48
Air Pollution and Turbulence: Modeling and Applications
where 0 i
B is another nondimensional constant. In terms of the frequency n, Equation
3.52 can be written as
−
πΦ
π =
ε
π
0
2
L
2
(2 )
2
i
i
B
n
n
(3.53)
and yields
−
= ε
2
L ( )
i
i
S n
B n
(3.54)
where =
π
0 2
i
i
B B
.
The use of the equations discussed above allows to obtain a theoretical expression for the fundamental scale factor β i . Following Corrsin (1963), let us integrate
Equations 3.51 and 3.54 to obtain an expression for the scale factor β i :
( )
∞
∞
−
⎛
⎞
α α
α α
ε
σ =
=
ε
=
⎜
⎟
⎝
⎠
π
π
∫
∫
E
E
2 3
2
2 35 3
2 3
2 3
E
3
( )d
( )
d
2 (2 )
2
i u
i u
i
i
n
n
U
S n n
U
n
n
n
(3.55)
and
∞
∞
ε
σ =
= ε
= π
∫
∫
L
L
0
2
L
L
2
L
d
1
d ( )
2
i
i
i
i
n
n
B
n
S n n
B
n
n
(3.56)
where n E and n L are, respectively, the Eulerian and Lagrangian initial frequencies
of the inertial subrange. These characteristic frequencies may be regarded as the
inverse of the timescales.
Equation 3.56 can also be rewritten in the form:
π
ε =
σ
2 3
2 3
2 3
4 3
L
L
2 3
0
(2 )
i
i
n
B
(3.57)
Since the mean energy dissipation rate ε is equal for both Lagrangian and Eulerian
frames of reference we can substitute Equation 3.57 in Equation 3.55 to obtain
σ
= α α
σ
4 3
2 3
2 3
L
E
2 3
2 3
2
L
0
3
2
i
i
i u
i
n
U
n
B
(3.58)
At this point, we are going to assume that the Lagrangian and Eulerian variances of
the turbulent velocity are equal σ = σ
2
L
(
)
i
i . Here, this equivalence will be identifi ed
by substituting both variances with σ i . Therefore, from Equation 3.58, the scale factor β i can be expressed as (Wandel and Kofoed-Hansen, 1962; Angell et al., 1971;
Pasquill, 1974; Hanna, 1981)
β =
=
= γ σ
L
E
L
i
i
i
i
T
n
U
T
n
(3.59)
© 2010 by Taylor and Francis Group, LLC
Air Pollution and Turbulence: Modeling and Applications
where 0 i
B is another nondimensional constant. In terms of the frequency n, Equation
3.52 can be written as
−
πΦ
π =
ε
π
0
2
L
2
(2 )
2
i
i
B
n
n
(3.53)
and yields
−
= ε
2
L ( )
i
i
S n
B n
(3.54)
where =
π
0 2
i
i
B B
.
The use of the equations discussed above allows to obtain a theoretical expression for the fundamental scale factor β i . Following Corrsin (1963), let us integrate
Equations 3.51 and 3.54 to obtain an expression for the scale factor β i :
( )
∞
∞
−
⎛
⎞
α α
α α
ε
σ =
=
ε
=
⎜
⎟
⎝
⎠
π
π
∫
∫
E
E
2 3
2
2 35 3
2 3
2 3
E
3
( )d
( )
d
2 (2 )
2
i u
i u
i
i
n
n
U
S n n
U
n
n
n
(3.55)
and
∞
∞
ε
σ =
= ε
= π
∫
∫
L
L
0
2
L
L
2
L
d
1
d ( )
2
i
i
i
i
n
n
B
n
S n n
B
n
n
(3.56)
where n E and n L are, respectively, the Eulerian and Lagrangian initial frequencies
of the inertial subrange. These characteristic frequencies may be regarded as the
inverse of the timescales.
Equation 3.56 can also be rewritten in the form:
π
ε =
σ
2 3
2 3
2 3
4 3
L
L
2 3
0
(2 )
i
i
n
B
(3.57)
Since the mean energy dissipation rate ε is equal for both Lagrangian and Eulerian
frames of reference we can substitute Equation 3.57 in Equation 3.55 to obtain
σ
= α α
σ
4 3
2 3
2 3
L
E
2 3
2 3
2
L
0
3
2
i
i
i u
i
n
U
n
B
(3.58)
At this point, we are going to assume that the Lagrangian and Eulerian variances of
the turbulent velocity are equal σ = σ
2
L
(
)
i
i . Here, this equivalence will be identifi ed
by substituting both variances with σ i . Therefore, from Equation 3.58, the scale factor β i can be expressed as (Wandel and Kofoed-Hansen, 1962; Angell et al., 1971;
Pasquill, 1974; Hanna, 1981)
β =
=
= γ σ
L
E
L
i
i
i
i
T
n
U
T
n
(3.59)
© 2010 by Taylor and Francis Group, LLC