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

Finite wing — the series

3 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Circulation across the span, for three planforms. How much circulation each part of the wing carries, plotted across the span. It has to reach zero at both tips, because a wing cannot carry circulation off its end, and the rate at which it falls is what determines the vorticity shed into the wake.

    The price of having ends

    An infinitely long wing in a fluid with no viscosity has no drag at all. Cut two ends into it and the drag appears — not from friction, which is still absent, but from the fact that circulation cannot be carried off the end of anything.

    part 1 · circulation
  2. Induced drag against aspect ratio, all at C_L = 0.6. Induced drag for wings of different aspect ratio, each flown at whatever incidence makes it carry the same lift coefficient. The drag falls as one over the aspect ratio, so the longest wing here pays a fraction of what the shortest does for exactly the same load.

    The span is the whole story

    Induced drag is the price of having ends, and the only thing that lowers it at a given lift is putting those ends further apart. Not more area, not a better section, not a cleverer planform — span, and everything else is a correction of a few per cent to it.

    part 2 · circulation
  3. Two theories, one composite, and the aspect ratio between them. The lift-curve slope against aspect ratio. Prandtl's lifting line is exact as the aspect ratio goes to infinity and Jones's slender-wing theory is exact as it goes to zero, and each is generous outside its own limit. Helmbold's formula reduces to both with no free constant, which is what a composite expansion is, and runs under them where they disagree.

    Where the line stops being a line

    Prandtl's lifting line replaces a wing with a single bound vortex and its trailing sheet, and the formula that comes out is the most quoted in low-speed aerodynamics. Solved numerically at aspect ratio one it returns its own closed form to sixteen decimals — and the answer is forty-one per cent too high.

    part 3 · circulation

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