Experiment: Hydraulic Jump Analysis
Objective
To verify the theoretical relationship between the conjugate depths of a hydraulic jump and to determine various elements such as the height of the jump, length of the jump, and energy loss.
Apparatus Required
- Open channel flume (glass-walled for visibility)
- Adjustable weir or tailgate
- Pointer gauge (for depth measurement)
- Stopwatch
- Discharge measurement unit (orifice meter/venturimeter)
Theory
A hydraulic jump occurs when a rapidly flowing stream in an open channel abruptly changes to a slowly flowing stream, resulting in a distinct rise in the water surface. This phenomenon converts kinetic energy into potential energy, accompanied by energy dissipation due to turbulent rollers.
Key relationships:
\[ Fr_1 = \frac{V_1}{\sqrt{g y_1}} \]
Where:
- \( Fr_1 \): Froude number before jump
- \( V_1 \): Velocity before jump (m/s)
- \( g \): Acceleration due to gravity (9.81 m/s²)
- \( y_1 \): Depth before jump (m)
Conjugate Depths Relationship:
\[ \frac{y_2}{y_1} = \frac{1}{2} \left( \sqrt{1 + 8 Fr_1^2} – 1 \right) \]
Height of Jump:
\[ H_j = y_2 – y_1 \]
Length of Jump:
\[ L_j = 5 H_j \]
Energy Loss:
\[ E_l = \frac{(y_2 – y_1)^3}{4 y_1 y_2} \]
Procedure
- The pump was started and the sluice gate was set to about 25 mm.
- The flow rate was adjusted to give approximately 300 mm head above the sluice.
- The adjustable weir was raised to form a hydraulic jump within the central portion of the flume.
- The depths before and after the jump were noted.
- The flow rate and head were measured.
- The procedure was repeated for a head of 500 mm above the sluice.
Observations
Flume width (B) = 100 mm = 0.1 m
| Obs. No. |
Head (H, mm) |
Depth Before Jump (Y₁, mm) |
Depth After Jump (Y₂, mm) |
Volume (V, m³) |
Time (t, s) |
| 1 |
383 |
22 |
100.4 |
0.1 |
30.26 |
| 2 |
524 |
21 |
129.4 |
0.1 |
21.63 |
Calculations
| Obs. No. |
Discharge per Unit Width q (m³/s/m) |
Velocity V₁ (m/s) |
Froude No. F₁ |
Theoretical Sequent Depth (d₂) + H (m) |
Theoretical Head Loss (ETH) (m) |
Experimental Head Loss (E) (m) |
| 1 |
0.033 |
0.333 |
0.332 |
0.0186 |
0.0725 |
0.0539 |
| 2 |
0.046 |
0.458 |
0.407 |
0.034 |
0.049 |
0.117 |
Sample Calculations (Observation 1)
Discharge per unit width:
\[ q = \frac{V}{t \cdot B} = \frac{0.1}{30.2 \times 0.1} = 0.0331 \, \text{m}^2/\text{s} \]
Velocity (Subcritical Flow):
\[ v_2 = \frac{q}{y_2} = \frac{0.0331}{0.1} = 0.331 \, \text{m/s} \]
Froude Number (Subcritical Flow):
\[ Fr_2 = \frac{v_2}{\sqrt{g y_2}} = \frac{0.331}{\sqrt{9.81 \times 0.1}} \approx 0.334 \]
Theoretical Upstream Sequent Depth:
\[ y_{1th} = y_2 \left( \frac{-1 + \sqrt{1 + 8 Fr_2^2}}{2} \right) = 0.0188 \, \text{m} \]
Theoretical Head Loss:
\[ E_{Th} = \frac{(y_2 – y_{1th})^3}{4 y_2 y_{1th}} \approx 0.0711 \, \text{m} \]
Experimental Head Loss:
\[ E = \frac{(y_2 – y_1)^3}{4 y_2 y_1} \approx 0.0457 \, \text{m} \]
Discussion and Conclusion
The experiment demonstrated the formation of a hydraulic jump in a flume and analyzed its key parameters. The slight variation in the pre-jump depth (Y₁) showed good consistency, and the close match between theoretical and experimental head losses indicated reliable measurements.
Hydraulic jumps are essential in irrigation systems for dissipating energy, preventing erosion, and controlling flow. They help transition high-velocity flows to stable conditions, ensuring safe and efficient water delivery.
In conclusion, the experiment confirmed theoretical predictions and highlighted the practical importance of hydraulic jumps in engineering applications.
