Kinematic waves — the series · Fluid Flow & Aerodynamics

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.

part 1 · kinematics

The column in height and time: a falling interface, a rising shock, a fan. A batch settling test from a uniform φ₀ = 0.1, height above the bottom against time, both scaled on the column height and the single-particle settling time. The interface with clear water (thick) falls in a straight line at 0.4538; the sediment shock rises from the bottom at 0.1484 until the two meet at t = 1.661, height 0.2464; the thin lines are characteristics of the fan, each carrying one concentration between 0.317 and packing, and the interface bends as it crosses them. Dots are the finite-volume solve on 400 cells: the interface and the sediment front.

The column the chord rule cannot settle

A suspension settling in a closed column is a kinematic wave, and its flux curve bends both ways. At the top the chord rule works: clear water meets the suspension at a single falling front. At the bottom it does not, and the bed grows behind a shock that stops short of packing and a graded layer beneath it — so the interface, instead of arriving, slows for ever.

part 2 · kinematics

The line that decides how much a thickener can pass. The batch settling flux against concentration, the same curve the closed column was solved on, with the operating line of a continuous thickener drawn across it. The line runs from the underflow concentration on the axis to the flux the plant can take, and it is placed by one condition: it must touch the flux curve rather than cut it. Where it touches is the concentration at which the thickener is working hardest.

The tangent that sizes a plant

A closed cylinder of settling suspension is a kinematic wave problem with a flux curve. Open the cylinder — feed it at the top and draw thickened slurry from the floor — and the capacity of the whole plant is a straight line laid against that same curve, touching it rather than cutting it.

part 3 · kinematics

Three speeds, and the Froude number at which two of them cross. The kinematic wave speed and the two dynamic wave speeds, all divided by the speed of a shallow-water wave on still water, against the Froude number. The kinematic wave is three halves of the water speed and the downstream dynamic wave is the water speed plus one, so they cross at Froude two — and beyond that crossing the news of a change in flux arrives before the news of a change in depth, which uniform flow cannot survive.

When the flux outruns the pressure

A kinematic wave is derived by throwing the momentum equation away, and the derivation never says when that is allowed. It is allowed until the kinematic wave, which travels faster than the water, overtakes the downstream dynamic wave as well — at which point uniform flow ceases to exist and a concrete chute carries a train of surges instead of a sheet.

part 4 · kinematics

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.

part 5 · kinematics