Crest width, cistern length of vertical drop, Q 1.55

Given data and assumptions

• $Q = 1.55$ m³/s, bed width $B = 1.1$ m, side slope 1:1, u/s FSL $= 106.00$ m, u/s bed $= 105.07$ m, d/s bed $= 104.32$ m.
• Full supply depth u/s $D_u = 0.929$ m, d/s $D_d = 0.929$ m (d/s FSL $= 105.247$ m); drop in water level $H_L = 0.750$ m.
• Bligh's coefficient $C = 7$; specific gravity of masonry/concrete $G = 2$.

1. Upstream flow conditions

$$\begin{aligned} A &= (B + mD_u)D_u = (1.1 + 1\times0.929)(0.929) = 1.885\ \text{m}^2 \\ v &= Q/A = 0.822\ \text{m/s}, \qquad h_a = \frac{v^2}{2g} = 0.034\ \text{m} \end{aligned}$$

2. Crest length, head and crest level

Discharge over a rectangular crest: $Q = C_d\sqrt{2g}\,L\,H^{3/2}$ with $C_d\sqrt{2g} = 1.838$, where $H$ is the total head over the crest (including $h_a$). Take crest length equal to the bed width, $L = B$:

$$H = \left(\frac{Q}{1.838\,L}\right)^{2/3} = \left(\frac{1.55}{1.838\times1.1}\right)^{2/3} = 0.838\ \text{m}$$ $$\text{Crest level} = FSL_u + h_a - H = 105.997 + 0.034 - 0.838 = 105.19\ \text{m}$$

Height of crest above d/s bed $d_c = 105.19 - 104.32 = 0.88$ m.

3. Cistern (energy dissipation)

Discharge per metre length of crest $q = Q/L = 1.409$ m²/s. Energy available above the d/s bed $E_1 = FSL_u + h_a - \text{d/s bed} = 1.713$ m. For a cistern of depth $X$ below the d/s bed, the jump must form with the conjugate depth $y_2$ equal to the tail water over the cistern, $y_2 = D_d + X$. Solving with $E_1 + X = y_1 + q^2/(2gy_1^2)$ and $y_2 = \frac{y_1}{2}(\sqrt{1+8F_1^2} - 1)$:

• Required $X = 0.247$ m. Provide $X = 0.25$ m (cistern floor level $= 104.07$ m).
• With this $X$: $y_1 = 0.242$ m, $F_1 = 3.77$, $y_2 = 1.176$ m.

Length of cistern (hydraulic jump length) $L_c = 5(y_2 - y_1) = 5(1.176 - 0.242) = 4.67$ m.

4. Crest (drop) wall

Top width (assumed) $T = 0.50$ m. Height of wall from cistern floor to crest $H_w = 105.19 - 104.07 = 1.13$ m. Bottom width for a gravity masonry wall:

$$B_w = \frac{H_w + H}{\sqrt{G}} = \frac{1.13 + 0.84}{\sqrt{2}} = 1.39\ \text{m}$$

Adopt bottom width $B_w = 1.39$ m (not less than the top width), with the d/s face vertical or slightly battered.

Summary

Answer: crest level 105.19 m, crest length 1.10 m, crest wall 0.50 m top and 1.39 m bottom width, cistern depth 0.25 m and length 4.67 m.

Sketch:

 FSL u/s ~~~~~~~~~~~|
                    |  crest
 u/s bed ___________|__          FSL d/s ~~~~
        |  u/s floor |  \  drop        ~~~~~
     d1 |            |   \_____cistern________ d/s bed
        |            |<-Lc->|   d/s floor   | d2
                     |____ floor thickness __|