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

Bell shape — the series

4 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Five quarters of the span, the same bending moment, 64/75 of the drag. The bell-loaded wing's induced drag and root bending moment against its span, both as fractions of an elliptic wing of unit span carrying the same lift. At equal span the bell is worse: its span efficiency is exactly three quarters. But its bending moment is exactly four fifths, and bending moment grows in proportion to span while drag falls as its square — so there is a span at which the bell has bought back the structure and is ahead on drag. It is at exactly five quarters, where the two curves are at 1 and 0.8533. Both numbers are rational and neither was put in by hand.

    The loading nobody used

    Elliptic loading is the least-drag answer to a question no aeroplane asks. Constrain the moment the lift makes about the wing root instead of the span, and a different curve comes out — five quarters of the span for sixty-four seventy-fifths of the drag, both exact — with an upwash over the outer wing and a yaw that turns the right way.

    part 1 · circulation
  2. A bending-moment wing has no best span until its tips are forbidden to push down. The least induced drag for the elliptic wing's lift and bending moment, against span, for a spar sized by its root moment and one sized by the moment along its whole span. Free, both fall without limit: at 1.7 times the span the root-sized optimum is 0.815 of the elliptic drag and the other 0.79, with the tips carrying downloads. With no section allowed to push down both stop where the free loading's tip load first reaches zero: at 1.225 and 0.8889 for the whole-span measure, and near 1.33 and 0.84375 for the root moment. Longer spans then leave their tips unloaded and gain nothing.

    A bending moment never chooses a span

    Hold a wing's lift and its bending moment and ask for the least induced drag, and the answer does not stop: the span grows for ever and the tips push down. Both celebrated answers — Prandtl's bell at √(3/2) of the span for 8/9 of the drag, and the root-moment wing at 4/3 for 27/32 — are the span at which the least-drag loading's tips stop lifting. The structure sets how much is saved; a condition nobody states sets where the saving ends.

    part 2 · circulation
  3. Counting the spar's weight stops the span, and the best wing then sheds a little load from its tips to grow five per cent. Total drag at a fixed payload, induced drag of the lift that carries the spar too, against span, both in units of the best elliptic wing's, for a wing whose area is held so that its spar thins as it lengthens. Elliptic, the best span carries a spar of a quarter of the lift; with the third harmonic free the best ratio is ε = -0.1356 (closed form (√33 − 9)/24 = -0.1356), the span 1.05 times and the drag 0.9575 times the ellipse's, with the spar still a quarter.

    A spar that must lift itself stops the span at a quarter of the weight

    Held only to a lift and a bending moment, the least-drag wing never stops growing. An aeroplane's wing stops because its spar has weight, and the weight has to be lifted. Count it, and the best span is the one at which the spar is exactly a quarter of the lift when the wing's area is fixed, and a third when its chord is — whatever the loading. The best loading then sheds some of its tip load, but not all the way to Prandtl's bell; with the chord held and nothing else charged, it is the bell exactly, and the more of the drag the wing's own area makes, the nearer it comes back to the ellipse.

    part 3 · circulation
  4. Inboard the spar is sized by its load, outboard by a stiffness floor its load cannot lower. The spar's weight per unit span along the semi-span, as a share of the payload's weight per unit of the reference span, at the best wing when the stiffness floor is 0.4: the weight the bending moment needs, and the floor. The floor governs the outer 53.1 per cent of the semi-span; the best loading has a third-harmonic ratio of -0.0993 and the spar is 32.1 per cent of the lift.

    A stiffness floor keeps the span and gives the tips their load back

    A spar sized by strength alone thins to nothing at the tips, and charged as lift it stops the best span where it weighs a quarter of everything the wing carries, with the tips a little unloaded. A real outer wing is sized by stiffness, and stiffness sets a floor under the spar's weight that no reduction of the tip load can lower. Put that floor in and the tips get their load back — the best loading slides from a sixth of the way to the bell towards the ellipse — the quarter does not survive, rising to two fifths as the floor spreads over the outer two thirds of the span, and the span itself barely moves. A weight per metre cannot stop a span.

    part 4 · circulation

All series