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GATE 2023 CE (CE1) – Question 43

Hydraulics · Gradually varied flow and water surface profiles · 2 marks · Multiple choice

A very wide rectangular channel carries a discharge (Q) of 70 m$^3$/s per meter width. Its bed slope changes from 0.0001 to 0.0009 at a point P, as shown in the figure (not to scale). The Manning's roughness coefficient of the channel is 0.01. What water surface profile(s) exist(s) near the point P?

a channel profile with the bed slope $S_o = 0.0001$ upstream of P and a steeper slope $S_o = 0.0009$ downstream, with the flow from left to right.
  1. M2 and S2
  2. M2 only
  3. S2 only
  4. S2 and hydraulic jump

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Show answer and explanation

Correct answer: (A) M2 and S2

Explanation

For a very wide channel $q = \frac{1}{n}y^{5/3}S^{1/2}$. The critical depth is $y_c = \left(\frac{70^2}{9.81}\right)^{1/3} = 7.94$ m. Upstream: $y_n^{5/3} = \frac{70 \times 0.01}{0.01} = 70$, so $y_n = 12.9$ m $> y_c$ (a mild slope). Downstream: $y_n^{5/3} = \frac{70 \times 0.01}{0.03} = 23.3$, so $y_n = 6.6$ m $< y_c$ (a steep slope). The flow passes through critical depth near P: upstream the depth falls from $y_n$ to $y_c$, giving an M2 curve, and downstream it falls from $y_c$ towards $y_n$, giving an S2 curve.