Category The Enigma of. the Aerofoil

The enigma of the aerofoil

In April 1997 Peter Galison and Alex Roland organized the conference “At­mospheric Flight in the Twentieth Century,” which was held at the Dibner Institute in Cambridge, Massachusetts. By a stroke of good fortune, and the generosity of the Dibner Institute, I was able to attend the meeting. My role was to act as an outside commentator. I was deeply impressed by the high quality of all of the papers that were presented, though I confess I was some­what daunted by the technical expertise of the contributors. The conference opened my eyes to a field of work, the history of aeronautics, that was new to me but which proved immediately attractive.1

One paper in the conference that caught my attention dealt with early British research in aerodynamics and the way in which, in Britain, the gulf between science and technology was bridged. The paper was titled “The Wind Tunnel and the Emergence of Aeronautical Research in Britain.”2 After the conference its author, Dr. Takehiko Hashimoto, kindly sent me the un­published Ph. D. thesis on which his paper had been based.3 Dr. Hashimoto’s main concern was with the role of those important individuals who act as mediators, middlemen, and “translators” between mathematicians and engi­neers. By comparing the development of British and American aerodynamics (and their respective responses to German aerodynamics after World War I), he reached the gratifying conclusion that the British had been somewhat more successful in this process of mediation than had the Americans. I say “gratifying” because I am British, and the British frequently take a pessimistic attitude toward their own technological capabilities and tend to assume that other countries always do things better. I did not pursue the theme of the mediator or middleman, but it was this work that prompted me to do the research presented here. Although we paint a somewhat different picture of certain people who feature in both of our studies, I express my indebtedness to Dr. Hashimoto and my appreciation of his work.

I began by following up some of Dr. Hashimoto’s references in the Pub­lic Record Office in London and soon found a set of research questions of my own that I wanted to answer, as well as evidence that there was material available with which to pursue them. My questions were these: In the early days of aviation, that is, in the early 1900s, there were rival accounts of how an aircraft wing provides “lift.” One account was supported by British ex­perts, while the other was mainly developed by German experts. This was well known to historians working in the field.4 These two theories of lift were also featured, though not in technical detail, in Dr. Hashimoto’s account.5 But I wanted to know (1) why the rivalry arose, (2) what sustained it for al­most twenty years, and (3) how it was resolved. These questions were not ad­dressed in Dr. Hashimoto’s work, nor had they been convincingly answered in any of the broader historical literature in the field. The present book sets out the conclusions that I eventually reached on these three questions.

My kind colleagues in the Science Studies Unit at the University of Ed­inburgh bore the disruptions caused by my research-related comings and goings with understanding and good humor. I am all too aware that my ac­tivities must have added to their own already considerable work load. Relief from teaching and administrative duties during crucial parts of the research was made possible by the Economic and Social Research Council (ESRC). I thank the Council for its financial support in the form of a project grant ESRC Res 000-23-0088. Grants specifically designed to offset the costs of publication came from two further sources: Trinity College, Cambridge, and the Royal Society of London. I thank the Master and Fellows of Trinity for their generosity, and I also express my appreciation for the continued sup­port of the Royal Society, in these financially straitened times, for work in the history of science.

The argument of my book involves a detailed comparison between British and German aerodynamic work, and this subject would have proven impos­sible to study without a number of lengthy visits to the Max-Planck-Institut fur Wissenschaftsgeschichte in Berlin. I must record my deep gratitude to Lorraine Daston and Hans-Jorg Rheinberger, the directors of Abteilung II and Abteilung III, respectively, and to Ursula Klein and Otto Sibum, who were directors of two of the independent research groups in the Institute. Their warm welcome and great generosity will never be forgotten, nor will the stimulus provided by the research environment they all worked so hard, and so successfully, to create. I also express particular thanks to Urs Schoe- pflin, the Institute librarian, and his dedicated team. They met my endless stream of requests and queries with unfailing professionalism, kindness, and scholarly understanding. Special mention must be made of one member of the library team, Monika Sommerer, who, in the final phases of writing the book, kindly began the work of approaching copyright holders for permission to reproduce the photographs and diagrams that illustrate my narrative.

One of the first things I did in Berlin was to make working translations of the main German technical papers that were relevant to the analysis. (By a “working translation” I mean something adequate for my own use rather than for public consumption.) Here I thank Marc Staudacher, a resourceful teacher of German and a professional translator, who spent many hours with me going over my attempts in order to check them and to explain points of grammar and meaning that were eluding me.

In developing the British side of the story I am indebted to the Royal Aeronautical Society in London for access to their unique collection of early aeronautical literature. I am deeply grateful to Brian Riddle, the librarian, who put this material, as well as his profound knowledge of the field, at my disposal. It was also through the good offices of Brian Riddle that I was able to make contact with Dr. Audrey Glauert of Clare Hall, Cambridge. Dr. Glauert generously made available to me material relating to her father and mother, both of whom played an important role in the development of aerodynamics and therefore feature prominently in my book. I hope I have been able to put that material to good use. The opportunity to talk with someone directly con­nected with the historical actors and episodes I was describing was a moving experience, and I express my gratitude to Dr. Glauert for her hospitality and kindness.

From its inception I have discussed my research project with Walter Vin – centi of the University of Stanford. I have benefited immeasurably from nu­merous and lengthy conversations drawing on his firsthand experience of aerodynamic research. His patience in discussing the arguments of the early technical papers and his willingness to read and comment so carefully on the first drafts of many of the chapters of this book have been invaluable to me in learning to find my way in this new field. It has been a privilege to be able to put my questions and problems to him and to be the recipient of his expert and thoughtful answers. Donald MacKenzie read and commented on a number of early draft chapters; later, drafts of the complete book were read by Barry Barnes, Celia Bloor, Michael Eckert, Jon Harwood, and Horst Nowacki. Not only their encouragement but also their critical comments have been invaluable, and I have made extensive alterations as a result of their suggestions. The responsibility for the defects that remain can only be laid at my doorstep.

In addition I have accumulated many other debts of gratitude for the help I have received in the course of the research—guidance to the literature and new sources, help in approaching and gaining access to archives, and numer­ous conversations on historiographical, methodological, and philosophical questions. I hope the following persons will forgive me if I do not mention individually their many and varied acts of kindness and generosity that, nev­ertheless, I so clearly remember. My sincere thanks to Andrew Barker, Jed Buchwald, Dianna Buchwald, Harry Collins, Ivan Crozier, Olivier Darrigol, David Edgerton, Heinz Fuetterer, Zae-Young Ghim, Judith Goodstein, Ivor Grattan-Guinness, John Henry, Dieter Hoffmann, Christoph Hoffmann, Marion Kazemi, Kevin Knox, Martin Kusch, Wolfgang Lefevre, David Mus- ker, Jurgen Renn, Simon Schaffer, Suman Seth, Steven Shapin, Skuli Sigur – dsen, Richard Staley, Nelson Studart, Steve Sturdy, Thomas Sturm, Annette Vogt, Andrew Warwick, and Richard Webb.

I have used material from the following archives and express my thanks to the archivists for permission to consult their holdings: Archives of the Cali­fornia Institute of Technology (Karman); Archiv zur Geschichte der Max – Planck-Gesellschaft (Prandtl); Churchill Archive Centre, Cambridge (Far – ren); Einstein Papers at Caltech (Einstein and Frank); Gottingen Archive of the Deutsche Gesellschaft fur Luft-und Raumfahrt (Prandtl); Library of the University of Cambridge (Tripos exam papers); National Library of Scotland (Haldane); Public Record Office (minutes of the ARC); Royal Aeronautical Society (Lanchester and Grey); Royal Air Force Museum, Hendon (Melvill Jones); St. John’s College, Cambridge (Jeffreys and Love); Trinity College, Cambridge (Taylor and Thomson); University of Coventry (Lanchester); and University of Edinburgh (A. R. Low).

The provenance of all photographic images and diagrams from published and unpublished sources is indicated in the caption along with an acknowl­edgment of copyright and permission to reproduce the material. In a few cases it proved impossible, despite every effort, to make contact with the holders of the copyright.

Finally I must mention my greatest debt. Throughout the research and the writing of this book I have benefited from the unstinting help of my wife. The book is dedicated to her. It is as good as I can make it, but it still seems little to give in return. I proffer it with the sentiment Wenig, aber mit Liebe.

Irrotational Flow and Laplace’s Equation

The motion of a fluid element involves three different kinds of change:

(1) translation, (2) strain, and (3) rotation. Translation involves change of position of the element, strain involves a deformation of the shape of the element, and rotation involves a change of angular orientation of the ele­ment. Rotation may seem to be an intuitively clear idea because the image that comes to mind is the rotation of a rigid body in which the fluid element is pictured as if it behaves like, say, a spinning ball. Sometimes fluid elements are indeed represented as spinning balls. Although shape is not really crucial, the picture of a sphere is sometimes invoked when explaining the striking result that a fluid element in an ideally inviscid fluid can never be made to rotate if it is not already rotating, nor can it be stopped from rotating if it is already in rotation. The rotation of an ideal fluid element can neither be created nor destroyed by, for example, the motion of a solid body that is immersed in, and surrounded by, a fluid. The argument is that, in a perfect fluid, neither the surrounding fluid nor such a moving body can exert any traction on the smooth surface of the element in order to change its exist­ing state of rotatory motion. It will be evident that, in light of this result, the origin of rotation becomes something of a mystery.19

Cowley and Levy, however, do not avail themselves of an intuitive pic­ture of fluid elements as rotating spheres of fluid. They opt for the more austere technical definition. Technically, the rotation of a fluid element (in two-dimensional flow) is defined as the average angular velocity of any two infinitesimal linear elements within the fluid element that are instantaneously perpendicular to one another. Mathematically this definition is expressed in the formula

1(dv du |

rotation = — ———- — .

2 ^dx dy )

The virtue of the technical definition is that commonsense comparisons tend to omit the possibility that the angular velocity of the two linear elements might cancel out so that, under some circumstances, rotation can be equal to zero by virtue of the deformation of the fluid element.20 A flow in which the quantity in the brackets in the previous formula is zero is called an ir – rotational flow.

Methodologically, the important point about the rotation of a fluid ele­ment is that by neglecting it, and restricting attention to irrotational flow, the mathematics is greatly simplified. Why is this? A glimpse into the reasons can be gained by taking another look at the stream function discussed in the pre­vious section. Consider the following expression involving the stream func­tion y. The expression is arrived at by differentiating у twice with respect to x and twice with respect to y and adding the result. Thus,

dy dy

dx2 + dy2′

It will be recalled that differentiating у once yields the velocity components of the flow and that the x and y components of the fluid velocity at a point are given by


u =—— and


= dy


Substituting these definitions of the velocity components in the expression under consideration gives


dx2 dy2 dx ^ dx J dy ^ dy J

dv du dx dy

The result of the substitution is precisely the expression that was used in the technical definition of the term “rotation.” It is in fact twice the rotation. If the rotation is zero, that is, if the flow is irrotational, then this term must be zero, and so, therefore, is the expression cited at the outset of the discussion. In other words, if the flow is irrotational, then the stream function у is gov­erned by the equation

dy+dy= 0

dx2 dy2

This equation is called Laplace’s equation. Although the equation itself may look far from simple, it is not difficult to appreciate that it is simpler than if the right-hand side were equated to some complicated function of x and y rather than to zero. Irrotational flow is thus a (relatively) simplified form of flow governed by Laplace’s equation.

Laplace’s equation is one of the most significant differential equations in the history of mathematical physics.21 The equation is often written as V2y = 0.22 The restriction to “irrotational” flow, which it signifies, not only simplified the mathematics, but it brought out the analogies between fluid flow and the results that had emerged or were emerging in other fields. Ir – rotational flow obeyed simple mathematical laws that were similar to those in areas such as the theory of gravitational force, the theory of heat, the theory of elasticity, and the theory of magnetism and electricity. Maxwell used the analogy, and Laplace’s equation, to shed light on the hydrodynamics of the flow of fluid through an orifice and the vena contracta, that is, the contraction shown by the jet of fluid a short distance from the orifice.23 Because of the electrical analogies, irrotational flows used to be called “electrical” flows. The interplay between hydrodynamics and the theory of electric phenomena was not only suggestive theoretically, but it was also exploited in the laboratory. In the interwar years it provided the basis of a laboratory-bench technique used by E. F. Relf at the National Physical Laboratory for graphically plotting the streamlines of the flow around objects with complicated shapes, such as aerofoils.24 The resulting representation was, of course, a representation of the flow as it would take place if the air were an ideal fluid.25

Neo-Newtonianism and the “Sweep” of a Wing

G. H. Bryan described the approach to lift adopted by the practical men as neo-Newtonian.64 The label accurately identified two salient features of their work. First, like everyone else, the practical men operated within the frame­work of Newton’s mechanics. Ultimately the wing must act on a mass of air, accelerating it downward, thus ensuring, in accordance with Newton’s third law of motion, that the wing suffered an equal and opposite reaction. This re­action was the ultimate source of the lift. Second, the practical men adopted a line of reasoning that was, in some respects, analogous to one that New­ton used in the Principia when he compared the forces exerted by a flowing fluid on a sphere and a cylinder “described on equal diameters.”65 Recall that Newton assumed that his fluid, or “rare medium,” consisted of a number of independent particles which would hit the sphere and cylinder and give up their momentum. (This was the model I previously likened to a shower of hailstones). The practical men greatly simplified the analysis by address­ing the case of a flat plate exposed to the uniform flow of this rare medium. While Newton was no doubt conscious of the distinction between his hypo­thetical fluid and real air, this difference tended to be blurred in some of the later aerodynamic discussions. By applying the reasoning to a simplified wing moving in air, the following argument was constructed.

Suppose that a flat plate has area A and is at an angle 0 to a uniform, horizontal flow of a fluid that was, like Newton’s, composed of independent particles. Suppose, further, that the collisions are inelastic so that the particles simply slide along the plane after impact. Let the particles in the main flow move at speed V units of distance per second. Then the volume of fluid strik­ing the plate each second is given by multiplying the vertical projection of the plate (A sin0) with the velocity V. The projection A sin0 was the “sweep” of the wing and sin0 was the “sweep factor.” Now multiply the volume AVsin0 by the density p (presumed to be the same as that of the air) to give the mass, and then multiply the mass by the velocity component normal to the plate (V sin0) to give the momentum exchanged per second. This is the source of the pressure P whose vertical component, P cos0, is the lift and whose hori­zontal component, P sin0, is the drag. This neo-Newtonian argument gave the formula for the resultant aerodynamic force P on the plate as

P = pAV2 sin2 0.

The formula was often called Newton’s sin2 law, although it is not to be found, in an explicit form, in the text of the Principia. There are two reasons for its absence. First, Newton was dealing with a curved surface not a flat plate, and second, his reasoning was geometrical in form, so that the trigonometric terms appear as geometrical ratios.66 The label is, however, a reasonable one. All of the subsequent work of the practical men involved versions of, and variations on, this formula.

For the range of angles relevant to aeronautics, sin0 is a small quantity, so its square is very small indeed. The Newtonian formula condemns any predicted lift to be small, except where the magnitude of A and V2 can offset the smallness of the squared sin0 term. On this analysis, lift would demand enormous velocities or unrealistic wing areas. Had the formula been true it would have rendered artificial flight a practical impossibility. It is little won­der that in his 1876 paper Rayleigh had expressed satisfaction that his own formula made pressure proportional to sin0 rather than, as in Newton’s for mula, to sin2 0. In following the Newtonian tradition the practical men inher­ited a serious problem and resorted to a variety of expedients in an attempt to overcome it. A number of examples will show how comprehensively they failed to meet this challenge.

Scope and Rigor

Consider the scope of Lanchester’s theory. His narrow focus on small angles of incidence was not shared by critics. Bairstow invoked standards of assess­ment appropriate to a wholly general theory of fluid resistance. Lanchester found this preposterous. He said it was like ignoring useful knowledge about how water flowed round a ship in normal motion because it did not also

explain the flow when it moved broadside. But it is not difficult to see how for the critics, if not for Lanchester, genuine knowledge of the one case also meant having knowledge of the other. Lanchester’s commonsense plea for theories of limited scope was at odds with the forms of generality routinely exhibited in classical hydrodynamics. This can be seen from the textbook treatment of the flow around an elliptical cylinder moving through a fluid. The elongated, elliptical cylinder bears a certain visual likeness to the plan of a boat, which was the case cited by Lanchester. Lanchester’s critics could point out that the mathematics of the flow does not single out, as being of special significance, any particular angle of inclination of the major axis of the ellipse to the di­rection of motion. It makes no difference to the mathematics whether the ellipse moves like a ship going forward or like a ship moving broadside, that is, awkwardly and inappropriately. Both motions are but special cases of the same general formula. Mathematically they merely depend on whether the real or the imaginary part of the complex potential is set to zero. This fact would have been familiar to any Cambridge student of hydrodynamics, or to anyone, such as Bairstow, Cowley, or Levy, schooled in a similar tradition. It was clearly not a significant reference point for Lanchester.3

The issue of scope also arose in another way. We have seen that circula­tion theory explained lift but not drag. The critics had rejected discontinuity theory because it could not yield accurate predictions of resistance, so on grounds of consistency circulation theory should be, and was, treated like­wise. The false prediction of zero drag was not lost on Lanchester, but it did not worry him in the way it did his critics. Unfortunately, Lanchester did not articulate a clear rationale for his stance, so the critics may have been tempted to see it as indicating a certain laxity on his part, compared to their own greater concern with truth and rigor.

There is some evidence that Cambridge physicists involved in aerodynam­ics were prone to misperceive the difference between their mental habits and those of engineers as the difference between rigor and sloppiness. Reflect­ing on his work as a physicist at Farnborough during the Great War, George Paget Thomson, the son of J. J. Thomson, drew attention to this cultural divide. Scientific work during wartime, said Thomson, “might properly be described as engineering.”4 He recalled how difficult it was for physicists to adopt the requisite point of view. As the author of Applied Aerodynamics, which had been well received by the “practical men,” Thomson could not be accused of lack of sympathy with engineers. But even he was inclined to exemplify, rather than bridge, the disciplinary divide he described. Thom­son spoke of the need for engineers to make up their minds on the basis of “insufficient evidence” and of the need to “compromise between conflicting requirements.” He concluded: “What is perhaps harder for the scientist to realize is the doctrine of ‘good enough.’ The better is the enemy of the good” (3).

Could it be that Lanchester, as an engineer, was prepared to accept the circulatory theory and perfect fluid theory because they were “good enough” for him even though they were not “good enough” for a physicist? The im­plication is that Lanchester, unlike his critics, was content with “insufficient evidence.” But there is another explanation of why a supporter of the circula­tion theory might find the lift-but-no-drag result an acceptable one. Rather than expressing a compromised standard of empirical accuracy, the response might simply embody a different standard and one that is not necessarily lower. The lift-without-drag result might be taken to be a true and accurate assertion about an “ideal wing,” that is, the sort of wing at which an engineer might aim. The result should perhaps be seen not as a false statement, but as an engineering ideal. This was not a defense explicitly offered by Lanchester, but as we shall see, it was how Ludwig Prandtl, a fellow pioneer of the circula­tion theory, expressed the matter.

Consider now the objection that the circulation is “arbitrary.” Both Kutta and Joukowsky were aware of the mathematical rationale behind this objec­tion, namely, that the theory contained no way of deducing the amount of circulation around a wing. Nevertheless they responded in a very different way to the British critics. They stipulated that the circulation be of precisely the amount necessary to ensure that the flow comes away smoothly from the trailing edge of a wing. The rear stagnation point must be on the trailing edge so that the flow does not have to wrap itself around a sharp corner. For a given angle of incidence, and a wing with a sharp trailing edge, this stipula­tion provides an unambiguous specification of the amount of circulation and is often called the Kutta condition. It derives its significance, and its nonarbi­trary nature, from the empirical fact that the flow of air over a (nonstalling) wing in a steady state settles down so that there is indeed an approximately smooth flow at the trailing edge. The Kutta condition tells the theorist what value of the circulation to assume, and thus what value of lift is predicted when this value is substituted into the formula Kp U, the lift equation.

Were Cowley and Levy, who made the complaint about arbitrariness, un­aware of this solution to the problem? The answer is that they were fully aware of the Kutta condition. In 1916 Levy had explicitly mentioned it in correspon­dence with Lanchester.5 In Levy’s view, however, Kutta’s proposal did not remove the arbitrary character of the amount of circulation. The argument presented to Lanchester was that, in reality, the trailing edge of a wing is not mathematically sharp but rounded. It therefore provides no mathematically unambiguous location for the rear stagnation point. The point on the curve that is selected for this role will itself be arbitrary. The amount of circulation needed to bring the stagnation point to this location will, therefore, also be arbitrary. The only thing that would remove this feature of the theory would be some means of deducing the circulation from first principles, given rel­evant data about the wing, for example, its shape and angle of incidence. No such method was known. For Cowley and Levy the word “arbitrary” clearly meant “not deducible from the basic equations of fluid dynamics.” They op­erated with a mathematical criterion and were looking for a mathematical solution to the problem, not an empirical one.

Section iii. the circular curved surface

Kutta now carried out the procedures for which he had prepared the ground. He began on the z-plane and specified the detailed geometry of the wing. It was to be an arc of a circle of radius r subtending an angle of 2a. This gave the coordinates of the endpoints A (the leading edge) and B (the trailing edge). The straight-line distance between A and B was the “chord,” and the highest point of the arc was to be 1/12 of the chord. Kutta chose to place this high­est point at the origin of the coordinate system. He then began the process of transformation. First he used a transformation in which every point was replaced by its reciprocal. Points on the z-plane were linked to those on the z’-plane by the formula z’= 1/z. This had the effect of turning the finite, cir­cular arc into what appeared to be two straight lines. One of them ran parallel to the positive part of the x-axis while the other ran parallel to the negative part of the x-axis. Both were at the same height above the axis. They started at equal distances from the y-axis (that is, there is a gap in the middle), and the lines went off to infinity in opposite directions.

It would have helped the reader of Kutta’s paper if, at this point, he had provided a diagram. Given the pedagogic values of the technische Hoch – schulen, he would surely have drawn pictures of such transformations on the blackboard when he presented them in lectures. Most mathematicians reading such a paper would sketch the appropriate figures, at least until the transformation had become routine for them. To help us follow Kutta’s argu­ment, I exploit an example of this practice. Sometime in the 1920s a young Cambridge mathematics graduate named Muriel Barker had occasion to work through Kutta’s article. She carefully wrote out the reasoning, some-

Section iii. the circular curved surface

mations to map the flow around a circular cylinder onto the flow around a circular arc representing the wing of Lilienthal’s glider. (By permission of Dr. Audrey Glauert)

times filling in the steps needed to get from one line to another. She also sketched the conformal transformations. These handwritten notes have sur­vived, and one page from them, containing the sketches, is reproduced here as figure 6.2. Muriel Barker will appear again, later in the story, when the reasons for her interest become apparent. For the moment her notes can help us follow Kutta’s thought processes.

On the top left of the page of the notes is a figure labeled z-plane. It is a drawing of the Kutta-Lilienthal wing with the leading edge labeled A and

Section iii. the circular curved surface

the trailing edge labeled B. The effect of the transformation z’ = i/z is shown next to it in the diagram, on the top right of the notes, labeled z’-plane. No­tice how the arc has become two straight lines and the leading and trailing edges A and B of the wing have become the endpoints A’ and B’ of the lines. Following his overall plan, Kutta next mapped these lines onto the t-plane where it would eventually link up with the transformed circle. This he did by using the Schwarz-Christoffel transformation. I have described how this transformation played an important role in the mathematical development of the theory of discontinuous flow. It was central to Greenhill’s massive report on this theory for the Advisory Committee for Aeronautics. Kutta used the transformation in a different way and in the service of the circulation theory. He needed it to construct the central arch of his mathematical bridge. The formula of the transformation can be seen about halfway down the page of notes in figure 6.2. It takes the form

The letter C is a constant, and a and b correspond to the endpoints of the wing. Immediately to the right of the formula is a sketch of the result of the transformation produced by applying this formula. The lines on the z’-plane have become the axis of the t-plane. The new line is shown as dotted in the figure, and the points corresponding to A’ and B’ have been marked in. All that was needed now was to work from the other end in order to map the circle onto the t-plane. The inferential bridge would then have been con­structed according to plan. The circle in the Z-plane is drawn on the bottom right-hand corner of the notes. The formula

Подпись: t = іZ+1

Z-1 is the transformation linking Z and t. This can be seen in the notes standing to the left of the drawing of the circle. Kutta’s aim might be described as getting from the figure at the bottom right to the figure at the top left of the page, but because he could see no way of doing this directly, he made the transition indirectly, by means of the other figures.

Coming back from the Barker notes to the original paper, we see that Kutta was now in a position to evaluate the constants in his formula in terms of the assumed velocity and direction of the free stream relative to the wing. He could also arrive at a value for the circulation on the assumption that the trailing edge is a stagnation point, that is, that the flow does not have to curl around the rear edge. This gave him the following expression for the
all-important circulation, which, in the notation used by Kutta, is 2ПС. The formula came out as

Circulation = 4nVr sin—sin I —+B,

2 ^ 2 H)

where V is the velocity, a the half angle of the arc that constitutes the wing, and в the angle of incidence of the wing to the free stream. The circulation is thus calculable from known or knowable quantities.

From Ground Effect to Biplanes

The law of Biot and Savart received a number of further aerodynamic appli­cations before the outbreak of World War I. All of these were published in the Zeitschrift fur Flugtechnik and came from the Gottingen group. Four of them were by Albert Betz and one by Carl Wieselsberger. I describe them briefly, keeping to the chronological order of their appearance.

In September 1912 Betz published some wind-channel results that showed that a wing operating in the vicinity of the ground would experience an in­crease in lift.61 Betz showed this by testing a model wing in a channel fit­ted with a false floor that could be raised or lowered. The phenomenon was an important one. Aircraft necessarily fly near the ground on landing and takeoff. Pilots were aware that there was a change in flying characteristics produced by these circumstances, but the nature of the change was little understood. This “ground effect” explains why an overloaded aircraft can sometimes take off with apparent success and then fail to gain height, with disastrous consequences. It also explains why some early aircraft could “fly” but never got more than a few feet above the ground.62 Betz also wanted to get a quantitative estimate of the effect of the walls of a wind channel on the measurements that were carried out in the course of experimentation. He showed that Prandtl’s new theory could lead to rough but quantitative pre­dictions that were confirmed by experiment. (The results were approximate, Betz suggested [220], because the “horseshoe” model ignored the downward motion of the trailing vortices.) Both of the subjects that Betz broached in his brief paper were to become a matter of enduring concern and research in subsequent years.

In January 1913, Betz published a second study, this time of the lift and resistance of a biplane.63 Whereas Foppl had used the Biot-Savart law to study the effect of the induced velocity on the tail wing, Betz now used the same approach to study the mutual interaction of wings that were positioned one above the other. The central point about the application of Prandtl’s approach to a biplane is that the trailing vortices from the upper wing will generate an induced resistance not only in the upper wing itself but also in the lower wing, while the trailing vortices from the lower wing will likewise affect both wings. Furthermore, if the wings are not located directly one above the other, the bound vortex corresponding to the wing itself (and not just the trailing vortices) will have to be taken into account when computing the induced velocity and induced drag on the other wing.

With the exception of Kutta’s second, 1911 paper, this work represented the first serious engagement with the theoretical aerodynamics of the biplane and the difficult problem of the mutual interaction of the different parts of an aircraft. It will be recalled that the “practical men” in Britain stressed holistic effects to justify their conviction that only the intuition of the engineer could cope with the problems of airplane design. Scientists and mathematicians, they said, simplified problems by studying one part at a time, which doomed them to failure. Such a procedure ignored the all-important effects of inter­action. Perhaps (had they known about it) the “practical men” would have been impressed to be told of the progress that was being made in Gottingen. Here engineers, such as Betz, were using the Biot-Savart law to put the study of interaction on a mathematical as well as an experimental basis.

Betz carried out wind-channel measurements of the lift and resistance of a set of two wings rigidly fastened into a biplane configuration. He studied (a) the effect of varying the distance apart of the wings, (b) the effect of giv­ing the wings different angles of incidence from one another (decalage), and (c) the effect of placing one wing ahead of the other (stagger). He found that the effects were small within the range he studied, though the most signifi­cant variable was the stagger of the wings. One of his practical concerns was to form some idea of the relative merits of monoplanes and biplanes. He summed up his results in four propositions: (1) A biplane arrangement with wings of equal span always has a less favorable ratio of lift to resistance than one of the wings taken separately. (2) A biplane can have advantages over a monoplane when the rest of the resistance of the aircraft, for example, a bulky fuselage, is taken into account. (3) A biplane is at an advantage if a high lift at low speeds is required. (4) The greatest maximum lift is obtained when

the upper wing of a biplane is placed ahead of the lower wing and is given a slightly smaller angle of incidence than the lower wing. All of these results, said Betz, were rendered intelligible by Prandtl’s theory, and the empirical graphs of lift and resistance were duly accompanied by theoretical curves calculated from the theory.64

In neither of his papers did Betz specifically mention, or illustrate the use of, the Biot-Savart law. He alluded to the horseshoe model but revealed none of the mathematics involved in his calculations. Like Foppl he prom­ised the reader that a fuller account was to follow from the pen of Prandtl himself. The Great War began in July 1914, but there seemed no immediate concern with secrecy. In a paper that appeared in August 1914, Wieselsberger preempted Prandtl and stated the Biot-Savart law explicitly and illustrated its application.65 He asked why birds often fly in a V formation. He did not man­age to answer the question, but he did succeed in laying out the basic ideas, and the basic mathematics, of Prandtl’s theory. In approaching the problem of formation flying, Wieselsberger ignored the beating wing motion involved in bird flight and treated birds as small airplanes. He then followed Prandtl and treated the airplane as a horseshoe vortex. By the use of the Biot-Savart law he showed that on either side of the horseshoe vortex there would be an updraft. This, he argued, allowed another wing, positioned to one side of the first wing, to operate at a more favorable angle of attack. This lowered the component of induced resistance in the direction of flight. On the basis of some plausible numerical assumptions, he made a quantitative estimate of the advantages to be derived from flying in the updraft of neighboring birds. His overall model, however, led to the conclusion that side-by-side flight would be just as efficient as the V formation.

In September 1914 Betz produced a study of wings with a sweepback and a twist at their ends,66 a configuration frequently used by designers of German aircraft at that time. The name Taube, or “dove,” was given to such machines. In Betz’s paper there was a passing reference to yet another formula attrib­uted to Prandtl and his new theory, though again no derivation was given. The formula concerned the minimum glide-angle that could be expected for a wing of given span and lift. The main result of Betz’s experiments on a range of Taube-style wings was to confirm the near optimum character of very sim­ple, rectangular wings. Having neither twist nor sweepback, such wings also had an economic and practical advantage: they were easy to construct. The glide coefficient (given by the ratio of resistance over lift) was not signifi­cantly improved by sweepback or twist, though Betz did find they improved longitudinal stability.

Perhaps because the promised theoretical paper from Prandtl was not forthcoming, Betz finally published his own account of the mathematics underlying his papers. Titled “Die gegenseitige Beeinflussung zweier Trag – flachen” (The mutual influence of two wings),67 the work appeared in the Zeitschrift fur Flugtechnik for October 1914. Betz concentrated on the case of the staggered biplane with wings of equal span where the upper wing was positioned ahead of the lower wing. Because the analysis proceeded on the assumption that each wing and vortex system could be represented by the simple “horseshoe” schema, the only real novelty in the paper lay in the more complex geometry of the computations, but the explicit development of the mathematics of the theory demonstrated its applicability to what was then a vitally important form of aircraft. It was clear that Prandtl and his colleagues now had a theory that could be used to predict the induced resistance of bi­planes, or triplanes, using only the wind-channel data for a single wing.

‘In the same year, 1914, Wieselsberger also published a survey article that described the state of knowledge in German aerodynamics with respect to lift and drag. It did not appear in the ZFM but in an Austrian journal, the Osterreichische Flug-zeitschrift.68 The article covered both two-dimensional and three-dimensional theory and took the reader through the work of Kutta, Joukowsky, Deimler, and Blumenthal and up to Prandtl’s horseshoe vortex. Wieselsberger’s survey effectively brought up to date an earlier survey by Reissner, of the TH in Aachen, which had laid stress on questions of stabil­ity and propeller theory.69

The international situation had been deteriorating throughout 1914, and British statesmen, such as Lord Haldane, became increasingly worried about the “war party” surrounding the German kaiser.70 With the threat of war, it was ever more important for European countries to monitor the technol­ogy of their potential enemies. If anyone had wanted to keep an eye on Ger­man aviation, the papers of Foppl, Betz, and Wieselsberger would have given them all they needed to know about the general state of scientific knowl­edge in the field of aerodynamics. These publications would have made clear that the circulation theory of lift was wholly taken for granted in Gottingen and the German-speaking world. Collectively, the publications showed that the theory had been developed to the point where it was being applied to problems of practical importance. Betz’s theoretical analysis of the biplane, however, was the last of the Gottingen research papers to appear in an open and accessible format. Thereafter they would be hidden away from public view in the Technische Berichte, published in individually numbered copies by the military authorities and marked Geheim—“secret.” In the meantime, the Gottingen results were in the public realm and were available to anyone in Cambridge or London who cared to study them.

Making the Horseshoe Model More Realistic

Prandtl never produced the promised article in the Zeitschrift fur Flugtech – nik. This was not because he harbored reservations about the approach. On the contrary, he was happy to produce accounts for general surveys, for ex­ample, in volume 4 of the Handworterbuch der Naturwissenschaften published in 1913. The handbook was an encyclopedic survey of the state of the natural sciences and contained articles by both Fuhrmann and Prandtl. Fuhrmann wrote on hydrostatics, and Prandtl wrote on fluid dynamics.71 In his contri­bution Prandtl gave an explicit account of the circulation theory and pre­sented a graph contrasting Kutta flow with Kirchhoff-Rayleigh flow (136). He also cited Lanchester’s work and gave a diagram (112) that laid out the qualita­tive basis of the horseshoe model, though the Biot-Savart law was not men­tioned by name. Why the hesitation? The simple horseshoe model was clearly in a provisional state and was still undergoing revision. It contained formal features that compromised both its empirical adequacy and its practical util­ity. Despite the successes of the theory, it would have been understandable if Prandtl had wanted to remove these limitations before presenting the ap­proach to a specialist readership. The time was hardly ripe for an authorita­tive presentation, which may explain the non-appearance of the article. Then the war intervened, and the form and level of presentation at which he seems to have been aiming were not achieved until 1918.

The problems with the “horseshoe” vortex were both mathematical and physical and were closely interconnected. Mathematically there was the dif­ficulty arising from the singularity in the Biot-Savart formula which has al­ready been remarked on, that is, the problem that arises when h = 0. The formula implied that the velocity of the downwash at the wingtips became infinite. The formula yields this result because of the uniformity of the vortex distribution implied by the model, that is, the constant value of the circula­tion along the bound vortex and hence along the span of the wing. This was a physically false picture. The existence of lift implies that there must be a greater pressure beneath the wing than above it, but the finite length of a real wing allows the air at high pressure beneath the wing to move round the tip to occupy the lower-pressure region above the wing. Such freedom of movement ensures that the pressure difference between the upper and lower surface will be zero at the tips. There will therefore be no lift at the tips and hence no circulation. Circulation cannot be constant along the span in the way that was assumed in the simple horseshoe model; it must fade away to zero at the tips.

Prandtl’s problem was to find a model with a more realistic lift distribu­tion along the span of the wing. His response was ingenious. He complicated the simple horseshoe model by introducing a number of horseshoe vortices laid out in the fashion indicated in figure 7.11. (A similar figure was used in an early article by Betz.)72 Starting from a single “horseshoe” whose span co­incided with the full span of the wing, he added others of smaller span. The parts of the vortex that lie along the span are to be thought of as piled on top of one another. In this way the constant distribution of circulation along the span is replaced by a variable, stepwise distribution with a maximum at the midpoint. The arrangement had the consequence that vortices now trailed from a number of points along the rear edge of the wing, rather than merely at the wingtips. This stepwise model, however, was only the starting point of Prandtl’s line of reasoning.

Prandtl did not simply introduce a number of horseshoe vortices such as the five in the diagram, or even 50 or 500. He introduced an infinite num­ber. He postulated an infinite number of vortices of infinitesimal strength. The vortices were infinitesimal for two reasons. First, an infinite number of vortices of finite strength would result in the absurdity of a wing with infinite circulation and infinite lift. Second, he needed the circulation and the lift at the tips to approach zero. A stepwise model with finite vortices would merely reproduce the problem that dogged the original. The vortices had to become infinitely small at the wingtips. Along the span of the wing the infinitesimal vortices were assumed to be compressed into a single line of bound vortic – ity (of varying strength) called the lifting line. These refinements made it possible to imagine a smooth, rather than stepwise, lift distribution that was amenable to mathematical treatment. To accord with the known facts, the

From Ground Effect to Biplanes

figure 7.11. Stepwise complication of the simple horseshoe model. Prandtl made the horseshoe model more realistic by multiplying the number of horseshoe vortices and imagining them stacked on top of one

another. From Tietjens 1931, 209. (By permission of Springer Science and Business Media)

smooth lift distribution had to have a maximum lift at the midpoint of the span and approach zero lift at the wingtips.

Having described Prandtl’s refined model in qualitative terms, I now show how he expressed these ideas in mathematical terms. This account will pre­pare the ground for the next two sections, which describe the technical and mathematical heart of the Gottingen achievement.

Suppose the wing has a span b and lies along the x-axis of a coordinate system so that it runs from x = —b/2 to x = +b/2. The distribution of the circu­lation can then be represented by Г(х). The symbol indicates that, for every value of x along the axis between the wingtips, there corresponds a specific value of Г, the circulation. Thus Г(о) is the value at x = 0, the origin, which, following convention, is taken as the center of the wingspan. It is known from experiment that the lift is at its maximum value at this central position. Because it plays an important role, it is customary to give the circulation at this point a special designation and write Г(о) = Г0. The lift and hence the circulation is zero at the tips, so that Г(—b/2) = 0 and Г(+Ь/2) = о. For the moment, and for the purpose of conveying the main outlines of Prandtl’s theory, the actual shape of the lift distribution need not be given in more detail than this. The mathematical shape described by the function Г(х) will, for the moment, remain unspecified, but it will be some smoothed-out ver­sion of the shape made by the stepwise lift distribution. The details are re­served for the next section. For the remainder of this section, the distribution is simply referred to as Г(х) so that the general structure of the mathematical reasoning can be rehearsed. My aim is to show, in general terms, how Prandtl used the Biot-Savart law to calculate the lift, the induced velocity, and the induced drag.

The first step was to relate each of the infinitesimal horseshoe vortices to the Biot-Savart law. The relevant version of the formula for a vortex of finite strength has already been stated, namely, w = —r/(4nh). Because the analysis was now to be applied to infinitesimal vortices, the formula became dw = —dr/(4nh). The goal was to calculate the downwash at some specified point on the wing with the coordinate, say, x = x’. All of the infinite number of trailing vortices (each coming away from the wing at some point with its own specific x-coordinate) will contribute to the downwash at the point x’. The Biot-Savart law gave the (infinitesimal) contribution dw made by each of these infinitesimal vortices. The perpendicular distance h in the formula needed to be re-expressed as (x’ — x). This was the distance between the point on the wing from which the infinitesimal vortex emerges and the point x’ at which the downwash was to be found. A process of integration that adds the contribution of all the infinitesimal trailing vortices would then give the total

downwash at X. A further calculation, and a further integration, was needed to get the downwash for the entire wing, that is, for all the points like x’ which lie along the span between x = – b/2 and x = +b/2.

The procedure that has just been sketched was based on the assumption that the quantity dr used in the Biot-Savart formula corresponded to the strength of the infinitesimal vortex at the arbitrary point x. How was this infinitesimal strength to be expressed? The answer was that the strength of the element of trailing vorticity issuing from a point x was equal to the change of vorticity on the wing at that point. This can be explained by going back to the stepwise model of a finite number of finite vortices that was shown in figure 7.11. First the outer horseshoe is put in place. Suppose this has strength Tj. Then the second horseshoe is added, which has strength Г2 and a slightly shorter span, then Г3 is added, which again has a slightly shorter span, and so on. Consider the two points on either side of the origin of the x-axis from which the trailing vortices of strength Г2 emerge. These are the points at which the distribution of circulation changes by an increase of the amount Г2. Thus the strength of vorticity trailing from the wing at that point equals the change in vorticity around the wing at that point.

This “strength equals change” rule holds even when there are an infinite number of infinitesimal horseshoe vortices. The distribution of circulation along the span is given by the curve r(x), so the change in circulation is the slope of the graph of r(x) multiplied by the distance over which the slope reaches. The slope is дГ/dx, and the distance is dx, so the change whose value is sought is dr = (дГ/dx) dx. This expression gave the strength of the cir­culation or vorticity to be entered into the formula for the Biot-Savart law. The infinitesimal contribution of the vorticity at x to the downwash at x’ was therefore

Подпись:(ЭГ / dx ^jdx 4n(x’ — x)

The total downwash at the point x’, designated by w(x’), is the integral of all of these infinitesimal contributions, summed over all the vortices issuing from the whole span of the wing. Thus,

Подпись:/X 1 +if2(dr / dx )dx w(x ) = -— I Л-Ж J

The above integral has a singularity at x = x’, when the denominator becomes zero, but the integration could be carried out in such a way as to avoid this problematic point.

Given the downwash it was then possible to calculate the induced angle of incidence at X. This angle, ф, follows from the value for w(x’) because it was simply the angle made by combining the downward induced velocity with the free-stream velocity. The ratio of the two speeds gave the tangent of the angle ф, but because the angle was small, the angle and tangent could be equated. The induced angle of incidence was

Подпись:w(x’) V ‘

The lift distribution could now be related to the overall lift and induced drag. Recall that for an infinite wing the flow at every cross section resembles that at every other cross section. The lift per unit length is constant and is given by the Kutta-Joukowsky formula as L = рГV. Prandtl took this formula to apply to each separate, infinitesimal element of a three-dimensional wing, with the proviso that the circulation would vary from element to element according to the distribution Г(х). The overall lift could then be represented by the integral of all the elementary lifts: dL(x) = р V r(x)dx. Thus,


Lift = pV J Г(х)dx.


Each point on the wing would generate an element of downward velocity and would thus be subject to a slight downward slope in the local flow. The ele­ment of lift dL(x) at that point would be tilted backward (relative to the main flow) so that the resultant force possesses a component opposing the motion. This was the induced drag. The induced drag at a given point x depended on the induced angle of incidence ф at that point. The component of induced drag resulting from the backward tilt equals dL(x) sinф(x). For small angles the sine of ф is equal to ф itself, so the element of induced drag was dL(x) ф^). Thus the total induced drag was given by the integral

+ b/2

Drag = pV J Г(х)p(x)dx.

— b/2

This relation could be expressed in terms of a coefficient of induced drag by dividing the value of the drag force itself by Уг р V2F, where F is the area of the wing. This gave the coefficient of induced drag as

2 +b/2

CD = — J T(x)<p(x)dx.


It will be evident from these formulas that a closely knit structure of theoretical relations was emerging in Gottingen which connected lift, drag, span, and the distribution of circulation along the span of a wing. For the purposes of exposition I have only presented this structure in a schematic form. The mathematical formulas just given all depend on the distribution of the circulation, Г(х), but the actual character of the function governing the distribution has remained unspecified. All that the above formulas entail is that if the distribution Г(х) is given, then the lift, the induced angle of incidence, and the induced drag can be calculated. Only when the distribu­tion is specified will the theory will have real content. The next question is: How was the distribution of lift and circulation found? How is Г(х) to be defined?

The Laws of Prandtl and the Laws of Nature

Prandtl was not vastly outstanding in any one field, but he was eminent in so many fields. He understood mathematics better than many mathematicians do.

max munk, “My Early Aerodynamic Research” (1981)1

After Glauert and McKinnon Wood had presented the reports on their Got­tingen visit, discussions continued in the Aeronautical Research Committee as the British experts sought to mobilize a collective response to the Ger­man wartime achievements. These (sometimes sharp) exchanges took place in the monthly meetings of the committee and its subcommittees that were held in London. The Cambridge contingent made the journey to London together by train and engaged in lively aeronautical debate en route. “I fear we must have been a pest to our fellow travellers,” recalled one.2 The upshot of the committee meetings are recorded not only in the minutes of their dis­cussions but also in the confidential technical reports circulated among the participants. The content of the technical reports sometimes surfaced in the published Reports and Memoranda issued by the committee and sometimes, in the case of especially important results, in leading scientific journals. A number of the main experiments done in this period appeared in the Philo­sophical Transactions of the Royal Society and in the Proceedings of the Royal Society. There were some significant and perplexing changes in the analysis of the experimental material as the data made the journey from the private to the public realm.

I have described how Taylor, in his 1914 Adams Prize essay, had dismissed Lanchester’s idea that the flow of air over a wing was describable in terms of a perfect fluid in irrotational motion with circulation. If Prandtl was right, then Taylor had been wrong. Led by Glauert, the postwar argument in the Aero­nautical Research Committee seemed to be going in Prandtl’s direction. The circulation theory was gaining ground. By 1923 Glauert felt able to write to Prandtl to tell him that his “aerofoil theory has certainly aroused much inter­est here and it would not be an exaggeration to say that it has revolutionised

many of our ideas.”3 But Taylor (see fig. 9.1) was not to be easily convinced that his earlier reservations had been misplaced. In the postwar discussions, he made it his job to scrutinize Glauert’s reasoning and to oppose it whenever he detected a logical gap or a questionable premise.

The General Argument

In paragraphs 185-99 of Philosophical Investigations, Wittgenstein imagined a pupil who is being taught to follow a simple rule, namely, the rule of “add 2.” The pupil must try to follow the rule and generate the rule-bound sequence of 2, 4, 6, 8, etc., by adding 2 to the previous member of the sequence. The pupil is taught by a familiar mixture of examples and explanations and is then encouraged to go on to produce further numbers in the sequence. Witt­genstein then imagined the pupil deviating from the expectations of the teacher and the other competent rule followers in the surrounding culture. On reaching 2,000, the pupil does not say 2,002, but 2,004, 2,008, etc. Witt­genstein studied the likely reactions of the competent rule followers and the resources at their disposal as they tried to get the deviant pupil to understand what the rule means and what the rule requires. It rapidly emerges from the analysis that just as the original rule might be misunderstood, so too might any further explanation. Such explanations also depend on a small number of illustrative examples that the pupil must use as a pattern to generate the new instances that are required. Thus, the injunction to go on in the same way merely pushes the problem back to that of defining “same.” Any attempt to furnish an algorithm for producing the next member of the sequence merely provides rules for following rules and the original problem of furnishing an adequate analysis repeats itself.10

Ultimately any attempt to furnish reasons, or to ensure that the pupil’s behavior is guided by what the rule requires, depends on the pupil react­ing normally and automatically to the training provided by those already ac­knowledged as competent. There is nothing else available, and all appeals to “reason,” “logic,” “meaning,” “implication” (as well as the concepts of “must” and “have to”) come down, in the end, to this. The basis of the pro­cess is participation in a shared practice. This applies not just to the pupil now learning the rule but to all who have ever learned it and all who now teach others and identify and correct their errors. The meaning of the rule and the compulsion of the rule depend on shared dispositions to react and shared dispositions to interact with one another in ways that ensure that in­dividual responses stay accountable and aligned. In other words, rules are conventions. To obey a rule, said Wittgenstein, is to follow a “custom” and to participate in an “institution.”

Wittgenstein did not mean this statement as a criticism of rule follow­ing or the practice of citing rules. The claim is not that rules are unreal or that rule following is a sham. He was not saying that rule followers have no reasons for what they do or, in reality, are acting for other reasons. His point was that the institution of rule following is a social reality, and his aim was to expose that reality to view. The institution is vital for, and constitutive of, cognitive and social order. Citing the rule as an explanation of why pupils act as they do, or why they should act thus and so, is the currency with which one rational, social agent interacts with another rational, social agent. The rules and reasons are actor’s categories. Invoking rules is not an idiom for describ­ing the interaction in causal terms; it is a means of acting causally within the situation. What Wittgenstein’s argument shows, and was meant to show, was that the actor’s account will not suffice if it is taken out of context and treated as if it were self-sufficient, that is, as an analyst’s account. The analyst needs to understand an interaction in terms of causes. This is why Wittgenstein ad­opted the sociological perspective. In his example he focused attention on the process of socialization. More generally he adopted the explanatory stance by invoking the concepts of convention, custom, and institution.11

Wittgenstein’s genius lay in identifying a simple example that epitomizes all the central points concerning the relation of reasons and causes. Rule following is the perfect example of rational compulsion, and Wittgenstein’s analysis can be related directly to the questions posed by critics of the Strong Program. Here, if anywhere, the case could be made for the operation of ra­tional rather than social causes. Doesn’t the rule provide a sufficient reason for the behavior of a competent rule follower? Can’t the rule better explain the behavior of the rule follower than any social causes can? It seems to give the critics everything they want. But, as Wittgenstein’s analysis shows, the critic’s case collapses into a regress of rules for following rules. For the purposes of analysis and explanation, the regress must be stopped and so the phenom­enon must be seen as a combination of psychological and sociological causes. The stated reasons and the appeal to reasons prove insufficient to explain what happens, and, for the scientific analyst, the actor’s account needs to be reconfigured in terms that acknowledge an all-pervasive and self-sufficient sociopsychological causation.

Despite the central role played by Wittgenstein’s argument in setting up the Strong Program, its critics insist on misreading the sociological approach as the claim that reason giving is spurious. Such a misreading is question begging. It amounts to making the assumption from the outset that “real” reasons and “good” reasons are not to be analyzed in terms of social causes. The categories of the “rational” and the “social” are set in opposition to one another. The critics cannot believe that rationality can be a sociological phe­nomenon in anything other than a trivial sense. They therefore counterpose the rational and the social and read the symmetry requirement of the Strong Program as an a priori exclusion of the power and presence of real reasons. From the critic’s standpoint it then seems easy to refute the sociological ap­proach. All that is needed is an example of an action authentically based on good reasons, for example, a case of rule following or a well-founded sci­entific inference. This, critics assume, refutes the preposterous generaliza­tion put forward by sociologists of knowledge. That the sociologist, following Wittgenstein, is actually challenging the dualistic assumptions upon which the critic’s argument rests is never considered.12

Although Wittgenstein’s example deals with a particular case, the lesson that can be drawn is wholly general. The lesson is that the familiar distinc­tion between “cognitive factors” and “social factors” is wrong. The injunc­tion to “disentangle” these two things is incoherent. The social factor cannot be considered “external” to the cognitive process; it is constitutive, and you cannot “disentangle” that which is constitutive. The cognitive factor in Witt­genstein’s example is, of course, the rule itself and the orientation to the rule. But what, when properly analyzed, is “the rule”? Is it something that mys­teriously exists “in advance” of the acts of following, like a rail stretching to infinity? No, said Wittgenstein, that is just a mythical picture. We must think in a different way. A rule is something that exists solely in virtue of the social practice of following the rule (just as, in economics, a currency exists solely in virtue of the practice of using the currency). The meaning and implica­tions of the rule only exist through being invoked by the actors to correct, challenge, justify, and explain the rule to one another in the course of their interactions. This is what Wittgenstein meant by calling a rule an “institu­tion.” The implication (though these are not Wittgenstein’s words) is that the rule, that is, the cognitive factor, is actually itself a social factor. Those who appeal to a combination of cognitive factors and social factors, as if they are two, qualitatively different kinds of things, are not being prudent; they are being muddled or metaphysical.13

The processes that Wittgenstein brilliantly distilled into his example are the same ones that occurred on a larger scale in my case study. That which is recognizably social, for example, the disciplinary identities, the institutional locations, the cultural traditions, the schools of thought, are not “external” to the reasoning processes that I have studied but are integral to them. They are constitutive of the step-by-step judgments by which the different bodies of knowledge were built up. Experts gave reasons to explain and justify their views and found that sometimes they were accepted and sometimes rejected. Facts and reasons that inclined the members of one group to orient in one direction inclined the members of the other group to orient in a different direction. As one would expect from Wittgenstein’s example, these accep­tances, rejections, indications, and orientations fell into patterns. The pat­terns form the customs, conventions, institutions and subcultures described in my story.14

Introduction: The Question to Be Answered

’Tis evident, that all the sciences have a relation, greater or less, to human nature; and that however wide any of them may seem to run from it, they still return back by one passage or another.

david hume, A Treatise of Human Nature (1739-40)1

Why do aircraft fly? How do the wings support the weight of the machine and its occupants? Even the most jaded passengers in the overcrowded airliners of the present day may experience some moments of wonder—or doubt—as the machine that is to transport them lifts itself off the runway. Because the action of the air on the wing cannot be seen, it is not easy to form an idea of what is happening. Some physical processes are at work that must generate powerful forces, but the nature of these processes, and the laws they obey, are not open to casual inspection. If the passengers looking out of the window really want an explanation of how a wing works, they must do what any lay person has to do and ask the experts. Unfortunately the answers that the ex­perts will give are likely to be highly technical. It will take patience by both parties if communication is not to break down. But given goodwill on both sides, the experts should be able to find some simplified formulations that will be useful to the nonexperts, and the nonexperts should be able to deepen their grasp of the problem.

In this book I discuss the question of why airplanes fly, but I approach the problem in a slightly unusual way. I describe the history behind the technical answer to the question about the cause of “lift,” that is, the lifting force on the wing. I analyze the path by which the experts, after much disagreement, ar­rived at the account they would now give. I am therefore not simply asserting that airplanes fly for this or that reason; I am asserting that they were under­stood to fly for this or that reason. I am interested in the fact that different and rival understandings were developed by different persons and in different places. I cannot speak as a professional in the field of aerodynamics; nor is my position exactly that of a layperson. I speak as a historian and sociologist of science who is poised between these categories.2

What are the specific questions that I am addressing and to which I hope to offer convincing answers? To identify them I first need to give some back­ground. The practical problem of building machines that can be flown, that is, the problem of “mechanical” or “artificial” flight, was solved in the final years of the nineteenth century and the early years of the twentieth century. In the 1890s Otto Lilienthal in Germany successfully built and flew what we today would call hang gliders. From 1903 to 1905 the Wright brothers in the United States showed that sustained and controlled powered flight was pos­sible and practical. What had long been called the “secret” of flight was now no longer a secret. But not all of the secret was revealed. Some parts of it remained hidden, and indeed, some parts are still hidden today. The practi­cal successes of the pioneer aviators still left unanswered the question of how a wing generated the lift forces that were necessary for flight. The pioneers mostly worked by trial and error. Some had experimented with models and taken measurements of lift and drag (the air resistance opposing the motion), but the measurements were sparse and unreliable.3 No deeper theoretical un­derstanding had prompted or significantly informed the early successes of the pioneers, nor had theory kept pace with the growth of practical under­standing. The action of the air on the wing remained an enigma.

A division of labor quickly established itself. Practical constructors con­tinued with their trial-and-error methods, while scientists and engineers be­gan to study the nature of the airflow and the relation between the flow and the forces that it would generate. For this purpose the scientists and engineers did not just perform experiments and build the requisite pieces of apparatus, such as wind channels. They also exploited the resources of a branch of ap­plied mathematics that was usually called hydrodynamics. The name “hydro­dynamics” makes it sound as if the theory was confined to the flow of water, but in reality it was a mathematical description that, with varying degrees of approximation, was applied to “fluids” in general, including air. Thus was born the new science of aerodynamics. The birth was accompanied by much travail. One problem was that the mathematical theory of fluid flow was im­mensely difficult. The need to work with this theory effectively excluded the participation of all but the most mathematically sophisticated persons, and this did not go down well with the practical constructors. The mathematical analysis also depended for its starting point on a range of assumptions and hypotheses, about both the nature of the air and the more or less invisible pattern of the flow of air over, under, and around the wing. Only when the flow was known and specified could the forces on the wing be calculated. Assumptions had to be made. The unavoidable need to base their investiga­tions on a set of assumptions proved to be deeply divisive. Different groups of experts adopted different assumptions and, for reasons I explain, stuck to them.

The first part of this historical story, the practical achievement of con­trolled flight, has been extensively discussed by historians. Pioneers, such as the Wright brothers, have been well served, and the attention given to them is both proper and understandable.4 The second part of the history, the de­velopment of the science of aerodynamics, is somewhat less developed as a historical theme, though a number of outstanding works have been written and published on the subject in recent years.5 The present book is a contribu­tion to this developing field in the history of science and technology.

In the early years of aviation there were two, rival theories that were in­tended to explain the origin and nature of the lift of a wing. They may be called, respectively, the discontinuity theory and the circulatory (or vortex) theory. The names derive from the particular character of the postulated flow of air around the wing. (I should mention that the circulatory theory is, in effect, the one that is accepted today.) My aim is to give a detailed account of how the advocates of the two theories developed their ideas and how they oriented themselves to, and engaged with, the empirical facts about flight. To do this I found that I also needed to understand how they oriented them­selves to, and engaged with, one another. I show that these two dimensions cannot be kept separate. This is why I have prefaced the work with the quo­tation from the famous Edinburgh historian and sociologist David Hume. The more one studies the technical details of the scientific work, the more evident it becomes that the social dimension of the activity is deeply impli­cated in these details. The more closely one analyses the technical reasoning, the more evident it becomes that the force of reason is a social force. The historical story that I have to tell about the emerging understanding of lift is, therefore, at one and the same time both a scientific and a sociological story. To understand the course taken by the science it is necessary to understand the role played by the social context, and to appreciate the role played by the social context it is necessary to deconstruct the technical and mathematical arguments.

In principle none of this should occasion surprise. Scientists and engi­neers do not operate as independent agents but as members of a group. They cannot achieve their status as scientists and engineers without being educated, and education is the transmission of a body of culture through the exercise of authority. Education is socialization.6 Scientists and engineers see them­selves as contributing to a certain discipline, as being members of certain institutions, as having loyalties to this laboratory or that tradition, as being students of A or rivals of B. Their activities would be impossible unless behav­ior were coordinated and concerted. For this the individuals concerned must be responsive to one another and in constant interaction. Their knowledge is necessarily shared knowledge, though, in its overall effects, the process of sharing can be divisive as well as unifying. The sharing is always what Hume would call a “confined” sharing.

All too frequently, when scientific and technical achievements become objects of commentary, analysis, or celebration, these simple truths are ob­scured. Academic culture is saturated with individualistic prejudices, which encourage us to trivialize the implications of the truth that science is a col­lective enterprise and that knowledge is a collective accomplishment. Phi­losophers of science actively encourage historians to distinguish between, on the one side, “cognitive,” “epistemic,” or “rational” factors and, on the other side, “social” factors. They enjoin the sociologist to “disentangle” scientific reasoning from “social influences” and to distinguish what is truly “internal” to science from what is truly “external.”7 These recommendations are treated as if they were preconditions of mental hygiene and based on self-evident truths. Historians and sociologists of science know better. They know that the problem of cognitive order is the problem of social order.8 These are not two things, even two things that are closely connected; they are one thing described from different points of view. The division of a historical narrative into “the cognitive” and “the social,” or “the rational” and “the social,” is wholly artificial. It is methodologically lazy and epistemologically naive.

I shall now briefly sketch the overall structure of the events I describe in this volume. Of the two theories of lift that I mentioned, one of them, the dis­continuity theory, was mainly developed in Britain. It was based on work by the eminent mathematical physicist Lord Rayleigh. The other, the circulatory theory, was mainly developed in Germany. It is associated primarily with the German engineer Ludwig Prandtl, although it had originally been proposed by the English engineer Frederick Lanchester. It rapidly became clear that the discontinuity theory was badly flawed because it only predicted about half of the observed amount of lift. At this point, shortly before the outbreak of World War I (or what the British call the Great War) in 1914, the British awareness of failure might have reasonably led them to turn their attention to the other theory, the theory of circulation. They did not do this. They knew about the theory but they dismissed it. At Cambridge, G. I. Taylor, for example, treated the discontinuity theory as a mathematical curiosity, but he also found Lanchester’s theory of circulation equally unacceptable. The reasons he gave to support this judgment were important and widely shared. Meanwhile the Germans embraced the idea of circulation and developed it in mathematical detail. The British also knew of this German reaction but still did not take the theory of circulation seriously. It was not until after the war ended in 1918 that the British began to take note. They found that the Germans had developed a mathematically expressed, empirically supported, and practically useful account of lift. Even then the British had serious res­ervations. The negative response had nothing to do with mere anti-German feeling. The British scientific experts were patriots, but, unlike some in the world of aviation, they were not bigots. Why then were they so reluctant to take the theory of circulation seriously? This is the main question addressed in the book.9

There are already candidate answers to this question in the literature, but they are answers of a different kind to the one I offer. The neglect of Lan – chester’s work became something of a scandal in the 1920s and 1930s, so it was natural that explanations and justifications were manufactured to account for it. Sir Richard Glazebrook, the head of the National Physical Laboratory, played an important role in British aviation during these years and was the source of one of the standard excuses, namely, that Lanchester did not pres­ent his ideas with sufficient mathematical clarity. Well into the midcentury, British experts in aerodynamics, who, along with Glazebrook, shared respon­sibility for the neglect of Lanchester’s ideas, were scratching their heads and wondering how they could have allowed themselves to get into this position. Clarity or no clarity, they had turned their backs on the right theory of lift and had become bogged down with the wrong one.

The retrospective accounts and excuses that have been given have been both fragmentary and feeble, though Lanchester’s biographer, P. W. Kings – ford, writing in 1960, still went along with a version of Glazebrook’s excuse.10 Other existing accounts merely tend to embellish the basic excuse by invok­ing the personal idiosyncrasies of the leading actors. The problem is ana­lyzed as a clash of personalities. It is true that some of those involved had strong characters as well as powerful intellects, and some of them could pass as colorful personalities. All this will become apparent in what follows. The psychology of those involved is clearly an integral part of the historical story, but such accounts miss the very thing that I want to emphasize and that I believe is essential for a proper analysis, namely, the interconnection of the sociological and technical dimensions. Only an account that is technically informed, and sensitive to the social processes built into the technical content of the aerodynamic work, will make sense of the history. I want to show that the real reasons for the resistance to the vortex or circulatory theory of lift were deep and interesting, but not really embarrassing at all.

Although I have posed the question of why the British resisted the the­ory of circulation, I do not believe it can be answered in isolation from the question of why the Germans embraced it. Both reactions should be seen as equally problematic. The historical record shows that the same type of causes were at work in both British and German aerodynamics. In both cases the ac­tors drew on the resources of their local culture and elaborated them in ways that were typical of their milieu and were encouraged by the institutions of which they were active members. Of course, the cultures and the institutions were subtly different. My explanation of the German behavior is thus of the same kind as my explanation of the British. The same variables are involved, but the variables have different values. Seen in this way the explanation pos­sesses a methodological characteristic that has been dubbed “symmetry.” Be­cause the point continues to be misunderstood, I should perhaps emphasize the words “same kind.” I am not saying that the very same causes were at work but that the same kinds of cause were in operation. Symmetry, in this sense, is now widely (though not universally) accepted as a methodologi­cal virtue in much historical and sociological work. Conversely, it is widely rejected as an error, or treated as a triviality, by philosophers. I hope that see­ing the symmetry principle in operation will help convey its meaning more effectively than merely trying to capture it in verbal formulas or justify it by abstract argument.

The overall plan of the book is as follows. In chapter 1 I start my account of the early British work in aerodynamics with the foundation of the con­troversial Advisory Committee for Aeronautics in 1909. The committee was presided over by Rayleigh. The frontispiece, taken from the Daily Graphic of May 13, 1909, shows some of the leading members of the committee striding purposefully into the War Office for their first meeting, and then emerging afterward looking somewhat more relaxed. The minutes of that important meeting are in the Public Record Office and reveal what they talked about in the interval between those two pictures.11 It is a matter of central concern throughout this book. Chapter 2 lays the foundation for understanding the two competing theories of lift by sketching the basic ideas of hydrodynam­ics and the idealized, mathematical apparatus that was used to describe the flow of air. A nontechnical summary is provided at the end of the chapter. In chapter 3, I introduce the discontinuity theory of lift and describe the British research program on lift and the frustrations that were encountered. Chap­ter 4 is devoted to the circulatory or vortex theory and describes the hostile reception accorded to Lanchester among British experts. I pay particular at­tention to the reasons that were advanced to justify the rejection. In chapter 5, I identify and contrast two different intellectual traditions that were brought to bear on the theory of lift. One of them was grounded in the mathematical physics cultivated in Britain and preeminently represented by the graduates of the Cambridge Mathematical Tripos. The other tradition, called technische Mechanik, or “technical mechanics,” was developed in the German technical colleges and was integral to Prandtl’s work on wing theory. Chapters 6 and 7 provide an account of the German development and extension of the circu­lation theory as worked out in Munich, Gottingen, Berlin, and Aachen. In chapters 8 and 9 there is a description of the British postwar response, which took the form of a period of intense experimentation; it also gave rise to some remarkable and revealing theoretical confrontations. What, exactly, did the experiments prove? The British did not find it easy to agree on the answer.

The divergence between British and German approaches was effectively ended in 1926 with the publication, by Cambridge University Press, of a text­book that became a classic statement of the circulation theory. The book was Hermann Glauert’s The Elements of Aerofoil and Airscrew Theory.12 Glauert, an Englishman of German extraction, was a brilliant Cambridge mathemati­cian who, in the 1920s, broke ranks and became a determined advocate of the circulation theory. As the title of Glauert’s book indicates, he did not just work on the theory of the aircraft wing, but he also addressed the theory of the propeller. This is a natural generalization. The cross section of a propel­ler has the form of an aerofoil, and a propeller can be thought of as a rapidly rotating wing. The “lift” of this “wing” becomes the thrust of the propeller, which overcomes the air resistance, or “drag,” as the aircraft moves through the air. Glauert’s book also dealt with the theory of the flow of air in the wind channel itself, that is, the device used to test both wings and propellers. This aspect of the overall theory was needed to ensure that aerodynamic experi­ments and tests were correctly interpreted. As always in science, experiments are made to test theories, but theories are needed to understand the experi – ments.13 The discussions of propellers and wind channels in Glauert’s book are important and deserve further historical study, but, on grounds of prac­ticality, I set aside both the aerodynamics of the propeller and the methodol­ogy of wind-channel tests in order to concentrate exclusively on the story of the wing itself.14

In the final chapter, chapter 10, I survey the course of the argument and consider objections to my analysis, particularly those that are bound to arise from its sociological character. I use the case study to challenge some of the negative and inaccurate stereotypes that still surround the sociology of scien­tific and technological knowledge. I also ask what lessons can be drawn from this episode in the history of aerodynamics. Does it carry a pessimistic mes­sage about British academic traditions and elitism? What does it tell us about the difference between Gottingen and Cambridge or between engineers and physicists? Finally, I ask what light the history of aerodynamics casts on the fraught arguments between historians, philosophers, and sociologists of sci­ence concerning relativism.15 Does the success of aviation show that relativ­ism must be false? I believe that, by drawing on this case study, some clear answers can be given to these questions, and they are the opposite of what may be expected.

During the writing of this book I had the great advantage of being able to make use of Andrew Warwick’s Masters of Theory: Cambridge and the Rise of Mathematical Physics.16 Although historians of British science had previously accorded significance to the tradition of intense mathematical training that was characteristic of late Victorian and Edwardian Cambridge, Warwick took this argument to a new level. By adopting a fresh standpoint he compellingly demonstrated the constitutive and positive role played by this pedagogic tra­dition in electromagnetic theory and the fundamental physics of the ether in the early 1900s.17

For me, one of the intriguing things about Warwick’s book is that the ac­tors in his story are, in a number of cases, also the actors in my story. What is more, his account of the resistance that some Cambridge mathematicians displayed to Einstein’s work runs in parallel with my story of the resistance to Prandtl’s work. Like Warwick I found that their mathematical training could exert a significant hold over the minds of Cambridge experts as they formu­lated their research problems. In many ways the study that I present here can be seen as corroborating the picture developed in Warwick’s book. Of course, shifting the area of investigation from the history of electromagnetism to the history of fluid mechanics throws up differences between the two studies, and not surprisingly there is some divergence in our conclusions. Whereas Warwick’s attention is mainly (though not exclusively) devoted to the British scene, my aim, from the outset, is that of comparing the British and German approaches to aerodynamics. Furthermore, on the British side, I follow the actors in my story as they move out of the cloisters of their Cambridge col­leges into a wider world of politics, economics, aviation technology, and war. If Warwick studied Cambridge mathematicians as masters of theory, I ask how they acquitted themselves as servants of practice.

Bernoulli’s Equation

The Euler equations permit the deduction of an important result known as Bernoulli’s law.26 Stated simply, Bernoulli’s law implies that the pressure of the fluid increases when the velocity decreases, and vice versa. There are technical restrictions imposed on its application, but the law has many practical uses in aerodynamics. It lies at the basis of an important measuring instrument used for determining the speed of flow of a real fluid such as air. The instru­ment is called a Pitot-static probe and is used, for example, in wind tunnels to establish the speed of flow. Furthermore, every aircraft is equipped with this device in order to determine the speed of flight. The instrument registers pressures, but it yields information about velocities in virtue of the relation given by Bernoulli’s law.

Stated quantitatively, the version of Bernoulli’s law to which I have re­ferred is

p+1 pV2 = H,

where H is a constant called Bernoulli’s constant. The formula only strictly applies to the steady, irrotational motion of an ideal fluid. It refers, in the first instance, to a single streamline and relates the pressure and velocity at any point on the streamline to the value of H that characterizes the streamline. In aeronautics all the streamlines can be taken to originate from a region of constant pressure and velocity, and then all of the streamlines have the same value of H. The Bernoulli constant has the same value for all parts of the flow, and its value can be established for the entire flow if it is known for any given point in the fluid. The first term in the equation, p, is called the static pressure. The second term is called the dynamic pressure, and their sum, H, the Bernoulli constant, is the sum of the static and dynamic pressures and is therefore called the total pressure. The formula indicates that as the velocity V increases at some point in the flow, the static pressure p goes down at that point because the two quantities, p and (1/2) p V2, must always add up to the same value. Furthermore, by knowing the density p and the value of p and H, we can calculate V, the speed of the flow. This is evident because the formula can be rearranged and restated as

Подпись: У =2(H – p)

Both the static pressure (p) and the total pressure (H ) of a flow can be mea­sured. Figure 2.3 shows a simple arrangement of tubes and manometers that

Bernoulli’s Equation

would yield measures of these quantities. The total pressure measurement (a) uses an open-ended tube. The static pressure measurement (b) uses a closed tube with a small hole in its side. The side hole is called the static tap. Both tubes are connected to their respective manometers. The third part of the figure (c) indicates how the two measuring devices can be unified to form a Pitot-static probe. In the combined instrument, the single manometer mea­sures the pressure difference (H – p) needed to establish the velocity.

Measuring the speed of flow by means of a Pitot-static probe can be ac­curate to about 0.1 percent, but it has a slow response rate and demands care and suitable conditions. The formula contains a term for the density of the
air, and density varies with altitude, a fact of importance when the instru­ment is used in an aircraft to measure speed. Furthermore, the Pitot probe itself can disturb the flow it is used to measure. Small faults such as a burr around the mouth of the static tap, or a misalignment of the probe, as well as turbulence in the flow, can significantly affect the readings.27 Conditions such as the formation of ice in and around the inlet holes can also falsify the instrument readings of an aircraft, and for this reason such devices are usually equipped with electrical heating elements. The formula underlying the use of the Pitot-static probe, which I have given, only applies to airflows that can be considered as incompressible and has to be modified to allow for compression effects for high-speed subsonic flight. Yet further modifications are needed to correct for the presence of shock waves at the nose of the Pitot tube as the speed of sound is approached.28