Nothing but the edge

Two theories of an aerofoil have appeared on this site and both have a shape they cannot handle. The conformal map is exact and works for the sections a particular map happens to produce. Thin-aerofoil theory works for any camber line and throws the thickness away.

A panel method has neither restriction. Cut the actual surface into straight segments, put an elementary flow on each, and choose the strengths so that nothing flows through the wall. The geometry can be anything — an aerofoil, a car, a ship’s hull, a wing with a flap hanging off it — and the answer is exact in the limit of small panels.

What makes it worth an essay is not that it works. It is what happens when the equations are counted.

What is on each panel #

The elementary flows are the two that a plane potential problem has, and they are the same two written twice.

Take a panel and work in its own frame, with the panel lying along the local x-axis from 0 to its length. A constant-strength source on it induces, at a point (x, z),

$$u = \frac{\sigma}{2\pi}\ln\frac{r_1}{r_2}, \qquad w = \frac{\sigma}{2\pi}\,\beta,$$

where r₁ and r₂ are the distances to the two ends and β is the angle the panel subtends at the point. A constant-strength vortex induces the same two integrals with the roles swapped and one sign changed. That is not a coincidence and is worth a sentence: the source and the vortex are the real and imaginary parts of the same complex potential, so their velocity fields are conjugate harmonic functions and the Cauchy–Riemann relations tie the two integrals together. One routine returns both.

The case that has to be right is a point on the panel. There, β is exactly π: the source pushes the point straight off the surface at half its strength, and the vortex, whose velocity is along the panel, contributes nothing to the normal direction at all. Both halves of that were checked before anything else was, on a circle, where every answer is known in closed form. Thirty-two panels reproduce $C_p = 1 - 4\sin^2\theta$ to fourteen decimal places and the drag comes out at 10⁻¹⁴ — d’Alembert’s zero, arriving from a numerical method that was never told about it.

Counting the equations #

The construction here is Hess and Smith’s, and its two ingredients are chosen for a reason worth stating. Each panel carries a source of its own strength, and every panel shares a single vortex strength between them.

Sources are what make a body have a thickness: a source distribution pushes the flow apart and closes it again, and a closed body needs the total source strength to be zero. Sources produce no circulation whatever — a source is symmetric, and the line integral of its velocity round any closed curve is zero. So the source distribution alone can make a shape and can never make lift.

The vortex is where the lift is, and it is shared rather than distributed because that is the minimum: one number, the circulation per unit length of surface, held constant round the section.

Now count. N panels give N tangency conditions — one at each panel’s midpoint, saying the normal velocity there is zero. The unknowns are N source strengths and one vortex strength: N + 1. The system is one row short.

The row that is missing #

It would be easy to read that as a construction error and go looking for another condition. There is another condition, and it is not a repair.

Dictate the circulation. Any value. Solve the N tangency equations for the N sources with that value held fixed. What comes back is a flow that satisfies mass conservation everywhere, satisfies the equations of motion everywhere, and does not leak through the aerofoil — the solver checks the last of those by evaluating the velocity a ten-millionth of a chord outside every panel and requiring the normal component to be under two parts in a thousand.

Done six times at six different circulations, it returns six different flows round the same body at the same incidence, with lift coefficients spanning 1.25. They are all correct.

That is not a defect of the method. It is the fact the Kutta condition exists to fix, and it is the one place on this site where that fact can be seen rather than argued. The exact conformal map hides it: the vortex strength is chosen at the moment the map is set up, by a line of code that puts the rear stagnation point at the trailing edge, and the indeterminacy has been resolved before anything is computed. A panel method has to decide, and the deciding is a row in a matrix.

The row is the Kutta condition written arithmetically: the tangential velocities on the two panels meeting at the trailing edge are equal and opposite. That closes the system, and the answer it picks out is the one marked on the figure.

The direction the family runs in #

The family has a direction, and it is worth looking at.

Solve twice with the circulation dictated at zero and at one, and subtract. The difference is a source distribution that compensates exactly for a unit of circulation, keeping the body a body. That vector is what the missing row would have to pin down.

It is not spread evenly. It is concentrated at the trailing edge, changes sign there, and is nearly nothing over most of the section. Which is why one sentence about the trailing edge is enough to fix the entire flow: the members of the family differ almost exclusively in a small region, and a statement about that region has enough leverage to select one of them.

Nothing in the theory required that. The null vector could have been a broad, gentle redistribution over the whole surface, in which case a condition at one point would be a weak and arbitrary way to choose. It is not, and the reason is the same reason the Kutta condition is physically right: the members of the family differ in how hard the flow has to turn round the sharp corner, and that is a local question.

Making it accurate, which is a separate problem #

Having the right equations is not having the right answer, and the gap between them is where most of the practical content of a panel method lives.

The test section is a Kármán–Trefftz aerofoil, which arrived on this site with this essay. It is the Joukowski map with the last step generalised so that the trailing edge closes at a finite angle rather than in a cusp, and it is exact in the same way — the lift comes out of the same argument about where the rear stagnation point sits.

The reason for going to the trouble is a fact about the method rather than about the shape, and it cost an afternoon to find. On a cusped trailing edge the two panels meeting at the tail are collinear, so the Kutta row is nearly a linear combination of the tangency rows beside it, the system is ill-conditioned, and the computed lift wanders as panels are added: 0.70, 0.60, 0.79 and 2.29 for the same section at 16, 24, 32 and 64 panels. A convergence study on that geometry measures the conditioning of a matrix and not the accuracy of a scheme. A cusp is not a hard case for a panel method; it is a degenerate one.

With a fifteen-degree trailing edge, the same code behaves.

The spacing measurement is the one a practitioner uses. Panels put down evenly along the arc spend most of themselves on the flat middle of the section and resolve the nose with a handful; cosine spacing clusters them where the curvature and the pressure gradient are. The exponents are 0.90 and 1.53, measured, and the gate requires both — the second to be above 1.4, the first to be between 0.7 and 1.2, and the difference to be at least 0.4.

There is a second requirement in that gate and it is the one that catches real errors: both sequences must be monotone. An error that wanders is the signature of conditioning rather than truncation, and that is exactly what the cusped section produced.

Two lifts, and the one to trust #

Every solve here produces the lift twice.

Kutta–Joukowski on the total circulation the vortex sheet carries: multiply the shared strength by the perimeter, and $C_l = 2\Gamma/Uc$.

And the pressure integrated round the surface: form $C_p$ at each panel from its own tangential velocity, resolve the force perpendicular to the free stream, and add up.

They share no arithmetic below the tangential velocities and they do not agree. At 16 panels they differ by 7.6 per cent; at 192 by 0.8. The gate requires the gap to be under 1.5 per cent at the finest and to have closed by a factor of four from the coarsest — which is to say it requires them to be converging on each other rather than merely being close.

The rate is the useful part. The circulation route inherits the method’s own convergence; the pressure route is an integral of a squared quantity over flat panels, and is first order. A panel code that reports lift by integrating the pressure distribution is throwing away most of its own accuracy, and that is a piece of practical advice that falls straight out of having computed the same number two ways.

A check nobody imposed, and what it says about the inside #

The construction asks for one thing at each collocation point and never asks that the body be closed. A closed body emits no net volume, so the source strengths weighted by their panel lengths must sum to zero — and nothing in the matrix says so.

It comes out anyway, and that makes it a free assertion of the kind this collection prefers: computable in one line from the solution vector, checked against a value that was never supplied, and loud when it fails. The two bugs it catches are the two most likely ones — a sign error in an influence coefficient, and a set of panel normals that are not consistently outward, which is the commonest geometry mistake and is invisible in every picture.

It also says something about what the solve has actually done. Setting the normal velocity to zero on the surface does not empty the interior: it leaves a perfectly good fictitious flow inside the body, which is simply ignored. The alternative is to impose a condition on the potential in there instead — hold it constant, so that the interior is genuinely at rest — and that formulation is what most production codes use, because it needs the potential rather than its gradient and generalises to three dimensions more cheaply. Same physics, different row, different arithmetic.

What a panel method is for, and what it is not #

It is worth being clear about where this sits, because a method that works for any geometry can look like a method that answers any question.

It computes an inviscid flow. There is no boundary layer, no wake, no separation, and therefore no drag: this machinery reproduces d’Alembert’s zero as faithfully as the conformal map does, and for the same reason. What it is good for is the pressure distribution, and a pressure distribution is exactly what a boundary-layer calculation needs as its input — which is how these methods were actually used, coupled to an integral boundary-layer solver, for about thirty years of aircraft design.

It has no free surface and no wake to find. Everything above assumed the geometry is given. A panel method can be asked about a wake whose shape is part of the answer — a rolled-up sheet behind a wing, a cavity behind a body — and then the system stops being linear, because the influence coefficients depend on where the unknown surface went. That is a different and considerably harder subject, and none of it is here.

It computes a two-dimensional flow here. The three-dimensional version — a vortex lattice, or a panel method on a whole aircraft — is the same idea with a harder influence coefficient, and it is what the Trefftz-plane machinery on this site is a distant relative of.

The trailing-edge angle has to be finite, as above.

And nothing about it is adaptive. The panels are placed by a rule chosen in advance; a method that put panels where the answer needed them would be better and would be a different subject.

There is one more property worth naming because it is the practical reason these methods survived into the era of grid solvers. A panel method discretises the boundary and not the space. A finite-volume calculation of the same flow needs cells filling the whole domain out to a far-field boundary somebody chose, and most of them are computing that nothing much is happening. A panel method has unknowns only on the surface, the far field is exact rather than truncated, and the cost scales with the area of the body rather than with the volume of air around it. That is a real advantage and it is why a panel code is still the right instrument for a first sweep over a hundred candidate geometries — and why it stopped being the right instrument the moment anybody wanted the viscous answer, which is a volume question and cannot be put on a surface.

What the model does not contain #

No viscosity anywhere, so no stall, no separation and no profile drag.

No compressibility. Everything here is incompressible. There are corrections that stretch a subsonic result — and this site owns the one that fails — but a panel method as built here has no Mach number in it at all.

The self-influence is a limit rather than a value. A panel’s own midpoint sits on the panel, where the source velocity is a jump: the calculation takes the outward-facing limit, which is a half of the strength. That is exact for a flat panel and is the piece of the arithmetic most likely to be wrong by a sign, which is why the whole machinery is checked against a circle first — where it reproduces $C_p = 1 - 4\sin^2\theta$ to machine precision at thirty-two panels.

The nodes must land on the true surface. Interpolating them along the chords of a finely sampled outline puts them slightly inside it, and that geometric error is second order in the sampling and swamps the method’s own convergence long before it is visible in a picture. On a circle it costs four digits at sixty-four panels. The panel geometry here therefore takes a parameterised curve rather than a list of points, and the difference between those two interfaces is the difference between measuring a scheme and measuring a polygon.

Who found it, and when #

Hess and Smith published the method at Douglas Aircraft in 1966, and the paper’s title — Calculation of potential flow about arbitrary bodies — is the whole claim. Before it, an aerodynamicist wanting a pressure distribution had a small catalogue of shapes with exact solutions and a great deal of ingenuity; after it, the shape was an input.

The date is worth dwelling on. Potential-flow theory was seventy years old and thought to be finished. What changed was not the theory but the arithmetic: an influence coefficient is a few transcendental functions, a hundred panels is a hundred-by-hundred solve, and neither is possible by hand and both are trivial for a machine that arrived in the early sixties. The method was not waiting on an idea; it was waiting on a computer, and when the computer came the ideas that had been sitting in Prandtl’s era turned into an industry.

The generalisation of the trailing edge is older and belongs to von Kármán and Trefftz, from 1918. It was invented for the opposite reason — because Joukowski’s cusp was thought to be an unphysical shape, not because it was numerically awkward — and it turned out to be the geometry a method invented half a century later needed.

Where the ladder goes next #

A method that works for arbitrary geometry can be asked about geometry the exact theories cannot touch, and the most interesting such geometry is two aerofoils at once. A wing with a slotted flap is the highest-lift device in ordinary use and it comes with a stock explanation about high-energy air being blown into a boundary layer. A solver with no boundary layer in it at all is exactly the instrument for testing that.