92 Computational Modelling in Hydraulic and Coastal Engineering
After some simple manipulation this equation takes the form
∂
∂
+
∂
∂
= −
∂
∂
+
−
u
t
u
u
x
g
y
x
g S S
o
e
(
)
(5.24)
where S o is the slope of the channel bed and S
g R
e
o
h
=
τ
ρ
is the slope of the
energy grade line, a function of the wall friction τ ο . By using the Chezy formula (Equation 5.3) for the wall friction, the operational form for S e becomes
S
u
C R
e
z h
=
2
2
(5.25)
For rectangular channels, Α = Βy and if the channel width is much bigger
than the depth (as in river cross-section geometry), R h = y.
For q L = 0 and by neglecting the nonlinear terms, the governing equations are written as
∂
∂
+
∂
∂
=
y
t B
Q
x
1
0
(5.26)
∂
∂
= −
∂
∂
+
−
u
t
g
y
x
g S S
o
e
(
)
(5.27)
In the case of steady flow, the continuity equation is reduced to Q = constant and the equilibrium equation takes the form
u
u
x
g
y
x
g S S
o
e
∂
∂
+
∂
∂
=
−
(
)
(5.28)
which after substitution of u = Q/A leads to the surface profile (Equation 5.17).
The applicability of the unsteady form is very wide, as it describes the
propagation of a transient flood wave along a natural stream (e.g. river) or
along an artificial canal (e.g. drainage canal).
For a rectangular channel of variable width, the linearized flood propagation model is comprised of the two linear hyperbolic equations of firstorder (Equations 5.26 and 5.27), the discharge relation
Q = uBy = uA
(5.29)
and the auxiliary conditions (i.e. initial and boundary).