---
title: "Critical flow conditions and 3 m to 2 m contraction"
subject: "Hydraulics"
chapter: 7
key: "semester_4/hydraulics:contraction-hump-conditions-critical-flow-and-bed-rise"
source: https://www.bctengineer.com/q/semester_4/hydraulics/contraction-hump-conditions-critical-flow-and-bed-rise
site: BCT Engineering
---

# Critical flow conditions and 3 m to 2 m contraction

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

- Asked in: 2070 Magh, 3+3+3 marks
- Video: [Critical flow condition in open channel hydraulics](https://www.youtube.com/watch?v=VF4CAetGaGc) — HydroMohsen
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/hydraulics/contraction-hump-conditions-critical-flow-and-bed-rise
- Whole chapter: https://www.bctengineer.com/md/semester_4/hydraulics/chapter-7

## Question

A 3 m wide rectangular channel carries 3 m³/s of water at a depth of 1 m. If the width is to be reduced to 2 m and bed raised by 10 cm, what would be the depth of flow in the contracted section? What maximum rise in the bed level of the contracted section is possible without affecting the depth of flow upstream of transition? Neglect loss of energy in transition. What would be the change in water surface elevations if the rise in bed is 30 cm?

## Short answer

(a) With 10 cm rise, contracted depth y2 = 0.74 m. (b) Maximum bed rise without affecting upstream depth is 0.133 m (13.3 cm). (c) For 30 cm rise, flow chokes, upstream depth rises to 1.18 m, critical depth 0.612 m occurs, and the water surface at the contraction is 0.912 m.

## Answer

#### Given data
$b_1=3$ m, $Q=3$ m³/s, $y_1=1$ m. Contracted width $b_2=2$ m. $q_1=1$ m²/s, $q_2=1.5$ m²/s.
$$
E_1=1+\frac{1^2}{19.62}=1.0510\ \text{m},\qquad y_{c2}=\sqrt[3]{\frac{1.5^2}{9.81}}=0.6121\ \text{m},\quad E_{min2}=0.9182\ \text{m}
$$

#### (a) Bed raised by 10 cm: depth in the contracted section
$$
E_2=E_1-0.10=0.9510\ \text{m}>E_{min2}
$$
so the flow is not choked (subcritical):
$$
y_2+\frac{1.5^2}{19.62y_2^2}=y_2+\frac{0.11468}{y_2^2}=0.9510\;\Rightarrow\;y_2=0.744\ \text{m}
$$

#### (b) Maximum rise without affecting the upstream depth
$$
\Delta z_{max}=E_1-E_{min2}=1.0510-0.9182=0.1328\ \text{m}\approx13.3\ \text{cm}
$$

#### (c) Rise of 30 cm ($>\Delta z_{max}$)
The flow is choked. Critical flow occurs in the contraction and the upstream depth rises to satisfy
$$
E_1'=E_{min2}+0.30=0.9182+0.30=1.2182\ \text{m}
$$
$$
y_1'+\frac{1}{19.62y_1'^2}=1.2182\;\Rightarrow\;y_1'=1.182\ \text{m}
$$
Water surface elevations (above the original upstream bed):

| Location | Before | After (30 cm rise) | Change |
|---|---|---|---|
| Upstream | 1.000 m | 1.182 m | rises by 0.182 m |
| Over the contraction | $0.10+0.744=0.844$ m (for 10 cm rise) | $0.30+0.612=0.912$ m | rises by about 0.068 m relative to the 10 cm case |

So the upstream water surface rises by 0.182 m (to 1.182 m), and the water surface at the contracted section is 0.912 m (critical depth 0.612 m), i.e. $1.182-0.912=0.270$ m below the new upstream surface.

**Answer: (a) $y_2=0.74$ m; (b) $\Delta z_{max}=0.133$ m; (c) the upstream depth rises from 1.00 m to 1.18 m, and critical flow ($y=0.61$ m) occurs in the contraction, where the surface is at 0.912 m (about 0.07 m higher than in the 10 cm case).**

## Check yourself

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

1. Narrowing from 3 m to 2 m changes q from 1 to
   A. 1.5 m^2/s
   B. 0.67 m^2/s
   C. 2 m^2/s
   D. 1 m^2/s
   Answer: A. q2 = Q/b2 = 3/2 = 1.5 m^2/s, raising yc2 to 0.612 m and Emin2 to 0.918 m. (Key point: Discharge per unit width)

2. With a 10 cm rise, E2 = 0.951 m exceeds Emin2, so the depth in the contraction is
   A. critical, 0.612 m
   B. about 0.744 m, the subcritical root
   C. 1.0 m
   D. 0.951 m
   Answer: B. Not choked, so the subcritical root of y + 0.11468/y^2 = 0.951 is 0.744 m. (Key point: Depth with 10 cm rise)

3. The maximum rise without affecting the upstream depth is
   A. 0.1328 m, equal to E1 - Emin2
   B. 0.30 m
   C. 0.612 m
   D. 0.0 m
   Answer: A. At the limit E2 = Emin2: 1.0510 - 0.9182 = 0.133 m. (Key point: Maximum rise)

4. With a 30 cm rise (greater than the maximum), what happens?
   A. Upstream depth is unchanged
   B. Upstream depth falls
   C. The flow becomes choked: critical depth in the contraction and the upstream depth rises to 1.18 m
   D. The hump has no effect
   Answer: C. E1' = Emin2 + 0.30 = 1.2182 m, giving y1' = 1.182 m and critical depth 0.612 m in the contraction. (Key point: 30 cm rise)

## Close variants set in other exams

- [Contraction and bed rise, 3.5 m channel (1 m depth)](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Acontraction-bed-rise-3-5m-wide-1m-depth) — Hydraulics, chapter 7, asked 2×

## Related in Hydraulics

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- [Contraction with lowered bed, 10 m³/s](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Acontraction-lowered-bed-3-5m-wide-10-m3s) — Hydraulics, chapter 7
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- [Hump and constriction: critical depth, 2 m channel](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Ahump-sidewall-constriction-critical-depth-2m-wide) — Hydraulics, chapter 7, asked 2×
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