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

Ideal flow — the series

5 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Ideal flow past a cylinder. 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.

    The theory that solves everything

    Throw away viscosity and assume nothing is spinning, and fluid mechanics collapses into a linear problem with closed-form answers. The price is one term, and the term turns out to matter more than everything kept.

    part 1 · inviscid
  2. Flow net — a stream past a cylinder. Two families of curves drawn over the same flow: the streamlines, along which the streamfunction is constant, and the equipotentials, along which the velocity potential is constant. They cross at right angles at every point, because they are the two parts of a single analytic function of position.

    One function instead of two

    A velocity field carries two numbers at every point and must satisfy two constraints. Both constraints can be solved once and for all by writing the whole flow as a single scalar function — and the two families of curves that function generates cross at right angles everywhere, for reasons that have nothing to do with fluids.

    part 2 · inviscid
  3. 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.

    part 3 · inviscid
  4. Four solved flows, and the minimum is on the surface in every one. Sampling the whole exterior of each body on a grid and comparing the lowest pressure found there with the lowest found on the surface. The surface wins by a margin that is not marginal — between 0.18 and 0.56 in pressure coefficient — and it wins for a reason rather than by luck: the pressure of an irrotational flow is superharmonic, and a superharmonic function has its minimum on a boundary.

    The lowest pressure is on the body

    In an ideal flow the minimum pressure is always on a surface — not usually, not for the shapes people draw, always. The proof is an identity about the velocity gradient, and the identity says exactly which flows are exempt.

    part 4 · inviscid
  5. Three answers to one question: what happens after a body is jerked into motion. The force following a step change in a body's velocity, for three models. The ideal one is a spike at the instant and nothing afterwards. The viscous one falls as the inverse square root of time and never reaches zero. The compressible one holds while the signal is still crossing the body and then settles.

    The theory with no memory in it

    Laplace's equation has no time in it, so an ideal flow's response to a body being jerked into motion is instantaneous and complete. Its indicial kernel is a spike and nothing afterwards. Beside it sit the two kernels that are not, and the comparison says which ingredient every memory in this collection came in through.

    part 5 · inviscid

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