Category The Enigma of. the Aerofoil

A significant part of the experimental evidence for the circulation theory came from Arthur Fage (1890-1977). Fage was a retiring man who had trained as an engineer at the Royal Dockyard School in Portsmouth. His fa­ther had been a coppersmith in the dockyards. Fage won a scholarship to the Royal College of Science in London and then moved to the National Physical Laboratory in October 1912 as a junior assistant in the aeronautics section. In 1915 he published The Aeroplane: A Concise Scientific Study.29 The book was “written to meet the requirements of engineers” (v). It embodied a wholly empirical approach and made no mention of either the discontinuity theory or the circulation theory. With his “infinite capacity for taking pains,” Fage “became progressively a better and better research scientist” and acquired the reputation of being one of the NPL’s most meticulous experimenters.30

After Bryant and Williams’ work on the infinite wing, the next step in the Research Committee’s plan was to test Prandtl’s account of the flow around a finite wing. This work was undertaken by Fage and L. F. G. Simmons.31 They tested a rectangular model wing with a 3-foot span and a chord of 0.5 foot that was set at an angle of incidence of 6° to a wind of 50 feet per second. The wing had a cross section known as the RAF 6a. This particular wing was chosen because it had been studied in earlier experiments at the NPL, and the lift coefficient and the distribution of lift along the span were already known. Fage and Simmons used a speed-and-direction meter to probe the space round the wing to build up a detailed, quantitative picture of the flow. They measured the flow as it cut a number of transverse planes across the wind channel at various distances behind the wing (see fig. 9.5). If the x-axis is taken as the longitudinal axis of the channel, then positions on these trans­verse planes would be defined in terms of y – and z-coordinates where the y-axis lay parallel to the span of the wing and the z-axis indicated positions above and below the level of the wing. These coordinate axes are shown in the figure. Notice that the origin is located at one of the wingtips. The transverse planes chosen for study were called A, B, C, and D. Distances were expressed in terms of the chord of the wing. Plane A, which featured very little in the subsequent discussion, lay at a distance of about half a chord in front of the wing. Planes B, C, and D, the main interest of the experimenters, lay respec­tively at distances of x = 0.57, x = 2.0, and x = 13.0 chords behind the wing. The aim was to measure the properties of the trailing vortices as they cut through planes B, C, and D. On the basis of previous work (such as Piercy’s), Fage and Simmons were by now confident of the existence of the vortices but their question was: Did these vortices behave quantitatively in the way that Prandtl had assumed?

For convenience the test wing was actually mounted vertically and the speed-and-direction meter was inserted through a hole in the floor of the wind channel. The meter could be moved up and down, parallel to the lead­ing and trailing edges. The tips of the wing were fastened to transverse run­ners on the roof and floor of the wind channel so that the wing could be made to slide from one side of the channel to the other without altering its angle of incidence. These two degrees of freedom (the wing going from side to side and the meter going up and down) allowed measurement to be made within

a given, transverse plane. After taking measurements in plane A, they had to remount the wing at a different distance from the meter to measure the flow in plane B, and similarly for planes C and D. The measurements were made with an instrument of a standard type developed at the NPL (see fig. 9.6). It consisted of four open-ended manometer tubes forming a head that could be tilted on its mounting as well as turned from side to side. The orientation of the head of the meter could be manipulated from outside the wind channel. When the head was directly aligned with the local flow, the pressure in the four tubes was equal, which allowed the direction of the air at that point to be read off. The speed of the air was given by the difference in pressure between one of the four tubes and a fifth open-ended tube pointing in the direction of the free flow.32

Fage and Simmons took hundreds of measurements in order to build up a quantitative picture of the flow, and they were able to draw a number of important conclusions. First, they found that the velocity components (u) in the direction of the main flow (that is, parallel to the x-axis) were very little changed by the air having passed over the wing. The circulation theory implies that the speed immediately above and below the wing was modified, but measurement showed that soon after its passage over the wing the u-component settled down to a steady value that only varied by about 1 percent. This (approximately) constant velocity could therefore be ignored when considering the flow in the transverse planes. Exploiting this fact, Fage and Simmons argued that the flow within each plane could be considered as a two-dimensional flow. This two-dimensional flow was taking place in the (y, z) plane, so the two components of speed in the flow were v (along the y-axis) and w (along the z-axis). Fage and Simmons proceeded to com­pute the stream functions, streamlines, and vorticity of this flow at a large

figure 9.6. Apparatus used by Fage and Simmons for detecting the speed and direction of the flow behind a wing. As shown, the neck of the apparatus is inserted through the floor of the wind tunnel. The head of the apparatus can be raised or lowered, turned from side to side, and also rotated. From Fage and Simmons 1926, 306. (By permission of the Royal Society of London) number of points in each transverse plane by using the values taken from their measurements.

By the term “vorticity” Fage and Simmons meant (dw / dy—dv / dz). This is the quantity previously introduced and defined (in chap. 2) as the measure of the rotation of a fluid element. The two terms “rotation” and “vorticity” were used interchangeably. Each of the two parts of the expression represents a rate of change and hence the slope of a graph. Thus the value of dw/dy at a given point refers to the rate of change of the speed w with distance along the y-axis at that point. A corresponding definition applies to dv/dz. Fage and Simmons had a sufficient number of velocity readings of v and w at a sufficient number of points to find the relevant slopes and rates of change. They were thus in a position to compute the rotation or vorticity at each of these points. Once numerical values of this expression were established for a large number of points on the planes B, C, and D, it was possible for the ex­perimenters to draw curves linking up the points of equal vorticity. The result was a striking picture of the flow.

Fage and Simmons’ figures show the contours of equal vorticity in each of the transverse planes behind one-half of the wing (see fig. 9.7). The wing is not shown on these diagrams but lies horizontally, obscured, as it were, by the vorticity. The wingtip would be on the right, and the center of the wing, which is off the diagram, would be on the left. The top figure refers to plane B immediately behind the wing; the other two refer to planes C and D, which are set farther back. Examination of the figures shows that initially (at about half a chord behind the wing) vorticity is spread in a nar­row sheet along the trailing edge but becomes more concentrated near the tip. At a distance of two chords behind the wing, the vorticity at the tip has becomes less intense and more spread out. These changes, said Fage and Simmons, demonstrate the unstable character of the vortex sheet and show that it rolls up in the way predicted by Prandtl. The third picture shows the rolling up to be almost complete. The vorticity is now confined to the area near the wingtip. Citing Lamb’s Hydrodynamics, Fage and Simmons state that the diagrams show that, in accordance with the classical mechanics of vortex motion, the vortices behave as if they are attracted to one another. Their centers are progressively displaced toward the center and away from the tips.

Fage and Simmons then used their data to test the quantitative aspects of the theory and to construct detailed connections between their observations and the mathematics of Prandtl’s picture. The central questions were whether the flow (outside the trailing vortices) was irrotational and whether the lift and circulation around the wing were connected to the trailing vorticity

according to the laws identified by Prandtl. To address the question of irrota – tional flow, Fage and Simmons computed the circulation around a sequence of rectangular contours of increasing size drawn on their transverse planes. They chose rectangular contours, with sides parallel to the y – and z-axes, be­cause they already had the relevant velocity components needed for the cal­culation. The circulation they sought was given by the quantity (vdy + wdz) calculated around the contour. The chosen rectangles all enclosed the same areas of vorticity in the wake of the wing. According to Stokes’ theorem the circulation around these contours gave a measure of the vortex strength they enclosed. The calculated values for the circulation were close to one another, and this, argued Fage and Simmons, showed that outside the vortex wake the flow was irrotational—that is, as the contours got bigger, no further rotating fluid elements, and hence no further vorticity, can have been enveloped by the contour.

This argument for the irrotational character of the flow outside the vor­tex wake had exactly the same form as Glauert’s inference when he gave his preliminary analysis of Bryant and Williams’ results. A number of contours of increasing size display the same amount of circulation, ergo the contours pass through an irrotational flow. This was the inference to which Taylor had taken exception. Had Fage and Simmons fallen into the same trap as Bry­ant and Williams? The answer is no. As a deductive inference the argument put forward by Fage and Simmons is not compelling (for the reasons that Taylor gave), but as an inductive inference the conclusion is plausible. Given their other computations dealing with vorticity in the flow, and the contours showing its distribution, these results reinforced the conclusion that the flow outside the wake was, indeed, irrotational.

The most important quantitative question was whether the strength of the vortices trailing from the three-dimensional wing had been correctly predicted by the proponents of the circulation theory. Fage and Simmons deduced that, according to Lanchester and Prandtl, the strength of vorticity leaving the semi-span of the wing should equal the circulation around the median section of the wing. If these two quantities could be found and com­pared, then the prediction could be tested. Fage and Simmons developed this argument mathematically, but the intuitive basis of the deduction can be seen immediately by inspecting figure 9.8.

The figure shows a version of the refined horseshoe vortex. The vortices that run along the span of the wing all pass through a contour around the me­dian section of the wing. The value of the circulation around this contour is determined by the vortices that run through it (from Stokes’ theorem). All of these vortices then peel off the trailing edge of the wing. Their total strength can be assessed by the circulation around another contour, namely, any con­tour that captures and includes them. For example it can be measured by the circulation around the rectangular contours drawn in the planes B, C, and D that have been previously discussed. Fage and Simmons compared these empirically based values with the predicted circulation around the median section.

How did Fage and Simmons evaluate the predicted circulation? It is the

quantity that has previously been called Г0. Recall that in chapter 7, it was shown that by making the assumption that the lift distribution was elliptical, Prandtl had given an expression that predicted the value of Г0 from known properties of the wing and the flow. Prandtl had shown that

2VSC.

г = L,

0 bn

where CL is the coefficient of lift, S is the area of the wing, b is the span, and V is the free-stream velocity. All of these were known quantities for the wing used in Fage and Simmons’ experiment. In this way they were able to compare the predicted value of Г0 and the observed results for the circula­tion around the vorticity coming away from the trailing edge. The two values should be the same. The theoretical prediction was corroborated for the flow up to two chord lengths behind the wing, that is, for planes B and C. On plane D, at a distance of thirteen chord lengths, the agreement was not so good. The measured vorticity was around 18 percent too small. Some of the vortic – ity appeared to have dissipated, but, within the limits of experimental error, Fage and Simmons declared that the match between theory and observation was a good one.

———– СглЬтти1 Ur*S of Y

——- – Фспіоиґ Uno of Ф

figure 9.9. The streamlines of a trailing vortex at three distances behind the tip of a finite wing. The position of the wing is shown on the diagrams with the wingtip on the right, behind the center of the vortex. The centerline of the wing is on the left. From Fage and Simmons 1926, 320-22. (By permission of the Royal Society of London)

Fage and Simmons not only drew the lines of equal vorticity (shown in fig. 9.7) but also drew the actual streamlines of the vortices that issued from the wings and passed through the planes B, C, and D. This, too, was an exer­cise that involved considerable computation. First they had to check that the preconditions for the existence of a stream function, y, were fulfilled. Was the continuity equation satisfied? Laboriously, they concluded that it was. They then identified the values of the stream function at a large number of points in the (y, z) plane in order to draw the streamlines y = constant. On grounds of symmetry they took the line y = 3, the center line of the wing, as the streamline y0. The pictures they produced from their measurements and computations were convincing. “The diagrams,” they said, “illustrate clearly the changes in the character of the airflow behind the aerofoil: close to the aerofoil, the contour lines are ovals with the ends pointed inwards; as the distance behind the aerofoil increases, the contour lines become more and more circular—indicating that the vorticity is becoming more concentrated” (320). The results are shown in figure 9.9.

Fage and Simmons had significantly strengthened the case for the cir­culation theory. They had effectively picked up the argument at the point where Pannel and Campbell had left it in 1916 when curiosity seems to have been swamped by collective incredulity. Unlike the conclusions of Bryant and Williams, the work of Fage and Simmons was not challenged by Taylor.33 Others, such as Lanchester and Foppl, had made qualitative observations of trailing vortices and, more recently, photographs had been taken, but now the vortices had been directly measured and linked to the mathematically defined relations of the circulation theory. Could it now be said that the entities that were subject to empirical measurement in the laboratory had been clearly identified as one and the same with the entities referred to in the circulation theory? If so, the work of Fage and Simmons would represent a qualitative change in the status of the circulation account of lift. What had previously been a theory could now, arguably, be counted as a directly verifi­able fact.

How can the Tripos tradition be invoked to explain both failure, in the case of lift, and success, in the case of stability? Should not the same cause have the same effect? The answer is that my causal explanation is not meant to ex­plain success and failure. It explains the preconditions of success and failure, that is, what succeeded or failed, and the response to success and failure. It concerns the path that the Cambridge scientists and their associates chose to follow, or declined to follow, rather than the consequences of the choice and what was found along the path that was selected. Neither the British nor the Germans knew what awaited them. Neither group could select their course of action in the knowledge of what would be successful or unsuccessful. Both had to take a gamble. In the case of the theory of lift, the British gamble failed and the German gamble succeeded.

Lanchester did not approve of the theoretical choices made by what he called the Cambridge school, but he had a clear understanding of the meth­odological gamble involved. He did not use the language of gambling, but as we have seen in his reflections on the role of the engineer, he chose the more cerebral metaphor of playing chess. The scientist and the engineer were like players in a game of chess who were confronting an opponent called Nature. As a player, Nature was subtle and her moves could not be easily predicted. Lanchester expressed the contingency of the outcome by saying that, at the outset, no one could identify which moves in the game were good ones and which were bad. A sound-looking move might turn out to be a mistake, and an apparent mistake might turn out to be the winning move. Only the un­known future would reveal this.

Lanchester’s metaphor can, and should, be taken further. It does not just apply to the opening moves in the game of research or to what is sometimes called “the context of discovery” as distinct from “the context of justifica­tion.” It applies to the entire course of research and development ranging from the origin of ideas to their acceptance and rejection. The game with Nature does not come to an end when results begin to emerge and when the chosen strategy of research starts to generate successes and failures. The uncertainties Lanchester identified at the outset of the game still inform the responses that have to be made to the feedback from experience. If the scien­tist or engineer scores a success, the question remains whether it will prove to be of enduring significance or short-lived. If there is a failure, does it indicate the need for a revision of the strategy or merely call for more resolve? These radical contingencies and choices can never be removed, and in one form or another, whether remarked or unremarked, they are present throughout sci­ence. Indeed, they are present in every single act of concept application. The game never comes to an end.49

The administrative structure that crystallized in Haldane’s mind was for a committee of ten or eleven, involving persons of the highest scientific talent, to address technical problems presented to them by the Admiralty and War Office. Unlike the proposed committee itself, these two old-established bod­ies would be responsible for commissioning and even constructing military airships and aircraft. The committee would analyze and define the scientific and technical problems encountered by these constructive branches of the military and would pass them on to the National Physical Laboratory (the NPL). The laboratory, which was based at Teddington just outside London, was to have a new department specializing in aeronautical experiments. This department would produce the answers to the questions posed by the Advi­sory Committee. Financially, the committee would be accountable not to the War Office or Admiralty but to the Treasury.

The structure that emerged conformed to this plan except for the addition of one more unit. In 1911 the former Balloon Factory at Farnborough, be­longing to the army and the home of Dunne and his supporters, was turned into the Aircraft Factory and then (in 1912) into the Royal Aircraft Factory (the RAF).27 After the Dunne episode it had been decided to drop aircraft research at Farnborough, but this resolution was now rescinded. It was thus determined, after some indecision on Haldane’s part, that new aircraft were to be designed by the government itself and built at its behest by private manufacturers.28

An organizational chart of Haldane’s arrangement would therefore take the form shown in figure 1.2. Problems passed from left to right on the chart,

from the Admiralty and War Office through the ACA to the National Physical Laboratory and the Royal Aircraft Factory. After experiments and tests had been completed, according to a schedule agreed on with the ACA, informa­tion and answers were passed back, from right to left on the chart, in the form of confidential technical reports. After these were discussed and agreed on by the ACA, and any required amendments had been made, the outcome was to be published in the form of a numbered series called Reports and Memoranda—a series that, over the years, ran into thousands and was to become famous for its depth and scientific authority. Each year the Advisory Committee presented an annual report containing an overview of its activi­ties to which was attached, as a technical appendix, a selection of the more important Memoranda.

With the passage of time, and the increased workload imposed on the ACA, the original committee was broken down into a number of subcom­mittees to which further experts were recruited from the universities, Farn – borough, and Teddington. Thus there was an Aerodynamics Sub-Committee, an Accidents Sub-Committee, an Engine Sub-Committee, a Meteorological Sub-Committee, and so on. Sometimes the subcommittees were further bro­ken down into panels, such as the Fluid Motion Panel, which was part of the Aerodynamics Sub-Committee. Such a structure may seem complicated and bureaucratic, but viewed with the benefit of hindsight, it proved highly effective.

Although many flows were discovered by the indirect method, there are di­rect methods for describing a flow. How, for example, does the mathema­tician manage to describe the flow around a straight barrier that is placed facing head-on into a uniform stream of ideal fluid? The flow in question is sketched in figure 2.5, again taken from Cowley and Levy’s book. (Because the flow is presumed to be symmetrical around the central streamline of the main flow, only the upper half of the flow need be considered. The central streamline can be treated as if it were a solid boundary.) How can the equa­tions for the streamlines ever be discovered if the mathematician does not have the good fortune to come across a function amenable to after-the-fact interpretation? The answer is by cleverly establishing a relationship between

figure 2.5. Ideal fluid flowing irrotationally around a barrier normal to the free stream. From Cowley and Levy 1918, 49.

this complicated flow problem and the simplest of all possible flow problems, namely, the uniform flow along a straight boundary. The method involves transforming the straight boundary into the shape desired, for example, the shape of a barrier that is sticking out at right angles into the flow. The process is carried out by means of what is called a conformal transformation.

First, I should explain the word “transformation.” Everyone is familiar with the process of redrawing a diagram on a different scale. Suppose a geo­metrical figure has been drawn on one piece of graph paper, and it is required that the figure be redrawn, to a different scale, on another piece of graph pa­per. A line three centimeters long in the original is to be, say, six centimeters in the new diagram. A circle of radius four centimeters is to become a circle of radius eight centimeters, etc. The rule, in this case, is to double the length of the straight lines. The original diagram has thus been subject to a very simple, linear “transformation.” Other, much more complicated transformations are possible. Not only might a transformation magnify the figure in the original, but it might shift it relative to the origin, or rotate it or even distort it in vari­ous ways, turning, say, a circle into an ellipse. This shift will depend on the particular transformation that is being followed, namely, the particular rule that relates the positions of points in the one figure to the points in the other figure. If two figures are related by a transformation, then, if we know one of the figures, along with the rule of transformation, we can construct the other figure. A figure can be subject to more than one transformation so that a fig­ure which results from one transformation can be transformed yet again.

Transformations are important in hydrodynamics for the following rea­sons. First, the rules governing many transformations can be embodied in mathematical formulas that are functions of a complex variable. These are the conformal transformations. Second, if the flow around one shape is known, and a formula of this kind is available to transform the shape into a new shape, then the flow around the new shape is known. Conformal trans­formations change the streamlines as well as the boundaries of the figure, modifying the shape of the flow to fit the new circumstances. Methodologi­cally this is important. It means that, given an appropriate transformation, it is possible to move from simple flows, with simple boundaries, to the descrip­tion of complicated flows with complicated boundaries. All this can be done once it has been established that the transformation maps the boundaries of the two flows on to one another. Cowley and Levy sum up the situation, tersely, as follows: “It must be noticed that as long as complex functions are dealt with, the hydrodynamical equations will be satisfied and it will only be necessary therefore to consider boundaries. If a functional relation exist­ing between two planes is such as to provide a correspondence between the boundaries in these planes it is the transformation required” (47). The “two planes” referred to in this quotation are, in effect, just the two pieces of graph paper I mentioned at the outset. In this case, however, the idea is that one plane (usually called the w-plane) has the boundaries of a simple flow drawn on it, while the other plane has the, transformed, boundaries of the more complicated flow. This is usually called the z-plane and the transformation, or the sequence of transformations, links the two planes.

The problem is to find the necessary rule, or rules, of transformation. Fortunately there are general theorems that deal with the subject of transfor­mation which can be put to use. For example, there is a powerful result called the Schwarz-Christoffel theorem which proved central to classical hydrody­namics and, as we shall see in later chapters, also played an important role in the history of aerodynamics. The Schwarz-Christoffel theorem is applicable to the present problem, namely, finding the flow around a barrier across the flow of the kind shown in figure 2.5. This theorem, used by Cowley and Levy in their book, transforms the interior of a closed polygon on one plane (the z-plane) into the upper half of another plane (usually called the t-plane) and turns the boundary of the polygon into the real axis of the t-plane. If the t-plane can then be related to the basic, simple flow along the horizontal axis in the w-plane, then the requisite connections have been made. The simple flow with its simple boundaries can be turned into the complicated flow. The bridge is symbolized by w=f(z). Although the details need not be described, I want to sketch the way the theorem is used. The first step is to explain where, and why, polygons come into the story.

The polygon is familiar from school geometry and is usually defined as a many-sided figure whose sides are straight lines. A “closed” polygon obvi­ously has an inside and an outside. The exterior angles must add up to four right angles. The interior angles add up to (n – 2)n, where n is the number of vertices. Thus a rectangle is a simple case of a closed polygon that has just four vertices and in which each of the four interior angles is also equal to n/2.

The Schwarz-Christoffel theorem is embodied in the following, daunting, formula:

dz a-1 —-1

– = A(t-ti)n (t-t2)n…

dt

The letter A represents a constant and a, p, . . . are the internal angles of the polygon. The numbers fi, t2, . . . are real numbers ranging from minus infinity to plus infinity, with one number for each vertex. In order to put the formula to work to transform a given polygon, it is necessary to insert the values for the interior angles of the polygon, a, P, etc., into the formula and to assign the vertices of the polygon to the positions fi, t2, etc. on the real axis of the t-plane. (Some of these assignments can be made arbitrarily, while some de­pend on the shape of the polygon. In a moment I shall show how Cowley and Levy made the assignment.) Having filled in the appropriate values in the formula, we must then integrate it, and the result is a function of a complex variable z = f( t).

Why is this result useful when the aim is to find the flow around a barrier? The answer is that the complicated boundary, represented by the barrier in figure 2.5, can be counted as a closed polygon for the purposes of the theo­rem, and this fact can be exploited to get the desired flow. Given the picture of a polygon that comes to mind from school geometry, such a designation seems counterintuitive. The streamline along the axis of symmetry combined with the barrier normal to the flow doesn’t look like the polygons drawn on a school blackboard. Clearly, the words “polygon” and “closed” have been given a wider meaning. The justification is that the sides of a polygon can be made “infinitely long,” and the vertices dispatched to “infinity,” provided that the appropriate conventions are still kept in place regarding what counts as the interior and the exterior of the polygon. In this extended sense a poly­gon can even take on the appearance of, say, a single straight line.32 Crucially, it can also take on the appearance of the boundary in figure 2.5 that repre­sents a straight barrier jutting out into a fluid flow.

How is the diagram of the barrier-as-polygon connected to the Schwarz – Christoffel transformation formula? Look at Cowley and Levi’s figure, that is, my figure 2.5. The “vertices” of the “polygon” are marked A, B, C, D, A’. Inspection of the figure shows that A and A’ are both located at “infinity.” The points B and D are at the front and back of the base of the barrier, while C is at the top of the barrier. The “internal” angles can also be located. In moving along the boundary the point B is the location of a right-angle turn at the front of the barrier, while at C there is a turn through 180° at the top edge of the barrier, and there is another right-angle turn at D on the rear face of the barrier. These are the angles a, p, etc. to be inserted into the formula. Cowley and Levy’s diagram also shows how they have assigned t-values to these verti­ces. The one assignment not shown in the figure is the point C, the top of the barrier, which is given the value t = 0.

Once these particular values have been inserted into the formula it is ready to be integrated. After integration the constant A in the formula, as well as the constants of integration, can be evaluated by using the initial and boundary conditions of the problem. Proceeding in this way gave Cowley and Levy a formula connecting z and t, namely,

z = U] (t2 -1).

The process is, however, not quite finished. The basic, simple flow itself now needs to be expressed in terms of the t-plane. The t-plane is an intermediary between the z – and w-planes. Only when the t-plane has been linked to the w-plane will the desired connection have been made. The general form of the simple flow on the w-plane and the boundaries on the t-plane suggest that the link will be a simple one having two constants and taking the general form w = at + b. Consideration of the velocity of the flow at a great distance from the barrier, and the disposition of the bounding streamlines, allows the constants to be evaluated. The transformation connecting w and t is then given by the formula w = l V t, where V is the free-stream velocity and l is the half-length of the plate.

Combining the two formulas by eliminating t gives the result that has been sought, the complex function expressing the flow around the barrier. The desired formula is

f (z) = V^z2 +12.

Separating out the imaginary part, y, gives an expression for the streamlines of the flow, and from this the velocities and pressures on the boundary can be calculated. The formula for y turns out to be a complicated one, but it allows the curves to be drawn by setting y = constant. The formula is

y4 + V2(x2 – y2 + l2)y2 – V2x2y2 = o.

Now the streamlines of the flow of an ideal fluid around a flat barrier placed head-on to the flow can be calculated and represented with mathematical precision.

The remarkable fact that functions of a complex variable such as f(z) = (z + l/z) and f(z) = Vy/z2 +12 are all descriptions of irrotational flows has un­doubtedly left its mark on the development of classical hydrodynamics.33

It also raises a question. Why should the functions of a complex variable, containing esoteric mathematical entities such as the square root of nega­tive numbers, yield pictures of fluid flows? Consider the formula for the flow around a circular cylinder. The formula itself, f(x) = (z + 1/z), is not remark­able and is familiar to any student of mathematics (and we meet it again in a later chapter). It is hardly surprising that the formula is to be found in G. H. Hardy’s famous, Tripos-oriented textbook A Course of Pure Mathematics, first published in 1908. It crops up in the miscellaneous examples at the end of the chapter on complex numbers.34 But Hardy’s student reader was set the purely mathematical task of proving that (z + 1/z) transforms concentric circles into confocal ellipses. There was no mention of streamlines. The formula merely provided the occasion for an exercise in analytical geometry. That is what is puzzling. What has geometry got to do with fluids?

Part of the answer is provided by noticing that the functions that de­scribe the complicated flows do so by virtue of being transformations of the simplest possible flow, namely, the uniform flow of an infinite fluid along a smooth, straight barrier. But that merely pushes the problem back. Why should mathematics furnish a description of even the simplest of fluid flows, and why should that applicability survive the transformations leading to the complicated cases of flows that go around circular cylinders and encounter barriers? Does it all, perhaps, hint at a preestablished harmony between math­ematics and nature? Metaphysical responses of this kind have a long history. Famously, Galileo declared that God wrote the Book of Nature and did so in the language of geometry.35 Such reactions should not be dismissed. They represent an attempt to address a real question, and they are not confined to the past. Even contemporary physicists have been struck by the “unreason­able” effectiveness of mathematics in the natural sciences. The implication is that something beyond reason is at work, something mysterious and even miraculous.36 In the present case, however, any hint of the noumenal will be quickly dispersed when the empirical track record of the theory of ideal fluids is examined. I now turn to this side of the matter.

Within the framework of Newton’s mechanics, the flow of air around an air­craft wing can only support the weight of the aircraft if the flow generates a force that is equal and opposite to that of the weight. In level flight the upward force, the “lift,” must be in equilibrium with the downward force of gravity. Expressed in terms of fluid dynamics, the lift must be the result of air pressure on the wings. There must be an overall pressure imbalance between the upper and lower surfaces of the wing. The pressure of the moving air on the upper surface of the wing pushes downward. This must be surpassed by the pressure on the lower surface of the wing which pushes upward. It is the excess of the upward over the downward pressure that constitutes the lift and is therefore the central fact to be explained. It cannot be assumed that the resultant downward pressures and the resultant upward pressures act through the same point. In general they will not; the pressures on the wing will not only have the capacity to produce a lift, but they will also generate a turning moment that causes the wing to pitch. These pitching moments played a significant role in the analysis of stability carried out by G. H. Bryan. In what follows, however, I am mainly concerned with the resultant lifting effect of the pressures on the wing. I have already introduced Bernoulli’s law which implies that, if the air behaves like an ideal fluid, then the faster the air flows over the wing the lower will be the pressure it exerts, and the slower the flow the higher the pressure. If it is also accepted that the airflow around a wing is not discontinuous Rayleigh flow but follows the surfaces of the wing, then the problem of lift is simplified. It reduces to that of explaining why the air immediately below the wing is moving more slowly than the air immedi­ately above the wing.

Here it is necessary to avoid a popular misconception. A cross section of a typical wing has a flat base and a curved upper surface. The airflow di­vides at the leading edge, and some air takes the upper route over the curved surface while some takes the lower route along the flat and straight surface. Looking at such a shape, one can easily imagine two molecules of air parting company at the leading edge and joining up with one another again at the trailing edge. Like two travelers they wave farewell at the parting of the ways and then shake hands when they meet up later. But the low road is straight while the high road is circuitous, so the traveler who took the high road must have sped along more swiftly in order to meet up with the traveler who took the shorter path. Is this how it is with the air? Equal transit time plus a path difference certainly implies a speed difference, but this is not the secret of the wing.3 The questionable assumption is that the traveling companions, that is, the two molecules, meet up again. There are decisive reasons why this theory cannot be right. First, the increase in speed necessary to pass over the curved, upper surface of the wing would not generate the observed amount of lift. The path difference is not great enough. Second, the theory would have the conse­quence that an aircraft could not fly upside down. Once inverted, the curved surface would become the lower surface. The theory would then imply that the aerodynamic force would reinforce gravity rather than counteract it. But aircraft can fly upside down, so the theory cannot be right.4 This false theory, based on path difference and equal transit time, must not be confused with the circulation theory of lift. The circulation theory offers a very different account of the speed differences above and below a wing, as I shall now explain.

The flow of air over the cross section of a wing is a complicated phenom­enon, but, argued the supporters of the circulation theory, it can be thought of as built up out of two, simple flows. These are (1) a steady wind of constant speed and direction, and (2) a swirling vortex that goes round and round a central point. The two components are shown separately in figure 4.1. I dis­cuss each component flow in turn and then explore the flow that arises when they are superimposed on one another. The steady wind arises from suppos­ing a steady, relative motion of the wing and the air. In reality the air is sta­tionary and the wing moves, but, as previously noted, aerodynamic processes are frequently described in terms of the situation in a wind channel where it is the air that moves. Let the steady wind have a constant speed V and move horizontally. At any given point the flow can be represented by a vector, that is, an arrow pointing in the direction of the flow whose length is proportional to the speed. All the vector arrows representing the steady wind are therefore of the same length and can be assumed to lie horizontally, as shown in the figure. The streamlines of the flow are then equally spaced horizontal lines.

+ V

STEADY

WIND

The vortex flow is more complicated, but the early work on aerodynamics was confined to a particularly simple form of vortex. Let the vortex swirl in a clockwise direction around a central point that is assumed to be in a fixed po­sition. Unlike a normal vortex, in water or air, this one is not carried along by the stream. This special sort of vortex came to be called a “bound” vortex. All the streamlines in the vortex flow have the form of concentric circles. The fluid elements at any given distance from the center of the vortex are assumed to move with the same, constant speed around one of these circles. The elements do not get drawn into the center of the vortex. This is expressed by saying that they have a constant “tangential” velocity and no “radial” velocity. Just as the velocity of the fluid elements in the steady wind can be represented by vec­tor arrows, the same can be done for the fluid elements in the vortex. In this case the arrow is a line whose length is proportional to the speed but whose direction always lies along a tangent to the circular streamline. The direction of the tangent varies as the fluid element proceeds around the streamline, although its length stays the same. Figure 4.1 shows a vortex with a clockwise rotation and gives the arrows of speed and direction at two important posi­tions. At what may be called the six o’clock position the arrows are horizontal and facing into the steady wind, while at the twelve o’clock position they are horizontal but point in the same direction as the uniform wind.

Typically one further important assumption was made about the struc­ture of the simple vortex. It was specified that the fluid elements that circle

around the vortex near the center move along their assigned path with greater speed than do fluid elements circling at a greater distance from the center. The speed drops off uniformly with distance from the center. The greater the radius of the streamline, the smaller the tangential velocity. The assumed relation can be expressed more precisely by saying that, for the kind of simple vortex under consideration, the speed of the flow (v) at any given point is “inversely proportional” to the radius (r) of the circular streamline that runs through that point. In mathematical terms the formula relating speed and radius is then v = k/r, where k is the constant of proportionality.

Now imagine that the constant wind and the vortex are superimposed. The two flows, which have hitherto been treated as separate cases, are now combined. What is the result? In reality, the mixing together of two flows, whether in water or air, is accompanied by all manner of eddying and tur­bulence produced by viscosity and other physical features of the fluid. In the analysis developed for aerodynamic purposes, all of these complications were put aside and an extremely simple process of combination was assumed to provide an adequate description. Because the two flows that are combined are steady, the new flow will also be steady and all that was necessary to describe it was a process called vector addition (see fig. 4.2).

R

RESULTANT VELOCITY

figure 4.2. At each point P in the combined flow of a steady wind and a vortex, two components are combined and determine the velocity of the resultant flow (a). This is done through vector addition, as shown in (b), which involves completing the parallelogram of velocities to give the speed and direction of the resultant velocity.

At any given point P in the new flow, there are two vector arrows to take into consideration. One, provided by the steady wind, is horizontal; the other, provided by the vortex, is at an angle determined by the position of P relative to the center of the vortex. P is located at some radial distance from the center of the vortex (this determines the speed), while the direction of the vortex component is determined by the direction of the tangent of the streamline that passes through P. A typical case is shown in figure 4.2a. The procedure needed to combine the effects of the two flows is shown in figure 4.2b. The resultant velocity is given by a geometrical construction called “completing the parallelogram,” whose intuitive meaning can be read directly off the diagram. Completing the parallelogram gives the speed and direction of the new flow at that point. A picture of the combined flow can be built up by the carrying out of this process at a large number of points. It will have the general appearance of figure 4.3, which is taken from Lanchester’s Aerodynamics.

The diagram shows that the streamlines are closer together above the cen­ter of the vortex than below, and this difference indicates a speed difference. The speed of flow above the vortex is greater than the speed below. How does this arise? To explain this occurrence it is sufficient to focus on two particu­larly important positions in the vortex, namely, the twelve o’clock and six o’clock positions, which are directly above and directly below the center of the vortex. Here the vector addition effectively reduces to simple arithmetical addition because there is no angle between the contributions of the two flows. At both points, but only at these points, the effect of the vortex is exactly aligned with the horizontal wind. At a point positioned some given distance r directly above the center of the vortex, the speeds of the two flows are going to add together to produce a flow with the speed V + v. At a diametrically opposite point, a distance r directly below the center, the two flows will op­pose one another to produce a reduced speed V – v. Elsewhere in the flow, at points not directly above or below the vortex, the contribution of the vortex component will augment the upper half of the flow, and diminish the lower half of the flow by less than v, but the general effect will still be present. Hence the spacing of the streamlines visible in Lanchester’s diagram.

The crucial step is the next one. The supporters of the circulatory theory supposed that, as it moves through the air, an aircraft wing (viewed in cross section) somehow generates a vortex effect around itself. There is, they ar­gued, a vortex “bound” to the position of the wing. The effect of the wing is to be represented by a vortex, even though the wing profile has an elongated shape, while the vortex is circular in form and centered on a geometrical point. Why a wing has this effect on the air and why it can be represented in this way were problems for the supporters of the theory, but they proceeded on the assumption that this was the case. They accepted that near the wing the flow could not look exactly like a combination of a steady wind and a vortex but that the picture became more accurate at a distance. Following the reasoning set out here, and applying it to the case of the wing, they argued that if the flow around the wing consisted (approximately) of a uniform flow combined with a vortex, then some of the air at a given distance above the wing would reach a maximum speed of V + v and some of the air at the same distance below it would drop to the speed V – v. Here was an explanation of the required speed differential in the flow over the wing, which in turn ac­counted for the pressure differential, and thus for the lift. Or, to be more pre­cise, here was an explanation of lift if the assumption is granted that the wing generates a vortex. But should this point be granted? The question epitomizes all the subsequent arguments over the circulatory theory.

Does the circulation theory imply that, during normal flight, molecules of air make a journey around the chord of the wing? No, this is not what its supporters were saying. Such a picture may be conjured up by abbreviated formulations, such as “lift is created by the circulation of air around a wing,” but these words depend on a technical meaning of the word “circulation” and do not mean what they may seem to mean. It is true that in an isolated vortex, such as a whirlwind, the air does indeed make a circular journey around the center of the vortex, but the theory does not require this to happen in the case of a wing delivering lift. The actual flow involves fluid elements curving up slightly to meet the leading edge of the wing. They then travel along the chord of the wing and leave with a slight downward inclination of the streamlines at the trailing edge. The claim is merely that during normal flight, the vortex exists as a component of this overall flow pattern.

In 1903 the Cambridge logician Bertrand Russell argued that “the compo­nent of any. . . vector sum, is not part of the resultant, which alone could be supposed to exist.”5 Russell (who was seventh wrangler in 1893) did not have aerodynamics in mind but was writing about the nature of mathematical con­cepts in general. His position suggests that only the resultant flow of air over a wing really exists, whereas the uniform flow and the vortex, being mere com­ponents, do not really exist. Such a conclusion does not do justice to what the supporters of the circulatory theory were saying. The component flows were meant to describe real tendencies existing in the resultant flow. These tendencies can be “supposed to exist” even when not manifesting themselves in isolation from other tendencies. This realistic way of speaking seems more natural than Russell’s formulation and better covers the range of empirical possibilities that would have been evident to those working in aerodynamics. First, the realistic idiom implies that if one contrived to bring a moving (and lift-generating) wing to a sudden halt in midair, then the circulating tendency would have nothing to modify it and would reveal itself in its full form.6 In these circumstances there would be air swirling around the wing. Second, as a general fact about vortex flow, if a very strong vortex is combined with a uniform wind, some of the air close to the center of the vortex actually will go around in a closed loop. (An examination of Lanchester’s diagram in fig. 4.3 shows that it represents a flow of this kind.) These considerations suggest that Russell was wrong and that the components of the vector addition can be as real as the resultant. Whether these real tendencies display themselves as independent phenomena is merely a matter of how strong they are relative to the other components.

Two characteristics have now been identified in the British response to the circulation theory of lift. First, there was a desire for theories of wide scope that embrace complex viscous phenomena beyond the reach of the theory. Second, there was a tendency to read Lanchester as contributing to an inviscid theory and therefore as committed to a simplified and unreal representation of fluid flow. Both of these indicate the importance that British experts at­tached to the distinction between real fluids and ideal fluids. Taylor insisted that fluid mechanics should have a firm basis in physics and dismissed the idealizations of classical hydrodynamics. Cowley and Levy described inviscid theory as fatally flawed and spoke of the need for a theory of viscous flow that would solve the problems of aerodynamics at a stroke. Bairstow agreed that it was fundamentally impossible to represent real fluids in terms of ideal fluids and duly turned to the study of viscous flow. What Bairstow had asserted with characteristic acerbity, Lamb had hinted at with characteristic restraint. The different objections and formulations all point to one conclusion. The distinction between viscous and inviscid fluids is to be seen as the axis around which British thinking revolved.16

It is important not to view this distinction as self-evident or something that was understood in the same way by all competent operators in the field of fluid dynamics. In reality it was treated differently in different institutional settings. How then should the distinction between viscous and inviscid fluids be understood? Formally, it centers on whether p, the symbol for viscosity in the Stokes equations, is to have a value of zero or of nonzero. Was p = 0, or p Ф 0? Logically it must be one or the other and it can’t be both. Empiri­cally, whether Stokes’ equations turn out to be true, and Euler’s false, (or vice versa), is something to be settled by reference to experiment. But these tru­isms do not tell us how to interpret the difference between putting p = 0 or p Ф 0; nor do they indicate what physical meaning is to be given to the mathematical limit when p ^ 0. They do not tell us whether the distinctions involved are qualitative or quantitative or whether the boundaries under dis­cussion are strong or weak or for what purposes they might be important or unimportant. This is the point. The conceptual boundary between viscous and inviscid fluids is more than merely formal. Rehearsing the elementary mathematical properties of the distinction does not tell us what methodologi­cal implications are attached to it by the scientists concerned. I shall now illustrate the broader, methodological significance of the distinction by refer­ence to Lamb’s own discussion of viscosity.

Lamb began his account of aerodynamics, in the 1916 edition, by point­ing out that the analysis of Kirchhoff-Rayleigh flow was the first attempt, “on exact theoretical lines,” to overcome the result that a perfect fluid exerts no resultant force on a body. He added: “The absence of resistance, properly so called, in such cases is often referred to by continental writers as the ‘paradox of d’Alembert’” (664). Why did Lamb think that “absence of resistance” was the more proper description? What was wrong with talking about a “para­dox”? The reasoning behind Lamb’s remark went back to the first edition of his book, where he had originally addressed the well-known discrepancies between the empirical facts of hydraulics and the mathematical deductions of hydrodynamic theory. He traced the problem back to “the unreality of one or more of the fundamental assumptions” of the theory (244). The empirically false conclusion about resistance came from an empirically false premise, namely, the inviscid character of the postulated fluid. However, d’Alembert’s reasoning was sound, and the logic of the situation was clear. An inviscid fluid is correctly characterized by the absence of resistance. This is how ideal fluids behave or would behave. It is a simple fact about them, and there is nothing paradoxical about it.

A paradox is more than a falsehood, even a blatant falsehood. A paradox must involve a seeming contradiction. Suppose that experiments on a fluid F showed that it exerts a resultant force on a submerged body, while a mathe­matical analysis of F entails a zero resultant. Suppose, further, that the experi­ments on F seemed wholly reliable and the mathematical analysis of F seemed wholly correct. That would be paradoxical. Contradictory specifications of F have been generated from sources that seem undeniable. This is not the case if the experiments refer to a real fluid Fr, and the mathematics refers to an ideal fluid F. There is now no single point of reference as there was with the “paradoxical” fluid F. Two conditions are thus required to make d’Alembert’s result a genuine paradox: (1) there must be two plausible specifications that exclude one another, and (2) the two specifications must be applied to one and the same fluid.

Lamb avoided paradox by treating the two specifications as referring to different things. He drew a boundary between the referent of the experiment and the referent of the theory and thus rejected condition (2). In eschewing the word “paradox,” Lamb’s language was meant to carry a methodological message. It was a way of saying that viscous fluids were one thing and perfect fluids were another and never should the two be confused. This was an ad­mirably straightforward position, but was it the only tenable position? To ad­dress this question I consider a line of reasoning advanced by Ludwig Prandtl and Georg Fuhrmann in Gottingen. It will become clear that these experts did not distinguish between ideal and real fluids in precisely the same way as their British counterparts did.

Kutta ended with some ideas about extending his mathematical methods to a variety of different wing profiles with rounded leading edges and with flaps attached to the trailing edge. He mentioned the need to develop a more gen­eral form of the Schwarz-Christoffel theorem and indicated the demanding amount of computational effort that would be involved, but Kutta did not feel that the limits of his approach had been reached and hinted at their fur­ther application to biplanes.

Kutta’s next paper, in 1911, utilized the same mathematical techniques as those adopted in 1910 but it dealt with more complicated cases.33 The analysis was generalized in two ways. First, Kutta showed how to apply his conformal transformations to an aerofoil whose cross section was composed of not one but two circular arcs in the form of a crescent or sickle shape. Such a sickle­shaped profile was used in the successful Antoinette monoplane, and von Mises has suggested that its practical use was prompted by Kutta’s analysis.34 Second, Kutta generalized the approach in order to describe the flow over a number of wings that could be arranged to make a biplane or a triplane or even a multiwing arrangement in the form of a “Venetian blind.” Again, all of these forms had actually been used, or experimented with, by those who tried to build flying machines.

Having made these two outstanding contributions to aerodynamics, Kutta fell silent. He never published anything again. The reason for the silence is unknown.

The wartime activities of the Gottingen group, and their colleagues in the technische Hochschulen, proceeded on a much broader front than I have so far described.92 On the theoretical side, building on Betz’s early papers, there were studies by Betz, Munk, and Prandtl on the aerodynamics of biplanes and triplanes. Using the apparatus of the Biot-Savart law they produced some general theorems that helped to guide the aircraft designer through the maze of possible multiplane configurations. Betz had proven that, for an unstag­gered biplane, the induced drag effects of the wings on one another would be equal. For a staggered arrangement it was now shown that the sum of the mutually induced drags was constant and independent of stagger, provided that the lifts, and their distribution, were not changed. (This condition could be satisfied by changing the angle of attack.) In general, it was shown that the best biplane configuration was one with wings of equal length, with the up­per wing ahead of the lower. The biplane work also confirmed the important finding that elliptical lift distributions provided a good approximation for wings with non-elliptical planforms.93

On the empirical side, the war effort called for wind-channel studies of the drag generated by different aircraft components such as undercarriages and machine-gun mountings, engine-cooling systems, and the ubiquitous struts and bracing wires of the period.94 Work was also done on the lift and drag of the fuselage, the interaction between the fuselage and the wing, the effect of dividing the wing, the forces on fins and rudders, and the empirical properties of the triplane configuration.95 Experiments were done to test the resistance of the nose shapes required by different engine types, for example, rotary as compared with in-line engines, and attempts were made to add rotating propellers to the wind-channel models to achieve realism.96 Some wind-channel tests were also done on models of complete aircraft.97 Munk and Cario continued the studies initiated by Foppl of the downwash behind a wing. Whereas Foppl had worked with the overall force exerted by the down – wash on the elevator, Munk and Cario studied the downwash in much more empirical detail, using fine silk threads to trace the local variations. They un­covered significant complexities in the flow and made clear the need for a more extended program of work.98

Numerous studies were carried out to measure the lift and resistance of individual wing profiles. These were mainly overseen by Munk and his col­laborator Erich Huckel.99 Significant efforts were made to ensure that the re­sults were intelligible to those who might use them in practice.100 An attempt was also made to introduce order into the vast amount of data that had accu­mulated for different aerofoils, though the classification remained largely at the empirical level.101 One trend was toward an interest in thicker rather than thinner aerofoils, something that surprised the British when they examined captured German aircraft.102 An example of these thick aerofoils was the Got­tingen 298 used on the famous Fokker triplane. The use of the 298 profile by the designer Anthony Fokker does not, however, appear to have been a con­sequence of Prandtl’s recommendation or scientific knowledge of its good lift and drag characteristics (characteristics that, a priori and wrongly, the British designers doubted). In fact, the aerofoil was introduced into the Fokker pro­duction line by their chief engineer Reinhold Platz on the basis of trial-and – error knowledge. Later, and unknown to the people at Fokker, it was tested in Gottingen, where it was given its designation.103

The Fokker episode indicates that there was a continuing gap between the “practical men” of Germany and those self-consciously developing science- based procedures and working in academic and government institutions. The alienation of the practical men was not purely a British phenomenon, though it seems to have been less acute as a problem for German aviation than for British. Evidence in the technical reports indicates that the members of the Gottingen school were themselves aware of this gap and found it frustrating. Max Munk addressed the issue directly in a brief report of October 15, 1917, titled “Spannweite und Luftwiderstand” (Span and air resistance).104 Refer­ring to the practical conversion formulas linking wings of different aspect ratio, Munk complained:

Die kurzlich von Betz veroffentlichen Prandtlschen Flugelformeln werden wohl, da sie auf theoretischen Grundlagen beruhen, in der Praxis nicht so freundlich aufgenommen werden, wie sie verdienen. Das ist sehr schade, denn die Formeln enthalten mehr und leisten Besseres als der Praktiker geneigt ist, ihnen zuzutrauen. (199)

The formulas of Prandtl’s wing theory that Betz has recently published will probably not be welcomed in the realm of practice as much as they deserve because they rest on theoretical grounds. This is a great shame because the formulas offer more and give better service than the practical man is inclined to believe.

Munk went on to give an explanation of the significance of the formulas for the aircraft designer and some simple, general rules for the rapid calculation of the induced resistance and angle of incidence. Despite this evidence of skepticism in certain quarters, there was no shortage of contract work to be done for individual aircraft firms during the war years. This is attested by the frequency with which such names as AEG, Aviatik, Rumpler, Siemens and Schuckert, and Zeppelin were mentioned in the technical reports. Despite the problems of communication between the representatives of theory and practice, Prandtl’s institute had achieved a central position in what would now be called the military-industrial complex of Wilhelmine Germany. If this development brought frustrations as well as the advantages of government support, it is clear that striking progress had been made in aerodynamics, both empirically and theoretically.

One of the more unusual aerofoils whose properties were reported on by Munk and his colleagues was designated as profile 95. Visually it stood out from the usual run of aerofoil shapes (see fig. 7.16). The aerofoil looked like the back of a cat when the animal stretched, and it was duly given the nick­name Katzenbuckelflache. The Gottingen tests showed that the performance characteristics of the “cat’s-back” profile 95 were notably poor. It was tested by Max Munk and Carl Pohlhausen in the course of a run of work on nearly one hundred aerofoils. The results were listed together in the Technische Ber – ichte of August 1917 and showed that the maximum-lift coefficient for each wing in this sequence was typically in the region of 130 or 140. The maxi­mum lift coefficient for profile 95, by contrast, was given as 95.2. Again, the maximum lift-to-drag ratio was typically 14 or 15, while the ratio for profile 95 was 10.8.105 The designer of the cat’s-back wing was the celebrated physicist Albert Einstein.106 In retrospect Einstein felt that his excursion into aerody­namics had been irresponsible—he used the word Leichtsinn. From 1915 to

1917, Einstein had been a consultant to two aircraft firms, LVG and Merkur, and an aircraft had been equipped with the Einstein wing. The test pilot for LVG, Paul Ehrhardt, barely managed to get the machine off the ground and gave his professional opinion on the wing by saying that the airplane flew like a pregnant duck.107 The Gottingen tests made the same point in more scientific terminology.

No account remains of how Einstein actually designed the wing, but some insight into his thought processes may be gained from an article he published in 1916 in Die Naturwissenschaften. Here he set out to explain, in elemen­tary terms, the basic principles of lift.108 How does a wing support an air­craft and why can birds glide through the air? Einstein declared, “Uber diese Frage herrscht vielfach Unklarheit; ja ich mufi sogar gestehen, dafi ich ihrer einfachsten Beantwortung auch in der Fachliteratur nirgends begegnet bin” (400) (There is a lot of obscurity surrounding these questions. Indeed, I must confess that I have never encountered a simple answer to them even in the specialist literature). This is a striking claim, given that Einstein was writing a number of years after the publications of Kutta, Joukowsky, and Prandtl.

Einstein drew an analogy between the flow of fluid through a pipe of vari­able cross section and its flow around a wing. As fluid passes along a pipe that gets narrower, the fluid speeds up. By Bernoulli’s law the pressure will be lower in the fast, narrow section than in the broader section. Einstein then invited the reader to consider a body of incompressible fluid with no signifi­cant viscosity (that is, a perfect fluid) which flowed horizontally but where the flow was divided by a thin, rigid, dividing wall. The wall was aligned with the flow except that it also had a curved section where the wall bulged up­ward. (See fig. 7.17, which is taken from Einstein’s paper.) The curved section looks a bit like a cat’s back and would appear to have the same shape as the underside of the Einstein wing tested at Gottingen. Einstein argued that the fluid above the dividing wall will behave like the fluid in a pipe when it en­counters a narrowing of the pipe and will speed up and exert a diminished pressure on the wall. The fluid below the wall will behave like the fluid in a pipe when it encounters a widening out of the pipe so it will slow down and increase the pressure on the wall. The fluid pressure pushing upward on the curved section of the dividing wall will thus be in excess of the pressure push­ing downward, so there will be a resultant force upward All that is necessary now is to imagine that most of the dividing wall has been removed, leaving behind just the curved section. This procedure, argued Einstein, will retain the features of the flow that generate the pressure difference and hence will represent a wing with lift.

Einstein’s argument rested on the assumption that the removal of all but

the curved portion of the dividing wall would leave the flow unchanged at the leading and trailing edge of the remaining arc. He appeared to take this as ob­vious: “Um diese Kraft zu erzeugen, braucht offenbar nur ein so grofies Stuck der Wand realisiert zu werden, als zur Erzeugung der wirksamen Ausbiegung der Flussigkeitsstromung erforderlich ist” (510) (To generate this force it is obviously only necessary for part of the wall to be real. It need only be suf­ficiently large to produce the effective curvature of the flow). It is puzzling that Einstein made no mention of circulation. Was he aware of the circulation theory of lift? This remains unclear, but given that he assumed the air to be a perfect fluid, it makes it all the more important to ask how he proposed to circumvent d’Alembert’s paradox. How did Einstein expect to get a lift force rather than a zero resultant?

Although Einstein wrote in a dismissive way about the aerodynamic liter­ature, he had, in effect, taken the discussion back to where Kutta started it in 1902. When, following Einstein’s instructions, all of the wall dividing the flow is removed except for the curved piece, what is left is essentially Kutta’s arc at a zero angle of incidence. Einstein did not specify that the curve he discussed in his article was precisely the arc of a circle, but his argument was offered as a general one. If it were right it would apply to Kutta’s arc. But it does not. The arc is a counterexample to what seems to be Einstein’s argument, that is, to any argument that depends on ideal fluids but does not make provision for circulation. In order to generate lift, and to generate the requisite speed dif­ferential between the upper and lower surfaces of the arc, a circulation must be postulated. In a continuous perfect fluid flow, without an independently specified circulation, such an arc would not produce lift. D’Alembert’s para­dox would come into play. Such an arc would not have its stagnation points on the leading and trailing edge. The formula for the complex velocity has singular points indicating infinite velocities for the ends of the arc.109

In his diagram Einstein put the stagnation points on the leading and trail­ing edges of the arc even though he did not explicitly invoke a circulation. He put them where they would have been, given the appropriate amount of circulation needed to avoid infinite velocities at the edges. He did this by sup­posing that the guiding effect of the dividing wall, smoothly leading the fluid toward, around, and then away from the arc, would still be present even when the wall was removed and only the arc was left. There are no grounds for this assumption. Perhaps the best that can be said is that Einstein had, in fact, made provision for circulation in his analysis, but had done so tacitly and by questionable means. In his published lectures, a few years later, Prandtl gave a diagram that was almost identical to Einstein’s but, in Prandtl’s case, the com­ponent of circulation in the flow was properly identified and made explicit.110 Prandtl’s diagram is reproduced as figure 7.18 for purposes of comparison with Einstein’s figure.

It would seem that Einstein had little knowledge of current developments in the field of aerodynamics. This episode is a salutary reminder of the differ­ence between fundamental physics and technical mechanics. Eminence in the former does not guarantee competence in the latter.

I want to dwell for a moment on the significance of this transition from the­ory to fact. It is a phase-change that has often taken place in the history of science. Assertions to the effect that, say, the blood circulates in the human body, or that water is made of hydrogen and oxygen, may once have been speculations but today can be taken as matters of fact. It would be a question­able use of language to keep calling them theories. How is this change of sta­tus, from theory to fact, best described? One attractive answer was provided in the form of a striking metaphor used by the Cambridge mathematician William Kingdon Clifford.34 Clifford, who had been a second wrangler in 1867, a fellow of Trinity, and a friend of Rayleigh, had wide-ranging interests in mathematical physics. In a lecture he gave in Manchester (some fifty years before the experiments that concern us), Clifford took as his example not hydrodynamics but the wave theory of light. This conception of light, he said, must now be accepted as fact. The difference between a theory and a demon­strated fact, he went on, “is something like this”:

If you suppose a man to have walked from Chorlton Town Hall down here say in ten minutes, the natural conclusion would be that he had walked along the Stretford Road. Now that theory would entirely account for all the facts, but at the same time the facts would not be proved by it. But suppose it happened to be winter time, with snow on the road, and that you could trace the man’s footsteps all along the road, then you would know that he had walked along that way. The sort of evidence we have to show that light does consist of waves transmitted through a medium is the sort of evidence that footsteps upon the snow make; it is not a theory which merely accounts for the facts, but it is a theory which can be reasoned back to from the facts without any other theory being possible. (117)

The thought is that if you can track a process in great detail, and see it de­velop step by step, then you can reach a stable understanding that is unique, unchallengeable, and enduring. Such an understanding, said Clifford, deals with facts, not theories.

Clifford’s metaphor may be a tempting one, but it cannot be wholly right, and it led Clifford himself astray. The development of physics showed that what he thought of as a demonstrated fact was actually a theory. The alleged impossibility of any alternative to the wave theory was refuted by the emer­gence of an alternative. In Clifford’s time the wave theory had superseded an older particle theory, but in 1905 Einstein once again postulated light par­ticles. These new-style particles or “photons” were invoked to explain the photoelectric effect, something that was proving difficult to understand in terms of waves. The photoelectric effect takes place when light is incident on a metal surface and releases electrons from the surface. How could the energy spread across a wave front be concentrated in the way necessary to release a charged particle? This was the problem Einstein’s theory was designed to answer. The light energy, he argued, was concentrated because light consists not of waves, but of particles, albeit particles with unusual properties.35 These developments took place after Clifford’s death. He cannot be blamed for not anticipating them, but they amount to a counterexample to Clifford’s argu­ment and show the need to introduce qualifications into his overconfident picture.

What was Clifford’s error? It was that of assimilating fundamental scien­tific inquiry to commonsense knowledge. While there are many similarities and connections, Clifford ignored a crucial difference. Everyone has seen, or could see, a man creating footprints in snow. The cause and the effect can be conjoined in experience, and both are open to inspection. This is the basis of subsequent inferences from footprints to their human causes and the basis of the conclusions that can be drawn about, say, the route someone had taken from Chorlton Town Hall to the location of Clifford’s lecture. The physicist, on the other hand, did not come to the wave theory of light by seeing light waves creating diffraction patterns or rainbows. The two things were not con­joined in experience in the way people and footprints have been conjoined. The inference to light waves did not have the same inductive basis as the com – monsense inference with which Clifford was comparing it.36

Clifford’s metaphor may have broken down for light waves, but it might still be applicable to Fage and Simmons’ achievement. It could be argued that Fage and Simmons were confronting the vortices and observing them bringing about their effects. Was not this conjunction precisely what the ex­periment was designed to expose? Even if the experimenters could not actu­ally see the flow of air, they could have made it visible, and others had done exactly that. In any case, the diagrams showing the streamlines of the vortices and the contour lines of equal vorticity allowed them to follow the path and development of the postulated vortices. The experimenters could set these diagrams side by side with the measured lift forces on the wing. Causal con­nections and correlations of phenomena that were originally speculative had, in a sense, been exposed to view, and the step-by-step progress of the cir­culation had been traced. Perhaps tracking the vortices through the pattern of measurements registered in Fage and Simmons’ diagrams was, after all, similar to tracking footprints in the snow.

If Clifford’s metaphor is applicable to the aerodynamic work, does this mean that the question “How does a wing produce lift?” can now be answered with the same level of certitude as the question about the man walking down the Stretford Road? In a sense, yes, it does. The phenomena of circulation, vortices, and lift had been made, or were on the way to being made, part of the routine and reality of daily life. At least, this was true for the laboratory life of some of the experimentalists working in this area. They were becoming increasingly familiar with the patterns in the data and the range of effects to be accounted for. Expectations were crystallizing, and experimenters were learning what they could take for granted. Techniques of calculation and pre­diction were becoming more confident and refined. What was once strange was becoming familiar and part of predictable, daily experience—like getting to know a new town. Learning to live with the theory of circulation was like learning to live in a new environment with new architectural styles and a new street plan. You want to get to Prof. Clifford’s lecture starting out from Chor – lton Town Hall? Then go down the Stretford Road! You want to calculate the induced drag? Then use Prandtl’s formula!37

Significantly, this was not yet how some of the most influential British experts saw the issue. They acknowledged that Fage and Simmons’ results represented a triumph of sorts for Lanchester, Prandtl, and Glauert, but they did not accept that the answer was now known to the question How is lift produced? On the contrary, they maintained that, despite the experimental advances and the increase in empirical knowledge, the answer to this ques­tion remained wholly unknown. Many questions, they acknowledged, had now been answered, but not this one. These experts were not simply being stubborn or blind in the face of mounting evidence, and their reaction un­derlines just how careful one must be in applying Clifford’s metaphor. It must be accepted that what looks like demonstrated fact from one point of view may appear less compelling or revealing from another point of view. This skeptical response to the mounting experimental evidence was articu­lated with great clarity by Richard Vynne Southwell (fig. 9.10). Southwell has already been mentioned in connection with the postwar contact with Prandtl and Gottingen, but it is appropriate to look more closely both at the man and at his response to the growing experimental literature.

In 1954 Philipp Frank published an article in the Scientific Monthly called “On the Variety of Reasons for the Acceptance of Scientific Theories.”50 He drew the striking conclusion that “the building of a scientific theory is not essentially different from the building of an airplane” (144). I will use Frank’s argument to comment on the theories developed in fluid dynamics and aero­dynamics, but first I should say a little about Frank himself.51 From 1912 to 1938 he was the professor of theoretical physics at the German University of Prague. A pupil of Boltzmann, Klein, and Hilbert, Frank had taken over the chair from Einstein when Einstein received the call to Zurich and then to Berlin. He had attended Einstein’s seminars in Prague, and Einstein strongly supported his appointment.

In their student days, before World War I, Frank and von Mises talked philosophy in their favorite Viennese coffeehouse and together played a seminal role in the formation of the Vienna Circle.52 In the interwar years, as established academics, Frank and von Mises jointly edited a book on the differential and integral equations of mechanics and physics, Die Differential – und Integralgleichungen der Mechanik und Physik,53 which brought together a range of distinguished contributors. Von Mises edited the first volume on mathematical methods, while Frank handled the second, more physically oriented, volume which included chapters by Noether, Oseen, Sommerfeld, Trefftz, and von Karman, who wrote on ideal fluid theory.54 The Frank-Mises collection, which was an update of a famous textbook by Riemann and Weber, established itself as a standard work in German-speaking Europe.55 In 1938 Frank was forced to leave Prague because of the threatening political situation in Europe, and he went to the United States. During and after World War II, he taught physics, mathematics, and the philosophy of science at Harvard.

Like that of von Mises, Frank’s philosophical position was self-consciously “positivist” in the priority given to empirical data and the secondary, instru­mental role given to theoretical constructs. Frank admired Ernst Mach as a representative of Enlightenment thinking, though his admiration was not uncritical, and he did not go along with Mach’s rejection of atomism.56 Much of Frank’s philosophical work was devoted to the analysis of relativity theory, quantum theory, and non-Euclidian geometry.57 He was a firm believer in the unity of science and rejected the idea that there was a fundamental divide between the natural and human sciences.58 He also insisted on the need to understand science as a sociological phenomenon. The sociology of science was part of “a general science of human behaviour” (140)—a theme central to the Scientific Monthly article.59

Frank asserted that most scientists, in their public statements, assume that two, and only two, considerations are relevant when assessing a scientific the­ory. These are (1) that the theory should explain the relevant facts generated by observation and (2) that it should possess the virtue of mathematical simplic­ity. Frank then noted that, historically, scientists (or those occupying the role we now identify as “ scientist”) have often used two further criteria. These are

(3) that the theory should be useful for technological purposes and (4) that it should have apparent implications for ethical and political questions. Does the theory encourage or undermine desirable patterns of behavior, either in society at large or in the community of scientists themselves? Such questions are often presented in a disguised form, for example, Is the theory consistent with common sense or received opinion or does it flout them? Common sense and received opinion, Frank argued, typically fuse together a picture of nature and a picture of society. The demand for consistency then becomes a form of social control that can be used for good or ill.

In Frank’s opinion it is naive to believe that theory assessment can be confined to the two, internal-seeming criteria. He offered three reasons. First, he noted that no theory has ever explained all of the observed facts that fall under its scope. Some selection always has to be made. Second, there is no unproblematic measure of simplicity. No theory has “perfect” simplic­ity. Simplicity will be judged differently from different, but equally rational, perspectives, depending on background knowledge, goals, and interests. Third, criteria (1) and (2) are frequently in competition with one another. The greater the number of facts that can be explained, or the greater the ac­curacy of the explanation, the more complicated the theory must be, while the simpler it is, the fewer are the facts that can be explained. Linear functions are simpler than functions of the second or higher degree, which is why phys­ics is full of laws that express simple proportionality, for example, Hooke’s law or Ohm’s law. “In all these cases,” wrote Frank, “there is no doubt that a nonlinear relationship would describe the facts in a more accurate way, but one tries to get along with a linear law as much as possible” (139-40). What is it to be: convenience or truth? Nothing within the boundaries of science itself, narrowly conceived, will yield the answer. This is why scientists have always moved outside criteria (1) and (2), and, consciously or unconsciously, invoked criteria of types (3) and (4).

These unavoidable choices and compromises tell us something about the status of any theory that is accepted by a group of scientists. “If we consider this point,” said Frank, “it is obvious that such a theory cannot be ‘the truth’” (144). But if the chosen theory is not “the truth,” what is it? Frank’s answer was that a theory must be understood to be “an instrument that serves to­ward some definite purpose” (144). It is an instrument that sometimes helps prediction and sometimes understanding. It can help us construct devices that save time and labor, and it sometimes helps to mediate a subtle form of social control. “A scientific theory is, in a sense, a tool that produces other tools according to a practical scheme” (144), he concluded. Like a tool, its connection to reality is not to be understood in terms of some static relation of depiction but in active and pragmatic terms. Its function is to give its us­ers a grip on reality and to allow them to pursue their projects and satisfy their needs—but it does so in diverse ways. It was at this point that Frank produced his comparison between assessing a theory in science and assessing a piece of technology, such as an airplane. Writing, surely, with the perfor­mance graphs of von Mises’ Fluglehre before his mind, he argued:

In the same way that we enjoy the beauty and elegance of an airplane, we also enjoy the “elegance” of the theory that makes the construction of the plane possible. In speaking about any actual machine, it is meaningless to ask whether the machine is “true” in the sense of its being “perfect.” We can ask only whether it is “good” or sufficiently “perfect” for a certain purpose. If we require speed as our purpose, the “perfect” airplane will differ from one that is “perfect” for the purposes of endurance. The result will be different again if

we chose safety. . . . It is impossible to design an airplane that fulfils all these purposes in a maximal way. (144)

It is the trade-off of one human purpose against another that gave Frank his central theme. Only by confronting this fact can the methods of science be understood scientifically. It is necessary to ask in the case of every scientific theory, as one asks in the case of the airplane, what determined the policy according to which these inescapable compromises are made and how well does the end product embody the policy? We must understand what Frank called, in his scientistic terminology, “the social conditions that produce the conditioned reflexes of the policy-makers” (144).60

In Frank’s terms, Lanchester’s metaphor of playing chess with nature as well as my sociological analysis are ways of describing scientific “policies.” Just as there were policy choices made over the relative importance of stabil­ity and maneuverability, and policy choices about how to distribute research effort between the theory of stability and the theory of lift, so within the pur­suit of a theory of lift there were policy decisions to be made. My analysis identifies one policy informing the Cambridge school and another policy guiding the Gottingen school. Again using Frank’s terms, the members of the respective schools constructed different technologies of understanding, that is, different theoretical “instruments.” Their policies, when construct­ing their theories, maximized different qualities and furthered different ends. The British wanted to construct a fundamental theory of lift, whereas the Germans aimed at engineering utility. Who were the “policy makers”? One might identify, say, Lord Rayleigh as the “policy maker” in Britain and Fe­lix Klein as the “policy maker” in Germany, but there is no need to assume that policy is made by individuals. Such a restriction would not correspond to Frank’s intentions; nor is it part of my analysis. Policies can emerge col­lectively. They can be tacitly present in the cultural traditions and research strategies of a scientific group. One could then say that everyone is a policy maker by virtue of their participation in the group, or one could say that the policy maker is the group itself. In my example the “social conditions” that determine the “conditioned reflexes of the policy-makers” reside in the divi­sion of labor between physicist and engineer.

One implication of Frank’s “policy” metaphor is that a stated policy need not correspond to an actual policy. The devious history of aircraft construc­tion in post-World War I Germany provides some obvious examples. Is this large aircraft really meant as an airliner or is it a bomber? Is this an aero­batic sports plane or a disguised fighter? Is all this enthusiasm for gliding just recreation or a way of training a future air force—and keeping the nation’s aerodynamic experts in a job?61 The difficulty of distinguishing a real from an apparent policy comes from the problematic relation between words and deeds. Sometimes the self-descriptions and methodological reflections of members of the Cambridge school could sound similar to those of German engineers. Both Lamb and Love occasionally invoked the ideas, and some­times the name, of Ernst Mach, but that did not make Lamb into a positivist, nor turn Love’s work on the theory of elasticity into technische Mechanik. Their real policy lay elsewhere.

In an address to the British Association in 1904, Lamb acknowledged that the basic concepts of physics, geometry, and mechanics were “contrivances,” “abstractions,” and “conventions.”62 But Lamb soon left behind this unchar­acteristic indulgence in philosophizing and turned the discussion back to the work of his old teacher, G. G. Stokes. He spoke warmly of “the simple and vigorous faith” that informed Stokes’ thinking.63 Lamb then raised the metaphysical question of what lay beyond science and justified faith in its methods. Why, as Lamb put it, does nature honor our checks? He gave no explicit answer, but the theological hint was obvious. Lamb also distanced himself and the Cambridge school from the “more recent tendencies” in ap­plied mathematics. He deplored the fragmentation of the field and regretted the passing of the large-scale monograph, which was a work of art, in favor of detailed, specialized papers. What differentiated the Cambridge school, he went on, related “not so much to subject-matter and method as to the gen­eral mental attitude towards the problems of nature” (425). It is this “general mental attitude” that constitutes the real policy.

How is an authentic “mental attitude” to be filtered out from misleading forms of self-description? The answer is: by looking at what is done and at the choices that are made. Words must be supported by actions. Bairstow, Cow­ley, Jeffreys, Lamb, Levy, Southwell, and Taylor not only gave their reasons for resisting the ideal fluid approach to lift, but they acted accordingly. This is why, in previous chapters, I have identified the mental attitude that informed the work of the Cambridge school and its associates as a confident, physics – based realism rather than a skeptical positivism. Stokes’ equations were not only said to be true, but they were treated as true. This was the attitude and policy that Love expressed by invoking the role of the “natural philosopher” rather than the engineer. And this was why Felix Klein, in his 1900 lecture on the special character of technical mechanics, could express admiration for Love’s treatise on elasticity and yet pass over it because it could not be taken as an example of technische Mechanik.64

Simplicity and the Kutta-Joukowsky Law

I now apply Frank’s ideas to the Kutta-Joukowsky law: L = p LT, where the lift (L) is equated to the product of the density (p), the speed (U), and the circulation (Г). The law is certainly simple, but what is the meaning of this simplicity? Is it a sign of the “deep” truth of the law and hence a quality that should command a special respect? The idea that nature is “governed” by simple mathematical laws is a familiar one—it goes back to the origins of modern science—but positivists have no time for this sort of talk.65 Frank could have pointed out that the simplicity, and apparent generality, of the Kutta-Joukowsky law derives not from its truth, but from its falsity and from everything that it leaves out of account. The law says nothing about the rela­tion between the shape of the aerofoil and the amount of lift. It contributes nothing to the problem of specifying the amount of circulation and (when used in conjunction with the Kutta condition) gives predictions for the lift that are consistently too high. The law cannot, in any direct or literal way, represent something deep within reality because its individual terms do not refer to reality. They refer to a nonexistent, ideal fluid under simplified flow conditions.

Frank would predict that if an attempt were made to repair the law, and make allowance for some of the factors that have been ignored, then the re­sult would no longer possess the impressive simplicity of the original. This was precisely what happened when, in 1921, Max Lagally of the technische Hochschule at Dresden, produced an extension of the Kutta-Joukowsky for­mula.66 Lagally exploited a result arrived at previously by Heinrich Blasius, one of Prandtl’s pupils, and this result needs to be explained first in order to make sense of Lagally’s formula. Blasius had developed a theorem, based on the theory of complex functions, that allowed the force components X and Y on a body to be written down as soon as the mathematical form of the flow of an ideal fluid over the body had been specified.67 In these terms, for a uniform, irrotational flow U along the x-axis, with a circulation Г, the Kutta – Joukowsky theorem takes the form

X – iY = ipU Г.

Here X, the force along the x-axis, represents the drag, while Y is the force along the y-axis and represents the lift. The letter i is a mathematical opera­tor. The right-hand side of the equation economically conveys the informa­tion that the drag is zero (because X = 0) and also that the lift obeys the Kutta-Joukowsky relation (because Y = pU Г). Blasius’ derivation of this re­sult depended on there being no complications in the flow. Lagally added some complications in order to see what effect they would have. In Lagally’s analysis the main flow has a horizontal component U and a vertical compo­nent V. More important, he assumed that there were an arbitrary number of sources and an arbitrary number of vortices in the fluid around the body. He specified that there were r sources located at the points ar where each source had a strength mr, and s vortices located at points cs where each vortex had vorticity Ks. When the formula was adjusted to allow for these conditions, it looked like this:

X — iY = —ipK(U — iV) + 2np^mr (u r — iv r + U — iV) — ip^Ks (us — ivs +U — iV),

where ur and vr are the components of velocity at ar (omitting the contribu­tion of mr) and us and vs are the velocity components at cs (omitting the con­tribution of Ks). The original Kutta-Joukowsky formula can be seen embed­ded in Lagally’s formula on the immediate right of the equality sign.68

If the original Kutta-Joukowsky relation could be admired for its elegance, like the sleek lines of a modern aircraft, can this be said of Lagally’s formula? I doubt if it attracted much praise on this score. But if the long formula really is an improvement on the short one, why shouldn’t it be seen as more beautiful? If we do not find it beautiful is it because we can’t imagine such complicated mathematical machinery “governing” reality? Frank and his fellow positivists would not want the question to be pursued in these metaphysical terms. They would say: If there is something important about the simplicity of the origi­nal formula L = p UT, then look for the utility that goes with simplicity. What does it contribute to the economy of thought? This question will expose the real attraction of simplicity and explain what might have been lost, along with what has been gained, by Lagally’s generalization.

Frank called a theory a tool that produces other tools according to a practi­cal scheme. He meant that the simple law provides a pattern, an exemplar, and a resource that is taken for granted in building up the more complex formula.69 This is how Lagally built his generalization, and if Frank is right, other scientists and engineers, interested in a different range of special condi­tions, will follow a similar path. This pattern fits what I have found. Recall the way Betz experimentally studied the deviations between the predictions of the circulatory theory and wind-tunnel observations. He sought to close the gap between theory and experiment by retaining the Kutta-Joukowsky law while relaxing the Kutta condition, that is, the understanding that the circula­tion is precisely the amount needed to position the rear stagnation point on the sharp trailing edge. Again, recall the later episode in which, prompted by the work of G. I. Taylor, the condition of contour independence was relaxed so that a “circulation” could be specified for a viscous flow. In both these examples the development exploited the same resource as Lagally, that is, the simple law was retained as a basic pattern. Simple laws are a shared resource and an accepted reference point. They are used when a group of scientists are striving to coordinate their behavior in order to construct a shared body of knowledge. They are salient solutions to coordination problems, which may explain the obscure “depth” attributed to them. The depth is a social, not a metaphysical, depth.70