Category The Enigma of. the Aerofoil

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