Concept

Shallow-water — where it appears

The equations for a layer of liquid whose depth is small against the lengths of its waves, so that its pressure is hydrostatic and its velocity uniform with depth. They carry gravity waves, bores and hydraulic jumps, and with bed friction they describe flow down rivers and chutes.

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

Surges merge down the chute until about four hundred time units, and then the train stops merging. The number of surges on a periodic chute 200 units long against time, after uniform flow at Froude number 3 is seeded with depth noise, at two resolutions, eddy viscosity 0.05. At the finer the count is 71, 55, 45, 45, 45 at times 100, 200, 400, 700 and 1000 — a spacing of 4.44; at the coarser, 63, 45, 32, 30, 30, a spacing of 6.67.

Roll waves stop merging at a spacing their fronts choose

On a steep chute the uniform flow breaks into a train of surges, and the train coarsens: a taller surge runs faster, catches the one ahead and swallows it, so the spacing grows down the chute. If the merging never stopped, the spacing in a photograph would only record how far the water had come. Marched down a long channel from noise, it stops. After a few hundred units of travel the surges stop catching each other and the train keeps its spacing for as long as it is followed. The spacing it keeps is not set by the chute or the noise but by the dissipation inside the surge fronts — which is the one part of the flow the shallow-water equations describe only roughly.

kinematics · Kinematic waves

Named alongside it

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

Froude number Characteristics Critical depth Dimensionless Hydraulic jump Hyperbolic Instability Kinematic wave Mach number Model limit Roll wave Specific energy

All concepts