Category The Enigma of. the Aerofoil

The Theory of Ideal Fluids

Physicists, chemists, physiologists, and engineers are all interested in air, and each group studies it from the perspective of its own discipline. In the history of each discipline there is a strand that represents the history of the chang­ing conceptions of the nature of air adopted by its practitioners. Sometimes aerodynamics is counted as a branch of physics and sometimes as a branch of engineering, but however it is classified, it is evident that it involved a determined attempt to relate the flow of air to the basic principles of me­chanics. The most important of these are the laws of motion first delineated by Newton, for example, the law that force equals mass times acceleration. The complexity of the air’s behavior, however, means that there is no unique way to connect the flow to the fundamental laws of Newtonian mechan­ics. How the relation is to be articulated depends on the model of air that is used.

Newton himself treated fluids in different ways at different times. When he was thinking about the pressure of the air in a container, he conjectured, for the purposes of calculation, that air was made up of static particles that repelled one another by a force that varied inversely with distance.4 This con­cept was a guess that explained some of the known facts, but it was a con­ception of the nature of air and gas that physicists later abandoned. In its place they adopted what is called the kinetic theory of gases in which it was assumed that a gas is made up of small, rapidly and randomly moving par­ticles. According to the kinetic theory, as developed by James Clerk Maxwell and others, gas pressure is not the effect of repulsion between the molecules of the gas but is identified with the repeated impact of the molecules on the walls of the container.5

When Newton was thinking of a flowing fluid impinging on the surface of an obstacle, he did not use his repulsion model but spoke, for mathematical purposes, simply of a “rare medium” and treated the fluid as made up of a lot of point masses or isolated particles that do not interact with one another.6 The fluid medium was treated as if it were like a lot of tiny hailstones (though this was not Newton’s comparison). Again, the model is not to be identified with the later kinetic theory of gases. The hailstone model, too, dropped by the wayside, though, as we shall see, in certain quarters it still played some role in early aerodynamics. The concept of a fluid that proved most influen­tial in hydrodynamics was different from either of the ideas used by Newton as well as being different from the kinetic theory of gases. The model that came to dominate hydrodynamics, and aerodynamics, was first developed in the eighteenth century by mathematicians such as d’Alembert, Lagrange, the two Bernoullis (father and son), and Euler. They thought of the air as a continuous medium.7

Because the aim was to be realistic, the hypothetical, continuous-fluid picture had to be endowed with, or shown to explain, as many of the actual properties of real fluids as possible. Thus air has density so the continuous fluid must also possess density. Density is usually represented by the Greek letter rho, written p. Empirically, density is defined as the ratio of mass (M) to volume (V), which holds for some finite volume. The number that results, p = M/V, represents an average which holds for that volume at that moment. To apply the concept to a theoretically continuous fluid requires the assump­tion that it makes sense to speak not merely of an average density but of the density at a point in the fluid, that is, the ratio of mass to volume as the volume under consideration shrinks to zero. If the air is actually made up of distinct molecules, then, strictly, the density will be zero in the space between the molecules and nonzero within the molecules, and neither of these values would qualify as values of the density of the fluid. This dilemma did not ap­pear to be a problem in practice, but it is a reminder that the relation between physical models of the air based on particles and physical models based on a continuum may, under some circumstances, prove problematic.8

Air is also compressible. The same mass can occupy different volumes at different pressures. For many of the purposes of aerodynamics, however, it can be assumed that the density stays the same. This is because (perhaps counter to intuition) the pressure changes involved in flight turn out to be small. The fluid continuum can then be treated as “incompressible.” This ap­proximation only becomes false when speeds approach the speed of sound, which is around 760 miles an hour. In the early days of aviation, when aircraft flew at about 70 miles an hour, compressibility was no problem for wing theory. Things were different for propellers. The tips of propellers moved at a much higher speed, and here compressibility effects began to make them­selves felt, but that part of the story I put aside.9

Another important attribute of a fluid is its viscosity, which refers to the sluggishness with which the fluid flows. If a body of fluid is thought of as made up of layers, then the viscosity can be said to arise from the internal friction between these layers. Pitch and treacle are highly viscous fluids, whereas water is not very viscous. Viscosity can be measured by experimen­tal arrangements involving the flow through narrow tubes. The results are summarized in terms of a coefficient of viscosity, which is usually repre­sented by the Greek letter mu, written |4. A highly viscous fluid will be given a high value of |4; a fluid with small viscosity will have a correspondingly small value of |4. Air only has a very slight viscosity. At the extreme, if there were a fluid that was completely free of viscosity, it would be necessary to write |4 = 0. In reality no such wholly inviscid fluids exist, but if the fiction of zero viscosity is combined with the fiction of total incompressibility, this concept can be taken to specify what might be called a “perfect” fluid or an “ideal” fluid.

The single most important fact to know about the historical develop­ment of wing theory and the aerodynamics of lift is that its mathematical basis lay in the theory of perfect fluids, that is, in a theory in which viscosity was apparently ignored and assumed to be zero. The assumption that air can be treated as an ideal fluid was the cause of much argument, doubt, and frustration, which becomes apparent in subsequent chapters, but its central, historical role is beyond dispute. What turned out to be the most striking developments in aerodynamics (as well as some failed attempts) depended on the idea that viscosity and compressibility were effectively zero. The at­tractions of this assumption were twofold. First, it seemed highly plausible, and second, it produced an enormous simplification in the mathematical task of describing the flow of a fluid. The exercise produced a set of partial differential equations that determined the velocity and pressure of the fluid, provided that the starting conditions of the flow and the solid boundaries that constrain it are specified. The equations were developed by imagining a small volume of fluid, called a fluid element, and identifying the forces on it. The forces derive from pressure imbalances on the surfaces of the fluid element.

Fluid elements, it must be stressed, are mathematical abstractions rather than material constituents of the fluid. They are not to be equated with the molecules that interest chemists and physicists or the particles that feature in the kinetic theory of gases. The equations of flow do not refer to the hidden, inner constitution of fluids. The reality that is described by the differential equations that govern fluid motion concerns the macrobehavior offluids rather than their microstructure. The abstract character of a fluid element is evident from the way it is typically represented by a small rectangle. The simple ge­ometry of the representation derives from the mathematical techniques that are being brought to bear on the flow. These are the techniques of the differ­ential and integral calculus.10 The concept of a fluid element is the means by which these techniques can be used to gain a purchase on reality. The differ­ential equations that were the outcome are called the Euler equations. They can be said to describe in a strict way the flow of an ideal fluid, but the hope was that they would also describe, albeit in an approximate way, the flow of a real fluid, air, whose viscosity is small but not actually zero.

To give a feel for the style of thinking that went into the classical hydro­dynamics of ideal fluids (and, later, into aerodynamics), I shall give a simple, textbook derivation of the Euler equations. It is the kind of derivation that was wholly familiar to many of the actors in my story, and certainly to those who worked in and for the Advisory Committee for Aeronautics. The discus­sion in the next section is therefore slightly more technical. It is based on the treatment given in one of the standard works of early British aerodynamics, namely, W. H. Cowley and H. Levy’s Aeronautics in Theory and Experiment that was published in 1918.11 Both Cowley and Levy worked at the National Physical Laboratory. Levy had graduated from Edinburgh in 1911, visited Got­tingen on a scholarship, and had then worked with Love in Oxford. Dur­ing the Great War he had been commissioned in the Royal Flying Corps but was seconded to the NPL. As a left-wing activist who wanted to unionize his fellow scientists, his relations with Glazebrook were not of the easiest. After the war Levy left to join the mathematics staff at Imperial College, where he was eventually awarded a chair.12 Cowley stayed at the NPL and worked on problems of drag reduction with R. J. Mitchell, who was design­ing the racing seaplanes that won the Schneider Trophy for Britain in 1929 and 1931.13

A Public Confrontation

In March 1915, Lanchester gave an exposition of his theory at the Institution of Automobile Engineers in London.62 In the audience of over 150 members and guests was a fellow member of the Advisory Committee, Mervin O’Gorman, as well as Leonard Bairstow of the National Physical Laboratory. Lanchester devoted the first part of the lecture to the theory of lift or “sustentation.”63 The presentation started from the observed differences in pressure be­tween the upper and lower surfaces of an aircraft wing. For maximum effi­ciency, argued Lanchester, the flow of air over the wing must conform closely to the surface of the wing. Conformability, rather than the separation charac­teristic of Kirchhoff-Rayleigh flow, was the central assumption. At the tip of the wing, however, complications enter into the story. The higher pressures on the lower surface cause the air to move around the tip from the lower to the upper surface. When combined with the motion of translation of the wing through the air, the circulating motion at the tips has two consequences. First, it gives the flow over the top of the wing an inwardly directed compo­nent, toward the center line, but an outwardly directed component on the lower surface. Second, at the tips themselves, the circulation is swept back­ward to form two trailing vortices coming away from the ends of the wings. To complete the dynamical system, argued Lanchester, the two trailing vorti­ces must be joined, along the length of the wing, by a vortex that has the wing itself as its solid core. The vortex provided the circulatory component of the flow around the wing and accounts for the velocity difference between the flow over the upper and lower surfaces. This in turn accounts for the pressure difference, and hence the lift.64

Lanchester combined his exposition with some methodological observa­tions. He began by distinguishing the theoretical approach to aerodynamics from the purely empirical approach and noted that the two methods can, to a great extent, be followed independent of one another. Nevertheless, he insisted that engineering needed theory and that experiment without theory was “inefficient.” When variables were effectively independent, simple em­pirical methods of keeping everything constant except one variable might suffice; when variables were dependent on one another, this method ob­scured the crucial connections. At the conclusion of his lecture he returned to these methodological points, saying, “It has not been found possible in the present paper to do more than give an outline of the theory of sustenta – tion, with sufficient examples and references to practice and experiment to illustrate the importance of the theoretical aspect of the subject as bearing on the experimental treatment; the latter has hitherto been dealt with almost without considerations of theory, and has degenerated into empiricism pure and simple” (207). Although Lanchester was making a general claim about the guiding role of theory, there can be little doubt that he had the neglect of his own theory in mind. This was certainly how he was understood by some of his audience.

Lanchester’s lecture impressed at least some of the practical men, and it was greeted by an enthusiastic editorial in Flight.65 The immediate reception by the audience was, however, mixed. Mervin O’Gorman began the discus­sion after the lecture by congratulating Lanchester on his freshness of outlook and went on to offer empirical support for Lanchester’s theory. Experiments had been done on full-sized wings at the Royal Aircraft Factory that demon­strated the predicted inward and outward flow on the respective upper and lower wing surfaces.

We fastened pieces of tape at one end of the upper surfaces of the leading edge of the tips of an aeroplane wing, and arranged a camera, worked by a Bowden wire, to photograph them in flight; they were not put there for the purpose indicated by the author, but we got exactly what he says we should get, and I am glad to confirm him so far. (228)

Leonard Bairstow (fig. 4.10) then rose and adopted a different tone. He an­nounced to the audience that he was not convinced by Lanchester’s ideas.

I quite agree with Mr. O’Gorman that the paper is extremely interesting, but I also find it extremely controversial, and I disagree with his final conclusions.

(229)

By “final conclusions” Bairstow was referring to Lanchester’s suggestion that aerodynamics had degenerated into pure empiricism. Bairstow took it personally:

Many references have been made in the paper to experimental work at the National Physical Laboratory, which work is generally under my charge, and the author has done his best to put the N. P.L. on its defence for not making practical application of his theory. (229)

Given that much of Bairstow’s work had been on stability, and had been guided by the theory developed by G. H. Bryan, it is easy to understand why the general criticism might have struck Bairstow as unjust. The work on sta­bility was certainly not mere empiricism. But Lanchester was talking about lift. Here the charge of empiricism was more plausible. For example, Joseph Petavel, a fellow member of the Advisory Committee and the future direc­tor of the National Physical Laboratory, had given the Howard Lectures in March and April of 1913 at the Royal Society of Arts. He had devoted them to aeronautics, but his treatment had been purely empirical.66 He simply pre­sented his audience with a stream of graphs and empirical coefficients. There was no mention of either the discontinuity theory or the theory of circula­tion. And had not Bairstow himself admitted the resort to empiricism when he had addressed the Aeronautical Society that same year?67

This was true, but all that Bairstow needed to claim to rationalize his po­sition was that Lanchester’s theory was not acceptable because it was a bad theory. He was saying, in effect, show me an adequate theory and I shall use it to guide my experiments, but as yet no such theory is on offer. Bairstow’s objection was that Lanchester’s theory covered some, but not all, of the facts that were of interest to the aeronautical engineer. Bairstow had come pre­pared to prove his point: “I will not pretend to follow the analytical steps between the author’s statements of the vortex theory and his applications, but I will deal with two experiments made at the N. P.L.” (229). With this heavy hint that Lanchester’s position lacked logical clarity, Bairstow proceeded to show the audience two photographs. They depicted a square, flat plate set at an angle of 40° to a stream of water. The water was injected with ink to make the flow visible. Both photographs were taken from above, the first being at a slow speed of flow, the second at a faster speed. Referring to the first pic­ture, Bairstow conceded that it looked to him like the flow that Lanchester had described and as it had been presented in a line drawing (called figure 6) in Lanchester’s talk. Two trailing vortices could be seen coming from the sides of the plate (which Bairstow described as a low-aspect-ratio wing). The higher speed flow, however, presented a very different appearance. If one

A Public Confrontation

figure 4.10. Leonard Bairstow (1880-1963). Bairstow was the principle of the Aerodynamics Division at the National Physical Laboratory, where he did extensive testing and development of G. H. Bryan’s work on stability. Bairstow was skeptical of the circulatory theory of lift and of any approach that ignored the viscosity of air. As a young man he had a reputation for intellectual pugnacity. (By permission of the Royal Society of London)

photograph fitted the theory, the other certainly didn’t. Introducing the first photograph Bairstow said: “The resemblance of this photograph to Fig.6 of the paper is very marked, and up to this point I am thoroughly in accord with the author as to the probable, and in fact almost certain, existence of the type of flow postulated in the early part of the paper” (230).

Moving on to the second picture with the more rapid flow, he added: “The type of flow is now very different from that to which the author’s theory applies. The fluid round the model aerofoil leaves it periodically in spinning loops. The spiral showing the spin inside the arch of one of the loops is very distinct” (230). He conceded that Lanchester’s theory might fit “the very best aerofoil that can be designed at its very best angle of incidence” (230), but the theory said nothing about the full range of significant flow patterns. The word “stall” was not used, but Bairstow’s argument was that Lanchester could not explain what happens when a wing stalls: “There appear, then, to be ex­ceptions to the author’s theory, or rather, there are cases of fluid motion of interest to aeronautical engineers which do not satisfy the conditions that the surface shall be conformable to the streams” (230).

Lanchester gave a robust reply. First, he put Bairstow in his place by re­minding him of their relative positions in the hierarchy of command. While Bairstow was in charge of much of the experimental work on aerodynamics at the NPL, he, Lanchester, was on the Advisory Committee for Aeronautics, which controlled that work. Would he, Lanchester, be denigrating the very institution for which he had responsibility?

Mr Bairstow has suggested that my paper is in some degree an attack on the National Physical Laboratory, or at least he states that I have done my best to put the Laboratory on its defence. I will say at the outset that the National Physical Laboratory is an institution for which I have the greatest possible respect, and I am happy to count amongst my friends members of the Labora­tory staff, whose work and whose capacity are too well known to be injured by friendly criticism. Beyond this, any criticism which is to be incidentally inferred as implied by my remarks is not only criticism of our own National Laboratory, but equally of every aerodynamic laboratory with whose records I happen to be acquainted. Finally, on this point, any destructive or detrimental criticism of the work being done in the aeronautical department of the N. P.L. must reflect adversely on myself, since I am a member of the Committee whose duty it is to direct or control the particular work in question. (241)

Having sorted out the status question, Lanchester turned to Bairstow’s photographs and the accusation that the circulation theory would only apply to a good aerofoil at the best angle of incidence. Is this really a fault asked Lanchester?

Put bluntly, my answer to this is that it is equivalent or analogous to saying that the theory of low speed ship resistance as based on streamline form, and skin friction, is invalid because it does not apply to a rectangular vessel such as a packing-case, and is only true if applied to the very best design of hull with the finest possible lines. (242)

If the theory applied to a few important facts that was triumph enough. All Bairstow’s photographs, Lanchester went on, dealt with flows outside the scope of his theory.

I consider it quite preposterous to suggest that my theory should be tested by its applicability to the case of a square plane at 40 degrees angle as to test the theory of streamline ships’ forms by tank experiments on a coffin or a cask of beer. (243)

Bairstow claimed that theories of wide scope served the interests of aero­nautical engineers, but Lanchester argued that they cut across, rather than expressed, the engineer’s pragmatic standards. Most practical solutions, said Lanchester, were narrow in scope. No one would expect to compute the “re­sistance of a ship in sidelong or diagonal motion through the water” by the same methods and equations “as those applicable in the ordinary way” (251).

Prandtl and the Boundary Layer

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

Prandtl and the Boundary Layer

figure 7.1. Ludwig Prandtl (1875-1953). Prandtl is shown ca. 1904 at the technische Hochschule in Hanover demonstrating his hand-driven water channel used to take flow pictures of boundary-layer phenomena.

Prandtl and the Boundary Layer

figure 7.2. Separation of boundary layer according to Prandtl. From Prandtl 1904, 487. (By permis­sion of Herr Helmut Vogel)

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 and the Boundary Layer

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

Prandtl and the Boundary Layer

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

A Conclusion and a Warning

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

Surfaces of Discontinuity

Consider the idealized model of the postcard experiment, that is, the stream­lines around a flat plate normal to the flow. The flow could take the form shown in figure 2.7 as well as that already shown in figure 2.5. Instead of curl­ing around the edges of the plate and moving down the back of the plate, the flow of ideal fluid can break away at the edges. Behind the plate the flow is not a mirror image of the flow in front of it but consists of a body of “dead air,” or dead fluid, bounded by the moving fluid which has met the plate, moved along the front face of the plate, and separated at the edges. The pressure in the “dead air” will be the same as that at a great distance from the plate and can be equated with the atmospheric pressure. In a real, viscous fluid, the moving fluid and the dead or stationary fluid would interact. There would be a transition layer, with a speed gradient created by the stationary fluid retarding the moving fluid while the moving fluid sought to drag the station­ary fluid along with it. In an ideal fluid there will be no such transition layer because there will be no traction between the two bodies of fluid. The free

Surfaces of Discontinuity

figure 2.7. Discontinuous flow of an ideal fluid around a barrier normal to the free stream. The surfaces of discontinuity or “free streamlines” represent the abrupt change between the moving fluid and the dead fluid behind the barrier.

stream will pass smoothly over the dead fluid so there will be a sudden transi­tion from fluid with zero velocity to fluid with a nonzero velocity. Mathema­ticians call this sudden transition a discontinuity in the velocity because there are no intermediate values. This term gives rise to the general label for flows of this kind, which are called discontinuous flows. The streamline that marks the mathematically sharp discontinuity between the moving and stationary bodies of fluid is called a free streamline. It is a line of intense vorticity along which the flow possesses rotation in the technical sense defined earlier in the chapter.

This attempt to make mathematical hydrodynamics more realistic was introduced by Helmholtz in 1868 in a paper titled “Uber discontinuirliche Flussigkeits-Bewegungen” (On discontinuous fluid motions).40 Helmholtz argued that all the flows that had produced d’Alembert’s paradox had de­pended not only on the assumption that the flow was inviscid but also on the assumption that the velocity distribution was continuous. Helmholtz explored flows involving surfaces of separation (Trennungsflache) or (what is mathematically equivalent) sheets of vorticity (Wirbelflache).41 Of course, said Helmholtz,

Die Existenz solcher Wirbelfaden ist fur eine ideale nicht reibende Flussigkeit eine mathematische Fiction, welche die Integration erleichtert. (220-21)

The existence of such a vortex sheet for an ideal inviscid fluid is a mathemati­cal fiction to make the integration [of the equations] easier.

But fiction or no fiction, Helmholtz had raised the hope that the glaringly false consequences of the standard picture of ideal-fluid flow could be avoided.

If a steady, discontinuous flow is to be possible, certain conditions must be satisfied. It must be the case that the static pressure on either side of the free streamline is the same, otherwise the flow pattern would not be in equi­librium and would modify itself. Since the flow at a great distance in front of the plate is assumed to have a constant speed V and to be at atmospheric pressure pa, while the dead air is also at atmospheric pressure, then the speed of the flow along the free streamlines that bound the dead air must also be V. This conclusion follows from Bernoulli’s equation relating speed and pres­sure. Bernoulli’s law also leads to the conclusion that a flow of this kind will generate a greater pressure on the front of the plate than on the back.

Consider the streamline that terminates at the stagnation point at the front of the plate. What is the pressure on the front of the plate at the stagna­tion point? Call the pressure p. Everywhere along the streamline that goes to the stagnation point, the static and dynamic pressure will sum to the same constant value, that of the Bernoulli constant or the total pressure head. The value of the constant, or the total pressure, at a distance from the plate is H = pa+ Уг p V2. On the plate, at the stagnation point, the speed is zero. There will be no dynamic pressure but only a static pressure that will equal the total pressure, therefore ps = H = pa + 4 p V2. The pressure produced by bringing the air to a standstill at the stagnation point thus exceeds the atmospheric pressure pa by the quantity Уі p V2. But the pressure on the back of the plate is also pa, so at this point there is an excess pressure on the front of the plate.

This argument only applies to the stagnation point, which is the point of maximum pressure on the front of the plate. What happens at other points on the front of the plate as the fluid moves away from the stagnation point and moves toward the edges? The fluid will speed up so its pressure will drop. But the pressure exerted by the moving fluid only drops to atmospheric pres­sure as it reaches the free stream velocity at the edges. It follows that, at all points on the front of the plate, there will be a higher pressure than the at­mospheric pressure on the rear of the plate. On this account, therefore, the forces on the plate do not cancel out, except at the very edges, and there is an overall resultant aerodynamic force on the plate.42

Discontinuous flows of this kind thus avoid the paradoxical-seeming zero-resultant outcome found by D’Alembert, but it is still necessary to ask whether the predicted forces are the right size. It is one thing to avoid a bla­tantly false outcome and another thing to do so by giving the right answer in quantitative terms. The question still remains: Do the forces predicted on the basis of discontinuous flow fully correspond to the observed forces? Quantitative knowledge of the forces on the plate calls for a quantitative knowledge of the speed and pressure of the flow along the front of the plate, not just at the stagnation point and the edges. Until this information could be provided, the picture was merely qualitative. Working independently of one another, Rayleigh and Kirchhoff provided testable answers.43

The quantitative analysis of discontinuous flows was not an easy task, but by the use of ingenious transformations, it proved possible to connect the discontinuous flow around a flat plate to the simple, uniform, horizon­tal flow. There was no guaranteed way to find the required steps leading to the simple flow. It called for a high order of puzzle-solving ingenuity. The character of the thinking required can be glimpsed from the first few steps of the process. Rayleigh and Kirchhoff noticed that in the original flow, the direction of the boundary streamline along the plate was known but not the velocity. For the free streamline, the reverse held: the velocity was known but not the direction. If the flow could be redrawn on a diagram where one axis was proportional to speed while the other axis was proportional to direction, then both parts of the streamline would be transformed into straight lines. This was a step toward the desired simplicity because the straight lines could be interpreted as “polygons” of the kind to which the Schwarz-Christoffel theorem could be applied. Neither Kirchhoff nor Rayleigh explicitly used the Schwarz-Christoffel theorem but used a number of ad hoc transformations to achieve the same goal.44 But once the formula describing the flow had been found, pressures and velocities could be calculated and quantitative predic­tions made.

Rayleigh’s achievement was to generalize Kirchhoff’s analysis, which dealt with plates that were normal to the flow, and give the analysis required for plates that were oblique to the flow. This classic result in hydrodynamics was published in 1878 and provided the starting point for the work of the Advi­sory Committee for Aeronautics when its members tried to explain the lift generated by an aircraft wing. The work was officially overseen by Rayleigh himself as president of the ACA. It was monitored on a day-to-day basis by Glazebrook and other mathematical physicists who were closely associated with Rayleigh. In the next chapter I describe this early British work on the lift and drag of a wing, which was based on the idea of discontinuous flow.

The Cambridge School

At 9.30 a. m., on August 24, 1912, Lamb took the chair of section III of the Fifth International Congress of Mathematicians that was being held in Cam­bridge.38 Section III was devoted to mechanics, physical mathematics, and astronomy. Lamb wanted to say a few words before getting down to business. He noted that, in spite of the subdivision of the field, the scope of the section was still a wide one. He then went on to offer a classification of the different styles of work that were to be represented. He also identified the predomi­nant style of what he called the “Cambridge school” within this typology. His words are revealing.

It has been said that there are two distinct classes of applied mathematicians; viz. those whose interest lies mainly in the purely mathematical aspect of the problems suggested by experience, and those to whom on the other hand analysis is only a means to an end, the interpretation and coordination of the phenomena of the world. May I suggest that there is at least one other and an intermediate class, of which the Cambridge school has furnished many examples, who find a kind of aesthetic interest in the reciprocal play of theory and experience, who delight to see the results of analysis verified in the flash of ripples over a pool, as well as in the stately evolutions of the planetary bodies, and who find a satisfaction, again, in the continual improvement and refine­ment of the analytical methods which physical problems have suggested and evoke? All these classes are represented in force here today; and we trust that by mutual intercourse, and by the discussions in this section, this Congress may contribute something to the advancement of that Science of Mechanics, in its widest sense, which we all have at heart. (1:51)

The tone may have been lofty but Lamb had a purpose. He was making a plea for the representatives of the different tendencies in the discipline to communicate and cooperate. The remarks suggest a background anxiety that there might be problems on this score, and Lamb may have known about the convoluted and acrimonious arguments between pure and applied math­ematicians that had been taking place in Germany. We must also remember that Lamb was addressing a gathering of men of powerful intellect, many with significant achievements behind them and reputations to make or break. Larmor, Levi-Civita, Darwin, Moulton, and Abraham were all in the audience, while the Gottingen laboratory was represented by the presence of two of Prandtl’s colleagues and former assistants, Theodore von Karman and Ludwig Foppl. All of these men played an active part in the session that fol­lowed. Given his unifying purpose, Lamb could not have risked caricaturing the different classes of mathematician.

The care with which Lamb would have chosen his words lends a particular interest to his description of the Cambridge school. Lamb saw the character­istic concern of its practitioners as lying between pure mathematics, on the one hand and, on the other, a purely instrumental view of mathematics, one in which its role was simply the interpretation and coordination of data. The point on which he placed the emphasis was that mathematical results should be verified by the interplay of theory and experience. Lamb obviously saw this process as more than mere success in the ordering of data. Truth and correspondence with reality were the central aims. He described this con­cern as “aesthetic”—a word chosen, surely, to portray an intellectual involve­ment that was dignified rather than merely useful. The emphasis on truth was certainly consistent with what Lamb had said elsewhere, for example, in the discussion of Stokes’ equations in his Hydrodynamics. Applied mathematics, as practiced at Cambridge, was to be justified by its capacity to portray the nature of physical reality, not by its employment of useful fictions.

Other Cambridge luminaries expressed themselves somewhat differently but conveyed a similar orientation. At a different session of the same con­ference the Cambridge mathematical physicist Joseph Larmor voiced senti­ments that reinforced Lamb’s message. Larmor asserted that the role of the mathematician and the physicist were essentially identical.39 A. E. H. Love had spoken out in support of Larmor at the conference.40 Love was reiterating a position already developed in his authoritative Treatise on the Mathemati­cal Theory of Elasticity.41 This volume contained a historical introduction in which tendencies and distinctions similar to those identified by Lamb were rehearsed and evaluated. Love declared, as one of his aims, that he wanted to make his book useful to engineers and this had led him “to undertake some rather laborious arithmetical computations” (v). But he also wanted to “em­phasise the bearing of the theory on general questions of Natural Philosophy” (v), and it was clear that this was where his heart lay. His historical comments were judicious, but he went out of his way to emphasize the non-utilitarian origins of the subject matter he was about to expound. Thus,

The history of the mathematical theory of Elasticity shows clearly that the development of the theory has not been guided exclusively by considerations of its utility for technical Mechanics. Most of the men by whose researches it has been founded and shaped have been more interested in Natural Philoso­phy than in material progress, in trying to understand the world than in try­ing to make it more comfortable. From this attitude of mind it may possibly have resulted that the theory has contributed less to the material advance of mankind than it might otherwise have done. Be this as it may, the intellectual gain which has accrued from the work of these men must be estimated very highly. (30)

Technical mechanics is to be distinguished from natural philosophy, and he, Love, was doing a species of natural philosophy. Any resulting failure to contribute to material progress did not seem to distress him unduly. He was more interested in the link with fundamental physics and in recounting the detailed discussions that had taken place over the number and meaning of the elastic constants. These had thrown light on “the nature of molecules and the mode of their interaction” (30). The wave theory of optics and the theory of the ether had benefited from advances in the theory of elasticity, as had, even, certain branches of pure mathematics. Though Love and Lamb expressed themselves differently, we see a similar distancing of applied math­ematics from issues of utility and an affirmation of the fundamental char­acter of the relation between mathematics and physical reality. G. I. Taylor’s demand, made a few years later in his Adams Prize essay, that applied math­ematics should have a firm basis in physics was the expression of a stance already endorsed by figures of authority on the Cambridge scene and already characteristic of the Cambridge school.42

The demand for a firm basis in physics had not always characterized what had passed as “mathematical physics” or “mixed mathematics” at Cambridge. Mathematicians of earlier generations had often been happy to see mathemat­ics arise from physical problems but had then developed it independently of experimental data or with only a loose or analogical link to physical reality. An example of this earlier phase, which was still evident as late as the 1870s, was James Challis’ Essay on the Mathematical Principles of Physics in which he offered a speculative, hydrodynamic cosmology.43 The closer connection between mathematics and real physics that Lamb and, later, Taylor were tak­ing for granted had originally been forged in the work of Stokes, Thomson, and Maxwell, who were critical of the earlier style.44 Lamb, however, still felt the need to express himself carefully when he said that the Cambridge school provided “many examples” of the intermediate path between an overly ab­stract and an overly utilitarian approach. He thus acknowledged a continuing diversity in Cambridge work. This should come as no surprise since tradi­tions, even vigorous traditions, will always encompass a range of positions as they change and develop. Rayleigh, like Lamb, spoke of “the Cambridge school,” and he too noted a certain inner complexity and development. In connection with Routh’s textbook on dynamics, Rayleigh took the view that the earlier editions had been overly abstract, whereas later editions evinced a closer engagement with genuine scientific problems.45 In other words Routh had shifted toward the position that Lamb, like Rayleigh himself, saw as the strong point of the Cambridge school.46

Scientific Intelligence: Fact and Fiction

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

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

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

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

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

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

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

Scientific Intelligence: Fact and Fiction

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

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

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

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

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

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

Scientific Intelligence: Fact and Fiction

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

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

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

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

3. but the multitude comprehended it not.

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

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

What had the Germans being doing during the war? At first, informa­tion filtered through to the British via the French and Americans. In 1919 the American National Advisory Committee for Aeronautics, the NACA, es­tablished an office at 10, rue Victorien Sardou in Paris. Their representative, William Knight, actively pursued a policy of information gathering. To the ir­ritation of military and diplomatic circles in Paris and Washington, he made contact with Prandtl and suggested that information on recent developments be shared. Knight contacted Prandtl on November 15, 1919, by letter and man­aged to get official agreement to visit him in April 1920. He made a second visit in the autumn of 1920.29 It appears to be due to the efforts made in Paris that the British Advisory Committee was furnished with translations of reports by A. Toussant and by Colonel Rene Dorand. Toussant was an engineer at the Aerotechnical Institute of the University of Paris at St. Cyr. In November 1919 he had produced a resume of the theoretical work done at Gottingen based on the Technische Berichte. Extracts of Toussant’s work were translated by the NACA, and these surfaced in February 1920 as a report designated Ae. Tech. 48 for consideration by Glazebrook and his colleagues.30 There was, however, little that Toussant could add to the information that was already at hand from the prewar abstracts. He gave mathematical formulas but none of the background reasoning. Dorand’s report was titled “The New Aerodynamical Laboratory at Gottingen.” It was translated as report T. 1516 in October 1920 and reached the Advisory Committee via the Inter-Allied Aeronautical Com­mission of Control.31 In Dorand’s report Prandtl was referred to through­out as “Proudet,” but it included three pages of blueprints of the Gottingen wind channels and provided technical details of the automatic speed con­trol and the measuring apparatus. Bairstow felt that there was nothing new in Dorand’s report.32 Others, however, noticed that Prandtl seemed to have solved certain problems that plagued the NPL channels and had “managed to achieve good velocity distribution in the working sections.”33

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

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

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

Objectivity and Reality

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

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

The example shows knowledge and skill tested by uncompromising, ex­ternal criteria. The pilots’ subjective feelings had to be mastered and their judgments subordinated to the objective demands of the situation. What, then, is “relative” about this episode and the knowledge and skills involved? The brief quotation from the Annual Review of Fluid Dynamics already indi­cates the lines on which an answer can be given. First, the relevant knowledge and beliefs were distributed unevenly across the groups mentioned in the Review article. The experts who worked theoretically, or who experimented with wind tunnels, had one opinion about vortex sheets; the pilots who chased the bombs over the fields of Kent are attributed with another opinion. Second, the character of the knowledge varied. The experts had a mathemati­cally refined understanding; the pilots had a rough-and-ready but practical sense of what they needed to do. What they lacked in rigor they made up in skill. Third, although the experts and the pilots were oriented to the same features of reality, they did not share a common language or common con­cepts. The article makes no mention of communication between pilots and aerodynamic experts on the matter, but it would almost certainly have been problematic. It had never been easy for pilots and aerodynamicists to talk to one another.82 Fourth, the range of circumstances that the members of the two groups took into account differed markedly. The experts operated in a world that was artfully controlled, shielded, and simplified; the pilots functioned in an environment saturated with complexity, interaction, noise, vibration, jolting, turbulence, and distraction.

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

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

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

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

Mathematicians versus Practical Men:. The Founding of the Advisory. Committee for Aeronautics

In the meantime every aeroplane is to be regarded as a collection of unsolved math­ematical problems; and it would have been quite easy for these problems to have been solved years ago, before the first aeroplane flew.

g. h. bryan, “Researches in Aeronautical Mathematics” (1916)1

The successful aeroplane, like many other pieces of mechanism, is a huge mass of compromise.

Howard t. wright, “Aeroplanes from an Engineers Point of View” (1912)2

The Advisory Committee for Aeronautics (the ACA) was founded in 1909. This Whitehall committee provided the scientific expertise that guided Brit­ish research in aeronautics in the crucial years up to, and during, the Great War of 1914-18. From the outset the ACA was, and was intended to be, the brains in the body of British aeronautics.3 It offered to the emerging field of aviation the expertise of some of the country’s leading scientists and engi­neers. In 1919 it was renamed the Aeronautical Research Committee, and in this form the committee, and its successors, continued to perform its guid­ing role for many years. After 1909 the institutional structure of aeronauti­cal research in Britain soon came to command respect abroad. When the United States government began to organize its own national research effort in aviation in 1915, it used the Advisory Committee as its model.4 The result­ing American National Advisory Committee for Aeronautics, the NACA, was later turned into NASA, the National Aeronautics and Space Administration. The British structure, however, was abolished by the Thatcher administration in 1980, some seventy years after its inception.5

If the Advisory Committee for Aeronautics was meant to offer the best, there were some in Britain, especially in the early years, who argued that, in fact, it gave the worst. For these critics the ACA held back the field of Brit­ish aeronautics and encouraged the wrong tendencies. The reason for these strongly divergent opinions was that aviation in general, and aeronautical sci­ence in particular, fell across some of the many cultural fault-lines running through British society. These fault lines were capable of unleashing powerful

and destructive forces. From the moment of its inception the Advisory Com­mittee was subject to the fraught relations, and conflicting interests, that divided those in government from those in industry; the representatives of the state from those seeking profit in the market place; the university-based academic scientist from the entrepreneur-engineer; the “mathematician” and “theorist” from the “practical man.” Throughout its entire life these struc­tural tensions dominated the context in which the ACA had to work.6

As The Lamps Were Going Out

Another of the practical men—and among the most interesting—was the designer A. R. Low, of Vickers, whose name has already been mentioned. In 1912 he had lectured at University College, London, on aerodynamics and, on March 4, 1914, with O’Gorman in the chair, gave a talk to the Aeronautical Society titled “The Rational Design of Aeroplanes.”82 Low discussed the same range of ideas that I have already identified in the writings of other practical men, but he did so with a lively, critical intelligence and a breadth of knowl­edge that makes his work stand out. Since he will play a significant role in the postwar story of the reception of Prandtl’s work, Low’s position in 1914 is worth appreciating.

Low argued that hydrodynamics is useful for providing a sound, gen­eral “outlook” on fluid flow but that in order to get “reasonably accurate numerical values we shall see. . . that we are thrown back on experimen­tal methods” (137). He showed his audience the diagrams from Greenhill’s report representing discontinuous flow around plates, some of which were normal to the flow and some at an angle to the flow. These were compared with photographs of real flows taken by the Russian expert in fluid dynamics, Dimitri Riabouchinsky. There was a “strong resemblance between the theo­retical boundary line between the stream and the back water, and the experi­mental boundary line between approximately steady flow and the region of marked turbulence” (138). But although Lord Rayleigh was “the first to give a formal value” for the reaction of a fluid on a barrier, his predicted value was only about half of the observed value for the small angles relevant to aerodynamics. An error of 50 percent is “quite intolerable to physicists and engineers” (137).

Low was clear that the idea of sweep was not the way forward. He did not use the word “sweep” but spoke of an “equivalent layer.” This approach, he said, introduced a number of variables, and there was no way of apportioning the energy losses between them. There were, said Low, an infinite number of possible ways of assigning the energy losses. Perhaps experimental methods, such as injecting colored dyes into the flow, could shed light on the question, but until then the picture was essentially arbitrary (140). Only the empiri­cal study of lift and resistance was left. Low then turned to a discussion of some experimental graphs and empirical formulas produced by Eiffel. It was known experimentally that, over a wide range, resistance varied as the square of the speed. The desired equations would then have the general form R = KAV2, where R is the force on the wing, Vis the velocity relative to the air, A is the area of the wing, and K is a constant that must be empirically determined. Given the number of variables involved, such as incidence, aspect ratio, and camber, Low observed that finding a formula that yielded the correct value would not be easy. A further complication was that the performance of the wing interacts with the flow round the rest of the machine. The point that Low wanted to stress was the “formidable series of special developments of engineering science” that were necessary before designers could be confident that any given set of drawings would turn into an airplane with predictable performance and air-worthy qualities. He concluded: “That nation will take the lead whose scientists and technical engineers, and whose works engineers, and whose pilots best understand each other and work together most cor­dially” (147). This plea for cooperation and coordination was timely and well meant, but Low must have known that none of these preconditions for taking the lead was satisfied. The divisive bigotry of C. G. Grey and his ilk put an end to any of the requisite cordiality.

The conclusion must be drawn that in 1914, on the eve of the Great War, none of the British workers in the field of aerodynamics, whether they were mathematicians or practical men, had any workable account of how an airplane could get off the ground. As the lamps of Europe were going out, vital parts of the new science of aeronautics were also shrouded in darkness.83 The mathematicians had a sophisticated theory that addressed the right ques­tions but, being based on the theory of discontinuous flow, gave disconcert­ingly wrong answers. The high-status and mathematically brilliant experts of the Advisory Committee were reduced to empiricism. The practical men were simply paddling in the shallows and, with the exception of a few, such as A. R. Low, appeared to be oblivious of the fact. Their ideas were vague, confused, and frequently failed to engage with either practical experience or experimental results. The mathematicians (unlike the practical men) could handle many of the problems about stability with confidence and rigor, but on the question of the origin and nature of lift, and the relation of lift to drag, they too had been effectively brought to a standstill.

The demands of the war years that followed seemed to discourage rather than encourage any fundamental reappraisal of the British approach. As far as the fundamental theory of lift and drag were concerned, British experts came out of the war little better than they went into it. In 1919 George Pagett Thomson summed up the situation as it had appeared to the British dur­ing those years. The terms of his assessment are sobering: “In spite of the enormous amount of work which has been done in aerodynamics and the allied science of hydrodynamics there is no satisfactory mathematical the­ory by which the forces on even the simplest bodies can be calculated with accuracy.”84 Thomson’s judgment was that for British experts, practice still ran ahead of theory, as it had at the beginning of the war and as it had from the earliest days of aviation. Throughout the war British aircraft could cer­tainly get off the ground. They flew and their wings worked. For this, both the trial-and-error methods of the practical men and the experimental work done by the Advisory Committee must be thanked. But why aircraft flew remained a mystery to those of a practical and a theoretical inclination alike. Only the most general principles of mechanics could be invoked by way of explanation, but these only indicated what, in terms of action and reaction, the wing must be doing, not how and why it did it. Probing the more specific workings of the aircraft wing remained in the realm of experiment, a process consisting of case-by-case empirical testing that was guided, or misguided, by intuition.85