The inside a flow does not decide

Every body on this site is built out of singularities that are not there. A doublet at the centre of a cylinder, a source and a sink inside a Rankine oval, a sheet of vortices along a camber line, a row of panels round an aerofoil. This collection has an essay about that already, and it establishes the thing a reader most needs: the interior field is fiction, and only the exterior is a solution.

What it does not say is that the fiction is not even unique.

The exterior determines nothing about the interior #

Take a circle and put a doublet at its centre; that is the standard solution and it has no higher multipoles at all. Now spread the same total doublet strength uniformly round a ring of radius $b$ inside the body. Outside that ring the field is exactly the point doublet’s, by the same argument that makes a spherical shell’s gravity indistinguishable from a point’s.

Spread it uniformly over a disc instead, and the same thing happens. Spread it over any distribution with circular symmetry, and again.

The measurement is unambiguous. Outside the distribution the ring and the disc agree with a point doublet to three parts in 10¹⁵, which is the arithmetic; inside they differ by everything — the ring has a field that reverses across it and the disc has one that goes to zero at the centre.

So the exterior flow — every value of the velocity at every point outside the body, which is as much data as a flow can carry — determines the interior singularity system not at all. It leaves an infinite-dimensional family, and the members of that family are not small perturbations of one another.

It is worth being clear what kind of statement that is, because it is stronger than the ones this subject usually makes about non-uniqueness. A boundary-value problem that is under-determined can often be closed by adding a condition: a hole in the domain leaves a one-parameter family and the Kutta condition picks a member of it. Here there is nothing to add. The exterior flow is already the complete solution of a well-posed problem, unique and smooth, and the family being described is a family of descriptions of that one solution rather than a family of solutions.

The distinction has a consequence for what a measurement could ever settle. No experiment anywhere in the fluid can distinguish two members of the family, because they produce the same field everywhere the fluid is. Choosing among them is not an empirical question and cannot be made into one. It is a question about which representation is easiest to compute with, and that is what the rest of this essay measures.

A discrete ring is not one of them, and fails at its own order #

It is worth separating the exact family from the near ones, because the near ones behave in a way that is itself a measurement.

$n$ equal doublets on a circle are not circularly symmetric. They carry the multipoles of order $n$, $2n$, $3n$ and upwards, and those decay as $(b/r)^n$. So a discrete ring agrees with a point doublet at a rate set by how many doublets it has, and the exponents fitted here come out at 6.04, 8.01 and 12.00 for rings of six, eight and twelve.

That is how one can tell the leak is the discreteness rather than anything else. An error that merely gets small is consistent with a dozen wrong explanations; an error falling at exactly the ring’s own order is consistent with one.

The same arithmetic explains a familiar practical rule. A source or vortex panel far from the point where the flow is being evaluated can be replaced by a point at its centroid, and the error of doing so falls at the panel’s own multipole order — which is why fast summation methods group distant panels and evaluate them once. The leak measured here is that approximation seen from the other end: the ring is the far-field grouping, and its error is the multipole it does not carry.

What the freedom costs #

The freedom would be a curiosity if nobody had to choose. Everybody who computes a flow round a body by placing singularities inside it does, and the choice has a price.

Thirty-six point sources are fitted to the boundary condition on an ellipse four fifths as tall as it is long, on an inner contour at a range of depths, by least squares against 288 collocation points. The error is measured against the exact solution — the ellipse’s surface speed is known in closed form — rather than against the fit’s own residual, so what is plotted is how wrong the answer is rather than how well the equations were satisfied.

There is a minimum, at seven tenths of the body, and it is sharp. Either side of it the error rises by orders.

The two ends fail for opposite reasons. Deep inside, every source looks the same from the surface: their normal-velocity kernels at the collocation points become nearly parallel, the least-squares matrix becomes singular to working precision, and the strengths blow up while the answer degrades. Close to the surface, each source dominates its own collocation point and the field between the points is not controlled, so the fit satisfies what it was asked and misbehaves where it was not asked.

The shape of the surface-speed comparison says which failure is which. The good placement lies on the exact curve everywhere. The deep one departs where the body is bluntest — near the nose and the tail, where the surface curvature is highest and the boundary condition varies fastest along it — because that is where reproducing the field needs the sources to be able to act locally, and from a fifth of the way in they cannot.

That is the general shape of the deep-end failure, and it is worth recognising because it does not look like a numerical failure. It looks like a smooth, plausible surface-speed distribution that is simply the wrong one, with no oscillation, no spike and nothing in it to suggest that the representation has run out of reach.

The coefficients are not bounded by the flow #

Two placements that both reproduce the exact surface speed to a part in ten thousand have source strengths differing by a factor of a hundred, and by their whole shape: one is a smooth front-to-back distribution, the other oscillates from source to source.

Across the whole admissible family the largest strength runs from 0.27 to seventy-seven thousand, for a unit free stream past a body of unit size. The condition number of the least-squares matrix spans thirteen decades.

That is worth separating from the ordinary observation that a badly conditioned problem is hard to solve, because nothing here is badly posed. The flow exists, it is unique, it is smooth, and it is given in closed form. What is badly conditioned is a choice — a representation of that flow — and the choice is invisible in every quantity a reader of the answer would look at.

At a fifth of the body’s size the thirty-six sources are linearly dependent to working precision and the factorisation meets a negative pivot. The amount of regularisation each placement needs is reported rather than applied quietly, because it is the measurement: a routine that silently added enough of it would turn all of this into a smooth curve with nothing in it.

Where the flow stops being a flow #

The deep end’s failure has a cause, and it is a piece of exact mathematics rather than a numerical observation.

The ellipse is the image of a circle under $z = \zeta + c^2/4\zeta$, and the flow past it is the flow past that circle carried through the map. The map’s derivative vanishes where $\zeta = \pm c/2$, which is $z = \pm c$: the foci. So the analytic continuation of the exterior flow into the interior is singular at two points whose distance from the centre is $\sqrt{A^2 - B^2}$, fixed by the shape and by nothing else.

For a circle, $A = B$ and the two foci merge at the centre. There is nothing to reach.

And that is exactly what the fits do. A circle is reproduced to a part in a hundred million at every depth tried, down to a twentieth of its radius. The ellipse’s error rises by a factor of nearly five hundred as the sources retreat inside its foci, because a representation sitting where the continued field is analytic is trying to reproduce, from there, a field that is not.

This is the same fact the far field’s forgetting is the outward-facing half of. There, information about the body dies as the distance grows and two shapes become indistinguishable. Here, information about the exterior does not reach inwards past a definite boundary, and inside it every representation is as good as every other.

The boundary condition is not the problem #

There is one more measurement, and it is the one that changes what a residual is worth.

Move the source ring outside the body, to 1.4 radii, and fit it exactly as before. It satisfies no-through-flow on the surface to seven parts in a hundred thousand — that is all the least-squares problem was ever asked for, and it does it well. The surface speed it produces is wrong by 1.8 free streams.

The field has point sources sitting in the fluid, where the speed runs away as the inverse distance and mass appears out of nothing: the flux out of a small circle round the strongest of them is 0.44, which is that source’s strength, because that is what a source is.

So the boundary condition is not the whole of the problem. What the exterior flow requires of a representation is that it be regular in the fluid, which is a condition on where the singularities are rather than on what their strengths are, and no amount of collocation tests it. A residual is not a proof.

Why the best placement is where it is #

The minimum at seven tenths is not a coincidence of this body and this number of sources, and it is worth saying what sets it.

The deep side is set by the continuation: the sources have to be able to represent a field whose analytic continuation is singular at the foci, at six tenths of the semi-major axis. The shallow side is set by the discretisation: adjacent sources have to be far enough from the collocation points that neither dominates its own, which means the spacing between sources must not be much larger than their distance from the surface. Thirty-six sources round this ellipse are spaced about a sixth of a semi-major axis apart, so they need to sit at least that far in.

Between six tenths and about nine tenths both conditions are met, and the fit reaches three parts in a hundred thousand. Outside that window one or the other fails, and the width of the window narrows as the number of sources rises — more sources are closer together, so they must be shallower, and the deep constraint has not moved.

That is why the practical answer for a general shape is to stop trying to find the window. Putting the singularities on the surface removes the deep constraint entirely and replaces the shallow one by the ordinary panel-size condition, at the cost of having to handle a singular kernel, which is what the standard panel formulations spend their machinery on.

What this means for the methods that use it #

The panel methods that this collection has already built — the discrete-vortex rule on a flat plate, the source and sink that make a Rankine oval, the inverse design that asks for a pressure and gets a shape — all choose a representation, and the choice is usually made by convention rather than by argument.

Three consequences follow, and none of them is an objection to the methods.

A converged surface pressure does not validate a strength distribution. They are different quantities, and one is a functional of the flow while the other is a coordinate on a family of representations of it. Reporting the strengths as though they were physical is reporting a coordinate.

Placing singularities on the surface is the disciplined choice. A surface distribution has no depth to get wrong, its conditioning is governed by the panel size rather than by a placement, and it is regular in the fluid by construction. That is why panel methods are written that way, and the reason is this page rather than tradition.

And a fit that has to be regularised is telling the truth about itself. The ridge those deep placements need is not a numerical inconvenience; it is the calculation reporting that the information required to pick a member of the family is not in the data.

The same freedom, seen three other ways #

The non-uniqueness here has relatives elsewhere in this collection, and putting them beside one another shows that it is a property of the mathematics rather than of panel methods.

A conformal map is a representation too. Mapping a circle to a wing produces the flow past the wing without placing a single singularity by hand, and the doublet at the circle’s centre becomes a distribution in the physical plane whose location nobody chose. It is another member of the same family, arrived at by a different route, and it agrees with all the others outside.

Superposition is the family’s construction rule. Flows add up, so any two representations differ by a field that is identically zero outside the body and non-zero inside — which is exactly the description of the freedom. There is one such field for every way of rearranging the interior singularities, and they form a linear space.

And the method of images is a placement decision that happens to be forced. The mirror that is a circle puts a singularity at a specific interior point because that is where it makes the boundary a streamline, and the position is determined by the geometry rather than chosen. That is the exception: it is the case where a condition outside the family — an exact boundary condition, satisfied identically rather than by least squares — reaches in and picks a member.

The pattern in all three is the one this collection keeps meeting. An exact statement about the outside is a constraint, the representation is what the constraint does not determine, and the residual freedom is a computable quantity rather than a matter of taste.

What a panel method’s convergence actually is #

There is a last observation worth making, because it changes how a convergence study of such a method should be read.

Refining a panel method means adding singularities, and adding singularities on a surface distribution makes the representation richer while keeping it regular in the fluid — the panels get smaller, the kernel stays integrable, and the answer converges. That is the ordinary and healthy case.

Refining an interior distribution does not behave that way. More sources at the same depth are closer together, so their kernels at the surface become more nearly parallel, so the conditioning degrades: the method converges towards the exact answer and towards an insoluble system at the same time. The error falls, then flattens, then rises as round-off takes over, and the best achievable accuracy is set by the conditioning rather than by the discretisation.

That is why a convergence study of a method of this kind has to plot the residual and the condition number, and why quoting a convergence rate from the early part of such a curve is quoting the part before the difficulty starts.

It is also the practical reason surface distributions won. Both representations describe the same flow, both are members of the family this essay is about, and only one of them gets better indefinitely when more effort is spent on it.

One more consequence, for anybody reading a published strength distribution. Papers on panel methods sometimes plot the source or vortex strengths along a body as though they were a result. They are a coordinate: a different placement, a different panel count or a different formulation gives a different plot of the same flow. What can be plotted as a result is the surface velocity, the surface pressure, or any quantity evaluated in the fluid — and the distinction is not pedantic, because two papers reporting incompatible strength distributions for the same body may be in perfect agreement about everything a reader cares about.

And a note on the number of sources. Thirty-six was chosen because it is enough to resolve the ellipse’s boundary condition and few enough that the conditioning at the deep end is visible rather than catastrophic. More sources sharpen every conclusion on this page and narrow the usable window, which is the trade the method has: refinement improves the representation and degrades the arithmetic at the same time.

What is not claimed #

The interior field is not being called wrong. It is not a field at all — there is no fluid there, the region inside a body is outside the solution — and the point of this page is that its non-uniqueness has consequences outside, in the conditioning and the coefficients, rather than that anybody was mistaken about the inside.

The ellipse’s focal result is two-dimensional and shape-specific. It follows from a conformal map and does not generalise to three dimensions or to an arbitrary section, where the singularities of the continued flow have to be located case by case and may be a surface rather than two points.

The condition numbers belong to this discretisation. Thirty-six sources, 288 collocation points, normal equations and a Cholesky factorisation. A least-squares solver working on the rectangular system directly would lose half as many digits, and one with column pivoting would report the rank deficiency rather than a negative pivot. The thirteen decades of conditioning are a property of the placement; how much of that reaches the answer is a property of the solver.

And none of this is a rate of convergence. The best placement here reaches three parts in a hundred thousand with thirty-six sources, which is a modest fit by the standards of the method. What is being measured is the variation across placements, not the accuracy attainable at the best one.