---
title: "Maximum obstruction height without upstream rise"
subject: "Hydraulics"
chapter: 7
key: "semester_4/hydraulics:obstruction-max-height-no-upstream-rise"
source: https://www.bctengineer.com/q/semester_4/hydraulics/obstruction-max-height-no-upstream-rise
site: BCT Engineering
---

# Maximum obstruction height without upstream rise

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

- Asked in: 2076 Bhadra, 7 marks
- Video: [Open channel flow-18 | channel transition hump | shubham sarathe](https://www.youtube.com/watch?v=SXy9LMLzviE) — EIQ
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/hydraulics/obstruction-max-height-no-upstream-rise
- Whole chapter: https://www.bctengineer.com/md/semester_4/hydraulics/chapter-7

## Question

A rectangular channel 2 m wide carries 3 m³/s of water at a flow depth of 1.5 m. What is the maximum height of the obstruction placed across the channel that will not cause a rise in the water surface upstream?

## Short answer

Upstream energy E1 =1.551 m, critical energy Emin = 1.5 yc = 0.9182 m (yc=0.612 m). Maximum height = E1 - Emin = 0.633 m. A higher obstruction would cause upstream rise.

## Answer

An obstruction (hump) of height $\Delta z$ across the bed does not affect the upstream depth as long as the specific energy over it is at least the minimum, $E_{min}$. The limiting height is when the flow over the obstruction is just critical.

#### Given data
$b=2$ m, $Q=3$ m³/s, $y_1=1.5$ m.
$$
q=\frac{3}{2}=1.5\ \text{m}^2/\text{s},\quad V_1=\frac{1.5}{1.5}=1.0\ \text{m/s}
$$
$$
E_1=y_1+\frac{V_1^2}{2g}=1.5+\frac{1.0^2}{19.62}=1.5510\ \text{m}
$$

#### Critical condition at the obstruction
$$
y_c=\sqrt[3]{\frac{q^2}{g}}=\sqrt[3]{\frac{1.5^2}{9.81}}=0.6121\ \text{m}
$$
$$
E_{min}=\frac32y_c=0.9182\ \text{m}
$$
For no rise upstream, $E_1=E_{min}+\Delta z_{max}$:
$$
\Delta z_{max}=E_1-E_{min}=1.5510-0.9182=0.633\ \text{m}
$$
**Answer: maximum height of the obstruction $=0.633$ m (about 63 cm).** A higher obstruction will cause the upstream water level to rise.

## Check yourself

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

1. The maximum obstruction height with no upstream rise occurs when
   A. flow over it is just critical, E1 = Emin + Δzmax
   B. the flow over it is supercritical
   C. the upstream depth equals critical depth
   D. Froude number upstream equals 1
   Answer: A. Any higher and available energy would fall below Emin, causing backwater. (Key point: Limiting condition)

2. With q = 1.5 m^2/s, yc is
   A. 1.5 m
   B. 0.612 m
   C. 0.918 m
   D. 1.224 m
   Answer: B. yc = (q^2/g)^(1/3) = (2.25/9.81)^(1/3) = 0.612 m. (Key point: Critical depth calculation)

3. What is Emin for this flow?
   A. 0.612 m
   B. 1.224 m
   C. 0.918 m
   D. 1.551 m
   Answer: C. Emin = 1.5 x 0.6121 = 0.918 m. (Key point: Minimum specific energy)

4. With E1 = 1.551 m, the maximum obstruction height is about
   A. 0.633 m
   B. 0.918 m
   C. 1.551 m
   D. 0.5 m
   Answer: A. Δzmax = E1 - Emin = 1.551 - 0.918 = 0.633 m. (Key point: Final result)

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