Hydraulic Jump Conjugate Depths

When supercritical flow has to become subcritical it cannot do so gradually, because the two states sit on opposite branches of the energy curve. It does it in a violent, turbulent, air-entraining step called a hydraulic jump, and the depth on the far side follows from momentum, not energy. Jean-Baptiste Belanger published the result in 1828: $y_2/y_1 = \ frac12(\sqrt{1+8Fr_1^2}-1)$. A 0.4 metre supercritical flow at $Fr_1 = 3$ jumps to $0.4 \ imes 3.772 = 1.51$ m.

Momentum rather than energy is the whole point. Across a jump the water surface is broken, the turbulence is intense, and energy is being destroyed at a rate you cannot compute in advance, so an energy balance has nothing to work with. Momentum is still conserved because the only external forces are the hydrostatic pressures on each face, and that closes the problem. This is one of the few places in engineering where the choice between the two conservation laws is not a matter of convenience.

What the equation will not tell you is where the jump sits, and that is what actually decides whether a stilling basin works. A jump forms wherever the downstream tailwater happens to match the sequent depth. If the tailwater is higher, the jump drowns and moves upstream, sometimes right up under the gate. If it is lower, the jump sweeps out of the basin and lands on unprotected channel, which is how spillway aprons fail. Designers therefore plot sequent depth against tailwater for the full range of discharges, not just the design flood. The Bureau of Reclamation's basin types are classified by $Fr_1$: below 1.7 there is barely a jump, 2.5 to 4.5 is the rough transition range that is deliberately avoided because the jump oscillates and sends waves downstream, and 4.5 to 9 is the well-behaved steady jump every textbook picture shows.