💦 Hydraulic Jump Calculator
Analyze the transition from rapid supercritical flow to tranquil subcritical flow — conjugate depths, energy dissipation, jump length, and specific energy curves.
📝 Configuration
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:
$P = \rho g Q h_L$
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 the inputs and click Compute to see results.
ℹ️ About Hydraulic Jumps
A hydraulic jump is a phenomenon in the science of hydraulics which is frequently observed in open channel flows such as rivers and spillways. When liquid at high velocity discharges into a zone of lower velocity, a rather abrupt rise occurs in the liquid surface.
Key features:
• **Supercritical upstream:** The incoming flow must have a Froude number Fr₁ > 1.0 (supercritical, rapid flow).
• **Subcritical downstream:** The outgoing flow has a Froude number Fr₂ < 1.0 (subcritical, tranquil flow).
• **Significant energy loss:** The transition causes heavy turbulence, dissipating energy as heat and sound, which is highly useful in spillway design to prevent erosion.
A hydraulic jump is a phenomenon in the science of hydraulics which is frequently observed in open channel flows such as rivers and spillways. When liquid at high velocity discharges into a zone of lower velocity, a rather abrupt rise occurs in the liquid surface.
Key features:
• **Supercritical upstream:** The incoming flow must have a Froude number Fr₁ > 1.0 (supercritical, rapid flow).
• **Subcritical downstream:** The outgoing flow has a Froude number Fr₂ < 1.0 (subcritical, tranquil flow).
• **Significant energy loss:** The transition causes heavy turbulence, dissipating energy as heat and sound, which is highly useful in spillway design to prevent erosion.
📘 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:
$$\frac{y_2}{y_1} = \frac{1}{2} \left( \sqrt{1 + 8 Fr_1^2} - 1 \right)$$
$$Fr_1 = \frac{V_1}{\sqrt{g y_1}}$$
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.
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.