Idealization, Abstraction, and Approximation

The “general mental attitude” (as Lamb would have called it) informing the German work on aerodynamics gradually gained ground in Britain in the interwar years and finally became routine during and after World War II. To convey this attitude I have appealed to the work of some of the main practi­tioners of technische Mechanik, and in this chapter I have supplemented these sources by drawing on Philipp Frank’s writings. To drive the point home, I introduce one final, representative thinker to characterize the methods of aerodynamics. The “general mental attitude” of modern aerodynamics is well captured in the work and writing of Dietrich Kuchemann, yet another of Prandtl’s distinguished pupils.93

Kuchemann came to Britain in 1945, immediately following World War II, after meeting McKinnon Wood in Gottingen. McKinnon Wood was then working for the Combined Intelligence Objectives Sub-Committee (CIOS). He was on a mission similar, but more formal and grimmer in tone, to the one he had undertaken with Glauert in 1921. He questioned Kuche­mann about the latter’s work on swept wings and the flow over engine ducts and fairings.94 Kuchemann then became one of the many German experts who, under varying pressures, began to work for their former enemies in the postwar years.95 His personal history thus illustrates the transnational char­acter of science and the contingencies that contribute to it. Kuchemann sub­sequently chose to stay in the United Kingdom and eventually took British citizenship. In 1963 he was elected a fellow of the Royal Society and in 1966 became the head of the Aerodynamics Department at Farnborough. His later trajectory was not unlike Glauert’s, but whereas Glauert started from Cam­bridge, Kuchemann started from Gottingen.

There is another parallel. At the beginning of my story I described the lectures on aeronautics that Sir George Greenhill had delivered at Imperial College, London, in 1910. In the early 1970s Kuchemann also gave a series of lectures at Imperial. Like Greenhill’s lectures, these were also turned into a book, but whereas Greenhill was transmitting the old style of Cambridge mathematical physics, Kuchemann was carrying with him the style of Got­tingen engineering. Kuchemann’s book was called The Aerodynamic Design of Aircraft.96 In it he offered the following cautionary words about the character of aerodynamic knowledge:

the most drastic simplifying assumptions must be made before we can even think about the flow of gases and arrive at equations which are amenable to treatment. Our whole science lives on highly idealised concepts and ingenious abstractions and approximations. We should remember this in all modesty at all times, especially when someone claims to have obtained “the right answer” or “the exact solution.” At the same time, we must acknowledge and admire the intuitive art of those scientists to whom we owe the many useful concepts and approximations with which we work. (23)

Even the most elementary and pervasive of all the concepts of fluid me­chanics, the “fluid element,” fits Kuchemann’s description. It is a tool of anal­ysis that facilitates a mathematical grasp of real fluids, but, taken in isolation, the concept is no more than a convenient fiction shaped by the demands of the differential calculus.97 Kuchemann’s message is clear, and it is in no way idiosyncratic.98 Aerodynamics demands modesty in the status attributed to it. Claims to possess right answers and exact solutions should be viewed with suspicion because they will soon need to be qualified. While acknowledging the highly mathematical character of aerodynamics, we must accept and em­brace both the intuitive and utilitarian character of the enterprise.

Everything that Kuchemann says about aerodynamics can be found in the examples I have described. It applies to Kutta’s mathematical arc, Joukowsky’s theoretical aerofoils with infinitely thin trailing edges, Prandtl’s bound vortex and his reduction of the wing to a lifting line, Betz’s expedient modification of the Kutta condition and his relaxed attitude to infinite velocities, Schlicht – ing’s comments about fluid in the boundary layer for which the laws of mo­tion are suspended in directions normal to the layer, and Glauert’s evanescent boundary layers. But Kuchemann’s description applies equally to the work of Stokes, Rayleigh, Greenhill, Bryan, Lamb, Taylor, and Bairstow, even if some of them, on occasion, might not have embraced this characterization of their methods as readily as those trained in the self-consciously engineering tradi­tion of technical mechanics.

In all of these instances, a detailed examination of the practices that were adopted reveals that the whole science of aerodynamics, both British and German, indeed lived on highly idealized concepts and ingenious abstrac­tions and approximations, and the more successful it was, the more radical the idealizations. If Kuchemann was right, these characteristics of aerody­namic knowledge were not merely a passing phase. The men and women I have studied did not resort to these expedients because of the immaturity of the field or because they had, for the moment, to be content with second best. Their manner of thinking and knowing was not one that would be left behind as if, today, aerodynamics rests upon a qualitatively different methodological basis than the one it rested on then. It does not.

Let me support this statement by two examples of recent work. First, con­sider the current status of the Navier-Stokes equations. To this day they have not been solved, and their properties remain shrouded in obscurity. There are no known, general, “closed-form” solutions to the Navier-Stokes equa­tions. There are approximate numerical solutions but none in the form of the analytical equations that were characteristic of classical hydrodynamics and Prandtl’s wing theory.99 The emergence, since the 1990s, of computational fluid dynamics, which is based on powerful digital computers, has not altered this situation. Computational fluid dynamics has made available numerical solutions to the Navier-Stokes equations and graphic representations of flow patterns over the wings and bodies of aircraft. These are enormously useful, but they have all been produced by programming techniques that embody the forms of idealization, abstraction, and approximation to which Kuche – mann was referring.100

My second example shows that Prandtl’s wing theory has retained its ca­pacity to inspire novel work. Although a product of the early twentieth cen­tury, it lies at the heart of some twenty-first century developments in aviation technology. These developments exploit some new solutions to Prandtl’s old equations.101 The solutions (which are analytic rather than merely numerical) take into account the possibility of creating a variable twist along the span of the wing. The equations show that, with the right distribution of twist, even wings that have (say) a rectangular planform can be made to generate the theoretical minimum amount of induced drag. Previously this minimum had been associated exclusively with an (untwisted) elliptical planform. The necessary twist required for minimum drag depends on gross weight, alti­tude, speed, and acceleration, but using modern technology, a wing could be built that automatically adjusted itself to these changing conditions. The result, over the course of a long flight, would be significant drag reduction and cost saving. These two pieces of evidence show that Kuchemann was not describing a passing phase in the emergence of aerodynamics but the endur­ing character of knowledge in this field and, perhaps, the unavoidable condi­tions of all practical cognition.

I have already argued, by reference to Frank, that if the concept of “rela­tivism” is to be used with precision, it can only mean one thing: a denial that there are any absolute truths. This is a necessary and sufficient condition for an account of knowledge to be identified as a form of relativism. A relativist can be comfortable with knowledge that is conjectural, inconsistent, expedi­ent, and partial, that is, with everything that science and technology really is. It is the nonrelativists for whom these facts about science are a cause of trou­ble. They can acknowledge them, but only as stations on the road to an abso­lute end point. But the familiar pragmatics of scientific work cannot belong to the realm of the absolute. It would be perverse to use the words “absolute truth” as a label for theories that are approximate or that get this bit right and that bit wrong or which depend on useful fictions and abstractions. It should be clear that Kuchemann is denying that practitioners in aerodynamics can ever make any claim to absolute certainty, the absolute status of their con­cepts, or absolute finality in their knowledge.102 It follows, given Kuchemann’s analysis, that aerodynamic knowledge must be understood in a relativist man­ner, namely, relative to all the contingencies of the “intuitive art” that enters into idealization, abstraction, approximation, and inductive inference.103