# Category The Enigma of. the Aerofoil

## Ludwig Prandtl and the Gottingen School

Der alte Gottinger Professor Dirichlet wurde sich wohl gefreut haben, wenn er dieses Resultat hatte sehen konnen; glaubte man doch gerade seine Potential-Theorie durch die einfache Tatsache, das ein Widerstand existiert, ad absurdum gefuhrt zu haben. j. ackeret, Das Rotorschiff und seine physikalischen Grundlagen (1925)1

The old Gottingen professor Dirichlet would have been so happy if he could have seen these results. People just believed that his theory of potential had been reduced to ab­surdity by the simple fact that there was resistance to motion.

The theory of lift may be divided into two parts: (1) the theory of the wing profile, that is, the wing sections of the kind studied by Kutta and Joukowsky, and (2) the theory of the planform of the wing. The planform is the shape of the wing when seen from above. Wings can be given very different planforms. The designer may chose a simple, rectangular shape or give the wing a more aesthetically pleasing curved leading or trailing edge. The wingtips may be rounded or square, and, most important of all, the wing may be made long and narrow (high aspect ratio) or short and stubby (low aspect ratio). It was known experimentally that some features of the aerodynamic performance of a wing depended on the profile, whereas others (such as the slope of the curve relating lift to angle of attack) depended on the planform and, particu­larly, the aspect ratio. Some of the features that depend on the profile were discussed in the last chapter, for example, the angle of attack at zero lift, the distribution of pressure along the chord, and the experimentally determined, but theoretically obscure, point of maximum lift. The minimum drag as well as the pitching moment were also found to depend on the profile. Now the discussion turns to the distribution of the lift along the span of the wing and the properties that a wing possesses in virtue of its finite length and the flow around the wingtips. Bringing order and understanding to these phenomena (and predicting unsuspected effects and relationships involving the aspect ratio of a wing) was the outstanding achievement of Ludwig Prandtl and his co-workers at the University of Gottingen.2 Before looking into the technical details of this achievement, I discuss the intellectual background of the work and its institutional context.3

## Plus Change

Prandtl’s boundary-layer theory provided the material that might give sub­stance and depth to the “scheme” of the wing theory. It suggested that inviscid approximations might be replaced by a more realistic account of the physics of viscous processes. The boundary layer became the focus of a sustained British research effort organized by the Fluid Motion Panel of the Aeronauti­cal Research Committee. The original intention was that Lamb would be the editor of the volumes that would draw the results together, although Fage, who clearly found Lamb’s work very demanding, put in a request that the mathematics should be kept as simple as possible. “Lamb’s Hydrodynamics,” said Fage in the course of a discussion of the proposed monograph, “was more suitable for the professional mathematician and was very difficult.”88 In the event, Lamb did not live to complete this task and it fell to Sidney Gold­stein. Goldstein had been a pupil of Jeffreys’ but he had also gone to Gottin­gen after the war to study with Prandtl.89 Despite continuing resistance, it is clear that the overall strategy that Prandtl had adopted in his Wright Lecture had been an appropriate one. He had engaged with the preoccupations of the British experts with viscous and eddying flow while reminding them of the intellectual resources that Gottingen had to offer.90 Writing to Prandtl, after the Wright Lecture, Major Low said that he had spoken to many mathemati­cians and physicists and they all said that “your paper will give a new direc­tion to aerodynamic research in this country.” Low identified the transition from laminar to turbulent flow as the special point of interest for the British audience.91

This concern with the boundary layer and turbulence became the new research front, and it was congenial territory for the British even though their head-on assault on the Navier-Stokes equations had proven frustrating. If the battle for circulation in the theory of lift was over, the war on turbulence in the boundary layer was about to begin.92 But even here the old worries were not far beneath the surface. On February 6, 1930, members of the Royal Aeronautical Society discussed a report titled “Modern Aerodynamical Re­search in Germany.”93 The report was presented by J. W. Maccoll, who had visited Gottingen and Aachen.94 Maccoll, who had a command of German, was a government scientist and was to hold the post of research officer in the Department of External Ballistics at the Woolwich Arsenal. He described in mathematical detail the original work on the laminar boundary layer and then the more recent work on the transition to turbulence. In the discussion that followed Maccoll’s paper, Bairstow identified what he saw as two fun­damentally different approaches to the current problems in fluid dynamics.

Bairstow declared that he had “been impressed by the extreme complication of the whole subject and the apparently little connection between the Ger­man methods of solution and the equations of motion of a viscous fluid. All would have noticed how often new variables were introduced into the equations to deal with failures of the original hypothesis. It seemed that the Germans were making an engineering attempt to get solutions of practical value and had little hope of solving the equations of motion in a sense that would satisfy Professor Lamb” (697).

Bairstow was describing, albeit in a one-sided way, the difference in ap­proach between a mathematically sophisticated engineer, adopting the meth­ods of technische Mechanik, and that of a mathematical physicist drawing on the finely honed traditions and research strategy of the Cambridge school. Bairstow might not have sat the Tripos, but he still took Prof. Lamb as his reference point.95 The difference in approach to which Bairstow was alluding, between the Cambridge and Gottingen traditions, has been present in one form or another throughout the story I have been telling. It was implicated in the original British dismissal of the circulatory theory, and it was central to the manner in which the theory was finally accepted by the British.96

In an article titled “Twenty-One Years’ Progress in Aerodynamic Science” which Bairstow published in 1930, the same year as the remarks just quoted, he surveyed the work that had been done since the creation of the Advisory Committee for Aeronautics in 1909. Bairstow invoked a revealing compar­ison to describe the discomfort that still surrounded the relation between the theory of viscous and inviscid fluids in aerodynamics. He likened the problem of reconciling the viscous and inviscid approaches to the problems that British physicists were experiencing in reconciling the wave and particle conceptions of light and of the electron. Two fundamentally different models were in use, but it was impossible to see how they could both be true.97 Bair­stow quoted the exasperated response to this situation of one of the country’s leading physicists, a response that mirrored, perhaps, the frustrations of Bair – stow’s own work on the Navier-Stokes equations. “Aerodynamic theory,” said Bairstow,

is now rather like the physical theory of light; Sir William Bragg recently said that physicists use the electron theory on Mondays, Wednesdays and Fridays, and the wave theory on alternate days. Both have uses but reconciliation of the two ideas has not yet been achieved. So it is in aeronautics. In our experi­mental work we assume that viscosity is an essential property of air and the building of a compressed-air tunnel is the latest expression of that belief. The practically useful theory of Prandtl comes from considering air as frictionless or inviscid. (29)

At the end of his survey Bairstow returned to this theme and defined his view of the prospects of aerodynamics in terms of this ambiguous and problematic image. We can be assured, he said, that aerodynamics has “a future compa­rable with that in electron theory” (30).

Despite Glauert’s efforts to renegotiate the conceptual distinction between perfect fluid theory and the theory of viscous fluids, it is clear that the lead­ing British mathematical physicists were in no hurry to abandon their view that the distinction was fundamental. The boundary separating the objects of the two theories was treated as ontologically rigid rather than methodologi­cally flexible. Eventually, though, by the mid – and late 1930s, what Glauert called the “true conception of a perfect fluid” appears to have filtered into British mathematical and experimental practice. It was not acknowledged explicitly, but it was implicit in the use of potential irrotational flow as an engineering ideal. By the 1940s its use for this purpose had become routine, for example, in estimating the role played by the viscous boundary layer.98 By this time the circulation theory of lift, and Prandtl’s wing theory, had already become an established part of British aerodynamics. The earlier insistence on a rigid conceptual boundary between ideal and real fluids nevertheless helps to explain why, when Prandtl’s wing theory was finally accepted by the Brit­ish, there was still a note of reservation. Prandtl’s theory may have been, as Bairstow conceded, “the best and most useful working hypothesis of modern times”—but it was still a working hypothesis.

For many years, one of the standard British textbooks in the field was Milne-Thomson’s Theoretical Aerodynamics." The book ran through four editions between 1947 and 1966 and contained the following, revealing obser­vation on the lifting-line theory. Following an explication of Lamb’s contrast between a scheme and a fundamental theory, Milne-Thomson said, “The student should be warned, however, that the investigation on which we are about to embark is one of discussing the deductions to be made from sche – matization of a very complicated state of affairs and that the ‘laws of Prandtl’ which will be used as a basis are not necessarily laws of nature” (191). Con­trasting the Laws of Prandtl with the Laws of Nature was just a picturesque way of saying what most British experts had felt all along. Prandtl’s work on the aerofoil was an exercise in engineering pragmatism rather than a contri­bution to a realistic and rigorous mathematical physics.

## New Approaches to Ideal Fluid Theory

Something was badly wrong with the picture of air behaving and moving as an ideal fluid. It was mathematically impressive but empirically defective. What exactly was wrong? Was it the assumption of zero viscosity itself that should be dropped or were there perhaps other, unnoticed, assumptions at work in the picture of the flow that might be the cause of the trouble? What about Laplace’s equation and the assumption of irrotational motion?

This question was addressed by a number of late nineteenth-century ex­perts whose investigations greatly deepened the understanding of ideal fluid theory. They began to explore some possibilities that previously had been neglected. But why did they not simply abandon ideal-fluid theory as em­pirically false and turn directly to the analysis of viscous fluids? Attempts were made to do this but with very limited success. The reason was that the mathematics of viscous fluids was so difficult. It was possible to write down the equations of motion of a viscous fluid by taking into account the trac­tion forces along the surface of the fluid element, but it was another matter to solve the equations. The full equations of viscous flow are now called the Navier-Stokes equations, though in Britain they used to be called just the Stokes equations. Of course, they contain a term involving the symbol Ц standing for the coefficient of viscosity. If the value of this coefficient is set at zero to symbolize the absence of viscosity, that is, Ц = 0, the Navier – Stokes equations turn back into the Euler equations that have been described earlier in this chapter. In a later chapter I look more closely at the status of the Navier-Stokes equations and the different responses to their seemingly intractable nature. For the moment it is only necessary to appreciate the problem they posed. No one could see how to solve and apply the equations except in a few simple cases. Because of their intractability, any attempt to avoid the impasse thrown up by the zero-resultant theorem had to be one that stayed with the Euler equations and thus within the confines of ideal- fluid theory.

The crucial insight that permitted the further development of ideal-fluid theory was provided by Helmholtz and Kirchhoff in Germany and Rayleigh in Britain. These men realized that the solutions to the Euler equations that gave the streamlines around an obstacle were not unique. More than one set of streamlines were possible and consistent with the equations. More than one kind of flow could satisfy the equations and meet the given boundary conditions. What is more, some of these flows could generate a resultant force. There were in fact two very different kinds of flow that might have this desired effect and, in principle, allow the zero-resultant outcome to be evaded. Rayleigh contributed to the study of both. Both approaches involved the limited introduction of fluid elements that possessed rotation and vor – ticity. The strict condition of irrotational motion was dropped. On one ap­proach this involved the introduction of just one singular point in the flow that rotated and constituted the center of a vortex. On the other approach a sheet or surface of vorticity was postulated. In both cases the remainder of the flow was still irrotational. These two approaches provided, respectively, the basis for the two different theories of lift that I mentioned in the introduction and called the circulatory or vortex theory of lift and the discontinuity theory of lift. Historically, the first of the two approaches to be developed in detail was the one that led to the discontinuity theory. I now introduce the ideas underlying this approach. The other approach and the other theory of lift are introduced in chapter 4.

## The Status of Stokes’ Equations

All real fluids are viscous, but not all viscous fluids are real. A mathematician may construct a model of a fluid which makes provision for viscosity, but it remains an open question whether any real fluid satisfies the specifications of the model. Lamb was very clear on this matter. He raised it in connection with the derivation of Stokes’ equations. All such derivations must start from assumptions, and these typically involve simplifications. A few years before Stokes’ work, Navier in France had arrived at these same equations and so they are often known as the Navier-Stokes equations.23 Navier, however, worked from assumptions about the supposed forces operating between the particles that made up fluids. Stokes is generally considered to have improved on this account by finding a way to avoid speculating about the ultimate particles of a fluid. He treated a fluid as a continuum and confined himself to considering the tangential stresses and shear forces on the sides of a fluid element. This modification avoided Navier’s assumptions but inevitably introduced others. What laws were obeyed by the stresses and forces? Stokes made the assump­tion that there was a linear relation between the shear force and the rate of shear. Lamb was careful to point this out to the reader.24 The assumption of linearity, he said, was exactly that—an assumption. He hastened to add that the assumption was plausible and the success of the equations, where they had been tested empirically, gave every reason to believe it was correct.

It will be noticed that the hypothesis made above that the stresses. . . are linear functions of the rate of stress. . . is of a purely tentative character, and that although there is considerable a priori probability that it will represent the facts accurately in the case of infinitely small motions, we have so far no assurance that it will hold generally. It was however pointed out by Reynolds that the equations based on this hypothesis have been put to a very severe test in the experiments of Poiseuille and others. . . . Considering the very wide range of values over which these experiments extend, we can hardly hesitate to accept the equations in question as a complete statement of the laws of viscosity. (571)

Assumptions had been made, but it turned out that the assumptions were correct. The Stokes equations were not approximations in competition with other approximations. Evidence, said Lamb, shows that the equations are to be accepted as a “complete statement” of the laws of the real-world phenom­enon of viscosity. In a word, the Stokes equations were true.

Given the immense authority behind this judgment, it can be difficult to realize that it was not necessitated by the facts. Lamb could have drawn a different conclusion. He was making a methodological choice and did not have to choose as he did. Others adopted a different stance toward Stokes’ equations and the experimental evidence that Lamb cited.25 I illustrate this point by reference to the work of the applied mathematician Richard von Mises (fig. 5.4). As well as his broad literary and philosophical interests, von Mises made important contributions to aerodynamic theory by generaliz­ing the mathematical technique for creating aerofoil shapes by conformal

 figure 5.4. Richard von Mises (1883-1953). A leading applied mathematician who worked extensively in fluid dynamics and aerodynamics, von Mises adopted an empiricist or “positivist” stance toward the equations of fluid dynamics and treated both the Euler and the Stokes equations as abstractions.

transformations.26 He also corresponded extensively with Prandtl about fluid dynamics and contributed to boundary-layer theory.27 Von Mises had lec­tured on aerodynamics to military aircrew in Berlin as early as 1913 and had himself learned to fly at Adlershof. Before the war von Mises held a chair at the University of Strassburg, where he published Elemente der technischen Hydromechanik (The elements of technical hydrodynamics).28 On the title page von Mises was styled as a Maschinenbau-Ingenieur, or “mechanical en­gineer.” During the Great War he returned to Vienna, served as a pilot and an instructor, and then worked on the design of a giant aircraft for which he had provided the wing profile.29 Toward the end of the war he published his military lectures in the form of a textbook, Fluglehre.30 The little book was warmly welcomed by Prandtl because it was written by someone who could handle both the scientific and the technical sides of the aeronautics.31

In 1909, in an article on the problems of technical hydromechanics, von Mises had made a proposal that was designed to rationalize the relation be­tween perfect fluid theory and the theory of viscous flow.32 He called it the “hydraulic hypothesis” and claimed that it was implicit in many of the practi­cal applications of hydrodynamics, even if it was not usually made explicit. Rather than emphasizing the fundamental difference between viscous and inviscid theory (for example, by saying that one referred to something real while the other referred to something unreal), the hydraulic hypothesis em­bodied the view that they were intimately connected. Von Mises still used the hypothesis many years later in his advanced textbook on aerodynamics, the Theory of Flight, first published in English in 1945.33

According to von Mises, so-called ideal fluids represent a process of av­eraging out the statistical fluctuations always present within real fluids. The implied relation between the ideal fluid and real fluid may be illustrated by an analogy. An element of an ideal fluid stands to the elements of a real fluid, that is, the molecules, in roughly the way that, say, the average taxpayer stands to the array of real taxpayers. The behavior of an element of perfect fluid mathematically encodes real information about a specified collection of real things, without itself constituting a further item in the collection. The only fluids are real fluids, just as the only taxpayers are real taxpayers. The concept of an ideal fluid is an instrument by which we talk about, reason about, and refer to real fluids. Indirectly, equations that involve ideal fluids have a real reference, just as statistical data about taxpayers have a real reference. The Euler equations capture the mean values of a statistically fluctuating reality.34 In one formulation von Mises put it like this:

the flow around an aerofoil in a wind tunnel is doubtless a turbulent flow of a viscous fluid. But if the small oscillations are disregarded, the remaining steady velocity values agree very well with those computed from the theory of perfect fluids. . . . The hydraulic hypothesis does not contend that the viscosity effects are negligible. On the contrary. . . the viscosity is responsible for the continual fluctuations or for the turbulent character of the motion. It is left undecided whether the instantaneous (fluctuating) velocities of the real fluid follow the Navier-Stokes equations or not. The hydraulic hypothesis states only that the mean velocity values satisfy, to a certain extent, the perfect-fluid equations. (84-85)

This wording comes from Theory of Flight and therefore dates from 1945, but it is entirely consistent with the original formulation of the hypothesis.

What, on this view, is the relation between Euler’s equations and Stokes’ equations? Von Mises’ answer is interesting. He insists that both are idealiza­tions. In neither case do their concepts have objects that are to be simply or directly identified with real fluids. Both have an indirect relationship. “It should be kept in mind that the ‘viscous fluid’ as well as the ‘perfect fluid’ are idealizations. In introducing the viscous fluid the presence of shearing stresses is admitted, and thus a broader hypothesis is used, which can be ex­pected to give a better approximation to reality. However, we are not entitled to call ‘real fluid’ what is still only an idealization” (76-77). The term “real fluid,” said von Mises, should only be used, “when reference is made to ob­served facts” (77). Real fluids are encountered in experiments and practical engineering. They always stand in contrast to the equations of the mathe­matician—a point that holds whether the equations describe a perfect or a viscous fluid. Both are idealizations, approximations, and constructions, and neither can be identified with reality.35 This was a very different position from that adopted by Lamb and his colleagues. Although Lamb acknowledged the idealization that entered into the construction of Stokes’ equations, he con­cluded that experiment had confirmed their truth. Such confirmation lifted the equations out of the realm of conjecture and put the stamp of reality on them. The idea that, formally, Stokes’ equations stood in the same relation­ship to real fluids as the equations of Euler would have blurred the funda­mental distinction that Lamb and his British colleagues wanted to make.

For certain purposes, some idealizations may be better than others. Von Mises acknowledged that, by taking into account the shearing stresses in the fluid, Stokes had offered a “broader hypothesis” and a “better approxima­tion” than that provided by a perfect fluid. This point must be handled with care. It is surely correct but it does not follow that Stokes’ equations will al­ways give a more accurate answer than that given by the Euler equations. It does not follow that a viscous fluid idealization will always outperform an inviscid idealization. Calling one a “better approximation” than the other may create a certain presumption to that effect, but, given that they are both idealizations, this should not be taken for granted. Von Mises’ own discus­sion of Poiseuille’s results provides a salutary reminder.

Poiseuille’s experiments concerned the uniform flow of a viscous fluid down a straight tube of circular cross section. The fluid will travel more quickly along the middle of the tube than it will closer to the perimeter, and it will be stationary on the walls of the tube itself. Of course, the fluid will have an average velocity, and this will depend on the pressure gradient. The velocity vector will always be parallel to the axis, so the flow is “laminar.” In these simple conditions it can be deduced from Stokes’ equations that the velocity will be distributed over the diameter of the tube in the form of a parabola. The shape of the parabola is determined by the result that the maxi­mum velocity, on the axis, turns out to be exactly twice that of the average velocity. Poiseuille established these facts experimentally, and it was Stokes’ ability to deduce them theoretically that Lamb cited as the grounds for the truth of his equations. But the deductions only hold good if the velocity of the flow is below a certain critical speed. Von Mises reported that, for air in a one-inch pipe, the critical speed is a little below 4 feet per second. Above that speed the analysis fails because the flow ceases to be laminar and becomes turbulent.

In turbulent flow the velocity distribution in the cross section of the pipe alters markedly. Instead of the parabolic distribution, a much flatter distri­bution prevails where the maximum is only a few percent higher than the average. Did this directly contradict Stokes’ equations? It remained unclear whether this behavior contradicted them or not. The relation to the equations could not be determined, and no one could predict the pattern of turbulent flow from them. But while the equations of viscous flow were no help, it was evident that the flat distribution looked strikingly similar to that predicted on the assumption of an inviscid fluid. A frictionless fluid would not adhere to the sides of the pipe, so the fluid there would not be retarded relative to that near the center. There would be no parabolic distribution of velocities but a uniform march forward on a straight front. And this is very nearly what happens in turbulent viscous flow. As von Mises put it in the Theory of Flight. “This uniform velocity distribution of the perfect fluid flow agrees much better with observations under turbulent conditions than the veloc­ity distribution of a laminar viscous flow” (83). The perfect fluid provided a better approximation to the complicated case of turbulent flow than did the equations of viscous laminar flow.

Nor did this superiority hold just for the case of a fluid in a pipe. Von Mises argued that it applied to other practically interesting flows such as those through curved channels and those with varying cross sections, and, as we have seen, to the flow in wind tunnels. “If the small fluctuations are disregarded and attention is given only to the average values at each point, there appears a marked resemblance to the irrotational flow pattern of a per­fect fluid. The mean values of the velocity are distributed very much like the instantaneous velocities in a perfect fluid” (84).

The hydraulic hypothesis was a particular expression of a more general view that von Mises adopted toward the state of mechanics in the early de­cades of the twentieth century. His understanding of both the Euler equations and the Stokes equations brought them into line with his views on probability theory and his understanding of the modern scientific picture of the world— “das naturwissenschaftliche Weltbild der Gegenwart.” He was impressed by current developments in quantum theory and understood them to mean that behind the differential equations of classical mechanics there lay a reality governed by statistical rather than causal laws.36

Whatever one makes of the hydraulic hypothesis and the ultimate indeter­minism of physical laws, the essential point that von Mises was making about the Stokes equations still holds good. Even if von Mises’ statistical interpreta­tion of the Euler equations were to be rejected, his claim that both the Euler and Stokes equations were idealizations would not be directly threatened and could be defended on independent grounds. The point was forcefully made by the American mathematician Garrett Birkhoff. In his book Hydrodynam­ics: A Study in Logic, Fact and Similitude,31 Birkhoff lists molecular dissocia­tion and ionization at hypersonic speeds, chemical kinetics, and sound atten­uation as some of the physical effects not covered by the equations. He also noted that “the first supersonic wind-tunnels were plagued by condensation shocks due to water vapor in the air—another ‘hidden variable’ ignored by the metaphysics of Navier and Stokes” (31). Impressive though they are, the Stokes equations are not a complete statement of the laws of viscosity. They should be seen, as von Mises saw them, as idealizations covering a very in­complete range of phenomena in a very partial manner.

I now put these divergent responses to perfect fluids, d’Alembert’s para­dox, and Stokes’ equations into context. I identify two divergent traditions of mathematical work, one British, the other German. The tradition with the strong boundary between ideal and real fluids might be called “Cambridge – style mathematical physics.” The other, with the weaker boundary between the real and the ideal, is the tradition of technical mechanics as it was devel­oped in the German system of technical colleges. I start with a characteriza­tion of the Cambridge approach and, again, take Horace Lamb as my refer­ence point.

## “We Have Nothing to Learn from the Hun”:. Realization Dawns

When I returned to Cambridge in 1919 I aimed to bridge the gap between Lamb and Prandtl.

g. i. taylor, “When Aeronautical Science Was Young" (1966)1

Oscar Wilde declared that if you tell the truth you are bound to be found out sooner or later.2 There is a corresponding view that applies to scientific theories. Given good faith and genuine curiosity, a true theory will eventu­ally prevail over false ones. These sentiments make for good aphorisms but the epistemology is questionable. Even if it were right, there would still be the need to understand the contingencies and complications of the historical path leading to the acceptance of a theory. My aim in the next two chapters is to describe some of the contingencies that bore upon the fortunes of the cir­culatory theory of lift in Britain after the Great War. I shall come back to the philosophical analysis of theory acceptance in the final chapter of the book, when all the relevant facts have been marshaled. I begin the present discus­sion with some observations about the flow of information between German and British experts before, during, and after the Great War.

## “The Phantom of Absolute Cognition”

The continuity between Frank’s ideas, developed in the 1930s, and the more recent work in the sociology of scientific knowledge was noted by the phi­losopher Thomas Uebel in his paper “Logical Empiricism and the Sociology of Knowledge: The Case of Neurath and Frank.”71 Uebel concluded (I think rightly) that Frank had anticipated all the methodological tenets of the Strong Program (147), but he insists that there is an important difference: the advo­cates of the Strong Program are “relativists,” whereas Frank “did not accept the relativism for which the Strong Programme is famous” (149). This state­ment is incorrect. The similarity does not break down at this point. Frank was also a relativist. I first want to establish this fact and then I shall use Frank’s relativism to illuminate some examples of aerodynamic knowledge.

Frank’s relativism was implicit, but clearly present, in his paper on the acceptance of theories, for example, in his assertion that there was no such thing as “perfect” simplicity. He meant that there is no absolute measure of simplicity that could exist in isolation from the circumstances and perspec­tives of the persons constructing and using the theory. If there is no absolute measure, then all measures must be relative, that is, relative to the contingen­cies and interests that structure the situation. Recall also the trade-off be­tween simplicity and predictive power. Frank said this meant there was no such thing as “the truth” because there was no absolute, final, or perfect com­promise. The relativist stance is epitomized by Frank’s comparison between assessing a theory and assessing an airplane. Talk about an “absolute aircraft” would be nonsense. All the virtues of an aircraft are relative to the aims and circumstances of the user. If the process of scientific thinking has an instru­mental character, and theories are technologies of thought, then talk about an absolute theory, or the absolute truth of a theory, is no less nonsensical.

Frank made his relativism explicit in a book called Relativity: A Richer Truth.72 Einstein wrote the introduction, and the book contains a number of examples drawn from Einstein’s work, but the book is not primarily about relativity theory. It is a discussion of the general status of scientific knowl­edge and its relation to broader cultural concerns. Frank’s purpose is much clearer in the title of the German edition, Wahrheit—Relativ oder absolut? (Truth—relative or absolute?),73 which poses the central question of the book. Does science have any place within it for absolutist claims? Frank said no. No theory, no formula, no observation report is final, perfect, beyond revision or fully understood. The world will always be too complicated to permit any knowledge claim to be treated as absolutely definitive. In devel­oping this argument Frank draws out the similarities between relativism in the theory of knowledge and relativism in the theory of ethics. Are there any moral principles that must be understood as having an absolute character? The claim is often made, but Frank argues that if close attention is paid to the actual employment of a moral principle, it always transpires that qualifica­tions and complications enter into their use. “For this so-called doctrine of the ‘relativity of truth’ is nothing more and nothing less than the admission that a complex state of affairs cannot be described in an oversimplified lan­guage. This plain fact cannot be denied by any creed. It cannot be altered or weakened by any plea or admonition on behalf of ‘absolute truth.’ The most ardent advocates of ‘absolute truth’ avail themselves of the doctrine of the ‘relativists’ whenever they have to face a real human issue” (52).

The book on relativism was written during the 1940s after Frank had left Prague. It was a response to a systematic attack on science by theological writ­ers in the United States. They blamed science for the ills of the time, such as the rise of fascism, the threat of communism, the decline in religious belief, and the loss of traditional values. The critics said that science encouraged relativism and relativism was inimical to responsible thinking. Frank con­fronted the attack head on. He did not seek to evade the charge by arguing that scientists were not relativists (and therefore not guilty); indeed, he said that scientists were relativists (and should be proud of the fact). The danger to rational thought and moral conduct came, he said, not from relativism but from absolutism. If we try to defend either science or society by making absolutist claims, we will merely find ourselves confronted by rival creeds making rival, absolute claims. If we take the issue outside the realm of reason, we must not be surprised if it is settled by the forces of unreason (21). Relativ­ism, he argued, is the only effective weapon against totalitarianism and has long been instrumental in the progress of knowledge. It has been made “a scapegoat for the failures in the fight for democratic values” (20).

Frank alluded to the many caustic things that critics said about relativism and then added, “this crusade has remained mostly on the surface of scientific discourse. In the depths, where the real battle for the progress of knowledge has been fought, this battle has proceeded under the very guidance of the doctrine of the ‘relativity of truth.’ The battle has not been influenced by the claim of an ‘absolute truth,’ since the legitimate place of this term in scientific discourse has yet to be found” (20-21). Notice that Frank placed the words “absolute truth” in quotation marks because, as a positivist, he would have been inclined to dismiss the words as meaningless. For him they had no real content and no real place in meaningful discourse. The claims of the absolut­ists were to be seen as similar to the claims of, say, the theologian. But if the best definition of relativism is simply the denial that there are any absolute truths, and if relativism is essentially the negation of absolutism, then relativ­ism is meaningless as well. The negation of a meaningless pseudoproposi­tion is also a meaningless pseudoproposition. Relativism would, likewise, be revealed as an attempt to say what cannot be said. This may explain why Frank also placed the words “relativity of truth” in quotation marks. There is much to recommend this analysis. It might be called the Tractatus view of relativism.74 Where, however, does this analysis leave Frank’s book? Does it not render the book meaningless and pointless? The answer is no. The reason is that absolutism, like theology, has practical consequences, and whatever the status of its propositions, the language is woven into the fabric of life. It provides an idiom in which things are done or not done. Even for the strictest positivist this penumbra of practical action has significance.

What is done, or not done, in the name of absolutism? The answer that Frank gave is clear. Absolutism inhibits the honest examination of the real practices of life and science. It is inimical to clear thinking about the human condition. The meaningful task of the relativist is grounded in this sphere. It is to be expressed by combating obscurantism and fantasy and by replacing them with opinions informed by empirical investigation. That is the “richer truth” referred to in the title of the English-language edition of Frank’s book. This down-to-earth orientation also provides the answer to another prob­lem that may appear to beset Frank’s relativist position. What is scientific knowledge supposed to be relative to? The answer is that it is relative to what­ever causes determine it. There are as many “relativities” as there are causes. That is the point: knowledge is part of the causal nexus, not something that transcends it. Knowledge is not a supernatural phenomenon, as it would have to be if it were to earn the title of “absolute.” Knowledge is a natural phenomenon and must be studied as such by historians, sociologists, and psychologists.

Frank’s relativism, and the relativist thrust of the positivist tradition, seems to have been forgotten.75 A number of prominent philosophers paid a moving tribute to Frank after his death in 1966, but they did not mention his relativism.76 In the course of this forgetting, a strange transformation has taken place. In his Kleines Lehrbuch des Positivismus, von Mises spoke of “the phantom of absolute cognition.”77 That phantom still stalks the intellectual landscape, but in Frank’s day it was scientists who were accused of relativism, whereas today it is scientists, or a vocal minority of scientists, who accuse others of relativism. From being the natural home of relativism, science has been polemically transformed into the abode of antirelativism and hence of absolutism. A significant role in this transformation has been played by phi­losophers of science who are today overwhelmingly, and often aggressively, antirelativist in their stance. The involvement of analytic philosophers should have ensured that the arguments for and against relativism were studied with clarity and precision. This has not happened. The philosophical discussion of relativism is markedly less precise today than when Frank addressed it fifty years ago and provided his simple and cogent formulation of what was at stake.

## Introduction: The Question to Be Answered

’Tis evident, that all the sciences have a relation, greater or less, to human nature; and that however wide any of them may seem to run from it, they still return back by one passage or another.

david hume, A Treatise of Human Nature (1739-40)1

Why do aircraft fly? How do the wings support the weight of the machine and its occupants? Even the most jaded passengers in the overcrowded airliners of the present day may experience some moments of wonder—or doubt—as the machine that is to transport them lifts itself off the runway. Because the action of the air on the wing cannot be seen, it is not easy to form an idea of what is happening. Some physical processes are at work that must generate powerful forces, but the nature of these processes, and the laws they obey, are not open to casual inspection. If the passengers looking out of the window really want an explanation of how a wing works, they must do what any lay person has to do and ask the experts. Unfortunately the answers that the ex­perts will give are likely to be highly technical. It will take patience by both parties if communication is not to break down. But given goodwill on both sides, the experts should be able to find some simplified formulations that will be useful to the nonexperts, and the nonexperts should be able to deepen their grasp of the problem.

In this book I discuss the question of why airplanes fly, but I approach the problem in a slightly unusual way. I describe the history behind the technical answer to the question about the cause of “lift,” that is, the lifting force on the wing. I analyze the path by which the experts, after much disagreement, ar­rived at the account they would now give. I am therefore not simply asserting that airplanes fly for this or that reason; I am asserting that they were under­stood to fly for this or that reason. I am interested in the fact that different and rival understandings were developed by different persons and in different places. I cannot speak as a professional in the field of aerodynamics; nor is my position exactly that of a layperson. I speak as a historian and sociologist of science who is poised between these categories.2

What are the specific questions that I am addressing and to which I hope to offer convincing answers? To identify them I first need to give some back­ground. The practical problem of building machines that can be flown, that is, the problem of “mechanical” or “artificial” flight, was solved in the final years of the nineteenth century and the early years of the twentieth century. In the 1890s Otto Lilienthal in Germany successfully built and flew what we today would call hang gliders. From 1903 to 1905 the Wright brothers in the United States showed that sustained and controlled powered flight was pos­sible and practical. What had long been called the “secret” of flight was now no longer a secret. But not all of the secret was revealed. Some parts of it remained hidden, and indeed, some parts are still hidden today. The practi­cal successes of the pioneer aviators still left unanswered the question of how a wing generated the lift forces that were necessary for flight. The pioneers mostly worked by trial and error. Some had experimented with models and taken measurements of lift and drag (the air resistance opposing the motion), but the measurements were sparse and unreliable.3 No deeper theoretical un­derstanding had prompted or significantly informed the early successes of the pioneers, nor had theory kept pace with the growth of practical under­standing. The action of the air on the wing remained an enigma.

A division of labor quickly established itself. Practical constructors con­tinued with their trial-and-error methods, while scientists and engineers be­gan to study the nature of the airflow and the relation between the flow and the forces that it would generate. For this purpose the scientists and engineers did not just perform experiments and build the requisite pieces of apparatus, such as wind channels. They also exploited the resources of a branch of ap­plied mathematics that was usually called hydrodynamics. The name “hydro­dynamics” makes it sound as if the theory was confined to the flow of water, but in reality it was a mathematical description that, with varying degrees of approximation, was applied to “fluids” in general, including air. Thus was born the new science of aerodynamics. The birth was accompanied by much travail. One problem was that the mathematical theory of fluid flow was im­mensely difficult. The need to work with this theory effectively excluded the participation of all but the most mathematically sophisticated persons, and this did not go down well with the practical constructors. The mathematical analysis also depended for its starting point on a range of assumptions and hypotheses, about both the nature of the air and the more or less invisible pattern of the flow of air over, under, and around the wing. Only when the flow was known and specified could the forces on the wing be calculated. Assumptions had to be made. The unavoidable need to base their investiga­tions on a set of assumptions proved to be deeply divisive. Different groups of experts adopted different assumptions and, for reasons I explain, stuck to them.

The first part of this historical story, the practical achievement of con­trolled flight, has been extensively discussed by historians. Pioneers, such as the Wright brothers, have been well served, and the attention given to them is both proper and understandable.4 The second part of the history, the de­velopment of the science of aerodynamics, is somewhat less developed as a historical theme, though a number of outstanding works have been written and published on the subject in recent years.5 The present book is a contribu­tion to this developing field in the history of science and technology.

In the early years of aviation there were two, rival theories that were in­tended to explain the origin and nature of the lift of a wing. They may be called, respectively, the discontinuity theory and the circulatory (or vortex) theory. The names derive from the particular character of the postulated flow of air around the wing. (I should mention that the circulatory theory is, in effect, the one that is accepted today.) My aim is to give a detailed account of how the advocates of the two theories developed their ideas and how they oriented themselves to, and engaged with, the empirical facts about flight. To do this I found that I also needed to understand how they oriented them­selves to, and engaged with, one another. I show that these two dimensions cannot be kept separate. This is why I have prefaced the work with the quo­tation from the famous Edinburgh historian and sociologist David Hume. The more one studies the technical details of the scientific work, the more evident it becomes that the social dimension of the activity is deeply impli­cated in these details. The more closely one analyses the technical reasoning, the more evident it becomes that the force of reason is a social force. The historical story that I have to tell about the emerging understanding of lift is, therefore, at one and the same time both a scientific and a sociological story. To understand the course taken by the science it is necessary to understand the role played by the social context, and to appreciate the role played by the social context it is necessary to deconstruct the technical and mathematical arguments.

In principle none of this should occasion surprise. Scientists and engi­neers do not operate as independent agents but as members of a group. They cannot achieve their status as scientists and engineers without being educated, and education is the transmission of a body of culture through the exercise of authority. Education is socialization.6 Scientists and engineers see them­selves as contributing to a certain discipline, as being members of certain institutions, as having loyalties to this laboratory or that tradition, as being students of A or rivals of B. Their activities would be impossible unless behav­ior were coordinated and concerted. For this the individuals concerned must be responsive to one another and in constant interaction. Their knowledge is necessarily shared knowledge, though, in its overall effects, the process of sharing can be divisive as well as unifying. The sharing is always what Hume would call a “confined” sharing.

All too frequently, when scientific and technical achievements become objects of commentary, analysis, or celebration, these simple truths are ob­scured. Academic culture is saturated with individualistic prejudices, which encourage us to trivialize the implications of the truth that science is a col­lective enterprise and that knowledge is a collective accomplishment. Phi­losophers of science actively encourage historians to distinguish between, on the one side, “cognitive,” “epistemic,” or “rational” factors and, on the other side, “social” factors. They enjoin the sociologist to “disentangle” scientific reasoning from “social influences” and to distinguish what is truly “internal” to science from what is truly “external.”7 These recommendations are treated as if they were preconditions of mental hygiene and based on self-evident truths. Historians and sociologists of science know better. They know that the problem of cognitive order is the problem of social order.8 These are not two things, even two things that are closely connected; they are one thing described from different points of view. The division of a historical narrative into “the cognitive” and “the social,” or “the rational” and “the social,” is wholly artificial. It is methodologically lazy and epistemologically naive.

I shall now briefly sketch the overall structure of the events I describe in this volume. Of the two theories of lift that I mentioned, one of them, the dis­continuity theory, was mainly developed in Britain. It was based on work by the eminent mathematical physicist Lord Rayleigh. The other, the circulatory theory, was mainly developed in Germany. It is associated primarily with the German engineer Ludwig Prandtl, although it had originally been proposed by the English engineer Frederick Lanchester. It rapidly became clear that the discontinuity theory was badly flawed because it only predicted about half of the observed amount of lift. At this point, shortly before the outbreak of World War I (or what the British call the Great War) in 1914, the British awareness of failure might have reasonably led them to turn their attention to the other theory, the theory of circulation. They did not do this. They knew about the theory but they dismissed it. At Cambridge, G. I. Taylor, for example, treated the discontinuity theory as a mathematical curiosity, but he also found Lanchester’s theory of circulation equally unacceptable. The reasons he gave to support this judgment were important and widely shared. Meanwhile the Germans embraced the idea of circulation and developed it in mathematical detail. The British also knew of this German reaction but still did not take the theory of circulation seriously. It was not until after the war ended in 1918 that the British began to take note. They found that the Germans had developed a mathematically expressed, empirically supported, and practically useful account of lift. Even then the British had serious res­ervations. The negative response had nothing to do with mere anti-German feeling. The British scientific experts were patriots, but, unlike some in the world of aviation, they were not bigots. Why then were they so reluctant to take the theory of circulation seriously? This is the main question addressed in the book.9

There are already candidate answers to this question in the literature, but they are answers of a different kind to the one I offer. The neglect of Lan – chester’s work became something of a scandal in the 1920s and 1930s, so it was natural that explanations and justifications were manufactured to account for it. Sir Richard Glazebrook, the head of the National Physical Laboratory, played an important role in British aviation during these years and was the source of one of the standard excuses, namely, that Lanchester did not pres­ent his ideas with sufficient mathematical clarity. Well into the midcentury, British experts in aerodynamics, who, along with Glazebrook, shared respon­sibility for the neglect of Lanchester’s ideas, were scratching their heads and wondering how they could have allowed themselves to get into this position. Clarity or no clarity, they had turned their backs on the right theory of lift and had become bogged down with the wrong one.

The retrospective accounts and excuses that have been given have been both fragmentary and feeble, though Lanchester’s biographer, P. W. Kings – ford, writing in 1960, still went along with a version of Glazebrook’s excuse.10 Other existing accounts merely tend to embellish the basic excuse by invok­ing the personal idiosyncrasies of the leading actors. The problem is ana­lyzed as a clash of personalities. It is true that some of those involved had strong characters as well as powerful intellects, and some of them could pass as colorful personalities. All this will become apparent in what follows. The psychology of those involved is clearly an integral part of the historical story, but such accounts miss the very thing that I want to emphasize and that I believe is essential for a proper analysis, namely, the interconnection of the sociological and technical dimensions. Only an account that is technically informed, and sensitive to the social processes built into the technical content of the aerodynamic work, will make sense of the history. I want to show that the real reasons for the resistance to the vortex or circulatory theory of lift were deep and interesting, but not really embarrassing at all.

Although I have posed the question of why the British resisted the the­ory of circulation, I do not believe it can be answered in isolation from the question of why the Germans embraced it. Both reactions should be seen as equally problematic. The historical record shows that the same type of causes were at work in both British and German aerodynamics. In both cases the ac­tors drew on the resources of their local culture and elaborated them in ways that were typical of their milieu and were encouraged by the institutions of which they were active members. Of course, the cultures and the institutions were subtly different. My explanation of the German behavior is thus of the same kind as my explanation of the British. The same variables are involved, but the variables have different values. Seen in this way the explanation pos­sesses a methodological characteristic that has been dubbed “symmetry.” Be­cause the point continues to be misunderstood, I should perhaps emphasize the words “same kind.” I am not saying that the very same causes were at work but that the same kinds of cause were in operation. Symmetry, in this sense, is now widely (though not universally) accepted as a methodologi­cal virtue in much historical and sociological work. Conversely, it is widely rejected as an error, or treated as a triviality, by philosophers. I hope that see­ing the symmetry principle in operation will help convey its meaning more effectively than merely trying to capture it in verbal formulas or justify it by abstract argument.

The overall plan of the book is as follows. In chapter 1 I start my account of the early British work in aerodynamics with the foundation of the con­troversial Advisory Committee for Aeronautics in 1909. The committee was presided over by Rayleigh. The frontispiece, taken from the Daily Graphic of May 13, 1909, shows some of the leading members of the committee striding purposefully into the War Office for their first meeting, and then emerging afterward looking somewhat more relaxed. The minutes of that important meeting are in the Public Record Office and reveal what they talked about in the interval between those two pictures.11 It is a matter of central concern throughout this book. Chapter 2 lays the foundation for understanding the two competing theories of lift by sketching the basic ideas of hydrodynam­ics and the idealized, mathematical apparatus that was used to describe the flow of air. A nontechnical summary is provided at the end of the chapter. In chapter 3, I introduce the discontinuity theory of lift and describe the British research program on lift and the frustrations that were encountered. Chap­ter 4 is devoted to the circulatory or vortex theory and describes the hostile reception accorded to Lanchester among British experts. I pay particular at­tention to the reasons that were advanced to justify the rejection. In chapter 5, I identify and contrast two different intellectual traditions that were brought to bear on the theory of lift. One of them was grounded in the mathematical physics cultivated in Britain and preeminently represented by the graduates of the Cambridge Mathematical Tripos. The other tradition, called technische Mechanik, or “technical mechanics,” was developed in the German technical colleges and was integral to Prandtl’s work on wing theory. Chapters 6 and 7 provide an account of the German development and extension of the circu­lation theory as worked out in Munich, Gottingen, Berlin, and Aachen. In chapters 8 and 9 there is a description of the British postwar response, which took the form of a period of intense experimentation; it also gave rise to some remarkable and revealing theoretical confrontations. What, exactly, did the experiments prove? The British did not find it easy to agree on the answer.

The divergence between British and German approaches was effectively ended in 1926 with the publication, by Cambridge University Press, of a text­book that became a classic statement of the circulation theory. The book was Hermann Glauert’s The Elements of Aerofoil and Airscrew Theory.12 Glauert, an Englishman of German extraction, was a brilliant Cambridge mathemati­cian who, in the 1920s, broke ranks and became a determined advocate of the circulation theory. As the title of Glauert’s book indicates, he did not just work on the theory of the aircraft wing, but he also addressed the theory of the propeller. This is a natural generalization. The cross section of a propel­ler has the form of an aerofoil, and a propeller can be thought of as a rapidly rotating wing. The “lift” of this “wing” becomes the thrust of the propeller, which overcomes the air resistance, or “drag,” as the aircraft moves through the air. Glauert’s book also dealt with the theory of the flow of air in the wind channel itself, that is, the device used to test both wings and propellers. This aspect of the overall theory was needed to ensure that aerodynamic experi­ments and tests were correctly interpreted. As always in science, experiments are made to test theories, but theories are needed to understand the experi – ments.13 The discussions of propellers and wind channels in Glauert’s book are important and deserve further historical study, but, on grounds of prac­ticality, I set aside both the aerodynamics of the propeller and the methodol­ogy of wind-channel tests in order to concentrate exclusively on the story of the wing itself.14

In the final chapter, chapter 10, I survey the course of the argument and consider objections to my analysis, particularly those that are bound to arise from its sociological character. I use the case study to challenge some of the negative and inaccurate stereotypes that still surround the sociology of scien­tific and technological knowledge. I also ask what lessons can be drawn from this episode in the history of aerodynamics. Does it carry a pessimistic mes­sage about British academic traditions and elitism? What does it tell us about the difference between Gottingen and Cambridge or between engineers and physicists? Finally, I ask what light the history of aerodynamics casts on the fraught arguments between historians, philosophers, and sociologists of sci­ence concerning relativism.15 Does the success of aviation show that relativ­ism must be false? I believe that, by drawing on this case study, some clear answers can be given to these questions, and they are the opposite of what may be expected.

During the writing of this book I had the great advantage of being able to make use of Andrew Warwick’s Masters of Theory: Cambridge and the Rise of Mathematical Physics.16 Although historians of British science had previously accorded significance to the tradition of intense mathematical training that was characteristic of late Victorian and Edwardian Cambridge, Warwick took this argument to a new level. By adopting a fresh standpoint he compellingly demonstrated the constitutive and positive role played by this pedagogic tra­dition in electromagnetic theory and the fundamental physics of the ether in the early 1900s.17

For me, one of the intriguing things about Warwick’s book is that the ac­tors in his story are, in a number of cases, also the actors in my story. What is more, his account of the resistance that some Cambridge mathematicians displayed to Einstein’s work runs in parallel with my story of the resistance to Prandtl’s work. Like Warwick I found that their mathematical training could exert a significant hold over the minds of Cambridge experts as they formu­lated their research problems. In many ways the study that I present here can be seen as corroborating the picture developed in Warwick’s book. Of course, shifting the area of investigation from the history of electromagnetism to the history of fluid mechanics throws up differences between the two studies, and not surprisingly there is some divergence in our conclusions. Whereas Warwick’s attention is mainly (though not exclusively) devoted to the British scene, my aim, from the outset, is that of comparing the British and German approaches to aerodynamics. Furthermore, on the British side, I follow the actors in my story as they move out of the cloisters of their Cambridge col­leges into a wider world of politics, economics, aviation technology, and war. If Warwick studied Cambridge mathematicians as masters of theory, I ask how they acquitted themselves as servants of practice.

## Four Practical Men

In 1907 Herbert Chatley, lecturer in applied mechanics at Portsmouth Tech­nical Institute, published The Problem of Flight: A Text-Book of Aerial Engi­neering. The book went through two further editions, in 1910 and 1921. In the preface Chatley explained that, in terms of mathematics, he followed the practice, “well established in engineering,” of omitting factors that appear unimportant: “The formulae are therefore ‘engineering formulae’ in the strict sense of the word, i. e. they are not the result of a deep mathematical analysis which it is, in the majority of cases, almost impossible to apply.”67

Chatley’s account of lift was eclectic. It involved elements of perfect fluid theory and discontinuity theory along with the Newtonian analysis and its problematic sin2 formula. To overcome the difficulties he added various em­pirical corrections. These cut across the deductive links between the formulas in a manner that might have been calculated to offend the sensibilities of a wrangler. He began by mentioning the continuous flow of a perfect fluid over an inclined plane. This, said Chatley, is described in Lamb’s Hydrody­namics, and its reality has been demonstrated by photographs taken by Prof. Hele-Shaw. (In fact Hele-Shaw’s photographs show the behavior of a viscous fluid in slow motion between two glass plates that are very close together. Stokes was able to show that, under these conditions, “creeping motion,” as it is called, provides an accurate simulation of the flow of a perfect fluid. The forces at work and the boundary conditions are different, but the pho­tographs show what a perfect fluid flow would look like.)68 Chatley went on to assert that at greater speeds this flow breaks down and is replaced by one showing surfaces of discontinuity enclosing pockets of turbulence on the rear of the plate. These eddies reduce the pressure. So far, Chatley’s qualitative pic­ture is approximately that of Kirchhoff-Rayleigh flow, combined with some of Kelvin’s ideas about the turbulence in the dead-water region.

Chatley then introduced some mathematics and deduced Newton’s sin2 formula in the manner just described. What Chatley meant by an engineer­ing formula became clear when he asserted (without giving evidence) that the effect of the eddying on the rear of the plate is to augment the pressure by half as much again. He therefore repeated the above formula, but now multiplied by 3/2, and called it Nmax.. Chatley claimed that this formula agrees “very fairly” with some experimental results of Coulomb, although it was conceded that, for small angles, experimenters disagree greatly. In what is presumably a reference to Rayleigh’s paper, he went on: “The latest results are almost unanimous in making the variations of thrust as sin0 and not as sin2 0. All these following expressions are thus divided by sin 0” (29). Thus the sin2 0 term in the Newtonian formula was simply altered to sin0. In the 1921 edition this abrupt step was justified by saying that the original formula is approximately correct for large angles of incidence (from 60° to 90°), but for small angles, “owing to the continuity of the air, sin0 must be substituted for sin2 0” (31). In the 1910 and 1921 editions the reader was told that the end result is “practically correct” for plane surfaces and, with “slight correction to the coefficients,” also applies to curved surfaces.

Algernon Berriman was the chief engineer at the Daimler works in Cov­entry and the technical editor of Flight. In 1911 both Flight and Aeronautics published accounts of his lectures titled “The Mathematics of the Cambered Plane,” and in 1913 he published Aviation: An Introduction to the Elements of Flight.69 This book was based on lectures he had given at the Northampton Polytechnic Institute. Berriman also started from the Newtonian idea of ac­tion and reaction. Lift came from the reaction on the wing of the mass of air that was, by some means, forced downward by the wing. Thus, “the wing in flight continually accelerates a mass of air downwards, and must derive a lift therefrom.”70 The basic formula is Force = Mass X Acceleration, but how is this formula to be applied? What is the mass of air that is involved? The original Newtonian picture must have underestimated this mass, hence the underestimation of the lift that can be generated.

Berriman assumed that the wing sweeps out, and pushes down, a greater area than is suggested by the simple geometry of an inclined plane. The wing exerts an influence on “all molecules within an indefinite proximity to the plane; in other words a stratum of air of indefinite depth.”71 Instead of stating that a wing of area A engages with a quantity of air determined by the frontal projection A sin0, the assumption was made that it sweeps out a larger area, AS, whose sweep factor S is typically much larger than sin 0. Berriman said that “practical considerations” gave reason to believe that “the effective sweep of a cambered plane may be defined in terms of the chord of the plane” (5). In other words, Berriman put S = 1. This equation was based on the experience of the pioneers and experimenters, such as Langley, who found by trial and error that they got the best results with a biplane when they positioned one wing about one chord length above the other.72 The reasoning was hardly rig­orous, but the assumption allowed Berriman to avoid the troublesome sine – squared term.

Although the concept of “sweep” was popular with the practical men, it had few practical advantages. All that could be done was to determine the lift empirically and then deduce that the quantity called sweep must have such and such a numerical value. No one could go in the other direction. There was no way to predict the lift from the sweep. One might argue, inductively, that similar wings will have similar sweeps, but one could also say that similar wings have similar lifts, so in practice nothing is gained by introducing the concept. G. H. Bryan, after expressing irritation with Berriman’s casual way with trigonometric formulas,73 identified the source of the difficulty:

there is no such thing as “sweep” except in Newton’s ideal medium of non­interfering particles satisfying the sine squared law. In a fluid medium the disturbance produced by a moving solid theoretically extends to an infinite distance, gradually decreasing as we go further off. Mr. Berriman’s “sweep” is, physically speaking, an impossibility. If, however, “sweep” is defined as the depth of a hypothetical column of air, the change of momentum in which would represent the pressure on the plane, then the introduction of this new quantity is only a useless and unnecessary complication. Instead of facilitating the determination of the unknown data of the problem, it merely replaces one variable which is physically intelligible and capable of experimental determi­nation by another variable satisfying neither of these conditions. (265)

The practical men never found a way to use the idea of sweep so that they could sustain what Kuhn called a “puzzle-solving tradition.”74

The work reported in Albert Thurston’s Elementary Aeronautics of 1911 was based on the hope of identifying the significant properties of the flow, such as the sweep of a wing, in an empirical manner.75 Thurston had worked for Sir Hiram Maxim and then became a lecturer in aeronautics at the East London Technical College. He took numerous photographs of airflows made visible by jets of smoke as they streamed past objects of various shapes. The objects ranged from rectangular blocks to aerofoil shapes, or “aero-curves” as Thurston called them. He concluded that the important qualitative factor in the flow over a wing was that the entry at the leading edge was smooth and avoided the “shock” detectable in the case of a simple, flat plate. The avoid­ance of shock was possible because of the rounded and slightly dipped front edge characteristic of a wing, a shape whose advantages had been discovered empirically by Horatio Phillips and Otto Lilienthal.76 The essential thing, ac­cording to Thurston, was to maintain a smooth “streamlined” flow. The at­tempt to impose sudden changes in the velocity of the air merely produces surfaces of discontinuity (20). The photographs showed that with a good, winglike shape at small angles of incidence, “the air divides at the front edge and hugs both sides as it passes along; its resistance to change of motion caus­ing a compression on the lower side of the plane and a rarefaction or suction on the upper side. As the inclination is increased a critical angle appears to be reached, after which the stream line ceases to follow the upper side and forms a surface of discontinuity with corresponding eddies” (21-24).

As with the work of Eden, Bairstow, and Melvill Jones at the NPL, such photographs revealed that the model of discontinuous flow was not going to provide a basis for understanding lift. The photographs did, though, support ideas about the extended sweep of a wing. Thurston avoided using the word “sweep” but referred to the “field” of a wing and, on the basis of his photo­graphs, asserted: “The air affected by an aeroplane [= wing], that is the field of an aeroplane, is greater than the air lying in its path. Thus. . . it will be seen that air, which is considerably above the front edge of the plane, is within the range of the plane, and is deflected downwards” (26-27).

Even with photographs of smoke traces in the flow, it proved impossible to identify the sweep in a quantitative way, and the underlying causes of the qualitative effects visible in the photographs remained obscure. Thurston, however, was convinced that the secret of good design, both for wings and other components, was attention to streamlining, that is, ensuring that the lines of flow in the immediate neighborhood of the body coincide with the surface of the body. Like Chatley, Thurston drew on the work of Hele-Shaw to show how streamline flow works and what it looks like.77

Frederick Handley Page also appealed to Hele-Shaw’s photographs. Hand­ley Page was one of the celebrated pioneers in the British aviation industry. Despite its early inability to deliver BE2s, his firm was later to achieve fame for its manufacture of large bomber and passenger aircraft. In April 1911 he gave a lecture at the Aeronautical Society titled “The Pressure on Plane and Curved Surfaces Moving through the Air.”78 He began by discussing the re­sult that had caused so much difficulty for the discontinuity theory, namely, that the formation of “dead” air behind the plate happens when it has passed the critical angle. Practical aeronautics by contrast, said Handley Page, deals with small angles of incidence where the flow hugs the back of the plate or aerofoil. There is then a maximum of lift to drift and a minimum of eddy disturbance. Commenting on a Hele-Shaw photograph of the flow of a per­fect fluid past an inclined plane, Handley Page said, “The air on meeting the plane divides into two streams. . . the streams meeting again at the back of the plane. At high velocities the eddies and turbulence at the rear of the plane completely obscure this, but up to the critical angle at which the ‘live’ air stream leaves the plane back, the effect is still the same” (48).

Handley Page, like Chatley and Thurston, took ideal fluid theory to pro­vide an accurate picture of real flow round a wing, even when there are no surfaces of discontinuity or other complications such as vortices in the flow. Hele-Shaw’s photographs, however, depicted d’Alembert’s “paradox” in ac­tion, not a wing delivering lift. The practical men were thus walking into the trap that Greenhill had identified in his lectures at Imperial. They were proposing a picture of the flow which the mathematician would immediately recognize as one that gave neither lift nor drag.

No one in the audience at the Aeronautical Society mentioned this prob­lem in the subsequent discussion. Even if the point had been raised it would have had little impact on Handley Page’s eclectic argument because, immedi­ately after this appeal to hydrodynamic theory, the perspective was changed. He adopted the neo-Newtonian approach but suggested refining the idea of sweep by dividing it into two parts. This modification revealed his real inter­est in the Hele-Shaw pictures. The portrayal of the flow at the leading edge suggested to Handley Page that two different processes were at work. There was the sweep associated with the flow upward from the stagnation point to the leading edge, and the sweep associated with the downward flow toward the rear edge. This complication enabled him to refine the mathematics of the sweep picture, but it did not get round Bryan’s objections: it merely doubled the number of unknowns. Handley Page still had to infer the total sweep from the observed lift and had no way to apportion the contributions of the two components of the sweep that he postulated.

Handley Page’s lecture was generally well received, though no mathemati­cians contributed to the discussion—if, indeed, any were present. Cooper, who had been so scathing about Greenhill, congratulated Handley Page on getting a formula that applied to experimental results: “it is not everybody,” he added, “who does that.” Cooper was either being polite or had failed to see how little had been achieved. He went on to say that he thought Handley Page’s analysis applied more to the flat plate, where the leading edge caused a “shock” in the flow, than it did to an aerofoil with its rounded, dipping front edge. Here, claimed Cooper, the stagnation point would be on the very front, not below the leading edge on the underside of the wing, as it was in Hele – Shaw’s picture of the plate. In what may have been meant, at least in part, as a response to this point, Handley Page said, “It seems to me that the entering front edge is only a kind of transformer. . . a curved plane is more efficient than a flat one because you have a more efficient transformer” (63). Unfortu­nately, no attempt was made to explain the metaphor of the “transformer.”

A few years after this exchange Handley Page introduced the famous, and commercially lucrative, Handley Page wing. The standard wing was modified by introducing a slot along the leading edge which changed the flow at the leading edge by directing air from underneath the front of the wing onto its upper surface. It was, in effect, a small extra aerofoil that ran along the leading edge of the main wing. The resulting change in the flow significantly increased the lift and delayed the stall.79 Could this innovation have been a result of the earlier conception of the leading edge as having a capacity to “transform” the flow? When Handley Page described his invention in the Aeronautical Journal, he made no mention of the metaphor of the transformer. Indeed, he never gave a clear account of the thought processes behind his invention, so the question must remain unanswered. (He had originally tried making slots that ran from the leading edge to the trailing edge, that is, along the chord of the wing rather than along the span. This suggests that the process of inven­tion was trial and error, rather than theory-led.) The value of the leading – edge slot is an example of that intriguing phenomenon “simultaneous dis­covery” and, almost predictably, gave rise to a priority dispute.80 The slot was developed independently in Germany by G. Lachmann, and for many years Thurston also argued his claim to be recognized as the inventor.81

## Section iv. the lift on the curved surface

Having arrived at the value of the circulation, Kutta immediately multiplied the circulation by the density of the air and the speed of flight to give the lift. He did not state the relevant formula, L = p V Г (where Г = circulation), but he used it implicitly. Here, then, was the lift on a Lilienthal-type wing specified in terms of known quantities: p, the density of the air; V, the speed of flight; r, the radius of the circular arc of the wing; b, the length of the wing; a, which was half the angle subtended by the arc; and p, the angle of incidence. The formula was

а (а і

Lift = 4npV2rbsin—sinI ~+P •

Kutta did not simply take the general lift formula p УГ for granted. He an­nounced that he was going to offer a general proof based on energy consid­erations, which he proceeded to do. The proof not only gave the magnitude of the resultant aerodynamic force as the product of density, velocity, and circulation, but it also carried the implication that the force must be at right angles to the direction assumed by the free stream at large distances from the wing.31 In other words: there was no drag. Given that Kutta was treating the air as an ideal fluid in irrotational motion, this result was a necessary conse­quence of his premises.

Kutta now had to confront a logical problem. If the fluid is perfect it will slide effortlessly over any material surface. This means that it can only exert a force normally to the surface. Consider a flat plate in a steady flow of ideal fluid and add a circulation around the plate. Suppose that the flow at a dis­tance from the plate is horizontal and that the plate has an angle of attack p to this flow. If the forces on the plate are normal to the plate, then won’t the resultant R be normal to it? It will be tilted back at an angle p to the verti­cal (see fig. 6.3). The resultant R will then have a drag component of R sinp. This contradicts the result of the general lift theorem, which Kutta had just proved, where the resultant is vertical, that is, normal to the flow but not

 ate a drag and contradict the Kutta-Joukowsky law of lift, according to which the resultant aerodynamic force must be normal to the flow, not the plate.

normal to the plate, so that the drag component is 0. Kutta was primarily considering an arc, not a flat plate, but the same result holds even though the geometry is more complicated. Much of the rest of his paper was spent exploring this apparent paradox.

Kutta said there was no contradiction because the force resulting from the normal pressures was not the only force at work. There must also be another force that operates on the very tip of the plate, hence his remark in the intro­duction when he said that the lift had two components. Kutta thus identified a suction force that was tangential to the surface at the leading edge. When this force is combined with the normal pressure forces, the resultant is verti­cal. The forward component of the suction counterbalances the backward component of the pressure forces to produce the zero-drag outcome. Again, the situation can be seen more simply with a flat plate. The tangential suction and the normal pressure forces on the plate are shown in figure 6.4. Intro­ducing the leading-edge suction restores consistency with the results of the kinetic energy proof that Kutta had provided for the law of lift.32

Kutta did not treat the leading-edge suction as a mere device to avoid a problem. He proceeded to investigate the flow field near the leading edge by introducing various approximations and assumptions about the shape of the streamlines. An idealized fluid flowing around an idealized, sharp edge would have an infinite speed. This would produce an infinitely large suction force concentrated on an infinitely small area, which suggests that the math­ematics would assume the indeterminate form ^/0. By reasoning that the approximate shape of the streamline would be that of a parabola, Kutta used the results he had already established to argue that the actual force would

converge to a determinate and finite value. He deduced that this value was exactly that which was required to turn the backward-leaning pressure resul­tant into a vertical lift and to give it the magnitude predicted by the general lift theorem.

## Glauert versus Taylor

There was clearly a desire by the members of the Aeronautical Research Committee to put the theory of circulation and Prandtl’s analysis of the finite wing to the test, but disagreement emerged about how to proceed. This gave rise to a sequence of technical reports in which Taylor and Glauert crossed swords. Part of the problem concerned experimental technique. A further difficulty was that Glauert was sensitive to the fundamental distinction be­tween the ideas underlying the two-dimensional picture of Kutta flow (that is, flow that is smooth at the trailing edge) and Prandtl’s three-dimensional picture of the wing as a lifting line with trailing vortices. Glauert wanted these ideas kept distinct, while other participants in the discussion ran these two ideas together and counted them as forming one single theory whose basic assumption was the irrotational character of the flow.

To explain what was at issue it is necessary to go back to December 1921 and the mathematical report submitted by Muriel Barker.4 She had suggested that the theoretical streamlines she had plotted for the flow over a Joukowsky aerofoil with circulation could be the basis for an experimental test: “it would be most instructive,” she had written, “if these same quantities could be ob­tained practically” (3). Miss Barker’s report and the question of what to do next were discussed by the Aerodynamics Sub-Committee and by the full Research Committee during February and March 1922.5 Should they follow her suggestion and place a model of a Joukowsky aerofoil in a wind channel or should they use a more practical aerofoil, for example, the RAF 15? If they used a real section then should they ask Miss Barker to generate the theoreti­cal streamlines by tedious computation or could a quicker method be found? Were mechanical or electrical methods of generating the theoretical stream­lines of comparable accuracy to those produced by the laborious calculations that would be needed? Lamb was in favor of using the Joukowsky profile and direct calculation. Southwell wanted to use a more realistic profile and a me­chanical method. He mentioned that Taylor had developed a piece of appara­tus that enabled him to use a soap film to model the potential surfaces of ideal fluid flow. Bairstow added that he and Sutton Pippard had devised graphical methods for solving Laplace’s equation.6 Then there was the possibility of using the techniques developed by Hele-Shaw derived from photographs of creeping flow. It was decided that Southwell and Taylor would report back on different analogue methods of producing theoretical streamlines.

Southwell started with his report T. 1696.7 He supported Muriel Barker’s suggestion that comparisons be made of theoretical and empirical stream­lines for an infinite wing, that is, where the model wing would reach right across the tunnel to exclude the effect of flow around the tips. In this way, said Southwell, “a direct check can be imposed upon one of the fundamental assumptions of the Prandtl theory” (2). Southwell then described the method developed by Taylor for simulating the streamlines and the bench-top ap­paratus that had been built.8 A soap film was stretched between the walls of a box while precise measurements were made of the position of the film. The film connected the outline of a small wing profile to other boundaries within the confines of the box. (These boundaries represented the walls of the wind tunnel.) Southwell explained how this technique could take into account the circulation as well as automatically correcting for the effect in the flow of the tunnel walls. “Using orthodox mathematical methods,” said Southwell, “it would appear that the problem thus presented is one of extreme difficulty”

(2) . Taylor, however, followed this up with a brief note, designated T. 1696a, in which he said that he had actually applied the soap-film method to a model aerofoil but had not taken the matter further.9 The small size of the apparatus prevented the measurements being made with the required accuracy. Tay­lor therefore backed the use of an electrical method, and eventually such a method was developed by E. F. Relf and formed the basis of the experimental comparisons that were later published.10

At this point Glauert intervened. In May 1922 he submitted his “Notes on the Flow Pattern round an Aerofoil” (T. 1696b).11 First, he took issue with Southwell’s claim that it would be difficult to allow for the influence of the channel walls by use of analytical methods. Glauert said that the effects could be represented in a simple way using standard mathematical techniques, the so-called method of images. He then went on to make some comments about the proposed experimental comparison involving an infinite wing and two­dimensional flow. It was important “to have a clear understanding of its bear­ing on the general question of aerofoil theory” (2). The implication was that some of the thinking behind the proposal lacked the requisite clarity. Not every test of the two-dimensional work was automatically a test of the three­dimensional claims, for example, the hypothesis that the flow over a wing is smooth at the trailing edge is not a necessary presupposition of Prandtl’s work. Prandtl used the idea that lift is proportional to circulation and that the circulation around a wing can be replaced by the circulation around a line vortex, that is, that the chord is negligible. But, said Glauert, no assump­tion is made “as to the relationship between the form and attitude of the aerofoil and the circulation round it, the analysis always being used only to estimate the behaviour of one aerofoil system from the known behaviour of another system of the same aerofoil section” (2). Taken in its own terms, he went on, the Prandtl theory has been applied “with considerable success” to three cases: (1) the effect of changes of aspect ratio, (2) the estimation of the behavior of multiplane structures on the basis of monoplane data, and (3) the description of flow patterns such as downwash. The comparison of predicted and observed data shows that the “agreement is reasonable.” This, Glauert insisted, constitutes “a satisfactory check of the fundamental equation” (3).

Glauert acknowledged that the hypothesis that the rear stagnation point is on the trailing edge overestimates the circulation and therefore the lift. It does so because of departures from the idealized condition of irrotational flow. The real flow detaches itself from the top surface of a wing before reach­ing the trailing edge and forms a “narrow, eddying wake behind the aerofoil.” Glauert had discussed this in his earlier report, “Aerofoil Theory,” but the committee seemed to be using the well-known facts about the existence of a turbulent wake as an objection to Prandtl’s work. If the wake really was to be a focus of interest, it would be necessary to make assumptions about the distribution of vorticity associated with “the contour of the aerofoil and in­side the wake region.” Prandtl’s aim was to give a first-order approximation for the flow at a distance from the aerofoil, and at points outside the wake. The vorticity of the aerofoil can then be concentrated at a point or, in the three-dimensional case, in a line, just as Prandtl assumed. It is legitimate un­der these circumstances to “ignore completely the series of alternative small vortices in the wake” (4). Glauert concluded by saying that the proposed ex­periment on an infinite wing would, indeed, illuminate the relation between aerofoil sections and the circulation round them, “but will not have any bear­ing on Prandtl’s aerofoil theory” (4).

Taylor did not agree. He produced a written reply, designated T. 1696c, in which he challenged both Glauert’s response to Southwell about mathemati­cal techniques and Glauert’s claim that the experiment would be irrelevant to Prandtl’s theory.12 On the latter point, Taylor declared that all the reasons Glauert “brings up to support his view were well known to most of the Com­mittee which discussed the proposed experiments and some of them were actually brought up in the discussion. It is curious, therefore, that Mr. Glau – ert should come to a view which is different from that of the members who proposed the experiments” (1).

Taylor said that the experiment on the infinite wing would constitute a test of Prandtl’s theory because the theory was based on the assumption

 figure 9.1. G. I. Taylor (1886-1975). Taylor, a Trinity mathematician, had dismissed Lanchester’s ap­proach in his Adams Prize essay of 1914. In the postwar years Taylor acted as an astute and creative critic of the new theories in aerodynamics and the experimental evidence advanced in their favor. (By permission of the Royal Society of London)

that the flow at a distance from the wing was irrotational. Glauert’s posi­tion, it seemed to Taylor, was that this assumption can be made a priori, but it cannot. It is an empirical matter, and the proposed experiment was designed to test it. Second, Glauert had said that the experimental evidence gathered so far had provided a satisfactory check on the fundamental equa­tions of the theory. Taylor replied that if “satisfactory” meant “sufficient” he could not agree. The fundamental equation L = p V Г, relating lift and circu­lation, might hold true for some body of data, and some experimental ar­rangement, but not for the reason that Prandtl had given, that is, not because the flow was irrotational. In fact, said Taylor, “there are an infinite num­ber of kinematically possible distributions of velocity for which this is the case, but only certain of them will correspond with irrotational motions”

(3) . Finally, Taylor turned to Prandtl’s assumption that the chord of the wing could be neglected. Again, insisted Taylor, this could not be assumed a pri­ori. “The assumption can only be justified by experiment or by calculation of the type indicated by Miss Barker or by the purely empirical method of comparing the results of Prandtl’s calculations with observed lifts and drags”

(4) . For these reasons, said Taylor, “I do not agree with the conclusions reached by Mr. Glauert.”