Ask for the pressure, and see what shape that is

Every aerodynamic calculation on this site so far has run the same way round. A shape is given, the condition that nothing flows through it is imposed on its surface, and the pressure comes out. That is the direct problem, it is what every solver here does, and it is not the order anybody designs in.

A designer knows what the pressure distribution has to do. It has to reach a certain suction to carry the lift; it has to hold that suction far enough back to carry it on a section thin enough to build; and then it has to recover to the trailing-edge pressure gently enough that the boundary layer survives the climb. None of those requirements is about coordinates. They are all about one curve, and the shape is whatever produces it.

So the useful question is the inverse one. Prescribe the boundary condition as a pressure, and solve for the boundary.

Nothing about the physics changes #

It is worth being clear about how small the change is. Laplace’s equation is the same equation. The free stream is the same free stream. The Kutta condition still picks the circulation, and the flow is still irrotational, incompressible and inviscid. What changes is which half of the pair — shape, pressure — is handed in and which is computed.

The consequences are out of all proportion to that. The direct problem always has an answer: any closed curve has a flow round it. The inverse problem usually has none, and finding out whether a particular ask has one is most of the work.

The circle already knows the answer #

The route through is the same one that turns a circle into a wing. In the circle plane the flow is known in closed form for ever: a unit circle in a stream at incidence $\alpha$, with the circulation the Kutta condition fixes, has surface speed

$$q_c(\theta) = 2U\left|\sin(\theta-\alpha) + \sin\alpha\right|.$$

A conformal map $z(\zeta)$ that is analytic outside the circle carries that flow to the flow round whatever shape it produces, and the two surface speeds are related by nothing more than the map’s own stretching:

$$q(\theta) = \frac{q_c(\theta)}{\left|dz/d\zeta\right|}.$$

Which means that prescribing the speed is prescribing the stretching. And the logarithm of an analytic function is analytic, so on the boundary its real and imaginary parts are a conjugate pair: give the real part and the imaginary part is not free, it is the conjugate series, computable in one transform. That gives $dz/d\zeta$ all the way round the circle, and the shape is its integral.

Three integrals, and what each of them means #

A prescribed speed distribution is not a shape. Three conditions have to hold before the integral above closes into a body at all, and each has a physical name rather than a mathematical one.

The first is that the mean of $\ln(q_c/q)$ round the circle must vanish. A non-zero mean is a scale factor on the map at infinity, which means the body it produces is sitting in a free stream of some speed other than the one it was designed against. The ask has silently changed the flight condition.

The second and third are that the circuit must close: $\oint (dz/d\theta)\, d\theta = 0$, which is two real conditions because $z$ is complex. Violated, the trailing edge does not come back to where it set off from, and what has been designed is not a body but a spiral.

That failure is not a small numerical residue to be tidied away. For the rooftop asked for above it is a tenth of a chord.

Most treatments state these as linearised conditions — that the first two Fourier coefficients of $\ln q$ must vanish — which is true to first order and hides what is being asked. The version computed here is the exact one: the closure gap is evaluated as the complex number it is, in chords, so a target that does not describe a body produces a picture of the body it does not describe.

The nearest ask that exists #

The correction has exactly as many degrees of freedom as there are conditions. A scale on the whole speed distribution moves the first; the two first harmonics of $\ln q$ move the other two. Solving for all three is one Newton step on a nearly linear system, and the residuals come out at $10^{-16}$.

What is interesting is not that it converges. It is how much it takes back.

The ask was for a rooftop with a mean pressure coefficient of $-0.632$ over the forward part of the upper surface, against the baseline’s $-0.249$. What survives closure is $-0.362$. The three integrals took back seventy per cent of the request, and they did it mostly through the scale term: the corrected target runs at 96.7 per cent of the speed that was asked for, everywhere.

That is the part of inverse design that no amount of computing power removes. The method does not fail on a target it cannot draw; it draws the nearest target it can, and the distance between the two is a property of the ask rather than of the solver.

The shape that comes back #

Two things about this figure are worth more than the shape itself.

The first is that nothing about the geometry was specified and everything about it changed. The ask was 28 per cent more speed over part of one surface. What came back is a thicker section with its maximum thickness moved forward and a different camber line. Thickness and camber are not independent knobs that a pressure distribution is built out of; they are what a pressure distribution looks like when it is drawn.

The second is the axis that is missing. The speed was prescribed against the circle’s parameter, and where any given station ends up along the chord is an output of the solve. A designer may ask for a pressure distribution and may not ask for it at a stated place. Moving the recovery point aft moves the shape that produced it, which moves where the recovery point is. The two are settled together or not at all, and that is not a limitation of this method — it is what having the boundary condition on the boundary means.

The check that could have failed #

Everything so far is one method agreeing with itself. The test worth running is to hand the designed shape to a solver that shares none of its arithmetic.

The residual disagreement is the forward method’s, not the inverse one’s, and that is established rather than asserted: at a hundred panels the worst gap is 0.070, at two hundred 0.038, at four hundred 0.020. Halving with each doubling is first-order convergence, which is exactly what a cusped trailing edge forces on any panel scheme — the two panels meeting at the tail are collinear there, so the Kutta row is nearly a combination of the tangency rows beside it, and the scheme is measuring its own conditioning.

What the trailing edge decides, and it is one number #

The sharp edge is a singularity of the map: $dz/d\zeta$ vanishes there, so $\ln|dz/d\zeta|$ is logarithmically infinite and no series in $\cos n\theta$ will represent it. The way through is to divide out the singular factor, which is known — an edge of interior angle $\tau$ needs $(1 - 1/\zeta)^{\kappa}$ with $\kappa = 1 - \tau/\pi$, and that function is analytic outside the circle, so it can be taken out before the transform and put back after.

Which $\kappa$ is not a setting. It is what the target asks for. A speed that is finite and non-zero at the trailing edge is asking for a sharp one; a speed that vanishes there is asking for a rounded body with a rear stagnation point. The sharpness of the trailing edge is chosen by one value on the pressure distribution, which is not how anybody would expect to be asked.

That value is the one place in the whole computation where the arithmetic is zero divided by zero. The circle’s speed vanishes at the trailing edge because the Kutta condition put a stagnation point there, and the map’s stretching vanishes there because the edge is sharp; the ratio is the finite speed a cusped edge has, and the sample computes it as zero over $10^{-12}$, which is zero. Left alone, that single value tells the inverse that the body is round. Every picture still looks like an aerofoil.

Asking for less, which is how it is actually done #

Seventy per cent of the request went back because the request covered the whole surface. That is worth dwelling on, because it makes the method look far more restrictive than it is, and the repair is a change in what is asked rather than in how it is solved.

The three conditions are three numbers. The target is a function — an infinite-dimensional object — and three constraints on an infinite-dimensional space leave an enormous amount of freedom. The trouble above is that the freedom was spent in the only place the solve could find it: uniformly, across every station at once, as a 3.3 per cent scale on the whole distribution. A scale on the speed everywhere is the least useful place a designer could possibly pay, because it is the one change that alters the lift coefficient and therefore the flight condition the section exists for.

So prescribe the pressure where the design intent lives, and leave the rest of the contour free. A designer cares about the rooftop and about the recovery behind it, which between them are perhaps two-thirds of the upper surface. Nothing on the lower surface aft of mid-chord is a design requirement in the same sense; it is structure, and it wants to be whatever it has to be. Hold the geometry fixed there, prescribe the speed forward, and the problem is no longer an inverse problem — it is a mixed one, with a pressure boundary condition on part of the boundary and a shape boundary condition on the rest.

That is the counting argument again, and it comes out cleanly. The closure conditions are solvability conditions on the data; the free segment supplies exactly the degrees of freedom they consume; and the residue that was distributed over the whole speed distribution is instead absorbed as a shape change in a region nobody specified. The design gets what it asked for where it asked for it, and pays somewhere it does not care about.

What is not free is whether the payment is affordable. The gap has to go somewhere, and pushing it into a short segment concentrates it: a closure error of a tenth of a chord spread over the whole outline is the mild global change drawn above, and the same error absorbed by the aft quarter of one surface is a shape change four times as violent there. Nothing in the method prevents it from coming back as a contour that crosses itself, or as a lower surface that passes through the upper one. The inverse problem’s refusals do not disappear when the ask is narrowed; they move, from this is not a body to this is a body nobody can build, and the second failure is harder to see because the integrals all close.

This is also the honest answer to why modern design software parameterises the geometry and optimises against a pressure target, rather than inverting the pressure directly. It looks like the crude option and it is the one that cannot be handed an impossible ask: every candidate the optimiser tries is a shape, so it closes by construction, and the target’s unreachability shows up as a residual in the objective rather than as a spiral. The price is search, which is thousands of direct solves against one inversion — Lighthill’s method answers in a single transform what an optimiser answers in a fortnight.

Both are computing the same distance. The inverse method reports it as the ask was 70 per cent too much, immediately and exactly; the optimiser reports it as a converged objective that stopped improving, and leaves the designer to work out whether the target was unreachable or the parameterisation was too poor to reach it. The advantage of the exact method is not speed. It is that a refusal comes back as a number rather than as a disappointment.

Where this stops describing anything #

Three limits, and the third is the one that matters most in practice.

It is ideal flow. There is no boundary layer anywhere in this calculation, so the recovery that the design exists to control is not being tested by anything. The whole purpose of asking for a particular pressure distribution is to keep a real layer attached, and this method cannot tell whether it does.

It is two-dimensional, and it is one operating point. A section designed for a rooftop at four degrees has some other pressure distribution at eight, and nothing here constrains it. Real design targets are a family of curves at several incidences, which turns the three conditions into an over-determined system and the Newton step into an optimisation.

And the parameterisation is the physics. The three constraints are three integrals of the target in the circle’s parameter, which is to say that whether a design exists depends on a variable the designer has no intuition about. The scale term is the worst of these: closure moved the entire speed distribution down by 3.3 per cent, which is a change in the lift coefficient, which is a change in the flight condition the section was being designed for.

Who found it, and what they did with it #

The method is Lighthill’s, published in 1945 while he was at Manchester, and it was not a mathematical exercise: the Royal Aircraft Establishment wanted laminar-flow sections, which are sections designed against a pressure distribution rather than against a shape, and the direct problem — a shape in, a pressure out — cannot produce one except by search. The three constraints appear in that paper as conditions on the Fourier coefficients of $\log$ of the velocity, in the linearised form, and they have been rediscovered in every generation of the subject since — in transonic design in the 1970s, where the same conditions appear as constraints on a hodograph, and in every modern optimiser, where they appear as the reason the geometry has to be parameterised rather than left free.

The surprising connection is with a different part of this site entirely. The closure conditions are solvability conditions for a boundary-value problem: the target is data, the shape is the answer, and not every set of data admits one. The same statement appears at a wall, where a body told to absorb mass produces a Neumann problem whose right-hand side is not orthogonal to the operator’s null vector, and no iteration can fix it. Two very different-looking refusals, and the same arithmetic underneath: an equation is a machine for turning some data into answers, and part of specifying it is saying which data.

Where the ladder goes next #

The obvious next rung is the loop rather than the pass: a target, a shape, a boundary-layer solve on that shape, and a modified target. That is the actual industrial method and it is a different subject, because its answer depends on a transition model this site does not have and would have to borrow.

The nearer one is the same inversion in three dimensions, where the closure conditions become conditions on a surface rather than on a curve and the whole thing stops being solvable in one transform. The reason is worth stating in advance: in two dimensions a boundary is a curve and a curve has one parameter, so prescribing a function on it is prescribing a function of one variable. The map is then an analytic function of one complex variable, and everything above follows from that single fact.