Problem Statement
The depth of flow of water, at a certain section of a rectangular channel of 5 m wide is 0.6 m. The discharge through the channel is 15 m³/s. If a hydraulic jump takes place on the downstream side, find the depth of flow after the jump.
Given Data & Constants
- Width of channel, \(B = 5 \, \text{m}\)
- Initial depth of flow, \(d_1 = 0.6 \, \text{m}\)
- Discharge, \(Q = 15 \, \text{m}^3/\text{s}\)
- Acceleration due to gravity, \(g = 9.81 \, \text{m/s}^2\)
Solution
1. Calculate Initial Velocity and Froude Number
First, we calculate the area of flow and the velocity before the jump.
$$ \text{Area of flow, } A_1 = B \times d_1 = 5 \times 0.6 = 3.0 \, \text{m}^2 $$ $$ \text{Velocity, } V_1 = \frac{Q}{A_1} = \frac{15}{3.0} = 5.0 \, \text{m/s} $$Next, we calculate the Froude number to confirm the flow is supercritical.
$$ \text{Froude Number, } Fr_1 = \frac{V_1}{\sqrt{g \cdot d_1}} = \frac{5.0}{\sqrt{9.81 \times 0.6}} \approx 2.06 $$2. Calculate the Depth of Flow After the Jump (\(d_2\))
Since the Froude number is greater than 1, the flow is supercritical and a hydraulic jump can occur. We use the hydraulic jump formula for a rectangular channel.
$$ d_2 = \frac{d_1}{2} \left[ \sqrt{1 + 8 Fr_1^2} - 1 \right] $$ $$ d_2 = \frac{0.6}{2} \left[ \sqrt{1 + 8 \times (2.06)^2} - 1 \right] $$ $$ d_2 = 0.3 \left[ \sqrt{1 + 8 \times 4.2436} - 1 \right] $$ $$ d_2 = 0.3 \left[ \sqrt{1 + 33.949} - 1 \right] = 0.3 [\sqrt{34.949} - 1] $$ $$ d_2 = 0.3 [5.912 - 1] = 0.3 \times 4.912 \approx 1.474 \, \text{m} $$The depth of flow after the hydraulic jump is approximately 1.47 m.
Explanation of a Hydraulic Jump
A hydraulic jump is a phenomenon that occurs in open channel flow when a high-velocity, shallow stream (supercritical flow) abruptly transitions to a low-velocity, deep stream (subcritical flow). This transition is characterized by a sudden rise in the water surface and significant turbulence and energy dissipation.
- Supercritical Flow: The initial condition where the flow is fast and shallow. This is indicated by a Froude Number (\(Fr\)) greater than 1.
- Subcritical Flow: The final condition where the flow is slow and deep, with a Froude Number less than 1.
The formula used is derived from the conservation of momentum principle across the jump. It allows us to directly calculate the resulting "sequent depth" (\(d_2\)) after the jump, based on the initial depth (\(d_1\)) and the initial Froude number.