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interface, and
q
bk
is the potential equilibrium transport rate of the
k
th size class of
bed load.
The bed material sorting in the 3-D model is handled with the same multiple-
layer approach used in the depth-averaged 2-D model. For example, the bed-material
gradation of the mixing layer is determined by Eq. (6.57).
For a well-posed solution, the bed-load transport rates and suspended-load con-
centrations of all size classes have to be specified at the inflow boundary. If the total
discharges of bed load and suspended load are specified at the inlet, their specific dis-
charges at each vertical grid line of the inlet can be determined using Eq. (6.63), and
then, the vertical distribution of local suspended-load concentration can be determined
according to the Rouse or Lane-Kalinske distribution.
The sediment boundary conditions at solid and outflow boundaries and initial
conditions in the 3-D model are similar to those in the depth-averaged 2-D model.
7.3.2 Discretization of sediment transport equations
To solve the suspended-load transport equation (7.43), the sediment settling term
∂(ω
sk
c
k
)/∂
z
can be treated as a source term or combined with the vertical convection
term. After considerable testing, Wu
et al
. (2000a) suggested the former approach
might be better. This term can be evaluated using the central or forward difference
scheme in the vertical direction. The central difference scheme has better accuracy, but
the forward difference scheme has better stability.
Eq. (7.43) can be discretized using the numerical methods introduced in Sections 4.2
and 4.3. The finite volume method is chosen here as an example. The discretized
suspended-load transport equation is
V
n
+
1
P
c
n
+
1
k
,
P
V
P
c
k
,
P
−
a
W
c
n
+
1
k
,
W
a
E
c
n
+
1
k
,
E
a
S
c
n
+
1
k
,
S
a
N
c
n
+
1
k
,
N
=
+
+
+
t
a
B
c
n
+
1
k
,
B
a
T
c
n
+
1
k
,
T
a
P
c
n
+
1
k
,
P
+
+
−
+
S
k
,
P
(7.51)
As presented by Wu
et al
. (2000a), boundary conditions (7.44) and (7.45) are
implemented by prescribing fluxes at the water and bed surfaces, respectively, which
are depicted in Fig. 7.4. An important choice has to be made as to the reference
level at which the equilibrium concentration
c
b
and hence the entrainment rate
E
b
are determined. In general, the reference level is set at the top of the bed-load
layer.
To determine the deposition rate
D
bk
, it is necessary to calculate the concentration
c
bk
at
z
∗
from
c
k
values at neighboring grid points. Wu
et al
. (2000a) assumed
that the concentration distribution between
z
=
z
b
+
δ
and the first grid (point 2 in
Fig. 7.4) is governed by the following equation, which is simplified from Eq. (7.43) by
ignoring the storage, convection, and horizontal diffusion effects:
=
z
b
+
δ
ε
+
ω
sk
c
k
∂
∂
s
∂
c
k
∂
=
0
(7.52)
z
z
