You will never understand lift. Forget it. You haven’t got a chance.
So I muttered to myself as I closed a fascinating book called The Enigma of the Aerofoil. The author, David Bloor, is an emeritus professor of the University of Edinburgh whose field is the sociology of science: how cultural and personal factors shape the acquisition and use of scientific knowledge.
Bloor’s story, which unfolds like a genial Foyle’s War-style slow-motion whodunit from the BBC, concerns the efforts of scientists and mathematicians in the ivory towers of England and Germany between 1909 and 1930 to understand how wings produce lift. Their work went on parallel to, and largely insulated from, that of manufacturers and aeronautical engineers, who built tens of thousands of airplanes without worrying about why wings worked. They did — that was all that mattered.
The sense in which scientists understand something is not the sense in which you and I do. We know what it feels like to stick a hand out the window of a moving car, to sail a boat or to carry an umbrella on a windy day. These are direct, elemental experiences, familiar since childhood, and we have no difficulty extrapolating them to the wings of airplanes. We understand lift by empathy: We feel it.
But that is not the kind of understanding that exercised the giant brains of Cambridge and Göttingen. They were not Impressionists. To them, understanding wings meant finding the mathematical laws that govern them and that would allow accurate prediction of their behavior. These laws were elusive. Not that much of the grunt work hadn’t already been done: Mathematical descriptions of the behavior of frictionless “ideal fluids” under highly prescribed conditions had been developed in the 18th century by Leonhard Euler and our old pal Daniel Bernoulli. But real air is not a frictionless fluid; there, so to speak, is the rub.
The basic difference is that an ideal fluid lacks viscosity, and air has it. In viscosity-free air there would be no drag and no lift. Any force produced as the wing pushed the air aside would be exactly neutralized as the air sprang back into place. Obviously, this was not what was happening, since real wings produce real lift and real drag. Where were they coming from?
In Bloor’s narrative, the British and the Germans approached the riddle from different angles. It was not just a matter of national character, however. Rather, it was that the British brains who worked on the problem were pure mathematicians in the last degree, mostly drawn from among the highest-performing survivors of Cambridge University’s merciless tripos examinations, while the Germans came from their country’s system of technical upper schools, in which it was not considered uncouth to take an interest in practical aspects of a problem.
The great difficulty concerned something called circulation theory. This is an approach that emerges from the readily observed fact that an airfoil slows the air passing below it and accelerates the air above it. The difference in speed bends the airflow behind the airfoil, angling it slightly downward. This small downward (or backward, in the case of a propeller) movement of a large amount of air — because the rapidly moving wing influences thousands of cubic feet of air per second — produces the reaction that we call lift or, in the case of a propeller, thrust.
It was possible, if you made the right set of assumptions, to get the circulation model to predict real lift quite accurately. The trick was to think of the flow of air past an airfoil as a circular motion superimposed upon a horizontal one. This was mathematically convenient, because methods existed for computing rotary and straight-line flows, and they could be combined. It was actually an Englishman, a plebeian polymath named Frederick Lanchester, who originally proposed a circulation theory of lift in the 1890s and connected the circulation to the rotation of the tip vortices. In modern terms, Lanchester pretty much had it right. But the tripos set looked down its collective nose at him; after all, Lanchester had gone to a technical school.
The stumbling block in the circulation approach for the British thinkers was that ideal-fluid theory provided no means of inducing a circulation in the first place. Ideal fluids were inherently circulation-free. Circulation was mathematically illegal. Thus, the fact that real wings displayed behavior that could be accounted for by a circulation theory, far from gratifying the Cambridge idealists, put them into a moral and intellectual dilemma. The worst of it was that they possessed a key that should in principle have unlocked the problem for them, but they could not get it to work. That key was a set of equations describing the behavior of a viscous fluid, which air really is. Unfortunately, these so-called Navier-Stokes equations were so complicated that no one could solve them.
German aeronautical research, centered at the University of Göttingen, achieved a better balance of the ideal and the practical. Less afflicted with the mathematical purism that paralyzed the British, it managed to build a bridge between the ideal and the real. The bridge was in many ways a makeshift, but it worked; you could get across it.
The German work was inaccessible to the British during the war. Only after the end of hostilities in 1918 did German technical publications begin to find their way to the English-speaking countries. It then slowly dawned on the British that, while they had been banging on a door that refused to open, the Germans had gone around the corner to find another one unlocked.
Göttingen’s central figure was Ludwig Prandtl, who possessed a remarkable gift for turning empirical observations into useful mathematical approximations. While the British obsessed about the extremely difficult problem of stalling — their grail was a theory that would account for the behavior of an airfoil at any angle of attack — Prandtl had focused his attention upon the small range of angles of attack that were actually of interest to the makers and flyers of airplanes. Within that range, he was able to deduce a number of simple formulas that still form the basis of aeronautical analysis and performance prediction today.
The contribution for which Prandtl is most famous is the concept of the boundary layer. Within the boundary layer, viscosity dominates; outside it, the air behaves in a way that closely approximates a frictionless “ideal fluid.” The presence of the boundary layer in effect changes the shape of an airfoil, and in doing so resolves the discrepancy between the ideal and the real. It is largely to Prandtl’s insights that we owe the modern ability to accurately simulate, with digital computers, the behavior of still-imaginary airplanes.
As late as 1925, one of the British theorists, Richard Southwell, told the Royal Aeronautical Society, “I suppose no problem is so fundamental as the question ‘Why does an aerofoil lift?’ We can hardly rest satisfied with the present position — which is, we have next to no idea. The general equations of motion for a viscous fluid … are about the most intractable equations in the whole of mathematical physics.” At the time Southwell said these words, the problem had been solved for all practical purposes. Still, he wasn’t wrong; in the terms of pure mathematical analysis in which the British had framed the problem, lift remained as mysterious as ever.
What makes Bloor’s book so interesting to me is the detailed and lively picture it paints of how the thinking of groups of people can be subtly circumscribed by certain preconceptions, conventions and mutual influences — “groupthink,” if you like — and by the way the problem is initially framed. For those top-hatted British boffins, however pigheaded and obtuse some of them may look in retrospect, were extremely brilliant men. England threw its best and brightest at the problem of lift, and they failed to unravel it.
Today we pilots are inclined to talk, and even argue, about lift in terms of a Newtonian action/reaction — the wing pushes air down, the air pushes back — or else in terms of the relationship between local flow velocities and surface pressures, which vary inversely in accordance with basic physical laws that were identified by Bernoulli, though they no more belong to him than gravitation does to Isaac Newton. These explanations, as far as they go, are correct, but they don’t go far. We are tots in a sandbox compared with the titans who laid siege to the stronghold of lift and were driven back. We do not understand lift; we merely talk about it.