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

Taper — the series

2 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. The rectangular wing stalls at the root; the tapered one stalls at the tip. Section lift coefficient across the half span for three planforms, each drawn at the wing incidence where its own worst section first reaches 1.5. A rectangular wing's peak is at the root, which is where a designer wants it: the stall starts inboard, ahead of the ailerons, and the pilot feels it. A tapered wing's peak has moved out to 0.62 of the semi-span — over the ailerons — because taper takes chord away from the tip faster than it takes circulation. The elliptic wing is the degenerate case: every section reaches the limit at once, which is elegant and is the worst possible stall behaviour.

    Which part of a wing stalls first

    A wing has one lift coefficient and its sections have a hundred, and no section is at the wing's. Which of them runs out first is decided by the planform, it decides whether the pilot keeps the ailerons, and the standard fix costs span efficiency in exact proportion to how much of it is applied.

    part 1 · circulation
  2. How flat the optimum is. The same curve near its maximum, with the bands within a tenth and a half of one per cent of the best shaded. Every taper ratio from 0.31 to 0.42 is within a tenth of a per cent of the optimum, and the whole band from 0.25 to 0.51 is within half a per cent. The optimum is exact, and choosing it rather than its neighbour buys nothing a wing can measure.

    The optimum that does not matter

    Elliptic loading gives the least induced drag, exactly, and the proof is one line: the penalty is a sum of squares. Which is also why the optimum is flat enough that a quarter of the design space sits inside a tenth of a per cent of it.

    part 2 · circulation

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