Geoscience Reference
In-Depth Information
6.4.2 Width-averaged 2-D sediment transport model
In the case of low sediment concentration, applying the width-averaged 2-D transport
equation (2.99) to the non-uniform suspended load yields
∂(
bC
k
)
∂
+
∂(
bU
x
C
k
)
∂
+
∂(
bU
z
C
k
)
∂
−
∂(
ω
sk
C
k
)
∂
b
t
x
z
z
bE
s
,
x
∂
bE
s
,
z
∂
=
∂
∂
C
k
∂
+
∂
∂
C
k
∂
+
S
ck
(
k
=
1, 2,
...
,
N
)
(6.123)
x
x
z
z
where
C
k
is the width-averaged concentration of the
k
th size class of suspended load;
and
E
s
,
x
and
E
s
,
y
are the effective diffusion (mixing) coefficients of sediment in the
longitudinal and vertical directions.
At the water surface, the net vertical sediment flux is zero. At the interface between
the bed-load and suspended-load layers, the deposition rate is
D
bk
=
ω
sk
C
bk
and the
entrainment rate
E
bk
is
z
=
z
b
+
δ
=
ω
sk
C
b
∗
k
E
s
,
z
∂
C
k
∂
E
bk
=−
(6.124)
z
where
C
b
∗
k
is the equilibrium concentration at the interface (reference level) deter-
mined using one of the existing formulas introduced in Section 3.5.2.
In the width-averaged 2-Dmodel, the bed-load transport is in fact a one-dimensional
process, which is determined by Eq. (5.28).
Extending Eq. (2.152) to the width-averaged 2-D model yields the bed change
equation:
∂
k
=
A
b
∂
1
L
(
p
m
)
(
1
−
B
b
(
D
bk
−
E
bk
)
+
Q
bk
−
Q
b
∗
k
)
(6.125)
t
where
B
b
is the channel width at the interface between the bed-load and suspended-load
zones.
Bed material sorting is simulated using the same multiple-layer model used in the
1-D model. For example, the bed-material gradation of the mixing layer is determined
by Eq. (5.32).
For a well-posed solution to the above set of equations, the vertical distribution of
the suspended-load concentration and the bed-load transport rate for each size class
have to be specified at the inflow boundary, and the suspended-load concentration
gradient in the flow direction is given zero at the outflow boundary.
The width-averaged suspended-load transport equation (6.123) is a typical
convection-diffusion equation, which can easily be solved by applying the numeri-
cal methods introduced in Sections 4.2 and 4.3. The near-bed sediment exchange in
the width-averaged 2-D model is similar to that in the 3-D model. The method for
handling it is discussed in Section 7.3.
Note that the width-averaged 2-D flow and sediment transport models described
above have been applied by Blumberg (1977), Karpik and Raithby (1990), Wu (1992),
