Jump at spillway toe; repelled and submerged jump

Given: $Q = 80\ \text{m}^3/\text{s}$, $b = 10$ m ($q = 8\ \text{m}^2/\text{s}$), crest 3 m above the downstream bed, head over crest $H = 2.47$ m. No energy loss on the spillway.

Pre-jump depth $y_1$

Total energy above the downstream bed:

$$E_0 = 3 + 2.47 = 5.47\ \text{m}$$

At the toe, $E_1 = E_0$:

$$y_1 + \frac{q^2}{2g\,y_1^2} = 5.47 \;\Rightarrow\; y_1 + \frac{3.262}{y_1^2} = 5.47 \;\Rightarrow\; y_1 = 0.839\ \text{m}$$ $$V_1 = \frac{8}{0.839} = 9.53\ \text{m/s},\qquad Fr_1 = \frac{9.53}{\sqrt{9.81 \times 0.839}} = 3.32$$

Post-jump (sequent) depth $y_2$

$$y_2 = \frac{0.839}{2}\left(\sqrt{1 + 8(3.32)^2} - 1\right) = 3.55\ \text{m}$$

The energy loss in the jump is $\dfrac{(y_2 - y_1)^3}{4y_1y_2} = 1.67$ m. The critical depth is $y_c = 1.87$ m.

Pre-jump depth = 0.84 m; post-jump depth = 3.55 m.

Repelled and submerged jumps

Compare the actual tailwater depth $y_t$ in the river with the sequent depth $y_2 = 3.55$ m:

For this flow, a repelled jump forms if the tailwater is below 3.55 m, and a submerged jump if it is above 3.55 m. In a stilling basin the tailwater is usually kept slightly above $y_2$ (by about 5 to 10%) by depressing the apron, to keep the jump stable on the apron.