Environmental Engineering Reference
In-Depth Information
Within a Lagrangian particle modeling framework, the behavior of the model
close to the boundaries and the correct estimation of the skewness of the velocity
field are of crucial importance to predict the concentration field close to the ground
(Dosio and Vilà-Guerau de Arellano, 2006). For instance, it is well known that
perfect reflection at the boundaries cannot be applied in non-Gaussian turbulence,
as the well-mixed condition requires that the distribution of particle velocities
crossing any level in a fixed time interval must be preserved. Thomson and
Montgomery (1994) proposed an exact solution for the case of positive turbu
lent velocity skewness observed in the convective boundary layer. Dosio and
Vilà-Guerau de Arellano (2006), and Mortarini et al. (2009) found that the skewness
of the relative mean concentration PDF is important to correctly predict the con-
centration field inside the canopy. Recently, Gailis et al. (2007) provided useful
parameterizations for fluctuating plume models inside a canopy, and suggested a
new analytical form for the relative intensity of concentration fluctuations.
We will analyze the influence of boundary conditions and skewness of the
particle relative mean distribution on a fluctuating plume model and a new ref-
lection condition which avoids over-estimation of the centroid PDF close to the
ground will be derived.
2. The Fluctuating Plume Model
The fluctuating plume model is based on the idea that absolute dispersion can be
divided in two different components: dispersion of the barycenter of the plume
and relative dispersion of the plume around its barycenter. The concentration
moments can be expressed as (Franzese, 2003):
∫
n
n
r
c
=
c
p
dz
(1)
m
m
where
p
m
is the barycentre PDF and
c
r
n
are the concentration moments
relative to the centroid position. Equation 1 is derived by writing the concentration
PDF,
pcx
,
(
, as the product of the barycentre position PDF,
p
m
z
,
z
( )
, and
( )
, and using the definition
c
r
n
=
c
∫
p
cr
cx
,
z
,
z
( )
dc
, where
x
is the downwind distance,
z
is the
vertical coordinate, and the subscript
m
refers to the barycentre. We use a
Lagrangian stochastich model to predict
p
m
the PDF of concentration relative to
z
m
,
p
cr
cx
,
z
,
z
m
z
,
z
( )
and we parametrise
(
)
.
p
cr
cx
,
z
,
z
m
