Category The Enigma of. the Aerofoil

In 1910 the Society of German Aeronautical Engineers began to publish the Zeitschrift fur Flugtechnik und Motorluftschiffahrt—the journal for aeronau­tics and motorized airship transport.45 The ZFM, as it was called, rapidly be­came the leading scientific publication in the field. There was no precise Brit­ish equivalent. The ZFM was more technical than, say, the Aeroplane or Flight and yet more accessible, and certainly more diverse, than the Reports and Memoranda of the Advisory Committee for Aeronautics. As well as scientific reports it contained general survey articles on the state of aviation, accounts of the latest exhibitions and meetings, and reviews of recent publications. There was, however, no close reporting of political controversies of the kind that was conspicuous in the British aeronautical press. Perhaps the nearest British publication was the Journal of the Aeronautical Society, but unlike the ZFM, this was not a routine vehicle for publishing research results.46 The lack of any British equivalent hints at the different ways aeronautical knowl­edge was integrated into the institutions of the two countries. Those who wrote for the ZFM communicated across boundaries between theorists and practitioners that seemed more difficult to overcome in Britain, while their silence in the domain of politics shows that there were other boundaries that remained higher in Germany than in Britain.

The advisory board of the ZFM was impressive. The journal was edited by the Berlin engineer Ansbert Vorreiter, and the scientific side was under the guidance of Ludwig Prandtl in Gottingen. Alongside Prandtl the board contained Carl Runge, also of Gottingen, along with Finsterwalder of the TH in Munich, Reissner of the TH in Aachen, and von Parseval and Bendemann from the TH in Charlottenburg. The masthead of the journal also carried the name of Dr. N. Joukowsky. His affiliation was given as the University of Moscow and the technische Hochschule of Moscow.

In the issue of November 26, 1910, the ZFM published the first part of a two-part article by Joukowsky titled “Uber die Konturen der Tragflachen der Drachenflieger” (On the shape of the wings of aircraft).47 The article has come to occupy a notable position in the history of aerodynamics. It is cited as the source of an important methodological shift in the mathematics of lift. The shift consisted in replacing Kutta’s complicated conformal transforma­tions with a single, simple transformation, now called the Joukowsky trans­formation. Not only was it simpler, but it produced a more realistic aerofoil shape. Kutta’s method had merely produced a geometrical arc. The arc was an adequate model of Lilienthal’s wing, but it did not capture the increas­ing use of wings with a rounded, rather than sharp, leading edge as well as a slender tailing edge. Kutta had established the logic of the process by which knowledge of the flow around a circular cylinder could be turned into knowl­edge about the flow around a wing. The next step was to refine and improve this method of analysis. It is in this connection that Joukowsky’s paper has, rightly, achieved the status of a classic.

A reader who is aware of its reputation, but who confronts Joukowsky’s paper with fresh eyes, might feel puzzled. Where is the bold simplifying stroke? The inner coherence of the mathematics of the infinite wing, so evi­dent in the textbooks that emerged a few years later, is not to be seen. The argument of the paper lacked clarity, and Joukowsky cited formulas without proof and used them without adequate explanation. There was also an edgy concern with issues of priority, particularly Russian priority, and some dis­tracting typographical errors. The formula in the theory of complex variables that is now called the Joukowsky transformation was not actually stated in the paper, although some of its immediate consequences were given a lim­ited application. But any inclination toward disappointment should be re­sisted. The smoothness of the later analysis is indeed absent from the paper, but that is because the later analysis was the work of others who learnt from Joukowsky and carried his ideas further. It was a collective, not an individual, accomplishment.

What was Joukowsky’s own contribution? I answer this question by giving an analysis of the argument of the 1910 paper. Joukowsky began by stating, without proof, two formulas for the lift, P, of an aerofoil that takes the form of a circular arc. The first was for an arc at zero angle of incidence; the second was for an arc at the arbitrary angle of incidence p. The formulas were

2 a 2 P = 4na sin— pV 2

and

P = 4na sin—sin+ pV2>

where V is the free stream velocity, p the density, a the radius of the circu­lar arc, and a is specified as half the angle subtended by the arc at the cen­ter of the circle. Clearly, the two expressions become the same when P = 0. Kutta published both of these formulas, the first, simpler one in 1902 and the second, more general one in 1910, the same year as Joukowsky’s paper.48 Jou­kowsky, however, said that his colleague Sergei Tschapligin had by this time already discovered the second formula.49

Next came a discussion of the general lift theorem L = p V Г. Here a full proof was provided. Joukowsky approached the problem in terms of the flow of momentum across a control surface. His proof was of a type that has now become standard in modern textbooks. Again, Joukowsky raised issues of priority. He allowed that Kutta discovered this theorem in his unpublished thesis of 1902 but pointed out that he, Joukowsky, in 1906, was the first to publish it.50 He also noted that Finsterwalder had accepted this priority claim.51 Joukowsky granted that Lanchester had been the first to explain the relation between two-dimensional and three-dimensional flow by introduc­ing the trailing vortices. At this point the main business of the paper was announced. In studying the problems of Kutta flow, said Joukowsky, he had found contours of a winglike form (“von flugelartiger Form”) that did not, like Kutta’s arc-wing, give rise to infinite velocities at the leading edge. The aim of the paper was to show how to construct these contours and to test their properties empirically:

Die Beschreibung der Konstruktion dieser Konturen und die experimentelle Untersuchung der ihnen entsprechenden Widerstandskrafte der Flussigkeit stellt den Inhalt dieser Arbeit dar. (283)

The content of the work can be represented as the description of the con­struction of these contours and the experimental study of the corresponding resistance forces of the fluid.

Joukowsky set out, step by step, a geometrical procedure for transforming a circle into the first of his two contours. Whereas Kutta had employed func­tions of a complex variable, Joukowsky took his readers back to the geometry lessons of the classroom. The procedure involved drawing circles and tan­gents, labeling significant points and angles in the figure, carrying out some careful measurements on the diagram, and then adding construction lines. To start the process, said Joukowsky, it is necessary to draw a circle whose center is labeled O and whose radius is a. Some arbitrary point E is then chosen which lies outside the circle, and from E two tangents are drawn. The angle enclosed by the tangents at E is called 2a. It is then required to draw a second, larger circle whose radius is called b. The larger circle does not share the same center O as the smaller circle. Rather, its position is determined by the requirement that it encloses the smaller circle but touches it so that it shares one of the tangents. It is this larger circle that is to be transformed into the aerofoil.

The next step was the addition of construction lines. These are needed to connect any specified point M on the larger circle to a corresponding point M’which will lie on the aerofoil. Joukowsky specified which lengths and an­gles to measure and explained how to use the results to arrive at the position of M’. By selecting, say, ten or twenty representative points around the circle, and following the instructions, the result is ten or twenty points that form an aerofoil shape. The more points that are transformed, the more accurately the outline of the wing emerges.

Joukowsky’s own finished diagram is reproduced here as figure 6.6. It looks complicated, but it is not difficult to identify the two main circles and the tangents, meeting at E, which were needed to start the construction. The resulting aerofoil shape can be discerned draped over the top of the diagram with its sharp tail at point C, on the left-hand side, and its rounded nose at M’on the right-hand side. The aerofoil that Joukowsky chose to construct for purposes of illustration has a marked camber and is very thick. This makes it look unrealistic, but such a degree of curvature and thickness is not intrinsic to the method. Joukowsky explained that the shape of the wing is determined by the three parameters, a, b, and a. As the circles are made larger or smaller and the point E is moved closer to, or farther from, the circles, so the shape of the wing is modified, and it can be made more rounded or more slender. In the limit, as b ^ a, and the larger circle comes ever closer to the smaller

stretches across the top of the figure, having its trailing edge near the letter C, on the left, and its leading edge near the letter M’, on the right. From Joukowsky 1910, 283. (By permission of Oldenbourg Wissen – schaftsverlag GmbH Munchen)

circle, the profile of the wing becomes so thin that it turns into an arc. In fact, it turns into Kutta’s arc.

Joukowsky then showed how to construct his second contour. He gave another set of instructions, this time involving trigonometry as well as geom­etry. Again the process started from two circles, one of radius a, and one of radius b, with b > a. The circles have their centers on the x-axis, and so their point of contact must also lie on the x-axis. Using the center of the smaller circle as the origin O, each point M on the larger circle can be specified by measuring the length r of the line joining O to M and the angle 0 between the line OM and the x-axis. Joukowsky gave the rules for transforming a point M into the corresponding point M’on the contour that is to be constructed. The rules gave the x – and y-coordinates of M’ in terms of the values of r and 0 that specified M. Thus,

Figure 6.7, taken from Joukowsky’s paper, shows that the larger circle b is transformed into a streamlined, rudderlike shape lying symmetrically along the x-axis. The thickness of the rudder depends on the relative size of the circles. As b ^ a, the rudder gets thinner and eventually turns into a straight line of length 2a lying along the x-axis. The arc and the line that constitute a sort of skeleton for the thicker shapes were referred to by Joukowsky as the “bases” of his contours.

The second, empirical, installment of Joukowsky’s paper was published in 1912, two years after the theoretical part. He focused attention on two, aero­dynamically important properties of his theoretical contours that could be made accessible to empirical testing. The two characteristics were (1) the angle of zero lift, that is, the small (and often negative) angle of incidence at which the wing first begins to produce lift, and (2) the slope of the graph when the coefficient of lift was plotted against the angle of incidence. Both of these angles could be deduced from the basic principles of the circulation theory. Their analysis proceeds in a similar way for all aerofoil shapes derived from a conformal transformation of a circle.52 This approach enabled Joukowsky to derive his predictions using Kutta’s lift formulas and then make experimental comparisons between models of Kutta’s arc-like wing and his own wings and rudders. The predictions applied (approximately) to all the profiles.

To address the angle of zero lift, consider again Kutta’s formula for the lift P on a circular arc at an angle в to a flow of velocity V. Kutta found that

where a is the radius of the circular arc of the wing and a is half of the angle

subtended by the arc. Assuming that the velocity V is not zero, then, if the lift is to be zero, the term sin (a/2 + P) must equal zero. In other words, P must equal – a/2. So – a/2 is the angle of zero lift, and it is determined by the ge­ometry of the wing. When Lilienthal selected a wing based on a circular arc, and decided that it should subtend an angle of 2a at the center of the circle, he was implicitly fixing the value of the angle of zero lift. More precisely, he was fixing the angle of zero lift, provided all the assumptions of the theoreti­cal analysis held true. It is striking that such a significant parameter should emerge so readily from the theory, and it was a consequence of the analysis that could be easily tested.

The other angle that interested Joukowsky was the slope of the lift- incidence curve. Joukowsky simplified Kutta’s formula by supposing that the arc of his wing could be treated as equivalent to two straight lines, one con­necting the trailing edge to the highest point of the arc, the other connecting the leading edge to the highest point. The length of the two lines was desig­nated /, and elementary trigonometry showed that / = 4a sin a/2. Substitut­ing this in Kutta’s formula for the lift P gave

P = np/sm^O. + ej V2.

Joukowsky then made two further changes to the formula. First, he replaced the lift by a coefficient of lift called Ky. This was done by dividing both sides of the above equation by V2 and /. Second, for small angles, the sine of an angle equals the angle itself (measured in radians). The equation then becomes

^+e •

Joukowsky noted that the angle (a/2 + P) represented the angle of incidence as measured from the line of zero lift. If the approximations are reasonable, and if the theory was on the right lines, this formula showed that a graph of the lift coefficient against angle of incidence should have the slope np. Jou – kowsky gave the slope the label K. So here was a second testable prediction. He worked out that for a temperature of 20° and an atmospheric pressure of 760 mm, the slope of the graph should be K = 0.39.

Joukowsky had built a wind tunnel in the TH in Moscow. The tunnel had a rectangular, working section of 150 X 30 cm and could achieve wind speeds of up to 22 m/sec. The wing sections under test were suspended ver­tically, with their ends close to the top and bottom of the tunnel, so that they approximated an infinite wing. The sections were rigidly fastened to a framework, and the forces were measured by the weights that were needed to counterbalance them and keep the framework in equilibrium. The wing and rudder contours to be tested had been constructed so that they accorded with the outcome of the geometrical transformations described in the earlier part of the paper. The wing form had been constructed geometrically using a small circle with radius a = 750 mm and with the larger circle of radius b = 762.5 mm and an angle a = 20°. This gave a much thinner and flatter section than the heavily cambered one shown in the diagram in the first installment of the paper. The more slender of the two rudder shapes was generated from two circles a = 250 mm and b = 260 mm, whereas the fatter model was based on two circles a = 250 mm and b = 270 mm.

Joukowsky’s graphs of his experimental measurements revealed the famil­iar pattern when lift and drag coefficients are plotted against the angle of in­cidence. The lift increased in a roughly linear fashion with angle of incidence up to about в = 15°, while the drag stayed low until about the same point and then increased rapidly. Joukowsky immediately noted that his coefficients of lift and drag had higher values than those reported by Eiffel for comparable shapes. This sort of discrepancy between the wind tunnels in different na­tional laboratories was to plague experimental work for many years. In this case Joukowsky suggested that the Moscow experiment approximated more closely the infinite wing assumed in the theoretical calculations. The impor­tant question, though, was whether his experimental graphs corroborated the theoretical predictions.

Joukowsky found that the angle of zero lift for his theoretically derived wing profile fitted more closely to the predicted value than did the Kutta – like arc that Joukowsky called its “basis” or skeleton. But even the model wings that were meant to conform to the Joukowsky profile did not achieve quite the predicted degree of lift. The wing ceased to give lift at -6°, and the circular arc that was its basis at around -4° compared with a theoretical value of (a/2) = -10°. Some of his computed values of the slope K, however, were very close to the predicted value where K = pn = 0.39. Thus he reported that K = 0.38 for the arc, K = 0.37 for the wing, but only K = 0.30 for the rudder.

The wind tunnel at the Moscow TH was soon to figure again in the pages of the Zeitschrift fur Flugtechnik. In June 1912, Joukowsky’s assistant G. S. Lou – kianoff published graphs showing the lift, drag, and center-of-pressure char­acteristics of the wing contours of seven types of aircraft that were currently flying with success: the Breguet, Antoinette, Wright, Bleriot, Farman, Hen – riot, and Nieuport machines.53 As von Mises observed, these early Moscow experiments gave a slope for the lift-incidence curve that closely corresponded to the theoretical value, though later experimenters found a slightly smaller value. In general, said von Mises, two-dimensional wing theory overestimates the slope by about 10 percent and underestimates the angle of zero lift by one or two degrees.54 But it was the theoretical achievement, rather than the experimental work, that proved most significant. Joukowsky’s aerofoils, the J-wings, as they were sometimes called, aroused an immediate and positive response in Germany. The interest in the theory was not abstract, aesthetic, or otherworldly. Joukowsky’s theoretical profiles became the focal point for a series of developments that brought the mathematical analysis of lift into intimate contact with both physical reality and engineering practice.

When I returned to Cambridge in 1919 I aimed to bridge the gap between Lamb and Prandtl.

g. i. taylor, “When Aeronautical Science Was Young" (1966)1

Oscar Wilde declared that if you tell the truth you are bound to be found out sooner or later.2 There is a corresponding view that applies to scientific theories. Given good faith and genuine curiosity, a true theory will eventu­ally prevail over false ones. These sentiments make for good aphorisms but the epistemology is questionable. Even if it were right, there would still be the need to understand the contingencies and complications of the historical path leading to the acceptance of a theory. My aim in the next two chapters is to describe some of the contingencies that bore upon the fortunes of the cir­culatory theory of lift in Britain after the Great War. I shall come back to the philosophical analysis of theory acceptance in the final chapter of the book, when all the relevant facts have been marshaled. I begin the present discus­sion with some observations about the flow of information between German and British experts before, during, and after the Great War.

Southwell entered Trinity in 1907 to read mechanical sciences. He was an engineer, but an engineer with impressive mathematical skills.38 In 1909 he was placed in the first class of part I of the Mathematical Tripos and in 1910 graduated with first-class honors in the Mechanical Sciences Tripos. He was coached by Pye and Webb, two of the best mathematical coaches of the time. On graduation he began research on elasticity theory and the strength of ma­terials and in 1912 became a fellow of Trinity. In 1914 Trinity offered Southwell the post of college lecturer in mathematics but he did not take up the offer because of the outbreak of war. He volunteered for the army and was sent to France. In 1915, however, he was brought back to work on airships for the

navy. In 1918 he was transferred to the newly created Royal Air Force, with the rank of major, and was sent to Farnborough in charge of the aerodynamic and structural department. After demobilization, and a brief return to Trin­ity, in 1920 he went to the National Physical Laboratory as superintendent of the Aerodynamics Department. He stayed at the NPL for five years and then returned again to Cambridge, where (unusual for an engineer) he was a fac­ulty lecturer in mathematics.

It was in the field of applied mathematics, rather than practical engineer­ing, that Southwell made his outstanding contribution. He developed novel mathematical techniques for the analysis of complex structures of the kind used in the building of airships. The technique was called “the relaxation of constraints” and depended on replacing the derivatives in the equations and boundary conditions by finite differences.39 Though the technique was initially developed to deal with engineering problems, Southwell later dem­onstrated its power as a general method of solving differential equations. Referring to the unavoidable complexities of practice, and the uncertainties

in data of whatever kind, he called his own Relaxation Method “an attempt to construct a ‘mathematics with a fringe.’”40 He was not only interested in elasticity and the strength of materials but also worked on viscous flow. Like Bairstow, Southwell started from Oseen’s approximation to the full equations of viscous flow and the developments provided by Lamb.41 In 1929 Southwell was offered the chair in engineering at Oxford, which he accepted after some hesitation but where he stayed until his retirement. Southwell had a lively sense of the different demands confronted by engineers and mathematical physicists, but it may be revealing that Glazebrook said of him that, although he was an Oxford professor, he was still a Cambridge man.42

As superintendent of the Aerodynamics Department at the NPL, South­well played a prominent role in the discussions that took place in the Aero­nautical Research Committee after the war when plans for future work were thrashed out. Southwell always placed great emphasis on fundamental scien­tific research. It was the long term, not the short term, that counted. Though an engineer by training, he defended the value of academic research of the kind so often attacked by the practical men. This came out clearly in the pol­icy discussions that took place in February 1921, devoted to the topic of “The Aeroplane of 1930.” The participants were invited to anticipate the character and needs of aviation in ten years’ time. Southwell wittily subverted the discussion by posing the question If we could know where we would be in ten years’ time, why wait? His point was that fundamental advances could not be predicted. He suspected that, whatever we said, we would be wrong.43 The most we can do is to be conscious of the gaps in existing knowledge and try to fill them. Consider, he said to the committee, the fundamental cause of the lift and drag on an aircraft wing: “We have much empirical data in regard to aerofoils, but our ignorance of the mechanisms by which their lift and drag are obtained has hitherto been almost absolute.” Here was a worthy focus for research: the true mechanisms of lift and drag must be identified.44

One might assume that Bryant and Williams’ experiments, as well as those of Fage and Simmons, were performed to identify the mechanisms that Southwell had in mind. But if this were so, we would expect that the results of the work (give or take Taylor’s reservations) would have been seen by South­well as furnishing the desired account of lift and drag. This was not how he saw them. The same sense of ignorance about fundamental causes still per­vaded Southwell’s thinking after this experimental work had been completed and after Glauert had begun to provide his superbly clear exposition and de­velopment of the circulation theory. The same pessimism that was expressed privately in committee in 1921 was expressed again, and publicly, some four years later in two lectures that Southwell gave in 1925. One of these lectures, on January 22, was to the Royal Aeronautical Society; the other, on August 28, was to the British Association meeting in Southampton.

The lecture to the RAeS was titled “Some Recent Work of the Aerodynam­ics Department” and was meant as a summary of the achievements of the department during the years of Southwell’s superintendence.45 His return to Trinity was an opportunity to take stock. Southwell began by welcoming the change from ad hoc wartime experimentation to programs of research guided by theory. Two main lines of theoretical concern were identified. First, there was the classical theory of stability, and Southwell described in detail the re­cent work of Relf and others. This had taken the experimental determination of the damping coefficients for roll, yaw, and pitch to new levels of sophistica­tion. The second set of theoretical concerns dealt with the fundamentals of fluid flow. For aerodynamics, said Southwell,

I suppose no problem is so fundamental as the question—why does an aero­foil lift? We can hardly rest satisfied with the present position—which is, that we have next to no idea. To answer the question completely would involve no less than the solution of the general equations of motion for a viscous fluid, and attacks on these equations have been made from all angles. Considering the energy expended, the results have been very small; but then, these are about the most intractable equations in the whole of mathematical physics. (154)

Southwell mentioned the role played in this (so far fruitless) endeavor by Bairstow, Cowley, and Levy and then moved on to the approach adopted by Prandtl, namely, using the inviscid theory of the “hydrodynamic textbooks” informally conjoined with the idea of a viscous boundary layer. In this way the “once discounted” classical theory of the perfect fluid had been “rein­stated” and could provide a close approximation to the truth when used “un­der proper control, and aided by assumptions based on physical intuition” (156). At the NPL, said Southwell, every opportunity had been taken to check the validity of Prandtl’s theory, and “in the main one must say, I think, that it has passed the ordeal with flying colours” (156). The most important tests “are those which Messrs. Fage, Bryant, Simmons and Williams have made” (156). Southwell explained that at the time of his lecture this work had not yet been published but it had confirmed the most important result, namely, “the theoretical relation between lift-coefficient and the circulation” (156).

At this point Southwell’s audience might have been puzzled. They were being told that Prandtl’s theory had passed the tests to which it had been subject with “flying colours,” and yet a moment before, Southwell had de­clared that experts had “next to no idea” how a wing produced lift. Didn’t these claims contradict one another? The answer is that Southwell’s argument was consistent but depended on a suppressed premise. For Southwell, the experiments of Fage and Simmons only justified the use of inviscid theory as a way of representing the real flow. They did not show that it truly described the flow. As far as Southwell was concerned, Fage and Simmons were not tracing footprints in the snow. In their experiments the imprint of reality had not been made in some familiar and reliable medium. Their analysis had used ideal fluid theory. The nature of the beast that left the footprints was still under discussion. The inviscid approach left it an open question whether the “actual flow” corresponded to the representation, and the most plausible answer was that it did not. The no-slip condition was violated by the inviscid representation, and Prandtl had assumed that the flow was steady. The eddies in the wake were neglected. The place to look to resolve these issues, Southwell concluded, was the boundary layer. It was this aspect of Prandtl’s work that really engaged Southwell. As he put it, “the conditions in this layer are the ultimate mystery of aerodynamics: somehow or other, in a film of air whose thickness is measured in thousandths of an inch, that circulation is generated which we have just seen to be the essential ingredient of ‘lift’” (158). Research should concentrate on the boundary layer. Theoreti­cally this required a deeper understanding of the equations of viscous flow; experimentally it called for the development of special instruments such as microscopic Pitot tubes to probe the boundary layer. Southwell mentioned that Muriel Glauert was working mathematically and experimentally on the calibration of such an instrument.46

Here was the explanation of Southwell’s apparently conflicting claims. Prandtl’s theory of the finite wing “worked,” but it could not be true because the mathematical analysis depended on false boundary conditions. This was the suppressed premise, which rendered the argument consistent. Although Prandtl’s wing theory could pass many tests, and even pass them with flying colors, it could not, by its very nature, answer the question that Southwell wanted to answer. In a very British way, he wanted to know how a viscous fluid generates lift. In the discussion after the lecture, in response to Major Low, Southwell said: “The really interesting part of Prandtl’s work was the work he had been doing subsequently in his study of the ‘boundary layer,’ because that work might ultimately explain why the assumptions which could not be correct could make such amazingly true predictions” (166).

In a lecture titled “Aeronautical Problems of the Past and of the Future,” delivered later in the same year, Southwell insisted that the aim of research was “not so much to achieve, as to understand.”41 Scientists should not be content with “achievement,” “unless it be the result of understanding’—something of which the “practical man” would never be persuaded (410). Understanding meant understanding based on a sound theory. Southwell identified three triumphs of British aeronautics that, in his opinion, met this condition. They were (1) the ability to build stable aircraft, (2) the analysis of the dangerous maneuver of spinning and its avoidance, and (3) the achievement of control in low-speed flight even after the aircraft had stalled. In all three cases, he argued, the end result had enormous practical value but the driving force had been the aim to understand. And it was mathematical analysis that had furnished the understanding.

The theory of lift was conspicuous by its absence from this list of tri­umphs. For Southwell, Prandtl’s wing theory was an achievement that was not yet informed by an adequate theoretical understanding. Bryant, Wil­liams, Fage, and Simmons were mentioned by name, and Southwell used diagrams taken from their papers. The role that he accorded the work, how­ever, was that of showing that the effects of viscosity can be ignored as far as the sliding of air on air is concerned but cannot be ignored very close to the surface of a wing or in the wake behind the wing. It is what happens in these regions that constitutes “the ultimate problem of hydrodynamics” (417). It was this “ultimate” problem that Southwell had in mind when he asked: Why does a wing generate lift? He was not denying the role of circulation, nor was he belittling the insights of Lanchester, Prandtl, or Glauert as they continued to develop the inviscid theory of lift. His point was that no one, following this route, could hope to explain the origin of circulation.48 Within inviscid theory, circulation had to be a postulate not a deduction.

Southwell’s skeptical position was endorsed by H. E. Wimperis, the quiet but influential director of scientific research at the Air Ministry.49 Wimperis had trained as an engineer in London and Cambridge and had sat the Me­chanical Sciences Tripos in 1890. During the Great War he had served as a scientist with the Royal Naval Air Service and had designed a bomb sight that carried his name. After the war he worked at Imperial College in a labora­tory financed by the Air Ministry. Along with Tizard, he was later to play an important role in the development of Britain’s radar defense system. In 1926 Wimperis, in his role as director of research, published a survey article in the Journal of the Royal Aeronautical Society called “The Relationship of Physics to Aeronautical Research.”50 One of Wimperis’ aims was to send the message that the Air Ministry and government were aware of the need for fundamental research. What, he asked, was engineering but applied physics? Government scientists at the National Physical Laboratory and Farnborough must have the freedom to pursue basic, physical problems. A second aim was to argue that this policy had already produced significant results. Here Wimperis cited, among other examples, the mathematical work that had been done on fluid flow and, in particular, the flow around a wing. It rapidly became clear, how­ever, that in Wimperis’ view, the approach based on inviscid theory was not an exercise in real physics but a mere preliminary to a genuine understanding of lift. On a classical hydrodynamic approach, he noted, the circulation must be added in an arbitrary way to the flow, and this only provides an “analogy with the lift force experienced by an aerofoil” (670). Admittedly there have been some successful predictions made “by the employment of this conven­tion” (670), but the theory becomes “somewhat far-fetched” in its account of what is happening on the surface of the wing. “Circulation,” said Wimperis, “must have a physical existence since velocity is greater above the wing than below; though this real circulation is a circulation with no slip, whereas the mathematical circulation has slip. Hence the rather amusing situation arises of adding to the mathematical study of streamlines a conventional motion which could not really arise in an inviscid fluid!” (670). Southwell was right, said Wimperis, in insisting that the real problem lay in discovering what was actually happening in the very thin, viscous layer close to the wing. This was a problem in physics rather than something that could be evaded by the use of mathematical conventions and unreal boundary conditions.

The continuity between Frank’s ideas, developed in the 1930s, and the more recent work in the sociology of scientific knowledge was noted by the phi­losopher Thomas Uebel in his paper “Logical Empiricism and the Sociology of Knowledge: The Case of Neurath and Frank.”71 Uebel concluded (I think rightly) that Frank had anticipated all the methodological tenets of the Strong Program (147), but he insists that there is an important difference: the advo­cates of the Strong Program are “relativists,” whereas Frank “did not accept the relativism for which the Strong Programme is famous” (149). This state­ment is incorrect. The similarity does not break down at this point. Frank was also a relativist. I first want to establish this fact and then I shall use Frank’s relativism to illuminate some examples of aerodynamic knowledge.

Frank’s relativism was implicit, but clearly present, in his paper on the acceptance of theories, for example, in his assertion that there was no such thing as “perfect” simplicity. He meant that there is no absolute measure of simplicity that could exist in isolation from the circumstances and perspec­tives of the persons constructing and using the theory. If there is no absolute measure, then all measures must be relative, that is, relative to the contingen­cies and interests that structure the situation. Recall also the trade-off be­tween simplicity and predictive power. Frank said this meant there was no such thing as “the truth” because there was no absolute, final, or perfect com­promise. The relativist stance is epitomized by Frank’s comparison between assessing a theory and assessing an airplane. Talk about an “absolute aircraft” would be nonsense. All the virtues of an aircraft are relative to the aims and circumstances of the user. If the process of scientific thinking has an instru­mental character, and theories are technologies of thought, then talk about an absolute theory, or the absolute truth of a theory, is no less nonsensical.

Frank made his relativism explicit in a book called Relativity: A Richer Truth.72 Einstein wrote the introduction, and the book contains a number of examples drawn from Einstein’s work, but the book is not primarily about relativity theory. It is a discussion of the general status of scientific knowl­edge and its relation to broader cultural concerns. Frank’s purpose is much clearer in the title of the German edition, Wahrheit—Relativ oder absolut? (Truth—relative or absolute?),73 which poses the central question of the book. Does science have any place within it for absolutist claims? Frank said no. No theory, no formula, no observation report is final, perfect, beyond revision or fully understood. The world will always be too complicated to permit any knowledge claim to be treated as absolutely definitive. In devel­oping this argument Frank draws out the similarities between relativism in the theory of knowledge and relativism in the theory of ethics. Are there any moral principles that must be understood as having an absolute character? The claim is often made, but Frank argues that if close attention is paid to the actual employment of a moral principle, it always transpires that qualifica­tions and complications enter into their use. “For this so-called doctrine of the ‘relativity of truth’ is nothing more and nothing less than the admission that a complex state of affairs cannot be described in an oversimplified lan­guage. This plain fact cannot be denied by any creed. It cannot be altered or weakened by any plea or admonition on behalf of ‘absolute truth.’ The most ardent advocates of ‘absolute truth’ avail themselves of the doctrine of the ‘relativists’ whenever they have to face a real human issue” (52).

The book on relativism was written during the 1940s after Frank had left Prague. It was a response to a systematic attack on science by theological writ­ers in the United States. They blamed science for the ills of the time, such as the rise of fascism, the threat of communism, the decline in religious belief, and the loss of traditional values. The critics said that science encouraged relativism and relativism was inimical to responsible thinking. Frank con­fronted the attack head on. He did not seek to evade the charge by arguing that scientists were not relativists (and therefore not guilty); indeed, he said that scientists were relativists (and should be proud of the fact). The danger to rational thought and moral conduct came, he said, not from relativism but from absolutism. If we try to defend either science or society by making absolutist claims, we will merely find ourselves confronted by rival creeds making rival, absolute claims. If we take the issue outside the realm of reason, we must not be surprised if it is settled by the forces of unreason (21). Relativ­ism, he argued, is the only effective weapon against totalitarianism and has long been instrumental in the progress of knowledge. It has been made “a scapegoat for the failures in the fight for democratic values” (20).

Frank alluded to the many caustic things that critics said about relativism and then added, “this crusade has remained mostly on the surface of scientific discourse. In the depths, where the real battle for the progress of knowledge has been fought, this battle has proceeded under the very guidance of the doctrine of the ‘relativity of truth.’ The battle has not been influenced by the claim of an ‘absolute truth,’ since the legitimate place of this term in scientific discourse has yet to be found” (20-21). Notice that Frank placed the words “absolute truth” in quotation marks because, as a positivist, he would have been inclined to dismiss the words as meaningless. For him they had no real content and no real place in meaningful discourse. The claims of the absolut­ists were to be seen as similar to the claims of, say, the theologian. But if the best definition of relativism is simply the denial that there are any absolute truths, and if relativism is essentially the negation of absolutism, then relativ­ism is meaningless as well. The negation of a meaningless pseudoproposi­tion is also a meaningless pseudoproposition. Relativism would, likewise, be revealed as an attempt to say what cannot be said. This may explain why Frank also placed the words “relativity of truth” in quotation marks. There is much to recommend this analysis. It might be called the Tractatus view of relativism.74 Where, however, does this analysis leave Frank’s book? Does it not render the book meaningless and pointless? The answer is no. The reason is that absolutism, like theology, has practical consequences, and whatever the status of its propositions, the language is woven into the fabric of life. It provides an idiom in which things are done or not done. Even for the strictest positivist this penumbra of practical action has significance.

What is done, or not done, in the name of absolutism? The answer that Frank gave is clear. Absolutism inhibits the honest examination of the real practices of life and science. It is inimical to clear thinking about the human condition. The meaningful task of the relativist is grounded in this sphere. It is to be expressed by combating obscurantism and fantasy and by replacing them with opinions informed by empirical investigation. That is the “richer truth” referred to in the title of the English-language edition of Frank’s book. This down-to-earth orientation also provides the answer to another prob­lem that may appear to beset Frank’s relativist position. What is scientific knowledge supposed to be relative to? The answer is that it is relative to what­ever causes determine it. There are as many “relativities” as there are causes. That is the point: knowledge is part of the causal nexus, not something that transcends it. Knowledge is not a supernatural phenomenon, as it would have to be if it were to earn the title of “absolute.” Knowledge is a natural phenomenon and must be studied as such by historians, sociologists, and psychologists.

Frank’s relativism, and the relativist thrust of the positivist tradition, seems to have been forgotten.75 A number of prominent philosophers paid a moving tribute to Frank after his death in 1966, but they did not mention his relativism.76 In the course of this forgetting, a strange transformation has taken place. In his Kleines Lehrbuch des Positivismus, von Mises spoke of “the phantom of absolute cognition.”77 That phantom still stalks the intellectual landscape, but in Frank’s day it was scientists who were accused of relativism, whereas today it is scientists, or a vocal minority of scientists, who accuse others of relativism. From being the natural home of relativism, science has been polemically transformed into the abode of antirelativism and hence of absolutism. A significant role in this transformation has been played by phi­losophers of science who are today overwhelmingly, and often aggressively, antirelativist in their stance. The involvement of analytic philosophers should have ensured that the arguments for and against relativism were studied with clarity and precision. This has not happened. The philosophical discussion of relativism is markedly less precise today than when Frank addressed it fifty years ago and provided his simple and cogent formulation of what was at stake.