Environmental Engineering Reference
In-Depth Information
where:
y
=
water depth (m)
t
=
time (s)
v
=
average velocity of the flow (m/s)
x
=
longitudinal distance (m)
Summarising, the continuity and dynamic equations for unsteady flow in
open channels read:
v
∂A
A
∂v
B
s
∂y
Continuity equation for unsteady
flow for a channel with general shape
∂x
+
∂x
+
∂t
=
0
(2.47)
∂y
∂x
+
v
g
∂v
∂x
+
1
g
∂v
∂t
+
Dynamic equation for
unsteady flow
S
f
−
S
o
=
0
(2.48)
The continuity and dynamic equation in this form are also known as the
de St. Venant equations.
From a combination of the dynamic equation and the continuity
equation follows:
gA
1
gA
3
∂y
Q
2
B
s
∂Q
∂t
−
2
QB
s
A
∂y
∂t
+
−
∂x
−
gAS
o
+
gAS
f
=
0
(2.49)
Combination of the dynamic and continuity equation for a canal with a
general shape
.
2.10 SOLUTION OF THE DE ST. VENANT EQUATIONS
The continuity and dynamic equation compose a set of gradually var-
ied, unsteady flow equations, which together form a complete dynamic
model of the flow. This complete model can provide accurate results for
an unsteady flow, but at the same time, the model can be very demanding
in view of the required computations. Moreover, the model is limited by
the assumptions made in the deduction of the de St. Venant equations
and the suppositions required for their application for specific problems,
e.g. assumptions regarding channel irregularities. The set of the two
simultaneous equations has to be solved for the two unknowns,
v
and
y
,
given appropriate boundary conditions and initial conditions.
At present three main numerical methods are available to solve the
de St. Venant equations, namely:
•
Finite differences (FD)
•
Method of characteristics (MOC)
•
Finite element method (FEM)
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