Sluice gate profiles and lengths by direct step

Given: $b = 4\ \text{m}$, $S_0 = 0.0008$, $n = 0.014$, $y_1 = 1.85\ \text{m}$ (upstream of gate), $y_2 = 0.35\ \text{m}$ (downstream of gate).

Discharge (energy equation across the gate, no loss):

$$y_1 + \frac{V_1^2}{2g} = y_2 + \frac{V_2^2}{2g} \;\Rightarrow\; Q = b\,y_1 y_2\sqrt{\frac{2g}{y_1 + y_2}} = 4(1.85)(0.35)\sqrt{\frac{19.62}{2.2}} = 7.735\ \text{m}^3/\text{s}$$

Critical and normal depths ($q = 1.934\ \text{m}^2/\text{s}$):

$$y_c = \left(\frac{q^2}{g}\right)^{1/3} = 0.725\ \text{m},\qquad y_n = 1.171\ \text{m}\ (\text{from Manning's equation, } R = A/P)$$

Since $y_n > y_c$, the slope is mild.

i) Profiles

• Upstream of the gate: $y = 1.85\ \text{m} > y_n$, so an M1 (backwater) curve, rising towards the gate.
• Downstream of the gate: $y = 0.35\ \text{m} < y_c$, so an M3 curve rising from the vena contracta. The sequent depth of 0.35 m is 1.311 m, which is greater than $y_n = 1.171$ m, so the jump is not at the gate. The M3 curve continues until its conjugate depth equals $y_n$ (at $y_j = 0.411\ \text{m}$), where a hydraulic jump to $y_n$ forms.

        M1                     gate
  ...........___----''''''''''|G|
  yn=1.171 ..........         | |  M3  J     yn
                              |_|____/'|----------

ii) Lengths by direct step

$$E = y + \frac{V^2}{2g},\quad S_f = \frac{n^2V^2}{R^{4/3}},\quad \Delta x = \frac{E_2 - E_1}{S_0 - \bar S_f}$$

Upstream (M1), $y$: 1.85 to 1.25 m in 3 steps of 0.20 m:

$$L_{up} = 348.3 + 420.7 + 681.3 \approx 1450\ \text{m (upstream of the gate)}$$

Downstream (M3), $y$: 0.35 to 0.411 m in 3 steps:

$$L_{down} = 5.46 + 5.39 + 5.30 \approx 16.1\ \text{m (downstream of the gate, up to the jump)}$$

Answer: M1 upstream (about 1450 m to within 7% of $y_n$) and M3 downstream (about 16 m, then a jump from 0.411 m to 1.171 m).