Geoscience Reference
In-Depth Information
where
A
P
is the area of the horizontal 2-D control volume.
The discretized equations (7.51) and (7.56) can be solved using the ADI or SIP
method described in Section 4.5.
It is important to note that as shown in Fig. 7.4, the first control volume near the bed
extends only to
z
for solving the suspended-load transport equation (7.43),
while it extends to the bed (
z
=
z
b
+
δ
z
b
) for solving the hydrodynamic equations. It is
cumbersome that the near-bed control volumes for flow and suspended-load calcula-
tions are not identical. A simpler practice is to set the top face of the near-bed control
volume at the lower limit of the suspended-load zone. This means that the bed-load
layer occupies the entire first control volume near the bed, and the computational
domain of suspended load starts from the second control volume. Moreover, the bed-
load transport rate
q
bk
in Eq. (7.46) can be replaced by the sediment concentration
in the bed-load layer:
c
2
k
=
=
. Here, the subscript “2” denotes the center of
the first control volume in Fig. 7.4, and
u
2
is the resultant flow velocity at this cell
center. This arrangement allows the bed-load equation (7.56) to be solved together
with the suspended-load equation (7.51) using a 3-D iteration solver.
The bed change equation (7.47) is discretized as
q
bk
/(δ
u
2
)
D
n
+
1
t
1
L
(
E
n
+
1
q
n
+
1
q
n
+
1
b
z
bk
,
P
=
bk
,
P
−
bk
,
P
+
bk
,
P
−
k
,
P
)
(7.57)
∗
1
−
p
m
If the overall sediment balance equation (7.48) is used to calculate the bed change,
it is integrated in the horizontal 2-D control volume, and the resulting discretized
equation is
q
bk
u
bk
−
dz
z
s
z
b
+
δ
(
q
n
+
1
bk
u
n
+
1
bk
1
c
k
−
c
n
+
1
k
z
bk
,
P
=
+
)
p
m
1
−
t
q
n
+
1
q
n
+
1
tk
,
e
q
n
+
1
tk
,
s
q
n
+
1
+
A
P
(
˜
tk
,
w
− ˜
+ ˜
− ˜
tk
,
n
)
(7.58)
(
1
−
p
m
)
q
n
+
1
q
n
+
1
q
n
+
1
tk
,
s
q
n
+
1
where
tk
,
n
are the total-load fluxes at faces
w
,
e
,
s
, and
n
.
The total change in bed elevation is determined by
˜
tk
,
w
,
˜
tk
,
e
,
˜
, and
˜
N
z
b
,
P
=
1
z
bk
,
P
(7.59)
k
=
After the bed change is calculated, the bed elevation is updated by
z
n
+
1
b
,
P
z
b
,
P
+
=
z
b
,
P
(7.60)
or by the following Lax-type scheme (van Rijn, 1987):
z
b
,
i
,
j
+
ψ
x
2
(
z
n
+
1
b
,
i
,
j
z
b
,
i
+
1,
j
+
z
b
,
i
−
1,
j
)
=
(
1
−
ψ
−
ψ
)
x
y
+
ψ
y
2
(
z
b
,
i
,
j
+
1
+
z
b
,
i
,
j
−
1
)
+
z
b
,
i
,
j
(7.61)
