One formula, and it does not ask what the shape is

What actually holds a wing up is the collection’s argument that lift comes from circulation, and lift with no wing at all is its cleanest demonstration: a spinning cylinder, with no camber and no trailing edge, lifting exactly as hard as its circulation says it should.

This essay is about how far that “exactly” goes, which is further than either of those needs, and about the price of it, which is that the theorem answers one question and refuses every other.

Six bodies, one number #

Kutta–Joukowski says the lift per unit span on any two-dimensional body in a steady, irrotational, incompressible stream is

$$L = \rho\,U\,\Gamma.$$

There is no shape in that formula. To find out whether there is a shape hiding in it, six bodies were built that share nothing else and set to a circulation of exactly two.

Two circles of different radii; two ellipses, one 4:1 and one almost round; and two Joukowski sections, one thick and uncambered at eight degrees, one thin and strongly cambered at four. The circles and ellipses have their circulation imposed as a free parameter, because a smooth body has no trailing edge and nothing to fix it; the sections have theirs fixed by the sharp edge, and the incidence that produces a circulation of two is one arcsine.

The pressure is then integrated round each of them.

Every one comes out at 2.00000000000, to three parts in a hundred billion. The chords span a factor of five and the thicknesses a factor of nine, and the lift does not move in the eleventh decimal place.

The same integrals resolved the other way give drags between four parts in $10^{16}$ and nine in $10^{12}$ of the lift. That is d’Alembert’s paradox turning up as a by-product of a calculation that was not looking for it, which is the most convincing way for it to turn up.

And it is exactly linear. Over circulations from a half to five the worst departure from $\rho U\Gamma$ is one part in ten billion, so this is not a small-disturbance result being used near its origin — the strongly cambered section at the top of the range is at eighteen degrees of incidence.

What “any body” actually covers #

It is worth being explicit about the class, because the theorem is usually met in a context that makes it look narrower than it is.

The hypotheses are: a closed contour, a steady flow, incompressible and irrotational outside the body, and a stream that is uniform far away. That is all. The body may be any shape at all, including one with corners, including one that is not convex, including one nobody would call an aerofoil. It need not be thin, it need not be at small incidence, and there is no expansion parameter anywhere in the derivation — which is what separates this result from thin aerofoil theory, where the shape enters through an approximation and the answer is a leading term.

The circulation may come from anywhere. On the sections it comes from the Kutta condition; on the circles and ellipses it is put in by hand, and physically it would come from spinning the body, which is a real experiment. The theorem does not ask.

What it does require is that the flow be irrotational outside the body and that the contour be closed round it. A body shedding a wake has vorticity outside it, and the theorem then applies to a contour that encloses the body and the wake together — which is the same bookkeeping the vortex a wing leaves behind is about, and is why the starting vortex has to be counted.

How the arithmetic has to be done #

Two details decide whether the eleven figures above are real, and both are worth a paragraph because both were got wrong first.

The surface speed must be evaluated on the surface. The general-purpose pressure integral in this site’s solver steps a small distance along the outward normal, because it is handed an outline and a velocity field and has no way to ask a mapped body for its own surface speed. Doing that here costs six parts in a thousand — a systematic error a hundred thousand times the one being measured, and one that would have been reported as the theorem’s. Every body above hands back the exact speed at a stated parameter, computed in the circle plane and divided by the derivative of its own conformal map.

And the panels must use the tangent rather than the chord. The force is $\oint p\,\mathbf{n}\,ds$ and $\mathbf{n}\,ds$ is exactly $(dy, -dx)$, so what the integral needs is the parametrisation’s own derivative. Replacing that by the chord between two panel corners makes the whole sum a trapezoid rule on the geometry, whose error is $(\pi/n)^2/6$ — 1.8 parts in ten million at three thousand panels, identical on every smooth body, and eight orders larger than the theorem’s own residual. With the tangent it is a midpoint rule on a smooth periodic integrand, which is spectrally accurate, and three thousand panels is round-off.

That is two ways to get a plausible answer that is about the instrument. Neither shows up as anything odd on screen.

What the theorem refuses to say #

Now the price. The six bodies agree about the lift and about nothing else whatever.

The four symmetric bodies carry exactly zero moment, by symmetry. The thick uncambered section carries $-0.237$ and the thin cambered one $-0.110$. The span is 0.24 in coefficient at a lift coefficient that is the same for all of them, so the moment is not a function of the circulation — and the moment is what decides the trim, the tail size and the torsional load, which is to say most of what a structure is designed for. Where the lift acts needs the next multipole in the far field, and this theorem has only the first.

Nor is the distribution. The symmetric bodies split their lift exactly in half between the front and rear of the chord; the thick section at incidence makes 85 per cent of it on the front half. Two wings with the same lift can load their spars completely differently, and nothing in $\rho U\Gamma$ distinguishes them.

And nor is the peak suction, which runs from 1.30 on the 4:1 ellipse to 12.47 on the thin cambered section — a factor of nine and a half at identical lift. That number decides whether the flow separates, whether the water cavitates, and in the end whether a real fluid will produce this circulation at all. The theorem is silent on all three.

So the exactness is a statement about a resultant and about nothing else. That is not a small thing — a resultant is what an aircraft weighs against — but it is one number out of the several a section has to deliver, and it is the only one this formula knows.

The one body it appears to fail on #

There is a seventh body, and it is instructive.

Run a flat plate — a Joukowski section of zero thickness — through the identical calculation and the surface integral comes up short. At nine degrees it finds 97.47 per cent of $\rho U\Gamma$; at eighteen and a half, 89.87 per cent.

The shortfall is exactly $\sin^2\alpha$, at every incidence, to one part in a hundred million.

That is not a coincidence and it is not a failure of the theorem. The pressure on a plate acts normal to the plate, so the surface integral returns the normal force $N$ and nothing else. The lift is $N\cos\alpha + S\sin\alpha$ with $S$ the leading-edge suction, and thin-aerofoil theory gives $S = N\tan\alpha$ — so what the integral misses is $N\sin\alpha\tan\alpha\cos\alpha$ over $N\cos\alpha + N\sin\alpha\tan\alpha$, which reduces to $\sin^2\alpha$ exactly. The missing force is a finite force from an infinite speed: unbounded suction acting over vanishing area at the sharp leading edge.

Two hundred and fifty-six times as many panels finds 97.46697 per cent instead of 97.46597. It converges, and it converges short, because no quadrature over a surface finds a delta function on that surface. The theorem is right and the instrument is not, which is a distinction the site’s own figure rules exist to keep visible.

The practical version of this is not academic: a panel method run on a sharp-edged section without a leading-edge suction correction under-predicts lift by exactly this amount, and the error grows as the square of incidence — invisible at two degrees, ten per cent at eighteen.

The same freedom, used the other way #

There is a reading of the six bodies that inverts the disappointment above into a design principle.

If the lift depends on the shape only through $\Gamma$, then a designer is free to choose the shape for everything else. The circulation needed is fixed by the weight and the speed; the section is then chosen to deliver that circulation with the gentlest suction peak, the most benign moment, the most useful thickness for a spar, or the best behaviour off design — and none of those choices costs any lift.

That is exactly what a section catalogue is: a list of shapes that all reach the same circulation, with their moments, their peak suctions and their stall behaviour tabulated because those are the numbers that differ. Read this way the theorem’s silence is the reason the catalogue exists, and the essay’s complaint and its usefulness are the same fact.

The one thing the freedom does not extend to is how much circulation is too much. The suction peak column in the table above is the constraint, and it is a constraint the theorem cannot see.

It survives compressibility, and dies of something else #

The hypothesis list says incompressible, and the theorem is tougher than that. In a steady subsonic irrotational flow of a compressible gas the lift is still exactly

$$L = \rho_\infty U_\infty \Gamma,$$

with the density and the speed those of the free stream and the circulation measured on a contour far out in the undisturbed air. The reason is the same contour argument: pushed far enough out the disturbance is small, the compressible corrections to the far field are higher multipoles, and the only surviving product is the cross term between the stream and the circulation.

That sharpens what compressibility actually does. It does not change the relation between circulation and lift. It changes the circulation a given shape carries at a given incidence — which is what a Prandtl–Glauert correction is a statement about, and why the correction belongs to the lift curve rather than to this formula.

Above Mach one it fails, and it fails for exactly the reason it worked. The far field is no longer undisturbed: waves run out to infinity at undiminished strength, the contour integral no longer collapses to one term, and what appears alongside the lift is a drag with no viscosity in it.

Why it is exact, in one paragraph #

The proof is a contour integral and it explains the indifference to shape.

Blasius’ theorem writes the force on a body as a contour integral of the square of the complex velocity, taken on any contour enclosing it — which is the same freedom that lets a wing’s reaction be found far from the wing. Push the contour out to where the body’s own detail has died away, and what is left of the velocity there is the free stream plus a circulation term plus a dipole term plus higher multipoles. Only one product in the square survives the integration: the cross term between the free stream and the circulation. The dipole and the multipoles are what the shape contributes, and every one of them integrates to zero against everything, including itself.

So the shape is not neglected. It is present in the field, it is present on every finite contour, and it cancels exactly.

The circle that is doing the work #

One line of the table above deserves separating out, because it is the whole argument in miniature.

The two circles differ only in radius: one is a unit circle and the other is 0.4 of that. Their displaced areas differ by a factor of six and a quarter, their surface lengths by two and a half, and their peak suctions by 1.56 — the smaller circle has to accelerate the flow harder to carry the same circulation round a shorter path. And their lifts are the same number to eleven decimal places.

There is no sense in which the bigger circle is “doing more”. The force is a property of the circulation and the free stream, and a body is a device for holding a circulation in place. That is the sentence the theorem is, and it is why a cylinder that has no wing-like feature at all lifts exactly as hard as one.

A note on what the six bodies cost to compute #

The exactness quoted here is eleven figures, and getting eleven figures out of a numerical integral is not free. It is worth saying where each of them came from, because the site’s habit is that a number without its arithmetic is a number to distrust.

Three of the eleven are the bodies: each hands back a closed-form surface speed rather than a sampled one, so no figure is lost to interpolation. Four are the quadrature: a midpoint rule on a periodic integrand converges faster than any power of the panel count, so three thousand panels is already at round-off and adding more changes nothing. Two are the geometry, which uses the analytic tangent rather than the chord. And the last two are the arithmetic itself — the residual quoted, three parts in $10^{11}$, is about where a sum of three thousand terms of ordinary size sits in double precision.

The point of the accounting is that the eleven figures are not evidence about fluid mechanics. They are evidence that the calculation is doing what it says. What is evidence about fluid mechanics is that six different calculations, on six different shapes, agree — and the agreement between two independent routes is worth more than the precision of either.

One sentence to carry away #

If a reader keeps one thing from this essay it should be the shape of the statement rather than the number: an exact result about a resultant is not an exact result about a distribution, and the two are routinely quoted as though they were one.

That is the same sentence the phase’s other exactness essays end on, arrived at from a different flow each time — a transport that is exactly cancelled and a profile that is not, an average dissipation that is exact and a local one that is a convention, a total lift that is exact and a moment that is free. The mathematics is the same in all of them, and so is the mistake.

What is not claimed #

Steady, two-dimensional, incompressible, irrotational. All four hypotheses are load-bearing and each has an essay’s worth of exceptions elsewhere in this collection: an accelerating body carries an added-mass force that is not in $\rho U\Gamma$, a finite wing needs the circulation integrated across the span, and above about Mach 0.3 the compressible correction is not a correction to this formula but a change to the equation it came from.

The circulations were imposed, not predicted. For the circles and ellipses the circulation is a free parameter set by hand; for the sections it comes from the Kutta condition. Nothing here says what circulation a real fluid will produce, which is the question the sharp edge decides answers and the one everything else depends on.

The suction peaks are ideal-flow values. A peak of 12.5 is a number no real section reaches, because the boundary layer separates first — so the ninefold spread above is a spread in what the bodies are asking of the fluid rather than in what they get.

The plate’s shortfall is a statement about the integral. It is not a statement that the plate makes less lift; the plate makes $\rho U\Gamma$, like everything else in the table.

And the six bodies are not a random sample. They were chosen to differ — in size, in fineness, in camber, in whether they have a sharp edge at all — and that is a demonstration rather than a survey. Nothing here rules out some exotic contour on which the calculation would behave differently; what rules it out is the contour-integral proof, and the six bodies are a check on the proof rather than a substitute for it.