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

Critical depth — where it appears

The depth at which an open channel carries a given discharge with the least specific energy, and at which the Froude number is one. Above it the flow is subcritical and controlled from downstream; below it, supercritical and controlled from upstream.

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

The one number that really is one. Three quantities against the Froude number. The upper line is the speed of a surface wave travelling downstream and the lower one the speed of the same wave travelling upstream, both in units of the wave speed itself; the second changes sign at Fr = 1 and not near it. That sign change is not a comparison of two term sizes going through unity — it is the moment a signal stops being able to reach upstream at all, so the equations change from elliptic to hyperbolic and the flow stops knowing what is ahead of it. The specific energy, drawn beneath, has its minimum at the same place, and for the same reason.

The number that really is one

Almost every threshold in this subject sits somewhere other than where its dimensionless group is one. The Froude number does not. At Fr = 1 a disturbance stops being able to travel upstream, the specific energy is least and the equations change type — three statements, one number, and no tolerance anywhere in it.

regimes · Froude 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.

Froude numberCharacteristicsConservationDimensionlessFlow meteringHydraulic jumpHydrostaticHyperbolicMach numberMeasurementModel limitMomentum theorem

All concepts