23.7
Hydraulic Jump: Problem Solving
Water flows through a rectangular channel at 6 meters per second. When a gate at the end of the channel suddenly closes, a wave called a hydraulic jump travels upstream at 2 meters per second.
In front of the jump, the wave's velocity combines the channel's initial flow, resulting in an effective upstream velocity of 8 meters per second. Behind the jump, the water flows at a speed of 2 meters per second.
Since the flow rate remains constant, the downstream-to-upstream velocity ratio is 1:4, so the downstream depth is four times the upstream depth.
To analyze the hydraulic jump, the Froude number, which relates flow inertia to gravity, is calculated as the upstream velocity divided by the square root of gravitational acceleration times the upstream depth.
Using this, the upstream depth is determined as 0.652 meters. Multiplying by four yields a downstream depth of 2.61 meters.
As a result, the depth ahead of the jump is 0.65 meters, and behind it, 2.61 meters.
To analyze a hydraulic jump in a rectangular channel with a flow speed of 6 meters per second, follow these steps:
- 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. - 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. - 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. - 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. - Determine Upstream Depth:
Calculate the upstream depth from the channel bottom to the water surface using the upstream velocity and Froude number. - Calculate Downstream Depth:
Since the downstream depth is four times the upstream depth, this value is calculated based on the upstream depth. - 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.
Water flows through a rectangular channel at 6 meters per second. When a gate at the end of the channel suddenly closes, a wave called a hydraulic jump travels upstream at 2 meters per second.
In front of the jump, the wave's velocity combines the channel's initial flow, resulting in an effective upstream velocity of 8 meters per second. Behind the jump, the water flows at a speed of 2 meters per second.
Since the flow rate remains constant, the downstream-to-upstream velocity ratio is 1:4, so the downstream depth is four times the upstream depth.
To analyze the hydraulic jump, the Froude number, which relates flow inertia to gravity, is calculated as the upstream velocity divided by the square root of gravitational acceleration times the upstream depth.
Using this, the upstream depth is determined as 0.652 meters. Multiplying by four yields a downstream depth of 2.61 meters.
As a result, the depth ahead of the jump is 0.65 meters, and behind it, 2.61 meters.
- Hydraulic Jump - A hydrodynamic phenomenon in fluid dynamics where pressure changes abruptly.
- Effective Upstream Velocity - The combined speed of the channel flow and the wave speed traveling upstream.
- Flow Velocities - The speeds of the fluid flow before and after a hydraulic jump.
- Velocity Ratio - The ratio of downstream to upstream velocity that impacts water depth.
- Froude Number - A dimensionless quantity used in fluid mechanics and wave theory, derived from flow characteristics.
- Define Hydraulic Jump - Understand what it means and how it is formed (e.g., hydraulic jump).
- Contrast Flow Velocities - Distinguish the flow speeds before and after a hydraulic jump (e.g., flow velocities).
- Explore Effective Upstream Velocity - Discuss how the wave speed and channel flow create effective upstream velocity (e.g., effective upstream velocity).
- Explain Velocity Ratio - Short description of how ratio of downstream to upstream velocity impacts water depth.
- Apply Froude Number - Elucidate its relevance in determining upstream depth in the context of hydraulic jumps.
- What is a hydraulic jump, and how does it impact flow velocities?
- How is the effective upstream velocity calculated, and why is it important?
- What is meant by the term 'velocity ratio,' and how does it relate to water depth?
From Chapter 23:
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