Geoscience Reference
In-Depth Information
be unity if the total turbulence kinetic energy is modeled. More discussion on these two
coefficients is given in Section 10.2.3. The values of other coefficients can be found in
Section 2.3.
Naot
et al
. (1996) and Neary (2003) applied the algebraic stress model and the
k
-
ω
turbulence model, respectively, in the simulation of flow in vegetated channels. The
details can be found in their papers.
Depth-averaged 2-D equations of flow in vegetated channels
Integrating Eqs. (10.37) and (10.38) over the flow depth yields the depth-integrated
2-D continuity andmomentum equations of flow in vegetated channels (Wu andWang,
2004b):
+
∂
[
ρ(
1
−
c
v
)
hU
y
]
∂
[
ρ(
1
−
c
v
)
h
]
+
∂
[
ρ(
1
−
c
v
)
hU
x
]
=
0
(10.43)
∂
t
∂
x
∂
y
hU
x
]
+
∂
[
ρ(
−
c
v
)
hU
y
U
x
]
∂
[
ρ(
1
−
c
v
)
hU
x
]
+
∂
[
ρ(
1
−
c
v
)
1
∂
∂
∂
t
x
y
h
∂
z
s
x
+
∂
[
(
1
−
c
v
)
h
(
T
xx
+
D
xx
)
]
+
∂
[
(
1
−
c
v
)
h
(
T
xy
+
D
xy
)
]
=−
ρ
g
(
1
−
c
v
)
∂
∂
x
∂
y
−
(
1
−
c
v
)τ
bx
−
N
a
F
dx
(10.44)
hU
y
]
+
∂
[
ρ(
1
−
c
v
)
∂
[
ρ(
1
−
c
v
)
hU
y
]
+
∂
[
ρ(
1
−
c
v
)
hU
x
U
y
]
∂
t
∂
x
∂
y
h
∂
z
s
+
∂
[
(
1
−
c
v
)
h
(
T
yx
+
D
yx
)
]
+
∂
[
(
1
−
c
v
)
h
(
T
yy
+
D
yy
)
]
=−
ρ
g
(
1
−
c
v
)
∂
y
∂
x
∂
y
−
(
1
−
c
v
)τ
by
−
N
a
F
dy
(10.45)
where
F
dx
and
F
dy
are the
x
- and
y
-components of the drag force on vegetation
exerted by the flow, defined in Eq. (10.7); and
c
v
is the depth-averaged volumetric
concentration of vegetation, defined in Eq. (10.4).
In the case of low vegetation density, Eqs. (10.43)-(10.45) are simplified as
+
∂(
hU
y
)
∂
h
t
+
∂(
hU
x
)
=
0
(10.46)
∂
∂
∂
x
y
hU
x
)
∂
+
∂(
hU
y
U
x
)
∂
∂(
hU
x
)
+
∂(
gh
∂
z
s
1
ρ
∂
[
h
(
T
xx
+
D
xx
)
]
=−
x
+
∂
∂
∂
t
x
y
x
1
ρ
∂
[
h
(
T
xy
+
D
xy
)
]
+
∂
y
1
ρ
τ
1
ρ
−
−
N
a
F
dx
(10.47)
bx
