---
title: "Hump and constriction: critical depth, 2 m channel"
subject: "Hydraulics"
chapter: 7
key: "semester_4/hydraulics:hump-sidewall-constriction-critical-depth-2m-wide"
source: https://www.bctengineer.com/q/semester_4/hydraulics/hump-sidewall-constriction-critical-depth-2m-wide
site: BCT Engineering
---

# Hump and constriction: critical depth, 2 m channel

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

- Asked in: 2071 Magh, 2+2+2 marks; 2068 Bhadra, 2+3+3+2 marks (asked 2 times)
- Video: [Change in depths due to hump in a rectangular channel | Open Channel Flow](https://www.youtube.com/watch?v=zyxOadh2BzM) — The Study Sphere
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/hydraulics/hump-sidewall-constriction-critical-depth-2m-wide
- Whole chapter: https://www.bctengineer.com/md/semester_4/hydraulics/chapter-7

## Question

A rectangular channel 2 m wide has a flow of 2.4 m³/s at a depth of 1.0 m. Determine if critical depth occurs (a) at the section where a hump of $\Delta Z$ = 20 cm high is installed across the bed, (b) a side wall constriction (no hump) reducing the channel width to 1.7 m, and (c) both the hump and side wall constriction combined. Will the upstream depth be affected for case (c)? If so, to what extent? Neglect head losses of the hump and constriction caused by friction, expansion and contraction.

## Short answer

For hump alone, E2 > Emin, so no critical depth; depth over hump 0.74 m. For constriction alone, E2 > Emin2, no critical depth; depth 0.96 m. With both, E2 < Emin2, critical depth 0.588 m occurs; upstream depth rises from 1.000 m to 1.010 m (10 mm increase).

## Answer

Critical depth occurs at a section when the available specific energy there is equal to the minimum specific energy for the discharge per unit width at that section: $E_2=E_{min}=\tfrac32y_c$, where $y_c=\sqrt[3]{q^2/g}$.

#### Approach flow
$Q=2.4$ m³/s, $b=2$ m, $y_1=1.0$ m.
$$
q_1=\frac{2.4}{2}=1.2\ \text{m}^2/\text{s},\quad V_1=1.2\ \text{m/s},\quad Fr_1=\frac{1.2}{\sqrt{9.81\times1}}=0.383\ (\text{subcritical})
$$
$$
E_1=y_1+\frac{V_1^2}{2g}=1+\frac{1.44}{19.62}=1.0734\ \text{m}
$$
$$
y_{c1}=\sqrt[3]{\frac{1.2^2}{9.81}}=0.5275\ \text{m},\qquad E_{min1}=1.5y_{c1}=0.7913\ \text{m}
$$

#### (a) Hump of 0.20 m (width unchanged)
Energy over hump $E_2=E_1-\Delta z=1.0734-0.20=0.8734$ m $>E_{min1}=0.7913$ m.

So **critical depth does not occur**. The flow stays subcritical over the hump, with depth found from $y+\dfrac{1.44}{19.62y^2}=0.8734$:
$$
y_2=0.739\ \text{m}\quad(>y_c=0.528\ \text{m})
$$
The upstream depth remains 1.0 m.

#### (b) Side-wall constriction to 1.7 m (no hump)
$$
q_2=\frac{2.4}{1.7}=1.4118\ \text{m}^2/\text{s},\quad y_{c2}=\sqrt[3]{\frac{q_2^2}{g}}=0.5879\ \text{m},\quad E_{min2}=0.8818\ \text{m}
$$
$E_2=E_1=1.0734>E_{min2}=0.8818$, so **critical depth does not occur**. Depth in the constriction (subcritical): $y_2=0.964$ m. Upstream depth unchanged.

#### (c) Hump and constriction together
Available energy at the section: $E_2=E_1-0.20=0.8734$ m, while the minimum required is $E_{min2}=0.8818$ m (for $q_2=1.412$).

$E_2<E_{min2}$, so **critical depth occurs** (choking) over the hump in the constriction, $y_c=0.588$ m. The channel cannot pass the flow at the original upstream energy, so the **upstream depth is affected**: it rises until
$$
E_1'=E_{min2}+\Delta z=0.8818+0.20=1.0818\ \text{m}
$$
$$
y_1'+\frac{1.2^2}{2g\,y_1'^2}=1.0818\;\Rightarrow\;y_1'=1.0098\ \text{m}
$$
Rise in upstream depth $=1.0098-1.000=0.0098$ m (about 10 mm).

**Answer: (a) not critical ($y=0.74$ m over hump); (b) not critical ($y=0.96$ m); (c) critical depth $0.588$ m occurs; upstream depth rises from 1.000 m to 1.010 m, about 10 mm.**

## Check yourself

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

1. How is it decided whether critical depth occurs at the hump or constriction section?
   A. Compare the available specific energy there with Emin = 1.5yc for the q at that section
   B. Compare the upstream depth with 1 m
   C. Check whether the velocity exceeds 1 m/s
   D. Compare the width with the depth
   Answer: A. Critical depth occurs when available energy E2 is equal to (or less than) Emin = 1.5yc for the discharge per unit width at that section. (Key point: Test for critical depth)

2. For the 0.20 m hump alone (E1 = 1.073 m, Emin = 0.791 m), what is the result?
   A. Critical depth occurs and upstream depth rises
   B. Critical depth occurs but upstream is unaffected
   C. Critical depth does not occur; E2 = 0.873 m exceeds Emin and the upstream depth stays 1.0 m
   D. The flow becomes supercritical over the hump
   Answer: C. E2 = 1.0734 - 0.20 = 0.8734 m > 0.7913 m, so flow stays subcritical and upstream is unaffected. (Key point: Hump alone, case (a))

3. Why does critical depth occur in case (c), though neither the hump nor the constriction alone causes it?
   A. The upstream velocity increases
   B. Energy is lost to friction
   C. The hump reduces E to 0.873 m while the narrower width raises Emin to 0.882 m, so E2 < Emin
   D. The width reduction lowers the discharge
   Answer: D. Combined, available energy (0.8734) is less than Emin2 (0.8818) for q2 = 1.412, so the section chokes. (Key point: Why combined case chokes)

4. In case (c), how is the new upstream depth found?
   A. Set E1' = Emin2 + Δz and solve y + q1^2/(2g y^2) = E1' for the subcritical root
   B. Set the upstream depth equal to critical depth 0.588 m
   C. Add the hump height 0.20 m to 1.0 m
   D. Keep 1.0 m because the discharge is unchanged
   Answer: A. Upstream energy must equal Emin2 + Δz = 1.0818 m, giving y1' of about 1.010 m, a rise of about 10 mm. (Key point: Upstream depth in choking)

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