Open Channel Flow - Sediment Transport in Irrigation Canals: A New Approach - page 40

of flow control move through a canal network. The structure of the net of
characteristics will improve the understanding of the numerical procedures
required for the practical solution of hydraulic phenomena in canal system.
Therefore a short explanation of the method of characteristics will be
given here; hydraulic reference topics present more details on unsteady
flow analysis.

v w * t

v w

v 1

v 2

y 1

Figure 2.15. Propagation of a
wave front in downstream
direction.

y 2

The set of continuity and momentum equations is very useful for
the formulation of solutions on the basis of finite differences, but a
transformation of the equations might help to understand the continuity
and dynamic e qua tions more easily. For the transformation the auxiliary
variable c = √ gD (Lagrange wave speed) will be used.

The transformation will use the de St. Venant equations and the
differential of the area A to x :

∂A
∂x =

∂ ( αB s y )
∂x

B s ∂y αy ∂B s

∂x

=

∂x +

The term αy ( ∂B s /∂x ) accounts for non-prismatic channels and ∂B s /∂x is
the change of the surface width in the flow direction. The coefficient α
is the Coriolis coefficient. The value of α depends on the geometry of
the cross-section. The value of α is about 0.5 for triangular and about
1.0 for rectangular cross-sections.
The continuity equation can be written as:

vB s ∂y

∂x + αyv ∂B s

A ∂v B s ∂y

∂x + ∂x + ∂t =

0

v B s
A

∂y
∂x +

αyv
A

∂B s
∂x +

∂v
∂x +

B s
A

∂y
∂t =

0

By definition the hydraulic depth D is A/B s
v
D

∂y
∂x +

∂v
∂x +

1
D

∂y
∂t +

αyv
A

∂B s
∂x =

0