HYD-09

Manning discharge

Q = (1/n) A R^{2/3} S^{1/2}. Uniform open-channel flow.

Reading speed

Open channelManning

Governing equation

Q=\dfrac1n A R^{2/3} S^{1/2}

$S \in \left[1\times 10^{-4}, 0.02\right]$

where

n: Roughness n (—)
A: Area (m²)
R: Hydraulic radius (m)
S: Slope (—)
Q: Discharge (m³/s)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

00:0 / 00:08

①

Governing law — red

Q=\dfrac1n A R^{2/3} S^{1/2}

$S \in \left[1\times 10^{-4}, 0.02\right]$

$S = 1\times 10^{-4}$ $Q = 2.652$

Along the curve

②

Domain of the variable — blue

S_{\min} = 1\times 10^{-4},\quad S_{\max} = 0.02

$S \in \left[1\times 10^{-4}, 0.02\right]$

$S = 1\times 10^{-4}$ $S = 1\times 10^{-4}$

Along the curve

③

Response curve — green

Q=\dfrac1n A R^{2/3} S^{1/2}

$S \in \left[1\times 10^{-4}, 0.02\right]$

$S = 1\times 10^{-4}$ $Q = 2.652$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}Q}{\mathrm{d}S}

$S \in \left[1\times 10^{-4}, 0.02\right]$

$S = 2.25\times 10^{-4}$ $Q^{\prime} = 9233.7$

Along the curve

Lecture brief

Historical brief

Open-channel and pipe flow were written by Chezy, Manning, Darcy and Weisbach in the nineteenth century, then Bakhmeteff and Bélanger on the hydraulic jump. The sheets compute conveyance, head loss and gradually varied profiles. This sheet (HYD-09 — Manning discharge) is the form associated with Manning. Working symbols: $n$ , $A$ , $R$ , $S$ $\rightarrow$ $Q$ . n is roughness; R = A/P hydraulic radius; S the friction slope.

Purpose

Purpose: compute

Q

from

n

A

R

S

in Hydraulics via

Q=\dfrac1n A R^{2/3} S^{1/2}

Q = (1/n) A R^{2/3} S^{1/2}. Uniform open-channel flow. Use it when a real hydraulics question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

n = 0.013\,\mathrm{—}

A = 4.000\,\mathrm{m^{2}}

R = 0.800\,\mathrm{m}

S = 0.001\,\mathrm{—}

, the governing relation

Q=\dfrac1n A R^{2/3} S^{1/2}

yields

Q = 8.385\,\mathrm{m^{3}/s}

. A prismatic channel, a water surface, a discharge. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Roughness n n0.013 —Area A4 m²Hydraulic radius R0.8 mSlope S0.001 —

Outputs

Discharge Q8.385 m³/s

Reading speed

Watch on YouTube

Free library

Full library

Free PDF / open book

YouTube channels

00:0 / 00:08

Narration of this film

A prismatic channel, a water surface, a discharge.

n is roughness; R = A/P hydraulic radius; S the friction slope.

Reading speed

Watch on YouTube