Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
Series

Lift — the series

8 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · circulation
  2. A Joukowski aerofoil at 8°. 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.

    The sharp edge decides

    Ideal flow round a wing admits infinitely many solutions, each with a different lift, and all of them exact. One extra requirement — that the air leaves the trailing edge instead of whipping round it — picks a single one.

    part 2 · circulation
  3. 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.

    The lift curve, and why it is a straight line

    Lift against angle of attack is a straight line, it does not pass through the origin, and its slope is very close to a number that has no business being there. All three facts fall out of the theory.

    part 3 · circulation
  4. Ideal flow past a cylinder with circulation -3.4. A uniform stream past a circular cylinder in a fluid with no viscosity. The solution is exact and closed-form: streamlines part at a stagnation point, run round the surface and close up perfectly behind, and the pressure recovers to exactly what it was in front.

    Lift with no wing at all

    A spinning cylinder has no camber, no aerofoil section and no trailing edge, and it lifts exactly as hard as its circulation says it should. Which settles what lift is caused by.

    part 4 · circulation
  5. The starting vortex, and the circulation it pays for. A wing that has just begun to lift, and the vortex it shed as it started. The circulation round the wing and the circulation round the shed vortex are equal and opposite, so a circuit large enough to contain both has no circulation at all — which is what Kelvin's theorem requires of a circuit that began at rest.

    The vortex a wing leaves behind

    A wing at rest has no circulation. A wing in flight has a great deal. Circulation round a circuit of fluid particles cannot change, so the difference had to come from somewhere — and it did, as an equal and opposite vortex dropped on the runway.

    part 5 · circulation
  6. Where the flow stops, at Γ = -9. A cylinder with circulation, with the points where the flow is at rest marked. As the circulation grows the two points slide round the surface towards each other, meet at the bottom, and then leave the body — after which there is nowhere on the surface where the air is at rest at all.

    How much circulation is too much

    Spin a cylinder faster and it lifts harder, with no limit in the equations. What does have a limit is the flow's willingness to stop anywhere on the surface — the two points where the air is at rest slide round towards each other, meet at the bottom, and leave the body altogether.

    part 6 · circulation
  7. The flap moves the curve and leaves its slope alone. Lift coefficient against incidence for several flap deflections. The lines are parallel: deflecting the flap gives the section lift at an incidence where it had none, and does not change how much lift each further degree of incidence buys.

    What a flap does, and what it does not

    Lowering a flap gives a wing lift at an incidence where it had none. It does not make the wing more responsive to being pitched — the lift curve moves sideways and its slope does not change, and those are two quite different things to buy.

    part 7 · circulation
  8. The six bodies, drawn at the same scale. Two circles, two ellipses and two Joukowski sections, each at the incidence that gives it a circulation of exactly two. There is no family resemblance and no common parameter; what they share is one number, and the theorem needs nothing else.

    One formula, and it does not ask what the shape is

    Kutta–Joukowski gives the lift of any two-dimensional body from one number. Six bodies with nothing else in common are put at that number here and come out with identical lifts — and with pitching moments, load distributions and suction peaks that are not even close.

    part 8 · circulation

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