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

Rayleigh’s Paper of 1876

In 1876 Rayleigh had published a paper called “On the Resistance of Fluids.”1 It contained one of the most striking results of classical hydrodynamics, which came to be reproduced in all the advanced treatises on the subject. By using conformal transformations Rayleigh had arrived at a formula for the

force exerted on an inclined, flat plate subject to a uniform flow of an ideal fluid (see fig. 3.1). The plate is at an angle a to a horizontal flow, and the fluid has a density p and a speed V. If the length from the leading to the trailing edge of the plate is l, then the resultant force R was

4 + n sin a

The resultant is perpendicular to the plate, that is, inclined backward at an angle a so that the vertical (lift) component would be R cosa and the hori­zontal (drag) component would be R sina. Rayleigh was also able to work out the position of the center of pressure, that is, the precise distance of the resul­tant force from the leading edge of the plate. The analysis was carried out in two dimensions, that is, the plate was assumed to be very long. The diagram thus represents a cross section in the middle of the plate, and what happens at the ends of the plate is ignored. The dead fluid can be seen to form a “wake” stretching downstream to an indefinite extent.

Подпись: - + J FIGURE 3.1. The discontinuous flow of an ideal fluid around an inclined plate is often called Rayleigh flow. Rayleigh saw this flow as a model of the flow around an aircraft wing and in 1876 calculated the resultant aerodynamic force on the plate.

At the time of its publication Rayleigh had presciently remarked that his result had interest because “it will be of vital importance in the problem of artificial flight” (431). The diagram in figure 3.1 is thus a drawing of a wing. Nearly thirty years before the success of the Wright brothers, Rayleigh had been attuned to the problem of explaining the lift of an aircraft wing and had offered a theoretical analysis, and perhaps even a solution, to the problem of how it generates lift. Rayleigh had the reputation, as a physicist, of being somewhat conservative.2 His forward-looking orientation to the problem of explaining lift is therefore all the more noteworthy, given that contemporaries such as Lord Kelvin were declaring that they had “not the smallest molecule

Rayleigh’s Paper of 1876

figure 3.2. John William Strutt, Lord Rayleigh (1842-1919). Rayleigh, who had made classic contribu­tions to fluid dynamics, became the first president of the Advisory Committee for Aeronautics. He held this post from the committee’s inception until his death in 1919. From Schuster 1921. (By permission of the Royal Society of London and the Trustees of the National Library of Scotland)

of faith in aerial navigation.”3 In 1909 the task the Advisory Committee set it­self was to see if, and how, Rayleigh’s early approach could be carried further. Greenhill, the mathematician on the committee, and himself an authority on hydrodynamics, set about consolidating and extending Rayleigh’s mathemat­ical analysis. Mallock and his collaborators, meanwhile, surveyed the work of other laboratories and used the resources of the NPL to gather relevant empirical data.

When Rayleigh (see fig. 3.2) first published his formula, there was little experimental data available on air or water resistance against which it could be tested. He had to rely on some old experiments made with an unsatisfac­tory type of apparatus called a whirling arm, which consisted of a horizontal beam or arm that rotated around a vertical axis. The arm had the test object at one end and a means (such as a spring) for measuring the force on it. The problem with an apparatus of this kind was that the test object was repeatedly exposed to the turbulence caused by its previous orbits. These experiments had, however, shown that resistance depended on the sine of the angle of incidence, and this result accorded with Rayleigh’s formula.

The first impression, in 1876, was that theory and experiment agreed

“remarkably well” (437). The impression did not last. By 1891, when Rayleigh reviewed Langley’s Experiments with Aeronautics for Nature, he knew that the experimentally determined relation between the angle of incidence and the aerodynamic force on a flat plate diverged from that stated in his formula.4 If the results are expressed as a graph and the lift force on the plate is plotted against angle of incidence, the curve that derives from Rayleigh’s theory is strikingly different from that derived from experimental measurements of these quantities.5

Rayleigh had a good idea where the trouble lay. It was a matter of what happened on the rear face of the plate. He cited experiments in which the pressure on the back of a plate had been measured. This was done by us­ing a hollow plate and making a hole in the rear surface. A thin pipe was led from the hole through the hollow plate and connected to a manometer. These measurements showed the presence of a suction effect. Rather than the assumed atmospheric pressure, there was a lowering of pressure, which was inconsistent with the model. As Rayleigh conceded, “It will naturally be asked whether any explanation can be offered of the divergence. . . from the theo­retical curve. . . . It seems probable that the cause lies in the suction operative, as a result of friction, at the back of the lamina. That the suction is a reality may be proved without much difficulty by using a hollow lamina. . . whose interior is connected with a manometer” (495). Rayleigh’s 1876 analysis, of course, was based on ideal fluid theory, and therefore the results of friction and viscosity had been ignored.6

Rayleigh had also ignored all the eddies in the flow that would be expected on the basis of simply observing, say, the flow of water past a barrier. Lord Kelvin (Sir William Thomson) objected in the pages of Nature that the “dead water” behind a barrier, or the “dead air” on the upper surface of the wing, was far from dead: it was full of turbulence and instability.7 But if the dead fluid wasn’t really dead, then the free streamlines would be unstable and the entire picture of the flow would be compromised. Rayleigh did not deny that there was a problem here but initially sought to play down its significance. Thus, “it was observed by Sir William Thomson at Glasgow, that motions involving a surface of separation are unstable. . . . But it may be doubted whether the calculations of resistance are materially affected by this circum­stance, as the pressures experienced must be nearly independent of what hap­pens at some distance in the rear of the obstacle, where the instability would first begin to manifest itself” (437). And there the matter was left.

In the analysis of the lift and drag of a wing, the situation was therefore very different from that which prevailed in the study of stability. The pro­gram of research begun in 1909 by the Advisory Committee was, from the outset, plagued with doubts and anomalies. Quantitatively, the predicted lift and drag were not accurately rendered by Rayleigh’s model, and qualitatively they did not seem to correspond to what was known about the physical fea­tures of the flow. Undeterred, the committee pressed on with their program of research.

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.

An Explanation

Understood in terms of schools, traditions, and disciplines, the question of accounting for the response to Lanchester’s work takes on a sociological rather than a psychological form. The psychological machinery of individual cogni­tion must underpin all sociological processes, but that does not mean that individualistic answers can be given to sociological questions. The explana­tion that is needed must involve the interaction of scientific subcultures and institutions and thus go beyond the personalities of those involved. As long as we atomize the process into a sequence of individual responses, determined by the idiosyncrasies of personality, we shall miss the significance of what happened to Lanchester. The central point that I have sought to demonstrate is that in Britain his work was actively assessed by members of a confident scientific subculture—a subculture from which Lanchester himself was ex­cluded. Its members were steeped in the achievements and exemplars of the Tripos tradition and the research of those who carried the tradition forward. The tradition and the line of research growing out of it were diffused through authoritative textbooks such as Lamb’s Hydrodynamics. This book and the growing body of work documented in successive editions was a reference point not only for those who had themselves sat the Senate House examina­tions, but also for those in Britain who trained at universities elsewhere.

As a by-product of this collective assimilation, Lanchester’s work was treated selectively. It was reinterpreted and restructured along lines that were familiar to the group responding to it. Such transformations are routine dur­ing processes of cultural assimilation. Social psychologists are familiar with the process and have given it a name. They call it conventionalization.73 In this chapter I have documented the conventionalization of Lanchester’s work to the norms and practices of Cambridge-style mathematical physics. The process of assimilation governed the way Lanchester’s work was understood. For a number of crucial years it was the precondition for the assessment and the rejection of Lanchester’s ideas.

The assimilation and conventionalization had the effect of simplifying the overall structure of Lanchester’s argument, introducing into it an exclusive emphasis on the behavior of ideal fluids that was not present in the original text. In itself this simplification could be represented either as a distortion or as an improvement in the formulation of the theory. It certainly increased the precision with which the central ideas of cyclic flow and lift were spelled out. In this respect there can be no doubt that the “mathematicians” saw more deeply into Lanchester’s work than did the general run of “practical men.” The mathematicians were doing for Lanchester what Maxwell had done pre­viously for Faraday. Qualitative ideas were cast into a mathematical form. This need have had no detrimental effect on the appreciation of Lanchester’s achievement. The mathematical reformulation could have been the starting point for work that carried the theory forward—as in Germany. In Britain it had the opposite effect and justified the rejection of Lanchester’s work.

The reason was that the local, scientific culture into which Lanchester’s work was assimilated had effectively abandoned ideal-fluid theory as a re­search topic and a research tool. The Euler equations of classical hydrody­namics described territory that had already been conquered, exploited, and left behind by the moving front of fundamental research. British mathemati­cal physicists were confident that ideal-fluid theory dealt only with a math­ematical fiction and not with a physical reality. It was material for examina­tion boards, not research committees. Interest had now moved to Stokes’ equations and the real behavior of viscous and turbulent fluids. By contrast, other experts from a different tradition, who were responsive to different imperatives, could respond very differently to the simplified, mathematical version of Lanchester’s cyclic theory. This is why it was the German engi­neers schooled in technische Mechanik who carried this approach forward. They too were professionally interested in viscosity and turbulence, but their background assumptions and engineering orientation encouraged them, and permitted them, to frame and partition the problems of aerodynamics and fluid dynamics in a different way to their British counterparts. The conse­quences of this orientation toward engineering is the subject of the next two chapters. These chapters contain a detailed discussion of the German work on aerodynamics and provide a further opportunity to see the tradition of technical mechanics in action and to explore its institutional context more deeply. They consolidate the picture that is beginning to emerge.

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.

Confrontation at the Royal Aeronautical Society

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

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

Relativism and Hypocrisy

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 Air as an Ideal Fluid:. Classical Hydrodynamics and the. Foundations of Aerodynamics

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 Memorandum

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

Greenhill’s Memorandum

figure 3.3. Sir Alfred George Greenhill (1847-1927). Fellow of Emmanuel College, Cambridge, and professor of mathematics at the Woolwich Arsenal. Greenhill was one of the founding members of the Advisory Committee for Aeronautics and wrote a detailed report on the theory of discontinuous flow as a basis for aerodynamic theory. Photo by J. W. Hicks, in H. F.B. 1928. (By permission of the Royal Society of London and the Trustees of the National Library of Scotland)

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