Turbulence and Dispersion of Contaminants in the Planetary Boundary Layer 47
−
= α ε
2 3 5 3
( )
t
E k
k
(3.46)
and
τ =
+ τ −
= α ε
τ
2
2 32 3
( ) [ (
)
( )]
( )
i
i
i
iS
D
v t
v t
c U
(3.47)
where
ε is the mean dissipation of energy per unit time per unit mass of fl uid (for a FDT
ε is the energy fl ux)
k = 2πn/U is the wave number
α t , α S , and c S are numerical constants
The Eulerian velocity structure function must be distinguished from the Lagrangian
velocity structure function that is described in terms of the variation in velocity of a
fl uid particle as it moves about in the turbulent fl ow. This variation can evidently
depend only on ε, which determines the local structure of the turbulence, and of course
on τ itself. Forming the only combination of ε and τ that has the correct dimensions,
we obtain for the Lagrangian velocity structure function the following relation:
τ =
+ τ −
= ετ
2
L
0
( ) [ (
)
( )]
i
i
i
i
D
v t
v t
C
(3.48)
where 0 i
C is the Kolmogorov constant. By the isotropy condition in the inertial subrange the 1-D spectra must have the same power law dependence on ε and k of
Equation 3.46, that is,
−
= α α ε
2 3 5 3
( )
i
i u
E k
k
(3.49)
where α u is determined experimentally to be about 0.5 ± 0.05 for the u-spectrum and
α i =1, 4/3, 4/3, for u, v, and w components, respectively (Monin and Yaglom, 1975;
Champagne et al., 1977; Sorbjan, 1989; Kaimal and Finnigan, 1994). The relation in
Equation 3.49 can be written as
2 3
5 3
1
( )
( )
i
i
i
u
E
U
U
U
−
ω
⎛ ⎞
Φ ω =
= α α ε
ω
⎜ ⎟
⎝ ⎠
(3.50)
and fi nally
−
α α
= πΦ π =
ε
π
2 3
5 3
2 3
( ) 2
(2 )
( )
(2 )
i u
i
i
S n
n
U n
(3.51)
In the inertial subrange, the Lagrangian energy spectrum Φ ω
L ( )
i
can depend only on
ε and ω. Consequently, dimensional consideration require that (Tennekes, 1981)
−
Φ ω =
εω
2
L
0
( )
i
i
B
(3.52)
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