💦 Hydraulic Jump Calculator

Analyze the transition from rapid supercritical flow to tranquil subcritical flow — conjugate depths, energy dissipation, jump length, and specific energy curves.

y₁ y₂ V₁ (Fast) V₂ (Slow) Supercritical (Fr₁ > 1) Subcritical (Fr₂ < 1) Jump Length L_j

Open Channel Hydraulic Jump

A hydraulic jump is an open-channel phenomenon where high-velocity supercritical flow transitions rapidly into subcritical flow. The water level rises abruptly, generating extreme turbulence and energy dissipation.

• Conjugate Depths: The initial depth ($y_1$) and post-jump depth ($y_2$) satisfy momentum conservation.
• Energy Loss ($h_L$): Viscous shear and turbulent rollers dissipate upstream kinetic head.
• Erosion Control: Jumps are designed downstream of spillways to absorb kinetic energy and protect riverbeds.

📝 Configuration

Channel Geometry

Flow Parameters

Hydraulic Jump Equations:

Conjugate Depth (Bélanger):
$y_2 = \frac{y_1}{2} \left(-1 + \sqrt{1 + 8 Fr_1^2}\right)$

Upstream Froude Number:
$Fr_1 = \frac{V_1}{\sqrt{g y_1}}$

Energy Loss:
$h_L = \frac{(y_2 - y_1)^3}{4 y_1 y_2}$

Power Dissipated:
$P = \rho g Q h_L$

📊 Results & Visualization

Configure inputs and click Compute Hydraulic Jump to view results.

📘 Calculation Methodology

Mathematical Model & Theory

A hydraulic jump occurs when a rapid, supercritical open-channel flow ($Fr_1 > 1.0$) transitions abruptly to a slower, subcritical flow ($Fr_2 < 1.0$), dissipating mechanical energy. The relationship between upstream depth ($y_1$) and downstream depth ($y_2$) is given by the Belanger equation:

$$\frac{y_2}{y_1} = \frac{1}{2} \left( \sqrt{1 + 8 Fr_1^2} - 1 \right)$$ $$Fr_1 = \frac{V_1}{\sqrt{g y_1}}$$

Assumptions

  • Rectangular, horizontal, and smooth channel bed (negligible slope and bed shear).
  • One-dimensional, steady flow conditions upstream and downstream of the jump.
  • Uniform velocity distribution at the boundary cross-sections.
  • Hydrostatic pressure distribution at the measurement stations.

Academic References

  1. Chaudhry, M. H.: Open-Channel Flow, Springer, 2nd Edition.
  2. French, R. H.: Open-Channel Hydraulics, Water Resources Publications.

Worked Engineering Example

Problem Statement:
A water flow at depth 0.2 m has an upstream Froude number $Fr_1 = 3.0$. Calculate the downstream conjugate depth $y_2$ after the hydraulic jump.

Step-by-step Solution:
1. Apply Belanger's equation:
$$\frac{y_2}{y_1} = \frac{1}{2} \left( \sqrt{1 + 8 \times 3.0^2} - 1 \right) = \frac{1}{2} (\sqrt{73} - 1) = \frac{1}{2} (8.544 - 1) = 3.772$$ 2. Calculate downstream depth $y_2$:
$$y_2 = 3.772 \times y_1 = 3.772 \times 0.2 = 0.754 \text{ m}$$
Final Result:
Downstream conjugate depth is 0.754 meters.