Category The Enigma of. the Aerofoil

A number of significant changes, both organizational and personal, took place in the higher reaches of British aeronautics at the end of the war. The size of the aeronautical section at the National Physical Laboratory had grown considerably during the conflict. Starting from three or four active workers in 1909, the section had expanded to around forty by the time of the armistice.39 Predictably, the return of peace meant that the budget was now to be cut back. Lord Rayleigh had died in 1919, and the Advisory Committee he had guided for a decade was formally dissolved and reconstituted as the Aeronautical Research Committee (ARC). The new committee held its first meeting on May 11, 1920.40 Glazebrook was given the job of restructuring it and preparing it for its new peacetime role. The National Physical Laboratory and the Aero­nautical Research Committee now came under the aegis of the newly formed Department of Scientific and Industrial Research (DSIR).41 Horace Lamb had been appointed to the Aerodynamics Sub-Committee in July 1918 and later joined the full committee.42 In an attempt to avoid the old hostility between the scientists and the manufacturers, there were now to be representatives of industry on the committee. J. D. North, of Boulton Paul, was appointed to the Aerodynamics Sub-Committee to represent the Society of British Aircraft Constructors. Bairstow left the National Physical Laboratory in 1917 and took up a post with the Air Board, the precursor to the Air Ministry, though he continued to serve on the new committee.43 Bairstow then moved again and become the Zaharoff Professor of Aviation at Imperial College, London. Sir Basil Zaharoff, who financed the chair, was an international arms dealer.44 Shortly afterward Emile Mond provided the money to set up a chair in aero­nautics at Cambridge in memory of his son killed flying on the western front. This chair was taken by Melvill Jones.45 Bairstow’s post as superintendent of the Aerodynamics Department at the NPL was taken over by Southwell, who moved from Farnborough to Teddington. Lanchester, who sometimes felt that Rayleigh was the only sympathetic member of the committee, left a year after Rayleigh’s death. Lanchester had been assiduous in his duties but had al­ready resigned from the Aerodynamics Sub-Committee in December 1918.46 Now that the emergency of the war was over he felt able to cite pressure of work as a basis for leaving. In his letter to the chairman of the Aerodynamics Sub-Committee he expressed “great pleasure in having been able to serve the committee,” but the retrospective account he gave of his departure from the full committee was very different in tone. He complained that he had been sidelined, snubbed, and deliberately edged out by Glazebrook.47

At the moment that Lanchester left the Whitehall scene feeling, justi­fiably, that his ideas had been ignored, moves were under way that would eventually lead to the triumph of the circulation theory he had pioneered. Two things happened. First, on November 13, 1920, Southwell received a let­ter from Prandtl, who sent him some up-to-date papers on wing theory and material from the Technische Berichte. Prandtl explained that his action had been prompted by his meeting with William Knight. Knight had apparently told Prandtl that Southwell wanted to get hold of information about develop­ments at Gottingen. Southwell replied on November 29 with thanks and ten­tatively asked Prandtl for details about his wind tunnel and the techniques for keeping the flow steady. He also stressed that the exchange with Prandtl had to be considered personal rather than official because of the British govern­ment’s policy of restricting formal contact with German institutions. Prandtl sent Southwell the required data about the air flow and said that more in­formation would soon be published in a volume to be titled Ergebnisse der Aerodynamischen Versuchsanstalt zu Gottingen.48

The second development was that two Farnborough scientists, Robert McKinnon Wood and Hermann Glauert, both members of the Chudleigh

set, were sent to Germany to report on the situation. McKinnon Wood, a product of the Cambridge Mechanical Sciences Tripos, was deputy director of the Aerodynamics Department at Farnborough and worked on propel­lers and the experimental side of aerodynamics.49 Hermann Glauert (fig. 8.3) had studied mathematics at Trinity College, Cambridge, where his circle of friends included David Pinsent, G. P. Thomson, and Ludwig Wittgenstein.50 He graduated with distinction in the first class of part II of the Tripos in 1913 and won an Isaac Newton studentship in 1914 and the Rayleigh Prize for mathematics in 1915.51 Glauert was born in Sheffield. His mother was an Englishwoman who had been born in Germany, while his father, a cutlery manufacturer, was German but had taken British citizenship. Originally spe­cializing in astronomy, Glauert published a number of papers on astronomi­cal topics at the beginning of the war and then, through a chance meeting with W. S. Farren, he was appointed to the staff of the Royal Aircraft Factory in 1916.52

Based in the Hotel Hessler in Charlottenburg, Glauert wrote on Janu­ary 21, 1921, to make contact with Prandtl and ask if was possible to arrange a

visit to Gottingen.53 The approach could not have been more different from Knight’s. Knight had sent a typed letter, in English, on elaborately headed NACA notepaper. The letter was replete with office reference numbers and subject headings and was introduced with a flourish of (questionable) diplo­matic credentials.54 Glauert penned his note in German on a modest sheet of unheaded paper. He introduced himself as a fellow of Trinity who worked at Farnborough and explained that he was very interested in the reports shown to him by his friend Herr Southwell. Could he and his friend Herr Wood, also of the Royal Aircraft Factory, come along next Monday? The visit duly took place but cannot have been an extended one because on February 2 Glauert was writing from Farnborough (again in German) to say that he and Wood were safely home after a thirty-six hour journey. All of the technical material that Prandtl had given them, he reported, had been carried over the border without difficulty. In return Glauert sent Prandtl a copy of Bairstow’s new Applied Aerodynamics.55 It was, he said, currently the best English book on aerodynamics.

Bairstow’s Applied Aerodynamics, published in 1920, offered a massive com­pilation of design data from the aerodynamic laboratories which, for Bair – stow, primarily meant from the National Physical Laboratory. It was com­prehensive and detailed but, as far as lift and drag were concerned, heavily empirical. The circulation theory of lift was placed on a par with the dis­continuity theory and rapidly dismissed. Both theories were said to be based on “special assumptions,” that is, ad hoc devices designed to get around the fatal zero-drag result. Kutta’s work, according to Bairstow, offered no more than a “somewhat complex and not very accurate empirical formula” (364). No account, he complained, was given by Kutta or Joukowsky of the critical angle of stall. Bairstow admitted that Joukowsky had found a way to avoid the infinite speeds at the leading edge of a wing profile, but he spoke of the Joukowsky transformation as if it were little more than a mathematical trick. Bairstow called it a “particular piece of analysis” and did not deem it suf­ficiently important to explain it to the reader. Prandtl must have perused Glauert’s gift with mixed feelings. He would have agreed with all of Bairstow’s facts but none of his evaluations. In particular, he would surely have dissented from Bairstow’s conclusion that it “appears to be fundamentally impossible to represent the motion of a real fluid accurately by any theory relating to an inviscid fluid” (361). Wasn’t this exactly what Prandtl and his colleagues had just done for the flow of air over a wing?

It is clear from the subsequent letters exchanged between Glauert and Prandtl that, during the visit, their conversations had been confined to tech­nical matters. Prandtl was now keen to discuss politics as well as aerodynam­ics. He was distressed by the economic and political situation, particularly the stance of the French and the severe reparations that were being demanded. Glauert expressed agreement with much that Prandtl said on these topics, for example, with the “absurd restrictions that have been placed on the develop­ment of German aviation,” but in his replies he encouraged Prandtl to see things in a less pessimistic light. Glauert explained that not everyone in Brit­ain agreed with these policies, and he thought there were strong economic reasons why, sooner or later, they would be modified. Writing now in English, he drew Prandtl’s attention to the influential arguments of the Cambridge economist John Maynard Keynes and mentioned that opposition to punitive sanctions was also part of the official policy of the Labour party.56 He even in­cluded a reassuring cutting from the correspondence columns of the Times.57 The concern was not just personal; it was also professional. Glauert was wor­ried that Prandtl would become so antagonized by allied policy that he would cease to take part in scientific exchanges. The anxiety was more than justified. The academic atmosphere in the immediate postwar years was a poisonous mixture of bitterness, intransigence, boycott, and counterboycott.58

The immediate result of the Glauert-Wood visit to Germany was the production of two confidential reports. In February 1921 McKinnon Wood produced Technical Report T. 1556, “The Aerodynamics Laboratory at Gottingen.”59 He argued (contrary to government policy) that Gottingen should be included in any future international trials that were envisaged to clarify the discrepancies that existed between the results of different labo­ratories.60 When the results of the British and Germans were compared, McKinnon Wood noted that the Gottingen channel gave the same lift and drag, but always for an angle of incidence smaller by about one degree. He also studied and reported on the complicated “three-moment” balance mechanism used to measure the aerodynamic forces. A blueprint of the bal­ance was included in the report. He judged that the Gottingen balances were “very inferior” in sensitivity to the British but then conceded that “our bal­ances and manometers are unnecessarily sensitive for most of the work for which they are required” (14). McKinnon Wood concluded by saying that “Mr. Glauert discussed Prandtl’s aerofoil theories with him and obtained some further papers. A discussion of these will be embodied in a separate paper (T. 1563) which Mr. Glauert is writing” (21). The “discussion” to which McKinnon Wood alluded turned out to be a piece of brilliant advocacy that was undoubtedly a major factor in undermining resistance to the circulation theory in Britain. Glauert had a gift for clear exposition and for seizing the essentials of the subject. Once he had accepted that Prandtl’s theory repre­sented the path to follow, Glauert produced a notable series of papers and reports explaining, testing, and developing the achievements of the Gottin­gen school. He corrected inadequate formulations and produced important extensions of the theory, as well as confronting the skeptics.

What made Glauert special? Why did he, with his impeccable Tripos background, strike out in a direction that had hitherto been unattractive to experts who shared with Glauert the intellectual culture of the “Cambridge school”? The “practical men” had always been divided over Lanchester’s the­ory of lift, but the “mathematicians” had been unanimous in their skepti­cism. Why did Glauert break ranks? No definitive answer can be given to this question, though one fact stands out and invites speculation. The Ger­man name, the German father, the command of the German language may have generated some affinity with the body of German work that was under consideration. These facts distinguished Glauert from British experts such as Bairstow who did not have the command of German that would have enabled them to read the literature or converse with Prandtl.61 (At that time Prandtl had little knowledge of English.) Of course, given the bitterness of the war, personal links to Germany might have had quite a different effect. Such links could be sources of difficulty, and there is evidence that they caused prob­lems, and some inner turmoil, for Glauert. Referring to the outbreak of war in 1914, Farren and Tizard said of Glauert: “His German descent was an em­barrassment to him, and he wisely decided to stay where such trivial matters did not assume the importance that they did elsewhere, and where he could work in peaceful surroundings, though not with a peaceful mind. His friends were far afield, and as time went on he became more and more restless and concerned with the difficulty of his position.”62

Some people in Glauert’s position would have kept their distance from all things German, and even shed their German name.63 One might specu­late that, in Glauert’s case, the balance in favor of the circulation theory was tipped by the opportunity to meet members of the Gottingen group. His visit to Germany enabled him to explore the mathematics of the circulation theory with Prandtl face-to-face. Others had visited Gottingen before the war, but British experts did not have a great deal of direct contact with their Ger­man counterparts.64 Even here it is necessary to be cautious about the im­pact of personal contact. Glauert was impressed by the circulation theory before he went to Germany. Doubts and qualifications dropped away after the visit to Gottingen and his advocacy became more confident, but he had begun to explore the theory through what he read in Southwell’s copies of the Technische Berichte. Farren and Tizard suggest that exposure to engineer’s shoptalk in the Chudleigh mess at Farnborough during the war made Glauert sympathetic to the needs of engineers and hence (one may suppose) to the theoretical approaches adopted by the engineers. Beyond this, little can be ventured in terms of explanation. It must simply be accepted that, equipped with the recent papers, Glauert made it his business to explore the Gottingen theory in great detail. He became committed to it, even when this led him to diverge from such authorities as Leonard Bairstow, Horace Lamb, R. V. Southwell, and G. I. Taylor.65

Glauert would have known that, however cogent he made his book, he could not meet all the demands of his intended audience. The diversity of interests that would inform the response would pull in opposing directions. The struc­tural tensions, present in British aerodynamics from the outset, were still at work. By its very nature, Glauert’s Elements of Aerofoil and Airscrew Theory could not satisfy the prejudices of both the practical engineer and the math­ematical physicist. Indeed, the book was designed to bring about a change of approach in both parties. Until it had worked its effect it was bound to be viewed with a certain reservation from both sides, even if, at the same time, its virtues were acknowledged.

The review that appeared in the Journal of the Royal Aeronautical Society was signed “A. R.L.” With its mixture of bluff praise and barbed comment, the review was such that no regular reader would have failed to recognize it as the work of Major A. R. Low.56 Glauert was identified as one (but only one) of the leading exponents of the Lanchester-Prandtl approach, and the Ele­ments was welcomed as (perhaps) the first full-length book on the subject in the English language. There was the hint that engineers might find Glauert’s discussion too advanced, and Low strongly recommended that students read a work by H. M. Martin “as an introduction to the volume under review, and, as well, for Mr. Martin’s mastery of elementary exposition” (167).57 Low simultaneously praised Glauert for the adequacy of his general references, which would “introduce the reader to the most important German work,” and criticized him for not citing Fuchs and Hopf’s Aerodynamik of 1922, which “quite evidently influenced the author, both as to selection and arrangement of materials” (168). The book by Fuchs and Hopf was broader in scope than the Elements, covering both lift and stability, and Low had reviewed it in the Aeroplane. While he had praised their treatment of lift, he had been scathing about their treatment of stability.58 Low acknowledged Glauert’s originality in using the method of images to arrive at a formula for tunnel interference effects in the case of rectangular (as distinct from circular) tunnels but quib­bled at an “unnecessary reference to Hobson’s Trigonometry” (168) to deal with a mathematical point that Low considered elementary.59 Low concluded his review by describing the Elements as “an important contribution to Eng­lish aerodynamic literature” and (high praise indeed) as a book that “should be of the greatest value to all designers of aircraft” (168).

Approaching the Elements from the direction of the mathematician rather than the engineer, R. V. Southwell reviewed it for the Mathematical Gazette.60 He praised Glauert’s “power of concise exposition” (394) and said the book gave “an admirable account of a fascinating theory.” It could be recommended as “indispensable to every student of modern aerodynamics” (395). But, as well as offering praise, it is clear that Southwell was taking care to locate Glauert’s achievement in a particular way. He began by noting that when Mr. Asquith first appointed the Advisory Committee for Aeronautics in 1909, “nothing seemed more certain than that aerodynamics must develop

as a purely empirical science” (394).61 Theoretical hydrodynamics was not sufficiently developed to take account of both inertial and viscous forces. Today, said Southwell, that “difficulty is not yet overcome, but it has been turned” (394). Prandtl’s wing theory is not “an exact theory,” but Prandtl “has supplied what for practice is almost as useful—a theory which can pre­dict” (394).

Southwell then clarified his distinction between exact and predictive theories. Consider Glauert’s chapter 8, which contained an account of skin friction and the origins of viscous drag. Southwell granted that Glauert’s brief treatment was appropriate given the limited purposes of the Elements but ex­pressed the hope that Glauert would produce a follow-up volume to expand the “somewhat slender outline” of the present chapter. The follow-up vol­ume would call for a “slightly altered arrangement” of the material. It appears to me, said Southwell, that

we ought to recognise not one “Prandtl theory,” but two. The first forms the main subject of the present work; its methods are those of the classical theory, and its assumptions, based on Lanchester’s picture of the flow pattern, are justified, ultimately, by its success in prediction. The second, which develops the notion of the “boundary layer” is more fundamental, more difficult, and (probably) less productive of concrete results than the first; its aim is to ex­plain the circulation round a lifting wing in terms of the known equations of viscous flow. (395)

The implication was that Glauert had adopted an “arrangement” of his material that did not adequately recognize the difference between the two theories. Glauert ran these two distinct theories together and aspired to a “combined” presentation. This may help the student, said Southwell, but it cannot do justice to either theory because “their methods are too distinct to permit a really satisfactory blend” (395). He did not believe that they could be combined in the way that Glauert wanted. “For the combined theory seeks to bring phenomena, in their very essence dependent on the viscosity of the fluid and its interaction with the solid boundary, within the scope of analysis which he knows is strictly applicable only to vortex motions existent through­out all time in a fluid devoid of all viscosity” (395). Southwell thus insisted on keeping apart what Glauert had sought to bring together. What the author had aspired to unify, the reviewer saw as incompatible. The themes invoked in Southwell’s review were familiar and characteristic of the British experts: there was the desire for a “fundamental” theory based on Stokes’ equations, a commitment to the “essential” difference between real and perfect fluids, and the appeal to the eternal character of vortices in a perfect fluid, that is, to Kel­

vin’s theorem. That Glauert, like Prandtl, was deliberately trying to overcome the idea that there is an “essential” difference between real and perfect fluids finds no recognition. Southwell acknowledged in Glauert’s unified presenta­tion not a principled methodological stance but a mere pedagogical expedi­ent, an “arrangement” of material to help students—and an arrangement that could not be sustained in the face of reality or in the pages of a more advanced treatise.

Southwell said that the views he expressed “imply no criticism” of Glau – ert’s book. The claim may look disingenuous but I think it should be accepted as authentic. The words would make sense if Southwell were reading Glau­ert’s book as an exercise in technology rather than physics. Once Prandtl’s wing theory was understood as no more than an instrument of prediction, as something that could be assessed using purely pragmatic criteria, then the real business of science could be thought of as proceeding in parallel to the technology. There would be no need for any quarrel between those engaged in the two distinct sorts of activity, provided they were kept apart and not confused with one another. Thus Southwell could honestly declare that he was not criticizing Glauert’s book but simply making it clear what manner of book it was, and what criteria were appropriate for its assessment.

This left just “one small detail” (395) that Southwell certainly wanted to criticize in an explicit way. He was worried about the imaginary roller bear­ings that Glauert interposed between a fluid and a material body or between two layers of fluid moving in different directions or with different speeds. References to roller bearings cropped up at a number of points in Glauert’s book, for example, on pages 95, 100, and 117, and were represented diagram­matically on page 131. Southwell thought such talk was misguided, and he implied that Glauert should know better. Vortices don’t behave like roller bearings, and it won’t help the beginner to understand “the purely math­ematical concept of vorticity” (395), that is, the technical definition of the rotation of a fluid element. Southwell’s point was that “vorticity,” as the term is used in fluid dynamics, can be present when nothing in the flow behaves like a “vortex,” as that term is used in common language, that is, nothing is swirling, rolling, or rotating. For example, mathematically, “vorticity” is pres­ent when two immediately adjacent layers of ideal fluid move horizontally with uniform but different speeds. All the fluid in the respective layers moves in straight lines, but for the mathematician, this phenomenon is equivalent to an infinitely thin sheet of vorticity between the layers. Talk of “roller bear­ings,” however, will produce an incorrect picture in the mind. The begin­ner “will misunderstand either the vortex sheet, or the action of roller bear­ings” (395). The harshest criticism was thus directed at Glauert’s engineering imagery. Southwell was not, in general, against visualization.62 The complaint was against the way that viscous processes and the viscous boundary layer were represented in nonviscous terms.63

Despite these reservations, the publication of Glauert’s Elements in 1926 represented the de facto victory of the circulation theory of lift among Brit­ish experts. The theory and references to Glauert’s exemplary account of it found their way into all subsequent treatises and textbooks, such as Lamb’s Hydrodynamics and Ramsey’s Treatise on Hydromechanics. The victory was, of course, underpinned by the steady accumulation of evidence from experi­mentalists such as Fage.64 The increasingly secure position of the circulation theory was, however, of a qualified kind. The victory was no simple rout of the opposition. The situation might be described with the use of political metaphors by saying that territory was conceded and new spheres of influ­ence agreed on. The power of the circulation theory had been demonstrated, and a certain zone of occupation was now recognized—though not the full legitimacy of what had taken place. The task now was to get on with life under the new dispensation. In the Great War, Germany may not have prevailed, but in the field of practical aerodynamics a new respect was accorded to the circulation theory and Prandtl’s wing theory. In 1927 Prandtl was invited to London to deliver the Wright Memorial Lecture to the Royal Aeronautical Society and to receive the Gold Medal of the society.65

There had been a previous suggestion that Prandtl might give a talk, which had been conveyed via Glauert in 1922. Prandtl had felt compelled to turn down the invitation, however, because of his lack of English.66 The Wright Lecture was a much grander affair, and Prandtl, who clearly appreciated the invitation, now felt better equipped to cope, though he still had some anxie­ties. In the preparatory exchange of letters with the chairman and the secre­tary of the society he fussed over what he should wear. Should he be in Frack, that is, tailed coat? In hesitant English he announced that “I have at this time English lessons and believe to be able up to the date of the lecture, to read the paper myself.”67 In the event, despite displaying the recommended tails, white tie, and white waistcoat, he only delivered the opening passages of the lecture and then called on the help of Major Low. Low, who had worked with Prandtl to translate the text, read the remainder.68

Those opening passages, however, touched on a matter of some delicacy. They concerned the origin of the theory of the aerofoil and the relative con­tributions of Prandtl and Lanchester. Who invented the theory and who should get the credit? Prandtl was diplomatic but forthright. He said that Lanchester had worked on the subject before he, Prandtl, had turned his at­tention to it and that Lanchester had independently obtained an important part of the theory. Prandtl insisted, however, that the ideas he used to build up his theory had occurred to him before he read Lanchester’s 1907 book. This prior understanding, he argued, may explain why “we in Germany were better able to understand Lanchester’s book when it appeared than you in England” (721). The truth, Prandtl went on, is that “Lanchester’s treatment is difficult to follow.” It makes “a very great demand on the reader’s intuitive perceptions,” and “only because we had been working on similar lines were we able to grasp Lanchester’s meaning at once” (721).

Is Prandtl here corroborating Glazebrook’s excuse for the British neglect of Lanchester? Surely not, though he certainly shared some of Glazebrook’s ideas about Lanchester’s work. Like Glazebrook, Prandtl did not countenance the possibility that it was the understanding of Lanchester, rather than the failure to understand him, that lay behind the British response. But, while go­ing along with part of Glazebrook’s story, Prandtl’s comments actually serve to accentuate the tensions between the different parts of Glazebrook’s excuse. They made it even more necessary to explain why the Germans were in a po­sition to grasp Lanchester’s meaning when, allegedly, the British had not been able to rise to the occasion. Glazebrook had excused one failure by citing another failure, and what Prandtl had to say aggravated rather than alleviated this logical weakness.69

My example of trailing vortices depended for its force on the difference be­tween the understanding of two groups of agents, the scientists and the pilots—the one group believing that trailing vortices had no practical sig­nificance, the other group knowing in a tacit and practical way that they did. What happens to the arguments for relativism when all parties know the same thing? This question is important because the universality of scientific under­standing is often taken to provide an adequate response to relativism. There is only one real world; the laws of nature are the same in London and Berlin; a true theory applies everywhere, and science knows no bounds of nation or race. “It is transnational and, despite what sociologists claim, independent of cultural milieu.”86 If science is independent of cultural milieu, then it cannot be relative to cultural milieu. Granted the premise, the conclusion follows, but my case study shows that the premise is false. The understanding of the phenomenon of lift was not the same in London and Berlin or Cambridge and Gottingen.

Such a response, based upon mere historical fact, is unlikely to satisfy the critics of relativism. It will be said that my study deals with a passing phase. Isn’t the important thing what happened after the episode that I described— when the truth emerged? The transnational character of science may take time to reveal itself, and progress may be inhibited by unfavorable social con­ditions, but universality triumphs in the end. It will be insufficient for the relativist to object that the antirelativist has shifted the discussion from what was the case to what ought to be the case or to what will be the case. The pic­ture of universal knowledge has force because there is, here and now, much science that is indeed transnational. This fact cannot be denied, so what can the relativist say?

Frank argued that relativism is consistent with universality. He said that the conditions leading to the spread of scientific knowledge were the very ones that ought to encourage a healthy relativism. As experience is broad­ened, the tendency to treat a belief as absolute will be undermined. Dogmati­cally held theories will encounter challenges, and a growing appreciation of the complexity of the world will undermine their apparently absolute status. Absolutism is parochialism—the cognitive equivalent of parish-pump pa­triotism. But the central reason why universalism is no threat to relativism is that the extent of a cultural milieu is purely contingent. In principle, a culture could be worldwide. The universal acceptance of a body of knowledge could only serve as a counterexample to relativism if universality indicates, or re­quires, absolute truth.

Let me explain this by an example. By the early 1930s Hermann Glauert had become a fellow of the Royal Society and the head of the Aerodynamics Department at Farnborough. He was at the height of his powers and had just finished a lengthy contribution on the theory of the propeller for William Durand’s multivolume synthesis of modern aerodynamic knowledge. Then tragedy struck. On August 4, 1934, a Saturday, Glauert took his three chil­dren for a walk across Laffan’s plain near Farnborough. The party stopped to watch some soldiers who were arranging an explosive charge to blow up a tree stump. The party stood, as required, at a safe distance some two hundred feet away, but the instructions they had been given were based on a misjudg – ment. Glauert was struck by a piece of debris from the explosion. No one else was hit, and his children were unhurt, but Glauert died instantly.87 In a dignified letter to Theodore von Karman, who had now moved from Aachen to the California Institute of Technology, Glauert’s wife (Muriel Barker) re­called the last time she and her husband had met von Karman. Along with G. I. Taylor they had all sat together, in the garden of Taylor’s Cambridge house, having tea and making plans for the future.88 Von Karman replied, in English: “The few people really interested in theoretical aerodynamics always felt as one family, and I am very proud to say that I had the feeling that your late husband and I were really friends, also beyond the common scientific interests.”89 Von Karman’s metaphor of the “family” to describe the relation­ship between leading members of the profession is striking. It resonates with, and lends support to, the theme of the transnational or universal character of science, though of course allowance must be made for the circumstances in which the expression was used. Perhaps it is the exchange of letters itself, rather than any particular choice of words, that should be considered the salient point. Former enemies in a bitter war are now consoling one another and affirming their solidarity. This epitomizes the increasingly cosmopolitan nature of the scientist’s world, at least as it was emerging, in the interwar years, in the field of aerodynamics.90

How is this emergence of a transnational science to be interpreted? There is a methodological choice to be made. In framing a response the choice lies between (1) invoking some form of inner necessity governing scientific progress and (2) settling for mere contingency. On the first approach it will be tacitly supposed that a “natural tendency” or telos is at work guiding the development. This idea will not recommend itself to an empirically minded analyst, who would therefore choose the second approach. Internationalism is to be analyzed strategically, not teleologically. The relevant comparisons are with the globalization of markets or the spread of the arms race.

Any move toward transnational knowledge should be interpreted in a wholly matter-of-fact manner. Sometimes scientists will reach out across na­tional boundaries, and sometimes they will not. It will depend on opportuni­ties and on perceived advantages and disadvantages and will vary with time and circumstance. (Recall Prandtl’s ambivalent reaction to cooperation in the immediate postwar years.) There will be no inner necessity at work, and references to the “transnational” character of science should not be accompa­nied by starry-eyed sentiments about Universal Truth. Why was von Karman in the United States? What was he doing at Caltech, and who was support­ing his research? 91 Each case needs to be examined by the historian for its particular features and causal structure. Thus in my study I found there was a phase when the reports of German work prepared for the Advisory Com­mittee for Aeronautics lay gathering dust, and there was another phase when copies of the Technische Berichte were sought with urgency. Both should be counted as equally natural. The universality of science and technology, or the absence of universality, depends on familiar, human realities. Some of these will be the brutal realities of war, power politics, and military and diplomatic strategy. Others will be the softer and more agreeable realities of the kind recalled in the exchange between Glauert’s young widow and von Karman— such as taking tea on a Cambridge lawn. These two levels, so different and yet so intimately connected, need to be brought together and linked to the calculations and experiments carried out at the research front. This is what I have sought to do.92

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.

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.

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.

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

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).

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

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.

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.

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

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).

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.