Category The Enigma of. the Aerofoil

Paying the Price of Simplification

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

Paying the Price of Simplification

figure 2.6. Continuous flow of an ideal fluid around an inclined plate. From Tietjens 1929, 161. (By permission of Springer Science and Business Media)

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.

Prandtl and Fuhrmann on Airship Resistance

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

‘SOLIDIFIED’ STREAM LINE

Prandtl and Fuhrmann on Airship Resistance

figure 5.2. A single source in a uniform flow of ideal fluid creates a flow pattern similar to that around a blunt-nosed object of infinite length. The surface of the object consists in an appropriate streamline of the flow that is imagined to be solidified.

‘SOLIDIFIED’ STREAM LINE

Prandtl and Fuhrmann on Airship Resistance

figure 5.3. Fuhrmann and Prandtl used a system of sources and sinks placed in a uniform flow of ideal fluid to simulate the flow around a closed solid. An appropriate distribution of sources and sinks produces streamlines similar to those over an airship.

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.

At the Eleventh Hour

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

Policies and Compromises

In 1954 Philipp Frank published an article in the Scientific Monthly called “On the Variety of Reasons for the Acceptance of Scientific Theories.”50 He drew the striking conclusion that “the building of a scientific theory is not essentially different from the building of an airplane” (144). I will use Frank’s argument to comment on the theories developed in fluid dynamics and aero­dynamics, but first I should say a little about Frank himself.51 From 1912 to 1938 he was the professor of theoretical physics at the German University of Prague. A pupil of Boltzmann, Klein, and Hilbert, Frank had taken over the chair from Einstein when Einstein received the call to Zurich and then to Berlin. He had attended Einstein’s seminars in Prague, and Einstein strongly supported his appointment.

In their student days, before World War I, Frank and von Mises talked philosophy in their favorite Viennese coffeehouse and together played a seminal role in the formation of the Vienna Circle.52 In the interwar years, as established academics, Frank and von Mises jointly edited a book on the differential and integral equations of mechanics and physics, Die Differential – und Integralgleichungen der Mechanik und Physik,53 which brought together a range of distinguished contributors. Von Mises edited the first volume on mathematical methods, while Frank handled the second, more physically oriented, volume which included chapters by Noether, Oseen, Sommerfeld, Trefftz, and von Karman, who wrote on ideal fluid theory.54 The Frank-Mises collection, which was an update of a famous textbook by Riemann and Weber, established itself as a standard work in German-speaking Europe.55 In 1938 Frank was forced to leave Prague because of the threatening political situation in Europe, and he went to the United States. During and after World War II, he taught physics, mathematics, and the philosophy of science at Harvard.

Like that of von Mises, Frank’s philosophical position was self-consciously “positivist” in the priority given to empirical data and the secondary, instru­mental role given to theoretical constructs. Frank admired Ernst Mach as a representative of Enlightenment thinking, though his admiration was not uncritical, and he did not go along with Mach’s rejection of atomism.56 Much of Frank’s philosophical work was devoted to the analysis of relativity theory, quantum theory, and non-Euclidian geometry.57 He was a firm believer in the unity of science and rejected the idea that there was a fundamental divide between the natural and human sciences.58 He also insisted on the need to understand science as a sociological phenomenon. The sociology of science was part of “a general science of human behaviour” (140)—a theme central to the Scientific Monthly article.59

Frank asserted that most scientists, in their public statements, assume that two, and only two, considerations are relevant when assessing a scientific the­ory. These are (1) that the theory should explain the relevant facts generated by observation and (2) that it should possess the virtue of mathematical simplic­ity. Frank then noted that, historically, scientists (or those occupying the role we now identify as “ scientist”) have often used two further criteria. These are

(3) that the theory should be useful for technological purposes and (4) that it should have apparent implications for ethical and political questions. Does the theory encourage or undermine desirable patterns of behavior, either in society at large or in the community of scientists themselves? Such questions are often presented in a disguised form, for example, Is the theory consistent with common sense or received opinion or does it flout them? Common sense and received opinion, Frank argued, typically fuse together a picture of nature and a picture of society. The demand for consistency then becomes a form of social control that can be used for good or ill.

In Frank’s opinion it is naive to believe that theory assessment can be confined to the two, internal-seeming criteria. He offered three reasons. First, he noted that no theory has ever explained all of the observed facts that fall under its scope. Some selection always has to be made. Second, there is no unproblematic measure of simplicity. No theory has “perfect” simplic­ity. Simplicity will be judged differently from different, but equally rational, perspectives, depending on background knowledge, goals, and interests. Third, criteria (1) and (2) are frequently in competition with one another. The greater the number of facts that can be explained, or the greater the ac­curacy of the explanation, the more complicated the theory must be, while the simpler it is, the fewer are the facts that can be explained. Linear functions are simpler than functions of the second or higher degree, which is why phys­ics is full of laws that express simple proportionality, for example, Hooke’s law or Ohm’s law. “In all these cases,” wrote Frank, “there is no doubt that a nonlinear relationship would describe the facts in a more accurate way, but one tries to get along with a linear law as much as possible” (139-40). What is it to be: convenience or truth? Nothing within the boundaries of science itself, narrowly conceived, will yield the answer. This is why scientists have always moved outside criteria (1) and (2), and, consciously or unconsciously, invoked criteria of types (3) and (4).

These unavoidable choices and compromises tell us something about the status of any theory that is accepted by a group of scientists. “If we consider this point,” said Frank, “it is obvious that such a theory cannot be ‘the truth’” (144). But if the chosen theory is not “the truth,” what is it? Frank’s answer was that a theory must be understood to be “an instrument that serves to­ward some definite purpose” (144). It is an instrument that sometimes helps prediction and sometimes understanding. It can help us construct devices that save time and labor, and it sometimes helps to mediate a subtle form of social control. “A scientific theory is, in a sense, a tool that produces other tools according to a practical scheme” (144), he concluded. Like a tool, its connection to reality is not to be understood in terms of some static relation of depiction but in active and pragmatic terms. Its function is to give its us­ers a grip on reality and to allow them to pursue their projects and satisfy their needs—but it does so in diverse ways. It was at this point that Frank produced his comparison between assessing a theory in science and assessing a piece of technology, such as an airplane. Writing, surely, with the perfor­mance graphs of von Mises’ Fluglehre before his mind, he argued:

In the same way that we enjoy the beauty and elegance of an airplane, we also enjoy the “elegance” of the theory that makes the construction of the plane possible. In speaking about any actual machine, it is meaningless to ask whether the machine is “true” in the sense of its being “perfect.” We can ask only whether it is “good” or sufficiently “perfect” for a certain purpose. If we require speed as our purpose, the “perfect” airplane will differ from one that is “perfect” for the purposes of endurance. The result will be different again if

we chose safety. . . . It is impossible to design an airplane that fulfils all these purposes in a maximal way. (144)

It is the trade-off of one human purpose against another that gave Frank his central theme. Only by confronting this fact can the methods of science be understood scientifically. It is necessary to ask in the case of every scientific theory, as one asks in the case of the airplane, what determined the policy according to which these inescapable compromises are made and how well does the end product embody the policy? We must understand what Frank called, in his scientistic terminology, “the social conditions that produce the conditioned reflexes of the policy-makers” (144).60

In Frank’s terms, Lanchester’s metaphor of playing chess with nature as well as my sociological analysis are ways of describing scientific “policies.” Just as there were policy choices made over the relative importance of stabil­ity and maneuverability, and policy choices about how to distribute research effort between the theory of stability and the theory of lift, so within the pur­suit of a theory of lift there were policy decisions to be made. My analysis identifies one policy informing the Cambridge school and another policy guiding the Gottingen school. Again using Frank’s terms, the members of the respective schools constructed different technologies of understanding, that is, different theoretical “instruments.” Their policies, when construct­ing their theories, maximized different qualities and furthered different ends. The British wanted to construct a fundamental theory of lift, whereas the Germans aimed at engineering utility. Who were the “policy makers”? One might identify, say, Lord Rayleigh as the “policy maker” in Britain and Fe­lix Klein as the “policy maker” in Germany, but there is no need to assume that policy is made by individuals. Such a restriction would not correspond to Frank’s intentions; nor is it part of my analysis. Policies can emerge col­lectively. They can be tacitly present in the cultural traditions and research strategies of a scientific group. One could then say that everyone is a policy maker by virtue of their participation in the group, or one could say that the policy maker is the group itself. In my example the “social conditions” that determine the “conditioned reflexes of the policy-makers” reside in the divi­sion of labor between physicist and engineer.

One implication of Frank’s “policy” metaphor is that a stated policy need not correspond to an actual policy. The devious history of aircraft construc­tion in post-World War I Germany provides some obvious examples. Is this large aircraft really meant as an airliner or is it a bomber? Is this an aero­batic sports plane or a disguised fighter? Is all this enthusiasm for gliding just recreation or a way of training a future air force—and keeping the nation’s aerodynamic experts in a job?61 The difficulty of distinguishing a real from an apparent policy comes from the problematic relation between words and deeds. Sometimes the self-descriptions and methodological reflections of members of the Cambridge school could sound similar to those of German engineers. Both Lamb and Love occasionally invoked the ideas, and some­times the name, of Ernst Mach, but that did not make Lamb into a positivist, nor turn Love’s work on the theory of elasticity into technische Mechanik. Their real policy lay elsewhere.

In an address to the British Association in 1904, Lamb acknowledged that the basic concepts of physics, geometry, and mechanics were “contrivances,” “abstractions,” and “conventions.”62 But Lamb soon left behind this unchar­acteristic indulgence in philosophizing and turned the discussion back to the work of his old teacher, G. G. Stokes. He spoke warmly of “the simple and vigorous faith” that informed Stokes’ thinking.63 Lamb then raised the metaphysical question of what lay beyond science and justified faith in its methods. Why, as Lamb put it, does nature honor our checks? He gave no explicit answer, but the theological hint was obvious. Lamb also distanced himself and the Cambridge school from the “more recent tendencies” in ap­plied mathematics. He deplored the fragmentation of the field and regretted the passing of the large-scale monograph, which was a work of art, in favor of detailed, specialized papers. What differentiated the Cambridge school, he went on, related “not so much to subject-matter and method as to the gen­eral mental attitude towards the problems of nature” (425). It is this “general mental attitude” that constitutes the real policy.

How is an authentic “mental attitude” to be filtered out from misleading forms of self-description? The answer is: by looking at what is done and at the choices that are made. Words must be supported by actions. Bairstow, Cow­ley, Jeffreys, Lamb, Levy, Southwell, and Taylor not only gave their reasons for resisting the ideal fluid approach to lift, but they acted accordingly. This is why, in previous chapters, I have identified the mental attitude that informed the work of the Cambridge school and its associates as a confident, physics – based realism rather than a skeptical positivism. Stokes’ equations were not only said to be true, but they were treated as true. This was the attitude and policy that Love expressed by invoking the role of the “natural philosopher” rather than the engineer. And this was why Felix Klein, in his 1900 lecture on the special character of technical mechanics, could express admiration for Love’s treatise on elasticity and yet pass over it because it could not be taken as an example of technische Mechanik.64

Simplicity and the Kutta-Joukowsky Law

I now apply Frank’s ideas to the Kutta-Joukowsky law: L = p LT, where the lift (L) is equated to the product of the density (p), the speed (U), and the circulation (Г). The law is certainly simple, but what is the meaning of this simplicity? Is it a sign of the “deep” truth of the law and hence a quality that should command a special respect? The idea that nature is “governed” by simple mathematical laws is a familiar one—it goes back to the origins of modern science—but positivists have no time for this sort of talk.65 Frank could have pointed out that the simplicity, and apparent generality, of the Kutta-Joukowsky law derives not from its truth, but from its falsity and from everything that it leaves out of account. The law says nothing about the rela­tion between the shape of the aerofoil and the amount of lift. It contributes nothing to the problem of specifying the amount of circulation and (when used in conjunction with the Kutta condition) gives predictions for the lift that are consistently too high. The law cannot, in any direct or literal way, represent something deep within reality because its individual terms do not refer to reality. They refer to a nonexistent, ideal fluid under simplified flow conditions.

Frank would predict that if an attempt were made to repair the law, and make allowance for some of the factors that have been ignored, then the re­sult would no longer possess the impressive simplicity of the original. This was precisely what happened when, in 1921, Max Lagally of the technische Hochschule at Dresden, produced an extension of the Kutta-Joukowsky for­mula.66 Lagally exploited a result arrived at previously by Heinrich Blasius, one of Prandtl’s pupils, and this result needs to be explained first in order to make sense of Lagally’s formula. Blasius had developed a theorem, based on the theory of complex functions, that allowed the force components X and Y on a body to be written down as soon as the mathematical form of the flow of an ideal fluid over the body had been specified.67 In these terms, for a uniform, irrotational flow U along the x-axis, with a circulation Г, the Kutta – Joukowsky theorem takes the form

X – iY = ipU Г.

Here X, the force along the x-axis, represents the drag, while Y is the force along the y-axis and represents the lift. The letter i is a mathematical opera­tor. The right-hand side of the equation economically conveys the informa­tion that the drag is zero (because X = 0) and also that the lift obeys the Kutta-Joukowsky relation (because Y = pU Г). Blasius’ derivation of this re­sult depended on there being no complications in the flow. Lagally added some complications in order to see what effect they would have. In Lagally’s analysis the main flow has a horizontal component U and a vertical compo­nent V. More important, he assumed that there were an arbitrary number of sources and an arbitrary number of vortices in the fluid around the body. He specified that there were r sources located at the points ar where each source had a strength mr, and s vortices located at points cs where each vortex had vorticity Ks. When the formula was adjusted to allow for these conditions, it looked like this:

X — iY = —ipK(U — iV) + 2np^mr (u r — iv r + U — iV) — ip^Ks (us — ivs +U — iV),

where ur and vr are the components of velocity at ar (omitting the contribu­tion of mr) and us and vs are the velocity components at cs (omitting the con­tribution of Ks). The original Kutta-Joukowsky formula can be seen embed­ded in Lagally’s formula on the immediate right of the equality sign.68

If the original Kutta-Joukowsky relation could be admired for its elegance, like the sleek lines of a modern aircraft, can this be said of Lagally’s formula? I doubt if it attracted much praise on this score. But if the long formula really is an improvement on the short one, why shouldn’t it be seen as more beautiful? If we do not find it beautiful is it because we can’t imagine such complicated mathematical machinery “governing” reality? Frank and his fellow positivists would not want the question to be pursued in these metaphysical terms. They would say: If there is something important about the simplicity of the origi­nal formula L = p UT, then look for the utility that goes with simplicity. What does it contribute to the economy of thought? This question will expose the real attraction of simplicity and explain what might have been lost, along with what has been gained, by Lagally’s generalization.

Frank called a theory a tool that produces other tools according to a practi­cal scheme. He meant that the simple law provides a pattern, an exemplar, and a resource that is taken for granted in building up the more complex formula.69 This is how Lagally built his generalization, and if Frank is right, other scientists and engineers, interested in a different range of special condi­tions, will follow a similar path. This pattern fits what I have found. Recall the way Betz experimentally studied the deviations between the predictions of the circulatory theory and wind-tunnel observations. He sought to close the gap between theory and experiment by retaining the Kutta-Joukowsky law while relaxing the Kutta condition, that is, the understanding that the circula­tion is precisely the amount needed to position the rear stagnation point on the sharp trailing edge. Again, recall the later episode in which, prompted by the work of G. I. Taylor, the condition of contour independence was relaxed so that a “circulation” could be specified for a viscous flow. In both these examples the development exploited the same resource as Lagally, that is, the simple law was retained as a basic pattern. Simple laws are a shared resource and an accepted reference point. They are used when a group of scientists are striving to coordinate their behavior in order to construct a shared body of knowledge. They are salient solutions to coordination problems, which may explain the obscure “depth” attributed to them. The depth is a social, not a metaphysical, depth.70

The enigma of the aerofoil

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

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

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

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

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

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

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

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

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

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

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

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

Neo-Newtonianism and the “Sweep” of a Wing

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

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

P = pAV2 sin2 0.

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

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

Section iii. the circular curved surface

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

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

Section iii. the circular curved surface

mations to map the flow around a circular cylinder onto the flow around a circular arc representing the wing of Lilienthal’s glider. (By permission of Dr. Audrey Glauert)

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

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

Section iii. the circular curved surface

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

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

Подпись: t = іZ+1

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

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

Circulation = 4nVr sin—sin I —+B,

2 ^ 2 H)

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

The Laws of Prandtl and the Laws of Nature

Prandtl was not vastly outstanding in any one field, but he was eminent in so many fields. He understood mathematics better than many mathematicians do.

max munk, “My Early Aerodynamic Research” (1981)1

After Glauert and McKinnon Wood had presented the reports on their Got­tingen visit, discussions continued in the Aeronautical Research Committee as the British experts sought to mobilize a collective response to the Ger­man wartime achievements. These (sometimes sharp) exchanges took place in the monthly meetings of the committee and its subcommittees that were held in London. The Cambridge contingent made the journey to London together by train and engaged in lively aeronautical debate en route. “I fear we must have been a pest to our fellow travellers,” recalled one.2 The upshot of the committee meetings are recorded not only in the minutes of their dis­cussions but also in the confidential technical reports circulated among the participants. The content of the technical reports sometimes surfaced in the published Reports and Memoranda issued by the committee and sometimes, in the case of especially important results, in leading scientific journals. A number of the main experiments done in this period appeared in the Philo­sophical Transactions of the Royal Society and in the Proceedings of the Royal Society. There were some significant and perplexing changes in the analysis of the experimental material as the data made the journey from the private to the public realm.

I have described how Taylor, in his 1914 Adams Prize essay, had dismissed Lanchester’s idea that the flow of air over a wing was describable in terms of a perfect fluid in irrotational motion with circulation. If Prandtl was right, then Taylor had been wrong. Led by Glauert, the postwar argument in the Aero­nautical Research Committee seemed to be going in Prandtl’s direction. The circulation theory was gaining ground. By 1923 Glauert felt able to write to Prandtl to tell him that his “aerofoil theory has certainly aroused much inter­est here and it would not be an exaggeration to say that it has revolutionised

many of our ideas.”3 But Taylor (see fig. 9.1) was not to be easily convinced that his earlier reservations had been misplaced. In the postwar discussions, he made it his job to scrutinize Glauert’s reasoning and to oppose it whenever he detected a logical gap or a questionable premise.

The Air as an Ideal Fluid:. Classical Hydrodynamics and the. Foundations of Aerodynamics

The following investigations proceed on the assumption that the fluid with which we deal may be treated as practically continuous and homogeneous in structure; i. e. we assume that the properties of the smallest portions into which we can conceive them to be divided are the same as those of the substance in bulk.

Horace lamb, Treatise on the Mathematical Theory of the Motion of Fluids (1879)1

Let me now prepare the ground for an account of the theory of lift and drag. The disputes over the correct analysis of lift and drag provide the central topic of this book. It was here that the scientists and engineers who addressed the new problems of aerodynamics called upon the highly mathematical techniques of what used to be called, simply, “hydrodynamics.” The modern label, which better captures the true generality of the subject, is “fluid dynam­ics.” Fluid dynamics provided the intellectual resources that were common to both the British and German work on lift and drag, although the stance toward that common heritage was often very different in the two cases. It is vital to have a secure sense of what the two groups of experts were disagree­ing about. The present chapter is a description of this common heritage and these shared resources. It is meant to provide background and orientation. In it I do my best to explain the basic concepts in simple terms, though this hardly does justice to the ideas and techniques that are mentioned. I sketch some of the initial, mathematical steps that went into their construction in order to convey something of the style and feel of the work. At the end of the chapter, I summarize the main points in nonmathematical terms.

Two members of the British Advisory Committee for Aeronautics—Lord Rayleigh and Sir George Greenhill—made important contributions to the field of hydrodynamics in the 1870s and 1880s. The numerous references to papers and results by Rayleigh and Greenhill in the standard textbooks of hydrodynamics of that time, for example, Lamb’s Hydrodynamics, attest to their prominence in the field.2 Rayleigh had arrived at some classical re­sults, which are described later in this chapter, and Greenhill had written

the authoritative article on hydrodynamics in the eleventh edition of the Encyclopaedia Britannica. Significantly the encyclopedia had two lengthy and detailed entries that dealt with fluid flow. One was the article titled “Hydro­mechanics” written by Greenhill; the other article was titled “Hydraulics” and was written by a distinguished engineer.3 The former presentation was filled with mathematics, while the latter was filled with descriptions and dia­grams of turbine machinery. The reason it was felt necessary to recognize this division of labor in drawing up the encyclopedia is relevant to my story and will become clear in what follows.

A Firm Basis in Physics

Further objections to the circulation theory came from G. I. Taylor, one of Cambridge’s most brilliant young applied mathematicians. I have already mentioned his Adams Prize essay of 1914.60 In that work Taylor did not con­fine himself to rejecting discontinuity theory; he also rejected the circulatory account of lift. Critical of the unreality of the textbook hydrodynamics that Bryan so admired, he argued that “the important thing in the earliest stages of a new theory in applied mathematics is to establish a firm basis in physics” (preface, 5). After describing the central idea of Rayleigh-Kirchhoff flow and pointing out its empirical shortcomings, Taylor turned briefly to Lanchester’s theory. This too was faulted because of its lack of a firm basis in physics. Tay­lor’s dismissal of Lanchester was swift: “Besides these [discontinuity] theories of the resistance of solids moving through fluids, Mr Lanchester has pro­posed the theory that a solid moving through a fluid is surrounded by an

irrotational motion with circulation. This theory, as far as I can see, has noth­ing to recommend it, beyond the mere fact that it does give an expression for the reaction between the fluid and the solid” (4-5).

All that was granted to the theory, in its two-dimensional form, was that it had the (minimal) virtue of avoiding d’Alembert’s paradox. It permitted the researcher to deduce “an expression” for the resultant force on the body, but that is all. The formula, however, was not, in Taylor’s opinion, grounded in a real physical process. The theory provided no understanding of the mechanism by which the circulation round the body could be created. The problem came from Kelvin’s proof that circulation can neither be created nor destroyed. If Lanchester’s theory was an exercise in perfect fluid theory, then the premises of the theory precluded the creation of the very circulation on which it depended. Setting a material body in motion in a stationary fluid would not create such a flow. An aircraft, starting from rest on the ground in still air, and moving with increasing speed along the runway, would never generate the lift necessary to get into the air (not, at least, if the air was mod­eled as an ideal fluid). This consequence put Lanchester’s theory in no less an embarrassing position than discontinuity theory. As far as it described any reality, discontinuity theory was a picture of a stalled wing, that is, of an aircraft dropping out of the sky. If Taylor was right, Lanchester’s theory was equally hopeless because it would leave the aircraft stranded on the ground and incapable of flight.

Taylor thought Lanchester’s theory was, if anything, worse than the ver­sion of perfect fluid theory that generates d’Alembert’s paradox, that is, the version in which the perfect fluid has neither discontinuities nor circulation. Referring to this version as the “ordinary” hydrodynamics of an irrotational fluid, Taylor said that it, at least, gave a rigorous picture of the flow that would arise if an object were moved in these hypothetical circumstances, though, of course, this picture bore “no relation whatever” to reality. “The advantages of the ordinary irrotational theory is that it does, at least, represent the motion that would ensue if the solid were moved from rest in an otherwise motion­less perfect fluid, and if there were perfect slipping at the surface. By taking irrotational circulation round the solid, Mr Lanchester loses the possibility of generating the motion from a state of rest by a movement of the solid” (5). Taylor drew the conclusion that “in searching for an explanation of the forces which act on solids moving through fluids, it is useless to confine one’s atten­tion to irrotational motion” (5).

The correct strategy, Taylor argued, is to address flows where the fluid elements possess rotation as a result of viscosity and friction (6). In this way turbulence and eddying might be brought into the picture so that a physi­cally realistic fluid dynamics could emerge. Taylor was aware that the direct deduction of turbulent and eddying flow, starting from the full Stokes equa­tions of viscous flow, presented insuperable obstacles. Progress would be im­possible “if one were to adhere strictly to the equations of motion, without any other assumptions” (11). He therefore proposed to begin by a “guess at some result which I think would probably come out as an intermediate step in the complete solution of the problem” (11). On the basis of this guess he would deduce consequences that could be tested by experiment, and if “the observations fit in with the calculation I then go back to the assumptions and try to deduce it from the equations of motion” (11-12).

Taylor’s reaction to Lanchester depended on his assimilating Lanchester’s analysis to the classical framework of perfect fluid theory, that is, to the equa­tions of Euler and Laplace’s equation. The brevity of the argument attests to the taken-for-granted character of this assimilation. It must have seemed ob­vious that this is what Lanchester was presupposing. There was no hesitation or qualification, nor any suggestion that alternative readings were available. Admittedly, due to the sudden onset of war, Taylor did not have Lanchester’s book in front of him.61 He was recalling the essential point of the theory, and this involved the irrotational flow of a perfect fluid with a circulation. As such, the theory fell under the scope of Kelvin’s theorem and hence could never cast light on the creation of the circulation.

Lanchester was aware of the theorem (which he called Lagrange’s theo­rem) that rotation and circulation within a continuous body of ideal fluid can be neither created nor destroyed. He even expressed the point with a striking analogy. Once created, he said, a vortex of perfect fluid, unlike a real vortex, would “pervade the world for all time like a disembodied spirit” (175). He knew this meant that an infinite (that is, two-dimensional) wing starting from rest and moving within an initially stationary ideal fluid cannot then generate a circulation. He was prepared to face the consequences. “It is, of course, con­ceivable,” he said, “that flight in an inviscid fluid is theoretically impossible” (172). As an engineer working with real fluids, such as air and water, he hardly expected mathematical idealizations to be accurate. The important thing was to learn what one could from the idealized case but not to be imposed on by it. As he remarked ruefully, “The inviscid fluid of Eulerian theory is a very peculiar substance on which to employ non-mathematical reasoning” (118). Discussing the “two parallel cylindrical vortices” that trail behind the tips of a finite wing, he accepted that the mechanics of their creation would not be illuminated by standard hydrodynamic theory: “for such vortex mo­tion would involve rotation, and could not be generated in a perfect fluid without involving a violation of Lagrange’s theorem. . . . In an actual fluid this objection has but little weight, owing to the influence of viscosity, and it is worthy of note that the somewhat inexact method of reasoning adopted in the foregoing demonstration seems to be peculiarly adapted, qualitatively speaking, for exploring the behaviour of real fluids, though rarely capable of giving quantitative results” (158). For Lanchester, the mathematical apparatus of classical hydrodynamics played a subsidiary and illustrative role. It was merely a way of representing some of the salient features of the flow. Nothing of this complex, if informal, dialectic linking ideal and real fluids found any recognition in Taylor’s characterization.

Taylor’s response to Lanchester remained unpublished, but it tells us something about the assumptions of some of Lanchester’s readers. If Taylor read the work in this way, then presumably others will have read it in a similar way. The case is different with the next objection. It was not made in private but was very public and was acted out before a large audience at one of the major professional institutions in London.