Coffee Spoons and Theology

Kelvin’s theorem did not categorically preclude circulation in a perfect fluid but asserted, conditionally, that it could only exist under certain circumstances. In Britain effort was put into making sure that the proof of Kelvin’s theorem was as rigorous as possible.71 In Germany the focus was subtly different: it was the scope of the theorem that attracted attention. In 1910 Felix Klein pub­lished a paper in the Zeitschrift fur Mathematik und Physik in which he argued

Coffee Spoons and Theology

figure 9.15. Klein’s coffee-spoon experiment. A surface, the “spoon,” is immersed in an ideal fluid and moved forward (a). It is then quickly removed (b), leaving behind a surface of discontinuity (shown in exaggerated form). The result is a vortex sheet and hence the creation of circulation.

that it was easy to create circulation in an ideal fluid—as easy as stirring a cup of coffee.72 If a thin, flat surface (the “spoon”) is partially inserted in a body of ideal fluid, moved forward, and then briskly removed, the result would be a vortex with a circulation around it—but, said Klein, Kelvin’s theorem would not be violated. The mechanics of the process are shown in figure 9.15. The motion of the surface has the effect of forcing the fluid to move down the front face and up the back face, as indicated by the arrows. Removing the sur­face then leaves two adjacent bodies of fluid moving in opposite directions. The result is a surface of discontinuity, that is, a sheet of vorticity, which then rolls up into a vortex. This does not contradict the theorem, argued Klein, because Kelvin’s proof assumed continuity of the fluid, and this precondi­tion is violated by the insertion and removal of the mathematically simpli­fied “coffee spoon.” The coffee-spoon experiment was not an exact prototype for Prandtl’s confluence argument in the Wright Lecture (because a wing is surrounded by air, not dipped into it), but it surely provided an analogical resource. Klein’s argument encouraged a tradition of critical assessment of Kelvin’s theorem. Further papers, in which the argument was extended and assessed, were written by Lagally, Jaffe, and Prandtl. Later contributions on this theme came from Betz and Ackeret.73 By contrast, Klein’s coffee-spoon paper received no mention in Lamb’s Hydrodynamics.

Prandtl was right to anticipate objections from “the mathematicians” or, at least, from some Cambridge mathematicians. For example, his defense of perfect fluid theory failed to convince the Cambridge mathematician Harold Jeffreys, who later became Plumian Professor of Astronomy and Experimen­tal Philosophy at Cambridge.74 Jeffreys (fig. 9.16) has not previously featured in the story and was not a specialist in aerodynamics. His primary contribu­tions were to geophysics, but he published creative mathematical work in an impressively wide range of subjects. Jeffreys, a notoriously withdrawn man, distinguished himself in part II of the Mathematical Tripos in 1913 and was elected a fellow of St. John’s College in 1914. He stayed at St. John’s for the rest of his life. During the Great War Jeffreys worked at the Cavendish labora­tory on gunnery and then on meteorology with Napier Shaw (who was on the Advisory Committee for Aeronautics). Like his friend G. I. Taylor, Jef­freys originally became interested in circulation and viscous eddies from a meteorological standpoint. In the 1920s, prompted by his lecturing commit­ments in applied mathematics at Cambridge, Jeffreys began a series of papers on fluid dynamics which made explicit contact with the work that had been done on circulation in aerodynamics.

The first in the series of papers, in 1925, was called “On the Circulation Theory of Aeroplane Lift.”75 Although an outsider in the field, Jeffreys sent a copy to Prandtl and received a somewhat formal reply. Prandtl clearly thought

Coffee Spoons and Theology

figure 9.16. Harold Jeffreys (1891-1989). Jeffreys was a powerful and wide-ranging applied math­ematician who originally approached fluid dynamics from the standpoint of meteorology. Like Taylor and Southwell, he argued that Kelvin’s theorem precluded the creation of circulation in an ideal fluid. (By permission of the Royal Society of London)

that Jeffreys needed to do his homework. He suggested Jeffreys read the 1904 paper on boundary-layer theory and the 1908 application of the theory by Blasius and duly enclosed the references.76 After something of a delay, Jeffreys acknowledged the response but said, rather untactfully, that he was too busy at the moment to follow up the references. He would get down to them as soon as he could.77 He added: “Of course it would not in the least surprise me to find that all the ideas in my paper had been anticipated, but they were not in any work I had seen & I thought it well that they should be published sim­ply because they were not well known in this country.” It may have been this exchange that gave Prandtl his sense of what topics needed to be addressed in the Wright Lecture and that helped him imagine the archetypal “mathemati­cian” resisting his account of the origin of circulation in an ideal fluid.

In 1930, three years after Prandtl’s Wright Lecture, and after discussions with Glauert and Taylor, Jeffreys published “The Wake in Fluid Flow Past a Solid.”78 Jeffreys started by noting that in many cases it was possible to ap­proximate the motion of a real fluid by a “cyclic irrotational motion, with local filaments of vorticity.” He instanced the work of Kutta and Joukowsky on two-dimensional flow and that of Lanchester and Prandtl on three­dimensional flow. But, he insisted: “The existence of cyclic motion is in dis­agreement with classical hydrodynamics, which predicts that there shall be no circulation about any circuit drawn in a fluid initially at rest or in uniform motion, and that there is no resultant thrust on a solid immersed in a steady uniform current” (376).

As far as Jeffreys was concerned, classical hydrodynamics had long “ceased to be a representation of the physical facts” (376). He agreed with the qual­itative explanation that Prandtl had advanced to show why a perfect fluid theory could be used to approximate a real flow at a distance from a solid boundary, but he did not accept Prandtl’s account of Kelvin’s theorem. For Jeffreys, classical hydrodynamics implied that a wing, starting to move from rest in a perfect fluid that was also at rest, could not generate circulation and lift. Prandtl had argued in his Wright Lecture that the generation of circula­tion and lift was consistent with Kelvin’s theorem; Jeffreys said it was not. Zero lift was the clear and inescapable consequence of the theorem in the case under discussion. Understanding the generation of lift required starting out with the theory of viscous flow. For Jeffreys, as previously for Bairstow, the problem was why ideal fluid theory seemed to work. Inquiry should not concentrate on explaining its numerous failures but on its few remarkable successes. “Considerable attention has been given to the reason why classical hydrodynamics fails to represent the experimental facts; but it appears to me
that these efforts arise from an incorrect point of view. . . the remarkable thing is not that classical hydrodynamics is often wrong, but that it is ever nearly right” (376).

Jeffreys’ way of addressing this question was to anchor the mathematics in physical processes and to make sure that what were really results in math­ematics were not treated as results in physics. Their physical application had to be justified, not taken for granted. Consider, for example, Kelvin’s theorem and the way it was used to explain the creation of circulation around a wing. The vortex that forms at the trailing edge, and then detaches itself, is said to cause the circulation around the wing. The circulation around the depart­ing vortex brings about the opposite circulation around the wing. How? The answer given by Prandtl and Glauert was that Kelvin’s theorem had to be satisfied. Jeffreys was not convinced by this answer, and surely he was right to be suspicious. If Kelvin’s theorem prohibits the creation of new circulation, why are two violations of the prohibition acceptable merely because they are violations in opposite directions? Things that cannot exist cannot cancel out. If it is illegal to drive down a certain street, two people may not drive down it and plead that the law was not broken because they were driving in opposite directions.

Jeffreys wanted to know why Kelvin’s theorem, which was a theorem about inviscid fluids, could be used in the course of an argument in which the role for viscosity had already been granted in order to explain the origin of circulation. This could only be justified if something equivalent to Kelvin’s result could be deduced starting from the premise of viscous flow. To explore this possibility, Jeffreys set himself the goal of deriving the rate of change of circulation with time for a viscous fluid. Kelvin’s theorem for an ideal fluid is expressed by writing d Г/ dt = 0, and Jeffreys wanted to know the value of d Г/ dt for a real, viscous fluid. The general circulation theorem for viscous flow that Jeffreys derived involved the integral of some five separate expressions, each of considerable complexity. For a uniform, incompressible fluid, how­ever, only one of the five terms survived. For aerodynamic purposes, Jeffreys was then able to replace Kelvin’s circulation theorem by the equation

Подпись: dr dt !vdjLdx, = Г vV2u dx,

C dxk ‘ Jc ‘ ‘ where Г is the circulation around the contour C moving with the fluid and V is the kinematic viscosity (that is, viscosity divided by density). In Jeffreys’ equation the three coordinate axes are represented not by x, y, z, but by X;

where i = 1, 2, 3, and the corresponding velocity components are given by щ. The summation convention is used for repeated suffixes, and the term £,ik is the vorticity, which is defined as

f _ duk_

k dXi j •

What did this new expression for dr/dt mean? Jeffreys followed an ear­lier discussion in Lamb’s Hydrodynamics and offered an explanation of the physical significance of the result as follows.79 The equation linking circula­tion and time, he said, can be recognized as one that represents a diffusion process. It shows that vorticity and the circulation it induces obey laws that are analogous to the laws governing the diffusion of temperature or density. From this analogy it follows that vorticity must diffuse outward from a solid boundary. Circulation cannot arise spontaneously within the body of viscous fluid itself. Before the diffusion process has carried the vorticity to regions distant from the boundary, the fluid in these distant regions shows no rate of change of vorticity with time. The rate of change of circulation around a contour therefore depends on the vorticity near the contour. There will there­fore be “no appreciable circulation except on contours part of which have passed near a solid boundary: in other words vorticity is negligible except in the wake” (380).

Jeffreys’ paper “The Wake in Fluid Flow Past a Solid” covered much of the same ground as Prandtl’s earlier but more qualitative treatment in the Wright Lecture, but it is clear that Jeffreys felt that only now had a proper basis been provided for the conclusions that had been advanced. He carefully investigated the orders of magnitude of the quantities involved in the diffu­sion of the vorticity. This analysis, he said, “constitutes the theoretical justi­fication of the ‘boundary-layer’ theory of Prandtl and his followers” (380). Jeffreys’ treatment converged with Prandtl’s but was offered as one “based on the physical properties of a real fluid and not on mathematical conceptions of vortex lines and tubes” (389). In a further paper, “The Equations of Viscous Motion and the Circulation Theorem,” Jeffreys made a similar claim about Prandtl’s account of the origin of circulation and the starting vortex that de­taches from the trailing edge.80 Only an understanding of viscous circulation, said Jeffreys, can provide the real “physical basis” needed for applying the theory of vorticity to real fluids.

Where did this leave Kelvin’s theorem and the (apparently) inconsistent use of that theorem by supporters of the circulation theory? Jeffrey’s position was that the diffusion picture showed that it was not really Kelvin’s theorem that was being invoked to explain the relation between the circulation around the detached vortex and the circulation around a wing. Rather, it was the the­orem for circulation in a viscous fluid that was really in play. Kelvin’s theorem dealt with inviscid fluids, but the counterpart theorem for viscous fluids, the diffusion equation, gave the same numerical result for the initial stages of the flow. “Thus,” Jeffreys stated, “the conditions assumed by classical hydrody­namics are reproduced, in the specified conditions, by the real fluid” (381).

Jeffreys was not alone in saying that Kelvin’s theorem clearly ruled out the creation of circulation by a wing in an ideal fluid. This had been Taylor’s position in 1914, and it was still Southwell’s position in 1930 when he gave the prestigious James Forrest Lecture.81 Southwell asserted that classical hy­drodynamics left the existence of circulation around a wing “an altogether amazing coincidence” (360). He added that the assumption of circulation was “rather theological” (361). The allusion was to Kelvin, for whom the eter­nal character of circulation and vortex rings indicated a divine origin. South­well, like Jeffreys, was unmoved by the first part of Prandtl’s Wright Lecture, dealing with Kelvin’s theorem and perfect fluid theory, but he was enthused by the second part on the boundary layer and the creation of vortices. South­well reproduced Prandtl’s photographs showing the control of the boundary layer by suction and showing how to make a divergent nozzle “run full.” He selected and emphasized the places where Prandtl’s concerns came closest to the long-standing British interest in viscous fluids and eddying flow. South­well further assimilated this aspect of Prandtl’s work to the British tradition by arguing that the analysis of backflow in the boundary layer was similar to Mallock’s work on reverse flow and eddies that was done in the early years of the Advisory Committee for Aeronautics.82

Lamb had also made gentle fun of the theory of circulation by exploit­ing the theological overtones of Kelvin’s theorem. In his Rouse Ball Lecture of 1924, titled “The Evolution of Mathematical Physics,” Lamb had said of perfect fluid theory that “this theory cannot tell us why an aeroplane needs power for its propulsion; nor, indeed, can it tell us how the aeroplane obtains its sustentation, unless by assuming certain circumstances to have been estab­lished at the Creation which, in all reverence, we find it hard to believe.”83 The “certain circumstances,” of course, were the provision of suitably adjusted values of the circulation. Every takeoff and landing, Lamb hinted, would re­quire divine anticipation and intervention. But if the tone was joking, the point was serious. Perfect fluid theory predicts zero drag and makes a mystery out of the origin of circulation. In the last edition of his Hydrodynamics, in 1932, Lamb returned to the problem of the origin of circulation and of un­derstanding how it resulted in a smooth flow being established at the trailing edge of a wing. He clearly felt that no satisfactory account had been given of this. He cannot have been convinced by what Prandtl and Glauert had to say, and his reference to Jeffreys’ efforts was noncommittal. Jeffreys may have deepened the discussion and clarified some of the physical principles, but it was still mathematically incomplete. Lamb summed up by saying: “It is still not altogether easy, in spite of the attempts that have been made, to trace out deductively the stages by which the result is established when the relative flow is started. Fortunately, some beautiful experiments with small-scale models in a tank come to our help. A vortex with counter-clockwise sense is first formed, and detached from the edge, and then passes down stream, leaving a complementary circulation around the aerofoil in the opposite sense” (691).

No one would have been deceived by the understatement. Lamb was say­ing that, by his standards, no one had given a satisfactory mathematical analy­sis of the processes by which a circulation was created. Prandtl may have been able to offer “beautiful photographs,” but that only meant that the analysis was still confined to the empirical level.84 Lamb was surely right. The circula­tory theory was accepted by the British without its supporters being able to offer a rigorous account of the origin of circulation. This had been a source of difficulty for the British from the outset. It was still a worry, but now, with varying degrees of unease, they appeared ready to live with the problem.

Lamb also remained skeptical when Prandtl and Glauert represented the surface of a body, such as a wing, by a sheet or line of “bound” vorticity. Lamb did not claim that the concept of a bound vortex was logically defec­tive, but, in responding to a paper that Glauert submitted to the Aeronau­tical Research Committee in 1929, he deemed it “artificial.” He succeeded in deducing Glauert’s results, concerning accelerated motion in two dimen­sions, by other more usual routes.85 Once again, there was tension between the advocates of two different approaches to applied mathematics: those who insisted that the mathematics described what they took to be physically real processes (and described them in a rigorous way) and those content with mathematical descriptions that were acknowledged to be expedient rather than physically true. Lamb never shifted from the view that he expressed in his presidential address to the British Association meeting in 1925.86 He spoke for many in British aerodynamics when he said that Prandtl had provided “the best available scheme for the forces on an aircraft” (14). The choice of the word “scheme” was meant to imply that Prandtl had failed to give a fun­damental account of the physical reality of the process.87