Weir design by Bligh's theory, 3200 m3/s

Data and assumptions

$Q = 3200$ m³/s, HFL before construction $= 185.0$ m, d/s bed $= 178.0$ m, canal FSL $= 184.0$ m, afflux $= 1.0$ m, $C = 12$, $f = 1.0$, safe exit gradient 1/6, regulator loss $= 0.5$ m, weir coefficient $1.7$, retrogression $0.5$ m. The u/s bed is assumed equal to the d/s bed (178.0 m).

1. Length of waterway (Lacey)

$$P = 4.75\sqrt{Q} = 4.75\sqrt{3200} = 268.7\ \text{m} \Rightarrow \text{adopt } L = 270\ \text{m}$$

2. Levels and crest

• Pond level $=$ FSL $+$ regulator loss $= 184.0 + 0.5 = 184.5$ m.
• HFL after construction $= 185.0 + 1.0 = 186.0$ m.
• Discharge per metre $q = 3200/270 = 11.85$ m³/s/m.
• Head over the crest from $Q = 1.7\,L\,H^{3/2}$:

$$H = \left(\frac{3200}{1.7\times 270}\right)^{2/3} = 3.65\ \text{m}$$

• Crest level (open flood) $= 186.0 - 3.65 = 182.35$ m. The pond level of 184.5 m is held by shutters or gates of height $184.5 - 182.35 = 2.15$ m above this crest.

3. Scour depth and pile depths

$$R = 1.35\left(\frac{q^2}{f}\right)^{1/3} = 1.35\left(\frac{11.85^2}{1.0}\right)^{1/3} = 7.02\ \text{m}$$

• U/s pile bottom $= 186.0 - 1.5R = 175.47$ m, so depth below bed $d_1 = 178.0 - 175.47 = 2.53$ m, adopt 2.5 m.
• D/s pile bottom $= 185.0 - 2R = 170.96$ m. The bed after retrogression is $178.0 - 0.5 = 177.5$ m, so $d_2 = 177.5 - 170.96 = 6.54$ m, adopt 6.5 m.

4. Length of impervious floor (Bligh)

Maximum static head $H_s = 184.5 - 177.5 = 7.0$ m. Total creep length

$$L_c = C H_s = 12\times 7.0 = 84\ \text{m}$$ $$L_c = 2d_1 + 2d_2 + b \Rightarrow b = 84 - 2(2.53 + 6.54) = 65.9\ \text{m}$$

Adopt a horizontal floor length $b \approx 66$ m between the piles.

5. Exit gradient check

$$\alpha = \frac{b}{d_2} = \frac{65.9}{6.54} = 10.08,\quad \lambda = 5.565,\quad G_E = \frac{H_s}{d_2}\cdot\frac{1}{\pi\sqrt{\lambda}} = \frac{7.0}{6.54}\times\frac{1}{\pi\sqrt{5.565}} = 0.145$$

$G_E = 0.145 < 1/6 = 0.167$, so the floor is safe against piping.

Answer: Waterway $\approx 270$ m; crest level $\approx 182.35$ m (shutters to 184.5 m); u/s pile $\approx 2.5$ m deep; d/s pile $\approx 6.5$ m deep; impervious floor $\approx 66$ m between piles (total creep 84 m); exit gradient $= 0.145$ (safe).