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Civil

Open-channel flow (Manning)

Flow rate in a river, ditch or culvert from its slope and roughness.

Ref: CIV-07
Rev: v1
Field: Civil
Result: Q [m³/s]

Q = \frac{1}{n} A {R_{h}}^{\frac{2}{3}} {S_{f}}^{\frac{1}{2}}, R_{h} = \frac{A}{P_{w}}

Q discharge in m³/s, n Manning roughness, A flow area, R_h hydraulic radius (area divided by wetted perimeter P_w), S_f friction slope, equal to the bed slope for steady uniform flow. SI units.

Water flows faster in a smooth channel, a steeper channel, and one with a large cross-section relative to its wetted edge. Manning’s equation turns those into a flow rate.

Calculator

Solve for

QDischargeComputed

m³/s

nManning roughness

0.010.1

AFlow area

m²

0.150

P_wWetted perimeter

0.540

S_fSlope

m/m

1e-40.05

Hydraulic radius R_h0.4 m
Mean velocity Q/A1.321 m/s

What drives the result

Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.

Discharge (m³/s) current value 1.585 m³/s

AFlow areaKey driver

0.0252794

S_fSlope

0.50111.2

P_wWetted perimeter

0.2825.23

nManning roughness

0.2062.06

Roughness n is an empirical estimate and is the largest uncertainty; engineers often check the sensitivity over a range of n.
Flow is inversely proportional to n, so a weedy channel (n = 0.035) carries about 2.7 times less than smooth concrete (0.013) of the same shape.
For the same area, the most efficient shape has the smallest wetted perimeter: a semicircle or half-hexagon.

Chart data as tables

Ranked variables

Variable	Typical range	Lowest result	Highest result	Swing (m³/s)	Local step	Change in result
A Flow area	0.1 to 50 m²	0.02519	793.6	793.6	+10%	+17.2 %
S_f Slope	1e-4 to 0.05 m/m	0.5011	11.21	10.7	+10%	+4.88 %
P_w Wetted perimeter	0.5 to 40 m	0.2818	5.233	4.951	+10%	−6.16 %
n Manning roughness	0.01 to 0.1	0.206	2.06	1.854	+undefined%	—

Sweep of Slope

Slope (m/m)	Discharge (m³/s)
1e-4	0.5011
2.175e-4	0.739
4.729e-4	1.09
0.001028	1.607
0.002236	2.37
0.004862	3.494
0.01057	5.153
0.02299	7.599
0.05	11.21

Sweep one variable

Vary

Discharge (m³/s)Hover, touch, or use the arrow keys on the chart

Variables and how each one contributes

Discharge

Result · m³/s

What the equation gives you. Everything else feeds it.

Manning roughness

· typical 0.01 to 0.1

Concrete about 0.013, earth channel 0.025, weedy river 0.035–0.1. Inversely proportional: the channel surface is the most uncertain input.

Flow area

m² · typical 0.1 to 50

Cross-sectional area of the water. Increases flow with A^(5/3) when the shape scales up with it.

P_w

Wetted perimeter

m · typical 0.5 to 40

Length of channel boundary in contact with water. More wetted edge means more friction and a lower hydraulic radius.

S_f

Slope

m/m · typical 1e-4 to 0.05

Bed slope (drop per unit length). Flow rises with the square root of the slope: 4× the slope doubles the flow.

Worked example with the values above

Hydraulic radius: R_h = A/P_w = 1.2 / 3 = 0.4 m. R_h^2/3 = 0.5429; S^1/2 = 0.03162.
Q = (1/0.013) × 1.2 × 0.5429 × 0.03162.
Q = 1.585 m³/s.

Assumptions and limits

Steady, uniform flow, so friction slope equals bed slope.
Prismatic channel, constant roughness.
Turbulent flow; SI form of the equation.

Common mistakes

Using the free-surface width instead of the wetted perimeter.
Using the US customary form (1.49/n) with SI units.
Applying to a short reach with changing depth.

Statistics and uncertainty

Roughness estimates from tables are typically uncertain by ±20–30%, which dominates all other errors. Slope errors are halved by the square root, and area is amplified by 5/3 power when the section scales.

Where it is used

Culvert and drainage design.
River crossing and fording assessments.
Flood-flow estimates from slope and depth.

Maths you need

Cross-section geometry: area and wetted perimeter
Exponents: powers 2/3 and 1/2

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