Geoscience Reference
In-Depth Information
representing the width-averaged quantities, is omitted:
∂(
bU
x
)
+
∂(
bU
z
)
=
0
(6.103)
∂
x
∂
z
bU
x
)
∂
∂(
bU
x
)
+
∂(
+
∂(
bU
z
U
x
)
b
1
ρ
∂
p
1
ρ
∂(
bT
xx
)
1
ρ
∂(
bT
xz
)
=−
x
+
+
∂
t
x
∂
z
∂
∂
x
∂
z
1
ρ
(
−
m
1
τ
+
m
2
τ
)
(6.104)
x
1
x
2
bU
z
)
∂
∂(
bU
z
)
+
∂(
bU
x
U
z
)
+
∂(
b
1
ρ
∂
p
1
ρ
∂(
bT
zx
)
1
ρ
∂(
bT
zz
)
=−
bg
−
z
+
+
∂
t
∂
x
z
∂
∂
x
∂
z
1
ρ
(
−
m
1
τ
+
m
2
τ
)
(6.105)
z
1
z
2
where
x
is the longitudinal coordinate;
z
is the vertical coordinate above a datum;
U
x
and
U
z
are the width-averaged flow velocities in the
x
- and
z
-directions, respectively;
p
is the width-averaged pressure; and
T
ij
(
i
,
j
=
x
,
z
)
are the width-averaged stresses:
)
∂
U
x
∂
2
3
ρ
T
xx
=
2
ρ(ν
+
ν
−
k
t
x
∂
U
x
∂
+
∂
U
z
∂
T
xz
=
T
zx
=
ρ(ν
+
ν
)
(6.106)
t
z
x
)
∂
U
z
∂
2
3
ρ
T
zz
=
2
ρ(ν
+
ν
−
k
t
z
where the eddy viscosity
t
can be determined using Prandtl's mixing length model
(2.48), the parabolic model (2.49), or Eq. (2.54) in the linear
k
-
ν
ε
turbulence models.
In the width-averaged 2-D
k
-
ε
turbulence models, the turbulent energy
k
and its
dissipation rate
ε
are determined by
ν
t
ν
t
∂
k
U
x
∂
k
U
z
∂
k
=
∂
∂
σ
k
∂
k
+
∂
∂
σ
k
∂
k
+
x
+
+
P
k
+
P
kl
−
ε
(6.107)
∂
t
∂
∂
z
x
∂
x
z
∂
z
ν
ν
2
∂ε
∂
U
x
∂ε
∂
U
z
∂ε
∂
=
∂
∂
∂ε
∂
+
∂
∂
∂ε
∂
c
ε
1
ε
k
(
c
ε
2
ε
t
σ
ε
t
σ
ε
+
x
+
+
P
k
+
c
l
P
kl
)
−
ε
t
z
x
x
z
z
k
(6.108)
2
where
P
k
is the production of turbulence due to shear, defined as
P
k
=
ν
t
[
(∂
U
x
/∂
)
+
2
x
2
2
2
(∂
U
z
/∂
z
)
+
(∂
U
x
/∂
z
+
∂
U
z
/∂
x
)
]
;
P
kl
accounts for the generation of turbulence due
to bank shear, modeled by
P
kl
=
c
kl
[
(
m
1
τ
+
m
2
τ
)
U
x
+
(
m
1
τ
+
m
2
τ
)
U
z
]
/ρ
; and
x
1
x
2
z
1
z
2
σ
k
, and
σ
ε
are coefficients. The values of
c
ε
1
,
c
ε
2
,
σ
k
, and
σ
ε
are listed
c
ε
1
,
c
ε
2
,
c
kl
,
c
ε
l
,
in Table 2.3, while
c
kl
and
c
l
are about 1.0 and 1.33, respectively.
If the flow depth is very small in comparison with the flow width, the bank shear
stresses
ε
τ
xi
and
τ
zi
are negligible; otherwise, they should be approximated by friction
