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# Manning formula

The Manning formula is an empirical formula for open channel flow, or flow driven by gravity. It was developed by the Irish engineer Robert Manning. For more than a hundred years, this formula lacked a theoretical derivation. Recently this formula was derived theoretically[1] using the phenomenological theory of turbulence.

The manning formula states:

$V = \frac{1}{n} R_h ^\frac{2}{3} \cdot S^\frac{1}{2}$

where:

V is the cross-sectional average velocity (m/s)
n is the Manning coefficient of roughness
Rh is the hydraulic radius (m)
S is the slope of the water surface or the linear hydraulic head loss (m/m) (S = hf / L)

When using English units, the 1/n may be simply replaced by 1.486/n in order to correct for different units, in which cases the units of metres above will be in feet.

$R_h = \frac{A}{P}$

where:

A is the cross sectional area of flow (m2)
P is wetted perimeter (m)

The hydraulic radius is not half the hydraulic diameter, despite what the similarity in the names may suggest. It is a function of the shape of the pipe, channel, or river in which the water is flowing. In wide rectangular channels, the hydraulic radius is approximated by the flow depth.

## Manning coefficient of roughness

The Manning coefficient of roughness, often denoted as n, is an empirically derived coefficient, which is dependent on many factors, including river-bottom roughness and sinuosity. Often the best method is to use photographs of river channels where n has been determined using Manning's formula.

Values typically range between 0.02 for smooth and straight rivers, to 0.075 for sinuous rivers and creeks with excess debris on the river bottom or river banks.

## References

1. ^ http://www.mechse.uiuc.edu/research/gioia/Art/manning.pdf