The part of the flow inside the body

Every flow in the exact theory is a harmonic function, and harmonic functions are the real parts of analytic ones. That is the whole reason the theory is tractable: it hands the subject the machinery of complex analysis, which is why one function instead of two can carry a whole velocity field and why from a circle to a wing can turn a solved problem into an unsolved one by a change of variable.

It also hands the subject something less often said. An analytic function does not stop at the boundary of the region it was defined on. It continues, and it keeps continuing until it meets a singularity.

So the flow past a body is defined outside it and exists inside it too, right up to the place where it breaks down. Where that place is decides three things a reader of this collection meets constantly: how many multipole terms the far field needs, how sharp a suction peak can get, and where a linearised theory stops approximating anything.

The circle theorem manufactures them, one for one #

Milne-Thomson’s circle theorem is the cleanest place to see it. The mirror that is a circle states it: given any flow whose singularities lie outside a circle of radius $a$, inserting the circle as a body is accomplished by adding $\overline{w(a^2/\bar z)}$ to the complex potential.

Read the added term. For every singularity of the original flow at $z_k$, it has one at the inverse point $a^2/\bar z_k$, which is inside the circle. Inserting a body does not merely bend the streamlines: it manufactures an interior singularity for each exterior one.

The case everybody knows is the uniform stream. Its singularity is at infinity, its inverse point is the centre, and the thing that appears there is the doublet — the $a^2/z$ in $w = U(z + a^2/z)$, which is the flow past a cylinder and which every reader has met without being told it is a singularity sitting at the origin.

The gap that closes, and the peak that goes with it #

A point vortex at distance $d$ from the centre of a cylinder of radius $a$ puts its image at $a^2/d$, with the opposite sign, plus one of the original sign at the centre. The image system closes the body to $4\times10^{-16}$, which is the check that it is the theorem’s.

Now bring the vortex in. The gap $d - a^2/d$ closes, and what the surface sees is two opposite singularities a vanishing distance apart — which is a dipole of diverging strength. Over the sweep from $d/a = 4$ to $d/a = 1.02$ the peak surface speed rises by a factor of 150 while the angular width of the peak at half its height falls by a factor of 39.

That is worth having in mind whenever a suction peak appears on this site. A sharp pressure feature on a surface is a singularity close to it on the other side, and the sharpness is the distance.

What the continuation is, and what it is not #

It is worth being precise about the claim, because “the flow inside the body” is a phrase that can be read two ways and only one of them is meant.

Nothing is flowing inside the body. There is no fluid there. What exists there is the analytic continuation of the complex potential: the unique analytic function that agrees with the flow’s potential in the region where the flow is defined, extended as far as it can be extended. Uniqueness is the point — an analytic function is determined by its values on any curve, so the continuation is not a choice anybody made about the interior, it is forced by the exterior.

That is why its singularities are informative. They are not modelling assumptions. They are properties of the exterior flow, expressed as facts about a region the exterior flow does not occupy, and they control the exterior flow’s behaviour near the boundary in the same way that the nearest pole of a function controls the convergence of its Taylor series.

The circle theorem’s arithmetic, in one line #

The theorem’s construction is short enough to write out and worth doing, because the inverse point is easy to state and easy to lose.

Given $w(z)$ analytic outside the circle $|z| = a$ apart from its own singularities, the combination

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

is real on $|z| = a$ — set $z = ae^{i\theta}$ and the second term is the conjugate of the first — so the circle is a streamline. The added term is analytic wherever $a^2/\bar z$ is outside the circle, and $a^2/\bar z$ is outside the circle exactly when $z$ is inside it. So the added term’s singularities are all interior, at the inverse points of the original’s.

A vortex at $d$ therefore acquires an image at $a^2/d$ of opposite sign, and — because the total circulation round the body must be preserved — one of the original sign at the centre. That third vortex is the one readers forget, and it is the one that keeps the body’s circulation at zero when the outside vortex is brought in from infinity.

The expansion that diverges on its own body #

The second consequence is about the far field, and it corrects something a reader could reasonably have inferred from what the far field remembers.

That essay establishes that three numbers survive the journey to infinity — a circulation, a net outflow and a dipole — and that the differences between bodies with the same ones die two orders faster. It is a statement about large $r$. The question here is how large.

The Rankine oval is the right test case because its expansion is exact and elementary. A source and a sink at $\mp s$ in a stream give

$$\frac{dw}{dz} = U - \frac{m}{\pi}\sum_{k \text{ odd}} \frac{s^k}{z^{k+1}},$$

with radius of convergence $|z| > s$. So the series converges outside the circle of radius $s$, and the question is whether the body is outside that circle.

For a fat oval it is. At a fineness ratio of 1.17 the waist sits at 1.87 against a source at 1, the whole surface is outside, and the series reaches one part in a hundred in seven terms and machine precision shortly after.

For a slender one it is not. At a fineness of 6.7 the waist is at 0.158 — inside the circle of singularities — and the series diverges on the body.

Not slowly, and not to the wrong answer. The relative error is 3.95 at one term, $4.4\times10^{16}$ at twenty-one and $6.2\times10^{64}$ at eighty-one. The crossover is at a fineness ratio of about 1.55, which is a very fat body indeed — so almost every slender shape in aerodynamics has a surface its own multipole expansion cannot reach.

This does not make the far-field argument wrong. It makes it a far-field argument.

The one that arrives on the nose #

The third consequence is the one that connects to thin-aerofoil theory, and it is the sharpest.

The Joukowski map $z = \zeta + c^2/\zeta$ has critical points at $\zeta = \pm c$, where $dz/d\zeta$ vanishes. One of them is the trailing edge, and the Kutta condition exists precisely to cancel the zero in the denominator there — that is what from a circle to a wing is doing when it picks the circulation.

The other maps to $z = -2c$, and that point is inside the body. Its distance from the leading edge is

$$\frac{4\mu^2}{1 + 2\mu},$$

where $\mu$ is the map’s offset, and the closed form is reproduced by the mapped profile to $10^{-9}$.

Against thickness that distance is a clean square: the exponent measured across a sweep from 1.3 to 20 per cent thickness is 2.004. A twelve per cent section is analytic only within 0.86 per cent of a chord of its own nose.

That is the leading-edge singularity of thin-aerofoil theory, before the theory is written down. The thin limit is the limit in which an interior singularity arrives on the surface, and a linearised theory that puts the vorticity on the chord line is a theory evaluated at the place the singularity has reached. The half that carries nothing computes what the thickness contributes to the lift — nothing, to $10^{-17}$ — and this is the companion statement about what it contributes to the validity: everything, near the nose.

A number worth carrying: the crossover fineness #

The fineness at which a Rankine oval’s waist reaches its own singularity circle is 1.55 — a body half again as long as it is thick.

That number is worth carrying because it is so much smaller than a reader would guess. The intuition that a multipole expansion works “outside the body” is correct for a sphere, a cylinder and a very fat oval, and wrong for everything aerodynamic. A wing section at twelve per cent thickness has a fineness of eight. A fuselage has thirty. A submarine hull has six. In every one of those the surface is well inside the circle containing whatever singularities generate the shape.

What saves practical far-field work is that it is done far away. A wake survey plane at several spans downstream is comfortably outside every circle in the problem, which is why where the lift reaction is can take a contour of any shape and get $\rho U\Gamma$ to eight figures. Move the contour onto the body and the same expansion has nothing to say.

What decides where they are #

There is a general rule underneath the three cases and it is worth stating, because it makes the answers predictable rather than surprising.

The singularities of the continued flow are wherever the body’s own construction put them. A body made of sources and sinks has them at the sources and sinks. A body made by a conformal map has them at the images of the map’s critical points and at the images of whatever the mapped flow had. A body made by the circle theorem has them at inverse points.

And a body specified only by its shape has them somewhere determined by the shape, in a place nobody chose. That is the case ask for the pressure is about from the other end: inverse design specifies the pressure distribution and solves for the shape, and the reason some prescribed distributions have no body is that they demand a singularity in the wrong place.

What this says about the exact theory’s reach #

The theory’s reputation is for being exact and irrelevant — closed-form answers to a problem with no viscosity in it. The picture that emerges here is different and more useful: it is exact, and it has a geometry of its own that is not the geometry of the body.

A body is a curve in the plane. The flow past it is an object with singularities in it, and the singularities sit at distances that have nothing to do with the body’s dimensions: $a^2/d$ for an image, $s$ for a source pair, $4\mu^2$ for a map. Every question about the flow’s behaviour at the surface is really a question about the distance from the surface to the nearest of them.

That is why a slender body is hard and a fat one is easy, in a theory where neither has any drag and both are solved in closed form.

The three distances, side by side #

Putting the three cases in one place makes the shape of the argument visible.

For the image system, the distance from the surface to the nearest singularity is $d - a^2/d$, which is the vortex’s own distance from the wall to leading order. The peak that results scales as the inverse of it, and the width as the distance itself: a singularity at a distance $\epsilon$ produces a feature of width $\epsilon$ and height $1/\epsilon$, which is the general behaviour of a dipole seen from close range.

For the oval, the distance is $t - s$ where $t$ is the half-thickness and $s$ the source position, and the sign of it decides whether an expansion exists at all rather than how good it is. That is a different kind of failure: not an approximation degrading, but a series with no meaning.

For the aerofoil, the distance is $4\mu^2/(1+2\mu)$, which is small because it is a square. The consequence is the one thin-aerofoil theory has always had, and stating it as a distance rather than as a singular integral makes it predictable: halve the thickness and the safe region near the nose shrinks by four.

Three mechanisms, one arithmetic.

The rule of thumb, in one line #

If a reader takes one thing away it should be this.

A pressure feature of width $w$ on a surface is a singularity a distance of about $w$ inside it, and its depth goes as $1/w$.

That converts a picture into a statement about the flow’s analytic structure without any computation. A broad suction hill over half a chord is a singularity half a chord in; a spike over one per cent of a chord is a singularity one per cent in, and it is a hundred times deeper. The vortex-near-a-cylinder sweep above is that rule measured — a factor of 150 in depth against a factor of 39 in width, which is close enough to a reciprocal for a rule of thumb.

It also explains why the trouble in aerodynamics is always at leading edges and never at trailing ones. A trailing edge has the Kutta condition cancelling the map’s zero; a leading edge has the other critical point sitting a distance $4\mu^2$ inside it, which is small and getting smaller as the section is thinned.

Limits recorded rather than smoothed over #

The oval’s expansion is about one centre. The series used here is about the origin, and its radius of convergence is the distance from the origin to the singularities. Expanding about a different point gives a different circle, and for a slender body a chain of local expansions can cover the surface where one global one cannot. What cannot be done is the thing the far-field argument suggests, which is to describe the whole surface with a few moments about one centre.

The nose result is for one family. The Joukowski map has a single thickness parameter and puts the maximum thickness at about a quarter chord for every member of the family, which is exactly the freedom the thickness essay says the collapse discards. That the depth goes as the square of the thickness for this family is computed; that it does so for every family is asserted.

Nothing here is a claim about real flow near a nose. A real leading edge at incidence has a boundary layer on it and often a separation bubble, and the singularity being close to the surface is a statement about the exact theory rather than about the air. When the flow lets go is where that question is asked properly.

And “the continued flow” is not a physical object. Nothing is flowing inside the body. What is inside is the analytic continuation of the function that describes the flow outside, which is a mathematical statement with physical consequences, and the consequences are the three measured above.

The pattern #

Each of the three measurements is a limit with a residue. Bring a vortex to a wall and the gap to its image closes, leaving a peak that diverges. Thin a body and its surface crosses the circle of its own singularities, leaving an expansion that diverges. Thin an aerofoil and the interior branch point reaches the nose, leaving a theory whose error near the leading edge does not go away.

In all three the limit is geometric and easy to picture, and the thing that survives it is the distance to a place inside the body where nothing is happening at all.