Hydraulic Jump Calculator

Upstream Flow Depth (\( y_1 \)):

Upstream Flow Velocity (\( v_1 \)):

Channel Width (\( B \)):

Upstream Froude Number (\( F_{r1} \)):

Jump Type:

Downstream Flow Depth (\( y_2 \)):

Downstream Flow Velocity (\( v_2 \)):

Head Loss (\( \Delta E \)):

Jump Height (\( h \)):

Jump Length (\( L \)):

Jump Efficiency (\( E_n \)):

1. What is Hydraulic Jump Calculator?

Definition: This calculator computes the characteristics of a hydraulic jump in a horizontal rectangular channel, including downstream depth, downstream velocity, head loss, jump height, jump length, and jump efficiency, based on upstream flow conditions.

Purpose: It is used in hydraulic engineering to analyze the transition from supercritical to subcritical flow, aiding in the design of energy dissipators, spillways, and water treatment systems.

2. How Does the Calculator Work?

The calculator uses the following equations for a horizontal rectangular channel:

Froude Number: \( F_{r1} = \frac{v_1}{\sqrt{g y_1}} \)
Conjugate Depth: \( \frac{y_2}{y_1} = \frac{1}{2} \left( \sqrt{1 + 8 F_{r1}^2} - 1 \right) \)
Discharge: \( Q = v_1 y_1 B \)
Downstream Velocity: \( v_2 = \frac{Q}{y_2 B} \)
Head Loss: \( \Delta E = \frac{(y_2 - y_1)^3}{4 y_1 y_2} \)
Jump Height: \( h = y_2 - y_1 \)
Jump Length: \( L = 220 y_1 \tanh\left(\frac{F_{r1} - 1}{22}\right) \)
Jump Efficiency: \( E_n = \frac{\left[\sqrt{8 F_{r1}^2 + 1}\right]^3 - 4 F_{r1}^2 + 1}{8 F_{r1}^2 (2 + F_{r1}^2)} \times 100\% \)

Where:

\( y_1 \): Upstream flow depth (m);
\( v_1 \): Upstream flow velocity (m/s);
\( B \): Channel width (m);
\( g \): Acceleration due to gravity (9.81 m/s²);
\( F_{r1} \): Upstream Froude number (dimensionless);
\( y_2 \): Downstream flow depth (m, converted to selected unit);
\( v_2 \): Downstream flow velocity (m/s, converted to selected unit);
\( \Delta E \): Head loss (m, converted to selected unit);
\( h \): Jump height (m, same unit as depth);
\( L \): Jump length (m, converted to selected unit);
\( E_n \): Jump efficiency (%).

Steps:

Enter the upstream flow depth (\( y_1 \)) with its unit.
Enter the upstream flow velocity (\( v_1 \)) with its unit.
Enter the channel width (\( B \)) with its unit.
Convert \( y_1 \), \( v_1 \), and \( B \) to SI units (m, m/s, m).
Calculate the upstream Froude number and display a warning if \( F_{r1} \leq 1 \), but proceed with calculations.
Determine the jump type based on \( F_{r1} \).
Calculate the downstream depth (\( y_2 \)) using the conjugate depth equation.
Calculate the discharge (\( Q \)) and downstream velocity (\( v_2 \)).
Calculate the head loss (\( \Delta E \)), jump height (\( h \)), jump length (\( L \)), and jump efficiency (\( E_n \)).
Convert results to the selected output units and display them, formatted in scientific notation if the absolute value is less than 0.001, otherwise with 5 decimal places.

3. Importance of Hydraulic Jump Calculation

Calculating hydraulic jump characteristics is crucial for:

Energy Dissipation: Designing stilling basins and spillways to reduce flow energy and prevent erosion downstream.
Mixing Processes: Enhancing mixing in water treatment plants by inducing turbulence.
Flow Control: Managing flow transitions in open channels to ensure stable subcritical flow downstream.

4. Using the Calculator

Example 1 (SI Units, Supercritical Flow): Calculate the hydraulic jump characteristics:

Upstream Flow Depth: \( y_1 = 0.5 \, \text{m} \);
Upstream Flow Velocity: \( v_1 = 3 \, \text{m/s} \);
Channel Width: \( B = 2 \, \text{m} \);
Froude Number: \( F_{r1} = \frac{v_1}{\sqrt{g y_1}} = \frac{3}{\sqrt{9.81 \times 0.5}} \approx 1.353 \);
Jump Type: Undular Jump (\( 1 < F_{r1} < 1.7 \));
Downstream Depth: \( \frac{y_2}{y_1} = \frac{1}{2} \left( \sqrt{1 + 8 \times 1.353^2} - 1 \right) \approx 0.676 \), so \( y_2 = 0.676 \times 0.5 \approx 0.338 \, \text{m} \);
Discharge: \( Q = 3 \times 0.5 \times 2 = 3 \, \text{m}^3/\text{s} \);
Downstream Velocity: \( v_2 = \frac{Q}{y_2 B} = \frac{3}{0.338 \times 2} \approx 4.438 \, \text{m/s} \);
Head Loss: \( \Delta E = \frac{(0.338 - 0.5)^3}{4 \times 0.5 \times 0.338} \approx -0.006 \, \text{m} \);
Jump Height: \( h = 0.338 - 0.5 = -0.162 \, \text{m} \);
Jump Length: \( L = 220 \times 0.5 \times \tanh\left(\frac{1.353 - 1}{22}\right) \approx 1.762 \, \text{m} \);
Jump Efficiency: \( E_n = \frac{\left[\sqrt{8 \times 1.353^2 + 1}\right]^3 - 4 \times 1.353^2 + 1}{8 \times 1.353^2 \times (2 + 1.353^2)} \times 100\% \approx 81.614\% \);
Result: \( y_2 = 0.33800 \, \text{m} \), \( v_2 = 4.43800 \, \text{m/s} \), \( \Delta E = -0.00600 \, \text{m} \), \( h = -0.16200 \, \text{m} \), \( L = 1.76200 \, \text{m} \), \( E_n = 81.61400\% \).

Example 2 (Mixed Units, Subcritical Flow with Warning): Calculate the hydraulic jump characteristics:

Upstream Flow Depth: \( y_1 = 12 \, \text{in} \);
Convert \( y_1 \): \( y_1 = 12 \times 0.0254 = 0.3048 \, \text{m} \);
Upstream Flow Velocity: \( v_1 = 2 \, \text{ft/s} \);
Convert \( v_1 \): \( v_1 = 2 \times 0.3048 = 0.6096 \, \text{m/s} \);
Channel Width: \( B = 5 \, \text{ft} \);
Convert \( B \): \( B = 5 \times 0.3048 = 1.524 \, \text{m} \);
Froude Number: \( F_{r1} = \frac{0.6096}{\sqrt{9.81 \times 0.3048}} \approx 0.352 \);
Warning: \( F_{r1} \leq 1 \), so a warning is displayed, but calculations proceed;
Jump Type: Undular Jump (\( F_{r1} < 1.7 \));
Downstream Depth: \( \frac{y_2}{y_1} = \frac{1}{2} \left( \sqrt{1 + 8 \times 0.352^2} - 1 \right) \approx 0.123 \), so \( y_2 = 0.123 \times 0.3048 \approx 0.0375 \, \text{m} \);
Convert \( y_2 \) to ft: \( y_2 = 0.0375 \times 3.28084 \approx 0.123 \, \text{ft} \);
Discharge: \( Q = 0.6096 \times 0.3048 \times 1.524 \approx 0.283 \, \text{m}^3/\text{s} \);
Downstream Velocity: \( v_2 = \frac{0.283}{0.0375 \times 1.524} \approx 4.953 \, \text{m/s} \);
Convert \( v_2 \) to ft/s: \( v_2 = 4.953 \times 3.28084 \approx 16.243 \, \text{ft/s} \);
Head Loss: \( \Delta E = \frac{(0.0375 - 0.3048)^3}{4 \times 0.3048 \times 0.0375} \approx -1.563 \, \text{m} \);
Convert \( \Delta E \) to ft: \( \Delta E = -1.563 \times 3.28084 \approx -5.128 \, \text{ft} \);
Jump Height: \( h = 0.0375 - 0.3048 = -0.2673 \, \text{m} \), or \( -0.2673 \times 3.28084 \approx -0.877 \, \text{ft} \);
Jump Length: \( L = 220 \times 0.3048 \times \tanh\left(\frac{0.352 - 1}{22}\right) \approx -1.975 \, \text{m} \);
Convert \( L \) to ft: \( L = -1.975 \times 3.28084 \approx -6.480 \, \text{ft} \);
Jump Efficiency: \( E_n = \frac{\left[\sqrt{8 \times 0.352^2 + 1}\right]^3 - 4 \times 0.352^2 + 1}{8 \times 0.352^2 \times (2 + 0.352^2)} \times 100\% \approx 97.614\% \);
Result: \( y_2 = 0.12300 \, \text{ft} \), \( v_2 = 16.24300 \, \text{ft/s} \), \( \Delta E = -5.12800 \, \text{ft} \), \( h = -0.87700 \, \text{ft} \), \( L = -6.48000 \, \text{ft} \), \( E_n = 97.61400\% \).

5. Frequently Asked Questions (FAQ)

Q: What is a hydraulic jump?
A: A hydraulic jump occurs when flow transitions from supercritical (\( F_r > 1 \)) to subcritical (\( F_r < 1 \)), resulting in an abrupt rise in water surface, turbulence, and energy dissipation.

Q: Why does the calculator show a warning for Froude number ≤ 1?
A: A hydraulic jump typically occurs when the upstream flow is supercritical (\( F_r > 1 \)). If \( F_r \leq 1 \), the flow may not produce a jump, but the calculator still provides results for educational purposes.

Q: What does the jump efficiency indicate?
A: Jump efficiency (\( E_n \)) represents the percentage of energy retained after the jump. A lower efficiency indicates greater energy dissipation, which is often desirable in hydraulic structures like stilling basins.