Environmental Engineering Reference
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lags when the velocity fluctuations become uncorre-
lated. Consequently, as t → ∞, both of the integral terms
in Equation (3.201) approach bounded constants such
that
3.5.2 Eulerian Approximation
The complicating factor in applying Equation (3.208) to
calculate the diffusion coefficient is that the diffusion
coefficient is expressed in terms of the Lagrangian
velocity covariance function, rather than the Eulerian
velocity covariance function that can be readily mea-
sured. In a statistically homogeneous velocity field, the
Eulerian velocity covariance function, C ij x ), is defined
by the relation
t
( )
C s ds K
=
1 ,
t
t
(3.202)
ij
0
and
t
( )
sC s ds K
=
2 ,
t
t
(3.203)
(3.209)
C ij
(
x
)
=
V
( )
x
V
′ +
(
x
x
),
ij
i
j
0
where the overbar indicates a time average (compared
with the ensemble average used in Lagrangian statis-
tics), Δ x is the separation between velocity measure-
ments, and
where K 1 and K 2 are constants that depend on the func-
tional form of C ij ( s ), and t is the time lag beyond which
the Lagrangian velocity fluctuations are uncorrelated.
Substituting Equations (3.202) and (3.203) into Equa-
tion (3.201) yields
V i ( x is the deviation of the i component of
t h e velocity at x from the (homogeneous) mean velocity,
V i , and is given by
σ ij t
( )
=
2
(
tK K
),
t
t
(3.204)
1
2
(3.210)
V
( )
x
=
V
( )
x
V
i
i
i
Taking the derivative of Equation (3.204) leads to the
useful result that
In most cases, temporal averages of Eulerian velocities
are equal to ensemble averages of Lagrangian velocities,
and such velocity fields are referred to as ergodic . The
relationship between the (measurable) Eulerian veloc-
ity covariance function, C ij x ), and the Lagrangian
velocity covariance function, C ij ( τ ), which determines
the diffusion coefficient in turbulent flows, is generally
unknown and varies with the characteristics of the flow
field. However, this relationship can be approximated
by invoking the frozen turbulence assumption (Thacker,
1977), where the time lag, τ , in the Lagrangian velocity
covariance function is related the spatial lag, Δ x , in the
Eulerian velocity covariance function by the relation
1
2
d
dt
i σ
(3.205)
=
K
,
t
t
1
which can also be written in the explicit form relating
the rate of growth of the variance of a contaminant
cloud to the covariance of the Lagrangian velocity field
1
2
d
dt
σ
ij
( )
(3.206)
=
C s ds
ij
0
The moment property of the diffusion equation ensures
that the diffusion coefficient, ε ij , is related to the rate of
growth of variance by the relation
x V
=
τ
(3.211)
where V is the mean flow velocity. The Lagrangian
velocity covariance function is then related to the Eule-
rian velocity covariance function by the approximate
relation
d
dt
σ
= 1
2
ij
(3.207)
ε
ij
and therefore the diffusion coefficient can be expressed
in terms of the Lagrangian velocity covariance function
as
(
)
( )
C ij
τ
C V
τ
(3.212)
ij
The combination of the approximate relation given by
Equation (3.212) and the exact relation given by Equa-
tion (3.208) can be used to estimate the diffusion coef-
icient, ε ij , from the (measurable) Eulerian velocity
covariance function, C ij , by the relation
0
( )
ε ij
=
ij C s ds
(3.208)
This equation was first derived by Taylor (1921), and has
been the cornerstone of many theories that relate the
diffusion coefficient to the characteristics of the ambient
velocity field (e.g., Chin and Wang, 1992).
C V
(
)
ε
τ τ
d
(3.213)
ij
ij
0
 
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