Contraction and bed rise, 3.5 m channel (1 m depth)

Given data

Upstream $b_1=3.5$ m, $y_1=1$ m, $Q=4$ m³/s; contracted $b_2=2.5$ m with bed raised $\Delta z=0.10$ m.

$$q_1=\frac{4}{3.5}=1.1429\ \text{m}^2/\text{s},\quad V_1=1.143\ \text{m/s},\quad E_1=1+\frac{1.143^2}{19.62}=1.0666\ \text{m}$$ $$q_2=\frac{4}{2.5}=1.6\ \text{m}^2/\text{s},\quad y_{c2}=\sqrt[3]{\frac{1.6^2}{9.81}}=0.6390\ \text{m},\quad E_{min2}=0.9586\ \text{m}$$

Depth in the contracted section

$$E_2=E_1-\Delta z=1.0666-0.10=0.9666\ \text{m}>E_{min2}=0.9586\ \text{m}$$

Flow is possible without choking; the flow remains subcritical:

$$y_2+\frac{1.6^2}{19.62y_2^2}=y_2+\frac{0.13048}{y_2^2}=0.9666\;\Rightarrow\;y_2=0.701\ \text{m}$$

Maximum rise in bed without affecting the upstream depth

The limit is when the contracted section has critical flow with $E_2=E_{min2}$:

$$\Delta z_{max}=E_1-E_{min2}=1.0666-0.9586=0.1080\ \text{m}\approx10.8\ \text{cm}$$

Answer: depth in the contracted section $=0.70$ m; maximum bed rise without affecting the upstream depth $=0.108$ m (about 10.8 cm).