Hump and constriction: critical depth, 1.8 m channel

Given: $Q = 2\ \text{m}^3/\text{s}$, $B = 1.8\ \text{m}$, $y_1 = 1.0\ \text{m}$. The approach flow is subcritical.

Approach specific energy

$$\begin{aligned} q_1 &= \frac{2}{1.8} = 1.111\ \text{m}^2/\text{s},\quad V_1 = 1.111\ \text{m/s} \\ E_1 &= 1.0 + \frac{1.111^2}{2 \times 9.81} = 1.063\ \text{m} \end{aligned}$$

Rule: critical depth occurs at the section only if the energy available there is not more than the minimum energy needed, $E_1 - \Delta z \le E_c = 1.5\,y_c$ (with $y_c = (q^2/g)^{1/3}$).

(a) Hump of 0.15 m

$$y_c = \left(\frac{1.111^2}{9.81}\right)^{1/3} = 0.501\ \text{m},\quad E_c = 0.752\ \text{m}$$

Available energy over the hump $= 1.063 - 0.15 = 0.913\ \text{m} > 0.752\ \text{m}$.

No critical depth. The flow stays subcritical over the hump with $y \approx 0.82\ \text{m}$.

(b) Side-wall contraction to 1.3 m

$$q_2 = \frac{2}{1.3} = 1.538\ \text{m}^2/\text{s},\quad y_c = \left(\frac{1.538^2}{9.81}\right)^{1/3} = 0.623\ \text{m},\quad E_c = 0.934\ \text{m}$$

Available energy $= 1.063\ \text{m} > 0.934\ \text{m}$.

No critical depth. The depth in the throat is about 0.92 m (subcritical).

(c) Hump and contraction together

$$E_c + \Delta z = 0.934 + 0.15 = 1.084\ \text{m} > E_1 = 1.063\ \text{m}$$

The approach flow does not have enough energy. Critical depth does occur at the throat ($y = y_c = 0.623\ \text{m}$) and the flow chokes. The upstream depth rises until $E_1' = 1.084\ \text{m}$:

$$y_1' + \frac{1.111^2}{2 g\, y_1'^2} = 1.084 \Rightarrow y_1' = 1.024\ \text{m}$$

Answer: (a) No, (b) No, (c) Yes. In (c) the upstream depth is raised from 1.00 m to about 1.02 m.