Our Ignorance Is Almost Absolute

Southwell entered Trinity in 1907 to read mechanical sciences. He was an engineer, but an engineer with impressive mathematical skills.38 In 1909 he was placed in the first class of part I of the Mathematical Tripos and in 1910 graduated with first-class honors in the Mechanical Sciences Tripos. He was coached by Pye and Webb, two of the best mathematical coaches of the time. On graduation he began research on elasticity theory and the strength of ma­terials and in 1912 became a fellow of Trinity. In 1914 Trinity offered Southwell the post of college lecturer in mathematics but he did not take up the offer because of the outbreak of war. He volunteered for the army and was sent to France. In 1915, however, he was brought back to work on airships for the

navy. In 1918 he was transferred to the newly created Royal Air Force, with the rank of major, and was sent to Farnborough in charge of the aerodynamic and structural department. After demobilization, and a brief return to Trin­ity, in 1920 he went to the National Physical Laboratory as superintendent of the Aerodynamics Department. He stayed at the NPL for five years and then returned again to Cambridge, where (unusual for an engineer) he was a fac­ulty lecturer in mathematics.

It was in the field of applied mathematics, rather than practical engineer­ing, that Southwell made his outstanding contribution. He developed novel mathematical techniques for the analysis of complex structures of the kind used in the building of airships. The technique was called “the relaxation of constraints” and depended on replacing the derivatives in the equations and boundary conditions by finite differences.39 Though the technique was initially developed to deal with engineering problems, Southwell later dem­onstrated its power as a general method of solving differential equations. Referring to the unavoidable complexities of practice, and the uncertainties

in data of whatever kind, he called his own Relaxation Method “an attempt to construct a ‘mathematics with a fringe.’”40 He was not only interested in elasticity and the strength of materials but also worked on viscous flow. Like Bairstow, Southwell started from Oseen’s approximation to the full equations of viscous flow and the developments provided by Lamb.41 In 1929 Southwell was offered the chair in engineering at Oxford, which he accepted after some hesitation but where he stayed until his retirement. Southwell had a lively sense of the different demands confronted by engineers and mathematical physicists, but it may be revealing that Glazebrook said of him that, although he was an Oxford professor, he was still a Cambridge man.42

As superintendent of the Aerodynamics Department at the NPL, South­well played a prominent role in the discussions that took place in the Aero­nautical Research Committee after the war when plans for future work were thrashed out. Southwell always placed great emphasis on fundamental scien­tific research. It was the long term, not the short term, that counted. Though an engineer by training, he defended the value of academic research of the kind so often attacked by the practical men. This came out clearly in the pol­icy discussions that took place in February 1921, devoted to the topic of “The Aeroplane of 1930.” The participants were invited to anticipate the character and needs of aviation in ten years’ time. Southwell wittily subverted the discussion by posing the question If we could know where we would be in ten years’ time, why wait? His point was that fundamental advances could not be predicted. He suspected that, whatever we said, we would be wrong.43 The most we can do is to be conscious of the gaps in existing knowledge and try to fill them. Consider, he said to the committee, the fundamental cause of the lift and drag on an aircraft wing: “We have much empirical data in regard to aerofoils, but our ignorance of the mechanisms by which their lift and drag are obtained has hitherto been almost absolute.” Here was a worthy focus for research: the true mechanisms of lift and drag must be identified.44

One might assume that Bryant and Williams’ experiments, as well as those of Fage and Simmons, were performed to identify the mechanisms that Southwell had in mind. But if this were so, we would expect that the results of the work (give or take Taylor’s reservations) would have been seen by South­well as furnishing the desired account of lift and drag. This was not how he saw them. The same sense of ignorance about fundamental causes still per­vaded Southwell’s thinking after this experimental work had been completed and after Glauert had begun to provide his superbly clear exposition and de­velopment of the circulation theory. The same pessimism that was expressed privately in committee in 1921 was expressed again, and publicly, some four years later in two lectures that Southwell gave in 1925. One of these lectures, on January 22, was to the Royal Aeronautical Society; the other, on August 28, was to the British Association meeting in Southampton.

The lecture to the RAeS was titled “Some Recent Work of the Aerodynam­ics Department” and was meant as a summary of the achievements of the department during the years of Southwell’s superintendence.45 His return to Trinity was an opportunity to take stock. Southwell began by welcoming the change from ad hoc wartime experimentation to programs of research guided by theory. Two main lines of theoretical concern were identified. First, there was the classical theory of stability, and Southwell described in detail the re­cent work of Relf and others. This had taken the experimental determination of the damping coefficients for roll, yaw, and pitch to new levels of sophistica­tion. The second set of theoretical concerns dealt with the fundamentals of fluid flow. For aerodynamics, said Southwell,

I suppose no problem is so fundamental as the question—why does an aero­foil lift? We can hardly rest satisfied with the present position—which is, that we have next to no idea. To answer the question completely would involve no less than the solution of the general equations of motion for a viscous fluid, and attacks on these equations have been made from all angles. Considering the energy expended, the results have been very small; but then, these are about the most intractable equations in the whole of mathematical physics. (154)

Southwell mentioned the role played in this (so far fruitless) endeavor by Bairstow, Cowley, and Levy and then moved on to the approach adopted by Prandtl, namely, using the inviscid theory of the “hydrodynamic textbooks” informally conjoined with the idea of a viscous boundary layer. In this way the “once discounted” classical theory of the perfect fluid had been “rein­stated” and could provide a close approximation to the truth when used “un­der proper control, and aided by assumptions based on physical intuition” (156). At the NPL, said Southwell, every opportunity had been taken to check the validity of Prandtl’s theory, and “in the main one must say, I think, that it has passed the ordeal with flying colours” (156). The most important tests “are those which Messrs. Fage, Bryant, Simmons and Williams have made” (156). Southwell explained that at the time of his lecture this work had not yet been published but it had confirmed the most important result, namely, “the theoretical relation between lift-coefficient and the circulation” (156).

At this point Southwell’s audience might have been puzzled. They were being told that Prandtl’s theory had passed the tests to which it had been subject with “flying colours,” and yet a moment before, Southwell had de­clared that experts had “next to no idea” how a wing produced lift. Didn’t these claims contradict one another? The answer is that Southwell’s argument was consistent but depended on a suppressed premise. For Southwell, the experiments of Fage and Simmons only justified the use of inviscid theory as a way of representing the real flow. They did not show that it truly described the flow. As far as Southwell was concerned, Fage and Simmons were not tracing footprints in the snow. In their experiments the imprint of reality had not been made in some familiar and reliable medium. Their analysis had used ideal fluid theory. The nature of the beast that left the footprints was still under discussion. The inviscid approach left it an open question whether the “actual flow” corresponded to the representation, and the most plausible answer was that it did not. The no-slip condition was violated by the inviscid representation, and Prandtl had assumed that the flow was steady. The eddies in the wake were neglected. The place to look to resolve these issues, Southwell concluded, was the boundary layer. It was this aspect of Prandtl’s work that really engaged Southwell. As he put it, “the conditions in this layer are the ultimate mystery of aerodynamics: somehow or other, in a film of air whose thickness is measured in thousandths of an inch, that circulation is generated which we have just seen to be the essential ingredient of ‘lift’” (158). Research should concentrate on the boundary layer. Theoreti­cally this required a deeper understanding of the equations of viscous flow; experimentally it called for the development of special instruments such as microscopic Pitot tubes to probe the boundary layer. Southwell mentioned that Muriel Glauert was working mathematically and experimentally on the calibration of such an instrument.46

Here was the explanation of Southwell’s apparently conflicting claims. Prandtl’s theory of the finite wing “worked,” but it could not be true because the mathematical analysis depended on false boundary conditions. This was the suppressed premise, which rendered the argument consistent. Although Prandtl’s wing theory could pass many tests, and even pass them with flying colors, it could not, by its very nature, answer the question that Southwell wanted to answer. In a very British way, he wanted to know how a viscous fluid generates lift. In the discussion after the lecture, in response to Major Low, Southwell said: “The really interesting part of Prandtl’s work was the work he had been doing subsequently in his study of the ‘boundary layer,’ because that work might ultimately explain why the assumptions which could not be correct could make such amazingly true predictions” (166).

In a lecture titled “Aeronautical Problems of the Past and of the Future,” delivered later in the same year, Southwell insisted that the aim of research was “not so much to achieve, as to understand.”41 Scientists should not be content with “achievement,” “unless it be the result of understanding’—something of which the “practical man” would never be persuaded (410). Understanding meant understanding based on a sound theory. Southwell identified three triumphs of British aeronautics that, in his opinion, met this condition. They were (1) the ability to build stable aircraft, (2) the analysis of the dangerous maneuver of spinning and its avoidance, and (3) the achievement of control in low-speed flight even after the aircraft had stalled. In all three cases, he argued, the end result had enormous practical value but the driving force had been the aim to understand. And it was mathematical analysis that had furnished the understanding.

The theory of lift was conspicuous by its absence from this list of tri­umphs. For Southwell, Prandtl’s wing theory was an achievement that was not yet informed by an adequate theoretical understanding. Bryant, Wil­liams, Fage, and Simmons were mentioned by name, and Southwell used diagrams taken from their papers. The role that he accorded the work, how­ever, was that of showing that the effects of viscosity can be ignored as far as the sliding of air on air is concerned but cannot be ignored very close to the surface of a wing or in the wake behind the wing. It is what happens in these regions that constitutes “the ultimate problem of hydrodynamics” (417). It was this “ultimate” problem that Southwell had in mind when he asked: Why does a wing generate lift? He was not denying the role of circulation, nor was he belittling the insights of Lanchester, Prandtl, or Glauert as they continued to develop the inviscid theory of lift. His point was that no one, following this route, could hope to explain the origin of circulation.48 Within inviscid theory, circulation had to be a postulate not a deduction.

Southwell’s skeptical position was endorsed by H. E. Wimperis, the quiet but influential director of scientific research at the Air Ministry.49 Wimperis had trained as an engineer in London and Cambridge and had sat the Me­chanical Sciences Tripos in 1890. During the Great War he had served as a scientist with the Royal Naval Air Service and had designed a bomb sight that carried his name. After the war he worked at Imperial College in a labora­tory financed by the Air Ministry. Along with Tizard, he was later to play an important role in the development of Britain’s radar defense system. In 1926 Wimperis, in his role as director of research, published a survey article in the Journal of the Royal Aeronautical Society called “The Relationship of Physics to Aeronautical Research.”50 One of Wimperis’ aims was to send the message that the Air Ministry and government were aware of the need for fundamental research. What, he asked, was engineering but applied physics? Government scientists at the National Physical Laboratory and Farnborough must have the freedom to pursue basic, physical problems. A second aim was to argue that this policy had already produced significant results. Here Wimperis cited, among other examples, the mathematical work that had been done on fluid flow and, in particular, the flow around a wing. It rapidly became clear, how­ever, that in Wimperis’ view, the approach based on inviscid theory was not an exercise in real physics but a mere preliminary to a genuine understanding of lift. On a classical hydrodynamic approach, he noted, the circulation must be added in an arbitrary way to the flow, and this only provides an “analogy with the lift force experienced by an aerofoil” (670). Admittedly there have been some successful predictions made “by the employment of this conven­tion” (670), but the theory becomes “somewhat far-fetched” in its account of what is happening on the surface of the wing. “Circulation,” said Wimperis, “must have a physical existence since velocity is greater above the wing than below; though this real circulation is a circulation with no slip, whereas the mathematical circulation has slip. Hence the rather amusing situation arises of adding to the mathematical study of streamlines a conventional motion which could not really arise in an inviscid fluid!” (670). Southwell was right, said Wimperis, in insisting that the real problem lay in discovering what was actually happening in the very thin, viscous layer close to the wing. This was a problem in physics rather than something that could be evaded by the use of mathematical conventions and unreal boundary conditions.