From a circle to a wing

The flow past a circular cylinder in an ideal fluid is completely known. It has been written down since the middle of the nineteenth century, it is short enough to fit on one line, and it describes an object nobody has ever wanted to fly.

The flow past a wing section is what everybody wants and nobody can write down from scratch. The gap between those two sentences is closed by a change of variable — and the remarkable part is not that the change of variable produces a wing-shaped outline, which is a matter of geometry, but that it hands over the solution along with the shape.

What a map has to preserve to be worth anything #

A map from one plane to another is just a rule assigning a point to a point, and almost all such rules are useless here. The one property that matters is that the map should be conformal: angles between curves must survive it. Two curves crossing at sixty degrees before the map must cross at sixty degrees after it.

The reason that is the property, and not any other, comes straight from what an ideal flow is. Such a flow is described by a function whose real and imaginary parts are the velocity potential and the streamfunction, and those two families of curves cross at right angles everywhere. A map that preserves angles takes that orthogonal net to another orthogonal net — which is to say, it takes a flow to a flow.

Any differentiable function of a complex variable is conformal wherever its derivative is not zero. That is the whole of the machinery, and it is why the subject reaches for complex analysis rather than for a fluid-specific trick.

It is worth being precise about what is being carried over, because “the solution maps across” can sound like a slogan. What maps across is the property of being a solution. Laplace’s equation says that a function has no local maxima or minima in the interior — every value is the average of its neighbours — and that property is preserved by any conformal map, because averaging over a small circle and averaging over the slightly stretched, slightly rotated ellipse the map sends it to give the same answer to leading order. A harmonic function composed with an analytic map is harmonic. That single sentence is the reason the construction works, and everything below is bookkeeping.

The particular map, and why this one #

The Joukowski transform is

$$\zeta = z + \frac{c^2}{z}$$

which is about as simple as a non-trivial map gets. Far from the origin it does almost nothing: the $c^2/z$ term dies away, so the uniform stream arriving from a long way off arrives unchanged in both planes. Close in, it folds the plane over itself in a way that turns circles into shapes with a point on them.

Everything then depends on which circle is fed to it. A circle centred exactly at the origin, of radius $c$, maps to a flat plate: the circle collapses onto the segment from $-2c$ to $2c$ of the real axis. Move the centre off the origin and the collapse is incomplete, and what comes out has thickness or camber or both.

What the solver computed, and how it was checked #

The circle used for the figures here has radius 1.1029 and its centre at $(-0.1, 0.08)$, and both of those numbers are consequences rather than choices. The centre is the two parameters — thickness and camber — and the radius is then forced, because the circle has to pass exactly through the critical point $z = c$. It does: the distance from $(-0.1, 0.08)$ to $(1, 0)$ is 1.1029 to four decimal places, which is the radius.

That requirement is the whole design. The point $z = c$ is where the map’s derivative vanishes, and a circle passing through it acquires a corner there. A circle that misses it produces a rounded, closed, perfectly respectable oval with no sharp edge anywhere — and therefore nothing for the Kutta condition to act on, no way to pick a circulation, and no lift.

The claim that the map is conformal everywhere except at that one point is checked by evaluating the derivative rather than by citing the theorem.

The trailing-edge probe is worth a note, because getting it wrong is easy and the wrong answer looks fine. The critical point is not at the top of the circle, or at its right-hand extremity, or at any other landmark: it is at the angle $-\beta$ measured from the centre, where $\beta$ is 4.1596° for this section. Probing at zero degrees instead returns $|d\zeta/dz| = 0.159$ — a small number, a plausible number, and a number belonging to a point where nothing whatever is wrong with the map.

The solution comes across for free #

With the geometry settled, the flow costs nothing. The velocity in the aerofoil plane is the velocity in the circle plane divided by the derivative of the map, which is one line of code and no new physics.

This is the part worth sitting with. The right-hand picture is a flow with a stagnation point on the underside, a fast run over the upper surface, and smooth departure from the sharp edge. It satisfies Laplace’s equation, conserves mass, and is tangent to a curved wing surface at every point of it. None of that was arranged. It is all inherited, because the left-hand picture had those properties and the map is angle-preserving.

The lift can then be extracted twice, by routes with nothing in common. Kutta–Joukowski gives $\rho U \Gamma$ from the circulation that the sharp edge selected: at six degrees, 2.4447. Walking the aerofoil’s surface, evaluating the pressure from Bernoulli and resolving perpendicular to the stream gives 2.4063. The two agree to 1.57%, and the residue is discretisation of a surface with a cusp on it rather than disagreement about the physics.

The circle’s centre is the design #

Because the two coordinates of the centre are the only freedom in the construction, the whole Joukowski family is two-dimensional, and each coordinate has a clean interpretation.

Moving the centre upstream, to negative $x$, thickens the section. The circle then encloses the critical point less symmetrically, and the fold that produces the trailing edge leaves more material behind it. An offset of 0.02 gives a section 2.6% of its chord thick, 0.08 gives 9.6%, and 0.16 gives 17.8% — very nearly proportional, which is a fact about the map rather than an assumption about aerofoils.

Moving the centre upward cambers it. That is the interesting coordinate, because camber is the thing that gives a wing lift at no incidence at all. The angle $\beta$ in the Kutta condition is exactly the angle of the centre as seen from the critical point, so the circulation is $-4\pi a U \sin(\alpha + \beta)$ and the lift vanishes not at zero incidence but at $\alpha = -\beta$. A camber offset of 0.06 puts $\beta$ at 3.122°, so that section makes $C_L = 0.374$ at zero incidence and nothing at all at −3.12°.

That last agreement is the kind worth more than a match to more decimal places would be. $\beta$ is an angle in the circle plane, defined before any flow exists; the zero-lift angle is a property of a lift curve computed by integrating pressures around a wing. They are the same number because the map carried one into the other.

The chord is worth a word too, since it is quoted throughout as 4.0334 and looks like an odd number to have chosen. Nothing chose it. A circle of radius $c$ centred at the origin maps to a segment four units long, so the family’s chord is near $4c$ for every member of it, and the small departures from exactly four — 4.002 for the thinnest section drawn, 4.078 for the thickest — are the thickness pushing the nose and tail apart. Every coefficient on this page is divided by the chord that came out of the map rather than by a nominal one, which matters more than it sounds: a 1.9% error in the chord is a 1.9% error in every lift coefficient computed from it, and it would be invisible in the picture.

Why this construction mattered so much historically #

Before the transform, aerofoil design was a matter of building shapes and testing them. After it, there was a two-parameter family of shapes whose flow was known in closed form, which meant the first serious question of aerodynamics — how does lift depend on the shape? — became a question that could be answered with algebra.

The answer it gave is the lift curve: straight, of slope near $2\pi$ per radian, displaced sideways by camber. That result outlived its own derivation. Modern sections are not Joukowski sections and have not been for a century, and every one of them is still described by a lift-curve slope near $2\pi$ and a zero-lift angle set by camber, because those properties turned out to belong to thin sections in general rather than to this particular family.

The transform also supplied the first honest account of where lift comes from. It does not appear from the shape: it appears from the circulation, and the circulation is whatever the sharp edge demands. A spinning cylinder with no wing shape at all lifts by the same rule, which is the cleanest demonstration that the shape was never the cause.

The same machinery run backwards #

Everything above goes one way: choose a circle, get a shape, get its flow. A designer wants the opposite. The useful question is not what does this section do but what section produces this behaviour — prescribe a pressure distribution, and find the geometry that delivers it.

Conformal mapping makes that tractable, and it is one of the few places where an inverse problem is easier than the direct one. The prescribed surface velocity lives on the aerofoil; pulled back to the circle it lives on a circle, where the mapping function’s magnitude and its argument are not independent — they are a conjugate pair, related by an integral over the circle. So specifying one of them determines the other, and the shape falls out of a transform rather than out of a search.

Theodorsen set that out in 1931 and Lighthill sharpened it in 1945, and it is how the sections that matter were actually designed. A laminar-flow section is not a shape somebody drew and then tested; it is the answer to keep the pressure falling to sixty per cent of chord. A supercritical section is the answer to hold a flat rooftop at a modest supersonic Mach number and recover gently.

And the target is not free. For the mapping to close into a single-valued profile, and for the flow far away to be the uniform stream that was asked for, the prescribed velocity distribution must satisfy a small number of integral constraints — three, in the standard formulation. A designer who sketches a pressure distribution by eye will violate them, and what comes back is not a slightly wrong aerofoil but a curve with a gap in it, or one whose free stream is at the wrong angle.

So the design freedom is genuine and it is not unrestricted: the space of achievable pressure distributions is a subspace, and a modern inverse-design code spends much of its effort projecting a designer’s wishes onto it.

What the picture cannot show #

The map is a statement about an inviscid, irrotational flow, and it inherits every one of that model’s failures wholesale. The mapped flow has no drag, because the circle’s flow has none and the map cannot manufacture any. It has no boundary layer, so the surface speeds near the trailing edge are higher than a real section ever achieves. And the cusp at the trailing edge is a mathematical idealisation: a real wing has a trailing edge of finite thickness, and the flow near it is decided by viscosity rather than by a condition imposed at a point.

There is also a limitation internal to the construction. The family is two-parameter, so it cannot produce a section with, say, a specified thickness distribution and a specified camber line. Every Joukowski section has its maximum thickness at a quarter of the chord and a cusped tail, whether that is wanted or not. The generalisations — Kármán–Trefftz and the rest — buy more freedom by complicating the map, and the modern answer is to give up closed forms entirely and solve the panel problem numerically, which is what the flap essay does.

Where the model stops #

Two boundaries are worth stating precisely.

The first is the trailing edge itself. At the critical point the derivative is zero, so the velocity in the aerofoil plane — which is the circle-plane velocity divided by that derivative — would be infinite unless the numerator vanishes there too. The Kutta condition is exactly the requirement that it does. Far from being an extra physical assumption bolted on, it is the condition that keeps the mapped solution finite, which is a much more satisfying way to arrive at it.

The second is that conformality is a local property. It preserves angles, and it does not preserve lengths, areas, or the relative spacing of streamlines. The net in the first figure is visibly stretched near the trailing edge and squashed far from it. Any quantity read off the mapped picture by measuring distances — a boundary-layer thickness, a curvature, a distance from the surface — has to be un-mapped before it means anything.

The one number the map does not fix #

There is a temptation, having seen the machinery work, to believe the construction settles everything about the section. It does not, and the exception is the one this whole site keeps returning to.

The map produces a family of shapes and, for each of them, a family of flows — one for every circulation. The circle plane admits any circulation at all: a vortex of any strength at the centre leaves the circle a streamline, so the mapped flow is a valid solution past the aerofoil for every one of them. What the map contributes is not the value of the circulation but the place where choosing it becomes unavoidable, namely the point where the derivative vanishes and the mapped velocity would otherwise be infinite.

So the sequence runs: the map hands over a one-parameter family of exact solutions, and finiteness at one point selects a member of it. Lift is the consequence of a regularity requirement, which is about as far from “the shape pushes air down” as an explanation can get, and it is the reason a shape with no circulation makes no lift however wing-like it looks.

Who found it, and when #

Nikolai Zhukovsky, whose name is transliterated a dozen ways and reaches English aerodynamics as Joukowski, published the transform’s aerodynamic use in 1910, having established the circulation theorem that bears his and Kutta’s names in 1906. Martin Kutta had arrived at the trailing-edge condition independently in 1902 while studying Lilienthal’s gliders.

The map itself is older and was not invented for fluids: it is a standard exercise in complex analysis, and its use in aerodynamics is a case of a subject discovering that its central problem had been solved in another department. That is worth noticing, because it is the reason the argument here is about angles and derivatives rather than about air.

Where the ladder goes next #

This rung is where the site’s ideal theory reaches its most useful form: a closed-form flow past a shape worth caring about. The rungs below it establish what an ideal flow is and why one function suffices. Above it, the theory gets used — to find where the lift acts, and to say what a flap does to a lift curve.

The honest sequel is the one that admits the closed form has run out. Real sections are not mapped circles, and the way modern thin-aerofoil theory works is to put a row of vortices on the camber line and solve a linear system — which is the same physics with the elegance traded for generality, and is exactly what the ground-effect and flap essays run on.