Geoscience Reference
In-Depth Information
Q
I
at cross-section
I
(outlet)
are derived from the downstream boundary condition, which can be a time series
of water stage, a stage-discharge rating curve, etc. If the water-stage time series is
specified,
z
n
+
1
s
,
I
To perform the second sweep, the values of
δ
h
I
and
δ
is known and the stage increment at cross-section
I
is
z
n
+
1
s
,
I
z
s
,
I
δ
h
I
=
−
(5.75)
and the discharge increment
Q
I
can then be determined using Eq. (5.69).
If a measured stage-discharge rating curve,
Q
δ
, is specified at the out-
let, a discretized equation can be obtained by applying the first-order Taylor series
expansion:
=
f
(
z
s
)
df
dz
s
δ
f
∗
−
Q
I
δ
Q
I
−
h
I
=
(5.76)
and the stage increment at point
I
can then be derived from Eqs. (5.69) and (5.76) as
Q
I
−
f
∗
T
I
+
δ
h
I
=
(5.77)
/
dz
s
−
df
S
I
In the outlet located in a nearly prismatic channel with a positive slope (
do
wnslope),
the flow can be assumed to be uniform; thus, a relation of
Q
)
√
S
0
exists, in
which
K
is the conveyance and
S
0
is the channel slope. In analogy to Eq. (5.77), the
following equation for the stage increment at point
I
can be derived:
=
K
(
z
s
K
∗
√
S
0
Q
I
−
T
I
+
δ
h
I
=
√
S
0
dK
∗
/
(5.78)
dz
s
−
S
I
If the outlet is controlled by an in-stream structure, such as spillway or weir, a free
overfall flow exists; thus, a stage-discharge rating curve
Q
can be obtained
using the critical flow condition near the brinkpoint, and then Eq. (5.77) can be
applied.
If a flood or tidal wave propagation is concerned, the outflow boundary must also
be non-reflective and able to damp out the waves. This type of outflow boundary
condition may be found in Hinatsu (1992) and others.
Therefore, the aforementioned two sweeps constitute an iteration step, yielding
=
f
(
z
s
)
Q
.
z
s
and
Q
∗
are then updated by
z
s
h
and
Q
∗
+
δ
δ
h
and
δ
+
δ
Q
. The iteration is
stopped when the solutions for
z
s
and
Q
∗
have converged (
δ
→
δ
→
h
0 and
Q
0). The
converged
z
s
and
Q
∗
are eventually given to
z
n
+
1
and
Q
n
+
1
.
s
Algorithm for a dendritic channel network
A dendritic (or tree-like) channel network includes tributaries and/or distributaries
without any loop. The previous double sweep algorithm can still be applied in the
