The constant a hole leaves behind

In a region with no holes in it, the exact theory is as well behaved as a subject can be. Laplace’s equation plus the normal velocity on the boundary has one solution, it depends continuously on the data, and there is nothing further to say.

Put a body in the flow and that stops being true, and the reason is not physical. It is topological. The region outside a body is doubly connected: there is a loop in it that cannot be shrunk to a point without leaving the region. Going once round such a loop, the velocity potential increases by the circulation, so the potential is not a single-valued function — and the circulation is not determined by anything in the problem.

Five solutions to the same problem #

The demonstration is short. Take a cylinder in a uniform stream and put a circulation on it. The surface speed is $2U\sin\theta + \Gamma/2\pi a$, and the flow is irrotational everywhere outside, tangent to the surface everywhere on it, and uniform at infinity, for every value of $\Gamma$.

Integrating the pressure over the surface gives a drag of $6\times10^{-16}$ at every one of them — which is no drag at all, holding as it must, since d’Alembert’s argument never mentioned the circulation — and a lift equal to $\rho U\Gamma$ to eight figures.

So a reader who wants to know what the lift is must supply something the theory does not have. What holds a wing up is the essay about what that something is; this one is about why it has to be supplied.

The pictures are worth reading for what they do not show. There is no sense in which one of these distributions is more natural than another, no residual to be minimised, no condition being nearly violated by four of them. The stagnation points move round the cylinder as the circulation is raised and leave it entirely once $|\Gamma| > 4\pi a U$ — the mirror that is a circle puts them where the image system does — and none of that is a failure of any kind. Five drawings, five solutions.

The count is the number of holes #

The generalisation is exact and it is worth stating because it makes the phenomenon predictable rather than surprising.

A domain’s first Betti number is the number of independent loops it has, and a potential flow in it has that many free circulations. One body, one constant. Two bodies, two constants — and they are independent, so a biplane’s two sections may carry any pair of circulations whatever without violating anything. A slotted aerofoil has three.

That is why the mirror that is a circle has to say what circulation the image system carries, and why every image construction in this collection is careful to add a compensating vortex at a body’s centre: the construction has to choose the circulation, and the choice is a separate statement from the tangency condition it is making.

Why the potential is the thing that fails #

It is worth being precise about which object is many-valued, because the velocity field is not.

The velocity is perfectly single-valued: at every point outside the body there is one velocity vector, and no ambiguity anywhere. What is many-valued is its potential. Integrating $\mathbf{u}\cdot d\mathbf{l}$ from a fixed point to a point $P$ gives an answer that depends on which way round the body the path went, and two paths on opposite sides differ by the circulation.

That is exactly the statement circulation is vorticity added up makes from the other end. Stokes’ theorem turns the circulation round a loop into the vorticity through it — and the theorem needs a surface spanning the loop that lies in the region. Round a body there is no such surface, so the theorem does not apply, and a flow can be irrotational everywhere and still have a circulation. The many-valued potential and the inapplicable theorem are the same fact.

The stream function, incidentally, has the opposite problem in the opposite case: it is many-valued when there is a net source, and single-valued when there is only circulation. One function instead of two is where the pair is introduced.

The energy separates, and the cross term is zero #

Once there is a family of solutions, the natural next question is which of them is special, and the natural way to ask it is with the energy.

Split the flow past a cylinder into its acyclic part — the stream and its doublet — and its cyclic part, $\Gamma/2\pi r$. Integrate the kinetic energy over the region between the body and a boundary at forty radii, and keep the cross term separately.

The cross term is $3.3\times10^{-16}$ of the cyclic energy, which is to say zero. It has to be: the acyclic azimuthal velocity goes as $\sin\theta$ and the cyclic one has no $\theta$ in it, so their product integrates to nothing round every circle.

The consequence is that the energy is a parabola in $\Gamma$ with no linear term,

$$E(\Gamma) = E(0) + \frac{\rho\Gamma^2}{4\pi}\ln\frac{R}{a},$$

checked here against a quadrature to one part in $10^4$ and against the closed form exactly. Doubling the circulation multiplies the cyclic part by four, to a part in a million.

Which means the least-energy flow has no circulation at all #

That is a sharper statement than it looks, and it is where the flow with the least energy needs a clause it is careful to include.

Kelvin’s minimum-energy theorem says the irrotational flow has less kinetic energy than any other solenoidal flow with the same normal velocity on the boundaries. Written like that it is false in a multiply connected region, because the irrotational flow with $\Gamma = 5$ and the irrotational flow with $\Gamma = 0$ both satisfy the boundary conditions and the second has less energy.

The theorem needs “and the same circulations”, and with that clause it is exactly right.

The clause is tested here rather than quoted. Take perturbations of the form $f(r)\sin(m\theta)$ with $f$ vanishing on both boundaries — so the normal velocity is unchanged everywhere and every circulation is untouched — and compute what they do. Each raises the energy, by 0.25, 0.41 and 0.68 per cent for $m = 1, 2, 3$; and each raises it by exactly its own energy, because its cross term with the flow is $10^{-16}$ of itself.

That is Kelvin’s theorem doing what it says, in the class it is about.

Two holes make a matrix #

With two bodies there are two constants, and the energy is a quadratic form in them.

Computing it needs the flow past two cylinders with prescribed circulations, which is done here by successive images: reflect each vortex in the other circle by the circle theorem, which adds an opposite one at the inverse point and an equal one at the centre, and iterate. The construction converges because the inverse points march inward, and it closes both bodies to $3.2\times10^{-12}$ at the closest separation tried.

The energy then comes off the boundaries rather than out of an area integral. For an irrotational flow with $\psi$ constant on each body,

$$E = \frac{\rho}{2}\oint \psi\,\frac{\partial\psi}{\partial n}\,ds = \frac{\rho}{2}\sum_i \Gamma_i(\psi_i - \psi_{\text{out}}),$$

so three loadings give the whole matrix. The self terms barely move as the bodies are separated and the mutual term falls as the logarithm of the distance, exactly as two circuits’ self and mutual inductances do, and for the same reason: both are energies of a field produced by a circulation round a hole.

What happens if the hole is closed up #

The last question is the one this run of essays is built on. The hole is what created the free constant; can it be removed?

$$E = \frac{\rho\Gamma^2}{4\pi}\ln\frac{R}{a}$$

says no, and says it in the strongest available way. As $a \to 0$ at fixed circulation the energy diverges — logarithmically, which is slow, but without bound.

The increment per natural logarithm of the shrink is 0.07957747, which is $1/4\pi$ to eight figures and does not vary by a thousandth of a per cent across the sweep.

So a doubly connected domain carrying circulation does not converge to a simply connected one. The limit exists as a limit of shapes and not as a limit of flows, and what stops it is an energy that is not there to be paid.

The same logarithm, three times in this collection #

That divergence is worth recognising because it turns up repeatedly and always means the same thing.

The energy a vortex cannot have is the direct case: a point vortex’s kinetic energy diverges logarithmically, and what makes point-vortex dynamics usable is that the divergent self-energies are the same in every configuration and cancel out of every difference.

The momentum with no value is the same disease in a different integral: the momentum of an unbounded ideal flow is conditionally convergent, so it is zero over a disc and something else over a box, and Kelvin’s impulse is the well-defined quantity that replaces it.

And the case here is the third. In all three the culprit is the same: a two-dimensional flow with net circulation has a velocity falling as $1/r$, and $\int (1/r)^2 r\,dr$ is a logarithm.

What a physicist would want to say, and why it does not work #

The natural objection at this point is that the free constant is an artefact of idealising: a real fluid has viscosity, viscosity fixes the circulation, and the whole difficulty is a consequence of throwing viscosity away.

That is true and it is not an answer, for two reasons.

The first is that the mechanism is not local to the body. Viscosity fixes the circulation by way of the starting process: a wing accelerated from rest sheds a starting vortex of equal and opposite circulation, and the reason it sheds one is that the flow round a sharp trailing edge would otherwise be singular. But once the starting vortex is far downstream and the flow is steady, the steady problem has forgotten all of that, and the free constant is free again. What the history left behind is a value, not a constraint.

The second is that the free constant is not a small effect to be corrected. It is the whole of the lift. A wing’s circulation is not a perturbation of some viscously determined value; it is the number the entire force depends on, and the exact theory declines to supply it. That is a much stronger statement than “the model neglects something”.

The right way to hold both facts together is the one this collection uses throughout: the exact theory is a machine for turning one number into every other number, and the one number has to come from somewhere else. The lift curve is that machine running.

The one place the constant is not free #

There is a case where the topology is the same and the constant is nevertheless determined, and it is worth naming so the rule is not over-applied.

A closed vortex system in an unbounded fluid, started from rest and evolving without any body in it, has zero total circulation round any large contour for ever. That is Kelvin’s theorem, and it does constrain: the sum of the circulations of everything present must vanish. What it does not do is fix any one of them individually, and it says nothing at all about a body’s circulation in a steady flow that has been going on for ever.

So the correct statement is a conservation law about a sum and not a determination of a value, and a reader who remembers Kelvin’s theorem as fixing the circulation of a wing is remembering the history rather than the theorem. What survives being wound up tests the theorem directly, by advecting a material loop through an exact solution until nothing about its shape is recognisable and finding the circulation unmoved to eight parts in $10^5$ while the perimeter grows by a factor of 5.7.

What the Kutta condition actually is, in this light #

It is worth being blunt about the standing.

The Kutta condition is not a theorem, it is not derived, and it does not follow from the exact theory. It is an additional physical statement — that the flow leaves a sharp trailing edge smoothly, because a real fluid cannot turn a sharp corner at infinite speed — imported to pick one member of a family the mathematics leaves open.

Nothing turns a sharp corner is the essay about the physical half of that: the exact theory’s velocity at a salient corner is infinite, and the infinity is what the condition is avoiding. This essay is the mathematical half — the reason there was a family to choose from, which is that somebody put a hole in the plane.

The two halves together are the whole of why lift is not a consequence of Euler’s equations.

What a numerical method has to do about it #

The free constant is not an abstraction for anybody writing a panel method, and it is worth saying how it shows up.

A panel method covers the body with source and vortex distributions and solves for their strengths from the tangency condition at a set of control points. With $N$ panels there are $N$ conditions and $N$ unknowns — and the resulting matrix is singular, because adding a uniform circulation to the distribution changes no normal velocity anywhere.

The standard fix is to replace one row of the system with a Kutta condition: equal and opposite tangential velocities at the two panels adjoining the trailing edge, or a zero vortex strength there. That is not a numerical trick; it is the extra physical statement being supplied, in the only place the linear algebra will accept one.

Ask for the pressure meets the same structure from the inverse direction, where prescribing a pressure distribution over-determines a shape and the resolution is again a condition at the trailing edge. In both cases the count of equations and unknowns is telling the same story the topology told above.

In three dimensions there is no hole to leave a constant behind #

The whole argument is a two-dimensional one, and running it in space produces the opposite conclusion — which is worth having, because it explains why a wing in three dimensions cannot simply be given a circulation.

The region outside a finite body in space is simply connected. Every closed loop in it can be shrunk to a point without leaving the fluid, so in an irrotational flow every circulation is zero. There is no free constant, one per hole, because there are no holes: a three-dimensional body does not obstruct a loop the way a two-dimensional one does.

So a finite wing in a strictly irrotational unbounded flow can carry no circulation and therefore no lift. The resolution is that the flow is not strictly irrotational: the wing trails a sheet of vorticity, and a loop enclosing the wing cannot be shrunk without crossing it. The wake supplies the hole the body does not have.

Which turns this essay’s residue into the reason for the price of having ends. In two dimensions a free constant is available for nothing; in three it has to be paid for with a wake, and the wake is where the drag comes from.

What is not computed here #

The two-cylinder energy uses an image series and a cut-off. The mutual term depends logarithmically on the outer boundary radius, so the numbers quoted are for a boundary at sixty radii and would change if it moved. What does not change is the difference between configurations at fixed total circulation, which is the physically meaningful quantity and the one the analogy with inductance is about.

Kelvin’s theorem is tested and not proved. Four perturbations from a two-parameter family are not a proof for all solenoidal fields; the computation shows the mechanism — orthogonality, hence a strictly positive addition — rather than establishing the theorem.

And the free constant is free only in the steady problem. In an unsteady flow started from rest, Kelvin’s circulation theorem does fix it: the circulation round a large material contour is zero for ever, so a circulation round the body must be paid for by an equal and opposite one somewhere in the fluid. What survives being wound up is the essay about that theorem, and it is the reason the starting vortex exists. The steady problem has forgotten the history, and the constant is free again.

The residue #

The limit here is the one that closes the hole, and it is the tidiest of them all because the residue is a single logarithm with an exact coefficient.

What survives is a number nobody chose: a free constant, one per hole, which the equations decline to determine and which the energy makes expensive to remove. Everything the subject says about lift is downstream of it.