---
title: "Verify jump downstream of sluice gate"
subject: "Hydraulics"
chapter: 9
key: "semester_4/hydraulics:sluice-gate-opening-jump-verify-5m-channel"
source: https://www.bctengineer.com/q/semester_4/hydraulics/sluice-gate-opening-jump-verify-5m-channel
site: BCT Engineering
---

# Verify jump downstream of sluice gate

Hydraulics · Chapter 9: Non-uniform rapidly varied flow (RVF)

- Asked in: 2068 Magh, 6 marks
- Video: [Numerical Problems on Hydraulic Jump](https://www.youtube.com/watch?v=nRCQwCMQQpY) — Vishwanatha Bhat
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/hydraulics/sluice-gate-opening-jump-verify-5m-channel
- Whole chapter: https://www.bctengineer.com/md/semester_4/hydraulics/chapter-9

## Question

A vertical sluice gate with an opening 0.67 m produces a downstream jet depth of 0.4 m when installed in a long rectangular channel 5 m wide conveying a steady discharge of 20 m³/s. Assuming that the flow downstream of the gate eventually returns to the uniform flow depth of 2.5 m, a) Verify that a hydraulic jump occurs. Assume $\alpha = \beta = 1$. b) If the downstream depth is increased to 3 m, analyze the flow conditions at the gate.

## Short answer

a) Jet depth 0.4 m, Fr=5.05; jump required. Sequent 2.66 m > tailwater 2.5 m, so jump forms just downstream. b) With tailwater 3 m > 2.66 m, submerged jump covers gate; upstream depth rises from 5.47 m to ~6.5 m.

## Answer

**Given:** $b = 5$ m, $Q = 20\ \text{m}^3/\text{s}$ ($q = 4\ \text{m}^2/\text{s}$), gate opening $a = 0.67$ m, jet depth $y_2 = 0.4$ m (so $C_c = 0.4/0.67 = 0.597$), downstream uniform depth $y_t = 2.5$ m, $\alpha = \beta = 1$.

**Upstream depth (free flow).** Energy equation across the gate:

$$
V_2 = \frac{4}{0.4} = 10\ \text{m/s},\quad E = 0.4 + \frac{10^2}{2g} = 5.497\ \text{m}
$$

$$
y_1 + \frac{(4/y_1)^2}{2g} = 5.497 \;\Rightarrow\; y_1 = 5.47\ \text{m}
$$

#### a) Does a jump occur?

The jet leaving the gate is supercritical:

$$
Fr_2 = \frac{10}{\sqrt{9.81 \times 0.4}} = 5.05
$$

The downstream uniform flow ($y_t = 2.5$ m, $V = 1.6$ m/s) is subcritical ($Fr = 0.32$). Flow must change from supercritical to subcritical, which is possible only through a **hydraulic jump**, so a jump occurs.

Its position depends on the sequent depth of the jet:

$$
y_2' = \frac{0.4}{2}\left(\sqrt{1 + 8(5.05)^2} - 1\right) = 2.66\ \text{m}
$$

Since the tailwater $2.5\ \text{m} < 2.66$ m, the tailwater is slightly too low, so the jump does not form at the vena contracta. It is swept a short distance **downstream** of the gate. The supercritical depth increases gradually (M3 type) until its sequent depth equals 2.5 m, and the jump then forms. The jet at the gate remains a free jet at 0.4 m, and the discharge equation for free flow applies.

#### b) Downstream depth increased to 3 m

Now $y_t = 3\ \text{m} > y_2' = 2.66$ m. The tailwater is too high, so the jump moves upstream, reaches the gate, and becomes a **submerged (drowned) jump** that covers the jet. The percent submergence is

$$
\frac{3 - 2.66}{2.66} = 12.7\%
$$

The gate is therefore **submerged**: the flow at the gate is no longer free, and the upstream depth rises. An approximate analysis keeps the jet velocity of 10 m/s and finds the effective pressure depth $y_x$ at the jet section from the momentum equation to the downstream section ($y_t = 3$ m, $V_t = 1.333$ m/s):

$$
\frac{\gamma b y_x^2}{2} = \frac{\gamma b y_t^2}{2} + \rho Q(V_t - V_2) = 220\,725 - 173\,333 = 47\,392\ \text{N}
$$

$$
y_x = 1.39\ \text{m}
$$

Energy from upstream to the jet section (with the pressure depth $y_x$):

$$
y_1' + \frac{(4/y_1')^2}{2g} = 1.39 + 5.097 = 6.49\ \text{m} \;\Rightarrow\; y_1' = 6.47\ \text{m}
$$

**Answer: (a) A jump occurs since $Fr_2 = 5.05$ and the downstream flow is subcritical; because 2.5 m < 2.66 m it forms just downstream of the gate. (b) With 3 m tailwater the jump is submerged against the gate, and the upstream depth rises from about 5.47 m to about 6.5 m (approximate analysis).**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Why does a hydraulic jump occur downstream of the gate in part (a)?
   A. The jet leaving the gate is supercritical (Fr = 5.05) but the downstream uniform flow is subcritical (Fr = 0.32)
   B. The upstream depth of 5.47 m is greater than the tailwater depth of 2.5 m
   C. The jet velocity of 10 m/s is greater than the downstream velocity of 1.6 m/s, which alone forces a jump
   D. The discharge per metre width is greater than the critical discharge of the channel
   Answer: A. Flow can change from supercritical to subcritical only through a hydraulic jump, and here the jet has Fr 5.05 while the tailwater flow has Fr 0.32. (Key point: Why a jump occurs)

2. The sequent depth of the 0.4 m jet with Fr2 = 5.05 is 2.66 m. What does this value represent?
   A. The critical depth of the channel
   B. The depth needed downstream for a jump to form right at the vena contracta
   C. The upstream depth just before the gate
   D. The uniform flow depth of the channel
   Answer: B. The sequent depth is the conjugate depth of the supercritical jet; if the tailwater equals it, the jump forms right at the jet section. (Key point: Sequent depth)

3. The tailwater is 2.5 m while the sequent depth is 2.66 m. Where does the jump form?
   A. Upstream of the gate, drowning it
   B. At the vena contracta exactly
   C. A short distance downstream of the gate, after the supercritical depth rises gradually
   D. No jump forms at all
   Answer: C. Tailwater is slightly lower than the sequent depth, so the jump is swept downstream until the sequent depth of the rising supercritical flow equals 2.5 m. (Key point: Jump location for 2.5 m tailwater)

4. When the downstream depth is raised to 3 m (more than 2.66 m), what happens at the gate?
   A. The jump moves downstream and the gate stays free
   B. The jump becomes submerged against the gate, so the gate flow is no longer free and the upstream depth rises
   C. The flow becomes critical at the gate
   D. Nothing changes because the jet depth stays at 0.4 m
   Answer: B. A tailwater higher than the sequent depth pushes the jump upstream until it drowns the jet, making the gate submerged and raising the upstream depth (about 5.47 m to 6.5 m). (Key point: Effect of 3 m tailwater)

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