Environmental Engineering Reference
In-Depth Information
where the overbar indicates an averaged value, and the
primed quantities represent random or uncertain devia-
tions from the averaged values. In accordance with the
definition of the primed quantities,
Combining Equations (3.186) and (3.187), and selecting
coordinate axes in the principal directions of the diffu-
sion coefficient tensor, ε
ij
yields
3
3
∂
∂
c
t
∂
∂
c
x
∂
∂
∂
∂
c
x
+
∑
∑
+
V
=
(
ε
+
D
)
S
m
(3.188)
(3.179)
c
′
( ,
x
)
t
=
0
i
i
m
x
i
i
i
i
=
1
i
=
1
(3.180)
v
i
( ,
′
x
)
t
=
0
(3.181)
where ε
i
is the principal component of ε
ij
in the
x
i
direc-
tion. This equation is called the
turbulent advection-
diffusion equa
ti
on
, and relates the average
tracer
concentration,
c
, to the average velocity field,
V
i
, with
the turbulent diffusion coefficient, ε
i
, accounting for the
effects of the random velocity fluctuations. The form of
the turbulent advection-diffusion equation is the same
as the form of the general advection-diffusion equation
given by Equation (3.15), with the understanding that
the concentration, velocity, and contaminant source
fields are averaged quantities, and the turbulent diffu-
sion coefficient is associated with random perturbations
in the ambient velocity field.
S
m
( ,
′
x
)
t
=
0
Expressing the general diffusion equation in the form
∂
∂
c
t
+ ∇⋅
(
V
)
c D c S
= ∇ +
2
m
(3.182)
and substituting Equations (3.176-3.178) into Equation
(3.182) yields
3
∂
∂
∂
∂
∑
(
)
(
)
+
(
)
c
+ ′
c
c
+ ′
c V v
+ ′
i
i
t
x
i
i
=
1
3
(3.183)
∂
∂
2
∑
(
)
(
)
+
=
D
c
+ ′
c
S
+ ′
S
m
m
2
x
i
3.5.1 Relationship of Turbulent Diffusion
Coefficient to Velocity Field
i
=
1
where
D
m
is the molecular diffusion coefficient. For
incompressible fluids, conservation of mass requires
that
In deriving the advection-diffusion equation for turbu-
lent flows, Equation (3.188), it was ass
um
ed that the
turbulent mass transport denoted by
v c
i
is Fickian,
as assumed in Equation (3.187). In this section, the
conditions under which this assumption is true are
identified, and the relationship between the turbulent
diffusion coefficient, ε
i
, and the ambient velocity field is
quantified.
Consider a tracer distributed in a fluid environment,
where the
x
i
coordinate of a tracer particle at a time
t
after release from location
x
i
= 0 is given by
′ ′
3
∂
∂
∑
x
V
i
=
0
(3.184)
i
i
1
=
3
∂
∂
∑
x
v
′ =
0
(3.185)
i
i
i
1
=
Combining Equations (3.183-3.185) and taking the
ensemble average of all terms yields
t
∫
( )
=
( )
x t
v
τ τ
d
(3.189)
i
i
3
3
3
2
∂
∂
c
t
∂
∂
c
x
∂
∂
∂
∂
c
x
0
∑
∑
∑
+
V
= −
v c D
′ ′ +
+
S
m
(3.186)
i
i
m
2
x
i
i
i
i
=
1
i
=
1
i
=
1
where
v
i
is the velocity of the contaminant particle at
time
τ
. The velocity field
v
i
(
τ
) is called the
Lagrangian
velocity field, and is related to the
Eulerian
velocity
field,
V
(
x
,
t
), by the relation
The quantity
v c
i
is called the
eddy correlation
and rep-
resents the net tracer mass flux associated with the
random velocity perturbations, a process that is funda-
mentally similar to the process of molecular diffusion,
although at a much larger scale. Taking this similarity a
step further, a
turbulent diffusion coefficient,
, ε
ij
, can be
defined to relate the mass fluxes associated with random
velocity perturbations to the gradient in the average
concentration as
′ ′
v
i
( )
τ
=
V x
( ( ), )
τ τ
(3.190)
where
x
is the position vector of a particle at time
τ
. If
the particle release is repeated many times from the
same initial location, each tracer particle would end up
at a different position due to the randomness of turbu-
lent velocity fluctuations encountered during its path.
The group of realizations of particle tracks is called an
ensemble
, and the spread of the tracer particles at a time
3
∂
∂
c
x
∑
v c
′ ′ = −
ε
(3.187)
i
ij
j
j
=
1
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