Geoscience Reference
In-Depth Information
hU
y
)
∂
+
∂(
∂(
hU
y
)
+
∂(
hU
x
U
y
)
∂
[
h
(
T
yx
+
D
yx
)
]
gh
∂
z
s
1
ρ
=−
y
+
∂
t
∂
x
y
∂
∂
x
1
ρ
∂
[
h
(
T
yy
+
D
yy
)
]
+
∂
y
1
ρ
τ
1
ρ
−
−
N
a
F
dy
(10.48)
by
R
1
/
s
,
in which
n
b
is the Manning roughness coefficient of the bed and
R
s
is the hydraulic
radius defined in Eqs. (10.17) and (10.22) or simply set as the flow depth
h
.
The depth-averaged stresses
T
ij
are calculated by Eq. (6.7), with the eddy viscosity
gn
b
/
The bed shear stresses
τ
bx
and
τ
by
are determined by Eq. (6.4), with
c
f
=
ν
t
determined using Eq. (2.54) and the turbulent energy
k
and its dissipation rate
ε
determined by
ν
ν
∂
k
U
x
∂
k
U
y
∂
k
y
=
∂
σ
k
∂
k
+
∂
∂
σ
k
∂
k
t
t
t
+
x
+
∂
∂
∂
∂
x
∂
x
y
∂
y
+
P
k
+
P
kv
+
P
kb
−
ε
(10.49)
ν
ν
∂ε
∂
U
x
∂ε
∂
U
y
∂ε
∂
y
=
∂
∂ε
∂
+
∂
∂
∂ε
∂
t
σ
ε
t
σ
ε
t
+
x
+
∂
x
x
y
y
2
1
ε
k
(
2
ε
+
c
P
k
+
c
f
ε
P
k
ν
)
+
P
−
c
(10.50)
ε
ε
b
ε
k
where
P
k
,
P
kb
, and
P
b
are defined in Eqs. (6.10) and (6.11); and
P
kv
is the generation of
turbulence due to vegetation, determined by
P
kv
=
ε
.
The dispersion momentum transports
D
ij
due to the non-uniformity of flow velocity
along the flow depth are determined using the models introduced in Section 6.3 for
curved channels, or combined with the turbulent stresses otherwise.
c
fk
N
a
(
F
dx
U
x
+
F
dy
U
y
)/
[
ρ(
1
−
c
v
)
]
1-D equations of flow in vegetated channels
Integrating Eqs. (10.43) and (10.44) over the channel width yields the 1-D continuity
and momentum equations in vegetated channels:
∂
[
ρ(
1
−
c
v
)
A
]
+
∂
[
ρ(
1
−
c
v
)
Q
]
=
ρ
q
l
(10.51)
∂
t
∂
x
Q
2
∂
[
ρ(
1
−
c
v
)
Q
]
+
∂
∂
ρβ(
1
−
c
v
)
A
∂
z
s
+
ρ
g
(
1
−
c
v
)
x
+
ρ
g
(
1
−
c
v
)
AS
f
=
ρ
q
l
v
x
∂
t
x
A
∂
(10.52)
where
c
v
is the volumetric concentration of vegetation averaged the cross-section; and
S
f
is the friction slope, including the effects of bed friction and vegetation drag:
B
Q
|
Q
|
S
f
=
S
fb
+
A
N
a
F
d
=
(10.53)
ρ
g
(
1
−
c
v
)
K
2
