---
title: "Partial profile with possible jump at A"
subject: "Hydraulics"
chapter: 8
key: "semester_4/hydraulics:hydraulic-jump-partial-profile-5m3s-point-a"
source: https://www.bctengineer.com/q/semester_4/hydraulics/hydraulic-jump-partial-profile-5m3s-point-a
site: BCT Engineering
---

# Partial profile with possible jump at A

Hydraulics · Chapter 8: Non-uniform gradually varied flow (GVF)

- Asked in: 2073 Bhadra, 5+5 marks
- Video: [Hydraulic Jump | Open Channel Flow | Hydraulics and Fluid Mechanics](https://www.youtube.com/watch?v=YqYmAf0NRPE) — The Content Paradox
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/hydraulics/hydraulic-jump-partial-profile-5m3s-point-a
- Whole chapter: https://www.bctengineer.com/md/semester_4/hydraulics/chapter-8

## Question

The partial water surface profile shown in figure below is for a rectangular channel of 3 m width in which water is flowing at a discharge of 5 m³/sec. a) Does a hydraulic jump occur in a channel? If so, is it located upstream or downstream at point A? b) Draw and name water surface profile. [Figure: a steep reach with flow depth 0.4 m joining a horizontal reach at point A; a sluice gate at the far end of the horizontal reach with 1.6 m depth]

## Short answer

Yes, a hydraulic jump occurs upstream of point A on the steep reach. The water surface profile consists of uniform supercritical flow at 0.4 m, a hydraulic jump to 1.007 m, an S1 curve rising to about 1.6 m at A, then an H2 curve on the horizontal reach to the gate.

## Answer

**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.

```text
 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.

## Check yourself

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

1. What sequent depth does 0.4 m supercritical flow (Fr = 2.10) require?
   A. 0.4 m
   B. 1.007 m
   C. 1.6 m
   D. 0.657 m
   Answer: B. y2 = (0.4/2)(sqrt(1+8Fr^2) - 1) = 1.007 m. (Key point: Sequent depth)

2. Why does the jump occur upstream of A on the steep reach?
   A. Tailwater (at least 1.6 m) exceeds the required sequent depth 1.007 m, pushing the jump upstream
   B. Tailwater is less than 1.007 m
   C. y = 0.4 m is subcritical
   D. Horizontal bed forces a jump at A
   Answer: A. Excess tailwater drowns the jump on the horizontal bed and moves it up the steep slope. (Key point: Jump location)

3. Which profile follows the jump on the steep reach?
   A. S2
   B. S1, rising to the depth at A
   C. S3
   D. M1
   Answer: B. Subcritical depth above yc on a steep slope is S1. (Key point: Profile after jump)

4. What profile applies on the horizontal reach?
   A. H3
   B. H1
   C. H2 with depth falling slightly in flow direction
   D. M2
   Answer: C. Subcritical flow on a horizontal bed with y > yc is H2; there is no zone 1 on a horizontal bed. (Key point: Horizontal reach)

## Related in Hydraulics

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