Manning's equation for uniform flow in open channels

In uniform flow the depth, area and velocity do not change along the channel. Gravity pulling water down the slope is exactly balanced by friction from the bed and sides.

Manning's formula

V = 1n R^2/3 S^1/2 Q = A V

V is the mean velocity (m/s), n Manning's roughness coefficient, S the bed slope and R the hydraulic radius:

R = AP

A is the flow area and P the wetted perimeter.

Typical n values

Surface	n
Smooth concrete lining	0.012 to 0.014
Brick in cement mortar	0.013 to 0.015
Earth canal, clean	0.018 to 0.025
Natural stream with weeds and stones	0.030 to 0.050

Worked example

A trapezoidal concrete-lined canal: bed width b = 3 m, side slopes 1.5H : 1V, depth y = 1.2 m, S = 1 in 2500, n = 0.015.

A = (b + 1.5 y) y = (3 + 1.8) × 1.2 = 5.76 m²

P = b + 2 y √(1 + 1.5²) = 3 + 2 × 1.2 × 1.803 = 7.33 m

R = 5.767.33 = 0.786 m

V = 10.015 × 0.786^2/3 × (4 × 10⁻⁴)^1/2 = 1.14 m/s

Q = 5.76 × 1.14 = 6.5 m³/s

Most economical section

For a given area, the best section has the least wetted perimeter. For a rectangle that means b = 2y (R = y/2). For a trapezoid, R = y/2 and the section fits around a semicircle.

Try different depths and slopes in our open channel tool, which also finds normal depth by trial.