Environmental Engineering Reference
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applied to extract, from the measured data, only the length-scales involved in the
meandering motion.
When the distance between centroid vertical position and a boundary is smaller
than the characteristic radius of the plume bulk, the barycentre is reflected.
Franzese (2003) and Mortarini et al. (2009) used perfect reflection, in this work,
not to over-estimate the centroid PDF near the ground, we extend the Thomson
and Montgomery (1994) boundary conditions to negative skewness, typical of
canopy flows. In good agreement with the results of Anfossi et al. (1997) in
convective turbulence, our numerical integration of the velocity PDF shows that
the reflected barycentre velocity is a function of the incident velocity and of the
velocity skewness. Thus an analytical function is found and is used as boundary
condition for the barycentre trajectories.
4. The PDF of Concentration in the Relative Frame
Yee et al. (1994) proposed the following form for
p
cr
, which has been used in
several other studies (e.g., Yee and Wilson, 2000; Luhar et al., 2000; Dosio and
Vilà-Guerau de Arellano, 2006; Gailis et al., 2007; Mortarini et al., 2009):
⎛
⎝
⎜
⎞
⎠
⎟
c
α−1
Γ
(
β
α
exp −
c
(
)
=
p
cr
cz
,
z
m
(4)
β
with
α=1
i
r
2
and
β=
c
r
α
, where
c
r
is the mean concentration in the
relative frame and
i
r
the intensity of concentration fluctuations. Substituting Eq. 4
in the definition of
c
r
n
gives:
)
(
=
Γ
n
+α
c
r
n
Γ
()
β
α
c
r
(5)
and finally substituting Eq. 5 in Eq. 1:
)
(
=
Γ
n
+α
H
∫
c
n
Γ
()
β
α
c
r
n
p
m
dz
m
(6)
0
Providing a functional form that express
α
and
β
as functions of the mean
concentration and intensity of concentration fluctuations in the relative frame, the
fluctuating plume model is able to reproduce all the concentration moments.
A simple parametrisation of
i
r
has been given by Gailis et al. (2007):
