In deriving the equation for the hydraulic jump in a rectangular channel in terms of conjugate depths and initial Froude number

Explanation:

• The hydraulic jump is analyzed using the continuity equation to ensure flow rate is constant before and after the jump.

• The momentum equation is applied to account for the change in momentum due to the sudden change in flow depth.

• The energy equation is generally not used because energy is not conserved across a hydraulic jump (energy loss occurs due to turbulence).

• Therefore, the derivation relies primarily on continuity and momentum principles.

 Additional Information

• A hydraulic jump is a sudden rise in water surface occurring when fast, shallow supercritical flow transitions to slower, deeper subcritical flow.

• This phenomenon is common downstream of spillways, sluice gates, and steep channels, and it helps dissipate excess energy in the flow.

• The continuity equation ensures that the flow discharge (volume per second) remains constant before and after the jump.

• The energy equation relates the specific energy (sum of potential and kinetic energy per unit weight) upstream and downstream, accounting for energy losses due to turbulence and mixing in the jump.

• By analyzing the depths before and after the jump (called conjugate depths), engineers can calculate the jump characteristics, such as the height of the water surface and energy dissipation.

• The Froude number, a dimensionless parameter comparing inertial to gravitational forces, determines the flow regime and the occurrence of the hydraulic jump (it occurs when flow transitions from supercritical, Froude number > 1, to subcritical, Froude number < 1).