Category The Enigma of. the Aerofoil

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After the Dunne episode it had been decided to drop aircraft research at Farnborough, but this resolution was now rescinded.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.

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

Writing in 1920, R. T. Glazebrook, the director of the National Physical Labo­ratory, recalled the events of 1909 and the inception of the Advisory Commit­tee.29 Mr. Haldane, he said, “appealed to Lord Rayleigh and myself to know if we could help at the National Physical Laboratory. A scheme of work was suggested, and at a meeting at the Admiralty at which Mr. McKenna, then First Lord, and Mr. Haldane were present, the details were agreed upon” (435). Haldane had made the acquaintance of John William Strutt (Lord Rayleigh) while they had worked together on an earlier committee, the Ex­plosives Committee, developing an improved and less corrosive propellant for the artillery.30 Rayleigh was a world-renowned physicist with formidable mathematical powers. He had published on aeronautical themes and had made fundamental contributions to fluid mechanics, the branch of physics that might explain how the flow of air over a wing could keep the machine in the air.31 Rayleigh was to become the president of the Advisory Committee for Aeronautics.

Given that Rayleigh was already the president of the National Physical Laboratory, so Glazebrook (fig. 1.3), as the director of the NPL, was an obvi­ous choice to work under him as the chairman of the Advisory Committee for Aeronautics. What of the other members? There were, of course, repre­sentatives of the Admiralty and the War Office. These were Major General Sir Charles Hadden for the army and Captain (later Admiral) R. H. S. Bacon

for the navy. (Haldane knew them both from the Committee of Imperial De­fence.) Bacon was soon to be joined (and then replaced) by Captain Mur­ray Seuter. Mervin O’Gorman, an engineer, joined the committee after his appointment as the new head of the Balloon Factory and then the Aircraft Factory.32 The remaining six members were Sir George Greenhill (mathema­tician), Dr. W. Napier Shaw (physicist and meteorologist), Horace Darwin (brother of Sir George Darwin and the founder of the Cambridge Scientific Instrument Company), H. R. A. Mallock (physicist), Prof. J. R. Petavel (en­gineer), and F. W. Lanchester (an engineer who had published a pioneering book on aerodynamics).33 The secretary of the committee was F. J. Selby, who had got to know Glazebrook when he went up to Trinity in 1888 and took classes from Glazebrook in physics and mathematics. In 1903 Selby joined the staff of the National Physical Laboratory and acted as Glazebrook’s personal secretary.34 The assistant secretary was J. L. Nayler, again of Cambridge and the NPL, who would coauthor a number of the early experimental reports.

Over fifty years after the founding of the ACA, Nayler gave a talk in which he recalled some of the personalities involved.35 He brought out clearly the closely knit character of the core group of scientists and the intellectual tra­dition to which they belonged. Glazebrook and Shaw, he said, had been as­sistants to Rayleigh in Cambridge, when Rayleigh had taken over the Caven­dish laboratory after the death of Clerk Maxwell. Glazebrook and Shaw had written textbooks of practical physics together, though “they both started as mathematicians.”36 Mallock had also worked as an assistant to Rayleigh.37 Nayler went on: “Seven out of the twelve were Fellows of the Royal Society and another became a Fellow later on, three were serving officers, two were heads of aeronautical establishments. Six, including the Secretary, began as wranglers, and five were trained engineers” (1045). The term “wrangler” is an old Cambridge label for a student who came at the top of the result list in the university’s highly competitive mathematical examination called the Math­ematical Tripos. The senior wrangler was at the very top of the list, followed by the second wrangler, and so on. Nayler notes that Rayleigh was a senior wrangler in 1865. Glazebrook and Shaw “were wranglers in the same year, 1876,” while Greenhill was “a second wrangler and Smith Prizeman” (1045). The award of the Smith’s Prize was an opportunity for the two or three top scorers in the Tripos to meet in a final contest in the battle for mathematical supremacy.

Nayler’s talk was given at Cambridge, which may explain some of the care taken to delineate the connections between the Advisory Committee mem­bers and the Mathematical Tripos. But we should not miss the significance or the specificity of the message. It would be difficult to overstate the centrality

of the Mathematical Tripos to the scientific life of Cambridge at the end of the nineteenth and the beginning of the twentieth centuries. Andrew Warwick’s impressive study Masters of Theory: Cambridge and the Rise of Mathematical Physics leaves no doubt about the intensity and brilliance of the Tripos tra­dition.38 In many areas of science in Cambridge, success in the Tripos was a precondition of scientific and academic preferment. The order of merit was published in the Times, and the senior wrangler of the year achieved celebrity status and, invariably, the offer of a college fellowship. Such success could only be achieved by intense coaching from one of a select band of brilliant but demanding tutors—men such William Hopkins, Edward John Routh, William Besant, R. R. Webb, and Robert Herman.39 Perhaps the greatest of all the coaches was Routh, whose caliber as a mathematician can be judged from the fact that, as a student, he had pushed James Clerk Maxwell into sec­ond place in the Tripos of 1854. Rayleigh had been coached by Routh, while Greenhill had been coached by Besant.

The top wranglers in turn would then become the examiners and the coaches for the next generation of students facing the rigors of the Senate House examinations. Lower-placed wranglers would often become school­masters and prepare their charges for mathematical scholarships to Cam­bridge, thus ensuring that each generation of students was better prepared than the last. This system increased the competition and forced up the stan­dards still further. So extraordinarily high did these standards become that examiners would use their current research and their latest discoveries as the basis for their questions. The most celebrated example was a result that subsequently played an important role in the mathematical underpinning of aerodynamics. The mathematical equation, usually called Stokes’ theorem, relating circulation and vorticity, first appeared in print in the Smith’s Prize examination of 1854. The candidates were required to prove the theorem. Stated in words, the theorem is that “the circulation round the edge of any finite surface is equal to the sum of the circulations round the boundaries of the infinitely small elements into which the surface may be divided.”40 (The meaning of the technical terms involved, such as “circulation,” are explained when the circulation theory of lift is introduced in chap. 4.)

The Tripos system certainly had its critics. It placed ambitious students under great strain, and many young minds found the demands too great. Critics also argued that the questions were too difficult for all but the best, and to rectify the problem various reforms and modifications were intro­duced over the years. The last order of merit list was published in 1909; there­after, candidates were placed in classes and the names listed alphabetically within the classes. There were also those who argued that the difficulty of the questions was because they were artificial and contrived so that their answers called for the mere mastery of technical tricks that had little educational value. This line taken in 1906 by G. H. Hardy, although his claims should be put in context. They were made in the course of arguments about the reform of the syllabus.41 Hardy was pressing the case for pure mathematics and rigorous foundations as against the applied mathematics and mathematical physics of the traditional Tripos.42 It was the more traditional form of the Tripos that informed the training of Rayleigh, Glazebrook, Shaw, Greenhill, and others, such as Horace Lamb, who later joined the Advisory Committee. Lamb had been coached by Routh and was second wrangler in 1872.

The scientific character of the Advisory Committee for Aeronautics, on which Haldane placed so much emphasis, was clearly weighted toward the Cambridge tradition of mathematical physics. In this connection, recall the three names that Haldane mentioned as possible leaders of the committee: Rayleigh, Moulton, and Darwin. Nayler noted that Rayleigh had been a se­nior wrangler, and to this one may add that Darwin, who had discussed mat­ters with Haldane on the Saturday before the Esher committee had met, was himself second wrangler in 1868.43 What about Lord Justice Moulton? He looks the odd one out. In fact John Fletcher Moulton was senior wrangler in 1868, soundly beating George Darwin into second place and achieving higher marks than any previous Tripos candidate.44 Rayleigh, Moulton, and Darwin had all been coached by Routh. It would be wrong to say that the ACA was, or was meant to be, simply a committee of wranglers. Lanchester was no math­ematician, nor were the military men, but it would not be an exaggeration to say that Cambridge wranglers were a powerful, and perhaps predominant, presence. Whether by accident or design the scientific culture of the Advi­sory Committee was, to a significant degree, the culture of the Mathematical Tripos.45

I have shown how, in order to generate and solve the equations of motion of an ideal fluid, all manner of simplifications had to be introduced. What price, if any, had to be paid for the advantages that the simplifications brought with them? What does it cost, for example, to bring the investigation under the scope of Laplace’s equation and confine the theory to motion in which fluid elements do not rotate? Those who developed classical hydrodynamics hoped that the price would not be a large one. The hope was reasonable because wa­ter and air were only slightly viscous. Unfortunately, the transition from small viscosity to zero viscosity sometimes had a very large effect on the analysis. In important respects the difference between real fluids and the theoretical behavior of a perfectly inviscid fluid was dramatic. The price of the approxi­mation was high, and it was extracted in a surprising way.

Suppose that you hold a small, rectangular piece of thin cardboard by one of its edges, for example, a picture postcard held by its shorter edge. Move the card rapidly through the air so that the card faces the flow head on (not edge­wise). It is easy to feel the resistance to the motion, and (within a Newtonian framework) this means feeling the force that the motion through the air ex­erts on the card. In the same terms, one can also see what must be the effect of the force by the way that the card bends. Experts in aerodynamics want to be able to calculate the magnitude and direction of the force that is so evidently present. A good theory would furnish them with an accurate description of the characteristics of the flow and an explanation of how the flow generates the forces. The theory should permit answers to questions such as: Does the air flow smoothly around the card? What are the streamlines like? Does the card leave a turbulent wake in the air? What happens at the edges of the card? How does the force vary with the angle at which the card is held, that is, with the “angle of attack”?

It turns out that if the air were a perfect fluid, there would be no resultant force at all on the card. The simplifications led to a mathematically sophis­ticated analysis but also to a manifestly false prediction. Why is this? The answer can be seen by looking at the form of the flows that are generated under the idealized conditions assumed by the mathematician. The (theoreti­cal) flow of an ideal fluid in irrotational motion over the postcard or lamina normal to the flow looks like the flow sketched in Cowley and Levy’s diagram shown in figure 2.5. In the diagram the card is represented as fixed, and the ideal fluid that stands in for the air is shown approaching the card and mov­ing around it. This example differs from the experiment in which the card moves through the air (which is assumed to be at rest), but scientists typically prefer to adopt this convention. The justification is that dynamically the two things are equivalent. As far as the forces are concerned, all that matters is the relative motion of the air and the obstacle. Pretending that the card is still and the air moves, rather than the other way round, turns out to be easier because seen from the standpoint of the card the flow is steady. It also makes the dia­gram fit more closely to experiments that are done in wind tunnels.

From figure 2.5 it can be seen that half of the air (or ideal fluid) that im­pinges on the front face moves away from B, the front stagnation point, up to the edge C, while the other half (not shown) will move down to the edge C. The fluid then curls sharply around the corner at each of these edges and approaches the point D (the rear stagnation point) and then continues on its way. The fluid farther from the plate follows a similar path to the fluid near the plate but with less abrupt changes of direction. At the stagnation points the lines representing the flow meet the surface of a body and can be thought of as splitting into two in order to follow the upper and lower contour of the body. At a stagnation point, mathematical consistency is preserved by tak­ing the direction of the flow to be indeterminate and the speed of the flow to be zero.

Inspection of the diagram for the steady flow around the flat plate as shown in figure 2.5 allows the direction and some indication of the speed of the flow to be read off. It can be seen that the flow moves rapidly around the edges of the plate. Inspection also shows that the flow is symmetrical about an axis that lies along the plate as well as being symmetrical about an axis that is normal to (that is, at right angles to) the plate. (A mathematician would spot the symmetry from the equation for the streamline because all the x – and y-terms appear as squares.) These symmetries have important consequences for the pressure that the flow exerts on the plate. According to Bernoulli’s law, as the fluid impinges on the plate and is brought to a halt, it exerts its maximum pressure. As it moves along the plate and gathers speed, it exerts a lesser pressure. Because the fluid is perfectly free of viscosity, there will be no tangential traction on the plate and all the forces will be normal to the plate. The pressure on both the front and the back will be high near the stagnation points and low near the edges of the plate. The symmetry of the flow around an axis along the plate means that the pressures exerted on the front of the plate will be of the same magnitude as the pressures exerted on the back of the plate. The pressures will be in opposite directions and will thus cancel out. There will be no resultant force.

The forces on a plate moving relative to a body of ideal fluid are there­fore fundamentally different from those on a plate (such as the postcard) moving relative to a mass of real air. Both experiment and everyday experi­ence stand in direct contradiction to the mathematical analysis. Treating a slightly viscous fluid (such as air) as if it were a wholly inviscid fluid may have seemed a small and reasonable approximation, but the effect is large. Neglecting a small amount of compressibility caused no trouble; neglecting a small amount of viscosity proved vastly more troublesome. The disconcert­ing conclusion that the resultant force is zero does not just apply to a flat plate running head-on against the flow. Consider again the flow around a circular cylinder. This was Cowley and Levy’s other textbook example and was shown

in figure 2.4. The closeness of the streamlines indicates that the flow speeds up as it passes the top and bottom of the circular cylinder. Since the fluid is free of any viscosity, all the pressure on the cylinder will be directed toward the center. The symmetry of the flow means that any pressure on the cylin­der will be directly counteracted by the pressure at the diametrically oppo­site point on the cylinder. Again, counter to all experimental evidence from cylinders in the flow of a real fluid, there will be no resultant force on the cylinder. In reality the pressure distribution is not the same on the front and rear faces.37

Do these results depend on the obstacle and the flow possessing symme­try? The answer is no. The results apply to objects of all shapes and orienta­tions. Consider the flow around a flat plate that is positioned not normal to the flow but at an oblique angle to the flow. The situation is represented in figure 2.6.

Such a flow introduces certain additional complexities into the analysis, but the outcome is still a zero resultant force. The extra complexity is that the forces at points on immediately opposite sides of the plate are not equal, which can be seen from the way the front and rear stagnation points are not directly opposite one another. The front stagnation point is near to, but be­neath, the leading edge, while the rear stagnation point is near to, but above, the trailing edge. As a result the plate is subject to what is called a “couple” that possesses a “turning moment.” A “couple” arises from two forces that are equal and opposite but act at different points. Here they exert leverage on the plate, causing it to rotate and to turn so that it lies across the stream. As the plate rotates, the two stagnation points move. The front stagnation point moves away from the leading edge toward the center point of the front of the plate. The rear stagnation point moves away from the trailing edge to the center of the back of the plate. When the plate is lying across the stream, the two stagnation points are directly opposite one another and there is no more leverage they can exert. For the inclined plate, then, there

is an effect produced by the forces, but it is still true that there is no resul­tant force. A force at a given point on one surface of the plate will still have a force of equal magnitude and opposite direction at some corresponding point on the other face. Overall, the forces will still sum to zero, as they did for the circular cylinder or the plate that was initially positioned normal to the flow.

Similar considerations apply to any complex shape and would therefore also apply to a shape chosen for an aircraft wing. If the air were an ideal fluid, there might be a turning effect exerted on the wing but there would be no resultant aerodynamic force. There would be neither lift nor drag. The zero-resultant theorem for plates and cylinders had been established many years before the practicality of mechanical flight had been demonstrated and had long been a source of some embarrassment. Interpreted as a pre­diction about either air or water, its falsity was evident, but it continued to haunt the theoretical development of aerodynamics. In France this old and well-established result was called d’Alembert’s paradox, in Germany it was called Dirichlet’s paradox, and in Britain it wasn’t called a paradox at all. Re­member that, for Cowley and Levy, the mathematics just defined the nature of the fluid. The zero-resultant theorem simply establishes that this is how an ideal fluid would behave were there such a thing. Interpreted in this way, the zero-resultant theorem brings out the difference between a real fluid and an ideal fluid, so it can be taken as a powerful demonstration of the unreality of ideal fluids.38 But if real and ideal fluids are so different, how are theory and experiment ever to be brought into relation to one another? This was a long-standing problem. Failure to resolve it had generated a sharp distinc­tion between hydrodynamics, which was largely a mathematical exercise, and hydraulics, which was largely an empirical practice—hence the two different chapters and the two different authors in the Encyclopaedia Britannica.39 Was aerodynamics to take the path of empirical hydraulics or the path of math­ematical hydrodynamics? There were strong social forces pulling in each of these opposing directions.