An Old Anomaly or a New Crisis?

How did the leading British aerodynamicists react? The theory of lift derived from the picture of discontinuous flow could not be right, at least, not as it stood. Was this situation to be treated (in Kuhn’s terms) as a “crisis”? Did it call for a radical response involving the use of new models and the creation of a wholly new approach to lift? Or was it no more than an “anomaly”? Was there still the possibility that some refinement of the old, discontinuity ap­proach would eventually allow the problem to be met? The answer is that the problem was seen as both an anomaly and a crisis, but it was seen in these different ways by different groups. The split was roughly along disciplinary lines. For the more purely mathematical contributors, the emerging results represented, or were collectively treated, as an anomaly rather than a crisis; for the experimentalists and physicists, even the mathematical physicists, the findings were more than merely anomalous: they were taken to herald a full­blown crisis.34

Greenhill, Bryan, and a number of other research mathematicians con­tinued to work on the problems of discontinuous flow. They aimed to refine the analysis by taking account of the curvature that was characteristic of the cross section of a wing. This involved complicating the Schwarz-Christoffel transformation, which applied to straight-sided figures. In 1914 G. H. Bryan and Robert Jones published a paper called “Discontinuous Fluid Motion Past a Bent Plane, with Special Reference to Aeroplane Problems.”35 They were able to align the analysis with some of the qualitative facts, for example, that a moderate degree of camber can increase lift without increasing drag. In 1915 J. G. Leathem (the sixth wrangler in 1894 and a fellow of St. John’s) published “Some Applications of Conformal Transformations to Problems of Hydrodynamics,” which was meant to put the introduction of “curve – factors” on a systematic basis.36 Also in 1915, Hyman Levy published “On the Resistance Experienced by a Body Moving in a Fluid,” in which he set out to link a discontinuity analysis to recent work on vortices by von Karman.37 In 1916 Greenhill published a substantial appendix to his R&M 19 which was titled “Theory of a Stream Line Past a Curved Wing.” Greenhill noted that a curved surface could be approximated by a large number of short, straight surfaces so that the way was open to work with a more realistic model of a wing. He added to his previous discussion by surveying the contributions to discontinuity theory of French and Italian mathematicians such as Bril – louin, Villat, Cissotti, and Levi-Civita.38 The same year, 1916, saw Levy’s “Dis­continuous Fluid Motion Past a Curved Boundary.”39 The investigation was again justified by its relevance to aerodynamics. The author asserted that “In aeronautics alone recent developments have shown the practical necessity for an effective discussion of the case where the plane is cambered” (285).

What kept the mathematicians at work?40 Partly, they hoped to bring dis­continuity theory into closer contact with experimental results, as indeed they did; it would be wrong, however, to overstate the optimism associated with this project. The main factor seems to have been the lack of any per­ceived alternative. The question they faced was whether there was any chance of digging beneath the equations of perfect fluid theory and making progress with the full, governing equations of viscous flow. If there was no chance, or very little chance, then it was reasonable to carry on as before. Consider the stance of Cowley and Levy. They concluded that the fatal flaw in the theory of discontinuous flow was that it depended on the assumptions of the theory of a perfect fluid, although, they argued, “it is remarkable. . . that the results ob­tained for the resistance are comparable at all with those derived from experi­ment” (65), and they speculated that perfect fluid flows with vortices might be used to simulate the flows that could be photographed in a turbulent, viscous fluid.41

While Cowley and Levy spoke of the mathematics of viscous flow as “not yet sufficiently developed” (75)—thus holding out hope—a more pessimistic induction is hinted at by G. H. Bryan’s remarks in the Mathematical Gazette of 1912.42 With a nod to Greenhill’s article on hydrodynamics in the Encyclo­paedia Britannica, Bryan said that the subject really consists in the study of certain partial differential equations and not “town water supply, resistance of ships, screw propellers and aeroplanes” (379). There was not much induce­ment for the mathematician to adapt his work to the needs of engineers. If he were going to do that, he might as well become an engineer “and give up most of his mathematics, relying on the introduction of constants or coefficients to save him from running his head against insoluble differential equations” (379). As for the hypothetical conditions that make hydrodynamics so unreal, these have “pretty well done their duty when they have been made use of to write down differential equations” (379). So it was not the empirical status of these conditions (that is, their falsity) that counted, but their power to help the mathematician frame tractable equations. The real issue was “remarkably simple”: if you give up ideal fluid theory, you get equations for which nobody can find the integrals, “at least, mathematicians have tried over and over in vain to find them” (379). And the same argument applied to the simplifying assumption of steady motion, which involved, and justified, ignoring “for example, eddy formation in the rear of planes” (380). Bryan’s position was the same as the one he adopted in his main field of research into stability. There the equations turned out to be accurate, but for Bryan, this was a bo­nus rather than something that was necessary for justifying the work. Even if inaccurate, he said, the analysis might still furnish a useful basis for the interpretation of experimental data.

Such was the reasoning by which a small number of high-status math­ematicians justified their continued elaboration of discontinuity theory and sustained a pessimistic form of “normal science.” For Bairstow, and others at

the NPL who were more experimentally inclined, the emerging problems in­dicated the end of the road for discontinuity theory. The theory was artificial and doomed to failure because it was grounded in the unreal conception of a perfect, frictionless fluid. What was needed was a return to the full equations of viscous flow and the attempt to develop new methods of approximation. For the moment, however, Bairstow accepted that the complex flow around a wing was beyond the comprehension of the mathematician.

In a lecture at the Aeronautical Society, on February 12, 1913, Bairstow asked what shape of aerofoil or strut would give the most lift or the least resistance.43 “A true theory of aerodynamics,” he said, “would answer these questions for us completely, but unfortunately for us the answers to such questions are beyond the reach of our present mathematical knowledge.” To reinforce the point Bairstow showed his audience a photograph that, he said, “illustrates a motion which has defied the mathematician” (117). The photo­graph was one of the NPL water-channel pictures showing a wing with the characteristic turbulent wake associated with a stall, that is, a wing exhibiting Kirchhoff-Rayleigh flow. Greenhill was in the audience—but he did not de­fend his mathematical model of discontinuous flow around a wing, nor did he challenge Bairstow’s conclusion. It is difficult to know how to interpret this disregard. Greenhill initiated the discussion that followed the lecture, but only to make jocular comments on the law of mechanical similarity. Ac­cording to Greenhill’s calculations, if angels existed in the form in which they are usually depicted, then they would have to be about the size of a bee.44 This lack of an explicit response to the shortcomings of the discontinuity theory was wholly characteristic. There had been no discussion of the de­mise of discontinuity theory at the meetings of the Advisory Committee (or none at which minutes were taken). Greenhill, though in regular attendance, appears to have made few contributions to the business of the committee. Anecdotal evidence reveals that Sir George was actually prone to fall asleep during meetings. On one such occasion Mervin O’Gorman, a talented artist, drew a sketch of the slumbering mathematician and left it on the table. The caption was from the well-known hymn: “There is a green hill far away.”45 But sleep patterns are not really very illuminating. Perhaps the reason for the reticence was simply that Greenhill, and his fellow wranglers on the commit­tee, shared something of Bryan’s pessimism. Without an analysis of viscous flow, the choice was between inviscid theory and empiricism—and inviscid theory had failed.

Bairstow’s slightly more optimistic view, that progress of some kind was possible with the equations of viscous flow, was shared by Geoffrey Ingram Taylor. Taylor came up to Trinity in 1905, took part I of the Mathematical

Tripos in 1907 and part II of the Natural Sciences Tripos in 1908. He was given a major scholarship at Trinity and in 1910 was elected to a fellowship. After war was declared on August 4, 1914, Taylor hurried to submit his dis­sertation for the Adams Prize, unsure whether it would be awarded because of the uncertainty of the international situation. He volunteered his services to the military on August 5, hoping to work in meteorology, but was immedi­ately drafted to Farnborough and later co-opted onto the Aerodynamics Sub­Committee of the ACA. Soon after his arrival at Farnborough, Taylor took his first flight. He flew with Edward Busk, on the day before Busk’s death. Later, after gaining his “wings” with the Royal Flying Corps, Taylor carried out numerous pieces of research, including measurements of the pressure distribution across the wing of a BE2. His work enabled comparisons to be made between wind-channel results on models and full scale data.46 In the early months of the war, however, he used every spare moment to bring his thesis to completion. It was titled “Turbulent Motion in Fluids.”47

The preface and introduction of the thesis were used to set out the current situation in fluid dynamics. Taylor’s position contrasted starkly with that of Bryan, who placed the emphasis on getting hold of some differential equa­tions rather than worrying about what had to be assumed or discarded en route. In Taylor’s view: “In no other branch of applied mathematics is the danger of neglecting the physical basis of the subject greater than it is in hy­drodynamics” (5). It has been possible, he said, to get “rigorous mathemati­cal solutions” in hydrodynamics, but they have no relation to what is found experimentally. Unfortunately, mathematicians adhere to these physically unrealistic assumptions simply because it makes the mathematics easier. As an example Taylor cited the theory of discontinuous flow. The theory is based on the assumption that the pressure in the dead-water region is the same as that of the undisturbed flow, when measurements show it is less than this. Referring to measurements made by Melvill Jones at the NPL, Taylor drew an unequivocal conclusion: “This, I think, finally disposes of the discontinuity theory, which . . . must now be placed among the curiosities of mathematics” (4). The rigor of the old methods, argued Taylor, was well worth sacrificing if there was a chance of explaining the turbulent behavior of real fluids—and this is what he set out to do in the remaining sections of the bulky thesis.48 His aim was to develop a new (statistical) theory of turbulence.

For Taylor, like Bairstow, discontinuity theory and Rayleigh flow were things of the past. The same judgment can be read into what was said, and what was not said, in an important lecture given by Glazebrook in June 1914. His title was “The Development of the Aeroplane,” and the aim was to de­scribe the achievements of the National Physical Laboratory. He mentioned the work of Stanton and Bairstow and explained how pressure measurements reveal the important role of the upper surface of the wing (from the outset, the weak point of the discontinuity theory). Glazebrook mentioned no names and did not make it explicit that the one candidate for the explanation of lift that had been taken seriously in British aerodynamics was now being quietly abandoned. All the attention was directed toward the successful work on sta­bility. The occasion of Glazebrook’s assessment was the second of the Wright Memorial Lectures. It was an important event so the assessment would have been carefully considered.49 The unspoken message was that, notwithstand­ing the opinion of a few of the older mathematicians, the discontinuity theory of lift was dead.