Geoscience Reference
In-Depth Information
Applying Eq. (5.46) in the reaches from cross-section 1 to cross-sections 2 and 3 yields
Q
2
|
β
1
Q
1
2
gA
1
+
=
β
2
Q
2
β
2
Q
2
2
gA
2
−
β
1
Q
1
+
x
12
2
Q
2
|
K
2
Q
1
|
Q
1
|
K
1
z
s
1
2
gA
2
+
z
s
2
+
+
λ
12
2
gA
1
(5.52)
Q
3
β
1
Q
1
2
gA
1
+
=
β
3
Q
3
β
3
Q
3
2
gA
3
−
β
1
Q
1
+
x
13
2
|
Q
3
|
Q
1
|
Q
1
|
z
s
1
2
gA
3
+
z
s
3
+
+
λ
13
K
3
K
1
2
gA
1
(5.53)
where
x
13
represent the lengths of the two reaches.
If the flow is subcritical, the water stages
z
s
2
and
z
s
3
at cross-sections 2 and 3 are
determined first by backwater calculations in channels 2 and 3. The water stage
z
s
1
at cross-section 1 can then be obtained by solving Eqs. (5.52) and (5.53). However,
because the ratio of flow discharges
Q
2
and
Q
3
is unknown, the following iteration
procedure is needed:
x
12
and
(1) Assume the flow discharges
Q
2
and
Q
3
that satisfy the continuity equation (5.51);
(2) Determine the water stages
z
s
2
and
z
s
3
at cross-sections 2 and 3 through backwater
calculations in channels 2 and 3, respectively;
(3) Calculate the water stage at cross-section 1, denoted as
z
s
1
, from cross-section 2
using Eq. (5.52), according to the procedure in Section 5.2.1.2;
(4) Calculate the water stage at cross-section 1, denoted as
z
s
1
, from cross-section 3
using Eq. (5.53), according to the procedure in Section 5.2.1.2;
(5) If
z
s
1
−
z
s
1
|
z
s
1
+
z
s
1
)/
2 as the water
stage
z
s
1
at cross-section 1, and stop iteration; otherwise, repeat from step (2) by
reducing
Q
2
and increasing
Q
3
if
z
s
1
>
|
is less than a reasonable tolerance, then set
(
z
s
1
, or increasing
Q
2
and reducing
Q
3
if
z
s
1
<
z
s
1
, until a convergent solution is obtained.
x
13
are very small, the water stages of the three cross-
sections at the split can be set the same. However, an iteration method similar to the
one described above is still needed to determine the ratio of flow discharges in the two
downstream channels.
If the flow is supercritical, the forewater calculation is carried out in channel 1
down to cross-section 1. The water stages at cross-sections 2 and 3 are calculated by
performing forewater calculations from cross-section 1 to cross-sections 2 and 3 using
Eqs. (5.52) and (5.53), respectively. In order to determine the ratio of flow discharges
Q
2
and
Q
3
, an additional equation is needed. The momentum balance equation at the
split is usually used. More details can be found in HEC (1997).
If the lengths
x
12
and
5.2.2 1-D unsteady flow calculation
5.2.2.1 Discretization of unsteady flow equations
The governing equations for unsteady open-channel flows are the St. Venant equations
(5.1) and (5.2). However, a variety of forms of these equations have been used in the
