Hydraulic jump definition and downstream Froude number

Hydraulic jump

A hydraulic jump is the sudden, turbulent rise of the water surface that occurs when a supercritical flow ($Fr > 1$) changes into subcritical flow ($Fr < 1$) in an open channel. It has a violent roller, strong mixing and large energy loss.

Why the energy principle is not used

The jump contains intense turbulence and eddies, and the energy dissipated (converted to heat) is unknown and cannot be written as a simple term. Hence the energy equation cannot be solved for the unknown depth. Instead the momentum equation is used, because the forces on the short control volume are known (the hydrostatic forces at the ends), while friction on the short, horizontal bed is negligible.

Numerical part

Given: $b = 5$ m, $Fr_1 = 3.5$, $y_1 = 1.2$ m.

Sequent depth

$$y_2 = \frac{y_1}{2}\left(\sqrt{1 + 8Fr_1^2} - 1\right) = 0.6\left(\sqrt{1 + 8(12.25)} - 1\right) = 5.37\ \text{m}$$

Velocities

$$V_1 = Fr_1\sqrt{g\,y_1} = 3.5\sqrt{9.81 \times 1.2} = 12.01\ \text{m/s}$$ $$V_2 = V_1\frac{y_1}{y_2} = 12.01 \times \frac{1.2}{5.37} = 2.68\ \text{m/s}$$

Froude number after the jump

$$Fr_2 = \frac{V_2}{\sqrt{g\,y_2}} = \frac{2.68}{\sqrt{9.81 \times 5.37}} = 0.37$$

Answer: $Fr_2 \approx 0.37$ (subcritical; $y_2 = 5.37$ m).