Open Channel Flow - Sediment Transport in Irrigation Canals: A New Approach - page 35

y

t

Figure 2.14. An unsteady flow
presented in the x , t and y
direction.

x

The dynamic equation
Unsteady flow analysis for open channels deals with the changes of
velocity and water depth with position and time. Here the depth y and
velocity v are the dependent variables. Remember that in steady flow the
gradient d E /d x represents the slope of the total energy line and is equal,
but opposite in sign to the friction slope S f =

v 2 / C 2 R . When the increase
in depth y and the downstream distance x are taken as positive, the equation
for the total energy may be written as:

αv 2
2 g

E = z + y +

d E
d x =

d z
d x +

d y
d x +

2 v
2 g

d v
d x

Remember that v f ( x , t ); hence the acceleration can be expressed by

partial differentials as:

=

d v
d x =

∂v
∂x +

∂v
∂t

∂t
∂x

∂v
∂x +

∂z
∂x +

∂y
∂x +

v
g

∂v
∂t

∂t
∂x

∂E
∂x =

∂x
∂t

∂t
δx =

1
v

v = or

∂y
∂x +

v
g

∂v
∂x +

1
g

∂v
∂t

− S f =− S o +