Category The Enigma of. the Aerofoil

In section II Kutta sketched his mathematical method and introduced some of the basic formulas. His aim was to use a number of conformal transfor­mations. The strategy was to exploit the known flow around a circular cylin­der by transforming the cylinder into the arc representing the cross section of Lilienthal’s wing. The streamlines around the cylinder would be trans­formed into the streamlines around the wing. What is more, any circulation that is ascribed to the flow around the cylinder would be transformed into a circulation around the wing. The two steps in the procedure are therefore (1) describing the flow around a circular cylinder and (2) finding the trans­formation to turn a circular cylinder into a shallow circular arc.

Here is the formula that Kutta gave to describe the most general flow round a circular cylinder, where the circle is inscribed on the Z – (zeta-) plane:

W ^z+Zj – і-c2 jV-Zj + і■c-logC-

Each of the three parts on the right-hand side of the formula characterized one aspect of the flow around the cylinder. The term q specified the component of flow along the horizontal axis of the Z-plane, while c2 gave the component of flow in the direction of the vertical axis. The term c toward the end of the formula (without a subscript) was the constant associated with the circulatory component of the flow, that is, the component of flow in concentric circles around the cylinder. Kutta did not use exactly the same formula as Rayleigh. Whereas Rayleigh worked with the stream function for the flow around a circular cylinder, Kutta worked with the more general complex-variable for­mula which captured both the streamlines and the potential lines.

The problem was to find the right transformation to apply to the circle. How could the circle be turned into a shape resembling Lilienthal’s wing pro­file? There are no rules for finding the right transformation, and Kutta did not spot any direct way to do this. He therefore had to proceed in a piecemeal way. He constructed the required transformation by combining known for­mulas that could act as intermediate steps and whose combination had the desired outcome. He built a mathematical bridge between the simple case of the circle and the difficult case of a winglike shape, but he did so by starting at both ends and meeting in the middle. In one direction he went from a circle represented on the Z-plane (which he called the “transcendental plane”) to another shape on the t-plane. In the other direction he went from a simpli­fied geometrical description of Lilienthal’s wing, on the z-plane (sometimes, today, called the “physical plane”), to another shape on an intermediate plane called the z’-plane, and finally from the z’-plane to the t-plane. The two procedures therefore mapped their respective starting points onto the same shape in the same plane, the t-plane. This was where they met. Kutta then had the required connection between the circle and the wing.

The insight that allowed Kutta to construct this transformational bridge was that he knew a transformation that would turn an arc of a circle into a straight line and another that would turn the exterior of a circle into the top half of an entire plane. As we saw in chapter 2, a straight line counts as a poly­gon, so once Kutta had a straight line he could use the Schwarz-Christoffel theorem to link the flow to the simple and basic case of the flow along a straight boundary.

The first published application and test of Prandtl’s approach was provided by Otto Foppl in the Zeitschrift fur Flugtechnik of July 19, 1911.57 Foppl had already produced a series of experimental reports using the Gottingen wind channel to test the resistance and lift of flat and curved plates.58 These studies indicated that the laws of resistance depended in a complicated way on the effects at the edges of the plates. It was clear that the move from an infinite to a finite wing would introduce significant new factors into the account of lift and drag. Prandtl had begun to identify these factors in his lectures. Foppl’s aim now was to test a quantitative prediction made by Prandtl on the basis of the new theory he was developing. The prediction kept away from the problematic singularity at the wingtips and concerned the angle at which the air would be moving downward at a specified distance behind the wing. It concerned the induced angle of incidence.

Foppl took for granted the qualitative picture of the two (straight-line) trailing vortices or “vortex plaits” (Wirbelzopfe). The reader of the Zeitschrift was assured that these had been rendered visible in the Gottingen wind chan­nel by introducing ammonia vapor into the flow (184). This “really existing flow” (“tatsachlich vorhandene Stromung”), said Foppl, was the empirical basis on which Prandtl had built his theory (184). The question was how the downwash and the tilt in the local flow could be generated and measured in the wind channel. First it would be necessary to introduce a model wing to generate the vortex system that was under study. It should then be possible to detect the downwash by introducing a flat plate at a distance behind the wing. The angle of the plate could be adjusted until it was aligned with the tilt of the airflow. When the plate was correctly aligned with the flow, there should be no lift force on it. This is the empirical clue giving the angle of the flow. It is important that the plate should be flat, because if it were curved or had the cross section of a normal aerofoil it would still generate a lift even when pointing directly into the local flow. The zero-lift position would not reveal the angle of the flow. With a flat plate, however, the observed angle of zero lift gives the actual tilt of the flow, and this can be compared with the predicted angle.

Having reviewed the logic of Foppl’s experimental design, we can now look at the details of the experiment and the connections that Foppl made with the realities of aircraft construction. Consider the choice of the distance between the wing and the flat plate. The distance used in the prediction and test was selected on the basis of practical considerations about current aircraft design. Increasingly, and unlike the early Wright machines, aircraft were being built with a control surface, called the elevator, located at some distance behind the main wing. The elevator controls the pitch of the aircraft. In the Wright Flyer, the elevator was at the front and the propellers at the rear. By 1911 designers typically put the propeller at the front and the elevator at the tail end of the fuselage. As Prandtl explained, according to the “horseshoe” model, if the el­evator is in a horizontal position behind the wing, it will experience a definite downthrust or negative lift (Abtrieb). This will only disappear if the elevator is rotated by a specific amount which depends on the circulation around the wing. If the elevator is a flat surface (in effect a moving tailplane), then for the reasons just given it will experience a zero-lift force when it is aligned with the downward inclination of the flow behind the wing.59 Foppl therefore built a model airplane with exactly this kind of adjustable elevator. The model is shown in figure 7.10.

The main wing was 60 X 12 cm and had a camber of 1/18, while the eleva­tor was a flat plate of 20 X 8 cm. Both were made of 2.3-mm-thick zinc. The elevator, which was rigidly attached to the wing by two struts, could be piv­oted about its leading edge and fixed at different angles relative to the airflow. There was a distance of 34 cm between the elevator and the main wing, that is, the line running along the span of the wing on which the bound vortex was supposed to be located.

Foppl’s experimental procedure involved four steps, each of them us­ing the Gottingen wind channel. First, Foppl removed the elevator from the model, leaving the main wing still connected to the two struts. He placed the wing and struts in the wind channel at a realistic angle of incidence of 4.6°. The channel was run at a single, fixed speed V, and the lift on the wing was measured. The next step was to reposition the wing (still without an eleva­tor) at a different angle of incidence. This time he chose 7.6°. Again the lift

was measured at the speed V. In both cases Foppl expressed the lift as a coef­ficient Za. (This involved dividing the lift force by the density, the area of the wing, and the square of the speed.) He now had two lift coefficients, one for each of the two angles of incidence. In preparation for the next part of the experiment, Foppl reattached the elevator in order to carry out two sequences of measurements on the whole model. In one sequence the elevator-wing system was suspended so that the angle of incidence of the main wing was
4.6°, while for the other sequence the main wing was at 7.6°. For each of these angles Foppl measured the overall lift of the combined system for a range of different elevator settings. He gave the elevator seven different settings, that is, seven different angles relative to the direction of the free flow. The angles of the elevator to the free airstream ranged from +30° to -10°.

To find the forces on the elevator alone, Foppl subtracted the lift mea­surement for the wing in isolation from that of the wing plus elevator. The remaining lift force (that is, the lift force on the elevator) was then cast into the form of a lift coefficient. This gave Foppl data that could be expressed in terms of two graphs in which the lift on the elevator was plotted against the angle of incidence of the elevator—one curve for each of the angles at which the main wing had been set to the free stream. The most important feature of these graphs was the point at which the curves passed through the x-axis, that is, the angle of the elevator when its lift coefficient was zero. This was the angle at which the elevator should be parallel with the downwash, that is, the local, downward flow of the air induced by the vortex system. The graphs indicated that when the main wing was at an angle of 4.6°, the zero-lift position of the elevator was 2.8°. When the main wing was at 7.6°, the zero – lift position of the elevator, and hence the angle of the downwash, was 4.3°. The question now was whether Prandtl’s theory could predict these angles of downwash from the main wing at the two angles of incidence that Foppl had selected for his test.

Foppl duly announced the predicted value of the angles that had been deduced from the theory—but he did not say on what basis the prediction had been made. He simply informed his readers that in his lectures Prof. Prandtl had derived a formula that gave the tangent of the predicted angle of downwash. The formula was stated, but the deduction that led to it was with­held. The tangent, Foppl said, was given by the ratio w/V, where, according to Prandtl,

w = Ь£а

V nl

As Foppl explained, the coefficients of lift ZA to be entered into the formula were the ones that had been determined experimentally for the isolated wing. All the other dimensions could be taken from the model itself. Thus, b was the chord of the main wing (12 cm), l was the span of the wing (60 cm), and x was the distance along the longitudinal axis of the model from the middle of the main wing to the middle of the elevator (34 cm). With these values for the tangent, the predicted angles themselves came out at 3.3° and 4.2°. Given
that the two measured angles of the downwash (derived from the zero-lift position of the elevator) were 2.8° and 4.3°, Foppl concluded that the result amounted to “a very good confirmation of the theory”—“eine sehr gute Bestatigung der Theorie” (184). The prediction derived from the horseshoe model was correct.

The force of this claim must have been somewhat diminished because the theory used to make the prediction was not revealed. Readers of the Zeitschrift would have known that something was afoot in Gottingen, but Foppl was not going to anticipate Prandtl and expound the theory. He merely said that Prof. Prandtl would soon publish his derivation of the formula in the ZFM. No such derivation was forthcoming, but, with the benefit of hindsight, an examination of the formula makes its origin easy to guess. The formula was simply the result of applying the Biot-Savart law to each of the three straight­line parts of the horseshoe vortex and then doing the trigonometry necessary to relate the formulas to Foppl’s model.60

R. V. Southwell, from the NPL, opened the discussion after the talks and sought to defuse the situation with good-natured praise for all of the speak­ers. Southwell wondered if the Stokes equations were quite as secure as Bair – stow assumed. He raised the possibility that the underlying physics might in­volve even more complications than those already expressed in the equations. He also reported that wind-tunnel experiments under way at the NPL seemed to be finding a value for the circulation around a wing that was similar to that predicted by Prandtl, though he, Southwell, doubted if the flow near the wing would correspond to that assumed by the circulatory theory. On the other hand, he was enthusiastic about Bairstow’s fundamental research program and fully supported the need to explain the success of Prandtl’s ap­proach. Bairstow’s own contribution to the discussion was a bland response to a Dutch speaker from the audience who had sketched some of the recent work at Aachen and Gottingen. Bairstow said he was glad to hear that Con­tinental workers were taking viscosity seriously. The discussion ended on a bizarre note when Sir George Greenhill proceeded to inform the audience that the modern approach to aerodynamics was based on a paper that Ray­leigh had written fifty years ago on the irregular flight of the tennis ball. This intervention was remarkable for two reasons. First, Greenhill was rewriting history and was expecting his audience to have forgotten all about the discon­tinuity theory of lift and his own, and Rayleigh’s, contribution to it. Second, it is unclear whether Greenhill had come into the lecture late or whether he had failed to register what had been said in his presence. Glazebrook had to draw Greenhill’s attention to the fact that he was repeating a version of what Major Low had just said. This done, Glazebrook thanked all of the speakers and promptly declared the meeting closed.

Despite the pointed criticism of the Advisory Committee, Glazebrook had chosen not to respond to Low. He might have been distracted by Greenhill’s odd behavior but, leaving psychology aside, there is another possible explana­tion for Glazebrook’s nonresponse. The Wednesday session was not the first time the issue of Lanchester had been raised at the Congress. Low and Glaze­brook had crossed swords on the previous Monday, June 25. It is possible that Glazebrook had decided he had said all he was prepared to say and was not going to be drawn out on the subject again. On that Monday, Glazebrook had given a paper titled “Standardisation of Methods of Research.”104 In the

discussion that followed he had encountered some criticism by Major Low about the reliability of wind-tunnel results. Low cited some negative remarks from G. P. Thomson’s book on aerodynamics and argued that wind-tunnel data needed to be corrected. The “Lanchester-Prandtl theory,” said Low, had shown how to make the corrections, and this theory would soon be the sub­ject of his own talk. Glazebrook, who did not like to air the problems of wind – tunnel research in public, suggested that Prof. Thomson had surely changed his mind. Then, perhaps alerted by Low’s mention of his forthcoming talk, Glazebrook added a comment that was not a direct response to anything that had actually been said. As if to head off trouble, Glazebrook launched into an apologia for the way Lanchester had been treated: “With regard to the refer­ence to the Prandtl theory, I trust there is no one here who will in any way depreciate the enormous value of the work done by Mr. Lanchester and of the suggestions he has made. But it was not until Prandtl put some such sugges­tions into mathematical form that it was possible to attach to them the kind of value they have now gained, or to give Mr. Lanchester all the credit and praise that we should desire to give for his work” (65). This preemptive state­ment may explain why, on the following Wednesday, Glazebrook remained silent. He had no wish to go round the issue again.

Glazebrook’s desire to give due, if belated, credit to Lanchester may be accepted at its face value, but as an excuse for the neglect of the circulation theory, his claim has three, obvious weaknesses. (1) To say that we can now see that Lanchester was doing something valuable because Prandtl has made it clear to us does not explain why the British could not have worked it out for themselves. (2) In reality, as I have argued, British mathematicians had no difficulty in seeing the underlying mathematical form of Lanchester’s ideas. It was not the obscurity of the relation to mathematics that was the cause of the trouble, but the opposite. British experts such a G. I. Taylor were very familiar with the mathematical form of the circulation theory. It was actually the underlying mathematical form (the potential flow of an inviscid fluid) that they rejected on the grounds that it could not refer to processes that were physically real. (3) When British mathematicians were presented with a developed mathematical expression of Lanchester’s theory, they still expe­rienced difficulty in coming to terms with it—witness, for example, G. H. Bryan’s negative review of Joukowsky, the responses of Lamb to Kutta, and Bairstow’s response to Prandtl and Betz. In all cases the work struck them as a problem rather than a solution. None of these three points is accounted for by Glazebrook’s version of the events. It is not difficult to see why a well – informed practical designer such as Low might have felt less than convinced by Glazebrook’s answer. No wonder he could not resist raising the matter again and putting Glazebrook on the spot.105

I have now looked at some of the discussions about aerodynamics that took place in Britain in the immediate postwar years. It is clear that the math­ematically sophisticated British experts did not take the view that “there was nothing to learn from the Hun.” They were learning and learning quickly, but there was disagreement about what, and how much, was to be learned. How, and on what terms, was the Gottingen work to be assimilated? While the arguments at the Royal Aeronautical Society and the International Con­gress were conducted in the public realm, there had been other arguments that were still running their parallel course behind the closed doors of com­mittee rooms. It is to these that I now return. In the next chapter I pick up the story of the discussions initiated in the Aeronautical Research Committee by Glauert’s resolve to champion the merits of the circulation theory and Prandtl’s theory of the finite wing.

One of the objections that critics have repeatedly directed against the Strong Program is that the commitment to causal, sociological explanation entails neglecting the role of reasons. The critics say that a Strong Program analysis involves disregarding the reasons that social actors themselves offer for their behavior. According to the critics these reasons can, on occasion, provide a sufficient explanation of the behavior and thus render redundant any attempt to construct a causal, sociological explanation. A recent example of this criti­cism is to be found in a 2006 paper by Sturm and Gigerenzer.7 The authors say: “Even after a strong sociological explanation has been given for the be­liefs of a scientist, it remains sensible to ask: Very fine, but how are these beliefs connected to the scientist’s justificatory reasons? Can these reasons perhaps explain better why the scientist acquired the relevant beliefs?” (144). As the wording indicates, these critics assume that the candidate sociological explanation will have been constructed without any significant reference to the agent’s reasons. The critics wish to make good this alleged lack, and they put their money on finding cases where nothing more than the scientist’s own reasoning is needed to explain some pattern of scientific belief. The agent’s reasons, they say, can be the cause of their beliefs, and a proper explanation of these beliefs should be in terms of these reasons (141).

Does the causal explanation that I have put forward to account for my findings in the history of aerodynamics depend on, or result in, a lack of

serious attention to the agent’s reasons? I hope it will be evident that such a complaint is groundless. The problem I have posed, the central problem of the book, is a problem about the reasons that were used to justify a certain scientific judgment. I have attended closely to the reasons given by the actors in the story and subjected them to a close analysis. I fear, however, that critics will dismiss my discussion of scientific reasoning as mere window dressing. Thus Sturm and Gigerenzer say, “Mentioning that scientists claim to employ certain reasons or reasoning standards, and mentioning which ones these are, is not the same as taking these standards seriously—seriously in the sense that they are acknowledged as causes of the scientist’s acceptance or rejection of claims” (142). For these critics, nothing short of treating reasons as self­sufficient explanations will count as taking them seriously. This I certainly have not done and will not do because it would corrupt the analysis. Despite this, attending to the actor’s reasons has played a central role, even though the analysis culminates in a causal explanation. The actor’s reasons are not merely mentioned; they are given a substantial role. Consider the stated reasons of­fered by the British to justify their rejection of the ideal fluid approach, for example, their complaint that the origin of circulation must forever remain a mystery. These reasons certainly illuminate the British rejection. When the reasons are examined, however, it becomes clear they do not adequately ex­plain the behavior of the British experts. This is not because the British had other “real” reasons. The inadequacy of the explanation is because their Ger­man counterparts understood the reasons that moved the British as well as the British did, but responded differently. Kelvin’s theorem was as familiar to Prandtl as it was to Taylor or Jeffreys. For me, therefore, taking reasons seri­ously, and assessing their causal role, requires asking why the same facts and the same theorems, that is, the same reasons, caused such divergent responses in the two groups of professionals.

Faced with a situation of this kind, where stated reasons underdetermine the response, should the historian look for a difference in the intellectual presuppositions behind the reasoning of the two groups? Perhaps this ap­proach could uncover hidden premises and lead to an explanation of the dif­ferent reactions. An appeal to presuppositions might keep the analysis within the realm of self-sufficient reasons in the way that the critics want. A search for presuppositions is certainly important in any historical analysis, and it has been a feature of my own procedure. As I have shown, such a search uncovers subtly different conceptions of an ideal fluid. The British experts treated an ideal fluid as a fluid of zero viscosity (p = 0), while their German counterparts treated it as the limit of a fluid of small viscosity (p ^ 0). The British drew a strong boundary between inviscid theory and viscous theory.

German-language work involved a weaker and differently positioned bound­ary. Thus von Mises was inclined to treat the objects of both the Euler and the Stokes equations as abstractions, while Prandtl was inclined to treat them both as realities. Despite their differences, both of these German-language thinkers placed viscous and inviscid fluids on a par with one another. This aligned the two against the more literal-minded realism of the British, who treated viscous fluids as real and inviscid fluids as unreal.

There can be no doubt, then, that identifying presuppositions of this kind deepens the analysis, but it still cannot furnish an explanation of the diver­gent responses of the British and the Germans. Presuppositions are simply reasons for reasons, so the real problem is postponed rather than solved. It merely leads to further questions: Where do the presuppositions come from? Why did the British and Germans have different presuppositions?

The point may be made in another way. I identified a sequence of judg­ments that informed the technical content of the aerodynamic knowledge of lift. At each point in the sequence the British experts jumped in one direction while the German experts jumped in the other. Such was the case regard­ing (1) the significance attached to the “arbitrary” value of the circulation,

(2) the meaning of the zero-drag result, (3) the importance of explaining the critical angle at which a wing stalled, (4) the reaction to the overoptimistic lift predictions derived from the theory of circulation, and (5) the problem of explaining the origin of the circulation around a wing within the confines of the theory of ideal fluids. The divergence of judgment on these questions was systematic, fundamental, and constitutive of the rival understandings of the two groups. It cannot be dismissed as a coincidence, but nor can it be ex­plained by the divergent reasons themselves. The deployment of reason is the problem, not the solution. The phenomenon calls for a causal explanation, and that is what I have given.

So far I have described my procedure in terms of an apparent transi­tion from reasons to causes. I have said, in effect, that my analysis may have started with reasons but it finished by my making an appeal to causes because reasons became equivocal. I justify the claim that the analysis is causal (and conforms to the Strong Program) by saying that an appeal to causes is, in the end, unavoidable. This argument is correct, but as a way of speaking it can generate problems. At best it is a provisional way to state the methodology behind the analysis.

The problem is that two modes of speech and two perspectives are in play: those of the actors and those of the analyst. Keeping both modes of speech in play may suggest that there are two different sorts of cause at work, namely, rational causes and sociological causes. Are we to conclude that reasons cause some of the behavior of the scientists under study but not all of it, so that the remainder has to be explained by sociological and nonrational causes? Does rationality provide a partial cause alongside other kinds of cause furnished, say, by the social context? Some such view may seem to be underwritten by the historian’s own investigative procedure or, at least, by the way the procedure is sometimes presented. First, it seems, reasons are examined and then, and only then, are sociological causes to be invoked (as if they were a residual cat­egory). But granted that the work of the historical analyst sometimes exhibits such a pattern, it would be wrong for the analyst to project this expository sequence into the picture of the historical actor and imagine that actors are, or may be, subject to a corresponding sequence of influences. I do not believe that a dualism of rational and sociological causes, which allegedly compete or alternate with one another, or supplement one another, can be the basis of a satisfactory perspective. It is eclectic and merely encourages baldly posed questions. Something more unified, and hence reductive, is called for. If the Strong Program is correct, then “rational causes,” which have so often been treated as sui generis, are really nothing more than a species of social causes.

Think of the confrontation between Lanchester and Bairstow when they clashed in public in 1915. Bairstow said that Lanchester could not explain why aircraft stalled, so his theory could not be taken seriously; Lanchester said that he did not need to explain this stalling because a theory of narrow scope could still be valuable within its limited domain. My claim is that, in tracing the arguments that Bairstow and Lanchester used against one another, a good historical analyst will, at the same time, be tracing the causal texture of their interaction. There will be no duality of rational and social causes and no tran­sition from one to the other. A properly historical account of the interaction between Lanchester and Bairstow will be in terms of social causation from the outset. A unified, social-causal perspective of this kind can be sustained if the analyst focuses relentlessly on the credibility that the participants and their audience attach to the arguments that are being advanced. Why did the failure to explain the onset of a stall worry Bairstow in a way that it did not worry Lanchester? Why was Bairstow’s concern shared by other British ex­perts but not, to the same degree, by Kutta, Prandtl, and other supporters of the circulatory theory in Germany? These are the questions that will expose the sociological basis of the power of reason, and these are the questions to which I have given answers.

The importance of credibility as a causal category, with its variable and distributed character, is at its most striking when the overall scene is brought into view, for example, the systematic divergence of the German and British responses to circulation. But actual or possible divergences of this kind are not confined to the large scale. They are a feature of every act of reason giving and every act of responding to reasons, whether interpersonal or intraper­sonal, whether public or private. This is because, on its own, invoking and formulating reasons can never be sufficient to render a belief causally intel­ligible or a course of action causally explicable. Things may not look this way from the actor’s point of view. Sometimes the reasons that are advances in the course of an interaction are accepted by other actors as sufficient justifica­tion or explanation. But I am giving the analyst’s perspective. I am speaking here from the point of view of a historian or sociologist who is committed to giving a causal analysis of a passage of interaction and behavior. Of course, critics say the claim that reasons are never sufficient is mere dogmatism or the result of an irresponsible generalization. Like others before them, Sturm and Gigerenzer see here nothing but a lack of prudence on the part of sup­porters of the Strong Program.8 They think it is more judicious to allow that reasons sometimes explain rather than never explain. In fact my claim is not dogmatic; it is made on the basis of a general and principled argument. The argument comes from Wittgenstein’s analysis of rule following and has ex­plicitly informed the Strong Program from the outset.9 Because the argument is so important, and so often misunderstood or ignored, I rehearse it here and then connect it to my overall analysis.

In April 1997 Peter Galison and Alex Roland organized the conference “At­mospheric Flight in the Twentieth Century,” which was held at the Dibner Institute in Cambridge, Massachusetts. By a stroke of good fortune, and the generosity of the Dibner Institute, I was able to attend the meeting. My role was to act as an outside commentator. I was deeply impressed by the high quality of all of the papers that were presented, though I confess I was some­what daunted by the technical expertise of the contributors. The conference opened my eyes to a field of work, the history of aeronautics, that was new to me but which proved immediately attractive.1

One paper in the conference that caught my attention dealt with early British research in aerodynamics and the way in which, in Britain, the gulf between science and technology was bridged. The paper was titled “The Wind Tunnel and the Emergence of Aeronautical Research in Britain.”2 After the conference its author, Dr. Takehiko Hashimoto, kindly sent me the un­published Ph. D. thesis on which his paper had been based.3 Dr. Hashimoto’s main concern was with the role of those important individuals who act as mediators, middlemen, and “translators” between mathematicians and engi­neers. By comparing the development of British and American aerodynamics (and their respective responses to German aerodynamics after World War I), he reached the gratifying conclusion that the British had been somewhat more successful in this process of mediation than had the Americans. I say “gratifying” because I am British, and the British frequently take a pessimistic attitude toward their own technological capabilities and tend to assume that other countries always do things better. I did not pursue the theme of the mediator or middleman, but it was this work that prompted me to do the research presented here. Although we paint a somewhat different picture of certain people who feature in both of our studies, I express my indebtedness to Dr. Hashimoto and my appreciation of his work.

I began by following up some of Dr. Hashimoto’s references in the Pub­lic Record Office in London and soon found a set of research questions of my own that I wanted to answer, as well as evidence that there was material available with which to pursue them. My questions were these: In the early days of aviation, that is, in the early 1900s, there were rival accounts of how an aircraft wing provides “lift.” One account was supported by British ex­perts, while the other was mainly developed by German experts. This was well known to historians working in the field.4 These two theories of lift were also featured, though not in technical detail, in Dr. Hashimoto’s account.5 But I wanted to know (1) why the rivalry arose, (2) what sustained it for al­most twenty years, and (3) how it was resolved. These questions were not ad­dressed in Dr. Hashimoto’s work, nor had they been convincingly answered in any of the broader historical literature in the field. The present book sets out the conclusions that I eventually reached on these three questions.

My kind colleagues in the Science Studies Unit at the University of Ed­inburgh bore the disruptions caused by my research-related comings and goings with understanding and good humor. I am all too aware that my ac­tivities must have added to their own already considerable work load. Relief from teaching and administrative duties during crucial parts of the research was made possible by the Economic and Social Research Council (ESRC). I thank the Council for its financial support in the form of a project grant ESRC Res 000-23-0088. Grants specifically designed to offset the costs of publication came from two further sources: Trinity College, Cambridge, and the Royal Society of London. I thank the Master and Fellows of Trinity for their generosity, and I also express my appreciation for the continued sup­port of the Royal Society, in these financially straitened times, for work in the history of science.

The argument of my book involves a detailed comparison between British and German aerodynamic work, and this subject would have proven impos­sible to study without a number of lengthy visits to the Max-Planck-Institut fur Wissenschaftsgeschichte in Berlin. I must record my deep gratitude to Lorraine Daston and Hans-Jorg Rheinberger, the directors of Abteilung II and Abteilung III, respectively, and to Ursula Klein and Otto Sibum, who were directors of two of the independent research groups in the Institute. Their warm welcome and great generosity will never be forgotten, nor will the stimulus provided by the research environment they all worked so hard, and so successfully, to create. I also express particular thanks to Urs Schoe- pflin, the Institute librarian, and his dedicated team. They met my endless stream of requests and queries with unfailing professionalism, kindness, and scholarly understanding. Special mention must be made of one member of the library team, Monika Sommerer, who, in the final phases of writing the book, kindly began the work of approaching copyright holders for permission to reproduce the photographs and diagrams that illustrate my narrative.

One of the first things I did in Berlin was to make working translations of the main German technical papers that were relevant to the analysis. (By a “working translation” I mean something adequate for my own use rather than for public consumption.) Here I thank Marc Staudacher, a resourceful teacher of German and a professional translator, who spent many hours with me going over my attempts in order to check them and to explain points of grammar and meaning that were eluding me.

In developing the British side of the story I am indebted to the Royal Aeronautical Society in London for access to their unique collection of early aeronautical literature. I am deeply grateful to Brian Riddle, the librarian, who put this material, as well as his profound knowledge of the field, at my disposal. It was also through the good offices of Brian Riddle that I was able to make contact with Dr. Audrey Glauert of Clare Hall, Cambridge. Dr. Glauert generously made available to me material relating to her father and mother, both of whom played an important role in the development of aerodynamics and therefore feature prominently in my book. I hope I have been able to put that material to good use. The opportunity to talk with someone directly con­nected with the historical actors and episodes I was describing was a moving experience, and I express my gratitude to Dr. Glauert for her hospitality and kindness.

From its inception I have discussed my research project with Walter Vin – centi of the University of Stanford. I have benefited immeasurably from nu­merous and lengthy conversations drawing on his firsthand experience of aerodynamic research. His patience in discussing the arguments of the early technical papers and his willingness to read and comment so carefully on the first drafts of many of the chapters of this book have been invaluable to me in learning to find my way in this new field. It has been a privilege to be able to put my questions and problems to him and to be the recipient of his expert and thoughtful answers. Donald MacKenzie read and commented on a number of early draft chapters; later, drafts of the complete book were read by Barry Barnes, Celia Bloor, Michael Eckert, Jon Harwood, and Horst Nowacki. Not only their encouragement but also their critical comments have been invaluable, and I have made extensive alterations as a result of their suggestions. The responsibility for the defects that remain can only be laid at my doorstep.

In addition I have accumulated many other debts of gratitude for the help I have received in the course of the research—guidance to the literature and new sources, help in approaching and gaining access to archives, and numer­ous conversations on historiographical, methodological, and philosophical questions. I hope the following persons will forgive me if I do not mention individually their many and varied acts of kindness and generosity that, nev­ertheless, I so clearly remember. My sincere thanks to Andrew Barker, Jed Buchwald, Dianna Buchwald, Harry Collins, Ivan Crozier, Olivier Darrigol, David Edgerton, Heinz Fuetterer, Zae-Young Ghim, Judith Goodstein, Ivor Grattan-Guinness, John Henry, Dieter Hoffmann, Christoph Hoffmann, Marion Kazemi, Kevin Knox, Martin Kusch, Wolfgang Lefevre, David Mus- ker, Jurgen Renn, Simon Schaffer, Suman Seth, Steven Shapin, Skuli Sigur – dsen, Richard Staley, Nelson Studart, Steve Sturdy, Thomas Sturm, Annette Vogt, Andrew Warwick, and Richard Webb.

I have used material from the following archives and express my thanks to the archivists for permission to consult their holdings: Archives of the Cali­fornia Institute of Technology (Karman); Archiv zur Geschichte der Max – Planck-Gesellschaft (Prandtl); Churchill Archive Centre, Cambridge (Far – ren); Einstein Papers at Caltech (Einstein and Frank); Gottingen Archive of the Deutsche Gesellschaft fur Luft-und Raumfahrt (Prandtl); Library of the University of Cambridge (Tripos exam papers); National Library of Scotland (Haldane); Public Record Office (minutes of the ARC); Royal Aeronautical Society (Lanchester and Grey); Royal Air Force Museum, Hendon (Melvill Jones); St. John’s College, Cambridge (Jeffreys and Love); Trinity College, Cambridge (Taylor and Thomson); University of Coventry (Lanchester); and University of Edinburgh (A. R. Low).

The provenance of all photographic images and diagrams from published and unpublished sources is indicated in the caption along with an acknowl­edgment of copyright and permission to reproduce the material. In a few cases it proved impossible, despite every effort, to make contact with the holders of the copyright.

Finally I must mention my greatest debt. Throughout the research and the writing of this book I have benefited from the unstinting help of my wife. The book is dedicated to her. It is as good as I can make it, but it still seems little to give in return. I proffer it with the sentiment Wenig, aber mit Liebe.

The motion of a fluid element involves three different kinds of change:

(1) translation, (2) strain, and (3) rotation. Translation involves change of position of the element, strain involves a deformation of the shape of the element, and rotation involves a change of angular orientation of the ele­ment. Rotation may seem to be an intuitively clear idea because the image that comes to mind is the rotation of a rigid body in which the fluid element is pictured as if it behaves like, say, a spinning ball. Sometimes fluid elements are indeed represented as spinning balls. Although shape is not really crucial, the picture of a sphere is sometimes invoked when explaining the striking result that a fluid element in an ideally inviscid fluid can never be made to rotate if it is not already rotating, nor can it be stopped from rotating if it is already in rotation. The rotation of an ideal fluid element can neither be created nor destroyed by, for example, the motion of a solid body that is immersed in, and surrounded by, a fluid. The argument is that, in a perfect fluid, neither the surrounding fluid nor such a moving body can exert any traction on the smooth surface of the element in order to change its exist­ing state of rotatory motion. It will be evident that, in light of this result, the origin of rotation becomes something of a mystery.19

Cowley and Levy, however, do not avail themselves of an intuitive pic­ture of fluid elements as rotating spheres of fluid. They opt for the more austere technical definition. Technically, the rotation of a fluid element (in two-dimensional flow) is defined as the average angular velocity of any two infinitesimal linear elements within the fluid element that are instantaneously perpendicular to one another. Mathematically this definition is expressed in the formula

1(dv du |

rotation = — ———- — .

2 ^dx dy )

The virtue of the technical definition is that commonsense comparisons tend to omit the possibility that the angular velocity of the two linear elements might cancel out so that, under some circumstances, rotation can be equal to zero by virtue of the deformation of the fluid element.20 A flow in which the quantity in the brackets in the previous formula is zero is called an ir – rotational flow.

Methodologically, the important point about the rotation of a fluid ele­ment is that by neglecting it, and restricting attention to irrotational flow, the mathematics is greatly simplified. Why is this? A glimpse into the reasons can be gained by taking another look at the stream function discussed in the pre­vious section. Consider the following expression involving the stream func­tion y. The expression is arrived at by differentiating у twice with respect to x and twice with respect to y and adding the result. Thus,

dy dy

dx2 + dy2′

It will be recalled that differentiating у once yields the velocity components of the flow and that the x and y components of the fluid velocity at a point are given by

dy

u =—— and

dy

= dy

dx

Substituting these definitions of the velocity components in the expression under consideration gives

dV+dV=+Af+d^VAf_—1

dx2 dy2 dx ^ dx J dy ^ dy J

dv du dx dy

The result of the substitution is precisely the expression that was used in the technical definition of the term “rotation.” It is in fact twice the rotation. If the rotation is zero, that is, if the flow is irrotational, then this term must be zero, and so, therefore, is the expression cited at the outset of the discussion. In other words, if the flow is irrotational, then the stream function у is gov­erned by the equation

dy+dy= 0

dx2 dy2

This equation is called Laplace’s equation. Although the equation itself may look far from simple, it is not difficult to appreciate that it is simpler than if the right-hand side were equated to some complicated function of x and y rather than to zero. Irrotational flow is thus a (relatively) simplified form of flow governed by Laplace’s equation.

Laplace’s equation is one of the most significant differential equations in the history of mathematical physics.21 The equation is often written as V2y = 0.22 The restriction to “irrotational” flow, which it signifies, not only simplified the mathematics, but it brought out the analogies between fluid flow and the results that had emerged or were emerging in other fields. Ir – rotational flow obeyed simple mathematical laws that were similar to those in areas such as the theory of gravitational force, the theory of heat, the theory of elasticity, and the theory of magnetism and electricity. Maxwell used the analogy, and Laplace’s equation, to shed light on the hydrodynamics of the flow of fluid through an orifice and the vena contracta, that is, the contraction shown by the jet of fluid a short distance from the orifice.23 Because of the electrical analogies, irrotational flows used to be called “electrical” flows. The interplay between hydrodynamics and the theory of electric phenomena was not only suggestive theoretically, but it was also exploited in the laboratory. In the interwar years it provided the basis of a laboratory-bench technique used by E. F. Relf at the National Physical Laboratory for graphically plotting the streamlines of the flow around objects with complicated shapes, such as aerofoils.24 The resulting representation was, of course, a representation of the flow as it would take place if the air were an ideal fluid.25

G. H. Bryan described the approach to lift adopted by the practical men as neo-Newtonian.64 The label accurately identified two salient features of their work. First, like everyone else, the practical men operated within the frame­work of Newton’s mechanics. Ultimately the wing must act on a mass of air, accelerating it downward, thus ensuring, in accordance with Newton’s third law of motion, that the wing suffered an equal and opposite reaction. This re­action was the ultimate source of the lift. Second, the practical men adopted a line of reasoning that was, in some respects, analogous to one that New­ton used in the Principia when he compared the forces exerted by a flowing fluid on a sphere and a cylinder “described on equal diameters.”65 Recall that Newton assumed that his fluid, or “rare medium,” consisted of a number of independent particles which would hit the sphere and cylinder and give up their momentum. (This was the model I previously likened to a shower of hailstones). The practical men greatly simplified the analysis by address­ing the case of a flat plate exposed to the uniform flow of this rare medium. While Newton was no doubt conscious of the distinction between his hypo­thetical fluid and real air, this difference tended to be blurred in some of the later aerodynamic discussions. By applying the reasoning to a simplified wing moving in air, the following argument was constructed.

Suppose that a flat plate has area A and is at an angle 0 to a uniform, horizontal flow of a fluid that was, like Newton’s, composed of independent particles. Suppose, further, that the collisions are inelastic so that the particles simply slide along the plane after impact. Let the particles in the main flow move at speed V units of distance per second. Then the volume of fluid strik­ing the plate each second is given by multiplying the vertical projection of the plate (A sin0) with the velocity V. The projection A sin0 was the “sweep” of the wing and sin0 was the “sweep factor.” Now multiply the volume AVsin0 by the density p (presumed to be the same as that of the air) to give the mass, and then multiply the mass by the velocity component normal to the plate (V sin0) to give the momentum exchanged per second. This is the source of the pressure P whose vertical component, P cos0, is the lift and whose hori­zontal component, P sin0, is the drag. This neo-Newtonian argument gave the formula for the resultant aerodynamic force P on the plate as

P = pAV2 sin2 0.

The formula was often called Newton’s sin2 law, although it is not to be found, in an explicit form, in the text of the Principia. There are two reasons for its absence. First, Newton was dealing with a curved surface not a flat plate, and second, his reasoning was geometrical in form, so that the trigonometric terms appear as geometrical ratios.66 The label is, however, a reasonable one. All of the subsequent work of the practical men involved versions of, and variations on, this formula.

For the range of angles relevant to aeronautics, sin0 is a small quantity, so its square is very small indeed. The Newtonian formula condemns any predicted lift to be small, except where the magnitude of A and V2 can offset the smallness of the squared sin0 term. On this analysis, lift would demand enormous velocities or unrealistic wing areas. Had the formula been true it would have rendered artificial flight a practical impossibility. It is little won­der that in his 1876 paper Rayleigh had expressed satisfaction that his own formula made pressure proportional to sin0 rather than, as in Newton’s for mula, to sin2 0. In following the Newtonian tradition the practical men inher­ited a serious problem and resorted to a variety of expedients in an attempt to overcome it. A number of examples will show how comprehensively they failed to meet this challenge.

Consider the scope of Lanchester’s theory. His narrow focus on small angles of incidence was not shared by critics. Bairstow invoked standards of assess­ment appropriate to a wholly general theory of fluid resistance. Lanchester found this preposterous. He said it was like ignoring useful knowledge about how water flowed round a ship in normal motion because it did not also

explain the flow when it moved broadside. But it is not difficult to see how for the critics, if not for Lanchester, genuine knowledge of the one case also meant having knowledge of the other. Lanchester’s commonsense plea for theories of limited scope was at odds with the forms of generality routinely exhibited in classical hydrodynamics. This can be seen from the textbook treatment of the flow around an elliptical cylinder moving through a fluid. The elongated, elliptical cylinder bears a certain visual likeness to the plan of a boat, which was the case cited by Lanchester. Lanchester’s critics could point out that the mathematics of the flow does not single out, as being of special significance, any particular angle of inclination of the major axis of the ellipse to the di­rection of motion. It makes no difference to the mathematics whether the ellipse moves like a ship going forward or like a ship moving broadside, that is, awkwardly and inappropriately. Both motions are but special cases of the same general formula. Mathematically they merely depend on whether the real or the imaginary part of the complex potential is set to zero. This fact would have been familiar to any Cambridge student of hydrodynamics, or to anyone, such as Bairstow, Cowley, or Levy, schooled in a similar tradition. It was clearly not a significant reference point for Lanchester.3

The issue of scope also arose in another way. We have seen that circula­tion theory explained lift but not drag. The critics had rejected discontinuity theory because it could not yield accurate predictions of resistance, so on grounds of consistency circulation theory should be, and was, treated like­wise. The false prediction of zero drag was not lost on Lanchester, but it did not worry him in the way it did his critics. Unfortunately, Lanchester did not articulate a clear rationale for his stance, so the critics may have been tempted to see it as indicating a certain laxity on his part, compared to their own greater concern with truth and rigor.

There is some evidence that Cambridge physicists involved in aerodynam­ics were prone to misperceive the difference between their mental habits and those of engineers as the difference between rigor and sloppiness. Reflect­ing on his work as a physicist at Farnborough during the Great War, George Paget Thomson, the son of J. J. Thomson, drew attention to this cultural divide. Scientific work during wartime, said Thomson, “might properly be described as engineering.”4 He recalled how difficult it was for physicists to adopt the requisite point of view. As the author of Applied Aerodynamics, which had been well received by the “practical men,” Thomson could not be accused of lack of sympathy with engineers. But even he was inclined to exemplify, rather than bridge, the disciplinary divide he described. Thom­son spoke of the need for engineers to make up their minds on the basis of “insufficient evidence” and of the need to “compromise between conflicting requirements.” He concluded: “What is perhaps harder for the scientist to realize is the doctrine of ‘good enough.’ The better is the enemy of the good” (3).

Could it be that Lanchester, as an engineer, was prepared to accept the circulatory theory and perfect fluid theory because they were “good enough” for him even though they were not “good enough” for a physicist? The im­plication is that Lanchester, unlike his critics, was content with “insufficient evidence.” But there is another explanation of why a supporter of the circula­tion theory might find the lift-but-no-drag result an acceptable one. Rather than expressing a compromised standard of empirical accuracy, the response might simply embody a different standard and one that is not necessarily lower. The lift-without-drag result might be taken to be a true and accurate assertion about an “ideal wing,” that is, the sort of wing at which an engineer might aim. The result should perhaps be seen not as a false statement, but as an engineering ideal. This was not a defense explicitly offered by Lanchester, but as we shall see, it was how Ludwig Prandtl, a fellow pioneer of the circula­tion theory, expressed the matter.

Consider now the objection that the circulation is “arbitrary.” Both Kutta and Joukowsky were aware of the mathematical rationale behind this objec­tion, namely, that the theory contained no way of deducing the amount of circulation around a wing. Nevertheless they responded in a very different way to the British critics. They stipulated that the circulation be of precisely the amount necessary to ensure that the flow comes away smoothly from the trailing edge of a wing. The rear stagnation point must be on the trailing edge so that the flow does not have to wrap itself around a sharp corner. For a given angle of incidence, and a wing with a sharp trailing edge, this stipula­tion provides an unambiguous specification of the amount of circulation and is often called the Kutta condition. It derives its significance, and its nonarbi­trary nature, from the empirical fact that the flow of air over a (nonstalling) wing in a steady state settles down so that there is indeed an approximately smooth flow at the trailing edge. The Kutta condition tells the theorist what value of the circulation to assume, and thus what value of lift is predicted when this value is substituted into the formula Kp U, the lift equation.

Were Cowley and Levy, who made the complaint about arbitrariness, un­aware of this solution to the problem? The answer is that they were fully aware of the Kutta condition. In 1916 Levy had explicitly mentioned it in correspon­dence with Lanchester.5 In Levy’s view, however, Kutta’s proposal did not remove the arbitrary character of the amount of circulation. The argument presented to Lanchester was that, in reality, the trailing edge of a wing is not mathematically sharp but rounded. It therefore provides no mathematically unambiguous location for the rear stagnation point. The point on the curve that is selected for this role will itself be arbitrary. The amount of circulation needed to bring the stagnation point to this location will, therefore, also be arbitrary. The only thing that would remove this feature of the theory would be some means of deducing the circulation from first principles, given rel­evant data about the wing, for example, its shape and angle of incidence. No such method was known. For Cowley and Levy the word “arbitrary” clearly meant “not deducible from the basic equations of fluid dynamics.” They op­erated with a mathematical criterion and were looking for a mathematical solution to the problem, not an empirical one.

Kutta now carried out the procedures for which he had prepared the ground. He began on the z-plane and specified the detailed geometry of the wing. It was to be an arc of a circle of radius r subtending an angle of 2a. This gave the coordinates of the endpoints A (the leading edge) and B (the trailing edge). The straight-line distance between A and B was the “chord,” and the highest point of the arc was to be 1/12 of the chord. Kutta chose to place this high­est point at the origin of the coordinate system. He then began the process of transformation. First he used a transformation in which every point was replaced by its reciprocal. Points on the z-plane were linked to those on the z’-plane by the formula z’= 1/z. This had the effect of turning the finite, cir­cular arc into what appeared to be two straight lines. One of them ran parallel to the positive part of the x-axis while the other ran parallel to the negative part of the x-axis. Both were at the same height above the axis. They started at equal distances from the y-axis (that is, there is a gap in the middle), and the lines went off to infinity in opposite directions.

It would have helped the reader of Kutta’s paper if, at this point, he had provided a diagram. Given the pedagogic values of the technische Hoch – schulen, he would surely have drawn pictures of such transformations on the blackboard when he presented them in lectures. Most mathematicians reading such a paper would sketch the appropriate figures, at least until the transformation had become routine for them. To help us follow Kutta’s argu­ment, I exploit an example of this practice. Sometime in the 1920s a young Cambridge mathematics graduate named Muriel Barker had occasion to work through Kutta’s article. She carefully wrote out the reasoning, some-

times filling in the steps needed to get from one line to another. She also sketched the conformal transformations. These handwritten notes have sur­vived, and one page from them, containing the sketches, is reproduced here as figure 6.2. Muriel Barker will appear again, later in the story, when the reasons for her interest become apparent. For the moment her notes can help us follow Kutta’s thought processes.

On the top left of the page of the notes is a figure labeled z-plane. It is a drawing of the Kutta-Lilienthal wing with the leading edge labeled A and

the trailing edge labeled B. The effect of the transformation z’ = i/z is shown next to it in the diagram, on the top right of the notes, labeled z’-plane. No­tice how the arc has become two straight lines and the leading and trailing edges A and B of the wing have become the endpoints A’ and B’ of the lines. Following his overall plan, Kutta next mapped these lines onto the t-plane where it would eventually link up with the transformed circle. This he did by using the Schwarz-Christoffel transformation. I have described how this transformation played an important role in the mathematical development of the theory of discontinuous flow. It was central to Greenhill’s massive report on this theory for the Advisory Committee for Aeronautics. Kutta used the transformation in a different way and in the service of the circulation theory. He needed it to construct the central arch of his mathematical bridge. The formula of the transformation can be seen about halfway down the page of notes in figure 6.2. It takes the form

The letter C is a constant, and a and b correspond to the endpoints of the wing. Immediately to the right of the formula is a sketch of the result of the transformation produced by applying this formula. The lines on the z’-plane have become the axis of the t-plane. The new line is shown as dotted in the figure, and the points corresponding to A’ and B’ have been marked in. All that was needed now was to work from the other end in order to map the circle onto the t-plane. The inferential bridge would then have been con­structed according to plan. The circle in the Z-plane is drawn on the bottom right-hand corner of the notes. The formula

Z+1

Z-1 is the transformation linking Z and t. This can be seen in the notes standing to the left of the drawing of the circle. Kutta’s aim might be described as getting from the figure at the bottom right to the figure at the top left of the page, but because he could see no way of doing this directly, he made the transition indirectly, by means of the other figures.

Coming back from the Barker notes to the original paper, we see that Kutta was now in a position to evaluate the constants in his formula in terms of the assumed velocity and direction of the free stream relative to the wing. He could also arrive at a value for the circulation on the assumption that the trailing edge is a stagnation point, that is, that the flow does not have to curl around the rear edge. This gave him the following expression for the
all-important circulation, which, in the notation used by Kutta, is 2ПС. The formula came out as

Circulation = 4nVr sin—sin I —+B,

2 ^ 2 H)

where V is the velocity, a the half angle of the arc that constitutes the wing, and в the angle of incidence of the wing to the free stream. The circulation is thus calculable from known or knowable quantities.

The law of Biot and Savart received a number of further aerodynamic appli­cations before the outbreak of World War I. All of these were published in the Zeitschrift fur Flugtechnik and came from the Gottingen group. Four of them were by Albert Betz and one by Carl Wieselsberger. I describe them briefly, keeping to the chronological order of their appearance.

In September 1912 Betz published some wind-channel results that showed that a wing operating in the vicinity of the ground would experience an in­crease in lift.61 Betz showed this by testing a model wing in a channel fit­ted with a false floor that could be raised or lowered. The phenomenon was an important one. Aircraft necessarily fly near the ground on landing and takeoff. Pilots were aware that there was a change in flying characteristics produced by these circumstances, but the nature of the change was little understood. This “ground effect” explains why an overloaded aircraft can sometimes take off with apparent success and then fail to gain height, with disastrous consequences. It also explains why some early aircraft could “fly” but never got more than a few feet above the ground.62 Betz also wanted to get a quantitative estimate of the effect of the walls of a wind channel on the measurements that were carried out in the course of experimentation. He showed that Prandtl’s new theory could lead to rough but quantitative pre­dictions that were confirmed by experiment. (The results were approximate, Betz suggested [220], because the “horseshoe” model ignored the downward motion of the trailing vortices.) Both of the subjects that Betz broached in his brief paper were to become a matter of enduring concern and research in subsequent years.

In January 1913, Betz published a second study, this time of the lift and resistance of a biplane.63 Whereas Foppl had used the Biot-Savart law to study the effect of the induced velocity on the tail wing, Betz now used the same approach to study the mutual interaction of wings that were positioned one above the other. The central point about the application of Prandtl’s approach to a biplane is that the trailing vortices from the upper wing will generate an induced resistance not only in the upper wing itself but also in the lower wing, while the trailing vortices from the lower wing will likewise affect both wings. Furthermore, if the wings are not located directly one above the other, the bound vortex corresponding to the wing itself (and not just the trailing vortices) will have to be taken into account when computing the induced velocity and induced drag on the other wing.

With the exception of Kutta’s second, 1911 paper, this work represented the first serious engagement with the theoretical aerodynamics of the biplane and the difficult problem of the mutual interaction of the different parts of an aircraft. It will be recalled that the “practical men” in Britain stressed holistic effects to justify their conviction that only the intuition of the engineer could cope with the problems of airplane design. Scientists and mathematicians, they said, simplified problems by studying one part at a time, which doomed them to failure. Such a procedure ignored the all-important effects of inter­action. Perhaps (had they known about it) the “practical men” would have been impressed to be told of the progress that was being made in Gottingen. Here engineers, such as Betz, were using the Biot-Savart law to put the study of interaction on a mathematical as well as an experimental basis.

Betz carried out wind-channel measurements of the lift and resistance of a set of two wings rigidly fastened into a biplane configuration. He studied (a) the effect of varying the distance apart of the wings, (b) the effect of giv­ing the wings different angles of incidence from one another (decalage), and (c) the effect of placing one wing ahead of the other (stagger). He found that the effects were small within the range he studied, though the most signifi­cant variable was the stagger of the wings. One of his practical concerns was to form some idea of the relative merits of monoplanes and biplanes. He summed up his results in four propositions: (1) A biplane arrangement with wings of equal span always has a less favorable ratio of lift to resistance than one of the wings taken separately. (2) A biplane can have advantages over a monoplane when the rest of the resistance of the aircraft, for example, a bulky fuselage, is taken into account. (3) A biplane is at an advantage if a high lift at low speeds is required. (4) The greatest maximum lift is obtained when

the upper wing of a biplane is placed ahead of the lower wing and is given a slightly smaller angle of incidence than the lower wing. All of these results, said Betz, were rendered intelligible by Prandtl’s theory, and the empirical graphs of lift and resistance were duly accompanied by theoretical curves calculated from the theory.64

In neither of his papers did Betz specifically mention, or illustrate the use of, the Biot-Savart law. He alluded to the horseshoe model but revealed none of the mathematics involved in his calculations. Like Foppl he prom­ised the reader that a fuller account was to follow from the pen of Prandtl himself. The Great War began in July 1914, but there seemed no immediate concern with secrecy. In a paper that appeared in August 1914, Wieselsberger preempted Prandtl and stated the Biot-Savart law explicitly and illustrated its application.65 He asked why birds often fly in a V formation. He did not man­age to answer the question, but he did succeed in laying out the basic ideas, and the basic mathematics, of Prandtl’s theory. In approaching the problem of formation flying, Wieselsberger ignored the beating wing motion involved in bird flight and treated birds as small airplanes. He then followed Prandtl and treated the airplane as a horseshoe vortex. By the use of the Biot-Savart law he showed that on either side of the horseshoe vortex there would be an updraft. This, he argued, allowed another wing, positioned to one side of the first wing, to operate at a more favorable angle of attack. This lowered the component of induced resistance in the direction of flight. On the basis of some plausible numerical assumptions, he made a quantitative estimate of the advantages to be derived from flying in the updraft of neighboring birds. His overall model, however, led to the conclusion that side-by-side flight would be just as efficient as the V formation.

In September 1914 Betz produced a study of wings with a sweepback and a twist at their ends,66 a configuration frequently used by designers of German aircraft at that time. The name Taube, or “dove,” was given to such machines. In Betz’s paper there was a passing reference to yet another formula attrib­uted to Prandtl and his new theory, though again no derivation was given. The formula concerned the minimum glide-angle that could be expected for a wing of given span and lift. The main result of Betz’s experiments on a range of Taube-style wings was to confirm the near optimum character of very sim­ple, rectangular wings. Having neither twist nor sweepback, such wings also had an economic and practical advantage: they were easy to construct. The glide coefficient (given by the ratio of resistance over lift) was not signifi­cantly improved by sweepback or twist, though Betz did find they improved longitudinal stability.

Perhaps because the promised theoretical paper from Prandtl was not forthcoming, Betz finally published his own account of the mathematics underlying his papers. Titled “Die gegenseitige Beeinflussung zweier Trag – flachen” (The mutual influence of two wings),67 the work appeared in the Zeitschrift fur Flugtechnik for October 1914. Betz concentrated on the case of the staggered biplane with wings of equal span where the upper wing was positioned ahead of the lower wing. Because the analysis proceeded on the assumption that each wing and vortex system could be represented by the simple “horseshoe” schema, the only real novelty in the paper lay in the more complex geometry of the computations, but the explicit development of the mathematics of the theory demonstrated its applicability to what was then a vitally important form of aircraft. It was clear that Prandtl and his colleagues now had a theory that could be used to predict the induced resistance of bi­planes, or triplanes, using only the wind-channel data for a single wing.

‘In the same year, 1914, Wieselsberger also published a survey article that described the state of knowledge in German aerodynamics with respect to lift and drag. It did not appear in the ZFM but in an Austrian journal, the Osterreichische Flug-zeitschrift.68 The article covered both two-dimensional and three-dimensional theory and took the reader through the work of Kutta, Joukowsky, Deimler, and Blumenthal and up to Prandtl’s horseshoe vortex. Wieselsberger’s survey effectively brought up to date an earlier survey by Reissner, of the TH in Aachen, which had laid stress on questions of stabil­ity and propeller theory.69

The international situation had been deteriorating throughout 1914, and British statesmen, such as Lord Haldane, became increasingly worried about the “war party” surrounding the German kaiser.70 With the threat of war, it was ever more important for European countries to monitor the technol­ogy of their potential enemies. If anyone had wanted to keep an eye on Ger­man aviation, the papers of Foppl, Betz, and Wieselsberger would have given them all they needed to know about the general state of scientific knowl­edge in the field of aerodynamics. These publications would have made clear that the circulation theory of lift was wholly taken for granted in Gottingen and the German-speaking world. Collectively, the publications showed that the theory had been developed to the point where it was being applied to problems of practical importance. Betz’s theoretical analysis of the biplane, however, was the last of the Gottingen research papers to appear in an open and accessible format. Thereafter they would be hidden away from public view in the Technische Berichte, published in individually numbered copies by the military authorities and marked Geheim—“secret.” In the meantime, the Gottingen results were in the public realm and were available to anyone in Cambridge or London who cared to study them.

Making the Horseshoe Model More Realistic

Prandtl never produced the promised article in the Zeitschrift fur Flugtech – nik. This was not because he harbored reservations about the approach. On the contrary, he was happy to produce accounts for general surveys, for ex­ample, in volume 4 of the Handworterbuch der Naturwissenschaften published in 1913. The handbook was an encyclopedic survey of the state of the natural sciences and contained articles by both Fuhrmann and Prandtl. Fuhrmann wrote on hydrostatics, and Prandtl wrote on fluid dynamics.71 In his contri­bution Prandtl gave an explicit account of the circulation theory and pre­sented a graph contrasting Kutta flow with Kirchhoff-Rayleigh flow (136). He also cited Lanchester’s work and gave a diagram (112) that laid out the qualita­tive basis of the horseshoe model, though the Biot-Savart law was not men­tioned by name. Why the hesitation? The simple horseshoe model was clearly in a provisional state and was still undergoing revision. It contained formal features that compromised both its empirical adequacy and its practical util­ity. Despite the successes of the theory, it would have been understandable if Prandtl had wanted to remove these limitations before presenting the ap­proach to a specialist readership. The time was hardly ripe for an authorita­tive presentation, which may explain the non-appearance of the article. Then the war intervened, and the form and level of presentation at which he seems to have been aiming were not achieved until 1918.

The problems with the “horseshoe” vortex were both mathematical and physical and were closely interconnected. Mathematically there was the dif­ficulty arising from the singularity in the Biot-Savart formula which has al­ready been remarked on, that is, the problem that arises when h = 0. The formula implied that the velocity of the downwash at the wingtips became infinite. The formula yields this result because of the uniformity of the vortex distribution implied by the model, that is, the constant value of the circula­tion along the bound vortex and hence along the span of the wing. This was a physically false picture. The existence of lift implies that there must be a greater pressure beneath the wing than above it, but the finite length of a real wing allows the air at high pressure beneath the wing to move round the tip to occupy the lower-pressure region above the wing. Such freedom of movement ensures that the pressure difference between the upper and lower surface will be zero at the tips. There will therefore be no lift at the tips and hence no circulation. Circulation cannot be constant along the span in the way that was assumed in the simple horseshoe model; it must fade away to zero at the tips.

Prandtl’s problem was to find a model with a more realistic lift distribu­tion along the span of the wing. His response was ingenious. He complicated the simple horseshoe model by introducing a number of horseshoe vortices laid out in the fashion indicated in figure 7.11. (A similar figure was used in an early article by Betz.)72 Starting from a single “horseshoe” whose span co­incided with the full span of the wing, he added others of smaller span. The parts of the vortex that lie along the span are to be thought of as piled on top of one another. In this way the constant distribution of circulation along the span is replaced by a variable, stepwise distribution with a maximum at the midpoint. The arrangement had the consequence that vortices now trailed from a number of points along the rear edge of the wing, rather than merely at the wingtips. This stepwise model, however, was only the starting point of Prandtl’s line of reasoning.

Prandtl did not simply introduce a number of horseshoe vortices such as the five in the diagram, or even 50 or 500. He introduced an infinite num­ber. He postulated an infinite number of vortices of infinitesimal strength. The vortices were infinitesimal for two reasons. First, an infinite number of vortices of finite strength would result in the absurdity of a wing with infinite circulation and infinite lift. Second, he needed the circulation and the lift at the tips to approach zero. A stepwise model with finite vortices would merely reproduce the problem that dogged the original. The vortices had to become infinitely small at the wingtips. Along the span of the wing the infinitesimal vortices were assumed to be compressed into a single line of bound vortic – ity (of varying strength) called the lifting line. These refinements made it possible to imagine a smooth, rather than stepwise, lift distribution that was amenable to mathematical treatment. To accord with the known facts, the

smooth lift distribution had to have a maximum lift at the midpoint of the span and approach zero lift at the wingtips.

Having described Prandtl’s refined model in qualitative terms, I now show how he expressed these ideas in mathematical terms. This account will pre­pare the ground for the next two sections, which describe the technical and mathematical heart of the Gottingen achievement.

Suppose the wing has a span b and lies along the x-axis of a coordinate system so that it runs from x = —b/2 to x = +b/2. The distribution of the circu­lation can then be represented by Г(х). The symbol indicates that, for every value of x along the axis between the wingtips, there corresponds a specific value of Г, the circulation. Thus Г(о) is the value at x = 0, the origin, which, following convention, is taken as the center of the wingspan. It is known from experiment that the lift is at its maximum value at this central position. Because it plays an important role, it is customary to give the circulation at this point a special designation and write Г(о) = Г0. The lift and hence the circulation is zero at the tips, so that Г(—b/2) = 0 and Г(+Ь/2) = о. For the moment, and for the purpose of conveying the main outlines of Prandtl’s theory, the actual shape of the lift distribution need not be given in more detail than this. The mathematical shape described by the function Г(х) will, for the moment, remain unspecified, but it will be some smoothed-out ver­sion of the shape made by the stepwise lift distribution. The details are re­served for the next section. For the remainder of this section, the distribution is simply referred to as Г(х) so that the general structure of the mathematical reasoning can be rehearsed. My aim is to show, in general terms, how Prandtl used the Biot-Savart law to calculate the lift, the induced velocity, and the induced drag.

The first step was to relate each of the infinitesimal horseshoe vortices to the Biot-Savart law. The relevant version of the formula for a vortex of finite strength has already been stated, namely, w = —r/(4nh). Because the analysis was now to be applied to infinitesimal vortices, the formula became dw = —dr/(4nh). The goal was to calculate the downwash at some specified point on the wing with the coordinate, say, x = x’. All of the infinite number of trailing vortices (each coming away from the wing at some point with its own specific x-coordinate) will contribute to the downwash at the point x’. The Biot-Savart law gave the (infinitesimal) contribution dw made by each of these infinitesimal vortices. The perpendicular distance h in the formula needed to be re-expressed as (x’ — x). This was the distance between the point on the wing from which the infinitesimal vortex emerges and the point x’ at which the downwash was to be found. A process of integration that adds the contribution of all the infinitesimal trailing vortices would then give the total

downwash at X. A further calculation, and a further integration, was needed to get the downwash for the entire wing, that is, for all the points like x’ which lie along the span between x = – b/2 and x = +b/2.

The procedure that has just been sketched was based on the assumption that the quantity dr used in the Biot-Savart formula corresponded to the strength of the infinitesimal vortex at the arbitrary point x. How was this infinitesimal strength to be expressed? The answer was that the strength of the element of trailing vorticity issuing from a point x was equal to the change of vorticity on the wing at that point. This can be explained by going back to the stepwise model of a finite number of finite vortices that was shown in figure 7.11. First the outer horseshoe is put in place. Suppose this has strength Tj. Then the second horseshoe is added, which has strength Г2 and a slightly shorter span, then Г3 is added, which again has a slightly shorter span, and so on. Consider the two points on either side of the origin of the x-axis from which the trailing vortices of strength Г2 emerge. These are the points at which the distribution of circulation changes by an increase of the amount Г2. Thus the strength of vorticity trailing from the wing at that point equals the change in vorticity around the wing at that point.

This “strength equals change” rule holds even when there are an infinite number of infinitesimal horseshoe vortices. The distribution of circulation along the span is given by the curve r(x), so the change in circulation is the slope of the graph of r(x) multiplied by the distance over which the slope reaches. The slope is дГ/dx, and the distance is dx, so the change whose value is sought is dr = (дГ/dx) dx. This expression gave the strength of the cir­culation or vorticity to be entered into the formula for the Biot-Savart law. The infinitesimal contribution of the vorticity at x to the downwash at x’ was therefore

(ЭГ / dx ^jdx 4n(x’ — x)

The total downwash at the point x’, designated by w(x’), is the integral of all of these infinitesimal contributions, summed over all the vortices issuing from the whole span of the wing. Thus,

/X 1 +if2(dr / dx )dx w(x ) = -— I Л-Ж J

The above integral has a singularity at x = x’, when the denominator becomes zero, but the integration could be carried out in such a way as to avoid this problematic point.

Given the downwash it was then possible to calculate the induced angle of incidence at X. This angle, ф, follows from the value for w(x’) because it was simply the angle made by combining the downward induced velocity with the free-stream velocity. The ratio of the two speeds gave the tangent of the angle ф, but because the angle was small, the angle and tangent could be equated. The induced angle of incidence was

w(x’) V ‘

The lift distribution could now be related to the overall lift and induced drag. Recall that for an infinite wing the flow at every cross section resembles that at every other cross section. The lift per unit length is constant and is given by the Kutta-Joukowsky formula as L = рГV. Prandtl took this formula to apply to each separate, infinitesimal element of a three-dimensional wing, with the proviso that the circulation would vary from element to element according to the distribution Г(х). The overall lift could then be represented by the integral of all the elementary lifts: dL(x) = р V r(x)dx. Thus,

+b/2

Lift = pV J Г(х)dx.

—b/2

Each point on the wing would generate an element of downward velocity and would thus be subject to a slight downward slope in the local flow. The ele­ment of lift dL(x) at that point would be tilted backward (relative to the main flow) so that the resultant force possesses a component opposing the motion. This was the induced drag. The induced drag at a given point x depended on the induced angle of incidence ф at that point. The component of induced drag resulting from the backward tilt equals dL(x) sinф(x). For small angles the sine of ф is equal to ф itself, so the element of induced drag was dL(x) ф^). Thus the total induced drag was given by the integral

+ b/2

Drag = pV J Г(х)p(x)dx.

— b/2

This relation could be expressed in terms of a coefficient of induced drag by dividing the value of the drag force itself by Уг р V2F, where F is the area of the wing. This gave the coefficient of induced drag as

2 +b/2

CD = — J T(x)<p(x)dx.

—b/2

It will be evident from these formulas that a closely knit structure of theoretical relations was emerging in Gottingen which connected lift, drag, span, and the distribution of circulation along the span of a wing. For the purposes of exposition I have only presented this structure in a schematic form. The mathematical formulas just given all depend on the distribution of the circulation, Г(х), but the actual character of the function governing the distribution has remained unspecified. All that the above formulas entail is that if the distribution Г(х) is given, then the lift, the induced angle of incidence, and the induced drag can be calculated. Only when the distribu­tion is specified will the theory will have real content. The next question is: How was the distribution of lift and circulation found? How is Г(х) to be defined?