Geoscience Reference
In-Depth Information
at the bottom face of the first cell closest to the bed is zero, the vertical fluxes at the
top faces of all cells along each vertical grid line are determined using Eq. (7.13) by
sweeping from 1 to
K
. At cell center
P
, the horizontal velocities are computed using
Eq. (7.39), and the vertical velocity can be calculated from the known vertical fluxes
at faces
b
and
t
.
7.3 3-D SEDIMENT TRANSPORT MODEL
7.3.1 Governing equations and boundary conditions
Lin and Falconer (1996), Olsen and Kjellesvig (1998), and Fang and Wang (2000)
developed 3-D suspended-load transport models, whereas Wang and Adeff (1986),
van Rijn (1987), Spasojevic and Holly (1993), Wu
et al
. (2000a), and Olsen (2003)
established 3-D total-load transport models. In general, a total-load transport model
divides the moving sediment into suspended load and bed load, hence the solution
domain into the bed-load layer with a thickness of
δ
and the suspended-load layer
with a thickness of
h
, as shown in Fig. 2.6. Applying Eq. (2.72) to non-uniform
suspended-load transport yields
−
δ
∂
+
∂
[
(
u
j
−
ω
sk
δ
)
c
k
]
=
∂
∂
s
∂
c
k
∂
c
k
j
3
ε
(
k
=
1, 2,
...
,
N
)
(7.43)
t
∂
x
j
x
j
∂
x
j
where
c
k
is the local concentration of the
k
th size class of suspended load, and
ε
s
is
the turbulent diffusivity of sediment.
At the water surface, condition (2.73) is written as
+
ω
sk
c
k
z
s
∂
c
k
∂
ε
z
s
=
0
(7.44)
z
=
A number of researchers (Wang and Adeff, 1986; Olsen and Kjellesvig, 1998; Olsen,
2003) have used the “concentration” boundary condition (2.74) at the lower limit
of the suspended-load layer for solving Eq. (7.43). This implies that the near-bed
concentration is typically defined as the equilibrium concentration and evaluated using
one of the empirical formulas introduced in Section 3.5.2. However, Celik and Rodi
(1988) suggested that this treatment is inadequate under non-equilibrium conditions.
In general, the “gradient” boundary condition (2.75) should be used at the interface
between the bed-load and suspended-load layers. Van Rijn (1987), Lin and Falconer
(1996), and Wu
et al
. (2000a) determined the deposition rate as
D
bk
=
ω
sk
c
bk
and the
entrainment rate as
z
=
z
b
+
δ
=
ω
sk
c
b
∗
k
s
∂
c
k
E
bk
=−
ε
(7.45)
∂
z
where
c
b
∗
k
is the equilibrium suspended-load concentration at the interface. Therefore,
the net sediment flux
D
bk
−
E
bk
=
ω
sk
(
c
bk
−
c
b
∗
k
)
is prescribed at the interface.
