210
Air Pollution and Turbulence: Modeling and Applications
Luhar et al. (1996) proposed a further more generalized closure, also based on the
skewness, but that has the correct property that their bi-Gaussian PDF collapses to a
Gaussian PDF in the zero skewness limit:
= σ
= − σ
=
1/3
,
,
(2/3)
A
A
B
B
w m
w
m
m
S
(8.24)
thus obtaining
1 2
2
2
(1
)
A
B
w
A
m
⎡
⎤
σ =
⎢
⎥
+
⎣
⎦
(8.25a)
1 2
2
2
(1
)
B
B
w
B
m
⎡
⎤
σ =
⎢
⎥
+
⎣
⎦
(8.25b)
1 2
1 1
2
4
r
A
r
⎡
⎤
⎛
⎞
=
−
⎢
⎥
⎜
⎟
⎝
⎠
+
⎢
⎥
⎣
⎦
(8.25c)
1
B
A
= −
(8.25d)
+
=
+
2 3 2
2
2 2
(1
)
(3
)
m S
r
m
m
(8.25e)
8.3.2 GRAM–CHARLIER PDF
The Gram–Charlier PDF, truncated to the fourth order (GC4), has the following
form (Kendall and Stuart, 1977):
(
)
2 2
3 3
4 4
( , )
1
2
x
e
P x z
C H C H
−
=
+
+
π
(8.26)
in which H 3 and H 4 are Hermite polynomials and C 3 and C 4 their coeffi cients, whose
expressions are
3
4
2
3
4
3
6
3
H x
x
H
x
x
= −
= −
+
(8.27a)
3
4
3
4
6,
(
3) 24
C
C
= μ
= μ −
(8.27b)
where
3
μ and
4
μ are the standardized moments of w and x = w/σ w .
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