Environmental Engineering Reference
In-Depth Information
t
after release is called turbulent diffusion. Let 〈
v
i
〉 be
the ensemble-averaged velocity experienced by the
released particles, which is independent of space and
time for statistically homogeneous velocity fields.
Further, let
x
i
be the coordinates relative to an origin
moving with the mean velocity 〈
v
i
〉, where
′
area =
K
1
′
x t
( )
=
x t
( )
−
v t
(3.191)
i
i
i
0
v
i
is the
i
component of the velocity relative to the
average velocity 〈
v
i
〉, where
If
′
separation (s)
Figure 3.22.
Typical Lagrangian covariance function.
′
v t
( )
=
v t
( )
−
v
(3.192)
i
i
i
If the Lagrangian velocity is statistically homogeneous,*
then the covariance between the velocities at times
τ
1
and
τ
2
depends only on the time interval
τ
2
−
τ
1
, and not
on the individual values of
τ
1
and
τ
2
. under these condi-
tions, Equation (3.197) can be written as
then substituting Equations (3.191) and (3.192) into
Equation (3.189) leads to
t
t
∫
∫
[
+
′
( )
]
′
( )
x
′
+
v t
=
v
v
τ
d
τ
=
v t
+
v
τ τ
d
i
i
i
i
i
i
0
0
(
)
=
′
(
)
′
(
)
(3.198)
C
τ
−
τ
v
τ
v
τ
(3.193)
ij
2
1
i
1
j
2
Substituting Equation (3.198) into Equation (3.196)
leads to
and therefore
t
∫
′
( )
=
′
( )
(3.194)
x t
v
τ τ
d
t
t
i
i
∫
∫
( )
=
(
)
σ
t
C
τ
−
τ
d d
τ τ
(3.199)
0
ij
ij
2
1
1
2
0
0
This expression describes the deviation of the location
of a tracer particle from the average position that par-
ticle would have at time
t
after an infinite number of
particle-track realizations. In studying the transport
process, our interest lies in the
variability
in the tracer
particle locations after a large number of realizations.
This variability is measured by the variance of the tracer
particle distribution,
σ
ij
(
t
), and is related to
Changing the variables in the integrand from (
τ
1
,
τ
2
) to
(
s
,
τ
), where
s
=
τ
2
−
τ
1
and
τ
= (
τ
1
+
τ
2
)/2, Equation
(3.199) can be integrated once to yield
t
2
0
( )
=
(
)
( )
σ
ij
t
t
−
s C s ds
(3.200)
ij
x
i
′
by
This equation provides a direct relationship between
the variance of the distribution of tracer particles (
σ
ij
)
and the statistics of the Lagrangian velocity field (
C
ij
)
causing the spread of tracer particles. This relationship
is fundamental to the study of turbulent diffusion,
since the statistics of the velocity field can usually be
estimated. Expanding the integral in Equation (3.200)
leads to
( )
=
′
( )
′
( )
σ
ij
t
x t x t
(3.195)
i
j
Combining Equations (3.194) and (3.195) leads to
t
t
∫
∫
′
( )
′
( )
σ
=
v
τ τ
d
v
τ τ
d
ij
i
j
0
0
t
t
∫
∫
′
(
)
′
(
)
=
v
τ
v
τ
d d
τ τ
(3.196)
i
1
j
2
1
2
t
t
0
0
∫
∫
( )
=
( )
( )
σ
ij
t
2
t C s ds
−
sC s ds
(3.201)
ij
ij
t
t
∫
∫
′
(
)
′
(
)
0
0
=
v
τ
v
τ
d d
τ τ
i
1
j
2
1
2
0
0
A typical Lagrangian velocity covariance function,
C
ij
(
s
), is shown in Figure 3.22, where a key feature of
C
ij
(
s
) is that the function approaches zero at large time
The integrand on the right-hand side of Equation (3.196)
is the covariance of the Lagrangian velocity field,
C
ij
(
τ
1
,
τ
2
), where
*
Statistical homogeneity means that the statistical properties of the
velocity time series are independent of time.
(
)
=
′
(
)
′
(
)
C
τ τ
,
v
τ
v
τ
(3.197)
ij
1
2
i
1
j
2
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