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

Open-channel — where it appears

A flow with a free surface exposed to the atmosphere, so that the depth is part of the solution rather than a boundary given in advance. Its behaviour is set by the Froude number: below one a disturbance can travel upstream, above one it cannot.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

The chord and the tangent, which are the two speeds. The flux of a conserved quantity against its own density, for a wide river and for traffic. At any point the slope of the chord from the origin is the speed the material moves at, and the slope of the tangent is the speed a disturbance moves at. They are the same number only if the curve is a straight line through the origin. For the river the tangent is five-thirds of the chord at every depth; for traffic the tangent turns negative above half the jam density while the chord never does.

A wave nothing in it travels with

A flood crest moves at five-thirds the speed of the water it is made of, at every depth, whatever the roughness and whatever the slope. A traffic wave moves backwards through cars that are all going forwards. Neither result contains a momentum equation.

kinematics · Kinematic waves The current that runs across the river. The transverse velocity through the depth in a bend, beside the downstream velocity that causes it. The surface water is thrown outward and the bed water is driven inward, the two cross at about half the depth, and the whole profile carries no net discharge — which it cannot, because the cross-section has to pass what it is given. What it does carry is the bed load, inward.

The current that runs across the river

The standard channel calculation is one-dimensional, and a river is not. Put a bend in it and the surface tilts by a few centimetres, which is invisible — and a helix forms, a few per cent of the main flow, which decides where the river will be in a hundred years.

applied · Open-channel The depth at the edge, against how much pressure is left there. The brink depth as a fraction of the critical depth, against the share of the hydrostatic pressure the brink is taken to retain. A free jet — no pressure at all — gives exactly two thirds, to 1e-16. The measured ratio of 0.715 is met at 40 per cent of hydrostatic, which is more residual pressure than a falling nappe looks as though it should have.

The depth at the edge is not the critical one

Every open-channel calculation assumes the pressure is hydrostatic, and usually says so nowhere. At a free overfall it is not, the assumption's failure can be measured, and the measurement is one number — a brink depth that is 0.715 of the critical depth where a hydrostatic reading would make it one.

applied · Open-channel

Named alongside it

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

ConservationModel limitMeasurementCharacteristicsConstitutive lawCritical depthCurvatureDimensionlessDiscontinuityFlow meteringFree surfaceFroude number

All concepts