Hydraulic Jump Sequent Depth Ratio
Hydraulic Jump Sequent Depth Ratio
A hydraulic jump is the abrupt, turbulent transition open-channel flow makes from fast, shallow supercritical flow to slow, deep subcritical flow — a common sight below spillways, sluice gates, and steep chute transitions — and it dissipates a large amount of kinetic energy in a short distance, which is exactly why engineers deliberately design stilling basins to force a jump to occur rather than let high-velocity flow continue downstream and erode the channel. The ratio of the depth after the jump to the depth before it depends only on the upstream Froude number, a dimensionless measure of how supercritical the incoming flow is, through a classical momentum-balance result that lets engineers predict the downstream water depth and jump strength directly from upstream flow conditions.
The sequent depth ratio is y2/y1 = 0.5·(sqrt(1+8·Fr1^2) - 1), where Fr1 is the upstream Froude number. where v1 and y1 are the upstream flow velocity and depth, g is gravitational acceleration, Fr1 is the upstream Froude number, and y2 is the downstream (sequent) depth.
Inputs
v_1 := 3 m/s=3 m/s
y_1 := 0.4 m=400 mm
g_h := 9.81 m/s^2=9.81 m/s^2
The Froude number compares the flow velocity to the speed of a small surface wave in that depth of water, and a value above 1 identifies the upstream flow as supercritical — the condition needed for a hydraulic jump to occur at all.
Fr_1 := v_1/sqrt(g_h·y_1)=1.51
This classical momentum-balance result gives the ratio of depths across the jump directly from the upstream Froude number, without needing to solve the momentum equation from scratch each time.
ratio_y := 0.5·(sqrt(1+8·Fr_1^2)-1)=1.7
Multiplying the ratio by the known upstream depth gives the actual downstream water depth the jump produces.
y_2 := ratio_y·y_1=679.74 mm
Results
An upstream Froude number around 1.5 with a downstream depth near 0.66 m describes a weak-to-moderate hydraulic jump, still well within the range where the jump forms cleanly and dissipates energy effectively without excessive spray or instability. If Fr1 were much higher — say above 4 or 5 — the jump would become a strong, highly turbulent "choppy" jump that dissipates far more energy but also demands a much longer, more robustly armored stilling basin to contain safely. This depth ratio is exactly the design check used to size a stilling basin's length and wall height downstream of a spillway or gate.
Assumptions & limitations
- Horizontal, rectangular channel of constant width across the jump.
- Hydrostatic pressure distribution assumed at both sections (standard momentum-equation assumption).
- Negligible boundary friction losses across the short jump length.
- Steady flow with no air entrainment effects included in the depth prediction.