Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
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Wedge flow — the series

3 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Four corners, and what the flow does in each. The local flow in corners of four different interior angles, drawn from the exact local solution ψ = r^(π/α) sin(πθ/α). The exponent of the speed is π/α − 1 and depends on nothing else: at a right angle the corner is stagnant, at a flat wall nothing happens, and at any angle greater than a straight line the speed has no bound at the corner. The last panel, at 360 degrees, is the flow round the edge of a plate.

    Nothing turns a sharp corner

    Near a corner the flow is fixed by the angle and by nothing else — not by the size of the corner, not by the flow far away, not by the fluid. The exponent is π/α − 1, and every sharp edge in aerodynamics is the one case where it comes out at minus one half.

    part 1 · inviscid
  2. Up a step the pressure rises to a stagnation at the foot and falls without limit at the lip, and the face feels both. The pressure coefficient along the wall of a right-angled forward-facing step of height h in ideal flow, against distance along the wall from the foot of the face in step heights: the floor ahead of it, the face, and the top beyond the lip (clipped at −3, where it goes to minus infinity). The face is pushed back by 0.6366 of a dynamic pressure times its height over its lower part and pulled forward by exactly as much near its lip.

    A step's face is pushed at its foot and pulled at its lip, equally

    Ideal flow running up a step has two corners in it, and the local theory of corners says what each does: the inside corner at the foot brings the flow to rest, the outside corner at the lip sends it round at infinite speed. Solved exactly, the step's face feels both — a push of 2/π of a dynamic pressure times its height spread over its lower four-fifths, and a pull concentrated at the lip — and they cancel exactly, at every ramp angle, because one reflection of the mapping exchanges the two corners. A step has no drag in ideal flow for the same reason a flat plate's lift has no drag: a finite force carried by an infinite speed.

    part 2 · inviscid
  3. Along a block's wall the pressure stagnates at each foot and plunges at each lip; between two lips close together it never recovers. The pressure coefficient along the wall unrolled, against the distance from the front foot in block heights: up the front face, along the top, down the back, for a block as long as it is tall and one six times as long; the lips' suction is infinite in ideal flow and is drawn to −6. On the cube-like block the pressure at the middle of its top is -1.89; on the long one, -0.365.

    A block's front face is pulled upstream by its own lip

    A forward-facing step's face is pushed at its foot and pulled at its lip by exactly equal amounts. Give the step a back — make it a block on the floor — and the balance goes. The two lips, a short top apart, each sit in the other's fast flow, and the suction at the front lip grows while the push at the foot hardly changes: a block as long as it is tall has its front face pulled upstream by three tenths of the dynamic pressure times its height, a block a quarter as long by two thirds. The back is pulled downstream by exactly as much, so the ideal block still has no drag; but every force that ideal flow assigns is now on the wrong face for separation to undo.

    part 3 · inviscid

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