Open Channel Flow - Sediment Transport in Irrigation Canals: A New Approach - page 25

friction and for uniform flow ( v constant) the component of

gravity in the flow direction balances the friction forces. These two
assumptions form the basis for the equations for uniform flow, including
the de Chézy formula.
When water flows in a channel, a force is developed that acts on the
boundary in the flow direction and is called the tractive force. In uniform
flow, the tractive force is equal to the component of the gravity force acting
on the control body parallel to the bottom (Simons & Sentruk, 1992).
The tractive force follows from the product of the shear stress and
the contact area. Assume that the average shear stress on the perimeter
P is τ . Next, the balance between the component of the gravity force
and the frictional resistance (tractive force) is used to derive that average
shear stress (see Figure 2.9). The shear stress τ follows from the weight
component (

= Q / A =

= ρgALS o ) and the frictional resistance ( = τPL ):

τ = ρgRS o (2.29)

(m)

y

(1)

Figure 2.9. Tractive force and
the distribution of the shear
stress in a trapezoidal channel.

gradient S 0

Tractive force on reach Distribution of shear stress

By definition the shear velocity u ∗ = √ τ/ρ and can be expressed as:

τ
ρ =

gRS o

u ∗ = (2.30)

For fully turbulent flow Prandtl showed that the shear stress τ is a
function of v 2 and the relationship between v and τ can be written as

Kv 2

τ = = ρgRS o . From this relation follows:

ρg
K RS o =

v 2 C 2 RS o = (2.31)

This relation is normally presented as the de Chézy equation:

C RS o v = (2.32)