Category The Enigma of. the Aerofoil

Kutta began with a nod not only to Lilienthal but to more recent develop­ments in aviation. These, he said, gave a great practical significance to the old, but difficult hydrodynamic problem of calculating the forces on a body immersed in a moving fluid such as air. Calculating the lift forces on a wing was particularly important. It should be possible to do this because the rel­evant flow can be understood (“aufgefafit werden kann”) as the superimpo­sition of a circulation and a steady stream. For an explanation of the basic ideas of the circulatory theory, Kutta directed the reader to the first volume of Lanchester’s Aerial Flight and a 1909 article by Finsterwalder. Finsterwalder’s article was titled “Die Aerodynamik als Grundlage der Luftschiffahrt” (Hy­drodynamics as the basis of airship flight), but it also dealt with the basics of the circulation theory of lift.29 Kutta’s choice of words, in saying that the flow can be so understood, simply indicates that if the flow is interpreted in this way, then the lift force becomes intelligible. There is, however, no reason to suppose that he doubted the reality of circulation. He was probably stepping carefully because of the highly artificial character of the concepts he was ap­plying to the problem: friction was being ignored, the flow was to be treated as two dimensional, and the fluid was taken to be free of vorticity (that is, the flow was irrotational).

Kutta then made three observations about his own earlier work. First, he said that in 1902 he had discovered a general theorem about lift which was re­discovered by Joukowsky in 1906. He thus made a priority claim for the result that lift is proportional to circulation and is given by the product of density, velocity, and circulation.30 Second, he said that the 1902 predictions about lift were supported by Lilienthal’s data but acknowledged that the discussion had been confined to wings at zero angle of incidence. This restriction would not apply to the more general analysis he was about to offer. Third, in his earlier account, the magnitude of the circulation could be fixed by specifying that the flow was to be smooth at both the leading and trailing edge. This was pos­sible because of the symmetry of the arc-like wing at zero angle of incidence. In the more general treatment, with an arc whose chord was at an angle to the flow, adjusting the circulation could only make the flow smooth at one edge, for example, the trailing edge. At the leading edge the fluid would divide, generally at a point on the lower surface, and some of it would be forced to flow around the leading edge. Because Kutta represented the wing by a geo­metrically thin line, this meant the fluid at the leading edge would achieve infinite speeds and pose a significant problem for the analysis.

Kutta indicated to the reader that the lift force on the wing would have to be broken down into two parts. Leaving the explanation until later, he stated that one part of the lift would be produced by pressure on the surface of the arc, while the other part could be represented as a tangential, suction force at the leading edge. He also signaled his intention to make his abstract, geo­metrical model of the wing more realistic by studying the effects of rounding off the leading edge.

Prandtl’s earliest publications on aeronautics did not deal with the circula­tion theory of lift in either its two – or three-dimensional form. He wrote mainly on airships, the general mechanical problems associated with build­ing an airplane, or the engineering that went into the construction of wind channels for testing models. Thus in June 1909 he lectured to the annual gen­eral meeting of the association of German engineers at Mainz on the signifi­cance of the model experiments to be done at Gottingen and the equipment that had been developed.47 September of the same year saw him speaking at the International Aeronautical Exhibition (ILA) in Frankfurt. Here Prandtl concentrated on the principles and preconditions of flight and the practical problems of achieving adequate lift and stability.48 In a series of articles in the new Zeitschrift fur Flugtechnik he laid out for the reader’s benefit those parts of mechanics of special relevance to aeronautics.49 His topics included gyroscopes, stability, and air resistance, with a mention of his own boundary – layer theory in connection with the phenomenon of separation and vortex formation. One of Prandtl’s concerns was to remove false ideas that contin­ued to cause trouble in the field. Many “inventors,” he explained, tried to achieve automatic stability in aircraft by some sort of pendulum device whose rationale was based on a misunderstanding. Prandtl also stressed that resis­tance depends as much on the suction effects behind a body as it does on what happens at its front face. From the outset he had no time for Rayleigh – Kirchhoff flow as an account of lift. He viewed it as wrong at the level of general principle.

In these articles for the ZFM, Prandtl recommended some of the relevant literature (64). Significantly he included both Lanchester’s book on aerody­namics and Lamb’s book on hydrodynamics. He thus symbolically conjoined what, in the homeland of those two authors, was proving so resistant to uni­fication. On the first page of the initial article in the ZFM series, Prandtl also nailed his colors to the mast regarding the principle of the unity of theory and practice. Not for him the slogans of the antimathematical movement, which treated theory, and especially mathematically based theory, as something in fundamental opposition to practice. Using words that closely echoed the for­

mulations adopted a decade earlier by August Foppl, in the preface to his Vorlesungen uber technische Mechanik, 50 Prandtl asserted:

Um nun meine Absichten naher zu kennzeichnen, mochte ich vorweg be – tonen, dafi ich das viel ausgesprochene Schlagwort vom Gegensatz zwischen Theorie und Praxis nicht gelten lassen will; der Gegensatz liegt fur mein Emp – finden zwischen guter und schlechter, richtig und unrichtig angewandter Theorie; eine gute Theorie is in Ubereinstimmung mit den Ergebnissen der praktischen Erfahrung, oder sie gibt zum mindesten wesentliche Zuge der Er – fahrungstatsachen wieder. (3)

In order to characterize my intentions more precisely, I should emphasize at the outset that I do not want to endorse the much used slogans about the oppo­sition of theory and practice. In my experience the contrast lies between theo­ries that are good or bad for application and between correctly or incorrectly applied theories. A good theory is in agreement with the results of practical experience or, at least, captures the essential thrust of the facts of experience.

Prandtl was not just striking attitudes. His work on the three-dimensional wing would meet the demanding requirement of applicability to engineering practice.

Prandtl tells us that it was in the winter semester of 1910-11 that he began to develop his own mathematical theory of the finite wing.51 It was treated in a course of lectures that he gave in Gottingen on the theme of aeronau­tics. The lectures were attended by Otto Foppl and, though the text of the lectures appears to have been lost, hints and mentions of their content are to be found in the early papers coming from the Gottingen group. The basic picture with which Prandtl worked was similar to the one already proposed by Lanchester, namely, a finite wing with circulation around it and with two vortices emerging from the wingtips. Prandtl’s aim was to make this picture amenable to a mathematical analysis, but to do so he had to resort to extreme simplification. He treated this complex three-dimensional, physical system as equivalent to three connected-line vortices. One vortex, called the bound vortex, represented the wing and was assumed to run along the span of the wing. The other two vortices, called free vortices or trailing vortices, extended back from the wingtips and were at right angles to the wing. The arrange­ment, as shown in figure 7.5, first appeared in a brief, qualitative account of the theory that Prandtl published in 1912.52 The two diagrams, which show the progressive abstraction of the picture, reappeared in Prandtl’s published lectures in the form shown here.

Physically the trailing vortices are assumed to extend right back from the wing in flight to the point where the aircraft left the ground. Mathematically

they are said to extend back “to infinity.” The three connected vortices thus form a single bent line of vorticity. A vortex line in an ideal fluid (whether straight or bent) has the same strength everywhere along its length, so the circulation Г around the wing, that is, around the bound vortex, also gives the vortex strength along the trailing vortices. For reasons examined later, this model was called, rather implausibly, the Hufeisen Schema, or “horseshoe schema.”

Prandtl’s task was to see how the presence of the two trailing vortices modified the flow in the vicinity of the wing. If the trailing vortices are ig­nored (as they are in the two-dimensional case), the law of lift is (using the usual notation) Lift = p V Г. Is this law preserved in the three-dimensional case? Surely some significant modification of the flow must occur, and this must have some consequences for the behavior of the wing—but what modi­fications and what consequences? Before I go into mathematical details, it may be useful to sketch the outcome of Prandtl’s analysis in qualitative terms. Using his simplified model, Prandtl was able to predict two unexpected and important effects.

First, Prandtl found that the effect of the trailing vortices ought to be the creation of a downwash at, and behind, the wing. The swirling air of the trail­ing vortices would (according to his mathematical analysis) influence the air in the vicinity of the wing in such a way as to give it a downward velocity component. The downward component would have the effect of tilting the airflow. The way that the tilt arises from the introduction of the downwash component is shown in figure 7.6a. The presence of the tilt means that the wing operates at a slightly lower angle of incidence relative to this new local flow. The effective angle of incidence is therefore reduced. Because (over the working range of the aerofoil) lift is proportional to the angle of incidence, the lift should be reduced. The same point may be expressed the other way round. In order to get the same lift per unit length in a finite aerofoil as in an infinite aerofoil of the same cross section, the angle of incidence relative to the main flow has to be increased.

The second prediction followed from the first. If the resultant lift force on the wing is at right angles to the local flow, and the local flow is tilted in the manner shown in figure 7.6b, then the resultant aerodynamic force will be tilted back relative to the main flow and hence to the direction of motion of the wing. The effect will be that the resultant force will now have a compo­nent that opposes the motion. There will be a drag. It is important to recall that this analysis was all done using ideal-fluid theory. The two-dimensional analysis showed how a flow involving a vortex and circulation could yield a lift but no drag, the three-dimensional analysis now predicts that it can pro­duce both a lift and a drag. For reasons that will become clear in a moment, Prandtl and his co-workers came to call this novel drag an “induced drag,” and the tilt of the local flow due to the downwash was called the “induced angle of incidence.” Induced drag was a form of drag that did not result from viscosity or skin friction or turbulence. It was produced by the very same inviscid mechanisms that generated the lift.

Having sketched these two initial predictions, I now look at their math­ematical derivation. How did the idealization of the “horseshoe” vortex help Prandtl to develop a mathematical description of the flow which led to these results? The answer lies in an analogy that exists between the hydrodynam­ics of a perfect fluid and electromagnetic phenomena. A line vortex is like a wire carrying an electric current which sets up, or “induces,” a magnetic field around it. The vortex in an ideal fluid does not, of course, induce a mag­netic field, it induces a velocity field. The velocity was exactly what Prandtl wanted to understand because once he had a mathematical expression for the velocities, he could deduce the pressures, and thus the forces on the wing. Prandtl’s use of the analogy was explicit. “Fur die Verteilung der Geschwindig – keit in der Umgebung irgendeines Wirbelgebildes besteht eine vollkommne

figure 7.6. In (a), the trailing vortices induce a downward component win the flow behind the wing. This tilts the airflow and effectively lowers the angle of incidence of the wing. The new “effective” angle of incidence is the original geometrical angle of the wing to the free stream minus the angle of the downwash produced by the tailing vortices. In (b), the tilt in the airflow means that the resultant aerodynamic force, which is at right angles to the local flow, is no longer at right angles to the free stream, that is, the direction of motion. This produces a drag component called the “induced drag.”

Analogie mit dem Magnetfeld eines stromdurchflossenen Leiters”53 (For the velocity distribution in the neighborhood of any such vortex formation there exists a complete analogy with the magnetic field around a current carrying conductor).

The analogy to which Prandtl referred was well known before the work on

aerodynamics and was discussed in standard textbooks in both electromag­netic theory and hydrodynamics. It was to be found in the books on Max­well’s theory and on technical mechanics written by Prandtl’s own teacher and father-in-law, August Foppl.54 The analogy is not a superficial one but exists at the level of the fundamental equations that can be used to describe the two different areas. Experts in electromagnetism, such as August Foppl, could easily describe the magnetic effects of a current-carrying wire shaped in the form of the idealized “horseshoe,” and Prandtl carried these results over to the corresponding system of line vortices.

The law of induction common to the hydrodynamic and electromagnetic cases is called the Biot-Savart law.55 The law may be explained by considering figure 7.7, which shows an infinite, straight-line vortex of strength Г. A small segment of the vortex is identified as ds. The point P lies at a distance r from the vortex element, and the line joining ds and P makes an angle 0 (theta) with the vortex. Following the electromagnetic analogy, there is a small com­ponent of velocity dw “induced” at the point P by the small element ds. Ac­cording to the Biot-Savart law, the component is

, rds. sind

dw = -—.

4nr

The velocity component is perpendicular to the plane determined by ds and the line r. This formula links infinitesimal quantities, and the causal relation between such infinitesimals that seems to be implied by the law occasioned some puzzlement. August Foppl had no time for such subtleties. Taken by itself, said Foppl, the formula has absolutely no meaning: “Es hat uberhaupt keinen Sinn.”56 Foppl’s point was that the real significance of the law only

emerges when it is integrated in order to give the finite effect of a finite length of vortex or, by extrapolation, the finite effect of a very long (and effectively infinite) vortex. This is what Prandtl needed in order to compute the effect of the, effectively infinite, trailing vortices.

As Prandtl first employed it, the Biot-Savart law was used to give the fi­nite velocity component w at an arbitrary point P in the vicinity of a finite, straight-line segment AB of a vortex (see fig. 7.8). If the strength of the vortex is Г, and the point P is at a perpendicular distance h from the line AB, then, after integration, the law now reveals that the contribution to the velocity of the finite segment is

Г

w = (cosa + cos в).

4nh

The angles a and в are the angles made by the lines joining the point P with the ends of the finite vortex segment under consideration. The direction of the velocity w at P is at right angles to the plane that passes through the points A, P, and B. Whether the velocity vector faces downward, into the page, or upward, depends on the sense of the circulation around AB. (If an observer who looks from B to A is confronted with a clockwise circulation, then the induced velocity vector points into the page and vice versa.) Notice that if the finite line segment AB is extended to infinity in both directions (so the angles a and в get smaller and smaller as the line gets bigger and bigger), then the velocity at point P should correspond to the velocity at a point situated a distance h from the center of a two-dimensional vortex, that is, a point vortex, of circulation Г. This is exactly what the formula provides. The expression (cos a + cos в) assumes the value 2 for a = в = 0, so that w = r/2nh.

Prandtl’s aim was to apply the Biot-Savart law to the “horseshoe” vortex because he was interested in the effect of the trailing vortices. The trailing vor­tices count as “semi-infinite” lines because they start from the wingtip and go to infinity in one direction. To understand Prandtl’s reasoning when he ap­plied the law to his horseshoe system, think of the arrangement in figure 7.8 modified in two ways until it turns into that in figure 7.9. First, A is moved to infinity so that a = 0 and cos a = 1. Second, the point B is moved inward until it coincides with the base of the perpendicular from P. This makes в = 90° so that cosP = 0. The formula then gives the value for the induced velocity w:

Г

w = .

4nh

Interpreted in terms of the horseshoe vortex model of the wing, this formula gives the contribution of one of the trailing vortices to the flow at a point on the wing that is distance h from the wingtip generating the vortex. The full downwash at any given point on the wing needs the contribution of both trailing vortices to be added together, but the formula reveals the mechanics of the process that generates the downwash. Prandtl spoke of a zusatzliche Abwartsgeschwindigkeit, an additional downward speed. Max Munk called the quantity w the “induced velocity,” and this name was taken over by the Prandtl school.

Sufficient has now been said to show how Prandtl was able to reach his predictions about the general effect of the trailing vortices, that is, (1) the

creation of a downwash, (2) the tilt that the downwash creates in the local flow, and (3) the resulting (induced) drag. One final point to notice about the formula for the induced velocity w of the downwash is that it takes the form of a fraction with h in the denominator. It contains a singularity at the point h = 0. For this value of h, the formula requires that the velocity w be infinite, which is physically impossible. Recall that h refers to the distance from the wingtip. This means the application of the Biot-Savart law to the horseshoe model of the three-dimensional wing breaks down for points close to the wingtip. Prandtl had made novel and important predictions, but, because of the singularity, the predictions carried with them a problem. Even if in many cases they were proven correct, they were based on a physically impossible model.

Major Low’s paper, the third Congress paper of the morning, was titled “The Circulation Theory of Lift, with an Example Worked out for an Albatross Wing-Profile.”99 The “albatross” of his title was not the bird but the name of a German aircraft company that had played a prominent role in the war.100 Low’s aim was to apply the circulation theory to an actual aircraft wing and to show the relation of the theory to drawing-office practice. He also wanted to straighten out one or two points of recent history. He began by reminding his listeners that the origin of the circulatory theory was grounded in the work of British physicists. Fifty years ago, said Low, Rayleigh had published his paper on the spin of the tennis ball and explained the force that made it veer by reference to the circulation around the ball. A similar idea, he said, was to be found in the work of P. G. Tait of Edinburgh (where Low had himself been a student, graduating with an honors degree in mathematics and natural philosophy in 1903).101 Low regaled his audience with a story about experi­ments, done in the dark cellars of the old Edinburgh University buildings, on the spin of golf balls. Tait was helped by his son, “the lamented Freddie Tait.” Freddie had been a professional golfer and, in the name of science, was required by his father to shoot golf balls through screens in order to trace their trajectory. On one occasion, in the gloom, he missed the screens. This resulted in the experimenters dodging around as the ball “ricocheted inter­minably off the walls of the cellar” (255).

This story was merely the disarming prelude to a point that was not in­tended to be amusing. Lanchester, Low went on, had boldly applied the idea of circulation to the wings of an aircraft and had given a thorough, descriptive account of the mechanism of flight. That was nearly twenty years ago. Why was it only now that the circulation theory was being taken seriously in the land of its origin? Low had an answer, and it was not a flattering one: “Had Rayleigh put forward the theory, how we should have vied with each other in the will to believe it, if not in power to understand it! But when it was offered by a man outside the circle of recognised physicists it was ignored” (255).

Leaving his audience to ponder this sociological point, Low went on to expound some of the basic techniques associated with the theory, confin­ing himself to “strictly graphical and descriptive” methods. First, he gave a graphical method for transforming a circle into Joukowsky profiles and then tackled the more difficult, inverse task of going from a given aerofoil back to a circle or a close approximation to a circle. As before, Low was conveying to his audience the content of recent German material, this time using a postwar publication by Geckeler.102 Low showed how to start from the Albatross wing and, using drawing-office methods, map it back to an approximate circle by a series of trial-and-error steps. “There now remains only a routine of laying off and measuring straight lines on the drawing board to determine the ve­locity and hence the pressure at every point of the field” (273). Low assumed a velocity U = 10 m/sec and an air density of p = 1.2 kg/m3. Using the formula L = p U Г, he derived the data to construct a theoretical curve for the Albatross wing relating lift to angle of incidence. Because the formula was based on the assumption of an infinite span, the curve could not be compared directly with wind-tunnel data derived from a finite wing. Low then appealed to the Gottingen transformation formulas relating wings of the same section but different aspect ratio. This allowed him to recast known experimental data on the Albatross wing into its equivalent for an infinite wing. Low now had two curves that linked lift and incidence for the Albatross wing, one curve coming from wind-tunnel tests, the other derived from the circulation theory. For the range of -5° to +10° the two graphs were close together. The theory was sup­ported by experiment. Given an arbitrary wing, a designer could now predict from the circulation theory the curve relating lift to angle of incidence at least up to the point of stall.

Having achieved his main goal, Low then returned to the theme with which he had begun. “In conclusion,” he said, “it is desired to call attention to the fact that this fundamental physical theory was first stated by an English writer, and then allowed to fall into complete neglect in the country of its origin, largely owing to the attitude taken up by some of Lanchester’s fellow members of the Advisory Committee for Aeronautics” (275). On this note the talk ended.

Calling the circulation account of lift a “fundamental physical theory” can only have been meant as a thrust at Bairstow, who had just explicitly denied it the status of being a fundamental theory. But the remarks blaming the Advisory Committee for Aeronautics for the neglect of Lanchester were even more pointed. The austere figure of Professor Sir Richard Glazebrook, who had been the chairman of the Advisory Committee, and who was thus the main focus of Low’s complaint, was present at the talk. In fact, he was more than present. He was presiding over the session at which Low had just delivered his paper.103

It is important to appreciate the difference between the professional perspec­tive of the sociologist and the perspective that prevails, and perhaps comes naturally, to social actors themselves in the course of everyday life. I refer to these as the “analyst’s perspective” and the “actor’s perspective.” The concerns of those engaged in sociological analysis are usually not identical to those of the social actors they study, though, of course, analysts themselves will some­times occupy the very roles that they investigate professionally. Conversely, sociological perspectives are sometimes invoked in the course of everyday interaction. (Major Low adopted such a stance when he speculated on what would have happened if Rayleigh had backed the circulation theory. He was reflecting on the role played in British aerodynamics by Rayleigh’s authority.) Despite this overlap and interweaving, it is the differences in the perspectives of the analyst and the actor that I want to emphasize.

In everyday life much of our curiosity centers on deviations from what normally happens or from (what we feel to be) our justified expectations. We want to know why things go wrong more than we want to know why they went right. Going right tends to be taken for granted. It is the failure of the airline to keep to its schedule that makes irate travelers demand to know the causes of the delay. They do not demand to know how and why a punctual departure was achieved. If they were to pose such a question, it would be heard as a hostile comment rather than a disinterested inquiry. The structure of everyday curiosity can be remarkably one-sided. Using a terminology that has become current in the sociology of science, such everyday curiosity may be described as “asymmetrical.” For the sociologist, however, the atypical is not the only thing that needs explaining. The typical is as interesting as the atypical, and the normal or the expected course of events is at least as im­portant as the deviations. The professional curiosity of the sociologist may therefore be called “symmetrical,” in contrast to the “asymmetry” of much commonsense curiosity.4

If an “asymmetrical” curiosity prompts us to ask for causes for half the story, then a “symmetrical” curiosity must lead us to demand causes for the whole story. If the commuters only want to know the causes of delay, then sociologists must risk the resentment prompted by their wanting to know the causes of nondelay. They must ask the questions others don’t ask or don’t want to answer. If sociologists were to study the workings of an airline, they would try to grasp its organizational features and see how its various parts related to one another. There are pilots and crew to be trained, maintenance schedules to be established, fuel supplies to be arranged, safety standards to be adhered to, duty rosters and wages to be negotiated, and shareholders to be satisfied. These dimensions of the organization would be common to all or most airlines, so the sociologist could construct a general model of such an organization and note the difference of practice between different instances of the model. This airline might devote twice as much time to safety training as that one; this one might repeatedly demand more flying hours between checks and repairs than that one; this one might meet its schedules by taking more risks. Such an investigative procedure would bring both the successful and the unsuccessful, the efficient and the inefficient, the safety conscious and the risk takers under the scope of the same model. By casting both sides of the story in the same terms, it is possible to use the different performances to probe the working of the general model, and hence to explore more deeply what it is to have a social organization capable of producing the range of ob­served outcomes.5

These considerations, drawn from the practice of general sociology, also apply to the sociology of knowledge. The central thrust of the Strong Pro­gram is that explanations in the sociology and history of science should be both “causal” and “symmetrical” in the sense that I have just explained. The same type of cause should explain the attractions of both true and false the­ories, and both successful and unsuccessful lines of work (where the judg­ments of truth and success derive from hindsight or are the analyst’s own). I have said that this approach has informed my case study, but when I asked “Why did the British resist the circulation theory?” I may seem to have ad­opted the asymmetrical stance of common sense. It is true that the question could be posed in a purely commonsense way. This, I suspect, is how Major Low meant it when he asked why Lanchester had been ignored. Despite his sociological insight, he primarily wanted someone to take the blame. Stated in isolation Low’s question is worded in a way that is consistent with either a symmetrical or an asymmetrical stance. What differentiates the two stances is the purpose behind the question and the distribution of curiosity informing the answer. The evidence I have presented indicates that it was local cultures, and the institutions sustaining them, that explain the reactions to the theory of circulation exhibited by both the British and the German experts. These

were the causes of the phenomena that I set out to explain, and the causes were of the same kind in the two cases. Cambridge was not Gottingen, but both were influential and brilliant research institutions. Mathematical phys­ics was not technical mechanics, but both were based on rich, mathemati­cal traditions. Lamb’s Hydrodynamics was not Foppl’s Vorlesungen, but both were much-used textbooks that, respectively, encouraged and transmitted their own characteristic, mental orientation.

The scientific study of a complex phenomenon, or the development of a complex technology, typically calls for the cooperation of specialists from a number of fields. The creation of the atomic bomb in Los Alamos involved physicists, chemists, metallurgists, engineers, and experts in fluid dynamics. An episode in the history of science and technology of the kind I have ana­lyzed likewise counts as a complex, real-world phenomenon, and its study will involve specialists from different fields, for example, historians, sociolo­gists, and psychologists. The psychologist studies the mental capacities nec­essary for learning about the world and becoming a competent member of society, perhaps even a member of a specialist subgroup—for example, a subgroup whose members are able to read Foppl’s textbook or sit the Tri­pos examination. The sociologist studies the social processes without which, ultimately, there would be no professional identities such as “the engineer” or “the physicist” and no institutions such as “the textbook,” “the examina­tion,” or “the university.”

It is evident that the episode I have described in my case study cannot be called “purely” sociological any more than it is “purely” psychological or “purely” a matter of grappling with the world. Likewise, the desire for a “complete” description or a “complete” explanation of the episode can be dismissed as utopian. But it is not unrealistic to hope for insights into parts of the problem, and some aspects of the episode may call for psychological study, while other aspects may call for sociological study. The one does not exclude the other. My emphasis on the sociological dimension is not a denial of the psychological dimension or any other naturalistic dimension. Rather, the emphasis on society arises because sociological variables are the ones most relevant to the question I am asking. British and German experts did not diverge because their basic cognitive faculties differed or because their personalities were different or because one group engaged with the material world while the other turned its back on it. As far as the present episode is concerned, they differed primarily because their education and professional lives were different. They worked in different disciplines and institutions whose traditions and reward structures diverged from one another.

One final feature of the sociological approach must be emphasized. It is central to my account that the actors involved were not detached intel­ligences moving in an abstract world of thoughts, theorems, and deductions. Nor did they move exclusively in a world of committee meetings, personal confrontations, status conflicts, and power struggles. These things were part of their world but not the whole of it. The experts in my story experimented in wind tunnels, built models, observed and measured the forces on them, flew airplanes, and sometimes died in them. The sociological variables to which I have drawn attention are not to be conceived in a way that excludes these practical, experimental activities or diminishes their importance. The sociological processes I have identified do not stand between people and the material environment with which they are engaged. Contrary to the claims repeatedly made by their critics, those who follow the Strong Program do not treat the social world as something to which scientists respond instead of re­sponding to the natural world. The cultures, institutions, and interests that I have identified did not block the active involvement with material reality but were the vehicle of that involvement and gave a specific meaning to it.6

To apply and solve the Euler equations, mathematicians had to introduce various techniques to relate them to specific flow problems. “As they stand,” said Cowley and Levy, “these equations are not very suitable for solution” (39). They need to be fleshed out. This was done by means of a variety of auxiliary concepts such as source, sink, vortex, and stream function. The gen­eral connotations of the labels “source,” “sink,” and “vortex” will be evident, and their mathematical idealization refines, but does not essentially alter, the everyday meaning of the word. A vortex is like a whirlwind around a central point. A source is a geometrical point at which fluid is created at a certain rate, and a sink is a geometrical point at which it disappears and is destroyed. The words “stream function,” however, do not have any obvious counterpart in common usage. In this section I describe briefly what they mean.

Imagine a coordinate system of x – and y-axes that is to be used to describe a flow of fluid. The value of the stream function at some point P is given by the amount of fluid that flows in unit time across a line drawn from the origin to P. To specify this quantity is to specify the value of the stream function. In hydrodynamics this value is usually designated by the Greek letter psi, y. Altering the position of the origin only alters the value of y by the same con­stant amount at all points in the flow. It follows from the definition that such a function has a simple relation to the velocity components of the flow, and this is the utility of the stream function. If u is the speed along the x-axis at P and v is the speed of flow along the y-axis, it can be shown that

u = -—, and
dy

dw

v ~~dX ■

Given the stream function, a process of differentiation gives the velocity com­ponents. Here is a simple example. The stream function for a uniform flow of speed U along the x-axis is

W = —Uy = —Ur sinft

The first expression is in Cartesian coordinates and the second is in polar co­ordinates, giving the value of у at the point (r, 0). In Cartesian coordinates, differentiating у with respect to x gives the correct answer v = 0, meaning that the flow has zero velocity along the y-axis. Differentiating with respect to y gives the speed u = U along the x-axis. Notice that putting у = c, a constant, gives a straight line parallel with the x-axis. Such a line can be called a stream­line of the flow. Later in the discussion it will become evident that, for all its simplicity, this flow plays a basic role in hydrodynamic reasoning. Logically, it provides the foundation of the edifice.

I have referred to a streamline of this basic flow, but what is a stream­line? In everyday language the words connote speed. Modern aircraft are “streamlined,” whereas aircraft in the period of the old Advisory Committee for Aeronautics, with their struts and protruding engines and undercarriage, were certainly not. This usage, and the idea of low-resistance, streamlined bodies, was already well established in early aerodynamics, even if it could not be realized in the construction of flying machines.17 The technical mean­ing of the term “streamline” in hydrodynamics, though related to this popu­lar meaning, is more specific. A streamline drawn through a point in a fluid flow is a line that conforms to the direction of motion of the fluid element that is located at that point at that moment in time. A moment later the point may be occupied by another fluid element with a different velocity. The pic­ture becomes much clearer if the flow is steady so that the speed and direction of the flow at a given point are constant over time. When the flow is steady, then streamlines will coincide with the path taken by the fluid element. Look­ing at the streamlines will give a picture of what the fluid elements are doing. Streamlines also indicate something about the speed of the flow. For steady incompressible flow they come closer together as the flow speeds up and be­come wider apart as the flow slows down.18

How does the mathematician identify streamlines in order to draw a dia­gram of a flow? The answer is by reference to the stream function. Once in possession of an expression у for the steam function of the flow, the math­ematician generates a series of curves by putting у = c, a constant, and giving

the constant a sequence of values q, c2, c3, etc. The curves are convention­ally plotted at equal intervals. These are the streamlines. As a simple example, recall the stream function for the uniform flow parallel to the x-axis—the basic flow. The formula for the stream function was у = – Uy. Putting у = (say) 0, 1, 2, 3, etc. gives the straight, horizontal lines y = 0, y = -1/U, y = -2/U, y = -3/U, etc. Notice that the greater the speed U, the smaller the gap between the lines. Because, by definition, a fluid element will not cross over a stream­line, then any streamline can be selected and interpreted as a solid boundary without this in any way changing the picture of the flow. (It is sometimes said that the fluid bounded by a streamline can be suddenly “frozen” or “solidi­fied” without altering the rest of the flow.) In the present case the line у = 0 can be selected for this treatment. The flow then becomes (that is, can now be regarded as) the uniform flow of an infinite ideal fluid along a flat, smooth wall located on the x-axis.

Other, more complicated, flows call for other, more complicated, formu­las for the stream function. For example, there are stream functions for the flow around point sources and point sinks and for vortices. The streamlines of sources and sinks radiate away from, or toward, their center point while the streamlines of a simple vortex are concentric circles. By the expedient of adding the stream functions, the flow can be found for combinations of sources, sinks, and vortices. Shortly I shall give the stream function and the streamlines for another, particularly important flow; for the moment, how­ever, the point to retain is that a streamline is specified by setting the stream function equal to a constant у = c.

The practical men did not like “scientific” aerodynamics.50 So what sort of aerodynamics did they like? I begin to answer this question by identifying what might be called their “practical epistemology.” Then I look in more de­tail at the accounts of lift that are to be found in books written for the design­ers of airplanes and in articles that appeared in the Aeroplane, Flight, and Aeronautics.

The epistemology of the practical man was intuitive and qualitative. It was formulated in conscious opposition to the pedantic concern with ac­curacy and irrelevant detail attributed to the despised figure of the mathema­tician.51 Reality must be grasped in all its complexity rather than simplified and broken down into imagined elements. In this sense their epistemology was holistic. It was also artistic. A good designer could rely on his eye, his experience, and his judgment. In a literature review in Aeronautics the editor said: “I don’t deny the infinitely valuable role of pure science, still less that of theory, but science should have some relation to practice, since it is its foster-mother. There is more than one aeroplane designer who knows just enough mathematics to make twice two work out at four, but he will turn out machines equal in performance to the best. We in this country know, as they do in the United States, of eminent designers who see a new type of machine rather than design it.”52

Grey made the point more bluntly with no genuflection in the direction of science: “Never mind what the scientists calculate. Trust the man who guesses, and guesses right.”53 The claim was that some designers have a track record of guessing rightly, and these are the people to trust. We may not be able to see how they do it, but we should not let this put us off. Trust rather than understanding lies at the root of things. This was indeed Grey’s view: there were not only unknown factors involved in the design of aircraft but there were actually unknowable factors, and this was something the “slide – rule scientists” could not grasp.54

The implication was that the reasons behind practical success will remain mysterious. This notion implied a species of intellectual pessimism or even nihilism. Such pessimism was not unusual among practical men and was sometimes echoed by those in the other camp. For example, writing as J. C., a reviewer of G. P. Thomson’s Applied Aerodynamics recommended the book to practical designers (even though it was the product of Farnborough) and said, “One of the first ideas that arises in the reading is the state of ignorance that still exists in aerodynamics; it is safe to say that we know practically noth­ing of the reasons for the experimental results that we find. The amazing thing is that we are able to make aeroplanes as well as we can.”55

At least two of these statements come from spokesmen of the practi­cal men rather than from designers themselves, but they seem to articulate a widely held view. Grey’s characteristic denunciations were repeated in a foreword he wrote in 1917 for the book Aeroplane Design by F. S. Barnwell, who was the chief designer at the British and Colonial Aeroplane Company. This firm, usually known as the Bristol Company, became famous during the Great War for the Bristol fighter, which was designed by Barnwell.56 Much harm had been done, said Grey, “both to the development of aeroplanes and to the good repute of genuine aeroplane designers by people who pose as ‘aeronautical experts’ on the strength of being able to turn out strings of incomprehensible calculations resulting from empirical formulae based on debatable figures acquired from inconclusive experiments carried out by persons of doubtful reliability on instruments of problematic accuracy.”57 If one asks what is left when all the hated calculations, experiments, and instru­ments have been swept away, the answer is intuition. This was Grey speaking, not Barnwell, so we cannot be sure that Barnwell endorsed it. Authors do not necessarily agree with what others say in the forewords of their books, but it is reasonable to expect general agreement.

W. H. Sayers, a strong critic of the National Physical Laboratory, was in­volved with the development of seaplanes during World War I. In an article written after the war, in 1922, called “The Arrest of Aerodynamic Develop­ment,” Sayers described the current conception and form of the airplane.58 It was, he said, “the hybrid product of two utterly different and independent methods of development.” From 1908 to 1914, its evolution was “the result almost entirely of individual adventure.” There were, he insisted, no wind – tunnel results worth mentioning, the mathematics of stability had no appar­ent connection with the facts, and even engineers regarded the airplane as a mechanical curiosity. “Individual designers worked, as artists worked, by a sort of inspiration as to what an aeroplane ought to be like, and built as nearly to their inspiration as the limited means, appliances and increasing knowledge they possessed would allow them” (138). Sayers went on to de­plore the degree of standardization that had set in with regard to design. This, he said, gave a spurious sense of understanding and control. In reality we did not know how to predict what would happen outside the limited range with which we had become familiar. Similarly, the laboratory workers had been in error in concentrating on simple bodies, especially “such simple bodies as might be used as components of the standard type of aeroplane” (138). The result, he said, was a bias toward an additive conception of the different aspects of design and a tendency to overlook large, qualitative effects such as the interference of different components.

Like many other practical men, Sayers was skeptical about model work.59 In his view, aerodynamicists did not yet know what dynamic “similarity” re­ally was, so that inferences from models remained doubtful. Full-scale ex­perimentation was the real basis of knowledge. Grey could be relied upon to give the relatively measured prose of Sayers, his frequent contributor, a more colorful rendering: “I would back any one of a dozen men I know to find out more about streamlines in a month at Brooklands, with the help of a borrowed racing car, a jobbing carpenter, and a spring-balance, than the combined efforts of the National Physical Laboratory, Chalais-Meuden, the Eiffel Tower, the laboratory at Kouchino, and the University of Gottingen have discovered since flying first attracted the attention of that section of hu­manity which the Americans expressively call ‘the high-brow.’”60

This cavalier dismissal of all the major aerodynamic laboratories of Eu­rope dramatizes the anti-intellectual strand in the epistemology. Not all of its expressions were so markedly of this character, but there is no denying a tendency in this direction. Nor can one deny a certain justice in the stance. If scientists have a tendency to simplify the complex and decompose it into its elements, where does this leave the designer who has to reassemble the ele­ments in novel ways? Even if simple principles can be discovered, it can still be unclear how these principles interact when they work together. Design is still a matter of judgment about their combination and compromise in their balance.

Grey’s dismissive attitude toward Gustav Eiffel’s work was not shared by all practical men. The impression created by articles and reviews in the tech­nical journals is that Eiffel was seen as an engineer who could be relied upon to operate in a practical way. If Eiffel’s large, empirical monograph, replete with tables of data, graphs and diagrams of airplanes, is laid side by side with

Greenhill’s mathematical report, there can be no more striking visual proof of the extremes of style that can be represented in aeronautical work. What is more, Eiffel’s work was frequently compared favorably with the experimental work of the NPL. Where the two laboratories diverged, the practical men backed Eiffel.

The reviewer of Eiffel’s La resistance de Fair et Faviation, for the Aero, in March 1911, was enthusiastic: “One is hardly going too far in describing this book as the most authoritative work on the subject that has yet appeared, and it is especially valuable in as much as the experiments have been evolved with an eye specially inclined toward their value in practical aeronautics. . . while experiments of a more purely academic interest have. . . been relegated to the background.” This, the reviewer continued, was strikingly different from the situation that “obtains in more than one experimental laboratory.”61

Writing in July 1916, the editor of Aeronautics invited readers to compare Eiffel, “working almost single handed,” with the National Physical Labora­tory: “It would not be unjust to say that Eiffel attains practical results, ne­glecting a slight margin of error, accounting probably 2 per cent. in extreme cases, which for the time being and for practical purposes is inappreciable. On the other hand, the N. P.L., in its beautiful work, seems rather to strive for the meticulous elimination of this negligible margin of error and passes by the major facts.” Ask Eiffel for the air resistance of, say, an airship hull and the job is done “in a couple of days,” while it would last “heaven knows how many weeks” at the N. P.L.62

The report of the Advisory Committee for 1911-12 noted that, between Eiffel’s laboratory and the NPL, there were differences of some 15 percent between the values of the lift coefficient for certain wings. The probable rea­son, it was said, was observational errors. The ACA resolved to investigate the matter and to ensure that a high degree of accuracy was maintained at Teddington. The “Editorial View” in the Aero was that to the “lay mind” such differences are “disquieting,” and the writer of the editorial chose to read the ACA’s response “almost as an acknowledgement of error on the part of Teddington.”63

For the engineer and the physicist are acquainted with exactly the same facts, but the manner in which they approach their subjects is quite different.

philipp frank, Relativity: A Richer Truth (1951)1

That it is Applied Physics is to me the most inspiring definition of engineering; and if this be true for engineering in general, as I think it is, especially true is it of aeronautics. h. e. wimperis, “The Relationship of Physics to Aeronautical Research" (1926)2

The circulation theory of lift was developed by Lanchester, who was an en­gineer. The reasons advanced against it were proposed by men such as G. I. Taylor who were not engineers but who worked in the British, and particularly the Cambridge, tradition of mathematical physics. This is a clue that needs to be followed up. If the objections were the expressions of a disciplinary standpoint, located at a specific time and place, then perhaps the resistance to the circulatory theory would be explicable as a clash of cultures, institutions, and practices. Such an explanation would not imply any devaluation of the reasons that were advanced against the circulatory theory. It would not be premised on the assumption that these reasons were not the real reasons for the resistance. On the contrary, the intention would be to take the objections against the theory in full seriousness and to probe further into them. To do this it is necessary to understand the sources of their credibility and why the reasons were deployed in precisely the way that they were. I shall now begin that process. By the end of the chapter I shall be in a position to outline a theory that could explain the negative character of the British response to Lanchester’s theory.

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.