Geoscience Reference
In-Depth Information
The 3-D
k
- and
ε
-equations considering the buoyancy effects are written as
(Rodi, 1993)
ν
∂
k
u
i
∂
x
i
=
∂
k
σ
k
∂
k
t
+
+
P
k
+
G
k
−
ε
(12.18)
∂
t
∂
∂
x
i
∂
x
i
ν
t
σ
ε
2
∂ε
∂
u
i
∂ε
∂
x
i
=
∂
∂ε
∂
c
ε
1
ε
k
(
c
ε
2
ε
+
+
P
k
+
c
ε
3
G
k
)
−
(12.19)
t
∂
x
i
x
i
k
where
c
3
is a coefficient; and
G
k
is the buoyancy product of
turbulence,
ε
determined by
ε
φ
∂φ
∂
G
k
=
β
g
i
(12.20)
x
i
where
is the volumetric expansion coefficient;
g
i
is the gravitational body force per
unit mass in the
x
i
-direction; and
β
φ
denotes the scalar quantity, such as temperature
and salinity.
When
G
k
>
0,
c
ε
3
is given a value of 1.0; and when
G
k
<
0,
c
ε
3
is in the range of
0-0.2 (ASCE Task Committee, 1988).
12.1.3 Effective diffusivities
The 1-D and 2-D heat and salinity transport equations should have dispersion terms
when they are derived by integrating the corresponding 3-D transport equations.
These dispersion terms are herein combined with the turbulent diffusion terms, so
the effective diffusivity (mixing coefficient) includes the molecular diffusivity
ε
m
, tur-
bulent diffusivity
ε
d
. Usually, the molecular diffusivity
is approximately equal to the kinematic viscosity of water. The turbulent diffusivity
is related to flow conditions, as described in Section 6.1. Normally, we have
ε
t
, and dispersion coefficient
t
.
The dispersion effect results from the non-uniformity of flow velocity and constituent
concentration along the flow depth and/or the channel width. Elder (1959) estimated
the longitudinal dispersivity as
ε
∝
ν
t
ε
=
5.86
hU
∗
(12.21)
d
ε
m
ε
t
ε
d
. The magnitude of the longitudinal effective
It can be seen that
diffusivity
E
is
E
=
ε
+
ε
+
ε
d
≈
6.0
hU
∗
(12.22)
m
t
Iwasa and Aya (1991) related the longitudinal effective diffusivity to the channel
width/depth ratio,
B
/
h
,as
2.0
B
h
1.5
E
hU
=
(12.23)
∗
