Category The Enigma of. the Aerofoil

Theory and Experiment

I now move from the context to the content of the Advisory Committee’s work to see how it carried out the research program it had originally set itself. Mr. Asquith assured the House of Commons on May 20, 1909, that the new committee would pursue the problems of aeronautics “by the application of both theoretical and experimental methods of research.”83 No significance

should be attached to the word order, placing theory before experiment, be­cause both found vigorous expression, although the relation between theory and experiment assumed very different forms in the different areas of the committee’s work.

Important tests on full-scale aircraft were carried out at Farnborough, but the main arena in which theory and experiment confronted one another was the wind channel (and sometimes the water channel) in which fluid flow over model wings and model aircraft could be observed and measured. The National Physical Laboratory already had a small water channel, and even a small vertical air channel, but the first task of the ACA was to oversee the construction of a better and more modern horizontal air channel to match those already known to be in use in Paris and Gottingen. By the end of the first year they were able to report on their plans to build a 4 X 4 X 20-foot channel with a draught of nearly 50 feet per second produced by a fan of 6 feet in diameter.84

Difficulty was experienced getting a steady flow, but by keeping the veloc­ity down to 30 feet per second, the flow was found to be “satisfactorily uni­form.” The measuring apparatus for registering the aerodynamic forces on various plates and models was also ready. It was now possible to measure the force component perpendicular to the flow (the lift) and that in the direction of the air current (the “drag” or “drift”). The apparatus could also be set up to determine centers of pressure, and the model could be adjusted to be at any angle with the current without stopping the flow.85

How was the apparatus to be used? Would it be employed to study the behavior of wings and other models in a purely empirical manner to build up an inductive knowledge of the regularities in their behavior? Or would it be used in a theory-testing manner for work that started not with the observable facts but with some theoretical conjecture? If the latter, what theories would be tested and where would they be found? The answer is that both strategies were present in the empirical work. Many of the measurements on model wings involved the highly empirical, and essentially inductive, engineering method of “parameter variation,” that is, systematically altering one factor at a time.86 For example, in one of the studies of a model biplane, the procedure involved keeping the sections, spans, chords, and the distance between the wings constant while altering the angle of stagger in order to try to isolate its effect on lift.87 But there were also bodies of important and sophisticated theoretical work waiting to be explored. The provenance of this theoretical work lay almost exclusively in the achievements of Cambridge mathemati­cal physics. Predictably, the orientation toward the fundamental, theoretical problems of aerodynamics was swept aside in 1914 by the demands of the war, which gave precedence to short-term, practical investigations. Before the cataclysm, however, in the period between 1909 and 1914, theory testing provided the focus for much of the research.

The theories in question concerned two general areas: (1) stability and control and (2) lift and drag. They therefore lay in two quite distinct areas of physics—one being grounded in rigid-body mechanics, the other in fluid dynamics. I consider them in turn, beginning, in this chapter, with the work on stability and, in the next chapter, moving to the fluid dynamics under­lying the theory of lift and drag.

Early British Work on Lift and Drag:. Rayleigh Flow versus the Aerodynamics of Intuition

To the scientist an aeroplane is merely a complex body moving through a fluid, and until he understands how a simple body moves he has no chance of understanding the fundamental principles of aeronautics.

g. i. taylor, “Scientific Method in Aeronautics" (1921)

The research agenda drawn up at the Admiralty and endorsed at the first meeting in the War Office accurately prefigured the approach that was to be adopted by the members of the Advisory Committee in their work on lift and drag. The immediate research aim was to provide a mathematical analysis that would predict the forces exerted on a flat or curved plate immersed at an angle to a flowing fluid. Of course, this was not the ultimate aim. The plate was to function as a simple model of an aircraft wing, and the mathematically idealized fluid, necessary to perform the calculations, was to act as a model of the air. To calculate the forces, researchers needed a precise and quantitative picture of the flow around the wing. What would that flow look like? For the British, the best available guess was provided by Rayleigh’s important work on discontinuous flow. Although the work was over thirty years old, and it was obvious to everyone that the analysis was highly idealized, it appeared to the Advisory Committee that here was the rational place to start. Initially, therefore, as far as lift was concerned, all the research effort of the ACA, both theoretical and experimental, went into studying the theory of discontinuity. I now describe this work and then, later in the chapter, contrast it with the ideas about lift put forward by the leading representative of the “practical men.” The contrast in style is stark.

Anonymity and Connectivity

Lanchester’s Aerodynamics was reviewed anonymously in Nature on August 18, 1908.46 The overall judgment was ungenerous and negative. No reader was likely to come away with the idea that the book contained striking insights into the nature of flight but instead that Lanchester was proposing a theory that was neither original nor successful. The theory was, perhaps, the product of a lively mind, but not a mind whose powers could be relied upon. The parts of the book that contained Lanchester’s most characteristic opinions were de­scribed by the reviewer as “the more shaky theoretical chapters” (338).

There was qualified praise for some of the more empirical sections, which described Lanchester’s experiments on viscosity and skin friction. The glider experiments, conceded the reviewer, gave results that were “remarkably con­sistent.” Lanchester’s account of the “chief methods and results of hydrody­namics,” which lay at the basis of his theory of lift, were described as “on the whole very clearly written,” but the reader was warned that Lanchester was “not, however, content to follow orthodox theory.” It was in chapter 4 of the book, noted the reviewer, that Lanchester “leaves behind the solid ground of orthodox theory” and “attempts to work out the motion of a curved lamina,” that is, a winglike surface (338). Furthermore, Lanchester’s originality was challenged: “It seems to us that the author is wrong in claiming to be the first to give a theory of the motion of curved surfaces, and [in claiming] that Lilienthal had only practical acquaintance with the curved form, for Lilien – thal clearly realised that the effect of curvature was to diminish eddy motion and to give an increased upward pressure due to the centrifugal force of the air. The theory has been worked out mathematically by Kutta, and his results are in fair agreement with Lilienthal’s experiments” (338).

The reviewer then turned to Lanchester’s own explanation of how a curved plate generates lift. It was introduced and dismissed in one sentence: “The author of the present volume attempts to work out the problem by applying the theory of cyclic motion to the motion of a surface in two dimensions, but it is difficult to see how this can have any application to the case of a lamina moving in free air” (338). Before looking into this expression of doubt I must address two preliminary points that concern the reviewer’s mention of Kutta. First, it looks as if the reviewer did not appreciate that Kutta had put forward a cyclic theory. Second, Kutta’s main contributions were published in 1910 and 1911, two or three years after the review. So what was the reviewer’s source?47

Other than personal contact, there were two possible sources of informa­tion. One was a brief account of his work that Kutta himself published in 1902 in the Illustrirte Aeronautische Mittheilungen.48 He gave his main results in the form of a complicated and opaque formula (not the simple product of density, circulation, and free-stream velocity). Kutta said that to reach the given formula he had used conformal transformations, but the assumptions behind his analysis were not explained. The other source was a footnote ref­erence to this article by Sebastian Finsterwalder, Kutta’s research supervisor at the technische Hochschule in Munich.49 Finsterwalder had contributed the article on aerodynamics to Felix Klein’s multivolume Encyklopadie der Math – ematischen Wissenschaften. The relevant volume had been published before the Nature review appeared. The cyclic character of Kutta’s theory was not apparent in the 1902 paper, though its relation to Lilienthal’s work was explic­it.50 The same holds true of the Finsterwalder reference: there was no men­tion of the role of circulation. If these were the sources used, it could account for the misleading way in which Kutta was invoked in the review.

Why did the reviewer find it “difficult to see” how an account of a two­dimensional, cyclic motion could have any application to the motion of a lamina in free air? The reasons behind the difficulty were not explained, so it is necessary to make a conjecture about the argument that was probably in the reviewer’s mind. The worry was about the move from two dimensions to three dimensions. Why should there be a problem about generalizing an ac­count of cyclic or vortex motion in this way? The answer lies in the properties of the space around the wing that mathematicians call “connectedness”—a topological theme with which all Cambridge-trained mathematicians would be familiar.51

Connectedness refers to the conditions under which a contour in the form of a closed loop can be shrunk into a point or stretched and distorted so that it coincides with another closed loop. A “simply-connected” space is one in which every closed loop can be changed into any other closed loop without going outside the space. A “multiply-connected” space is one that is divided by barriers so that it ceases to be true that any two arbitrary loops can be made to coincide. Now a loop enclosing the infinitely long wing cannot be unhooked from it. It can be transformed into any other loop that is itself already around the wing, but it cannot be transformed into a loop that does not go around the wing. The space around an infinite wing is thus “doubly connected,” while the space around a finite wing is “simply connected.”

The move from a two-dimensional analysis to a three-dimensional analy­sis thus involves a move from a multiply connected space to a simply con­nected space. But why should this matter? A mathematically sophisticated re­viewer will have known that, in a simply connected region, the only possible form of irrotational motion is acyclic.52 In an acyclic motion there is no circu­lation and hence no lift. The reviewer seems to have assumed that Lanchester was exploiting a special, topological feature of two-dimensional flow but was then illegitimately applying the analysis to the three-dimensional case.53 This assumption may explain why it was “difficult to see” how a theory of cyclic motion in a surface of two dimensions could have any application to a lamina moving in free air, that is, in three dimensions.

Was Lanchester’s work really vitiated by these considerations? The answer is no. If this was the reviewer’s argument, it was wrong. Lanchester had at­tended with some care to issues of connectivity. He stated explicitly that “we are consequently confined, in an inviscid atmosphere, strictly to the case where the aerofoil is of infinite extent, for a cyclic motion is only possible in a multiply connected region” (162).

How did Lanchester, having formulated the topological problem for himself, get round it? He needed some way to render the space of the three­dimensional case multiply connected. Lanchester did this by appeal to the trailing vortices issuing from the wingtips and reaching back to the ground. This method divided the space in such a way as to destroy its simple con – nectivity.54 In figure 81 of his book (175), Lanchester gave a clear diagram of the vortices reaching back from the wingtips to the ground. But if Lanchester had anticipated and solved this problem, there was still another issue left. If circulation now makes mathematical sense, there is still the physical problem of how it gets started. Lanchester conceded that, as long as the atmosphere was viewed as an inviscid fluid, his vortices could be neither created nor de­stroyed. Such a system, he said, “in a fluid that is truly inviscid would be un – creatable and indestructible” (174). His response was to appeal to the viscos­ity of real fluids: “In dealing with a real fluid the problem becomes modified; we are no longer under the same rigid conditions as to the connectivity of the region” (175). Lanchester’s remarks were perceptive, but the problem of the creation and destruction of vortices, and thus the problem of how circulation could arise, would continue to haunt the theory.


Lanchester came to loath Bairstow and what he called “the Cambridge School”—a group to which he had no hesitation in assigning Bairstow, de­spite the latter’s London provenance.63 Unlike the positive comments he made about the National Physical Laboratory in 1915, in later years Lanchester ex­pressed resentment at the lack of support he had received from that quarter and identified the majority of those working there as effectively belonging to the “Cambridge School.” In a memorandum written in 1936, in which he sought recognition from the Air Ministry for his contribution, Lanchester expressed himself with some bitterness: “The trouble is, or arose from the fact, that with the exception of Lord Rayleigh, the N. P.L. did not take my work seriously. . . . They fell into the error, and for this Leonard Bairstow was mainly to blame, of casting doubt on my work, I believe because my methods did not appeal to them in view of their training. They mostly belonged to the Cambridge School, whereas I was the product of the Royal College of Science (then the Normal School of Science)” (19-20).64 He recalled that, on more than one occasion, Bairstow had asserted, during meetings of the Advisory Committee for Aeronautics, that “we do not believe in your theories” (20). In an earlier letter of 1931 to Capt. J. L. Pritchard, the secretary of the Royal Aeronautical Society, Lanchester referred to “that man Bairstow who would have nothing of the vortex or cyclic theory and took every occasion when I was a member of the Advisory Committee to laugh and jeer at it.” 65

The minutes of the Advisory Committee do not contain any specific re­cord of episodes of this kind.66 Whether those writing the minutes drew a veil over such exchanges or whether Lanchester’s memory was at fault is im­possible to determine. Nevertheless there is no reason to doubt the essential accuracy of Lanchester’s account, and the minutes contain clear evidence of Bairstow’s opposition. There is also ample corroboration in the public realm. As J. L. Nayler, the secretary of the committee, put it, in his early years Bair­stow was “a dominant and almost pugilistic character.”67 In another letter to Pritchard, Lanchester left no doubt as to where he placed the blame for the opposition to his work. “The whole thing,” he asserted, “originated with Bairstow backed up by Glazebrook.”68

The personalized focus of this explanation has been taken up by others. This was the line taken by J. A. D. Ackroyd in his Lanchester Lecture of 1992. After giving an authoritative account of Lanchester’s contributions to aerody­namics, Ackroyd posed the question of why there was so little interest in the circulatory theory. “Perhaps part of the problem,” he suggested, “lay in the personalities involved.” 69 Ackroyd, however, did not place all the emphasis on Bairstow’s personality but noted the role of Lanchester’s own strong per­sonality and his inclination to be critical of Cambridge and London graduates and the work of the NPL. Perhaps, Ackroyd concluded, there was a mutual an­tipathy between the persons involved. In developing this argument, Ackroyd cited and endorsed the psychologically oriented explanation that had been advanced some years previously by the eminent applied mathematician Sir Graham Sutton FRS. Sutton pointed to what he called Lanchester’s “isolation” and put this down to Lanchester having been one of the great “individualists” of science. “Throughout his life he remained an individualist, perhaps the last and possibly the greatest lone worker that aerodynamics will ever see.”70

The clash of personalities must be part of the story, but can this really be the explanation of the opposition to the cyclic theory? I do not believe that it can. Consider the role of Bairstow’s personality. In the survey that I gave of the reasons advanced against Lanchester, it is clear that Bairstow’s arguments were aligned with those offered by others, such as Taylor, Cowley, Levy, and Lamb. Later I shall add more names to this list. I have seen no evidence that suggests they shared Bairstow’s main personality characteristic, that is, his aggressiveness. They had their own, quite different, personalities. Levy, for example, always said Cambridge was an unattractive place where the math­ematical traditions were too “pure” for his tastes. With his Jewish and Scot­tish working-class background, he said he did not feel socially or politically comfortable in Cambridge and declined the chance to do postgraduate work there. Levy’s class consciousness and bitterness at the blighted lives he had witnessed in the slums never left him.71 After graduating from Edinburgh, however, Levy used his scholarship funds to visit Gottingen (where he met von Karman) and then took himself to Oxford to work with the Cambridge – trained Love. The relation between Levy’s personal feelings and this career trajectory is not easy to fathom,72 but perhaps we do not need to understand such matters. What can be said about all these diverse and complex person­alities is that they all took a similar stance on the central, technical problems that were in question. They shared professional opinions and judgments, not individual personality traits. The explanation in terms of personality, there­fore, breaks down. The candidate cause (personality) varies, but the effect (resistance to Lanchester’s ideas) stays the same. This means that we must look elsewhere for the real cause.

What, in any case, would be the basis of an account that rested on an appeal to personality? No one believes that certain psychological types are selectively attracted to this, that, or the other preferred pattern of fluid flow, whether viscid or inviscid. Those who invoke “personality” generally do so in order to explain the disruption of a process of rational assessment that (it is assumed) would otherwise have proceeded in a different way. It is offered as a way of explaining why things went wrong. It is meant to explain why a theory was rejected when it should have been accepted, and the answer is found in individual psychological traits. But given that the assessment of Lanchester actually rested on the appeal to shared standards, common to a group of otherwise diverse individuals, this explanatory approach bypasses the most salient feature of the episode. Its outstanding characteristic was its systematic and shared nature. It had the character of a concerted action by a group.

A further point needs to be stressed. An examination of the technical ar­guments that were used against Lanchester suggests that the response to his work was not a disruption in the rational working of science but a routine ex­ample of it. It was orderly, consistent, and reasoned and drew upon a refined body of received opinion and technique. It is true that some of the complex­ity was factored out of Lanchester’s text, but that again was a consistent and shared feature of the response, not an individual variable. Personality played its part, but only by giving a different tone, and a different degree of intensity, to the expression of a central core of repeated, and overlapping, argumenta­tion. The common content of the arguments derived not from individual psychology but from participation in a shared scientific culture.

Lanchester himself hinted at an explanation of this kind. As well as his explicit and angry psychological account, focused on his irritation with Bair – stow, there was also an implicit, more sociological dimension to his account of the resistance to his theory. This aspect surfaced in his reference to the “Cambridge School” and the common background of training of the scien­tists at the NPL. We should also recall his 1917 discussion of the organizational characteristics of well-conducted aeronautical research. This, too, can be read for its bearing on the resistance encountered by Lanchester’s work. His central preoccupation was that the different parties to the process of research should confine themselves to their proper spheres of competence. No good would come, he argued, of mathematicians and physicists encroaching on territory outside the (narrow) limits of their expertise. What could have been in Lanchester’s mind? What examples of invasive physicists might he have cited? The public confrontation with Bairstow, two years previously, when they clashed over the proper scope of a theory of lift could not have been far beneath the surface. Was it necessary to find a universal law of nature, as Bair – stow wanted, or would a specialized, practically oriented approach suffice, as Lanchester believed? Whether or not this was the example in Lanchester’s mind, it illustrates the general problem to which he was referring, namely, the problem of the division of labor.

The division of labor generates a diversity of specialized perspectives and localized forms of knowledge. Professional subgroups and disciplinary divi­sions such as those between mathematical physics and technical mechanics are instances of this general phenomenon. What happens when the product of one of these subgroups and perspectives is assessed from the standpoint of another, different subgroup with a different perspective? We have here all the preconditions for a small-scale culture clash. Has the knowledge claim been properly understood, or has it been misinterpreted? Is a contribution to one project being assessed (deliberately or unwittingly) by criteria more appro­priate to another project? If I am right, this is exactly what happened when Lanchester’s work was assessed so negatively by the “Cambridge School,” and it was this problem (although it was not the only problem) that Lanchester was addressing when he discussed the proper organization of aerodynamic research.

Generalizing the Mathematics

Not only is a perfect fluid a theoretical substitute for a real fluid, but the geometry of the Joukowsky profile is a theoretical substitute for a real aero­foil section. The mathematics of the Joukowsky transformation of a circle always gives a profile with some highly unrealistic properties. At the trailing edge, the tangents to the upper and lower surfaces of the wing coincide with one another. The trailing edge is like an infinitely thin blade. No engineer would design such a wing, and no workshop could produce one. At most they could produce an approximation of the kind that the Gottingen workshops must have produced for Betz. This raised a question: Could the mathemati­cal advantages of the Joukowsky transformation be retained while avoiding the unrealizable features of the profile? Could a transformation be found that turned a circle into a winglike profile whose trailing edge met at some speci­fied, nonzero, angle? The answer to these questions is yes.

Once again it was members of the technische Hochschule at Aachen who provided the answers. In 1918 Theodore von Karman and Erich Trefftz showed that this job could be done by a transformation from a circle in the Z-plane to a wing profile in the z-plane that took the form

z – kl _ (Z-l)k z + kl ~ (Z + l)k ’

where k is a constant less than 2 and l is the length that featured in the pre­vious discussions by Blumenthal. Whereas the Joukowsky transformation effectively draws a wing profile that has a circular arc as “skeleton,” the

Подпись: FIGURE 6.14. Karman-Trefftz profile. Unlike the Joukowsky profile, in the Karman-Trefftz profile the upper and lower surfaces meet at a nonzero angle at the trailing edge. The “skeleton” of the aerofoil is not the arc of a circle but a crescent. This family of profiles avoids the unreality of the Joukowsky aerofoils, which have an infinitely thin trailing edge.

Karman-Trefftz transformation produces an aerofoil that has a crescent or sickle shape, made up of two circular arcs, as its “skeleton.”61 Just as the trail­ing edge of the Joukowsky profile shades into, and becomes, the single arc of its skeleton, the Kutta arc, so the trailing edge of the new profile combines with, and becomes, the endpoint of the crescent (see fig. 6.14).

Generalizing the Mathematics

Von Mises pointed out that the Karman-Trefftz formula is a close relative of the Joukowsky formula.62 Starting with a Joukowsky transformation in the form

he showed the link in three simple steps. First, subtract the quantity 2І from both sides. Second, write down the Joukowsky formula again and, this time, add 2І to both sides. Third, form the quotient of these two expressions. The result is another version of the Joukowsky transformation that looks like this:

z – 2l = (Z-1 )2 z + 2l _ (Z +1)2′

The Joukowsky transformation, with its knife-blade trailing edge, is thus a special case of the Karman-Trefftz transformation, that is, the case where the exponent is k = 2. Replacing the exponent 2 by a value of k where k < 2, gives the formula for a transformation that generates an aerofoil with a more real­istic trailing edge. As k gets smaller, the angle at the trailing edge gets larger.

Von Karman and Trefftz ended their paper by posing the following ques­tion: Given some arbitrary, but plausible, aerofoil shape, is it possible to dis­cover a transformation that will relate it to a circle and thus allow the flow to be predicted? It is one thing to be given, or to discover, a transformation that will go from a circle to an aerofoil-like shape, but starting with an aerofoil and trying to find the transformation is quite a different matter. This is the

question that an aircraft designer would pose. What will be the properties of the wing if it is built like this rather than like that?

Von Karman and Trefftz argued that if a conformal transformation is applied in reverse to some given profile, it may not turn it back into a circle but will turn it into a shape that is not greatly removed from a circle. They then offered a transformation that would, to an adequate degree of accuracy, turn this near circle into a better circle. They thus began to address the way in which ideal-fluid theory could be applied not just to a few favored “theoreti­cal” aerofoils, but to any shape that might come from the drawing board of a designer—shapes that would be strongly influenced by the contingencies of the construction process.

The Karman-Trefftz transformation showed how to avoid the unrealistic cusp at the trailing edge of the Joukowsky profile, but it did this at the price of a certain complexity. Betz argued that the extraordinary simplicity of the orig­inal Joukowsky transformation was worth preserving. The Karman-Trefftz transformation, he said, was difficult to use in practice. He then exhibited a much simpler way to achieve a finite angle at the trailing edge by a modi­fication of the original graphical method used by Blumenthal and Trefftz. The modification produced a profile with a slightly rounded rear edge, and this again raised the problem of the position of the rear stagnation point. How was the circulation to be determined? Betz declared that from a practi­cal point of view this indeterminacy was of no great significance because the real circulation was always smaller than the theoretical prediction. In reality, even the usual Joukowsky profiles do not unambiguously determine the cir­culation: “also auch bei gewohnlichen Schukowsky Profilen nicht eindeutig bestimmt ist.”63 Betz suggested that some point on the rounded edge could be designated to play the role of the sharp edge of the original profile when calculating the circulation.

The cusp on the trailing edge of the Joukowsky profile was not the only problem. There were other respects in which this family of aerofoil shapes differed from those which experience and practice were beginning to favor. Typically, Joukowsky profiles were too rounded and bulky at the front and too thin at the back, even when the zero angle of the trailing edge was avoided. Also, the maximum camber lies near the center of the chord rather than, as was preferred in practice, in the first third of the chord. How were these prob­lems addressed? In a series of articles in the Zeitschrift fur Flugtechnik, begin­ning in 1917, Richard von Mises suggested a generalization of the Joukowsky transformation that could yield aerofoils that met almost any specifications of their geometrical properties. Such aerofoils could be designed in a way that

Generalizing the Mathematics

avoided the faults identified in the original Joukowsky profiles.64 Von Mises explored transformations of the following kind:

Any aerofoil could be described given a sufficient number of terms in this sequence.65 The Joukowsky transformation was a special case of the formula for which n = l. Von Mises also wanted to show how the parameters that governed the conformal transformation of the circle were related to the aero­dynamic characteristics of the resulting wing. For example, he showed how to construct a profile in which the resultant aerodynamic force always acts through the same point of the wing, a point that came to be called the aero­dynamic center of the wing. The result was that the pitching moment of the wing was zero for all moderate angles of attack (that is, for the straight-line part of the curve relating lift to angle of attack). This was a property of poten­tial importance for the stability and handling properties of an aircraft. The general shape of a von Mises profile is shown in figure 6.15. Like the Karman – Trefftz profiles, it avoids the cusp at the trailing edge, but in addition it is characterized by a shallow S-shape with a slight upturn at the rear edge.

Glauert Makes the Case

On March 30, 1920, before he went to Germany, Glauert had presented his “Notes on the German Aerofoil Theory” to a meeting of the Aerodynam­ics Sub-Committee which included Glazebrook, Greenhill, and Lamb. The notes amounted to a brief overview and assessment of two conversion for­mulas. Part 1 of the paper dealt with the formula linking the aerodynamic characteristics of monoplane wings of different aspect ratios, while part 2 concerned the link between monoplane wings and biplane configurations. Glauert stated the relevant formulas without proof and simply said that they were taken from “German Technical Reports.” In his comments to the sub­committee, when introducing the notes, he added that he had not yet been able to locate the papers giving the theory on which the formulas were based. His aim was to marshal some empirical data to find out if the formulas gave the right answers. He concluded that in some cases they did but in some cases they did not. In general, the transformation formulas discussed in part 2 of the notes seemed problematic, while those in part 1 worked well for predict­ing the induced drag but badly for predicting the induced angle of incidence. Because the (good) result for drag is theoretically dependent on the (bad) result for angle of incidence, Glauert declared himself puzzled.

These notes provided the basis for an article that Glauert published soon afterward in the short-lived journal Aircraft Engineering.66 The article gives further insight into the status he accorded to Prandtl’s theory before the Got­tingen visit. Glauert put it like this: “Good agreement is not obtained for the angle of incidence, and as the theory estimates the change in drag from the effective change in incidence, it is evident that the basis of the theory cannot be regarded as quite satisfactory. The form of the expression found for the induced drag has a certain theoretical justification, but it is probably safer to regard the results as empirical formulae which are confirmed by experi­mental results” (161). The tone of this conclusion, in which Prandtl’s results were accorded the status of mere “empirical formulae,” contrasted with that adopted after his visit with McKinnon Wood to Gottingen and his talk to Prandtl.

In February 1921, after his return from Gottingen, Glauert produced his report T. 1563 on the outcome of his talk with Prandtl. The report was titled simply “Aerofoil Theory” and was based on six sources, all by Betz, Munk, and Prandtl.67 These sources included Prandtl’s Tragflugeltheorie and the Gottingen dissertations of his two assistants. Glauert’s aim was to give “an account of the development of the theory and of the main results contained in the original papers” (2). He divided his report into five sections: (1) aero­foils of infinite span, (2) the finite monoplane wing, (3) special cases of the monoplane wing, (4) biplane wing structures, and (5) the influence of walls and the free boundaries of a stream on the flow in a wind channel. What fol­lowed was one of the most lucid accounts that has ever been given of the ba­sics of the subject. Farren and Tizard refer to the “faultless style” of Glauert’s exposition.68 Although some of the same reservations were carried over from the earlier “Notes on the German Aerofoil Theory,” for example, the empiri­cal weakness of the prediction of the induced angles of incidence, these were not deemed to be of great practical importance compared to the accurate predictions of induced drag. Furthermore, the fuller treatment of the relation between monoplanes and biplanes had removed some of the earlier doubts. In the light of further analysis, Glauert now concluded that “the theoretical formulae may be accepted as giving a reasonably accurate method of predict­ing the biplane characteristics from those of the monoplane” (26).

How was Glauert’s report received? What, for example, did the Aerody­namics Sub-Committee make of it? At meeting 38 of the subcommittee on April 5, 1921, minute 375(b) records that “Prof. Lamb remarked that he had read the report with great interest and considered it a very valuable addition to aerodynamic theory.” Lamb did, however, say that he found the vortex lines difficult to visualize, and J. D. North suggested that the relevant dia­grams were to be found in Lanchester’s book.69 (Whether Lamb found those diagrams acceptable or whether, like Prandtl, he thought they were wrong, is not recorded.) Although one may wonder about the identity of the (implied) prior theory, to which Prandtl’s theory was a “valuable addition,” Lamb’s re­sponse may seem positive enough. There is, however, a second version of Lamb’s reaction which must put a question mark over this positive inter­pretation. The second version is given in the minutes of the full Aeronauti­cal Research Committee that met for its tenth meeting a few days later, on Tuesday, April 12, 1921, at the Royal Society. (Lamb now served on both the Aerodynamics Sub-Committee and the full research committee.) Minute 111 of the full committee meeting deals with the business of the subcommittee and refers to Glauert’s “Aerofoil Theory” as “report (ii).” It reads as follows: “The report (ii) was stated by Professor Lamb to form a good basis for the commencement of work on the development of an aerofoil theory. Professor Bairstow expressed his dissent.”

Had Lamb moderated an earlier, more positive response or did the later minutes simply capture nuances that were lost in the earlier summary? When one recalls the highly qualified wording that Lamb had used in his Hydrody­namics, when describing Kutta’s work, the later minute seems closer to the authentic voice of this cautious spokesman of the Cambridge school. Either way, the full Aeronautical Research Committee did not receive Glauert’s ac­count of the Gottingen work with open arms. Bairstow was clearly not im­pressed by what he was hearing of Prandtl’s achievements, and Lamb’s ap­parent support now had so many qualifications that it is difficult to decide whether he was really being supportive or not. To say that something is a “basis” for a “commencement” of a “development” is not to say a great deal.

Undeterred by this response Glauert presented a second report in May 1921 called “Some Applications of the Vortex Theory of Aerofoils,” which dealt with both wing theory and propeller theory. (I confine myself to the former.) Glauert was clearly in no mood to compromise and began by assert­ing that his previous paper had led to “a satisfactory theory for correlating the lift and drag of different wing structures and for determining the effect of changes of aspect ratio.” His aim now was to see whether it gave an accurate picture of the flow of air in the vicinity of the wing. Glauert’s talk of the the­ory “correlating” data suggests he may have still been concerned lest Prandtl’s approach merely provided empirical formulas rather than a physically true account of the actual air flow. His intention was to address this anxiety by comparing the calculated and observed “downwash” of air at three locations in the vicinity of a wing: (1) above or below the center of the wing, (2) behind the (main) wing in the region of the tailplane, and (3) at the wingtips.70

Before making the comparison Glauert entered a caveat. Prandtl’s theory rested on drastic simplifications, and these would necessarily preclude it giv­ing an accurate picture of certain features of the flow. First, the wing was replaced by the abstraction of a “lifting line.” For both the simple horseshoe model and the refined model, with a varied distribution of circulation along the span, the chord of the wing was neglected. So the flow close to the wing could not possibly be described accurately. Second, where a vortex sheet was assumed to be issuing from the trailing edge, the sheet would roll up, so the flow behind the wing would have a different character at different distances. As a partial response to this second problem Glauert performed his calcu­lations of the downwash in two different ways: (1) on the assumption of a constant distribution of lift (the simple horseshoe model) and (2) on the as­sumption of an elliptical distribution of lift (the refined horseshoe model). He argued that the rectangular wing used in the experimental tests would have a lift distribution somewhere between these two extremes. Further­more, the trailing-vortex system near the wing would be more like the re­fined model, whereas the system at a distance would be more like the simpler model. Glauert argued that provided the tests were not carried out too near the wing, or too far behind it, the theory ought to give a reliable picture of the surrounding airflow.

The first of the three tests used downwash data taken from a BE2E biplane with its wings at an angle of incidence of 6°. Measurements were made along an axis that was normal to the wing at its midpoint. For distances away from the wing of greater than one and a half times the chord, it was found that the predictions based on a uniform loading agreed fairly well.71 Other results, however, using wind-channel data from a monoplane wing with an RAF 6 section at an incidence of 3°, showed a downwash that was much greater than predicted. The second test measured downwash along the longitudinal axis, that is, at a number of points toward, and beyond, where the tailplane is typi­cally located. Observed values of the downwash were progressively smaller than those predicted by the elliptical distribution of lift but larger than those to be expected on uniform distribution. The trend of the results was roughly right but not the numerical values. The third test concerned the flow at the wingtips, and theoretical calculations were compared with wind-channel measurements made on a model Bristol fighter. This time, only calculations based on the elliptical distribution were used. (A uniform distribution was ruled out because it failed to represent the fact that lift falls to zero at the tips.) The predictions agreed with the observations in showing that, as one moves along the span of the wing, downwash decreases toward the tips and turns into an upwash beyond the tips.72

Glauert admitted that he was perplexed by the mixed results of the first test but deemed the results for the flow round the wingtips “quite good.” The theory represented the flow “with reasonable accuracy,” especially given all the approximations involved.73 The results for downwash on the longitudinal axis obviously took Glauert into the area studied by Foppl in the very first published test of Prandtl’s theory. Glauert concluded that the theory “cannot be used in any simple manner to predict the angle of downwash behind the wings”—which is exactly what Foppl had tried to do. The operative words, though, are “in any simple manner.” Glauert pointed out that the inaccuracy probably arose because the vortex sheet behind the wing was unstable and so the theory must be made more complicated to allow for this effect. He then noted that Prandtl had offered some suggestions about how to describe the rolling up of the vortex sheet. These promised to bring calculation and obser­vation back into alignment.

The study of the downwash behind a wing structure had held the promise of giving “a direct method of testing the underlying assumptions of the the­ory,” but it proved to be a complicated phenomenon and generated a lengthy and ramified program of experimental and theoretical investigation.74 Glau – ert was clearly sensitive to the problematic character of the empirical data and the complex relation between theoretical calculation and experimental measurement. It is also clear that he did not treat the empirical difficulties confronting Prandtl’s theory as refutations of the theory. He saw them as challenges that called for its further development. In a quiet but determined way Glauert shouldered the burden of developing the theory mathematically, and he did so, for a while, almost single-handedly. Farren and Tizard said Glauert was a “bonny fighter” in argument and worthy of any opponent, but they remembered him as a man of “essential modesty and gentleness.”75 This characterization accords with the calm and nonpolemical character of every­thing he wrote. Not all of those who came to support Prandtl shared these character traits. One who did not was the redoubtable Major Low.

Negotiating Kelvin’s Theorem

Prandtl’s lecture had the title “The Generation of Vortices in Fluids of Small Viscosity.” The choice of subject matter is revealing. Prandtl used the oppor­tunity to address the two problems that most worried the British. First, how did circulation arise? Second, why did perfect fluid theory, though false, work in practice? Prandtl argued that these problems can be resolved by a care­ful analysis of what is, and what is not, implied by the theorems of classical hydrodynamics.

Consider Kelvin’s theorem, which, in Prandtl’s words, asserted that “in a homogeneous, frictionless fluid the circulation around every closed fluid line is invariable with time.” What did Prandtl mean by a “fluid line”? A closed fluid line is not just a closed geometrical line imaginatively and arbitrarily projected into the fluid. It is meant to be a line that is always made up of the same fluid elements. “Let us suppose a ‘fluid line’ to be a line composed permanently of the same fluid particles” (722). If the whole of a body of fluid is at rest, then the circulation around any such circuit is zero, and it follows from Kelvin’s result that it will stay zero for all time. Prandtl then invited his listeners to imagine a body, such as a wing or strut, surrounded by perfect

Negotiating Kelvin’s Theorem

figure 9.13. Stationary fluid around a strut subdivided into a mesh of fluid lines. From Prandtl 1927c, 723. (By permission of the Royal Aeronautical Society Library)

fluid where the fluid is at rest. The fluid around the object is supposed to be subdivided into a mesh of small circuits in the manner shown in figure 9.13, where each circuit is a fluid line.

Kelvin’s theorem implies that the circulation around each circuit or fluid line remains zero, and this apparently leads to the conclusion that rotation cannot appear anywhere. From this it would seem to follow that lift is pre­cluded, as the British critics of the circulation theory always argued. For Prandtl, however, “this conclusion is premature” (722). He went on: “We must first ascertain whether every point of the fluid set in motion is actually enclosed by the lines which in the state of rest were closed, if our conclusion is to be permissible. But closer investigation shows that it is possible to give instances in which this is not the case” (722).

How is this possible? Where do the rotating fluid elements come from? Prandtl’s answer was based on what happens when the body moves through the fluid “so that the upper and lower streams flow together, or as we shall say become confluent, at the sharp rear edge of the body” (723). Such a motion is shown in figure 9.14.

The network of circuits is divided by a surface “along which our conclu­sion as to the absence of vorticity is no longer applicable” (723). This surface is indicated by the dotted line in the figure. The vorticity does not arise be­cause some material element of fluid is set in rotation; it comes from the rela­tive motion of two adjacent bodies of fluid. Prandtl did not try to challenge Kelvin’s theorem by finding a drop of perfect fluid that somehow escaped the division into closed circuits; rather, he was exploiting the fact that rota­tion (defined technically) can exist without anything (or any finite thing) rotating. Kelvin’s theorem does not imply that “rotation cannot appear any­where”; it has a more specific, and limited, meaning. A vortex sheet, Prandtl insisted, can arise from confluence, and do so “without contradicting Kelvin’s theorem” (723).

Negotiating Kelvin’s Theorem

figure 9.14. Strut in motion with confluence at rear edge. Prandtl argued that circulation can be created in an ideal fluid by confluence without violating Kelvin’s theorem. From Prandtl 1927c, 723. (By permission of the Royal Aeronautical Society Library)

Confluence, according to Prandtl, can also generate the circulation around a wing. In an attempt to show how it can do this, he set out the sequence of events in the first few moments after a wing has begun to move through still air. Prandtl referred to the lower surface of the wing as the “pressure” side and the upper surface as the “suction” side. “During accelerations,” he said,

the velocity at the rear edge is greater on the pressure side than on the suc­tion side, since the path along the pressure side is shorter. Consequently, after confluence a discontinuous distribution of velocity is set up which effectively constitutes a sheet of intense vorticity; the surface of discontinuity then begins to roll up into a spiral. The circulation for each circuit enclosing the wing and the surface of discontinuity still remain zero, from which it may be inferred that the circulation around the wing is equal and opposite to the circulation of the vortex produced by rolling up of the sheet. This is the method of genera­tion of the circulation around the wing. (723)

It cannot be said that the argument is entirely clear.70 Prandtl acknowl­edged that “the mathematician” would object and insist that an inviscid fluid would flow around the trailing edge. There would be a stagnation point on the upper (suction) surface of the wing rather than a vortex sheet coming away from the trailing edge. This would be so even if the trailing edge were sharp and the perfect fluid had to move at an infinite speed to get round the corner. The result would be no circulation and no lift. This must be the correct flow, the mathematicians would say, because it would be “everywhere irrotational in accord with the theoretical laws for a flow produced from a state of rest!” (723). For the mathematicians, this flow alone would be consistent with the theorems of Lagrange and Kelvin. On Prandtl’s reading the theorems do not carry this implication. He argued that both his proposed flow (with circu­lation) and the mathematician’s flow (without circulation) were consistent with the classical theorems of hydrodynamics. But, he insisted, only the flow with circulation and the smooth confluence at the trailing edge is physically realizable. This was really all Prandtl needed. Even if the process by which circulation was generated remained obscure, it was the logical possibility of circulation, and the logical right to postulate it, that really mattered for the perfect fluid approach.

Prandtl then turned to his second topic: Why did perfect fluid theory work? Going back to his discovery of the boundary layer in 1904, he explained that the thickness of the layer was inversely proportional to the square root of the viscosity and that it generated a tangential friction proportional to the 3/2 power of the speed and the square root of the viscosity. “But,” he said, “there is something of more importance to us here” (725). The boundary layer is the cause of the formation of vortices. Kelvin’s theorem only applies to flows devoid of viscous forces, and so, in motion starting from rest, the circulation will remain practically zero in fluid circuits that do not pass, or have not passed, through a boundary layer. This is why real fluids, such as air, behave like perfect fluids in irrotational motion at a distance from a solid body. By means of photographs and a film Prandtl then demonstrated that the bound­ary layer could be manipulated, for example, removed by suction, and he showed that this procedure had a dramatic effect on the flow. He concluded his lecture with a discussion of the role of turbulence in the boundary layer and explained that an increase in turbulence could reduce drag. This allowed Prandtl to clinch his justification for using perfect fluid theory: “We thus get the unique characteristic that it is precisely these turbulent flows of low resis­tance around bodies which can be so closely represented by the theory of the perfect liquid” (739).

No one could say that Prandtl had evaded the arguments of his British critics. He had confronted their doubts, but had he dispelled them? The util­ity of Prandtl’s wing theory had been largely conceded, but did the Wright Lecture remove the residual worries about its theoretical basis? The imme­diate answer was that it did not. Partly this may have been because of the difficulty in following certain steps in Prandtl’s line of thought, but a deeper reason lay in the divergent readings of Kelvin’s theorem.

Idealization, Abstraction, and Approximation

The “general mental attitude” (as Lamb would have called it) informing the German work on aerodynamics gradually gained ground in Britain in the interwar years and finally became routine during and after World War II. To convey this attitude I have appealed to the work of some of the main practi­tioners of technische Mechanik, and in this chapter I have supplemented these sources by drawing on Philipp Frank’s writings. To drive the point home, I introduce one final, representative thinker to characterize the methods of aerodynamics. The “general mental attitude” of modern aerodynamics is well captured in the work and writing of Dietrich Kuchemann, yet another of Prandtl’s distinguished pupils.93

Kuchemann came to Britain in 1945, immediately following World War II, after meeting McKinnon Wood in Gottingen. McKinnon Wood was then working for the Combined Intelligence Objectives Sub-Committee (CIOS). He was on a mission similar, but more formal and grimmer in tone, to the one he had undertaken with Glauert in 1921. He questioned Kuche­mann about the latter’s work on swept wings and the flow over engine ducts and fairings.94 Kuchemann then became one of the many German experts who, under varying pressures, began to work for their former enemies in the postwar years.95 His personal history thus illustrates the transnational char­acter of science and the contingencies that contribute to it. Kuchemann sub­sequently chose to stay in the United Kingdom and eventually took British citizenship. In 1963 he was elected a fellow of the Royal Society and in 1966 became the head of the Aerodynamics Department at Farnborough. His later trajectory was not unlike Glauert’s, but whereas Glauert started from Cam­bridge, Kuchemann started from Gottingen.

There is another parallel. At the beginning of my story I described the lectures on aeronautics that Sir George Greenhill had delivered at Imperial College, London, in 1910. In the early 1970s Kuchemann also gave a series of lectures at Imperial. Like Greenhill’s lectures, these were also turned into a book, but whereas Greenhill was transmitting the old style of Cambridge mathematical physics, Kuchemann was carrying with him the style of Got­tingen engineering. Kuchemann’s book was called The Aerodynamic Design of Aircraft.96 In it he offered the following cautionary words about the character of aerodynamic knowledge:

the most drastic simplifying assumptions must be made before we can even think about the flow of gases and arrive at equations which are amenable to treatment. Our whole science lives on highly idealised concepts and ingenious abstractions and approximations. We should remember this in all modesty at all times, especially when someone claims to have obtained “the right answer” or “the exact solution.” At the same time, we must acknowledge and admire the intuitive art of those scientists to whom we owe the many useful concepts and approximations with which we work. (23)

Even the most elementary and pervasive of all the concepts of fluid me­chanics, the “fluid element,” fits Kuchemann’s description. It is a tool of anal­ysis that facilitates a mathematical grasp of real fluids, but, taken in isolation, the concept is no more than a convenient fiction shaped by the demands of the differential calculus.97 Kuchemann’s message is clear, and it is in no way idiosyncratic.98 Aerodynamics demands modesty in the status attributed to it. Claims to possess right answers and exact solutions should be viewed with suspicion because they will soon need to be qualified. While acknowledging the highly mathematical character of aerodynamics, we must accept and em­brace both the intuitive and utilitarian character of the enterprise.

Everything that Kuchemann says about aerodynamics can be found in the examples I have described. It applies to Kutta’s mathematical arc, Joukowsky’s theoretical aerofoils with infinitely thin trailing edges, Prandtl’s bound vortex and his reduction of the wing to a lifting line, Betz’s expedient modification of the Kutta condition and his relaxed attitude to infinite velocities, Schlicht – ing’s comments about fluid in the boundary layer for which the laws of mo­tion are suspended in directions normal to the layer, and Glauert’s evanescent boundary layers. But Kuchemann’s description applies equally to the work of Stokes, Rayleigh, Greenhill, Bryan, Lamb, Taylor, and Bairstow, even if some of them, on occasion, might not have embraced this characterization of their methods as readily as those trained in the self-consciously engineering tradi­tion of technical mechanics.

In all of these instances, a detailed examination of the practices that were adopted reveals that the whole science of aerodynamics, both British and German, indeed lived on highly idealized concepts and ingenious abstrac­tions and approximations, and the more successful it was, the more radical the idealizations. If Kuchemann was right, these characteristics of aerody­namic knowledge were not merely a passing phase. The men and women I have studied did not resort to these expedients because of the immaturity of the field or because they had, for the moment, to be content with second best. Their manner of thinking and knowing was not one that would be left behind as if, today, aerodynamics rests upon a qualitatively different methodological basis than the one it rested on then. It does not.

Let me support this statement by two examples of recent work. First, con­sider the current status of the Navier-Stokes equations. To this day they have not been solved, and their properties remain shrouded in obscurity. There are no known, general, “closed-form” solutions to the Navier-Stokes equa­tions. There are approximate numerical solutions but none in the form of the analytical equations that were characteristic of classical hydrodynamics and Prandtl’s wing theory.99 The emergence, since the 1990s, of computational fluid dynamics, which is based on powerful digital computers, has not altered this situation. Computational fluid dynamics has made available numerical solutions to the Navier-Stokes equations and graphic representations of flow patterns over the wings and bodies of aircraft. These are enormously useful, but they have all been produced by programming techniques that embody the forms of idealization, abstraction, and approximation to which Kuche – mann was referring.100

My second example shows that Prandtl’s wing theory has retained its ca­pacity to inspire novel work. Although a product of the early twentieth cen­tury, it lies at the heart of some twenty-first century developments in aviation technology. These developments exploit some new solutions to Prandtl’s old equations.101 The solutions (which are analytic rather than merely numerical) take into account the possibility of creating a variable twist along the span of the wing. The equations show that, with the right distribution of twist, even wings that have (say) a rectangular planform can be made to generate the theoretical minimum amount of induced drag. Previously this minimum had been associated exclusively with an (untwisted) elliptical planform. The necessary twist required for minimum drag depends on gross weight, alti­tude, speed, and acceleration, but using modern technology, a wing could be built that automatically adjusted itself to these changing conditions. The result, over the course of a long flight, would be significant drag reduction and cost saving. These two pieces of evidence show that Kuchemann was not describing a passing phase in the emergence of aerodynamics but the endur­ing character of knowledge in this field and, perhaps, the unavoidable condi­tions of all practical cognition.

I have already argued, by reference to Frank, that if the concept of “rela­tivism” is to be used with precision, it can only mean one thing: a denial that there are any absolute truths. This is a necessary and sufficient condition for an account of knowledge to be identified as a form of relativism. A relativist can be comfortable with knowledge that is conjectural, inconsistent, expedi­ent, and partial, that is, with everything that science and technology really is. It is the nonrelativists for whom these facts about science are a cause of trou­ble. They can acknowledge them, but only as stations on the road to an abso­lute end point. But the familiar pragmatics of scientific work cannot belong to the realm of the absolute. It would be perverse to use the words “absolute truth” as a label for theories that are approximate or that get this bit right and that bit wrong or which depend on useful fictions and abstractions. It should be clear that Kuchemann is denying that practitioners in aerodynamics can ever make any claim to absolute certainty, the absolute status of their con­cepts, or absolute finality in their knowledge.102 It follows, given Kuchemann’s analysis, that aerodynamic knowledge must be understood in a relativist man­ner, namely, relative to all the contingencies of the “intuitive art” that enters into idealization, abstraction, approximation, and inductive inference.103

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.