Category The Enigma of. the Aerofoil

The Personnel of the ACA

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

The Personnel of the ACA

figure 1.3. Richard Tetley Glazebrook (1854-1935). Glazebrook was chairman of the Advisory Com­mittee for Aeronautics from its inception in 1909. A long-standing colleague of Rayleigh’s, he was also the head of the National Physical Laboratory. (By permission of the Royal Society of London)

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

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.

The Mathematical Definition of Circulation

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

The Mathematical Definition of Circulation
Подпись: (a)

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.

Подпись: a FIGURE 4.6. The circulation around an aerofoil is defined as the value of the integral |qcos0 ds where the integration is taken around any contour enclosing the aerofoil. The integration adds all the infinitesimal components of speed along the tangent.

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

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


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.


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.

Locating Kutta

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 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

Footprints in the Snow?

I want to dwell for a moment on the significance of this transition from the­ory to fact. It is a phase-change that has often taken place in the history of science. Assertions to the effect that, say, the blood circulates in the human body, or that water is made of hydrogen and oxygen, may once have been speculations but today can be taken as matters of fact. It would be a question­able use of language to keep calling them theories. How is this change of sta­tus, from theory to fact, best described? One attractive answer was provided in the form of a striking metaphor used by the Cambridge mathematician William Kingdon Clifford.34 Clifford, who had been a second wrangler in 1867, a fellow of Trinity, and a friend of Rayleigh, had wide-ranging interests in mathematical physics. In a lecture he gave in Manchester (some fifty years before the experiments that concern us), Clifford took as his example not hydrodynamics but the wave theory of light. This conception of light, he said, must now be accepted as fact. The difference between a theory and a demon­strated fact, he went on, “is something like this”:

If you suppose a man to have walked from Chorlton Town Hall down here say in ten minutes, the natural conclusion would be that he had walked along the Stretford Road. Now that theory would entirely account for all the facts, but at the same time the facts would not be proved by it. But suppose it happened to be winter time, with snow on the road, and that you could trace the man’s footsteps all along the road, then you would know that he had walked along that way. The sort of evidence we have to show that light does consist of waves transmitted through a medium is the sort of evidence that footsteps upon the snow make; it is not a theory which merely accounts for the facts, but it is a theory which can be reasoned back to from the facts without any other theory being possible. (117)

The thought is that if you can track a process in great detail, and see it de­velop step by step, then you can reach a stable understanding that is unique, unchallengeable, and enduring. Such an understanding, said Clifford, deals with facts, not theories.

Clifford’s metaphor may be a tempting one, but it cannot be wholly right, and it led Clifford himself astray. The development of physics showed that what he thought of as a demonstrated fact was actually a theory. The alleged impossibility of any alternative to the wave theory was refuted by the emer­gence of an alternative. In Clifford’s time the wave theory had superseded an older particle theory, but in 1905 Einstein once again postulated light par­ticles. These new-style particles or “photons” were invoked to explain the photoelectric effect, something that was proving difficult to understand in terms of waves. The photoelectric effect takes place when light is incident on a metal surface and releases electrons from the surface. How could the energy spread across a wave front be concentrated in the way necessary to release a charged particle? This was the problem Einstein’s theory was designed to answer. The light energy, he argued, was concentrated because light consists not of waves, but of particles, albeit particles with unusual properties.35 These developments took place after Clifford’s death. He cannot be blamed for not anticipating them, but they amount to a counterexample to Clifford’s argu­ment and show the need to introduce qualifications into his overconfident picture.

What was Clifford’s error? It was that of assimilating fundamental scien­tific inquiry to commonsense knowledge. While there are many similarities and connections, Clifford ignored a crucial difference. Everyone has seen, or could see, a man creating footprints in snow. The cause and the effect can be conjoined in experience, and both are open to inspection. This is the basis of subsequent inferences from footprints to their human causes and the basis of the conclusions that can be drawn about, say, the route someone had taken from Chorlton Town Hall to the location of Clifford’s lecture. The physicist, on the other hand, did not come to the wave theory of light by seeing light waves creating diffraction patterns or rainbows. The two things were not con­joined in experience in the way people and footprints have been conjoined. The inference to light waves did not have the same inductive basis as the com – monsense inference with which Clifford was comparing it.36

Clifford’s metaphor may have broken down for light waves, but it might still be applicable to Fage and Simmons’ achievement. It could be argued that Fage and Simmons were confronting the vortices and observing them bringing about their effects. Was not this conjunction precisely what the ex­periment was designed to expose? Even if the experimenters could not actu­ally see the flow of air, they could have made it visible, and others had done exactly that. In any case, the diagrams showing the streamlines of the vortices and the contour lines of equal vorticity allowed them to follow the path and development of the postulated vortices. The experimenters could set these diagrams side by side with the measured lift forces on the wing. Causal con­nections and correlations of phenomena that were originally speculative had, in a sense, been exposed to view, and the step-by-step progress of the cir­culation had been traced. Perhaps tracking the vortices through the pattern of measurements registered in Fage and Simmons’ diagrams was, after all, similar to tracking footprints in the snow.

If Clifford’s metaphor is applicable to the aerodynamic work, does this mean that the question “How does a wing produce lift?” can now be answered with the same level of certitude as the question about the man walking down the Stretford Road? In a sense, yes, it does. The phenomena of circulation, vortices, and lift had been made, or were on the way to being made, part of the routine and reality of daily life. At least, this was true for the laboratory life of some of the experimentalists working in this area. They were becoming increasingly familiar with the patterns in the data and the range of effects to be accounted for. Expectations were crystallizing, and experimenters were learning what they could take for granted. Techniques of calculation and pre­diction were becoming more confident and refined. What was once strange was becoming familiar and part of predictable, daily experience—like getting to know a new town. Learning to live with the theory of circulation was like learning to live in a new environment with new architectural styles and a new street plan. You want to get to Prof. Clifford’s lecture starting out from Chor – lton Town Hall? Then go down the Stretford Road! You want to calculate the induced drag? Then use Prandtl’s formula!37

Significantly, this was not yet how some of the most influential British experts saw the issue. They acknowledged that Fage and Simmons’ results represented a triumph of sorts for Lanchester, Prandtl, and Glauert, but they did not accept that the answer was now known to the question How is lift produced? On the contrary, they maintained that, despite the experimental advances and the increase in empirical knowledge, the answer to this ques­tion remained wholly unknown. Many questions, they acknowledged, had now been answered, but not this one. These experts were not simply being stubborn or blind in the face of mounting evidence, and their reaction un­derlines just how careful one must be in applying Clifford’s metaphor. It must be accepted that what looks like demonstrated fact from one point of view may appear less compelling or revealing from another point of view. This skeptical response to the mounting experimental evidence was articu­lated with great clarity by Richard Vynne Southwell (fig. 9.10). Southwell has already been mentioned in connection with the postwar contact with Prandtl and Gottingen, but it is appropriate to look more closely both at the man and at his response to the growing experimental literature.

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 Research Agenda

What was the scheme of work that was drawn up in the meeting at the Ad­miralty mentioned by Glazebrook? Some indication was given in the House of Commons when questions were taken on the founding of the commit­tee. More details, however, can be gleaned from a document titled “Prelim­inary Draft for Programme of Possible Experimental Work,” dated June 1, 1909, and used as the basis for an interim report during the first year of the committee’s activities.46 The program of work was divided into six sections: I, “General Questions on Aerodynamics”; II, “Questions Especially Relating to Aeroplanes”; III, “Propeller Experiments”; IV, “Motors”; V, “Questions

Especially Related to Airships”; and VI, “Meteorology.” For our purposes, only the first two sections are of interest; within them, some fourteen distinct topics were identified. They repay scrutiny and call for some comment. The list of topics was as follows:

I. General Questions on Aerodynamics

1. Determination of the vertical and horizontal components of the forces on inclined planes in a horizontal current of air, especially for small angles of inclination to the current.

2. Determination of surface friction on plates exposed to currents of air.

3. Centre of pressure for inclined planes.

4. Distribution of pressure on inclined planes.

5. Pressure components, distribution of pressure for curved surfaces of various forms.

6. Resistance to motion of bodies of different shapes = long and short cylinders &c.

7. Combination of planes: effect on pressure components of various ar­rangements of two or more planes.

II. Questions Especially Relating to Aeroplanes

8. Resistance components of aeroplane models.

9. Resistance of struts and connections.

10. Resistance of different stabilising planes both horizontal and vertical.

11.Problems connected with stability.

(i) Mathematical investigation of stability.

(ii) The stability of aero curves of different section and plane.

(iii) Effect of stabilising planes.

(iv) Effect of sudden action.

(v) Effect of gusts of wind.

(vi) Investigations as to stability of models for different dispositions of weight etc.

12. Materials for aircraft construction.

13. Consideration of different forms of aeroplane: monoplane, biplane, etc.

14. Other forms of heavier than air machines, helicopter, etc.

The list conveys the range of problems confronting the committee but also something of its priorities and preferences. The emphasis to be placed on sta­bility stands out clearly in the degree of definition accorded to the problem, which is carefully divided into six subsections. Likewise, the scientific style of the approach is clear. Experimentally, a significant amount of the work was to be done with models while, theoretically, the complexity of the real flying machine was replaced by simplified concepts such as planes and cylinders and centers of pressure.47 The operation of an aircraft was being assimilated to the abstract categories familiar to the committee from their Tripos text­books on mechanics and hydrodynamics.

New Approaches to Ideal Fluid Theory

Something was badly wrong with the picture of air behaving and moving as an ideal fluid. It was mathematically impressive but empirically defective. What exactly was wrong? Was it the assumption of zero viscosity itself that should be dropped or were there perhaps other, unnoticed, assumptions at work in the picture of the flow that might be the cause of the trouble? What about Laplace’s equation and the assumption of irrotational motion?

This question was addressed by a number of late nineteenth-century ex­perts whose investigations greatly deepened the understanding of ideal fluid theory. They began to explore some possibilities that previously had been neglected. But why did they not simply abandon ideal-fluid theory as em­pirically false and turn directly to the analysis of viscous fluids? Attempts were made to do this but with very limited success. The reason was that the mathematics of viscous fluids was so difficult. It was possible to write down the equations of motion of a viscous fluid by taking into account the trac­tion forces along the surface of the fluid element, but it was another matter to solve the equations. The full equations of viscous flow are now called the Navier-Stokes equations, though in Britain they used to be called just the Stokes equations. Of course, they contain a term involving the symbol Ц standing for the coefficient of viscosity. If the value of this coefficient is set at zero to symbolize the absence of viscosity, that is, Ц = 0, the Navier – Stokes equations turn back into the Euler equations that have been described earlier in this chapter. In a later chapter I look more closely at the status of the Navier-Stokes equations and the different responses to their seemingly intractable nature. For the moment it is only necessary to appreciate the problem they posed. No one could see how to solve and apply the equations except in a few simple cases. Because of their intractability, any attempt to avoid the impasse thrown up by the zero-resultant theorem had to be one that stayed with the Euler equations and thus within the confines of ideal- fluid theory.

The crucial insight that permitted the further development of ideal-fluid theory was provided by Helmholtz and Kirchhoff in Germany and Rayleigh in Britain. These men realized that the solutions to the Euler equations that gave the streamlines around an obstacle were not unique. More than one set of streamlines were possible and consistent with the equations. More than one kind of flow could satisfy the equations and meet the given boundary conditions. What is more, some of these flows could generate a resultant force. There were in fact two very different kinds of flow that might have this desired effect and, in principle, allow the zero-resultant outcome to be evaded. Rayleigh contributed to the study of both. Both approaches involved the limited introduction of fluid elements that possessed rotation and vor – ticity. The strict condition of irrotational motion was dropped. On one ap­proach this involved the introduction of just one singular point in the flow that rotated and constituted the center of a vortex. On the other approach a sheet or surface of vorticity was postulated. In both cases the remainder of the flow was still irrotational. These two approaches provided, respectively, the basis for the two different theories of lift that I mentioned in the introduction and called the circulatory or vortex theory of lift and the discontinuity theory of lift. Historically, the first of the two approaches to be developed in detail was the one that led to the discontinuity theory. I now introduce the ideas underlying this approach. The other approach and the other theory of lift are introduced in chapter 4.