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(a)
Q
x
A
y
S o
v
P
(b)
B
q
W
h
S o
x
v
Figure 5.2 Schematic diagram for surface flows with slope S o and distance x measured along the slope:
(a) one-dimensional representation of open channel flow with discharge Q, cross-sectional area A, wetted
perimeter P, average velocity v and average depth y; (b) one-dimensional representation of overland flow as
a sheet flow with specific discharge q, width W, average velocity v and average depth h.
considered as a convenient approximation to the full three-dimensional flows. One-dimensional solutions
must use average cross-sectional velocities as solution variables, even for the case of overbank flow for
the channel and variable depth overland flows (Figure 5.2). The one-dimensional case may be described
by the equations developed by the Barre de St. Venant (1797-1886). These equations assume that the
flow may be described in terms of average cross-sectional velocities and depths and are developed from
the balances of both mass and momentum in the flow. Thus for a flow of average velocity v , of average
depth h , in a cross-sectional area, A , and wetted perimeter P , on a bed slope of S o , and lateral inflow per
unit length of slope or channel of q , the mass balance equation may be written as:
∂A
∂t
A ∂v
v ∂A
∂x
=−
∂x
+
q
(5.4)
and the momentum balance equation, assuming that water is incompressible, as:
∂Av
∂t
∂Av 2
∂x
∂Agh
∂x
f
2 g v 2
+
+
=
gAS o
gP
(5.5)
where f is the Darcy-Weisbach uniform roughness coefficient.
The St. Venant equations are a fully dynamic or dynamic wave description of the flow that can be
used in routing flood waves and hydrographs down a channel or in a reach of a channel network. The
derivation of these equations is given in Box 5.6, together with explanations of simplified versions known
as the diffusion wave and kinematic wave approximations, resulting from neglecting different terms in
Equation (5.5). As in the case of the Richards equation, solution of the St. Venant equations for cases
 
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