Vertical drop design, Q 5.1 m3/s

Given data and assumptions

• $Q = 5.1$ m³/s, bed width $B = 2.5$ m, side slope 1:1, u/s bed $= 255.15$ m, d/s bed $= 254.45$ m.
• U/s FSL $=$ bed $+$ normal depth $= 255.15 + 0.95 = 256.10$ m; d/s FSL $= 254.45 + 0.95 = 255.40$ m. Full supply depth $D_u = D_d = 0.950$ m; drop in water level $H_L = 0.700$ m.
• Assumed (not given): Bligh's coefficient $C = 6$ (sandy loam soil), specific gravity of masonry $G = 2.24$, and $C_d = 0.415$ for a rectangular crest.

1. Upstream flow conditions

$$\begin{aligned} A &= (B + mD_u)D_u = (2.5 + 1\times0.950)(0.950) = 3.278\ \text{m}^2 \\ v &= Q/A = 1.556\ \text{m/s}, \qquad h_a = \frac{v^2}{2g} = 0.123\ \text{m} \end{aligned}$$

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, $L = B$:

$$H = \left(\frac{Q}{1.838\,L}\right)^{2/3} = \left(\frac{5.1}{1.838\times2.5}\right)^{2/3} = 1.072\ \text{m}$$ $$\text{Crest level} = FSL_u + h_a - H = 256.100 + 0.123 - 1.072 = 255.15\ \text{m}$$

Height of crest above d/s bed $d_c = 255.15 - 254.45 = 0.70$ m.

3. Cistern (energy dissipation)

Discharge per metre length of crest $q = Q/L = 2.040$ m²/s. Energy available above the d/s bed $E_1 = FSL_u + h_a - \text{d/s bed} = 1.773$ 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 $X = 0.494$ m. Provide $X = 0.50$ m (cistern floor level $= 253.95$ m).
• With this $X$: $y_1 = 0.330$ m, $F_1 = 3.43$, $y_2 = 1.446$ m.

Length of cistern (hydraulic jump length) $L_c = 5(y_2 - y_1) = 5(1.446 - 0.330) = 5.58$ m.

4. Crest (drop) wall

Top width (assumed) $T = 0.50$ m; check $0.55\sqrt{d_c} = 0.55\sqrt{0.70} = 0.46$ m $< 0.50$ m, so it is adequate. Height of wall from cistern floor to crest $H_w = 255.15 - 253.95 = 1.20$ m. Bottom width for a gravity masonry wall:

$$B_w = \frac{H_w + H}{\sqrt{G}} = \frac{1.20 + 1.07}{\sqrt{2.24}} = 1.52\ \text{m}$$

Adopt bottom width $B_w = 1.52$ m (not less than the top width), with the d/s face vertical or slightly battered.

5. Length of impervious floor (Bligh's theory)

Total creep length required $L = C\,H_L = 6\times0.700 = 4.20$ m. Cut-offs: u/s $d_1 = 0.6$ m, d/s $d_2 = 0.9$ m. Downstream floor length (Bligh) with $q$ in m²/s:

$$L_d = 2.21\,C\sqrt{H_L\sqrt{q/75}} = 2.21\times6\sqrt{0.700\sqrt{2.040/75}} = 4.51\ \text{m}$$

The d/s floor (measured from the toe of the wall) must also contain the cistern, so adopt $L_d = 5.58$ m. Creep already provided by cut-offs and wall $= 2(d_1 + d_2) + B_w = 4.52$ m, so the u/s floor needed is

$$b_u = L - [2(d_1+d_2) + B_w + L_d] = -5.90\ \text{m}$$

Bligh's creep is already satisfied, so a nominal u/s floor of 1.0 m is provided. Actual creep $= 11.10$ m $\ge 4.20$ m, safe. Total floor length $= b_u + B_w + L_d = 8.10$ m.

6. Thickness of the floor

Pressure head at the toe of the wall: creep to that point $= 3.72$ m, so residual head $h_1 = H_L\,\dfrac{L_{act} - 3.72}{L_{act}} = 0.465$ m. At the end of the cistern the creep is 9.30 m and $h_2 = 0.114$ m.

$$t_1 = \frac{4}{3}\frac{h_1}{G-1} = \frac{4}{3}\times\frac{0.465}{1.24} = 0.50\ \text{m}$$

Adopt floor thickness 0.50 m under the cistern near the wall, tapering to 0.30 m at the end of the floor (minimum 0.30 m).

Summary

Answer: crest level 255.15 m, crest length 2.50 m, crest wall 0.50 m top and 1.52 m bottom width, cistern depth 0.50 m and length 5.58 m, floor length 8.10 m, floor thickness 0.50 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 __|