The mirror that is a circle

The method of images is the cheapest trick in the subject, and this collection has already used it on a flat wall: reflect every singularity in the plane, reverse the sign of the vortices, and the plane becomes a streamline by symmetry. Nothing is solved. The boundary condition is satisfied by accident and then observed.

The obvious question is whether the trick survives bending the mirror, and the answer — which took until 1940 and is one line long — is that it does, with two changes. The reflection becomes an inversion, and a vortex picks up a second image that has nothing to do with the wall.

The theorem #

If $w = f(z)$ is the complex potential of a flow with no singularities inside $|z| = a$, then

$$w(z) \;=\; f(z) \;+\; \overline{f\!\left(\frac{a^2}{\bar z}\right)}$$

is a flow with the same singularities outside the circle and with $|z| = a$ as a streamline.

That is Milne-Thomson’s circle theorem, and the proof is two observations. On $|z| = a$ the point $a^2/\bar z$ is $z$, so the second term is the conjugate of the first and the sum is real — which is to say $\psi = 0$, which is to say the circle is a streamline. And the second term’s singularities are the images of the first’s under $z \mapsto a^2/\bar z$, which maps the exterior into the interior, so nothing has been added to the region the flow is in.

Everything hard about the boundary is done by a substitution. No integral equation, no panel method, no solve of any kind.

The cylinder everybody knows is an image system #

Before the vortex, the simplest possible test: hand the theorem a uniform stream.

$f(z) = Uz$ has no singularities anywhere inside the circle, so the theorem applies. Its image is $\overline{f(a^2/\bar z)} = Ua^2/z$, and the sum is

$$w = U\left(z + \frac{a^2}{z}\right),$$

which is the flow past a cylinder that this collection has been using from the beginning. The doublet nobody put there is the image of the stream in the circle. It was arrived at originally by adding a stream to a doublet and noticing that a circular streamline appeared; the theorem says the doublet was never a choice.

Where the image goes #

For a point vortex of strength $\Gamma$ at distance $d$ from the centre, unwrapping the formula gives two images:

• $-\Gamma$ at the inverse point $a^2/d$, on the same ray;
• $+\Gamma$ at the centre.

The first is the reflection, with the sign reversed exactly as it is in a flat mirror. The second is the surprise, and it is worth being clear about what it is for.

Take a loop that encircles the cylinder and nothing else. Its circulation is the sum of the strengths inside it, which is $-\Gamma + \Gamma = 0$. The image at the centre is what makes the circulation round the body zero, and without it the construction would still satisfy the wall condition and would describe a cylinder with a circulation of $-\Gamma$ round it.

Which raises the question the rest of this essay is about: what decides which of those is right?

Nothing about the wall decides it #

Add a vortex of any strength whatever at the centre. It produces a purely azimuthal flow, which is tangent to every circle centred there, so the wall condition is untouched.

So the circle theorem’s answer is one member of a one-parameter family, all of them exact solutions of Laplace’s equation with the same body and the same boundary condition, and they are genuinely different flows. The surface speed differs. The pressure differs. The force on the cylinder differs, by $\rho U \Gamma_c$ for the part of the flow that has a stream in it.

This is the same hole the panel method finds when it comes out with one more unknown than it has equations, and the same one the minimum-energy theorem cannot close in a region with a body in it. The boundary data determines an ideal flow up to one number per hole in the region, and the number is not a fact about the flow now.

It is a fact about how the flow was made, and the instrument that supplies it is Kelvin’s circulation theorem: if the fluid started at rest, the circulation round any material loop was zero and stays zero, so the circulation round a loop that has always enclosed the cylinder is zero, and the centre image is the right one. Change the history — spin the cylinder up before the vortex arrives, or start it in a fluid that was already rotating — and a different member of the family is the answer.

What the vortex does next #

A point vortex has no velocity of its own. Its own field is antisymmetric about it and there is nothing at its centre to be carried by, so everything it does, something else did. Here the something else is its own images.

Evaluating the two images at the vortex’s position gives, in closed form,

$$V = \frac{\Gamma a^2}{2\pi d\,(d^2 - a^2)},$$

directed along the wall. So the vortex circles the cylinder, at a speed that diverges as it approaches the surface and falls off as $1/d^3$ far away. At two radii out with unit circulation the period of one orbit is 474 time units, which is slow — a vortex near a body is not violently attracted to it, it drifts round.

Two features of that expression deserve reading rather than glancing at. The $a^2$ in the numerator says the effect is a property of the body’s area rather than of its perimeter, which is why a thin plate barely disturbs a passing vortex and a fat one carries it a long way round. And the $d^2 - a^2$ in the denominator is the gap in disguise: writing $d = a + g$ makes it $g(2a + g)$, so for a small gap the speed goes as $1/g$ — the same inverse-gap law a vortex has near a flat wall, arrived at from a formula that contains no wall.

The sign is worth a sentence. The image is of the opposite sign, so the pair behaves like a counter-rotating pair and translates — which, since the image is constrained to the inverse point, means the real vortex travels round the body rather than into it. A vortex near a wall does not stick to it. It runs along it.

What the theorem needs #

One hypothesis, and it is the one that decides whether a construction is legitimate: the flow being reflected must have no singularities inside the circle.

The reason is the direction the inversion runs. A singularity at $z_0$ outside produces its image at $a^2/\bar z_0$ inside, where it does no harm because the fluid is not there. A singularity inside produces an image outside — in the fluid — and the construction has added something to the flow region that nobody asked for.

The practical version: a source placed inside the body is not a way of modelling a body that blows. It is a way of putting a second source in the stream at a place decided by arithmetic.

The flat wall as a limit #

The two constructions have to agree when the circle is large enough, and how they agree is a measurement rather than a hope.

Hold the gap $g$ between the vortex and the surface fixed and let the radius grow. The inverse point is at $a^2/(a+g)$, so its depth below the surface is

$$a + g - \frac{a^2}{a+g} = 2g\left(1 - \frac{g}{2a} + \cdots\right),$$

against the flat mirror’s $2g$. The discrepancy is the gap divided by twice the radius — one part in eight hundred for a cylinder two hundred gaps across — and the centre image, at distance $a$, is by then contributing a velocity of order $\Gamma/2\pi a$, which vanishes in the same limit.

That is the honest version of the statement that a large body looks locally flat. It looks flat to within the ratio of the length being used to the radius of curvature, and the two constructions are one expansion apart rather than two different physical pictures.

What it is for #

The circle theorem is not a curiosity about circles. It is the tool for every problem in which a body sits in a field that somebody else made.

A wing near the ground is the flat-wall case and is priced elsewhere. A wing in a wind tunnel is a pair of flat walls, and reflecting between them gives an infinite stack of images whose sum is the blockage correction. A blade passing another blade’s shed vortex is this problem, with the cylinder standing in for the second blade’s thickness.

And the one that reaches furthest: the circle theorem plus a conformal map solves the same problem for any shape. Map the body to a circle, apply the theorem, map back. The whole of Joukowski’s construction is that sentence, and it works because a conformal map carries streamlines to streamlines and leaves circulation alone.

A sphere is not a circle, and the difference is an integral #

The flat mirror generalises in two directions and only one of them stays this simple. Bending the wall in the plane gives the theorem above, one line long. Going up a dimension does not.

A plane wall in three dimensions still reflects: a source at height $h$ above a plane has a source at $-h$, exactly as in two, and the argument is the same symmetry argument. A sphere is a different matter. Weiss’s theorem of 1944 gives the image system for a sphere of radius $a$, and it is not a singularity at the inverse point. It is a singularity at the inverse point plus a distribution smeared along the segment joining that point to the centre — an integral, with a weight that falls off along the line, rather than a set of images that could be drawn as dots.

The reason is worth having, because it explains why the two-dimensional theory of this collection is as strong as it is. The circle theorem works because $\overline{f(a^2/\bar z)}$ is again an analytic function, so reflecting a harmonic function in a circle produces a harmonic function of the same kind. In three dimensions the corresponding operation — the Kelvin transform — carries a harmonic function to a harmonic function only after multiplying by $a/r$, and that factor is what smears a point into a line. The inversion is still the right map; what it no longer does is take a point singularity to a point singularity.

The practical consequence is the one this collection keeps meeting from other directions. A two-dimensional body in a field somebody else made can be solved by writing down a formula; a three-dimensional one is a solve. That is the same wall the lifting line meets against a real wing, and the same reason the inverse problem stops being a single transform when the boundary becomes a surface: a curve has one parameter and a surface has two, and analyticity of one complex variable is a property with no counterpart in the second case.

Two circles, and images that never stop #

The other direction the theorem does not survive intact is having a second body.

Put a vortex between two cylinders. Reflecting it in the first gives an image inside the first; that image is outside the second, so it must be reflected there too; and that image is outside the first. The construction never terminates. What comes out is an infinite sequence of images, each the inversion of the previous one in the other circle, with strengths that do not decay geometrically in any obvious way.

The sequence does converge, and where it converges is the interesting part: successive inversions in two circles drive the images towards the two limit points of the pair — the two points that inversion in either circle leaves fixed as a set, which are real when the circles do not intersect. Everything that is difficult about two bodies near each other is contained in how fast that sequence converges, and it converges slowly when the gap is small, which is exactly when the interference matters most.

That is the same shape as the wind-tunnel stack: a wing between two walls is an infinite column of images, and the correction is a sum rather than a term. The difference is that a stack of plane images is a lattice with a closed-form sum, and a chain of circular inversions is not — so the two-cylinder problem, which looks like the circle theorem applied twice, is the point at which the method stops being cheaper than solving.

What the picture cannot show #

The vortex has no core. It is a point, its velocity field is unbounded at its own position, and its kinetic energy is infinite in any region containing it. Everything above is a statement about the flow outside a small disc that has been quietly excluded, and the exclusion is invisible in every figure.

There is no boundary layer on the cylinder. A real vortex passing a real body induces a pressure gradient over the surface, and the layer under it responds — thickening, sometimes separating, and shedding vorticity of its own that the image system knows nothing about. The construction is exact for a fluid with no viscosity and is a first term for anything else.

And the surface speed is not what a probe would read. The figures print the tangential velocity at the wall, which for a real fluid is zero. What the ideal solution supplies is the speed at the edge of the layer, which is the number the boundary-layer problem is driven by and is not the number at the wall.

Who found it, and when #

The plane image goes back to Kelvin and Maxwell in electrostatics and was in fluid mechanics well before 1900. The circle theorem is L. M. Milne-Thomson’s, published in 1940 in a four-page note — three centuries after the problem of a body in a stream was first posed, and after generations of textbooks had done the vortex-and-cylinder case by writing down the answer and verifying it.

It is worth asking why it took so long, because the answer is not that anybody lacked the algebra. The vortex-and-cylinder solution was known: it is in Lamb, arrived at by writing down the two images and verifying that the boundary condition holds. What was missing was the observation that the verification is unnecessary — that the substitution $z \mapsto a^2/\bar z$ makes the circle a streamline for any flow at all, so the construction is a theorem about the region rather than a result about one arrangement of singularities. A century of textbooks solved the instance and did not state the rule.

The surprising connection is with a piece of pure geometry that predates the fluid mechanics by two thousand years. Inversion in a circle — the map taking a point at distance $r$ to one at $a^2/r$ on the same ray — is the transformation Apollonius used to turn problems about circles into problems about lines, and it is the same map here, doing the same job: it takes the awkward boundary to a place where it is not awkward. That the map that solves a geometry problem should also solve a boundary-value problem is not a coincidence; it is because both are statements about a function that is harmonic outside a circle, and there is only one such family.

Where the ladder goes next #

Below this rung are the flat wall it generalises and the two vortices it puts near a body.

Beside it is what happens when four of them are in the room at once, which is where the exactness of every formula on this page stops being any help at all.

And above it is the question this rung raised and did not settle: what the boundary conditions do not decide, and what supplies the missing number in each of the cases where they do not.