Geoscience Reference
In-Depth Information
stage is specified at the outlet. Therefore, the flow discharge in the solution domain can
be calculated easily in a forewater sweep by applying mass continuity, and the water
stage can then be determined by backwater calculation using Eq. (5.42). Because of
its nonlinearity, Eq. (5.42) needs to be solved iteratively.
Define the following function:
Q
i
+
1
=
β
i
+
1
Q
i
+
1
2
gA
i
+
1
−
β
i
Q
i
+
x
i
+
1
/
2
2
|
Q
i
+
1
|
Q
i
|
Q
i
|
F
2
gA
i
+
z
s
,
i
+
1
−
z
s
,
i
+
K
i
+
1
K
i
(5.47)
1 have been
obtained from the previous calculation in the reach between cross-sections
i
Because
z
s
,
i
+
1
and the corresponding
A
i
+
1
and
K
i
+
1
at cross-section
i
+
+
1 and
i
2, or from the given water stage at the outlet, now the problem is determining
z
s
,
i
and the corresponding
A
i
and
K
i
by ensuring
F
+
=
0. The following bisection method
is often used:
(1) Find a segment
[
Z
lower
,
Z
upper
]
in which the solution of
z
s
,
i
exists,
i.e.,
0, with
F
upper
and
F
lower
being the values of
F
corresponding to
Z
upper
and
Z
lower
, respectively;
(2) Set
Z
middle
F
upper
F
lower
<
=
(
Z
upper
+
Z
lower
)/
2 and calculate
F
middle
, the value of
F
correspond-
ing to
Z
middle
;
(3) If
F
middle
=
0 (or less than a certain tolerance),
Z
middle
is the solution of
z
s
,
i
and
then stop iteration; otherwise, if
F
middle
F
lower
<
0, then set
Z
upper
=
Z
middle
, and
if
F
upper
F
middle
<
0 , then set
Z
lower
=
Z
middle
;
(4) If
Z
upper
2
to be the solution of
z
s
,
i
and stop iteration; otherwise, repeat from step (2) until
the convergent solution is obtained.
−
Z
lower
is less than a reasonable tolerance, then set
(
Z
upper
+
Z
lower
)/
Note that the search in step (1) for the lower and upper bounds
Z
lower
and
Z
upper
of
the initial segment where the solution exists can start from either the channel thalweg
elevation or
z
s
,
i
+
1
. The search starting from the thalweg is upward only, whereas the
search starting from
z
s
,
i
+
1
must be conducted upward and downward. The former
search is simpler and can guarantee the solution.
For supercritical flow, both flow discharge and water stage are usually specified
at the inlet. Therefore, the water stage in the solution domain can be determined by
forewater calculation using Eq. (5.42). Similarly, Eq. (5.42) must be solved using an
iteration method, such as the bisection method. The difference is only that
z
s
,
i
and the
corresponding
A
i
and
K
i
are known while
z
s
,
i
+
1
and the corresponding
A
i
+
1
and
K
i
+
1
are unknown.
For flow in mixed regimes, the entire computational domain is divided into subdo-
mains according to the flow regimes, and then the previous methods are used to solve
the subcritical and supercritical flows in all subdomains individually. Usually, internal
boundary conditions should be applied in the transition regions between subdomains.
Because the energy equation (5.40) may not be applicable in regions with hydraulic
jumps, internal boundary conditions should be derived from the momentum equation
instead, which may be found in Chow (1959) and HEC (1997).
