---
title: "Conditions for critical flow in open channel"
subject: "Hydraulics"
chapter: 7
key: "semester_4/hydraulics:critical-flow-conditions-open-channel"
source: https://www.bctengineer.com/q/semester_4/hydraulics/critical-flow-conditions-open-channel
site: BCT Engineering
---

# Conditions for critical flow in open channel

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

- Asked in: 2070 Magh, 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/critical-flow-conditions-open-channel
- Whole chapter: https://www.bctengineer.com/md/semester_4/hydraulics/chapter-7

## Question

What are the different conditions to be fulfilled when flow is critical in open channel?

## Short answer

Froude number Fr = 1, specific energy is minimum for the given discharge, and discharge is maximum for a given specific energy. Velocity head equals half the hydraulic depth; for rectangular channels, yc = (q²/g)^(1/3). The flow is unstable and small disturbances cannot travel upstream.

## Answer

When flow in an open channel is critical, the following conditions are satisfied.

1. **Froude number equals unity:** $Fr=\dfrac{V}{\sqrt{gD_h}}=1$, where $D_h=A/T$ is the hydraulic depth. Velocity equals the celerity of a small gravity wave.
2. **Specific energy is minimum** for the given discharge: $\dfrac{dE}{dy}=0\Rightarrow\dfrac{Q^2T}{gA^3}=1$.
3. **Discharge is maximum** for a given specific energy.
4. **Specific force (momentum function) is minimum** for the given discharge.
5. **Velocity head equals half the hydraulic depth:** $\dfrac{V^2}{2g}=\dfrac{A}{2T}$. For rectangular channels, $\dfrac{V_c^2}{2g}=\dfrac{y_c}{2}$ and $E_{min}=\tfrac32y_c$.
6. **Critical depth** for a rectangular channel is $y_c=\sqrt[3]{q^2/g}$ and for other sections is found from $\dfrac{A^3}{T}=\dfrac{Q^2}{g}$ (section factor $Z=Q/\sqrt g$).
7. The slope of the channel equals the critical slope $S_c$ (for uniform flow at critical depth): $S_c=\dfrac{gn^2D_h}{R^{4/3}}$ or $\dfrac{g P_c}{C^2T_c}$.
8. The flow is unstable: the water surface is wavy and sensitive to small changes in energy or roughness, and the depth changes rapidly. The flow is at the boundary between subcritical and supercritical states.
9. Disturbances (small waves) cannot travel upstream.

## Check yourself

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

1. At critical flow the Froude number is
   A. less than 1
   B. exactly 1, so V equals the celerity of a small gravity wave
   C. greater than 1
   D. zero
   Answer: B. Fr = V/sqrt(g A/T) = 1 means velocity equals wave celerity. (Key point: Froude number)

2. At critical flow, for a given discharge and for a given specific energy respectively,
   A. E is maximum; Q is minimum
   B. E is minimum; Q is maximum
   C. E is minimum; Q is minimum
   D. E is maximum; Q is maximum
   Answer: B. Critical depth gives the minimum specific energy for a given Q and the maximum Q for a given E. (Key point: Energy and discharge)

3. At critical flow the velocity head equals
   A. the depth
   B. twice the hydraulic depth
   C. half the hydraulic depth, A/(2T)
   D. zero
   Answer: C. V^2/2g = A/2T, so in a rectangular channel Emin = y + y/2 = 1.5yc. (Key point: Velocity head)

4. What is true of small disturbances in critical flow?
   A. They travel upstream faster
   B. They cannot travel upstream
   C. They die at once
   D. They are unaffected by flow
   Answer: B. Wave speed equals flow velocity, so waves stay stationary and cannot propagate upstream. (Key point: Wave behaviour)

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