Steady Uniform flow: The Chezy and Manning Equation

Steady Uniform flow

When uniform flow occurs gravitational forces exactly balance the frictional resistance forces which apply as a shear force along the boundary (channel bed and walls).

Considering the above diagram, the gravity force resolved in the direction of flow is Gravity orce = rgAL sinq

and the boundary shear force resolved in the direction of flow is

shar force = t_oPL

In uniform flow these balance

t_oPL = rgALsinq

Considering a channel of small slope, (as channel slopes for unifor and gradually varied flow seldom exceed about 1 in 50) then

So

sinq ~ tanq = S_o

t_o  = rgA S_o / P  = rgRS_o

1.The Chezy equation

If an estimate of ? o can be made then we can make use of Equation.

If we assume the state of rough turbulent flow then we can also make the assumption the shear force is proportional to the flow velocity squared i.e.

t_o      a  V²

t_o = KV²

 Substituting this into equation gives

V = Rt(pg/K . RS_o)

Or grouping the constants together as one equal to C

V=C. Rt(RS_o)

This is the Chezy equation and the C the 'Chezy C'

Because the K is not constant the C is not constant but depends on Reynolds number and boundary roughness (see discussion in previous section).

The relationship between C and is easily seen be substituting equation 1.9 into the Darcy- Wiesbach equation written for open channels and is

C = Rt(2g/f)

2 The Manning equation

A very many studies have been made of the evaluation of C for different natural and manmade channels. These have resulted in today most practising engineers use some form of this relationship to give C:

C = R^1/4/n

This is known as Manning's formula, and the n as Manning's n . Substituting equation 1.9 in to 1.10 gives velocity of uniform flow:

Or in terms of discharge

V=R^2/3So^1/2 / n

Q = 1/n . A^5/3 / P^2/3 . So^1/2

Note:

Several other names have been associated with the derivation of this formula - or ones similar and consequently in some countries the same equation is named after one of these people. Some of these names are; Strickler, Gauckler, Kutter, Gauguillet and Hagen.

The Manning's n is also numerically identical to the Kutter n .

The Manning equation has the great benefits that it is simple, accurate and now due to it long extensive practical use, there exists a wealth of publicly available values of n for a very wide range of channels.

Below is a table of a few typical values of Manning's n

Conveyance

Channel conveyance, K , is a measure of the carrying capacity of a channel. The K is really an agglomeration of several terms in the Chezy or Manning's equation:

Q=KS_o^1/2

K=ACR^1/2

Use of conveyance may be made when calculating discharge and stage in compound channels and also calculating the energy and momentum coefficients in this situation.