Three successive slopes: profiles and length

Given: $b = 7\ \text{m}$, $Q = 8\ \text{m}^3/\text{s}$ ($q = 1.143\ \text{m}^2/\text{s}$). Slope 1: $S_{01} = 0.00032$, $n_1 = 0.011$. Slope 2: $S_{02} = 0.0911$, $n_2 = 0.021$. Slope 3: $S_{03} = 0.0002$, with $y = 1.2$ m at its downstream end. $n_3$ is not given, so $n_3 = n_1 = 0.011$ is assumed.

Critical and normal depths

$$y_c = \left(\frac{1.143^2}{9.81}\right)^{1/3} = 0.511\ \text{m}$$

i) Longitudinal profiles

• Reach 1 (mild): the flow upstream is uniform at 0.886 m. At the break to the steep slope the flow passes through critical depth, so the surface falls as an M2 curve from $y_n$ to $y_c$ at the end of the reach.
• Reach 2 (steep): the flow starts at $y_c$ and falls as an S2 curve to the uniform supercritical depth of $0.224$ m.
• Reach 3 (mild): the depth at the downstream end is $1.2\ \text{m} > y_{n3}$, so an M1 backwater curve rises downstream from near $y_{n3} = 1.033$ m to 1.2 m.
• Jump: the sequent depth of the supercritical flow (0.224 m) is $0.983$ m, which is less than the depth at the head of reach 3 ($\approx 1.033$ m). The tailwater is too deep for the jump to form on the mild reach, so it is pushed up onto the steep reach 2. After the jump an S1 curve rises from about 0.98 m to about 1.03 m at the break. Because $S_{02}$ is very steep, this S1 curve is short.

 R1 (mild)     |  R2 (steep)          | R3 (mild)
 -- yn=0.886   |                      |            1.2 m
   \__ M2      |\ S2          J  S1   |  M1  _____---
 yc  '-.. - - -|  '-._ yn=0.224 /'''''|__---'

ii) Length of the M2 profile in reach 1

The M2 curve approaches $y_n$ only asymptotically, so it is taken from $0.98\,y_{n1} = 0.868\ \text{m}$ to $y_c = 0.511$ m in 3 equal steps (the control is at the downstream end).

$$\Delta x = \frac{E_2 - E_1}{S_0 - \bar S_f},\qquad S_f = \frac{n^2V^2}{R^{4/3}}$$ $$L = 748.6 + 171.2 + 30.7 \approx 950\ \text{m}$$

Answer: Profile sequence: M2 (reach 1), S2, hydraulic jump and S1 (reach 2), M1 (reach 3). The M2 profile in slope 1 is about 950 m long (measured from $0.98\,y_n$ to $y_c$, in the flow direction).