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

Biplane — where it appears

Two lifting surfaces one above the other, whose induced drag is less than one wing of the same total lift and more than one of the same total span. The saving depends on the gap between them and not at all on the stagger.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Four aeroplanes, one number. Four biplanes with the same gap and the same loadings, staggered by nothing, by four tenths, by nine tenths and by one and six tenths of a chord. Their induced drags agree to the last bit of the arithmetic — the calculation cannot even express the stagger, because the Trefftz plane is a cross-section and everything drawn here projects onto the same one. That is Munk's stagger theorem, and stating it as the drag is unchanged understates it: there is no place in the computation where the stagger could be entered.

Two wings and it does not matter where

Move one wing of a biplane a chord forward and the induced drag does not change. Not approximately, not to a good approximation — the calculation that gives the induced drag has nowhere to put the stagger, because everything projects onto the same cross-section of the wake.

circulation · Biplane A closed wake, and the loading of least drag on it. The wake of a box wing in the plane that decides its drag, with the circulation of least induced drag drawn as a thickness along it. The horizontal members carry a loading close to elliptic and the vertical ones carry a share that lifts nothing — they contribute no lift, since lift is Γ dy and dy is zero on a vertical, and they change the drag by changing where the wake's vorticity is. At a gap of 20 per cent of span this system costs 67.1 per cent of what a single wing of the same span and lift would.

A wake that closes on itself

Prandtl's best wing system is a rectangle, not a wing. Solving for the loading on a closed wake turns up a circulation that costs nothing and does nothing — a gauge freedom in the middle of an optimisation — and a drag that keeps falling with no floor under it.

circulation · Box wing

Named alongside it

The objects these essays reach for when they reach for this one.

CirculationInduced dragSpan loadingThe Trefftz planeBoundary conditionConstraintDownwashGauge freedomInterferenceModel limitMunk staggerMunk's stagger theorem

All concepts