Slope change 1/120 to 1/1500, jump and profile length

Given: wide rectangular channel, $q = 3.8\ \text{m}^2/\text{s}$, $n = 0.012$, $S_{01} = 1/120 = 0.00833$, $S_{02} = 1/1500 = 0.000667$. For a wide channel $R \approx y$.

Normal and critical depths

$$y_n = \left(\frac{nq}{\sqrt{S_0}}\right)^{3/5},\qquad y_c = \left(\frac{q^2}{g}\right)^{1/3} = \left(\frac{3.8^2}{9.81}\right)^{1/3} = 1.138\ \text{m}$$ $$y_{n1} = \left(\frac{0.012 \times 3.8}{\sqrt{0.00833}}\right)^{0.6} = 0.659\ \text{m}\ (< y_c: \text{steep})$$ $$y_{n2} = \left(\frac{0.012 \times 3.8}{\sqrt{0.000667}}\right)^{0.6} = 1.407\ \text{m}\ (> y_c: \text{mild})$$

Sketch

The upstream slope is steep (supercritical, $y_{n1} = 0.659$ m) and the downstream slope is mild (subcritical, $y_{n2} = 1.407$ m). Supercritical flow must change to subcritical through a hydraulic jump, with an M3 curve before it (see the jump location below).

 STEEP              |  MILD
 ---- CDL 1.138 - - | - - - - - - - - - - - -
                    |            J  _____ yn2=1.407
 ______ yn1=0.659 __|__M3_____/'''|

Does a jump occur, and where?

Sequent depth of $y_{n1}$:

$$Fr_1 = \frac{3.8/0.659}{\sqrt{9.81 \times 0.659}} = 2.27,\qquad y_2 = \frac{y_{n1}}{2}\left(\sqrt{1 + 8Fr_1^2} - 1\right) = 1.809\ \text{m}$$

Since $y_2 = 1.809\ \text{m} > y_{n2} = 1.407\ \text{m}$, the downstream depth is too low to force a jump at the break. A jump does occur, on the downstream (mild) slope. Flow follows an M3 curve from $y_{n1} = 0.659$ m upwards until its conjugate depth equals $y_{n2}$:

$$y_j:\ \frac{y_j}{2}\left(\sqrt{1 + 8\frac{q^2}{g y_j^3}} - 1\right) = 1.407\ \Rightarrow\ y_j = 0.905\ \text{m}$$

The jump takes the depth from 0.905 m to 1.407 m.

Length of the M3 profile (direct step, 2 steps, mild slope $S_0 = 1/1500$)

$$E = y + \frac{V^2}{2g},\quad S_f = \frac{n^2q^2}{y^{10/3}},\quad \Delta x = \frac{E_2 - E_1}{S_0 - \bar S_f}$$

Steps: $y = 0.659 \to 0.782 \to 0.905$ m.

$$L = 62.7 + 57.8 = 120.5\ \text{m}$$

Answer: A jump occurs on the downstream (mild) slope, from 0.905 m to 1.407 m. The M3 profile from the break to the jump is about 120 m long.