Hump and constriction: critical depth, 2 m channel

Critical depth occurs at a section when the available specific energy there is equal to the minimum specific energy for the discharge per unit width at that section: $E_2=E_{min}=\tfrac32y_c$, where $y_c=\sqrt[3]{q^2/g}$.

Approach flow

$Q=2.4$ m³/s, $b=2$ m, $y_1=1.0$ m.

$$q_1=\frac{2.4}{2}=1.2\ \text{m}^2/\text{s},\quad V_1=1.2\ \text{m/s},\quad Fr_1=\frac{1.2}{\sqrt{9.81\times1}}=0.383\ (\text{subcritical})$$ $$E_1=y_1+\frac{V_1^2}{2g}=1+\frac{1.44}{19.62}=1.0734\ \text{m}$$ $$y_{c1}=\sqrt[3]{\frac{1.2^2}{9.81}}=0.5275\ \text{m},\qquad E_{min1}=1.5y_{c1}=0.7913\ \text{m}$$

(a) Hump of 0.20 m (width unchanged)

Energy over hump $E_2=E_1-\Delta z=1.0734-0.20=0.8734$ m $>E_{min1}=0.7913$ m.

So critical depth does not occur. The flow stays subcritical over the hump, with depth found from $y+\dfrac{1.44}{19.62y^2}=0.8734$:

$$y_2=0.739\ \text{m}\quad(>y_c=0.528\ \text{m})$$

The upstream depth remains 1.0 m.

(b) Side-wall constriction to 1.7 m (no hump)

$$q_2=\frac{2.4}{1.7}=1.4118\ \text{m}^2/\text{s},\quad y_{c2}=\sqrt[3]{\frac{q_2^2}{g}}=0.5879\ \text{m},\quad E_{min2}=0.8818\ \text{m}$$

$E_2=E_1=1.0734>E_{min2}=0.8818$, so critical depth does not occur. Depth in the constriction (subcritical): $y_2=0.964$ m. Upstream depth unchanged.

(c) Hump and constriction together

Available energy at the section: $E_2=E_1-0.20=0.8734$ m, while the minimum required is $E_{min2}=0.8818$ m (for $q_2=1.412$).

$E_2<E_{min2}$, so critical depth occurs (choking) over the hump in the constriction, $y_c=0.588$ m. The channel cannot pass the flow at the original upstream energy, so the upstream depth is affected: it rises until

$$E_1'=E_{min2}+\Delta z=0.8818+0.20=1.0818\ \text{m}$$ $$y_1'+\frac{1.2^2}{2g\,y_1'^2}=1.0818\;\Rightarrow\;y_1'=1.0098\ \text{m}$$

Rise in upstream depth $=1.0098-1.000=0.0098$ m (about 10 mm).

Answer: (a) not critical ($y=0.74$ m over hump); (b) not critical ($y=0.96$ m); (c) critical depth $0.588$ m occurs; upstream depth rises from 1.000 m to 1.010 m, about 10 mm.