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

Free-streamline — the series

3 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. The Kirchhoff flow past a flat plate. A uniform stream meeting a flat plate held across it, with two streamlines leaving the edges and never returning. Between them is a wake of fluid at rest at a constant pressure. The equations solved are the same equations that give d'Alembert's paradox for a closed body, and this flow has a drag coefficient of 0.8798.

    Drag in the theory that forbids it

    d'Alembert's paradox is a theorem about flows that close behind the body. Stop requiring that, let two streamlines leave the edges and never come back, and the same equations — no viscosity, no vorticity — produce a drag coefficient of 0.8798.

    part 1 · inviscid
  2. The hodograph plane, where the unknown boundary is the known one. The same flow drawn in the plane of its own velocity, ζ = (u − iv)/U. The plate, whose shape is known in the physical plane, becomes a segment of the imaginary axis; the axis of symmetry becomes a segment of the real one; and the free streamline — whose shape nobody knows — becomes an arc of the unit circle, because the speed on it is exactly the free stream. The unknown and the known have changed places, which is why the problem can be solved at all.

    Where the unknown boundary is the known one

    A free surface is the hardest kind of boundary — its shape is part of the answer, so the region the problem is posed in is not known until the problem is solved. Draw the same flow in the plane of its own velocity and the shape becomes an arc of a circle, known in advance and exactly.

    part 2 · inviscid
  3. Stalled from its leading edge, a flat plate keeps a quarter of its attached lift slope. The normal force on a flat plate over the dynamic pressure on its chord against incidence: Kirchhoff's stalled flow, by quadrature through its hodograph, with Rayleigh's closed form 2π sin α/(4 + π sin α); and the attached flat plate's 2π sin α. At 5° the stalled plate carries 0.128 against the attached 0.548; at 20°, 0.423; at 90°, 0.8798, Kirchhoff's drag.

    A stalled plate keeps a quarter of its lift slope and moves its centre aft

    Kirchhoff's free streamlines gave a plate held across a stream a drag with no viscosity in it. Tilt the plate, let the flow leave both its edges, and the same construction is a wing that has stalled from its leading edge. Rayleigh solved it in 1876, and the hodograph confirms him to seven figures: near zero incidence the stalled plate lifts at a quarter of the attached plate's rate, its centre of pressure sits at five sixteenths of the chord instead of a quarter, and its force is normal to it, so its lift-to-drag ratio is the cotangent of its incidence. A real stalled wing lifts more, by a factor the wake's pressure decides.

    part 3 · inviscid

All series