Partial profile with possible jump at A

Given: $b = 3\ \text{m}$, $Q = 5\ \text{m}^3/\text{s}$ ($q = 1.667\ \text{m}^2/\text{s}$). A steep reach carries flow at $y = 0.4$ m (taken as the uniform supercritical depth) and joins a horizontal reach at A. At the end of the horizontal reach a sluice gate holds a depth of 1.6 m.

Supercritical flow

$$V_1 = \frac{1.667}{0.4} = 4.167\ \text{m/s},\qquad Fr_1 = \frac{4.167}{\sqrt{9.81 \times 0.4}} = 2.10$$

Sequent (conjugate) depth

$$y_2 = \frac{y_1}{2}\left(\sqrt{1 + 8Fr_1^2} - 1\right) = \frac{0.4}{2}\left(\sqrt{1 + 8(4.424)} - 1\right) = 1.007\ \text{m}$$

a) Does a jump occur, and where?

The critical depth is $y_c = (1.667^2/9.81)^{1/3} = 0.657$ m, so flow at 0.4 m is supercritical and the flow behind the gate (1.6 m) is subcritical. A jump must occur between them.

On a horizontal bed the subcritical depth does not decrease upstream from the gate (an H2 curve, which falls only slightly downstream because of friction), so the depth at A is at least 1.6 m. This is greater than the sequent depth required by the 0.4 m flow (1.007 m). The tailwater is too deep to allow the jump to stay on the horizontal bed, so it moves upstream.

Yes, a hydraulic jump occurs, located upstream of A, on the steep reach.

b) Water surface profile

1. Steep reach, far upstream: uniform supercritical flow at $y_n = 0.4$ m.
2. A hydraulic jump on the steep reach, taking the depth from 0.4 m to 1.007 m.
3. After the jump, an S1 curve (depth above $y_c$ on a steep slope) rising to the depth at A (about 1.6 m).
4. Horizontal reach: an H2 curve (subcritical, $y > y_c$), whose depth decreases slightly in the flow direction to 1.6 m at the gate.

 STEEP                      A   HORIZONTAL
 ---- yn=0.4 ----.    J S1 _.___ H2 ___ 1.6 m |gate
                  '--|--'''        ''-----   | |
                                             |_|

Answer: (a) The jump occurs on the steep reach, upstream of A (0.4 m to 1.007 m). (b) Uniform flow, jump, S1, then H2.