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
figure 2.6. Continuous flow of an ideal fluid around an inclined plate. From Tietjens 1929, 161. (By permission of Springer Science and Business Media)
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.