Category The Enigma of. the Aerofoil

The way Kutta’s creative achievement was reconfigured in terms of the Jou – kowsky transformation, and then subsumed under a sequence of ever more general results, is striking. But generality alone was certainly not the driving force of the development that I have described. The goals that were being pur­sued were not abstract ones. Kutta, Joukowsky, Deimler, Blumenthal, Trefftz, Betz, von Karman, and von Mises were confronting mathematical puzzles, but their puzzle solving operated within a set of identifiable parameters, and those parameters were set by the practicalities of aeronautics. These men were all aiming to make their mathematical tools work for them so that the ideas involved could be brought into closer contact with the problems faced by

engineers who designed wings and built aircraft. Their tools were abstract ones (ideal-fluid theory, conformal mapping, geometry and mechanics), but they were harnessed to engineering goals and exploited or modified accordingly.

The stance the German, or German-language, experts took toward their mathematical apparatus was neither that of the pure mathematician nor that of the physicist. Neither rigor nor purity were central concerns, nor was it their primary goal to test the physical truth of their assumptions. They tested their conclusions for utility rather than their assumptions for truth. Expedi­ency was a prominent characteristic of their mathematical and experimental activity. When Betz looked for deviations between theory and experiment, he was tracking the scope of his approach, not trying to expose its falsity (which he took for granted). While no one directly asserted the literal truth of ide­al-fluid theory (though Prandtl came close), no one evinced much anxiety about its evident falsity either. Not a single author, in any of the papers de­scribed here, even mentioned the problem of how a circulation might arise in an ideal fluid. It was an issue of which they were aware, but it was not a stumbling block.

The particular blend of mathematics and engineering that was visible in Kutta’s 1910 paper was sustained throughout all the subsequent developments that have been examined in this chapter. The most vital ingredient in the blend was the orientation toward specific artifacts and the engineering problems associated with them. There is no evidence throughout the developments I have described that practitioners felt the need to make a choice between mathematics and their practical concerns. On the contrary: the former was seen as a vehicle for expressing the latter. Those working in aerodynamics were confident in their ability to combine mathematics and practicality. The continuity and homogeneity of their work suggest an increasingly secure dis­ciplinary identity. Workers in aerodynamics were beginning to form an intel­lectual community, and they had an institutional basis. Finsterwalder called their discipline “modern” applied mathematics. I have followed August Foppl and brought it under the rubric of technische Mechanik.

The particular form of the unity of theory and practice embodied in tech – nische Mechanik was eloquently affirmed in a lecture given in 1914 by Arthur Proll of the TH in Danzig.66 Speaking at a meeting of the recently formed Wissenschaftliche Gesellschaft fur Flugtechnik, Proll chose as his topic “Luft – fahrt und Mechanik” (Aeronautics and mechanics). Proll surveyed a wide range of topics, including stability and the strength of materials, but he began with the work on lift that had started with Kutta. He described the basic ideas of the circulation theory and reproduced the flow diagrams worked out by Deimler. For Proll this was a clear illustration of how a “good” theory can work hand in hand with practical concerns (“wie eine ‘gute’ Theorie mit der Praxis derart Hand in Hand arbeiten kann”). Responding to the rhetoric of the antimathematical movement, he went on:

Der Kampf ums Dasein mit den Erfordernissen des praktischen Lebens legt auch der wissenschaftlichen Spekulation gewisse Fesseln an und zwingt sie, Uberflussiges oder Unsicheres uber Bord zu werfen. Das ist eine erste gute Frucht der gegenseitigen Verstandigung von Theorie und Praxis, und eine solche finden wir auch hier bei der Aerodynamik vor. (95)

The struggle for existence and the demands of practical life impose certain constraints on scientific speculation and force us to throw overboard what is superfluous or insecure. This is the first fruit of the mutual understanding of theory and practice and it is what we actually find here in aerodynamics.

PrOll was not simply reporting a sequence of results in his field. He was mak­ing the case for a certain style of work and the methodology that it involved. He was celebrating the utility of technical mechanics in the face of familiar criticisms and characterizing that utility by using the slogan of the unity of theory and practice. He was saying what that unity meant for the practitio­ners of technical mechanics.67 This was not lost on his audience, and not all of them accepted his understanding of that unity. Not everyone with an interest in aeronautics was a specialist in technical mechanics, and for them Proll’s claims were not necessarily congenial ones.

On member of the audience was Prof. Friedrich Ahlborn, whose interest in hydrodynamics was empirical not mathematical. Ahlborn was a specialist in, and a pioneer of, the photography of fluid flows.68 For Ahlborn the math­ematics of ideal fluids was just the plaything of theorists who did not realize that experiment alone would yield understanding. In the discussion follow­ing Proll’s lecture, Ahlborn was the first on his feet in order to explain these facts to the assembled company. The work Proll had just described, he said, was mere theory and could be ignored. Ahlborn’s remarks about the Prandtl – Fuhrmann work on airships were scathing. As for the new Joukowsky aero­foils, Ahlborn warned aeronautical engineers that they should not assume that they will make good wings. Only experiment could establish that.69 Proll, he implied, had ignored experiment. Prandtl, who was also in the audience, sprang to Proll’s defense. The lecture, he insisted, had not been one-sided. Proll’s theme was the unity of theory and practice in aeronautics and that, surely, implied the unity of theory and experiment. If Ahlborn was not con­vinced, he, Prandtl, was.

To those who were outside the culture of technical mechanics, the work done by the insiders could seem of little value. This did not just apply to those, like Ahlborn, with no mathematical aptitude. It also applied to those whose mathematical expertise was beyond question, for example, to Cambridge-trained mathematical physicists. As G. H. Bryan had made clear in his review of Joukowsky’s book, the methods that had proven so fertile in the hands of Blumenthal at Aachen, or Betz at Gottingen, were of no interest to him. They seemed too elementary to be of any value, and they appeared to have nothing to teach a good Tripos man. British experts complained that the Kutta condition was arbitrary and, in any case, could not be applied to a rounded edge. Betz, by contrast, felt free to experiment with different posi­tions of the stagnation point and to explore the flow over a rounded and realistic trailing edge. The mathematically precise position of the stagnation point, he argued, was not of great practical significance. The British, unlike their German counterparts, were greatly exercised by the problem of how a circulation could ever arise in an ideal fluid. But where the German group, in one institutional setting, had surged forward and constructed a cumulative, puzzle-solving, and practically oriented tradition, the British mathematicians, in a different institutional setting, turned their backs on the opportunity, and they felt entirely justified in doing so.

From 1922 the Royal Aeronautical Society (RAeS) became the main public forum in London for informed, and sometimes sharp, debate over the merits of the Prandtl theory. This was a matter of deliberate policy. Bairstow had become president of the RAeS and, at an ordinary general meeting on No­vember 2, 1922, reported that the council of the society felt that there should be more opportunity for the expert discussion of technical subjects. Talks to the society need not be kept accessible to a general audience and could be prolonged over more than one session. An obvious topic for such treatment, said Bairstow, was Prandtl’s theory. To start the ball rolling Bairstow invited Major Low to give his “Review of Airscrew Theories.”76

Major Low was Archibald Low, the designer from Vickers, who has al­ready been mentioned in relation to the conflicts between the manufacturers and the Royal Aircraft Factory. Low’s role in that dispute showed that he was not a divisive man, but he always had definite opinions and was prepared to speak his mind. He belonged to the section of practical men inclined to be sympathetic rather than hostile to Lanchester. Having constantly defended the National Physical Laboratory and the Factory from its detractors, he said he now felt justified in offering some outside criticism. In the journal Aero­nautics Low had earlier expressed the view that “there was a tendency on the part of the official circle of aero-dynamic science in this country to think they were absolutely ‘it’ and that there was very little outside. . . . That was a dan­gerous attitude of mind to get into. He believed we no longer had anything like the supreme position of advantage.”77 Low had acquired his rank of ma­jor during the war and was now employed by the Air Ministry. He was based in the library of the ministry and was engaged on translation work. Low was later to become a member of the Fluid Motion Panel of the Aeronautical Re­search Committee. Although his mathematical expertise was not comparable to that of, say, Glauert, his contributions were deemed interesting by authori­ties such as Taylor and Southwell.78 As Bairstow said, when introducing Low, their speaker had earned the reputation of being “very interesting and very contentious.” Laughter greeted this remark, but it may have been nervous laughter.

Low used his talk as an excuse to lay out the basis of the circulation theory and Prandtl’s work. He had a command of the German literature and could not resist taking Bairstow to task for the inadequacy of the foreign references in the latter’s recently published Applied Aerodynamics. Low described for his audience some of the German papers that had been available for a number of years but had lain neglected. He described the basic geometry of confor­mal transformation and sketched the main results of the work on the infinite wing. He then gave a qualitative account of Prandtl’s theory of the finite wing and reported that the transformation formulas, linking wings of different as­pect ratio, had been confirmed experimentally. Here Low quoted the first volume of the Ergebnisse that Prandtl had mentioned in his exchange of let­ters with Southwell.

In the course of the talk it became clear that Low wanted to force Bairstow and others to acknowledge their culpability for neglecting the circulation theory. They had disregarded Lanchester and left it to the Germans to de­velop insights that Lanchester had published in 1907. Lanchester had shown “remarkable insight into the physics of a problem that had baffled scientists of the last century. Had our physicists followed up his ideas, this country might have shared in the work” (43). Low went on to make a comparative observation. He noted that Lanchester’s work on the theory of lift had been ignored in this country while being known in Germany. By contrast, G. H. Bryan’s work on the theory of stability had been fully appreciated in Britain but had made much less impact in Germany. To illustrate his claim Low cited Joukowsky’s acknowledgment of Lanchester in the Zeitschrift fur Flugtechnik and contrasted it with a reproach by Reissner, directed at his fellow country­men, for their neglect of Bryan. As Low put it:

Although not till recently honoured in his own country, Lanchester has had very full recognition in Germany, unlike Bryan, who is generally ignored. In Joukowsky’s words, “Lanchester’s distinguished service is the elucidation of the transition from plates of infinite span. . . to finite span in simply con­nected space” (Z. f.F. u.M., 1910, p. 282). Compare this with Reissner’s reproof to German writers, “Bryan’s highly distinguished service in first (1904) putting the problem of aeroplane stability in complete mathematical form should not be ignored in citing names” (Jahrbuch d. Wiss. Gesell. f. Luftfahrt, 1915-16, p. 141). (43 – 44)

Knowing how these two quoted sources should be interpreted is obviously no easy matter, but Low’s point is an interesting one. Perhaps the strengths and weaknesses of the two nations complemented each other. Any overall assessment of British and German aerodynamics should take this possibility into account.79

In the lengthy discussion after Low’s talk, Bairstow declared that he would speak “mainly as a critic of the Prandtl theory” (62). Bairstow admitted that he was impressed by the way Prandtl had brought experiments on aerofoils of different aspect ratio into agreement and by Betz’s success in bringing cal­culated and measured pressure distributions into alignment. Overall his po­sition was that Prandtl’s theory connected together a great number of facts. It was “a very good empirical theory,” but, he told his audience, they should not think of “scrapping all their previous work.” Prandtl’s theory “was not sufficiently well established” (62). Bairstow declared himself surprised that Low had got through the whole of his lecture “without mentioning a fun­damental property of air on which its motion depends, viz., its viscosity” (63). This brought Bairstow to what he called his fundamental objection to Prandtl’s theory: “They could have various theories which were good or de­fective in various proportions, but ultimately if they were going to deal with a real physical problem they must come back as the basis to physical ideas. They had in the equations given by Stokes, and the experiments of Poiseuille and Stanton, very strong experimental indications that these equations were sufficient to account for the phenomena, whether it was a steady flow or an eddying flow. These equations did not appear in the Prandtl theory” (63).

In what was presumably a reference to the boundary-layer equations, Bairstow said that Prandtl gave “other equations” but that nobody knew what relation they had to the Stokes equations. The Stokes equations were cur­rently the subject of research by a group at Imperial College. The members of this group “naturally looked for the source of the circulation of which the Prandtl theory makes use, [but] without finding it. In the solution of Stokes’ equations it appeared there was no circulation, i. e., the motion of a viscous fluid around a body moving in it was free of circulation. He knew of no natural mechanism which could produce circulation in a viscous fluid and that seemed to him to make a great difference to one’s appreciation of the Prandtl theory” (63). Prandtl’s theory, said Bairstow, apparently speak­ing of both the theory of the boundary layer and the aircraft wing, was not a “fundamental theory” in the way that Stokes’ equations were fundamental. He concluded by suggesting that both Lanchester and Prandtl were aware of these limitations and knew that they had not provided the last word in aero­foil or propeller theory. The “ultimate solution,” insisted Bairstow, must be along other lines.

There was no way in which Low could match the technical authority of this attack, but he was not lost for a tart rejoinder. Casting himself in the role of the “engineer,” responding to Bairstow the “pure scientist,” he said he had no objection to providing scientists with endowments and facilities to allow them to pursue their “strictly abstract studies.” But who knows when, if ever, these studies will bear fruit? As an engineer “he did not intend to wait for them on this occasion” (65). With the benefit of hindsight one cannot deny that Low had a point.80

Kelvin’s theorem did not categorically preclude circulation in a perfect fluid but asserted, conditionally, that it could only exist under certain circumstances. In Britain effort was put into making sure that the proof of Kelvin’s theorem was as rigorous as possible.71 In Germany the focus was subtly different: it was the scope of the theorem that attracted attention. In 1910 Felix Klein pub­lished a paper in the Zeitschrift fur Mathematik und Physik in which he argued

that it was easy to create circulation in an ideal fluid—as easy as stirring a cup of coffee.72 If a thin, flat surface (the “spoon”) is partially inserted in a body of ideal fluid, moved forward, and then briskly removed, the result would be a vortex with a circulation around it—but, said Klein, Kelvin’s theorem would not be violated. The mechanics of the process are shown in figure 9.15. The motion of the surface has the effect of forcing the fluid to move down the front face and up the back face, as indicated by the arrows. Removing the sur­face then leaves two adjacent bodies of fluid moving in opposite directions. The result is a surface of discontinuity, that is, a sheet of vorticity, which then rolls up into a vortex. This does not contradict the theorem, argued Klein, because Kelvin’s proof assumed continuity of the fluid, and this precondi­tion is violated by the insertion and removal of the mathematically simpli­fied “coffee spoon.” The coffee-spoon experiment was not an exact prototype for Prandtl’s confluence argument in the Wright Lecture (because a wing is surrounded by air, not dipped into it), but it surely provided an analogical resource. Klein’s argument encouraged a tradition of critical assessment of Kelvin’s theorem. Further papers, in which the argument was extended and assessed, were written by Lagally, Jaffe, and Prandtl. Later contributions on this theme came from Betz and Ackeret.73 By contrast, Klein’s coffee-spoon paper received no mention in Lamb’s Hydrodynamics.

Prandtl was right to anticipate objections from “the mathematicians” or, at least, from some Cambridge mathematicians. For example, his defense of perfect fluid theory failed to convince the Cambridge mathematician Harold Jeffreys, who later became Plumian Professor of Astronomy and Experimen­tal Philosophy at Cambridge.74 Jeffreys (fig. 9.16) has not previously featured in the story and was not a specialist in aerodynamics. His primary contribu­tions were to geophysics, but he published creative mathematical work in an impressively wide range of subjects. Jeffreys, a notoriously withdrawn man, distinguished himself in part II of the Mathematical Tripos in 1913 and was elected a fellow of St. John’s College in 1914. He stayed at St. John’s for the rest of his life. During the Great War Jeffreys worked at the Cavendish labora­tory on gunnery and then on meteorology with Napier Shaw (who was on the Advisory Committee for Aeronautics). Like his friend G. I. Taylor, Jef­freys originally became interested in circulation and viscous eddies from a meteorological standpoint. In the 1920s, prompted by his lecturing commit­ments in applied mathematics at Cambridge, Jeffreys began a series of papers on fluid dynamics which made explicit contact with the work that had been done on circulation in aerodynamics.

The first in the series of papers, in 1925, was called “On the Circulation Theory of Aeroplane Lift.”75 Although an outsider in the field, Jeffreys sent a copy to Prandtl and received a somewhat formal reply. Prandtl clearly thought

that Jeffreys needed to do his homework. He suggested Jeffreys read the 1904 paper on boundary-layer theory and the 1908 application of the theory by Blasius and duly enclosed the references.76 After something of a delay, Jeffreys acknowledged the response but said, rather untactfully, that he was too busy at the moment to follow up the references. He would get down to them as soon as he could.77 He added: “Of course it would not in the least surprise me to find that all the ideas in my paper had been anticipated, but they were not in any work I had seen & I thought it well that they should be published sim­ply because they were not well known in this country.” It may have been this exchange that gave Prandtl his sense of what topics needed to be addressed in the Wright Lecture and that helped him imagine the archetypal “mathemati­cian” resisting his account of the origin of circulation in an ideal fluid.

In 1930, three years after Prandtl’s Wright Lecture, and after discussions with Glauert and Taylor, Jeffreys published “The Wake in Fluid Flow Past a Solid.”78 Jeffreys started by noting that in many cases it was possible to ap­proximate the motion of a real fluid by a “cyclic irrotational motion, with local filaments of vorticity.” He instanced the work of Kutta and Joukowsky on two-dimensional flow and that of Lanchester and Prandtl on three­dimensional flow. But, he insisted: “The existence of cyclic motion is in dis­agreement with classical hydrodynamics, which predicts that there shall be no circulation about any circuit drawn in a fluid initially at rest or in uniform motion, and that there is no resultant thrust on a solid immersed in a steady uniform current” (376).

As far as Jeffreys was concerned, classical hydrodynamics had long “ceased to be a representation of the physical facts” (376). He agreed with the qual­itative explanation that Prandtl had advanced to show why a perfect fluid theory could be used to approximate a real flow at a distance from a solid boundary, but he did not accept Prandtl’s account of Kelvin’s theorem. For Jeffreys, classical hydrodynamics implied that a wing, starting to move from rest in a perfect fluid that was also at rest, could not generate circulation and lift. Prandtl had argued in his Wright Lecture that the generation of circula­tion and lift was consistent with Kelvin’s theorem; Jeffreys said it was not. Zero lift was the clear and inescapable consequence of the theorem in the case under discussion. Understanding the generation of lift required starting out with the theory of viscous flow. For Jeffreys, as previously for Bairstow, the problem was why ideal fluid theory seemed to work. Inquiry should not concentrate on explaining its numerous failures but on its few remarkable successes. “Considerable attention has been given to the reason why classical hydrodynamics fails to represent the experimental facts; but it appears to me
that these efforts arise from an incorrect point of view. . . the remarkable thing is not that classical hydrodynamics is often wrong, but that it is ever nearly right” (376).

Jeffreys’ way of addressing this question was to anchor the mathematics in physical processes and to make sure that what were really results in math­ematics were not treated as results in physics. Their physical application had to be justified, not taken for granted. Consider, for example, Kelvin’s theorem and the way it was used to explain the creation of circulation around a wing. The vortex that forms at the trailing edge, and then detaches itself, is said to cause the circulation around the wing. The circulation around the depart­ing vortex brings about the opposite circulation around the wing. How? The answer given by Prandtl and Glauert was that Kelvin’s theorem had to be satisfied. Jeffreys was not convinced by this answer, and surely he was right to be suspicious. If Kelvin’s theorem prohibits the creation of new circulation, why are two violations of the prohibition acceptable merely because they are violations in opposite directions? Things that cannot exist cannot cancel out. If it is illegal to drive down a certain street, two people may not drive down it and plead that the law was not broken because they were driving in opposite directions.

Jeffreys wanted to know why Kelvin’s theorem, which was a theorem about inviscid fluids, could be used in the course of an argument in which the role for viscosity had already been granted in order to explain the origin of circulation. This could only be justified if something equivalent to Kelvin’s result could be deduced starting from the premise of viscous flow. To explore this possibility, Jeffreys set himself the goal of deriving the rate of change of circulation with time for a viscous fluid. Kelvin’s theorem for an ideal fluid is expressed by writing d Г/ dt = 0, and Jeffreys wanted to know the value of d Г/ dt for a real, viscous fluid. The general circulation theorem for viscous flow that Jeffreys derived involved the integral of some five separate expressions, each of considerable complexity. For a uniform, incompressible fluid, how­ever, only one of the five terms survived. For aerodynamic purposes, Jeffreys was then able to replace Kelvin’s circulation theorem by the equation

!vdjLdx, = Г vV2u dx,

C dxk ‘ Jc ‘ ‘ where Г is the circulation around the contour C moving with the fluid and V is the kinematic viscosity (that is, viscosity divided by density). In Jeffreys’ equation the three coordinate axes are represented not by x, y, z, but by X;

where i = 1, 2, 3, and the corresponding velocity components are given by щ. The summation convention is used for repeated suffixes, and the term £,ik is the vorticity, which is defined as

f _ duk_

k dXi j •

What did this new expression for dr/dt mean? Jeffreys followed an ear­lier discussion in Lamb’s Hydrodynamics and offered an explanation of the physical significance of the result as follows.79 The equation linking circula­tion and time, he said, can be recognized as one that represents a diffusion process. It shows that vorticity and the circulation it induces obey laws that are analogous to the laws governing the diffusion of temperature or density. From this analogy it follows that vorticity must diffuse outward from a solid boundary. Circulation cannot arise spontaneously within the body of viscous fluid itself. Before the diffusion process has carried the vorticity to regions distant from the boundary, the fluid in these distant regions shows no rate of change of vorticity with time. The rate of change of circulation around a contour therefore depends on the vorticity near the contour. There will there­fore be “no appreciable circulation except on contours part of which have passed near a solid boundary: in other words vorticity is negligible except in the wake” (380).

Jeffreys’ paper “The Wake in Fluid Flow Past a Solid” covered much of the same ground as Prandtl’s earlier but more qualitative treatment in the Wright Lecture, but it is clear that Jeffreys felt that only now had a proper basis been provided for the conclusions that had been advanced. He carefully investigated the orders of magnitude of the quantities involved in the diffu­sion of the vorticity. This analysis, he said, “constitutes the theoretical justi­fication of the ‘boundary-layer’ theory of Prandtl and his followers” (380). Jeffreys’ treatment converged with Prandtl’s but was offered as one “based on the physical properties of a real fluid and not on mathematical conceptions of vortex lines and tubes” (389). In a further paper, “The Equations of Viscous Motion and the Circulation Theorem,” Jeffreys made a similar claim about Prandtl’s account of the origin of circulation and the starting vortex that de­taches from the trailing edge.80 Only an understanding of viscous circulation, said Jeffreys, can provide the real “physical basis” needed for applying the theory of vorticity to real fluids.

Where did this leave Kelvin’s theorem and the (apparently) inconsistent use of that theorem by supporters of the circulation theory? Jeffrey’s position was that the diffusion picture showed that it was not really Kelvin’s theorem that was being invoked to explain the relation between the circulation around the detached vortex and the circulation around a wing. Rather, it was the the­orem for circulation in a viscous fluid that was really in play. Kelvin’s theorem dealt with inviscid fluids, but the counterpart theorem for viscous fluids, the diffusion equation, gave the same numerical result for the initial stages of the flow. “Thus,” Jeffreys stated, “the conditions assumed by classical hydrody­namics are reproduced, in the specified conditions, by the real fluid” (381).

Jeffreys was not alone in saying that Kelvin’s theorem clearly ruled out the creation of circulation by a wing in an ideal fluid. This had been Taylor’s position in 1914, and it was still Southwell’s position in 1930 when he gave the prestigious James Forrest Lecture.81 Southwell asserted that classical hy­drodynamics left the existence of circulation around a wing “an altogether amazing coincidence” (360). He added that the assumption of circulation was “rather theological” (361). The allusion was to Kelvin, for whom the eter­nal character of circulation and vortex rings indicated a divine origin. South­well, like Jeffreys, was unmoved by the first part of Prandtl’s Wright Lecture, dealing with Kelvin’s theorem and perfect fluid theory, but he was enthused by the second part on the boundary layer and the creation of vortices. South­well reproduced Prandtl’s photographs showing the control of the boundary layer by suction and showing how to make a divergent nozzle “run full.” He selected and emphasized the places where Prandtl’s concerns came closest to the long-standing British interest in viscous fluids and eddying flow. South­well further assimilated this aspect of Prandtl’s work to the British tradition by arguing that the analysis of backflow in the boundary layer was similar to Mallock’s work on reverse flow and eddies that was done in the early years of the Advisory Committee for Aeronautics.82

Lamb had also made gentle fun of the theory of circulation by exploit­ing the theological overtones of Kelvin’s theorem. In his Rouse Ball Lecture of 1924, titled “The Evolution of Mathematical Physics,” Lamb had said of perfect fluid theory that “this theory cannot tell us why an aeroplane needs power for its propulsion; nor, indeed, can it tell us how the aeroplane obtains its sustentation, unless by assuming certain circumstances to have been estab­lished at the Creation which, in all reverence, we find it hard to believe.”83 The “certain circumstances,” of course, were the provision of suitably adjusted values of the circulation. Every takeoff and landing, Lamb hinted, would re­quire divine anticipation and intervention. But if the tone was joking, the point was serious. Perfect fluid theory predicts zero drag and makes a mystery out of the origin of circulation. In the last edition of his Hydrodynamics, in 1932, Lamb returned to the problem of the origin of circulation and of un­derstanding how it resulted in a smooth flow being established at the trailing edge of a wing. He clearly felt that no satisfactory account had been given of this. He cannot have been convinced by what Prandtl and Glauert had to say, and his reference to Jeffreys’ efforts was noncommittal. Jeffreys may have deepened the discussion and clarified some of the physical principles, but it was still mathematically incomplete. Lamb summed up by saying: “It is still not altogether easy, in spite of the attempts that have been made, to trace out deductively the stages by which the result is established when the relative flow is started. Fortunately, some beautiful experiments with small-scale models in a tank come to our help. A vortex with counter-clockwise sense is first formed, and detached from the edge, and then passes down stream, leaving a complementary circulation around the aerofoil in the opposite sense” (691).

No one would have been deceived by the understatement. Lamb was say­ing that, by his standards, no one had given a satisfactory mathematical analy­sis of the processes by which a circulation was created. Prandtl may have been able to offer “beautiful photographs,” but that only meant that the analysis was still confined to the empirical level.84 Lamb was surely right. The circula­tory theory was accepted by the British without its supporters being able to offer a rigorous account of the origin of circulation. This had been a source of difficulty for the British from the outset. It was still a worry, but now, with varying degrees of unease, they appeared ready to live with the problem.

Lamb also remained skeptical when Prandtl and Glauert represented the surface of a body, such as a wing, by a sheet or line of “bound” vorticity. Lamb did not claim that the concept of a bound vortex was logically defec­tive, but, in responding to a paper that Glauert submitted to the Aeronau­tical Research Committee in 1929, he deemed it “artificial.” He succeeded in deducing Glauert’s results, concerning accelerated motion in two dimen­sions, by other more usual routes.85 Once again, there was tension between the advocates of two different approaches to applied mathematics: those who insisted that the mathematics described what they took to be physically real processes (and described them in a rigorous way) and those content with mathematical descriptions that were acknowledged to be expedient rather than physically true. Lamb never shifted from the view that he expressed in his presidential address to the British Association meeting in 1925.86 He spoke for many in British aerodynamics when he said that Prandtl had provided “the best available scheme for the forces on an aircraft” (14). The choice of the word “scheme” was meant to imply that Prandtl had failed to give a fun­damental account of the physical reality of the process.87

The conclusion that an understanding of aerodynamics supports, rather than undermines, a relativist analysis of knowledge will sound strange to ears ac­customed to the incessant academic rhetoric directed against relativism.104 Antirelativists confidently use aerodynamics in their attacks on what they see as the debilitating evil of relativism. Airplanes have become a veritable symbol of the absurdity of the relativist position. Consider the following challenge issued by the well-known scientist Richard Dawkins.105 Dawkins who, after his scientific career, went on to occupy a chair in the Public Understanding of Science at Oxford, has said, “Show me a cultural relativist at thirty thou­sand feet and I’ll show you a hypocrite. Airplanes built according to scientific principles work. They stay aloft, and they get you to a chosen destination. Airplanes built to tribal or mythological specifications, such as the dummy planes of the cargo cults in jungle clearings or the beeswaxed wings of Icarus, don’t” (32). Dawkins calls this a “knock-down argument,” and I suspect that most philosophers agree with him.106 In reality it is nothing of the kind. Who are the targets to be knocked down? They are identified as “cultural relativ­ists” and then defined as people who believe that “science has no more claim to truth than tribal myth” (31). There may be persons who cannot distinguish science from myth, but no relativist is committed to such a position simply by virtue of being a relativist. Dawkins has simply chosen an easy target, namely, a foolish version of relativism, and omits to mention that a foolish absolutist might also believe that science has no more truth than a tribal myth. But to reject foolish versions of relativism is not to refute relativism.

Dawkins acknowledges that there are “sensible” people “who, confus­ingly, also call themselves cultural relativists.” The reader is told that these “sensible” relativists believe that “you cannot understand a culture if you try to interpret its beliefs in terms of your own culture” (32). This is indeed a reasonable position, but notice that, on this definition, the sensible relativist stance puts no critical pressure on Dawkins’ position. “Sensible” relativism, so defined, can be conceded by an absolutist (as it is conceded by Dawkins) without any inconvenience. An absolutist can allow that some things are rela­tive as long as not everything is relative. Thus the claim that the meanings of concepts are relative to culture can be accepted because (the antirelativist will say) it is the truth-status of the propositions conveyed by those meanings that really counts, and this, it will be claimed, is not relative. The real argument is therefore not about meaning, it is about truth—as Frank saw clearly.

Dawkins says that airplanes built according to scientific principles work and stay aloft, while those built according to mythological principles don’t. This assertion makes no allowance for the fact that flying machines were mostly developed on a trial-and-error basis by practical men whose stance was often unscientific. Nor does Dawkins allow for cases like the Davis wing. The Davis wing was used on thousands of Consolidated B-24 bombers dur­ing World War II. The aerofoil was produced by the inventor David R. Davis according to a procedure he kept secret. The wing section went into produc­tion because, to everyone’s surprise, it outperformed rivals when tested in the wind tunnel at the California Institute of Technology—von Karman’s home base. When, later, the secret of its design was revealed, it turned out to have no intelligible relation to the laws of fluid dynamics. The procedure was pseu­doscientific hocus-pocus.107

Nor does Dawkins say which scientific principles are supposed to be play­ing the star role in his version of the history of aviation. The discontinuity theory of lift was based on scientific principles and, at one stage, was sup­ported by no less a figure than Rayleigh, but that wasn’t much help. Einstein may have regretted his involvement in aviation, but he was deploying the same formidable scientific intellect that had proven so successful in other areas. On the one side, then, there is the Davis wing, which was unscientific but worked, while on the other side there is the Einstein wing, which was sci­entific but didn’t work. The procedures of science are neither necessary nor sufficient for success.

Dawkins makes mock of what he calls “tribal” science and paints a picture of pathetic, nonflying, cargo-cult replicas of aircraft to drive home the point. He assumes that “tribes” do not have real science, and real science does not have “tribes.” The historical episode that I have studied could be read as a counterexample to this questionable assumption. In a nontechnical sense of “tribe,” I have examined the different practices and rituals of two, scientific tribes. One of these tribes lived on the banks of the river Cam and was called the Cambridge school; the other lived on the banks of the river Leine and was called the Gottingen school.108

At no point does Dawkins grasp the nettle. Relativists deny that humans are in possession of any absolute truths. If Dawkins rejects relativism and uses aerodynamics as his leading example, he must think aerodynamics is a case of absolute truth. Can he really think this? If not, he had better find another example or become a relativist. There is no middle way— other than obscu­rantism and evasion.109 Where does this leave Dawkins’ challenge? Show me a cultural relativist at thirty thousand feet and I’ll show you a hypocrite. Dawkins is committed to a proposition of the form “All As are Bs,” hence “Show me an A and I’ll show you a B.” A necessary and sufficient condition for refuting a claim of this form is to produce an A that is not a B. A sufficient reply would therefore be to introduce Dawkins to Dietrich Kuchemann. Here is someone who knows all about the reality of flying and yet is a relativist about the very science that it involves. Of course, Kuchemann would not normally be called a “cultural relativist,” but I have explained why he must be counted as a bold and unequivocal relativist—and that is what the argument is all about.

The authority behind Kuchemann’s observations about the methods of aerodynamics will be evident. As an aerodynamicist he had the reputation of being one of the best of his generation. His work at Farnborough was de­voted to the aerodynamics of transonic and supersonic flight. Would Daw­kins really dare to impute intellectual hypocrisy to the man who discovered the novel aerodynamic principles embodied in the remarkable wing of the supersonic Concorde?110 Of course, while Concorde was an aerodynamic tri­umph, everyone knows that it was also an economic disaster.111 This makes it a resonant symbol for many things—but the weakness of relativism certainly isn’t one of them. The rise and fall of the Concorde project demands a rela­tivist analysis. It was cases of this kind that Frank had in mind when he said that strengths and weaknesses trade off against one another. This was why he cited the design of airplanes to remind his readers that even the best piece of technology cannot simultaneously meet all human demands at once—and why he then used an airplane as a metaphor for the relativity of scientific knowledge in general.

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?