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
n
m , P
z bk , P + δ
δ
z b , P )
p n + 1
bk , P =
(6.70)
n
1
m , P
+
δ
 
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