Category The Enigma of. the Aerofoil

Prandtl’s boundary-layer theory provided the material that might give sub­stance and depth to the “scheme” of the wing theory. It suggested that inviscid approximations might be replaced by a more realistic account of the physics of viscous processes. The boundary layer became the focus of a sustained British research effort organized by the Fluid Motion Panel of the Aeronauti­cal Research Committee. The original intention was that Lamb would be the editor of the volumes that would draw the results together, although Fage, who clearly found Lamb’s work very demanding, put in a request that the mathematics should be kept as simple as possible. “Lamb’s Hydrodynamics,” said Fage in the course of a discussion of the proposed monograph, “was more suitable for the professional mathematician and was very difficult.”88 In the event, Lamb did not live to complete this task and it fell to Sidney Gold­stein. Goldstein had been a pupil of Jeffreys’ but he had also gone to Gottin­gen after the war to study with Prandtl.89 Despite continuing resistance, it is clear that the overall strategy that Prandtl had adopted in his Wright Lecture had been an appropriate one. He had engaged with the preoccupations of the British experts with viscous and eddying flow while reminding them of the intellectual resources that Gottingen had to offer.90 Writing to Prandtl, after the Wright Lecture, Major Low said that he had spoken to many mathemati­cians and physicists and they all said that “your paper will give a new direc­tion to aerodynamic research in this country.” Low identified the transition from laminar to turbulent flow as the special point of interest for the British audience.91

This concern with the boundary layer and turbulence became the new research front, and it was congenial territory for the British even though their head-on assault on the Navier-Stokes equations had proven frustrating. If the battle for circulation in the theory of lift was over, the war on turbulence in the boundary layer was about to begin.92 But even here the old worries were not far beneath the surface. On February 6, 1930, members of the Royal Aeronautical Society discussed a report titled “Modern Aerodynamical Re­search in Germany.”93 The report was presented by J. W. Maccoll, who had visited Gottingen and Aachen.94 Maccoll, who had a command of German, was a government scientist and was to hold the post of research officer in the Department of External Ballistics at the Woolwich Arsenal. He described in mathematical detail the original work on the laminar boundary layer and then the more recent work on the transition to turbulence. In the discussion that followed Maccoll’s paper, Bairstow identified what he saw as two fun­damentally different approaches to the current problems in fluid dynamics.

Bairstow declared that he had “been impressed by the extreme complication of the whole subject and the apparently little connection between the Ger­man methods of solution and the equations of motion of a viscous fluid. All would have noticed how often new variables were introduced into the equations to deal with failures of the original hypothesis. It seemed that the Germans were making an engineering attempt to get solutions of practical value and had little hope of solving the equations of motion in a sense that would satisfy Professor Lamb” (697).

Bairstow was describing, albeit in a one-sided way, the difference in ap­proach between a mathematically sophisticated engineer, adopting the meth­ods of technische Mechanik, and that of a mathematical physicist drawing on the finely honed traditions and research strategy of the Cambridge school. Bairstow might not have sat the Tripos, but he still took Prof. Lamb as his reference point.95 The difference in approach to which Bairstow was alluding, between the Cambridge and Gottingen traditions, has been present in one form or another throughout the story I have been telling. It was implicated in the original British dismissal of the circulatory theory, and it was central to the manner in which the theory was finally accepted by the British.96

In an article titled “Twenty-One Years’ Progress in Aerodynamic Science” which Bairstow published in 1930, the same year as the remarks just quoted, he surveyed the work that had been done since the creation of the Advisory Committee for Aeronautics in 1909. Bairstow invoked a revealing compar­ison to describe the discomfort that still surrounded the relation between the theory of viscous and inviscid fluids in aerodynamics. He likened the problem of reconciling the viscous and inviscid approaches to the problems that British physicists were experiencing in reconciling the wave and particle conceptions of light and of the electron. Two fundamentally different models were in use, but it was impossible to see how they could both be true.97 Bair­stow quoted the exasperated response to this situation of one of the country’s leading physicists, a response that mirrored, perhaps, the frustrations of Bair – stow’s own work on the Navier-Stokes equations. “Aerodynamic theory,” said Bairstow,

is now rather like the physical theory of light; Sir William Bragg recently said that physicists use the electron theory on Mondays, Wednesdays and Fridays, and the wave theory on alternate days. Both have uses but reconciliation of the two ideas has not yet been achieved. So it is in aeronautics. In our experi­mental work we assume that viscosity is an essential property of air and the building of a compressed-air tunnel is the latest expression of that belief. The practically useful theory of Prandtl comes from considering air as frictionless or inviscid. (29)

At the end of his survey Bairstow returned to this theme and defined his view of the prospects of aerodynamics in terms of this ambiguous and problematic image. We can be assured, he said, that aerodynamics has “a future compa­rable with that in electron theory” (30).

Despite Glauert’s efforts to renegotiate the conceptual distinction between perfect fluid theory and the theory of viscous fluids, it is clear that the lead­ing British mathematical physicists were in no hurry to abandon their view that the distinction was fundamental. The boundary separating the objects of the two theories was treated as ontologically rigid rather than methodologi­cally flexible. Eventually, though, by the mid – and late 1930s, what Glauert called the “true conception of a perfect fluid” appears to have filtered into British mathematical and experimental practice. It was not acknowledged explicitly, but it was implicit in the use of potential irrotational flow as an engineering ideal. By the 1940s its use for this purpose had become routine, for example, in estimating the role played by the viscous boundary layer.98 By this time the circulation theory of lift, and Prandtl’s wing theory, had already become an established part of British aerodynamics. The earlier insistence on a rigid conceptual boundary between ideal and real fluids nevertheless helps to explain why, when Prandtl’s wing theory was finally accepted by the Brit­ish, there was still a note of reservation. Prandtl’s theory may have been, as Bairstow conceded, “the best and most useful working hypothesis of modern times”—but it was still a working hypothesis.

For many years, one of the standard British textbooks in the field was Milne-Thomson’s Theoretical Aerodynamics." The book ran through four editions between 1947 and 1966 and contained the following, revealing obser­vation on the lifting-line theory. Following an explication of Lamb’s contrast between a scheme and a fundamental theory, Milne-Thomson said, “The student should be warned, however, that the investigation on which we are about to embark is one of discussing the deductions to be made from sche – matization of a very complicated state of affairs and that the ‘laws of Prandtl’ which will be used as a basis are not necessarily laws of nature” (191). Con­trasting the Laws of Prandtl with the Laws of Nature was just a picturesque way of saying what most British experts had felt all along. Prandtl’s work on the aerofoil was an exercise in engineering pragmatism rather than a contri­bution to a realistic and rigorous mathematical physics.

In my study of the difference between the British and German responses to the circulatory theory of lift I have followed out the implications of Frank’s comparison. I have tried to dig into what he called “the depths,” where, as he rightly said, “the real battle for the progress of knowledge has been fought.”112 I have engaged with the details of the scientific and technical argumentation over the theory of lift because it is here, in these details, that both the social character of knowledge and the consequent relativity of knowledge find their most revealing expression. The story reminds us of the sheer contingency and unpredictability of the outcome of any research enterprise and shows how complex and fine-grained that contingency can be. It shows the vital and ineradicable role played by cultural traditions and the institutions that transmit these traditions. And, as Frank predicted, nowhere in the analysis of scientific discourse was a legitimate place found for the term “absolute truth.”

This insight is in constant danger of being forgotten or obscured by the false friends of science. The relativity of all scientific concepts to culture and society is deemed unacceptable by the self-appointed guardians of knowledge who claim to “take reflective responsibility, as it were, for the normativity of our most fundamental cognitive categories.”113 Historians and sociologists, like experimental psychologists and anthropologists, have always known that it is not normative posturing but close and careful empirical studies of cogni­tion that are needed. Strange though it may seem, this principle needs special emphasis when the cognition in question is that of scientists and engineers. The practices of scientists and engineers must be studied in a hard, factual light as natural phenomena that belong to the material world of cause and effect. Only studies conducted in this spirit can carry the analysis beyond ide­ology and propaganda and lay the basis for a proper, public understanding of science and technology.114

Detailed empirical studies always need a methodological context, and at­tention must always be given to the broader framework in which they are un­derstood. Thus one may legitimately ask where all the intellectually brilliant activity of the men and women I have studied is to be positioned in relation to the grand philosophical categories of Progress, Reality, and Truth. Now that the facts of the case study are at hand, the answers are not difficult to supply, though their implications may be disturbing.

Let me take each category in turn. That the work of the German engineers constituted technical progress is beyond doubt, and it is this which eventually had its impact on the British. The German work had utility and practicality relative to goals and interests shared by the experts of the two nations. The British led the way in the study of stability, but when it came to the study of lift and drag, failure and frustration took its toll on the British experts. The practical rewards and opportunities offered by the German approach eventu­ally tempted even the strictest to compromise their principled commitment to theories with a firm basis in physical reality. The theory of circulation and Prandtl’s theory of the finite wing allowed the experts to do things that they wanted to do, and that fact alone was, in its own way, rewarding. If the at­traction of moving forward could not rationally compel a change of mind on important theoretical questions, it encouraged a pragmatic accommodation.

That everyone in the field of aerodynamics, British and German, was, each in his or her own way, grappling with reality is also evident, and this was wholly taken for granted in all of the reasoning of the actors I have described. Although one may question the extent of Sir George Greenhill’s connection with reality in his notorious Reports and Memoranda No. 19, on discontinu­ous flow and free streamlines, this work did not set the pattern for the future reports of the Advisory Committee. As a group the British were no less con­cerned than their German counterparts with understanding the real perfor­mance of real airplanes under real conditions of use. And wasn’t Sir George himself acting as the spokesman of the practical realists when he (correctly) took G. H. Bryan to task for neglecting the gyroscopic effect of the engine and propeller in his analysis of stability?

Engagement with reality may be common ground, but my example shows that there are different ways of grappling with reality. It also shows that these different modes of engagement are social modes belonging to, and sustained by, different groups with different local traditions. Such differences can di­vide groups that otherwise share much by way of a common culture, as did the British and German experts in aerodynamics. Even more important, the example shows that there are no independent methodological principles by which these different forms of engagement could be reliably and usefully as­sessed. Such principles as emerged in the episode were themselves integral to the forms of engagement they were used to justify. They were rationalizations of existing practices and institutions. That there are different ways of engag­ing with the world may seem obvious; that the only ground available to the actors for justifying their choices is question begging is perhaps less obvious. But obvious or not, it follows directly from the fundamentally social charac­ter of cognition. This is one reason, though not the only reason, why the so­ciologically minded David Hume was right when he said that all the sciences have a relation to human nature and that “however wide any of them may seem to run from it, they still return back by one passage or another.”115

What, finally, is to be said about truth? The progress in aerodynamics made in the technische Hochschulen and the University of Gottingen derived from the use of a theory of perfect fluids in potential motion. The theory dealt with an idealization and a simplification. This theory was dismissed in Cambridge and London as physically false and logically self-defeating. It was false because it denied the viscosity of the air and self-defeating because cir­culation was unchanging with respect to time, and its origin was beyond the reach of the theory. The premises of the British objections were true and the reasoning based on them was sound, but the conclusions led to failure rather than success.

The German advances in the understanding of lift and the properties of wings depended on the use of abstract and unreal concepts that were some­times employed with questionable logic. Progress in aerodynamics thus de­pended on the triumph of falsity over truth. Everyone knows that false prem­ises can sometimes lead to true conclusions and that evidence can sometimes support false theories, but the story of the aerofoil involved more than this. The successful strategy involved the deliberate use of known falsehoods poised in artful balance with accepted truths. The supporters of the theory of circulation showed how simple falsehoods could yield dependable conclu­sions when dealing with a complex and otherwise intractable reality. This is the real enigma of the aerofoil.

The enigma would hold no surprises for Hume. It would simply be an expression of what he called “the whimsical condition of mankind.”116 The lesson Hume learned from the study of history and society was that “the ul­timate springs and principles” of the natural world will never be accessible to the human mind. The utmost that reason can achieve is the simplifica­tion of complexity. Humans live and operate in a world of limited experience dominated, necessarily but beneficially, by custom, convention, habit, and utility. Hume acknowledged that “the philosophical truth of any proposi­tion by no means depends on its tendency to promote society” but argued that we should be neither surprised nor unduly alarmed if truths (or sup­posed truths) that lack utility sometimes “yield to errors which are salutary and advantageous” (279). The story I have told deals with a technology that may seem remote from the world whose problems exercised Hume, but the central fact to emerge in my story, the fact I have called the real enigma of the aerofoil, can be understood in the humane, skeptical, and sophisticated terms he offered. Among the British it was an accepted truth that the air was a viscous fluid governed by Stokes’ equations. In the field of aeronautics that truth, if truth it be, yielded to the erroneous but salutary and advantageous picture of the air as an inviscid fluid governed by Euler’s equations.

Are there general lessons to be learned? Not if aerodynamics is a special case, but I do not think it is a special case. The conclusions reached in this case study surely can be generalized.117 What, then, should be concluded? Individual developments in the sciences will differ in their details, but what Frank had to tell us about the compromises involved in the design of air­planes applies (and was meant to apply) to the technology and instruments of all thinking. There are always compromises to be made. The warning given by von Mises against the phantom of absolute cognition will always be rel­evant. And there will always be a role in science and engineering for the blunt advocacy of a Major Low and the rapier responses of a G. I. Taylor. Above all, what Kuchemann had to say about the idealizations of aerodynamics cap­tured the essence of the creative work of Lanchester, Prandtl, and Glauert. But idealizations are salutary and advantageous falsehoods which play a vital role in all science, pure as well as applied. In stressing the role of idealization,

Kuchemann may have identified a feature of cognition that is more salient in engineering than in physics—but it is the engine of progress in all fields. Those who point to the airplane as a symbol of the truth of science, the power of technology, and the reality of knowledge are therefore right—but do they know what they are saying? The enigma of the aerofoil is the enigma of all knowledge.

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

Greenhill’s contribution was not confined to the daunting R&M 19. As well as working on the mathematics of gyroscopes and problems of airship stability, in 1910 and 1911 he gave a series of lectures at the Imperial College of Science and Technology. The course was published a year later in a book titled The Dynamics of Mechanical Flight.21 Greenhill explained that the lift of a wing depended crucially on “the opening out of the stream lines” (40) behind the wing. This occurrence would be the expected effect of the surfaces of discon­tinuity enclosing the “dead” air region above and behind the wing. Greenhill went on to contrast the truth, as he saw it, of the picture of discontinuous flow with the error of certain popular conceptions about the flow of air over a wing. (The two different ideas of the flow are represented in figs. 13 and 14 in his book.) The passage in which he contrasts them is a revealing one: “A popular figure of the stream lines past a cambered wing as here in Fig. 13, showing no such broadening, would imply at once to our eye an absence of all thrust and lift; the figure should be more like Fig. 14” (41). The diagrams to which Greenhill was referring are shown as my figures 3.4 and 3.5 (with Greenhill’s numbering identified in the captions).

Greenhill’s second diagram that was meant to describe the correct flow indicates turbulence in the “dead” air, though his mathematical analysis does not make provision for this. The embellishment seems to be a concession to

1912, 41.

Kelvin. We know why Greenhill believed that there was lift in the case of the discontinuous flow, because of the reasoning set out by Rayleigh, but why was it obvious to Greenhill’s eyes that the popular flow picture, my figure 3.4, would be devoid of all thrust and lift? The reasoning may have been that with­out surfaces of discontinuity, the flow must have the character of the original, continuous flow of an ideal, frictionless fluid—with the “paradoxical” result of zero-resultant force. It looks as if Greenhill took the diagram of smooth, streamlined flow over a wing to imply that the air was being treated as a con­tinuous, ideal fluid in irrotational motion. In other words, it was taken as a flow in which there would be no resultant and where d’Alembert’s paradox would be applicable.

If this was the reasoning, then two significant details of Greenhill’s first drawing were wrong. The flow is not pictured accurately at the leading or trailing edge. The front stagnation point should be below the leading edge, while the rear stagnation point should be on the upper surface of the wing in front of the trailing edge. Instead the air is shown coming away smoothly from the trailing edge itself. Greenhill would certainly have noticed this error, and he gave the correct form of the diagram for a flat plate on page 47 of his book. He presumably put the inaccuracy down to the approximate character of the “popular” representation, as he had in an earlier criticism of drawings of leading-edge flow (22). In any event, he seems to have taken the popular diagram as an attempt to depict the kind of idealized, continuous perfect fluid flow where, as any mathematician would know, all the forces (except the turning couple) canceled out.

Although Greenhill made more effort in the book than in the ACA re­port to bring real aircraft into the discussion, it was still full of examples and mathematical technicalities of questionable relevance. Predictably, it did not go down well with the practical men. The anonymous reviewer for Aero­nautics, who had apparently attended the lectures, said the book confirmed the earlier impression that the calculations were really aimed at providing a diverting recreation for the mathematical mind. “Practical value they lack wholly; the data on which Sir George Greenhill’s mathematical excursions are based are theoretical without fail.”22

G. H. Bryan’s review was very different.23 Up to the present, said Bryan, there had been a lack of understanding about the role of mathematics in aeronautics. The subject has failed to attract our best mathematicians, while “practical men” make claims “in utter disregard for the fundamental prin­ciples of elementary mathematics and physics” (264). Under these “chaotic conditions” it would be useful to have a work “by so reliable a mathematical authority” as Greenhill (265). Bryan acknowledged the presence of drastic simplifications involved in Greenhill’s approach but insisted that, in spite of these shortcomings, “the theory of discontinuous motion affords the best opening to the study of pressures on planes from the mathematical stand­point” (266). He listed the “great mathematicians” who had developed the theory, but noted that it had only been applied to flat plates and not yet to bent or cambered planes (which would make better models of the aerofoils in practical use). Some calculations of this kind, the reader was told, were now under way. Bryan did not once ask if the theory of discontinuous flow gave empirically adequate answers. Rayleigh, of course, knew that, as the theory stood, it did not give the right answers, and so did his experimentally inclined colleague Mallock. I now turn from the mathematical to the experimental study of discontinuous flow to see how matters were carried forward on this front.

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

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).

Wilhelm Martin Kutta (fig. 6.1) was born in Pitschen in Upper Silesia in 1867. He lost both parents at an early age and was brought up in the household of an uncle in Breslau. After attending the university in Breslau from 1885 to 1889, he went to the University of Munich, where he studied from 1891 to 1894. Kutta went on to achieve a lasting place in the history of applied

figure 6.i. Martin Wilhelm Kutta (1867-1941). In 1910 and 1911 Kutta published and extended an analysis of the flow of air around the wing of Lilienthal’s glider that he had worked out in 1902 in a dis­sertation at the technische Hochschule in Munich. Kutta assumed that the flow contained a circulation and showed how to link the flow around the wing to the simpler and already solved problem of the flow around a circular cylinder. He was then able to make a plausible prediction of the lift of the wing. After these pioneering papers, Kutta published nothing more. (By permission of the Universitatsarchiv Stuttgart) mathematics for two reasons. First, in his doctoral work of 1900, he developed a numerical method for solving ordinary differential equations. This has be­come known as the Kutta-Runge method and is to be found in all textbooks on the subject.2 Second, he produced a pioneering paper on aerodynamics which appeared in 1910,3 with further developments published in 1911. These papers were based on methods he had developed in his Habilitationschrift of 1902, which he wrote at the technische Hochschule in Munich.4 (This institu­tion is often referred to by its initials as the THM and, for brevity, I follow this practice.) Unfortunately no copies of the Habilitationschrift appear to have survived.5 From the brief summary that was published in 1902, however, it seems to have been the first, mathematical analysis of lift that was based on the circulation theory.6

Kutta was a conscientious teacher who, over the years, introduced hun­dreds of engineering students to the methods of applied mathematics. His mathematical knowledge was said to be of enormous scope and his help was frequently requested by colleagues. He had a deep knowledge of history, literature, and music, a command of languages, including Arabic, and was widely traveled. He never married, however, and was something of a recluse. A colleague of long-standing, Friedrich Pfeiffer, who had been a student un­der Kutta at the THM, wrote an obituary for Kutta after the Second World War.7 In the article, Pfeiffer recalls that Kutta would typically sit alone in the most remote corner of the Mathematical Institute at Munich. After Kutta’s retirement, said Pfeiffer, he and other colleagues would sometimes encounter Kutta, though this happened infrequently. The lack of contact was put down to Kutta’s reticence. When they did meet, Pfeiffer was unhappy with what he found. In later years, he said, Kutta obviously lacked a loving and caring hand (“wie sehr ihm eine liebende und sorgende Hand fehlte”). He went on:

Oft habe ich Kuttas Leben reich und beneidenswert gefunden wegen seiner Aufgeschlossenheit fur so viele Seiten menschlichen Geisteslebens, oft aber fand ich es auch arm und bedauernswert in seiner Einsamkeit und Zuruck – gezogenheit. (56)

I have often found Kutta’s life rich and enviable because of his openness to so many aspects of human cultural life, but I have also often found it poor and rather sad in its solitariness and seclusion.

How were things really, asked Pfeiffer, and did not know the answer. But if Kutta’s inner life was closed to his colleagues, and must be closed to us, his work is open to inspection. Seclusion notwithstanding, he published work that bore the stamp of a time and a place. It was the product of a specific, professional milieu.

Kutta’s career as an academic began in 1894 when he became a teaching assistant in higher mathematics at the THM. Like all the technische Hoch – schulen, the THM had experienced long-standing tensions over the role to be played by mathematics in the training of engineers. How much mathematics should be on the syllabus? What sort of mathematics should be offered, at what level, and who should teach it? These tensions have now been the subject of close, historical study, and thanks to this work there is much about the overall structure of the situation, as well as the particular circumstances in Munich, that can be sketched with some confidence. It is thus possible to form a pic­ture of the context in which Kutta came to do his work on aerodynamics.

Three points must stand out in any general overview. First, the technische Hochschulen (or THs) tended to recruit their mathematics teachers from the universities and, when they were good, lose them again to the universi­ties. This mixture of policy and necessity carried with it certain problems. From the mid-1850s, university mathematics in Germany had been increas­ingly dominated by a concern with rigor and so-called pure mathematics.8 Although the THs provided jobs for mathematicians, those who took the jobs often had their eyes focused on matters that fell outside the concerns of the THs. Their teaching, like their research, was abstract and lacked relevance to engineering. Justifiably, this caused resentment among the engineers, with the result that mathematics appointments often turned into a struggle be­tween different factions in the TH.9

Second, and predictably, engineers were not a homogeneous group. Some engineers wanted to use mathematics as the model on which to construct a “science” of engineering and the nature of machines. The aim was to create a body of knowledge that was general, abstract, and deductive. This movement, which was designed to improve the status of engineering, was associated par­ticularly with the names of Franz Reuleaux and Franz Grashof and achieved considerable influence during the 1870s and 1880s.10 These tendencies in the direction of purity and rigor by one part of the profession provoked an angry reaction in the 1890s from some other parts of the profession. The reaction took the form of an antimathematical movement (Anti-mathematische Be – wegung) led by Alois Riedler at the TH in Charlottenburg. Riedler presented the issue as one of the very survival of Germany in a world where technologi­cal effort must go hand-in-hand with commercial activity and efficient social organization. In this struggle for existence (“Kampf ums Dasein”) there was no place for the speculations of the unproductive classes, whether they be literary or mathematical. The practical men who backed Riedler (the Prakti – kerfraktion) argued that mathematical teaching should be cut down to what was, in their opinion, immediately useful.11

Third, and finally, in 1899, in a measure backed by Kaiser Wilhelm II, the THs were finally granted the right to issue doctoral degrees, hitherto the prerogative of the universities. As a consequence the status, influence, and size of these technical institutions increased steadily in the years leading up to the First World War. The engineering profession was, in many ways, still a divided and fractious body, but in the course of the expansion, the anti­mathematical movement lost much of its force. The alliance of industry and sophisticated science became increasingly acknowledged as an economic and military necessity. The emergence of aviation and the rapid uptake of this subject in the THs helped to consolidate the position of the applied math­ematician and swing the pendulum back to a less hostile stance toward math­ematically formulated theory.12

In his important study of engineers in German society, New Profession, Old Order, Kees Gispen quotes, and expresses agreement with, “a certain Friedrich Bendemann,” writing in 1907, who commented on this swing back and forth between theory and practice and declared that it was time to redress the present imbalance and reintroduce more theoretical training.13 Though Gispen does not mention it, the Herr Bendemann in question, who had re­ceived his doctorate from the TH in Charlottenburg, was a significant force in the aeronautical world. He was a specialist in aircraft engines and propel­lers. In 1912 he was to become the director of the Deutsche Versuchsanstalt fur Luftfahrt at Adlershof outside Berlin.14 Bendemann’s 1907 comments were a direct, and face-to-face, riposte to Riedler. They suggest the growing con­fidence of the aeronautical community in the THs in the face of old schisms and old campaigns.15 Those involved with airships and airplanes were begin­ning to think of aeronautics as a natural home for what von Parseval called the “gebildete Ingenieure,” that is, the educated or cultivated engineer whose thinking, by definition, combined both theory and practice.16

Kutta’s career thus began amid some of the more acrimonious attacks on mathematicians, but he was fortunate to be sheltered from the worst ex­cesses of the Theorie-Praxis-Streit by the special situation in Munich.17 The mathematicians at the THM had long made efforts (though with varying de­grees of determination and success) to accommodate the needs of engineers. They had cultivated a geometrical, visual, and concrete mode of teaching. The trend had started when Felix Klein held a chair at the THM and was continued by his successor Walther von Dyck, who was appointed in 1884 at the age of twenty-seven.18 Von Dyck had been Klein’s pupil and remained a friend and confidant. It has been said that von Dyck played an analogous role in South Germany to Klein’s role in North Germany.19 Von Dyck wanted the THM to be an institution of high scientific merit as well as being tech­nologically oriented. He was able to call upon the support of mathematically sophisticated members of the more technical departments at Munich, such as August Foppl, who likewise had no time for the simple Praktikers.

Kutta was von Dyck’s teaching assistant and frequently took on his classes when von Dyck became involved, as he increasingly did, with running the THM. Kutta also worked with Sebastian Finsterwalder (1862-1951), who held a mathematics chair at the THM. Finsterwalder was significantly more ori­ented to applied work than von Dyck and has been called “der Prototyp des ‘Technik-Mathematikers’”—the prototype of the technologically oriented mathematician.20 As early as 1893 Finsterwalder was giving lecture courses on the application of differential equations to the problems of technology. He was also an aeronautical enthusiast and a member of the local ballooning club.21 It was Finsterwalder who suggested that the topic of Kutta’s Habili – tationschrift should be the mathematical analysis of the flow of air over an aircraft wing. This may be guessed from Kutta’s thanks to Finsterwalder, but the colleague who wrote Kutta’s obituary endorsed the point.22 He said that the stimulus for the chosen topic would, in any case, be clear:

das ist aber fur denjenigen auch klar, der die Jahre kurz nach 1900 im Ma – thematischen Intitut der T. H. Munchen miterlebte. Von Finsterwalders re­gem Interesse an den aerodynamischen Grundlagen der damals in den ersten Anfangen stehenden Luftfahrt wurden auch die jungeren Krafte am Institut angesteckt. Ich denke noch daran, mit welchem Interesse Photographien der ersten Fluge—heute wurde man bescheidener sagen: Sprunge—die Farmen mit seinem Aeroplan bei Paris ausfuhrte, studiert und ausgemessen wurden, Photographien, die Finsterwalder mitbrachte: es wird so 1906 oder 1907 ge – wesen sein. (50)

quite clear to anyone who had been at the Mathematical Institute at the TH Munich in the years after 1900. Finsterwalder’s avid interest in the aerody­namic basis of the first beginnings of aviation at that time also infected the younger people at the institute. I think of the interest with which the pho­tographs of the first flights—today one would more modestly say jumps— were studied and measured. These photographs of Farman with his airplane in Paris, which Finsterwalder brought back with him, would have been in 1906 or 1907.

Finsterwalder’s suggestion to Kutta must have been made some four or five years before the episode with the photographs recalled by Pfeiffer, and thus before the first powered flights had been made. At this earlier date Fin – sterwalder would have been preparing his chapter on aerodynamics for Felix Klein’s encyclopedia of the mathematical sciences.23 The aeronautical adven­tures that were attracting attention at that time were the experiments with hang gliders of the kind pioneered by the engineer Otto Lilienthal. Lilienthal had been killed in a flying accident in 1896 but had left a legacy of both en­thusiasm and information. The information was in his book Der Vogelflug als Grundlage der Fliegerkunst published in 1889.24 Kutta was explicit about the connection between his work and Lilienthal’s machines in both the 1902 account and the 1910 paper.25 The link is clearly evident in the circular arc that Kutta took as his representation of a wing profile. This was not only a mathematical simplification; it also corresponded to the profile used by Lilienthal.26

After his successful Habilitationschrift Kutta continued as teaching assis­tant in the TH Munich until 1907. He then became an extra-ordinary profes­sor (that is, an associate professor) in the same institution. In 1909 he moved on to become an extra-ordinary professor at the University of Jena, and in 1910 was appointed as an ordinary professor (a full professor) at the TH in Aachen. Finally in 1911, the year of his second paper on the circulation theory, he settled down as an ordinary professor at the TH in Stuttgart, where he stayed until his retirement. After his two papers on aerodynamics, in 1910 and 1911, he published nothing more, although he did not retire until 1935 and lived until 1944.

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

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

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

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

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

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

and

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

I dx j

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

du + dv 0 dx dy

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