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

Open-channel — the series

5 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Two depths, and a gap the model will not describe. The surface either side of a hydraulic jump at an arriving Froude number of 5.05. Both depths are exact consequences of the momentum balance. The distance between them is not: the shallow-water model has no length scale in it and cannot say how far the transition takes, so the region between the two levels is left blank and the six-depth rule of thumb beside it is somebody's measurement rather than this site's result.

    The shock in a river

    Shallow water is a gas whose ratio of specific heats is two. The white water below a weir is a shock wave, momentum is conserved across it exactly, energy is not, and one direction is forbidden for the same reason an expansion shock is forbidden — which makes the analogy exact to first order and wrong at the second.

    part 1 · applied
  2. Two depths for the same energy, and one for the least. Specific energy against depth for a discharge of 0.5 square metres per second per metre of width. Every energy above the minimum is carried by two different depths — one fast and shallow, one slow and deep — and the minimum is carried by exactly one. That depth is the critical depth, the Froude number there is one, and the least energy is three halves of it; all three are found here by search and checked against their closed forms.

    The depth that costs least

    For a given flow there are two depths that carry it at any energy above a floor, and exactly one at the floor. That one depth is where the Froude number is one, the least energy is exactly three halves of it, and a bump in the bed that asks for more than the flow has does not thin the water — it backs it up.

    part 2 · applied
  3. The backwater curve behind a weir. The depth at the control is 1.6 times the normal depth; integrating upstream the profile relaxes onto normal depth over about seven hundred metres and stays there. That relaxation is the reach forgetting the weir, and nothing about the weir survives past it.

    The section that decides the river

    A reach of open channel has a normal depth and a critical depth, and the water has neither. What it has is a profile obeying a first-order equation, which needs exactly one condition — and whether that condition belongs at the upstream end or the downstream end is not a choice, because the equation is stable in one direction and unstable in the other.

    part 3 · applied
  4. 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.

    part 4 · applied
  5. 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.

    part 5 · applied

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