---
title: "Jump at spillway toe; repelled and submerged jump"
subject: "Hydraulics"
chapter: 9
key: "semester_4/hydraulics:jump-at-toe-of-spillway-repelled-submerged"
source: https://www.bctengineer.com/q/semester_4/hydraulics/jump-at-toe-of-spillway-repelled-submerged
site: BCT Engineering
---

# Jump at spillway toe; repelled and submerged jump

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

- Asked in: 2068 Bhadra, 6 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/hydraulics/jump-at-toe-of-spillway-repelled-submerged
- Whole chapter: https://www.bctengineer.com/md/semester_4/hydraulics/chapter-9

## Question

Find the pre jump and post jump heights of the hydraulic jump formed at the toe of the spillway. Neglect energy loss due to flow over spillway. Height of the crest above D/S bed level = 3 m; Discharge = 80 m³/s; Width of the canal = 10.0 m; Head over the crest level = 2.47 m. Explain the formation condition of repelled and submerged jump for the above flow condition.

## Short answer

Energy from reservoir (5.47 m above bed) gives toe depth 0.84 m, sequent depth 3.55 m. Repelled jump if tailwater < 3.55 m (swept downstream); submerged jump if tailwater > 3.55 m (drowned at toe).

## Answer

**Given:** $Q = 80\ \text{m}^3/\text{s}$, $b = 10$ m ($q = 8\ \text{m}^2/\text{s}$), crest 3 m above the downstream bed, head over crest $H = 2.47$ m. No energy loss on the spillway.

#### Pre-jump depth $y_1$

Total energy above the downstream bed:

$$
E_0 = 3 + 2.47 = 5.47\ \text{m}
$$

At the toe, $E_1 = E_0$:

$$
y_1 + \frac{q^2}{2g\,y_1^2} = 5.47 \;\Rightarrow\; y_1 + \frac{3.262}{y_1^2} = 5.47 \;\Rightarrow\; y_1 = 0.839\ \text{m}
$$

$$
V_1 = \frac{8}{0.839} = 9.53\ \text{m/s},\qquad Fr_1 = \frac{9.53}{\sqrt{9.81 \times 0.839}} = 3.32
$$

#### Post-jump (sequent) depth $y_2$

$$
y_2 = \frac{0.839}{2}\left(\sqrt{1 + 8(3.32)^2} - 1\right) = 3.55\ \text{m}
$$

The energy loss in the jump is $\dfrac{(y_2 - y_1)^3}{4y_1y_2} = 1.67$ m. The critical depth is $y_c = 1.87$ m.

**Pre-jump depth = 0.84 m; post-jump depth = 3.55 m.**

#### Repelled and submerged jumps

Compare the actual tailwater depth $y_t$ in the river with the sequent depth $y_2 = 3.55$ m:

| Condition | Type of jump | What happens |
|---|---|---|
| $y_t = 3.55$ m | Jump at the toe (ideal) | the jump forms right at the foot of the spillway |
| $y_t < 3.55$ m | **Repelled jump** | tailwater too low; the jump is swept downstream of the toe, with high-velocity flow over the apron and risk of scour |
| $y_t > 3.55$ m | **Submerged (drowned) jump** | tailwater too high; the jump is pushed up against the toe and the jet is drowned, so energy dissipation is poor and the spillway toe is covered |

For this flow, a repelled jump forms if the tailwater is below 3.55 m, and a submerged jump if it is above 3.55 m. In a stilling basin the tailwater is usually kept slightly above $y_2$ (by about 5 to 10%) by depressing the apron, to keep the jump stable on the apron.

## Check yourself

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

1. How is y1 at the toe found?
   A. From the head over the crest only
   B. From critical depth
   C. From E1 = crest height + head = 5.47 m, solved with y1 + q^2/(2g y1^2) = 5.47
   D. From Manning's equation
   Answer: C. No energy is lost on the spillway, giving y1 = 0.84 m. (Key point: Pre-jump depth)

2. How is y2 found?
   A. Belanger equation with y1 = 0.839 m and Fr1 = 3.32, giving 3.55 m
   B. Equal energy at the toe
   C. y2 = yc
   D. y2 = 2 y1
   Answer: A. The jump follows momentum, not energy. (Key point: Post-jump depth)

3. When does a repelled jump form?
   A. Tailwater above y2
   B. Tailwater below the sequent depth y2, so the jump moves downstream of the toe
   C. Tailwater equal to y2
   D. Tailwater equal to yc
   Answer: B. A low tailwater cannot hold the jump at the toe, leaving high velocity over the apron. (Key point: Repelled jump)

4. When does a submerged jump form?
   A. Tailwater below y2
   B. Tailwater equal to y2
   C. Tailwater below yc
   D. Tailwater above the sequent depth y2, which drowns the jump with poor energy dissipation
   Answer: D. High tailwater drowns the jump, so dissipation is poor. (Key point: Submerged jump)

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