---
title: "Contraction and bed raise, 16 m³/s"
subject: "Hydraulics"
chapter: 7
key: "semester_4/hydraulics:contraction-bed-raised-16-m3s-surface-change"
source: https://www.bctengineer.com/q/semester_4/hydraulics/contraction-bed-raised-16-m3s-surface-change
site: BCT Engineering
---

# Contraction and bed raise, 16 m³/s

Hydraulics · Chapter 7: Energy and Momentum Principles in Open channel flow

- Asked in: 2068 Magh, 6 marks
- Video: [Open Channel Flow-20 | Channel Width Reduction | Shubham Sarathe](https://www.youtube.com/watch?v=V123w0PH0I8) — EIQ
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/hydraulics/contraction-bed-raised-16-m3s-surface-change
- Whole chapter: https://www.bctengineer.com/md/semester_4/hydraulics/chapter-7

## Question

A discharge of 16 m³/s flows with depth of 2 m in a 4 m wide rectangular channel. At a downstream section, the width is reduced to 3.5 m and the channel bed is raised by 0.35 m. To what extent will the surface elevation be affected by these changes?

## Short answer

The flow chokes because the approach energy (2.204 m) is less than the required E_c + Δz (2.280 m). Upstream depth rises by 0.09 m to 2.09 m. At the contracted, raised section, critical depth 1.29 m occurs; the water surface drops 0.36 m below the original level.

## Answer

**Given:** $Q = 16\ \text{m}^3/\text{s}$, $b_1 = 4\ \text{m}$, $y_1 = 2\ \text{m}$; at the downstream section $b_2 = 3.5\ \text{m}$ and the bed is raised by $\Delta z = 0.35\ \text{m}$.

**Upstream:**

$$
V_1 = \frac{16}{4 \times 2} = 2\ \text{m/s},\quad Fr_1 = \frac{2}{\sqrt{9.81 \times 2}} = 0.45,\quad E_1 = 2 + \frac{2^2}{2g} = 2.204\ \text{m}
$$

**At the contracted, raised section (check for choking):**

$$
q_2 = \frac{16}{3.5} = 4.571\ \text{m}^2/\text{s},\quad y_c = \left(\frac{4.571^2}{9.81}\right)^{1/3} = 1.287\ \text{m},\quad E_c = 1.930\ \text{m}
$$

Minimum upstream energy needed $= E_c + \Delta z = 1.930 + 0.35 = 2.280\ \text{m}$.

Since $2.280 > E_1 = 2.204\ \text{m}$, the approach flow has too little energy. **The flow is choked**: critical depth forms at the section and the upstream depth must rise.

**New upstream depth:**

$$
y_1' + \frac{(16/4)^2}{2g\,y_1'^2} = 2.280 \;\Rightarrow\; y_1' = 2.094\ \text{m}
$$

**Effect on the water surface:**

| Location | Change |
|---|---|
| Upstream of the contraction | Surface rises by $2.094 - 2.000 = 0.094\ \text{m}$ (backwater) |
| At the contracted, raised section | Depth $= y_c = 1.287\ \text{m}$; surface is $0.35 + 1.287 = 1.637\ \text{m}$ above the old bed level, i.e. 0.363 m below the original surface |

**Answer:** The surface is affected: it rises about **0.09 m** upstream (to 2.09 m depth) and drops to critical depth 1.29 m over the raised, narrowed section.

## Check yourself

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

1. How is it decided that the flow is choked?
   A. Fr1 = 0.45 is below 1
   B. yc exceeds y1
   C. The minimum upstream energy Ec + dz = 2.280 m exceeds E1 = 2.204 m
   D. The width ratio is less than 0.9
   Answer: C. The approach flow lacks the energy to pass the raised, narrowed section, so it chokes. (Key point: Choking test)

2. At the narrowed section, yc is computed from:
   A. q1 = 4 m^2/s
   B. q2 = 16/3.5 = 4.571 m^2/s, giving yc = 1.287 m
   C. V1 = 2 m/s
   D. E1/2
   Answer: B. The contracted width raises q to 4.571 m^2/s, giving yc = (q^2/g)^(1/3) = 1.287 m. (Key point: Critical depth at throat)

3. What equation fixes the new upstream depth y1'?
   A. y1' = yc + dz
   B. y1' = 2 yc
   C. Specific energy at upstream with q1 = 4 m^2/s equals 2.280 m
   D. y1' = E1 + dz
   Answer: C. Upstream energy must rise to Ec + dz = 2.280 m using q1 = 16/4, giving y1' = 2.094 m. (Key point: New upstream depth)

4. What is the overall effect on the water surface?
   A. Surface falls 0.09 m upstream
   B. No effect anywhere
   C. Surface rises about 0.09 m upstream and depth over the section is critical at 1.29 m
   D. Surface rises 0.35 m everywhere
   Answer: C. Backwater of 2.094 - 2.000 = 0.094 m forms upstream, and the throat flows at yc = 1.287 m. (Key point: Surface effect)

## Related in Hydraulics

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