Saying and Showing

In accordance with the Royal Aeronautical Society’s policy of sustained, tech­nical discussion, Low’s paper was followed, later in the year, by other material relating to the Prandtl theory. On November 16, 1922, R. McKinnon Wood gave a paper titled “The Co-Relation of Model and Full Scale Work.”81 Like Low, McKinnon Wood also took the opportunity to describe the basis of the circulation theory and, like Low, found himself confronting Bairstow. It transpired in the discussion that Bairstow was engaged in experiments at the NPL to work out where viscous and nonviscous flows differed in the case of aerofoil shapes. Bairstow said he had no doubt that the theory of nonviscous flow would yield results that could be tested, but “he did not expect to find the circulation at all in the experiments” (499).

There were also two talks given to the society which were devoted to wind-tunnel studies of the vortex system behind a wing both by N. A. V. Piercy of the East London College.82 Piercy had been a colleague and collabo­rator of Thurstone but worked at a much more sophisticated level both em­pirically and theoretically. Using the college’s wind tunnel, Piercy produced detailed measurements of the airflow both behind the wing and in the region of the wingtips. It was clear that there were vortex structures to be mapped, and these corresponded, at least qualitatively, to the expectations created by

Lanchester’s and Prandtl’s work. Although they were broadly supportive of the circulatory picture, the results were actually understood by Piercy to sup­port Bairstow’s suspicion that too little weight had been given to the role of viscosity.

Piercy was very conscious of the empirical variability of the phenomena under study in his wind tunnel. He argued that the vortex effect behind a wing sometimes achieved its maximum value after the wing had stalled and thus after the lift (and, presumably, the circulation) had dropped away. How could this be explained on the Lanchester-Prandtl theory? He made three suggestions, none of which could be easily accommodated within the circula­tory theory as it stood. First, he wondered whether, during a stall, the wingtip vortices continue to exist but are not joined together by a vortex that lies along the span of the wing. This would produce the effect to which he was referring, namely, wingtip vortices without lift or with diminished lift. But if the vortices can exist without circulation around the wing at, and beyond, the angle of stall, surely “it is not necessary for them to be so joined at a smaller angle” (502). Second, even if it were the case that the two wingtip vortices are still joined (in some fashion) after the stall, “they may be joined in such a manner as not to give cyclic lift” (502). “As a third alternative,” said Piercy, we may “suppose that cyclic lift may be destroyed to a considerable extent by viscous effects,” but then, “it seems reasonable to conclude that cyclic lift is not immune from viscosity at smaller angles” (502).

Piercy was well aware that supporters of the circulation theory had always sought to draw a line between normal flight at low angles of incidence and the phenomenon of stall at high angles of incidence. Their position was that a good theory of the former did not have to explain the latter. An explanation was desirable but not necessary. This had been a central part of the argument between Lanchester and Bairstow in 1915. Piercy sided with Bairstow on this matter. He dismissed the defense as an evasion. It was, he said, “beside the point.” “The question is whether we can afford to neglect at 8 deg. incidence, say, a factor so powerful as to be able to overthrow the vortex system at, say, 16 deg. Should we not rather conclude that at any angle viscosity is playing an essential and important role in the whole system of flow?” (502). The long­standing British concern with stalling, and the desire for a unified and realis­tic theory of broad scope, was still in play.

The next paper in the 1923 volume of the Royal Aeronautical Society jour­nal which discussed the circulation theory appeared under the name Glauert, but it came from Muriel Glauert, Hermann Glauert’s new wife. The paper was called “Two-Dimensional Aerofoil Theory” and was based on a technical report written some two years earlier for the Aeronautical Research Commit-

tee, but under the name Muriel Barker, not Muriel Glauert.83 Muriel Barker worked for the Royal Aircraft Establishment and was the holder of the post­graduate Bathurst Studentship in Aeronautics at Cambridge. Her notes on Kutta, which were mentioned and used in a previous chapter, were prob­ably made when gathering material for writing the original technical report. The discussion of two-dimensional aerofoil theory for the RAeS journal was based on the assumption, rejected by Bairstow and many others, that inviscid methods are legitimate. To develop this starting point, Muriel Glauert intro­duced a general theorem due to Ludwig Bieberbach.84 The theorem showed that there was one and only one conformal transformation of the form:

(where b1, b2 , . . . were complex) which would map the space around a shape, such as an aerofoil, in the z-plane into the space round a circle in the Z-plane, leaving the region at infinity unchanged. She then worked through, in math­ematical detail, the special case of this theorem provided by the Joukowsky transformation and dealt with circular arcs, Joukowsky aerofoils, double circular arcs, struts, Karman-Trefftz profiles, von Mises profiles, and Tref- ftz’s graphical methods. Muriel Glauert’s paper makes it clear that the British had been doing their homework. They had now brought themselves up to date and absorbed all the mathematical techniques and results of the German work on the two-dimensional wing that I described in chapter 6.

Next to be published in the sequence of Prandtl-oriented discussion pa­pers was one by Hermann Glauert himself, titled “Theoretical Relationships for the Lift and Drag of an Aerofoil Structure.”85 At first glance Glauert’s pa­per has the appearance of being no more than an elementary treatment of the circulatory theory—far less mathematical, for example, than Muriel Glau­ert’s paper. Unlike his wife’s paper, or his own technical reports for the Aero­nautical Research Committee, the present paper was not replete with math­ematical formulas. The appearance of simplicity, however, is misleading. The paper may have been essentially qualitative in its argument but it was in no way elementary. It was sharply focused on difficult problems, but the prob­lems in question were methodological ones. It dealt with the orientation that was needed to appreciate Prandtl’s approach—the very thing that divided Glauert from his mathematically sophisticated British contemporaries. They did not need to be convinced of the mathematics; they needed to understand the mathematics in a different way.

The solution of a physical problem in aerodynamics, said Glauert, can be analyzed into three steps. First, certain assumptions must be made about
what quantities can be neglected, for example, gravity, compressibility, and viscosity. Only rarely is it necessary to take into account the full complexity of a phenomenon. Second, the physical system, in its simplified form, must be expressed in mathematical terms, for example, a differential equation and its boundary conditions. Third, the mathematical symbols must be manipulated until they yield numerical results that can be tested experimentally or used for some practical purpose. This third step, said Glauert, must not be misun­derstood. It is where some of the greatest difficulties arise because the math­ematical problems may be insurmountable. At this stage it may be necessary to simplify further the initial, physical assumptions or to confine attention to a limited range of cases, such as small deviations from known motions. It is important to remember, said Glauert, that “in no case are these assump­tions absolutely rigid” (512). Glauert’s three steps are not simply sequential: what happens during the third step can feed back into what was called the first step.

Glauert then rehearsed the assumptions that were made by proponents of the circulation theory, that is, the assumption that the air could be represented as a perfect fluid with neither compressibility nor viscosity; the assumption that the fluid flow is irrotational; and the need to postulate a circulation to avoid a zero resultant force or to resort to the theory of discontinuous flow. “In view of this discussion,” said Glauert, “it appears that no satisfactory so­lution of an aerodynamic problem is to be expected when the effects of com­pressibility and viscosity are neglected, and it becomes necessary to consider the effect of these two factors” (513). This sentence is a striking one. It appears to concede all the points made by the critics of the circulation theory. Is this not exactly what Leonard Bairstow would say? Is not Glauert here following the line that led the young Taylor to dismiss Lanchester? Given Glauert’s ac­complishments as a stylist, however, both the import and the impact of these words would have been carefully weighed. He would not have inadvertently conceded too much or expressed himself inaccurately on such an important question.

How could Glauert grant that no satisfactory solution can be expected if viscosity is neglected without also granting the dismissive conclusions drawn by the critics of the circulation theory? The answer hinges on what it is to “ne­glect” compressibility and viscosity and what it is to “consider” their effects. Is this something done at the outset, in step 1 of the methodology? Or is it done at step 3, not as a sweeping assumption but as a technique for making the mathematics tractable? The vital but subtle methodological point that Glau – ert was making can be expressed like this: viscosity cannot, indeed, be wholly neglected but, contrary to first appearances, that does not preclude the use of perfect fluid theory. There were ways of operating with the mathematics of a perfect fluid that involved consideration of viscosity. The acts of consideration that were in question could not be stated in the inviscid equations themselves but would be shown in how they were deployed and interpreted.

Glauert explained that two important facts about viscosity must be ac­commodated. First, there is the no-slip condition, which stands in contrast to the perfect fluid property of finite slip. Second, viscous forces are propor­tional to the rate of change of velocity and hence are important close to a body such as a wing but become negligible at large distances. These are the physical facts for which approximations must be found. They cannot be dis­missed in step 1 of the sequence of steps Glauert had described. That would indeed amount to a decision to “neglect” them, and it is known that this produces the empirically false result of a zero resultant force. Rather than neglecting these two facts, their reality must be taken into account by a jus­tifiable approximation, an approximation of the kind introduced in step 3. Glauert’s development of this point deserves to be quoted in full. Notice the specific meaning he attached to the word “ignore” in the quoted passage and the implied contrast between ignoring something (in step 1) and approximat­ing its properties (in step 3):

It is known that the solution obtained by ignoring the viscosity is unsatisfac­tory, but it is by no means obvious that the limiting solution obtained as the viscosity tends to zero is the same as the solution for zero viscosity. In particu­lar, in the case of a body with a sharp edge, there is a region where the velocity gradient tends to infinity, and where the viscous forces will be of the same order of magnitude as the dynamic forces, however small the viscosity. On the other hand, the layer round the body in which viscosity is of importance can be conceived as of zero thickness in the limit, and this conception is equiva­lent to allowing slip on the surface of the body. It appears, therefore, that the non-viscous equations will be the same as the limit of the viscous equations, except in the region of sharp edges. (514)

The argument was that under the right conditions the equations of in­viscid flow are legitimate approximations to the viscous equations and their use does not amount to “ignoring” or “neglecting” the viscous properties of the flow. The crucial requirement is that the inviscid flow must be one that can be understood as a limiting case of a viscous flow. Glauert appears to have carried this crucial lesson away with him from his conversations with Prandtl. The Royal Aeronautical Society paper was therefore not merely an elementary exposition of the theory of lift; rather, it was an attempt to con­front the habits of thinking that had justified the systematic neglect of the in­viscid approach by British experts. Up to this point the conviction of British mathematical experts that ideal-fluid theory was false, and ultimately useless for aerodynamics, had carried almost everything before it. Only the theory of viscous flow dealt with reality. Ideal-fluid theory may provide some re­sidual mathematical challenges, and some suggestive analogies, but it could not be taken seriously as a means for directly engaging with reality. Glauert was challenging this assumption. He sent a copy of his paper to Prandtl along with copies of Piercy’s two papers. Of Piercy’s work he remarked that the ex­perimental results were interesting but expressed doubt about the theoretical interpretation: “His experimental results are of considerable value, but his interpretation of them leaves a good deal to be desired.” Glauert described his own piece as a “short note I wrote in justification of the principles underlying the vortex theory of aerofoils.” 86 The question was: could Glauert shift the way his contemporaries understand those underlying principles?