Category The Enigma of. the Aerofoil

If Prandtl had never turned his attention to wing theory he would still have occupied a significant position in the history of fluid dynamics. In 1904, at the International Congress of Mathematicians, held that year in Heidelberg, Prandtl had delivered a brief paper called “Uber Flussigkeitsbewegung bei sehr kleiner Reibung” (On fluid motion in fluids with very small friction).4 In this paper he introduced the now famous concept of the boundary layer. At the time, the full significance of the work escaped most of the audience, though not Felix Klein.5 Much later the Heidelberg paper came to be seen as one of the most important contributions to science that was made during the twentieth century.6 It has been likened in its impact to Einstein’s 1905 paper on the theory of relativity.7 The significance of Prandtl’s work was that it provided a bridge—a long-sought-for bridge—that connected the behav­ior of real, viscous fluids and the unreal, inviscid fluid of previous math­ematical theory. There had always been a gap between the Stokes equations, which appeared to be true but unsolvable, and the Euler equations, which were known to be solvable but untrue. This logical gap had profound meth­odological consequences. It attenuated the link between the mathematical hydrodynamics of the lecture theater and the engineering hydraulics of the workshop. It undermined hope in the unity of theory and practice. Prandtl’s boundary-layer theory restored that hope. Figure 7.1 shows Prandtl at work on his boundary-layer research.

The theory of the boundary layer can be broken down into four parts: (1) an underlying physical model, (2) an implied technology of control, (3) a mathematical formulation of the model and the technology, and (4) a heuris­tic resource. I briefly describe each of these dimensions of the theory.

The physical model expressed the idea that, in a fluid of small viscosity, the effects of viscosity arise in, and are often confined to, a thin layer that is in contact with a solid boundary. In the vicinity of the boundary, the fluid layer possesses a sharp velocity gradient. On the actual surface of the body along which the fluid is moving (for example, a wing or the walls of a chan­nel), the fluid is stationary. A short distance away it achieves the velocity of the free stream. The velocity gradient in the Ubergangsschicht, or transition layer as Prandtl called it, is shown diagrammatically in figure 7.2 (taken from the 1904 paper). As long as the fluid within the layer has the kinetic energy to overcome any adverse pressure gradient, then the boundary layer will con­form to the surface along which it is flowing. If it meets too great a pressure, then a backflow will set in and the flow will separate from the surface. This process is shown in Prandtl’s diagram. The intense vorticity of the fluid in

the boundary layer will then diffuse into the surrounding flow and alter its general character.

The boundary-layer theory thus encompassed the phenomenon of flow separation, which had intrigued Prandtl from his early days as an engineer in industry when he had worked on suction machinery.8 For Prandtl, as an engineer, the question was how to stop separation and improve the ef­ficiency of the suction effect. A significant part of the 1904 paper implicitly bore upon this engineering problem because it was devoted to the question of boundary-layer control. Prandtl reasoned that if the boundary layer could be removed, then it could not detach itself and modify the rest of the flow. He therefore constructed an apparatus to explore this effect. It consisted of
a hollow cylinder with a slit along one side. The cylinder was inserted in a flow of water and, by means of a suction pump, some of the fluid from the boundary layer was drawn through the slit. The result was that on the side of the cylinder with the slit, the remaining flow stayed close to the surface of the cylinder. As predicted, it did not detach itself and cause vorticity and turbulence in the surrounding fluid. Prandtl presented his Heidelberg audi­ence with photographs of this process to show them the difference made by the intervention.9

Prandtl was able to express the ideas underlying this process in a math­ematical form. He gave the equations of motion for the fluid elements in the boundary layer. He did so by reflecting on the orders of magnitude of the forces and accelerations of the flow in the boundary layer as the viscosity approached zero.10 This line of thought told him which quantities could be ignored in the original Stokes equations governing viscous fluids. It led to a simplification of the equations that did not involve wholly ignoring either the viscous forces or the inertial forces. It proved possible to keep them both in play. Prandtl thus managed to simplify the Stokes equations without simpli­fying them too much. Consider the two-dimensional flow of an incompress­ible fluid in a boundary layer that flows horizontally, that is, along the x-axis. After his simplification Prandtl was left with two equations that described the flow of fluid in the boundary layer by specifying the respective velocity com­ponents, u and v, in the x and y directions. If p is the density, p the pressure, and p the viscosity, then Prandtl was able to write

and

On the basis of these two equations Prandtl worked out an approximate, but reasonable, value for the drag on a horizontal plate acting as the solid bound­ary along which the fluid was flowing. He was also able to arrive at an expres­sion giving the thickness of the boundary layer and show that the thickness approached zero as the viscosity approached zero. In 1908, in a Ph. D. thesis supervised by Prandtl, Blasius fully solved the boundary-layer equations for the case of the flat plate and improved on the original estimate of the drag.11 Other Gottingen doctoral students—Boltze, Hiemenz, and Toepfer—refined Blasius’ procedure and extended the analysis to circular cylinders and bodies
of rotation.12 Although work on the boundary layer began slowly and, for a decade, was confined to Gottingen and the circle around Prandtl, the theory gradually became the focus of extensive empirical and theoretical research in Europe and America. The idea of the boundary layer eventually found appli­cation in every branch of technology where fluid dynamics plays a role.13

Given this idea’s wide applicability, it is worth noting some of the logical characteristics of Prandtl’s equations and reflecting on their methodological status. I have written the equations in a way that brings out their similarities and differences with the Euler equations and the Stokes equations. It is easy to see that the first equation is more complicated than the corresponding Eu­ler equation but simpler than the corresponding Stokes equation. But notice in particular the second, and shorter, of the above equations. It indicates that, given the approximations that are in play, there is a zero rate of change of pressure perpendicular to the plate. The pressure is constant along the y-axis as it cuts through the boundary layer. Clearly, Prandtl’s picture of the bound­ary layer involved some ruthless idealizations. This fact was emphasized by Hermann Schlichting, another of Prandtl’s pupils, who would later write an authoritative monograph on the boundary layer.14 Commenting explicitly on the second of the above equations, Schlichting said:

Die hieraus folgende Vernachlassigung der Bewegungsgleichung senkrecht zur Wand kann physikalisch auch so ausgesprochen werden, dafi ein Teilchen der Grenzschicht fur seine Bewegung in der Querrichtung weder mit Masse behaftet ist noch eine Verzogerung durch Reibung erfahrt. Es is klar, dafi man bei so tief greifenden Veranderungen der Bewegungsgleichungen erwarten mufi, dafi ihre Losungen einige mathematische Besonderheiten aufweisen, und dafi man auch nicht in allen Fallen Ubereinstimmung der beobachteten und berechneten Stromungsvorgange erwarten kann. (121)

The disregard of the equation of motion at right angles to the wall that results from this can be expressed in physical terms by saying that, in its transverse motion, a fluid particle in the boundary layer has no mass and experiences no frictional retardation. It is clear that with such far-reaching changes in the equations of motion one must expect that their solutions will show some mathematical peculiarities and that one cannot in all cases expect agreement between the observed and calculated flow processes.

The fluid particles in the boundary layer, as described by Prandtl’s equa­tions, have zero mass and zero friction in the direction transverse to the layer. Clearly no one believes that a real, physical object could satisfy these specifications, at least not given all the assumptions about the world taken for granted by physicists. Thus Prandtl portrayed the fluid in his boundary layer in terms that are reminiscent of the idealized fluid of classical hydro­dynamics. Euler’s equations of inviscid flow generated false empirical pre­dictions, and these errors were usually explained by noting that the equa­tions neglected friction, whether between the fluid elements themselves or between the fluid and solid boundaries. One might therefore expect that a determined effort would be made to remove all such idealizations and un­realities concerning friction in the course of producing the improved, more realistic, boundary-layer equations. This appears not to have been the case. As far as friction is concerned, the particles of fluid in the boundary layer are hardly less exotic than the particles of an ideal fluid. More will be said later about the way in which idealization is an enduring feature of scientific progress in fluid dynamics.

Not only did Prandtl’s boundary-layer equations involve physical unreali­ties, but the reasoning that generated them involved mathematical assump­tions for which no justifications were given. Certain mathematical questions had been passed over, for example, questions about the existence and unique­ness of solutions to the equations and the convergence of the approximation techniques that were employed. This left the precise relation between Prandtl’s equations and Stokes’ equations unclear. As one mathematician noted, even fifty years after the introduction of the boundary-layer equations, this de­ductive obscurity had still not been dispelled. But, he added, there has been a tendency to disregard it because of the great, practical success of Prandtl’s contribution.15

The boundary-layer equations, as such, played no explicit part in the mathematical apparatus employed in the early Gottingen aerodynamic work. The mathematics that Prandtl actually used for his theory of the finite wing was confined to the Euler equations of inviscid flow, but the idea of the boundary layer was always in the background and undoubtedly played a heuristic role.16 The interpretation of the theoretical results depended on qualitative reasoning that appealed to boundary-layer theory. For example, postulating the existence of the boundary layer effectively divided the fluid into two parts. One part demanded recognition of its viscosity, while the other could be treated as if it were an inviscid fluid. If the flow sticks closely to the surface of a solid body, and there is no separation, then the bulk of the flow can be treated as an exercise in ideal-fluid theory. This was the basis of Prandtl’s claims, discussed earlier, that for streamlined bodies the theory of perfect fluids had been dramatically confirmed. The viscosity assumed to be present in the boundary layer also provided a resource for explaining the ori­gin of the circulation around a wing. The viscous fluid in the boundary layer possesses vorticity, so that if fluid from the layer were to diffuse into the free stream, this occurrence might modify the overall structure of the flow and introduce a component of circulation, even if the circulating flow were then attributed to a perfect fluid.

The model of the boundary layer was itself subject to development both theoretically and experimentally. At first it had been assumed that the flow within the layer had a laminar character. Later, Prandtl relaxed this assump­tion and explored the idea of a turbulent boundary layer. Because turbu­lence implied an increased exchange of energy between the slower-moving boundary layer and the faster-moving free stream, a turbulent boundary layer would possess more energy than a laminar boundary layer because it would have absorbed energy from the free stream. The increased energy de­lays the separation that occurs when the boundary layer runs out of energy and brakes away from, say, the surface of the wing. The delay means the flow conforms more closely to the surface of the wing. This lowers the pressure drag and thus brings the behavior of the air closer to that of a perfect fluid. The idea of boundary-layer turbulence also explained some intriguing dis­parities between the wind-channel measurements of the resistance of spheres made in Gottingen and those from Eiffel’s laboratory in Paris. Strangely, in Paris resistance coefficients for spheres were about half the value of those in Gottingen: 0.088 compared with 0.22. In the course of a review of Eiffel’s wind-channel results, which were otherwise comparable with those in Got­tingen, Otto Foppl concluded that, in the case of the resistance of spheres, there was obviously some mistake in the French work: “Bei der Bestimmung des Widerstands einer Kugel ist offenbar ein Fehler unterlaufen.”17

Prandtl, however, was able to explain the result without attributing a mis­take to Eiffel. Rather than a trivial error, the anomaly indicated the presence of something deep. Prandtl argued that in Gottingen the flow in the wind channel was less turbulent than in Eiffel’s channel. He deliberately increased the turbulence in the Gottingen channel by means of a wire mesh and re­produced Eiffel’s results. What is more, Prandtl argued that the boundary layer itself may have been laminar in Gottingen, whereas in Paris it had been turbulent. This analysis was then subject to an ingenious experimental test in the Gottingen wind channel. Just as Prandtl had introduced the original idea of the boundary layer alongside a demonstration of how to remove the layer by suction, so he now showed how to manipulate the turbulence of the layer. He (counterintuitively) reduced the resistance of a sphere by wrapping a trip wire around it to render the boundary layer turbulent. Photographs taken by Wieselsberger provided further corroboration. Not only was the measured resistance reduced, but the introduction of smoke into the flow showed the separation points pushed toward the back of the sphere. The tur­bulent boundary layer must be clinging to the sphere longer than the laminar layer. In both of these cases, that of the laminar and the turbulent boundary layer, Prandtl’s engineering mind linked a novel theoretical idea to a novel technology of intervention.18

The International Air Congress for the year 1923 was held in London. It pro­vided a further occasion for assessing the advances that had been made in aeronautics during the war years and for addressing unresolved problems. It was a highly visible platform on which the supporters and opponents of the circulatory theory could express their opinions and, in some cases, air their grievances. In the morning session of Wednesday, June 27, there were three speakers: Leonard Bairstow, Hermann Glauert, and Archibald Low.

The first to speak was Bairstow, whose talk was titled “The Fundamen­tals of Fluid Motion in Relation to Aeronautics.”87 Bairstow was explicit: his aim was nothing less than the mathematical deduction of all the main facts about a wing from Stokes’ equations and the known boundary conditions. The work of Stanton and Pannell had shown that eddying motion did not compromise the no-slip condition and had established kinematic viscosity as the only important variable.88 “These experiments appear to me,” said Bairstow, “to remove all doubt as to the correctness of the equations of mo­tion of a viscous fluid as propounded by Stokes and the essential boundary conditions which give a definite solution to the differential equations. The range of these equations covers all those problems in which viscosity and compressibility are taken into account, and from them should follow all the consequences which we know as lift, drag etc. by mathematical argument and without recourse to experiment. Such a theory is fundamental” (240-41). The boundary conditions were empirical matters, but thereafter everything should follow deductively: lift, drag, changes in center of pressure, the onset of turbulent flow and stalling characteristics, along with a host of other re­suits important to the designer of an aircraft. Confronted by an aerodynamic problem the response would not be “let us experiment” but “let us calculate.” This was what it meant to possess a “fundamental” theory.

Bairstow was not being naive. He, as well as anyone, knew the problems standing in the way of any such employment of the Stokes equations. But he insisted that these difficulties had to be confronted because only in this way could aerodynamic theory be given a proper foundation in physical reality. Until aerodynamic results could be derived from the Stokes equations, they lacked a true and reliable foundation. They could be no more than makeshift approximations combined with ad hoc appeals to experimental findings. For Bairstow this was not an intellectually acceptable state of affairs. Bairstow did not deny that the success of Prandtl’s theory was “striking.” What wor­ried him was that Prandtl made this “start without reference to fundamental theory” (241). If Prandtl’s approach was legitimate, then it must be the case that it can be related to the Stokes equations.

If the successes of the circulation theory could no longer be denied, Bair – stow now said it was those very successes that constituted the problem. The theory worked, but why did it work? What might, at first, have appeared to be the strength of the circulation theory—that it worked—was now identified as a source of worry. “The questions which naturally arise,” said Bairstow, are “(i) Why does the circulation theory apply with a sufficient degree of approx­imation in some cases and what is the fundamental criterion of its applicabil­ity? (ii) Is further progress possible along the same lines?” (242). There might seem to be an obvious response to the second question. Why not just try and see what happens? In Bairstow’s opinion, however, the rational thing to do was to seek guidance from the fundamental equations in advance, rather than resort to trial and error. But it was the first question that provided the most characteristic expression of Bairstow’s position. The inviscid model of air was physically false. The appearance of truth must be explained away by showing why a viscous fluid sometimes behaves like an inviscid fluid. This capacity to appear inviscid should be deducible from the Stokes equations.89 Bairstow therefore proceeded to lay out for his audience some of the mathematics of viscous flow.

In the course of his discussion Bairstow remarked that the postulation of a boundary layer was an attempt to respond to the “essential failure” of the (inviscid) theory to meet the boundary conditions, that is, the condition of no slip. He conceded that this move, that is, postulating a boundary layer, “does not present an impassable barrier to acceptance,” but he insisted that difficulties begin “when the region is defined as of infinitesimal width” (244). If Bairstow was going to countenance a boundary layer at all, it had to be an empirically real layer with a finite depth, not the mathematical fiction of an infinitesimally thin layer. Experimentally, he said, the infinitesimally thin boundary layer was “unacceptable at the trailing edge,” where there was a clear wake; and, in any case, the inviscid approach to lift that it appeared to sanction (that is, the Kutta-Joukowsky formula making lift proportional to circulation) “leads to an estimate of lift which is 25 per cent too great” (244).

At the time he gave his talk to the Air Congress, a program of work and publication was under way designed to carry Bairstow toward his fundamen­tal goal. Money had been acquired from the Department of Scientific and In­dustrial Research to pay for two assistants, Miss Cave and Miss Lang, to work under Bairstow’s guidance at Imperial College. One paper from the team had already appeared: Bairstow, Cave, and Lang’s “The Two-Dimensional Slow Motion of Viscous Fluids.”90 In the same year as the Congress these three authors also published “The Resistance of a Cylinder Moving in a Viscous Fluid.”91 In the latter paper Bairstow explained that the purpose “was to pre­pare the ground for a solution of the complete equations of motion for very general boundary forms, and steps are now being taken toward that end” (384). In the event, he did not actually address the complete equations but fol­lowed Stokes and Lamb in using an approximation. Stokes had simplified his own equations by doing the opposite of Euler and had neglected everything but viscosity. All the inertial terms had been dropped and only the viscous terms retained. This had enabled him to arrive at equations that described, for example, the very slow motion of a very small sphere in a very viscous fluid. The formula was accurate near the sphere but failed at a large distance from the sphere. Stokes also drew attention to the fact that his approximation could not be applied to two-dimensional flow. It could not be made to work for the two-dimensional case such as a circular cylinder.92

The Swedish mathematician Carl Wilhelm Oseen had proposed another approximation for the full viscous equations in 1910.93 These produced the same results as Stokes’ analysis in the neighborhood of a sphere but differed at large distances. In 1911 Lamb had published a paper in which he drew at­tention to Oseen’s approach and had simplified the working.94 Lamb also showed how to apply Oseen’s approximate form of the Stokes equations to the case of a circular cylinder. He showed how it could be extended to the two-dimensional case in a way that had proven impossible using Stokes’ own simplification. It was Lamb’s work that provided Bairstow and his team with their method. “The line of attack adopted by us,” said Bairstow, “was sug­gested by Lamb’s treatment of the circular cylinder” (385-86).

Bairstow was able to generalize Lamb’s result for the circular cylinder to an ellipse. Most of the resulting formulas could be evaluated without resort to graphical or mechanical methods, but these could not be avoided when ana­lyzing shapes such as cross sections of wings and struts. In the case of a wing Bairstow was not able to reach a determinate result, but, as he put it, at least the problem “has been attacked and a method of solution indicated” (384). In their second paper Bairstow, Cave, and Lang offered a complicated, gen­eral formula for the lift of a wing shape, that is, an expression for the vertical component Ry of the resistance. The implications of the formula, however, were not clear. Bairstow could only say, “Except in the case of symmetry it is not obvious that Ry will vanish, but rather that a lift may be expected” (419). The computations needed to get this result were considerable, but despite all the expenditure of effort he had done no more than demonstrate the possibil­ity of a lift.

Bairstow had sent a copy of the collaborative 1923 paper, on the resistance of a cylinder in a viscous fluid, to Prandtl. Prandtl replied on October 17, in German. He expressed polite interest but said that he had certain doubts about Bairstow’s calculations. There followed two closely typed pages of technical objections. Prandtl demonstrated that Oseen’s approximation, and Bairstow’s use of it, was only acceptable in the vicinity of the cylinder when the Reynolds number was small, that is, when the flow was very viscous. He identified the precise equations in Bairstow’s paper that were inadequate and explained why they failed when the Reynolds number was large and the vis­cosity therefore relatively small, that is, in the cases that were relevant to aero­dynamics. He signed off, rather abruptly, after recommending that Bairstow acquaint himself with the theoretical work of Blasius and the experimental measurements of Wieselsberger.95

My question at the beginning of this volume was: Why did British experts in aerodynamics resist the circulatory theory of lift when their German coun­terparts embraced it and developed it into a useful and predictive theory? My answer has been: Because the British placed aerodynamics in the hands of mathematical physicists while the Germans placed it in the hands of math­ematically sophisticated engineers. More specifically, my answer points to a divergence between the culture of mathematical physics developed out of the Cambridge Tripos tradition and the culture of technical mechanics devel­oped in the German technical colleges.

This abbreviated version of my argument and my conclusion is correct, but a condensed formulation of this kind carries with it certain dangers. It may invite, and may seem to permit, assimilation into a familiar, broader narrative, which destroys its real significance. Thus it may appear that the “moral” of the story is that (at least for a time) certain social prejudices en­couraged resistance to a novel scientific theory and led to scientific evidence being ignored or overridden by social interests and cultural inertia. Accord­ing to this stereotype the story came to an end when “rational factors” or “epistemic factors” eventually overcame “social factors” and science was able to continue on its way—a little sadder and wiser, perhaps, but still securely on the path of progress.

Is there really any danger of the episode that I have described in so much detail being trivialized in this way? I fear there is.100 In one form or another, the narrative framework I have just sketched is widely accepted. It has nu­merous defenders in the academic world who confidently recommend it for its alleged realism and rectitude. It is deemed realistic because no one who adopts this view need deny that science is a complicated business. Scientists are, after all, human. Sometimes the personality or the metaphysical beliefs of a scientist may imprint themselves on a historical episode. Sometimes politi­cal interests and ideologies will intervene to complicate the development of a subject and perhaps even distort and corrupt a line of scientific inquiry. What worldly person would ever want to deny that this can happen? But who could approve of these things or, after sober reflection, think that they represent the full story of scientific progress? The intrusions of extra-scientific interests must therefore be exposed as deviations from an ideal that is characteristic of science at its rational, impersonal, and objective best. As well as personal and social contingencies (the argument goes on), it is vital to acknowledge that there are rational principles that, ultimately, stand outside the historical process and outside society. These represent the normative standards that sci­ence must embody if it is to achieve its goal. Fortunately the norms of rational thinking are realized with sufficient frequency that science manages to do its proper job. The norms ensure that the Voice of Reason and the Voice of Nature are heard. With due effort, and a degree of good fortune, this is how science actually works. The rest (the deviations and failings) merely provide a human-interest story of which, perhaps, too much has been made.101

Doesn’t the episode I have described fit into this stereotype? The dispute over the circulation theory ended because the evidence had become too strong to resist. Isn’t that really all there was to it? The British experts were initially too impressed by the great name of Rayleigh, and their resistance to the circulation theory was not a credit to their rationality. Eventually, though none too soon, they came round. Ultimately, therefore, evidence and reason triumphed over prejudice, tradition, and inertia. Reality stubbornly thwarted vested interests, and rationality subverted conventional habits and complacent expectations. Knowledge triumphed over Society. Isn’t this how my story ends?

The answer is no. This is not the story, and it is not how the story ends. Such a framework does not do justice to even half of the story I have told. In reality the end of the story is of a piece with its beginning and its middle. There was continuity both in the particular parameters of the episode I have described as well as in the general epistemological principles that ran through it. The supporters of the circulation theory never provided an adequate ac­count of the origin of circulation, and the critics never deduced the aero­dynamics of a wing from Stokes’ equations. Nor were there any qualitative differences in the relations linking knowledge to society and to the mate­rial world at the end of the story compared with the beginning of the story. There were changes of many kinds throughout the course of the episode, but they were not changes in the fundamentals of cognition or the modes of its expression. Fundamental social processes were operating in the same, principled way before, during, and after the episode described, and they are operating in the same way today. Society was not an intruder that was even­tually dispelled or an alien force that had to be subordinated to the norms of rationality or the voice of nature. There was no Manichean struggle between the Social and the Rational.

Trivializing versions of how the story ends may appeal to propagandists who want to spin simple moral tales, but to the historian and sociologist such tales indicate that the complexities of the episode are being edited out and its structure distorted. This danger is amplified if only a summary version of the story is retained in the memory. To offset this tendency I want to make explicit the methodological framework in which the story should be located, and I want to defend this framework against trivializing objections and mis­guided alternatives. Such is the function of the discussions in the final chap­ter. The aim is to keep the details of the story alive and its structure intact while, at the same time, reflecting on its broader significance.102

The equations of motion for an ideal fluid were derived from two basic state­ments. In a sense, said Cowley and Levy, “these two statements and all that they involve are a definition of the nature of the fluid.” Furthermore, “all de­ductions regarding its behavior can only be a recasting of these [statements] into a new but equivalent form” (37). The two statements were as follows:

(1) The mass of fluid in any region remains constant. This is called the condi­tion of “continuity.” And (2) the motion of every fluid element is consistent with Newton’s Laws of Motion. To spell out the first of these principles, con­sider a small volume through which the fluid flows. This is represented in cross section in figure 2.1 by the rectangle ABCD, with sides labeled dx and dy. Before going further I should say something about the conventions used in such diagrams in hydrodynamics. The d and 8 (delta) symbols indicate that the lengths are not just “small” in a commonsense manner of speaking but have been, or will be, made “infinitely small” in the course of the mathemati­cal reasoning. They will be subject to a “limiting process” in which it is as­sumed that they can be made ever smaller without the shrinkage demand­ing any significant changes in the pattern of reasoning (which is essentially what Lamb meant in the passage quoted at the head of the chapter). The reason why the volume can be represented by an area is because the volume is assumed to be of unit depth, so the number 1, representing the depth, is present but can be suppressed. Diagrams of this kind thus amount to a two­dimensional cross section of the flow, and the situation portrayed is routinely referred to as a two-dimensional flow.14

Two-dimensional flow diagrams do not allow any representation of what happens at the edges of the figure other than those shown in the cross section. The parts of the object that go into, and out of, the page are not shown. In describing a situation in this truncated way, the mathematician assumes that nothing significant happens at the edges that are not represented. In reality this is not true, so all discussions of two-dimensional flow are by their na­ture simplifications. These simplifications will become especially significant when the object under discussion is a wing and the cross section takes on the shape of an aerofoil. In the literature on aerodynamics the simplified, two-dimensional diagram of the flow is then often called a diagram of the flow around an “infinite” wing. This usage can be disconcerting, but it is simply a way of saying that the flow shown in the picture is representative of what goes on in the central parts of the wing. The wingtips are assumed to be

sufficiently far from the action that they do not interfere in any way and can be ignored. The literature on hydrodynamics and aerodynamics is full of ref­erences to infinity. The word “infinity” can nearly always be read as meaning either “so far away that it causes no disturbance” or “so far away that it can be considered to be undisturbed.”

Having dealt with these terminological matters, I now come back to the small volume represented in figure 2.1.15 The fluid is assumed to be incom­pressible with constant density p, so in a given time interval the mass of fluid flowing into the volume always equals the mass flowing out. In the x-direction the speed of flow into the volume is designated by u and the speed of outflow by ^u+d~^x j. The symbol du/dx means “the rate of change of u with x,” so the expression in parentheses refers to the original speed plus the change of speed. The change can be positive or negative. The mass entering the control volume per unit time is puby and the mass leaving is pi u+—Sx | Sy. The

I dx j

same procedure is repeated for the flow in the y-direction. In each case the quantity of fluid entering and leaving the control volume is obtained by mul­tiplying the speeds by the density of the fluid and dimensions of the face crossed by the flow. Mathematically, the condition of continuity is then ex­pressed by summing all these quantities and equating the sum to zero. Fluid in must equal fluid out, with zero shortfall. When the expression for this zero sum is simplified, the equation that results takes the form

du + dv 0 dx dy

This expression states, in a mathematical form, the condition of continuity.

Reports and Memoranda No. 16, of September 1909, was a note prepared by Mallock on experiments that involved moving flat and curved plates through a tank of water and measuring the forces on them.24 The plates were rect­angles whose sides were in the ratios of 2:1, 1:1, and 1:2 and were inclined at various angles between 0° and 90° to the direction of motion. Measurements were taken at various speeds between 150 and 500 feet per minute. The plates were attached to scales in order to read off the forces exerted on them. The resultant force on the plate was broken down into a resistance force (the drag directly opposing the motion) and a lateral force at right angles to the direc­tion of travel (a lift force). A graph of the results showed that the measure­ments “differ greatly from the values calculated on the assumption that the pressure on the rear surface is uniform and equal to that of the fluid at a dis­tance” (40), that is, calculated on the assumption that there was “dead water” behind the plate. Mallock’s explanation for the disparity was the same as Ray­leigh’s.25 There can, said Mallock, “be no doubt that the distribution of the pressure over the rear surface and its difference from the pressure prevailing in the fluid at a distance, account both for the peculiarities of the resistance and the lateral thrust curves” (40).

There was a negative pressure at the back of the plate, and Mallock sug­gested that the cause of this suction effect was eddy formation: “The fact of negative pressure being found on the down-stream side of the surface is ul­timately connected with the formation of eddies. . . nor will a satisfactory explanation of such features. . . be obtained until the formation of eddies under varying conditions has been investigated” (40). These ideas had al­ready found expression in one of the very first Reports and Memoranda, that of May 19, 1909, titled “Memorandum on General Questions to Be Studied.” Much of this publication was devoted to the results of several series of experi­ments Mallock had performed on plates in currents of air.26 He described the general form of the curves relating angle of incidence to lift and drag. The lift increases in a roughly linear fashion but only up to a certain, critical angle. After this it declines more or less sharply. In the region of the critical angle, he noted, the airflow had an unsteady, oscillating character accompanied by the formation of eddies.

Mallock went on to make the following significant remark about the dif­ference between perfect fluids and real fluids: “The details of eddy formation are also important. The eddy in a real fluid differs greatly from the ideal vor­tex ring. The latter is a separate entity which could not be made, and, if exist­ing, could not be destroyed. The eddies in real fluids are composite structures in which layers of originally different velocities are wound up together” (22). Mallock was arguing that the turbulence that characterizes real flows can­not be modeled by the kind of vortices that can exist in a perfect fluid. He was implicitly citing one of the most important results of classical hydrody­namics. Kelvin had provided a rigorous proof of a result that had long been known in one form or another, namely, that vortex or rotational motion in an ideal fluid can be neither created nor destroyed. The governing equations of ideal flow did not allow a vortex to be generated (for example, by the mo­tion of a solid body immersed within the fluid), nor could any existing vortex be damped down or dissipated. Mallock concluded that an adequate under­standing of the flow of a real fluid (such as air), around a real object (such as a wing), cannot be achieved using a theory of an ideal, perfectly inviscid, fluid. Like Rayleigh, he now believed it was necessary to take account of viscosity.

A further criticism of Kirchhoff-Rayleigh flow was contained in Reports and Memoranda No. 24, dated April 21, 1910. This was written by T. E. Stan­ton, the head of the Engineering Department at the NPL, and his assistant Leonard Bairstow.27 The aim of the research was to discover the relative ef­ficiency of the various rudders and lifting surfaces that were used on airships. Could these be modeled as flat planes set at an angle to the flow, and could Rayleigh’s formula be used to predict the forces on them? Recall that Rayleigh had deduced the formula for the distance of the center of pressure of the resultant aerodynamic force from the leading edge of the plate. Stanton and Bairstow carried out measurements in the air channel to test this. They found that “the experimentally determined position of the centre of pressure for the pattern of rudders used are in all cases ahead of the theoretical position for a thin plate” (75). Here was a second blow to the utility of Kirchhoff-Rayleigh flow. First it underestimated the force on wing; now it emerged that it located that force in the wrong place.

The next step was to try to render the flow around wings and plates vis­ible and to capture them in photographs. The photographs were provided for the Advisory Committee in 1912 by C. G. Eden in Reports and Memoranda No. 58.28 Eden injected a mixture of aniline and toluene into the flow of water in a small water channel, a 3 X 4-inch cross section, containing a small, 1-inch­wide model wing that could be given various angles of incidence. The flow had a speed of 1 inch per second and could be illuminated by means of an arc lamp. The specific gravity of the injected fluid was adjusted to match that of the water so that it did not alter the character of the flow. The optical proper­ties of the oil were such that at an angle of about 70° to the incident beam, the oil drops appeared as bright dots and could be photographed. Commenting on the resulting photographs, Eden described the effect of putting the wing at different angles of incidence, saying, “At small angles up to 9° no eddies are formed, but it will be seen that between 9° and 10° a change takes place, and at 10° there is a small ‘dead water’ region at the back of the plate and the eddying of the flow in the wake is clearly shown” (99). At first it may seem that the photographs were evidence in favor of Kirchhoff-Rayleigh flow. Eden detected eddies that the theory could not explain, but he reported a clearly visible region of “dead water,” which is the most characteristic feature of the flow. The real significance of the photographs, however, came out when they were related to model wings as they were being studied in the wind channel.

Bairstow and Melvill Jones produced two papers in 1912 that had a direct bearing on the meaning of the photographs. The first, Reports and Mem­oranda No. 53, was a general assessment of the properties of aerofoils, “as deduced from the results of various aeronautical laboratories.”29 These in-

cluded Eiffel’s near Paris and Prandtl’s in Gottingen. These results, said Bair – stow and Jones, made it possible to identify the salient features of aircraft wings, both qualitatively and quantitatively. Suppose a wing is aligned with the general direction of airflow. It generates a lift force at right angles to the direction of flow, and as the angle of incidence is slowly increased the lift increases in an approximately linear manner. At a certain point, around 10° to 15°, the lift reaches a maximum and then declines, sometimes very sharply. This was Mallock’s “critical” angle. Finally, because the wing produces lift when it is horizontal, the position of zero lift must occur when the wing has a slight negative angle of incidence. The general form of the graph relating lift to incidence, as described by Bairstow and Melvill Jones, is indicated in figure 3.6.

In their next report, “Experiments on Models of Aeroplane Wings,” of March 1912, Bairstow and Melvill Jones made two important moves.30 First, they reinforced the evidence that the factors regulating the amount of lift in a wing are precisely those that are excluded by the discontinuity theory, namely, the shape of the upper surface. Second, they made the crucial link to Eden’s photographs. To show the significance of the upper surface, they used a series of aerofoil sections with a flat base but (1) varied the camber of the upper surface and then (2) altered the position along the chord of the maximum ordinate. (The camber was measured by the ratio of the maximum height of the curved upper surface to the chord.) They demonstrated how changes in camber altered the ratio of the lift to the drag. Thus, starting from a nearly flat wing, an increase in camber at first increased and then decreased this ratio, the maximum being for a wing with a curvature of 0.05. The op­timum position of the highest point in the upper surface turned out to be one-third of the chord from the leading edge. Variation of the curvature of the lower surface of the wing, by contrast, produced little effect on the lift or drag. Thus the systematic accumulation of inductive evidence about the be­havior of wings consolidated Rayleigh’s early doubts about his classic results. The most significant aerodynamic processes on a wing had, on his analysis, been located on the wrong surface.

Eden’s photographs showed a winglike shape at various angles below and above the critical angle. At small angles there were no eddies, so the flow was smooth and stayed close to the surface of the plate or wing. Around 9°, eddies formed behind the plate and the smooth flow broke down. Here the illuminated oil drops stayed as spots of light and did not show up as lines because they did not move during the exposure time. As Bairstow and Melvill Jones put it, “It seems clear that the alterations in pressure at the critical angle are due to the sudden breakdown in the character of the fluid flow in the neighbourhood of this angle, and in this connection the photographs . . . are of interest. . . . It will be noted that above the critical angle the fluid near the upper surface is practically ‘dead’” (10).

The implication was that dead air does indeed form behind a wing, as in Rayleigh flow—but only after the wing has passed the critical angle. Bairstow, Melvill Jones, and Eden were all aware that this reasoning had a weakness. The experiment was in water while the conclusion was about air. The conditions of similarity, required for a confident inference from the one phenomenon to the other, were not strictly satisfied.31 The fact remained that the photographs that looked most like Kirchhoff-Rayleigh flow showed a wing that was over the critical angle.32 This information pointed to an important but disconcert­ing conclusion. The mathematical studies devoted to Kirchhoff-Rayleigh flow were not describing at all how a wing can be so effective in producing lift. In as far as this model of flow over a wing approximated to reality, it was actually describing the breakdown of the pattern that was required for the efficient working of a wing. If they described anything, the formulas of Kirchhoff – Rayleigh flow were the mathematics of a stall, when lift fails. They described the situation when an aircraft was about to fall out of the sky.33

As well as the empirical support briefly mentioned by O’Gorman, some fur­ther evidence favorable to Lanchester emerged in the following year. It came from Bairstow’s own laboratory and arose from the attempt to clarify some disconcerting experimental results about control surfaces. With the con­struction of ever-larger aircraft, the forces that pilots had to exert on the con­trols became correspondingly greater. To overcome this problem the controls were “balanced,” that is, part of the area of the control surface was positioned in front of the hinge around which it turned so that some of the aerodynamic forces worked with, rather than against, the pilot. An experimental study of balanced controls was carried out at the National Physical Laboratory in early 1916 by John Robert Pannell and Norman Robert Campbell. Pannell, who had been on the staff since 1906, was the senior assistant in the Aerodynamics Department. He was a familiar figure who bicycled to work every morning, arriving on wet days with an umbrella held aloft in one hand while steering with the other. His main concern was with tests on full-size airship, and he was to die in 1921 when the R38 met with disaster on its trial flight.68 The balancing experiments, however, were conducted using the “flaps” on an air­craft wing. “Flap” was the old name for the lateral control surfaces, today called ailerons, which allow the pilot to bank and roll the aircraft.

Pannell’s co-worker in these experiments was a Cambridge experimental physicist who had played a prominent role in early debates about relativity theory.69 Campbell, who had been seconded to the NPL for war work, was in the process of writing a book on scientific method, Physics: The Elements, which was published after the war and was to prove an influential work in the philosophy of science. Campbell argued for the importance of models in scientific inference and theory construction.70 On this occasion the models that interested Campbell were not models of the atom or the electromagnetic field but scale models of the wings of a 110-foot-span biplane that was under construction at the Royal Aircraft Factory.71

The “flaps” of the projected aircraft ran along the rear edge of the outer portion of the wings but also included the tips of the wings themselves. When the part of the control surface that was on the trailing edge was lowered, then the part of the wingtip that was connected to it, and that was in front of the axel on which it pivoted, went up. The whole tip of the wing was thus part of the flap. This construction was meant to give the desired balance. Pannell and

Campbell wanted to find the proportion of the area that should be in front of the axel. The model wings were placed vertically in the 4 X 4-foot tunnel at a wind speed of 40 feet per second. Different proportions of fore and aft area were tested with the wings set at 0°, +4°, and +12° to the wind and with the flaps (and tips) put at a variety of angles relative to the main wings.

It proved impossible to find a fully satisfactory balance. Frustratingly, there was no ratio of the areas, fore and aft of the pivot, that fully balanced over the desired range of angles. Also, when looked at in detail, the results had some odd features. On occasion, where the experimenters had expected to be able to detect forces at work on the wingtip, there weren’t any: “In particular it was found that when the main planes of a biplane were inclined at +12° to the wind, there was no moment on the portion of the wing flap forward of the hinge, if this flap was inclined at an angle of -5° to the wind.”72 This result suggested that the flow of air near the wingtip was itself at a negative angle to the undisturbed flow. Given the conventions for designating angles positive or negative, this meant that the air near the wingtip was moving upward rela­tive to the wing. The air was going round the tips from what, during normal flight, would count as the lower to the upper surface.

In order to shed light on this, Pannell and Campbell conducted a further, qualitative investigation of the flow near the tips.73 Using a direction and ve­locity meter to plot the velocity components of the moving air, they found what they called “a very simple and obvious explanation” for their “remark­able results.” It became clear that air was indeed flowing round the wingtips from the lower to the upper surface. This was associated with a movement of air along the span of the wing, that is, not just from the leading to the trailing edge of the wing but lengthwise along it. There was a component of outward movement, toward the tips, on the lower surface and an inward movement, away from the tips, on the upper surface. Pannell and Campbell argued that

The presence of this flow round the wing tips affords, in outline at least, an explanation of the result on the balancing of wing flaps. . . . For, if there is a marked flow near the wing tip directed from the lower to the upper surface, a plane parallel to this flow will experience no wind force. Now it was precisely when the balancing flap was inclined at a negative angle to the wind, so that its plane lay along a flow having a component from the under to the upper side of the plane, that the experiments indicated that there was no force on it. (141)

The qualitative study also addressed the component of flow along the chord of the wing, that is, in the direction of flight rather than around the tips or along the span. Measurements were taken with the direction and ve­locity meter to build up a picture of the disturbed flow. Next, the steady, undisturbed flow was subtracted from it. If the resultant flow was made up of two parts, the free stream plus a circulation, this subtraction would expose the circulatory component. This is exactly what it did. In the experimenter’s words, it emerged that “the component of the wind disturbance which is par­allel to the direction of flight is in the direction of flight almost everywhere below the wing, and in the opposite direction everywhere above the wing. There is therefore some indication of the cyclical motion of the air round the wing in the vertical plane of flight which has been assumed by Mr. Lanchester in his discussion of the theory of the aerofoil” (142). Although the full path of the circulation had not been traced, those parts of it above and below the wing had been factored out and, as it were, exposed to view. Here again was evidence for the reality of the circulatory component that was central to Lanchester’s theory.

Unlike the 1902 report of his work, Kutta’s 1910 paper was long and detailed. It was called “Uber eine mit den Grundlagen des Flugproblems in Beziehung stehende zweidimensionale Stromung” (On a two-dimensional flow relevant to the fundamental problem of flight).27 All the basic reasoning in the paper and much of the detailed working were made explicit. It covered some fifty pages of the proceedings of the Royal Bavarian Academy of Science and in­volved mathematical techniques that ranged from the general and abstract to the concrete and numerical.28 Using Kutta’s own headings, I shall go over the seven sections of the paper to convey the structure of the argument.

Prandtl wrote his paper on the boundary layer while holding the chair of mechanics at the technische Hochschule in Hanover. Felix Klein (fig. 7.3) soon arranged for him to be called to Gottingen. In subsequent years Klein contin­ued to use his contacts with powerful government ministers, such as Friedrich Althoff, to support Prandtl and his work. Although his greatest mathematical achievements were now behind him, it was during the period 1890 -1914 that Klein was at the height of his influence as an academic politician.19 The in­stitutional structures created in Gottingen by Klein provided the context for Prandtl’s aeronautical work. It is hardly surprising that Klein always remained a figure whom Prandtl revered.20

In the years from the turn of the century and through the period of the

Weimar Republic, Gottingen was a place of extraordinary intellectual bril­liance in the fields of mathematics and physics. The university was the home of not only Felix Klein but David Hilbert, Hermann Minkowsky, and Her­mann Weyl. They all played leading roles in the sometimes tense discussions that took place over the mathematical foundations of general relativity and the geometrical nature of space, time, and matter. It is right that these aspects of the Gottingen scene have attracted the attention of historians of science and have been subject to detailed analysis.21 Nevertheless, for my purposes it is important to ensure that this aspect of Gottingen does not dazzle us. It must not obscure the very different character of the work done by Prandtl and his school.

The two greatest Gottingen mathematicians of that time, Klein and Hil­bert, represented divergent intellectual tendencies. Hilbert’s work can be seen as formal and abstract, whereas Klein’s was more concrete and intui­tive. Hilbert has been described as a “modernist,” while Klein has been pre­sented as a “countermodernist.”22 The differences between the work of the two men are a matter of continuing discussion among historians of math­ematics. The sharpness of the contrast between them has, perhaps, been blunted by recent scholarship, and the degree of overlap in their interests is now better appreciated.23 Despite this, something of the older polarity still remains. Both men wanted to increase unity in the field of mathematics, but they sought unity in different ways and aspired to a qualitatively dif­ferent kind of unity. Hilbert looked for unity through powerful methods and results that would radically reconfigure what had gone before. Klein sought unity through an encyclopedic and cumulative arrangement of re­sults. He was opposed to the bitter sectarianism of the competing “schools” of mathematicians in the German universities and sought to cure it not by the victory of one tendency, but by an ordering of mathematical knowledge that was at once inclusive and hierarchical. The historian of mathemat­ics David Rowe was surely right when he treated Klein’s great, collective project—the Encyklopadie der mathematischen Wissenschaften—as a reveal­ing expression of Klein’s conception of mathematical knowledge and its proper organization.24

Klein’s tireless organizational activities have been subject to detailed anal­ysis by, for example, Manegold, Pyenson, Rowe, Schubring, and Tobies.25 The main outlines of the story, as it bears on Gottingen aerodynamics, can be quickly sketched. After his visit, as a representative of the Prussian govern­ment, to the World’s Fair in Chicago in 1893, Klein became convinced of the need for German universities to attach more importance to applied science and technology. By 1897 he had been able to persuade the government and the University of Gottingen to create an institute for applied physics. In 1898 Klein and Henry von Bottinger, the vigorous and persuasive chairman of the Bayer chemical company, founded the Gottinger Vereinigung zur Forderung der angewandten Physik und Mathematik. This was a novel institutional ve­hicle by which commercial firms could finance applied research in the uni­versity.26 On this basis, in 1905, the original Institute for Applied Physics grew into the Institute for Applied Mathematics and Mechanics under Prandtl and Runge.27

When Klein was pressing the authorities to appoint Prandtl, he was ex­plicit about the engineering connection.28 He introduced Prandtl as someone “who combines the expert knowledge of the engineer with a mastery of the apparatus of mathematics and has a strong power of intuition combined with a great originality of thought” (“der mit der Sachkenntnis des Ingenieurs und der Beherrschung des mathematischen Apparatus eine starke Kraft der Intuition und eine grofie Originalitat des Denkens verbindet”; 232). It was Klein who prompted Prandtl to bring problems connected with airships and aerodynamics within the scope of his new institute.29 The Motorluftschiff – Studiengesellschaft, the Society for the Study of Motorized Airships, which had been founded in July 1906, provided the finance for setting up a model­testing facility at Gottingen, the Modellversuchsanstalt, with its wind channel designed by Prandtl and constructed by Fuhrmann. The channel was given a test run in December 1908 and went into operation in January 1910. In Octo­ber 1913 the model-testing facility was taken over by the university.30

There is no doubt that Prandtl was in close contact with all the leading mathematicians and physicists at Gottingen. He went with them on their af­ternoon walks; he plotted with them over local academic politics; he sat at the “high table” along with Klein and Hilbert during the intimidating meetings of the Gottingen Mathematical Society.31 He was a friend and neighbor of Karl Schwarzschild, who was at work solving the field equations of Einstein’s general theory of relativity.32 Unlike that of his colleagues, Prandtl’s intellec­tual life was not dominated by the struggles over the nature of gravity or the structure of matter. His mathematical heritage lay elsewhere, in the technis – che Mechanik of the engineering tradition and August Foppl’s textbooks.33 The significance of this tradition is conveyed with some force by the lengthy account that Runge and Prandtl wrote in 1906 describing the history and work of their Institute for Applied Mathematics and Mechanics.34 It is full of enthusiastic and detailed specifications of pumps, boilers, condensers, dyna­mos, diesels, turbines, and electrically driven ventilators (which were used in the aerodynamic experiments). No wonder their colleagues made sly jokes about the Department of Lubricating Oil.35 This was not the world of the philosopher-mathematician or the physicist-as-cosmologist, any more than it was the world of Horace Lamb, Augustus Love, or Lord Rayleigh. Certainly the Cambridge mathematicians would not have chosen the words with which Runge and Prandtl ended their article:

Die technischen Wissenschaften sind reich an Kapiteln, deren volles Verstand – nis eine tiefe mathematische Bildung erfordert. Der Unterricht setzt sich zum Ziel, die Entwicklung der mathematischen Methoden zu vereinigen mit dem vollen Verstandnis der praktischen Probleme in dem Umfang und in der Fas – sung, wie sie sich dem ausubenden Ingenieur darbieten. (280)

Technical science is rich in material whose full understanding demands a deep mathematical education. The goal of the teaching [at the institute] is to unite the development of mathematical methods and a complete understanding of practical problems in as far as they concern, and in the manner they present themselves to, practicing engineers.

This was neither the voice of Tripos-oriented Cambridge, nor the voice of the mathematical Gottingen that has attracted the lion’s share of the historian’s attention, but it was the voice of Prandtl’s Gottingen. There is no problem in acknowledging this difference as long as it is recognized that the University of Gottingen was not a unified intellectual environment. Of course, there was overlap between its different parts both personally and institutionally: Runge had worked on atomic spectra and von Karman was a friend of Max Born’s and collaborated with him on the quantum theory of specific heats.36 But it was engineering that defined the orientation of the aerodynamic work, and this orientation had an institutional niche in Gottingen thanks to Klein’s efforts.37

Institutional plurality was wholly consistent with Klein’s vision and prac­tice. Mathematics, for Klein, always had an integrating function in science, but that function was to be discharged in diverse ways. It required coordi­nation and cooperation, but it did not require that everyone have the same preoccupations. That would have run counter to the encyclopedic outlook that informed Klein’s organizational plans. Klein had himself lectured on me­chanics at Gottingen and had tried to offset the tendency toward excessive mathematical abstraction. In doing this he had, as von Mises put it, restored “the essential but almost lost connection with ‘technical mechanics.’”38 In 1900 Klein gave a general lecture titled “Ueber technische Mechanik” in which he sought to capture the special qualities of the discipline.39 Like Au­gust Foppl, Klein asserted that practitioners of technical mechanics had their own Fragestellung, that is, their own way of posing and answering questions. It involved subtle judgments and made unique demands. In particular Klein noted the problematic relation between technologically oriented mathemat­ics and the established knowledge of basic physical principles: “Es ist vielfach nicht moglich, die Erscheinungen mit den Principien oder Grundgleichun – gen der classischen Mechanik in luckenlosen Zusammenhang zu bringen” (28) (Very often it is impossible to bring phenomena into a rigorous relation­ship to the principles and fundamental equations of classical mechanics). In the eyes of some of Klein’s technological critics, these sympathetic method­ological insights were not enough. Nor were Klein’s Gottingen seminars on elasticity, graphical statics, and hydrodynamics always deemed a success. The implied commitment to technology, said the critics, was inadequate. This was the line taken by the engineer H. Lorenz, who had been brought to Gottingen by Klein but who had left in disappointment and, in a flurry of anti-Klein criticism, had taken himself off to the TH at Danzig.40 Klein was always fight­ing on two fronts. On the one side were those within the universities who argued that technology was not a fit subject for a truly academic institution, and on the other side were those within the technische Hochschulen who saw technology as the special preserve of these institutions. Both sides resented Klein’s suggestion that the universities involve themselves with applied sci­ence and technology.

Klein’s attempts to meet these conflicting accusations were not always suc­cessful. He eventually got his way, but not without frustrations and compro­mise. He supported the technische Hochschulen in their fight for the right to grant doctoral degrees, but there were delicate issues of status involved. Klein tried to find a division of labor in which different roles were to be played by the different institutions. On one occasion, in 1895, he sought to convey his vision by means of a military metaphor. He spoke of the universities as providing the Generalstabsoffiziere der Technik or “general-staff officers” of technology, while the products of the technical institutions would consti­tute the “frontline officers.”41 Perhaps Klein thought that the heroic image of frontline combat would make the suggestion acceptable to colleagues in the technische Hochschulen. If so, he was wrong. The implied disparities of rank and status were too blatant to ignore.

One of those who took offense was August Foppl. Without naming Klein as the source of the military metaphor, Foppl pointedly remarked that if one were going to speak in these terms, then each different arm of the military service (Waffengattung) should be accorded equal value. Were they not all equal as comrades in arms? The route to the top, and promotion to high command, should depend on the qualities of the individual, not the particu­lar branch of the service in which they happened to be trained.42 But while Foppl bridled at the condescension, his son-in-law was a beneficiary, and exemplification, of Klein’s rank ordering. Prandtl’s institute was devoted to fundamental questions in the field of technology, and he soon occupied a position in which he could influence the strategy of aerodynamic research. Prandtl was indeed a Generalstabsoffizier, and the banner under which his army marched was inscribed with the word “Engineer.”

The point is confirmed by the qualifications of those whom Prandtl re­cruited to serve with him. They prefaced their names with their engineering diplomas and styled themselves Dipl.-Ing. Fuhrmann, Dipl.-Ing Foppl, Dipl.- Ing. Betz, Dipl.-Ing. Wieselsberger, and Dipl.-Ing. Munk.43 Theodore von Karman (fig. 7.4) also had an engineering training and was a graduate of the Royal Technical University of Budapest. He had originally come to Gottingen to do research with Prandtl on elasticity and the strength of materials but, as he was fond of recounting, turned to hydrodynamics through reflecting theoretically on the experimental difficulties of his colleague Karl Hiemenz.44 This was the origin of von Karman’s work on the stability of vortex motion and his mathematical analysis of the shedding of vortices that formed a flow pattern called a Karman vortex street.45 A number of Prandtl’s inner circle

eventually left Gottingen to take up positions in the technische Hochschulen. Foppl, Wieselsberger, and von Karman all went to Aachen, making it a pow­erful rival to Gottingen. Aachen was not Gottingen, but von Karman was not a humble man.46 Without doubt, he and his colleagues still saw themselves as ranking members of the general staff of aeronautical technology.

Glauert’s contribution to the London Congress was a paper titled “Some As­pects of Modern Aerofoil Theory.”96 It covered much of the same ground as the methodological paper given to the RAeS, but with the addition of tech­nical results about propeller theory and wind-tunnel corrections. Though deeply opposed to Bairstow’s view, Glauert did not directly attack what had just been said. Instead he quietly sought to outflank it by demonstrating that the concern with “fundamentals,” as Bairstow conceived it, was out of touch with events at the front line of active research.

Glauert began by pointing out that the study of the forces and moments on a body in motion through a viscous fluid was beset by complexity and prog­ress had been slow. But a “modified form of the classical hydrodynamics” was proving successful. “The present paper,” Glauert went on, “is concerned only with the problem of aerofoil structures, whose essential characteristic is that they give a relatively large lift force at right angles to the direction of motion at the expense of a relatively small drag force retarding the motion” (245). A few minutes earlier Bairstow had drawn attention to the “limited” scope of application of the circulatory theory as a point of criticism and as an unac­ceptable feature of the work of Kutta, Joukowsky, and Prandtl. Right at the outset of his talk Glauert was doing the opposite. He was drawing attention to the limited focus of the work as a wholly-taken-for-granted feature that in no way told against it. Glauert was implicitly making an engineering-style response to Bairstow of exactly the kind that Lanchester had made, explicitly, in the 1915 confrontation.

Having led his audience, nonmathematically, through the main develop­ments in aerofoil theory, Glauert concluded by saying that the most impor­tant feature of the “modern” approach was that it “presents us with a point of view” with which to examine new problems. It provides us with a small number of theoretical conceptions “which serve to bind into a single unity a multitude of experimental results” (255). Clearly this point of view was dif­ferent from that adopted by Bairstow—and Glauert’s idea of unity was not Bairstow’s. For Bairstow unity meant deducibility from the Stokes equations; for Glauert it meant linking experimental results by adopting the modern, methodological standpoint. Glauert expressed himself in an interesting way. Referring to the moment when the boundary layer became infinitely thin, he said, “The effect of the evanescent viscosity is represented in the non-viscous solution by the possibility of a circulation round the aerofoil” (246).

The word “evanescent” means “passing quickly from sight or memory.” It was also the old Newtonian word for describing the infinitesimal quanti­ties that entered into the differential calculus. Infinitesimals were “evanescent quantities” that were neither zero nor nonzero but poised on the very brink of vanishing. Newton and his followers spoke in this way because they did not possess the modern concept of a limiting process.97 Glauert, of course, did possess it. His idea was to contrast the limiting value of, say, /(p) as p ^ 0 with the value of /(0) , that is, the value of the function at p = 0. These can be different. Glauert’s “evanescent viscosity” was thus viscosity on the point of reaching the limit zero, but his language was meant to register a method­ological as well as a mathematical point. It signalizes the difference between deciding that viscosity is zero in step 1 of his three-step methodology, and accepting that it may be treated as zero in step 3.

Bairstow wanted to know why ideal-fluid theory sometimes worked. Glauert’s answer was that it works when the ideal flow is a limiting case of a flow that would take place in a fluid of small viscosity. This answer was not one that Bairstow was prepared to accept. Recall that Bairstow saw no “im­passable barrier” to the idea of a boundary layer, but, he said, difficulties be­gan when the region was said to have an infinitesimal width (244). Bairstow did not want the relationship between inviscid and viscous flows to hinge on a limiting process, and certainly not on a limiting process that was carried out informally. He did not want Glauert’s “evanescent” viscosity; he wanted what, in his own mind, would count as physically real viscosity.98

Menschen haben geurteilt, ein Konig konne Regen machen; wir sagen dies widersprache aller Erfahrung. Heute urteilt man, Aeroplan, Radio, etc. seien Mittel zur Annaherung der Volker und Ausbreiting von Kultur.

Men have judged that a king can make rain; we say this contradicts all experience. Today they judge that aeroplanes and the radio etc. are means for the closer contact of peoples and the spread of culture.

ludwig Wittgenstein, Uber Gewifiheit, On Certainty (1969)1

The British resistance to the circulatory theory of lift casts light on a number of wider themes, some methodological, some cultural, and some philosophi­cal. Here I take the opportunity afforded by the case study to pose and answer some of the questions of this kind that have been raised. The first thing is to make explicit the methodological principles that have informed my own inquiry. The present study is not only an exercise in the history of science and technology; it is also a contribution to the sociology of knowledge. My approach has been that of the Strong Program in the sociology of knowl­edge, and so I start this chapter with a brief account and defense of the main features of the program and the perspective it is designed to encourage. It is a perspective very different from the naive, philosophical narratives I identi­fied at the end of the last chapter and that I mentioned in the introduction.2 I then address two broader questions. First, was the resistance to the circula­tory theory of lift an all-too-typical example of British failings in the field of technological innovation? I argue against this pessimistic reading. Second, what about the controversial topic of “relativism”? Aviation, as a successful and impressive technology, is often cited as a quick and decisive refutation of relativism. I believe this line of antirelativist argument is groundless and obscures the most striking characteristics of aerodynamic knowledge.

In this chapter I make use of the writings of the Viennese physicist Philipp Frank. Frank was a specialist in modern physics, but he wrote some half dozen papers on fluid dynamics and aeronautical topics.3 He knew many of the leading experts in these fields and was a lifelong friend of Richard von Mises. Frank is relevant to my discussion for two reasons. First, he compared the

way a scientific theory is assessed to the way the performance of an aircraft is assessed. I follow Frank in this aeronautical comparison and then close the circle by applying his comparison directly to theories in aerodynamics. Sec­ond, Frank’s work provides a valuable resource when discussing relativism. I use Frank’s simple and forthright definition of the word “relativism” to struc­ture my own discussion.