Geoscience Reference
In-Depth Information
vegetation elements. Time- and space-averaging the Navier-Stokes equations yields the
3-D governing equations for flow in vegetated channels:
∂
[
ρ(
1
−
c
v
0
)
]
+
∂
[
ρ(
1
−
c
v
0
)
u
i
]
=
0
(10.37)
∂
t
∂
x
i
∂
[
ρ(
1
−
c
v
0
)
u
i
]
+
∂
[
ρ(
1
−
c
v
0
)
u
i
u
j
]
)
∂
p
=
ρ(
1
−
c
v
0
)
F
i
−
(
1
−
c
v
0
∂
t
∂
x
j
∂
x
i
+
∂
[
(
1
−
c
v
0
)τ
]
ij
−
N
a
f
di
(10.38)
∂
x
j
where
f
di
(
are the components of drag force per unit vegetation height,
defined in Eq. (10.8); and
c
v
0
is the local volumetric concentration of vegetation,
defined in Eq. (10.5).
Compared with the equations derived by Shimizu and Tsujimoto (1994) and Lopez
and Garcia (2001), Eqs. (10.37) and (10.38) include the vegetation concentration
c
v
0
,
which changes in time and space due to the heterogeneity and seasonal growth and
death of vegetation as well as the change of flow conditions. This should be particularly
important in the case of high vegetation density. In the case of low vegetation density,
1
i
=
1, 2, 3
)
−
c
v
0
≈
1 and, thus, Eqs. (10.37) and (10.38) can be simplified as
∂
u
i
x
i
=
0
(10.39)
∂
∂
u
i
∂
+
∂(
u
i
u
j
)
1
ρ
∂
p
1
ρ
∂τ
1
ρ
ij
=
F
i
−
x
i
+
x
j
−
N
a
f
di
(10.40)
t
∂
x
j
∂
∂
τ
ij
include the effects of molecular viscosity, turbulence, and non-
uniformity of flow velocity around vegetation elements. The last effect causes
dispersion, which is often combined with the turbulent effect. Thus, the stresses are
calculated using the Boussinesq assumption (7.3), with the eddy viscosity
The stresses
ν
t
deter-
mined by Eq. (2.54) and the turbulent energy
k
and its dissipation rate
ε
determined
by (Shimizu and Tsujimoto, 1994)
ν
∂
k
u
j
∂
x
j
=
∂
k
σ
k
∂
k
t
+
+
P
k
+
P
kv
−
ε
(10.41)
∂
t
∂
∂
x
j
∂
x
j
ν
2
∂ε
∂
u
j
∂ε
∂
x
j
=
∂
∂ε
∂
c
ε
1
ε
k
(
c
ε
2
ε
t
σ
ε
t
+
+
P
k
+
c
f
ε
P
kv
)
−
(10.42)
∂
x
j
x
j
k
where
P
k
is the production of turbulence by shear, defined in Eq. (2.52); and
P
kv
is the generation of turbulence due to vegetation, determined by
P
kv
=
c
fk
N
a
f
di
u
i
/
[
ρ(
1
−
c
v
0
)
]
. Shimizu and Tsujimoto (1994) selected coefficients
c
fk
=
0.07
and
c
f
ε
=
0.16 based on calibration, whereas Lopez and Garcia (2001) determined
coefficients
c
fk
=
1.0 and
c
f
ε
=
1.33 based on a theoretical argument that
c
fk
should
