Geoscience Reference
In-Depth Information
where
C
k
is the cross-section-averaged concentration of the
k
th size class of sus-
pended load.
In the 3-D and depth-averaged 2-D models, the transport equation of bed load in
vegetated channels is
+
∂
[
α
by
(
1
−
c
vb
)
q
bk
]
∂
[
(
1
−
c
vb
)
q
bk
/
u
bk
]
+
∂
[
α
bx
(
1
−
c
vb
)
q
bk
]
∂
t
∂
x
∂
y
1
L
(
=
−
c
vb
)(
q
b
∗
k
−
q
bk
)
1
(10.64)
where
c
vb
is the volumetric concentration of vegetation in the bed-load zone.
In the 1-D model, the transport equation of bed load in vegetated channels is
∂
[
(
1
−
c
vb
)
Q
bk
/
U
bk
]
+
∂
[
(
1
−
c
vb
)
Q
bk
]
1
L
(
=
1
−
c
vb
)(
Q
b
∗
k
−
Q
bk
)
+
q
lbk
∂
∂
t
x
(10.65)
where
Q
bk
is the transport rate of the
k
th size class of bed load.
For low vegetation density, Eqs. (10.61)-(10.65) can be simplified by eliminating
the vegetation concentration.
The vegetation densities in the water column and channel bed might be different, but
this complexity is ignored in the present models. Therefore, the bed change and bed
material sorting equations in vegetated channels are the same as those in non-vegetated
channels. For example, in the depth-averaged 2-D model, the bed change and the
bed-material gradation in the mixing layer in vegetated channels are determined by
Eqs. (2.159) and (2.161), respectively.
To close the sediment transport models introduced above, the sediment transport
capacities need to be computed using empirical formulas. As described in Section
10.1.3, the work of Okabe
et al
. (1997) suggests that the Ashida-Michiue (1972)
bed-load formula can be used in vegetated channels. According to Wu
et al
. (2005),
the Wu
et al
. (2000b) formula can be extended to determine the sediment transport
capacity in the case of emergent vegetation. In the case of submerged vegetation, the
total bottom shear stress determined using Eq. (10.16) with Eq. (10.22) for
R
s
includes
both the shears from the bed and at the top of vegetation elements. Considering only
the shear on the bed affects the transport of bed load, the total bottom shear stress
should be modified by multiplying a factor
h
v
/
h
when the bed-load transport capacity
is computed. This factor
h
v
h
is derived by considering Eq. (10.20) and the first term
on the right-hand side of Eq. (10.21). However, this modification is not necessary in
the 3-D model that directly calculates the bed shear stress.
The aforementioned sediment transport equations in vegetated channels can be
solved by straightforwardly extending the numerical methods presented in Chapters
5-7 for sediment transport models in non-vegetated channels. The factor 1
/
−
c
v
can be
eliminated from the left-hand sides of Eqs. (10.61)-(10.65), or treated by replacing
ρ
with
in the finite volume method, as described in Section 10.2.2. The details
can be found in Wu and Wang (2004b) and Wu
et al
. (2005).
ρ(
1
−
c
v
)
