Understanding Hydraulic Jump | Civil Engineering Optional Notes for UPSC PDF Download

In wide open-channel flows, liquid is only confined by a lower solid boundary and its upper surface is exposed to the atmosphere. A control-volume analysis can be performed on a section of an open channel flow to balance inlet and outlet transport of mass and momentum (Fig 2). If the velocities are assumed uniform at the inlet and outlet of the control volume (V₁ and V₂ respectively) with corresponding liquid depths H₁ and H₂, then a steady mass flow balance reduces to:

The x-direction momentum analysis of this control volume balances forces from hydrostatic pressure (due to fluid depth) with the inlet and outlet momentum flow rates (Eqn. 2). The pressure forces act inward on the two sides of the control volume, and are equal to the specific gravity of the liquid (liquid density times gravitational acceleration: ρg), multiplied by the average liquid depth on each side (H₁/2, H₂/2), multiplied the height over which the pressure acts on each side (H₁, H₂). This results in the quadratic expression on the left side of Eqn. 2. The momentum flow rates through each side (Eqn. 2, right side) are equal to the mass flow rates of liquid through the control volume (in: , out: ) multiplied by the fluid velocities (V₁, V₂).

Eqn. 1 can be substituted into Eqn. 2 to eliminate V₂. The Froude number () can also be substituted in, which represents the relative strength of inflow fluid momentum to hydrostatic forces. The resulting expression can be stated as:

This cubic equation has three solutions. One is H1 = H2, which gives the normal open-channel behavior (inlet depth = outlet depth). A second solution gives a negative liquid level, which is unphysical, and can be eliminated. The remaining solution allows for an increase in depth (hydraulic jump) or a decrease in depth (hydraulic depression), depending on the inlet Froude number. If the inlet Froude number (Fr₁) is greater than one, the flow is called supercritical (unstable) and has high mechanical energy (kinetic + gravitational potential energy). In this case, a hydraulic jump can form spontaneously or due to some disturbance to the flow. The hydraulic jump dissipates mechanical energy into heat, significantly reducing the kinetic energy and slightly increasing the potential energy of the flow. The resulting outlet height is given by Eqn. 4 (a solution to Eqn. 3). A hydraulic depression cannot occur if Fr₁ > 1 because it would increase mechanical energy of the flow, violating the second law of thermodynamics.

The strength of hydraulic jumps increases with inlet Froude numbers. As Fr₁ increases, the magnitude of H₂/H₁ increases and a greater portion of inlet kinetic energy is dissipated as heat [1].

Figure 2: Control volume of a section of an open-channel flow containing a hydraulic jump. Inlet and out mass and momentum flow rates per unit width are indicated. Hydrostatic forces per unit width indicated in lower diagram.