Near the downstream end of a river spillway, a hydraulic jump often forms, as illustrated in Fig.

Question:

Near the downstream end of a river spillway, a hydraulic jump often forms, as illustrated in Fig. P5.123 and Video V10.11. The velocity of the channel flow is reduced abruptly across the jump. Using the conservation of mass and linear momentum principles, derive the following expression for \(h_{2}\),

\[ h_{2}=-\frac{h_{1}}{2}+\sqrt{\left(\frac{h_{1}}{2}\right)^{2}+\frac{2 V_{1}^{2} h_{1}}{g}} \]

The loss of available energy across the jump can also be determined if energy conservation is considered. Derive the loss expression

\[ \text { jump loss }=\frac{g\left(h_{2}-h_{1}\right)^{3}}{4 h_{1} h_{2}} \]

Figure P5.123