The Hydraulic Jump: Flow Behavior and Energy Dissipation

The Physics of a Hydraulic Jump

A hydraulic jump occurs when a rapid, high-velocity supercritical flow ($Fr > 1$) transitions abruptly into a slow, tranquil subcritical flow ($Fr < 1$). This rapid transition results in a sudden rise in water level, severe turbulence, air entrainment, and significant energy dissipation.

Momentum and Conjugate Depths

Because a hydraulic jump involves high internal friction and turbulence, energy is not conserved across the jump. Therefore, engineers apply the conservation of momentum equation to calculate the relationship between the initial depth ($y_1$) and the post-jump depth ($y_2$), known as conjugate depths. For a rectangular channel, this is expressed as:

$$rac{y_2}{y_1} = rac{1}{2} left( sqrt{1 + 8Fr_1^2} – 1
ight)$$

Where $Fr_1$ is the Froude number of the incoming supercritical flow.

Energy Dissipation efficiency

The energy loss ($Delta E$) in the jump is the difference between specific energies before and after the jump:

$$Delta E = E_1 – E_2 = rac{(y_2 – y_1)^3}{4y_1y_2}$$

In dam design, hydraulic jumps are intentionally induced in stilling basins at the base of spillways. This dissipates the kinetic energy of floodwaters, preventing dangerous scour and erosion at the foundation of hydraulic structures.