Geoscience Reference
In-Depth Information
solution of unsteady flows in this type of channel network, provided that a certain
computational order is respected.
A dendritic network of three channels, shown in Fig. 5.11, is used as illustration.
Suppose that the forward sweep starts from point 1 of channel
A
, at which a boundary
condition, such as the time series of flow discharge or water stage, is given. The
recurrence coefficients are calculated along channel
A
using Eqs. (5.71) and (5.72). At
the last point of channel
A
, the following relation is obtained:
δ
Q
A
,
M
=
S
A
,
M
δ
h
A
,
M
+
T
A
,
M
(5.79)
where the subscript
A
denotes channel
A
, and
M
denotes the last point in channel
A
.
Figure 5.11
Dendritic network with three channels.
The forward sweep in channel
B
is also carried out from the first to the last point.
A boundary condition should be given at the first point, while the following relation
is obtained at the last point,
N
:
δ
Q
B
,
N
=
S
B
,
N
δ
h
B
,
N
+
T
B
,
N
(5.80)
where the coefficients
S
B
,
N
and
T
B
,
N
are determined using Eqs. (5.71) and (5.72).
Now, let us consider how to handle the junction. For convenience, the three cross-
sections at the junction are located very close together. Therefore, it can be assumed
that the water stages at the three cross-sections are equal, and the flow discharge
at the downstream cross-section is equal to the sum of those at the two upstream
cross-sections:
z
n
+
1
sA
,
M
z
n
+
1
sB
,
N
z
n
+
1
sC
,1
=
=
(5.81)
Q
n
+
1
C
,1
Q
n
+
1
A
,
M
Q
n
+
1
B
,
N
=
+
(5.82)
Eqs. (5.81) and (5.82) are the compatibility conditions at the junction. Substituting
Eqs. (5.58) and (5.61) into Eqs. (5.81) and (5.82) yields
z
sC
,1
−
z
sA
,
M
δ
h
A
,
M
−
δ
h
C
,1
=
(5.83)
z
sC
,1
−
z
sB
,
N
δ
h
B
,
N
−
δ
h
C
,1
=
(5.84)
