Concept

Joukowski — where it appears

The conformal map that turns a circle into an aerofoil shape with a sharp trailing edge. It carries the exact solution with it, which is why this collection can compute a wing's lift in closed form rather than by panelling it.

Named by 8 essays across 4 fields — each of them below, with the objects they name alongside it.

A Joukowski aerofoil at 6°. A cambered aerofoil in a uniform stream. The circulation is not chosen: it is whatever value makes the flow leave the sharp trailing edge smoothly, and that single condition fixes the lift.

What actually holds a wing up

Not the shape, and not the story about air meeting up again behind. A wing lifts because there is circulation round it, and the sharp trailing edge is what decides how much.

circulation · Lift

The circle plane and the aerofoil plane. A circle with a polar net around it, and the same net after the Joukowski map. Curves that crossed at right angles still cross at right angles everywhere except at the single point where the map's derivative vanishes, and that point is the sharp trailing edge.

From a circle to a wing

The flow past a circular cylinder is known exactly and is of no interest to anybody who wants to fly. A change of variable turns that circle into a wing section — and, because the change of variable preserves angles, it carries the whole solution across with it. Nothing is solved twice.

inviscid · Ideal flow

The stretch of surface that is boiling. The pressure coefficient along both surfaces of a section at 4 degrees, computed from the same potential-flow solution as the other ideal-flow aerofoil figures. The horizontal line is the vapour pressure at a cavitation number of 1: wherever the suction curve is above it, the liquid there has been pulled below its vapour pressure and is boiling at whatever temperature it happens to be. This section cavitates at any σ below 1.428.

When a body tears the water

A propeller blade moving fast enough pulls the pressure at its own surface below the vapour pressure of the liquid, and the water boils at whatever temperature it happens to be. Where that happens is decided by an inviscid calculation of the pressure along the blade.

applied · Cavitation

Three answers for the lift-curve slope, with three different signs. Thin-aerofoil theory has no thickness term at all and returns 2π for every section. The exact potential solution rises: 2π(1 + 0.766 t/c), measured off the Joukowski map. Real sections do the opposite, because the boundary layer thickens towards the trailing edge and decambers the section. Two of these curves are computed here; the third is what measurement says.

The half that carries nothing

Thin-aerofoil theory splits a section into a camber line that carries all the lift and a thickness distribution that carries none — at any incidence, exactly none. That is very nearly true, and what it discards decides the peak suction, the critical Mach number and where the boundary layer gives up.

circulation · Thickness

The trailing-edge speed against circulation, for a sharp edge and a round one. Sampled one grid point off the trailing edge. The sharp edge is singular at every circulation but one: the speed there is about U at the Kutta value and 11.3U half a Kutta circulation away, and it grows without bound as the sample approaches the edge. The round edge has no such point. That is the whole of the Kutta condition's justification, and it needs the corner.

The condition that can be bought

Ideal flow round a closed body has one solution for every circulation, and the Kutta condition picks one. Its justification is entirely the sharp edge: every other circulation puts an infinite velocity there. Take the corner away and nothing chooses — which is not a curiosity, it is what a circulation-control aerofoil is.

circulation · Circulation control

The singularity a thin aerofoil drives onto its own nose. The Joukowski map has a critical point that maps to a place inside the body, a distance 4 mu²/(1 + 2 mu) from the leading edge. Against thickness that distance is a clean square: a twelve per cent section is analytic only within eight thousandths of a chord of its own nose, and the thin-aerofoil limit is the limit in which the singularity arrives on the surface.

The part of the flow inside the body

A potential flow outside a body is an analytic function, and an analytic function does not stop at the boundary it was defined on. It continues inward until it meets a singularity — and every body in this collection has at least one inside it, in a place that decides how the flow behaves outside.

inviscid · Singularities

The parcel over the top arrives first, and never waits for the other. Two parcels released together far upstream, a whisker above and below the streamline that divides at the nose of a Joukowski section 11.8% thick at 4°. Both crawl past the nose; then the upper one is carried over the top faster and reaches the far line 1.13 time units ahead — 0.279 chord-transit times — against the 1.59 that the circulation divided by the speed squared predicts. Behind the section they travel on side by side at the stream's speed and the gap between them never closes.

The upper parcel leads by the circulation

Transit time looked like a quantity that could not be measured: a parcel released near the dividing streamline crawls past the nose for as long as one likes. The difference between two parcels' transit times, one either side of that streamline, does not crawl. It converges, and on a thin section it is the circulation divided by the square of the speed — the lift, measured in seconds. The upper parcel arrives first by that much, and behind the wing the two never close the gap.

misconceptions · Equal transit

Added linearly, the parts give the section's suction peak a quarter too weak; carried through Bernoulli, most of it comes back. Pressure coefficient, suction upward, on the upper and lower surfaces of the 12 per cent section with 4 per cent camber at 4°: exact; the parts' disturbance speeds summed and the pressure taken linearly; and the same sum carried through Bernoulli. At a tenth of the chord on the upper surface the exact −Cp is 1.39, the linear sum 1.06 and the Bernoulli sum 1.34.

Thickness carries no load of its own and still moves the load forward

Thin-aerofoil theory builds a section's pressures from three pieces computed apart — its thickness, its camber, its incidence — and adds them. Every piece can be computed exactly on a Joukowski section, and so can the section they add up to. Added, the parts give a suction peak a quarter too weak and a loading that sits too far aft, and they miss nine per cent of the lift: thickness, which lifts nothing by itself, multiplies what camber and incidence make by 1 + 0.77 t/c. Carry the summed velocities through Bernoulli instead of adding the pressures, and most of the error goes; the part that stays is the cross term between thickness and lift, and it is first order in the lift.

inviscid · Superposition

Named alongside it

The objects these essays reach for when they reach for this one.

Conformal map Kutta condition Model limit Potential flow Thin-aerofoil theory Camber Circulation Stagnation point Lift Lift curve slope Pressure coefficient Separation

All concepts