the details of which are unknown other than its possession of a wing area of 27.9 m2 . This, when unloaded and tested as a glider, may have achieved a L/D ratio as high as 7 or 8. When carrying a man running (Cayley, 1810) . . . forward with it, with his full speed, taking advantage of a gentle breeze in front, it would bear upward so strongly as scarcely to allow him to touch the ground; and would frequently lift him up, and convey him several yards together. Much of the above work provides the material for Cayley’s great triple paper on ‘aerial navigation’ (Cayley, 1809, 1810) which ushers in the science of aeronautics. Its publication was prompted by misleading reports of the ornithopter ‘flights’ of Jacob Degen (1761–1848) in which, as it later transpired, he had been carried aloft beneath a hydrogen balloon. Additional items in the paper include Cayley’s astute estimates of the power requirements for flight based on his assessment of bird performance, suggestions for flapper propulsion systems, and designs for wing structures incorporating wire braced hollow tubular spars for strength and rigidity. As to a cambered wing’s ability to produce an enhanced lifting effect, he presciently argues in the triple paper that ‘a slight vacuity’ is created on the upper surface a little aft of the leading edge. Moreover, the curved shape of the wing’s section removes upflow at the leading edge whilst turning the flow downward at the trailing edge. Here we meet for the first time some of the rudiments of the understanding that a wing creates lift by first destroying upward flow momentum around the nose, then imparting downward momentum at the tail. The reaction on the wing, caused by this overall downward momentum change imposed on the flow, is the wing’s lift force. All this, unbeknown to Cayley, is the consequence of the wing’s ‘bound’ vortex. Other aerodynamic arguments are, however, instructive of Cayley’s further limitations at this stage of his career. Whilst recognizing now that the centre of pressure lies forward of a wing’s mid-chord at low incidence, and that these two locations should coincide at 90° incidence, he therefore assumes a monotonic behaviour on the part of the centre of pressure in which it moves aft with increasing incidence. As remarked earlier, modern wing theory, in contrast, teaches that with increasing incidence the centre of pressure moves forward asymptotically to the quarter chord point. While Cayley may have been correct in his assessment of the centre of pressure’s movement


on certain of his plate-like very low aspect ratio wings, as later results obtained from flat plates at incidence were to show, his prediction is not correct in the context of the cambered, higher aspect ratio wings developed later. The Wright brothers were the first to realize this error through the kite-testing of their No. 2 glider in 1901 (MacFarland, 1953 ). However, Cayley’s argument built on this misconception is that the tail-less aeroplane is longitudinally inherently stable, the tailplane function being merely that of steering and re-trimming for different flight speeds. The vertical fin also provides merely a steering action; there is no recognition here of the fixed fin’s provision of weathercock lateral stability. As to other features of lateral stability, although Cayley recognizes the benefit of wing dihedral his argument for it is over-simplistic. The triple paper ends with Cayley’s thoughts on streamlining perhaps drawn from Du Buat (1786), the only authority at that time to have recognized something of its significance: It has been found by experiment, that the shape of the hinder part of the spindle is of as much importance as that of the front, in diminishing resistance. This arises from the partial vacuity created behind the obstructing body. If there be no solid to fill up this space, a deficiency of hydrostatic pressure exists within it, and is transferred to the spindle. This is seen distinctly near the rudder of a ship in full sail, where the water is much below the level of the surrounding sea. The cause here, being more evident, and uniform in its nature, may probably be obviated with better success; in as much as this portion of the spindle may not differ essentially from the simple cone. I fear however, that the whole of this subject is of so dark a nature, as to be more usefully investigated by experiment than by reasoning. . . at which point he turns to his trout shape of Fig. 4.11. As mentioned in Chapter 1, this ‘deficiency of hydrostatic pressure’ occurs as a result of boundary layer separation before the rear of a badly streamlined body is reached. Often, as here in his linking of pressure drop to water depression at a ship’s rear, Cayley connects such keen observation to rational argument so as to reach sensible conclusions. The ‘dark nature’ of the subject, however, was not to be illuminated until Prandtl (1904) introduced the boundary layer concept. Much of Cayley’s work then became devoted to flapper propulsion methods, airships, and indeed his other multifarious activities described by Pritchard (1961). The cause of ‘aerial navigation’ by aeroplane was then left to two engineers involved in the lace industry, William Samuel Henson (1812–1888) and John Stringfellow (1799–1883). Gibbs-Smith (1985) and Penrose (1988) give details of their activities. Henson’s design for his ‘Aerial Steam Carriage’ (Fig. 4.12) emerged in 1843 and clearly followed Cayley’s teaching. Yet in certain features Henson’s thinking went significantly beyond Cayley’s conception. In Fig. 4.12 can be seen the airscrew propulsion, a feature which Cayley, with but one exception, never advocated for his aeroplanes. The high aspect ratio wing, again avoided by Cayley on structural grounds, was to be carried on three spars, the outer ones being hollow and tapered while attached to the wire-braced king posts. Attached also to the spars were shaped and cambered ribs covered above and below by oiled silk. The vertical rudder was separated from the elevator tailplane. The wing span was to be about 46 m, the wing area 418 m2 and power was to be supplied by a 19–22 kW steam engine (Ballantyne and Pritchard,

37

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Fig. 4.12 Henson’s ‘Aerial Steam Carriage’, 1843 (Ballantyne and Pritchard, 1956).

1956; Gibbs-Smith, 1985 ). It was intended that the machine would be launched on its tricycle undercarriage down a ramp so as to supplement engine power at take off. The sad story of Henson’s involvement in attempts to form a premature Aerial Transit Company, to convey passengers all over the globe, is told in detail by Penrose (1988). Although the design was never built, a small steam engine-powered model of 6 m span was built and tested, unsuccessfully, between 1845 and 1847 at Bala Down near Chard. Apparently, structural failure was a persistent problem. Henson had charge of the design and construction of the model while his friend, Stringfellow, contributed to the engine. Discouraged by the model’s failure, Henson emigrated to the United States in 1848. Stringfellow, in contrast, persisted, building a small steam-powered model of about 3.25 m span. A reproduction of it is shown in Fig. 4.13, the model being carried on an overhead wire from which it was launched in tests in a disused lace factory in Chard in 1848 and later from sheltered ground in the Cremorne Gardens, London. How successful these tests were remains a subject of debate although Penrose (1988) argues that at least some of the Chard tests may have been successful. Moreover, Penrose (1988) goes on to cite evidence for Stringfellow’s success in 1851 with a larger steam-powered model of 3.66 m span.

Fig. 4.13 Science Museum replica of Stringfellow’s powered flying machine, 1848.


Fig. 4.14 Cayley’s triplane ‘Boy Carrier’, 1849 (Gibbs-Smith, 1962).

Henson’s design of 1843 excited much comment, and criticism, not least from Cayley himself who clearly suspected the structural integrity of the high aspect ratio wing. Cayley (1843) suggests in its place the use of smaller span multiplane configurations, thereby providing the basis of the biplanes and triplanes of the future. Thus in his old age Cayley took up the cause of the aeroplane again, by 1849 constructing and testing a flapper-powered triplane (Fig. 4.14) having a wing area of 31.4 m2 and mass 59 kg. In a later paper, the full text of which can be found in Pritchard (1961), Cayley describes glider tests of what may well have been this triplane and what has now become popularly known as ‘the boy-carrier’: The balance and steerage was ascertained, and a boy of about ten years of age was floated off the ground for several yards on descending a hill, and also for about the same space by some persons pulling the apparatus against a very slight breeze by a rope. Apart from the wing configuration, a further essential feature to note is the duplicated tail unit, a feature which also appears in the later design of Cayley (1852) (Fig. 4.15). Although there is no indication that Cayley built a glider to this design, he is specific on its constructional features and anticipated performance. Clearly, this glider, which Cayley (1852) terms a ‘governable parachute’, is steered by the lower tail unit. Concerning the upper tail unit, however, Cayley (1852) says (our explanatory additions in brackets) The larger one [the upper tail unit] is, when it has once been adjusted so as to give a straight and steady steerage, to be permanently secured in that position. It gives the most steady and secure course when slightly elevated [at negative incidence], which also tends to secure the parachute from pitching, should it be exposed to an eddy of wind, and, together with the weight of the car, immediately restores the horizontal position. Although there is still no recognition of the upper fixed fin’s role in lateral stability, here at last we have Cayley’s recognition of the stability requirement for a fixed

39

40

Early Developments of Modern Aerodynamics B

A

E

C F

B

A

B

Fig. 4.15 ‘Governable Parachute’ (Cayley, 1852).

tailplane. Moreover, the upper fixed tailplane is to be set at slight negative incidence so as to place it in a downloaded condition, the usual outcome of trim and stability considerations on the modern aeroplane. As if to emphasize this newly recognized stability consideration, we also find the following (Cayley, 1852): The centre of gravity of the car and its load must be so much in advance of the centre of resistance of the main sail as will incline it downward in front in an angle of 5° or 6° with the horizon. A few trials as to the required position of the permanent rudder, so as to adjust the angle of descent to be steady, and not subject to alternate rises and falls, will soon enable more extensive ranges to be made. The downloaded tailplane is consistent with this recommendation for a forward centre of gravity, all of which probably produced a longitudinally stable aeroplane. As to range, Cayley (1852) anticipates a horizontal progress of about five or six times the glider’s initial release altitude. This implies gliding angles expected to lie between 9.5° and 11.3° , or a L/D ratio lying between 5 and 6. This is achievable, he implies, with the low aspect ratio wing, the recommended area of this and the rudder being 46.5 m2 , the structural mass 68 kg. For lateral stability a dihedral angle between 8° and 10° is recommended. The final record of Cayley’s involvement with the aeroplane is the recollections of his granddaughter, Mrs Dora Thompson, when an old lady. She recalled witnessing, at about the age of nine, the testing of some form of flying machine at Brompton Dale around 1853. The sole occupant of the machine was Cayley’s coachman who, upon landing, leaped out in some agitation, crying (Gibbs-Smith, 1962): Please Sir George, I wish to give notice, I was hired to drive and not to fly.


No detailed description of this coachman-carrier survives but Gibbs-Smith (1962) suggests that it was probably similar to the boy-carrier, but perhaps incorporating the monoplane wing of the ‘governable parachute’. Cayley died on 15 December 1857, a little before his 84th birthday. His remains lie in the family vault within the Parish Church of All Saints, Brompton, at which in 1802 William Wordsworth had married the local lady, Mary Hutchinson. With typically British understatement, to put the most diplomatic of interpretations on what some might see as base neglect, a blue plaque at his supposed birth place of Paradise House in Scarborough and a small tablet at Brompton Hall are the only national memorials to this great man’s efforts. Sadly, too, as related by Gibbs-Smith (1985 ), at Cayley’s death much of the impetus for aeroplane development passed to France, where people such as Félix Du Temple (1823–1890), Victor Tatin (1843–1913) and, particularly, Alphonse Pénaud (1850–1880) made significant headway in the development of flying models.

41

5 Nineteenth-century developments in the understanding of fluid flow In the early 1840s, as Cayley renewed his interest in the aeroplane, the British became apprised of Continental developments in flow theory largely through extensions provided by two eminent academics, William Thomson (later, Lord Kelvin) (1824–1907) and George Gabriel Stokes (1819–1903). Initially, Thomson’s interests centred largely on electric field problems. However, since these are governed by Laplace’s equation, his methods (the method of images and his inversion theorem being examples) had immediate application to inviscid flow theory, as Stokes and others quickly realized. As to Stokes’s own contributions, for the analysis of axisymmetric flows Stokes (1842) introduces the stream function which now bears his name. Later he investigated a number of unsteady inviscid flows about cylinders and spheres (Stokes, 1849), using line and point doublets (the combination of Euler’s sources and sinks described in Chapter 1) as appropriate. Acknowledging Stokes’s use of these mathematical singularities, William John Macquorn Rankine (1820–1872) employed a uniform array of doublets set between two ‘foci’ so as to obtain flow solutions about oval shapes – the Rankine Ovals (see Fig. 1.6b) – which, he believed, might be of use in ship hull design (Rankine, 1864). He appeared to be unaware that a more elementary matched source–sink pair located at his foci would serve equally well (see Fig. 1.6b). However, this over-complex approach nonetheless provides an early example of the powerful so-called method of distributed singularities. Subsequently, Rankine (1868) showed how more complicated shapes could be obtained from doublet distributions between four foci for both two-dimensional and axisymmetric cases and that streamline patterns could be formed by ingenious geometrical construction. Meanwhile, in 1850 it appears that Thomson had communicated to Stokes what became popularly known as the Stokes Theorem, Stokes then setting the theorem in the Smith’s Prize Examination for 1854. As described in Chapter 1, this theorem links circulation to the vorticity within the circulation circuit. From the viewpoint of our survey, perhaps Stokes’s greatest triumphs were his derivations of what we now call the Navier–Stokes equations governing viscous flows (Stokes, 1845) and of their solution for the slow steady streaming flow about a sphere (Stokes, 1851). All this was not achieved entirely in isolation, however. In 1822 Baron Augustin-Louis Cauchy (1789–1857) had announced his generalization of the pressure concept which adds tangential shear stresses to the normal (pressure) stress hitherto assumed to be acting on material elements (the earliest published version of this is in Cauchy, 1823). From this he had gone on to derive the equations of motion for a fluid


subjected to such a stress system. From assumed molecular models of a fluid, ClaudeLouis-Marie-Henri Navier (1785–1836) (Navier, 1823) and Siméon-Denis Poisson (1781–1840) (Poisson, 1831) had, in effect, obtained relations for such stresses so as to derive forms of the Navier–Stokes equations. Later, Adéhmar-Jean-Claude Barré de Saint-Venant (1797–1886) (Saint–Venant, 1843) arrived at the Navier–Stokes equations along lines very similar to those of Stokes; regrettably, Saint-Venant has received little credit for this achievement. In his paper Stokes (1845), initially unaware of Saint-Venant’s work, obtains the equations by first examining the kinematic conditions governing the relative fluid motion about a point, from which he deduces that . . . the most general instantaneous motion of an elementary portion of a fluid is compounded of a motion of translation, a motion of rotation, a motion of uniform dilatation, and two motions of shifting . . ., these last motions later being identified as the rates of strain. Similarly, the term ‘strain’ is not used by Stokes in his related discussion on elastic solids which closes the paper. In this respect, Stokes (1845) here confirms general conclusions stated by Cauchy (1843) to the effect that any change of shape of a material element can be decomposed into strains and rotations. However, near the beginning of his paper Stokes (1845) states his basic assumption: That the difference between the pressure on a plane in a given direction passing through any point P of a fluid in motion and the pressure which would exist in all directions about P if the fluid in its neighbourhood were in a state of relative equilibrium depends only on the relative motion of the fluid immediately about P; and that the relative motion due to any motion of rotation may be eliminated without affecting the differences of the pressures above mentioned. With rotation, or vorticity, thus ruled out as having an effect, Stokes then proposes general linear relations between these ‘pressures’ (following Cauchy, we would say stresses) and the above motions of dilatation and shifting in which the principle of isotropy is embedded. This principle requires that the material has no preferred direction for its change of shape. Stokes further necessarily introduces into these relations two coefficients, and , which are properties of the fluid but which remain thus far unidentified. However (Stokes, 1845), . . . we may at once put D 0 if we assume that in the case of a uniform motion of dilatation the pressure at any instant depends only on the actual density and temperature at that instant and not on the rate at which the former changes with the time. This, then, is Stokes’s additional assumption concerning what we now call the bulk viscosity, . The crucial result to emerge from all this is that the (viscous) stresses are equal to the product of and various expressions related to the flow field’s velocity gradients, these expressions representing the rates of strain in the fluid. It was an over-simplified form of these expressions – merely the velocity gradient itself – which Newton (1687) had employed in his incorrect analysis of the viscous vortex problem.

43

44


Interestingly, Stokes (1845) uses d’Alembert’s Principle to obtain the final Navier–Stokes equations of motion, an indication of the Lagrangian provenance of the dynamic principle invariably invoked on the Continent at that time, in contrast to the Newtonian–Eulerian basis employed nowadays. In addition to a study of the effect of on the attenuation of sound waves, a few simple flows are then analysed as further illustrations of the use of these equations: ž flow in a rotating annulus, Stokes noting and correcting Newton’s error in the viscous vortex problem; ž flow in an open channel and in a circular pipe, in the latter case Stokes allowing the possibility of slipping flow at the pipe’s surface since not all experiments agree with his prediction of discharge rate (the cause of this, later realized, was turbulence in the pipe flow). However, in his later massive paper on the application of these equations Stokes (1851) uses the no-slip condition throughout. As the title of that paper suggests, his main intention is to resolve previously puzzling anomalies observed in the decaying motions of pendulums; his conclusions support the findings of Galilei (1638) for pendulums oscillating at very low speeds. However, other related problems are attacked: ž the flows created by an oscillating plate of infinite extent and by the impulsive motion of such a plate; ž the slow motions of a sphere and a cylinder, Stokes, in the latter case, running into the mathematical problem resolved finally by the modern method of matched asymptotic expansions. The sphere case is also used to predict the terminal velocity of droplets in a cloud but, for us, the crucial result is that the sphere’s drag, D, is shown to be proportional to the sphere’s speed V, radius a, and : D D 6aV.

5.1

This result is therefore not only the first to overcome the steady flow d’Alembert Zero Drag Paradox but also suggests an intimate connection between drag and . The fact that it does not agree with the V2 common rule will be returned to presently. As to an explanation for the physical cause of , this had to await the development of the kinetic theory of gases by James Clerk Maxwell (1831–1879) (Maxwell, 1860). From this Maxwell was able to show that , by now labelled the ‘coefficient of internal friction’, is independent of density, contrary to what Stokes (1845) had supposed. In his subsequent experimental determinations of , now identified at last as the viscosity coefficient, Maxwell (1866) remarks that, within the accuracy of his measurements, the no-slip condition appears to hold at the solid surfaces of his apparatus. Other consequences of viscosity’s action had already begun to be explored by this time, although viscosity’s subtlety of action prevented it, as yet, from being revealed as the culprit. In fact, all of these consequences are caused by the ability of viscous shear stresses to create vorticity within subject fluid elements. After passing briskly through flow solutions satisfying Laplace’s equation, and therefore exhibiting zero vorticity, Euler had become far more intrigued by flows possessing finite vorticity (see Truesdell, 1954). In this he presaged the more thorough investigation of Hermann Ludwig Ferdinand von Helmholtz (1821–1894). The latter not only coined the term


‘velocity potential’ for the mathematical function satisfying Laplace’s equation – the function discovered by Euler and later explored by Lagrange and Laplace – but also gave the name ‘vorticity’ to the rotation of fluid elements in flows for which no velocity potential exists (von Helmholtz, 1858). His exploration of vorticity’s behaviour produced the vortex theorems which now bear his name; of particular relevance to what came later is his proof that, in crude terms, a vortex cannot end. Eleven years later, these theorems came to be summarized by the single theorem of Thomson (1869) (now usually referred to as Kelvin’s Theorem). This states that the circulation around any closed curve containing a given set of fluid particles remains constant with time provided the external force field acting is conservative. The latter condition therefore includes the actions of pressure and gravity, but not those of viscosity. Put very crudely, and as described in Chapter 1, circulation is the multiple of the flow velocity around a complete circuit and the length of that circuit. Meanwhile, the observation by von Helmholtz (1868) that a jet does not immediately diffuse into a surrounding quiescent fluid, but retains well-defined boundaries, caused him to suggest the idea of discontinuous motion across such boundaries. Such discontinuities are represented by infinitesimally thin vorticity sheets, leaving the flow itself to be analysed by the potential theory centred on Laplace’s equation. This flow model was adapted by Gustav Robert Kirchhoff (1824–1887) to analyse the flow about a normal plate, the surfaces of discontinuity emanating from the plate’s edges so as to curve away downstream (Kirchhoff, 1869). The fluid lying aft of the plate and trapped between these two surfaces Kirchhoff assumed to be at rest, the pressure in this region therefore having its ambient value. His flow solution yielded the pressure distribution on the forward face of the plate and hence the plate’s CD value, albeit that for a plate of infinite span, or infinite aspect ratio. Although he extended his analysis to the angled plate, he did not complete its solution. This was accomplished by John William Strutt, Lord Rayleigh (1842–1919), whose solution (Rayleigh, 1876) predicts the rearward shift of the plate’s centre of pressure with increasing incidence referred to earlier. The result, however, like that of Kirchhoff, is for a plate of infinite aspect ratio. Moreover, both solutions underestimate CD due to the assumption of the ‘dead air’ region aft of the plate. In reality, due to the re-circulating flow there, the pressure is less than ambient, as Du Buat (1786) had measured in his normal plate experiments and as Cayley (1810) had surmised from his observation of a ship’s wake. Rayleigh (1876) sought to redress the solutions’ deficiencies here through a comparison of his theoretical results and those obtained experimentally by Vince (1798). His panacea was to adopt a correction factor based on the normal plate case. However, this procedure does not address the problem that, whereas the theory is for a plate of infinite aspect ratio, Vince’s results are for plates of aspect ratios near unity (square, or nearly square, plates). Nonetheless, these Kirchhoff/Rayleigh solutions also broke the stranglehold of d’Alembert’s Paradox. Later they were to suggest an important concept: certain types of flow fields dominated by viscous effects might yet be analysed successfully by relatively simple non-viscous methods. The key to an understanding of the Kirchhoff/Rayleigh case was the subsequent realization that the plate’s sharp edges there enforce boundary layer separation. Thus the spinning (vortical) cast off boundary layer fluid is the physical source of the assumed vorticity sheets and the consequent discontinuous motion.

45

46


E F

a

b

(a)

w

r

l

y

v

u Z

r G

F E

A

u m

c

b a B

(b)

b

Fig. 5.1 Rotating cylinder in air jet (Magnus, 1852).

A further stimulus to this approach of modelling viscosity-dominated flows by inviscid methods was provided by the slightly earlier experiments of Heinrich Gustav Magnus (1802–1870). Motivated by the continuing problem of explaining the wayward motion of spinning shot, Magnus (1852) describes experiments in which a spinning cylinder is exposed to an air jet. The jet is provided by the small hand-driven centrifugal fan shown in Fig. 5.1a,b. With this apparatus two different experiments are conducted. In the first (Fig. 5.1a), the cylinder shown is flanked by the movable vanes a, b. When the cylinder is rotated Magnus (1852) observes that, on the cylinder’s side where its rotation has the same direction as that of the jet current, the vane on that side is deflected towards the cylinder’s surface. On the cylinder’s other side, where the jet and the rotation are in opposition, the vane there, in contrast, deflects away from the cylinder’s surface. These vane deflections Magnus (1852) immediately links


to local pressure changes near the sides of the cylinder (our explanatory addition in brackets): . . . when it [the cylinder] rotates, the velocity, and consequently the decrease of pressure is greater on the side which moves in the same direction as the current than on the other, where these motions have opposite directions. To confirm this belief, Magnus (1852) then performed his second experiment (Fig. 5.1b) in which a cylinder is suspended at the end of the beam yz shown. The beam itself is here free to rotate about its vertical suspension thread. The cylinder can be made to spin about its vertical axis by pulling on a silk thread wrapped about that axis, a sharp pull at this thread giving the cylinder a rotation which persists, he says, for two or three minutes. The fan’s jet is then directed to play on the now-rotating cylinder. As the cylinder begins to move laterally, its support beam yz turning with it, the jet can be made to follow this movement by rotating the baseboard AB (Fig. 5.1b) carrying the fan. The observed lateral movement of the cylinder is both sustainable and large (Magnus, 1852): If, by turning the board AB , the centrifugal fan was made to follow the cylinder, the latter moved laterally as long as its rotation continued to be pretty strong, often describing a complete circle. Moreover (Magnus, 1852), When the cylinder rotated without the current of air being directed against it, the former remained stationary. It was also stationary when the current was directed against it during its non-rotation. If the current, however, struck against the rotating cylinder, the latter, together with the beam yz, moved towards the side on which the air, by rotation, and that in the current, had a like direction. . . This, then, is the Magnus Effect caused, as Magnus (1852) concludes, by a lateral pressure difference generated by the body’s rotation, the lower pressure occurring on the body’s side at which the additional flow due to rotation augments the stream velocity. Although Magnus (1852) makes no appeal to the Bernoulli equation (3.3), the above-mentioned lateral pressure difference follows immediately from that equation once the additional velocities due to rotation are accepted. It is here, then, that the earlier confusions of the artillery experts are finally resolved; Robins (see Wilson,1761; Chapter 4), for example, had argued on the basis of lateral ‘resistance’, not pressure, differences across his bodies. It is to the experiments of Magnus (1852) that Rayleigh (1877) refers in his analytical investigation of the case observed by Newton (1672), that of the curved flight of the sliced tennis ball. In this short paper, only three pages long, Rayleigh (1877) analyses the flow about a circular cylinder rather than the more complicated case of a sphere. He imposes on the cylinder’s usual potential flow a vortex motion (see Fig. 1.7b), the vortex used being located at the cylinder’s centre. He refers to the fluid as ‘perfect. . .without molecular rotation’; in other words, he takes the fluid to be inviscid and the flow irrotational. As to the generation of this vortex in practice, Rayleigh (1877) says (our change in brackets)

47

48


if the [cylinder] rotate, the friction between the solid surface and the adjacent air will generate a sort of whirlpool of rotating air, whose effect may be to modify the force due to the stream. The step as yet missing here is to connect surface ‘friction’ – the catch-all culprit since at least the time of Daniel Bernoulli – with the effects of intense shear generated by viscosity. Finding the cylinder’s pressure distribution via the (unstated) Bernoulli equation and his potential flow solution for velocity, Rayleigh (1877) then calculates the value of the lateral force experienced by the cylinder. This he shows to be proportional to the free stream velocity and to ‘the velocity of the motion of circulation’. The result therefore comes close to a statement of the later Kutta–Zhukovskii theorem for this particular case. As to surface friction itself, three years earlier William Froude (1810–1879) had reported on his extensive experiments in water, these being carried out on behalf of the Admiralty, from which he had begun to deduce similarity laws (Froude, 1874). A more evident perception of viscosity’s role emerged from the experimental work of Reynolds (1883) on the stability of water flow in pipes. Osborne Reynolds (1842–1912) became the University of Manchester’s first professor of engineering, one of the earliest appointments in this field in Britain. His original apparatus (Fig. 5.2) still exists and is frequently demonstrated to students and visitors. Reynolds (1883) describes the fate of the coloured flow filament introduced at the pipe’s entry as follows:

Fig. 5.2 The Reynolds tank (Reynolds, 1883).


(a)

(b)

(c)

Fig. 5.3 Transition to turbulence in pipe flow (Reynolds, 1883).

When the velocities were sufficiently low, the streak of colour extended in a beautiful straight line across the tube, Fig. 5.3a . . . As the velocity was increased by small stages, at some point in the tube . . . the colour band would all at once mix up with the surrounding water, and fill the rest of the tube with a mass of coloured water, as in Fig. 5.3b . . . On viewing the tube by the light of an electric spark, the mass of colour resolved itself into a mass of more or less distinct curls showing eddies, as in Fig. 5.3c. Using three pipe diameters, different flow speeds and various water temperatures so as to change the viscosity, Reynolds found that this observed transition occurred naturally when the value of the non-dimensional quantity Vd/ (V the pipe’s mean flow velocity, d the pipe diameter, the kinematic viscosity equal to /) reached about 13 000. However, when the entry flow to the pipe was severely disturbed – he connected the pipe to the mains – the numerical value of Vd/ could be as low as about 2000. The random eddies which Reynolds observed had already been named ‘turbulence’ in 1877 by Sir William Thomson and by 1908 the German physicist Arnold Sommerfeld (1868–1951) (see von Karman, 1954) was to suggest the name ‘Reynolds number’ for the non-dimensional quantity which had characterized transition in the pipe experiments. In a later paper, Reynolds (1895) goes on to analyse the consequences of random flow fluctuations about a mean motion, thereby introducing the effective additional stresses of turbulent flow now known as the Reynolds stresses. As to the eventual recognition of the all-pervasive role of viscosity, on a number of occasions Rayleigh (1892, 1910; see also the Appendix to Zahm, 1904) used dimensional arguments to show that in the most general terms the common rule (3.1) needed modification so as to become R / V2 Sfn Re . Here Re is the Reynolds number given by Re D

Vl ,

5.2

49

50

Early Developments of Modern Aerodynamics CD 100

10

1

0.1 10−1

100

101

102

103

104 105 Vd Re = n

106

Fig. 5.4 Drag coefficient, CD ¾ Reynolds number, Re, for a sphere.

l being a characteristic length scale for the body. This conclusion gave experimentalists the clear message that they should explore Reynolds number dependencies in their investigations to determine such non-dimensional coefficients as CD and CL . According to this, the drag coefficient for a sphere, for example, should depend only on the Reynolds number. Over the succeeding years the experimental data gathered produced the graph shown in Fig. 5.4 which substantially confirmed this conclusion. One crucial point to emerge from this is that CD does not have a constant value, as Newton, Robins and many of their successors had supposed. As it turned out, many of the experimental investigations on spheres had been conducted, by chance, over a Reynolds number range covering the more or less horizontal stretch of the CD graph. A further point concerns the behaviour of the CD graph of Fig. 5.4 for Reynolds numbers less than unity. The form of the graph suggests that here CD / 1/Re. This behaviour provides a resolution of those awkward questions surrounding the slow speed pendulum experiments of Galilei (1638) and Newton (1687), as well as the theoretical drag result of Stokes (1851) for spheres at low speed, all of which gave drag proportional to V, not V2 . Taking the Stokes (1851) result, equation (5.1), and re-casting it in terms of a drag coefficient, we obtain CD D

24 , Re

Re D

Vd ,

d being the sphere’s diameter. In this particular case, then, the general function of Re contained in the relation (5.2) here becomes simply the reciprocal of Re. Moreover, all such low speed cases emerge as, more fundamentally, very low Reynolds number cases. Luckily, from the viewpoint of aeronautics, the very low Reynolds number range is well below that encountered in flight.

6 Practical developments in aeronautics While the basic scientific investigations mentioned thus far did much to lay the foundations of our modern understanding of fluid flow, little had emerged from them to encourage and instruct aspiring constructors of flying machines. For such persons Cayley had laid the foundations, yet at his death in 1857 few in Britain showed an inclination to carry his programme forward. However, in 1866 the Aeronautical Society (now the Royal Aeronautical Society) was founded in London, its first lecture being given by Francis Herbert Wenham (1824–1908), an engineer well versed in the practice of heated air, gas and steam engines as well as in the use of the propeller. As he makes clear in his lecture, reprinted in Pritchard (1958), he, like Cayley before him, had conducted whirling arm experiments on flat plates at incidence and had come to the conclusion that, at small incidences, the centre of pressure lies forward. On this slender basis, supplemented by his own considerable observation of bird flight, he strongly favours the use of high aspect ratio wings. By 1870, for lack of any theoretical guidance, the Aeronautical Society had decided on experimental investigations. For these Wenham, together with John Browning, directed the construction of the world’s first wind tunnel, albeit a primitive arrangement of a 3 m long duct, 0.46 m square in section, through which air was driven by a steam engine-powered fan. In it four plates of various shapes were tested. The results announced in the Society’s annual report for 1871 (see Pritchard, 1957 ) indicated the best values for L/D at small incidences. Some ten years later Horatio Phillips (1845–1924) constructed the second wind tunnel, this, astonishingly, being driven by steam ejection (Fig. 6.1). In it cambered sections (Fig. 6.2) were tested, the highest L/D ratio of 10 being achieved with section No. 5 at 15° incidence. Regrettably, data at this incidence only were released so that no indication of the variation of lift with incidence change could be gained. However, attaching a ribbon to the undersurface of section No. 6, Phillips (1884) found backflow there as indicated by the arrow in the figure. Although his reasoning was unsound, Phillips (1884) concluded correctly that overpressure exists beneath the wing while, more significantly, and as Cayley (1810) had guessed, suction is present on the upper surface. Later, he reported even better results (Phillips, 1891) using the cambered section shown at the bottom in Fig. 6.2 which possessed a rather thicker leading edge. Far more extensive results with similarly shaped sections (Fig. 6.3) were provided by Otto Lilienthal (1848–1896) in Germany (Lilienthal, 1889) so that the ‘dipping front edge’ section, as Phillips termed these cambered aerofoils, at last gained ascendancy.

B

3'.6"

2'.6"

2. 0"

A

Fig. 9.

B

8

Boiler

12"

A

D

D

12'.0" 7' 10"

Section on line A,B B A A

B


A

6'.0"

Wind guage

A 6"

52

D

B 6" p

c w

Fig. 6.1 The second wind tunnel, driven by steam ejector (Phillips, 1884). No. 1.

No. 2. No. 3.

No. 5. No. 6.

No. 4.

a

A

E

Fig. 6.2 Cambered wing sections (Phillips, 1884, 1891).

Lilienthal’s data had considerable influence both on the aeroplanes designed around the turn of the century and on the future development of aerodynamic theory. The influence on the latter topic has been touched on already in Chapter 1. However, Lilienthal’s more immediate influence was through his generous and widespread provision of his data, which he appeared to have used successfully in the construction of his own gliders. His invention of the hang-glider (Gibbs-Smith, 1985 ) enabled him to gain

Practical developments in aeronautics

0.5 m 2

of motion

0.4 m

Direction

1.8 m

Direction of motion

Fig. 6.3 Cambered sections and wing planform (Lilienthal, 1889).

extensive flight experience, despite the fact that control was restricted to leg movement alone. As importantly, through the relatively new medium of the printed photograph, he was seen to be successful (Fig. 6.4). However, there was a major difficulty with his basic aerodynamic data, which were obtained by two different methods (Ackroyd, 1984; Anderson, 1997 ). The first employed a whirling arm, the second a natural wind balance which therefore necessitated the use of an anemometer to measure wind speed. Lilienthal’s anemometer consisted of a plate exposed to the wind. The plate was restrained by a spring which compressed under the action of the plate’s drag force, the spring compression then providing a measure of that force. Equation (3.1) was then employed to convert the measured drag force into wind speed, an assumed value of CD for plates being selected. For this calibration of his instrument Lilienthal used Rouse’s overlarge value of CD for plates quoted by Smeaton (1759), the consequence being that he consistently underestimated wind speed. As a result, he could not obtain agreement between his two data sets. Wary of the errors inherent in the use of whirling arms, regrettably he favoured the incorrect natural wind data, giving greater prominence to them in his publications. It was the Wrights (see MacFarland, 1953 ) who first realized this error, through the flight testing of their No. 2 glider in 1901, and it was this

53

54


Fig. 6.4 Lilienthal gliding, 1896.

realization which prompted them to build their own wind tunnel so as to embark on their extensive aerodynamic tests of 1901/1902. However, their results did not become available until MacFarland (1953) published their collected papers. The CL values obtained from the Lilienthal (1889) data are shown in Fig. 6.5. For the whirling arm tests three camber values were used, 1/12, 1/25 and 1/40, camber being measured as the maximum height of the wing’s arch divided by the wing chord. Fig. 6.5 shows the lift advantage of increasing camber. Also included in Fig. 6.5 are the CL values obtained from the natural wind tests, in this case one value of camber only, 1/12, being used by Lilienthal (1889). However, the latter data have been subjected to two corrections before being included in Fig. 6.5. The first was carried out by Lilienthal (1889) himself. He discovered that the wind moving over the flat plain on which his wind balance was sited rose upward by 3° and therefore added this angle to his wing incidence measurements. He was not to know that the natural wind boundary layer was the cause of this effect. Nonetheless, the correction is reasonable under the circumstances. The second correction is that used by Ackroyd (1984) and adjusts the CD value used in the calibration of the Lilienthal (1889) anemometer. The result is that the corrected natural wind and whirling arm data show rather better agreement than Lilienthal achieved, at least at the smaller incidences. At higher incidences the whirling arm data themselves are likely to be in error since the higher lifts here induce a significant downward movement in the air, causing errors in true incidence. However, the essential point, not lost on both Kutta and Zhukovskii, is that the Lilienthal (1889) data cover a wider incidence range than any other data available at that time and show the marked improvement in CL , and indeed in L/D, of the cambered wing compared with flat surfaces. Lilienthal built and flew sixteen versions of his hang-glider, three of which were biplanes, before being killed as a result of a stall in 1896. Nonetheless, his considerable successes before his death encouraged others elsewhere to follow his example. His sole emulator in Britain was Percy Sinclair Pilcher (1866–1899) (Jarrett, 1987 ), who obtained gliding experience with Lilienthal near Berlin in 1895. Adopting a wing

Practical developments in aeronautics Whirling arm camber 1/12

1.6

1.4 1/25

1.2

1/40 1.0

CL

Corrected natural wind 1/12 camber

0.8

0.6

0.4

0.2

−10

−5

0

5 α°

10

15

20

25

Fig. 6.5 Whirling arm and natural wind results; lift coefficient, CL ¾ incidence, ˛ (Lilienthal, 1889).

construction method similar to that of Lilienthal, he built four hang-gliders. The last of these, the ‘Hawk’ built in 1895/96, gave him significant success. However, during a demonstration glide in 1899 part of the ‘Hawk’s’ tail assembly failed, Pilcher being killed as a result of the subsequent crash. For their Lake Michigan flights of 1896 Octave Chanute (1832–1910) and Augustus Moore Herring (1867–1926) (GibbsSmith, 1985; Crouch, 1981 ) built and flew successfully a hang-glider having biplane wings cross-braced for the first time by the Pratt truss wire rigging system. As significantly, however, they included provision for the pilot to exercise a greater degree of longitudinal control by sliding his whole body along rails. In contrast, the Wrights, Wilbur (1867–1912) and Orville (1871–1948), decided on aerodynamic control alone for their gliders and powered machines, inventing their major contribution of roll control, their helicoidal warping system, for this purpose (see MacFarland, 1953 ). In addition to the pioneers of the glider, there were those who chose what they believed to be a more direct route to sustained flight. At this point it is useful to remind readers of the perceptive distinction drawn by Gibbs-Smith (1985) between ‘chauffeurs’ and ‘airmen’. Lilienthal, Pilcher, Chanute, Herring and the Wrights all exhibited the ‘airman’s’ desire to get into the air, there to learn the processes of control through the less demanding experience of glider flight. In contrast, those who espoused the direct route – Maxim in Britain, Langley in the United States, Ader in France, all lacking gliding experience, all showing little understanding of control – all took the ‘chauffeur’s’ view that they might drive directly into the air so as to achieve powered sustainable flight.

55

56


Clément Ader (1841–1925) had produced the first powered take-off from level ´ ground, followed by a hop flight of 50 m, in his steam engine-powered ‘Eole’ of 1890. However, subsequent attempts at sustained flight appear to have shown little improvement (Gibbs-Smith, 1968 ). After extensive whirling arm and wind tunnel tests on mildly cambered surfaces, Sir Hiram Stevens Maxim (1840–1916) began construction of his machine in 1893 at Baldwyns Park, Kent (Maxim, 1908). Tested in the following year, this massive machine had the wingspan of a Vulcan bomber and was powered by two 130 kW steam engines, each driving propellers of 5.4 m diameter. It ran along a twin-railed track but, should take off occur, it would be restrained to rise no more than about 0.6 m by additional outer rails with which the machine’s outrigger wheels would engage from beneath. On its third run an outrigger wheel’s axletree broke and the machine suffered significant damage. Repairs enabled a continuation of the tests until 1895, at which point Maxim abandoned the project. Samuel Pierpont Langley (1834–1906), the august Secretary of the Smithsonian Institution and therefore holder of the most prestigious scientific appointment in the United States, also conducted extensive whirling arm tests, initially on flat plates only (Langley, 1891), but then realized the lift advantages of camber. Earlier and rather inadequate experiments with rubber-powered flying models, from which, he said, ‘it was possible to learn little’, caused him nonetheless to retain their tandem wing arrangement for his series of ‘aerodromes’, as he called his later far larger models (Crouch, 1981 ). Most of these proved disappointing but the re-built Aerodrome No. 5 flew successfully, although describing a wandering circular path, for about 1 km in 1896. Launching was by catapult from a houseboat on the Potomac River, near Washington, DC. Despite endless difficulties with it, this launching system was adopted for a man-carrying fullscale machine, the ‘Great Aerodrome’, based on the successful model No. 5. Moreover, the machines all suffered from wing distortion in flight, the probable cause of No. 5’s wandering flight and a problem created by Langley’s near-obsession with low structural weight. This was to bedevil the tests of the man-carrying ‘Great Aerodrome’, the construction of which was completed in 1903. Charles Matthew Manly (1876–1927), Langley’s assistant, agreed to be its pilot on the two occasions it was catapulted over the Potomac. Both ended in disaster, wing distortion being a contributory cause, and Manly was lucky to survive the ensuing plunges into the river. These events effectively ended Langley’s seventeen-year quest to achieve powered, controllable flight. The world, well aware of the great Langley’s failure, was then in no mood to accept the garbled report a matter of days later of the success at Kitty Hawk, North Carolina, by two young men, reputedly merely bicycle mechanics, from Dayton, Ohio. This, then, was the backdrop to the new aerodynamic theory’s emergence as the twentieth century began. It did not augur well: two of the mere handful of gliding pioneers dead as a direct result of their endeavours, and all of the pioneers of powered flight meeting resounding failure. This depressing picture was to change dramatically, of course, once those two young men from Dayton, the Wrights, began their public demonstrations of powered flight. But these did not occur until 1908 so that the Wrights’ achievements are not entirely relevant to our story. Moreover, by 1908 the essential first steps in the new theory had already been taken, and these, as indicated earlier, had been inspired by the work of Lilienthal. Yet even before the turn of the century, in Britain one man working entirely alone, Frederick William Lanchester, had constructed an aerodynamic theory which presaged much of what was to come.

7 Frederick William Lanchester (1868–1946) According to Kingsford (1960), Lanchester’s active interest in flight began in 1891. Lanchester (1907) himself tells us that he . . . first formulated his theory in 1892, the basis being the study of the special case of an aeroplane of infinite lateral breadth. Nomenclature being far from settled at that time, Lanchester’s term ‘aeroplane’ here means a wing of infinite span. Lanchester’s public announcement of his interest in flight seems to have been the lecture which he gave before the Birmingham Natural History and Philosophical Society in 1894, one topic being his lift theory for the wing of infinite span (Kingsford, 1960 ). Lanchester (1907) mentions that, in the years 1894–95, he had taken the further step of incorporating in this lift theory his ideas concerning the flow in the tip regions of wings of finite span. As to subsequent events, he explains that A more complete account of this work formed the subject matter of a paper offered to the Physical Society of London, but rejected (3 September 1897). Lanchester (1907) then adds that, in the present work, . . . the main argument and demonstration are taken without substantial alteration from the rejected paper, the subsequent work being a revision of the theory on more orthodox hydrodynamic lines. Regrettably, the texts of both the Birmingham lecture and the rejected paper have been lost. Nonetheless, on the basis of the above quotations and information given by Lanchester (1926) regarding the figures used in the Birmingham lecture, it has been possible to resurrect what are believed to be the essential elements of Lanchester’s arguments for finite and infinite span wings as they developed in the period 1894–1907 (Ackroyd, 1992, 1996 ). To deal first with wings of infinite span, Lanchester’s argument is that the flow in this case (Fig. 7.1) is subjected to a force field which is symmetric fore and aft. In terms of the classical potential flow theory, as Lanchester was later to realize, this amounts to the assumption that the flow is both inviscid and irrotational. As Lanchester argues, energy is thus conserved so that flow streamlines must return to their initial

58


Fig. 7.1 Flow about a wing of infinite span at zero incidence, c.1894 (Lanchester, 1907).

levels and the air must return to its initial velocity. According to this idealized model, then, the wing will experience no pressure contribution to its overall drag. However, at these early stages of its development, the argument is that the wing rides on its own supporting wave. As we might put it now, the wing is seen as being rather like a surfboard riding the crest of a wave, yet a crest created and shaped by the shape of the board – or wing – itself. As to the best shape for such a wing’s section (Lanchester, 1907): It would appear that any appropriate smoothly curved form, whose leading and trailing angles are conformable to the lines of flow, might be regarded as fulfilling the necessary conditions, the essential feature evidently being that neither edge shall give rise to a surface of discontinuity. As will emerge presently, Lanchester’s ideas on the conformability of flows and the cause of surfaces of discontinuity spring from his independent discovery of the boundary layer concept. However, the essential point is that Lanchester sees the wing’s supporting wave as providing an explanation for the efficacy in lift generation of the ‘dipping front edge’ cambered sections devised and investigated by Phillips (1884, 1891) and Lilienthal (1889). As to an analysis of this flow field, it seems likely that Lanchester turned initially to the ‘rare medium’ concept of Newton (1687) (see the exposition of Lanchester, 1907 here). Recognizing that this leads inevitably to the sine squared incidence relation (3.2), whereas more recent experimental data indicate a sine relationship, Lanchester adopts a crude modification in which a definable streamtube of fluid is assumed to be alone affected by the wing. The further assumption is that this streamtube has a cross-sectional area which is independent of incidence (Fig. 7.2). This area Lanchester (1907) terms the ‘sweep area’ of the wing. He notes that the experiments of Langley (1891) on superposed parallel plates set at various distances apart suggest that the factor (see Fig. 7.2; this factor is not to be confused with the bulk viscosity, , of Chapter 5) might be near unity in value. Lanchester, himself, suggests that might lie between 1 and 2. In order to estimate the consequences of the interaction between this streamtube and the wing, Lanchester uses an adaptation of the method developed for propeller

Frederick William Lanchester (1868–1946) Platform area S

Sweep area KS

V

a

Fig. 7.2 Lanchester’s model of a wing flow showing the ‘sweep area’ S, c.1894. β1

V

Vu

βt

Vd V

Fig. 7.3 Lanchester’s ‘conformable flow’ about a wing, c.1894.

theory by Rankine (1865, 1867), William Froude (1878) and his son Robert Edmund Froude (1846–1924) (Froude, 1889). The method employs mass and momentum flux arguments which, in the latter respect, Lanchester (1907) tells us has been dubbed ‘the doctrine of the continuous communication of momentum’. As to the interaction itself, Lanchester’s argument is that the air meets the wing’s leading edge ‘conformably’ (see Fig. 7.3), the rather crude assumption being made that the airflow meets the leading edge with its initial free stream velocity V, then subsequently leaves the trailing edge in like manner. Consequently, according to this mathematical model, the wing removes the leading edge’s upward component of V, Vu , and then imparts a downward component of V, Vd , at the trailing edge. A point worth noting here is that Lanchester realizes – and it seems that, apart from Cayley’s imperfect grasp of the idea, he was the first to do so – that a wing’s lift is not merely the reactive consequence of an ability to push air downward at the trailing edge, but also the result of an ability to destroy the induced upward momentum of the air approaching the leading edge. As yet, however, this upward flow approaching the leading edge is seen as being created by the supposed supporting wave shown in Fig. 7.1. Nonetheless, with these assumptions Lanchester (1907) is able to obtain an equation for the lift, L. Although Lanchester (1907) gives no results specifically for the case of the cambered circular arc wing of infinite span set at zero incidence, as shown in Fig. 7.1, it is possible to make an estimate of the lift from the data he records. When set in terms of the modern form of the lift coefficient,

59

60


w

Fig. 7.4 ‘Vortex fringes’ at the lateral edges of a dropped plate of finite span, c.1894 (Lanchester, 1907).

CL , this can be written as

h . CL ³ 4.8 4 c

Here, h/c is the camber (the height of the arc, h, divided by the wing chord, c). In contrast, the accurate result obtained by Kutta (1902), using potential theory, is h . 7.1 CL D 4 c Thus Lanchester’s method in this case results in a significant over-estimation of lift. Lanchester’s interest, however, is in the performance of wings of finite span. The argument which Lanchester (1907) gives for this case begins with considerations of a horizontal plate dropped vertically in still air (Fig. 7.4). The air must circulate around the plate’s edges so as to form what Lanchester calls the ‘vortex fringes’ shown. Now superimpose on the plate a forward motion. The vortices are still present at the lateral edges of the plate, but we again obtain the situation in which the air incident at the plate’s leading edge is in a state of upward motion relative to the plate, whereas, aft of the plate, the air is in downward motion. Once again, the lift is then not merely the reactive consequence of causing this downward motion of the air, but the reaction is enhanced by the plate’s destruction of the air’s incident upward motion. As to the existence of the ‘vortex fringes’, or trailing vortices, Lanchester (1907) recommends that this can be demonstrated experimentally . . . by the employment of an aerofoil under water and inverted. . .the vortices being evidenced by the dimples in the surface of the water. Thus Lanchester (1907) regards the flow field . . . as in the main consisting of two parallel cylindrical vortices. . .which are being continually formed at the flank extremities, whose energy is continually dissipated in the wake of the advancing aerofoil. Consequently, in the act of sustaining itself, the wing must inevitably create continuously the swirling kinetic energy of these flanking vortices. Therefore, in


creating this kinetic energy, the wing must perform work. And to do work requires that motion must take place against a restraining force. This, as Lanchester (1907) puts it, is ‘aerodynamic resistance’ or, in the modern terms used in Chapter 1, induced drag. As yet, however, our modern view of events is not quite complete. Because the kinetic energy of the flanking vortices is continually being dissipated by viscous action in the wake of the wing (Lanchester, 1907), From another point of view, this loss of energy may be looked upon as a gradual spreading out and dissipation of the wave on the crest of which the aerofoil rides, and it becomes necessary that the aerofoil should constantly renew the diminished wave energy in order to maintain sufficient amplitude and support the given load. Thus at this stage in the development of the argument the wing itself is not yet seen as creating its own lifting bound vortex, but is still conceived as riding along on Lanchester’s ‘supporting wave’. Lanchester (1897) repeats this conception and later he (Lanchester, 1926) confirms that this was his initial view. Nonetheless, with this flow model Lanchester (1907) is able to estimate the lift and ‘aerodynamic resistance’ (induced drag) of a wing of finite span. To calculate lift, as indicated earlier, he applies momentum flux arguments, whereas for induced drag, Di , the principle employed is that the rate at which work is done by Di must be equal to the rate of change of the flow’s kinetic energy as it passes the wing. The details of the argument can be found in Ackroyd (1992) where it is shown that, when put into modern notation, the result for the rate of change of CL with incidence ˛ is ∂CL D 2 ε C 1 , ∂˛ whereas the expression for the induced drag coefficient, CDi , is CDi D

C2L . 4 1 C ε / 1 ε

Here εD

7.2

7.3

ˇl ˇt

is the ratio of the leading and trailing edge angles of the wing (see Fig. 7.3). Lanchester (1907) supplies data on the factors and ε as functions of wing aspect ratio, A, where span 2 . area These factors, however, have been estimated largely from results obtained from flat plates. Nonetheless, it is informative to compare the relations (7.2) and (7.3) with those later to emerge from the now-accepted lift theory of Prandtl (1918, 1919) for wings of elliptic planform: AD

2A ∂CL D , ∂˛ 2CA CDi D

C2L . A

7.4 7.5

61

62

Early Developments of Modern Aerodynamics 6

5

Prandtl ∂CL ∂α 4 Lanchester

3

2 4

6

A

8

10

12

Fig. 7.5 Comparison of the Lanchester (1907) and Prandtl (1918) results for lift curve slope, ∂CL/∂˛ ¾ aspect ratio, A (Ackroyd, 1992).

A comparison of these results is given in Figs 7.5 and 7.6, and here it is seen that, whereas Lanchester underestimates lift, induced drag prediction is rather more successful. At some point in the development of his argument, perhaps prompted by the Physical Society’s rejection of his paper, Lanchester began a revision of his theory, as he says, ‘on more orthodox hydrodynamic lines’. As a result, the reader of Lanchester (1907) meets with the somewhat disconcerting experience of finding the ‘supporting wave’ explanation of lift being replaced midway through the text by one in which a ‘cyclic system’, a bound vortex, takes on that role. Clearly, the upflow and downflow induced by such a vortex system (see, for example, Fig. 7.1) also fits his earlier picture of the ‘supporting wave’. However, at this stage it is not entirely clear that he sees the circulation of his bound vortex as being equal to the circulations of his trailing vortices, the requirement which had emerged from the vortex theorems of von Helmholtz (1858). It may be that Lanchester’s grasp of the concept of a bound vortex as the cause of lift generation pre-dates the use made by Kutta (1902) of this concept. Nonetheless, it must be emphasized that, in his calculations for the prediction of wing performance, Lanchester (1907) makes virtually no use of the concept. Indeed, whereas in the period 1897–1907 the model of the flow field has been developed, principally by the incorporation of a bound vortex, the calculation procedure may not have changed. It is not until later that Lanchester (1915) makes clear his understanding of the connection between the bound and trailing vortices, remarking that (our emphasis in bold italics) . . . the author regards the two trailed vortices as a definite proof of the existence of a cyclic component of equal strength in the motion surrounding the aerofoil itself.

Frederick William Lanchester (1868–1946) 40

ΠA

30

4k(1 + ) 1− ∋ ∋

ΠA, 4k(1 + ) 20 1− ∋ ∋

10 4

6

A

8

10

12

Fig. 7.6 Comparison of the Lanchester (1907) and Prandtl (1918) induced drag relations. Since Lanchester’s result (equation (7.3)) has the same form as that of Prandtl (equation (7.5)), the variation with aspect ratio, A, of the denominator of Prandtl’s equation, A, is compared with Lanchester’s equivalent quantity, 4 1 C ε / 1 ε . Here ε is the ratio of the wing’s leading and trailing edge angles selected for a particular aspect ratio and is the ‘sweep area’ factor (Fig. 7.2) also dependent upon aspect ratio (Ackroyd, 1992).

Yet, whereas Prandtl had immediately, and much earlier, seen this as crucial to his mathematical modelling of the flow, his theory emerging in definitive form in Prandtl (1918, 1919), even now Lanchester (1915) makes little use of this aspect of the more precise and detailed physical model he has developed. Nonetheless, although much of this revision emerges largely as a refined version of that produced in 1907, Lanchester (1915) successfully modifies his analysis of induced drag so as to arrive at expression (7.5), identical to that given later by Prandtl (1918, 1919). All this being accepted, it has nonetheless to be admitted that, throughout his career, Lanchester demonstrates a remarkable feel for aerodynamics. As early as 1894, using the arguments described above, he constructed a wing having the section shown in Fig. 7.7. This wing, moreover, had an elliptic planform (Fig. 7.8) with the relatively high aspect ratio of 13. When tested later at the National Physical Laboratory (Lanchester, 1913), a three-quarter scale model of it produced the highest lift to drag ratio then known, a value of 17.1 at 4° incidence. Note, however, that, while Lanchester uses a section shape of generally streamlined form, the leading edge has the sharp contour adopted since antiquity in the belief that this edge must cut the impinging flow. As to his choice of planform geometry, Lanchester (1907) provides a rather speculative yet perceptive argument for this. He argues that, from the viewpoint of reducing aerodynamic resistance, it is better not to have the kinetic energy of the trailing vortices concentrated at the tips but to spread this out along the span of the wing. He ends with the comment that bird wings have . . . ordinates that approximate more or less closely to those of an ellipse.

63

64

Early Developments of Modern Aerodynamics Mid-section

3′′ Section half wing length

Plan-form,- ellipse. Aspect ratio, 13:1

Fig. 7.7 Sections of the Lanchester wing, c.1894 (Lanchester, 1907).

Fig. 7.8 F. W. Lanchester with rubber-powered model having a wing of elliptic planform, 1894.

Presumably, the planform of the wing of 1894 was obtained on this basis. It is instructive to add at this point that one of the major results to emerge from the wing theory of Prandtl (1918, 1919) is that a wing of whatever aspect ratio, but having this specific planform, possesses minimum induced drag at that aspect ratio. Underpinning much of Lanchester’s thinking on flight is his recognition of viscosity’s vital role in this. Indeed, independently of Prandtl (1904), Lanchester (1907) also proposes a boundary layer concept. Regrettably, no precise date can be given to its


inception, all that seems to be known being that, in 1905, Lanchester was carrying out experiments with glider models, his specific intention being to estimate skin friction. Throughout his first major publication on aerodynamics, Lanchester (1907) is intent on emphasizing the fundamental importance of viscosity in all fluid motions. At this point in history, it seems that Lanchester and Prandtl were virtually alone in adopting this crucial view. Lanchester (1907) introduces the viscosity concept by way of the hypothesis of Newton (see Chapter 2) and Maxwell’s work mentioned in Chapter 5. In particular, Maxwell’s simple model of a viscous shear flow (Fig. 1.1, see Chapter 1) is used to demonstrate the essential principles involved. Here, Lanchester (1907) takes it as ‘well established’ that the no-slip condition must apply to the two solid surfaces involved. He shows that the resistances experienced by both plates are proportional to the velocity gradient V/h which is constant throughout the flow. In this case, then, the surface resistances are proportional to V. Lanchester (1907) then turns to the problem which is intended to serve as a simple model for his discussion of wing drag and in which the wing-like shape is reduced to its bare essentials. This problem is that of a flat plate set at zero incidence to a uniform stream of speed V. Lanchester (1907) argues that the simple proportionality for viscous resistance obtained for the Maxwell problem now should no longer apply: If, when the velocity V increases, we suppose that the layer of fluid affected to any given degree becomes thinner, and vice versa, it is clear that the viscous forces will rise in a greater ratio than directly as V, for the velocity gradient will be steeper, also the inertia forces will be less, for the mass of fluid to be set in motion will be less. On the basis of this penetrating insight, Lanchester (1907) then begins his analysis of what is, in fact, the boundary layer problem independently attacked earlier by Prandtl (1904). A detailed review of Lanchester’s method, together with its rather important omissions, is given by Ackroyd (1992). Essentially, Lanchester (1907) balances the viscous forces experienced within the plate’s boundary layer with the fluid momentum defect in the layer. From this he deduces correctly that the viscous drag, Df , experienced by the plate has the form Df / V3/2 . 7.6 In doing so, Lanchester produces a result which is tantamount to the correct statement that the thickness of the layer is proportional to V1/2 . Although he does not comment on this, this result supports his statement, quoted above, to the effect that as V increases the layer thickness decreases. Lanchester (1907) recognizes that the frictional drag, Df , must also depend on other quantities such as fluid density, viscosity, and some characteristic dimension of the plate. So as to introduce these quantities into his statement of proportionality, relation (7.6), he uses the technique of dimensional analysis. However, rather than use the plate’s length, l, as the characteristic length dimension, he chooses to use the square root of the plate’s area, S. As Ackroyd (1992) shows, the result for Df , when expressed in modern notation, is l2 1/2 2Df / CDf D , Re S V2 S

65

66


where Re D

Vl

is the Reynolds number based on the plate length, l. This can be compared with the result already obtained by Prandtl (1904): CDf / Re1/2 .

7.7

Here Prandtl (1904) not only recognizes that it is l, not S1/2 , which is the important length dimension, since the length l characterizes the growth of the boundary layer along the plate, but he is also able to produce a (rather inaccurate) value for the constant of proportionality in the relation (7.7). Lanchester (1907) then turns to the experiments of Reynolds (1883) (see Chapter 5) so as to introduce the concept of dynamic similarity in flows. Again using dimensional analysis, he demonstrates that c VD l . . . may be taken as the general equation connecting all similar systems of flow in viscous fluids. This amounts to the statement that the Reynolds number, Re, must be the same for all situations of dynamic similarity (in effect, Re D c). However, Lanchester’s subsequent arguments based on this idea are far less clearly expressed than the cogent, concise relation (5.2) of Chapter 5 given by Lord Rayleigh (1910). Having dealt with laminar viscous motion and obtained the relation (7.6) for skin friction, Lanchester (1907) then considers the effects of turbulence. For higher flow velocities he argues that the effects of turbulence should dominate, thus increasing skin friction above the laminar flow value. For turbulent flow, then, he suggests that the skin friction relation should take the form Df / Vm (m greater than 3/2, the laminar index, but no greater than 2). As evidence for this he cites the experimental results obtained in water which Beaufoy (1834) provides (m between 1.7 and 1.8) as well as those of Froude (1874) (mean value of m D 1.92). For practical application in the largely turbulent flows anticipated in flight, Lanchester (1907) therefore suggests a relation of the form Df / V2 . 7.8 This he takes as representing what he terms ‘direct resistance’, as distinct from the ‘aerodynamic resistance’ (induced drag) described earlier. In his discussions on the effects of viscosity, Lanchester (1907) reflects, no doubt unknowingly, a number of points central to the seminal paper of Prandtl (1904). Thus, in one highly perceptive remark Lanchester (1907) suggests that, at the instant at which a body begins to move through a viscous fluid at rest, an irrotational flow field is generated. Subsequently, however, other flow regimes can develop due to viscous action. His later comments indicate that, in particular, he sees such action as the cause of the detachment of surfaces of discontinuity, or, as we would say, boundary layer


separation. Such a process, he recognizes, thereby changes the whole character of the flow field. This Lanchester (1907) links to the direct resistance experienced: In all probability also the V2 law, in cases involving other than pure skin-friction, is closely associated with the phenomenon of discontinuity. A system of flow of the discontinuous type is almost certainly accompanied by resistance following the V2 law. As to the mechanism by which such discontinuous flow is generated, Lanchester (1907) once again demonstrates a remarkable feel for the problem as he links this to the behaviour of what he calls ‘the inert layer’, or boundary layer (our explanatory additions in brackets): Now the surface of a body possesses regions of greater and regions of less pressure, and this inert layer will be steadily pushed along the surface from the regions of greater pressure to those of less. Therefore, taking the case of a normal plane [a flat plate held perpendicularly to the approaching stream], the surface current of fluid so formed will be available to ‘inflate’ the surfaces of hydrodynamic flow [the inviscid, irrotational flow field exterior to the boundary layer] in the region of the edges, almost as if the edges of the plane were emitting fluid by volatilization. This inflation of the surfaces of flow in regions of least pressure can be conceived to continue until the combined inflated region becomes one whole, the ‘dead water’ [the slower moving wake], occupying the space in the rear of the plane. Similarly for other forms of body. This might indicate that Lanchester (1907) here suggests the not unreasonable criterion that boundary layer separation is to be expected near pressure minima. However, in discussing his observation of the flow about a circular cylinder, Lanchester (1907) remarks merely that the . . . discontinuity arises from a line some distance in front of the plane of maximum section. Here, as in Chapter 1, it is appropriate to remark that the wide wake created by this very badly streamlined shape so severely distorts the inviscid flow field that boundary layer separation occurs, as Lanchester notes, before even the maximum section has been reached. Having emphasized the drag penalty of discontinuous motion, Lanchester (1907) then suggests the following perceptive definition of streamline form: A streamline body is one that in its motion through a fluid does not give rise to a surface of discontinuity. In this context he discusses a variety of fish shapes, around which the flow field is, as he puts it, ‘conformable’, or, as we would say, boundary layer separation occurs only at the tail of the body. In certain circumstances, however, he recognizes that conformability may not be maintained. Thus, in discussing the flow about the arched plate sections which form the basis of his wing theory, Lanchester (1907) shows three

67

68


(a)

(b)

(c)

Fig. 7.9 Sketches of ‘non-conformable’ flow about arched plates (Lanchester, 1907).

perceptive illustrations (Fig. 7.9) of non-conformability. In Fig. 7.9a the section is at slightly too great an incidence so that, although upper surface leading edge separation occurs, re-attachment is envisaged aft of that edge. In Fig. 7.9b the incidence is too great for re-attachment to be possible, whereas in Fig. 7.9c the section is at too low an incidence and undersurface leading edge separation is predicted. A consequence here, as Phillips (1884) had already detected (see Fig. 6.2), is backflow immediately aft of the undersurface leading edge. One final result obtained by Lanchester (1907) must be mentioned, and this concerns his refutation of what had become known as ‘Langley’s Law’. Langley (1891) had argued that viscous effects in flight would be so small as to be of no account. From his flat plate results he had argued that, the faster one flies, the less is the drag. Lanchester (1907), in contrast, recognizes that the ‘inert layer’s’ skin friction cannot be ignored and, according to the relation (7.8), this must increase as V2 . However, the contribution from ‘aerodynamic resistance’ (induced drag) which follows from the result (7.3) behaves as 1/V2 , a term such as this alone being that which Langley (1891) had retained. Lanchester (1907) adds these two contributions so as to obtain the total drag experienced by an aeroplane: D D Df C Di D AV2 C

B . V2

For an aeroplane in steady level flight at constant altitude the quantities A and B are both constants. Clearly, this relationship indicates a minimum drag condition which Lanchester (1907) demonstrates by the use of calculus. At this minimum drag speed,


V1 D B/A 1/4 , the skin friction and induced drags are equal to each other. At speeds in excess of V1 skin friction drag dominates that of induced drag (and ‘Langley’s Law’), whereas the reverse is the case for speeds below V1 . Lanchester (1907) then goes on to show by similar methods that an aeroplane will also possess a further speed, V2 D V1 /31/4 , at which the engine power is a minimum, the induced drag being here three times the skin friction drag. Going on to consider gliding flight in still air, Lanchester (1907) then obtains the now-accepted expressions for minimum gliding angle and minimum sink rate. All this is astonishing when one takes account of the fact that, at that time, the powered aeroplane was generally perceived as having barely left the ground, and that the surviving successful gliding pioneers – Augustus Herring and the Wrights – could be counted on the fingers of one hand. The scene is now set for the work of Martin Wilhelm Kutta (1867–1944), Nikolai Egorovitch Zhukovskii (1847–1921) and Ludwig Prandtl (1875–1953) to enter the story. As remarked in Chapter 1, it was Kutta’s research supervisor at Munich, Sebastian Finsterwalder (1862–1951), who encouraged him to attempt the lift theory which now bears both his name and that of Zhukovskii. As we shall see, two further characters are to be added to our cast, namely Sergei Alekseevich Chaplygin (1869–1942), the outstanding mathematician and Zhukovskii’s colleague in Moscow, and Paul Richard Heinrich Blasius (1883–1970), one of Prandtl’s earliest doctoral students at Göttingen.

69

8 Lift forces in flowing fluids Dr W. M. Kutta (1902) If one looks for the disturbances experienced by a fixed body immersed in a flowing fluid caused by the fluid motion and starts from the basic equations of hydrodynamics, the calculation leads to the result that pressure forces acting on the body cancel each other out if incompressible, inviscid and irrotational flow is assumed. Since this does not agree with experience it may be deduced that it must be the behaviour of vortices or vortex sheets (sheets of discontinuity), possibly friction as well, which characterize the real flow and these are responsible for the pressure forces actually occurring. Certain flow problems have been analysed accordingly by Helmholtz, Kirchhoff and others, assuming such vortex sheets. If the above statement is generally valid for three-dimensional problems, then it is not necessarily the case when the problem is set in two dimensions, that is, when there is a uniform flow in the X direction and one immerses an infinitely long body of arbitrary cross-sectional shape perpendicular to the flow. One obtains an approximation to this theoretical case in practice with a body of finite length and indeed it is in general quite sufficient to use a body which has a length three to four times the lateral dimension. Ultimately, the effect will also be similar on the whole if one replaces the body with one which is approximately cylindrical. The behaviour of the flow in the three-dimensional zone moreover may be understood in this way: additional vortex lines develop at the ends of the body and illustrate the transition to the two-dimensional case. The two-dimensional problem specified has an exact mathematical solution in many cases. If one takes the cross-section of the body to be a shallow arc of a circle with a subtended angle of 2˛ (Fig. 8.1), therefore the immersed body as a segment of a hollow cylinder generated from two arcs, one comes to a form which Otto Lilienthal has considered (Der Vogelflug als Grundlage der Fliegekunst, 1889, Fig. 33). The assumption that the flow is irrotational requires the flow at the edges at the end of the cylindrical element to be in the direction of the tangent to the arc cross section. One will accordingly come to the perception that the flow will develop as indicated in Fig. 8.1; a type of suction will therefore occur at the end surfaces. One can immediately recognize from the Figure that the flow velocity below point A will be smaller than the velocity of the undisturbed free stream and above point A it will be larger and as a consequence of this the hydrodynamic pressure will be higher below A than above A. Hence, a pressure difference will be present, which acts on the body from the lower (concave) side and therefore produces lift. If one investigates the problem with an exact calculation, instead of using an approximate overview, which is given here to facilitate an understanding, it shows that the

Lift forces in flowing fluids y C

A

x B

D 2α

Fig. 8.1

solution can be found by the method of conformal mapping, and it gives the following description after completing the calculation: 4V1 b2 wD 1 C b2 2 ZD

1 ; Z0

1 b2 2

1 b2 C 2t arccos 1 C b2

    2 tC1  1 1 0 2   C Z D b C 1 b i , 2  2 21 b  b t  tC1

C

1 C tb2 t



˛ b D tan 4

.

b2 t Here Z D x C iy gives the coordinates of a point and w D ϕ C i the stream function at this point. t is a complex intermediate parameter, V the velocity of the undisturbed flow, 2˛ the subtended angle of the arc forming the cross-section, therefore tan

˛ height of arc D . 2 half the chord length

p Finally, i D 1. One obtains from this, for example, the velocity of the flow above A and below A as ˛ 2 ˛ 2 and V 1 sin . V 1 C sin 2 2 The total pressure experienced by the cylindrical shell is, for unit length with a cylinder radius of 1, ˛ 4 sin2 V2 2

71

72


and for unit area 2

2˛ 2 sin V . ˛ 2

Here represents the density of the fluid. If one wants to apply these formulae to aerodynamics one must, to begin with, summarize the most important assumptions made. First, friction is disregarded. Then, the length of the cylindrical shell is taken to be infinitely long. It has already been mentioned that the errors arising from this for shells which are several times as long as they are wide are not very significant. Thirdly, the fluid is assumed to be incompressible; a further correction will therefore be added as a consequence of the compressibility of air. However, this error may be shown to be extremely small, since the pressure differences arising, for example, in the following numerical examples amount to scarcely 1/2000 atmosphere. They can therefore scarcely alter the flow field to a measurable degree given the relatively large differences in velocity occurring. The last assumption was that no vortices arise at the edges. This assumption is completely false if the angle 2˛ is large. It is obvious, for example, that for 2˛ greater than 180° it will be precisely the existence of vortices and a discontinuity layer (a type of separation of the fluid at the edge) that is the main characteristic of the phenomenon. On the other hand it is also to be expected of course for small angles of ˛ (and therefore very shallow shells), that the disturbance arising at the edge through the generation of vortices will no longer be so substantial. Our formulae will therefore be completely inapplicable for large values of the angle ˛, but they can give the chief characteristics of the flow field for small ˛. Let us compare the pressure obtained with the formula with the results of the observations of Lilienthal. The measurements made by him in the wind (Tables V and VI 1 of his book) refer to curved surfaces, in which the height of the arc amounted to 12 of the width of the arc. The corresponding value of ˛ is given by tan

˛ 1 D ; 2 6

˛ D 18° 550 .

In the formula is represented by D

1.293 1 Ð , 9.81 1 C 0.00273T

where T designates the temperature, and one obtains the lift in kilograms, if the velocity V of the undisturbed flow is introduced in metres. Hence we obtain the lift for V D 10 m 6.78 kg per square metre of the surface. 1 C 0.0273T Therefore, assuming the air temperature to be 0° C during the observations: 6.78 kg. With the more probable assumption of an air temperature of 20° C: 6.42 kg. According to Lilienthal’s experiments (Table VI, one should also compare Moedebeck, Taschenbuch für Flugtechniker und Luftschiffer, Chapter VIII) the resulting buoyancy is (angle of attack here is equal to zero) 4.95 kg per square metre of the surface.

Lift forces in flowing fluids

The discrepancy from the value advanced theoretically amounts to about 27% to 22%. It is smaller than we might have expected, given the simplifying assumptions which were at least somewhat dubious, particularly since, although ˛ D 18° 550 is not large, it is still not exactly very small. Incidentally, one is not permitted to ascribe any great accuracy to the figure of Lilienthal. The lift force for a horizontal angle of zero given by him in Table V was 7.15 kg per square metre. The figure of 4.95 appeared because Lilienthal assumed a rising wind direction of at least 3° inclination, which naturally was based partly on estimation. Unfortunately no observations exist for flatter arcs, with which one could hope to find from the outset still better agreement with the formula. The resistances measured by Lilienthal by rotating the surface (Tables I to IV) are, according to his own opinion, less in accordance with the actual conditions of a wind current. They are carried out for arcs 1 1 1 of height 40 , 25 , and 12 of the width and show an increase in the lift with the curvature, as our formula requires, even if the increase is not quite so great quantitatively as in 1 , therefore our angle ˛ is the formula. If the camber is significantly greater than 12 significantly greater than 20° , the effect according to Lilienthal is still smaller. We can conclude that the suction effect of the edges does not take place any longer in a non-disruptive way, as assumed in Fig. 8.1, but creates vortex sheets which emanate from the edges. If the shell under observation glides steeply downwards, with its chord in the gliding direction, then one obtains a horizontal and a vertical component of the calculated pressure. The horizontal component tries to accelerate the horizontal motion; if it is more powerful than the friction then this acceleration in the horizontal direction really comes to the fore, as Lilienthal had already noticed. After all, one is reminded by this that the calculations carried out relate only to the case when the angle of attack is zero. We want to say a few more words over the problem which occurs with several surfaces simultaneously immersed in a flowing fluid. A number of conclusions may already be drawn from the observation of the disturbances to the flow in the case of one surface. It may be shown from the expression given above for the stream function w that the magnitude of the velocity of the flow lateral from the plate (in the neighbourhood of B, Fig. 8.1) displays an almost vanishing discrepancy from the free stream velocity, while the deviation in the direction is still noticeable. Above or below the surface (at C or D) the magnitude of the velocity disturbances also extend perceptibly. I have not been able to complete mathematically the case of two curved surfaces installed near each other, whereas there was success at least for particular cases, when curved surfaces are placed vertically above one another (Fig. 8.2). The solution, which is rather laborious to evaluate arithmetically, was completed only in two cases, one of which corresponds to Fig. 8.2. The two shells drawn, with half angles ˛ D 45° and ˇ D 34° 480 and radii of 1 and 0.76, would experience pressure forces of 1.84V2 and 0.85V2 if they were located separately in the flow field. If they are installed above each other as in Fig. 8.2, then the pressure force on the upper shell is 2.01V2 and on the lower one 0.24V2 . We see that the pressure on the upper shell has increased somewhat, while that on the lower shell has decreased very substantially. We can express this by saying that the lower shell is sheltered from the flow by the upper one. The total pressure on both shells is 2.25V2 , compared with 2.69V2 which one would obtain if the shells were positioned very distant from one another. Only half the

73

74


2a 2b

Fig. 8.2

extra lift, which ought to have been gained by adding the smaller shell to the larger one, has actually been achieved with such a close arrangement of the shells to one another. Certainly the example has been chosen to be rather unfavourable in order to make the effect striking; the angles ˛ and ˇ are too large for vortex generation at the edges not to become noticeable. Figs 8.1 and 8.2 have been correspondingly drawn in terms of the principal features and the quantitative disturbances, without nevertheless claiming any accuracy in particular.

Commentary on Kutta (1902) In the original published version of this paper the two figures were transposed. This has been corrected in the translation printed here. The clear intention of this seminal paper by Kutta (1902) is to demonstrate to those involved in the practical pursuit of manned flight that the enhanced lifting effect of the cambered surfaces tested and used by Lilienthal (1889) can now be predicted theoretically, and with reasonable accuracy. With this audience in mind, Kutta (1902) does not trouble to provide a detailed exposition of his lift theory. Consequently, the key ideas – the inclusion of a vortex flow bound to the wing, the vortex strength being determined by the physical condition of smooth flow at the trailing edge – remain, as yet, unstated. Kutta (1902) begins by reminding us, essentially, of the inevitable result of the d’Alembert Paradox: for inviscid, irrotational steady flows the drag force is precisely zero. However, the finite forces experienced in reality can be attributed to the presence of vortices or vortex sheets (sheets of discontinuity) and to friction. Here he recalls the work of von Helmholtz (1868) and Kirchhoff (1869) in which vortex sheets are used in attempts to model real flows so as to calculate finite forces. He then concentrates attention on the two-dimensional case of a body of infinite spanwise extent, arguing, somewhat unrealistically as it turns out, that its flow field

Lift forces in flowing fluids

might be nearly achieved in reality by a body of span three to four times its chord. The particular two-dimensional section shape investigated is formed from the circular arc of small camber shown in Fig. 8.1, the arc subtending an angle of 2˛. It is this section shape, he reminds us, that Lilienthal (1889) has investigated in his experiments. Although Kutta (1902) cannot have been aware of Lanchester’s thus far unpublished work (Lanchester, 1907; see Chapter 7, Fig. 7.1), it is precisely this section shape which Lanchester uses in his own earlier approximate analyses of wing flows. As shown in Fig. 8.1, Kutta (1902) takes the flow to be everywhere tangential to the arc’s surface. He particularly emphasizes this condition at the leading and trailing edges. He points out that this flow will then inevitably create a lower than ambient pressure on the upper surface, whereas a higher than ambient pressure will be generated on the lower surface. Consequently, this flow field will generate a lift force. Such a situation, he states, has been analysed by irrotational flow theory. He gives little indication of how this has been done, merely quoting the final form for what is called the complex potential function for the complete flow field, w D ϕ C i . Here is the d’Alembert stream function (see Chapter 3) and ϕ is that function p named by von Helmholtz (1858) individually as the velocity potential (see Chapter 5), i being 1. Both ϕ and satisfy Laplace’s equation. As becomes clear from his later complete exposition (Kutta, 1910), to obtain this expression for w Kutta (1902) begins with the solution for the circular cylinder flow plus vortex of, for example, Rayleigh (1877) (see Fig. 1.7b). By the process of what is called conformal mapping, he is able to distort the cylinder so as to form it into the required circular arc shape, the surrounding streamline pattern being also distorted by the transformation process. It is then necessary to make an assumption as to the value of the vortex circulation to be used. This value he determines by insisting that the flow leave the arc’s trailing edge tangentially, a condition which in reality, but as yet unknown to Kutta (1902), is imposed by boundary layer separation at the sharp trailing edge (see Chapter 1). As it happens, for this zero incidence case only, Kutta’s assumption also ensures that the flow meets the leading edge tangentially. To prove his earlier point, he quotes the results for the flow velocities obtained at mid-chord on the upper and lower surfaces; clearly the value of the velocity on the upper surface is higher than that on the lower surface. Having obtained the velocity field, he can then calculate the arc’s pressure distribution using the Bernoulli equation (3.3). From this the expression for the lift force is obtained, the result being quoted in the form 4V2 sin2 ˛ 2 for a circular arc of unit radius. With the use of a little geometry, however, this result can be re-written in terms of the lift coefficient, CL , as h CL D 4 . c

8.1

Here h is the height of the arc, chord c, and the result is that quoted earlier as equation (7.1). Before beginning a comparison with the experimental results of Lilienthal (1889), Kutta (1902) reminds us of his initial assumptions: friction is ignored, the arc is of infinite length (infinite aspect ratio), the flow is incompressible so that, in effect, Mach number effects can be ignored. He then adds that the arc angle 2˛ must not be sufficiently large that vortex sheets are generated in the flow. In other words the camber must remain small and the flow field remain attached.

75

76


Kutta (1902) compares his result largely with the lift measurements obtained in a natural wind by Lilienthal (1889) for a wing of 1/12 camber only (see Chapter 6). These data Lilienthal (1889) had corrected solely for the 3° upward inclination of the wind to the ground on which his balance had been placed. Kutta notes Lilienthal’s preference for these data, rather than those obtained with the whirling arm and which covered cambers of 1/40 and 1/25 as well as 1/12. Kutta (1902) finds that his theoretical result for a wing at zero incidence exceeds that measured by Lilienthal (1889) by about 22% to 27%, a difference which he does not take to be too significant in view of the camber value used. However, in this comparison Kutta (1902) is unaware of two crucial points. The first concerns the serious error in Lilienthal’s natural wind data caused by the incorrect calibration of his anemometer (see Chapter 6). The second point concerns Kutta’s belief that his theoretical result for a wing of infinite aspect ratio can be compared more or less directly with that for a wing of moderate aspect ratio. All of the Lilienthal results were obtained using the ‘Zulu shield’ planform shape of Fig. 6.3, having an aspect ratio of 6.48. A correction for this effect can be obtained from the later results of Prandtl (1918, 1919) (see equation (7.4)). The corrected version of the Lilienthal (1889) natural wind data, adjusted for the error in Lilienthal’s anemometer, is shown in Fig. 6.5. Thus, to deal in terms of CL rather than the more cumbersome procedure Kutta (1902) adopts, the following results for arc section wings of 1/12 camber at zero incidence emerge:

Wing of infinite aspect ratio (equation (8.1)) Wing of aspect ratio 6.48 (equation (8.1) corrected by equation (7.4)) Lilienthal natural wind, corrected, Fig. 6.5 Lilienthal whirling arm, Fig. 6.5

CL 1.05 0.76 0.46 0.32

Thus a comparison between the more realistic theoretical CL value of 0.76 and an experimental result of 0.46 (or 0.32 from the whirling arm) is not quite as encouraging as that supposed by Kutta (1902). The discrepancy can largely be attributed to the poor boundary layer behaviour on the cambered arc shape of wing, a behaviour suggested in the prescient sketches of Lanchester (1907) (see Fig. 7.9) in which the flow does not generally remain attached. After a brief description of gliding flight, Kutta (1902) then turns to the case shown in Fig. 8.2, that of a biplane arrangement of two cambered arcs having different chords. Again, the details of his theoretical calculations for this case are omitted. However, his results show that, whereas the lift on the upper wing is enhanced by the proximity of the lower wing’s vortex, the reverse is the case for the lower wing. The net effect is that overall lift is reduced compared with the case in which the two wings are placed at a great distance apart. This example of wing interference effects foreshadows the more extensive examination of the problem given by Prandtl (1918, 1919).

9 On the motion of fluids with very little friction L. Prandtl from Hannover (1904) Classical hydrodynamics deals predominantly with the motion of inviscid fluids. With viscous fluids, one possesses the differential equation, and its formulation is well validated by physical observations. As for solutions of this differential equation, one has only those for which the inertia of the fluid is neglected or at least plays no role, apart from one-dimensional problems, such as given by Lord Rayleigh1 and others. The solution is still awaited for two and three-dimensional problems in which both friction and inertia are considered. The reason for this doubtless lies in the irksome characteristics of the differential equation. In the vector symbolism of Gibbs2 this reads ∂v 9.1 C v Ð rv C r V C p D kr2 v ∂t (v velocity, density, V force function, p pressure, k friction constant). In addition there is the continuity equation; for incompressible fluids, which alone are to be considered here, this becomes simply div v D 0. It is easy to conclude from the differential equation that for sufficiently slow and also slowly changing motion the expression multiplied by density is small compared with other terms, so that the influence of inertia may then be neglected with sufficient accuracy. On the other hand with sufficiently speedy motion the quadratic term in v (v Ð rv, the change in velocity as a consequence of a change in position) becomes large enough for the viscous effect kr2 v to appear quite unimportant. The latter is almost always true for the cases of fluid motion under consideration in engineering. This suggests, therefore, that we should simply use the equation for inviscid fluids in these cases. Nevertheless one is aware that the well-known solutions of this equation mostly agree very badly with experience; I remind you only of Dirichlet’s sphere, which, according to the theory, is supposed to move without resistance. Now, I have set myself the task of systematically investigating the laws of motion of a fluid in which friction is assumed to be very small. The friction should be so small 1 Proceedings 2aÐb

Lond. Math. Soc. vol. 11, p. 57 (Papers vol. I, p. 474) ∂ Cj ∂ Ck ∂ scalar product, a ð b vector product, r Hamilton differential operator r D i ∂x ∂y ∂z

78


that it may be neglected everywhere, except where large variations in velocity arise or where an accumulating effect of friction occurs. This approach has turned out to be very fruitful in that, on the one hand, one arrives at mathematical formulations which enable the problem to be mastered, and, on the other hand, the agreement with observation promises to be very satisfactory. To mention one case immediately: if, for example, in the steady state flow around a sphere, one moves from the motion with friction to the boundary of inviscid flow, one obtains something quite different from the Dirichlet motion. The Dirichlet motion is merely an initial condition which is immediately destroyed through the effect of even a little friction. I now pass on to individual questions. The force on a unit cube arising from friction is K D kr2 v.

9.2

If one defines the vorticity with ω D 12 rot v, then, according to a well-known vector transformation and considering that div v D 0 : K D 2k rot ω. From this it emerges without further analysis that if ω D 0 then K D 0. That means that however strong the friction, vortex-free movement represents a possible motion. Nevertheless, if it is not preserved in certain cases the reason is that rotational fluid is penetrating into the irrotational zone from the periphery. With any periodic or cyclical motion the effect of friction can mount up over a long time period, even when it is very small. From this one must require for steady state conditions that the work done by K, that is, the line integral K Ð ds along every streamline, is zero for every full cycle in cyclical motion. With spatially periodic flows one has P K Ð ds D V2 C p2 V1 C p1 . For two-dimensional motion in which a stream function exists3 , a general expression for the distribution of vorticity may be derived with the help of Helmholtz’s vortex laws. In plane motion one obtains4 V2 C p2 V1 C p1 ∂ω D ; P ∂ 2k v Ð ds For closed streamlines this is equal to zero. Therefore the simple result is obtained that inside a region of closed streamlines the vorticity takes up a constant value. For axisymmetric motion with flow in meridional planes the vorticity is proportional to the radius: ω D cr. This gives a force K D 4kc in the direction of the axis. By far the most important question in the problem area is the behaviour of fluids at the walls of solid bodies. One does sufficient justice to the physical processes in 3 Encyklop¨ adie

der mathem. Wissenschaft. IV, vol. 14, p. 7. to Helmholtz the vorticity of an element is constant if the length along the axis of the vortex is constant. Therefore ω is constant in steady plane flow along every stream line D const.. Therefore ω D f and 4 According

K Ð ds D 2k

rot ! Ð ds D 2kf0

rot y Ð ds D 2kf0

v Ð ds

On the motion of fluids with very little friction

the boundary layer between the fluid and the solid body if one assumes that the fluid adheres to the wall and that the velocity there is zero or correspondingly equal to the velocity of the body. Now, if the friction is very small and the flow path along the wall is not too long, then the velocity will already have its normal value very close to the wall. The abrupt changes in velocity produce considerable effects in the narrow transition layer in spite of the small viscosity. One best deals with this problem if one makes systematic simplifications to the general differential equation. If one takes k to be small in the second order, then the thickness of the transition layer becomes small in the first order, and similarly the normal velocity component. Transverse pressure differences can be neglected, similarly any possible curvature of the streamlines. The pressure distribution in our transition layer will be imposed from the free stream. For a plane flow, which alone I have analysed up to now, one obtains the following differential equation for steady state conditions ∂u ∂u dp ∂2 u u Cv C D k 2; ∂x ∂y dx ∂y where x is the tangential direction, y is the normal direction and u and v are the corresponding velocity components. In addition ∂u ∂v C D 0. ∂x ∂y If, as usual, dp/dx is given throughout, and furthermore the variation of u for the initial cross-section of the flow, then every problem of this kind may be mastered numerically, in that one can obtain from every value of u the corresponding ∂u/∂x by quadrature. With this and the help of one of the familiar approximate methods5 one can repeatedly move a step at a time in the x direction. Of course a difficulty exists with various singularities arising at solid boundaries. The simplest case of the flow situations considered here is the one in which water flows alongpa thin flat plate. A reduction in the variables is possible here; one can put u D fy/ x. One comes up with a formula for the flow resistance using a numerical result of the resulting differential equation R D 1.1 Ð Ð Ð b klu03 (b width, l length of the plate, u0 the velocity of the undisturbed water opposite the plate). Fig. 9.1 gives the distribution of u.

u

uo

Fig. 9.1 5 Zeitschrift

f. Math. u. Physik, vol. 46, p. 435 (Kutta).

y x

79

80


Y (enlarged) u

X

Fig. 9.2

The most important applicable result of these investigations, however, is that in certain instances the fluid flow separates from the wall at a position fixed completely by external conditions (see Fig. 9.2). Consequently a fluid layer, set rotating as a result of friction at the wall, moves out into the free stream and there it plays the same role as Helmholtz’s separation layers, effecting a complete transformation of the motion. With a change in the friction constant k, thepthickness of the vortex layer merely changes (it is proportional to the magnitude of kl/u). Everything else remains unchanged; if one wants, one can therefore move across to the wall with k D 0 and still retain the same flow pattern. As further discussion will show, the necessary condition for the separation of the flow stream is that an increasing pressure is present along the wall in the direction of flow. The magnitude that this pressure increase must have in particular cases can only be inferred from the numerical solution of the problem, which is still to be made. One can argue, as a plausible basis for separation of the flow, that in the event of a pressure rise the kinetic energy of the free steam is in part converted into potential energy. The transition layers, however, have lost a great deal of their kinetic energy; they no longer possess enough to penetrate into the region of higher pressure and hence will avoid this laterally. From the preceding comments the analysis of a particular flow process therefore splits into two interactive parts. On the one hand, one has the free stream which can be treated as frictionless according to Helmholtz’s vorticity laws. On the other hand there are the transition layers at the solid boundaries, the motion of which is governed by the free stream, but which on their part give the free stream its characteristics by emitting vortex layers. I have attempted in a few cases to follow the process more precisely by drawing streamlines; for all that, the results do not make any pretence of quantitative accuracy. In the sketches one can profitably use the fact that streamlines form an orthogonal flow map with the lines of constant velocity potential, provided that the motion is vortex-free. Figs 9.3 and 9.4 show in two stages the onset of the motion around a wall projecting into the flow. The initial vortex-free motion is quickly transformed by the (dashed) separation layer, which originates from the edge of the obstruction and unwinds spirally. The vortex moves further and further away and leaves tranquil water behind the eventually stationary separation layer. How the analogous process takes place with a circular cylinder is to be seen in Figs 9.5 and 9.6; the fluid layers which are caused to rotate by friction are again marked by dashes. The separation sheets also extend here in the steady state ad infinitum. As is known these separation sheets are unstable; if a small sinusoidal disturbance is present,


Fig. 9.3

Fig. 9.4

Fig. 9.5

Fig. 9.6

81

82


the motion develops as represented in Figs 9.7 and 9.8. One sees how clearly separate vortices form through the interlinking of fluid streams. The vortex layer is rolled up inside this vortex, as shown in Fig. 9.9. The lines in this figure are not streamlines, but those which would be obtained by adding coloured liquid. I want now to report briefly on experiments which I have undertaken for comparison with theory. The experimental apparatus (shown in Fig. 9.10 in elevation and plan) consists of a 1 12 m long tank with an intermediate partition. The water is set in motion by a paddle wheel and it enters the upper section well-ordered, calmed and reasonably vortex-free via a guiding device a and four sieves b; the object to be investigated is

Fig. 9.7

Fig. 9.8

Fig. 9.9

b

c

a

c

Fig. 9.10


installed at c. A mineral (micaceous iron ore) in the form of delicate pieces of glistening foil is suspended in the water. By this means all somewhat distorted locations in the water stand out, especially vortices, as a result of a characteristic glow which is brought about by the orientation of the pieces of foil found there. The photographs (Fig. 9.11) placed together on the plate are obtained in this way. In all cases the flow is from left to right. Numbers 1 to 4 deal with the motion at a wall projecting into the flow. One recognizes the separation sheet which emanates from the edge; it is still very small in 1, is already concealed by strong disturbances in 2, the vortex extends over the whole picture in 3, and 4 shows the ‘permanent condition’. One also notes a disturbance above the wall; since a higher pressure dominates in the corner as a consequence of the stoppage of the water flow, in the course of time the fluid flow also separates from the wall here. The various streaks which are visible in the ‘vortex-free’ part of the flow (especially in 1 and 2) are due to the fact that the motion of the liquid was not completely calm initially. Numbers 5 and 6 give the flow around a circularly curved obstruction, or, if one prefers, the flow through a channel which continuously narrows and then widens again. Number 5 shows a stage shortly after the onset of the motion. The upper separation layer has wound up into a spiral, the lower one has been stretched lengthways and has broken down into very regular vortices. One notices on the convex side near the right-hand end the start of a separating stream;

Fig. 9.11

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

83

84


number 6 shows the permanent situation, in which the stream separates approximately at the narrowest cross-section. Numbers 7–10 show the flow around a circular cylindrical obstruction (a post). Number 7 shows the onset of separation, 8 and 9 further stages. A streak is visible between the two vortices; this consists of water which had belonged to the transition layer before the onset of separation. Number 10 shows the permanent condition. The train of whirling water behind the cylinder oscillates to and fro, hence the unsymmetrical instantaneous shape. The cylinder contains a manufactured slit running longitudinally. If one positions this as in nos. 11 and 12 and sucks water out of the inside of the cylinder with a tube, one can then intercept the transition layer on one side. When it is absent, its effect, the separation, must also be absent. One sees only one vortex and the streak in number 11, which corresponds chronologically to number 9. In number 12 (permanent condition) the flow is attached closely to the wall of the cylinder up to the slit, although, as one sees, only a diminishing part of the water enters the inside of the cylinder. Instead a separation layer has now formed on the flat outer wall of the tank (a first indication of this phenomenon is already to be seen in number 11). Since the velocity must decrease in the expanding flow passage and hence the pressure rises6 , the conditions for flow separation from the wall are given, so that this surprising phenomenon is also based on the theory presented.

Commentary on Prandtl (1904) As was the intention of Kutta (1902), it is not Prandtl’s purpose here to give a detailed mathematical exposition of his analyses but rather, in this case, to interest mathematicians in the practical applications of his ideas. In this respect, it appears that Prandtl (1904) was not immediately entirely successful. The brief overview of the theory of the boundary layer which he provides is clearly some form of approximation to the full Navier–Stokes equations (see Chapter 5), yet the precise mathematical nature of that approximation is unclear. This, no doubt, remained a source of disquiet for the mathematicians of his, and indeed later, generations. To them, an approximation can only be seen as valid provided that it can be improved upon, and by rational mathematical analyses capable of giving successively better approximations using, for example, formal series expansion methods. In this strictest of mathematical senses the nature of the boundary layer approximation did not become clarified until the 1950s (see, for example, Van Dyke, 1964). At that time the formal method of matched asymptotic expansions evolved so as to provide higher order corrections to the results of boundary layer theory. Nonetheless, during the period of the 1920s the spectacularly successful results already obtained by Prandtl and his co-workers began to encourage others elsewhere to adopt the concept of the thin viscous boundary layer. More immediately, the result of Prandtl’s presentation of his paper on the boundary layer to the Heidelberg Congress of 1904 resulted in his move from the Technische Hochschule, Hannover, to the prestigious University of Göttingen. This move was effected largely at the hands of Göttingen’s Christian Felix Klein (1849–1925) who recognized the importance of Prandtl’s work. The intriguing history surrounding this move and the 6 1 v2 2

C V C p D const. on every streamline.


progressive adoption of Prandtl’s thinking, particularly in the United States, is related by Hanle (1982). Despite this paper by Prandtl (1904) being the origin of boundary layer theory, the specific term boundary layer is used once only, the term transition layer being the more usual term employed. Presumably Prandtl later dropped the latter term so as to avoid confusion with the phenomenon of turbulent transition in the boundary layer. The paper, as originally published, is a mere eight pages long yet it contains much of the basic ideas to be exploited by Prandtl and his Göttingen associates over at least the next decade. Prandtl (1904) begins by emphasizing the intractability of the full Navier–Stokes equations for viscous flow. Such solutions as are available are largely those for flows in which the fluid acceleration terms (the inertia terms) can be, for one reason or another, ignored. The few exceptions to this are those problems of unsteady flows which otherwise depend upon a single spatial ordinate, the spatial accelerations nonetheless being zero. Prandtl (1904) cites the case analysed by Rayleigh (1880) in which two parallel uniform streams are suddenly brought in contact, his solution for the vorticity showing how the initially discontinuous motion is smoothed away by the diffusive action of viscosity. Prandtl could equally well have cited the related case, the first to be analysed successfully (Stokes, 1851), of the viscous fluid flow caused by the impulsive motion of a flat plate along its length. All such solutions, in fact, are similar in mathematical form to cases of heat diffusion governed by the so-called heat conduction equation, the solutions of which usually emerge in terms of the so-called error function. Prandtl (1904) gives as his equation (9.1) the full Navier–Stokes equations for an incompressible fluid of constant viscosity. These are stated in vector form, his factor k being the viscosity coefficient usually written, in this book as elsewhere, as . Also stated in vector form is the incompressible continuity equation, i.e. div v D 0, v being the velocity vector. No doubt having in mind the known solution for the slow steady motion of a sphere in a viscous fluid (Stokes, 1851), Prandtl (1904) reminds us that in this case also the acceleration terms of equation (9.1) can be ignored. For higher speed flows, in contrast, the viscous terms in k can ‘appear quite unimportant’. However, neglecting these terms in the case, yet again, of sphere flow ends in the usual dilemma of the d’Alembert Zero Drag Paradox. The solution for the incompressible, inviscid flow about a sphere to which Prandtl (1904) specifically refers here is that due to Dirichlet (1852). However, essentially the same solution, using a point doublet in a uniform stream, had been given already by Stokes (1849) (see Chapter 5). The purpose of the present paper, Prandtl (1904) says, is to report the very fruitful results of his analytical investigation of viscous flows in which the viscosity coefficient has a small value. In this connection, the one rather important statement missing from Prandtl’s otherwise detailed description of flows of fluids possessing small viscosity is that the flow Reynolds number is therefore large, that the boundary layer is a high Reynolds number phenomenon. However, pre-empting his detailed description of the boundary layer concept, he suggests that in such cases the effects of viscosity can be ignored everywhere except those regions in which large velocity variations occur. He then returns to the sphere problem, pointing out that the regions of large velocity variation lie close to the surface, whereas the flow exterior to such regions is, in effect, inviscid. In such circumstances, he says, the resulting flow field can be quite different

85

86


to that given by the entirely inviscid Dirichlet solution. He then makes precisely the point later emphasized by Lanchester (1907), that at the instant at which a flow starts from rest the inviscid solution applies but this is rapidly destroyed by the effects of viscosity. Although he does not say so at this point, it later becomes clear that this destruction is brought about by boundary layer separation. Prandtl (1904) then turns to the role of viscosity in the creation of vorticity, showing that vorticity so created can penetrate the irrotational, inviscid flow field. In a shift of emphasis, he goes on to provide two vorticity theorems for the special case of viscous flows with closed streamlines. In more recent times Batchelor (1956a, b) has provided exhaustive treatments of such cases while being apparently unaware of Prandtl’s theorems. The crucial behaviour of viscous flows in the proximity of solid surfaces is then addressed. The no-slip condition at the solid surface itself is asserted, the resulting rapid change in velocity through the thin boundary layer thus producing ‘considerable effects’ despite the smallness of the viscosity coefficient. Prandtl (1904) then provides a brief order of magnitude argument to show that the boundary layer thickness will be small, as will be the velocity component perpendicular to the surface. The pressure, moreover, does not change through the boundary layer and is therefore that imposed by the inviscid flow exterior to the boundary layer. Prandtl (1904) then gives the now-accepted mathematical form of the boundary layer momentum equation, together with its associated continuity equation, for two-dimensional steady flow. He points out that, given the pressure distribution imposed by the exterior inviscid flow, in principle the boundary layer equations can be solved by integration. For this he recommends the fourth-order procedure now known as the Runge–Kutta method. Martin Wilhelm Kutta we have met already, of course, whereas the mathematician Carl David Tolmé Runge (1856–1927) was Prandtl’s friend and colleague at Hannover who moved to Göttingen at the same time as Prandtl and who was shortly to translate Lanchester (1907) into German at Prandtl’s behest. As to specific solutions of the boundary layer equations, Prandtl (1904) reports that he has already obtained a numerical solution for the case of the thin flat plate set at zero incidence. This he has been able to do because, in this specific case, the partial differential equations of the boundary layer reduce to an ordinary differential form when written in terms of the single independent variable y/x 1/2 . Here y is the ordinate normal to the surface whereas x is the ordinate along the surface. This characteristic, now known as self-similar behaviour, was to be exploited with immense success in a number of the boundary layer problems later addressed, most notably the flows over wedge-shaped bodies calculated by Falkner and Skan (1931). As to the flat plate problem itself, Prandtl (1904) shows in Fig. 9.1 the variation through the boundary layer of the velocity component, u, parallel to the surface. He is also able to calculate the drag caused by the boundary layer on both surfaces of the plate. When translated into the drag coefficient CD , his result (the relation for R) takes the form CD D 2.2Re1/2 .

9.3

Here the Reynolds number Re is based on the plate length l and the exterior inviscid flow velocity u0 . Through later and more accurate calculations Prandtl’s numerical factor of 2.2 was corrected to produce the now-accepted value of 2.656. As remarked


in Chapter 7, Lanchester (1907) was later to produce a result somewhat similar in form to equation (9.3) but without the advantage of Prandtl’s numerical factor. Prandtl (1904) rightly sees that the most important application of the boundary layer concept is to provide an explanation for the drag obtained on badly streamlined shapes. Using a kinetic energy argument he thus describes the process of boundary layer separation. He illustrates the idea through the use of his Fig. 9.2 in which the small value of the ordinate normal to the surface, Y, has been enlarged so as to make the details of the separation process visible. He emphasizes that the boundary layer’s separation position is determined entirely by the pressure variation imposed by the exterior inviscid flow and that separation can occur at those parts of the flow field suffering progressive rises in pressure. Separation thus results in the vortical separated fluid moving laterally to the surface, thereby providing the source of the vorticity sheets, modelled as discontinuous motion by von Helmholtz (1868), Kirchhoff (1869) and Rayleigh (1876), which can change completely the character of the flow. He then again stresses the flow model already implied, that of an inviscid, irrotational exterior flow and a thin surface boundary layer whose motion is governed by the exterior flow but which can change the character of that exterior flow by the emission of vorticity sheets. As an indication of the consequences of separation, Prandtl (1904) then provides sketches of the separation processes at the sharp edge of a flat plate normal to a stream (Figs 9.3 and 9.4) and at the rear of a circular cylinder (Figs 9.5 and 9.6). In such cases he illustrates (Figs 9.7, 9.8 and 9.9) how the vortical layers, inherently unstable, thus wind themselves into well-defined vortices. He then turns to a description of his experiments conducted in a small water channel, the flow being driven by a paddle wheel (Fig. 9.10). Tiny shards of micaceous iron ore seeded in the flow are used to reveal the flow patterns. His descriptions of the photographic records of these flow patterns require no further explanation here. Two points, however, are worthy of note. The first is the poor flow quality displayed by the Kutta-like cambered plate (Photographs 5 and 6). This feature is suggested in the sketches of Lanchester (1907) (Fig. 7.9) and has been referred to already as an explanation for the rather poor agreement between the theoretical lift result of Kutta (1902) and the experimental results of Lilienthal (1889). The other point to note is Prandtl’s illustration of the suppression of boundary layer separation through the use of surface suction (Photographs 11 and 12).

87

10 On annexed vortices Professor N. E. Zhukovskii (1906a) Section 1. We shall consider the motion of an unbounded mass of incompressible fluid, composed of translational motion and a motion due to the effect of a number of parallel linear vortices. For simplicity a plane perpendicular to the axes of the vortices will be considered, which reduces the problem to a two-dimensional one. In such a flow we will have a number of closed streamlines embracing one or more vortex areas. These closed streamlines will be assumed to be the walls of rigid bodies placed into an unbounded fluid flow, and we shall choose them in such a way that the flow outside the bodies is free of vortices and may be considered as a steady flow in two dimensions. The vortices characterizing the flow will be referred to as the vortices annexed to the rigid bodies, and our purpose is to determine the hydrodynamic forces acting upon the bodies. Section 2. It is known that an unbounded irrotational flow of an incompressible fluid moving over a motionless rigid body, with no discontinuities appearing in the velocity field, exerts pressure forces on the body, the sum of which being projected upon any axis proves to be nought. This may be easily proven with the help of the first three equations given by Kirchhoff1 . Setting p D q D r D 0,

u D const,

v D const,

w D const,

we find that the projections of the resultant forces acting upon the body are in fact zeros. Independently of Kirchhoff’s equations this result may be obtained by making use of the first of the following formulae: ∂f ∂f ∂f ∂f p cos ˛ d D cos ˛ C cos ˇ C cos d, 10.1 ∂x ∂x ∂y ∂z px cos ˇ y cos ˛ d

D

∂f ∂f x y ∂y ∂x

∂f ∂f ∂f cos ˛ C cos ˇ C cos d. 10.2 ∂x ∂y ∂z

Here the integration is performed over all the elements d of the boundary surfaces; ˛, ˇ, are the angles made by the normal to these surfaces directed inside the fluid mass with the coordinate axes, f is the velocity potential and p is the hydrodynamic 1 Vorlesungen

u¨ ber mathematische Physik, Vorl. 19.

On annexed vortices

pressure, being related to the fluid density via the formula 2 2 ∂f 2 ∂f ∂f C C . p D const 2 ∂x ∂y ∂z

10.3

Let us show that formulae (10.1) and (10.2) are valid. For this purpose we use the transformation of a surface integral into the volume one: ∂ ∂ ∂ l cos ˛ C m cos ˇ C n cos d D l dx dy dz, Cm Cn ∂x ∂y ∂z 10.4 where the double integration applies to the boundary surfaces while the triple integration is performed over the volume bounded by these surfaces; functions , l, m, n are finite and continuous inside this volume. We choose D p,

l D 1,

m D 0,

nD0

and write, taking into account that r2 f D 0, ∂f ∂ ∂f ∂f ∂ ∂f p cos ˛ d D C C ∂x ∂x ∂x ∂y ∂y ∂x ∂ ∂f ∂f ∂ ∂f ∂f D C C ∂x ∂x ∂x ∂y ∂y ∂x

∂f ∂ ∂f dx dy dz ∂z ∂z ∂x ∂ ∂f ∂f dx dy dz. ∂z ∂z ∂x

For the last triple integral we apply formula (10.4) again, this time with D

∂f , ∂x

lD

∂f , ∂x

mD

∂f , ∂y

nD

∂f . ∂z

This results in formula (10.1). In order to derive formula (10.2) we set in formula (10.4) D p,

l D y,

m D x,

n D 0.

We have px cos ˇ y cos ˛ d

∂ ∂f ∂ ∂f ∂f ∂ ∂f ∂ ∂f ∂f D x y C x y ∂x ∂x ∂y ∂x ∂x ∂y ∂y ∂y ∂y ∂x ∂f ∂ ∂f ∂ ∂f C x y dx dy dz ∂z ∂z ∂y ∂z ∂x ∂f ∂f ∂ ∂f ∂f ∂ ∂f ∂f x C x D y y ∂x ∂x ∂y ∂x ∂y ∂y ∂y ∂x ∂f ∂ ∂f ∂f C x dx dy dz y ∂z ∂z ∂y ∂x

89

90


∂f ∂f ∂ ∂f ∂f ∂ ∂f ∂f y y x C x ∂x ∂x ∂y ∂x ∂y ∂y ∂y ∂x ∂ ∂f ∂f ∂f C x dx dy dz. y ∂z ∂z ∂y ∂x

D

If formula (10.4) is applied here to the last triple integral with Dx

∂f ∂f y , ∂y ∂x

lD

∂f , ∂x

mD

∂f , ∂y

nD

∂f , ∂z

then formula (10.2) will be obtained. In order to prove that the sum of the projections upon the Ox-axis of all the pressure forces a fluid flow exerts upon a rigid body is zero2 , let us apply formula (10.1) to the fluid mass confined between the body and a sphere of an infinitely large radius R with its centre situated at the origin of the coordinate system. We denote the sum of the projections of the pressure forces upon the axis Ox by Px and write f D ux C vy C wz C ϕ,

10.5

where the derivatives of ϕ with respect to the coordinates are small quantities of an 1/R3 order. Then we can see that the first part of the formula (10.1) applied to the body surface gives Px . Being applied to the sphere it appears to be nought by formulae (10.3) and (10.5). As far as the second part of formula (10.1) is concerned, it is nought since the body surface is a stream surface. If applied to the sphere it also equals nought by formula (10.5). Section 3. We now turn to the two-dimensional motion mentioned in Section 1. Let us choose the Oz-axis to be parallel to the axes of the linear vortices and apply formulae (10.1) and (10.2) to the fluid confined between two planes drawn parallel to the xy-plane and separated from one another by a unit length measure, the surfaces of the cylinders representing rigid bodies with vortices inside them and the surface of a circular cylinder of large enough radius R whose axis coincides with the Oz-axis. Since the right and lefthand sides of formulae (10.1) and (10.2) vanish when applied to the surfaces parallel to the xy-plane, in the case under consideration formulae (10.1) and (10.2) may be represented via single integrals: ∂f ∂f ∂f p cos ˛ ds D 10.6 cos ˛ C cos ˇ ds, ∂x ∂x ∂y ∂f ∂f ∂f ∂f y cos ˛ C cos ˇ ds, 10.7 px cos ˇ y cos ˛ ds D x ∂y ∂x ∂x ∂y 2 ∂f 2 ∂f p D const C . 10.8 2 ∂x ∂y Here the integration extends to all the contours of the bodies placed in the flow and the circle of infinitely large radius R. As far as formula (10.5) is concerned, for a flow 2 The proof of this statement has been given by me in the article: ‘Reaction of in-flowing and outgoing fluid’.

On annexed vortices 0

x

y

Fig. 10.1

in two dimensions it may be written as f D ux C vy C ϕ, y yn 1

kn arctan , ϕD x xn

10.9

where kn is the strength of the nth vortex area, while xn , yn are the coordinates of its centre. Let P be the resultant of the pressure forces acting on all the bodies in the flow (Fig. 10.1). We shall try to find the projection Px of P upon the Ox-axis with the help of formula (10.6). With the integration covering the contours of all the bodies, the left-hand side of formula (10.6) gives Px ; its right-hand side is zero. On the contour of the infinitely large circle the derivatives ∂ϕ/∂x and ∂ϕ/∂y are small quantities of order 1/R; hence attention in the integrals should be given only to the products of these derivatives with the velocity components u and v. We have ∂ϕ ∂ϕ ∂ϕ Px D u Cv cos ˛ ds C uC ∂x ∂y ∂x ∂ϕ ∂ϕ ð uC cos ˛ C v C cos ˇ ds ∂x ∂y ∂ϕ ∂ϕ ∂ϕ D u Cv cos ˛ ds C uC u cos ˛ C v cos ˇ ds ∂x ∂y ∂x ∂ϕ ∂ϕ ∂ϕ D u Cv cos ˛ ds C u cos ˛ C v cos ˇ ds ∂x ∂y ∂x ∂ϕ ∂ϕ D v cos ˇ cos ˛ ds, ∂x ∂y where the integration has to be carried out along the circle of radius R.

91

92


The quantity in the brackets is the velocity vector circulation in the fluid motion due to the vortices. By the Stokes theorem, the circulation is equal to the double sum of the strengths of all the vortex elements. Hence, we find Px , and analogously Py ,

kn , Py D 2 u kn . 10.10 Px D 2 v According to these formulae the force P, by the magnitude and direction, is equal to the resultant of the parallel forces which may be obtained by multiplying the flow velocity by 2kn and rotating the resulting vectors through a right angle in the direction opposite to that of the vortices’ rotation. We shall now show that the force P is not just the same magnitude and direction as the resultant of the above mentioned parallel forces, but does in fact coincide with the latter. For this purpose we turn to formula (10.7). Let L be the sum of the moments of the pressure forces acting on the bodies and calculated with respect to the origin of the coordinate system. Applied to the bodies, the left-hand side of formula (10.7) gives L, while the right-hand side proves to be zero. On the circle of radius R the left-hand side of this formula, apparently, is nought; as far as the right-hand side is concerned, we shall express it in the polar coordinates 2 ∂f ∂f d!, 10.11 L D R ∂! ∂R 0 where ! is the angle made by the radius-vector R with the Ox-axis. From Fig. 10.2 it may be seen that for the centre M of a vortex area with the coordinates xn , yn and a point N on the circle of radius R with the coordinates x, y, the following relation holds: arctan

y yn υ sin! !n xn sin ! yn cos ! ! D$D D . x xn R R

Hence, it follows from formulae (10.9) that 1 xn sin ! yn cos ! C ! kn . f D Ru cos ! C v sin ! C R x

0

q

R

Jh d

M

m

N

y

Fig. 10.2

10.12

On annexed vortices

Substituting (10.12) into formula (10.11) and discarding the terms which become zero when the limits of integration are substituted, one has

kn xn u C yn v. 10.13 L D 2 If we draw a vector from the point M in the direction of the flow velocity whose magnitude equals the flow velocity multiplied by 2kn and turn it in the counterclockwise direction through a right angle (we assume kn to be positive), then the projections of the vector on the Ox and Oy axes will be 2 kn v and 2 kn u respectively. The moment of this vector with respect to the coordinate origin is xn 2kn u yn 2kn v D 2kn xn u C yn v. This shows that the quantity L given by formula (10.13) is the sum of the moments of the above described vectors, with respect to the coordinate origin. By this the proof of our statement has been completed. Section 4. In Section 3 we determined the resultant of the pressure forces acting on all the bodies in the flow with annexed vortices. Let us now try to find out what are the individual forces acting upon each of the bodies in the flow (Fig. 10.1). For this purpose we shall consider an imaginary flow inside the body contour. Let us draw small circles of radii rn around all the vortices inside the body; the centres of the circles are chosen to coincide with the vortices’ axes. Now formula (10.7) may be applied to the region between the body contour and the circles. The left-hand side of the formula, being applied to the body contour, gives the moment L of the pressure forces exerted by the external flow (due to the pressure continuity) on the body contour; the right-hand side of the formula equals zero on the body contour. Based on this we can write: 2 2 ∂f ∂f C x cos ˇ y cos ˛ ds LD 2 ∂x ∂y ∂f ∂f ∂f ∂f x 10.14 y cos ˛ C cos ˇ ds, ∂y ∂x ∂x ∂y where the integration extends to all the circles of radii rn . When performing the integration we shall represent the function f near the centres of the vortices in the form kn y yn , 10.15 arctan x xn where un and vn are the velocity components of the nth vortex centre (with respect to the fluid in the immediate vicinity of the vortex). The right-hand side of the formula (10.14) being applied to one of the circles of radius rn gives ∂ϕn ∂ϕn C vn un x cos ˇ y cos ˛ ds ∂x ∂y ∂ϕn ∂ϕn y x un cos ˛ C vn cos ˇ ds ∂y ∂x ∂ϕn ∂ϕn ∂ϕn ∂ϕn D un xn cos ˇ cos ˛ ds C vn yn cos ˇ cos ˛ ds ∂x ∂y ∂x ∂y f D un x C vn y C ϕn ,

D 2 kn un xn C vn yn .

ϕn D

93

94


Hence, it follows from formula (10.14) that

kn un xn C vn yn . L D 2

10.16

Therefore the moment L is expressed by the sum of the moments of the forces produced by all the vortices annexed to the body; each individual force is calculated by multiplying the velocity of the vortex centre by the fluid density and the double strength of the vortex, and then rotating the resulting vector through the right angle in the direction opposite to the vortex rotation. As this conclusion remains valid with an arbitrarily chosen coordinate origin, one has to conclude that the force acting upon each of the bodies in the flow is the resultant of the above described rotated vectors. It is obvious that the sum of all these forces gives the total resultant force obtained in Section 3. Hence, using formulae (10.10) it follows that

u kn D un kn ,

v kn D vn kn , where the summation extends to all the vortices. These equalities yield the well-known result of rectilinear and uniform motion of the ‘vortex mass centre’ in a flow without stationary walls:

un kn vn kn uD

, vD

. 10.17 kn kn On the other hand, if using formula (10.16) we compute the sum of the moments of all the forces acting on all the bodies with the annexed vortices and compare the result with the value L given by the formula (10.13), we will find that

u kn xn C v kn yn D un kn xn C vn kn yn . This equality may be rewritten, using (10.17), as

kn fun uxn ut C vn vyn vtg

∂ ∂ xn ut Ð xn ut C yn vt Ð yn vt D 0, D kn ∂t ∂t leading to the well-known integral

kn fxn ut2 C yn vt2 g D const,

10.18

which expresses the conservation of the moment of inertia of all the vortices with respect to the centre moving with the freestream velocity, under the assumption that there are no rigid walls in the flow. Section 5. As a first example, let us consider the two-dimensional fluid flow defined by the following function of the complex variable z D x C iy: F D uz with z1 D iy1 .

ik z z1 DfCi , ln z1

10.19

On annexed vortices

Here the potential function f and the stream function

are given by     

k y y1 arctan , x x 2 C y y1 2  k  . D uy ln  2 y1

f D ux C

10.20

Comparison of the first of formulae (10.20) with formula (10.8) shows that, in the flow we are dealing with, there is one vortex with coordinates 0, y1 . The velocity components in this flow are  ∂f k y y1  , Du  ∂x x 2 C y y1 2  10.21  k ∂f x   D . ∂y x 2 C y y1 2 If we want the stagnation point to be at the coordinate origin, we have to set k D uy1 .

10.22

Assuming u and y1 positive, we have a negative value for k which means that the vortex rotates in the counter-clockwise direction (Fig. 10.3). The equation for the streamlines, as follows from the second of formulae (10.20) and formula (10.22), has the form x 2 C y y1 2 D y12 e

y 2 2 y C uy 1 1

.

10.23

Choosing D 0 we obtain the streamline passing through the coordinate origin (the position x D y D 0 satisfies equation (10.22) with D 0) and forming a loop at this point; when varies from 0 to 1 we will have open streamlines outflanking the loop from below with their ends stretching to infinity; when, on the other hand, varies from 0 to 1 we will have for each value of two streamlines, one open and situated

0

x

B A

Py

Fig. 10.3

95

96


above the loop, the other closed and situated inside the loop surrounding the vortex centre. Since for large negative values of we can ignore the term 2y/y1 in formula (10.23), the closed streamlines tend to circles as the centre of the vortex is approached. Any of these closed lines, as well as the loop itself, may be assumed to be the contour of a cylindrical body placed in the flow. The resultant of the pressure forces acting on such body will be, as our general theory suggests, directed along the Oy-axis in its positive direction and will have the value Py D 2 ku D 2 y1 u2 .

10.24

This formula can be verified by means of the direct calculation of the hydrodynamic pressure force assuming the above mentioned loop to be the rigid body contour. We have 2 u V2 Py D p cos ˇ ds D V2 dx, dx D 2 2 where V is the fluid velocity on the contour of the loop. Let us use instead of the integration variable x the variable y by setting dx D

dx dy, dy

and taking into account that the integral along the entire loop may be replaced by the double integral along the half-loop OBA: y2 y2 dx V2 dy D x dV2 , 10.25 Py D dy 0 0 where y2 is the height of the loop. Combining formulae (10.21) and (10.22) we can find the value of V2 to be 2 2 k2 ku ∂f ∂f 1 y y1 2 V D C D u2 C 2 2 2 2 2 ∂x ∂y x C y y1 2 x C y y1 y12 2y y1 y1 2 Du 1C 2 C 2 . x C y y1 2 x C y y1 2 According to formula (10.23), we have here x 2 C y y1 2 D y12 e

Therefore 2

V Du

2

1Ce

y 2y

1

and correspondingly dV2 D 4u2 e

2

y 2 y

1

.

y 2 1 , y1 y y1

y dy . y1 y1

10.26

On annexed vortices

From formula (10.26) we find

x D y1

e

y 2 y

1

2 y 1 . y1

Substituting these into formula (10.25) yields y 2 y2 y y y dy 2 y 1 e 1 1 e y1 . Py D 4 u y1 y1 y1 y1 0 Introducing the new integration variable y y 1 e y1 , tD y1 y

dt D e y1 we have

y dy , y1 y1

Py D 4 u2 y1

t1

1 t2 dt.

t0

Now the limits of the integration t0 and t1 should be found. Setting x D 0 in equation (10.26) we find y  y y1 D šy1 e y1 ,  y 10.27 y 1 e y1 D š1;  y1 whence t0 D 1, t1 D C1. Therefore C1 1 t2 dt D 2 u2 y1 , Py D 4 u2 y1 1

which was to be proved. Formulae (10.27) allow the ratio y2 /y1 to be found. We have: y y2 2 D 1 C e y1 , y1

whence

1 y2 0, q1 < 0 – the type of pressure distribution required in subsection 3 of the introduction is thus already yielded by the first two powers. It is thence to be expected that also $0 and $1 , even if they are not exactly quantitatively correct, will already represent the effect of the separation. The next approximation is also calculated in one of the problems which is to be treated later in a similar way, and it proved there the validity of the limit of the first two powers. 2. We have for $0 and $1 the equations: k 2 $ 0 0 $0 $000 D q02 C $0000 , $00 $10 $0 $100 C 3$10 $00 $1 $000 D 4q0 q1 C

k 000 $ . 1

119

120


Here we can again establish the form in which q0 , q1 , k and go in from mechanical similarity. Also the first two terms here show universal significance, in a manner of speaking. We therefore write uN D q0 x š q1 x 3

D $0 x š $1 x 3

and introduce the following dimensions for x, y, $0 and $1 :

q1 q0 2 2 q0 y 0 D x D $0 1 D $1 . D q0 2k kq0 kq12 We then obtain:

uN D

q03 š 3 q1

kq02 0 š 1 3 2 q1

1 q03 0 š 10 3 etc. uD 2 q1 0 D

0 and 1 satisfy the differential equations as functions of : 2

0 0 0 000 D 4 C 0000 400 10 3000 1 0 100 D 16 C 1000 . Boundary conditions: for

D0:

for

D1

00 D 0

0 D 0

1 D 0

00 D 2

10 D 0 10 D 2

3. For 0 we apply the power series: D

1 ˛&C1 b&

&!

&D2

& .

Introduction into the differential equation yields: b2 D 1, arbitrarily, since ˛ is already a constant of integration; ˛4 ˇ3 D 4, since we have not considered the inhomogeneity of the equation for 0 in the estimate of the integration constant ˛ and so ˛ also appears in this equation in an abnormal way; b4 D 0: the curvature of the velocity profile does not change to begin with, since friction is in advance of inertia in its effect; from m D 5 onwards: bm D

m3 &D2

m3 &1

m3 &

b& bm1& .

Boundary layers in fluids with small friction

The coefficients of these recursion formulae can be calculated, like all numbers composed in this way from binomial coefficients, from a scheme similar to Pascal’s triangle, the start of which is the following:

m 0

m 3

−1 −1

4

−1

5 −1

6 −1

7 8

−1

−3

−4

3

4

5

6

1 1

0 −2

−5

2

1 0

−1

−2

1

1 2

2 0

1 3

5

1 4

1

and in which every term is the sum of those standing above it. Only the framed part comes into consideration according to upper limits of the sums. The first 13 coefficients are (in part see above) 4 ˛4

b2 D 1

b3 D

b5 D 1

b6 D 2b3

b7 D 2b32

b8 D 1

b9 D 4b3

b10 D 16b32

b12 D 181b3

b13 D 840b32 .

b11 D 27 16b33

b4 D 0

4. Apart from ˛ two further integration constants appear through the asymptotic approximation, which we, as in the previous problem, would have to connect to the power series calculated here. For our purpose (calculation of the separation position) it is sufficient, however, to know ˛ and we will see, as mentioned above, that ˛ may be calculated with sufficient accuracy with the help of the power series alone. We put Z0 D 1/˛0 , H D ˛ and sketch dZ0 /dH as a function of H from the power series. The derivative itself still depends on b3 D 4/˛4 and should, for the right value of ˛, approach the asymptote 2 dZ0 D 2. dH ˛ For other values of ˛ it certainly does not approach any asymptote and thus the method even becomes rather sensitive for the determination of ˛, as shown in Fig. 11.6 nearby. We obtain ˛ D 1.515, the last digit no longer being certain. 5. The calculation of 1 from the above linear equation and boundary conditions occurs in an analogous way. The power series is: 1 D υ Ð

1 c& & &! &D2

121

122

Early Developments of Modern Aerodynamics dZo dH 1,0

2 = 0,85 a2 0,87 0,88 0,90

0,87

2 = 1,00 a2

0,5

H

2,0

1,0

Fig. 11.6

c2 D 1, because υ is already the constant of integration; υc3 D 16; c4 D 0; and for m 5: m3 m 3 m3 m3 cm D 3 C4 ˛&C1 b& cm1& . &2 &1 & &D2

The coefficients here in these formulae can also be calculated, as above, from a scheme in which the first line m D 3 consists of the numbers 1, C4, 3, while the others follow from addition: m

0

m 3

−1 −1

4 −1

5 −1

7 8

−1

1 0

−1

7

−2

4

4

5

6

7

−3 −5

2 −3

8 15

3

−3

1

6 7

2 −3

3 2

−1

6

1 +4

−11

5

−3 −8 −11

−3 −3

The first coefficients become: c2 D 1;

υc3 D 16;

c8 D 17˛6 ;

c4 D 0;

c5 D 4˛3 ;

c9 D 30˛6 c3 224˛3 ;

c11 D 2048c3 C 294˛9 ;

c6 D 6˛3 c3 8;

c7 D 32c3 ;

c10 D 576˛3 c3 256;

c12 D 783˛9 c3 5092˛6 ;

c13 D 17392˛6 c3 C 59648˛3 ;

c14 D 221952˛3 c3 315˛12 136192;

c15 D 11025˛12 c3 1024000c3 54864˛9 ;

c16 D 174168˛9 c3 221296˛6 .


6. Also we do not need the asymptotic approximation here but determine the integration constant υ from the fact that 10 must have the asymptote 10 D 2. To begin with, the terms of the power series for 10 which are independent of c3 and those which are multiplied by c3 are plotted in Fig. 11.7 as curves A and B, so that: 10 D υ Ð A 16 Ð B 10 is then drawn for various values of υ and already one recognizes from this curve that at D 1.6 the convergence of our series is rather bad despite the large number of calculated coefficients c. For all that, the terms show a satisfactory trend if one combines equal powers of ˛, so that the series are nevertheless usable. We are still ζ′f 6,0

5,0

4,0

3,0 d = 8,30 8,25 8,20 2,0

1,0

A B h 0,5

Fig. 11.7

1,0

1,2

1,4 1,5 1,6 1,7

123

124


able to recognize that the correct value of υ lies between 8.20 and 8.30. In the first instance, the curve increases very strongly and then approaches our asymptote from above. This strong influence on u near D 0 in comparison with D 1 also allows u to change sign at the boundary in the case of separation sooner than farther out. 7. We come now to the calculation of the separation point and we remember the comment (subsection 1) that the results are quantitatively not exact, since we have considered only the first and third powers of x. The separation point [] is determined by:

0D

∂u D ∂y

q04 00 [] š 100 []3 at 8kq1 0

D0

or according to subsections 3 and 5: ˛3 š υ[]2 D 0. According to subsections 4 to 6, ˛ D 1.515, υ D 8.25; therefore the coordinate of the separation point for the case of the lower sign, which alone interests us here, is: [] D 0.65. Therefore

q0 [x] D 0.65 . q1

Moreover uN D q0 x q1 x 3 . The maximum velocity (minimum pressure) consequently occurs where q0 x D 0.577 , q1 p while the velocity would become zero in the outer flow only for x D 1 Ð q0 /q1 . According to this the separation point therefore lies at about 12% of the total length of the boundary layer after the pressure maximum. The numbers obtained are independent of the friction constant, density and a proportional increase in all velocities. 8. According to the drawing given by Prandtl (see subsection 3 of the Introduction) the streamline corresponding to D 0 leaves the boundary at a particular angle. We calculate this in the following way: in the proximity of the separation point the formulation of the expression given in subsection 2 for reads:

kq02 1 000 D Ð 0 [] 1000 []3 3 C 3000 3100 []2 []2 2 q1 3! D 0 now yields for the bifurcating stream line: 3υ[]2 ˛3 D 11.5 D3 [] 16[]3 4[]


or in the non-reduced coordinates: y D 11.5 Ð x [x]

2kq1 . q02

These formulae are limited by some uncertainty since firstly we have calculated only two terms in the formulation for and secondly because the higher derivatives, which indeed also represent more subtle processes, are always calculated less exactly than the earlier ones.

III Emergence of the boundary layer and the separation position with a sudden onset of motion from rest 1. The two previous problems concerned steady flows. At this stage we turn ourselves to the problem of the formation of boundary layers: a cylinder of arbitrary cross-section in a fluid at rest will be suddenly set in motion and continuously maintained at constant velocity from t D 0. To begin with the state of potential flow will appear under the sole action of the pressure distribution produced by the impulse; the thickness of the boundary layer is initially zero, as far as the sudden velocity distribution can actually be specified. The formation of the boundary layer occurs in the first instance as a result of friction, after that through the convective terms. We will obtain the result that after a particular time the separation starts on the back side of the body and advances slowly from there. Since our basic equations refer only to thin boundary layers, only the start of the separation process will, of course, be represented by them, just as the previous problem concerned the boundary layer only up to the separation position. 2. The equations to be used are: ∂u ∂uN ∂2 u ∂u ∂u Cu Cv D uN C, 2 ∂t ∂x ∂y ∂x ∂y ∂ ∂ vD . ∂y ∂x Here , represents k/ to simplify matters. The potential flow which appears first of all yields the boundary value uN as a function of x. Since the process for t D 0 is singular, the type of formulation is temporarily still unknown, it is established only through successive approximation. To begin with (for small t) friction has the main influence on the changes and hence we write for the first approximation u0 : ∂2 u0 ∂u0 D, 2. ∂t ∂y The integral of this equation 2Nu 2 u0 D p e d - 0 y D p 2 ,t uD

125

126


satisfies our boundary conditions, giving a disappearing boundary layer for t D 0 and agreeing with the outer flow u0 D uN for y D 1. We will obtain the following approximation by inserting u0 into the convective terms while the time and friction terms contain u D u0 C u1 . Thus the following equation for u1 emerges: ∂u1 ∂uN ∂2 u1 D , 2 C uN ∂t ∂x ∂y

function of

We satisfy this equation from mechanical similarity by the expression: ∂uN Ð f ∂x which also does not contradict the boundary condition u1 D 0 for y D 0 and y D 1. After further considerations, which in particular concern the entry of x, we succeed in representing u by a series of powers of t, in which the coefficients are functions of and therefore also contain t as well. These functions also still depend on x, on this occasion x enters the differential equations only as a parameter. 3. After this the expression for results: u1 D tuN

1 p D 2 ,t Ð t $ x D0

y D p 2 ,t

uD

1

t

D0

∂$ ∂

and from this the differential equations for $: &1 ∂2 $& ∂$& ∂$&1 ∂2 $ ∂3 $& ∂$ ∂2 $&1 C 2 2 4& D4 ∂ ∂ ∂x∂ ∂x ∂3 ∂ ∂2 D0 4Nu ∂∂xuN still appears on the right-hand side for & D 1. We limit ourselves, as in the previous section, to the first two terms and put $0 D uN 0

$1 D uN

so that therefore u D uN 00 C tuN

∂uN 1 , ∂x

∂uN 0 . ∂x 1

We obtain the equations for 0 and 1 : 0000 C 2000 D 0 2

1000 C 2100 410 D 4 0 0 0 000 1. Boundary conditions: for

for

D0:

D1:

0 D 0

00 D 0

1 D 0

10 D 0

00 D 1

10 D 0.


4. The solutions of the above differential equations, which we shall also need with the following problem, are gained by quadrature, if the homogeneous equations are integrated. I obtained the last integrals from the following consideration: the homogeneous parts of our equations arise from the time and friction terms, which together form the heat conduction equation ∂2 u ∂u D k 2. ∂t ∂y As one can recognize from considerations of similarity there exist from this equation integrals of the form: y un D tn fn D p , 2 kt for which fn satisfies the differential equation f00n C 2f0n 4nfn D 0, which therefore possesses the above form. So, for example (see above), 2 u0 D f0 D p -

2

e d. 1

At D 0 u0 D 1 for

t>0

u0 D 0 for

t < 0.

At D 0, i.e. at y D 0, un is proportional to tn and must therefore be represented in the following form by superposition of solutions for u0 : 1 t y y n1 un D n u0 Ð t0 dt0 D n u0 Ð t0n1 dt0 . 2 kt t0 2 kt t0 0 0 Then for y D 0 this is equal to

t

n 0

t0n1 dt0 D tn .

To evaluate the integral we put t t0 D /, 0 y p un D n u0 Ð t /n1 d/. 2 k/ t If we introduce into this y p D , 2 kt tD

1 y2 , 4k 2

/D

1 y2 , 4k 2

y p D , 2 k/ d/ D

1 y2 d, 2k 3

127

128


we eventually obtain: 4n un D p tn 2n -

1

1 3

1 1 2 2

n1

2

eϑ dϑ.

d 1

Calculation of these integrals using the binomial theorem and the procedure of partial integration mentioned earlier eventually yields: n & 2 n & 2 2& fn D e d 2& 1 Ð Ð Ð 3 Ð 1 1 &D0   n 1     2 n  n  & 2 &C C 1 21 e . &D 2& 1 Ð Ð Ð 2& 2 C 1   D1    The other integral is algebraic and indeed equals the above factor of n & 2 n & fn D 2& . 2& 1 Ð Ð Ð 3 Ð 1 &D0

1

2

e d:

5. 0 is determined after this in the following way. By considering the boundary conditions 2 2 0 0 D 1 C p e d. - 1 From this by integration 2 1 0 D p C C p We still need:

1 2 2 e d C e . 2 1

2 2 000 D p e . -

The second differential equation (second order for 10 ) then has the form: 16 2 8 2 00 00 0 1 C 21 41 D p e d p e - 1 2 16 1 22 1 2 2 2 2 C e d e e d e C e . 2 2 1 1 The integral of the homogeneous equation for 10 is, according to subsection 4, 2 2 2 2 f1 D ˛2 C 1 C ˇ e C 2 C 1 e d . 1


The integral of the inhomogeneous equation would at this stage be obtained by quadrature. However, we can also use the fact that with two differentiations the differential equation is eventually transformed into 00000 C 2 0000 D function of which as an equation of essentially first order is easier to integrate. It then becomes 2 2 0000 D e eC Ð [function of ] Ð d. 1

Since after two differentiations the forcing function of our differential equation contains 2 2 e in every term, eC drops out and we can first integrate 0000 and finally . For 2 amongst the integrals there exist only functions, which at most contain e twice and apart from that powers of , and which are to be integrated several times, which may be carried out using the methods stated in the earlier subsection I.6. The result of the very extensive calculation reads 2 6 2 2 2 2 2 2 10 D e e d C 22 1 e d C e2 1 1 1 4 4 2 2 2 C p e p e d e 3- 1 2 2 C ˛22 C 1 C ˇ e C 22 C 1 e d , 100

2 2 D 22 1e -

1

e

2

1

8 d C -

e

2

2

d

1

1 8 2 2 2 p 22 C 3e C e C 4˛ C ˇ 2e 3-

2 22 e 2 C 4 e d .

1

The equation here could be integrated in closed form in spite of its relationship with the above steady-state problem due to the fact that ∂u/∂t is simpler than u∂u/∂x even though both possess the order of derivative appropriate to ‘heat conduction characteristics’. The determination of ˛ and ˇ from the boundary conditions 10 D 0 for D 0 and D 1 yields: 4 3 ˛ D 0, ˇ D p C 3/2 . - 36. To calculate the separation position we have: 1 ∂uN 00 ∂u 00 D p uN 0 C [t]Nu 1 0D ∂y ∂x 2 ,t

at D 0.

Now 2 000 D p , -

8 2 100 D p C 3/2 . - 3-

129

130


We obtain as a condition for the separation time [t]: 4 ∂uN 1C 1C [t] D 0. 3∂x ∂uN /∂x must therefore be negative. The separation occurs at the earliest where ∂uN /∂x is at its greatest in absolute terms. The result is valid for cylindrical bodies of arbitrary cross-section. uN is the corresponding potential flow.

IV Formation of the separation position with motion uniformly accelerating from rest 1. An objection can be put forward to the physical basis of the previous problem, in that the continuity of the fluid could be invalidated by the sudden impulse. Hence the solution of the problem is given here for the case of an immersed body subjected to a constant acceleration from time t D 0 onwards. Then uN D t wx,

1 ∂p ∂w ∂uN ∂uN D C uN D w C t2 w . ∂x ∂t ∂x ∂x

2. From similar considerations to those used previously we make the solution of our differential equation ∂u ∂u ∂u 1 ∂p ∂2 u Cu Cv D C, 2 ∂t ∂x ∂y ∂x ∂y the formulation: 1 p t2C1 $2C1 x, D 2 ,t Ð D0

uD

1

t2C1

D0

∂$2C1 , ∂

y D p . 2 ,t By substitution into the basic equation we obtain: ∂2 $2%C1 ∂$2%C1 ∂3 $2%C1 C 2 42% C 1 3 2 ∂ ∂ ∂ %1 ∂$2&C1 ∂2 $2%2&1 ∂$2%2&1 ∂2 $2&C1 . D4 ∂ ∂x∂y ∂x ∂2 &D0 For % D 0, 4w is added to the right-hand side, for % D 1, 4w∂w/∂x is added. Here we also limit the calculation of the condition to the first two terms; in so doing


it should be mentioned that also just the two terms of the pressure w C t2 w∂w/∂x are taken into consideration. The forcing function of the next differential equations contains only earlier coefficients of the formulation. We shall nevertheless still calculate the coefficients of the next term in the final equation which yields us the separation position. We can introduce the dependence of $1 and $3 on x in the following way: $1 D w1 ,

$3 D w

∂w 3 . ∂x

The differential equations for the are then: ∂3 1 ∂2 1 ∂1 C 2 4 D 4, ∂ ∂3 ∂2 ∂2 3 ∂3 ∂3 3 D 4 C 4 C 2 12 3 2 ∂ ∂ ∂ Boundary conditions: 1 D 0, at D 0 : 3 D 0, at D 1 :

∂1 D 1, ∂

∂1 D 0   ∂ ∂3 D 0  ∂ ∂3 D0 ∂

∂1 ∂

2

∂2 1 1 2 . ∂

from

uD0 , vD0

from

u D tw.

3. We can immediately write down ∂1 /∂ from the general solutions of the type of differential equations in question given in subsection III.4, since the inhomogeneous term 4 is satisfied by ∂1 /∂ D 1. We obtain 1 by integration using the methods which have already been applied several times (subsection I.6): 4 ∂2 1 2 2 D p e C 2 e d , ∂2 1 ∂1 2 2 2 2 e d , D 1 C p e C 1 C 2 ∂ 1 2 2 2 3 2 C 3 C 2 e d . 1 D C p 1 C 1 C e 3 1 These functions are drawn quantitatively in Fig. 11.8. A table of values is to be found further on in subsection 6. The forcing function on the right-hand side of the second equation is according to this: 16 2 16 2 2 p e d C 4 e d C 2e 3- 1 1 16 2 2 22 3 2 C 2 C e C 4 C 4 e e d 31 2 2 C3 C 44 e d . 1

131

132


2,26 2,0

ζ′1

1,0

ζ′′1 ζ1 η 0,25

0,5

1,0

Fig. 11.8

4. The integration of the second equation also succeeds here in closed form using the same methods which led to the goal in subsection III.5. An expression with undetermined coefficients is specifically to be recommended for the part of the forcing 2 function which is quadratic in e : 2 2 4 22 3 5 2 C d C e C f e e d a C b C c e 1

2

4

6

C g C h C i C k

e

2

2

d

.

1 2

This expression fails if the forcing function contains terms which go beyond 6 e2 , 2 2 2 2 5 e 1 e d, 4 1 e d (see subsection III.5). The coefficients are determined from linear equations. The remaining parts of ∂3 /∂ are easier to calculate. 3 and ∂2 3 /∂2 then result from integration and differentiation. If the integration constants are also determined correctly we finally obtain the result: ∂2 3 4 2 32 2 2 2 D p e e d C C 23 e2 2 1533 ∂ 1 C1 C 22 C 84 e

2

2

e d C 6 C 83 C 85 1

C

1 5

16 5 p p 6 - 45 -3 3

1

16 C 362 C 84 e

5

C60 C 80 C 16

e 1

2

d ,

2

2

e d

2


4 ∂3 16 2 2 2 D p e d e C 2 e d ∂ 153 - 1 1 1 2 2 2 C 8 C 2 C 24 e2 C 24 C 83 C 85 e e d 91 2 16 1 5 2 p p C9 C 182 C 124 C 86 e d C 15 6 - 45 -3 1 3 5 2 2 4 6 2 ð 33 C 28 C 4 e C 15 C 90 C 60 C 8 e d 1

and from this by integration: 2 8 2 2 2 2 3 D p e C 2 e d e C 1 C 22 e d 153 1 1 1 2 2 C 49 C 113 C 105 e2 C 768 e2 d 3151 2 2 C 537 C 1982 C 644 C 406 e e d 1

3

5

7

C315 C 210 C 84 C 40

e

2

2

d

1

C

1 105

16 5 p p 6 - 45 -3

2 24 C 872 C 404 C 46 e

C

C105 C 2103 C 845 C 87

2

e d 1

9 128 128 p p p C 3 105 2- 14 1575 -

.

These three functions are drawn in Fig. 11.9, they rigorously satisfy the differential equations and the boundary conditions for the formulation coefficients $. 5. The condition for the separation position has the form: 2 ∂u 1 t ∂2 1 2 ∂w ∂ 3 0D D Ct w ∂y yD0 2 , ∂x ∂2 ∂2 D0 which gives from the above formulae 4 ∂2 1 Dp ∂2

∂2 3 256 31 p . D p 15 - 225 -3 ∂2

We would therefore obtain the following equation for the separation time [t]: 31 ∂w 64 1C [t]2 D 0. 60 225∂x

133

134


1,0 0, 964

0,5

ζ′3 0,138

ζ3 0,25

0,50

1,0

η

ζ′′23

Fig. 11.9

The next term in the separation equation ∂u/∂y D 0 would read: ∂2 $5 1 p t5 2 , 2 ,t ∂ and in order to be able to take this into consideration, we most certainly do not want the whole development for $5 but calculate the coefficient in this separation equation. The development term $5 satisfies the equation: ∂2 $5 ∂$5 ∂3 $5 C 2 20 3 2 ∂ ∂ ∂ ∂$1 ∂2 $3 ∂$3 ∂2 $1 ∂$1 ∂2 $3 ∂$3 ∂2 $1 D4 C . ∂ ∂x∂ ∂x ∂2 ∂ ∂x∂ ∂x ∂2 The entry of x in $1 and $3 is known and by evaluating the right-hand side one convinces oneself that $5 has the form: 2 ∂w ∂2 w $5 D w 5 C w2 2 5a . ∂x ∂x We obtain as a condition for separation, since tw (see above) cancels out: 2 2 2 ∂2 1 ∂ 5 2 ∂w ∂ 3 4 ∂w Ct Ct 2 2 ∂x ∂ ∂x ∂ D0 ∂2 D0 D0 2 2 4 ∂ w ∂ 5a Ct w 2 D 0. ∂x ∂2 D0


6. The calculation of the coefficients ∂2 5 ∂2 5a and ∂2 D0 ∂2 D0 is still to be performed. For $5 we have the differential equation: ∂1 ∂3 ∂3 5 ∂2 5 ∂5 ∂2 3 ∂2 1 D8 41 2 43 2 D f C 2 2 20 3 ∂ ∂ ∂ ∂ ∂ ∂ ∂ and the boundary conditions: 5 D 0,

∂5 D 0 for ∂

∂5 D0 ∂

D 0;

for D 1.

The forcing function f is known from the functions written down above. We calculate the desired coefficient ∂2 5 ∂2 D0 using Green’s method in the following way: 1 1 3 ∂5 ∂2 5 ∂5 ∂2 5 ∂ϑ ∂5 ∂ 5 C 2ϑ ϑ C 2 2 20 d D ϑ 2 ∂ ∂ ∂ ∂ ∂3 ∂ ∂ 0 0 1 2 ∂5 ∂ ϑ ∂ϑ 20ϑ d C 2 ∂ ∂2 ∂ 0 We now let ϑ satisfy the adjoint differential equation: ∂2 ϑ ∂ϑ 22ϑ D 0, 2 2 ∂ ∂ and the boundary conditions: ϑ1 D 0

ϑ0 D 1

and so we obtain:

∂2 5 ∂2

D0

1

ϑ Ð f Ð d,

D 0

f is already given above; the influence coefficient ϑ (Green’s function) results from integrating the differential equation for it: ϑ D

! 2 p 2895 C 52803 C 23525 C 3527 C 169 945 2

4

6

8

10

C945 C 9450 C 12600 C 5040 C 720 C 32 e

2

e 1

2

d .

135

136


The behaviour of ϑ is found in Fig. 11.10, also that of the product ϑ Ð f. The area enclosed by this last curve yields the desired coefficient. In the same way we have equations to calculate [∂2 5a /∂2 ]D0 : ∂3 5a ∂2 5a ∂5a ∂1 ∂3 ∂2 1 D4 3 2 D g C 2 2 20 ∂ ∂ ∂ ∂3 ∂ ∂ 1 ∂2 5a D ϑ Ð g Ð d ∂2 D0 0 ϑ Ð g is drawn in Fig. 11.10 from the values given below. The calculated values are as follows: 0

1 ∂1 /∂ ∂2 1 /∂2 3 ∂3 /∂ ∂2 3 /∂2 f ϑ ϑÐf g ϑÐg

0.25

0.50

1.00

1.50

1

0 0.061 0.211 0.638 – 0.376 0 0.450 0.720 0.943 – 1 2.26 1.396 0.799 0.201 0.035 0 0 0.022 0.060 0.115 – 0.138 0 0.137 0.150 0.020 – 0 0.964 0.231 0.092 0.156 0.05 0 0 0.315 0.750 0.457 – 0 1 0.327 0.112 0.018 – 0 0 0.103 0.084 0.008 – 0 0 0.124 0.240 0.016 – 0 0 0.041 0.027 0.0003 – 0

The areas enclosed by the two curves are approximately: ∂2 5 ∂2 5a D 0.058; D 0.023. ∂2 D0 ∂2 D0 7. Consequently we obtain the equation for separation: 2 4 256 ∂w 31 ∂w p p C [t]2 p [t]4 Ð 0.058 ∂x 15 - 225 -3 ∂x [t]4 w or 1 C 0.427 Ð [t]2

∂w 0.026 Ð [t]4 ∂x

∂w ∂x

∂2 w Ð 0.023 D 0 ∂x 2

2

0.01 Ð [t]4 w

∂2 w D 0. ∂x 2

What is more, since the new correction term to be added is negative, the existence of the zero point seems to be certain. The position and time of the separation follow from the earlier approximation (without the term just calculated) [t]2

∂w D 2.34. ∂x


0,5

0

0,25 ϑg

0,50

1,0

η

ϑf ϑ

−0,5

−1,0

Fig. 11.10

With the newly calculated correction we obtain for the case of a cylinder symmetrical to the flow direction, in which w∂2 w/∂x 2 D 0 for the point at the back where separation begins: ∂w [t]2 D 2.08. ∂x Therefore the error is approximately 10%. Hereafter the quality of the approximation may very well also be judged for the other problems, where we have taken only the first two powers.

V

Application of the results of the separation problems to the circular cylinder

1. With a circular cylinder uN D 2V sin

x R

x/R is named the reduced coordinate X, V is the velocity with which the parallel stream flows towards the right, or correspondingly the cylinder is moved to the left. Inpthe steady state case the separation occurs according to subsection II.7 at xsep D 0.65 q0 /q1 , the maximum velocity at q0 xmax D 0.577 , q1

137

138

Early Developments of Modern Aerodynamics x Xy

yy

K

X

y

Fig. 11.11

in which uN D q0 x q1 x 3 . If one takes the normal power series of a sine, then 2V 2V , q1 D R 6R3 D 91° 150 ; xmax D 1.41 Ð R, q0 D

xsep D 1.59 Ð R,

Xsep

Xmax D 81° .

In contrast, if one approximates the sine in the interval 0–- using the method of least squares, then 2V Ð 0.856, R D 1.97 Ð R,

q0 D xsep

xmax D 1.75 Ð R,

2V Ð 0.093 R3 D 113° ,

q1 D Xsep

Xmax D 101° .

In any case, according to our calculation the separation point lies at about 11–12% of the total length of the boundary length after the maximum in velocity. To be sure this statement makes no claim on accuracy since only the first two powers of x were considered. Moreover, experimental records of the pressure drop show that the conditions near separation are on the whole scarcely attainable by development from the starting point of the boundary layer, since they are too strongly influenced by the pressure distribution of the body of vortices behind the cylinder. Hence the previous calculations merely have the aim of showing that separation is actually predicted by the hydrodynamic equations. The further development of calculation methods from this promises success, particularly for the more important problems of rotating bodies. 2. If the cylinder is suddenly set in motion with a constant velocity, then uN D 2V sin X,

∂uN 2V D cos X. ∂x R


The separation time [t] is, according to subsection III.6, given by: 4 ∂u R 1C [t] D 1, [t] D 0.35 . 3∂x V cos X The separation begins at X D -, cos X D 1 at the time: R t0 D 0.35 . V Up to that time the cylinder has therefore travelled a distance: S D Vt0 D 0.35 Ð R. All of this is independent of velocity, density and friction coefficient (of course friction is assumed to be small). 3. With a constant acceleration: uN D t wx D 2Vt sin

x R

where V is now the acceleration of the cylinder in the flow. The separation time for the onset of separation is (subsection IV.7): [t]2

∂w D 2.34 ∂x

or [t]2 D 1.17

R V cos X

to

to

D 2.08

D 1.04

R . V cos X

The distance travelled by the cylinder is: SD

1 2 Vt . 2

At the start of separation X D -: S D 0.59 Ð R

to

D 0.52 Ð R.

4. We further calculate the resistance which the cylinder experiences with constant acceleration. The stress components are: ∂v ∂u ∂v . Xy D k C Yy D p C 2k , ∂y ∂y ∂x As a consequence of the smallness of the friction, ∂v/∂y and ∂v/∂x drop out compared with ∂u/∂y and there remains as the force in the direction of the outer flow: - ∂u KD2ÐB p cos X C k sin X R dX, ∂y 0 where B is the width of the layer (the height of the part of the cylinder submerged).

139

140


We calculate the part due to pressure in this way: p cos X dX D 2BR2 Kpressure D 2BR 0

Now ∂p D ∂X

-

0

∂uN ∂uN C uN ∂t ∂x

uN D tw,

∂p sin X dX. ∂x

∂w D wCt w ∂x 2

,

w D 2V sin X.

The second term drops out during the integration, the first yields: Kpressure D 2- BR2 V and therefore an increase in the inertia due to the doubled contribution of the displaced fluid. The part due to friction is 2 2kBR ∂2 1 3 ∂w ∂ 3 Kfriction D p tw 2 C t w sin X dX. ∂x ∂2 ∂ 2 ,t 0 Here , D k/ . The second term also drops out here because ∂2 1 /∂2 and ∂2 3 /∂2 are indeed only constants and we obtain: Kfriction D 4 - kt Ð B RV. 5. In order to give a picture of the flow conditions corresponding to our formulae, the streamlines should in conclusion be drawn for a particular stage of the motion of the uniformly accelerated cylinder. The parameters R, V and , are arbitrary and so we must introduce reduced quantities for x, y, t, and u so that R, V and , drop out of our formulae. This happens with:

2 R 4 R, x D RX, tD T, yD Y, V V p p 1 D R3 ,2 V, u D RV U. Using the formulae (see IV.2 and V.3): p 3/2 2 ∂w D 2 ,t w 1 C t 3 , ∂x ∂1 ∂w ∂3 , u D tw C t2 ∂ ∂x ∂ y x D p , w D 2V sin , R 2 ,t

t2

∂w Vt2 x D2 cos ∂x R R


Fig. 11.12

the following reduced equations emerge:  D 4T3/2 sin X Ð 1 C 2T2 cos X Ð 3 , ∂1 U D 2T sin X C 2T2 cos Xfrac∂3 ∂ , ∂ Y D p , 2 T out of which we must first of all construct the curve for  for a particular time T p and for a series of values of the coordinate X as a function of Y D 2 T Ð , in order then to be able to read off the position of the value  D const. from these curves. In Fig. 11.12 the cylinder is drawn from X D -/2 to X D -. The separation time is given by 2T2 cos X D 2.34, therefore the beginning of separation by T D 1.08. In Fig. 11.12 2T2 D 5, therefore T is chosen to be 1.58. For the time chosen the separation point has already advanced up to more than 60° on the cylinder, in spite of this the boundary layer is still extremely thin, the dimensions correspond to the values 2 2 R D 10 cm, , D 0.01 cm s (water), V D 0.1 cm/s , therefore a very small acceleration. According to this t D 15.8 s.

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142


For V D 10 cm/s2 , one obtains the above picture according to the reduction p formulae after 1.58 s. The boundary layer thickness would be reduced by the ratio 1 : 10.

Commentary on Blasius (1908) In this astonishing paper by Blasius (1908), one finds a detailed exposition of many of the ideas and mathematical methods required for the further exploitation of the boundary layer theory for laminar flows which was to occur over at least the next seventy years. The paper is astonishing in at least two respects. First there is the sheer range of problems attacked. Not only is the well-known case of flow about a flat plate analysed, an analysis for which the paper is justly famous, but also addressed are the cases of a blunt-nosed symmetric cylinder in steady flow, a similar cylinder when impulsively started from rest and, finally, when accelerated uniformly from rest. In these last three cases Blasius (1908) is anxious to demonstrate the efficacy of the new boundary layer theory in the provision of accurate estimates of the boundary layer separation process. A second source of astonishment, at least for the newcomer to the subject, is the analytical level of the mathematics required. Much of this involves series expansion methods of considerable complexity. Blasius’s purpose here is to demonstrate that, despite their pervasive complexity, the boundary layer solutions he obtains are mathematically well behaved. All of this – both the range of problems capable of being attacked and the evident good behaviour of their solutions – bodes well for the future of the new boundary layer theory. In his Introduction Blasius (1908) recapitulates the basic arguments for the boundary layer to be found in Prandtl (1904). In addition, he provides a more detailed order of magnitude argument for the derivation of the boundary layer equations themselves and, for the first time, a statement of the behaviour of the stream function in the locality of a separation point. The first problem to be attacked, that of the flat plate at zero incidence, cannot exhibit separation. Despite its absence, he says that he has included this case because it demonstrates ‘the calculation methods to be used later’. This case, of course, together with some of the main features of its solution, had been touched on already by Prandtl (1904). Even so, Blasius (1908) gives a detailed exposition of the formation of what we would now call the self-similarity variables for this zero pressure gradient case. All of this produces the crucial step, already indicated in Prandtl (1904), of reducing the partial differential equations of the boundary layer to a single ordinary differential equation. This is the famous Blasius equation, here stated in the form 000 C 00 D 0 in which is the non-dimensional stream function. Primes refer to differentiation with respect to the non-dimensional stretched y variable (essentially y/x 1/2 as Prandtl (1904) had stated). Here y is the ordinate perpendicular to the plate’s surface, x lying along that surface and measured from the plate’s leading edge. The Blasius equation is clearly third order, and non-linear. Moreover, an added complication is the split nature of the boundary conditions; at D 0 (the plate surface) both and 0 are zero whereas it is as tends to infinity (the boundary layer’s outer edge) that 0 must tend to 2. Even a near-century after the equation’s first statement no analytical solution to this equation has been found although, with the massive computer power now available, numerical solution is the most straightforward option. Lacking the latter, Blasius (1908)


embarks on solution by series methods. For the small values of near the plate he develops a series expansion in powers of , its coefficients cn being determined from the form of the Blasius equation. This expansion also involves the unknown constant ˛ which, as it turns out, is essentially the plate’s shear stress. A further asymptotic series expansion is then developed for large values of . This is based upon the observation that, since 0 D 2 at the outer edge of the boundary layer, then for large must behave as D 2 C constant C 1 C successively higher approximations. The unknown constant here, emerging eventually in the form of ˇ, is the second unknown to be found by matching of the two series. The terms of the asymptotic series, led by 1 itself, are obtained by substitution into the Blasius equation. These turn out to involve increasingly complicated variants of the error function. These two series expansions for essentially 0 and 00 Blasius (1908) matches at an intermediate value of so as to obtain values for ˛ and ˇ. In doing so, he introduces the new variables Z and X and thereby provides the basic idea behind the so-called homological property possessed by the Blasius equation (see, for example, Rosenhead, 1963 ). This property has particularly aided numerical solution in more recent years, enabling a solution to be obtained by a single integration. However, having obtained a numerical value for ˛, Blasius (1908) is thus able to give the result for wall shear stress. This, when integrated over both surfaces of the plate, corrects the result given by Prandtl (1904) (the relation (9.3)) and yields in modern terms CD D 2.654Re1/2 . 11.1 More modern calculations give the numerical coefficient as 2.656. For his second problem, Blasius (1908) analyses the flow about a symmetric cylinder, taking the inviscid flow velocity to vary in ascending odd powers of x, x being the ordinate measured along the cylinder surface from its forward stagnation point. He carries out a related expansion for the boundary layer stream function, obtaining ordinary differential equations for this in terms of the stretched y variable here written as . The leading ordinary differential equation for the stream function here, 0 , valid in the region x small (close to the stagnation point), is again third order, non-linear and possesses the split boundary conditions found earlier for the flat plate. As it happens, this boundary layer equation, valid in the locality of a stagnation point, also satisfies the complete Navier–Stokes equations and was to be investigated in more detail later by Blasius’s Göttingen doctoral colleague Karl Hiemenz (Hiemenz, 1911). As to its solution, Blasius (1908) obtains an approximation based solely on a power series expansion strictly valid for small values of the stretched y ordinate. A similar procedure is used to solve the linear equation for the stream function perturbation 1 . From the results of these two equations alone, those for 0 and 1 , Blasius (1908) then obtains an estimate of the separation position. It will be realized that in this investigation Blasius (1908) has approximated the inviscid flow velocity variation by terms in x and x 3 only, terminating his series expansions at that point. In later investigations (see, for example, Schlichting, 1968 ) such series expansion methods have been taken to higher powers of x in attempts to improve accuracy. However, it should be remarked that improved series convergence and even greater accuracy have been achieved by the method of Görtler (1957), in which a stretched x variable has also been employed. The third problem investigated is that of a cylinder impulsively started from rest. At the instant at which motion begins Blasius (1908) argues correctly that the potential

143

144


solution represents the complete inviscid flow field exterior to the boundary layer. At this instant the boundary layer’s temporal acceleration is balanced solely by the viscous effect. This leads, as a first approximation, to an equation having the form of the heat conduction equation, giving a solution in terms of the error function. Subsequently, further approximations are obtained by series expansions which introduce the x dependence of the solution. From these Blasius (1908) is able to find the separation position, this being first encountered at the position at which the derivative with respect to x of the potential flow velocity has its maximum value. The fourth problem is related to the previous one, but now, more realistically, the cylinder is accelerated uniformly from rest. No doubt guided by the form of the variables found in the previous problem, Blasius (1908) is able to construct a series expansion for the boundary layer stream function in this case. Again, these series generate increasingly complicated variants of the error function, yet, even so, Blasius (1908) is able to obtain the criteria governing separation. Subsequent developments in the understanding of these two unsteady flow cases are described in detail by Schlichting (1968). In the final section of his paper, Blasius (1908) applies his findings to the case of a circular cylinder. The potential flow solution for the exterior velocity in this case has a form which varies as sin Rx , R being the cylinder’s radius, whereas in the steady flow boundary layer problem this has been approximated by the series containing terms in x and x 3 only. Nonetheless, rather cunningly Blasius (1908) is able to estimate from this that separation should occur at an angle lying between 101° and 113° from the nose. As subsequent analyses were to show (see Schlichting, 1968 ), as a first attempt this is remarkably accurate. The one fly in the ointment here is that the assumed potential flow solution begins to depart from reality even before the cylinder’s maximum thickness has been reached and becomes highly inaccurate, in particular, around the rear of the cylinder. All this is due to the gross nature of the boundary layer separation, this producing such a wide wake as to distort considerably the inviscid flow field. This difficulty was to be addressed by Hiemenz (1911), using his experimentally measured exterior velocity variation. His boundary layer solution indicates separation at 82° from the nose whereas experiment gives an angle of 81° . As to the two unsteady flow cases of the circular cylinder, again reference should be made to Schlichting (1968) in order to see subsequent developments. Nonetheless, the Blasius (1908) solutions remain remarkable for their time and provide a realistic case for the separation first appearing at the cylinder’s rear as motion begins, the separation position subsequently moving forward towards the position given by the steady flow calculations. All of this substantiates the belief expressed by both Prandtl (1904) and Lanchester (1907) that, at the instant at which motion begins, potential flow is established. Subsequently, however, this situation can become significantly modified by a boundary layer separation process which moves around the surface until finally settling at a fixed position.

12 On a two-dimensional flow related to the fundamentals of the problems of flight W. M. Kutta (1910)

1

Introduction

The flow and pressure phenomena which are observed at immersed bodies in moving fluids (in particular, in air) have for some time offered hydrodynamic theory a subject which is not at all easy. These questions have also achieved great practical significance since Otto Lilienthal’s feats and the newer development and solution of the problem of flight. The explanation and calculation of the lift forces occurring and the point where they act is of particular importance as well as the power required for maintaining the motion of the body. The origin of dynamic lift with a uniform motion of the body requires the existence of a flow in the surroundings of the same, which can be understood as the superposition of the velocities of two motions, of which one avoids the body in the simplest way, while the other circulates around it1 . The necessity of a power requirement follows only from a consideration of the outer or inner fluid friction. It is possible after excluding a linear rotating region of the fluid to represent the flow in the remaining part by a complex velocity potential. By excluding the vortex axes this part becomes a multiply-connected volume and the lift forces are related closely to the circulation of the fluid or the time periods of the velocity potential in this volume. The indirect effect of the developing vortex, which results from this, can produce a substantial contribution to the lift and it will in any case show far more steady characteristics in contrast to the very rapidly changing vortex appearance itself. With regard to dynamic lift effects, the most important types of a body immersed in a flowing fluid are long flat plates placed at an angle to the flow and slightly curved cylindrical shells, which experience lift forces even if the chord of their crosssection lies parallel to the flow. I have analysed the latter particular problem in my habilitation script of 1902, along with others stimulated by my distinguished teacher, Professor Dr Finsterwalder, while I was laying the foundation of the derivation for 1 See Lanchester, Aerial Flight vol 1, also Finsterwalder, Die Aerodynamik als Grundlage der Luftschiffahrt (Aerodynamics as the basis for airship travel), Verhandlungen der Schweizer Naturforschenden Gesellschaft (Proceedings of the Swiss Society of Natural Science), 92nd Annual Assembly, Lausanne, Vol 1, p 12, 1909.

146


an infinitely long cylindrical shell which has the cross-section of a circular arc of angle 2˛ and which is transverse to the flow. Here the flow problem becomes a twodimensional one and the explicit consideration of the vortices attached to the narrow sides drops out of the mathematical analysis (the sides are thought to be infinitely remote). The numerical results obtained agreed reasonably (for small values of the angle ˛) with the numbers obtained from the experiments of Lilienthal. A general, newly obtained theorem, advanced at that time and since then by Zhukovskii2 , implies that the lift force for every cylindrical body is at right angles to the flow direction and its magnitude is proportional to the component of velocity existing at right angles to the radius, which at infinity is close to constant velocity (it is small in the first order), therefore it is proportional to the parameter today described as the circulation. This following proposition, which comes from the principle of kinetic energy, excludes a power requirement with frictionless flow in vortex-free regions, even if they are coherent and repeated. The analysis was not published in detail, only a part of the main results was given briefly in the Aeronautical Communications of 1902, p. 133. Following renewed encouragement from Professor Finsterwalder I have taken up the investigation again in the last few months and as a result I have also found flow patterns for angled flows over plates and shells, which agree in the essential trends and in the numerical results with those to be expected from the experiments. At the same time the method allows flows to be calculated around still more cylinder shapes of practical interest. The aid to solving the flow problem of the two dimensional type is now, as then, the conformal mapping of pieces of the surface, whether it is on the half plane or on the surface region of the plane obtained by cutting out a circle. The second immediately allows the most general desired flow with velocity potential around a plate or shell to be obtained from the familiar most universal flow around a full circular cylinder, as explained in paragraph II. As in this case, that is also indeterminate because of the still arbitrary circulation, as, to be precise, an arbitrary constant enters the solution. The particular solution is then determined in the problems submitted from the physical requirement that no infinitely large fluid velocities should appear, as, for all that, are yielded by the general solution for the sharp edges and corners under consideration here. In the problems analysed in 1902, in particular the flow against the shell with the free stream velocity parallel to the chord, a solution was clearly set up successfully on the basis of the symmetrical characteristics of the problem through a suitable choice of circulation constant, which satisfies that requirement at the front and rear edges. In the same way such a solution may be given for cylinder shapes with only one sharp edge (for instance the rear edge). In contrast, an infinitely fast flow around an edge can be avoided in problems of an angled flow against a plate or shell only at one edge – we call it the rear edge – and be replaced by an exit flow with finite velocity; the solution is unequivocal. Besides, a lift force results which may be separated into two components. The first component results from the overall pressure of the fluid on the elements of the shell; it is directed towards the centre of the shell. The second component represents a suction effect on the edge (the front edge), around which the flow has an infinite velocity, and hence which experiences an infinitely negative force per unit area in the direction 2 Bulletin

de l’institut aerodynamique de Koutschino, Fascicule I, Petersburg 1906.

On a two-dimensional flow related to the fundamentals of the problems of flight

of the tangents to the shell, even if only on an infinitely narrow edge. An extremely insignificant rounding off of the edge, which will not change the first component of the lift substantially, is sufficient to remove the physically inadmissible negative pressures and excessive velocities, without, first of all, particularly influencing the suction force at the edge. In contrast, it may be shown that the velocity gradients arising at the edge or the rounded surface reach excessive values if the curved surface does not have significant dimensions. Then we may no longer disregard the effect of the inner friction of the fluid at this place. This effect, as may still be expressed in isolation, will decrease velocities substantially at the critical edge and with them the suction effect, indeed making the suction effect so slight as to as good as extinguish it. If this is the case we find, apart from the amount of power required for the movement of the shell resulting from outer friction at the shell, a still further amount caused in the manner indicated by inner friction. We consider that minimizing this as much as possible and maintaining the suction effect as undamaged as possible is the essential purpose of thickening the front face, as shown by the wings of birds and also by Lilienthal’s kites.

2

General expression

If one cuts the surface of the unit circle drawn around the point D 0 out of the complex plane D C i, the remaining infinite surface area is simply continuous in the sense of function theory, since indeed the infinitely distant point is not to be seen as a boundary. The transformation of the part of the surface on the positive complex half plane t ensues in the interior without singularities using tDi

C1 ; 1

D

tCi ti

and t are complex variables; the point t D i in the t plane corresponds to the infinitely remote point in the plane. On the other hand, the treatment of the two-dimensional irrotational flow problem in the surface under consideration is to be viewed as doubly continuous, and in so far as a flow condition will be given for infinity, since this is to be understood as a boundary condition, the infinitely distant point appears as a boundary. Accordingly the solution of the flow problem produces a periodic velocity potential and the period appears with the ‘circulation’ as an arbitrary constant. The solution is still not determined by the velocity at infinity, just as the vortex lines of arbitrary total vortex strength may be simulated in the accompanying three-dimensional problem of the circular cylinder. Closed streamlines can be present, but it is not in general necessary; in the problems treated later it is not the case. Let the stream function be W D U C iV, where V D const. for the system of the streamlines and U is the velocity potential. If u and v are the velocity components of the flow at the point of the surface, measured in the and direction, then as is known dW ∂W ∂U ∂V ∂U ∂U D D Ci D i D u iv. d ∂ ∂ ∂ ∂ ∂

147

148


The most general solution is given by 1 1 W D c1 C i Ð c2 C i Ð c ln , where c1 , c2 , c are assumed to be real constants. The first term for itself alone represents a flow with a velocity of c1 at infinity in the positive direction; the streamline on the axis splits into two at D 1, follows a semicircle on both sides and reunites at D C1. Symmetry in the flow pattern exists about the and axes. The second term alone represents a flow which has a velocity of c2 at infinity in the positive direction; the streamline on the axis splits at D i and recombines at D Ci. Lastly, the third term gives a flow in circles around D 0; the velocity at infinity is zero, 2c is the size of the circulation; c will therefore become the arbitrary constant. We transfer to another plane, given by z D x C iy in complex representation. Out of this an arbitrary closed surface is to be cut out and removed; this can also be shrunk onto a doubly calculated curved surface; the most general flow is wanted in the remaining surface. To do this one has only to transform the new surface onto that which was previously considered, free of singularities on the inside, and in doing so take care that the infinitely remote point in corresponds to the infinitely remote point in z. To achieve this purpose one will have to transform the surface z onto the positive half-plane t introduced earlier. Then let z D 1 be transformed into t D t0 . The transformation contains three real constants; one uses two of these in order to make t0 D i, through which the correspondence between z D 1 and D 1 is produced; the third is not of substantial significance; it merely produces in a quite insignificant rotation of the figure about the point D 0. If z D Ft is the desired transformation function, then the most general desired stream function results for the surface z by elimination of t and (this, of course, does not actually have to be carried out) using C1 1 1 1 ic2 C i Ð c ln W D c1 C tDi

z D Ft or, what is the same, t D z. One finds the velocity of the flow infinitely remote from z through dW dW dW d dt D dt D Ð Ð for z D 1, D 1, t D i dz dz d dt dz dt 12 d D c1 ic2 D c1 ic2 Ð k, Ð 2i dz D1 zD1

where the last factor, abbreviated as k and equal to d/dzzD1 , will in general become finite although of the form [1 Ð 0].


If the velocity V infinitely remote from z is prescribed at an angle ˇ to the x axis, therefore with components in the x and y directions of V1 D V Ð cos ˇ, V2 D v sin ˇ, then c1 and c2 may be determined from V1 iV2 D c1 ic2 k. In contrast the arbitrary circulation constant c still remains optionally at one’s disposal; there are still an infinite number of solutions. Now we will certainly expect that among the possible steady state solutions found particular ones will stand out as a result of a greater stability of the flow pattern, therefore the tendency exists for just these to be produced in the presence of disturbances. Which particular solutions these are depends on the particular form of the boundary curve; in the case of the circle we will be inclined to expect the symmetrical solution, therefore c D 0. For all that we shall not in general be surprised to obtain an oscillation of the flow pattern about that especially stable pattern – unless the special form of the boundary curve favours quite particularly one solution before all others and indeed not only with regard to geometrical relationships, like symmetry, but with regard to the actual physical flow conditions. Such a case of particular preference occurs when the boundary curve of the surface z shows an obtuse angle ˛ externally at one of its points (in practice instead of this also a very small curvature inwards on the closed curve). If this is the point z1 where W D W1 then as you know the formulation of W at the point z1 gives ˛ ˛ z z1 D W W1 C0 C power series in W W1 , consequently

dW dz

D

1 W W1

˛ [D0 C power series],

where ˛ > . It follows that the flow velocity at that point is in general infinitely large. This irritating detail can be removed if the constant D0 is set to zero. Then there is therefore a value of the circulation c which, while it makes D0 zero, represents a physically excellent solution of that kind and this solution is clearly determined. Besides z1 will in general become a dividing point of a streamline. I will draw attention to such cases in the last paragraph. If however there are several such obtuse boundary positions it will no longer in general be possible to prevent infinite values of the velocity everywhere by choice of a circulation constant; on the contrary that will be possible at only one place. We will see (in Section 5) that it requires only an extremely minute rounding off, which in itself is practically not to be avoided, in order to prevent the velocity exceeding a physically permitted size at the remaining places and the pressure reduction produced by it leading to inadmissible negative pressures. On the other hand, greater rounding off will be needed in order to render the very large velocity gradient occurring at the critical point as innocuous as possible and with this to reduce the effects of the otherwise considerable and non-negligible internal friction of the fluid to minute proportions. Of course, several of the type of singular boundary locations can also exceptionally and in individual special cases be simultaneously made innocuous, for example in cases of symmetry, by a suitable choice of the one constant c.

149

150


3

The arched shell of circular form

The example which I want to treat concerns the case of a cylindrical shell of infinite length, which has the cross-section of an arc of a circle with a subtended angle of 2˛, and which is immersed in the flowing fluid. The boundary curve z is therefore to be a circular arc of radius r, in which the centre of arc O is to lie at the point z D 0; the arc is supposed to touch the x axis there. Call the middle point of the circle M. Then the end points of the arc A and B have the coordinates: A : x1 D r Ð sin ˛

B : x2 D Cr Ð sin ˛

y1 D r Ð 1 cos ˛ therefore z1 D 2r sin

˛ ˛ ˛ cos C i sin 2 2 2

y2 D r Ð 1 cos ˛

z2 D 2r sin

˛ ˛ ˛ cos i sin . 2 2 2

The transformation onto the positive half-plane of t succeeds by means of z0 with z D 1/z0 , so that the points corresponding to A and B in z0 are: 1 ˛ 1 ˛ A0 : z 0 D i cot ; B0 : z 0 D i C cot 2r 2 2r 2 z0 D 0 corresponds to the point z D 1; z0 D 1 to z D 0. Then one has further to apply the well-known Christoffel–Schwarz formula for transforming polygons bounded by straight lines and obtains here: t at b dz0 DC dt t2 where C, a and b are real constants. From this it follows that

ab z0 D C t a C b ln t C C0 . t The following should correspond: tDa tDb tDi

1 2r 1 z0 D 2r z0 D 0.

and z0 D

˛ 2 ˛ i cot 2 i C cot

That yields b D a; a D cot45° ˛/4 D cot υ, if 45° ˛/4 is abbreviated by υ, and finally:  ˛ cot ˛ 1 i  2   C C z0 D t Ð tan 45° ˛ 4r 4 2r t Ð tan 45° 4

1 i tan 2υ t Ð tan υ C C . D 4r t Ð tan υ 2r


If we add the formula for W, in which is expressed in t, the following arrives: W D 2c1

t2 1 t C 4c2 2 C 2c arctan t. t2 C 1 t C1

This formula contains the desired general flow. For many calculations it is recommended that instead of t a new working parameter is introduced by t D tan /2). From this it follows that W D 2c1 cos C 2c2 sin C c Ð

tan 2υ i 0 z D tan Ð tan υ C cot Ð cot υ C ; 4r 2 2 2r

zD

1 . z0

The flow velocity at point z results from dW dW dW 0 D dt D dt0 Ð z 2 D dz dz dz dt dt This gives dW 2c t 1 t2 C 4c2 C , D 8c1 2 2 dt 1 C t 1 C t2 2 1 C t2

dW dW 0 d d 2 D 0 Ð z . dz dz d d

dz0 1 tan 2υ tan υ 2 , D dt 4r t tan υ  

cot υ  dz0 tan 2υ  tan υ D  . d 4r cos2 sin2 2 2 First of all we apply the assembled formulae to express c1 and c2 from the given flow velocity V1 iV2 D Vcos ˇ i sin ˇ therefore for t D i, or D i Ð 1. For t D i C ε dW D 2c1 sin d

C 2c2 cos

C c,

1 dW D 2 [2ic1 2c2 c Ð iε C terms in ε2 . . .] dt ε 2r cos 2υ [1 C i Ð ε cos2 υ C terms in ε2 . . .] zD ε 2r cos 2υ dz D [1 C terms in ε2 . . .]. dt ε2 Hence finally ci ic1 C c2 C ε dW 2 C terms in ε2 . . . D dt r cos 2υ ic1 C c2 ci 1 D C C terms in 2 . . . r cos 2υ z z Since this should give V1 iV2 for z D 1 we obtain c1 D V2 Ð r cos 2υ D V Ð r cos 2υ sin ˇ, c2 D CV1 Ð r cos 2υ D CV Ð r cos 2υ cos ˇ.

151

152


We note that in the formulation for dW/dz at infinity in terms of powers of 1/z the coefficient of 1/z is equal to c Ð i. The constant c remains, as foreseen, arbitrarily at our disposal. The introduction of the values obtained for c1 and c2 into W, omitting an additional arbitrary constant, gives ˛ sin ˇ C t cos ˇ Ð C 2c arctan t 2 1 C t2 ˛ D 2rV sin Ð sin C ˇ C c Ð . 2

W D 4r sin

We now further calculate the velocities dW/dz near the edges A and B, that is, for ˛ C C k, t D a C ε D cot υ C ε, D 2 2 ˛ C C k. and t D Ca C ε D C cot υ C ε, DC 2 2 For the surroundings of point B it follows that ˛ dW D 2rV sin cos C ˇ C c d 2 ˛ ˛ k2 ˛ ˛ C ˇ C k cos Cˇ sin C ˇ Ð Ð Ð C c, sin D 2rV sin 2 2 2 2 2  ˛ ˛  ˛ 3 2 sin i C cot i C cot k k 2 C 2  2 Ð Ð Ð C z0 D  , 2 ˛ 3 ˛ 2r 4r cos cos 2 2 ˛ ˛ z D r [sin ˛ i 1 cos ˛] r Ð tan cos ˛ i sin ˛ k 2 C k 3 tan C Ð Ð Ð , 2 2 ˛ ˛ dz D r Ð tan cos ˛ i sin ˛ 2k C 3k 2 tan C Ð Ð Ð . d 2 2 One recognizes that dW/dz only then remains finite for k D 0, and therefore at point B, if the term in dW/d which is independent of k disappears. Therefore ˛ ˛ 2rV sin sin C ˇ C c D 0. 2 2 From this, c and the circulation 2c are clearly determined, as foreseen. The final formula for W reads:

˛ ˛ t cos ˇ C sin ˇ W D 4rV sin C sin Ð arctan t C ˇ 2 2 1 C t2 or W D 2rV sin

˛ ˛ Cˇ Ð sin C ˇ C sin . 2 2


Again returning to point B, we introduce  the distance from the arc of the different point, which is about k from B in the working parameter , and obtain

From this

 1 2 sin ˛ i cos ˛ C Ð Ð Ð cos ˛ i sin ˛ C r 2 r2 ˛ ˛ D tan cos ˛ i sin ˛ k 2 C k 3 tan C Ð Ð Ð . 2 2 3/2  1  1 kD ÐÐÐ C terms in ˛ r tan 2 r r 2

and finally ˛



dW ˛  D V cos cos C ˇ 1 dz 2 2

˛ 1 ˛ 1  3 tan C tan C ˇ Ð 2 r tan ˛ 2 2 2 

C terms in

  Ð Ð Ð Ð [cos ˛ C i sin ˛]. r

The last bracket is only a direction factor. The formula yields in particular for point B itself  D 0 the finite velocity ˛ ˛ V cos cos Cˇ 2 2 in the direction of the circular arc at B; it therefore represents, if positive, a flow away from the shell for ˛ ˛ 4 arctan

1 , 4C

the pressure force P exceeds P1 , and when 2˛2 D 4 arctan 1/2) it becomes equal to P2 , i.e. greater than P0 . The first of these values is approximately equal to 32° , the second 36° . Let us now evaluate the lift force produced in the air by a one square metre of the plate surface. At speed 14 m/s (50 kph) it equals (approximately) 2 Ð 1.292 Ð 196 D 5.3 kg, 48 g in metres 3 This statement is valid provided the angle between the tangent to the arc and the chord nowhere exceeds /2. 4 When this paper was in press Prof. N. E. Zhukovskii drew my attention to the note by W. M. Kutta ‘Auftriebskräfte in strömenden Flüssigkeiten’, Illustr. Aëron. Mitt., Strassburg, 3, 1902, where the solution of the same problem was presented. Formulae derived by Kutta for the stream function have a different (more complicated) form, but an expression for the pressure force completely coincides with (13.14).

On the pressure exerted by a plane-parallel flow on obstructing bodies (aeroplane theory)

if the plate is bent such that the central angle equals 15° ; with the angle 30° this force is approximately two times larger. The following table shows the pressure P per square metre (in kilograms) at the flow speed 14 m/s for different values of the central angle 2˛ and the corresponding values of the ratio of the arc depth to the chord s/2c D k.

2˛

k

P

15° 30° 32° 36°

1 : 30.5 1 : 15.2 1 : 14.2 1 : 12.6

5.3 10.6 11.4 12.8

Let us now determine the velocity distribution along the plate. According to formulae (13.11) and (13.12) v D j1 hei 2 j D 1 C h2 2h cos . v0

Consequently, the maximum velocity has the value v0 1 C h 2 , and the minimum is v0 1 h 2 ; the velocities at the leading and trailing edges are equal, and can be found by setting cos D h; this gives v0 1 h2 , where h D sin˛/2 . As may be easily seen for small h the velocity changes within a quite narrow range. We shall conclude this section by deriving formulae which solve the problem by means of mapping the exterior of the arc in the z-plane and of the corresponding region in the w-plane onto the upper half-plane of the complex variable u or onto an infinite strip bounded by two parallel straight lines. These formulae will be used for solving other problems in the following sections. It may be easily seen that with z D 4ain

1 , u C n i u n C i

z Daa

u C n C i u n i , u C n i u n C i

13.15

u dz du D 8ain u C n i 2 u n C i 2 we map the flow region in the z-plane onto the upper half of the u-plane such that the infinite point of the real axis corresponds to the lower edge of the arc which is assumed to coincide with the coordinate origin in the z-plane; the Oy-axis is tangent to the arc at this point. To show the correspondence of points in the two planes they are marked by the same letters in Figs 13.5 and 13.6. Constant n in the above formulae has a simple geometrical value, namely n D tan˛/2 . Function dw/du is expressed via the new variable u by the formula dw u , DK 2 du u C n i u C n C i 2

13.16

195

196


A

y

B D

Z

2a

E x

O

Fig. 13.5 η

u u = i −n z=∞ 0

B

E u = i +n z=α A

D

ξ 0

Fig. 13.6

where K may be determined from the condition

Ki

Ki unCi 2 dw

i0 D v0 e D Ð D 1 C in 2 .

dz 1 8na uDnCi uCnCi 8na Hence KD

8nav0 , 1 C n2

ei0 D iei˛ ,

0 D

3 ˛, 2

which shows that the flow at the infinite point is directed parallel to the chord AO as was to be expected. The complex variable w is defined in the u-plane as

K iCn uCni in wD C in ln . 13.17 4 uCni uCnCi uCnCi The circulation around the infinite point u D n C i, z D 1 is CD

nK ˛ n2 av0 D 4av0 sin2 D 4av0 h2 , D 4 2 2 2 1Cn

which brings us again to formula (13.13) for the pressure force.


If we, finally, assume that the region in the u-plane has the form of an infinite strip between two parallel straight lines separated by 2 then the required mapping onto the z-plane is given by Neumann’s isothermic transformation z D c tan

uC˛ , 2

dz c D uC˛. du 2 cos2 2

13.18

Here the real part of u D , C i- varies from 0 to 2, and the imaginary part from 1 to C1. The convex side of the plate corresponds to , D 0, the concave side to , D 2. At point A at the lower edge of the plate - D 1; the upper edge C corresponds to - D C1. The image of the infinite point z D 1 is situated at u D ˛, and z D 0 maps into u D 2 ˛ (point O); at the centre of the arc u D C ˛ (point E). For dw du and w the following expressions can be easily derived ˛ cos2 dw D icv0 2 , u ˛ 2 du 2 cos sin 2 2  u ˛Cu sin sin ˛   2 4 w D icv0  u ˛ C tan 2 ln ˛ u ; cos sin sin 2 2 4 hence, using (13.18)

 2 ˛ uC˛     cos cos dw 2 2 D iv0 u ˛  .  dz  cos sin  2 2 y

E

α

0D B

x

A

Fig. 13.7 ξ

u C η

Fig. 13.8

D 0 E B

A

13.19

197

198


In particular, at z D 1, u D ˛ dw D iv0 D v0 ei0 , dz

0 D

. 2

The circulation around the infinite point z D 1, u D ˛ is defined by formula (13.19) as being ˛ ˛ C D icv0 tan 2i D 2cv0 tan , 2 2 and the components of the pressure force may be found from the equation Y C iX D Cv0 ei0 D 2iv20 c tan

˛ D 2iv20 s, 2

which fully agrees with formula (13.13).

5

The pressure force acting on a complex wing

Having solved the problem of the flow past a segment of a circular cylinder, we shall now turn to the analysis of other forms of cylinders which allow relaxation of the restriction on the direction of the flow at infinity. As was mentioned earlier, the theory is free of this restriction if the number of corner points on the obstacle surface with acute angles facing the flow is less than two. A quite simple and elegant example of this sort of solution may be presented for the flow past a complex wing which is composed of the arc shaped plate, considered in the previous section, with the addition of a head in the form of a circular cylinder placed on one of the arc’s edges. A sketch of the perpendicular cross-section of this wing together with the anticipated form of the streamlines is shown in Fig. 13.9. The head radius should be assumed small as it is intended to represent a natural bluntness of the wing’s leading edge; in reality it is never mathematically sharp. Due to the

y

C z E

α

ODB L M

Fig. 13.9

K

x

On the pressure exerted by a plane-parallel flow on obstructing bodies (aeroplane theory) K M

η

B ∞

u

L ξ

E 0 D

Fig. 13.10

head an additional parameter may be introduced in function w, being dependent on the location of the critical point M where the streamline coming in contact with the wing splits into two parts. We can map the region in the z-plane occupied by the moving fluid onto a semi-strip in the plane of complex variable u by using the Neumann transformation z D c tan

uC˛ , 2

13.20

as in formula (13.18). Along the right-hand side of the arc CBK the real part of u D , C i- is null; along the circle KML the imaginary part - D b, where b is a rather large positive quantity; on the left-hand side LDC of the arc , D 2; finally, at point C at the trailing edge - D C1. The flow considered in the previous section corresponds to b D 1. The dependence of w D ϕ C i on u may be most easily found by first mapping w onto the half-plane of an auxiliary variable s which then should be mapped onto the region in the u-plane. To perform the s to u mapping it is sufficient to set u C ib 2 , sD b cosh 2 cos

13.21

in which case the infinite point z D 1,

uD˛

maps into the point ˛ ib ˛ ˛ b 2 s D s0 D m C in D D sin C i cos tanh , b 2 2 2 cosh 2 sin

whence ˛ m D sin , 2

n D cos

˛ b tanh . 2 2

13.22

Now, taking into account the properties of the singular point s0 and observing the condition that w should be real on the real axis in the s-plane, we can express w and

199

200


dw/ds by the formulae wD

As m C B s m in C iK ln , 2 2 s m C in s m C n

A 2Kn 2s m [As m C B] dw D . 2 2 ds s m C n [s m 2 C n2 ]2

13.23

Since u C ib 1 sin 2 c dz ds D D , uC˛, b du 2 du 2 cos2 cosh 2 2 ds u C ˛ 1 u C ib cos2 D sin b dz 2 2 c cosh 2

13.24

we find that b uC˛ u C ib cosh2 cos2 sin ds 2 2 2 D c s3 dz 3 u C ib cos 2 is finite at s D 1, - D 1. Consequently, ds/dz is an order s3 quantity, and to make it finite at this point one has to set the coefficient of 1/s2 in the expansion of dw/ds to zero. As a result we have A 2Kn 2A D 0,

KD

A , 2n

13.25

and therefore An2 Bs m dw . D2 ds [s m 2 C n2 ]2

13.26

Let us now consider the flow velocity and its direction at infinity. If we denote by N the quantity b An2 C Bm cosh , 13.27 ND B 2 then from (13.26) and (13.24) we can find using (13.21) lim

z!1

 u C ˛ 2 ˛ ib cos ˛ ib   2 2 N C sin lim   . u!˛ s m 2 C n2 2 2 b c cosh 2

2B cos D

dw D v0 ei0 dz


Substituting m and n we easily find u C ib ˛ C ib b cos cosh s m in D cos 2 2 2 uC˛ u C ˛ C 2ib D 2 sin sin ; 4 4 u C ib ˛ ib b cos cosh s m C in D cos 2 2 2 u C ˛ C 2ib uC˛ sin . D 2 sin 4 4 Hence u u C 2ib b ˛ ˛ 2 2 cosh [s m C n ] D cos sin cos , sin 2 2 2 2 2 2

uC˛ b i cosh2 2 2 lim . D ˛ ˛ ib b u!˛ s m 2 C n2 2 cos cos sinh 2 2 2 cos

Thus, the limit under question is written as

v0 e

i0

b ˛ ib B coth2 N C sin dw 2 2 . D D lim ˛ ib z!1 dz 2 ˛ 2c cos cos 2 2

13.28

Consequently   ˛  2 b  C sinh   2 b 2 ˛ 2 2  B D 2v0 cos 2 tanh 2 ,   b ˛ 2 ˛ 2 b   2  C sinh N C 2N sin cosh C sin  2 2 2 2      b ˛   cos sinh b ˛ 2 2 0 D 1 C 2, tan 2 D tan 2 tanh 2 , tan 1 D b,  ˛   N C sin cosh    2 2     b b ˛    sinh N sin C cosh   2 2 2  . tan 0 D  ˛  b ˛  cos N cosh C sin  2 2 2

cos2

13.29

From the formula for tan 0 it is clear that when the coefficient N varies from C1 to 1, the angle 0 assumes different values between 2 and C 2. Let us now turn to the location of the critical point M and analyse the velocity distribution on the wing surface.

201

202


As far as point M is concerned, its position is defined by the root of equation (13.26), which defines the corresponding point in the u-plane by means of the following equation N C cos

u C ib D 0, 2

13.30

where N is defined by the incidence angle. When 1 < N < 1, point M is situated on the circular head of the wing. Indeed, under this condition the root of equation (13.30) is written as cos

u D , ib,

, D N, 2

and for the limit values š1 of the coefficient N the critical point coincides with either point L or point K. If N assumes a value outside these limits then the critical point M finds itself on the concave side of the wing for N > 0 and on the convex side otherwise. The flow velocity on the convex side of the wing may be found by calculating the modulus of dw/dz with -Cb cosh 2 , sD b cosh 2 where - varies from b to 1. For the concave side we have to set -Cb 2 . b cosh 2

cosh sD

Using formulae (13.24), (13.26) and (13.27) we find -Cb -Cb ˛ p cosh š sin cosh š N sinh 2 B2 b 2 2 2 2 cosh2 vD , c 2 - C 2b ˛ ˛ 2 cosh Ý sin cosh Ý sin 2 2 2 2 where the upper and lower signs correspond to the convex and concave sides of the wing respectively. At the trailing edge - D 1 and p 2 B2 b v D v1 D cosh2 eb c 2 ˛ b cos2 C sinh2 ˛ b 2 2 D 4 cos2 sinh2 eb v0 . 2 2 ˛ b ˛ b N2 C 2N sin cosh C sin2 C sinh2 2 2 2 2


If b D 1 then the head disappears, and the above formula reduces to the already known form ˛ v1 D v0 cos2 D v0 1 h2 . 2 As may be easily seen, the velocity on the plate surface does not differ significantly from v1 or v0 . As far as the velocity on the head surface is concerned, it may be found by setting , cos 2 sD b cosh 2 in the formula for the modulus of dw/dz, yielding , , 2 ,C˛ 2 b p C sinh cos N C cos sin 2 2 B b 2 2 2 2 vD cosh2 . c 2 , ˛ b 2 , ˛ b cos2 2 cos sin cosh C sin2 C sinh2 2 2 2 2 2 2 This velocity assumes very large values at some points on the head surface if the head is rather small (b is large) and the flow direction at infinity differs from being parallel to the wing chord. In the air flow or, what is the same, when a wing moves through the air this velocity may reach the speed of sound, and then a steady flow becomes impossible (see my paper ‘On gas jets’). This situation may arise if N is an order coshb/2 quantity. As follows from the above expression the velocity attains its maximum at a value of , close to . We shall therefore evaluate v at , D and assume it to be approximately equal to vmax . Using formulae (13.29) we can bring, after rather complicated manipulations, the expression for vmax to the following simple form 2 b 2 b 2 ˛ C cos N sinh sinh ˛ 2 2 2 . vmax D vj,D D 4v0 cos cos 0 b ˛ 2 2 b 2 ˛ N cos C sin sinh C sin 2 2 2 2 Neglecting sin˛/2 which is small compared to the quantities of order cosh2 b/2 , we further find vmax D 4 cos

b b ˛ ˛ cosh v0 cos 0 D 4 cos cosh v0 sin ˇ, 2 2 2 2

where ˇ is the angle between the flow direction at infinity and the chord of the plate. Let us now imagine a wing with a small bluntness of the leading edge. The wing will be modelled by a circular arc with the central angle 30° and 2 m long chord, and having a circular head of 1 cm in diameter. It may be easily seen that the geometric value of b is defined by the equation sinh b D

c ,

203

204


where c is the half-chord and the radius of the head; in the case considered they have (in centimetres) the values 100 and 0.5. Thus sinh b D 200, and we have approximately cosh

b D 10.025, 2

whence, vmax D 40.65v0 sin ˇ.

Let us now determine the conditions under which vmax might reach the speed of sound corresponding to the state of gas at the location where this maximum is observed. We shall assume that the gas density and pressure are related to each other by the adiabatic law. Denoting the speed of sound by V we have v2 6 p0 6 p V2 C 0. C D 6 1 2 6 1 0 2

Hence, taking into account that

we find

p V2 D 6 ,

VD

26 p0 6 1 C v20 D 6 C 1 0 6 C1

7p0 v2 C 0. 60 6

If we disregard the second term in the root compared to a larger first term then for normal atmospheric conditions we will have V D 302 m/s. Setting vmax D V and using the above formula for vmax yields (approximately) v0 sin ˇ D 7.4 m/s.

For v0 < 7.4 the flow of the type considered is possible for any value of the angle as defined by the formulae (13.29) up to its limit value 0 D 2, ˇ D /2 2. However, if v0 significantly exceeds 7.4 m/s, then additional restrictions on the incidence angle arise. For example, for normal aeroplane speed v0 D 14 m/s the limit value for angle ˇ appears to be 32° . The sharper the leading edge the smaller appears to be the limit angle. If, for example, we reduce the head radius to D 0.25 cm keeping the chord of the wing unchanged, then b sinh b D 400, cosh D 14.15, 2 and the limit angle appears to be slightly less than 23° . With the wing incidence exceeding the limit value the continuity of the flow field can no longer be preserved.


Let us now turn to the calculation of the lifting force. The circulation in the clockwise direction around point s D s0 D m C in may be found from equations (13.23) and (13.25) to be b N m cosh A 2 B, C D 2 D b 2n 3 n cosh 2 and after substituting m and n from (13.22) ˛ b cosh 2 2 cosh2 b B. C D ˛ b 2 cos3 sinh3 2 2 N C sin

Using this expression in the main formula Y C iX D Cv0 ei0 , and replacing B by its value defined by the first of equations (13.29) yields ˛ b cosh 2 2 Y C iX D ˛ b cos sinh 2 2 ˛ b cos2 C sinh2 2 2 ei0 . ð ˛ ˛ b b N2 C 2N sin cosh C sin2 C sinh2 2 2 2 2 2cv20

N C sin

Taking into account that ˛ b ˛ b ˛ b C sinh2 D cos2 cosh2 C sin2 sinh2 , 2 2 2 2 2 2 b ˛ ˛ b N2 C 2N sin cosh C sin2 C sinh2 2 2 2 2 2 ˛ ˛ b b D N C sin cosh C cos2 sinh2 , 2 2 2 2 cos2

and using formulae (13.29), we can reduce the above expression to the following simple form ˛ cos 1 i0 Y C iX D 2cv20 tan Ð e . 2 sin 2 Finally, introducing angle ˇ made by the flow at infinity with the wing chord, and taking into account that 1 D 0 2 D ˇ 2, 2

205

206


we have Y C iX D 2cv20 tan The full lift force PD

˛ sinˇ C 2 i0 Ð e . 2 sin 2

X2 C Y2 ,

is perpendicular to the flow, as usual, and is defined by the formula P D 2cv20 tan

˛ sinˇ C 2 Ð , 2 sin 2

13.31

which for ˇ D 0 coincides with the lift force produced by the circular arc wing without the head, because c tan˛/2 is the depth of the arc. Thus, if the chord is parallel to the flow direction at infinity or, what is the same, the wing moves through still air in the direction parallel to its chord (when the continuous steady form of the flow past the wing without the head is possible), then the lifting pressure force does not depend on the size of the head installed on the leading edge. The second important conclusion which may be drawn from formula (13.31) consists in the following: the lift force drops very fast if the wing inclines forward; it becomes zero when ˇ D 2, and then turns into an opposite force pressing upon the wing from above. This critical angle depends on the shape of the wing solely. It is interesting that one can easily plot angle 2 geometrically. Indeed, as has been already noticed sinh b D

c ;

where D OF Ð tan 6 D c Ð tan 6, and therefore sinh b D cot 6,

tanh

cosh b D

1 , sin 6

sinh b cos 6 b D D . 2 cosh b C 1 1 C sin 6

c , we have Taking into account that OO0 D cos 6 tanh

b D 2

c 1 c D D , 1 OO0 C OH C tan 6 cos 6

and using (13.29) ˛ c tan ˛ b 2 D OG . tan 2 D tan tanh D 2 2 OH OH

On the pressure exerted by a plane-parallel flow on obstructing bodies (aeroplane theory) x G y

µ

H′ C

β

tγ

0

0′ H A F

Fig. 13.11

Thus, placing on the chord Oy point H0 which is symmetrical to H with respect to O and connecting H0 with G we find 2 D 6 GH0 O; the pressure force is given by the formula P D 2v20 GH0 sin 9, where 9 is the angle between the flow direction (shown in Fig. 13.11 by the arrow) and GH0 . The theoretical results presented above are valid as long as the steady flow of the type considered remains possible. The latter requires that angle 0 should not exceed the critical value. If we return to the example of the wing with the central angle 2˛ D 30° , we can expect the lifting force to grow gradually according to the sinusoidal law while the incidence angle ˇ is confined between 7° 300 and C23° for the wing with leading edge thickness less than 1/400 of the half-chord (v0 D 14 m/s). For ˇ close to the critical value the real flow near the head will differ significantly from the flow of an incompressible fluid. This, however, can hardly influence the resultant pressure force, because of the smallness of the head. For ˇ > 23° on the upper surface of the wing a region of still fluid will probably form behind the head. It moves together with the wing, and in the case of viscous fluid it takes the form of an accompanying vortex. For ˇ close to a right angle, the flow is expected to assume a separated form of the normal Rayleigh type; in this case the lifting force should be significantly smaller as compared to the value it reaches at the critical incidence angle. In the separated flow with ˇ D /2 it is given by the formula P1 D 2kv20 c, where k lies between 0.44 and 0.5. On the other hand, for ˇ D 23° we have P D 2v20 c Ð tan 7° 300

sin 30° 300 sin 7° 300

which means that P/P1 is close to 3. Thus the variation of the pressure force with the incidence angle should have the form sketched in Fig. 13.12.

207

208

Early Developments of Modern Aerodynamics P

0

23°

90°

β

Fig. 13.12

6

A wing with one cuspidal point

Having performed a detailed study of a particular form of the wing which was chosen because it gives a simple and clear description of the flow field, we shall now move to other wing forms resembling the bird’s wings with smooth contour and only one corner point at the trailing edge which is a cuspidal point. The simplest example of such a wing may be constructed by mapping the flow region in the z-plane onto the upper half-plane of the complex variable u, and using for function z and its derivative the following expressions dz u C iε , D 8ain 2 du u C n i [u n C i1 C 2ε ]2

13.32

z D 4ain u C n i [u 1 n C i1 C 2ε ] , which for small ε differ only slightly from (13.15) The corresponding flow is characterized by the functions

wD

N 4

dw D N ub , du u C n i 2 u C n C i 2

nCbi nCbCi uCni C n C b i ln uCni uCnCi uCnCi

13.33

Constant N is defined by the condition at infinity

dw

iN dw

i0 D v e D D i n b [i1 C ε n] 0 dz zD1 dz uDnCi 8an or

3

v0 ei 2 0 D

N [1 C in C b ][1 C ε C in], 8an

whence ND

8anv0 [1 C n C

b 2 ][1

n , tan 2 D 1Cε

C

ε 2

C

n2 ]

,

3 0 D ˇ C 2, 2

13.34

tan ˇ D n C b.

Let us now determine the form of the wing we are dealing with. For this purpose we need to separate the real and imaginary parts in the second of equations (13.32).


Assuming u real we have x C iy D 4ain

1 , u2 C 1 n2 C 2ε C 2iεu C n C εn

and therefore on the wing contour x D 8an

εu C n1 C ε , u2 C 1 C 2ε n2 2 C 4[εu C n 1 C ε ]2

y D 4an

u2 C 1 C 2ε n2 , u2 C 1 C 2ε n2 2 C 4[εu C n 1 C ε ]2

x 2 C y 2 D 16a2 n2

1 , u2 C 1 C 2ε n2 2 C 4[εu C n 1 C ε ]2

x D , D 2εu C 2n1 C ε , x C y2 y 4an 2 D - D u2 C 1 C 2ε n2 . x C y2 4an

2

13.35

This proves that the contour of the wing represents the inversion of the parabola whose equation is 4ε2 - 1 2ε C n2 D [, 2n1 C ε ]2 . Fig. 13.13 clarifies the matter. Corresponding points of the parabola and the wing’s contour are marked in this figure by the same letters; point O corresponds to the parabola’s infinite point. If ε D 0 then the parabola turns into a straight line parallel to its axis, and the cross-section of the wing becomes a circular arc with the central angle 2˛ chosen such that ˛ n D tan . 2 We, therefore, return to the theory presented in Section 4. Point O of the wing contour is a cuspidal point of the first kind, and a small portion OKA of the contour is the inversion of the semi-infinite arc A0 K0 1 of the parabola. The wing contour lies within O K

D

x, ξ

K′

A′

A

y, η

F B′ C′ C

B

u 0

Fig. 13.13

A

B

D

0

209

210


angle 6 KOC which is made of two rays drawn tangentially to the wing from the origin O. Denoting this angle by 26, we find n 2 C ε2 . tan 6 D 1 C 2ε We also see that tan 6 KOY D

ε2 n2 C ε2 1 C 2ε C n1 C ε

n2 C ε2 1 C 2ε C n1 C ε , 1 C 2ε n2 KOY C 6 COY D 26, tan 6 COY D

6

,

13.36

the first of these angles being very small for small ε. There are two inflection points on the wing contour, one lies near the origin and the other near the point corresponding the apex of the parabola. The values of the auxiliary variable at these points may be determined from the following fourth order equation for u 3εu4 C 8n1 C ε u3 6ε1 C 2ε n2 u2 ε 1 C 2ε n2 2 4εn2 1 C ε 2 4ε3 1 C 2ε n2 D 0, which has two real roots, one is negative and rather large (for small ε), the other is close to zero and positive. If n D 18 and ε D 0.01, then the second root equals approximately 0.236, and the thickness of the wing at this location appears to be 0.004a. Thus the wing proves to be almost ideally thin. The components of the pressure force acting on the wing may be found based on the general rule; using (13.33), (13.34) we have Y C iX D

N n C b v0 ei0 . 2 y

µ β+µ

β−µ

x

Fig. 13.14


Alternatively it may be written as Y C iX D 4av20 sin 2 sin ˇei0 PD

p

D 4av20 sin 2 sin ˇieiˇC2 , X2 C Y2 D

4av20

sin 2 sin ˇ.

      

13.37

It may be easily seen from the second of equations (13.34) that ˇ C 2 is the angle made by the velocity vector at infinity with the negative Oy-axis. As far as 2 is concerned, it is determined by the equation tan 2 D

n . 1Cε

When compared with (13.36), this formula shows that 2 approximately equals 6. Therefore, if we consider the angle made by two rays drawn from the origin tangentially to the wing, then the bisector of this angle will have the direction making angle ˇ with the flow at infinity. This is precisely the angle involved in formula (13.37) for the pressure force. The character of this formula remains the same as in the case of the complex wing considered in Section 5. Turning to the velocity distribution on the wing contour, we find from (13.32) and (13.33) u b 2 u n 2 C 1 C 2ε 2 . v D v0 u C n 2 C 1 ε2 C u 2 As may be easily seen, for small ε the velocity grows when u approaches zero. Therefore, to approximate the maximum value, we set u D 0 which yields vmax D

v0 b . ε

If ε D 0.01 and, as before, v0 D 14 m/s, then using for vmax the speed of sound 302 m/s, we find b D 0.2. Then for n D 1/8 it follows from tan ˇ D n C b D 0.336 that the limiting incidence is close to ˇ D 19° . The angle made by the tangent to the lower surface of the wing with the flow direction at infinity proves to be 12° (19° 2 D 12° ), and the lifting force turns to nought at ˇ D 0. Thus all the flow properties mentioned when analysing the complex wing in Section 5, hold in this case as well. In particular, one can notice that the range of possible flow directions becomes wider with a thickening of the wing, i.e. with increase of ε and decrease of the velocity v0 . The dependence of the resultant pressure force on the incidence is also expected to be the same.

7

Other wing forms

The analysis in the previous section may be easily extended to a rather wide class of wing forms. For this purpose, keeping formula (13.33) unchanged, i.e.

211

212


ub dw DN , du u C n i 2 u C n C i 2 we shall assume that dz u C iε fu D 8ain du u C n i 2 [u n0 C i1 C 2ε0 ]2 D 8ain

Fu , u C n i 2

13.38

where function fu is finite and continuous in the entire upper half plane of the complex variable u. For example, we can assume p

fu D

u C gm C ihm qm , u C km C ilm rm mD1

13.39

! ! qm D rm , and choose arbitrarily real coefficients g, h, k, l, q, r with the where only restriction that h and l were positive. As point u D i n has to be a simple pole of function z, the condition F0 i n D 0

should hold, which defines the coefficients n0 and , 0 via n, ,, g, h, k, l, q, r. The value of N is determined from the condition

dw

Ni n C b i i0 v0 e D D ,

dz 8an 4Fi n zD1

leading to N D 32an

jFi n j 1 C n C b 2

v0 ,

3 0 D arctann C b arg [iFi n ] D ˇ C 2, 2 where, as before, ˇ D arctann C b . The lifting force may now be calculated Y C iX D

Nn C b v0 ei0 D 4av20 9 sin ˇ ei0 , 2

13.40

with 9 D 4njFi n j. Again the phenomenon remains unchanged in character. For each particular problem when the wing shape is given, one has to choose the coefficients in (13.39) to suit appropriately the contour of the wing. An interesting example is a wing with toothed concave surface of the type shown in Fig. 13.15. It may be obtained if the coefficients gm and km are assumed negative and growing with index m, such that km is confined between gm and gm1 ; the coefficients hm and

On the pressure exerted by a plane-parallel flow on obstructing bodies (aeroplane theory) y

0 x

Fig. 13.15

lm should be positive and small; all the exponents qm and rm should also be chosen positive. In particular, if qm D rm D 1/2, then the teeth will have a rectangular form with slightly smoothed corners. Constants n0 and ε0 may be assumed to coincide with n and ε provided that the rest of the coefficients are such that F0 i n D 0, i.e.

p "

" 1 1 D . i1 C hm n C gm i1 C lm n C km mD1 mD1 p

The expression for 9 now takes the form # $ p $ n 1 C hm 2 C n gm 2 4 % 9D 2 2 1 C ε 2 C n2 mD1 1 C lm C n km Based on the above discussion it may easily be seen that we can choose the coefficients at our disposal such that the radical in this formula would significantly exceed unity. Thus, comparing formula (13.40) with (13.37), we see that appropriately shaped teeth on the lower surface of the wing can produce an increase of the lift force.

8

The flow past a plate shaped into a circular arc in the presence of a vortex

In Section 4 the flow past a circular arc was considered, and it was demonstrated that a continuous solution is possible only if the direction of the flow at infinity is parallel to the chord subtending the arc. Now we shall show that in addition to this solution another one is possible. In this second solution the velocity field remains continuous for different flow directions at infinity thanks to the presence of a standing vortex. The strength of the vortex and its position with respect to the plate change depending on the velocity and direction of the flow at infinity. While the presence of the vortex does not lead to an additional pressure force acting on the wing, as the analysis of Section 3 shows, it nevertheless affects the force indirectly by changing the circulation along a contour encircling the wing. As in Section 4 we shall use the conformal mapping of the region occupied by the fluid in the z-plane onto the interior of the circle of unit radius in the plane of the

213

214


auxiliary complex variable u. According to formulae (13.6) and (13.7) to perform this mapping one has to set zDa

hu , uhu 1

dz 1 C u2 2hu , D ah 2 du u hu 1 2

13.41

where h D sin˛/2 and ˛ is a half of the central angle subtending the arc. As far as the function w is concerned, we shall now use instead of formula (13.9) the following expression to define this function & ' eiˇ u ei iˇ w D v0 ahi . 13.42 e u C 2A ln u C B ln u u ei With ˇ D 0, A D h, B D 0 this expression reduces to (13.9). The point u D u0 D ei corresponds to the vortex centre (0 < < 1). The constant ˇ has a very simple geometrical meaning. Indeed, & ' 2A eiˇ 1 2 dw iˇ Ce , 13.43 D v0 ahi CB du u u2 u2 ei C ei 2 C 1 u and using (13.41)

dw

D v0 ei0 D iv0 eiˇ , dz zD1, uD0

whence 0 D /2 ˇ. Consequently, ˇ is the angle made by the direction of the flow at infinity (or, which is the same, by the direction of the plate motion in a still air) with the chord of the plate. Coefficients A and B have to be determined from the condition that dw/du D 0 at the points where dz/du D 0. This gives sin ˇ [1 C 2 2 cos 9 ][1 C 2 2 cos C 9 ] , sin 1 2 ( ) sin ˇ 1 C 2 A D h cos ˇ C h cos , sin 2

BD

13.44

where 9 D /2 ˛/2 . Substituting (13.44) into formula (13.43) we can further express this formula in the form 2N 2 2 iˇ iˇ u 1 C u 2hu u e Ce dw ( ) D v0 ahi , 13.45 du 2 C 1 2 2 i i u u e Ce u where 2N D 1 C 2

sinˇ C sin ˇ 2h . sin sin

13.46


If we set ˇ D 0 and suppose that is different from 0 and , then formula (13.45) turns into (13.10) and the theory of Section 4 is reproduced. Let us now suppose that ˇ is not equal to zero, and try to find and from the condition that the vortex is motionless. As was shown in Section 3 this condition requires that

dw dw

Rés D 0, du dz uDu0 or, what is the same, d du

&

dw u u0 du

dz du

'

D 0 at u D u0 .

Using (13.41) and (13.45) we find 2N u 1 C u2 2huhu 1 u2 eiˇ C eiˇ d D 0 at u D u0 , du uu ei with u0 D ei . Having performed the differentiation, we replace u by ei . This yields N 1 e h h C D 0, C C 2 i 1 2 h ei e C ei 2h 2 eiˇ C eiˇ 2N i

2eiˇ 2

and using the expression for N ei h 1 ei h h C C 2 2 i C 2 i i 2 i 1 h e e C e 2h e C ei 2h

13.47

2 sinˇ C 1 C 1 sin ˇ 2 ei C ei 2h D 0.

From the complex equation (13.47) one can deduce two real equations for and . One of them is obtained by setting the imaginary part of the expression on the left-hand side of (13.47) to zero. The second equation is most conveniently derived by multiplying (13.47) by ei C 2 ei 2h and separating again the imaginary part of the resulting equation. After quite simple manipulations we arrive at the following two equations 1 2 2 h sinˇ C 1 h2 sin ˇ D 1 2 2

1 C 2 2h cos 1 C 2 2h cos 2 42 1 h2 sin2 D1

13.48

1 h2 C h2 1 2 2 , 1 C h2 2 2h cos

which serve to determine the position of the vortex as a function of the flow direction at infinity.

215

216


As far as the lifting force is concerned, its components are calculated using formula (13.4) Y C iX D Cv0 ei0 , where C is the circulation around point u D 0 in the clockwise direction. For the flow at hand C D 2v0 ah Ð 2A. Using the equation cot 2 D

1 2 cot , 2h sin

13.49

to introduce the auxiliary angle 2, we can easily find that the coefficient A may be expressed in the form sinˇ C 2 , ADh sin 2 and for the resultant pressure force we have P D 4v20 ah2

sinˇ C 2 . sin 2

13.50

Thus, if ˇ D 0 while 2 is not zero ( differs from 0 and ), then (13.50) reduces to (13.13). We can further notice that formula (13.50) coincides with (13.31) since 2ah2 D c tan˛/2 is the depth of the arc. The difference, however, is that now angle 2 depends on the direction of the flow at infinity. The relationships established by formulae (13.48), (13.49) and (13.50) are quite complicated, and for this reason the general behaviour of the flow remains unclear. It is, therefore, desirable to perform the calculations at least for a particular geometry. We shall do it for the value of h defined by

h 1

h2

D tan

˛ 1 D , 2 8

1 hD p , 65

˛ D 7° 70 3000 , 2

2˛ D 28° 300 ,

which corresponds to the arc depth s of one sixteenth of the wing chord: s D c Ð tan

˛ 1 D 2c; 2 16

this value is quite close to that of a real aeroplane wing. To start with, let us reformulate equations (13.48) by introducing a new variable t defined by h1 C 2 2 cos 1 h2 t D 13.51 1 2 and setting for short s D 1 2 ,

kD

h 1 h2

.


As a result equations (13.48) take the form p

s2 1s

Ðk

1 C k2

t ks 2 s C kst sin C ˇ D ks D . sin ˇ 1 C kst 1 C t2

13.52

From (13.51) and (13.49) we have 1 [st 2k C ks], cos D p 2 1 C k2 1 s 2s p cot 2 D cot . 2h sin 1 s

13.53

From equations (13.48) we can find that cos is confined between 1 and h. Replacing k in (13.52) by the particular value 1/8 we have s2 1 C 2t2 8t 8s 8 C t3 2t C 128 D 0 which defines s as a function of t. We shall now assign to t different numerical values and calculate s, , ˇ and 2 one after another. It should be noted that due to restrictions upon s and cos we have to choose t > 3.06. When t is close to this value all the quantities change very fast, but then they start to vary more gradually. As the accuracy of calculations was not a major concern in this study, we used for this purpose the logarithmic tables, and the results presented below serve to demonstrate the trends in the flow behaviour:

1 2 3 4 5 6 7 8 9 10 11

t

s

ˇ

2

3.064 3.07 3.1 3.25 3.35 3.5 4 4.5 5 8 1

0.51709 0.51451 0.50191 0.44391 0.40952 0.36373 0.25012 0.16944 0.13027 0.03175 0

177° 260 172° 560 164° 10 146° 90 139° 140 131° 390 116° 340 106° 530 102° 500 90° 140 ˛ D 82° 520 3000 2 2

0° 360 2500 1° 370 2° 490 5° 570 6° 130 5° 580 5° 40 2° 290 1° 410 0° 140 3700 0

160 500 440 1° 390 3° 270 4° 60 4° 480 5° 560 6° 330 6° 420 7° 40 ˛ D 7° 70 3000 2

Corresponding to that the ratio sinˇ C 2 / sin 2 decreases monotonically from 3.2 to 1. We see that the flow of this type is possible only at small positive values of the incidence angle; the maximum value of ˇ is slightly larger than 6° . Within the permitted

217

218

Early Developments of Modern Aerodynamics x

0 y

Fig. 13.16

interval from 0 to ˇmax two solutions exist with two different positions of the vortex and, correspondingly, two different forms of the flow. For one of these solutions the lift force proves to decay (approximately by 20%) when ˇ increases from 0 to 6° . As far as I know this never happens in observations; most probably, the reason is that due to instability of the vortex, this form of the flow cannot be observed in practice, and the corresponding solution has theoretical meaning only. A sketch of the streamline pattern is shown in Fig. 13.16. The vortex is situated near the convex side of the plate. There are also two critical points on this side of the wing where the velocity turns to zero, as may be easily deduced from the presented solution.

9

Calculation of the rotating moment

The question of the magnitude of the rotating moment is an important part of the described theory. Of special interest is the case when the incidence angle reaches the critical value and the lift force becomes zero. Let us denote by xN , yN the coordinates of the centre of the pressure forces acting on the wing. The pressure force components acting on an element of the wing surface can be expressed as v2 dy v2 dx X0 D ds , Y0 D ds . 2 ds 2 ds Here the term with constant pressure has been disregarded as it gives no contribution to the resultant force. Therefore the centre of the pressure forces, which is defined by the equation " Nx C iy Y N C iX D x C iy Y0 C iX0 , may be found as a result of the integration along the contour of the wing dy dx i zv v ds. Nx C iy Y N C iX D 2 ds ds Since on this contour v ds D dw,

v

dx dy i ds ds

D

dw , dz


we can finally write Nx C iy Y N C iX D xN Y yX N C iNx X C yY N D D 2

du z dz

dw du

2

2

z

dw dw dz

13.54

du,

where the contour of integration should be followed in the z-plane in the counterclockwise direction, i.e. from the x-axis to the y-axis. The direction of integration in the u-plane depends on the conformal mapping employed. In all the examples considered in this study the mapping changes the direction of the integration; in the u-plane it should be performed in the clockwise direction. From formula (13.54) we can conclude that the rotating moment is given by the following simple formula M D

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