Bodies made out of nothing

The usual way of setting up a flow problem is to decide on a shape and then find the flow that fits round it. That is hard: the shape appears in the boundary condition, and boundary conditions on awkward curves are exactly what makes partial differential equations difficult.

The inverse method throws the order away. Add up some elementary flows, look at the streamlines that come out, and if one of them happens to be closed, declare it to be a body. It sounds like cheating and it is entirely rigorous, because in this theory a wall does exactly one thing — nothing passes through it — and a streamline does exactly that too.

What a source is, and what it is not #

A source is a point from which fluid appears at a fixed rate, spreading out equally in all directions. Its velocity falls off as one over distance, which is the rate at which a fixed flow rate has to thin out as the circle it is crossing gets bigger.

It is not a physical object and there is no need to pretend otherwise. Fluid does not appear from nowhere. What a source is, is a solution of Laplace’s equation that is well behaved everywhere except at one point, and the point is going to end up hidden inside something — which is the whole technique. The same applies to the vortex, the doublet, and every other elementary piece.

The rule that makes it work is linearity: Laplace’s equation adds, so any number of these can be laid on top of a uniform stream and the sum is still a solution. Where the singularities end up buried, the flow outside them is a perfectly ordinary field that satisfies everything it is supposed to.

Reading the strength is worth doing once. A source of strength m delivers m units of volume per unit time per unit depth. Set it in a stream of speed U and there is one distance at which the source’s outward push exactly balances the stream’s inward one — that is the stagnation point, and it sits at −m/2πU. Every dimension of what follows is built from those two numbers.

There is one more thing a source is not, and it is worth saying because the word invites the mistake. A source is not a jet. A jet has momentum in a direction and entrains the fluid around it; a source has no preferred direction at all and no momentum whatever, because it pushes equally hard in every direction and the pushes cancel. The asymmetry in every figure below comes from the stream, never from the source.

A body with a front and no back #

Take one source, one stream, and nothing else.

Far upstream the stream wins and the flow is essentially uniform. Close to the source the source wins. Between them is the stagnation point, and through it runs a streamline that divides the fluid that came from upstream from the fluid the source produced. That dividing streamline never closes — the source keeps producing and there is nothing to take it away — so it opens out downstream forever.

Both of those numbers are worth deriving rather than reading off. The nose is where the stream and the source cancel, which is the balance above. The far-field width is a mass balance: the source emits m, and far downstream that fluid is travelling at very nearly the stream speed U, so it needs a channel of width m/U — half of it above the axis and half below, hence m/2U on each side.

For the values drawn, m = 1 and U = 1, so the nose is at −0.159 and the half-width tends to 0.500. The figure prints both, and both come out of the same closed form the surface was traced from.

A half-body is a real shape rather than a curiosity. It is the front of a bridge pier, the nose of a long fairing, the leading edge of anything much longer than it is thick — and the whole point of the construction is that the pressure distribution over that nose is available exactly, with no computation beyond arithmetic.

Its outline is also available exactly, which the oval’s is not, and the reason is worth a line. On the surface the streamfunction equals m/2, and writing that out in polar coordinates gives y = (m/2U)(1 − θ/π) directly. Every point of the surface follows from one angle without any solving at all — so the curve in the figure above is not a traced approximation but the shape itself, evaluated.

The two dashed lines are the asymptote rather than a measurement, and drawn as an asymptote deliberately. The body approaches m/2U and never reaches it, which an arrow labelled with a width would misrepresent as an achieved dimension.

Closing the back #

To get a finite body, take the fluid away again. A sink is a source of negative strength, and a source and sink of equal strength in a stream produce a closed dividing streamline: everything the source emits, the sink swallows, and the fluid in between never leaves.

The result is the Rankine oval, and its proportions are set by one parameter — how far apart the two singularities are. Close together and it is nearly circular. Far apart and it stretches into a long slender shape with almost parallel sides.

The stagnation points are the two ends, and their positions come out in closed form:

$$x_s = \pm\sqrt{s^2 + \frac{m s}{\pi U}}$$

where 2s is the separation. The width does not: finding it means solving for where the streamfunction vanishes off the axis, and there is no closed form for that.

That distinction matters more than it looks. The half-body’s surface could be written down. The oval’s cannot, and has to be found numerically — which means the figure is drawing the output of a root-finder, and a root-finder can be wrong.

What the solver computed, and how it was checked #

The oval’s outline is found by bisection. At each station across the body, the streamfunction is evaluated just above the axis and far above it; the first is negative and the second positive, so a root is bracketed, and ninety bisection steps close on it.

The check is the one that makes the whole inverse method honest. The outline is handed to the tangency assertion, which walks it and measures the component of velocity perpendicular to it at every point. If the root-finding were wrong — a bracket that occasionally caught the wrong root, a convergence tolerance too loose, an off-by-one in the cosine spacing — the traced curve would not be a streamline, the flow would visibly pass through it, and the figure would still look completely convincing.

The worst normal component over the oval’s 240-point outline is 8.4 × 10⁻⁶ of the local speed. Over the half-body’s analytic outline it is 1.9 × 10⁻⁵, slightly worse, and worse for an interesting reason: that curve is traced in the angular variable, so its points bunch near the nose and spread near the tail, and the finite-difference tangent used by the check is a poorer approximation where they are far apart. The check is measuring its own discretisation there, not the field.

Both are far below the 5 × 10⁻³ tolerance the gate applies, and the gate has a rejection test beside them: handed a uniform flow and a circle, the same assertion refuses at a normal component of 1.

The limit that produces a cylinder #

Now push the source and sink together, keeping the product of strength and separation fixed.

In the limit the pair becomes a doublet, and the body it makes is a circle. The radius is √(ms/πU), and the figure’s four panels show the measured half-width converging on it: 27.9% away at a separation of 1.6, then 16.8%, then 9.4%, then 3.0% at a separation of 0.12.

This is the same cylinder the ideal theory starts from, and arriving at it this way says something the direct derivation does not. A cylinder in a stream is not a special case that happens to have a tidy solution; it is the degenerate end of a one-parameter family of shapes, all of which have solutions of the same kind. The tidiness is inherited, not intrinsic.

It also makes the doublet less mysterious. A doublet is usually introduced as a source and sink brought together with strength going to infinity, which is a limit that sounds like a piece of sleight of hand. Watching the ovals round out into a circle makes it a shape converging on a shape.

The pressure distribution, which is the actual product #

Calling these shapes bodies makes the outline sound like the deliverable, and it is not. Nobody needs another oval. What the construction hands over free is the pressure everywhere on the surface, in closed form, and that is the thing a designer would otherwise pay for.

For the half-body it is one line. The speed on the surface follows from adding the stream to the source’s own contribution, and the pressure follows from Bernoulli. Three features come out of it.

At the nose the speed is zero and the pressure coefficient is exactly one, as it is at any stagnation point in any incompressible flow — which is a check rather than a result.

Round the shoulder the flow overshoots. The maximum speed is about 1.26 times the free stream, giving a minimum pressure coefficient near −0.59, and it occurs about two-thirds of the way out to the body’s eventual half-width rather than at the nose or at the widest point.

And then the pressure recovers, from that minimum back towards free-stream, over the whole of the infinite remaining length.

That last feature is why the half-body survives and the oval does not. A pressure recovery is what a boundary layer has to climb, and the half-body spreads half a dynamic pressure of it over an unbounded distance — the gentlest possible ramp, which any real layer will follow. The oval has the same overshoot and must give it all back before its tail, over a finite length that gets shorter as the body gets fatter.

So the family is not one of shapes at all. It is a family of pressure recoveries, and the shape is the incidental thing the singularities happened to draw round it.

What the picture cannot show #

The dividing streamline is a wall to this theory and to nothing else.

Fluid does not cross it, so the flow outside is exactly what it would be if the region inside were solid. But the region inside is not solid. It is fluid, and in the ovals above it is fluid in vigorous motion — running from the source to the sink through the middle of the body — and none of that is drawn, because none of it affects anything outside.

That is fine as far as the pressure distribution goes, and it is not fine if the question is about anything that touches the surface. There is no boundary layer here, because there is no wall for one to form on, and consequently no skin friction and no possibility of separation. The figures show a body that no real fluid would produce.

The inverse method has one further limitation that is more practical than conceptual, and it is why it was eventually abandoned. The shape cannot be chosen. A designer with a particular aerofoil in mind cannot get it by adding sources; they get whatever the sources happen to produce, and adjusting the shape means guessing at strengths and looking at the result. Panel methods solved this by turning it round again — distribute unknown singularity strengths over a specified surface and solve for the strengths that make it a streamline — and that is a linear system rather than a search.

Where the model stops #

Everything drawn here is ideal flow, so it inherits the whole of that model’s failures. The most conspicuous is that all these bodies have exactly zero drag, which is d’Alembert’s paradox and is wrong by all of it.

The oval is a particularly good illustration of why. Its flow is fore-and-aft symmetric: the pressure distribution over the front half is the mirror image of the one over the back, so every push forward is matched by a push backward. In a real fluid the flow separates somewhere on the back half and the pressure there never recovers, which is where the drag comes from. Nothing about the construction above can produce that, because the sink is there to guarantee the flow closes up.

The half-body is subtler and slightly better behaved. It has no back at all, so there is no cancellation to be spurious about, and its nose pressure distribution is genuinely useful — which is why it survives in practice where the oval does not.

There is a second limitation, less often mentioned, and it is about which ovals are worth trusting. The pressure over the widest part of a slender oval falls a long way and then has to recover over the rear half, and the steeper that recovery the sooner a real boundary layer will give up. A short fat oval asks for a violent recovery and separates almost at once; a long thin one asks for a gentle one and can genuinely stay attached most of the way. So the family drawn above spans shapes the ideal theory describes well and shapes it describes not at all, and nothing in the ideal solution distinguishes them. Only the gradient does, and reading it off is a separate calculation.

That is the honest summary of the inverse method as a design tool: it produces shapes and their exact inviscid pressure distributions at essentially no cost, and it says nothing whatever about whether a real fluid will follow them.

Who found it, and when #

William Rankine described these shapes in the 1860s, in work aimed at ships rather than at aeroplanes, which did not yet exist. The name has stuck to both the half-body and the oval, and Rankine’s own interest was in the stream lines of a hull moving through water and the wave pattern that came with them.

The inverse method itself is older and more diffuse — it is implicit in anything anyone did with Laplace’s equation from the 1780s on — and it was standard practice for the whole of the nineteenth century, because it was the only method available that did not require solving anything. Whole catalogues of body shapes were assembled by putting singularities in a row and drawing what came out.

The catalogues are gone and the technique is not. Every panel method, every vortex-lattice code, every boundary-element solver in current use is the same idea with the search replaced by a matrix inversion: represent the body by singularities, and let the surface be a consequence.

There is a wider point in that survival, and it is the one worth taking away. The inverse method looks like a trick played to avoid a hard boundary condition, and it is; but the reason it keeps being reinvented is that it changes what has to be stored. A field described by a handful of singularities is described by a handful of numbers, wherever it is evaluated, at whatever resolution is wanted. A field described on a grid is described by the grid. For three centuries the first has been the cheaper representation, and it is still the cheaper representation for anything smooth.

Where the ladder goes next #

Next rungs on this anchor: distributed singularities, where a line of sources of varying strength replaces a discrete pair and any slender shape becomes reachable; the panel method proper, which inverts the problem to solve for strengths on a given surface; the method of images, which is the same construction used to produce a boundary rather than a body; and three-dimensional bodies of revolution, where a line of sources on an axis produces a shape and the algebra survives the extra dimension.

Then across to the flow net, which is where the streamfunction that defines all these bodies comes from, and to the exact theory, whose cylinder turns out to be the last member of this family rather than the first.