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

Reciprocity — where it appears

An exact identity between two Stokes flows in the same region of fluid. Each one's stresses do the same work on the other's velocities as the other's do on its. It is used to obtain a force without solving for the flow that produces it, and it fails as soon as inertia is present.

Named by 2 essays across 2 fields — 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 sphere does not go at the speed of the flow it is in. How far a force-free sphere on the axis of a round tube lags the fluid at its own centre, against its size. The lag is two-thirds of the square of the size ratio and comes from the reciprocal theorem in one line: the sphere moves at the average of the ambient flow over its own surface, and a parabolic profile is slower everywhere on that surface than at the middle. At a fifth of the tube's radius the lag is 2.7 per cent, which is the difference between a tracer and a thing being measured.

A force without the flow that makes it

A sphere carried along by a flow does not travel at the speed of the fluid at its centre. It travels at the average of the flow over its own surface — and getting that result needs no solution of the flow around the sphere at all, only the answer to a completely different problem that everybody already knows.

viscous · Reciprocity

Named alongside it

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

BiplaneBoundary conditionCirculationConstraintCreeping flowDragInduced dragInterferenceLinearityMeasurementModel limitMunk's stagger theorem

All concepts