Category The Enigma of. the Aerofoil

The following investigations proceed on the assumption that the fluid with which we deal may be treated as practically continuous and homogeneous in structure; i. e. we assume that the properties of the smallest portions into which we can conceive them to be divided are the same as those of the substance in bulk.

Horace lamb, Treatise on the Mathematical Theory of the Motion of Fluids (1879)1

Let me now prepare the ground for an account of the theory of lift and drag. The disputes over the correct analysis of lift and drag provide the central topic of this book. It was here that the scientists and engineers who addressed the new problems of aerodynamics called upon the highly mathematical techniques of what used to be called, simply, “hydrodynamics.” The modern label, which better captures the true generality of the subject, is “fluid dynam­ics.” Fluid dynamics provided the intellectual resources that were common to both the British and German work on lift and drag, although the stance toward that common heritage was often very different in the two cases. It is vital to have a secure sense of what the two groups of experts were disagree­ing about. The present chapter is a description of this common heritage and these shared resources. It is meant to provide background and orientation. In it I do my best to explain the basic concepts in simple terms, though this hardly does justice to the ideas and techniques that are mentioned. I sketch some of the initial, mathematical steps that went into their construction in order to convey something of the style and feel of the work. At the end of the chapter, I summarize the main points in nonmathematical terms.

Two members of the British Advisory Committee for Aeronautics—Lord Rayleigh and Sir George Greenhill—made important contributions to the field of hydrodynamics in the 1870s and 1880s. The numerous references to papers and results by Rayleigh and Greenhill in the standard textbooks of hydrodynamics of that time, for example, Lamb’s Hydrodynamics, attest to their prominence in the field.2 Rayleigh had arrived at some classical re­sults, which are described later in this chapter, and Greenhill had written

the authoritative article on hydrodynamics in the eleventh edition of the Encyclopaedia Britannica. Significantly the encyclopedia had two lengthy and detailed entries that dealt with fluid flow. One was the article titled “Hydro­mechanics” written by Greenhill; the other article was titled “Hydraulics” and was written by a distinguished engineer.3 The former presentation was filled with mathematics, while the latter was filled with descriptions and dia­grams of turbine machinery. The reason it was felt necessary to recognize this division of labor in drawing up the encyclopedia is relevant to my story and will become clear in what follows.

Greenhill’s task was to carry Rayleigh’s mathematical analysis forward. The result was the Advisory Committee’s lengthy Reports and Memoranda No. 19, published in 1910. Its full title was “Report on the Theory of a Stream Line Past a Plane Barrier and of the Discontinuity Arising at the Edge, with an Application of the Theory to an Aeroplane.”8 (The word “aeroplane” here means “wing.”) Greenhill addressed neither the empirical shortcomings of Rayleigh’s model (acknowledged by Rayleigh himself), nor the issue of instability and turbulence in the flow that had been raised by Kelvin (and brushed aside by Rayleigh). He discharged his duty by assembling everything that was known about the mathematics of discontinuous flow. As Greenhill put it, “the object of the present report is to make a collection of all such problems solved so far, and to introduce a further simplification into the treatment” (3). It was not unsolved problems but solved problems and their further refinement that engaged Greenhill’s attention. Particular attention was given in the report to the work of two other Cambridge mathemati­cians, Michell and Love. J. H. Michell was fourth wrangler in 1887. He be­came a fellow of Trinity in 1890 and a fellow of the Royal Society in 1902. In 1890 Michell had written a seminal paper titled “On the Theory of Free Stream Lines.”9 A. E. H. Love’s Cambridge credentials have been mentioned in chapter 1. His paper “On the Theory of Discontinuous Fluid Motions in Two Dimensions” was published in 1891 and provided a development of Michell’s work.10

Michell and Love introduced two new methods into the repertoire for turning free streamlines into simple straight line flow. One method was to make a transformation by taking logarithms. Such a transformation has the effect of turning the arc of a circle into a straight line. It led to a new dia­gram of the flow which had the angle of flow as one axis and the logarithm of the reciprocal of the speed as its other axis. In this way the streamlines were turned into a polygon in the extended, mathematical sense of the word. The other contribution was to make explicit use of the Schwarz-Christoffel theorem, which put the process of finding the necessary transformations on a more systematic basis. Compared with Rayleigh’s original discussion, the

number of transformations had been increased, but the new approach gave the analysis a more routine character, and Greenhill (see fig. 3.3) applied it, indefatigably, to case after case.

There is no doubting the mathematical sophistication of the material that Greenhill gathered together. The report was a virtuoso display of wrangler skills. One could say that it was Tripos aeronautics in full flight—but for one oddity. Where were the airplanes? The cases he collected appear to have little to do with aeronautics. Apart from comparisons with electrical phenomena, drawing heavily on the work of Maxwell and J. J. Thomson, the bulk of the examples treated themes such as the flow of water through orifices, spouts, and mouthpieces. Jets of water impinge on plates, and water flows through channels, around barriers, past piers, and over weirs. Walls, bridges, and pil­lars feature more prominently than flying machines. The puzzle is not the extent to which the discussion deals with water rather than air; these can properly be dealt with together. The problem is why this particular range of examples has been introduced. Given that Greenhill had little to offer that might directly strengthen the connection between Rayleigh’s mathematics and the wings of aircraft, what did he take himself to be doing?

The clue lies in the diagrams. All the cases that Greenhill discussed could be reduced to simple configurations of straight lines. They were all shapes that could be turned into the “polygons” needed for the application of the Schwarz-Christoffel theorem. Interpreting them as “mouthpieces,” “reser­voirs,” “weirs,” “piers,” and the like was distinctly post hoc. This was espe­cially true in the few cases where an attempt was made to link the diagram to aeronautics. Thus Greenhill gave an analysis, analogous to Rayleigh’s, of a flow against an inclined plane, but the main line in the diagram was posi­tioned between two further lines, one above it and one below it. This, said Greenhill, “may be taken to represent a rudder boxed in” between two planes (17). A variant of this figure effectively dispensed with the upper line by lo­cating it at infinity, “so that the analysis will serve for an aeroplane flying horizontally near the ground” (20). The words “may be taken to represent” and “will serve for” reveal the derivative character of the interpretation. The “indirect method” was at work. The examples had been gathered, not because of their relation to wings or aircraft, but because of their relation to a cer­tain, favored mathematical technique. Relevance to what Greenhill called the “analytical method” of the report—which from the outset he identified as the deployment of the Schwarz-Christoffel theorem—was the real principle of selection.11

Greenhill’s report to the Advisory Committee was modeled on a species of document characteristic of Cambridge mathematics and perhaps unique to it. It had become customary for the examiners of the Mathematical Tripos to publish the questions they had set in the previous year, compiling books of problems along with their approved solutions. This practice contributed to a cumulative archive of mathematical work which progressively deepened and refined the Tripos tradition.12 The archive was vital for the coaches in honing the skills of the next cohort of would-be wranglers. It enabled them to identify the main theorems that would be tested so that they could teach their students to recognize all the possible applications of the result, however diverse the fields, and disguised in outer form, they might be. Routh had published such a collection in i860 when he had acted not only as a coach but also as an examiner.13 The most famous collection was Joseph Wolsten – holme’s Mathematical Problems of 1867.14 Greenhill himself, after his stint as a Tripos examiner, had published Solutions of the Cambridge Senate-House Problems and Riders for the Year 1875.15 The 1910 report was just such a col­lection of problems and solutions. One hears the voice of the conscientious coach as Greenhill provided useful hints to his readers to help them avoid errors and traps. “The signs are changed when the area is to the left hand,” warned Greenhill, “so it is useful to employ an independent check of the sign” (5). “It simplifies the work to take i = <x>” (6). Again, “we introduce an angle ф, not to be confused with ф the velocity function” (10).

Although its contribution to aeronautics was close to zero, Greenhill’s R&M 19 soon joined the papers of Michell and Love in the list of canonical sources that were cited in Horace Lamb’s Hydrodynamics.16 A. S. Ramsey, of Magdalene College, who had been seventh wrangler in 1889, did not mention Greenhill by name but introduced his extensive discussion of discontinuous flow in his 1913 Treatise on Hydromechanics by saying, “such problems have recently acquired a new interest because of their relation to Aerodynamics.”17 Others of a more practical bent were less appreciative. The review in the Aeronautical Journal for 1911 was signed “B. G.C.”—presumably Bertram G. Cooper, who was to become the editor in 1913. Cooper was exasperated by Greenhill’s report: there were 96 large-format (“foolscap”) pages of text and 13 sheets of diagrams with, “on the average, about 8 lines of the vernacular to each fsc. page, the rest being mathematical equations.”18

It would doubtless be expecting too much of human nature to ask that the mathematician and the practical man should make up their minds to co­operate. Only, however, by a reasonable combination of the methods of both can the best results be obtained. If, therefore, the Advisory Committee were to lay their heads together and produce a volume giving a quantitative compari­son between solutions of problems as calculated mathematically and as obtained by actual experiment, they would clear the ground enormously, and inciden­tally would do something towards fulfilling the function which the average man (doubtless from the depths of his ignorance) considers that they exist to perform. The publication of an expensive work, such as this, giving no results or deductions in English, is highly to be regretted. (94)

The anonymous reviewer in Flight was equally aghast, calling it the “most extraordinary book yet published relating to the subject of aeronautics.”19 It would be unintelligible to 9,999 out of every 10,000 potential readers. Would some other member of the Advisory Committee, asked the reviewer, please write a nonmathematical report explaining the “practical deductions” to be drawn from Greenhill’s work? The reviewers in the technical journals clearly believed that the Advisory Committee was throwing down the gauntlet to the practical men, and their reaction was predictable. It would seem, however, that these robust responses from the nonmathematical reviewers had an ef­fect. Subsequent publications, when not entirely empirical, typically involved a comparison of theory and experiment. Nothing quite like Greenhill’s report was seen again.20

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.