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.
Next, we calculate the Froude number to confirm the flow is supercritical.
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.
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.
