Geoscience Reference
In-Depth Information
increments
δ
h
at all nodes as unknowns:
[
]{
δ
}={
}
S
h
b
(5.102)
where
[
S
]
is the coefficient matrix with
m
×
m
elements,
{
δ
h
}
is the vector of
m
{
}
unknowns, and
is the vector with
m
elements holding all free terms. Here,
m
is the total number of nodes.
The system represented by Eq. (5.102) can be solved using a matrix inversion tech-
nique. Once the stage increments are solved at the nodes, Eqs. (5.94)-(5.96) are used
to determine the discharge increments at the ends of each channel, and Eqs. (5.88) and
(5.89) are used to compute
b
δ
h
i
+
1
and
δ
Q
i
+
1
for all intermediate points in a generalized
return sweep.
In principle, Eq. (5.102) can be solved using any matrix inversion technique. How-
ever, because the matrix may be quite large, a direct inversion computation can be
expensive. An iterative inverse computation may also have trouble when the matrix
loses diagonal dominance. The block tri-diagonal matrix solution technique suggested
by Mahmood and Yevjevich (1975) has been found to be very efficient in the solution
of equation system (5.102). The details can be found in that reference.
5.2.2.4 Treatment of hydraulic structures as internal
boundaries
Because of their complexity, it is almost impossible to simulate the detailed flow pat-
terns around in-stream hydraulic structures, such as culverts, bridge crossings, drop
structures, weirs, sluice gates, spillways, and measuring flumes, using a 1-D model.
Simplifications must be made to obtain a feasible solution. The storage effect of the
flow at a hydraulic structure is usually neglected, so the same flow discharge is imposed
at its upstream and downstream ends:
Q
up
=
Q
down
(5.103)
which can be expanded as
Q
down
−
Q
up
δ
Q
up
−
δ
Q
down
=
(5.104)
The water stage at the hydraulic structure is often determined using a stage-discharge
relation, which is related to whether the flow is upstream or downstream controlled.
The upstream control flow is treated as a free overfall flow that is critical, while the
downstream control flow is treated as an orifice-like flow. For the upstream control
flow, the critical flow condition implies
g
A
c
B
c
Q
=
A
c
(5.105)
where
A
c
and
B
c
are the area and top width of flow at the structure, respectively. Both
are functions of flow depth. Thus, the following general stage-discharge relation can
