Scope and Rigor

Consider the scope of Lanchester’s theory. His narrow focus on small angles of incidence was not shared by critics. Bairstow invoked standards of assess­ment appropriate to a wholly general theory of fluid resistance. Lanchester found this preposterous. He said it was like ignoring useful knowledge about how water flowed round a ship in normal motion because it did not also

explain the flow when it moved broadside. But it is not difficult to see how for the critics, if not for Lanchester, genuine knowledge of the one case also meant having knowledge of the other. Lanchester’s commonsense plea for theories of limited scope was at odds with the forms of generality routinely exhibited in classical hydrodynamics. This can be seen from the textbook treatment of the flow around an elliptical cylinder moving through a fluid. The elongated, elliptical cylinder bears a certain visual likeness to the plan of a boat, which was the case cited by Lanchester. Lanchester’s critics could point out that the mathematics of the flow does not single out, as being of special significance, any particular angle of inclination of the major axis of the ellipse to the di­rection of motion. It makes no difference to the mathematics whether the ellipse moves like a ship going forward or like a ship moving broadside, that is, awkwardly and inappropriately. Both motions are but special cases of the same general formula. Mathematically they merely depend on whether the real or the imaginary part of the complex potential is set to zero. This fact would have been familiar to any Cambridge student of hydrodynamics, or to anyone, such as Bairstow, Cowley, or Levy, schooled in a similar tradition. It was clearly not a significant reference point for Lanchester.3

The issue of scope also arose in another way. We have seen that circula­tion theory explained lift but not drag. The critics had rejected discontinuity theory because it could not yield accurate predictions of resistance, so on grounds of consistency circulation theory should be, and was, treated like­wise. The false prediction of zero drag was not lost on Lanchester, but it did not worry him in the way it did his critics. Unfortunately, Lanchester did not articulate a clear rationale for his stance, so the critics may have been tempted to see it as indicating a certain laxity on his part, compared to their own greater concern with truth and rigor.

There is some evidence that Cambridge physicists involved in aerodynam­ics were prone to misperceive the difference between their mental habits and those of engineers as the difference between rigor and sloppiness. Reflect­ing on his work as a physicist at Farnborough during the Great War, George Paget Thomson, the son of J. J. Thomson, drew attention to this cultural divide. Scientific work during wartime, said Thomson, “might properly be described as engineering.”4 He recalled how difficult it was for physicists to adopt the requisite point of view. As the author of Applied Aerodynamics, which had been well received by the “practical men,” Thomson could not be accused of lack of sympathy with engineers. But even he was inclined to exemplify, rather than bridge, the disciplinary divide he described. Thom­son spoke of the need for engineers to make up their minds on the basis of “insufficient evidence” and of the need to “compromise between conflicting requirements.” He concluded: “What is perhaps harder for the scientist to realize is the doctrine of ‘good enough.’ The better is the enemy of the good” (3).

Could it be that Lanchester, as an engineer, was prepared to accept the circulatory theory and perfect fluid theory because they were “good enough” for him even though they were not “good enough” for a physicist? The im­plication is that Lanchester, unlike his critics, was content with “insufficient evidence.” But there is another explanation of why a supporter of the circula­tion theory might find the lift-but-no-drag result an acceptable one. Rather than expressing a compromised standard of empirical accuracy, the response might simply embody a different standard and one that is not necessarily lower. The lift-without-drag result might be taken to be a true and accurate assertion about an “ideal wing,” that is, the sort of wing at which an engineer might aim. The result should perhaps be seen not as a false statement, but as an engineering ideal. This was not a defense explicitly offered by Lanchester, but as we shall see, it was how Ludwig Prandtl, a fellow pioneer of the circula­tion theory, expressed the matter.

Consider now the objection that the circulation is “arbitrary.” Both Kutta and Joukowsky were aware of the mathematical rationale behind this objec­tion, namely, that the theory contained no way of deducing the amount of circulation around a wing. Nevertheless they responded in a very different way to the British critics. They stipulated that the circulation be of precisely the amount necessary to ensure that the flow comes away smoothly from the trailing edge of a wing. The rear stagnation point must be on the trailing edge so that the flow does not have to wrap itself around a sharp corner. For a given angle of incidence, and a wing with a sharp trailing edge, this stipula­tion provides an unambiguous specification of the amount of circulation and is often called the Kutta condition. It derives its significance, and its nonarbi­trary nature, from the empirical fact that the flow of air over a (nonstalling) wing in a steady state settles down so that there is indeed an approximately smooth flow at the trailing edge. The Kutta condition tells the theorist what value of the circulation to assume, and thus what value of lift is predicted when this value is substituted into the formula Kp U, the lift equation.

Were Cowley and Levy, who made the complaint about arbitrariness, un­aware of this solution to the problem? The answer is that they were fully aware of the Kutta condition. In 1916 Levy had explicitly mentioned it in correspon­dence with Lanchester.5 In Levy’s view, however, Kutta’s proposal did not remove the arbitrary character of the amount of circulation. The argument presented to Lanchester was that, in reality, the trailing edge of a wing is not mathematically sharp but rounded. It therefore provides no mathematically unambiguous location for the rear stagnation point. The point on the curve that is selected for this role will itself be arbitrary. The amount of circulation needed to bring the stagnation point to this location will, therefore, also be arbitrary. The only thing that would remove this feature of the theory would be some means of deducing the circulation from first principles, given rel­evant data about the wing, for example, its shape and angle of incidence. No such method was known. For Cowley and Levy the word “arbitrary” clearly meant “not deducible from the basic equations of fluid dynamics.” They op­erated with a mathematical criterion and were looking for a mathematical solution to the problem, not an empirical one.