Category The Enigma of. the Aerofoil

Further objections to the circulation theory came from G. I. Taylor, one of Cambridge’s most brilliant young applied mathematicians. I have already mentioned his Adams Prize essay of 1914.60 In that work Taylor did not con­fine himself to rejecting discontinuity theory; he also rejected the circulatory account of lift. Critical of the unreality of the textbook hydrodynamics that Bryan so admired, he argued that “the important thing in the earliest stages of a new theory in applied mathematics is to establish a firm basis in physics” (preface, 5). After describing the central idea of Rayleigh-Kirchhoff flow and pointing out its empirical shortcomings, Taylor turned briefly to Lanchester’s theory. This too was faulted because of its lack of a firm basis in physics. Tay­lor’s dismissal of Lanchester was swift: “Besides these [discontinuity] theories of the resistance of solids moving through fluids, Mr Lanchester has pro­posed the theory that a solid moving through a fluid is surrounded by an

irrotational motion with circulation. This theory, as far as I can see, has noth­ing to recommend it, beyond the mere fact that it does give an expression for the reaction between the fluid and the solid” (4-5).

All that was granted to the theory, in its two-dimensional form, was that it had the (minimal) virtue of avoiding d’Alembert’s paradox. It permitted the researcher to deduce “an expression” for the resultant force on the body, but that is all. The formula, however, was not, in Taylor’s opinion, grounded in a real physical process. The theory provided no understanding of the mechanism by which the circulation round the body could be created. The problem came from Kelvin’s proof that circulation can neither be created nor destroyed. If Lanchester’s theory was an exercise in perfect fluid theory, then the premises of the theory precluded the creation of the very circulation on which it depended. Setting a material body in motion in a stationary fluid would not create such a flow. An aircraft, starting from rest on the ground in still air, and moving with increasing speed along the runway, would never generate the lift necessary to get into the air (not, at least, if the air was mod­eled as an ideal fluid). This consequence put Lanchester’s theory in no less an embarrassing position than discontinuity theory. As far as it described any reality, discontinuity theory was a picture of a stalled wing, that is, of an aircraft dropping out of the sky. If Taylor was right, Lanchester’s theory was equally hopeless because it would leave the aircraft stranded on the ground and incapable of flight.

Taylor thought Lanchester’s theory was, if anything, worse than the ver­sion of perfect fluid theory that generates d’Alembert’s paradox, that is, the version in which the perfect fluid has neither discontinuities nor circulation. Referring to this version as the “ordinary” hydrodynamics of an irrotational fluid, Taylor said that it, at least, gave a rigorous picture of the flow that would arise if an object were moved in these hypothetical circumstances, though, of course, this picture bore “no relation whatever” to reality. “The advantages of the ordinary irrotational theory is that it does, at least, represent the motion that would ensue if the solid were moved from rest in an otherwise motion­less perfect fluid, and if there were perfect slipping at the surface. By taking irrotational circulation round the solid, Mr Lanchester loses the possibility of generating the motion from a state of rest by a movement of the solid” (5). Taylor drew the conclusion that “in searching for an explanation of the forces which act on solids moving through fluids, it is useless to confine one’s atten­tion to irrotational motion” (5).

The correct strategy, Taylor argued, is to address flows where the fluid elements possess rotation as a result of viscosity and friction (6). In this way turbulence and eddying might be brought into the picture so that a physi­cally realistic fluid dynamics could emerge. Taylor was aware that the direct deduction of turbulent and eddying flow, starting from the full Stokes equa­tions of viscous flow, presented insuperable obstacles. Progress would be im­possible “if one were to adhere strictly to the equations of motion, without any other assumptions” (11). He therefore proposed to begin by a “guess at some result which I think would probably come out as an intermediate step in the complete solution of the problem” (11). On the basis of this guess he would deduce consequences that could be tested by experiment, and if “the observations fit in with the calculation I then go back to the assumptions and try to deduce it from the equations of motion” (11-12).

Taylor’s reaction to Lanchester depended on his assimilating Lanchester’s analysis to the classical framework of perfect fluid theory, that is, to the equa­tions of Euler and Laplace’s equation. The brevity of the argument attests to the taken-for-granted character of this assimilation. It must have seemed ob­vious that this is what Lanchester was presupposing. There was no hesitation or qualification, nor any suggestion that alternative readings were available. Admittedly, due to the sudden onset of war, Taylor did not have Lanchester’s book in front of him.61 He was recalling the essential point of the theory, and this involved the irrotational flow of a perfect fluid with a circulation. As such, the theory fell under the scope of Kelvin’s theorem and hence could never cast light on the creation of the circulation.

Lanchester was aware of the theorem (which he called Lagrange’s theo­rem) that rotation and circulation within a continuous body of ideal fluid can be neither created nor destroyed. He even expressed the point with a striking analogy. Once created, he said, a vortex of perfect fluid, unlike a real vortex, would “pervade the world for all time like a disembodied spirit” (175). He knew this meant that an infinite (that is, two-dimensional) wing starting from rest and moving within an initially stationary ideal fluid cannot then generate a circulation. He was prepared to face the consequences. “It is, of course, con­ceivable,” he said, “that flight in an inviscid fluid is theoretically impossible” (172). As an engineer working with real fluids, such as air and water, he hardly expected mathematical idealizations to be accurate. The important thing was to learn what one could from the idealized case but not to be imposed on by it. As he remarked ruefully, “The inviscid fluid of Eulerian theory is a very peculiar substance on which to employ non-mathematical reasoning” (118). Discussing the “two parallel cylindrical vortices” that trail behind the tips of a finite wing, he accepted that the mechanics of their creation would not be illuminated by standard hydrodynamic theory: “for such vortex mo­tion would involve rotation, and could not be generated in a perfect fluid without involving a violation of Lagrange’s theorem. . . . In an actual fluid this objection has but little weight, owing to the influence of viscosity, and it is worthy of note that the somewhat inexact method of reasoning adopted in the foregoing demonstration seems to be peculiarly adapted, qualitatively speaking, for exploring the behaviour of real fluids, though rarely capable of giving quantitative results” (158). For Lanchester, the mathematical apparatus of classical hydrodynamics played a subsidiary and illustrative role. It was merely a way of representing some of the salient features of the flow. Nothing of this complex, if informal, dialectic linking ideal and real fluids found any recognition in Taylor’s characterization.

Taylor’s response to Lanchester remained unpublished, but it tells us something about the assumptions of some of Lanchester’s readers. If Taylor read the work in this way, then presumably others will have read it in a similar way. The case is different with the next objection. It was not made in private but was very public and was acted out before a large audience at one of the major professional institutions in London.

Die Stromungs – und Druckerscheinungen, wie sie in bewegten Flussigkeiten, insbe – sondere auch der Luft, an den dareinversenkten Korpern beobachtet werden, haben schon seit langerer Zeit der hydrodynamischen Theorie einen viel bearbeiteten, nicht ganz einfachen Gegenstand geboten. Seit Otto Lilienthals Errungenschaften, und der neueren Entwicklung und Losung des Flugproblems haben diese Fragen auch grofie praktische Bedeutung erlangt.

w. m. kutta, “ Uber eine mit den Grundlagen des Flugproblems in Beziehung stehende zweidimensionale Stromung’ (1910)1

The flow and pressure phenomena, as they can be observed on bodies immersed in a moving fluid, particularly the air, have long provided for hydrodynamic theory a much worked on, but far from simple, object of study. Since Otto Lilienthal’s achieve­ments and recent developments in solving the problem of flight, these questions have acquired great practical significance.

In the next two chapters I show technische Mechanik in action by giving an overview of the early German (or German-language) development of the cir­culatory theory. In this chapter I deal with the “infinite wing” paradigm, that is, with an analysis deliberately confined to a two-dimensional cross section of the flow in which the wingtips are ignored. I then devote the next chapter to the more realistic theory dealing with a wing of finite span and the three­dimensional flow around it. It was Wilhelm Kutta in Munich who triggered the striking progress in the field of two-dimensional flow that was made in Germany before and during the Great War. His work is my starting point. Where Rayleigh used a simple, flat plane as a model of a wing, Kutta used a shallow, circular arc. Both men treated the air as an inviscid fluid, but where Rayleigh postulated a flow with surfaces of discontinuity, Kutta postulated an irrotational flow with circulation. Joukowsky, a Russian who published in German, then showed how to simplify and generalize Kutta’s reasoning. A variety of other workers in Gottingen, Aachen, and Berlin, starting from Kutta’s and Joukowsky’s publications, carried the experimental and theoreti­cal analysis yet further. Appreciating why these developments constitute an exercise in technical mechanics, rather than mathematical physics, requires

engaging with the details of the scientific reasoning. As a first step I place Kutta and his achievement in their institutional setting.

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

In accordance with the Royal Aeronautical Society’s policy of sustained, tech­nical discussion, Low’s paper was followed, later in the year, by other material relating to the Prandtl theory. On November 16, 1922, R. McKinnon Wood gave a paper titled “The Co-Relation of Model and Full Scale Work.”81 Like Low, McKinnon Wood also took the opportunity to describe the basis of the circulation theory and, like Low, found himself confronting Bairstow. It transpired in the discussion that Bairstow was engaged in experiments at the NPL to work out where viscous and nonviscous flows differed in the case of aerofoil shapes. Bairstow said he had no doubt that the theory of nonviscous flow would yield results that could be tested, but “he did not expect to find the circulation at all in the experiments” (499).

There were also two talks given to the society which were devoted to wind-tunnel studies of the vortex system behind a wing both by N. A. V. Piercy of the East London College.82 Piercy had been a colleague and collabo­rator of Thurstone but worked at a much more sophisticated level both em­pirically and theoretically. Using the college’s wind tunnel, Piercy produced detailed measurements of the airflow both behind the wing and in the region of the wingtips. It was clear that there were vortex structures to be mapped, and these corresponded, at least qualitatively, to the expectations created by

Lanchester’s and Prandtl’s work. Although they were broadly supportive of the circulatory picture, the results were actually understood by Piercy to sup­port Bairstow’s suspicion that too little weight had been given to the role of viscosity.

Piercy was very conscious of the empirical variability of the phenomena under study in his wind tunnel. He argued that the vortex effect behind a wing sometimes achieved its maximum value after the wing had stalled and thus after the lift (and, presumably, the circulation) had dropped away. How could this be explained on the Lanchester-Prandtl theory? He made three suggestions, none of which could be easily accommodated within the circula­tory theory as it stood. First, he wondered whether, during a stall, the wingtip vortices continue to exist but are not joined together by a vortex that lies along the span of the wing. This would produce the effect to which he was referring, namely, wingtip vortices without lift or with diminished lift. But if the vortices can exist without circulation around the wing at, and beyond, the angle of stall, surely “it is not necessary for them to be so joined at a smaller angle” (502). Second, even if it were the case that the two wingtip vortices are still joined (in some fashion) after the stall, “they may be joined in such a manner as not to give cyclic lift” (502). “As a third alternative,” said Piercy, we may “suppose that cyclic lift may be destroyed to a considerable extent by viscous effects,” but then, “it seems reasonable to conclude that cyclic lift is not immune from viscosity at smaller angles” (502).

Piercy was well aware that supporters of the circulation theory had always sought to draw a line between normal flight at low angles of incidence and the phenomenon of stall at high angles of incidence. Their position was that a good theory of the former did not have to explain the latter. An explanation was desirable but not necessary. This had been a central part of the argument between Lanchester and Bairstow in 1915. Piercy sided with Bairstow on this matter. He dismissed the defense as an evasion. It was, he said, “beside the point.” “The question is whether we can afford to neglect at 8 deg. incidence, say, a factor so powerful as to be able to overthrow the vortex system at, say, 16 deg. Should we not rather conclude that at any angle viscosity is playing an essential and important role in the whole system of flow?” (502). The long­standing British concern with stalling, and the desire for a unified and realis­tic theory of broad scope, was still in play.

The next paper in the 1923 volume of the Royal Aeronautical Society jour­nal which discussed the circulation theory appeared under the name Glauert, but it came from Muriel Glauert, Hermann Glauert’s new wife. The paper was called “Two-Dimensional Aerofoil Theory” and was based on a technical report written some two years earlier for the Aeronautical Research Commit-

tee, but under the name Muriel Barker, not Muriel Glauert.83 Muriel Barker worked for the Royal Aircraft Establishment and was the holder of the post­graduate Bathurst Studentship in Aeronautics at Cambridge. Her notes on Kutta, which were mentioned and used in a previous chapter, were prob­ably made when gathering material for writing the original technical report. The discussion of two-dimensional aerofoil theory for the RAeS journal was based on the assumption, rejected by Bairstow and many others, that inviscid methods are legitimate. To develop this starting point, Muriel Glauert intro­duced a general theorem due to Ludwig Bieberbach.84 The theorem showed that there was one and only one conformal transformation of the form:

(where b1, b2 , . . . were complex) which would map the space around a shape, such as an aerofoil, in the z-plane into the space round a circle in the Z-plane, leaving the region at infinity unchanged. She then worked through, in math­ematical detail, the special case of this theorem provided by the Joukowsky transformation and dealt with circular arcs, Joukowsky aerofoils, double circular arcs, struts, Karman-Trefftz profiles, von Mises profiles, and Tref- ftz’s graphical methods. Muriel Glauert’s paper makes it clear that the British had been doing their homework. They had now brought themselves up to date and absorbed all the mathematical techniques and results of the German work on the two-dimensional wing that I described in chapter 6.

Next to be published in the sequence of Prandtl-oriented discussion pa­pers was one by Hermann Glauert himself, titled “Theoretical Relationships for the Lift and Drag of an Aerofoil Structure.”85 At first glance Glauert’s pa­per has the appearance of being no more than an elementary treatment of the circulatory theory—far less mathematical, for example, than Muriel Glau­ert’s paper. Unlike his wife’s paper, or his own technical reports for the Aero­nautical Research Committee, the present paper was not replete with math­ematical formulas. The appearance of simplicity, however, is misleading. The paper may have been essentially qualitative in its argument but it was in no way elementary. It was sharply focused on difficult problems, but the prob­lems in question were methodological ones. It dealt with the orientation that was needed to appreciate Prandtl’s approach—the very thing that divided Glauert from his mathematically sophisticated British contemporaries. They did not need to be convinced of the mathematics; they needed to understand the mathematics in a different way.

The solution of a physical problem in aerodynamics, said Glauert, can be analyzed into three steps. First, certain assumptions must be made about
what quantities can be neglected, for example, gravity, compressibility, and viscosity. Only rarely is it necessary to take into account the full complexity of a phenomenon. Second, the physical system, in its simplified form, must be expressed in mathematical terms, for example, a differential equation and its boundary conditions. Third, the mathematical symbols must be manipulated until they yield numerical results that can be tested experimentally or used for some practical purpose. This third step, said Glauert, must not be misun­derstood. It is where some of the greatest difficulties arise because the math­ematical problems may be insurmountable. At this stage it may be necessary to simplify further the initial, physical assumptions or to confine attention to a limited range of cases, such as small deviations from known motions. It is important to remember, said Glauert, that “in no case are these assump­tions absolutely rigid” (512). Glauert’s three steps are not simply sequential: what happens during the third step can feed back into what was called the first step.

Glauert then rehearsed the assumptions that were made by proponents of the circulation theory, that is, the assumption that the air could be represented as a perfect fluid with neither compressibility nor viscosity; the assumption that the fluid flow is irrotational; and the need to postulate a circulation to avoid a zero resultant force or to resort to the theory of discontinuous flow. “In view of this discussion,” said Glauert, “it appears that no satisfactory so­lution of an aerodynamic problem is to be expected when the effects of com­pressibility and viscosity are neglected, and it becomes necessary to consider the effect of these two factors” (513). This sentence is a striking one. It appears to concede all the points made by the critics of the circulation theory. Is this not exactly what Leonard Bairstow would say? Is not Glauert here following the line that led the young Taylor to dismiss Lanchester? Given Glauert’s ac­complishments as a stylist, however, both the import and the impact of these words would have been carefully weighed. He would not have inadvertently conceded too much or expressed himself inaccurately on such an important question.

How could Glauert grant that no satisfactory solution can be expected if viscosity is neglected without also granting the dismissive conclusions drawn by the critics of the circulation theory? The answer hinges on what it is to “ne­glect” compressibility and viscosity and what it is to “consider” their effects. Is this something done at the outset, in step 1 of the methodology? Or is it done at step 3, not as a sweeping assumption but as a technique for making the mathematics tractable? The vital but subtle methodological point that Glau – ert was making can be expressed like this: viscosity cannot, indeed, be wholly neglected but, contrary to first appearances, that does not preclude the use of perfect fluid theory. There were ways of operating with the mathematics of a perfect fluid that involved consideration of viscosity. The acts of consideration that were in question could not be stated in the inviscid equations themselves but would be shown in how they were deployed and interpreted.

Glauert explained that two important facts about viscosity must be ac­commodated. First, there is the no-slip condition, which stands in contrast to the perfect fluid property of finite slip. Second, viscous forces are propor­tional to the rate of change of velocity and hence are important close to a body such as a wing but become negligible at large distances. These are the physical facts for which approximations must be found. They cannot be dis­missed in step 1 of the sequence of steps Glauert had described. That would indeed amount to a decision to “neglect” them, and it is known that this produces the empirically false result of a zero resultant force. Rather than neglecting these two facts, their reality must be taken into account by a jus­tifiable approximation, an approximation of the kind introduced in step 3. Glauert’s development of this point deserves to be quoted in full. Notice the specific meaning he attached to the word “ignore” in the quoted passage and the implied contrast between ignoring something (in step 1) and approximat­ing its properties (in step 3):

It is known that the solution obtained by ignoring the viscosity is unsatisfac­tory, but it is by no means obvious that the limiting solution obtained as the viscosity tends to zero is the same as the solution for zero viscosity. In particu­lar, in the case of a body with a sharp edge, there is a region where the velocity gradient tends to infinity, and where the viscous forces will be of the same order of magnitude as the dynamic forces, however small the viscosity. On the other hand, the layer round the body in which viscosity is of importance can be conceived as of zero thickness in the limit, and this conception is equiva­lent to allowing slip on the surface of the body. It appears, therefore, that the non-viscous equations will be the same as the limit of the viscous equations, except in the region of sharp edges. (514)

The argument was that under the right conditions the equations of in­viscid flow are legitimate approximations to the viscous equations and their use does not amount to “ignoring” or “neglecting” the viscous properties of the flow. The crucial requirement is that the inviscid flow must be one that can be understood as a limiting case of a viscous flow. Glauert appears to have carried this crucial lesson away with him from his conversations with Prandtl. The Royal Aeronautical Society paper was therefore not merely an elementary exposition of the theory of lift; rather, it was an attempt to con­front the habits of thinking that had justified the systematic neglect of the in­viscid approach by British experts. Up to this point the conviction of British mathematical experts that ideal-fluid theory was false, and ultimately useless for aerodynamics, had carried almost everything before it. Only the theory of viscous flow dealt with reality. Ideal-fluid theory may provide some re­sidual mathematical challenges, and some suggestive analogies, but it could not be taken seriously as a means for directly engaging with reality. Glauert was challenging this assumption. He sent a copy of his paper to Prandtl along with copies of Piercy’s two papers. Of Piercy’s work he remarked that the ex­perimental results were interesting but expressed doubt about the theoretical interpretation: “His experimental results are of considerable value, but his interpretation of them leaves a good deal to be desired.” Glauert described his own piece as a “short note I wrote in justification of the principles underlying the vortex theory of aerofoils.” 86 The question was: could Glauert shift the way his contemporaries understand those underlying principles?

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.

In my study of the difference between the British and German responses to the circulatory theory of lift I have followed out the implications of Frank’s comparison. I have tried to dig into what he called “the depths,” where, as he rightly said, “the real battle for the progress of knowledge has been fought.”112 I have engaged with the details of the scientific and technical argumentation over the theory of lift because it is here, in these details, that both the social character of knowledge and the consequent relativity of knowledge find their most revealing expression. The story reminds us of the sheer contingency and unpredictability of the outcome of any research enterprise and shows how complex and fine-grained that contingency can be. It shows the vital and ineradicable role played by cultural traditions and the institutions that transmit these traditions. And, as Frank predicted, nowhere in the analysis of scientific discourse was a legitimate place found for the term “absolute truth.”

This insight is in constant danger of being forgotten or obscured by the false friends of science. The relativity of all scientific concepts to culture and society is deemed unacceptable by the self-appointed guardians of knowledge who claim to “take reflective responsibility, as it were, for the normativity of our most fundamental cognitive categories.”113 Historians and sociologists, like experimental psychologists and anthropologists, have always known that it is not normative posturing but close and careful empirical studies of cogni­tion that are needed. Strange though it may seem, this principle needs special emphasis when the cognition in question is that of scientists and engineers. The practices of scientists and engineers must be studied in a hard, factual light as natural phenomena that belong to the material world of cause and effect. Only studies conducted in this spirit can carry the analysis beyond ide­ology and propaganda and lay the basis for a proper, public understanding of science and technology.114

Detailed empirical studies always need a methodological context, and at­tention must always be given to the broader framework in which they are un­derstood. Thus one may legitimately ask where all the intellectually brilliant activity of the men and women I have studied is to be positioned in relation to the grand philosophical categories of Progress, Reality, and Truth. Now that the facts of the case study are at hand, the answers are not difficult to supply, though their implications may be disturbing.

Let me take each category in turn. That the work of the German engineers constituted technical progress is beyond doubt, and it is this which eventually had its impact on the British. The German work had utility and practicality relative to goals and interests shared by the experts of the two nations. The British led the way in the study of stability, but when it came to the study of lift and drag, failure and frustration took its toll on the British experts. The practical rewards and opportunities offered by the German approach eventu­ally tempted even the strictest to compromise their principled commitment to theories with a firm basis in physical reality. The theory of circulation and Prandtl’s theory of the finite wing allowed the experts to do things that they wanted to do, and that fact alone was, in its own way, rewarding. If the at­traction of moving forward could not rationally compel a change of mind on important theoretical questions, it encouraged a pragmatic accommodation.

That everyone in the field of aerodynamics, British and German, was, each in his or her own way, grappling with reality is also evident, and this was wholly taken for granted in all of the reasoning of the actors I have described. Although one may question the extent of Sir George Greenhill’s connection with reality in his notorious Reports and Memoranda No. 19, on discontinu­ous flow and free streamlines, this work did not set the pattern for the future reports of the Advisory Committee. As a group the British were no less con­cerned than their German counterparts with understanding the real perfor­mance of real airplanes under real conditions of use. And wasn’t Sir George himself acting as the spokesman of the practical realists when he (correctly) took G. H. Bryan to task for neglecting the gyroscopic effect of the engine and propeller in his analysis of stability?

Engagement with reality may be common ground, but my example shows that there are different ways of grappling with reality. It also shows that these different modes of engagement are social modes belonging to, and sustained by, different groups with different local traditions. Such differences can di­vide groups that otherwise share much by way of a common culture, as did the British and German experts in aerodynamics. Even more important, the example shows that there are no independent methodological principles by which these different forms of engagement could be reliably and usefully as­sessed. Such principles as emerged in the episode were themselves integral to the forms of engagement they were used to justify. They were rationalizations of existing practices and institutions. That there are different ways of engag­ing with the world may seem obvious; that the only ground available to the actors for justifying their choices is question begging is perhaps less obvious. But obvious or not, it follows directly from the fundamentally social charac­ter of cognition. This is one reason, though not the only reason, why the so­ciologically minded David Hume was right when he said that all the sciences have a relation 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.”115

What, finally, is to be said about truth? The progress in aerodynamics made in the technische Hochschulen and the University of Gottingen derived from the use of a theory of perfect fluids in potential motion. The theory dealt with an idealization and a simplification. This theory was dismissed in Cambridge and London as physically false and logically self-defeating. It was false because it denied the viscosity of the air and self-defeating because cir­culation was unchanging with respect to time, and its origin was beyond the reach of the theory. The premises of the British objections were true and the reasoning based on them was sound, but the conclusions led to failure rather than success.

The German advances in the understanding of lift and the properties of wings depended on the use of abstract and unreal concepts that were some­times employed with questionable logic. Progress in aerodynamics thus de­pended on the triumph of falsity over truth. Everyone knows that false prem­ises can sometimes lead to true conclusions and that evidence can sometimes support false theories, but the story of the aerofoil involved more than this. The successful strategy involved the deliberate use of known falsehoods poised in artful balance with accepted truths. The supporters of the theory of circulation showed how simple falsehoods could yield dependable conclu­sions when dealing with a complex and otherwise intractable reality. This is the real enigma of the aerofoil.

The enigma would hold no surprises for Hume. It would simply be an expression of what he called “the whimsical condition of mankind.”116 The lesson Hume learned from the study of history and society was that “the ul­timate springs and principles” of the natural world will never be accessible to the human mind. The utmost that reason can achieve is the simplifica­tion of complexity. Humans live and operate in a world of limited experience dominated, necessarily but beneficially, by custom, convention, habit, and utility. Hume acknowledged that “the philosophical truth of any proposi­tion by no means depends on its tendency to promote society” but argued that we should be neither surprised nor unduly alarmed if truths (or sup­posed truths) that lack utility sometimes “yield to errors which are salutary and advantageous” (279). The story I have told deals with a technology that may seem remote from the world whose problems exercised Hume, but the central fact to emerge in my story, the fact I have called the real enigma of the aerofoil, can be understood in the humane, skeptical, and sophisticated terms he offered. Among the British it was an accepted truth that the air was a viscous fluid governed by Stokes’ equations. In the field of aeronautics that truth, if truth it be, yielded to the erroneous but salutary and advantageous picture of the air as an inviscid fluid governed by Euler’s equations.

Are there general lessons to be learned? Not if aerodynamics is a special case, but I do not think it is a special case. The conclusions reached in this case study surely can be generalized.117 What, then, should be concluded? Individual developments in the sciences will differ in their details, but what Frank had to tell us about the compromises involved in the design of air­planes applies (and was meant to apply) to the technology and instruments of all thinking. There are always compromises to be made. The warning given by von Mises against the phantom of absolute cognition will always be rel­evant. And there will always be a role in science and engineering for the blunt advocacy of a Major Low and the rapier responses of a G. I. Taylor. Above all, what Kuchemann had to say about the idealizations of aerodynamics cap­tured the essence of the creative work of Lanchester, Prandtl, and Glauert. But idealizations are salutary and advantageous falsehoods which play a vital role in all science, pure as well as applied. In stressing the role of idealization,

Kuchemann may have identified a feature of cognition that is more salient in engineering than in physics—but it is the engine of progress in all fields. Those who point to the airplane as a symbol of the truth of science, the power of technology, and the reality of knowledge are therefore right—but do they know what they are saying? The enigma of the aerofoil is the enigma of all knowledge.

Physicists, chemists, physiologists, and engineers are all interested in air, and each group studies it from the perspective of its own discipline. In the history of each discipline there is a strand that represents the history of the chang­ing conceptions of the nature of air adopted by its practitioners. Sometimes aerodynamics is counted as a branch of physics and sometimes as a branch of engineering, but however it is classified, it is evident that it involved a determined attempt to relate the flow of air to the basic principles of me­chanics. The most important of these are the laws of motion first delineated by Newton, for example, the law that force equals mass times acceleration. The complexity of the air’s behavior, however, means that there is no unique way to connect the flow to the fundamental laws of Newtonian mechan­ics. How the relation is to be articulated depends on the model of air that is used.

Newton himself treated fluids in different ways at different times. When he was thinking about the pressure of the air in a container, he conjectured, for the purposes of calculation, that air was made up of static particles that repelled one another by a force that varied inversely with distance.4 This con­cept was a guess that explained some of the known facts, but it was a con­ception of the nature of air and gas that physicists later abandoned. In its place they adopted what is called the kinetic theory of gases in which it was assumed that a gas is made up of small, rapidly and randomly moving par­ticles. According to the kinetic theory, as developed by James Clerk Maxwell and others, gas pressure is not the effect of repulsion between the molecules of the gas but is identified with the repeated impact of the molecules on the walls of the container.5

When Newton was thinking of a flowing fluid impinging on the surface of an obstacle, he did not use his repulsion model but spoke, for mathematical purposes, simply of a “rare medium” and treated the fluid as made up of a lot of point masses or isolated particles that do not interact with one another.6 The fluid medium was treated as if it were like a lot of tiny hailstones (though this was not Newton’s comparison). Again, the model is not to be identified with the later kinetic theory of gases. The hailstone model, too, dropped by the wayside, though, as we shall see, in certain quarters it still played some role in early aerodynamics. The concept of a fluid that proved most influen­tial in hydrodynamics was different from either of the ideas used by Newton as well as being different from the kinetic theory of gases. The model that came to dominate hydrodynamics, and aerodynamics, was first developed in the eighteenth century by mathematicians such as d’Alembert, Lagrange, the two Bernoullis (father and son), and Euler. They thought of the air as a continuous medium.7

Because the aim was to be realistic, the hypothetical, continuous-fluid picture had to be endowed with, or shown to explain, as many of the actual properties of real fluids as possible. Thus air has density so the continuous fluid must also possess density. Density is usually represented by the Greek letter rho, written p. Empirically, density is defined as the ratio of mass (M) to volume (V), which holds for some finite volume. The number that results, p = M/V, represents an average which holds for that volume at that moment. To apply the concept to a theoretically continuous fluid requires the assump­tion that it makes sense to speak not merely of an average density but of the density at a point in the fluid, that is, the ratio of mass to volume as the volume under consideration shrinks to zero. If the air is actually made up of distinct molecules, then, strictly, the density will be zero in the space between the molecules and nonzero within the molecules, and neither of these values would qualify as values of the density of the fluid. This dilemma did not ap­pear to be a problem in practice, but it is a reminder that the relation between physical models of the air based on particles and physical models based on a continuum may, under some circumstances, prove problematic.8

Air is also compressible. The same mass can occupy different volumes at different pressures. For many of the purposes of aerodynamics, however, it can be assumed that the density stays the same. This is because (perhaps counter to intuition) the pressure changes involved in flight turn out to be small. The fluid continuum can then be treated as “incompressible.” This ap­proximation only becomes false when speeds approach the speed of sound, which is around 760 miles an hour. In the early days of aviation, when aircraft flew at about 70 miles an hour, compressibility was no problem for wing theory. Things were different for propellers. The tips of propellers moved at a much higher speed, and here compressibility effects began to make them­selves felt, but that part of the story I put aside.9

Another important attribute of a fluid is its viscosity, which refers to the sluggishness with which the fluid flows. If a body of fluid is thought of as made up of layers, then the viscosity can be said to arise from the internal friction between these layers. Pitch and treacle are highly viscous fluids, whereas water is not very viscous. Viscosity can be measured by experimen­tal arrangements involving the flow through narrow tubes. The results are summarized in terms of a coefficient of viscosity, which is usually repre­sented by the Greek letter mu, written |4. A highly viscous fluid will be given a high value of |4; a fluid with small viscosity will have a correspondingly small value of |4. Air only has a very slight viscosity. At the extreme, if there were a fluid that was completely free of viscosity, it would be necessary to write |4 = 0. In reality no such wholly inviscid fluids exist, but if the fiction of zero viscosity is combined with the fiction of total incompressibility, this concept can be taken to specify what might be called a “perfect” fluid or an “ideal” fluid.

The single most important fact to know about the historical develop­ment of wing theory and the aerodynamics of lift is that its mathematical basis lay in the theory of perfect fluids, that is, in a theory in which viscosity was apparently ignored and assumed to be zero. The assumption that air can be treated as an ideal fluid was the cause of much argument, doubt, and frustration, which becomes apparent in subsequent chapters, but its central, historical role is beyond dispute. What turned out to be the most striking developments in aerodynamics (as well as some failed attempts) depended on the idea that viscosity and compressibility were effectively zero. The at­tractions of this assumption were twofold. First, it seemed highly plausible, and second, it produced an enormous simplification in the mathematical task of describing the flow of a fluid. The exercise produced a set of partial differential equations that determined the velocity and pressure of the fluid, provided that the starting conditions of the flow and the solid boundaries that constrain it are specified. The equations were developed by imagining a small volume of fluid, called a fluid element, and identifying the forces on it. The forces derive from pressure imbalances on the surfaces of the fluid element.

Fluid elements, it must be stressed, are mathematical abstractions rather than material constituents of the fluid. They are not to be equated with the molecules that interest chemists and physicists or the particles that feature in the kinetic theory of gases. The equations of flow do not refer to the hidden, inner constitution of fluids. The reality that is described by the differential equations that govern fluid motion concerns the macrobehavior offluids rather than their microstructure. The abstract character of a fluid element is evident from the way it is typically represented by a small rectangle. The simple ge­ometry of the representation derives from the mathematical techniques that are being brought to bear on the flow. These are the techniques of the differ­ential and integral calculus.10 The concept of a fluid element is the means by which these techniques can be used to gain a purchase on reality. The differ­ential equations that were the outcome are called the Euler equations. They can be said to describe in a strict way the flow of an ideal fluid, but the hope was that they would also describe, albeit in an approximate way, the flow of a real fluid, air, whose viscosity is small but not actually zero.

To give a feel for the style of thinking that went into the classical hydro­dynamics of ideal fluids (and, later, into aerodynamics), I shall give a simple, textbook derivation of the Euler equations. It is the kind of derivation that was wholly familiar to many of the actors in my story, and certainly to those who worked in and for the Advisory Committee for Aeronautics. The discus­sion in the next section is therefore slightly more technical. It is based on the treatment given in one of the standard works of early British aerodynamics, namely, W. H. Cowley and H. Levy’s Aeronautics in Theory and Experiment that was published in 1918.11 Both Cowley and Levy worked at the National Physical Laboratory. Levy had graduated from Edinburgh in 1911, visited Got­tingen on a scholarship, and had then worked with Love in Oxford. Dur­ing the Great War he had been commissioned in the Royal Flying Corps but was seconded to the NPL. As a left-wing activist who wanted to unionize his fellow scientists, his relations with Glazebrook were not of the easiest. After the war Levy left to join the mathematics staff at Imperial College, where he was eventually awarded a chair.12 Cowley stayed at the NPL and worked on problems of drag reduction with R. J. Mitchell, who was design­ing the racing seaplanes that won the Schneider Trophy for Britain in 1929 and 1931.13

Greenhill’s contribution was not confined to the daunting R&M 19. As well as working on the mathematics of gyroscopes and problems of airship stability, in 1910 and 1911 he gave a series of lectures at the Imperial College of Science and Technology. The course was published a year later in a book titled The Dynamics of Mechanical Flight.21 Greenhill explained that the lift of a wing depended crucially on “the opening out of the stream lines” (40) behind the wing. This occurrence would be the expected effect of the surfaces of discon­tinuity enclosing the “dead” air region above and behind the wing. Greenhill went on to contrast the truth, as he saw it, of the picture of discontinuous flow with the error of certain popular conceptions about the flow of air over a wing. (The two different ideas of the flow are represented in figs. 13 and 14 in his book.) The passage in which he contrasts them is a revealing one: “A popular figure of the stream lines past a cambered wing as here in Fig. 13, showing no such broadening, would imply at once to our eye an absence of all thrust and lift; the figure should be more like Fig. 14” (41). The diagrams to which Greenhill was referring are shown as my figures 3.4 and 3.5 (with Greenhill’s numbering identified in the captions).

Greenhill’s second diagram that was meant to describe the correct flow indicates turbulence in the “dead” air, though his mathematical analysis does not make provision for this. The embellishment seems to be a concession to

1912, 41.

Kelvin. We know why Greenhill believed that there was lift in the case of the discontinuous flow, because of the reasoning set out by Rayleigh, but why was it obvious to Greenhill’s eyes that the popular flow picture, my figure 3.4, would be devoid of all thrust and lift? The reasoning may have been that with­out surfaces of discontinuity, the flow must have the character of the original, continuous flow of an ideal, frictionless fluid—with the “paradoxical” result of zero-resultant force. It looks as if Greenhill took the diagram of smooth, streamlined flow over a wing to imply that the air was being treated as a con­tinuous, ideal fluid in irrotational motion. In other words, it was taken as a flow in which there would be no resultant and where d’Alembert’s paradox would be applicable.

If this was the reasoning, then two significant details of Greenhill’s first drawing were wrong. The flow is not pictured accurately at the leading or trailing edge. The front stagnation point should be below the leading edge, while the rear stagnation point should be on the upper surface of the wing in front of the trailing edge. Instead the air is shown coming away smoothly from the trailing edge itself. Greenhill would certainly have noticed this error, and he gave the correct form of the diagram for a flat plate on page 47 of his book. He presumably put the inaccuracy down to the approximate character of the “popular” representation, as he had in an earlier criticism of drawings of leading-edge flow (22). In any event, he seems to have taken the popular diagram as an attempt to depict the kind of idealized, continuous perfect fluid flow where, as any mathematician would know, all the forces (except the turning couple) canceled out.

Although Greenhill made more effort in the book than in the ACA re­port to bring real aircraft into the discussion, it was still full of examples and mathematical technicalities of questionable relevance. Predictably, it did not go down well with the practical men. The anonymous reviewer for Aero­nautics, who had apparently attended the lectures, said the book confirmed the earlier impression that the calculations were really aimed at providing a diverting recreation for the mathematical mind. “Practical value they lack wholly; the data on which Sir George Greenhill’s mathematical excursions are based are theoretical without fail.”22

G. H. Bryan’s review was very different.23 Up to the present, said Bryan, there had been a lack of understanding about the role of mathematics in aeronautics. The subject has failed to attract our best mathematicians, while “practical men” make claims “in utter disregard for the fundamental prin­ciples of elementary mathematics and physics” (264). Under these “chaotic conditions” it would be useful to have a work “by so reliable a mathematical authority” as Greenhill (265). Bryan acknowledged the presence of drastic simplifications involved in Greenhill’s approach but insisted that, in spite of these shortcomings, “the theory of discontinuous motion affords the best opening to the study of pressures on planes from the mathematical stand­point” (266). He listed the “great mathematicians” who had developed the theory, but noted that it had only been applied to flat plates and not yet to bent or cambered planes (which would make better models of the aerofoils in practical use). Some calculations of this kind, the reader was told, were now under way. Bryan did not once ask if the theory of discontinuous flow gave empirically adequate answers. Rayleigh, of course, knew that, as the theory stood, it did not give the right answers, and so did his experimentally inclined colleague Mallock. I now turn from the mathematical to the experimental study of discontinuous flow to see how matters were carried forward on this front.

In March 1915, Lanchester gave an exposition of his theory at the Institution of Automobile Engineers in London.62 In the audience of over 150 members and guests was a fellow member of the Advisory Committee, Mervin O’Gorman, as well as Leonard Bairstow of the National Physical Laboratory. Lanchester devoted the first part of the lecture to the theory of lift or “sustentation.”63 The presentation started from the observed differences in pressure be­tween the upper and lower surfaces of an aircraft wing. For maximum effi­ciency, argued Lanchester, the flow of air over the wing must conform closely to the surface of the wing. Conformability, rather than the separation charac­teristic of Kirchhoff-Rayleigh flow, was the central assumption. At the tip of the wing, however, complications enter into the story. The higher pressures on the lower surface cause the air to move around the tip from the lower to the upper surface. When combined with the motion of translation of the wing through the air, the circulating motion at the tips has two consequences. First, it gives the flow over the top of the wing an inwardly directed compo­nent, toward the center line, but an outwardly directed component on the lower surface. Second, at the tips themselves, the circulation is swept back­ward to form two trailing vortices coming away from the ends of the wings. To complete the dynamical system, argued Lanchester, the two trailing vorti­ces must be joined, along the length of the wing, by a vortex that has the wing itself as its solid core. The vortex provided the circulatory component of the flow around the wing and accounts for the velocity difference between the flow over the upper and lower surfaces. This in turn accounts for the pressure difference, and hence the lift.64

Lanchester combined his exposition with some methodological observa­tions. He began by distinguishing the theoretical approach to aerodynamics from the purely empirical approach and noted that the two methods can, to a great extent, be followed independent of one another. Nevertheless, he insisted that engineering needed theory and that experiment without theory was “inefficient.” When variables were effectively independent, simple em­pirical methods of keeping everything constant except one variable might suffice; when variables were dependent on one another, this method ob­scured the crucial connections. At the conclusion of his lecture he returned to these methodological points, saying, “It has not been found possible in the present paper to do more than give an outline of the theory of sustenta – tion, with sufficient examples and references to practice and experiment to illustrate the importance of the theoretical aspect of the subject as bearing on the experimental treatment; the latter has hitherto been dealt with almost without considerations of theory, and has degenerated into empiricism pure and simple” (207). Although Lanchester was making a general claim about the guiding role of theory, there can be little doubt that he had the neglect of his own theory in mind. This was certainly how he was understood by some of his audience.

Lanchester’s lecture impressed at least some of the practical men, and it was greeted by an enthusiastic editorial in Flight.65 The immediate reception by the audience was, however, mixed. Mervin O’Gorman began the discus­sion after the lecture by congratulating Lanchester on his freshness of outlook and went on to offer empirical support for Lanchester’s theory. Experiments had been done on full-sized wings at the Royal Aircraft Factory that demon­strated the predicted inward and outward flow on the respective upper and lower wing surfaces.

We fastened pieces of tape at one end of the upper surfaces of the leading edge of the tips of an aeroplane wing, and arranged a camera, worked by a Bowden wire, to photograph them in flight; they were not put there for the purpose indicated by the author, but we got exactly what he says we should get, and I am glad to confirm him so far. (228)

Leonard Bairstow (fig. 4.10) then rose and adopted a different tone. He an­nounced to the audience that he was not convinced by Lanchester’s ideas.

I quite agree with Mr. O’Gorman that the paper is extremely interesting, but I also find it extremely controversial, and I disagree with his final conclusions.

(229)

By “final conclusions” Bairstow was referring to Lanchester’s suggestion that aerodynamics had degenerated into pure empiricism. Bairstow took it personally:

Many references have been made in the paper to experimental work at the National Physical Laboratory, which work is generally under my charge, and the author has done his best to put the N. P.L. on its defence for not making practical application of his theory. (229)

Given that much of Bairstow’s work had been on stability, and had been guided by the theory developed by G. H. Bryan, it is easy to understand why the general criticism might have struck Bairstow as unjust. The work on sta­bility was certainly not mere empiricism. But Lanchester was talking about lift. Here the charge of empiricism was more plausible. For example, Joseph Petavel, a fellow member of the Advisory Committee and the future direc­tor of the National Physical Laboratory, had given the Howard Lectures in March and April of 1913 at the Royal Society of Arts. He had devoted them to aeronautics, but his treatment had been purely empirical.66 He simply pre­sented his audience with a stream of graphs and empirical coefficients. There was no mention of either the discontinuity theory or the theory of circula­tion. And had not Bairstow himself admitted the resort to empiricism when he had addressed the Aeronautical Society that same year?67

This was true, but all that Bairstow needed to claim to rationalize his po­sition was that Lanchester’s theory was not acceptable because it was a bad theory. He was saying, in effect, show me an adequate theory and I shall use it to guide my experiments, but as yet no such theory is on offer. Bairstow’s objection was that Lanchester’s theory covered some, but not all, of the facts that were of interest to the aeronautical engineer. Bairstow had come pre­pared to prove his point: “I will not pretend to follow the analytical steps between the author’s statements of the vortex theory and his applications, but I will deal with two experiments made at the N. P.L.” (229). With this heavy hint that Lanchester’s position lacked logical clarity, Bairstow proceeded to show the audience two photographs. They depicted a square, flat plate set at an angle of 40° to a stream of water. The water was injected with ink to make the flow visible. Both photographs were taken from above, the first being at a slow speed of flow, the second at a faster speed. Referring to the first pic­ture, Bairstow conceded that it looked to him like the flow that Lanchester had described and as it had been presented in a line drawing (called figure 6) in Lanchester’s talk. Two trailing vortices could be seen coming from the sides of the plate (which Bairstow described as a low-aspect-ratio wing). The higher speed flow, however, presented a very different appearance. If one

photograph fitted the theory, the other certainly didn’t. Introducing the first photograph Bairstow said: “The resemblance of this photograph to Fig.6 of the paper is very marked, and up to this point I am thoroughly in accord with the author as to the probable, and in fact almost certain, existence of the type of flow postulated in the early part of the paper” (230).

Moving on to the second picture with the more rapid flow, he added: “The type of flow is now very different from that to which the author’s theory applies. The fluid round the model aerofoil leaves it periodically in spinning loops. The spiral showing the spin inside the arch of one of the loops is very distinct” (230). He conceded that Lanchester’s theory might fit “the very best aerofoil that can be designed at its very best angle of incidence” (230), but the theory said nothing about the full range of significant flow patterns. The word “stall” was not used, but Bairstow’s argument was that Lanchester could not explain what happens when a wing stalls: “There appear, then, to be ex­ceptions to the author’s theory, or rather, there are cases of fluid motion of interest to aeronautical engineers which do not satisfy the conditions that the surface shall be conformable to the streams” (230).

Lanchester gave a robust reply. First, he put Bairstow in his place by re­minding him of their relative positions in the hierarchy of command. While Bairstow was in charge of much of the experimental work on aerodynamics at the NPL, he, Lanchester, was on the Advisory Committee for Aeronautics, which controlled that work. Would he, Lanchester, be denigrating the very institution for which he had responsibility?

Mr Bairstow has suggested that my paper is in some degree an attack on the National Physical Laboratory, or at least he states that I have done my best to put the Laboratory on its defence. I will say at the outset that the National Physical Laboratory is an institution for which I have the greatest possible respect, and I am happy to count amongst my friends members of the Labora­tory staff, whose work and whose capacity are too well known to be injured by friendly criticism. Beyond this, any criticism which is to be incidentally inferred as implied by my remarks is not only criticism of our own National Laboratory, but equally of every aerodynamic laboratory with whose records I happen to be acquainted. Finally, on this point, any destructive or detrimental criticism of the work being done in the aeronautical department of the N. P.L. must reflect adversely on myself, since I am a member of the Committee whose duty it is to direct or control the particular work in question. (241)

Having sorted out the status question, Lanchester turned to Bairstow’s photographs and the accusation that the circulation theory would only apply to a good aerofoil at the best angle of incidence. Is this really a fault asked Lanchester?

Put bluntly, my answer to this is that it is equivalent or analogous to saying that the theory of low speed ship resistance as based on streamline form, and skin friction, is invalid because it does not apply to a rectangular vessel such as a packing-case, and is only true if applied to the very best design of hull with the finest possible lines. (242)

If the theory applied to a few important facts that was triumph enough. All Bairstow’s photographs, Lanchester went on, dealt with flows outside the scope of his theory.

I consider it quite preposterous to suggest that my theory should be tested by its applicability to the case of a square plane at 40 degrees angle as to test the theory of streamline ships’ forms by tank experiments on a coffin or a cask of beer. (243)

Bairstow claimed that theories of wide scope served the interests of aero­nautical engineers, but Lanchester argued that they cut across, rather than expressed, the engineer’s pragmatic standards. Most practical solutions, said Lanchester, were narrow in scope. No one would expect to compute the “re­sistance of a ship in sidelong or diagonal motion through the water” by the same methods and equations “as those applicable in the ordinary way” (251).

Wilhelm Martin Kutta (fig. 6.1) was born in Pitschen in Upper Silesia in 1867. He lost both parents at an early age and was brought up in the household of an uncle in Breslau. After attending the university in Breslau from 1885 to 1889, he went to the University of Munich, where he studied from 1891 to 1894. Kutta went on to achieve a lasting place in the history of applied

figure 6.i. Martin Wilhelm Kutta (1867-1941). In 1910 and 1911 Kutta published and extended an analysis of the flow of air around the wing of Lilienthal’s glider that he had worked out in 1902 in a dis­sertation at the technische Hochschule in Munich. Kutta assumed that the flow contained a circulation and showed how to link the flow around the wing to the simpler and already solved problem of the flow around a circular cylinder. He was then able to make a plausible prediction of the lift of the wing. After these pioneering papers, Kutta published nothing more. (By permission of the Universitatsarchiv Stuttgart) mathematics for two reasons. First, in his doctoral work of 1900, he developed a numerical method for solving ordinary differential equations. This has be­come known as the Kutta-Runge method and is to be found in all textbooks on the subject.2 Second, he produced a pioneering paper on aerodynamics which appeared in 1910,3 with further developments published in 1911. These papers were based on methods he had developed in his Habilitationschrift of 1902, which he wrote at the technische Hochschule in Munich.4 (This institu­tion is often referred to by its initials as the THM and, for brevity, I follow this practice.) Unfortunately no copies of the Habilitationschrift appear to have survived.5 From the brief summary that was published in 1902, however, it seems to have been the first, mathematical analysis of lift that was based on the circulation theory.6

Kutta was a conscientious teacher who, over the years, introduced hun­dreds of engineering students to the methods of applied mathematics. His mathematical knowledge was said to be of enormous scope and his help was frequently requested by colleagues. He had a deep knowledge of history, literature, and music, a command of languages, including Arabic, and was widely traveled. He never married, however, and was something of a recluse. A colleague of long-standing, Friedrich Pfeiffer, who had been a student un­der Kutta at the THM, wrote an obituary for Kutta after the Second World War.7 In the article, Pfeiffer recalls that Kutta would typically sit alone in the most remote corner of the Mathematical Institute at Munich. After Kutta’s retirement, said Pfeiffer, he and other colleagues would sometimes encounter Kutta, though this happened infrequently. The lack of contact was put down to Kutta’s reticence. When they did meet, Pfeiffer was unhappy with what he found. In later years, he said, Kutta obviously lacked a loving and caring hand (“wie sehr ihm eine liebende und sorgende Hand fehlte”). He went on:

Oft habe ich Kuttas Leben reich und beneidenswert gefunden wegen seiner Aufgeschlossenheit fur so viele Seiten menschlichen Geisteslebens, oft aber fand ich es auch arm und bedauernswert in seiner Einsamkeit und Zuruck – gezogenheit. (56)

I have often found Kutta’s life rich and enviable because of his openness to so many aspects of human cultural life, but I have also often found it poor and rather sad in its solitariness and seclusion.

How were things really, asked Pfeiffer, and did not know the answer. But if Kutta’s inner life was closed to his colleagues, and must be closed to us, his work is open to inspection. Seclusion notwithstanding, he published work that bore the stamp of a time and a place. It was the product of a specific, professional milieu.

Kutta’s career as an academic began in 1894 when he became a teaching assistant in higher mathematics at the THM. Like all the technische Hoch – schulen, the THM had experienced long-standing tensions over the role to be played by mathematics in the training of engineers. How much mathematics should be on the syllabus? What sort of mathematics should be offered, at what level, and who should teach it? These tensions have now been the subject of close, historical study, and thanks to this work there is much about the overall structure of the situation, as well as the particular circumstances in Munich, that can be sketched with some confidence. It is thus possible to form a pic­ture of the context in which Kutta came to do his work on aerodynamics.

Three points must stand out in any general overview. First, the technische Hochschulen (or THs) tended to recruit their mathematics teachers from the universities and, when they were good, lose them again to the universi­ties. This mixture of policy and necessity carried with it certain problems. From the mid-1850s, university mathematics in Germany had been increas­ingly dominated by a concern with rigor and so-called pure mathematics.8 Although the THs provided jobs for mathematicians, those who took the jobs often had their eyes focused on matters that fell outside the concerns of the THs. Their teaching, like their research, was abstract and lacked relevance to engineering. Justifiably, this caused resentment among the engineers, with the result that mathematics appointments often turned into a struggle be­tween different factions in the TH.9

Second, and predictably, engineers were not a homogeneous group. Some engineers wanted to use mathematics as the model on which to construct a “science” of engineering and the nature of machines. The aim was to create a body of knowledge that was general, abstract, and deductive. This movement, which was designed to improve the status of engineering, was associated par­ticularly with the names of Franz Reuleaux and Franz Grashof and achieved considerable influence during the 1870s and 1880s.10 These tendencies in the direction of purity and rigor by one part of the profession provoked an angry reaction in the 1890s from some other parts of the profession. The reaction took the form of an antimathematical movement (Anti-mathematische Be – wegung) led by Alois Riedler at the TH in Charlottenburg. Riedler presented the issue as one of the very survival of Germany in a world where technologi­cal effort must go hand-in-hand with commercial activity and efficient social organization. In this struggle for existence (“Kampf ums Dasein”) there was no place for the speculations of the unproductive classes, whether they be literary or mathematical. The practical men who backed Riedler (the Prakti – kerfraktion) argued that mathematical teaching should be cut down to what was, in their opinion, immediately useful.11

Third, and finally, in 1899, in a measure backed by Kaiser Wilhelm II, the THs were finally granted the right to issue doctoral degrees, hitherto the prerogative of the universities. As a consequence the status, influence, and size of these technical institutions increased steadily in the years leading up to the First World War. The engineering profession was, in many ways, still a divided and fractious body, but in the course of the expansion, the anti­mathematical movement lost much of its force. The alliance of industry and sophisticated science became increasingly acknowledged as an economic and military necessity. The emergence of aviation and the rapid uptake of this subject in the THs helped to consolidate the position of the applied math­ematician and swing the pendulum back to a less hostile stance toward math­ematically formulated theory.12

In his important study of engineers in German society, New Profession, Old Order, Kees Gispen quotes, and expresses agreement with, “a certain Friedrich Bendemann,” writing in 1907, who commented on this swing back and forth between theory and practice and declared that it was time to redress the present imbalance and reintroduce more theoretical training.13 Though Gispen does not mention it, the Herr Bendemann in question, who had re­ceived his doctorate from the TH in Charlottenburg, was a significant force in the aeronautical world. He was a specialist in aircraft engines and propel­lers. In 1912 he was to become the director of the Deutsche Versuchsanstalt fur Luftfahrt at Adlershof outside Berlin.14 Bendemann’s 1907 comments were a direct, and face-to-face, riposte to Riedler. They suggest the growing con­fidence of the aeronautical community in the THs in the face of old schisms and old campaigns.15 Those involved with airships and airplanes were begin­ning to think of aeronautics as a natural home for what von Parseval called the “gebildete Ingenieure,” that is, the educated or cultivated engineer whose thinking, by definition, combined both theory and practice.16

Kutta’s career thus began amid some of the more acrimonious attacks on mathematicians, but he was fortunate to be sheltered from the worst ex­cesses of the Theorie-Praxis-Streit by the special situation in Munich.17 The mathematicians at the THM had long made efforts (though with varying de­grees of determination and success) to accommodate the needs of engineers. They had cultivated a geometrical, visual, and concrete mode of teaching. The trend had started when Felix Klein held a chair at the THM and was continued by his successor Walther von Dyck, who was appointed in 1884 at the age of twenty-seven.18 Von Dyck had been Klein’s pupil and remained a friend and confidant. It has been said that von Dyck played an analogous role in South Germany to Klein’s role in North Germany.19 Von Dyck wanted the THM to be an institution of high scientific merit as well as being tech­nologically oriented. He was able to call upon the support of mathematically sophisticated members of the more technical departments at Munich, such as August Foppl, who likewise had no time for the simple Praktikers.

Kutta was von Dyck’s teaching assistant and frequently took on his classes when von Dyck became involved, as he increasingly did, with running the THM. Kutta also worked with Sebastian Finsterwalder (1862-1951), who held a mathematics chair at the THM. Finsterwalder was significantly more ori­ented to applied work than von Dyck and has been called “der Prototyp des ‘Technik-Mathematikers’”—the prototype of the technologically oriented mathematician.20 As early as 1893 Finsterwalder was giving lecture courses on the application of differential equations to the problems of technology. He was also an aeronautical enthusiast and a member of the local ballooning club.21 It was Finsterwalder who suggested that the topic of Kutta’s Habili – tationschrift should be the mathematical analysis of the flow of air over an aircraft wing. This may be guessed from Kutta’s thanks to Finsterwalder, but the colleague who wrote Kutta’s obituary endorsed the point.22 He said that the stimulus for the chosen topic would, in any case, be clear:

das ist aber fur denjenigen auch klar, der die Jahre kurz nach 1900 im Ma – thematischen Intitut der T. H. Munchen miterlebte. Von Finsterwalders re­gem Interesse an den aerodynamischen Grundlagen der damals in den ersten Anfangen stehenden Luftfahrt wurden auch die jungeren Krafte am Institut angesteckt. Ich denke noch daran, mit welchem Interesse Photographien der ersten Fluge—heute wurde man bescheidener sagen: Sprunge—die Farmen mit seinem Aeroplan bei Paris ausfuhrte, studiert und ausgemessen wurden, Photographien, die Finsterwalder mitbrachte: es wird so 1906 oder 1907 ge – wesen sein. (50)

quite clear to anyone who had been at the Mathematical Institute at the TH Munich in the years after 1900. Finsterwalder’s avid interest in the aerody­namic basis of the first beginnings of aviation at that time also infected the younger people at the institute. I think of the interest with which the pho­tographs of the first flights—today one would more modestly say jumps— were studied and measured. These photographs of Farman with his airplane in Paris, which Finsterwalder brought back with him, would have been in 1906 or 1907.

Finsterwalder’s suggestion to Kutta must have been made some four or five years before the episode with the photographs recalled by Pfeiffer, and thus before the first powered flights had been made. At this earlier date Fin – sterwalder would have been preparing his chapter on aerodynamics for Felix Klein’s encyclopedia of the mathematical sciences.23 The aeronautical adven­tures that were attracting attention at that time were the experiments with hang gliders of the kind pioneered by the engineer Otto Lilienthal. Lilienthal had been killed in a flying accident in 1896 but had left a legacy of both en­thusiasm and information. The information was in his book Der Vogelflug als Grundlage der Fliegerkunst published in 1889.24 Kutta was explicit about the connection between his work and Lilienthal’s machines in both the 1902 account and the 1910 paper.25 The link is clearly evident in the circular arc that Kutta took as his representation of a wing profile. This was not only a mathematical simplification; it also corresponded to the profile used by Lilienthal.26

After his successful Habilitationschrift Kutta continued as teaching assis­tant in the TH Munich until 1907. He then became an extra-ordinary profes­sor (that is, an associate professor) in the same institution. In 1909 he moved on to become an extra-ordinary professor at the University of Jena, and in 1910 was appointed as an ordinary professor (a full professor) at the TH in Aachen. Finally in 1911, the year of his second paper on the circulation theory, he settled down as an ordinary professor at the TH in Stuttgart, where he stayed until his retirement. After his two papers on aerodynamics, in 1910 and 1911, he published nothing more, although he did not retire until 1935 and lived until 1944.