水跃问题求解

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.

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.

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.

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.

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.

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.

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.