Category The Enigma of. the Aerofoil

Before we look at the early Reports and Memoranda generated by this re­search program, something should be said about the public and professional reception that was given to Haldane’s new committee. If there was a cautious welcome in some quarters, elsewhere bitter disappointment was expressed that the commercial manufacturers of aircraft and the pioneers of flight (who were usually the same men) were not represented. To these critics the ACA was just a committee of professors, not of producers. Even the inclusion of Lanchester did not satisfy the critics. He had written books on airplanes, but these were dismissed as theoretical works. He had not built airplanes, only motorcars (and some of the critics didn’t even like his cars). Where were the names of Britain’s aeronautical pioneers, such as Handley Page, Fairey, Roe, Rolls, Short, or Grahame-White?48

To prepare the ground for the prime minister’s announcement of the Ad­visory Committee, Haldane had written on May 4, 1909, to the newspaper magnate Lord Northcliffe, who had been agitating for government action. “We have,” Haldane said, “at last elaborated our plans for the foundation of a system of Aerial Navigation.” The government had created “a real scientific Department of State” for its study. In his reply of May 9, Northcliffe was dismissive. He gracelessly declared that the composition of the committee “is one of the most lamentable things I have read in connection with our national organisation.” He conceded that Rayleigh was a good choice as chairman, but “the Committee should certainly include the names of some of the now numerous English practical exponents.” As for Lanchester, he was known to be critical of the Wright machine,49 about which Northcliffe was enthu­siastic, and was “the same Mr Lanchester, I understand, who is responsible for. . . one of the most complicated motor cars we have ever had.” This was Northcliffe writing to a member of the Cabinet; when corresponding with his political cronies he simply referred to the Advisory Committee as Haldane’s “collection of primeval men.”50

In reply to Northcliffe (on May 18) Haldane said that the advice he had been given had convinced him “of what I was very ready to be convinced, that here as in other things we English are far behind in scientific knowledge. The men you mention are not scientific men nor are they competent to work out great principles: they are very able constructors and men of business. But in this big affair much more than that is needed.”51 Predictably, this response failed to mollify Northcliffe and his friends, such as J. L. Garvin of the Ob­server (“too many theorists”; May 7). Nor were they alone in their negative response. The Tory Morning Post of May 7 had declared that “too much value has been attached to the purely theoretical side, while no evidence is forth­coming that the practical side will be advanced at all.” In an interview for the Post, the aviation and motoring enthusiast Lord Montague said that “the Commission is composed of theoretical and official people as distinct from practical men. . . . I do not recognize the name of any man on it of actual practical experience.”52 The journal Flight joined in and got its revenge for its failed prediction over the Wright brothers’ contract: “It is a bad system to encumber enterprise by establishing ‘Boards of Opinion.’ The opinions of the practical men who are doing the work are worth more to the nation than those of a miscellaneous collection of scientists.”53

A more positive response to the committee was to be found in a short article in the pages of Nature on May 13, 1909.54 It was by the brilliant and opinionated mathematician George Hartley Bryan, himself a wrangler and a former fellow of Peterhouse College.55 Bryan welcomed the creation of the Advisory Committee, saying: “It is clear. . . that mathematical and physi­cal investigations are to receive a large share of attention, and that the mere building of aeroplanes and experience in manipulating them are not to in­terfere with the less enticing and no less important work of finding out the fundamental principles underlying their construction” (313). The problem of stability, he noted with satisfaction, had been singled out for attention, though the “mathematics of this problem are pretty complicated” (313). Bryan was not surprised that newspapers were complaining that the committee was too theoretical in its orientation and that the “practical man” was not properly represented. The real problem, said Bryan, was not too much theory but too many publications that contained equations and algebraic symbols written by people who did not understand mathematics. “Indeed, in many cases it is the ‘practical man’ who revels in the excessive use and abuse of formulae, and the mathematician and physicist who would like to bring themselves in touch with practical problems are consequently deterred from reading such litera­ture” (314). There was an urgent need, Bryan concluded, “for a clear division of labour between the practical man and the physicist” (314). The failure to create such a division, he argued, had already cost England the loss of its chemical and optical industries, and France had a long head start in automo­biles. Now at last there was a chance to make up the ground in aeronautics.

Bryan’s reaction was just the kind that Haldane would have been hoping for, though the reference to the “mere building of aeroplanes” was hardly politic. These two initial responses—that of Northcliffe and his allies and that of Bryan—indicate the tension surrounding the ACA. They also serve to introduce some of the labels that were used at the time to signalize the differ­ent and opposed parties. The term “practical man” does not refer to a cloth – capped artisan but primarily to engineers and entrepreneurs, and included, for example, the Hon. Charles Stewart Rolls. The label was a badge of honor intended to mark the contrast with university academics, civil servants, and others with no direct involvement in market processes.56 I follow out some of the further expressions and consequences of this social divide.

Consider the idealized model of the postcard experiment, that is, the stream­lines around a flat plate normal to the flow. The flow could take the form shown in figure 2.7 as well as that already shown in figure 2.5. Instead of curl­ing around the edges of the plate and moving down the back of the plate, the flow of ideal fluid can break away at the edges. Behind the plate the flow is not a mirror image of the flow in front of it but consists of a body of “dead air,” or dead fluid, bounded by the moving fluid which has met the plate, moved along the front face of the plate, and separated at the edges. The pressure in the “dead air” will be the same as that at a great distance from the plate and can be equated with the atmospheric pressure. In a real, viscous fluid, the moving fluid and the dead or stationary fluid would interact. There would be a transition layer, with a speed gradient created by the stationary fluid retarding the moving fluid while the moving fluid sought to drag the station­ary fluid along with it. In an ideal fluid there will be no such transition layer because there will be no traction between the two bodies of fluid. The free

stream will pass smoothly over the dead fluid so there will be a sudden transi­tion from fluid with zero velocity to fluid with a nonzero velocity. Mathema­ticians call this sudden transition a discontinuity in the velocity because there are no intermediate values. This term gives rise to the general label for flows of this kind, which are called discontinuous flows. The streamline that marks the mathematically sharp discontinuity between the moving and stationary bodies of fluid is called a free streamline. It is a line of intense vorticity along which the flow possesses rotation in the technical sense defined earlier in the chapter.

This attempt to make mathematical hydrodynamics more realistic was introduced by Helmholtz in 1868 in a paper titled “Uber discontinuirliche Flussigkeits-Bewegungen” (On discontinuous fluid motions).40 Helmholtz argued that all the flows that had produced d’Alembert’s paradox had de­pended not only on the assumption that the flow was inviscid but also on the assumption that the velocity distribution was continuous. Helmholtz explored flows involving surfaces of separation (Trennungsflache) or (what is mathematically equivalent) sheets of vorticity (Wirbelflache).41 Of course, said Helmholtz,

Die Existenz solcher Wirbelfaden ist fur eine ideale nicht reibende Flussigkeit eine mathematische Fiction, welche die Integration erleichtert. (220-21)

The existence of such a vortex sheet for an ideal inviscid fluid is a mathemati­cal fiction to make the integration [of the equations] easier.

But fiction or no fiction, Helmholtz had raised the hope that the glaringly false consequences of the standard picture of ideal-fluid flow could be avoided.

If a steady, discontinuous flow is to be possible, certain conditions must be satisfied. It must be the case that the static pressure on either side of the free streamline is the same, otherwise the flow pattern would not be in equi­librium and would modify itself. Since the flow at a great distance in front of the plate is assumed to have a constant speed V and to be at atmospheric pressure pa, while the dead air is also at atmospheric pressure, then the speed of the flow along the free streamlines that bound the dead air must also be V. This conclusion follows from Bernoulli’s equation relating speed and pres­sure. Bernoulli’s law also leads to the conclusion that a flow of this kind will generate a greater pressure on the front of the plate than on the back.

Consider the streamline that terminates at the stagnation point at the front of the plate. What is the pressure on the front of the plate at the stagna­tion point? Call the pressure p. Everywhere along the streamline that goes to the stagnation point, the static and dynamic pressure will sum to the same constant value, that of the Bernoulli constant or the total pressure head. The value of the constant, or the total pressure, at a distance from the plate is H = pa+ Уг p V2. On the plate, at the stagnation point, the speed is zero. There will be no dynamic pressure but only a static pressure that will equal the total pressure, therefore ps = H = pa + 4 p V2. The pressure produced by bringing the air to a standstill at the stagnation point thus exceeds the atmospheric pressure pa by the quantity Уі p V2. But the pressure on the back of the plate is also pa, so at this point there is an excess pressure on the front of the plate.

This argument only applies to the stagnation point, which is the point of maximum pressure on the front of the plate. What happens at other points on the front of the plate as the fluid moves away from the stagnation point and moves toward the edges? The fluid will speed up so its pressure will drop. But the pressure exerted by the moving fluid only drops to atmospheric pres­sure as it reaches the free stream velocity at the edges. It follows that, at all points on the front of the plate, there will be a higher pressure than the at­mospheric pressure on the rear of the plate. On this account, therefore, the forces on the plate do not cancel out, except at the very edges, and there is an overall resultant aerodynamic force on the plate.42

Discontinuous flows of this kind thus avoid the paradoxical-seeming zero-resultant outcome found by D’Alembert, but it is still necessary to ask whether the predicted forces are the right size. It is one thing to avoid a bla­tantly false outcome and another thing to do so by giving the right answer in quantitative terms. The question still remains: Do the forces predicted on the basis of discontinuous flow fully correspond to the observed forces? Quantitative knowledge of the forces on the plate calls for a quantitative knowledge of the speed and pressure of the flow along the front of the plate, not just at the stagnation point and the edges. Until this information could be provided, the picture was merely qualitative. Working independently of one another, Rayleigh and Kirchhoff provided testable answers.43

The quantitative analysis of discontinuous flows was not an easy task, but by the use of ingenious transformations, it proved possible to connect the discontinuous flow around a flat plate to the simple, uniform, horizon­tal flow. There was no guaranteed way to find the required steps leading to the simple flow. It called for a high order of puzzle-solving ingenuity. The character of the thinking required can be glimpsed from the first few steps of the process. Rayleigh and Kirchhoff noticed that in the original flow, the direction of the boundary streamline along the plate was known but not the velocity. For the free streamline, the reverse held: the velocity was known but not the direction. If the flow could be redrawn on a diagram where one axis was proportional to speed while the other axis was proportional to direction, then both parts of the streamline would be transformed into straight lines. This was a step toward the desired simplicity because the straight lines could be interpreted as “polygons” of the kind to which the Schwarz-Christoffel theorem could be applied. Neither Kirchhoff nor Rayleigh explicitly used the Schwarz-Christoffel theorem but used a number of ad hoc transformations to achieve the same goal.44 But once the formula describing the flow had been found, pressures and velocities could be calculated and quantitative predic­tions made.

Rayleigh’s achievement was to generalize Kirchhoff’s analysis, which dealt with plates that were normal to the flow, and give the analysis required for plates that were oblique to the flow. This classic result in hydrodynamics was published in 1878 and provided the starting point for the work of the Advi­sory Committee for Aeronautics when its members tried to explain the lift generated by an aircraft wing. The work was officially overseen by Rayleigh himself as president of the ACA. It was monitored on a day-to-day basis by Glazebrook and other mathematical physicists who were closely associated with Rayleigh. In the next chapter I describe this early British work on the lift and drag of a wing, which was based on the idea of discontinuous flow.

Frederick William Lanchester (fig. 4.7) was born in 1868, the son of an archi­tect.10 He was educated at the Royal College of Science and Finsbury Techni­cal College. He began work in 1889 with a company making gas engines, and in 1895 he began to develop his own motorcar. Until 1919 he was managing director and then the consultant engineer of the Lanchester Motor Company Ltd. He was responsible for some important patents for devices that suc­cessfully reduced the vibration that plagued early engines. During this time Lanchester had also been working on the problems of artificial flight and experimenting with model gliders. In 1907 he assembled the ideas about lift that he had been developing since the mid-1890s and published them in the form of a bulky volume called Aerodynamics, the first of a two-volume Trea­tise on Aerial Flight.11 This work is now recognized as the locus classicus of the circulation theory of lift, though it does not read like a modern textbook on aerodynamics. The circulation theory is only one strand in the argument that had evolved over some dozen years and had been changed to bring it more into line with the concepts used in, for example, Lamb’s Hydrodynamics.12 The precise character of the changes and the form in which the theory was

first conceived are not known, though they may, to some extent, be guessed from the variations in the uneven text.

Aerodynamics shows the traces of at least five interwoven lines of argu­ment: (1) an evolutionary perspective, (2) the concept of a wing being carried on a wavelike airflow, (3) a quasi-Newtonian idea of the “sweep” of a wing, (4) examples of the theory of discontinuous flow, and (5) versions of the theory of circulation or the vortex theory. Although Lanchester devoted all of chapter 3 of his book to an exposition of basic hydrodynamic ideas, the assimilation was incomplete. He did not avail himself of the mathematical formula expressing the circulation as an integral, though he did accept the ideas behind it. Furthermore, his use of the word “circulation” was not con­sistent. It was often used informally to refer to fluid that was displaced by a body and pushed from the front to the rear.13 Tracking the word “circula­tion” in Lanchester’s text does not necessarily reveal those places where the circulation theory of lift was being developed. Terminologically, Lanchester preferred to speak of “cyclic flow.”

The theoretical centerpiece of the 1907 book was chapter 4, called “Wing Form and Motion in the Periptery.” The word “periptery” was coined by Lanchester to refer to the characteristic form of airflow in the vicinity of a lift­ing surface. The chapter began with an evolutionary argument and a criticism of an existing theory of lift. In order to perform its biological function, argued Lanchester, the wing of a bird must have evolved into a shape that conforms to the pattern of airflow necessary to provide lift. It should therefore be pos­sible to read off this pattern of flow from the shape of the wing. All such natu­rally occurring wings show a similar “design and construction” that involves an arched profile and a slight downward inclination of the front edge. From this Lanchester inferred that the air must be moving upward as it approaches the leading edge of the wing and downward as it leaves the trailing edge.

The advantages of the dipping front edge was first recognized by Horatio Phillips, who made it the subject of two patents in 1884 and 1891, but, said Lanchester, Phillips gave an incorrect account of it. According to Phillips the air impinged on the sloping, upper surface of the leading edge and was deflected upward, off the surface of the wing, leaving a partial vacuum on the upper surface. Lanchester rejected this in favor of an explanation based on principles drawn from Newton’s mechanics. The central point, he said, was the exchange of momentum. The air, which was rising at the front of the wing, had to have the vertical component of its motion reduced to zero. The air then had to be given a downward direction, and thus supplied with another vertical component of motion, but this time in the opposite direc­tion. It was important, said Lanchester, that during this process the flow of air remained conformable to the shape of the wing and that no surfaces of discontinuity were created.

These ideas were developed by means of a thought experiment involv­ing the fall of a flat plate. The plate was to fall so that it presented its full surface-area to the direction of its descent. During the fall the air would be pushed around the edges from the lower to the upper surface. “There is at first a circulation of air round the edge of the plane from the under to the upper surface, forming a kind of vortex fringe” (145). (Notice that here the word “circulation” refers to the air being literally displaced from the front to the back of the object.) Lanchester then supposed the falling plate was given a horizontal velocity. This, he said, made the case equivalent to an inclined plane moving horizontally. If, following Newton, air is treated as if it is made up of independent particles, the analysis gives the wrong answers. Lanchester concluded that it was necessary to take account of the continuous nature of the fluid medium but knew of no way to do this for the flow under consid­eration. This led him to introduce a (more or less) arbitrary “sweep” factor to define the amount of air that was involved in the momentum exchange. It was Lanchester’s references to “sweep,” rather than circulation, that were picked up by the practical men.

Lanchester then explored a number of different approaches. The reader was told that “the peripteroid system may be regarded as a fixed wave” (156), though this idea was nowhere adequately explained. It seems to have been part of an early version of the theory. A few pages later Lanchester explained that because the disturbances in the neighborhood of the aerofoil possess an­gular momentum, it can be inferred that the flow comprises a cyclic motion. Lanchester went on: “The problem, then, from the hydrodynamic stand­point, resolves itself into the study of cyclic motion superposed on a transla­tion” (162). He then used the formulas of mathematical hydrodynamics to plot the streamlines and potential lines for the flow around a circular cylinder on the assumption that the flow contained a circulation or cyclic component. Graphical methods were used to establish that the imbalance of pressures furnished a lift force. The flow depicted in the plot did not look like a picture of the flow round a wing, because there was a circular cylinder, rather than a winglike profile, at the center of the action, but, said Lanchester, we may look upon this figure “as representing in section a theoretical wing-form, or aerofoil, appropriate to an inviscid fluid” (163). He justified this statement by observing that from “the hydrodynamic standpoint,” that is, with a perfect fluid, the shape of the aerofoil section is irrelevant.

Lanchester then moved from perfect fluids to real fluids and from the in­finite wing to the wing of finite length where the behavior of the air at the tips had to be considered. Finite wings could be understood by supposing that the cyclic flow extends beyond the wingtips in the form of two vortices issuing from the ends of the wing and trailing behind it. The trailing vortices could be assumed to extend back to the point on the ground from which the aircraft took off. Such a picture meets the requirement, first identified by Helmholtz, that a vortex can only end on a surface of the fluid. Lanchester acknowledged that, because of Kelvin’s theorem, the creation of such vortices in a perfect fluid presented a problem. He argued that viscosity had to be invoked to start the process, but inviscid theory could be used for the subsequent description. He also noted that the two trailing vortices would interact with one another. As Lanchester put it, “We have seen. . . that the lateral terminations of the aerofoil give rise to vortex cylinders. . . . Such a supposition presents no dif­ficulties in a viscous fluid. . . . Now we know that two parallel vortices, such as we have here, possessed of opposite rotation. . . will precess downwards as fast as they are formed” (173). Lanchester then referred his readers to the diagram that is reproduced here as figure 4.8.

Lanchester now had a qualitative account of lift that fulfilled the following conditions: (1) it was based on “cyclic” flow, that is, a flow around a vortex with circulation; (2) it was applicable to a finite wing; (3) it identified the role of trailing vortices; and (4) it made appeal to the viscosity of air as well as to conceptions derived from ideal fluid theory.14

At 9.30 a. m., on August 24, 1912, Lamb took the chair of section III of the Fifth International Congress of Mathematicians that was being held in Cam­bridge.38 Section III was devoted to mechanics, physical mathematics, and astronomy. Lamb wanted to say a few words before getting down to business. He noted that, in spite of the subdivision of the field, the scope of the section was still a wide one. He then went on to offer a classification of the different styles of work that were to be represented. He also identified the predomi­nant style of what he called the “Cambridge school” within this typology. His words are revealing.

It has been said that there are two distinct classes of applied mathematicians; viz. those whose interest lies mainly in the purely mathematical aspect of the problems suggested by experience, and those to whom on the other hand analysis is only a means to an end, the interpretation and coordination of the phenomena of the world. May I suggest that there is at least one other and an intermediate class, of which the Cambridge school has furnished many examples, who find a kind of aesthetic interest in the reciprocal play of theory and experience, who delight to see the results of analysis verified in the flash of ripples over a pool, as well as in the stately evolutions of the planetary bodies, and who find a satisfaction, again, in the continual improvement and refine­ment of the analytical methods which physical problems have suggested and evoke? All these classes are represented in force here today; and we trust that by mutual intercourse, and by the discussions in this section, this Congress may contribute something to the advancement of that Science of Mechanics, in its widest sense, which we all have at heart. (1:51)

The tone may have been lofty but Lamb had a purpose. He was making a plea for the representatives of the different tendencies in the discipline to communicate and cooperate. The remarks suggest a background anxiety that there might be problems on this score, and Lamb may have known about the convoluted and acrimonious arguments between pure and applied math­ematicians that had been taking place in Germany. We must also remember that Lamb was addressing a gathering of men of powerful intellect, many with significant achievements behind them and reputations to make or break. Larmor, Levi-Civita, Darwin, Moulton, and Abraham were all in the audience, while the Gottingen laboratory was represented by the presence of two of Prandtl’s colleagues and former assistants, Theodore von Karman and Ludwig Foppl. All of these men played an active part in the session that fol­lowed. Given his unifying purpose, Lamb could not have risked caricaturing the different classes of mathematician.

The care with which Lamb would have chosen his words lends a particular interest to his description of the Cambridge school. Lamb saw the character­istic concern of its practitioners as lying between pure mathematics, on the one hand and, on the other, a purely instrumental view of mathematics, one in which its role was simply the interpretation and coordination of data. The point on which he placed the emphasis was that mathematical results should be verified by the interplay of theory and experience. Lamb obviously saw this process as more than mere success in the ordering of data. Truth and correspondence with reality were the central aims. He described this con­cern as “aesthetic”—a word chosen, surely, to portray an intellectual involve­ment that was dignified rather than merely useful. The emphasis on truth was certainly consistent with what Lamb had said elsewhere, for example, in the discussion of Stokes’ equations in his Hydrodynamics. Applied mathematics, as practiced at Cambridge, was to be justified by its capacity to portray the nature of physical reality, not by its employment of useful fictions.

Other Cambridge luminaries expressed themselves somewhat differently but conveyed a similar orientation. At a different session of the same con­ference the Cambridge mathematical physicist Joseph Larmor voiced senti­ments that reinforced Lamb’s message. Larmor asserted that the role of the mathematician and the physicist were essentially identical.39 A. E. H. Love had spoken out in support of Larmor at the conference.40 Love was reiterating a position already developed in his authoritative Treatise on the Mathemati­cal Theory of Elasticity.41 This volume contained a historical introduction in which tendencies and distinctions similar to those identified by Lamb were rehearsed and evaluated. Love declared, as one of his aims, that he wanted to make his book useful to engineers and this had led him “to undertake some rather laborious arithmetical computations” (v). But he also wanted to “em­phasise the bearing of the theory on general questions of Natural Philosophy” (v), and it was clear that this was where his heart lay. His historical comments were judicious, but he went out of his way to emphasize the non-utilitarian origins of the subject matter he was about to expound. Thus,

The history of the mathematical theory of Elasticity shows clearly that the development of the theory has not been guided exclusively by considerations of its utility for technical Mechanics. Most of the men by whose researches it has been founded and shaped have been more interested in Natural Philoso­phy than in material progress, in trying to understand the world than in try­ing to make it more comfortable. From this attitude of mind it may possibly have resulted that the theory has contributed less to the material advance of mankind than it might otherwise have done. Be this as it may, the intellectual gain which has accrued from the work of these men must be estimated very highly. (30)

Technical mechanics is to be distinguished from natural philosophy, and he, Love, was doing a species of natural philosophy. Any resulting failure to contribute to material progress did not seem to distress him unduly. He was more interested in the link with fundamental physics and in recounting the detailed discussions that had taken place over the number and meaning of the elastic constants. These had thrown light on “the nature of molecules and the mode of their interaction” (30). The wave theory of optics and the theory of the ether had benefited from advances in the theory of elasticity, as had, even, certain branches of pure mathematics. Though Love and Lamb expressed themselves differently, we see a similar distancing of applied math­ematics from issues of utility and an affirmation of the fundamental char­acter of the relation between mathematics and physical reality. G. I. Taylor’s demand, made a few years later in his Adams Prize essay, that applied math­ematics should have a firm basis in physics was the expression of a stance already endorsed by figures of authority on the Cambridge scene and already characteristic of the Cambridge school.42

The demand for a firm basis in physics had not always characterized what had passed as “mathematical physics” or “mixed mathematics” at Cambridge. Mathematicians of earlier generations had often been happy to see mathemat­ics arise from physical problems but had then developed it independently of experimental data or with only a loose or analogical link to physical reality. An example of this earlier phase, which was still evident as late as the 1870s, was James Challis’ Essay on the Mathematical Principles of Physics in which he offered a speculative, hydrodynamic cosmology.43 The closer connection between mathematics and real physics that Lamb and, later, Taylor were tak­ing for granted had originally been forged in the work of Stokes, Thomson, and Maxwell, who were critical of the earlier style.44 Lamb, however, still felt the need to express himself carefully when he said that the Cambridge school provided “many examples” of the intermediate path between an overly ab­stract and an overly utilitarian approach. He thus acknowledged a continuing diversity in Cambridge work. This should come as no surprise since tradi­tions, even vigorous traditions, will always encompass a range of positions as they change and develop. Rayleigh, like Lamb, spoke of “the Cambridge school,” and he too noted a certain inner complexity and development. In connection with Routh’s textbook on dynamics, Rayleigh took the view that the earlier editions had been overly abstract, whereas later editions evinced a closer engagement with genuine scientific problems.45 In other words Routh had shifted toward the position that Lamb, like Rayleigh himself, saw as the strong point of the Cambridge school.46

Otto Blumenthal had been Hilbert’s first doctoral student at Gottingen and continued to help Hilbert edit the distinguished journal Mathematische An – nalen.55 In the winter semester of 1911-12 Blumenthal, now at the TH in Aachen, gave a course of lectures on the hydrodynamic basis of flight. He de­scribed, mathematically, the irrotational flow of an ideal fluid over a range of different Joukowsky profiles. Along with his colleagues at Aachen, Karl Toep- fer and Erich Trefftz, he drew up diagrams of the precise shape of the profiles. The result of the joint work was published in two papers in the ZFM for 1913. The main paper, by Blumenthal, was titled “Uber die Druckverteilung langs Joukowskischer Tragflachen” (On the pressure distribution along Joukowsky wings).56 It was followed by a short note by Trefftz giving a simplified geo­metrical method for drawing Joukowsky profiles and a graphical technique for rapidly computing the predicted air velocities, and hence pressures, on the surface of the wing.57

Blumenthal began by drawing attention to a unifying principle that had not emerged in Joukowsky’s original paper. Joukowsky had used two geo­metrical constructions. The first, which was the more complicated, gener­ated the wing profile, while the second, which was simpler, generated the symmetrical rudder. Blumenthal pointed out that only the second of the two constructions need be used. What is more, the process could be represented by a simple mathematical formula. This formula was the version of the Jou – kowsky transformation that was to achieve such fame.58 The formula can yield wing shapes and curved, Kutta-like arcs as well as rudder shapes and flat plates. Only one transformation, not two, was needed. It was all a matter of the position of the circle on the coordinate system of the plane that was to be transformed. The totality of Joukowsky contours, said Blumenthal, could be generated by the set of all circles that can be drawn on the Z = + i П plane

that pass through = – I/2, provided they either pass through, or contain, the point = +I/2. All that is required is that the circles are then subject to the transformation:

Those circles that pass through both = – I/2 and = +I/2 will have their cen­ters on the n-axis and will generate arcs similar to Kutta’s wing. The one circle in this family that has its center precisely at the origin, and hence has the line from = – I/2 to = +I/2 as its diameter, will be transformed into the straight line that is the limiting case of the arc. Wing shapes will be generated by all of the (off-center) circles whose circumference passes through = – I/2 but contains = +I/2, that is, which are sufficiently large that the circumference goes around the point = +I/2. The sharp trailing edge of the wing will be the transformation of the point = —/2, and the curved leading edge will go round the transformation of the point = +I/2. As a point moves around the circumference of such an off-center circle, the transformation will trace out the curve of an aerofoil shape with a rounded nose and an elongated tail.59 These, said Blumenthal, are “the Joukowsky figures in the proper sense” (“die Joukowskischen Figuren im eigentlichen Sinne”; 125).

It was Blumenthal who provided the unity lacking in Joukowsky’s original paper but which, today, is so often taken for granted. But Blumenthal’s aim was not merely to achieve a formal unity. He was bringing the generation of Joukowsky figures under intuitive control in order to facilitate their practical use. He isolated the features of the construction process that had an aero­dynamically significant effect on the overall geometry of the wing. Where Joukowsky had merely said that the geometrical construction of the wing de­pended on an angle and two lengths, Blumenthal identified the results of the choices that are to be made.

Blumenthal referred his readers to the diagram reproduced here as fig­ure 6.8. The circle in the figure has center M and passes through the point H, which is at a distance I/2 from the origin O. (Notice that I/2 featured in the formula that Blumenthal chose to specify the transformation.) The off-center circle in the diagram is to be transformed by means of the Joukowsky formula and turned into a wing profile. The point H (sometimes called the “pole” of the transformation) is to be transformed into the all-important trailing edge. The radial line from M to H cuts the vertical axis at a point labeled M’. Then, explained Blumenthal, the distance OM’ (labeled f/2) controls the height of curvature of the wing, while the distance M’M (labeled 6) controls the thick­ness of the wing. In general, if the center of the circle to be transformed is on
the positive vertical axis, the result is one of Kutta’s arcs; if the center is on the positive horizontal axis, the result is a symmetrical rudderlike figure; if it is somewhere in between (as in fig. 6.8), the result will be a curved profile of the characteristic Joukowsky type. How curved and how rounded will depend on the factors that Blumenthal had just identified.

Blumenthal gave four examples of Joukowsky profiles to show the effects of modifying these parameters, that is, the effect of moving the center of the circle while ensuring that its circumference still passed through H. Thus the curvature parameter (expressed as the ratio f/l) was given the value of 0, 1/10, 1/5, and 1/5 (again), while the thickness parameter (expressed as the ratio 8/l) was set at 1/10, 1/20, 1/20 (again), and 1/50. The effect of these choices was clearly visible as the Joukowsky profiles that he illustrated went from a sym­metrical shape to a markedly curved shape and from fat to thin.

From the velocity q of the flow (provided by Trefftz’s speedy method of graphical calculation), the pressure on the surface of the aerofoil could be

computed, and this led to the most striking feature of Blumenthal’s paper. Each of the four Joukowsky profiles that he had constructed was accompa­nied by a graph showing the theoretical pressure distribution on the upper and lower surfaces. (In all cases Blumenthal assumed that the profile was at an angle of incidence of 6°.) One of his profiles and its accompanying pressure graph is shown in figure 6.9. Summing the areas enclosed by the graphs gave a quantity proportional to the resultant lift. Blumenthal discussed each aero­foil in turn, pointing out the significance of the predicted pressure distribu­tion and its dependence on the parameters of the profile. Some features were common and stood out very clearly in the graphs, for example, the greater contribution of the suction effect on the upper surface of the wing compared to the pressure effect on the lower surface. Others were special to one shape, for example, the presence of a small suction effect even on the lower surface of the symmetrical (rudderlike) aerofoil and the very high speeds at the lead­ing edge of the thinnest profile.

Looking back to the period of the Great War, after some sixty years, Max Munk expressed the belief that the aeronautical work he had carried out in Gottingen had rapidly fallen into the hands of the Allies. According to Munk, the secret Technische Berichte “were translated in England a week after appear­ance and distributed there and in the U. S.”3 Exactly how this feat of espionage was performed Munk did not say. Similar stories have been related about the flow of sensitive information in the other direction, from the Allies to the Germans. I have already mentioned the secret testing of the Dunne biplane in the Scottish Highlands before the war. This was said to have attracted the at­tention of numerous German “spies,” though these stories surely owed more

to John Buchan than to reality.4 A more sober counterpart to Munk’s beliefs is provided by J. L. Nayler, one of the secretaries to the Advisory Committee. Also speaking retrospectively, he said that the wartime Reports and Memo­randa produced in Farnborough and Teddington eventually found their way into German hands. Nayler, though, suggested that this took months rather than weeks.5 Perhaps British spies were just superior to German spies.6

The truth was almost certainly more pedestrian than these claims sug­gest. There is no evidence that agents acting on behalf of the British gov­ernment got their hands on any information about the wartime Gottingen work and passed it on to their masters in Whitehall or their allies in Paris and Washington. There appears to have been no successful espionage activ­ity. It is not the speed with which information traveled that is striking but its slowness. When information did travel, the channels were overt and obvious rather than mysterious.7 The war had the predictable effect of attenuating the flow of technical information between different national groups, but even during the prewar years, with no military or diplomatic impediments, the flow was surprisingly limited. It is important to identify where the restriction lay. It did not arise because of what might be called material or external fac­tors, such as censorship, but because of more subtle, cultural constraints. It was not the physical inaccessibility of reports, journals, or books that caused the problem. What counted was the response, on the intellectual level, even when they were accessible. For example, both Sir George Greenhill and G. H. Bryan were present at the congress in Heidelberg, in 1904, when Prandtl presented his revolutionary, boundary-layer paper.8 Bryan explicitly men­tioned Prandtl’s contribution in his postconference report for Nature, but he ignored its mathematical content entirely and confined his comments to the experiments and photographs.9 It is difficult to resist the conclusion that if such important matters can be passed over in these circumstances, then even if there had been “spies” reporting back to the British Advisory Committee, their efforts would have been wasted.

To reinforce this claim I start with some other prewar events and look at the information that members of the Advisory Committee had available to them about their German counterparts. From the outset the committee, and the Whitehall apparatus that supported it, accepted the principle that it was important to monitor the work of foreign experts. Haldane stressed the point in Parliament, and the theme was picked up by the aeronautical press.10 The commitment to gathering intelligence was made apparent in three ways. First, the preliminary documentation of the committee, when it was estab­lished in 1909, included what was, in effect, a reading list for the committee members. The list cited some twenty-two works by French, German, Ital-

ian, and American writers. The German authors included Ahlborn, Finster – walder, and Lilienthal.11 Second, the sequence of Reports and Memoranda issued by the committee began with a description of the program of German airship research. It was presented by Rear Admiral Bacon at the very first meeting of the Advisory Committee on May 12, 1909.12 R&M 1 consisted of translated extracts from the publications of the German Society for the Study of Airships and included a lengthy quotation from Prandtl.13 There was men­tion of Prandtl’s wind channel, his experiments on model airships, and, in­triguingly, a passing reference was made to his “hydraulic machine” (shown earlier in fig. 7.1). This was the apparatus used to take his boundary-layer photographs. There was, however, no mention of the mathematical theory. Third, and most important of all, the committee was provided with a series of summaries of foreign papers from leading journals such as the Zeitschrift fur Flugtechnik. A steady stream of these summaries was published in the period between the founding of the committee and the outbreak of the Great War, when such material was immediately withdrawn from public circulation.14

A measure of the size of the intelligence initiative can be gathered by count­ing the number of such abstracts published yearly in the annual report of the Advisory Committee. Such a procedure can only provide an approximate measure of the potential flow of information because it does not take account of the different scope of the individual publications, but it gives some guide. Figure 8.1 charts the year-by-year production of summaries and abstracts of foreign-language publications that were made available to the committee.15 Two things stand out. First, the size of the effort put into tracking foreign work was clearly considerable. Second, there was a consistently high level of attention given to German work, amounting on average to identifying and abstracting some eighteen items per year for a period of six years.

Moving from the quantitative to the qualitative character of the informa­tion, it is important to know which authors the committee deemed interest­ing. The answer is that Prandtl and his collaborators were prominent among them. In December 1910, Glazebrook, as chairman, explicitly drew the Got­tingen work to the attention of the members of the Advisory Committee.16 In August 1913, in preparation for a forthcoming visit to the laboratory in Teddington, Prandtl sent a number of his papers to the National Physical Laboratory (NPL) and received acknowledgment from Selby, the secretary.17 Thus, by one route or another, all of the major prewar work of the Gottin­gen school had been made available, including accounts of the wind channel and the airship work but also material directly concerned with the circulation theory of lift. In addition there were abstracts of papers of indirect interest

because of their significance for fluid dynamics in general. More specifically, among the papers summarized, sometimes at length, were those of Foppl on the resistance of flat and curved plates (abstracts 93, 94, 97, 98, 118, and 131), Fuhrmann on the resistance of different airship models (abstracts 95, 96, and 127), and Prandtl’s classic study of the flow of air over a sphere in which he in­troduced turbulence into the boundary layer by means of a trip wire (abstract 234). Of those explicitly related to the idea of circulation and Prandtl’s wing theory, accounts were given of Foppl’s 1911 study of the downwash behind a wing (abstract 128, but incorrectly attributed to Fuhrmann); Wieselsberger’s 1914 study of formation flying in birds (abstract 276); the 1914 paper by Betz on the interaction of biplane wings (abstract 279); Joukowsky’s pioneering 1910 article (abstract 299); Blumenthal’s 1913 paper on the pressure distribu­tion along a Joukowsky aerofoil (abstract 301); and Trefftz’s 1913 graphical construction of a Joukowsky aerofoil (abstract 302).

The principal mathematical formulas associated with the circulation theory in both its two – and three-dimensional forms were also to be found in the abstracts. Thus the basic law of lift, linking density, velocity, and cir­culation, L = pvr, was stated, as was the law of Biot-Savart, which was the basis of the three-dimensional development of the theory. The abstracts pro­vided everything that was needed to show that the circulation theory was capable of mathematical development and was more than a mere collection of impressionistic ideas. The abstracts gave clear, documentary evidence of the progress that the German engineers were making. It would appear that the circulation theory was there for the taking. Nevertheless, the availability of the abstracts generated no more enthusiasm for the theory of circulation in its mathematical form than did Lanchester’s original publication with its more intuitive treatment of the subject.

Why might this be? Obviously, the abstracts had no power to force them­selves on anyone’s attention. They were things to be used selectively and were subject to the filtering effects of interpretation, both in their composition and their evaluation. Thus Glazebrook’s act in drawing attention to the Gottingen work was probably indicative of his enduring concern with discrepancies be­tween the results of different wind channels and the fundamental problems shared by the NPL and Gottingen in the interpretation of their findings. Gla – zebrook was acutely aware that such problems would be grist to the mill of the “practical men” and was anxious lest they be used to persuade the govern­ment to cut the budget of the NPL.18 Furthermore, the precise content of the abstracts reveals the way that reported work may be glossed so that certain as­pects of it are given salience at the expense of other readings. Take, for exam­ple, the account given in abstract 131, which was devoted to Foppl’s 1910 paper in the Jahrbuch der Motorluftschiff-Studiengesellschaft.19 This paper contained a comparison of Rayleigh flow with Kutta’s theory of circulatory flow. After summarizing the contents of the paper, the abstract writer drew the conclu­sion that neither approach to the flow over an inclined plate was satisfactory. What was needed was an understanding of certain subtle, viscous effects. “It is suggested that Kutta’s theory throws some light on the experimental results, and in some respects, qualitatively, is in fair agreement with the experiments. At present, however, no entirely satisfactory theory seems to be possible until more is known of the nature of the air flow, the main differences being due to the difficulty of including the frictional effects” (257). The need to include frictional effects was, of course, an abiding theme in the British work. The abstract writer then went on to single out, as the “most striking result” of Foppl’s investigation, “the discontinuity in lift and drift coefficients within the region from 38° to 42°” (258). All the attention was thus directed toward extremely difficult, fundamental, and unstable features of the flow that lay far outside the typical working range of an aerofoil. Once again, the British were drawn to the phenomenon of stalling. The focus was on all the things that could not be understood on the basis of inviscid flow at small angles of inci­dence rather than on what could be achieved using perfect fluid theory over a limited range. Thus the abstract and summary itself prefigured the selective tendencies and implicit evaluations that worked against the circulatory theory.

The prewar information about German thinking on aerodynamics was rich but unexploited, whereas during the war, the pressure of short-term work added to the tendency to pass over the significance of the German theoretical approach.20 What of the pattern of information flow, and the reaction to it, immediately after the cessation of hostilities? In some quarters in Britain, the outcome of the war produced a jingoistic complacency. Such sentiments were exemplified by C. G. Grey, editor of the Aeroplane, when he said in 1918: “We have nothing to learn from the Hun in aerodynamics.”21 This boast was a continuation of a commonplace theme in the aeronautical press, which, throughout the war, dismissed German inventiveness, originality, and skill.22 Such vulgarity was largely absent from the writing of the more technically so­phisticated members of the British aerodynamic community, who had, if not an admiration, at least a healthy respect for German achievements.23 Among the members and associates of the Advisory Committee there was an under­standable degree of self-congratulation as the war drew to a close, but it was modest in tone.24

The first reaction to the outbreak of peace by the scientists at the Royal Aircraft Establishment (formerly the Royal Aircraft Factory) was to poke fun at themselves and their critics. The period immediately after the cease-fire, between November and December 1918, saw the production of a light-hearted work titled “The Book of Aeron: Revelations of Abah the Experimenter.”25 This undergraduate-style spoof was composed by the remarkable group of young men who had been recruited by Mervin O’Gorman. Many of them were billeted in a large house in Farnborough called Chudleigh (see fig. 8.2).

The house acted as a mess for the RAE and had originally been organized by Major F. M. Green, who, until 1916, had been the engineer in charge of designs.26

The “Book of Aeron,” circulated as an internal report, had been written by a committee that included R. V. Southwell and Hermann Glauert. It was couched in mock, Old Testament terms, with ancient Egyptian overtones, and told the story of the Land of Rae (the RAE), its ruler Bah Sto (Bairstow), and the wicked scribe Grae (C. G. Grey). The Abah of the subtitle appears to be a reference to the designation “Department BA” in which the experi­mental work was conducted at Farnborough. Naturally, the clash between the aircraft manufacturers (“the merchants”) and the Farnborough scientists (“the men of Rae”) played a significant role in the story. The following pas­sage conveys the spirit of the enterprise:

2. And the men of Rae built air chariots for their king, and brought forth new chariots of diverse sorts; and to each chariot did they give a letter and a num­ber, that the wise men might learn their habits:

3. but the multitude comprehended it not.

4. Then murmured the merchants one to another saying, Why strive the men of Rae so furiously against us? For the king goeth to war with their chariots, and behold our chariots are cast into the pit. (3)

After the relief of the armistice, and the lessening of tension, came the serious business of taking stock. Just what had been achieved during the war? What had been sacrificed, scientifically, because of the demands of the war effort? What, if anything, was to be learned from newly accessible German literature? Bairstow and other committee members rapidly let it be known that they deplored the cutback in basic research that had been caused by the war. In terms of fundamentals, they argued, the period of rapid achievement in aerodynamics had been before the war. It was now time to get back to deeper questions and that meant solving Stokes’ equations of viscous flow. Pure mathematicians may have given up on this task, but new techniques, perhaps using graphical methods, might be developed for this purpose. In a confidential report of November 1918,27 mapping out a program of work for 1919-20, the argument was put like this: “General research in fluid motion has been discontinued during the war and it is very desirable that it should be resumed at the earliest possible moment. It is proposed as soon as the opportunity occurs to continue the study of the motion of viscous fluids to which considerable attention has already been given” (9-10). These senti­ments represented the beginnings of a campaign by Bairstow and Glazebrook to channel more resources from short-term to basic research. They argued, in letters to the Times, that government figures for the aeronautics budget con­flated the spending on development with that on research proper. This made the expenditure on fundamental work, which was vital for future technology, appear larger than it really was.28

What had the Germans being doing during the war? At first, informa­tion filtered through to the British via the French and Americans. In 1919 the American National Advisory Committee for Aeronautics, the NACA, es­tablished an office at 10, rue Victorien Sardou in Paris. Their representative, William Knight, actively pursued a policy of information gathering. To the ir­ritation of military and diplomatic circles in Paris and Washington, he made contact with Prandtl and suggested that information on recent developments be shared. Knight contacted Prandtl on November 15, 1919, by letter and man­aged to get official agreement to visit him in April 1920. He made a second visit in the autumn of 1920.29 It appears to be due to the efforts made in Paris that the British Advisory Committee was furnished with translations of reports by A. Toussant and by Colonel Rene Dorand. Toussant was an engineer at the Aerotechnical Institute of the University of Paris at St. Cyr. In November 1919 he had produced a resume of the theoretical work done at Gottingen based on the Technische Berichte. Extracts of Toussant’s work were translated by the NACA, and these surfaced in February 1920 as a report designated Ae. Tech. 48 for consideration by Glazebrook and his colleagues.30 There was, however, little that Toussant could add to the information that was already at hand from the prewar abstracts. He gave mathematical formulas but none of the background reasoning. Dorand’s report was titled “The New Aerodynamical Laboratory at Gottingen.” It was translated as report T. 1516 in October 1920 and reached the Advisory Committee via the Inter-Allied Aeronautical Com­mission of Control.31 In Dorand’s report Prandtl was referred to through­out as “Proudet,” but it included three pages of blueprints of the Gottingen wind channels and provided technical details of the automatic speed con­trol and the measuring apparatus. Bairstow felt that there was nothing new in Dorand’s report.32 Others, however, noticed that Prandtl seemed to have solved certain problems that plagued the NPL channels and had “managed to achieve good velocity distribution in the working sections.”33

Both the British and the Americans were keen to locate and translate cop­ies of the Technische Berichte. At a meeting that took place on July 13, 1920, at the Royal Society, minute 28 records Treasury authority for the “employment of abstractors to make abstracts of German technical reports.”34 Across the Atlantic, Joseph Ames, who was the chairman of the Committee on Publica­tions and Intelligence, and one of the founding members of the American National Advisory Committee for Aeronautics, wrote from the NACA head­quarters to J. C. Hunsaker at the Navy Department in Washington, D. C., on October 15, 1920:

It is with great pleasure that I am informing you that the National Advisory Committee for Aeronautics has been successful in obtaining a number of sets of “Technische Berichte” and we are mailing you under separate cover vol­umes No.1, 2 and 3. The Committee is also forwarding a carefully prepared translation of the index of the first three volumes with a list of the symbols used. . . . The importance of the information contained in the “Technische Berichte” cannot be over-estimated and it is the desire of the Committee that all research laboratories and individuals interested in aeronautical research should become familiar with the results of the aeronautical research carried on in Germany during the War.35

The success of this search appears to have been due to Knight’s persistence. He used Prandtl’s good offices to approach the German publishers but had to overcome the numerous obstacles arising from the immediate postwar currency and customs restrictions.36 Meanwhile the U. S. National Advisory Committee continued negotiations with the British about the translation and abstraction of the reports.37 The enthusiastic terms of Ames’ letter, and the continuing efforts by both the British and the Americans, provide a sufficient basis for rejecting Max Munk’s claim that the reports were in the possession of the allies soon after their completion. Had this intelligence already been gathered, all the postwar concern would have been unnecessary.38

In the German-speaking world the circulation theory of lift and Prandtl’s wing theory found their way into the textbooks during the Great War. Rich­ard Grammel, who taught mechanics at the technische Hochschule in Danzig, led the way in 1917 with his Die hydrodynamischen Grundlagen des Fluges5 In 1919, immediately after the war, Arthur Proll of the TH in Hannover, pub­lished Flugtechnik: Grundlagen des Kunstfluges.52 H. G. Bader published his Grundlagen der Flugtechnik in 1920, and in 1922 Richard Fuchs of Berlin and Ludwig Hopf of Aachen produced their comprehensive Aerodynamik.53 The content and level of these books contrasted markedly with what was available on the British textbook scene. As described earlier, both Cowley and Levy’s Aeronautics in Theory and Experiment of 1918 and Bairstow’s Applied Aero­dynamics of 1920 dismissed the circulatory theory, whereas G. P. Thomson’s Applied Aerodynamics of 1919 was almost purely empirical. Thomson spoke of the “complete failure” (26) of mathematical hydrodynamics to account for lift. He concluded that an account of lift required an understanding of eddies and turbulent motion: “This is the solution we want for aerodynamics, and not that found by the ordinary mathematical method” (32). In 1926 the situa­tion changed radically when Cambridge University Press published Hermann

Glauert’s Elements of Aerofoil and Airscrew Theory.54 This work showed the power of the “ordinary mathematical method” of which Thomson, like most of his Cambridge companions, had despaired. Glauert’s Elements proved to be an outstanding work of exposition which, even today, some eighty years later, is still confidently recommended to students.55

The book consisted of seventeen brief chapters and surveyed all the main themes of modern aerofoil theory for a reader with no previous knowledge of fluid mechanics (though a significant degree of mathematical competence was presupposed). The first chapter described the main facts to be explained, while chapters 2-5 outlined the theory of perfect fluids. Chapters 6 and 7 in­troduced the theory of conformal transformation and the specific properties of the Joukowsky transformation. Chapter 8 dealt with viscosity and drag. Here Glauert introduced the Stokes equation for viscous flow and informally derived Prandtl’s boundary-layer approximation. Chapter 9 was called “The Basis of Aerofoil Theory,” and in it Glauert sought to bring together the ap­parently antithetical ideas of viscous and inviscid flow into a practical synthe­sis. His aim was “to obtain the true conception of a perfect fluid” (127). The form of the desired synthesis was described by Glauert as follows: “The vis­cosity must be retained in the equations of motion and the flow of a perfect fluid must be obtained by making the viscosity indefinitely small” (117).

When analyzing the motion of an object in a fluid, the concept of a per­fect fluid must be deployed in a way that retains the effects of viscosity. If a perfect fluid is defined as a fluid devoid of viscosity, such a requirement is contradictory: how can the viscosity, which is excluded by definition, also be “retained”? Here, once again, Glauert sought to convey his novel, Gottin – gen-style methodology. He argued that the requirement he formulated can be satisfied even though the boundary conditions, the way the fluid behaves at a solid boundary, are wholly different for a perfect fluid and for a viscous fluid. (For a perfect fluid the boundary conditions are that it cannot penetrate the solid boundary but it can slide smoothly along it; for a viscous fluid the conditions are zero penetration of the boundary and zero slip along it.) The trick needed to retain viscosity in the equations of motion is to start with a viscous boundary layer of finite thickness around the object and imagine it to become an infinitely thin sheet of vorticity. The boundary layer is a real phenomenon belonging to viscous fluids, and a vortex sheet is an idealization appropriate to perfect fluids. The connecting link that allows viscosity to be “retained” is that

in the limit the boundary layer becomes a vortex sheet surrounding the sur­face of the body and the vortices of this sheet act as roller bearings between

the surface of the body and the genera! mass of the fluid. The conception of a perfect fluid with a vortex sheet surrounding the surface of the body therefore represents the limiting conditions of a viscous fluid when the viscosity tends to zero, and the existence of the vortex sheet implies that the perfect fluid so­lution need not satisfy the condition of zero slip at the boundary. (117-18)

If the cross section of a wing is drawn in two dimensions, the line that traces its profile is to be thought of as made up of an infinite number of points acting like infinitesimal roller bearings. These rollers are said to be rotating fluid elements, that is, “fluid elements in vortical motion” (119), and they stand in for the boundary layer. The rotating fluid elements pass along the surface of the body and finally leave it to pass downstream in the wake. Glauert linked this picture to von Karman’s work on the so-called vortex street that exists behind a bluff body placed in a fluid flow. Clearly Glauert was introducing a significant degree of idealization, but even this degree would soon be surpassed. Having replaced the boundary layer on the wing surface by a sheet of vorticity, the chord and profile of the wing was then ignored altogether. In two dimensions the wing was reduced to a single point, that is, to the “cross section” of a line of vorticity imagined to be perpendicular to the page on which the figure is drawn.

How did Glauert explain the origin of the circulation around a wing? The discussion of this sensitive and difficult question was located in a short sec­tion of chapter 9. Combining candor with British understatement, Glauert introduced the issue as follows: “The process by which the circulation round an aerofoil develops as the aerofoil starts from rest presents certain theoreti­cal difficulties, since the process would be impossible in a perfect fluid, and it is again necessary to consider the limiting condition as the viscosity tends to zero” (121). Circulation is impossible if the analysis starts with p = 0 and con­fines itself to this condition. To overcome the difficulty it is necessary to start by considering p Ф 0 and then make the transition from viscous to nonvis­cous flow by imagining that p ^ 0. Here within the one sentence we see the British and German conceptions of an ideal fluid directly juxtaposed. Glauert then proceeded to offer a qualitative account of the required transition. The analysis effectively hinged on two diagrams of the flow at the trailing edge of a wing. I reproduce the diagrams in figure 9.11.

At low speeds, as the aerofoil starts from rest, the air behaves like an ideal fluid and curves round the trailing edge as shown in (a) in figure 9.11. There is a stagnation point S on the upper surface not far from the trailing edge. Glauert argued that, as the velocity increases, the streamlines coming from the undersurface are unable to turn round the trailing edge “owing to the large viscous forces brought into action by the high velocity gradient” (121).

The flow thus breaks away from the trailing edge in the manner shown in (b). The result is that “a vortex is formed between the trailing edge and the old stagnation point S” (121). When the vortex has reached a certain stage of development, it breaks away and floats downstream in the wake of the wing.

Although Glauert does not make the point explicitly, it is obviously im­portant for the argument that the vortex that detaches itself and moves down­stream is rotating in the correct direction. Its direction of rotation determines the direction of the circulation in any contour that surrounds it. The infor­mation about the direction in which the vortex rotates is contained in Glau – ert’s diagram rather than his text. The diagram shows that the vortex rotates in a counterclockwise direction. Why counterclockwise? The presumption must be that the flow that initially went round the trailing edge and then progressively failed to navigate the sharp corner is moving more rapidly than the flow that comes away from the stagnation point. The speed difference of the adjacent bodies of fluid would constitute a surface of discontinuity and hence a surface of vorticity—and the differences, in this case, would produce a counterclockwise vortex.

The next step in the argument was equally crucial. In the diagram a vortex has detached itself from the wing, and this, the argument goes, generates an equal and opposite circulation around the wing. The detached vortex had a counterclockwise circulation, so the circulation around the wing will be clockwise, and this produces the speed differential postulated by the circula­tion theory. But why does this process create an equal and opposite circula­tion? The answer given by Prandtl and Glauert was that such an outcome is required in order that Kelvin’s theorem is satisfied. Kelvin’s theorem initially looked as if it would rule out the onset of circulation entirely. If there is zero circulation at the onset of movement, there will be zero circulation at all later times. But the enemy is now converted into an ally. Glauert explained that the circulation around a large contour, enclosing both the wing and the detached vortex, will indeed stay zero. This is necessary to satisfy Kelvin’s theorem, but the theorem can be satisfied by virtue of two opposing vortices whose

respective circulations cancel each other out. The argument is represented diagrammatically in figure 9.12, again taken from Glauert. The large contour is called ABCD. The vortex is called E, and the circulation around the wing is K, while the circulation around the vortex is, accordingly, -K. As Glauert put it: “the circulation round any large contour ABCD which surrounded the aerofoil initially was and must remain zero, and as this contour includes the vortex E there must be a circulation K round the aerofoil which is exactly equal and opposite to the circulation round the vortex E” (121). The detached vortex floats away on the free stream with velocity V, leaving the wing with circulation K and lift given by the Kutta-Joukowsky law L = pKV. Glauert’s argument was moving rapidly at this point, but though potentially puzzling, this sequence of steps is now to be found in all standard textbooks.

Having prepared the ground in the first nine chapters, Glauert moved on to discuss the aerofoil in three dimensions. In chapter 10 he introduced the mathematics of the simple and refined horseshoe vortex system as well as the important concepts of induced velocity and induced drag. In chapters 11-13 he dealt with the effects of varying the aspect ratio of a wing and generalized the results to biplanes. In chapter 14 he discussed wind-tunnel corrections, while in the final chapters, 15 and 16, he applied the circulation theory to the airscrew.

Time and again critics attempt to refute relativism by drawing attention to the objectivity of what is known in both science and daily life.78 Such attempts are misguided. The only kind of counterexample that could refute relativism would be an example of absolute knowledge. Proof or evidence of objectiv­ity will not suffice unless the objectivity in question can be shown to be an absolute objectivity. The demand for objectivity is legitimate, but it is meant to preclude subjectivity, and subjectivism is not the same as relativism. The subjective-objective distinction is one thing, the relative-absolute distinc­tion is another thing, and the two should not be conflated. Frank was ad­mirably clear on this point and knew that his defense of relativism was not an attack on objectivity. He (rightly) believed in both the relativity and the objectivity of scientific knowledge.79

Rather than explore this theme in an abstract way, let me take an example from aerodynamics. The example, which concerns the rolling up of the vor­tex sheet behind a wing, is designed to show the objective (that is, nonsub­jective) character of knowledge at its most dramatic. The question is: Can the example be understood in relativist terms? Here is the example. In the spring of 1944, at a crucial stage of the Second World War, London was at­tacked by V-1 flying bombs. The V-1 was a large bomb fitted with small wings and a ram-jet engine, and it flew some 300 miles per hour. The bombs were launched from sites on the French and Dutch coasts by means of a shallow ramp that pointed in the direction of the target. After the bomb had trav­eled a predetermined distance, its engine was switched off by an onboard device that simultaneously altered the trim of the wings, causing the bomb to fall to the ground and explode. In an effort to stem the attacks, the pi­lots of the Royal Air Force chased after the bombs and tried to bring them down in open country where they would do less harm. It was not possible to close in on the bomb to shoot it down because of the danger that the re­sulting explosion would destroy the attacking aircraft. Some pilots therefore developed the technique of flying close to the bomb, making use of the air­flow behind the wing of their aircraft to flip the missile on its side so that it would drop to the ground.80 This technique did not involve direct, metal-on – metal contact with the V-1 but, it has been argued, exploited the rolling up of the vortex sheet behind the wing of the aircraft. It was the rotating air of the vortex that turned the missile on its side. According to an article in the An­nual Review of Fluid Dynamics in 1998, the rolling up of the trailing vortices behind a wing of high-aspect ratio was, for a long time, considered to be a matter of little practical importance by experts in aerodynamics. The experts acknowledged its existence but not its utility. But, says the author, if the theo­rists ignored the significance of the roll up, “fighter pilots who used their own vortices to topple V-1 flying bombs had another opinion.”81

The example shows knowledge and skill tested by uncompromising, ex­ternal criteria. The pilots’ subjective feelings had to be mastered and their judgments subordinated to the objective demands of the situation. What, then, is “relative” about this episode and the knowledge and skills involved? The brief quotation from the Annual Review of Fluid Dynamics already indi­cates the lines on which an answer can be given. First, the relevant knowledge and beliefs were distributed unevenly across the groups mentioned in the Review article. The experts who worked theoretically, or who experimented with wind tunnels, had one opinion about vortex sheets; the pilots who chased the bombs over the fields of Kent are attributed with another opinion. Second, the character of the knowledge varied. The experts had a mathemati­cally refined understanding; the pilots had a rough-and-ready but practical sense of what they needed to do. What they lacked in rigor they made up in skill. Third, although the experts and the pilots were oriented to the same features of reality, they did not share a common language or common con­cepts. The article makes no mention of communication between pilots and aerodynamic experts on the matter, but it would almost certainly have been problematic. It had never been easy for pilots and aerodynamicists to talk to one another.82 Fourth, the range of circumstances that the members of the two groups took into account differed markedly. The experts operated in a world that was artfully controlled, shielded, and simplified; the pilots functioned in an environment saturated with complexity, interaction, noise, vibration, jolting, turbulence, and distraction.

The conclusion must be that although both groups were actively engaged with a reality that was largely independent of their subjective will, the qual­ity of that engagement was different. In both cases their understanding was objective rather than subjective, but it was also to be seen as relative to their standpoints. In neither case did it have an absolute character. In developing the argument of his book, Frank was therefore right to insist that the doctrine of the relativity of truth “does not imperil by any means the ‘objectivity’ of truth” (21).

It may be objected that the pilots had causal knowledge of reality. Theo­ries may come and go, and verbal accounts may vary, but don’t actions and interventions put an agent into direct contact with reality? This, it may be said, proves that there is a way of grasping reality that is not merely relative. But does it? The critic who takes this line must confront and answer the ques­tion What is supposed to be absolute about the knowledge of causes and the exploitation of this knowledge? The correct answer is that there is nothing absolute about causal knowledge. This conclusion ought to be well known because it was established over two hundred years ago by David Hume in his Treatise of Human Nature. Hume gave a relativist analysis whose essential points remain unchallenged to this day. His argument was that all knowledge about causes, for example, that A causes B, whether expressed in words or ac­tions, is inductive knowledge based on experience. Inductive inferences, said Hume, can never be given an absolute justification. Inductive knowledge is irremediably relative.83

The limited, fallible, and relative character of practical knowledge can be generalized from the example of the flying bombs to my entire case study. The pattern of flow over a wing described by Lanchester, Kutta, Joukowsky, and Prandtl is not the only one that can render mechanical flight possible. Their picture, which is now called “classical aerodynamics,” and whose his­tory I have been describing, rests on the principle that the separation of the flow from the surface of the wing must be minimized. Flow separation, it was assumed, always leads to a breakdown of the lift. It has now been dis­covered that flow separation can be both exploited and controlled in a way that actually generates lift. Leading-edge vortices, and even shockwaves, can be exploited to create lift.84 This was not realized until many years after the events I have described. As one authority put it, “We must realise, however, that Prandtl’s is but one of many possible bases of wing theory and there can be no doubt that more comprehensive assumptions will eventually be

developed for this interesting type of physical flow.” 85 Until the late 1950s, all of the technical knowledge in aerodynamics concerning lift had been de­veloped on what can now be seen as a narrow basis. What the future holds is always unknowable, but the more recently acquired, broader perspective serves to expose the hitherto unappreciated relativities of past achievements. But we should not allow ourselves to think that, as these historical relativi­ties are exposed, knowledge progressively sheds its relative character and moves closer to something absolute. To cherish such a picture is to indulge in metaphysics.

Two related complaints came together in the arguments that were mobilized on behalf of the “practical man.” First, it was said that the designers of air­planes did not need mathematical knowledge and that persons who did pos­sess such knowledge were mere theorists who were ill equipped to deal with real problems. The National Physical Laboratory, it was said, was staffed by mathematicians and theorists rather than engineers with practical experience. Second, those who were in the employ of the government led cushioned and subsidized lives that protected them from the bracing rigors of the market. This complaint included the staff at the NPL but was particularly directed at the Royal Aircraft Factory.

Within a year of the ACA’s founding, the aeronautical press was asking for evidence that the committee’s work was bearing fruit. The anonymous writer of an editorial in Aeronautics in 1910 lamented that “it is too late to renew criticism of the composition of the Committee or of the limitations placed on its work; it will be sufficient once again to place on record our opinion that in both respects the Committee is bound largely to be a failure so far as results of immediate practical value are concerned.”57 In an article on aeronautical research in the Aeroplane of August 31, 1911, P. K. Turner, a regular contributor, said it was time that theory and practice were brought together. Was this not why institutions such as the National Physical Labora­tory had been set up? “But it appears that the workers at these institutions, like the monks of old, are growing fat and useless; and of all the shameful wastes perpetuated in our alleged civilisation, the worst, in my eyes, is an equipped factory, laboratory, or office, where owing to the incompetence of those in charge or the laziness of their subordinates or both, or vice versa, nothing is done.”58 The issues came to a head at the military air trials of 1912 held on Salisbury Plain. Private contractors, British and foreign, were invited to enter their aircraft into a competition to see which ones best met the per­formance criteria laid down by the military. The competition was organized by O’Gorman, though the terms of the competition precluded the govern­ment Factory from formally entering its own designs. Although not an of­ficial competitor, a Factory model, a tractor biplane called the BE2, was put through its paces during the trials and informally faced the same tests as the others. The designation BE came from a system of classification developed by O’Gorman. The E stood for “experimental”; the B for “Bleriot-type” and referred to the position of the engine at the front of the aircraft. Pusher air­craft with the engine behind the pilot were given the designation F after the pioneer Henri Farnam. The BE2 was designed by Geoffrey de Havilland, who had been taken onto the staff of the Factory after successfully building some aircraft of his own. The BE2’s performance manifestly outclassed that of the entries from private firms. It was reasonably stable, made good speed, and de Havilland even set a new altitude record with the BE2 during the trials. The official winner was a machine entered by Cody, but the War Office proceeded to ignore the competition and focused its interest on the superior machine, even though it had been precluded from official entry. It offered contracts to private constructors to build not their own machines, but twelve of the government-designed BE2s. In an improved form this machine became, for a number of years, the mainstay of the Royal Flying Corps. Figure 1.4 shows a side elevation of the BE2A.

For the government’s critics the policy of contracting out government de­signs constituted an outrage. It was denounced as an attack on private enter­prise that strangled all design initiative.59 But private enterprise had put up a miserable showing at the air trials. As one eyewitness noted: “Of the seventeen British aeroplanes that were nominally in evidence, at least seven of the newer

makes were either unfinished or untested on the opening day, and thus some of the very firms for whose benefit the trials had, in a measure, been orga­nized, spoiled their own chances in competition with the older constructors who, for the most part, had entered well-tried models.”60 An editorial in the Aero for September 1912 admitted: “It is undoubtedly a fact that the majority of our home manufacturers have not gained in reputation through partici­pating in the military trials.”61 The same point was conceded by an editorial in Flight in which it was acknowledged that the BE2 was “one of the best fly­ers ever produced.” Of the firms that were granted a contract to produce the BE2, the editorial continued, “not everyone could have as readily justified a similar demand for its own machines on demonstrated merits in the Military trials.”62 The War Office decision was reasonable; if anything it was more ac­commodating of the sensibilities of the private manufacturers than it should have been. Looking back from 1917, an editorial in the journal Aeronautics ac­knowledged that “before the War there was in the whole country not a single decently organised aircraft manufacturing firm.”63 For example, the Handley Page Company had accepted orders to produce five of the twelve BE2s, but it failed to deliver even this small number. Only three of the machines had been delivered by 1914.64 If the editorial in Aeronautics is to be believed, the situation at Handley Page was the rule rather than the exception; not a single manufacturer of the BE2s made its delivery on time. The editorial went on: “We are not blind to the faults of the Royal Aircraft Factory, which are of a nature which seems inseparable from any state-owned institution. Nor do we ignore the fact that the Factory was bitterly detested and thoroughly dis­trusted by the industry at large. But truth compels us to recognise the fact that the industry was chiefly responsible for its own grievances” (185). The writer of this passage was probably J. H. Ledeboer, the editor of Aeronautics. The self-critical tone was hardly typical of most of the polemics unleashed against the Royal Aircraft Factory, though the assumption that government institutions would be inferior to those of the business world certainly was.

In Parliament the aircraft manufacturers had the support of, among oth­ers, William Joynson-Hicks and Arthur Hamilton Lee, both Conservative MPs, and Noel Pemberton Billing, an independent MP and founder of the Supermarine company. Billing had conducted his theatrical campaign for election on the basis of his commitment to, and knowledge of, aviation and all matters relating to it. The various critics of the government did not always agree with one another, but their combined voice was loud and persistent. Week after week, and year after year in the pages of the aeronautical journals, in Parliament, and in the right-wing press, they directed their anger and con­tempt at the Advisory Committee, the National Physical Laboratory, and the

Royal Aircraft Factory. Every setback, every accident, every tragedy was used as a stick with which to beat the government and as proof of the inferiority of government design and construction compared with private enterprise.

The BE2, and everyone associated with it, became the objects of a cam­paign of denigration. As so often happened in the early years of aviation, acci­dents occurred, and a number of persons flying the BE2 were tragically killed. One was Lt. Desmond Arthur, whose BE2 broke up in the air at Montrose at 7:30 a. m. on May 27, 1913. Lt. Arthur was a friend of C. G. Grey, the editor of the Aeroplane, and the loss fed Grey’s state of permanent anger against the government.65 Another victim was E. T. “Teddy” Busk, a brilliant engineer­ing graduate from Cambridge who had joined the Royal Aircraft Factory in the summer of 1912. Busk had been conducting a program of experiments designed to improve the stability of the BE2 when the machine he was pilot­ing caught fire.66 This accident, and the other fatalities, provided the critics with the excuse for which they were looking. “The Victims of Science” was the headline in the Aeroplane of March 19, 1914.67 Grey (see fig. 1.5) exploited the opportunity to the full. He argued that it was the scientific approach to airplane design that had killed these unfortunate men. He wrote: “I submit

that if the Department of Military Aeronautics will hold an enquiry into the design and construction of Mark BE2 biplanes and will take the evidence of workshop foreman and practical constructors—apart from the scientists and theoreticians—among contractors who are building the BEs they will obtain sufficient criticism to condemn almost every distinctive feature of the BE— provided always they can guarantee that in the event of the practical men speaking their minds they will not jeopardise their firm’s chances of obtain­ing further orders” (320).

Grey was an accomplished polemicist and he took care to cover himself lest the criticisms he was confidently predicting were not forthcoming. He implied that this could only mean that sinister, government forces were sup­pressing them. Having secured his line of retreat, Grey then asserted that the deaths had been caused by criminal negligence and he knew who the crimi­nals were: “Those responsible are the people, if you please, who have ‘the best brains in the world,’ and through whom aeroplane design is to excel. These are the people who base their calculations on the theories of the armchair airmen of the National Physical Laboratory” (321). When the Aeronautical Society opened a subscription to honor Busk, Grey accused it of exploiting the young man’s death. He had an unpleasant talent for criticizing others for what he was doing himself.68

Political attacks on the aeronautical establishment became even more in­tense after the start of the war in 1914. The summer of 1915 saw the Fokker Scourge. Anthony Fokker, a Dutch designer working for the Germans, had developed a forward-firing machine gun, synchronized to fire through the propeller disc. He fitted it to an otherwise undistinguished monoplane, and the new arrangement marked the emergence of the specialized “fighter air­craft.” It gave the Germans a marked advantage and, for a while, increased the losses of British pilots and machines. The BE2, whose stability compro­mised its maneuverability, was no match for the Fokker Eindecker—not, at least, when the Fokker was flown by the particularly skilled pilots to whom it was selectively assigned. By any standards this issue was one of importance for a country at war. Rhetorically, however, it became another opportunity to voice the interests of the aircraft manufacturers. In the House of Commons, Pemberton Billing denounced the government and military authorities as “murderers.” His claim was that if the young men of the Royal Flying Corps had been given machines designed and built by private firms rather than gov­ernment agencies, they would be alive today.

It would be a study in its own right to trace all the twists and turns of the protracted, political campaign conducted by Billing, Grey, and the man­ufacturers, and it would be no easy matter to decide, in every case, which complaints had substance and which were unscrupulous exaggerations and self-serving falsehoods. Given the seriousness of Billing’s allegations, and the place in which they were made, it was inevitable that official inquiries had to be launched. Two issues had to be unraveled: (1) was the Royal Flying Corps conducting its military business properly? and (2) was the Royal Aircraft Factory dealing improperly with the private manufacturers? The Burbridge Committee addressed the first problem and the Bailhache Committee the second. During the course of these inquiries, Pemberton Billing’s behavior became so eccentric and evasive that even former supporters began to back away. Flight, which had previously welcomed his election, decided that the talk of the “deadly Fokker” was a gross exaggeration and suggested that there must be some ulterior motive.69 Soon the editorials were dismissing his in­dictments as “irresponsible” and “sensational” rather than “the measured views of a man in earnest for the welfare of his country.”70

The official inquiries could find no basis in fact for Pemberton Billing’s accusations against the Royal Flying Corps, but he and the other critics ef­fectively “won” the argument against the Royal Aircraft Factory.71 The report conceded that there had been inefficiencies. The tepid defense of the Fac­tory meant that the interests represented by the critics ultimately prevailed.72 Flight, which had frequently been supportive of the Advisory Committee, the National Physical Laboratory, and the Factory, now concluded that the rights of the manufacturers had indeed been encroached upon.73 The edito­rial column proudly affirmed the principle that private enterprise was always superior to government, and then promptly asked for government subsidy for the aircraft industry.74 The government acceded to the critical pressure of the manufacturers and the Aircraft Factory was turned exclusively toward re­search rather than design. O’Gorman was removed, and his team of designers and engineers dispersed into the private sector. An impartial assessment of the rights and wrongs of the issue would, however, have to note that, before the restriction on its activities, the personnel of the Royal Aircraft Factory had produced one of the most outstanding fighter aircraft of the war, Henry Folland’s SE5. (In O’Gorman’s nomenclature the S stood for “scout” and the E for “experimental.”) Folland and his colleagues had skillfully balanced the competing demands of stability and maneuverability to produce one of the most formidable fighting machines of the war and an aircraft that was a match for anything its pilots might meet.75

The critics also “won” in that, by 1915, they had managed to hound Hal­dane out of office.76 Haldane was denounced in the right-wing press as a pro­German sympathizer. He was said to have opposed and delayed the dispatch of the British Expeditionary Force to France in 1914 and to have known of the

German war plans without informing his Cabinet colleagues. There was not a shred of truth in any of these allegations. Indeed, it was only thanks to Hal­dane’s earlier army reforms that the country had a viable expeditionary force at all. The charges even alluded to a secret wife in Germany and to Haldane being an illegitimate half brother of the kaiser. It was ludicrous and vile but it worked, and the “reptile aeronautical press,” as O’Gorman justifiably called it, played its part in the affair with enthusiasm.77 The episode must count as one of the most disreputable in twentieth-century British politics.78

During the Great War enormous social pressure was placed on men to con­tribute to the war effort and not to shirk their patriotic duty to lay down their lives on the field of battle. Pacifists and critics of the war were reviled. This practice was routine in the aeronautical press.79 Grey was happy to mobilize the hatred of “trench-dodgers” and use it against those who, instead of be­ing at the front, were working at Farnborough and Teddington. He reprinted an article from the Times (under the title “The Farnborough ‘Funk-Hole’”) asking why the fit young men seen coming in and out of the Aircraft Factory were not in France.80 The theme was taken up again when reviewing the Ad­visory Committee’s report for 1917. As usual, said Grey, the report is devoted to the glorification of the National Physical Laboratory, though he noted with approval that two “practical men of proven merit” had been brought on to the Engine Sub-Committee and the Light Alloys Sub-Committee.81 He was, however, censorious of those who were performing basic, hydrodynamic ex­periments, for example, those involving water tanks. Making play with the word “tank,” Grey declared that “if some of these able-bodied young men were to take a course of experimental work in motor-tanks at the front they would confer greater benefits on their native land” (315). Meanwhile Grey was corresponding with Winston Churchill to plead for his own exemption from the inconvenience of wartime obligations. Judging from surviving let­ters, now in the archives of the Royal Aeronautical Society, Churchill, though brief and formal in his responses, duly obliged, and through his intervention Grey got the exemptions for which he had asked.82 Dulce et decorum est pro patria mori.

The main points of this chapter may be summarized by the following ten items. The list begins with a resume of some of the terminology of the field. This terminology is taken for granted in the subsequent discussion.

1. A flow is called a two-dimensional flow when it can be drawn in cross section and the drawing taken as representative of the flow at any other cross section. Thus if the flow around a barrier, or some other obstacle, is drawn in two dimensions, this ignores the complications introduced into the flow by what happens at the edges not shown in the picture (that is, below the page and above the page). If the drawing shows the cross section of, say, a wing, then the picture does not portray what is happening at the wingtips, that is, the third dimension of the situation. This absence can be justified if the wing is very long and the immediate concern is with the flow at parts of the wing that are distant from the tips. The diagram can then adequately represent the flow around the central sections of the wing. A wing that is long enough to justify this approximation is often called an infinite wing. The word “infinite” is much used in hydrodynamics. References to, for example, “the flow at in­finity” usually mean the flow as it is at a great distance from some disturbance so that the effects of the disturbance can be ignored.

2. The main theoretical resource used in early aerodynamics came from classical hydrodynamics. Hydrodynamics offered a mathematically sophis­ticated theory of the flow of an “ideal” fluid, that is, a fluid that was incom­pressible and also completely devoid of viscosity. Of these two idealizations, the most contentious was the neglect of viscosity. The differential equations that govern the flow of an ideal fluid are called the Euler equations. These equations give the speed of the flow at a specified position and time. To make it easier to solve these equations, mathematicians introduced two further ide­alizations. First, it was often assumed that the flow was steady. This meant that rate of change with time was zero and could be ignored. Second, it was assumed that the fluid elements did not rotate. There was an emphasis on irrotational flows because it simplified the mathematics. Unfortunately the benefit of mathematical simplicity was purchased at the cost of making the flows being analyzed less than realistic as models of real fluid flows.

3. Fluid elements (that is, the small volumes of fluid whose velocities and rotations are under study) are not to be identified with molecules or atoms or material particles, although occasionally such identifications seem to have been made. Fluid elements are mathematical abstractions that enable the methods of the differential and integral calculus to be applied to fluids.

4. One logical consequence that can be derived from the Euler equations is a highly useful result called Bernoulli’s law. With the assumption of a steady, irrotational, and incompressible flow, the law takes on a simple form. It states that the pressure and the velocity at a point in the flow are related by a simple law that implies that as the speed increases the pressure will decrease, and as the speed decreases the pressure increases. Speed and pressure trade off against one another. The use of the law makes it important to distinguish between three different meanings that are attached to the word “pressure.” There is (i) static pressure, (ii) dynamic pressure, and (iii) total pressure. Total pressure equals the sum of static pressure and dynamic pressure. Static pressure is the pressure on the sides of a pipe or the surface of a wing. Total pressure is the pressure felt when a body of fluid is brought to a standstill. Dynamic pressure is the name given to the quantity V2 p V2 where p is the density of the fluid and V is the speed of flow. In the simplified conditions dealt with in early aerodynamics, the total pressure can be considered to have a constant value. As speed V increases and hence dynamic pressure increases, then static pressure must go down. Care is needed to ensure its correct ap­plication, but Bernoulli’s law plays an important role in (i) calculating the forces on an object that is immersed in a flowing fluid, for example, a wing in a stream of air, and (ii) understanding the operation of instruments such as the Pitot probe, which registers total and static pressure and (via Bernoulli’s law) permits the computation of velocities.

5. The restriction to irrotational flow permitted the mathematical descrip­tion of a wide variety of two-dimensional flows such as the flow of a steady stream around a circular cylinder and the flow around a barrier facing head – on into the stream. The streamlines of these flows could be drawn on the ba­sis of the formula (called the stream function) that furnished the mathemati­cal description of the flow. In a steady flow (but not in an unsteady flow) the streamlines give the path taken by the fluid elements. As well as streamlines a flow can be described by what are called lines of equal potential. These are orthogonal to the streamlines except at points called stagnation points, which are points where streamlines come to a halt on the surface of a body. Stream­lines and potential lines can be switched in the sense that the potential lines can be interpreted as the streamlines of a new flow. The old streamlines then become the potential lines of the new flow. Just as the streamlines are speci­fied by the stream function so the potential lines are specified by a potential function.

6. The possibility of arriving at a mathematical description of a flow was greatly improved because a large number of familiar mathematical functions (called functions of a complex variable) turned out to be interpretable as possible fluid flows. The geometrical patterns generated by these functions (that is, the lines of the curves plotted on graph paper) could be read as the patterns made by a flowing fluid and the boundaries that constrain them. Ex­ploring a function and then giving it an after-the-fact interpretation in terms of a flow of ideal fluid was called the indirect method of arriving at the equa­tions of the flow. I illustrated this process by means of a function that could be understood as describing the flow of a uniform stream around a circular cylinder. The example, along with the overall presentation of the material in this chapter, was taken from one of the standard textbooks of the World War I period, namely, Cowley and Levy’s Aeronautics in Theory and Experiment published in 1918.

7. A more direct line of attack was sometimes available to the mathemati­cian in search of a mathematical description of a flow pattern. This method involved constructing a set of equations that related the flow to be under­stood to a very simple flow that was already understood, for example, the uniform flow along a straight boundary. If the boundaries of the simple flow could be transformed into the boundaries of the more complicated flow, then the methods of transformation would also turn the simple streamlines into the more complicated streamlines of the desired flow. A particular set of transformations called conformal transformations played a central role in this process. Many such transformations had been studied as exercises in pure mathematics and geometry but were found to be important resources in the study of fluid flow. One such important transformation was called the Schwarz-Christoffel theorem.

8. The main problem with the hydrodynamics of an ideal fluid was that, although it became mathematically sophisticated, it appeared to provide no resources for explaining the resistance that an object experiences when placed in the flow of a real fluid such as water or air. When an ideal fluid flows around, say, a flat plate or a circular cylinder, the flow exerts no resul­tant force on the object. This is often called d’Alembert’s paradox, although whether it is a paradox in the true sense of the word is examined in more detail later. What is beyond dispute is that the result presented a problem for anyone who wanted to understand the air by likening it to an ideal fluid. The use of the theory of ideal fluids led to the false result of zero resistance or zero drag.

9. One possible response to this “paradoxical” result would be to reject ideal fluid theory as useless for the study of real fluids such as air. Why not develop a more realistic hydrodynamic theory devoted to viscous fluids? This project was begun, and the equations of motion of a viscous fluid were for­mulated. They are now called the Navier-Stokes equations. (The British just called them the Stokes equations.) Frustratingly they could only be solved in a few very simple cases, which gave special significance to the search for new ways to make ideal-fluid theory more realistic. In principle there were two ways to do this—hence the existence of two competing theories of lift. Only one of these ways (called the theory of discontinuous flow) was described in this chapter. The alternative, the vortex theory, is discussed in chapter 4.

10. The theory of discontinuous flow was proposed by Helmholtz and carried forward by Kirchhoff and Rayleigh. Helmholtz argued that the “para­doxical” result of zero resistance or drag arose because an ideal fluid could wrap itself around an object and exert pressure from all sides in a way that canceled out any resultant force. The discontinuous flow approach exploited the possibility that there could be discontinuities in the velocity of different bodies of ideal fluid that were in direct contact with one another. The flow was assumed to break away from the edges of an obstacle and create a wake behind it. The wake would be “dead water” or “dead air,” and the main body of ideal fluid would flow past it. (The assumption here is that the body is stationary and the fluid moving. This is the situation of a model airplane in a wind tunnel.) Such a flow pattern in an ideal fluid, with a wake of dead fluid, turned out to be compatible with the Euler equations. Furthermore, it could be established that, given such a discontinuous flow (see fig. 2.7), the pressure on the front face of an object would be greater than the pressure of the dead fluid on the rear. The forces did not cancel out and d’Alembert’s paradox was avoided. If the resultant force proved large enough, here was a theory that could, in principle, explain the lift of a wing as well as the resistance to mo­tion, the “drag.” Such was Rayleigh’s idea for explaining the lift on an aircraft wing, and it was taken up by the British Advisory Committee for Aeronau­tics. The results that emerged from the theoretical and experimental study of this model are described in the next chapter.