23.7: Hydraulic Jump: Problem Solving

To analyze a hydraulic jump in a rectangular channel with a flow speed of 6 meters per second, follow these steps:

1. Calculate Effective Upstream Velocity:
   When the downstream gate closes, a hydraulic jump forms, traveling upstream at 2 meters per second. This wave speed combines with the initial channel flow velocity, creating an effective upstream velocity.
2. Identify Flow Velocities Before and After the Hydraulic Jump:
   Upstream of the hydraulic jump, the effective flow velocity includes both the channel flow and wave speed. Downstream, the flow matches the wave speed, which is slower than the initial upstream velocity.
3. Establish Velocity Ratio for Depth Calculation:
   Because the flow rate remains constant, the ratio of downstream to upstream velocity impacts the depth of the water. In this example, the downstream depth is four times the upstream depth.
4. Calculate the Froude Number:
   The Froude number, which compares flow inertia to gravitational forces, is calculated using upstream flow characteristics. This number helps determine the upstream depth.
5. Determine Upstream Depth:
   Calculate the upstream depth from the channel bottom to the water surface using the upstream velocity and Froude number.
6. Calculate Downstream Depth:
   Since the downstream depth is four times the upstream depth, this value is calculated based on the upstream depth.
7. Summarize Depths Across the Hydraulic Jump:
   The results show that the water depth ahead of the hydraulic jump is significantly shallower than behind it, reflecting changes in depth due to flow conditions and wave behavior.

These steps provide a clear method for analyzing the hydraulic jump, illustrating how flow characteristics affect water depth across the channel.