# Category The Enigma of. the Aerofoil

## Blumenthal Brings Unity

Otto Blumenthal had been Hilbert’s first doctoral student at Gottingen and continued to help Hilbert edit the distinguished journal Mathematische An – nalen.55 In the winter semester of 1911-12 Blumenthal, now at the TH in Aachen, gave a course of lectures on the hydrodynamic basis of flight. He de­scribed, mathematically, the irrotational flow of an ideal fluid over a range of different Joukowsky profiles. Along with his colleagues at Aachen, Karl Toep- fer and Erich Trefftz, he drew up diagrams of the precise shape of the profiles. The result of the joint work was published in two papers in the ZFM for 1913. The main paper, by Blumenthal, was titled “Uber die Druckverteilung langs Joukowskischer Tragflachen” (On the pressure distribution along Joukowsky wings).56 It was followed by a short note by Trefftz giving a simplified geo­metrical method for drawing Joukowsky profiles and a graphical technique for rapidly computing the predicted air velocities, and hence pressures, on the surface of the wing.57

Blumenthal began by drawing attention to a unifying principle that had not emerged in Joukowsky’s original paper. Joukowsky had used two geo­metrical constructions. The first, which was the more complicated, gener­ated the wing profile, while the second, which was simpler, generated the symmetrical rudder. Blumenthal pointed out that only the second of the two constructions need be used. What is more, the process could be represented by a simple mathematical formula. This formula was the version of the Jou – kowsky transformation that was to achieve such fame.58 The formula can yield wing shapes and curved, Kutta-like arcs as well as rudder shapes and flat plates. Only one transformation, not two, was needed. It was all a matter of the position of the circle on the coordinate system of the plane that was to be transformed. The totality of Joukowsky contours, said Blumenthal, could be generated by the set of all circles that can be drawn on the Z = + i П plane

that pass through = – I/2, provided they either pass through, or contain, the point = +I/2. All that is required is that the circles are then subject to the transformation:

Those circles that pass through both = – I/2 and = +I/2 will have their cen­ters on the n-axis and will generate arcs similar to Kutta’s wing. The one circle in this family that has its center precisely at the origin, and hence has the line from = – I/2 to = +I/2 as its diameter, will be transformed into the straight line that is the limiting case of the arc. Wing shapes will be generated by all of the (off-center) circles whose circumference passes through = – I/2 but contains = +I/2, that is, which are sufficiently large that the circumference goes around the point = +I/2. The sharp trailing edge of the wing will be the transformation of the point = —/2, and the curved leading edge will go round the transformation of the point = +I/2. As a point moves around the circumference of such an off-center circle, the transformation will trace out the curve of an aerofoil shape with a rounded nose and an elongated tail.59 These, said Blumenthal, are “the Joukowsky figures in the proper sense” (“die Joukowskischen Figuren im eigentlichen Sinne”; 125).

It was Blumenthal who provided the unity lacking in Joukowsky’s original paper but which, today, is so often taken for granted. But Blumenthal’s aim was not merely to achieve a formal unity. He was bringing the generation of Joukowsky figures under intuitive control in order to facilitate their practical use. He isolated the features of the construction process that had an aero­dynamically significant effect on the overall geometry of the wing. Where Joukowsky had merely said that the geometrical construction of the wing de­pended on an angle and two lengths, Blumenthal identified the results of the choices that are to be made.

Blumenthal referred his readers to the diagram reproduced here as fig­ure 6.8. The circle in the figure has center M and passes through the point H, which is at a distance I/2 from the origin O. (Notice that I/2 featured in the formula that Blumenthal chose to specify the transformation.) The off-center circle in the diagram is to be transformed by means of the Joukowsky formula and turned into a wing profile. The point H (sometimes called the “pole” of the transformation) is to be transformed into the all-important trailing edge. The radial line from M to H cuts the vertical axis at a point labeled M’. Then, explained Blumenthal, the distance OM’ (labeled f/2) controls the height of curvature of the wing, while the distance M’M (labeled 6) controls the thick­ness of the wing. In general, if the center of the circle to be transformed is on
the positive vertical axis, the result is one of Kutta’s arcs; if the center is on the positive horizontal axis, the result is a symmetrical rudderlike figure; if it is somewhere in between (as in fig. 6.8), the result will be a curved profile of the characteristic Joukowsky type. How curved and how rounded will depend on the factors that Blumenthal had just identified.

Blumenthal gave four examples of Joukowsky profiles to show the effects of modifying these parameters, that is, the effect of moving the center of the circle while ensuring that its circumference still passed through H. Thus the curvature parameter (expressed as the ratio f/l) was given the value of 0, 1/10, 1/5, and 1/5 (again), while the thickness parameter (expressed as the ratio 8/l) was set at 1/10, 1/20, 1/20 (again), and 1/50. The effect of these choices was clearly visible as the Joukowsky profiles that he illustrated went from a sym­metrical shape to a markedly curved shape and from fat to thin.

From the velocity q of the flow (provided by Trefftz’s speedy method of graphical calculation), the pressure on the surface of the aerofoil could be

 figure 6.8. Blumenthal identified the unifying principle behind Joukowskys separate treatments of the arc, the symmetrical rudder shape and the curved winglike shape. All of the shapes came from the same transformation formula applied to a circle that passed through a fixed point H on the y-axis. The shape produced depended on the position of M, the center of the circle. From Blumenthal 1913, 126. (By permission of Oldenbourg Wissenschaftsverlag GmbH Munchen)

 figure 6.9. One of Blumenthal s theoretically predicted pressure distributions along the upper and lower surface of a Joukowsky profile. The part of the graph above the dotted line shows the underpres­sure (the suction effect) on the upper surface of the wing. The lower graph shows the overpressure on the lower surface of the wing. From Blumenthal 1913, 128. (By permission of Oldenbourg Wissenschaftsverlag GmbH Munchen)

computed, and this led to the most striking feature of Blumenthal’s paper. Each of the four Joukowsky profiles that he had constructed was accompa­nied by a graph showing the theoretical pressure distribution on the upper and lower surfaces. (In all cases Blumenthal assumed that the profile was at an angle of incidence of 6°.) One of his profiles and its accompanying pressure graph is shown in figure 6.9. Summing the areas enclosed by the graphs gave a quantity proportional to the resultant lift. Blumenthal discussed each aero­foil in turn, pointing out the significance of the predicted pressure distribu­tion and its dependence on the parameters of the profile. Some features were common and stood out very clearly in the graphs, for example, the greater contribution of the suction effect on the upper surface of the wing compared to the pressure effect on the lower surface. Others were special to one shape, for example, the presence of a small suction effect even on the lower surface of the symmetrical (rudderlike) aerofoil and the very high speeds at the lead­ing edge of the thinnest profile.

## Scientific Intelligence: Fact and Fiction

Looking back to the period of the Great War, after some sixty years, Max Munk expressed the belief that the aeronautical work he had carried out in Gottingen had rapidly fallen into the hands of the Allies. According to Munk, the secret Technische Berichte “were translated in England a week after appear­ance and distributed there and in the U. S.”3 Exactly how this feat of espionage was performed Munk did not say. Similar stories have been related about the flow of sensitive information in the other direction, from the Allies to the Germans. I have already mentioned the secret testing of the Dunne biplane in the Scottish Highlands before the war. This was said to have attracted the at­tention of numerous German “spies,” though these stories surely owed more

to John Buchan than to reality.4 A more sober counterpart to Munk’s beliefs is provided by J. L. Nayler, one of the secretaries to the Advisory Committee. Also speaking retrospectively, he said that the wartime Reports and Memo­randa produced in Farnborough and Teddington eventually found their way into German hands. Nayler, though, suggested that this took months rather than weeks.5 Perhaps British spies were just superior to German spies.6

The truth was almost certainly more pedestrian than these claims sug­gest. There is no evidence that agents acting on behalf of the British gov­ernment got their hands on any information about the wartime Gottingen work and passed it on to their masters in Whitehall or their allies in Paris and Washington. There appears to have been no successful espionage activ­ity. It is not the speed with which information traveled that is striking but its slowness. When information did travel, the channels were overt and obvious rather than mysterious.7 The war had the predictable effect of attenuating the flow of technical information between different national groups, but even during the prewar years, with no military or diplomatic impediments, the flow was surprisingly limited. It is important to identify where the restriction lay. It did not arise because of what might be called material or external fac­tors, such as censorship, but because of more subtle, cultural constraints. It was not the physical inaccessibility of reports, journals, or books that caused the problem. What counted was the response, on the intellectual level, even when they were accessible. For example, both Sir George Greenhill and G. H. Bryan were present at the congress in Heidelberg, in 1904, when Prandtl presented his revolutionary, boundary-layer paper.8 Bryan explicitly men­tioned Prandtl’s contribution in his postconference report for Nature, but he ignored its mathematical content entirely and confined his comments to the experiments and photographs.9 It is difficult to resist the conclusion that if such important matters can be passed over in these circumstances, then even if there had been “spies” reporting back to the British Advisory Committee, their efforts would have been wasted.

To reinforce this claim I start with some other prewar events and look at the information that members of the Advisory Committee had available to them about their German counterparts. From the outset the committee, and the Whitehall apparatus that supported it, accepted the principle that it was important to monitor the work of foreign experts. Haldane stressed the point in Parliament, and the theme was picked up by the aeronautical press.10 The commitment to gathering intelligence was made apparent in three ways. First, the preliminary documentation of the committee, when it was estab­lished in 1909, included what was, in effect, a reading list for the committee members. The list cited some twenty-two works by French, German, Ital-

ian, and American writers. The German authors included Ahlborn, Finster – walder, and Lilienthal.11 Second, the sequence of Reports and Memoranda issued by the committee began with a description of the program of German airship research. It was presented by Rear Admiral Bacon at the very first meeting of the Advisory Committee on May 12, 1909.12 R&M 1 consisted of translated extracts from the publications of the German Society for the Study of Airships and included a lengthy quotation from Prandtl.13 There was men­tion of Prandtl’s wind channel, his experiments on model airships, and, in­triguingly, a passing reference was made to his “hydraulic machine” (shown earlier in fig. 7.1). This was the apparatus used to take his boundary-layer photographs. There was, however, no mention of the mathematical theory. Third, and most important of all, the committee was provided with a series of summaries of foreign papers from leading journals such as the Zeitschrift fur Flugtechnik. A steady stream of these summaries was published in the period between the founding of the committee and the outbreak of the Great War, when such material was immediately withdrawn from public circulation.14

A measure of the size of the intelligence initiative can be gathered by count­ing the number of such abstracts published yearly in the annual report of the Advisory Committee. Such a procedure can only provide an approximate measure of the potential flow of information because it does not take account of the different scope of the individual publications, but it gives some guide. Figure 8.1 charts the year-by-year production of summaries and abstracts of foreign-language publications that were made available to the committee.15 Two things stand out. First, the size of the effort put into tracking foreign work was clearly considerable. Second, there was a consistently high level of attention given to German work, amounting on average to identifying and abstracting some eighteen items per year for a period of six years.

Moving from the quantitative to the qualitative character of the informa­tion, it is important to know which authors the committee deemed interest­ing. The answer is that Prandtl and his collaborators were prominent among them. In December 1910, Glazebrook, as chairman, explicitly drew the Got­tingen work to the attention of the members of the Advisory Committee.16 In August 1913, in preparation for a forthcoming visit to the laboratory in Teddington, Prandtl sent a number of his papers to the National Physical Laboratory (NPL) and received acknowledgment from Selby, the secretary.17 Thus, by one route or another, all of the major prewar work of the Gottin­gen school had been made available, including accounts of the wind channel and the airship work but also material directly concerned with the circulation theory of lift. In addition there were abstracts of papers of indirect interest

 figure 8.i. The number of abstracts of foreign works made available to members of the Advisory Com­mittee for Aeronautics in the years before the Great War. Data from the committee’s annual reports.

because of their significance for fluid dynamics in general. More specifically, among the papers summarized, sometimes at length, were those of Foppl on the resistance of flat and curved plates (abstracts 93, 94, 97, 98, 118, and 131), Fuhrmann on the resistance of different airship models (abstracts 95, 96, and 127), and Prandtl’s classic study of the flow of air over a sphere in which he in­troduced turbulence into the boundary layer by means of a trip wire (abstract 234). Of those explicitly related to the idea of circulation and Prandtl’s wing theory, accounts were given of Foppl’s 1911 study of the downwash behind a wing (abstract 128, but incorrectly attributed to Fuhrmann); Wieselsberger’s 1914 study of formation flying in birds (abstract 276); the 1914 paper by Betz on the interaction of biplane wings (abstract 279); Joukowsky’s pioneering 1910 article (abstract 299); Blumenthal’s 1913 paper on the pressure distribu­tion along a Joukowsky aerofoil (abstract 301); and Trefftz’s 1913 graphical construction of a Joukowsky aerofoil (abstract 302).

The principal mathematical formulas associated with the circulation theory in both its two – and three-dimensional forms were also to be found in the abstracts. Thus the basic law of lift, linking density, velocity, and cir­culation, L = pvr, was stated, as was the law of Biot-Savart, which was the basis of the three-dimensional development of the theory. The abstracts pro­vided everything that was needed to show that the circulation theory was capable of mathematical development and was more than a mere collection of impressionistic ideas. The abstracts gave clear, documentary evidence of the progress that the German engineers were making. It would appear that the circulation theory was there for the taking. Nevertheless, the availability of the abstracts generated no more enthusiasm for the theory of circulation in its mathematical form than did Lanchester’s original publication with its more intuitive treatment of the subject.

Why might this be? Obviously, the abstracts had no power to force them­selves on anyone’s attention. They were things to be used selectively and were subject to the filtering effects of interpretation, both in their composition and their evaluation. Thus Glazebrook’s act in drawing attention to the Gottingen work was probably indicative of his enduring concern with discrepancies be­tween the results of different wind channels and the fundamental problems shared by the NPL and Gottingen in the interpretation of their findings. Gla – zebrook was acutely aware that such problems would be grist to the mill of the “practical men” and was anxious lest they be used to persuade the govern­ment to cut the budget of the NPL.18 Furthermore, the precise content of the abstracts reveals the way that reported work may be glossed so that certain as­pects of it are given salience at the expense of other readings. Take, for exam­ple, the account given in abstract 131, which was devoted to Foppl’s 1910 paper in the Jahrbuch der Motorluftschiff-Studiengesellschaft.19 This paper contained a comparison of Rayleigh flow with Kutta’s theory of circulatory flow. After summarizing the contents of the paper, the abstract writer drew the conclu­sion that neither approach to the flow over an inclined plate was satisfactory. What was needed was an understanding of certain subtle, viscous effects. “It is suggested that Kutta’s theory throws some light on the experimental results, and in some respects, qualitatively, is in fair agreement with the experiments. At present, however, no entirely satisfactory theory seems to be possible until more is known of the nature of the air flow, the main differences being due to the difficulty of including the frictional effects” (257). The need to include frictional effects was, of course, an abiding theme in the British work. The abstract writer then went on to single out, as the “most striking result” of Foppl’s investigation, “the discontinuity in lift and drift coefficients within the region from 38° to 42°” (258). All the attention was thus directed toward extremely difficult, fundamental, and unstable features of the flow that lay far outside the typical working range of an aerofoil. Once again, the British were drawn to the phenomenon of stalling. The focus was on all the things that could not be understood on the basis of inviscid flow at small angles of inci­dence rather than on what could be achieved using perfect fluid theory over a limited range. Thus the abstract and summary itself prefigured the selective tendencies and implicit evaluations that worked against the circulatory theory.

The prewar information about German thinking on aerodynamics was rich but unexploited, whereas during the war, the pressure of short-term work added to the tendency to pass over the significance of the German theoretical approach.20 What of the pattern of information flow, and the reaction to it, immediately after the cessation of hostilities? In some quarters in Britain, the outcome of the war produced a jingoistic complacency. Such sentiments were exemplified by C. G. Grey, editor of the Aeroplane, when he said in 1918: “We have nothing to learn from the Hun in aerodynamics.”21 This boast was a continuation of a commonplace theme in the aeronautical press, which, throughout the war, dismissed German inventiveness, originality, and skill.22 Such vulgarity was largely absent from the writing of the more technically so­phisticated members of the British aerodynamic community, who had, if not an admiration, at least a healthy respect for German achievements.23 Among the members and associates of the Advisory Committee there was an under­standable degree of self-congratulation as the war drew to a close, but it was modest in tone.24

The first reaction to the outbreak of peace by the scientists at the Royal Aircraft Establishment (formerly the Royal Aircraft Factory) was to poke fun at themselves and their critics. The period immediately after the cease-fire, between November and December 1918, saw the production of a light-hearted work titled “The Book of Aeron: Revelations of Abah the Experimenter.”25 This undergraduate-style spoof was composed by the remarkable group of young men who had been recruited by Mervin O’Gorman. Many of them were billeted in a large house in Farnborough called Chudleigh (see fig. 8.2).

 figure 8.2. The Chudleigh set. Hermann Glauert is seated on the plinth on the extreme left. George Paget Thomson, in uniform, is standing behind Glauert. David Pinsent (left) and Robert McKinnon Wood (right) are seated on the lower steps at the front of the group. W. S. Farren, in uniform, is directly behind McKinnon Wood. F. A. Lindemann is standing behind Thomson, and Frank Aston is seated on the right- hand plinth. David Pinsent was killed in a flying accident on May 8, 1918. From Birkenhead 1961.

The house acted as a mess for the RAE and had originally been organized by Major F. M. Green, who, until 1916, had been the engineer in charge of designs.26

The “Book of Aeron,” circulated as an internal report, had been written by a committee that included R. V. Southwell and Hermann Glauert. It was couched in mock, Old Testament terms, with ancient Egyptian overtones, and told the story of the Land of Rae (the RAE), its ruler Bah Sto (Bairstow), and the wicked scribe Grae (C. G. Grey). The Abah of the subtitle appears to be a reference to the designation “Department BA” in which the experi­mental work was conducted at Farnborough. Naturally, the clash between the aircraft manufacturers (“the merchants”) and the Farnborough scientists (“the men of Rae”) played a significant role in the story. The following pas­sage conveys the spirit of the enterprise:

2. And the men of Rae built air chariots for their king, and brought forth new chariots of diverse sorts; and to each chariot did they give a letter and a num­ber, that the wise men might learn their habits:

3. but the multitude comprehended it not.

4. Then murmured the merchants one to another saying, Why strive the men of Rae so furiously against us? For the king goeth to war with their chariots, and behold our chariots are cast into the pit. (3)

After the relief of the armistice, and the lessening of tension, came the serious business of taking stock. Just what had been achieved during the war? What had been sacrificed, scientifically, because of the demands of the war effort? What, if anything, was to be learned from newly accessible German literature? Bairstow and other committee members rapidly let it be known that they deplored the cutback in basic research that had been caused by the war. In terms of fundamentals, they argued, the period of rapid achievement in aerodynamics had been before the war. It was now time to get back to deeper questions and that meant solving Stokes’ equations of viscous flow. Pure mathematicians may have given up on this task, but new techniques, perhaps using graphical methods, might be developed for this purpose. In a confidential report of November 1918,27 mapping out a program of work for 1919-20, the argument was put like this: “General research in fluid motion has been discontinued during the war and it is very desirable that it should be resumed at the earliest possible moment. It is proposed as soon as the opportunity occurs to continue the study of the motion of viscous fluids to which considerable attention has already been given” (9-10). These senti­ments represented the beginnings of a campaign by Bairstow and Glazebrook to channel more resources from short-term to basic research. They argued, in letters to the Times, that government figures for the aeronautics budget con­flated the spending on development with that on research proper. This made the expenditure on fundamental work, which was vital for future technology, appear larger than it really was.28

Both the British and the Americans were keen to locate and translate cop­ies of the Technische Berichte. At a meeting that took place on July 13, 1920, at the Royal Society, minute 28 records Treasury authority for the “employment of abstractors to make abstracts of German technical reports.”34 Across the Atlantic, Joseph Ames, who was the chairman of the Committee on Publica­tions and Intelligence, and one of the founding members of the American National Advisory Committee for Aeronautics, wrote from the NACA head­quarters to J. C. Hunsaker at the Navy Department in Washington, D. C., on October 15, 1920:

It is with great pleasure that I am informing you that the National Advisory Committee for Aeronautics has been successful in obtaining a number of sets of “Technische Berichte” and we are mailing you under separate cover vol­umes No.1, 2 and 3. The Committee is also forwarding a carefully prepared translation of the index of the first three volumes with a list of the symbols used. . . . The importance of the information contained in the “Technische Berichte” cannot be over-estimated and it is the desire of the Committee that all research laboratories and individuals interested in aeronautical research should become familiar with the results of the aeronautical research carried on in Germany during the War.35

The success of this search appears to have been due to Knight’s persistence. He used Prandtl’s good offices to approach the German publishers but had to overcome the numerous obstacles arising from the immediate postwar currency and customs restrictions.36 Meanwhile the U. S. National Advisory Committee continued negotiations with the British about the translation and abstraction of the reports.37 The enthusiastic terms of Ames’ letter, and the continuing efforts by both the British and the Americans, provide a sufficient basis for rejecting Max Munk’s claim that the reports were in the possession of the allies soon after their completion. Had this intelligence already been gathered, all the postwar concern would have been unnecessary.38

## Glauert’s Textbook

In the German-speaking world the circulation theory of lift and Prandtl’s wing theory found their way into the textbooks during the Great War. Rich­ard Grammel, who taught mechanics at the technische Hochschule in Danzig, led the way in 1917 with his Die hydrodynamischen Grundlagen des Fluges5 In 1919, immediately after the war, Arthur Proll of the TH in Hannover, pub­lished Flugtechnik: Grundlagen des Kunstfluges.52 H. G. Bader published his Grundlagen der Flugtechnik in 1920, and in 1922 Richard Fuchs of Berlin and Ludwig Hopf of Aachen produced their comprehensive Aerodynamik.53 The content and level of these books contrasted markedly with what was available on the British textbook scene. As described earlier, both Cowley and Levy’s Aeronautics in Theory and Experiment of 1918 and Bairstow’s Applied Aero­dynamics of 1920 dismissed the circulatory theory, whereas G. P. Thomson’s Applied Aerodynamics of 1919 was almost purely empirical. Thomson spoke of the “complete failure” (26) of mathematical hydrodynamics to account for lift. He concluded that an account of lift required an understanding of eddies and turbulent motion: “This is the solution we want for aerodynamics, and not that found by the ordinary mathematical method” (32). In 1926 the situa­tion changed radically when Cambridge University Press published Hermann

Glauert’s Elements of Aerofoil and Airscrew Theory.54 This work showed the power of the “ordinary mathematical method” of which Thomson, like most of his Cambridge companions, had despaired. Glauert’s Elements proved to be an outstanding work of exposition which, even today, some eighty years later, is still confidently recommended to students.55

The book consisted of seventeen brief chapters and surveyed all the main themes of modern aerofoil theory for a reader with no previous knowledge of fluid mechanics (though a significant degree of mathematical competence was presupposed). The first chapter described the main facts to be explained, while chapters 2-5 outlined the theory of perfect fluids. Chapters 6 and 7 in­troduced the theory of conformal transformation and the specific properties of the Joukowsky transformation. Chapter 8 dealt with viscosity and drag. Here Glauert introduced the Stokes equation for viscous flow and informally derived Prandtl’s boundary-layer approximation. Chapter 9 was called “The Basis of Aerofoil Theory,” and in it Glauert sought to bring together the ap­parently antithetical ideas of viscous and inviscid flow into a practical synthe­sis. His aim was “to obtain the true conception of a perfect fluid” (127). The form of the desired synthesis was described by Glauert as follows: “The vis­cosity must be retained in the equations of motion and the flow of a perfect fluid must be obtained by making the viscosity indefinitely small” (117).

When analyzing the motion of an object in a fluid, the concept of a per­fect fluid must be deployed in a way that retains the effects of viscosity. If a perfect fluid is defined as a fluid devoid of viscosity, such a requirement is contradictory: how can the viscosity, which is excluded by definition, also be “retained”? Here, once again, Glauert sought to convey his novel, Gottin – gen-style methodology. He argued that the requirement he formulated can be satisfied even though the boundary conditions, the way the fluid behaves at a solid boundary, are wholly different for a perfect fluid and for a viscous fluid. (For a perfect fluid the boundary conditions are that it cannot penetrate the solid boundary but it can slide smoothly along it; for a viscous fluid the conditions are zero penetration of the boundary and zero slip along it.) The trick needed to retain viscosity in the equations of motion is to start with a viscous boundary layer of finite thickness around the object and imagine it to become an infinitely thin sheet of vorticity. The boundary layer is a real phenomenon belonging to viscous fluids, and a vortex sheet is an idealization appropriate to perfect fluids. The connecting link that allows viscosity to be “retained” is that

in the limit the boundary layer becomes a vortex sheet surrounding the sur­face of the body and the vortices of this sheet act as roller bearings between

the surface of the body and the genera! mass of the fluid. The conception of a perfect fluid with a vortex sheet surrounding the surface of the body therefore represents the limiting conditions of a viscous fluid when the viscosity tends to zero, and the existence of the vortex sheet implies that the perfect fluid so­lution need not satisfy the condition of zero slip at the boundary. (117-18)

If the cross section of a wing is drawn in two dimensions, the line that traces its profile is to be thought of as made up of an infinite number of points acting like infinitesimal roller bearings. These rollers are said to be rotating fluid elements, that is, “fluid elements in vortical motion” (119), and they stand in for the boundary layer. The rotating fluid elements pass along the surface of the body and finally leave it to pass downstream in the wake. Glauert linked this picture to von Karman’s work on the so-called vortex street that exists behind a bluff body placed in a fluid flow. Clearly Glauert was introducing a significant degree of idealization, but even this degree would soon be surpassed. Having replaced the boundary layer on the wing surface by a sheet of vorticity, the chord and profile of the wing was then ignored altogether. In two dimensions the wing was reduced to a single point, that is, to the “cross section” of a line of vorticity imagined to be perpendicular to the page on which the figure is drawn.

How did Glauert explain the origin of the circulation around a wing? The discussion of this sensitive and difficult question was located in a short sec­tion of chapter 9. Combining candor with British understatement, Glauert introduced the issue as follows: “The process by which the circulation round an aerofoil develops as the aerofoil starts from rest presents certain theoreti­cal difficulties, since the process would be impossible in a perfect fluid, and it is again necessary to consider the limiting condition as the viscosity tends to zero” (121). Circulation is impossible if the analysis starts with p = 0 and con­fines itself to this condition. To overcome the difficulty it is necessary to start by considering p Ф 0 and then make the transition from viscous to nonvis­cous flow by imagining that p ^ 0. Here within the one sentence we see the British and German conceptions of an ideal fluid directly juxtaposed. Glauert then proceeded to offer a qualitative account of the required transition. The analysis effectively hinged on two diagrams of the flow at the trailing edge of a wing. I reproduce the diagrams in figure 9.11.

At low speeds, as the aerofoil starts from rest, the air behaves like an ideal fluid and curves round the trailing edge as shown in (a) in figure 9.11. There is a stagnation point S on the upper surface not far from the trailing edge. Glauert argued that, as the velocity increases, the streamlines coming from the undersurface are unable to turn round the trailing edge “owing to the large viscous forces brought into action by the high velocity gradient” (121).

 figure 9.11. The initial moment in the creation of a vortex (a). The vortex detaches itself from the trailing edge and floats downstream (b), leaving behind an opposing circulatory tendency. From Glauert 1926, 121. (By permission of Cambridge University Press)

The flow thus breaks away from the trailing edge in the manner shown in (b). The result is that “a vortex is formed between the trailing edge and the old stagnation point S” (121). When the vortex has reached a certain stage of development, it breaks away and floats downstream in the wake of the wing.

Although Glauert does not make the point explicitly, it is obviously im­portant for the argument that the vortex that detaches itself and moves down­stream is rotating in the correct direction. Its direction of rotation determines the direction of the circulation in any contour that surrounds it. The infor­mation about the direction in which the vortex rotates is contained in Glau – ert’s diagram rather than his text. The diagram shows that the vortex rotates in a counterclockwise direction. Why counterclockwise? The presumption must be that the flow that initially went round the trailing edge and then progressively failed to navigate the sharp corner is moving more rapidly than the flow that comes away from the stagnation point. The speed difference of the adjacent bodies of fluid would constitute a surface of discontinuity and hence a surface of vorticity—and the differences, in this case, would produce a counterclockwise vortex.

The next step in the argument was equally crucial. In the diagram a vortex has detached itself from the wing, and this, the argument goes, generates an equal and opposite circulation around the wing. The detached vortex had a counterclockwise circulation, so the circulation around the wing will be clockwise, and this produces the speed differential postulated by the circula­tion theory. But why does this process create an equal and opposite circula­tion? The answer given by Prandtl and Glauert was that such an outcome is required in order that Kelvin’s theorem is satisfied. Kelvin’s theorem initially looked as if it would rule out the onset of circulation entirely. If there is zero circulation at the onset of movement, there will be zero circulation at all later times. But the enemy is now converted into an ally. Glauert explained that the circulation around a large contour, enclosing both the wing and the detached vortex, will indeed stay zero. This is necessary to satisfy Kelvin’s theorem, but the theorem can be satisfied by virtue of two opposing vortices whose

 figure 9.12. The creation of circulation around a wing (around contour ABD) counterbalances the circulation (around the contour BCD) created by the vortex which detaches itself from the trailing edge. The two opposing circulations sum to zero, and thus the flow is said to conform to Kelvin’s theorem. Kelvin’s theorem entails that if there is no circulation in the flow of an ideal fluid when motion begins, then there can be no circulation at a later time. From Glauert 1926, 121. (By permission of Cambridge University Press)

respective circulations cancel each other out. The argument is represented diagrammatically in figure 9.12, again taken from Glauert. The large contour is called ABCD. The vortex is called E, and the circulation around the wing is K, while the circulation around the vortex is, accordingly, -K. As Glauert put it: “the circulation round any large contour ABCD which surrounded the aerofoil initially was and must remain zero, and as this contour includes the vortex E there must be a circulation K round the aerofoil which is exactly equal and opposite to the circulation round the vortex E” (121). The detached vortex floats away on the free stream with velocity V, leaving the wing with circulation K and lift given by the Kutta-Joukowsky law L = pKV. Glauert’s argument was moving rapidly at this point, but though potentially puzzling, this sequence of steps is now to be found in all standard textbooks.

Having prepared the ground in the first nine chapters, Glauert moved on to discuss the aerofoil in three dimensions. In chapter 10 he introduced the mathematics of the simple and refined horseshoe vortex system as well as the important concepts of induced velocity and induced drag. In chapters 11-13 he dealt with the effects of varying the aspect ratio of a wing and generalized the results to biplanes. In chapter 14 he discussed wind-tunnel corrections, while in the final chapters, 15 and 16, he applied the circulation theory to the airscrew.

## Objectivity and Reality

Time and again critics attempt to refute relativism by drawing attention to the objectivity of what is known in both science and daily life.78 Such attempts are misguided. The only kind of counterexample that could refute relativism would be an example of absolute knowledge. Proof or evidence of objectiv­ity will not suffice unless the objectivity in question can be shown to be an absolute objectivity. The demand for objectivity is legitimate, but it is meant to preclude subjectivity, and subjectivism is not the same as relativism. The subjective-objective distinction is one thing, the relative-absolute distinc­tion is another thing, and the two should not be conflated. Frank was ad­mirably clear on this point and knew that his defense of relativism was not an attack on objectivity. He (rightly) believed in both the relativity and the objectivity of scientific knowledge.79

Rather than explore this theme in an abstract way, let me take an example from aerodynamics. The example, which concerns the rolling up of the vor­tex sheet behind a wing, is designed to show the objective (that is, nonsub­jective) character of knowledge at its most dramatic. The question is: Can the example be understood in relativist terms? Here is the example. In the spring of 1944, at a crucial stage of the Second World War, London was at­tacked by V-1 flying bombs. The V-1 was a large bomb fitted with small wings and a ram-jet engine, and it flew some 300 miles per hour. The bombs were launched from sites on the French and Dutch coasts by means of a shallow ramp that pointed in the direction of the target. After the bomb had trav­eled a predetermined distance, its engine was switched off by an onboard device that simultaneously altered the trim of the wings, causing the bomb to fall to the ground and explode. In an effort to stem the attacks, the pi­lots of the Royal Air Force chased after the bombs and tried to bring them down in open country where they would do less harm. It was not possible to close in on the bomb to shoot it down because of the danger that the re­sulting explosion would destroy the attacking aircraft. Some pilots therefore developed the technique of flying close to the bomb, making use of the air­flow behind the wing of their aircraft to flip the missile on its side so that it would drop to the ground.80 This technique did not involve direct, metal-on – metal contact with the V-1 but, it has been argued, exploited the rolling up of the vortex sheet behind the wing of the aircraft. It was the rotating air of the vortex that turned the missile on its side. According to an article in the An­nual Review of Fluid Dynamics in 1998, the rolling up of the trailing vortices behind a wing of high-aspect ratio was, for a long time, considered to be a matter of little practical importance by experts in aerodynamics. The experts acknowledged its existence but not its utility. But, says the author, if the theo­rists ignored the significance of the roll up, “fighter pilots who used their own vortices to topple V-1 flying bombs had another opinion.”81

The conclusion must be that although both groups were actively engaged with a reality that was largely independent of their subjective will, the qual­ity of that engagement was different. In both cases their understanding was objective rather than subjective, but it was also to be seen as relative to their standpoints. In neither case did it have an absolute character. In developing the argument of his book, Frank was therefore right to insist that the doctrine of the relativity of truth “does not imperil by any means the ‘objectivity’ of truth” (21).

It may be objected that the pilots had causal knowledge of reality. Theo­ries may come and go, and verbal accounts may vary, but don’t actions and interventions put an agent into direct contact with reality? This, it may be said, proves that there is a way of grasping reality that is not merely relative. But does it? The critic who takes this line must confront and answer the ques­tion What is supposed to be absolute about the knowledge of causes and the exploitation of this knowledge? The correct answer is that there is nothing absolute about causal knowledge. This conclusion ought to be well known because it was established over two hundred years ago by David Hume in his Treatise of Human Nature. Hume gave a relativist analysis whose essential points remain unchallenged to this day. His argument was that all knowledge about causes, for example, that A causes B, whether expressed in words or ac­tions, is inductive knowledge based on experience. Inductive inferences, said Hume, can never be given an absolute justification. Inductive knowledge is irremediably relative.83

The limited, fallible, and relative character of practical knowledge can be generalized from the example of the flying bombs to my entire case study. The pattern of flow over a wing described by Lanchester, Kutta, Joukowsky, and Prandtl is not the only one that can render mechanical flight possible. Their picture, which is now called “classical aerodynamics,” and whose his­tory I have been describing, rests on the principle that the separation of the flow from the surface of the wing must be minimized. Flow separation, it was assumed, always leads to a breakdown of the lift. It has now been dis­covered that flow separation can be both exploited and controlled in a way that actually generates lift. Leading-edge vortices, and even shockwaves, can be exploited to create lift.84 This was not realized until many years after the events I have described. As one authority put it, “We must realise, however, that Prandtl’s is but one of many possible bases of wing theory and there can be no doubt that more comprehensive assumptions will eventually be

developed for this interesting type of physical flow.” 85 Until the late 1950s, all of the technical knowledge in aerodynamics concerning lift had been de­veloped on what can now be seen as a narrow basis. What the future holds is always unknowable, but the more recently acquired, broader perspective serves to expose the hitherto unappreciated relativities of past achievements. But we should not allow ourselves to think that, as these historical relativi­ties are exposed, knowledge progressively sheds its relative character and moves closer to something absolute. To cherish such a picture is to indulge in metaphysics.

## The “Reptile Aeronautical Press”

Two related complaints came together in the arguments that were mobilized on behalf of the “practical man.” First, it was said that the designers of air­planes did not need mathematical knowledge and that persons who did pos­sess such knowledge were mere theorists who were ill equipped to deal with real problems. The National Physical Laboratory, it was said, was staffed by mathematicians and theorists rather than engineers with practical experience. Second, those who were in the employ of the government led cushioned and subsidized lives that protected them from the bracing rigors of the market. This complaint included the staff at the NPL but was particularly directed at the Royal Aircraft Factory.

Within a year of the ACA’s founding, the aeronautical press was asking for evidence that the committee’s work was bearing fruit. The anonymous writer of an editorial in Aeronautics in 1910 lamented that “it is too late to renew criticism of the composition of the Committee or of the limitations placed on its work; it will be sufficient once again to place on record our opinion that in both respects the Committee is bound largely to be a failure so far as results of immediate practical value are concerned.”57 In an article on aeronautical research in the Aeroplane of August 31, 1911, P. K. Turner, a regular contributor, said it was time that theory and practice were brought together. Was this not why institutions such as the National Physical Labora­tory had been set up? “But it appears that the workers at these institutions, like the monks of old, are growing fat and useless; and of all the shameful wastes perpetuated in our alleged civilisation, the worst, in my eyes, is an equipped factory, laboratory, or office, where owing to the incompetence of those in charge or the laziness of their subordinates or both, or vice versa, nothing is done.”58 The issues came to a head at the military air trials of 1912 held on Salisbury Plain. Private contractors, British and foreign, were invited to enter their aircraft into a competition to see which ones best met the per­formance criteria laid down by the military. The competition was organized by O’Gorman, though the terms of the competition precluded the govern­ment Factory from formally entering its own designs. Although not an of­ficial competitor, a Factory model, a tractor biplane called the BE2, was put through its paces during the trials and informally faced the same tests as the others. The designation BE came from a system of classification developed by O’Gorman. The E stood for “experimental”; the B for “Bleriot-type” and referred to the position of the engine at the front of the aircraft. Pusher air­craft with the engine behind the pilot were given the designation F after the pioneer Henri Farnam. The BE2 was designed by Geoffrey de Havilland, who had been taken onto the staff of the Factory after successfully building some aircraft of his own. The BE2’s performance manifestly outclassed that of the entries from private firms. It was reasonably stable, made good speed, and de Havilland even set a new altitude record with the BE2 during the trials. The official winner was a machine entered by Cody, but the War Office proceeded to ignore the competition and focused its interest on the superior machine, even though it had been precluded from official entry. It offered contracts to private constructors to build not their own machines, but twelve of the government-designed BE2s. In an improved form this machine became, for a number of years, the mainstay of the Royal Flying Corps. Figure 1.4 shows a side elevation of the BE2A.

For the government’s critics the policy of contracting out government de­signs constituted an outrage. It was denounced as an attack on private enter­prise that strangled all design initiative.59 But private enterprise had put up a miserable showing at the air trials. As one eyewitness noted: “Of the seventeen British aeroplanes that were nominally in evidence, at least seven of the newer

 figure 1.4. The BE2 was developed by the Royal Aircraft Factory and played a central but contro­versial role in the British war effort. The highly stable BE2 has been called one of the most interesting airplanes ever built. This side elevation is from Cowley and Levy 1918.

makes were either unfinished or untested on the opening day, and thus some of the very firms for whose benefit the trials had, in a measure, been orga­nized, spoiled their own chances in competition with the older constructors who, for the most part, had entered well-tried models.”60 An editorial in the Aero for September 1912 admitted: “It is undoubtedly a fact that the majority of our home manufacturers have not gained in reputation through partici­pating in the military trials.”61 The same point was conceded by an editorial in Flight in which it was acknowledged that the BE2 was “one of the best fly­ers ever produced.” Of the firms that were granted a contract to produce the BE2, the editorial continued, “not everyone could have as readily justified a similar demand for its own machines on demonstrated merits in the Military trials.”62 The War Office decision was reasonable; if anything it was more ac­commodating of the sensibilities of the private manufacturers than it should have been. Looking back from 1917, an editorial in the journal Aeronautics ac­knowledged that “before the War there was in the whole country not a single decently organised aircraft manufacturing firm.”63 For example, the Handley Page Company had accepted orders to produce five of the twelve BE2s, but it failed to deliver even this small number. Only three of the machines had been delivered by 1914.64 If the editorial in Aeronautics is to be believed, the situation at Handley Page was the rule rather than the exception; not a single manufacturer of the BE2s made its delivery on time. The editorial went on: “We are not blind to the faults of the Royal Aircraft Factory, which are of a nature which seems inseparable from any state-owned institution. Nor do we ignore the fact that the Factory was bitterly detested and thoroughly dis­trusted by the industry at large. But truth compels us to recognise the fact that the industry was chiefly responsible for its own grievances” (185). The writer of this passage was probably J. H. Ledeboer, the editor of Aeronautics. The self-critical tone was hardly typical of most of the polemics unleashed against the Royal Aircraft Factory, though the assumption that government institutions would be inferior to those of the business world certainly was.

In Parliament the aircraft manufacturers had the support of, among oth­ers, William Joynson-Hicks and Arthur Hamilton Lee, both Conservative MPs, and Noel Pemberton Billing, an independent MP and founder of the Supermarine company. Billing had conducted his theatrical campaign for election on the basis of his commitment to, and knowledge of, aviation and all matters relating to it. The various critics of the government did not always agree with one another, but their combined voice was loud and persistent. Week after week, and year after year in the pages of the aeronautical journals, in Parliament, and in the right-wing press, they directed their anger and con­tempt at the Advisory Committee, the National Physical Laboratory, and the

Royal Aircraft Factory. Every setback, every accident, every tragedy was used as a stick with which to beat the government and as proof of the inferiority of government design and construction compared with private enterprise.

The BE2, and everyone associated with it, became the objects of a cam­paign of denigration. As so often happened in the early years of aviation, acci­dents occurred, and a number of persons flying the BE2 were tragically killed. One was Lt. Desmond Arthur, whose BE2 broke up in the air at Montrose at 7:30 a. m. on May 27, 1913. Lt. Arthur was a friend of C. G. Grey, the editor of the Aeroplane, and the loss fed Grey’s state of permanent anger against the government.65 Another victim was E. T. “Teddy” Busk, a brilliant engineer­ing graduate from Cambridge who had joined the Royal Aircraft Factory in the summer of 1912. Busk had been conducting a program of experiments designed to improve the stability of the BE2 when the machine he was pilot­ing caught fire.66 This accident, and the other fatalities, provided the critics with the excuse for which they were looking. “The Victims of Science” was the headline in the Aeroplane of March 19, 1914.67 Grey (see fig. 1.5) exploited the opportunity to the full. He argued that it was the scientific approach to airplane design that had killed these unfortunate men. He wrote: “I submit

 figure 1.5. C. G. Grey, editor of the Aeroplane and vehement critic of the Advisory Committee for Aeronautics. (By permission of the Royal Aeronautical Society Library)

that if the Department of Military Aeronautics will hold an enquiry into the design and construction of Mark BE2 biplanes and will take the evidence of workshop foreman and practical constructors—apart from the scientists and theoreticians—among contractors who are building the BEs they will obtain sufficient criticism to condemn almost every distinctive feature of the BE— provided always they can guarantee that in the event of the practical men speaking their minds they will not jeopardise their firm’s chances of obtain­ing further orders” (320).

Grey was an accomplished polemicist and he took care to cover himself lest the criticisms he was confidently predicting were not forthcoming. He implied that this could only mean that sinister, government forces were sup­pressing them. Having secured his line of retreat, Grey then asserted that the deaths had been caused by criminal negligence and he knew who the crimi­nals were: “Those responsible are the people, if you please, who have ‘the best brains in the world,’ and through whom aeroplane design is to excel. These are the people who base their calculations on the theories of the armchair airmen of the National Physical Laboratory” (321). When the Aeronautical Society opened a subscription to honor Busk, Grey accused it of exploiting the young man’s death. He had an unpleasant talent for criticizing others for what he was doing himself.68

Political attacks on the aeronautical establishment became even more in­tense after the start of the war in 1914. The summer of 1915 saw the Fokker Scourge. Anthony Fokker, a Dutch designer working for the Germans, had developed a forward-firing machine gun, synchronized to fire through the propeller disc. He fitted it to an otherwise undistinguished monoplane, and the new arrangement marked the emergence of the specialized “fighter air­craft.” It gave the Germans a marked advantage and, for a while, increased the losses of British pilots and machines. The BE2, whose stability compro­mised its maneuverability, was no match for the Fokker Eindecker—not, at least, when the Fokker was flown by the particularly skilled pilots to whom it was selectively assigned. By any standards this issue was one of importance for a country at war. Rhetorically, however, it became another opportunity to voice the interests of the aircraft manufacturers. In the House of Commons, Pemberton Billing denounced the government and military authorities as “murderers.” His claim was that if the young men of the Royal Flying Corps had been given machines designed and built by private firms rather than gov­ernment agencies, they would be alive today.

It would be a study in its own right to trace all the twists and turns of the protracted, political campaign conducted by Billing, Grey, and the man­ufacturers, and it would be no easy matter to decide, in every case, which complaints had substance and which were unscrupulous exaggerations and self-serving falsehoods. Given the seriousness of Billing’s allegations, and the place in which they were made, it was inevitable that official inquiries had to be launched. Two issues had to be unraveled: (1) was the Royal Flying Corps conducting its military business properly? and (2) was the Royal Aircraft Factory dealing improperly with the private manufacturers? The Burbridge Committee addressed the first problem and the Bailhache Committee the second. During the course of these inquiries, Pemberton Billing’s behavior became so eccentric and evasive that even former supporters began to back away. Flight, which had previously welcomed his election, decided that the talk of the “deadly Fokker” was a gross exaggeration and suggested that there must be some ulterior motive.69 Soon the editorials were dismissing his in­dictments as “irresponsible” and “sensational” rather than “the measured views of a man in earnest for the welfare of his country.”70

The official inquiries could find no basis in fact for Pemberton Billing’s accusations against the Royal Flying Corps, but he and the other critics ef­fectively “won” the argument against the Royal Aircraft Factory.71 The report conceded that there had been inefficiencies. The tepid defense of the Fac­tory meant that the interests represented by the critics ultimately prevailed.72 Flight, which had frequently been supportive of the Advisory Committee, the National Physical Laboratory, and the Factory, now concluded that the rights of the manufacturers had indeed been encroached upon.73 The edito­rial column proudly affirmed the principle that private enterprise was always superior to government, and then promptly asked for government subsidy for the aircraft industry.74 The government acceded to the critical pressure of the manufacturers and the Aircraft Factory was turned exclusively toward re­search rather than design. O’Gorman was removed, and his team of designers and engineers dispersed into the private sector. An impartial assessment of the rights and wrongs of the issue would, however, have to note that, before the restriction on its activities, the personnel of the Royal Aircraft Factory had produced one of the most outstanding fighter aircraft of the war, Henry Folland’s SE5. (In O’Gorman’s nomenclature the S stood for “scout” and the E for “experimental.”) Folland and his colleagues had skillfully balanced the competing demands of stability and maneuverability to produce one of the most formidable fighting machines of the war and an aircraft that was a match for anything its pilots might meet.75

The critics also “won” in that, by 1915, they had managed to hound Hal­dane out of office.76 Haldane was denounced in the right-wing press as a pro­German sympathizer. He was said to have opposed and delayed the dispatch of the British Expeditionary Force to France in 1914 and to have known of the

German war plans without informing his Cabinet colleagues. There was not a shred of truth in any of these allegations. Indeed, it was only thanks to Hal­dane’s earlier army reforms that the country had a viable expeditionary force at all. The charges even alluded to a secret wife in Germany and to Haldane being an illegitimate half brother of the kaiser. It was ludicrous and vile but it worked, and the “reptile aeronautical press,” as O’Gorman justifiably called it, played its part in the affair with enthusiasm.77 The episode must count as one of the most disreputable in twentieth-century British politics.78

During the Great War enormous social pressure was placed on men to con­tribute to the war effort and not to shirk their patriotic duty to lay down their lives on the field of battle. Pacifists and critics of the war were reviled. This practice was routine in the aeronautical press.79 Grey was happy to mobilize the hatred of “trench-dodgers” and use it against those who, instead of be­ing at the front, were working at Farnborough and Teddington. He reprinted an article from the Times (under the title “The Farnborough ‘Funk-Hole’”) asking why the fit young men seen coming in and out of the Aircraft Factory were not in France.80 The theme was taken up again when reviewing the Ad­visory Committee’s report for 1917. As usual, said Grey, the report is devoted to the glorification of the National Physical Laboratory, though he noted with approval that two “practical men of proven merit” had been brought on to the Engine Sub-Committee and the Light Alloys Sub-Committee.81 He was, however, censorious of those who were performing basic, hydrodynamic ex­periments, for example, those involving water tanks. Making play with the word “tank,” Grey declared that “if some of these able-bodied young men were to take a course of experimental work in motor-tanks at the front they would confer greater benefits on their native land” (315). Meanwhile Grey was corresponding with Winston Churchill to plead for his own exemption from the inconvenience of wartime obligations. Judging from surviving let­ters, now in the archives of the Royal Aeronautical Society, Churchill, though brief and formal in his responses, duly obliged, and through his intervention Grey got the exemptions for which he had asked.82 Dulce et decorum est pro patria mori.

## Nonmathematical Summary

The main points of this chapter may be summarized by the following ten items. The list begins with a resume of some of the terminology of the field. This terminology is taken for granted in the subsequent discussion.

1. A flow is called a two-dimensional flow when it can be drawn in cross section and the drawing taken as representative of the flow at any other cross section. Thus if the flow around a barrier, or some other obstacle, is drawn in two dimensions, this ignores the complications introduced into the flow by what happens at the edges not shown in the picture (that is, below the page and above the page). If the drawing shows the cross section of, say, a wing, then the picture does not portray what is happening at the wingtips, that is, the third dimension of the situation. This absence can be justified if the wing is very long and the immediate concern is with the flow at parts of the wing that are distant from the tips. The diagram can then adequately represent the flow around the central sections of the wing. A wing that is long enough to justify this approximation is often called an infinite wing. The word “infinite” is much used in hydrodynamics. References to, for example, “the flow at in­finity” usually mean the flow as it is at a great distance from some disturbance so that the effects of the disturbance can be ignored.

2. The main theoretical resource used in early aerodynamics came from classical hydrodynamics. Hydrodynamics offered a mathematically sophis­ticated theory of the flow of an “ideal” fluid, that is, a fluid that was incom­pressible and also completely devoid of viscosity. Of these two idealizations, the most contentious was the neglect of viscosity. The differential equations that govern the flow of an ideal fluid are called the Euler equations. These equations give the speed of the flow at a specified position and time. To make it easier to solve these equations, mathematicians introduced two further ide­alizations. First, it was often assumed that the flow was steady. This meant that rate of change with time was zero and could be ignored. Second, it was assumed that the fluid elements did not rotate. There was an emphasis on irrotational flows because it simplified the mathematics. Unfortunately the benefit of mathematical simplicity was purchased at the cost of making the flows being analyzed less than realistic as models of real fluid flows.

3. Fluid elements (that is, the small volumes of fluid whose velocities and rotations are under study) are not to be identified with molecules or atoms or material particles, although occasionally such identifications seem to have been made. Fluid elements are mathematical abstractions that enable the methods of the differential and integral calculus to be applied to fluids.

4. One logical consequence that can be derived from the Euler equations is a highly useful result called Bernoulli’s law. With the assumption of a steady, irrotational, and incompressible flow, the law takes on a simple form. It states that the pressure and the velocity at a point in the flow are related by a simple law that implies that as the speed increases the pressure will decrease, and as the speed decreases the pressure increases. Speed and pressure trade off against one another. The use of the law makes it important to distinguish between three different meanings that are attached to the word “pressure.” There is (i) static pressure, (ii) dynamic pressure, and (iii) total pressure. Total pressure equals the sum of static pressure and dynamic pressure. Static pressure is the pressure on the sides of a pipe or the surface of a wing. Total pressure is the pressure felt when a body of fluid is brought to a standstill. Dynamic pressure is the name given to the quantity V2 p V2 where p is the density of the fluid and V is the speed of flow. In the simplified conditions dealt with in early aerodynamics, the total pressure can be considered to have a constant value. As speed V increases and hence dynamic pressure increases, then static pressure must go down. Care is needed to ensure its correct ap­plication, but Bernoulli’s law plays an important role in (i) calculating the forces on an object that is immersed in a flowing fluid, for example, a wing in a stream of air, and (ii) understanding the operation of instruments such as the Pitot probe, which registers total and static pressure and (via Bernoulli’s law) permits the computation of velocities.

5. The restriction to irrotational flow permitted the mathematical descrip­tion of a wide variety of two-dimensional flows such as the flow of a steady stream around a circular cylinder and the flow around a barrier facing head – on into the stream. The streamlines of these flows could be drawn on the ba­sis of the formula (called the stream function) that furnished the mathemati­cal description of the flow. In a steady flow (but not in an unsteady flow) the streamlines give the path taken by the fluid elements. As well as streamlines a flow can be described by what are called lines of equal potential. These are orthogonal to the streamlines except at points called stagnation points, which are points where streamlines come to a halt on the surface of a body. Stream­lines and potential lines can be switched in the sense that the potential lines can be interpreted as the streamlines of a new flow. The old streamlines then become the potential lines of the new flow. Just as the streamlines are speci­fied by the stream function so the potential lines are specified by a potential function.

6. The possibility of arriving at a mathematical description of a flow was greatly improved because a large number of familiar mathematical functions (called functions of a complex variable) turned out to be interpretable as possible fluid flows. The geometrical patterns generated by these functions (that is, the lines of the curves plotted on graph paper) could be read as the patterns made by a flowing fluid and the boundaries that constrain them. Ex­ploring a function and then giving it an after-the-fact interpretation in terms of a flow of ideal fluid was called the indirect method of arriving at the equa­tions of the flow. I illustrated this process by means of a function that could be understood as describing the flow of a uniform stream around a circular cylinder. The example, along with the overall presentation of the material in this chapter, was taken from one of the standard textbooks of the World War I period, namely, Cowley and Levy’s Aeronautics in Theory and Experiment published in 1918.

7. A more direct line of attack was sometimes available to the mathemati­cian in search of a mathematical description of a flow pattern. This method involved constructing a set of equations that related the flow to be under­stood to a very simple flow that was already understood, for example, the uniform flow along a straight boundary. If the boundaries of the simple flow could be transformed into the boundaries of the more complicated flow, then the methods of transformation would also turn the simple streamlines into the more complicated streamlines of the desired flow. A particular set of transformations called conformal transformations played a central role in this process. Many such transformations had been studied as exercises in pure mathematics and geometry but were found to be important resources in the study of fluid flow. One such important transformation was called the Schwarz-Christoffel theorem.

8. The main problem with the hydrodynamics of an ideal fluid was that, although it became mathematically sophisticated, it appeared to provide no resources for explaining the resistance that an object experiences when placed in the flow of a real fluid such as water or air. When an ideal fluid flows around, say, a flat plate or a circular cylinder, the flow exerts no resul­tant force on the object. This is often called d’Alembert’s paradox, although whether it is a paradox in the true sense of the word is examined in more detail later. What is beyond dispute is that the result presented a problem for anyone who wanted to understand the air by likening it to an ideal fluid. The use of the theory of ideal fluids led to the false result of zero resistance or zero drag.

9. One possible response to this “paradoxical” result would be to reject ideal fluid theory as useless for the study of real fluids such as air. Why not develop a more realistic hydrodynamic theory devoted to viscous fluids? This project was begun, and the equations of motion of a viscous fluid were for­mulated. They are now called the Navier-Stokes equations. (The British just called them the Stokes equations.) Frustratingly they could only be solved in a few very simple cases, which gave special significance to the search for new ways to make ideal-fluid theory more realistic. In principle there were two ways to do this—hence the existence of two competing theories of lift. Only one of these ways (called the theory of discontinuous flow) was described in this chapter. The alternative, the vortex theory, is discussed in chapter 4.

10. The theory of discontinuous flow was proposed by Helmholtz and carried forward by Kirchhoff and Rayleigh. Helmholtz argued that the “para­doxical” result of zero resistance or drag arose because an ideal fluid could wrap itself around an object and exert pressure from all sides in a way that canceled out any resultant force. The discontinuous flow approach exploited the possibility that there could be discontinuities in the velocity of different bodies of ideal fluid that were in direct contact with one another. The flow was assumed to break away from the edges of an obstacle and create a wake behind it. The wake would be “dead water” or “dead air,” and the main body of ideal fluid would flow past it. (The assumption here is that the body is stationary and the fluid moving. This is the situation of a model airplane in a wind tunnel.) Such a flow pattern in an ideal fluid, with a wake of dead fluid, turned out to be compatible with the Euler equations. Furthermore, it could be established that, given such a discontinuous flow (see fig. 2.7), the pressure on the front face of an object would be greater than the pressure of the dead fluid on the rear. The forces did not cancel out and d’Alembert’s paradox was avoided. If the resultant force proved large enough, here was a theory that could, in principle, explain the lift of a wing as well as the resistance to mo­tion, the “drag.” Such was Rayleigh’s idea for explaining the lift on an aircraft wing, and it was taken up by the British Advisory Committee for Aeronau­tics. The results that emerged from the theoretical and experimental study of this model are described in the next chapter.

## Anyone for Tennis?

Lanchester offered no quantitative expression for the dependence of lift on circulation. He presented many of his main results in a verbal or pictorial form. A modern textbook of aerodynamics would give a mathematical for­mula to express the central ideas of Lanchester’s theory of lift. The formula would be

Lift = рГУ,

where p is the density of the air, Г is circulation (that is, the value of the integral described earlier), and V is the speed of the wing relative to the un­disturbed air.

It is a matter of some historical interest that this important formula had been published in 1877 by Lord Rayleigh. Rayleigh did not, however, offer it as a law that might govern the lift on an aircraft wing. He reserved that role for the very different formula he had published in the previous year describ­ing the pressure on a flat plate created by a discontinuous flow. How and why did Rayleigh arrive at the рГУ formula? I want to look into this matter and then come back to Lanchester.

The short paper in which the formula appeared was titled “On the Ir­regular Flight of a Tennis Ball.”15 In it Rayleigh posed the question of why a spinning ball sometimes veers to one side in flight: “It is well known to tennis players that a rapidly rotating ball in moving through the air will often deviate considerably from the vertical plane. There is no difficulty in so projecting a ball against a vertical wall that after rebounding obliquely it shall come back in the air and strike the same wall again” (344).

In posing his question Rayleigh would have been thinking of the old game of royal or real tennis, which was played indoors and involved a net strung between two walls. In this form of tennis the players are permitted to use the walls and hence produce the effect that Rayleigh mentioned.16 Real tennis courts only existed in a few country houses and ancient universities, but there were (and still are) courts in Cambridge. It was Rayleigh’s chief recreation as an undergraduate.17 Rayleigh rejected the idea that the trajectory of the ball deviated from a straight line because the ball “rolled” on a cushion of con­densed air that had formed in front of it. That, he noted, would produce a deflection in the wrong direction. The correct answer, he said, had been given in a qualitative form by the Berlin scientist Gustav Magnus in 1852. Magnus had been interested in artillery shells rather than the more gentlemanly tennis ball.18 The present paper, said Rayleigh, was designed to supplement Magnus’ answer with a mathematical formulation.

To simplify the problem Rayleigh assumed that the ball was stationary and the air flowed uniformly past it. If the ball is not spinning, all the forces, frictional and otherwise, will be in the direction of the air stream. To produce a deviation in the trajectory there must be a lateral force. This force depends on the rotation of the ball, which, by friction between the ball and the air, produces what Rayleigh called “a sort of whirlpool of rotating air” (344). If the whirlpool is combined with the uniform stream, the velocities oppose one another on one side of the ball and augment one another on the other side. This velocity difference produces a pressure difference and hence a side force. This time the force is in the correct direction. “The only weak place in the argument,” said Rayleigh, “is in the last step, in which it is assumed that the pressure is greatest on the side where the velocity is least. The law that a diminished pressure accompanies an increased velocity is only generally true on the assumption that the fluid is frictionless and unacted upon by external forces; whereas, in the present case, friction is the immediate cause of the whirlpool motion” (345). The law relating diminished pressure with increased velocity was Bernoulli’s law, and Rayleigh was rightly sensitive to
the preconditions of its legitimate use. Despite the questionable character of this step in the argument Rayleigh continued to develop the analysis in terms of a circular cylinder “round which a perfect fluid circulates without molecu­lar rotation.” What was meant by “molecular” here? Rayleigh was explicitly working with an ideal or perfect fluid, so the “molecules” would actually be fluid elements. Rayleigh would not have been confusing them with the mol­ecules of the chemist. He was simply postulating an irrotational flow of an incompressible, inviscid fluid with circulation.

Rayleigh proceeded to write down the stream function for the flow he was postulating. He was able to do this by adding together the stream functions for the two components of the flow, namely, the stream functions for a uni­form flow round a cylinder and for a vortex round a point. The formula he used was, in his notation,

r sine + ySlogr,

where r and 0 are the polar coordinates of any point in the fluid and a is the radius of the cylinder. The two symbols a and в are constants proportional to the velocity of the free stream and the circulation, respectively. Examina­tion of Rayleigh’s formula shows that it can be related to the examples I have discussed in previous sections. The first part is the stream function that was discussed in chapter 2 for the flow around a circular cylinder. The second part, involving the log term, was introduced earlier in this chapter as the stream function of a vortex.

Rayleigh used this formula to arrive at an expression for the velocity and hence the pressure at any point on the surface of the cylinder. The lateral component was integrated around the circular cylinder to give the resultant lateral force. The expression that Rayleigh arrived at was

2n

I (P – P0 )a sinede = -2mxp,

0

where p is the pressure at the surface of the cylinder and p0 is the pressure at infinity. The other symbols have the meanings already assigned. The conclu­sion indicated by the final term of this equation was that the lateral force causing the irregular flight of the tennis ball was “proportional both to the velocity of the motion of circulation, and also to the velocity with which the cylinder moves” (346). This is essentially the result on which Lanchester’s analysis depended, and the right-hand side of the above equation is equiva­
lent to the law relating lift to density, circulation, and velocity. The minus sign on the right-hand side of the above equation indicates direction, the circulation is Г = 2пв, while the free-stream velocity is V = a. Rayleigh took the density of the air to be of unit value, which explains why the term p did not appear in his final formula.

Having arrived at the lateral force on the tennis ball, Rayleigh asserted without argument (but citing the authority of Kelvin) that, under the sim­plified conditions of the example, the trajectory of the tennis ball would be along the arc of a circle, the motion having the same direction as the circula­tion.19 Rayleigh’s analysis was taken up and extended by Greenhill in a paper published in 1880 in which the equation of the trajectory was deduced in an explicit way.20 Further, whereas Rayleigh had ignored external forces such as gravity, Greenhill reintroduced this complication and showed that its effect was to give the trajectory the form not of a circle but of a trochoid.21

How seriously did Rayleigh take his analysis of the tennis ball and, by implication, the artillery shell? His aim in the paper, he explained, had been to “solve a problem which has sufficient relation to practice to be of interest, while its mathematical conditions are simple enough to allow of an exact so­lution” (345). These words reveal the element of judgment that entered into the exercise. Relevance was being traded with simplicity. A balance was being struck, but Rayleigh evinced some doubts about the terms of the exchange. His final words in the paper referred back to what he had called the weak links in the argument. The behavior of a real fluid would give rise to a wake behind the ball, and this aspect of the flow finds no recognition in the analysis. Ray­leigh thus ended his paper with a warning: “It must not be forgotten that the motion of an actual fluid would differ materially from that supposed in the preceding calculation in consequence of the unwillingness of stream-lines to close in at the stern of an obstacle, but this circumstance would have more bearing on the force in the direction of motion than on the lateral compo­nent” (346). The informed reader of Rayleigh’s paper would know that there was a mathematical approach that could represent this “unwillingness” of the streamlines to close in behind an obstacle. Such an analysis was provided by the theory of discontinuous flow and free streamlines. Perhaps this would do more justice to the complex reality of the case. Explicitly, then, the reader was given a circulation analysis but, implicitly, was being directed toward the theory of discontinuous flows.22

Lanchester also had an account of the swerving tennis ball in his book on aerodynamics. His account did not invoke the circulation theory but ap­pealed to the discontinuity theory, that is, to the picture of Kirchhoff-Rayleigh

 figure 4.9. Lanchester’s account of why a spinning ball generates lift. Notice the presence of the dead air behind the ball and hence the appeal to the discontinuity theory. From Lanchester 1907, 43.

flow. Lanchester’s discussion of the spinning ball was given in a section of his book titled “Examples Illustrating Effects of Discontinuous Motion.” The diagram he used to illustrate his analysis is given here in figure 4.9. The figure shows a ball moving through a viscous fluid, such as air, where the direction of motion is shown by the arrow A, and the rotation by arrow B. In this case the ball will veer upward. Like Rayleigh, Lanchester drew attention to the in­crease in the speed of the air on one side (here the top) and its diminution on the other side (the bottom). Where Rayleigh merely hinted, Lanchester was explicit that the flow round the ball creates dead fluid behind it. He argued for this concept on experimental grounds. The effect of the spin was then said to modify the shape of the dead air, as is shown in the figure. Lanchester claimed that at the top of the dead-air region, the spin helps to expel the dead air. At the bottom, the spin helps to draw in the dead air and spread it along the lower surface of the ball. This action was said to have the effect of giving the ball an upward thrust.

In retrospect, familiarity with the circulation theory of lift makes it easy to assimilate the case of the spinning tennis ball and the case of the wing and to see the possibilities inherent in treating both of them as involving circulation. It is evident that for Rayleigh and Lanchester the situation was not clear cut and both inclined toward an eclectic account. In broad terms it can be said that Rayleigh used the discontinuity theory to explain wings but the circula­tion theory to explain tennis balls. Lanchester did the opposite and used dis­continuity theory to explain the swerve of tennis balls and circulation theory to explain the lift of wings.

Professor or Practical Man?

Not long after its publication, the first volume of Lanchester’s Treatise be­came the topic of a seminar discussion in Gottingen, and Lanchester was approached for permission to produce a German translation. He received a letter to that effect from Prof. Carl David Runge. At first Lanchester had no idea who Runge was, but inquiries soon revealed that he was an emi­nent applied mathematician. He had done work on methods of numerical approximation used to solve differential equations as well as having discov­ered empirical formulas to predict spectral lines.23 Runge had been brought from the technische Hochschule in Hanover to the university in Gottingen at the behest of Felix Klein. During the session of 1907-8 Runge had joined Prandtl in giving a seminar on hydro – and aeromechanics. They discussed Lanchester’s book, and Runge penned his letter to Lanchester in March 1908. In September, Lanchester visited Runge in Gottingen to discuss the transla­tion, and they became lifelong friends. Runge was impressed by Lanchester’s vigor and humor, which were very different from the demeanor of the aca­demic Englishmen he had met before. Lanchester was a guest of Runge and his wife and amused their young sons by flying model gliders. Lanchester recalled that one evening Prandtl and von Parseval (the airship designer)24 “blew in” for a beer. A convivial time was had, though communication was limited by Lanchester’s lack of German and Prandtl’s lack of English. Runge, by contrast, spoke English with a perfection that amazed Lanchester.25 The German translation of the Treatise was a collaborative effort by Runge and his wife. The first volume was published in 1909.26

Lanchester’s appointment to the Advisory Committee for Aeronautics in 1909 is not difficult to understand in general terms—though exactly how it came about remains unknown. He had published a substantial work on aero­nautics and was an engineer with a proven record of innovation. His knowl­edge of engine design was considerable, and the difficulty of producing light, powerful, and reliable engines remained a major stumbling block to progress in aviation. Lanchester grumbled retrospectively that Glazebrook thought that this was his only area of expertise: “he always treated me as if I was ap­pointed purely for my knowledge of the internal combustion engine.” In the hope of being taken seriously on a broader front, Lanchester had distributed six copies of his treatise Aerial Flight to his fellow committee members. Ray­leigh responded with an appreciative letter and gave Lanchester four volumes of his collected papers in return. Of the rest, said Lanchester, only one ac­knowledged the gift, while the others “ignored it entirely.”27 Despite these slights Lanchester worked hard, both for the main committee and for the two subcommittees on which he served: the Engine Sub-Committee and the Aerodynamics Sub-Committee. His biographer described how Lanchester would attend some forty to fifty meetings a year as well as taking on the re­sponsibility of writing a significant number of papers published as Reports and Memoranda.28

While Lanchester’s appointment to the ACA is intelligible, his position on the committee was anomalous. He was a representative of the world of practical engineering on an administrative and scientific body dominated by influential Cambridge academics. He was not the only engineer or the only nonacademic or, indeed, the only non-Cambridge man on the com­mittee, but (the military representatives aside) the others were more closely integrated into academic life. Joseph Petavel was a professor of engineering at Manchester and Horace Darwin, of the Cambridge Scientific Instrument Company, had close connections with the university. This left O’Gorman and Lanchester as the only authentic examples of what the British called “practi­cal men,” but this did not spare them from the hostility that was directed against the “theoreticians” of the Advisory Committee.

Lanchester’s treatise was published before his appointment to the hated committee, so how was it received in the aeronautical and engineering press? The first volume was reviewed in Aeronautics in February 1908. The review was unsigned but was probably written by the editor, J. H. Ledeboer. The reviewer recommended Lanchester’s book for thorough study by all those who take an interest in aeronautical problems, “since it includes a mass of valuable theory.” The aircraft designer could gain “many a hint” from the book. The reviewer did not, however, identify the cyclic theory of lift among the “mass.”29 The anonymous reviewer in the Aeronautical Journal likewise greeted Lanchester’s first volume as a valuable, but difficult, contribution to aerodynamical theory and also failed to convey to the reader the cyclic char­acter of the flow postulated in Lanchester’s book.30

The same pattern was repeated in the anonymous reviews published in the Engineer, Engineering, and the Times engineering supplement.31 Florid congratulations were offered by the reviewer, but there was a signal failure to convey the actual content of the theory of lift. The words “circulation” or “cy­clic” did not feature in any of the discussions. There were also warning signs suggesting that the reviewers might have opposed the theory had they grasped it. Thus, the reviewer in the Times spoke of “doubtful hydrodynamic analo­gies” (6), while the reviewer in the Engineer noted that “Evidently the author has a deep admiration for the work of Horace Lamb” (617) and went on to say that Lanchester “somewhat naively” described how he had added the hydro­dynamic interpretation to his original ideas in order to give it a more secure basis. The naivety, presumably, lay in Lanchester’s failure to realize that, in certain quarters, invoking Lamb would reduce rather than increase credibil­ity. The reviewer in Engineering also took issue with Lanchester’s diagrams: “The diagram given on page 145 seems to indicate that particles originally in contact with the lower surface of the plane finally reach the upper by describ­ing circular paths round the edge. The true condition of affairs is, we believe, much more nearly represented by a diagram given later in the volume—viz. on page 266” (461). The complaint is revealing. As Lanchester had explained on the page before the diagram favored by the reviewer, at that point in the discussion “the flow is of the Rayleigh-Kirchhoff type” (265). The reviewer was responding to, and endorsing, a diagram that shows a discontinuous flow—precisely the picture that Lanchester was challenging.

Most of the “practical men” who read Lanchester initially extracted from him not the idea of circulation but the neo-Newtonian idea of “sweep.” The realization that circulation played a central role in Lanchester’s thinking came later. For example, Arthur W. Judge in his eclectic Properties of Aerofoils and Aerodynamic Bodies followed Lanchester in giving an account of both circula­tion and sweep, including a reproduction of some of the diagrams showing trailing vortices.32 Judge’s book was published in 1917, rather than being an immediate reaction, so there was more time for Lanchester to have dissemi­nated his ideas.33 Nevertheless, Judge did little to integrate the circulatory theory with empirical data, despite his references to some of the Gottingen publications.34

The second volume of Lanchester’s treatise was reviewed in Aeronautics a year after the first volume, and this time the hand of the editor was sig­naled by the initials J. H. L.35 He expressed “whole-hearted admiration” for the work but soon took the discussion into the vexed domain of the relation between theory and practice. “The present work once again raises the oft – debated question of the advantages in aeronautical research of the theoretical or mathematical and practical methods” (6). In the reviewer’s opinion, “vic­tory” rested with the practical methods. Admittedly, he went on, people such as Bryan, Rayleigh, “and, we may now add, Lanchester” show the path along which research is to proceed, but the fact remained that “the most success­ful machine of all,” that of the Wright brothers, was built without the aid of mathematical formulas. This, then, was, “the main reproach we could make against Mr. Lanchester’s work: it is far too theoretical” (6).

The designer J. D. North had apparently arrived at a similar conclusion. He later recalled that he had not found Lanchester’s book as useful as the more empirical, data-rich volumes of Eiffel.36 This opinion was not universal, but clearly many people saw Lanchester in this way, that is, as a representative of an overly theoretical approach. In February 1914, Herbert Chatley wrote an article in Aeronautics designed to head off the reading of Lanchester’s book as just another example of mathematical “x-chasing.”37 Chatley acknowledged that there was controversy in the technical press about the value of applying mathematics to aeronautical problems. One dimension of the problem, he said, was generational. Older engineers had been trained practically and re­sented the attempts of younger colleagues “to dominate engineering practice with mathematical theorems.” Chatley wanted to do justice to both sides and to get the competing parties to see that the problem did not have a black or white answer. The issue, he said, was always one of degree.

Chatley illustrated his conclusion by an example: “There are three books published in England by men with big reputations dealing with the theory of aviation, viz. Lanchester’s ‘Aerial Flight,’ Bryan’s ‘Stability in Aviation,’ and Greenhill’s ‘Dynamics of Mechanical Flight.’ To those unfamiliar with math­ematics these three books are classed together and, I imagine, are regarded by the majority of practical aeronautical engineers as beyond the limits of practi­cal application” (46). It was a mistake, he said, to treat these books as three of a kind. Lanchester’s book was not obscure and was of the greatest use to designers; Bryan’s book was, indeed, mathematically much more demanding, but it was genuinely engaged with real problems; Greenhill’s book, however, was quite different. It was even more mathematically difficult than Bryan’s but “very little of it has at present any application to practical problems.” As well as making discriminations of this kind, said Chatley, both sides, the prac­tical men and the mathematicians, must recognize the weakness in their own positions. The former must give up reliance on “mechanical instinct,” while the latter must “assimilate more complex conditions” rather than “elaborate hypotheses on too simple foundations” (46).

Chatley was seeking a compromise, but this desire was wholly foreign to C. G. Grey, editor of the Aeroplane. On September 12, 1917, Grey described Lanchester as “several government officials rolled into one, and a mathemati­cian, and an advisory expert, and several other equally exalted things at the same time.”38 Grey took pleasure in baiting Lanchester, who held no academic post, by calling him “Professor.”39 While granting that Lanchester’s books had their “lucid moments,” Grey claimed that, overall, they were “utterly ununderstandable. . . to the ordinary man, and even to some trained math­ematicians.” He called on Lanchester to mend his ways: “No! Mr. Lanchester. Forget your professorship for a while. Be a practical engineer” (158). In an article headed “On a Professor,” Grey described Lanchester’s main publica­tions in unflattering terms: “He has written two ponderous tomes, entitled ‘Aerodynamics’ and ‘Aerodonetics’ respectively, which profess to be math­ematical expositions of the theory of flying. On these volumes his reputation in aeronautics was founded. Not being a mathematician myself I can only ac­cept the statement of others better informed, who say that these expositions are not mathematics in the accepted sense of the term, but some science of symbols invented by Professor Lanchester.”40

Now, a year later, Grey was repeating the old complaints. Notice, said Lanchester, how those making the attacks were always “wily enough to avoid putting them in a form which would permit of their being challenged in the Courts” (51). They always expressed themselves in a vague and general way. They talked of “appalling waste,” “official designers,” a “little clique,” etc. It was a case, said Lanchester, of simply throwing mud in the hope that some­thing would stick. The ground of the attack, as before, was the allegation of official incompetence and favoritism. Even the old BE type, an aircraft cur­rently being replaced, was still under attack. “Its present survival in active service,” said Lanchester, “is talked about as an offence attributable to some interested and evil-disposed persons in authority. The whole tenor of these long abusive articles is based upon the existence of machines, any machines whatever, of official design, when (it is alleged) far better machines of propri­etary design are available” (51).

This was the nerve of the issue: the existence of official designs. The ques­tion with which we are faced, said Lanchester, “is why and to whom is the said design an offence?” (52). Had these machines failed to perform their re­quired duties? Were there any better designs available when the BE2C was put into manufacture? The answer to both questions, said Lanchester, “is clearly in the negative” (52). The BE type had been the backbone of the reconnais­sance service for three years of war. It had to bear the blame for its shortcom­ings as a fighter and for being not quite the equal in speed and climb to the Fokker, but, insisted Lanchester, it had acquitted itself “in a manner which is little short of astonishing” (52).

Lanchester then addressed Grey’s complaints about engine shortages and the mobilization of manufacturing capacity in order to improve aircraft sup­ply. In Lanchester’s opinion, none of Grey’s dramatic remedies could stand up to scrutiny. Grey had no understanding of real conditions, and his ideas about economics were naive. Grey was responding to an inevitable feature of large-scale production but was treating it as if it were a defect that could be swept away. Thus, said Lanchester,

He condemns offhand everything which is not the best. If a new machine is produced to-day which is better in its performance. . . on trial than some­thing in service, then the latter is at once condemned, and the people who tolerate its use are obvious fools or incompetent rogues! It is spoken of as murder still to employ the B. E.2c for reconnaissance, or an 80-h. p. scout as a fighter. Mr. Grey has evidently had no experience of quantity production, otherwise he would know that it is always the case, so long as there is progress, that the latest and the best designs will not be available in sufficient quantity. It cannot in the nature of things be otherwise. (53)

Grey may have been the mouthpiece of the aircraft manufacturers, but in Lanchester’s opinion he betrayed no understanding of the system of manu­facture and development.

Lanchester returned to these themes later in 1917, in the December issue of Flying. The title of Lanchester’s article was “The Foundation Stones,”44 and he set out to explain the scientific and research basis on which airplane de­sign depended. The airplane as we know it today, argued Lanchester, doesn’t merely depend on the experience of the draughtsman, the intuition of the designer, or the cunning of the craftsman but “is the ultimate outcome of scientific research” (354). Theory, both mathematical and otherwise, played a role, while engineering and physical research, carried out in the workshop, the laboratory, and with full-scale aircraft, consolidated the results already explored theoretically. Lanchester’s aim, however, was not simply to draw attention to the diversity of the scientific foundations of aeronautics. The different contributions had to be properly coordinated: “In order to secure continued progress it is vitally necessary that forces of many different kinds should act in concert, and nothing but evil can result from the worker in any particular field trying to encroach on other fields” (354). Lanchester listed the various contributors and delimited the proper scope of their activities. The mathematician may be useful but is “utterly incompetent” when it comes to “specifying or designing an aeroplane” (345). The mathematician’s field is im­portant but narrow. Unfortunately, many mathematicians are men of “child­like simplicity” and fail to understand that there is anything beyond their symbols. Physicists also have a vital role but “only one degree less narrow” than that of the mathematician. “For the physicist to invade territory outside his own ken would certainly not lead to satisfactory results” (354). The engi­neer is responsible for design and construction. “His job is frequently that of a buffer.” He has to listen to both the mathematician and the physicist but he must also be responsive to the requirements of the user, and pay attention to economic considerations and, in times of war, to shortages of materials and the problems of substituting one material for another. The pilot also had a legitimate claim to be heard, as did the military and naval authorities who ordered the aircraft and were responsible for its deployment.

Lanchester had no illusions about a system of this kind. As a realist he knew there would always be clashes of interest. They were inevitable and could not be wished away, but they were not always bad in their effect. In a tilt at the press campaigns he declared: “Now there are many who think that fric­tion in the management of our Air Services as between one department and another, or between different departments and manufacturers, or between the military authorities at the Front and those at home, etc., is essentially a sign of bad management and muddle. No such idea is justified” (354).

Theoretically, if a single “autocrat” were in control, friction might be avoided, but even then the autocrat would have to defer to those with spe­cialized knowledge. In reality the answer lay in the proper division of labor and clear, long-term policies. “The greater part of the difficulties that have arisen in the past have been due to two causes, one a failure to define prop­erly the spheres or fields of activity of the different contributory factors; the other an absence of foresight and far-sighted policy, both in detail and in the gross” (354).

As an example of a failure of foresight, Lanchester pointed to the attacks on mathematicians and physicists and their condemnation as impractical. At times these specialists may have made themselves too prominent, but their contributions are indispensable. The best way to achieve the desired har­mony between the different factions “can only come from each man putting up as stout a case as he can for his particular views,” while being prepared to listen to the views of others. Such negotiations have something of the char­acter of a game of chess. We should not forget that when a mistake is made, its consequences “may often be so remote as to defy immediate analysis,” or that the winning move “may have every appearance at the time it is made of being a blunder” (355).

In Lanchester’s opinion, “indiscriminate Press criticism and stump ora­tory” only served to distract from the serious business of the war (355). Jour­nalists who thought they were omniscient, he went on, merely traded on the ignorance of their fellows. The proper way to proceed was shown by the late E. T. Busk, who acted as a “bridge between the scientific man, the engineer and the pilot” (356). In the period leading up to the war, opinion was divided on the value of the inherent stability of an aircraft. Physicists and mathema­ticians, said Lanchester, strongly supported stability, while many pilots had been against it (as had the Wright brothers). The military authorities didn’t know what to do. Busk resolved this dispute and “proved the value of in­herent stability” (356). Lanchester accepted that since Busk’s work, in 1914, knowledge of how to balance the relative virtues of stability and instability had been deepened. Despite these subsequent advances, the “BE2C machine was the immediate outcome of his work, and it is worthy of note that every Zeppelin brought down in night flying in this country was brought down by a machine of this type. For a long time it was the only machine which could safely be sent up or flown in the dark. Beyond this, for the first twelve months of the war, and more (up to the spring of 1916), nearly two thirds of the total enemy aeroplanes brought down on the Flanders front were brought down by the BE2C” (356).

Lanchester was the only member of the Advisory Committee for Aero­nautics to engage in public with the likes of Grey and Pemberton Billing. He was appalled at their activities and had the civic courage to say so. He was also disturbed by the mounting evidence that Billing was conducting his campaign with the help of inside information provided by a member of the Advisory Committee itself. That member, Lanchester concluded, was the naval rep­resentative Murray Seuter. Lanchester complained in letters to A. J. Balfour and G. A. Steel at the Admiralty that Seuter was a slacker who did not pull his weight and abused his position of trust. Lanchester’s suspicions had been aroused when he had shown Seuter an article he intended to publish in one journal but had then decided to publish elsewhere. Within an hour of showing the document to Seuter, one of Billing’s supporters had rung up the original (that is, the wrong) publisher to raise objections. “I think,” said Lanchester pointedly, “Commodore Seuter should be given a change of air.”45

Although some practical men treated Lanchester as if he were a mere theoretician, in his own understanding he was very much the engineer. He had a sharp sense of the proper relationship of the engineer to the mathema­tician and physicist and of the differences in their perspective. I shall now look at how this sense of difference was reciprocated by mathematicians and physicists themselves. I review, in roughly chronological order, a sequence of critical responses to the circulatory theory of lift, as they were advanced by the high-status, mathematically sophisticated British experts in aerodynam­ics, and I identify all of the main objections. Running against the flow of these objections were some experimental results that might have worked in Lanchester’s favor, but, strangely, these had little impact. Why this was so is part of the problem to be addressed.

## Foppl’s Vorlesungen

The influential vision of the turn-of-the-century Cambridge school of math­ematical physics, as Lamb, Love, and others presented it, stood in contrast to the German idea of technical mechanics. This body of work came out of the great system of German technical colleges or technische Hochschulen, such as that at Charlottenburg, or in Munich where Prandtl had studied, or Hanover where both Prandtl and Runge had taught before their call to Gottingen. A representative example of this style of work is provided by August Foppl’s influential lectures on technical mechanics. His multivolume and vastly pop­ular Vorlesungen uber technische Mechanik was published in many editions around the turn of the century. Foppl originally worked in industry and had spent a number of years teaching in a trade school. He later rose to become the professor of theoretical mechanics at the Munich Hochschule and the di­rector of their materials laboratory.47 A versatile mathematician, Foppl had written the first book introducing Maxwell’s work on electromagnetism into Germany. The book was later revised and coauthored with the experimental physicist Max Abraham, and it is known that one student who was influenced by it was the young Einstein.48 Foppl had been Prandtl’s teacher at Munich and had supervised his doctoral research on the buckling of loaded beams.49 In 1909 Prandtl married Foppl’s eldest daughter, Gertrud. In Prandtl’s biog­raphy, written by his own daughter, there is evidence of a certain tension between Prandtl and his father-in-law, occasioned by the older man’s au­thoritarian attitudes, but there was no lack of scientific respect.50 Ludwig Foppl, who, along with Abraham and von Karman, had been in the audience at Cambridge when Lamb spoke, was one of August Foppl’s two sons. The other son, Otto Foppl, worked with Prandtl on wind-tunnel experiments in Gottingen. Some of Otto Foppl’s work is discussed in a later chapter; for the moment, however, the concern is with August Foppl (fig. 5.5).

What did the many readers of Foppl’s published lectures on technische Mechanik learn about the status of their field as they imbibed its carefully graded and expertly presented content? First, they learned that mathematics was a means to an end, rather than an end in itself. In the introduction to the first volume, Foppl wrote that mechanics makes extensive use of mathematics, but as an auxiliary. Mathematical techniques, he said, were simply the clothing in which the body of knowledge was garbed. The point was reiterated at the beginning of the more mathematically demanding third volume, but this time with more stress on just how important, on occasion, these aids could be:

Analytische Entwicklungen betrachte ich immer nur als ein Mittel zur Er – kenntnis des inneren Zusammenhangs der Thatsachen. Wer auf sie verzichten wollte, wurde das scharfste und zuverlassigste Werkzeug zur Verarbeitung der Beobachtungsthatsachen aus der Hand geben. (1900a, viii)

I only consider analytical processes as a means for understanding the intimate interconnections of the facts. Those who want to renounce them are letting go of the sharpest and most dependable tools for working with the facts of observation.

Mathematics provided what Foppl called Hilfsmittel and Werkzeuge, “aids” and “instruments.” Foppl’s language is important here. There had been in-

 figure 5.5. August Foppl (1854-1924) was a versatile mathematician and a professor at the technische Hochschule in Munich. He was the author of an extremely influential textbook on technical mechanics that was based on his Munich lectures. He was also the father-in-law of Ludwig Prandtl. Photograph from Baseler et al. 1924.

tense, not to say wearisome, debate in German academic circles over whether mathematics was to be seen as a Hilfswissenschaft or as a Grundwissenschaft with respect to technology.51 Was it an auxiliary to, or a foundation of, tech­nology? The debate was really a coded argument over the status of math­ematicians in the technical college system and their role in the education of engineers. Foppl was signaling that mathematicians had to earn their living by making themselves useful to engineers. The function of mathematics was to further technology and engineering.

The first chapter of volume 1 of the Vorlesungen was devoted to the ori­gin and goals of mechanics. Foppl acknowledged that mechanics was part of physics and, like all the sciences, was grounded in experience. To grasp experience, he argued, it was always necessary to work with simplified, easily imagined “pictures” (Bilder) of reality. The ideas of a point particle and a rigid body were two such pictures. Both were valuable and had their appropriate range of application, but they must not be mistaken for physical realities.52 Foppl also drew a distinction between the Naturforscher and the Techniker— the natural scientist and the engineer. His book was for the latter, not the former, and dealt with a mode of knowledge having special characteristics that differentiated it from natural science in general.

Bei der technischen Mechanik tritt als bestimmender Beweggrund fur ihre Fassung zu der Absicht einer Erforschung der Wirklichkeit. . . noch die an – dere Absicht, ihre Lehren nutzbringend in der Technik zu verwerthen. (11)

In the case of technical mechanics there is a definite motive for its approach over and above the intention to investigate reality. . . and that further inten­tion is that its theories be usefully applicable in technology.

Foppl was just the sort of utilitarian, applied mathematician from whom Lamb and Love had distanced themselves. Indeed, Lamb’s typology might have been expressly contrived to ensure that the Cambridge school did not get caught in the cross-fire between the champions of mathematics as Hilfswis- senschaft and as Grundwissenschaft. Be this as it may, Foppl certainly didn’t have the Cambridge tone. His technical mechanics was not natural philoso­phy. Furthermore, Foppl differentiated technical mechanics from mechan­ics in general because there are many cases when the general doctrines of mechanics do not, or do not yet, provide rigorous answers to the questions that have to be confronted by the engineer. Natural scientists and engineers, he said, stand in a wholly different relationship to these cases: “solchen Fallen steht aber der Naturforscher anders gegenuber als der Techniker” (11). The engineer must produce an answer and must forge concepts to deal with the problem. The natural scientist can wait for inspiration or more information; the engineer cannot:

Der Techniker dagegen steht unter dem Zwange der Nothwendigkeit; er muss ohne Zogern handeln, wenn ihm irgend eine Erscheinung hemmend oder fordernd in den Weg tritt, und er muss sich daher unbedingt auf irgend eine Art, so gut es eben gehen will, eine theoretische Auffassung davon zurechtle – gen. (11-12)

The engineer, by contrast, is subject to the force of necessity. He must, with­out delay, deal with the matter when some phenomenon interferes and inter­poses itself in his path. He must, in some way or other, arrive at a theoretical understanding of it as best he can.

The demands of this enforced creativity may generate concepts that do not meet the logical demands of existing mechanics. Here, said Foppl, was the deep reason for separating out technical mechanics as a special branch of knowledge—“diese Absonderung der technischen Mechanik als eines besonderen Zweiges der Wissenschaft” (11). Its practitioners must have the freedom to develop concepts of their own, and these might be distinct from those acceptable in the more reflective and leisurely branches of knowledge. For example, the application of hydrodynamic theory to turbines developed by Prasil and H. Lorenz, depended on certain artifices or tricks (Kunstgriffe) involving the idea of “forced accelerations.”53 The approach had been con­troversial, but Foppl defended it. He went on to say that in such cases subse­quent developments in science might permit a reconciliation. The anomalous concepts, special to engineering and technical mechanics, might be absorbed back into the main body of knowledge. But this was an open question, some­thing for the unspecified future rather than the urgent present. His main concern was to emphasize the restless force running through the scientific life of modern technology, which was, he said, like the life force in a tree that continually generated new branches.

In the peroration rounding off the introductory chapter, however, Foppl suddenly changed the metaphor. The life force of a tree was replaced by an­other kind of force. Knowledge was power, said Foppl, the power of a mod­ern technological state. The bucolic image was replaced by a military one. Those who first possessed the right theory (“die richtige Theorie”) might be able to intervene in nature at will. In this way, said Foppl, science, put at the disposal of humanity and its peoples, is the most powerful of all weapons— “und darum ist die Wissenschaft die gewaltigste Waffe, die Menschen und Volkern zu Gebote steht” (12).

What did Foppl understand by the “right” theory? His normative stan­dards were predictably active and pragmatic. Like many in his position in the technische Hochschulen, Foppl was fighting a war on two fronts. On one side were “humanistic” critics. These were usually outside the technical col­lege system, or only passing through it on their way to posts in universities. In as far as they wanted mathematics at all, they wanted it “pure.” On the other side were critics, often from within the technical colleges, who placed all the emphasis on practicality and were suspicious of any form of higher mathematics.54 These were the counterpart of the “practical men” in Britain who were so hostile to the work of the Advisory Committee for Aeronautics. In responding to the local, German, variant, Foppl dismissed such people as mere Praktiker.55 He had no time for them or their slogans about the conflict between theory and practice—“dem Gegensatze zwischen Theorie und Praxis” (3:vii.). For Foppl there was no such conflict:

Diese Behauptung lasse ich aber auf dem Gebiete der technischen Mechanik durchaus nicht gelten; hier kann nur von einem Gegensatze zwischen falscher oder unvollstandiger Theorie und der richtigen Theorie die Rede sein. Die richtige Theorie ist immer in Ubereinstimmung mit der Praxis. (3:vii)

I do not admit this claim as having any validity in the realm of technical me­chanics. Here one can only speak of the conflict between false or incomplete theories and the right theory. The right theory is always in agreement with practice.

Did Foppl mean that a theory was practical because it was right, or that it counted as right because it worked in practice? Taken in isolation, his wording was ambiguous. If, however, we recall Foppl’s insistence on the overpower­ing, practical necessities that dominate the life of the engineer—the “Zwang der Nothwendigkeit”—then the formal ambiguity can be resolved. In Foppl’s world, practice was the effective criterion not of an abstract and future truth but of acceptability and viability for the pressing moment. In the colleges of Cambridge, if Lamb is to be believed, the pursuit of truth had an aesthetic character. In the colleges of technology a theory was counted as right if, and only if, it worked.

Others in the field of applied mathematics in Germany may have made the case in different words, but Foppl’s general orientation toward engi­neering represented a widely held view. For example, in 1921 Richard von Mises started a new journal for applied mathematics—the Zeitschrift fur angewandte Mathematik und Mechanik. Prandtl and von Mises had been in correspondence after the war on new institutional arrangements for encour­aging applied mathematics. Prandtl mentioned that he and von Karman had been discussing the founding of a society to promote technical mechanics, “eine Vereinigung fur technische Mechanik,” and these exchanges were part of the process that culminated in the journal.56 On the first page of the new publication von Mises set out his conception of the task and goals of the dis­cipline and the role of the journal. He would have been conscious of stepping into a long-standing discussion about the role of mathematics in the German academic world but he had no desire to equivocate. He insisted that the core of the journal would be devoted to mechanics whose cultivation, he said, to­day lies almost exclusively in the hands of engineers, “deren Pflege heute fast ausschliefilich in den Handen der Ingenieure ruht.”57

Mathematics, said von Mises, covered a wide spectrum of activities so that the partition between pure and applied mathematics was a relative one, lo­cated differently by different practitioners. Each would count what was (so to speak) on their “left” as pure and what was on their “right” as applied. But there was not only this dimension to consider: the very content of mathemat­ics itself changed with time as new areas (for example, the concept of prob­ability) were brought within the scope of quantitative analysis. We must, said von Mises, accept this “two-fold relativity” in the identity of applied math­ematics. To overcome the definitional problems this created, he concluded that a practical, rather than a theoretical, specification of the field was called for. Applied mathematics and mechanics were to be defined as what was done, at that time, by scientifically oriented engineers. Thus,

Angesichts dieses Tatbestandes zweifacher Relativitat der Begriffsabgrenzung mussen wir nun eine praktische Erklarung dafur suchen, was wir hier im Fol – genden unter “Angewandter Mathematik” verstehen wollen. Es ist selbstver – standlich, dafi wir uns auf den Boden der Gegenwart stellen, und es sei hinzu – gefugt: auf den Standpunkt des wissenschaftlich arbeitenden Ingenieurs. (3)

Given the facts of this twofold relativity of the conceptual boundary, we must now seek for a practical explanation of what, in the following, we want to un­derstand by “applied mathematics.” It will be obvious that we take our stand on the basis of the present and, let it be added, on the standpoint of the scien­tific work of the engineer.

Given this mixture of divergent tendencies and common ground, it is not surprising that the members of the respective traditions did not them­selves always have a clear awareness of the relations between them. This was epitomized in an exchange of letters that took place between G. I. Taylor and Ludwig Prandtl a number of years after the events described here. By the time the letters were written, in the 1930s, the magnitude of Prandtl’s contribu­tions had become known and widely admired. Taylor had written to say that he thought Prandtl deserved the Nobel Prize in physics. Prandtl’s response, in a letter of November 30, 1935, was not only becomingly modest but was also culturally revealing.61 He said that what he had done would not count as physics in Germany. Rather, it was a contribution to Mechanik.

Nach der in Deutschland ublichen Einteilung der Wissenschaften wenigstens wird die Mechanik heutzutage nicht mehr als ein Teil der Physik betrachtet, sondern steht als selbstandiges Gebiet zwischen der Mathematik und den Ingenieurwissenschaften.

At least according to the division of the sciences that is usual in Germany today, mechanics is no longer considered to be part of physics. Rather, it stands as an independent area between mathematics and the engineering sciences.

While Taylor now assimilated Prandtl’s work to physics, the Germans saw it as something distinct from physics and as standing between mathematics and engineering.62 The different stance toward engineering and its demands perhaps sheds light on why the circulation theory was actively resisted in Britain but accepted and developed in Germany. Before taking this argument further, however, I look at what Lanchester himself said to explain the rough ride given to his work. Lanchester’s account will give me an opportunity to look at another variable whose explanatory potential needs to be assessed, namely, the personalities of the main actors.

## Betz on Pressure Distributions

Knowing the predicted pressure distribution along a specified aerofoil opens up the possibility of subjecting the circulation theory of lift to a demand­ing empirical test. Does the predicted distribution correspond to the real

 figure 6.io. Ludwig Prandtl (left) and Albert Betz (right) standing in front of one of the Gottingen wind tunnels. In the early years of aerodynamics, Betz was second only to Prandtl in the scope of his theoretical and experimental investigations. (By permission of Zentrales Archiv, Deutsche Zentrum fur Luft und Raumfahrt)

distribution in as far as it can be measured? In 1915, two years after Blumen – thal’s theoretical analysis, a detailed experimental study was published in the Zeitschrift fur Flugtechnik that was designed to answer this question. The pa­per was by Albert Betz, Prandtl’s close collaborator in Gottingen (fig. 6.10). It was called “Untersuchung einer Schukowskyschen Tragflache” (An investiga­tion of a Joukowsky wing).60

Betz used one of Blumenthal’s profiles and worked with a model wing that had a span of 50 cm and a chord of 20 cm. It had a curvature (f/l) of 1/10 and a thickness ration (8/l) of 1/20 and so corresponded exactly to the second of the four profiles described by Blumenthal (that is, the one shown in fig. 6.9). Betz’s aim was to use wind-tunnel data to test Blumenthal’s predicted lift and pressure distribution.

The model wing-section was manufactured from metal plate in the form of an airtight, hollow body and made to conform as precisely as possible to the theoretical, Joukowsky profile. Following Fuhrmann’s work on model airships, the wing was fitted with bore holes and the hollow interior was connected by a thin pipe leading from the wingtip to a manometer. This enabled pressure measurements to be taken at a number of points on the surface along the chord of the wing. Measurements were taken with one hole at a time exposed while the other holes were smoothly plugged. The line of bore holes was not positioned at the center of the span but was displaced a few centimeters to one side. This was to avoid interference to the flow of air over the holes from the strut that had to be attached to the wing in order to hold it rigidly in place in the wind tunnel. As well as the pressure, Betz also needed to know the overall lift and drag of the wing. For this the wing had to be suspended on a balance so that force measurements could be made. Dur­ing this phase of the experiment, the pipes leading to the manometer were disconnected and all the bore holes plugged.

It was necessary to make sure that the experimental arrangement pro­vided a good approximation to the infinite wing presupposed in the math­ematics. Joukowsky had simply made his wing section run from the top to the bottom of the shallow Moscow wind tunnel. This is the basis of all attempts to realize a two-dimensional flow, but Betz put a lot of effort into refining the technique. His aim was to make the test section as free as possible from disturbing effects produced by the walls of the tunnel and the join between the walls and the ends of the wing. An elaborate system of auxiliary sidewalls, gaskets, and seals was designed and tested to ensure a uniform flow across the experimental cross section of the wing. Once he had an acceptable approxi­mation to two-dimensional flow, Betz’s apparatus gave him two sets of data: (1) direct measurements of lift and drag and (2i) pressure measurements dis­tributed over the surface of the wing.

The direct measurements of lift and drag showed the familiar pattern, which partially conformed to, and partially violated, theoretical expectations. The observed lift increased in the predicted, linear fashion with angle of in­cidence, but only from about -9° up to about +10°, at which point the wing stalled. Even over this range the predicted lift was significantly higher than the observed lift. In fact, the observed lift was only about 75 percent of the predicted value. And, of course, there was an observed drag when, theoreti­cally, it should have been zero. There was also another general feature of the flow that Betz observed. Theoretically each angle of incidence should cor­respond to one, and only one, value of the lift. Betz found that if he took a sequence of readings in which the angle of incidence was increased in a stepwise fashion, and another sequence in which it was decreased, a given angle might correspond to one value of the lift in one sequence and another value in the other. There were two values of the lift corresponding to each of these angles, not one. This effect was particularly noticeable above the stalling angle. Thus, at +15°, the coefficient of lift had the values of 0.68 and 0.55. Such a phenomenon fell wholly outside the scope of the theory.

The most important results, however, were those relating to the distri­bution of pressure. Here Betz’s graphs of the manometer readings showed a definite similarity to the theoretical graphs prepared by Blumenthal and his colleagues at Aachen. Betz’s results are shown in figure 6.11. Note that

 figure 6.ii. Betzs pressure graphs for a Joukowsky aerofoil at 6°. Theoretical predictions are indicated by the broken line and experimental results by the solid line. There is a similarity between prediction and observation, but Betz wanted to improve the fit. From Betz 1915, 176. (By permission of Oldenbourg Wis – senschaftsverlag GmbH Munchen)

Betz used a convention different than Blumenthal’s when drawing his dia­grams, and the data for the upper side of the wing are now placed below the base line. Significantly, the general shape of the graph derived from theory and that of the graph derived from experiment were the same. Betz, how­ever, pressed the comparison into greater detail. He was interested in getting information about the residual deviations between theory and reality, “die Abweichungen der Theorie von der Wirklichkeit” (173). This was the stated purpose of the experiment. He therefore drew attention to where the empiri­cal distribution differed from the theoretical distribution. The areas under the empirical graphs were clearly not the same as those under the theoretical graphs. These areas were proportional to the lift, and the theoretical area was significantly greater than the observed area. This was consistent with the fact, mentioned earlier, that the directly measured lift was less than the theoreti­cally predicted lift. What was the cause of the difference, and what should be done in response to it?

Like Fuhrmann, Betz located the source of the difference between the the­oretical and empirical flow in the tail region. Theoretically, if the Kutta condi­tion is satisfied and the wing profile is that of a pure Joukowsky contour, and if the air acts like an ideal fluid, then the flow along the upper surface will meet the flow along the lower surface in a smooth way at the trailing edge. In reality, however, the air did not behave in this way at the trailing edge, so Betz made a conjecture. It was a characteristically Gottingen conjecture (see fig. 6.12).

Betz suggested that, although the flow of air along the lower surface runs smoothly along the common tangent, the flow along the upper surface does not. Rather, it detaches from the upper surface before reaching the trailing edge, and this leaves a gap between the two flows. The intervening space be­tween the flows, said Betz, constitutes a turbulent wake filled with “Karman vortices” (177).

Betz argued that the effect of this separation is twofold. First, it disrupts the pressure relations in the vicinity of the trailing edge and disturbs the equi­librium between the forward-pointing and backward-pointing components of the pressure distribution. Since it is this equilibrium that generates the zero drag of an (ideally) efficient aerofoil, the disturbance must be a contributory cause of the observed drag. Second, the vortices in the wake draw off energy, and this has the effect of lowering the circulation around the aerofoil and hence diminishes the lift. Betz conceded that it was difficult to analyze these processes rigorously but suggested a simple (and intriguing) way to model the situation using the resources of inviscid theory. He proposed rejecting the Kutta condition and relocating the stagnation point. Whereas Kutta had used the position of the stagnation point to fix the amount of circulation, Betz reversed the process. He used the amount of circulation to fix the stag­nation point.

Betz began with the value for the lift at a given angle of attack that he had found in his experiment. He then inserted the value into the basic Kutta-

 figure 6.12. Wake formation near trailing edge was cited to explain the difference between predicted and observed results. To correct for this error in prediction, Betz abandoned the Kutta condition that the rear stagnation point should be at the trailing edge. From Betz 1915, 177. (By permission of Oldenbourg Wissenschaftsverlag GmbH Munchen)

Joukowsky theorem L = p V Г. Because he knew the density and velocity of air in the wind tunnel, the formula allowed him to deduce the value of the circulation. He then used this empirical value of the circulation to tell him where the rear stagnation point must be relocated according to the theory of inviscid flow. (Betz was working with a Joukowsky profile, so this point could be calculated by transforming the flow around a circular cylinder.) Given that the empirical value of the circulation is lower than the theoretical value, the stagnation point is on the top surface of the aerofoil, rather than precisely at the trailing edge. It was then possible to replot the theoretical speed and pres­sure distributions and compare them afresh with the empirical curves. The question was whether the revised location of the stagnation point brought the empirical and theoretical graphs onto closer accord. The method automati­cally equalizes the areas under the empirical and theoretical curves but the question remains: do the distributions agree? Betz’s revised graph, superim­posed on the empirical curve, is shown in figure 6.13. The improvement is clear. The graphs are now almost identical. Betz pronounced the agreement to be “extraordinarily good.” As he put it,

Sehen wir von der nachsten Umgebung der Hinterkante ab, fur die ja Voraus – setzungen der theoretischen Stroming vollstandig andere sind wie in Wirk – lichkeit, so mussen wir die Ubereinstimmung der beiden Kurven aufieror – dentlich gut bezeichnen. (177)

If we disregard the vicinity of the rear edge, where indeed the presuppositions of the theoretical flow wholly differ from reality, then we would have to char­acterize the agreement of the two curves as extraordinarily good.

Betz had thus brought theory and experiment into closer accord. His proce­dure involved interweaving them in an interesting way, and the methodologi­cal principles implicit in the process are worth looking at with some care.

The Kutta condition had become established as one of the central as­sumptions of the circulation theory. Could Betz really afford to modify its role in this way? The overriding advantage of the Kutta condition was that it avoided the need for the ideal fluid to move around the trailing edge at an in­finite, and physically impossible, speed. How did Betz justify the reintroduc­tion of infinite speeds given that the need to avoid them was routinely cited whenever the Kutta condition was invoked? The point was addressed explic­itly in the paper. The argument was as follows: Since perfect fluid theory is known to be unrealistic from the outset, said Betz, one more piece of unreal­ity hardly matters. The entire approach is an artifice, so this should not be disturbing. The infinite speeds are just the way that the complicated, physical

 figure 6.13. Betzs revised pressure graphs after modification of the Kutta condition. The observed and predicted curves are now closer. From Betz 1915, 178. (By permission of Oldenbourg Wissenschafts – verlag GmbH Munchen)

processes at the trailing edge receive some manner of recognition within the terms of the theory. Their presence in the analysis simply indicates that fur­ther assumptions need to be introduced to mediate between reality and the idealized picture. Outside the wake, the flow can be reasonably modeled by ideal-fluid theory, and the presence of the wake can be taken into account by lowering the value of the circulation. This approach can be seen as the first step toward a better account of the phenomenon. As Betz put it,

Dafi dabei ein Umstromen der scharfen Hinterkante stattfinden mufite, was praktisch unmoglich ist, braucht uns nicht zu storen, da ja die Stromungen. . . bis zu dem gemeinsamen Ablosungspunkt nur ein theoretischer Ersatz sind fur die in Wirklichkeit vorhandenene Wirbelbewegung. (177)

That this would have to result in a physically impossible flow round the sharp trailing edge need not disturb us. This is because the flow. . . up to the com­mon point of separation is only a theoretical substitute for the vortex motions that exist in reality.

## Postwar Contact with Gottingen

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

 figure 8.3. Hermann Glauert (1892-1934). Glauert, an Englishman of German extraction, was a Cam­bridge mathematician and fellow of Trinity. He worked at the Royal Aircraft Factory in Farnborough during the Great War and visited Prandtl in Gottingen soon after the war’s end. He became an advocate of the circulation theory of lift and Prandtl’s theory of the finite wing. (By permission of the Royal Society of London)

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

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

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

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

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

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

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

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