2D Hydraulic Jump Flume: Sluice Gate, Froude Number, Conjugate Depths and Energy Loss Lab

Presets

Live readouts

Graphs

Free surface, energy line and critical depth along the flume

Specific energy against depth: the two depths and the loss

Momentum function: the conjugate depths share one value

Jump toe position and Froude number against time

What is happening now

Starting.

Numerical checks

running...

How to use

Start with Classic stationary jump and watch the yellow marker. Then (1) raise the tailwater slider: the jump walks upstream and finally drowns against the gate; (2) lower it: the jump is swept out; (3) close the gate (smaller opening) to raise the Froude number and watch graph 2: the energy lost grows; (4) tilt the bed or roughen it and see the jump move. Hover over the flume for local depth, speed and Froude number. Space pauses, R resets.

The science

A hydraulic jump is the abrupt change from fast, shallow (supercritical) flow to slow, deep (subcritical) flow, seen below sluice gates, spillways and in a kitchen sink. The lab solves the one-dimensional Saint-Venant (shallow-water) equations in conservative form with a finite-volume Rusanov flux, so the jump appears as a captured shock without any special treatment:

∂h/∂t + ∂q/∂x = 0 ∂q/∂t + ∂(q²/h + g h²/2)/∂x = g h S₀ − g n² q|q| / h^7/3

h is depth, q = u h the discharge per metre of width, S₀ the bed slope and n the Manning roughness. Under the sluice gate the water contracts to y₁ = C_c a with C_c = 0.61 and the discharge follows from energy, q = y₁√(2g(H₀ − y₁)). The Froude number Fr = u/√(g h) separates the regimes: Fr > 1 supercritical, Fr < 1 subcritical.

Why the jump has to exist. At a given discharge the specific energy E = y + q²/(2 g y²) has a minimum at the critical depth y_c = (q²/g)^1/3; every E above it has two depths, one fast and one slow (graph 2). A jump is the only way back from the fast branch to the slow one if the tailwater forces a deep subcritical flow, but it does not conserve energy; it conserves momentum M = q²/(g y) + y²/2 (graph 3). Equating M at both sides gives the Belanger relation and the energy loss:

y₂ = (y₁/2)(√(1 + 8 Fr₁²) − 1) ΔE = (y₂ − y₁)³ / (4 y₁ y₂) L_j ≈ 6 y₂

Where the jump stands is set by the tailwater: when the downstream depth equals the conjugate depth of the local supercritical flow, the jump forms there. Too much tailwater drowns the jump against the gate; too little sweeps it away. Engineers use exactly this to design stilling basins: they want the jump to stay on a concrete apron and burn off the energy of the spillway flow (the dissipated power per metre is ρ g q ΔE).

Honest simplifications. The model is depth-averaged, hydrostatic and one-dimensional: it has no surface roller, air entrainment, turbulence or undular waves, so in a real jump the toe is a turbulent, aerated roller and weak jumps (Fr1 below about 1.7) are undular. The jump length 6 y2 is an empirical rule, and Manning friction is a simple closure. The numerical shock is smeared over a few cells (0.03 m each), which does not change the conserved jump relations; the in-page checks confirm that. Scale: a laboratory flume, not a dam spillway.