The International Air Congress of 1923

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