SeriesCalc Logo

SeriesCalc

🌊 Hydraulic Jump Calculator

Calculate sequent depth and energy loss in an open channel hydraulic jump.

Inputs

Results

Froude Number (Fr₁)
0.00
Sequent Depth (y₂)
- ft
Energy Loss (ΔE)
-
ft

Hydraulic Jump Profile

Subcritical Flow (No Jump)

Calculator Description

A hydraulic jump is the abrupt rise of the water surface in an open channel where a fast, shallow (supercritical) flow suddenly transitions to a slow, deep (subcritical) flow. The turbulence and eddies produced in this transition dissipate a significant amount of energy.

What this calculator finds

This calculator determines the upstream Froude number (Fr₁), the depth after the jump (y₂, the sequent depth), and the energy loss (ΔE) dissipated in the jump, from the initial depth and velocity.

Why it matters

  • Designing stilling basins downstream of spillways and gates to safely dissipate flow energy
  • Predicting the location and length of the jump to prevent scour
  • Determining the flow regime (super-/subcritical) and computing downstream depth in open channels

Formula

Hydraulic Jump Equations

The relation between the depths before and after the jump is derived from momentum conservation and depends only on the upstream Froude number (the Bélanger equation). The energy loss follows from the difference between the two depths.

Fr1=v1g y1Fr_1 = \dfrac{v_1}{\sqrt{g\,y_1}}
y2=y12[1+8 Fr12−1]y_2 = \frac{y_1}{2}\left[\sqrt{1 + 8\,Fr_1^2} - 1\right]
ΔE=(y2−y1)34 y1 y2\Delta E = \dfrac{(y_2 - y_1)^3}{4\,y_1\,y_2}
  • Fr₁ — Upstream (pre-jump) Froude number (dimensionless)
  • v₁ — Velocity before the jump [m/s]
  • y₁, y₂ — Depth before the jump / sequent depth after the jump [m]
  • g — Acceleration of gravity (9.81 m/s²)
  • ΔE — Energy (head) loss per unit weight [m]

How the formula works

  • A jump occurs only when Fr₁ > 1 (supercritical). If Fr₁ ≤ 1, no jump forms.
  • The larger Fr₁ is, the greater the sequent depth y₂ and the more sharply the energy loss ΔE grows.
  • Energy loss scales with the cube of (y₂ − y₁), so stronger jumps dissipate dramatically more energy.

Worked example

For y₁ = 0.3 m and v₁ = 6 m/s, Fr₁ = 6 / √(9.81 × 0.3) ≈ 3.5 (supercritical). The sequent depth is y₂ = (0.3/2)·[√(1 + 8×3.5²) − 1] ≈ 1.35 m, and the energy loss is ΔE = (1.35 − 0.3)³ / (4 × 0.3 × 1.35) ≈ 0.71 m.

Useful Tips

Practical tips

  • The jump type depends on Fr₁: 1.7–2.5 is a weak jump, 2.5–4.5 an oscillating jump, and 4.5–9.0 a steady jump that is ideal for stilling basins.
  • The jump length is often estimated as roughly 6·y₂ to size stilling basins.

Limitations & cautions

  • The equations assume a horizontal, rectangular channel; steep slopes or other cross-sections require corrections.
  • Bed friction is neglected, so friction losses must be considered separately over long reaches.
  • Oscillating jumps (Fr₁ 2.5–4.5) send waves downstream that can cause erosion and are best avoided in design.