Manning's Equation

Equivalent forms

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

v = \frac{k}{n}R^{2/3}S^{1/2}

One roughness coefficient n captures everything from glass-smooth concrete to a weedy creek.

Symbol	Name	SI unit	Dimension	Range
$n$	Manning Roughness Empirical channel roughness coefficient	s/m^{1/3}	T·L⁻¹ᐟ³	0.008 – 0.1
$R$	Hydraulic Radius Flow cross-sectional area divided by wetted perimeter	m	L	0.01 – 10
$S$	Channel Slope Bed slope (energy gradient), rise over run	m/m	1	0.00001 – 0.1
$v$	Flow Velocityoutput Mean flow velocity in the channel	m/s	L·T⁻¹	0 – 20

Unit systems

Symbol	SI	CGS	Imperial
$n$	s/m^{1/3}	s/cm^{1/3}	s/ft^{1/3}
$R$	m	cm	ft
$S$	m/m	cm/cm	ft/ft
$v$	m/s	cm/s	ft/s

Where it holds

Empirical, for steady, uniform, fully turbulent open-channel flow. SI form uses

k = 1.0

; US-customary uses

k = 1.486

. Less accurate for very shallow, laminar, or steeply non-uniform flows. Manning n is tabulated, not derived.

Dimensional analysis

[(1/n)R^{2/3}S^{1/2}] with $[n] = T\cdot L^{-1}$ ᐟ $^{3}$ gives $(L^{1}$ ᐟ $^{3}\cdot T^{-1})(L^{2}$ ᐟ $^{3})(1) = L\cdot T^{-1} \checkmark$ (velocity)

Discovery

Robert Manning · 1889

Robert Manning, an Irish engineer, distilled decades of open-channel measurements into a simple empirical velocity formula. Philippe Gauckler (1867) and others reached similar forms, but Manning's version became the worldwide standard for hydraulics.

Try this

How do engineers predict how fast a river, canal, or storm drain will carry water?

An open concrete channel (Manning n = 0.013) has a hydraulic radius of 0.5 m and a bed slope of 0.001. How fast does the water flow?

Research status: stable

Real-world applications

Sizing storm sewers, culverts, and drainage canals
River discharge and flood-stage prediction
Irrigation canal and aqueduct design
Hydraulic modeling for environmental and civil engineering

Common misconceptions

Manning's n is not dimensionless and is not a pure property of the material alone — it lumps roughness, vegetation, and channel irregularity
The formula is for open-channel (free-surface) flow, not pressurized pipe flow, where Darcy-Weisbach is preferred
The numerical constant differs between SI (1.0) and US units (1.486); using the wrong one is a classic error

Experimental verification

Over a century of river-gauging and laboratory-flume measurements underpin the standard tables of Manning'

s n (\approx 0.011

for smooth concrete,

\approx 0.035

for natural streams,

\approx 0.10

for dense floodplain vegetation). Discharge predictions match gauged flows within engineering tolerance.

Derivation

Manning's equation is empirical, but it can be motivated from the Chézy formula

v = C\sqrt{RS}

, where balancing gravity along the slope

(\rho gRS

per unit area) against turbulent wall shear

(\propto \rho v^{2})

gives

v \propto \sqrt{RS}

.

Manning's data showed the Chézy coefficient C itself scales as R^{1/6}/n, yielding

v = (1/n)R^{2/3}S^{1/2}

.

Limiting cases

S \to 0

(flat channel)⟶

v \to 0

With no slope there is no gravitational driving force; the water stands still.

n large (very rough/weedy)⟶ v smallHigh roughness dissipates energy and slows the flow.

R large (deep, efficient section)⟶ v rises as R^{2/3}Deeper flow has proportionally less wetted-perimeter drag, so it speeds up.

What if…

What if the channel becomes choked with weeds (n rises from 0.013 to 0.05)?

Velocity drops by the factor $0.013/0.05 \approx 0.26$ , so the channel carries roughly a quarter of the flow — a major flood risk in unmaintained drains.

What if you double the slope?

Velocity rises $by \sqrt{2} \approx 1.41\times (v \propto S^{1/2})$ , so steeper channels flow faster but can become erosive and supercritical.

1

Velocity in a concrete channel

Given ·

n:: 0.013
R:: 0.5
S:: 0.001

Find · v

Steps

$R^{2/3} = 0.5^{0.667} \approx 0.63$
$S^{1/2} = 0.001^{0.5} \approx 0.0316$
$v = (1/0.013) \times 0.63 \times 0.0316 \approx 1.53\,\mathrm{m/s}$

Result ·

v = (1/0.013)(0.5)^{2/3}(0.001)^{1/2} = 76.9 \times 0.63 \times 0.0316 \approx 1.53\,\mathrm{m/s}

Darcy-Weisbach Equation→Froude Number→Reynolds Number→