Geoscience Reference
In-Depth Information
where
a
E
,
a
W
,
a
N
,
a
S
, and
a
P
are coefficients; and
S
ck
,
P
includes the cross-derivative
diffusion terms.
The bed-load transport equation (6.54) is integrated over the control volume shown
in Fig. 4.21, with the convection terms discretized using the first-order upwind scheme
or the QUICK scheme. The discretized bed-load transport equation is
q
n
+
1
bk
,
P
u
n
+
1
q
bk
,
P
u
bk
,
P
A
P
a
q
a
q
a
q
a
q
a
q
W
q
n
+
1
E
q
n
+
1
S
q
n
+
1
N
q
n
+
1
P
q
n
+
1
bk
,
P
−
=
+
+
+
−
bk
,
W
bk
,
E
bk
,
S
bk
,
N
bk
,
P
t
+
A
P
L
(
q
n
+
1
b
q
n
+
1
k
,
P
−
bk
,
P
)
(6.66)
∗
Eqs. (6.65) and (6.66) can be iteratively solved using the Gauss-Seidel, ADI, or SIP
method.
Note that the coefficient
a
P
in the discretized suspended-load transport equation
(6.65) includes the term
F
e
F
s
, as shown in Eq. (4.135). This term can
be treated using the discretized continuity equation (6.23) for better stability. How-
ever, the coefficient
a
q
−
F
w
+
F
n
−
P
in the discretized bed-load transport equation (6.66) cannot
be treated thus. An alternative is to define a quantity
C
bk
, substitute this
relation into Eq. (6.54), and discretize the new bed-load transport equation in terms of
C
bk
as the dependent variable. The coefficient
a
P
in the resulting discretized equation
has the term
F
e
=
q
bk
/(
Uh
)
F
s
, which can then be treated using Eq. (6.23).
To ensure mass conservation, the discretizations of the exchange terms in the bed
change equation (6.55) and in the suspended-load and bed-load transport equations
(6.53) and (6.54) should be consistent. Thus, Eq. (6.55) is discretized as
−
F
w
+
F
n
−
z
bk
,
P
=
αω
sk
t
t
C
n
+
1
k
,
P
C
n
+
1
∗
q
n
+
1
q
n
+
1
b
p
m
(
−
k
,
P
)
+
L
(
bk
,
P
−
k
,
P
)
(6.67)
∗
1
−
(
1
−
p
m
)
where
z
bk
is the change in bed elevation due to the
k
th size class of sediment at time
step
t
.
After the fractional change in bed elevation has been calculated, the total change is
obtained as
N
z
b
,
P
=
1
z
bk
,
P
(6.68)
k
=
and the bed elevation is then updated by
z
n
+
1
b
,
P
z
b
,
P
+
=
z
b
,
P
(6.69)
The bed material sorting equation (6.57) is discretized as
1
m
,
P
n
+
m
,
P
p
bk
,
P
+
n
p
∗
bk
,
P
(δ
n
m
,
P
z
bk
,
P
+
δ
−
δ
−
z
b
,
P
)
p
n
+
1
bk
,
P
=
(6.70)
n
1
m
,
P
+
δ