GATE 2019 CE (CE2) – Question 58
At a spillway foot water depth is 23 cm and speed 8.1 m/s. Flow enters a long wide rectangular channel as an M3 profile, with bed slope 1/1800 and Manning n = 0.015. A hydraulic jump occurs downstream. Take g = 9.81 m/s². Find depth y1 just before the jump (m), to two decimal places.
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Correct answer: .41 to .43
Explanation
Unit discharge is $q=0.23(8.1)=1.863$ m²/s. The long downstream subcritical reach is taken at normal depth. Wide-channel Manning gives $y_n=[qn/\sqrt S]^{3/5}=1.1076$ m. Conjugate depths conserve specific force: $$\frac{y_1^2}2+\frac{q^2}{gy_1}=\frac{y_n^2}2+\frac{q^2}{gy_n}.$$ The smaller positive root is **y₁ ≈ 0.42 m**. This keyed model assumes normal depth downstream; the 0.23 m spillway-foot depth increases along the M3 profile before the jump.