23.7
Решение задач на гидравлический прыжок
Чтобы проанализировать гидравлический прыжок в прямоугольном канале со скоростью течения 6 метров в секунду, выполните следующие действия:
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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.
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Q1: What happens to water velocity when a hydraulic jump forms in a channel?
When a gate closes, a hydraulic jump travels upstream, combining the initial channel flow with wave speed to create an effective upstream velocity. Upstream of the jump, water moves at 8 meters per second, while downstream it slows to 2 meters per second. This velocity change is fundamental to understanding rapidly varying flow conditions in open channels.
Q2: How does the Froude number help determine water depth in a hydraulic jump?
The Froude number relates flow inertia to gravitational forces and is calculated using upstream velocity divided by the square root of gravitational acceleration times upstream depth. This dimensionless number allows engineers to determine the upstream depth from known flow conditions, which then enables calculation of downstream depth using the velocity ratio.
Q3: Why is the downstream depth four times greater than the upstream depth?
Since flow rate remains constant through the channel, the downstream-to-upstream velocity ratio is 1:4. When velocity decreases by a factor of four, depth must increase proportionally to maintain constant flow rate. In this example, upstream depth is 0.652 meters, making downstream depth 2.61 meters.
Q4: What is the effective upstream velocity in a hydraulic jump problem?
The effective upstream velocity combines the initial channel flow velocity with the hydraulic jump's wave speed traveling upstream. When water flows at 6 meters per second and the jump travels at 2 meters per second, the effective upstream velocity becomes 8 meters per second, representing the relative motion of water approaching the jump.
Q5: How do you calculate upstream depth using flow velocity and gravitational acceleration?
Using the Froude number formula, upstream depth is derived from upstream velocity and gravitational acceleration. With an effective upstream velocity of 8 meters per second and applying the Froude number relationship, the upstream depth is calculated as 0.652 meters. This value then serves as the basis for determining downstream depth.
Q6: What role does wave speed play in forming a hydraulic jump?
Wave speed represents the velocity at which the hydraulic jump travels upstream when a downstream gate closes. At 2 meters per second in this example, the wave speed combines with the channel's initial flow to create the effective upstream velocity. Behind the jump, water flows at the wave speed, creating the characteristic depth change.
Q7: How does constant flow rate relate to velocity and depth changes across a hydraulic jump?
Constant flow rate means discharge remains unchanged across the jump. Since discharge equals velocity multiplied by depth, when velocity decreases downstream, depth must increase proportionally. The 1:4 velocity ratio directly produces a 4:1 depth ratio, demonstrating the inverse relationship between velocity and depth in energy considerations open channel flow.