Category The Enigma of. the Aerofoil

Kutta ended with some ideas about extending his mathematical methods to a variety of different wing profiles with rounded leading edges and with flaps attached to the trailing edge. He mentioned the need to develop a more gen­eral form of the Schwarz-Christoffel theorem and indicated the demanding amount of computational effort that would be involved, but Kutta did not feel that the limits of his approach had been reached and hinted at their fur­ther application to biplanes.

Kutta’s next paper, in 1911, utilized the same mathematical techniques as those adopted in 1910 but it dealt with more complicated cases.33 The analysis was generalized in two ways. First, Kutta showed how to apply his conformal transformations to an aerofoil whose cross section was composed of not one but two circular arcs in the form of a crescent or sickle shape. Such a sickle­shaped profile was used in the successful Antoinette monoplane, and von Mises has suggested that its practical use was prompted by Kutta’s analysis.34 Second, Kutta generalized the approach in order to describe the flow over a number of wings that could be arranged to make a biplane or a triplane or even a multiwing arrangement in the form of a “Venetian blind.” Again, all of these forms had actually been used, or experimented with, by those who tried to build flying machines.

Having made these two outstanding contributions to aerodynamics, Kutta fell silent. He never published anything again. The reason for the silence is unknown.

The wartime activities of the Gottingen group, and their colleagues in the technische Hochschulen, proceeded on a much broader front than I have so far described.92 On the theoretical side, building on Betz’s early papers, there were studies by Betz, Munk, and Prandtl on the aerodynamics of biplanes and triplanes. Using the apparatus of the Biot-Savart law they produced some general theorems that helped to guide the aircraft designer through the maze of possible multiplane configurations. Betz had proven that, for an unstag­gered biplane, the induced drag effects of the wings on one another would be equal. For a staggered arrangement it was now shown that the sum of the mutually induced drags was constant and independent of stagger, provided that the lifts, and their distribution, were not changed. (This condition could be satisfied by changing the angle of attack.) In general, it was shown that the best biplane configuration was one with wings of equal length, with the up­per wing ahead of the lower. The biplane work also confirmed the important finding that elliptical lift distributions provided a good approximation for wings with non-elliptical planforms.93

On the empirical side, the war effort called for wind-channel studies of the drag generated by different aircraft components such as undercarriages and machine-gun mountings, engine-cooling systems, and the ubiquitous struts and bracing wires of the period.94 Work was also done on the lift and drag of the fuselage, the interaction between the fuselage and the wing, the effect of dividing the wing, the forces on fins and rudders, and the empirical properties of the triplane configuration.95 Experiments were done to test the resistance of the nose shapes required by different engine types, for example, rotary as compared with in-line engines, and attempts were made to add rotating propellers to the wind-channel models to achieve realism.96 Some wind-channel tests were also done on models of complete aircraft.97 Munk and Cario continued the studies initiated by Foppl of the downwash behind a wing. Whereas Foppl had worked with the overall force exerted by the down – wash on the elevator, Munk and Cario studied the downwash in much more empirical detail, using fine silk threads to trace the local variations. They un­covered significant complexities in the flow and made clear the need for a more extended program of work.98

Numerous studies were carried out to measure the lift and resistance of individual wing profiles. These were mainly overseen by Munk and his col­laborator Erich Huckel.99 Significant efforts were made to ensure that the re­sults were intelligible to those who might use them in practice.100 An attempt was also made to introduce order into the vast amount of data that had accu­mulated for different aerofoils, though the classification remained largely at the empirical level.101 One trend was toward an interest in thicker rather than thinner aerofoils, something that surprised the British when they examined captured German aircraft.102 An example of these thick aerofoils was the Got­tingen 298 used on the famous Fokker triplane. The use of the 298 profile by the designer Anthony Fokker does not, however, appear to have been a con­sequence of Prandtl’s recommendation or scientific knowledge of its good lift and drag characteristics (characteristics that, a priori and wrongly, the British designers doubted). In fact, the aerofoil was introduced into the Fokker pro­duction line by their chief engineer Reinhold Platz on the basis of trial-and – error knowledge. Later, and unknown to the people at Fokker, it was tested in Gottingen, where it was given its designation.103

The Fokker episode indicates that there was a continuing gap between the “practical men” of Germany and those self-consciously developing science- based procedures and working in academic and government institutions. The alienation of the practical men was not purely a British phenomenon, though it seems to have been less acute as a problem for German aviation than for British. Evidence in the technical reports indicates that the members of the Gottingen school were themselves aware of this gap and found it frustrating. Max Munk addressed the issue directly in a brief report of October 15, 1917, titled “Spannweite und Luftwiderstand” (Span and air resistance).104 Refer­ring to the practical conversion formulas linking wings of different aspect ratio, Munk complained:

Die kurzlich von Betz veroffentlichen Prandtlschen Flugelformeln werden wohl, da sie auf theoretischen Grundlagen beruhen, in der Praxis nicht so freundlich aufgenommen werden, wie sie verdienen. Das ist sehr schade, denn die Formeln enthalten mehr und leisten Besseres als der Praktiker geneigt ist, ihnen zuzutrauen. (199)

The formulas of Prandtl’s wing theory that Betz has recently published will probably not be welcomed in the realm of practice as much as they deserve because they rest on theoretical grounds. This is a great shame because the formulas offer more and give better service than the practical man is inclined to believe.

Munk went on to give an explanation of the significance of the formulas for the aircraft designer and some simple, general rules for the rapid calculation of the induced resistance and angle of incidence. Despite this evidence of skepticism in certain quarters, there was no shortage of contract work to be done for individual aircraft firms during the war years. This is attested by the frequency with which such names as AEG, Aviatik, Rumpler, Siemens and Schuckert, and Zeppelin were mentioned in the technical reports. Despite the problems of communication between the representatives of theory and practice, Prandtl’s institute had achieved a central position in what would now be called the military-industrial complex of Wilhelmine Germany. If this development brought frustrations as well as the advantages of government support, it is clear that striking progress had been made in aerodynamics, both empirically and theoretically.

One of the more unusual aerofoils whose properties were reported on by Munk and his colleagues was designated as profile 95. Visually it stood out from the usual run of aerofoil shapes (see fig. 7.16). The aerofoil looked like the back of a cat when the animal stretched, and it was duly given the nick­name Katzenbuckelflache. The Gottingen tests showed that the performance characteristics of the “cat’s-back” profile 95 were notably poor. It was tested by Max Munk and Carl Pohlhausen in the course of a run of work on nearly one hundred aerofoils. The results were listed together in the Technische Ber – ichte of August 1917 and showed that the maximum-lift coefficient for each wing in this sequence was typically in the region of 130 or 140. The maxi­mum lift coefficient for profile 95, by contrast, was given as 95.2. Again, the maximum lift-to-drag ratio was typically 14 or 15, while the ratio for profile 95 was 10.8.105 The designer of the cat’s-back wing was the celebrated physicist Albert Einstein.106 In retrospect Einstein felt that his excursion into aerody­namics had been irresponsible—he used the word Leichtsinn. From 1915 to

1917, Einstein had been a consultant to two aircraft firms, LVG and Merkur, and an aircraft had been equipped with the Einstein wing. The test pilot for LVG, Paul Ehrhardt, barely managed to get the machine off the ground and gave his professional opinion on the wing by saying that the airplane flew like a pregnant duck.107 The Gottingen tests made the same point in more scientific terminology.

No account remains of how Einstein actually designed the wing, but some insight into his thought processes may be gained from an article he published in 1916 in Die Naturwissenschaften. Here he set out to explain, in elemen­tary terms, the basic principles of lift.108 How does a wing support an air­craft and why can birds glide through the air? Einstein declared, “Uber diese Frage herrscht vielfach Unklarheit; ja ich mufi sogar gestehen, dafi ich ihrer einfachsten Beantwortung auch in der Fachliteratur nirgends begegnet bin” (400) (There is a lot of obscurity surrounding these questions. Indeed, I must confess that I have never encountered a simple answer to them even in the specialist literature). This is a striking claim, given that Einstein was writing a number of years after the publications of Kutta, Joukowsky, and Prandtl.

Einstein drew an analogy between the flow of fluid through a pipe of vari­able cross section and its flow around a wing. As fluid passes along a pipe that gets narrower, the fluid speeds up. By Bernoulli’s law the pressure will be lower in the fast, narrow section than in the broader section. Einstein then invited the reader to consider a body of incompressible fluid with no signifi­cant viscosity (that is, a perfect fluid) which flowed horizontally but where the flow was divided by a thin, rigid, dividing wall. The wall was aligned with the flow except that it also had a curved section where the wall bulged up­ward. (See fig. 7.17, which is taken from Einstein’s paper.) The curved section looks a bit like a cat’s back and would appear to have the same shape as the underside of the Einstein wing tested at Gottingen. Einstein argued that the fluid above the dividing wall will behave like the fluid in a pipe when it en­counters a narrowing of the pipe and will speed up and exert a diminished pressure on the wall. The fluid below the wall will behave like the fluid in a pipe when it encounters a widening out of the pipe so it will slow down and increase the pressure on the wall. The fluid pressure pushing upward on the curved section of the dividing wall will thus be in excess of the pressure push­ing downward, so there will be a resultant force upward All that is necessary now is to imagine that most of the dividing wall has been removed, leaving behind just the curved section. This procedure, argued Einstein, will retain the features of the flow that generate the pressure difference and hence will represent a wing with lift.

Einstein’s argument rested on the assumption that the removal of all but

the curved portion of the dividing wall would leave the flow unchanged at the leading and trailing edge of the remaining arc. He appeared to take this as ob­vious: “Um diese Kraft zu erzeugen, braucht offenbar nur ein so grofies Stuck der Wand realisiert zu werden, als zur Erzeugung der wirksamen Ausbiegung der Flussigkeitsstromung erforderlich ist” (510) (To generate this force it is obviously only necessary for part of the wall to be real. It need only be suf­ficiently large to produce the effective curvature of the flow). It is puzzling that Einstein made no mention of circulation. Was he aware of the circulation theory of lift? This remains unclear, but given that he assumed the air to be a perfect fluid, it makes it all the more important to ask how he proposed to circumvent d’Alembert’s paradox. How did Einstein expect to get a lift force rather than a zero resultant?

Although Einstein wrote in a dismissive way about the aerodynamic liter­ature, he had, in effect, taken the discussion back to where Kutta started it in 1902. When, following Einstein’s instructions, all of the wall dividing the flow is removed except for the curved piece, what is left is essentially Kutta’s arc at a zero angle of incidence. Einstein did not specify that the curve he discussed in his article was precisely the arc of a circle, but his argument was offered as a general one. If it were right it would apply to Kutta’s arc. But it does not. The arc is a counterexample to what seems to be Einstein’s argument, that is, to any argument that depends on ideal fluids but does not make provision for circulation. In order to generate lift, and to generate the requisite speed dif­ferential between the upper and lower surfaces of the arc, a circulation must be postulated. In a continuous perfect fluid flow, without an independently specified circulation, such an arc would not produce lift. D’Alembert’s para­dox would come into play. Such an arc would not have its stagnation points on the leading and trailing edge. The formula for the complex velocity has singular points indicating infinite velocities for the ends of the arc.109

In his diagram Einstein put the stagnation points on the leading and trail­ing edges of the arc even though he did not explicitly invoke a circulation. He put them where they would have been, given the appropriate amount of circulation needed to avoid infinite velocities at the edges. He did this by sup­posing that the guiding effect of the dividing wall, smoothly leading the fluid toward, around, and then away from the arc, would still be present even when the wall was removed and only the arc was left. There are no grounds for this assumption. Perhaps the best that can be said is that Einstein had, in fact, made provision for circulation in his analysis, but had done so tacitly and by questionable means. In his published lectures, a few years later, Prandtl gave a diagram that was almost identical to Einstein’s but, in Prandtl’s case, the com­ponent of circulation in the flow was properly identified and made explicit.110 Prandtl’s diagram is reproduced as figure 7.18 for purposes of comparison with Einstein’s figure.

It would seem that Einstein had little knowledge of current developments in the field of aerodynamics. This episode is a salutary reminder of the differ­ence between fundamental physics and technical mechanics. Eminence in the former does not guarantee competence in the latter.

A significant part of the experimental evidence for the circulation theory came from Arthur Fage (1890-1977). Fage was a retiring man who had trained as an engineer at the Royal Dockyard School in Portsmouth. His fa­ther had been a coppersmith in the dockyards. Fage won a scholarship to the Royal College of Science in London and then moved to the National Physical Laboratory in October 1912 as a junior assistant in the aeronautics section. In 1915 he published The Aeroplane: A Concise Scientific Study.29 The book was “written to meet the requirements of engineers” (v). It embodied a wholly empirical approach and made no mention of either the discontinuity theory or the circulation theory. With his “infinite capacity for taking pains,” Fage “became progressively a better and better research scientist” and acquired the reputation of being one of the NPL’s most meticulous experimenters.30

After Bryant and Williams’ work on the infinite wing, the next step in the Research Committee’s plan was to test Prandtl’s account of the flow around a finite wing. This work was undertaken by Fage and L. F. G. Simmons.31 They tested a rectangular model wing with a 3-foot span and a chord of 0.5 foot that was set at an angle of incidence of 6° to a wind of 50 feet per second. The wing had a cross section known as the RAF 6a. This particular wing was chosen because it had been studied in earlier experiments at the NPL, and the lift coefficient and the distribution of lift along the span were already known. Fage and Simmons used a speed-and-direction meter to probe the space round the wing to build up a detailed, quantitative picture of the flow. They measured the flow as it cut a number of transverse planes across the wind channel at various distances behind the wing (see fig. 9.5). If the x-axis is taken as the longitudinal axis of the channel, then positions on these trans­verse planes would be defined in terms of y – and z-coordinates where the y-axis lay parallel to the span of the wing and the z-axis indicated positions above and below the level of the wing. These coordinate axes are shown in the figure. Notice that the origin is located at one of the wingtips. The transverse planes chosen for study were called A, B, C, and D. Distances were expressed in terms of the chord of the wing. Plane A, which featured very little in the subsequent discussion, lay at a distance of about half a chord in front of the wing. Planes B, C, and D, the main interest of the experimenters, lay respec­tively at distances of x = 0.57, x = 2.0, and x = 13.0 chords behind the wing. The aim was to measure the properties of the trailing vortices as they cut through planes B, C, and D. On the basis of previous work (such as Piercy’s), Fage and Simmons were by now confident of the existence of the vortices but their question was: Did these vortices behave quantitatively in the way that Prandtl had assumed?

For convenience the test wing was actually mounted vertically and the speed-and-direction meter was inserted through a hole in the floor of the wind channel. The meter could be moved up and down, parallel to the lead­ing and trailing edges. The tips of the wing were fastened to transverse run­ners on the roof and floor of the wind channel so that the wing could be made to slide from one side of the channel to the other without altering its angle of incidence. These two degrees of freedom (the wing going from side to side and the meter going up and down) allowed measurement to be made within

a given, transverse plane. After taking measurements in plane A, they had to remount the wing at a different distance from the meter to measure the flow in plane B, and similarly for planes C and D. The measurements were made with an instrument of a standard type developed at the NPL (see fig. 9.6). It consisted of four open-ended manometer tubes forming a head that could be tilted on its mounting as well as turned from side to side. The orientation of the head of the meter could be manipulated from outside the wind channel. When the head was directly aligned with the local flow, the pressure in the four tubes was equal, which allowed the direction of the air at that point to be read off. The speed of the air was given by the difference in pressure between one of the four tubes and a fifth open-ended tube pointing in the direction of the free flow.32

Fage and Simmons took hundreds of measurements in order to build up a quantitative picture of the flow, and they were able to draw a number of important conclusions. First, they found that the velocity components (u) in the direction of the main flow (that is, parallel to the x-axis) were very little changed by the air having passed over the wing. The circulation theory implies that the speed immediately above and below the wing was modified, but measurement showed that soon after its passage over the wing the u-component settled down to a steady value that only varied by about 1 percent. This (approximately) constant velocity could therefore be ignored when considering the flow in the transverse planes. Exploiting this fact, Fage and Simmons argued that the flow within each plane could be considered as a two-dimensional flow. This two-dimensional flow was taking place in the (y, z) plane, so the two components of speed in the flow were v (along the y-axis) and w (along the z-axis). Fage and Simmons proceeded to com­pute the stream functions, streamlines, and vorticity of this flow at a large

figure 9.6. Apparatus used by Fage and Simmons for detecting the speed and direction of the flow behind a wing. As shown, the neck of the apparatus is inserted through the floor of the wind tunnel. The head of the apparatus can be raised or lowered, turned from side to side, and also rotated. From Fage and Simmons 1926, 306. (By permission of the Royal Society of London) number of points in each transverse plane by using the values taken from their measurements.

By the term “vorticity” Fage and Simmons meant (dw / dy—dv / dz). This is the quantity previously introduced and defined (in chap. 2) as the measure of the rotation of a fluid element. The two terms “rotation” and “vorticity” were used interchangeably. Each of the two parts of the expression represents a rate of change and hence the slope of a graph. Thus the value of dw/dy at a given point refers to the rate of change of the speed w with distance along the y-axis at that point. A corresponding definition applies to dv/dz. Fage and Simmons had a sufficient number of velocity readings of v and w at a sufficient number of points to find the relevant slopes and rates of change. They were thus in a position to compute the rotation or vorticity at each of these points. Once numerical values of this expression were established for a large number of points on the planes B, C, and D, it was possible for the ex­perimenters to draw curves linking up the points of equal vorticity. The result was a striking picture of the flow.

Fage and Simmons’ figures show the contours of equal vorticity in each of the transverse planes behind one-half of the wing (see fig. 9.7). The wing is not shown on these diagrams but lies horizontally, obscured, as it were, by the vorticity. The wingtip would be on the right, and the center of the wing, which is off the diagram, would be on the left. The top figure refers to plane B immediately behind the wing; the other two refer to planes C and D, which are set farther back. Examination of the figures shows that initially (at about half a chord behind the wing) vorticity is spread in a nar­row sheet along the trailing edge but becomes more concentrated near the tip. At a distance of two chords behind the wing, the vorticity at the tip has becomes less intense and more spread out. These changes, said Fage and Simmons, demonstrate the unstable character of the vortex sheet and show that it rolls up in the way predicted by Prandtl. The third picture shows the rolling up to be almost complete. The vorticity is now confined to the area near the wingtip. Citing Lamb’s Hydrodynamics, Fage and Simmons state that the diagrams show that, in accordance with the classical mechanics of vortex motion, the vortices behave as if they are attracted to one another. Their centers are progressively displaced toward the center and away from the tips.

Fage and Simmons then used their data to test the quantitative aspects of the theory and to construct detailed connections between their observations and the mathematics of Prandtl’s picture. The central questions were whether the flow (outside the trailing vortices) was irrotational and whether the lift and circulation around the wing were connected to the trailing vorticity

according to the laws identified by Prandtl. To address the question of irrota – tional flow, Fage and Simmons computed the circulation around a sequence of rectangular contours of increasing size drawn on their transverse planes. They chose rectangular contours, with sides parallel to the y – and z-axes, be­cause they already had the relevant velocity components needed for the cal­culation. The circulation they sought was given by the quantity (vdy + wdz) calculated around the contour. The chosen rectangles all enclosed the same areas of vorticity in the wake of the wing. According to Stokes’ theorem the circulation around these contours gave a measure of the vortex strength they enclosed. The calculated values for the circulation were close to one another, and this, argued Fage and Simmons, showed that outside the vortex wake the flow was irrotational—that is, as the contours got bigger, no further rotating fluid elements, and hence no further vorticity, can have been enveloped by the contour.

This argument for the irrotational character of the flow outside the vor­tex wake had exactly the same form as Glauert’s inference when he gave his preliminary analysis of Bryant and Williams’ results. A number of contours of increasing size display the same amount of circulation, ergo the contours pass through an irrotational flow. This was the inference to which Taylor had taken exception. Had Fage and Simmons fallen into the same trap as Bry­ant and Williams? The answer is no. As a deductive inference the argument put forward by Fage and Simmons is not compelling (for the reasons that Taylor gave), but as an inductive inference the conclusion is plausible. Given their other computations dealing with vorticity in the flow, and the contours showing its distribution, these results reinforced the conclusion that the flow outside the wake was, indeed, irrotational.

The most important quantitative question was whether the strength of the vortices trailing from the three-dimensional wing had been correctly predicted by the proponents of the circulation theory. Fage and Simmons deduced that, according to Lanchester and Prandtl, the strength of vorticity leaving the semi-span of the wing should equal the circulation around the median section of the wing. If these two quantities could be found and com­pared, then the prediction could be tested. Fage and Simmons developed this argument mathematically, but the intuitive basis of the deduction can be seen immediately by inspecting figure 9.8.

The figure shows a version of the refined horseshoe vortex. The vortices that run along the span of the wing all pass through a contour around the me­dian section of the wing. The value of the circulation around this contour is determined by the vortices that run through it (from Stokes’ theorem). All of these vortices then peel off the trailing edge of the wing. Their total strength can be assessed by the circulation around another contour, namely, any con­tour that captures and includes them. For example it can be measured by the circulation around the rectangular contours drawn in the planes B, C, and D that have been previously discussed. Fage and Simmons compared these empirically based values with the predicted circulation around the median section.

How did Fage and Simmons evaluate the predicted circulation? It is the

quantity that has previously been called Г0. Recall that in chapter 7, it was shown that by making the assumption that the lift distribution was elliptical, Prandtl had given an expression that predicted the value of Г0 from known properties of the wing and the flow. Prandtl had shown that

2VSC.

г = L,

0 bn

where CL is the coefficient of lift, S is the area of the wing, b is the span, and V is the free-stream velocity. All of these were known quantities for the wing used in Fage and Simmons’ experiment. In this way they were able to compare the predicted value of Г0 and the observed results for the circula­tion around the vorticity coming away from the trailing edge. The two values should be the same. The theoretical prediction was corroborated for the flow up to two chord lengths behind the wing, that is, for planes B and C. On plane D, at a distance of thirteen chord lengths, the agreement was not so good. The measured vorticity was around 18 percent too small. Some of the vortic – ity appeared to have dissipated, but, within the limits of experimental error, Fage and Simmons declared that the match between theory and observation was a good one.

———– СглЬтти1 Ur*S of Y

——- – Фспіоиґ Uno of Ф

figure 9.9. The streamlines of a trailing vortex at three distances behind the tip of a finite wing. The position of the wing is shown on the diagrams with the wingtip on the right, behind the center of the vortex. The centerline of the wing is on the left. From Fage and Simmons 1926, 320-22. (By permission of the Royal Society of London)

Fage and Simmons not only drew the lines of equal vorticity (shown in fig. 9.7) but also drew the actual streamlines of the vortices that issued from the wings and passed through the planes B, C, and D. This, too, was an exer­cise that involved considerable computation. First they had to check that the preconditions for the existence of a stream function, y, were fulfilled. Was the continuity equation satisfied? Laboriously, they concluded that it was. They then identified the values of the stream function at a large number of points in the (y, z) plane in order to draw the streamlines y = constant. On grounds of symmetry they took the line y = 3, the center line of the wing, as the streamline y0. The pictures they produced from their measurements and computations were convincing. “The diagrams,” they said, “illustrate clearly the changes in the character of the airflow behind the aerofoil: close to the aerofoil, the contour lines are ovals with the ends pointed inwards; as the distance behind the aerofoil increases, the contour lines become more and more circular—indicating that the vorticity is becoming more concentrated” (320). The results are shown in figure 9.9.

Fage and Simmons had significantly strengthened the case for the cir­culation theory. They had effectively picked up the argument at the point where Pannel and Campbell had left it in 1916 when curiosity seems to have been swamped by collective incredulity. Unlike the conclusions of Bryant and Williams, the work of Fage and Simmons was not challenged by Taylor.33 Others, such as Lanchester and Foppl, had made qualitative observations of trailing vortices and, more recently, photographs had been taken, but now the vortices had been directly measured and linked to the mathematically defined relations of the circulation theory. Could it now be said that the entities that were subject to empirical measurement in the laboratory had been clearly identified as one and the same with the entities referred to in the circulation theory? If so, the work of Fage and Simmons would represent a qualitative change in the status of the circulation account of lift. What had previously been a theory could now, arguably, be counted as a directly verifi­able fact.

How can the Tripos tradition be invoked to explain both failure, in the case of lift, and success, in the case of stability? Should not the same cause have the same effect? The answer is that my causal explanation is not meant to ex­plain success and failure. It explains the preconditions of success and failure, that is, what succeeded or failed, and the response to success and failure. It concerns the path that the Cambridge scientists and their associates chose to follow, or declined to follow, rather than the consequences of the choice and what was found along the path that was selected. Neither the British nor the Germans knew what awaited them. Neither group could select their course of action in the knowledge of what would be successful or unsuccessful. Both had to take a gamble. In the case of the theory of lift, the British gamble failed and the German gamble succeeded.

Lanchester did not approve of the theoretical choices made by what he called the Cambridge school, but he had a clear understanding of the meth­odological gamble involved. He did not use the language of gambling, but as we have seen in his reflections on the role of the engineer, he chose the more cerebral metaphor of playing chess. The scientist and the engineer were like players in a game of chess who were confronting an opponent called Nature. As a player, Nature was subtle and her moves could not be easily predicted. Lanchester expressed the contingency of the outcome by saying that, at the outset, no one could identify which moves in the game were good ones and which were bad. A sound-looking move might turn out to be a mistake, and an apparent mistake might turn out to be the winning move. Only the un­known future would reveal this.

Lanchester’s metaphor can, and should, be taken further. It does not just apply to the opening moves in the game of research or to what is sometimes called “the context of discovery” as distinct from “the context of justifica­tion.” It applies to the entire course of research and development ranging from the origin of ideas to their acceptance and rejection. The game with Nature does not come to an end when results begin to emerge and when the chosen strategy of research starts to generate successes and failures. The uncertainties Lanchester identified at the outset of the game still inform the responses that have to be made to the feedback from experience. If the scien­tist or engineer scores a success, the question remains whether it will prove to be of enduring significance or short-lived. If there is a failure, does it indicate the need for a revision of the strategy or merely call for more resolve? These radical contingencies and choices can never be removed, and in one form or another, whether remarked or unremarked, they are present throughout sci­ence. Indeed, they are present in every single act of concept application. The game never comes to an end.49

Writing in 1920, R. T. Glazebrook, the director of the National Physical Labo­ratory, recalled the events of 1909 and the inception of the Advisory Commit­tee.29 Mr. Haldane, he said, “appealed to Lord Rayleigh and myself to know if we could help at the National Physical Laboratory. A scheme of work was suggested, and at a meeting at the Admiralty at which Mr. McKenna, then First Lord, and Mr. Haldane were present, the details were agreed upon” (435). Haldane had made the acquaintance of John William Strutt (Lord Rayleigh) while they had worked together on an earlier committee, the Ex­plosives Committee, developing an improved and less corrosive propellant for the artillery.30 Rayleigh was a world-renowned physicist with formidable mathematical powers. He had published on aeronautical themes and had made fundamental contributions to fluid mechanics, the branch of physics that might explain how the flow of air over a wing could keep the machine in the air.31 Rayleigh was to become the president of the Advisory Committee for Aeronautics.

Given that Rayleigh was already the president of the National Physical Laboratory, so Glazebrook (fig. 1.3), as the director of the NPL, was an obvi­ous choice to work under him as the chairman of the Advisory Committee for Aeronautics. What of the other members? There were, of course, repre­sentatives of the Admiralty and the War Office. These were Major General Sir Charles Hadden for the army and Captain (later Admiral) R. H. S. Bacon

for the navy. (Haldane knew them both from the Committee of Imperial De­fence.) Bacon was soon to be joined (and then replaced) by Captain Mur­ray Seuter. Mervin O’Gorman, an engineer, joined the committee after his appointment as the new head of the Balloon Factory and then the Aircraft Factory.32 The remaining six members were Sir George Greenhill (mathema­tician), Dr. W. Napier Shaw (physicist and meteorologist), Horace Darwin (brother of Sir George Darwin and the founder of the Cambridge Scientific Instrument Company), H. R. A. Mallock (physicist), Prof. J. R. Petavel (en­gineer), and F. W. Lanchester (an engineer who had published a pioneering book on aerodynamics).33 The secretary of the committee was F. J. Selby, who had got to know Glazebrook when he went up to Trinity in 1888 and took classes from Glazebrook in physics and mathematics. In 1903 Selby joined the staff of the National Physical Laboratory and acted as Glazebrook’s personal secretary.34 The assistant secretary was J. L. Nayler, again of Cambridge and the NPL, who would coauthor a number of the early experimental reports.

Over fifty years after the founding of the ACA, Nayler gave a talk in which he recalled some of the personalities involved.35 He brought out clearly the closely knit character of the core group of scientists and the intellectual tra­dition to which they belonged. Glazebrook and Shaw, he said, had been as­sistants to Rayleigh in Cambridge, when Rayleigh had taken over the Caven­dish laboratory after the death of Clerk Maxwell. Glazebrook and Shaw had written textbooks of practical physics together, though “they both started as mathematicians.”36 Mallock had also worked as an assistant to Rayleigh.37 Nayler went on: “Seven out of the twelve were Fellows of the Royal Society and another became a Fellow later on, three were serving officers, two were heads of aeronautical establishments. Six, including the Secretary, began as wranglers, and five were trained engineers” (1045). The term “wrangler” is an old Cambridge label for a student who came at the top of the result list in the university’s highly competitive mathematical examination called the Math­ematical Tripos. The senior wrangler was at the very top of the list, followed by the second wrangler, and so on. Nayler notes that Rayleigh was a senior wrangler in 1865. Glazebrook and Shaw “were wranglers in the same year, 1876,” while Greenhill was “a second wrangler and Smith Prizeman” (1045). The award of the Smith’s Prize was an opportunity for the two or three top scorers in the Tripos to meet in a final contest in the battle for mathematical supremacy.

Nayler’s talk was given at Cambridge, which may explain some of the care taken to delineate the connections between the Advisory Committee mem­bers and the Mathematical Tripos. But we should not miss the significance or the specificity of the message. It would be difficult to overstate the centrality

of the Mathematical Tripos to the scientific life of Cambridge at the end of the nineteenth and the beginning of the twentieth centuries. Andrew Warwick’s impressive study Masters of Theory: Cambridge and the Rise of Mathematical Physics leaves no doubt about the intensity and brilliance of the Tripos tra­dition.38 In many areas of science in Cambridge, success in the Tripos was a precondition of scientific and academic preferment. The order of merit was published in the Times, and the senior wrangler of the year achieved celebrity status and, invariably, the offer of a college fellowship. Such success could only be achieved by intense coaching from one of a select band of brilliant but demanding tutors—men such William Hopkins, Edward John Routh, William Besant, R. R. Webb, and Robert Herman.39 Perhaps the greatest of all the coaches was Routh, whose caliber as a mathematician can be judged from the fact that, as a student, he had pushed James Clerk Maxwell into sec­ond place in the Tripos of 1854. Rayleigh had been coached by Routh, while Greenhill had been coached by Besant.

The top wranglers in turn would then become the examiners and the coaches for the next generation of students facing the rigors of the Senate House examinations. Lower-placed wranglers would often become school­masters and prepare their charges for mathematical scholarships to Cam­bridge, thus ensuring that each generation of students was better prepared than the last. This system increased the competition and forced up the stan­dards still further. So extraordinarily high did these standards become that examiners would use their current research and their latest discoveries as the basis for their questions. The most celebrated example was a result that subsequently played an important role in the mathematical underpinning of aerodynamics. The mathematical equation, usually called Stokes’ theorem, relating circulation and vorticity, first appeared in print in the Smith’s Prize examination of 1854. The candidates were required to prove the theorem. Stated in words, the theorem is that “the circulation round the edge of any finite surface is equal to the sum of the circulations round the boundaries of the infinitely small elements into which the surface may be divided.”40 (The meaning of the technical terms involved, such as “circulation,” are explained when the circulation theory of lift is introduced in chap. 4.)

The Tripos system certainly had its critics. It placed ambitious students under great strain, and many young minds found the demands too great. Critics also argued that the questions were too difficult for all but the best, and to rectify the problem various reforms and modifications were intro­duced over the years. The last order of merit list was published in 1909; there­after, candidates were placed in classes and the names listed alphabetically within the classes. There were also those who argued that the difficulty of the questions was because they were artificial and contrived so that their answers called for the mere mastery of technical tricks that had little educational value. This line taken in 1906 by G. H. Hardy, although his claims should be put in context. They were made in the course of arguments about the reform of the syllabus.41 Hardy was pressing the case for pure mathematics and rigorous foundations as against the applied mathematics and mathematical physics of the traditional Tripos.42 It was the more traditional form of the Tripos that informed the training of Rayleigh, Glazebrook, Shaw, Greenhill, and others, such as Horace Lamb, who later joined the Advisory Committee. Lamb had been coached by Routh and was second wrangler in 1872.

The scientific character of the Advisory Committee for Aeronautics, on which Haldane placed so much emphasis, was clearly weighted toward the Cambridge tradition of mathematical physics. In this connection, recall the three names that Haldane mentioned as possible leaders of the committee: Rayleigh, Moulton, and Darwin. Nayler noted that Rayleigh had been a se­nior wrangler, and to this one may add that Darwin, who had discussed mat­ters with Haldane on the Saturday before the Esher committee had met, was himself second wrangler in 1868.43 What about Lord Justice Moulton? He looks the odd one out. In fact John Fletcher Moulton was senior wrangler in 1868, soundly beating George Darwin into second place and achieving higher marks than any previous Tripos candidate.44 Rayleigh, Moulton, and Darwin had all been coached by Routh. It would be wrong to say that the ACA was, or was meant to be, simply a committee of wranglers. Lanchester was no math­ematician, nor were the military men, but it would not be an exaggeration to say that Cambridge wranglers were a powerful, and perhaps predominant, presence. Whether by accident or design the scientific culture of the Advi­sory Committee was, to a significant degree, the culture of the Mathematical Tripos.45

I have shown how, in order to generate and solve the equations of motion of an ideal fluid, all manner of simplifications had to be introduced. What price, if any, had to be paid for the advantages that the simplifications brought with them? What does it cost, for example, to bring the investigation under the scope of Laplace’s equation and confine the theory to motion in which fluid elements do not rotate? Those who developed classical hydrodynamics hoped that the price would not be a large one. The hope was reasonable because wa­ter and air were only slightly viscous. Unfortunately, the transition from small viscosity to zero viscosity sometimes had a very large effect on the analysis. In important respects the difference between real fluids and the theoretical behavior of a perfectly inviscid fluid was dramatic. The price of the approxi­mation was high, and it was extracted in a surprising way.

Suppose that you hold a small, rectangular piece of thin cardboard by one of its edges, for example, a picture postcard held by its shorter edge. Move the card rapidly through the air so that the card faces the flow head on (not edge­wise). It is easy to feel the resistance to the motion, and (within a Newtonian framework) this means feeling the force that the motion through the air ex­erts on the card. In the same terms, one can also see what must be the effect of the force by the way that the card bends. Experts in aerodynamics want to be able to calculate the magnitude and direction of the force that is so evidently present. A good theory would furnish them with an accurate description of the characteristics of the flow and an explanation of how the flow generates the forces. The theory should permit answers to questions such as: Does the air flow smoothly around the card? What are the streamlines like? Does the card leave a turbulent wake in the air? What happens at the edges of the card? How does the force vary with the angle at which the card is held, that is, with the “angle of attack”?

It turns out that if the air were a perfect fluid, there would be no resultant force at all on the card. The simplifications led to a mathematically sophis­ticated analysis but also to a manifestly false prediction. Why is this? The answer can be seen by looking at the form of the flows that are generated under the idealized conditions assumed by the mathematician. The (theoreti­cal) flow of an ideal fluid in irrotational motion over the postcard or lamina normal to the flow looks like the flow sketched in Cowley and Levy’s diagram shown in figure 2.5. In the diagram the card is represented as fixed, and the ideal fluid that stands in for the air is shown approaching the card and mov­ing around it. This example differs from the experiment in which the card moves through the air (which is assumed to be at rest), but scientists typically prefer to adopt this convention. The justification is that dynamically the two things are equivalent. As far as the forces are concerned, all that matters is the relative motion of the air and the obstacle. Pretending that the card is still and the air moves, rather than the other way round, turns out to be easier because seen from the standpoint of the card the flow is steady. It also makes the dia­gram fit more closely to experiments that are done in wind tunnels.

From figure 2.5 it can be seen that half of the air (or ideal fluid) that im­pinges on the front face moves away from B, the front stagnation point, up to the edge C, while the other half (not shown) will move down to the edge C. The fluid then curls sharply around the corner at each of these edges and approaches the point D (the rear stagnation point) and then continues on its way. The fluid farther from the plate follows a similar path to the fluid near the plate but with less abrupt changes of direction. At the stagnation points the lines representing the flow meet the surface of a body and can be thought of as splitting into two in order to follow the upper and lower contour of the body. At a stagnation point, mathematical consistency is preserved by tak­ing the direction of the flow to be indeterminate and the speed of the flow to be zero.

Inspection of the diagram for the steady flow around the flat plate as shown in figure 2.5 allows the direction and some indication of the speed of the flow to be read off. It can be seen that the flow moves rapidly around the edges of the plate. Inspection also shows that the flow is symmetrical about an axis that lies along the plate as well as being symmetrical about an axis that is normal to (that is, at right angles to) the plate. (A mathematician would spot the symmetry from the equation for the streamline because all the x – and y-terms appear as squares.) These symmetries have important consequences for the pressure that the flow exerts on the plate. According to Bernoulli’s law, as the fluid impinges on the plate and is brought to a halt, it exerts its maximum pressure. As it moves along the plate and gathers speed, it exerts a lesser pressure. Because the fluid is perfectly free of viscosity, there will be no tangential traction on the plate and all the forces will be normal to the plate. The pressure on both the front and the back will be high near the stagnation points and low near the edges of the plate. The symmetry of the flow around an axis along the plate means that the pressures exerted on the front of the plate will be of the same magnitude as the pressures exerted on the back of the plate. The pressures will be in opposite directions and will thus cancel out. There will be no resultant force.

The forces on a plate moving relative to a body of ideal fluid are there­fore fundamentally different from those on a plate (such as the postcard) moving relative to a mass of real air. Both experiment and everyday experi­ence stand in direct contradiction to the mathematical analysis. Treating a slightly viscous fluid (such as air) as if it were a wholly inviscid fluid may have seemed a small and reasonable approximation, but the effect is large. Neglecting a small amount of compressibility caused no trouble; neglecting a small amount of viscosity proved vastly more troublesome. The disconcert­ing conclusion that the resultant force is zero does not just apply to a flat plate running head-on against the flow. Consider again the flow around a circular cylinder. This was Cowley and Levy’s other textbook example and was shown

in figure 2.4. The closeness of the streamlines indicates that the flow speeds up as it passes the top and bottom of the circular cylinder. Since the fluid is free of any viscosity, all the pressure on the cylinder will be directed toward the center. The symmetry of the flow means that any pressure on the cylin­der will be directly counteracted by the pressure at the diametrically oppo­site point on the cylinder. Again, counter to all experimental evidence from cylinders in the flow of a real fluid, there will be no resultant force on the cylinder. In reality the pressure distribution is not the same on the front and rear faces.37

Do these results depend on the obstacle and the flow possessing symme­try? The answer is no. The results apply to objects of all shapes and orienta­tions. Consider the flow around a flat plate that is positioned not normal to the flow but at an oblique angle to the flow. The situation is represented in figure 2.6.

Such a flow introduces certain additional complexities into the analysis, but the outcome is still a zero resultant force. The extra complexity is that the forces at points on immediately opposite sides of the plate are not equal, which can be seen from the way the front and rear stagnation points are not directly opposite one another. The front stagnation point is near to, but be­neath, the leading edge, while the rear stagnation point is near to, but above, the trailing edge. As a result the plate is subject to what is called a “couple” that possesses a “turning moment.” A “couple” arises from two forces that are equal and opposite but act at different points. Here they exert leverage on the plate, causing it to rotate and to turn so that it lies across the stream. As the plate rotates, the two stagnation points move. The front stagnation point moves away from the leading edge toward the center point of the front of the plate. The rear stagnation point moves away from the trailing edge to the center of the back of the plate. When the plate is lying across the stream, the two stagnation points are directly opposite one another and there is no more leverage they can exert. For the inclined plate, then, there

is an effect produced by the forces, but it is still true that there is no resul­tant force. A force at a given point on one surface of the plate will still have a force of equal magnitude and opposite direction at some corresponding point on the other face. Overall, the forces will still sum to zero, as they did for the circular cylinder or the plate that was initially positioned normal to the flow.

Similar considerations apply to any complex shape and would therefore also apply to a shape chosen for an aircraft wing. If the air were an ideal fluid, there might be a turning effect exerted on the wing but there would be no resultant aerodynamic force. There would be neither lift nor drag. The zero-resultant theorem for plates and cylinders had been established many years before the practicality of mechanical flight had been demonstrated and had long been a source of some embarrassment. Interpreted as a pre­diction about either air or water, its falsity was evident, but it continued to haunt the theoretical development of aerodynamics. In France this old and well-established result was called d’Alembert’s paradox, in Germany it was called Dirichlet’s paradox, and in Britain it wasn’t called a paradox at all. Re­member that, for Cowley and Levy, the mathematics just defined the nature of the fluid. The zero-resultant theorem simply establishes that this is how an ideal fluid would behave were there such a thing. Interpreted in this way, the zero-resultant theorem brings out the difference between a real fluid and an ideal fluid, so it can be taken as a powerful demonstration of the unreality of ideal fluids.38 But if real and ideal fluids are so different, how are theory and experiment ever to be brought into relation to one another? This was a long-standing problem. Failure to resolve it had generated a sharp distinc­tion between hydrodynamics, which was largely a mathematical exercise, and hydraulics, which was largely an empirical practice—hence the two different chapters and the two different authors in the Encyclopaedia Britannica.39 Was aerodynamics to take the path of empirical hydraulics or the path of math­ematical hydrodynamics? There were strong social forces pulling in each of these opposing directions.

The behavior of a simple vortex, of the kind just described, can be used to introduce the mathematical definition of the quantity called “the circulation” of a flow.7 So far I have used the concept of circulation in an intuitive way, but for the purposes of aerodynamics it was important to deploy a precise definition. To follow the history of the dispute over the circulatory theory of lift, readers must grasp the general features of the more precise, mathemati­cal concept. I introduce them in two stages. First I confine the discussion to a simple, isolated vortex of the kind already introduced and then I move to a more general definition that can be used to detect and measure circulation in complicated flows where the vortex is just one component. The ideas dis­cussed in this section are now standard, textbook material, but that was not always the case. They found no place in Cowley and Levy’s textbook of 1918. These authors were still under the influence of the discontinuity theory of lift and did not take the circulation theory seriously.

Consider again the family of simple vortices where the streamlines are concentric circles of radius r and the uniform speed of flow along the stream­lines (the tangential velocity v) diminishes in proportion to the distance r of the streamline from the center of the vortex. This model has already been expressed by the formula v = k/r, where k is the constant of proportional­ity. Three features of this family of vortices are immediately evident. First, the product of v and r, that is, the speed along a streamline multiplied by the radius of the streamline, always has the same value, namely, k. Second, as the radius r gets very small, the velocity v gets very large. Mathematically, as r ^ 0, v ^ ro. Third, as r gets very large, the velocity of the flow gets very small. In the limit, as r ^ ^, then v ^ 0. The flow is effectively stationary at great distances from the center of the vortex. All vortices of this simple kind therefore cover the entire range of velocities from infinite velocity at the cen­ter to zero velocity at the distant periphery. The situation at the very center of the vortex, at r = 0, where the flow rotates at infinite speed, is obviously physi­cally unrealizable. It indicates the abstract character of the vortex model. The problematic point at r = 0 is called a “singular point” of the formula.

Intuitively some vortices are said to go faster than others or to be stron­ger than others. How are these distinctions to be expressed given that the simple k/r formula covers the full range of velocities from zero to infinity? The answer is that at any given distance from the center, the speed of flow of some vortices will be greater than others at that distance. In order to make the distinctions that are desired, the speed of flow must always be related to the distance from the center. This requirement must be built into any definition of the strength of a vortex. Such a relation is embodied in the constant k that has already been introduced in the formula relating tangential speed v and radius r. The constant k characterizes a vortex and distinguishes the faster from the slower vortices. For any given distance r from the center, the bigger the value of k the faster the vortex goes.

Suppose now that the word “circulation” is introduced with the aim of describing a vortex. The aim is to discriminate between them so that a fast, strong vortex has a “greater circulation” than a slower, weaker vortex. One plausible candidate for the definition of the concept would be to equate cir­culation with the value of k. Historically the chosen definition was close to this but not identical. The measure of circulation that was chosen made it proportional to k but not equal to k. It was decided that the circulation was to be 2nk. Stated in words, the circulation around a simple vortex was defined as the product of the tangential velocity v along some streamline and the circumference of the streamline, which is 2ПГ. This (provisional) definition refers to the speed along a contour multiplied by the length of the contour. Using the symbol Г (gamma) for the circulation, the circulation around a simple vortex is given by the formula Г = v 2ПГ or by the formula Г = 2nk. The value of the circulation around a given vortex has the property that it comes out the same whatever streamline is used to arrive at the value.

The next step is to generalize the definition so that it is not tied to flows with circular streamlines. The move toward generality exploited the fact, noted earlier, that in a simple vortex it does not matter which circular stream­line is used to arrive at the value of the circulation. The circulation could be computed in part by using the speed vj along an arc of the circle of radius r1 and in part by using the speed v2 along an arc of the streamline of radius r2 (see fig. 4.4a). The jump from one arc to the other could be taken along an appropriate radius. The jump would not contribute to, or alter, the circula­tion because all the speed in the vortex is tangential and the speed along any radius is zero. As long as the two circular arcs were combined so as to make a closed circuit around the vortex, the value of the circulation would be no different from the value calculated using a single circular streamline as the relevant contour. And what holds for jumping between two different circular streamlines holds for jumping between three or four or any number. This too makes no difference to the value attributed to the circulation. In the limit, the jumps can be imagined to be so numerous, and the streamlines can be imag­ined to be so close, that the contour along which the circulation is computed could weave any path through the field of flow (see fig. 4.4b). The upshot is that the contour can be arbitrary. All that matters is that it forms a closed loop around the center of the vortex.

In the original definition of circulation around a simple vortex, where the contours were concentric circles, the speed used in the calculation was the tangential speed along the circumference. When the contour is arbitrary, the speed used in the computation must be the speed along this arbitrary contour. Stated generally, at any point P on the contour, the flow can be as­sumed to have a speed q and a direction that makes an angle 0 to the contour (see fig. 4.5). To compute the circulation, the component of speed along the contour, namely, q cos0, must be used. The contribution to the circulation at that point is dГ = q cos0 ds. Summing or integrating this quantity around the closed contour gives the circulation around the contour. This leads to the general definition of the circulation Г around a closed contour C as

figure 4.4. In (a) the product of the speed and circumference is the same for all streamlines of the vortex. The value of the circulation can therefore be computed using a contour that jumps from one streamline to another. Part (b) shows how this fact can be exploited and any contour used to compute the circulation. An arbitrary contour can be thought of as a limiting case involving an infinite number of jumps. The strength of the circulation is thus contour-independent provided that the contour encloses the vortex.

figure 4.5. The speed of the flow at P is q. The component of q along the curve RS is q cos0. The “flow” along RS is defined as Jqcos0ds where the integration is taken along the contour RS. If the contour is extended to form a closed loop, the flow counts as the “circulation” around the closed contour.

Г= I qcosOds

The previous, and provisional, definition of the circulation around a sim­ple vortex can now be seen as a special case of this more general definition. In the special case, C is a circle of radius r. The speed q is equal to the constant tangential velocity v and always lies along the contour so that cos0 = cos 0 = 1.

These constant quantities can be taken outside the integral sign, leaving a line integral which equals the circumference 2ПГ. Thus, as before, the circulation around the vortex is Г = v2TCr.

The general definition identifies a circulation in a flow even when the vor­tex merely acts as one component. If there are no vortices in the flow, the cir­culation is zero. The important point for aerodynamics, and the circulation theory of lift, is that the measure of circulation should be independent of the contour so the selection of the contour can be arbitrary, as in figure 4.6. All contours that form a closed loop containing an aerofoil, and the vortex that it is assumed to generate, should yield the same measure for the circulation around it. A large circle could be drawn around a two-dimensional aerofoil, or a rectangle could be used, and the numerical value of the integral along these contours should be the same.

Where the integral produces different values for different contours, the concept of “the circulation of the flow” is not well defined. This can happen, for example, when the fluid is viscous. Contour-independence will prove to be of great significance later in the story, when I describe how, in the 1920s, the personnel of the National Physical Laboratory attempted to conduct ex­periments to establish, once and for all, whether the circulation theory gave the right account of the lift of a wing moving through air.8

Georg Fuhrmann had been Prandtl’s pupil at the technische Hochschule in Ha­nover, where his ability in the course on technical mechanics caught Prandtl’s attention. After completing his training as an engineer in 1907, Fuhrmann joined Prandtl at Gottingen and played a significant part in setting up the wind tunnel. It was Fuhrmann in 1910 who wrote the review of the German translation of Lanchester for the Zeitschrift fur Flugtechnik.1 His warm rec­ommendation stood in marked contrast to the coldness of the Nature re­view. In 1911 Fuhrmann carried out important theoretical and experimental research on the resistance of model airships. He was to die in action in the first few weeks of the war in 1914.18

The experiments were designed to compare the predictions of ideal-fluid theory with wind-tunnel measurements.19 To make his theoretical predic­tions Fuhrmann had used a standard technique from classical hydrodynam­ics in which complex flows were built up from simpler flows, for example, from an array of sources and sinks, each of which can be represented by a simple velocity potential. He assumed a theoretical distribution of sources and sinks in a uniform flow of perfect fluid and arranged them in such a way that they gave rise to airship-like configurations of streamlines. The basis of Fuhrmann’s procedure can be conveyed intuitively by examining figures 5.2 and 5.3. If a source is combined with a uniform flow, then the fluid from the source pushes the free stream aside as shown in figure 5.2. At the same time the streamlines radiating out from the source are distorted and bent back. The streamlines of the new flow coincide with the streamlines of the flow around a long, blunt-nosed body. Selecting an appropriate streamline of the new flow and imagining that it is suddenly solidified gives the surface of the body.

If now a line of sinks is introduced directly downstream of the source and spread along the axis of the body, the overall flow is modified once again, as in figure 5.3. The fluid injected into the flow at one point by the source is now drawn out of the flow at other points by the sinks. If the intake of the

sinks equals the output of the source, the streamlines of the combined flow can close up again. While the obstacle represented in figure 5.2 was infinitely long, the obstacle in figure 5.3 is of finite length and resembles the outline of the hull of an airship. Fuhrmann used Bernoulli’s law to calculate the pres­sures at various points on its surface. These surface pressures, in accord with d’Alembert’s paradox, summed to zero, but it was the distribution of pres­sure that was the focus of interest.

By assuming different distributions of sources and sinks, Fuhrmann could produce theoretical configurations representing different shapes of airship, for example, some with blunter noses or longer tails than others. He worked out the streamlines for six different shapes. The next step was to construct a set of hollow model airships, made out of metal, which accurately conformed to these theoretically generated shapes. Fuhrmann placed the models in the Gottingen wind tunnel and measured the pressures at a number of points on their surface. He did this by means of small holes in the surface that were con­nected to a manometer. Next, after removing the holes and piping, the mod­els were suspended from wires and attached to scales so that wind-tunnel measurements could be made to find their total drag.

Careful corrections had to be made to allow for both the resistance and the stretching of the supports. Fuhrmann conceived of the drag on the airship as divided into two parts: the pressure drag and the friction drag. The pres­sure drag was the result of pressures normal to the surface; the friction drag was tangential. Normal pressures could be generated by a perfect fluid, but it takes a viscous fluid to create a tangential traction. Fuhrmann reached three important conclusions. First, the graph of the observed pressure distribution of the air flow was very close to that predicted from ideal-fluid theory except at the very tail of the airship models. The only exception was a blunt-nosed model, where there was deviation from the predicted pressure at the nose as well as the tail. Second, models with a rounded nose, slender body, and long tapered tail had astonishingly low resistance, for example, at 10 meters per second they had less than one-twentieth of the resistance of a sphere of the same volume. Third, nearly all of this small, residual drag could be accounted for by the frictional drag of the air on the surface. Even given the slight devia­tion at the tail and, of course, the effect of the air in immediate contact with the surface of the airship, the air behaved like an ideal fluid.

In the period immediately after the Great War, Prandtl wrote an account of the Gottingen airship work for the American National Advisory Com­mittee for Aeronautics. It appeared in 1923 in English as the NACA Report No. 116.20 The first part of the report included a survey of ideal-fluid theory, and the second part began with an account of the resistance measurements on model airships carried out by Fuhrmann. Prandtl declared that the agree­ment between theory and experiment in Fuhrmann’s work had given them “the stimulus to seek further relations between theoretical hydrodynamics and practical aeronautics” (174). “Theoretical hydrodynamics,” here, meant perfect fluid theory. Even more striking was how Prandtl described the char­acter of the agreement that had so encouraged them. “The theoretical theo­rem that in the ideal fluid the resistance is zero,” he said, “receives in this a brilliant confirmation by experiment” (174).

Prandtl’s British counterparts such as Bairstow, Cowley, Lamb, Levy, and Taylor did not speak in this way. For the British, a “theoretical theorem” would be the result of deduction from the premises of the theory and would be something to be judged by logical, not experimental, criteria. It described an ideal fluid not a real fluid. Even if ideal-fluid theory could, on occasion, generate an empirically correct answer, this would only be because false premises can sometimes produce true conclusions. Properly speaking, exper­iment could never provide a “brilliant confirmation” of what was essentially a mathematical theorem, and certainly not of a theorem that referred to an acknowledged mathematical fiction. The first would be wholly unnecessary and the second wholly impossible.

Prandtl’s enthusiastic formulation was slightly qualified when Fuhrmann’s experiments were discussed in his Gottingen lectures, which were published a few years later.21 In 1931 Prandtl described Fuhrmann’s result as follows:

Diese Tatsache kann man bis zu einem gewissen Grade als einen experimen – telien Nachweis ansehen fur den Satz der klassischen Hydrodynamik, dafi in einer reibungslosen Flussigkeit der Widerstand eines bewegten (hier aller – dings stromlinienformigen!) Korpers Null ist. (153)

Up to a certain degree one can regard this fact as an experimental proof of the classical hydrodynamic theorem that the resistance of a moving body (at least, a streamlined one!) in a frictionless fluid is zero.

The brilliant confirmation had become a proof up to a “certain degree,” but when Prandtl came to spell out the basis of this more qualified judgment, it is clear that this did not bring him nearer to the British position. In his lectures, Prandtl dealt with Fuhrmann’s work in a section devoted to bod­ies of small resistance. He introduced the section by identifying the area in which inviscid theory has a legitimate application to the real world. It was, he said, an area of great technological significance and included airships, aircraft wings, and propellers.

Wahrend die klassische Hydrodynamik der reibungslosen Flussigkeit durch – weg in allen denjenigen Fallen versagt, in denen es sich um Stromungsvor – gange mit betrachtlichem Widerstand handelt, lasst sie sich mit Vorteil an – wenden bei Flussigkeitsbewegungen mit geringem Widerstand. In den meisten praktischen Fallen—so besonders in der Flugtechnik und im Luftschiffbau— handelt es sich aber darum, den meist schadlichen Widerstand auf ein Min – destmafi zu bringen, so dafi gerade hier ein grosses Anwendungsgebiet der Methoden der Hydrodynamik reibungsloser Flussigkeiten vorliegt. Auf die – sen Umstand ist es zuruckzufuhren, dass die Flugtechnik und Luftschiffahrt so ausserordentlich durch die neueren Untersuchungen der Luftbewegungen (Luft aufgefasst als reibungslose Flussigkeit) gefordert wurde—wir erinnern nur an die Ausbildung der gunstigsten Luftschifform, an die Tragflugel – und Propellertheorie—und dass umgekehrt die praktischen Probleme der Flug – technik der Theorie eine grosse Anzahl dankbarer Fragestellungen gegeben haben. (150-51)

While the classical hydrodynamics of a frictionless fluid always fails in those cases where the flow must cope with a considerable resistance, it can be ap­plied with advantage to fluid motion with small resistance. In most practical cases—particularly in aviation and the construction of airships—it is a mat­ter of bringing the most damaging forms of resistance down to a minimum. It is precisely here that there lies a large field for the application of the methods of the hydrodynamics of frictionless fluids. It is for this reason that aviation and airship travel received such benefit from the new investigations into the flow of air (where the air was conceived as a frictionless fluid)—one calls to mind the development of the most satisfactory shapes for airships, and wing and propeller theory. Conversely, the practical problems of aviation have pre­sented to the theory a large number of fruitful questions.

This passage gives Prandtl’s argument for conceiving air as a friction­less fluid. Lamb kept the two things separate, putting one in a box marked

“real” and the other in a box marked “ideal.” Prandtl put them both in the same box.

Giving theory and experiment the same referent is necessary for turning d’Alembert’s result from a mere theorem into a genuine paradox. This was why Lamb adopted the strategy of assigning them separate referents. In giving them the same referent, did Prandtl intend to generate or embrace a paradox? Or, if this was not his intention, was it the unwitting consequence of his posi­tion? The answer is neither. Prandtl’s stance was not paradoxical. He avoided paradox, but he did so by rejecting precondition (1) rather than, as Lamb did, precondition (2). To create a paradox it is necessary that two sources of infor­mation about a common object contradict one another. Prandtl said they did not contradict one another. The theory predicted zero resistance—and this was (very nearly) what was found by experiment.

The situation here was not, as G. P. Thomson might suspect, a case of en­gineers working in the realm of “good enough.” It was the opposite. Prandtl and Fuhrmann found they could use the theory of ideal fluids to design air­ships that were very close approximations to the zero resistance entailed by the theory of perfect fluids. It helped them to identify the places where smooth flow was breaking down so that they could reduce it further. Their efforts were informed by an ideal they were striving to attain. The ideal was not kept distinct from practice, or set in opposition to it, but was integral to it and gave practice its direction and purpose. Max Munk, a distinguished pupil of Prandtl, looking back over some seventy years, recalled the Prandtl and Fuhrmann experiments and clearly thought that their methodological significance had not been properly appreciated.22 Munk said: “The wind tunnel was asked whether the actual pressure distribution was sufficiently equal to the one computed for a perfect fluid. It was asked whether the study of the motion of a perfect fluid was helpful for practical aerodynamics. The wind tunnel answered with a loud Yes. This was a very great achievement of Prandtl, one for which he did not get enough credit” (1). Munk did not indicate who had been reluctant to give due credit, but it is clear that, had he wished to do so, he could have pointed to the British stance and the method­ological assumptions behind it.

Although Kutta’s 1902 thesis appears to have been lost, the historian Ulf Hashagen has discovered the examiner’s report in the archives of the THM. Hashagen draws attention to the revealing way in which Kutta’s work was described.35 Finsterwalder, who acted as both adviser and examiner, said that Kutta’s thesis was solidly constructed, industrious, and skillful. The calcula­tions presented many difficulties, and these required a detailed knowledge of the theory of functions. But, he went on,

Erfreulich ist aber auch, dafi die Aufgabe noch ein besonderes praktisches In – teresse besitzt—wie denn die von Kutta ins Auge gefafite kunftige Lehrtatig – keit gerade auf die Anwendungen der Mathematik sich beziehen soll. Dabei zeigt die gegenwartige Arbeit, wie auch die fruheren, recht guten von ihm verfafiten Abhandlungen und Kuttas ganzer Studiengang, dafi er die “Anwen­dungen der Mathematik” in dem modernen Sinne kennt, welcher sich mit wirklich aktuellen Fragen der Physik und Mechanik beschaftigt, statt—wie dies fruher ublich war—nur eine physikalische Einkleidung rein mathemati – scher Untersuchung unter “Angewandter Mathematik” zu verstehen. (257)

It was also good that the task had a specific practical interest—so it will be just the thing for Kutta to use in the future teaching that he intends to do on the application of mathematics. Like the very good earlier papers that he has authored, and indeed like his whole course of study, this work shows that he knows what it is to “apply mathematics,” in the modern sense of those words.

This means getting involved with real questions in physics and mechanics

rather than—as used to be the case earlier—dressing up pure mathematical

investigations in physical clothing and calling it “applied mathematics.”

Some mathematicians had been less than genuine in their attempt to accom­modate, or be seen to accommodate, the demands for relevance coming from the engineers in the technische Hochschulen. Finsterwalder wanted a real en­gagement. For Finsterwalder, applied mathematics in the “modern sense” would truly embody the ideals of a scientifically oriented technology, and Kutta’s aerodynamic work was held up as an exemplary case of the study of mechanics pursued in this spirit.36

Finsterwalder’s description of the modern spirit sometimes took on a de – tectably marshal air, as did that of his respected Munich colleague August Foppl.37 The period around 1910 was one of diplomatic crises and inter­national tension. It was also a time of intense public interest in aviation. There was the triumphant, but troubled, development of Germany’s giant Zeppelin airships and Bleriot’s dramatic flight across the English Channel.38 The military significance of these events would not have been lost on the at­tentive Finsterwalder. In the 1909 article on the scientific basis of aeronautics, cited by Kutta, Finsterwalder had already spoken ominously of “the demands of the time” (“Forderungen der Zeit”) and the “honorable contest of nations” (“edlen Wettstreit der Nationen”; 32). These were euphemisms for war. Fin- sterwalder had also noted, pointedly, that in aeronautical matters, unlike nautical ones, all nations could participate equally (“alle Staaten gleichmafiig beteiligt sind”; 31). Given Germany’s well-publicized naval arms race with Britain, this comparison could not have been a casual one. It carried the sug­gestion that what was proving expensive and difficult for Germany in the maritime sphere might be achieved more easily in the sphere of aeronautics because all nations had the same starting point. No wonder the worldly Fin- sterwalder was now encouraging Kutta to take up aerodynamics once again and prepare an extended version of his old research for publication.

Kutta’s work has achieved a classic status, and he has been accorded epon­ymous honor. The law relating lift and circulation bears his name, and the condition of smooth flow at the trailing edge is frequently referred to as the Kutta condition.39 His identification of a wing with a geometrical arc can be seen as a precursor to what is called the theory of thin aerofoils.40 But Kutta’s mathematical techniques have not entered the textbook tradition. His work is never described in his own terms; it is always reworked by means of later techniques. Finsterwalder did, however, encourage Wilhelm Deimler, an assistant in Munich, to calculate and publish the precise pattern of stream­lines around Kutta’s arc-like wing.41 Apart from this, and one unpublished doctoral dissertation, there appears to have been little by way of follow up.42

Looking at Kutta in retrospect, we may also remark that the empirical support he claimed is open to question. The quality of the data he used was not good. Kutta was aware of some of these problems, though not all of them. One obvious problem was that Kutta’s theory was two-dimensional, whereas Lilienthal’s data were three-dimensional. Lilienthal’s experiments were also conducted in a natural wind and therefore depended on the calibration of the anemometer. Unfortunately, Lilienthal was depending on inaccurate data from other experimenters to enable him to read the wind speed. Finally, when Kutta applied his own theory to a flat plate, he assumed a smooth flow in which the air stayed close to the surface, but the flat-plate data from Lang­ley and others did not satisfy this condition.

These shortcomings have been identified in Early Developments of Modern Aerodynamics by Ackroyd, Axcell, and Ruban. The book provides a transla­tion of Kutta’s papers of 1902 and 1910 and a valuable commentary from the standpoint of today’s aerodynamics.43 The commentary contains a brief, and negative, evaluation of Kutta’s discussion of the rounding off of the leading edge. Kutta is said to have embarked on this project “rather fruitlessly, as subsequent events were to prove” (185). Looking back with the benefit of hindsight this is fair comment. For the purpose of locating Kutta historically, however, it is important not to pass over the clue that this expenditure of ef­fort can give us, however misguided it now seems. In this part of his paper Kutta was engaging with a question that confronted those who were design­ing and building wings. What sort of leading edge should they give the wing? Fruitful or fruitless, Kutta’s attempt to grapple with this question provides evidence of the engineering orientation that Finsterwalder wanted. Although today’s reader may be tempted to hurry past these sections of Kutta’s paper, for Finsterwalder they were evidence of the intimate relation between Kutta’s mathematics and technology.44

The point deserves emphasis. If we are to understand Kutta historically, we must keep in mind the following characteristics of his work: (a) its focus on a specific, technical artifact, namely, the wing of Lilienthal’s glider; (b) his concern with issues of optimization and trade-off, for example, with regard to the amount of rounding off of the leading edge; (c) his willingness to use highly artificial theoretical tools such as perfect fluid theory; (d) his aware­ness of the limitations of these conceptual tools but a willingness to postpone asking and answering certain questions, for example, about the origin of the circulation; and (e) a determination to bring the theory, however unrealis­tic, into direct contact with data at the numerical level. This combination of mathematical methods with an engineering orientation placed Kutta’s work in the category that Finsterwalder called “modern” applied mathemat­ics. Also, when dealing with turbulence and other complications in the flow, Kutta, in his 1910 paper, used a version of what von Mises called the hydraulic hypothesis. Kutta went on:

Dennoch halte ich es fur moglich, dafi diese komplizierten Erscheinungen sich uber das hier beschriebene Stromungsbild nur superponieren, und die durchschnittlichen Druck-und Geschwindigkeitsverteilung—besonders die erstere—der geschilderten nahe steht. (51)

I therefore think it is possible that these complicated phenomena are merely superimposed on the flow formations described here, and the average pres­sure and speed distributions, especially the former, are close to those that have been presented.

Pioneering work also raises questions about its origins. Consider the break­through Kutta made in his 1902 Habilitationschrift. His appeal to the circula­tion theory of lift was made some years before the publication of Lanchester’s work. How then did Kutta come to make the link between circulation and lift? From where did he get his ideas? Of course, Finsterwalder may have pointed out the role of circulation when he suggested the research topic to Kutta, but this conjecture only postpones the problem. How did Finsterwalder get the idea?

The proper response is to see that there is something wrong with the ques­tion. Neither Kutta nor Finsterwalder needed to think up the idea. It was com­mon knowledge that a force can be generated by combining a uniform, recti­linear flow with a circulating flow. This had long been known in Cambridge and would have been equally well known in Munich. It was not the availabil­ity of the idea that constituted a problem; it was the willingness to use it. The thing that needs explaining is not how the idea was generated but why some people saw it as useful while others saw only trouble and futility. It is the mo­bilization of action that constitutes the real puzzle, not the origin of ideas.

Kutta clearly felt free to use the simple model of circulation in a perfect, irrotational fluid. He did not feel compelled to stop in his tracks because he could not explain how the circulation might arise. He knew that in using the circulation model he had to make physically false assumptions about viscos­ity. He might have been, but was not, deterred from going ahead on this basis. He might have concluded that he should have been devoting his efforts exclu­sively to a theory of viscous fluids, but he did not. This is the point. Kutta was prepared to go ahead, while others, such as G. I. Taylor in Cambridge, saw no point in taking this route. This was no personal idiosyncrasy on Kutta’s part, any more than the opposite response was a personal idiosyncrasy on Taylor’s part. Kutta was simply going with the flow or, at least, with the local flow. He was in a context where this course of action made sense and, if attended by a measure of success, would produce rewards rather than puzzlement.

Recall that August Foppl spoke of the necessity for the engineer to get answers, even if it meant using ideas that could not be justified within the broader framework of established, physical knowledge. One such scientifi­cally unjustifiable idea was that of the perfect, inviscid fluid. Here we have an explanation of the prima facie oddity that it was engineers who displayed the greater tolerance of ideal fluids, while the physicists proved intolerant of them. The engineer’s commitment to practicality does not mean that fictional concepts have to be shunned. Rather, it is the physicist’s commitment to truth that makes such fictions so unpalatable. In this respect, the engineer’s sense of necessity, and the shared awareness of this within the profession and its supporting institutions, generated a certain freedom of action. It meant that a pragmatic move or an expedient step did not meet with immediate criticism, provided its engineering utility could be demonstrated. On the contrary, if such a move contributed to the engineer’s project, it would be met with en­couragement. Encouragement was precisely what Finsterwalder gave to Kutta. Kutta’s willingness to venture where others would not venture becomes intel­ligible when set in the institutional context of technische Mechanik.

At eleven o’clock, on November 11, 1918, a cease-fire was declared on the west­ern front. The “war to end all wars” was over. European civilization would never be the same again, nor would aerodynamics. The military situation did not take Prandtl wholly by surprise, and he had already begun to explore the possibilities for a program of peacetime research.111 The comprehensive collapse of the German military effort meant that financial support for the Gottingen institute would all but disappear. At the height of the war Prandtl’s institute had employed around fifty people, but now some 60 percent of its staff had to be dismissed.112 At the very moment when the institutional and financial arrangements that had sustained German aerodynamic work were crumbling, the full scope of the Gottingen achievement was coming into view.

Toward the end of the war Prandtl finally brought together the overall theoretical picture that had been hinted at, and promised, but never pro­duced, for the readers of the Zeitschrift fur Flugtechnik. Even then it was not the readers of the Zeitschrift who would be the immediate beneficiaries. On April 18, 1918, Prandtl gave a comprehensive, and confidential, lecture on the theory of the lift and drag of an aircraft wing to the annual meeting of the Wisssenschaftlichen Gesellschaft fur Luftfahrt in Hamburg.113 A few months later, on July 26, 1918, Prandtl presented the first part of his classic paper “Tragflugeltheorie” to the Gottingen Academy of Science.114 It is the theory of the wing as laid out in these papers that I have described in this chapter.

In presenting his theory to the Academy, Prandtl prefaced the aerody­namic work with a highly abstract sketch of fluid-dynamic principles. Per­haps in deference to Hilbert and the Gottingen fashion for formal axiom systems he even offered two new “axioms” to be added to classical hydro­dynamics. (Axiom I stated that vortex layers can arise at lines of confluence. Axiom II was that infinite speeds cannot arise at protruding sharp edges of the body, or, if they do, only in the most limited way possible.) It is difficult to avoid the suspicion that these embellishments were added because Prandtl was conscious of addressing a high-status, scientific audience rather than an audience of engineers. Moritz Epple points out that the “axioms” Prandtl introduced do not justify the approximation processes that he used, nor do they operate as axioms in the way that Hilbert would understand them.115 The reference to “axioms” appears to have more to do with style than sub­stance. Furthermore, the general principles of fluid mechanics presented at the outset needed the help of drastic approximations before the theory of the wing could be presented in a recognizable and useful manner. As Prandtl introduced these approximations into his exposition, so the tone of the talk to the Academy changed. There was a shift from abstract principle to con­crete practice; from science to engineering; and from classical mechanics to technische Mechanik.116