The hydraulic jump: sequent depths and energy loss
How supercritical flow suddenly turns subcritical, how to find the depth after the jump, and why jumps are used to protect spillways.
When fast, shallow (supercritical) water meets deeper, slower water, it cannot change depth smoothly. It rises abruptly in a turbulent roller called a hydraulic jump, and a lot of energy is lost as heat and noise.
Sequent depths
The depth before the jump, y₁, and after it, y₂, are called sequent (conjugate) depths. Applying the momentum equation to a rectangular channel gives the Bélanger equation:
y₂y₁ = 12 ( √(1 + 8 Fr₁²) − 1 )
Energy loss
ΔE = (y₂ − y₁)³4 y₁ y₂
Worked example
Water leaves a sluice gate at y₁ = 0.4 m with V₁ = 8 m/s.
Fr₁ = 8√(9.81 × 0.4) = 4.04
y₂ = 0.42 ( √(1 + 8 × 4.04²) − 1 ) = 2.09 m
ΔE = (2.09 − 0.40)³4 × 0.40 × 2.09 = 1.44 m
The specific energy before the jump was 0.4 + 8²/(2 × 9.81) = 3.66 m, so the jump destroys about 39% of it.
Classification by Fr₁
| Fr₁ | Jump type |
|---|---|
| 1 to 1.7 | Undular |
| 1.7 to 2.5 | Weak |
| 2.5 to 4.5 | Oscillating (avoid: waves travel downstream) |
| 4.5 to 9 | Steady, best for energy dissipation |
| above 9 | Strong, rough water surface |
Where engineers use it
Water coming down a dam spillway is fast enough to scour the riverbed. A stilling basin makes it jump in a lined, controlled place, so the energy is lost there and not by digging out the foundation. The jump also mixes chemicals well in water treatment plants.
Quick check
A hydraulic jump can form only when the incoming flow is: