Category The Enigma of. the Aerofoil

Stability and Routh’s Discriminant

As the minutes of their first meeting show, G. H. Bryan had been in touch with the Advisory Committee and, though not a member, was considered central to their effort to understand stability.88 Bryan (see fig. 1.6) was a versa­tile applied mathematician who wrote on thermodynamics and fluid dynam­ics but had become interested in aeronautics through contact with Sir Hiram

Stability and Routh’s Discriminant

figure i.6. George Hartley Bryan (1864-1928), a British pioneer in the analysis of aircraft stability. Bryan applied the mathematical techniques that had been developed by his Cambridge coach Edward Routh. (By permission of the Royal Society of London)

Maxim and the pioneer glider flyer Percy Pilcher.89 Bryan had been publishing calculations on stability since 1904 and was, without doubt, the leading Brit­ish authority in the field.90 It was failure to understand stability, he argued, that led to so many fatal accidents.91 Lanchester had written on stability, but in Bryan’s eyes, and judged by Tripos standards, this work “certainly appears wanting in rigour.”92 Lanchester’s approach was original, conceded Bryan, and he avoided the errors that had vitiated many other attempts, but he did not deduce his conclusions from clearly stated assumptions. Describing how he had arrived at his own, highly mathematical, analysis Bryan recalled that “about the year 1903 I noticed that if a glider or other body is moving in a resisting medium, such as air, in a vertical plane with respect to which it is symmetrical, the small oscillations about steady motion in that plane are determined by a biquadratic equation; and Prof. Love directed my attention to the condition of stability given by Routh.”93 A quadratic equation has the form ax 2 + bx + c = 0, whereas a biquadratic of the kind referred to by Bryan has a term in x 4 and takes the form ax4 + bx3 + cx2 + dx + e = 0. When Bryan said that he “noticed” that the oscillations of a glider were determined by a biquadratic equation, he did not mean that he drew this conclusion simply by looking at a model glider in flight. He meant that he noticed this math­ematical fact in the course of using Newton’s laws to write down the general equations of motion of a body, such as an airplane, moving with a specified velocity and subject to specified forces such as gravity, lift, and drag.

Bryan posed the following question: If an airplane was in steady flight and hence in dynamic equilibrium, and was then subject to small, disturbing forces, such as a gust of wind or a sudden alteration of the control surfaces, what would happen? Would the disturbance die away or would it get bigger and bigger? If the disturbance died away, the machine would count as stable; if the result was that the disturbance became amplified and disruptive, then the machine would count as unstable. He treated the airplane as a rigid body subject to forces of acceleration and rotation. Given the force of gravity and the aerodynamic forces to which it was subject, how did this mechanical sys­tem respond? What sort of longitudinal or lateral oscillations would follow from the disturbance? At this point Prof. Love stepped in. A. E. H. Love, a student at St. John’s, was second wrangler in 1885 and first Smith’s prizeman in 1887. He later became professor of mathematics at Oxford.94 Love appar­ently reminded Bryan that the techniques and concepts he needed to answer his question about stability had already been worked out by Routh, who had been Bryan’s old Cambridge coach. E. J. Routh’s Adams Prize essay of 1877 and his textbook, Dynamics of a System of Rigid Bodies, contained a general analysis of stability for mechanical systems. Both of these works had shown the importance of a mathematical device that came to be known as Routh’s discriminant, an expression whose negative or positive value indicated the stability or instability of the system under analysis.95

Following Routh’s methods, and citing Routh’s results, Bryan was able to reduce the problem of the stability of an aircraft subject to small disturbances to the behavior of an equation of the general form

AX4 + BX3 + CX2 + DX + E = o,

where X was the symbol for the modulus of decay or the strength of the damp­ing tendency on the oscillations that were being investigated. (This equation in X was the “biquadratic” that Bryan “noticed.”) The coefficients A, B, C, etc., in Bryan’s equation were complicated mathematical expressions involv­ing terms that were called “resistance derivatives” and “rotary derivatives.” These described the rate of change of the various forces, and their leverage on the aircraft, relative to its varying conditions of speed and orientation. The values of the derivatives, and hence the values of the coefficients A, B, etc., depend on the details of the particular machine. They could not be calculated from first principles but could be given numerical values on the basis of ap­propriate measurements made on models in a wind channel.

An examination of the four roots, that is, the values of X that satisfy the equation, would determine whether the machine were stable. As Bryan put it, “the small oscillations. . . are determined by an equation of the fourth degree, so the conditions for stable steady motion are those obtained by Routh.”96 Routh had discovered the general result that the stability of an oscillating system required that the coefficients A, B, C, D, and E should all be positive and that the quantity BCD – AD2 – EB2 should also be positive. This latter expression was called Routh’s discriminant. Abstract though it was, it cast light on design features that unwittingly rendered many aircraft dangerous to fly and prone to accidents. The proper mathematical understanding of an aircraft in terms of this equation, argued Bryan, could diminish the risks. In his 1904 paper he had recommended that mathematical investigations should be carried out on any “aerial machines that may be designed or constructed” (115) before they take to the air. Like Haldane, he had no reservations about asserting the priority of theory over practice.97 Bryan’s studies culminated in 1911 in a treatise titled Stability in Aviation

Rayleigh used to say that when he hit a hard mathematical problem he would pick up pen and paper, call to mind his old coach, and “write it out for Routh.”99 This may also have been Bryan’s procedure. That he too was writing it out for Routh is suggested by the way he echoed the title of Routh’s book when he projected a second volume to follow from his own 1911 book.

Bryan intended to call the combined, two-volume work The Rigid Dynam­ics of Aeroplane Motions. The aim was to carry the analysis into much more difficult problems, such as that of circling and helical flight, which would generate an equation with terms involving X to the power eight.100 Whatever the underlying psychological processes, however, there can be no doubt that the skills honed in the Tripos classes and coaching rooms of Cambridge were about to be given a new application, and one whose potential importance would be inestimable.

If he were given the right empirical data about an aircraft, Bryan was in a position to make predictions about its stability. Now the question became: Were those predictions correct? It was not evident, a priori, that even Bryan’s sophisticated mathematics would capture the complex reality of the behav­ior of a real aircraft. At a discussion at the Aeronautical Society, Greenhill, with considerable experience in ballistics to back up his words, expressed his concern that gyroscopic effects such as those from the engine and propeller had been neglected. “I must confess it alarms me,” he said in response to an exposition of the theory by E. H. Harper, a co-worker of Bryan’s, “that w, p, q, have no influence on u, v, r, especially with gyroscopic influence,” where the first three letters referred to rotations around the axes of the aircraft and the latter three to velocities of translation along those axes. Greenhill could not resist a further dig at Bryan by adding that of course the pioneers of flight “could not wait for the solution of a differential equation or its determinantel quartic.” Greenhill’s reservations could only have emboldened the “practical men” in the audience, who also suspected that all manner of simplifications must have been introduced into the calculations. Bryan’s colleague and rep­resentative was questioned closely by Handley Page and others. What about the tangential forces on the wings? Would this approach be of help designing a new machine rather than comparing two given machines?101

Such suspicions were shared by the reviewers of Stability in Aviation in the scientific press. The review in Nature was signed W. H.W.102 The writer was clearly impressed by the book but drew attention to the problematic relation between mathematics and reality, and to Bryan’s uncompromising attitude. The reviewer quoted the following passage, observing dryly, “it strikes the keynote of the book itself.” In this book, said Bryan,

attention is concentrated on the mathematical aspect of the problem for sev­eral reasons. In the first place, there is no obvious alternative between de­veloping the mathematical theory fairly thoroughly and leaving it altogether alone; any attempt at a via media would probably lead to erroneous conclu­sions. In the second place, the formulae arrived at, even in the simplest cases, are such that it is difficult to see how they could be established without a mathematical theory. In the third place, there is probably no lack of com­petent workers in the practical and experimental side of aviation, and under these conditions it is evident that the balance between theory and practice can be improved by throwing as much weight as possible on the mathematical side of the scale.103

Bryan’s position, first stated in his 1904 paper, was that even if the analysis was wrong, provided it was not too wrong, it would provide a “basis of com­parison” and the means for interpreting experimental results “in their true light” (100). As for the problem created for his theory by gyroscopic effects, of the kind that worried Greenhill, Bryan took the view that the fault was with reality not with his theory: “surely it may be left to practical men to get rid of these objectionable influences by proper balancing.”104 This attitude was precisely what worried W. H.W.

W. H.W. was probably Sir William H. White, FRS, an expert in naval ar­chitecture. If so, then the reviewer and Bryan had crossed swords before. At a heated meeting of the British Association in 1910, White had taken Bryan to task for insisting that mathematicians and practical men should stick to their own, separate spheres of activity. The report of this confrontation, as given in the Aero, is worth quoting:

The advocacy of watertight compartments, so to speak, drew from Sir William White a strong protest against drawing any such sharp demarcation, for he conceived the existence of an engineer who was a mathematician and a math­ematician who was an engineer. Sir William White was also somewhat severe on a suggestion made by Dr. Bryan that had the mathematical problems been sufficiently studied many, if not all, of the unfortunate fatal accidents to flying men would have been avoided, and that the practical man’s refusal to work on these lines rendered the accidents the results of foolhardiness rather than bravery.105

Another reviewer of Bryan’s book, this time in the Mathematical Gazette, went into the presuppositions behind the analysis of stability in some detail and remarked:

the author is obliged to make a series of assumptions—that the air resistance on the planes are linear functions of the small changes in linear and angu­lar velocities; that in steady motion they are proportional to the square of the velocity; that they are normal to the planes; that they are proportional to sin a; that the angle of attack a is small; that the pressure on an element of a narrow plane is independent of the motion of neighbouring elements, etc. Methods of approximation are also at times employed to simplify the alge­bra. The cumulative effect of small inaccuracies in each assumption may be considerable.106

Only experiment would reveal if the approximations were cumulative and failed to cancel out. If this were so, then the predictions would fail, however elegant the mathematics and however pure its Tripos pedigree.

A young scientist called Leonard Bairstow—a product of London Uni­versity rather than Cambridge—led the wind-channel work at the NPL that gave empirical content to Bryan’s equations. Working with Nayler and Ben­nett Melvill Jones, a Cambridge engineering graduate, Bairstow provided the data needed to attach values to the coefficients in the equations and hence to check on the viability of the assumptions behind the calculations.107 The mea­surements were delicate, involving the timing of oscillations on models of complete aircraft supported on a spindle, and damped by a spring, when they were exposed, respectively, to still and moving streams of air. As might be ex­pected with difficult experiments, there were problems behind the scenes that were not always apparent in the published reports. As the aircraft designer J. D. North pointed out, “torsional oscillations in the spindle connecting the model with the indicating or recording apparatus” was a disturbing fac­tor and gave rise to “varying results with different moments of inertia of the apparatus.”108 Despite these complications, Bairstow’s experiments seemed to show the models in the wind channel behaving in the manner predicted from Bryan’s equations. There was a gratifying coordination between experiment and theory.

Because both the experiments and the theory concerned small distur­bances, the results necessarily had their limitations, and the scope of the agreement between fact and theory was still open to discussion. Bairstow vigorously defended the work on stability by insisting that some, at least, of the limitations were “more apparent than real.” Consider, for example, “the necessity for assuming infinitesimally small disturbances from the path of flight.” A similar assumption had to be made, said Bairstow, invoking one of the classic achievements of mechanics, when setting up the differential equation for the motion of a simple pendulum. But the solution can then be “applied to oscillations of finite magnitude, without sacrificing any great proportion of accuracy.”109 The appeal to infinitesimal motions does not viti­ate the empirical significance of the inquiry. Rhetorically this was a powerful comparison, and the move from infinitesimal to small, finite disturbances can be justified by the analogy. Cautious persons, however, would note that this argument still left the move from small finite disturbances to large finite disturbances unaccounted for. The inference from the stability of an aircraft under small disturbing forces to its stability when confronted with larger forces therefore remained problematic. Bairstow’s colleague Melvill Jones, who worked on control during slow flying and stalling, and who was strongly supportive of the stability research program, nevertheless acknowledged that Bryan’s equations became inapplicable under these circumstances.110 Some experts also remained troubled by the points raised by the reviewers—that the forces and couples were assumed to depend on linear and angular veloci­ties but not on accelerations.111

The most visible symbol of the British preoccupation with the problem of stability was the excellent BE2, the machine subject to so much hatred in the aviation press. Even here it could not be asserted that de Havilland’s original machine had been stable because it had been designed according to Bryan’s equations. It had not. The aircraft had been the result of good judgment and had then been further improved and, in the form of the BE2C, rendered in­herently stable by subsequent trial and error. This result had been achieved not just by mathematics and the wind tunnel, but also through the danger­ous flying experiments of Busk and his colleagues. Furthermore, whatever may be true of the relation between Bryan’s equations and small-scale mod­els, J. D. North continued to argue that the relation between these equations and full-size aircraft remained problematic. Speaking specifically of the lon­gitudinal damping of the BE2, North said that it was “the only rotary de­rivative deduced from quantitative results” and insisted that even then it had “not shown good agreement with the estimated figures.”112 Added to reser­vations of this kind was a more general issue. Stability was only one of the competing virtues that might be desirable in a design. Maneuverability was another, and often incompatible, demand. Contrary to the critics, Bairstow always maintained that, scientifically and technically, the BE2C was “one of the most interesting aeroplanes ever built.”113 Despite the confidence of men such as Bairstow, however, the politics of stability would not be resolved in the laboratory.

The undoubted achievement of a greater understanding of stability was sufficient to impress at least one practical man, Archibald Reith Low, of Vick­ers (see fig. 1.7). Low was himself a pilot and had designed the Vickers “Gun Bus” of 1913, a machine that earned the accolade of being the first purely military aircraft.114 Low had been to an evening lecture given by Bairstow to the Aeronautical Society on January 21, 1914. Bairstow reported on the NPL’s stability work and illustrated the findings with model gliders.115 In the discus­sion after the demonstration Low, who had previously expressed reservations about Bryan, affirmed his enthusiastic conversion. The NPL work, he said, “constituted a triumphant vindication of Professor Bryan and the Advisory Committee.” He promised (generously if not perhaps entirely seriously) to spend the next couple of years digesting the theory of small oscillations and learning about Routh’s discriminant. Low recalled that, despite the advances

Stability and Routh’s Discriminant

figure 1.7. Grahame-White type 10 aero-charabanc, 1912. A. R. Low is seated third from the left; J. D. North is fourth from the right. (By permission of the Royal Aeronautical Society Library)

he had made, Bryan had been laughed out of a British Association meeting by “so called ‘practical’ engineers.” Low also expressed the hope that the “igno­rant agitation” in the press would be stopped by the dawning realization on the part of those responsible that “there were problems in aviation that they had not begun to be able to understand.”116

Certainly the scientists directly responsible for mastering the problem of stability were in no doubt about the value of their achievements—even if there was more work yet to be done. It was clear to them that mathemati­cians could now contribute to the design of inherently stable aircraft (and they were beginning to convince at least some practical men). In 1915 the Aeronautical Society awarded Bryan their Gold Medal and, if the subsequent history of aeronautics is to be the judge, the honor was well deserved: Bryan’s equations are still used.117 Understandably, Bryan was deeply grateful to those who had rendered his theory applicable. In a letter of February 21, 1916, he said it was an “extraordinary feat” that Bairstow and E. T. Busk and their colleagues had got inherent stability “into a sufficiently practical form to be incorporated into military aeroplanes.” But, he went on, in the present war­time conditions it was necessary for everyone to keep working in both pure and applied research. In the prewar days, “Reissner and Bader were running us pretty hard on the mathematical side,” so no one could be complacent. He ended with a warning: “the Germans are probably putting their best brains into improving their aeroplanes.”118 Bryan was right, and his sentiments did not fall on deaf ears. As Greenhill had said in 1914 in the pages of Nature, this was a “Mathematical War.119 Despite the scoffing directed at mathematicians, the exponents of scientific aerodynamics were proud of their contribution to the understanding of stability and the progress that had been made. “It can­not be regarded otherwise,” said W. L. Cowley and H. Levy, two of the lead­ing experts at the National Physical Laboratory, “than in the light of a signal triumph for mathematical science.”120

G. H. Bryan Reviews Joukowsky

Who was Lanchester’s anonymous reviewer? 55 The most likely candidate was G. H. Bryan. There are three reasons for drawing this conclusion. First, Bryan had been involved with Klein’s mathematical encyclopedia (contributing the article on thermodynamics) and so was in a position to have come across the mention of Kutta in that work. Second, Bryan was the usual reviewer on aeronautical topics used by Nature and was later to review the second volume of Lanchester’s treatise.56 Third, there is a piece of internal evidence. The 1908 review of Lanchester broached one of Bryan’s pet themes: the dependence of aerodynamics on hydrodynamics and the more fundamental status of hydro­dynamics compared to that of the new, would-be discipline. As far as math­ematical theory was concerned, said the reviewer, “aerodynamics as applied to problems of flight does not differ from hydrodynamics” (337). This denial of the independent status of aerodynamics was taken up again a few years later in a review of Joukowsky’s work that appeared in Nature, under Bryan’s name, on February 15, 1917.57

Joukowsky had already published a German-language account of the cir­culation theory in the Zeitschrift fur Flugtechnik for 1910 and 1912.58 In 1916 a book-length exposition of Joukowsky’s seminal work, based on his lectures, appeared in French under the title Aerodynamique, and it was this that Bryan reviewed.59 The subject matter of Joukowsky’s book, insisted Bryan, was not of a sufficiently distinct character to form the nucleus of a new science— aerodynamics. It was “hydrodynamics pure and unadulterated” (465). Bryan also pointed out that there were two ways of “reconciling the existence of a pressure on a moving lamina with the properties of a perfect fluid.” One was by assuming a circulation, and this, he said, appeared to be the basis of Joukowsky’s work. The other, “which has now been greatly elaborated in this country,” was the theory of discontinuous motion. “Of this theory,” sniffed Bryan, “Prof. Joukowski’s treatment is practically nil” (465).

Bryan did not explain why Joukowsky should have discussed the discon­tinuity theory. Though Bryan still adhered to it, most British experts had abandoned it, so some justification for the reproach would have been appro­priate. Nor did Bryan say what might be wrong with the circulation theory. The absence of any detailed engagement with the theory suggests that it was simply considered to be a nonstarter and that the reviewer believed he could count on his readers’ agreement in this matter. But if this part of the argu­ment was implicit, other parts were explicit. Bryan insisted at some length that Joukowsky’s book was of an elementary nature from which little was to be learned—except, that is, by a certain class of engineer. “According to the usual conventions in this country,” said Bryan, “practical and experimental considerations regarding the motion of fluids are classified under the des­ignation of hydraulics” (465). He went on to insist that both hydraulics and hydrodynamics should form the basis of a good engineering education:

It is very important that engineering students who are proposing to take up aeronautical work should be equipped with a knowledge of the necessary hy­drodynamics and hydraulics, and Prof. Joukowski’s lectures were probably admirably adapted to the students in his classes. But the book goes only a very little way towards covering the subject-matter contained in the English treatises on hydrodynamics of more than thirty years ago, with their chapters on sources, doublets, and images, motion in rotating cylinders in the form of lemniscates and cardioids, motions of a solid in a liquid, tides and waves, and detailed treatment of discontinuous motion in two dimensions. (465)

The subjects mentioned by Bryan look suspiciously like the syllabus of an aspiring wrangler. This suspicion is confirmed when Bryan goes on to recommend that any “advanced student” revisit the standard, English trea­tises for “a thorough grounding in hydrodynamics” rather than rely on the “more superficial and fragmentary treatment of the same subject” offered by Joukowsky. Both the tone and the content of Bryan’s review suggest that Jou – kowsky’s book was not taken seriously in its own terms but was being judged as a Tripos textbook in hydrodynamics—and found wanting. It might do for the engineering students in Joukowsky’s technical-college classes, but it would not get anyone through their Senate House examinations.

Theory and Practice

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.

Coffee Spoons and Theology

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

Coffee Spoons and Theology

figure 9.15. Klein’s coffee-spoon experiment. A surface, the “spoon,” is immersed in an ideal fluid and moved forward (a). It is then quickly removed (b), leaving behind a surface of discontinuity (shown in exaggerated form). The result is a vortex sheet and hence the creation of circulation.

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

Coffee Spoons and Theology

figure 9.16. Harold Jeffreys (1891-1989). Jeffreys was a powerful and wide-ranging applied math­ematician who originally approached fluid dynamics from the standpoint of meteorology. Like Taylor and Southwell, he argued that Kelvin’s theorem precluded the creation of circulation in an ideal fluid. (By permission of the Royal Society of London)

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

Подпись: dr dt !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