Geoscience Reference
In-Depth Information
)
∂
U
y
∂
2
3
ρ
T
yy
=
2
ρ(ν
+
ν
y
−
k
(6.7)
t
where
t
is the eddy viscosity that needs to be
determined using a turbulence model. Introduced below are the choices for determining
ν
ν
is the kinematic viscosity of water, and
ν
t
, including the depth-averaged parabolic model, modified mixing length model, and
three depth-averaged linear
k
-
turbulence models.
Averaging the parabolic eddy viscosity equation (2.49) over the flow depth yields
ε
ν
=
α
1
U
∗
h
(6.8)
t
where
/6. However,
it has been given various values in practice, because of the anisotropic structures of
turbulence in horizontal and vertical directions and the effects of dispersion. According
to experiments by Elder (1959),
α
1
is an empirical coefficient. Theoretically,
α
1
should be equal to
κ
1
is about 0.23 for the longitudinal turbulent diffu-
sion in laboratory channels. For transverse turbulent diffusion, Fischer
et al
. (1979)
proposed that
α
1
is about 0.15 in laboratory channels and 0.6 (0.3-1.0) in irregular
waterways with weak meanders.
Eq. (6.8) is applicable in the region of main flow. Because the influence of horizontal
shear is ignored, significant errors may arise when Eq. (6.8) is applied in regions close
to rigid sidewalls. Improvement can be achieved through a combination of Eq. (6.8)
and Prandtl's mixing length theory:
α
l
h
|
S
2
2
ν
=
(α
0
U
∗
h
)
+
(
|
)
(6.9)
t
|
S
2
2
2
1
/
2
;
where
|=[
2
(∂
U
x
/∂
x
)
+
2
(∂
U
y
/∂
y
)
+
(∂
U
x
/∂
y
+
∂
U
y
/∂
x
)
]
α
0
is an empirical
coefficient similar to
α
1
in Eq. (6.8) and has a value of about
κ/
6; and
l
h
is the
, with
y
being the
distance to the nearest wall and
c
m
an empirical coefficient ranging between 0.4 and
1.2 (Wu
et al
., 2004b).
Rastogi and Rodi (1978) established a depth-averaged
k
-
y
,
c
m
h
horizontal mixing length, determined using
l
h
=
κ
min
(
)
ε
turbulence model through
depth-integration of the 3-D standard
k
-
t
is still deter-
mined by Eq. (2.54), whereas the depth-averaged turbulent energy
k
and its dissipation
rate
ε
model. The eddy viscosity
ν
ε
are calculated using the following transport equations:
ν
ν
∂
k
U
x
∂
k
U
y
∂
k
y
=
∂
σ
k
∂
k
+
∂
∂
σ
k
∂
k
t
t
+
x
+
+
P
k
+
P
kb
−
ε
(6.10)
∂
t
∂
∂
∂
x
∂
x
y
∂
y
ν
ν
2
∂ε
∂
U
x
∂ε
∂
U
y
∂ε
∂
y
=
∂
∂ε
∂
+
∂
∂
∂ε
∂
c
ε
1
ε
k
P
k
+
c
ε
2
ε
t
σ
ε
t
σ
ε
+
x
+
+
P
−
ε
b
t
∂
x
x
y
y
k
(6.11)
where
P
k
is the production of turbulence due to the horizontal velocity gradients,
defined as
P
k
|
S
2
; and
P
kb
and
P
b
are the source terms, including all terms
originating from non-uniformity of vertical profiles. The main contribution to
P
kb
=
ν
|
t
ε
