An Outline of Lagrangian Stochastic Dispersion Models
205
of the fi rst two or three, sometimes four, moments of the probability density function
(PDF) of the three wind velocity components pertaining to the point at which each
particle is. This input information is given on a regular grid and comes either from
measurements or from parameterizations appropriate to the actual stability conditions (unstable, neutral, stable), the type of site (fl at or complex terrain, coast, etc.),
the time, and space scales considered.
Thanks to the present computer power, the use of LSDM is growing even as
regulatory models (see, for instance, the offi cial German Federal Environmental
Agency air dispersion model—AUSTAL2000: Luft TA 2002).
8.2 GENERALITIES ON LAGRANGIAN STOCHASTIC MODELS
Thomson (1987) showed that the criterion for selecting the correct model for the
diffusion of scalars in a turbulent fl ow is the “well-mixed condition”: particles that
are initially uniformly distributed must remain so. The LSDM that are based on
the generalized Langevin equation satisfy the well-mixed condition. It is worth
noting that these models have a unique solution in one dimension only, while they
do not have a unique solution in two- or three-dimensional fl ows (Sawford and
Guest, 1988).
The position of each particle, at each time step, is obtained by numerically integrating the following 3-D equations (Thomson, 1987; Anfossi and Physick, 2005;
Ferrero, 2005):
=
+
d
( , , )
( , , )d ( )
i
i
i j
j
u a x u t
b x u t W t
(8.2)
=
⋅
d ( )
( ) d
i
i
x t
u t t
(8.3)
where
x i is the position vector of each particle
u i its corresponding Lagrangian velocity vector
dW j is the incremental Wiener process that is Gaussian with zero mean and a variance of dt (random velocity fl uctuation)
The term a i of Equation 8.2 is a deterministic term, representing the friction force
exerted by the fl ow on the particle, and the term b i,j is a stochastic term, representing
the random accelerations caused by pressure fl uctuations. b i,j is obtained from the
Kolmogorov theory of local isotropy in the inertial subrange (Monin and Yaglom,
1965; Thomson, 1987) and has the following expression:
= δ
⋅ ε
0
ij
ij
b
C
(8.4)
where
ε is the dissipation rate of turbulent kinetic energy
C 0 is a numerical constant
© 2010 by Taylor and Francis Group, LLC
205
of the fi rst two or three, sometimes four, moments of the probability density function
(PDF) of the three wind velocity components pertaining to the point at which each
particle is. This input information is given on a regular grid and comes either from
measurements or from parameterizations appropriate to the actual stability conditions (unstable, neutral, stable), the type of site (fl at or complex terrain, coast, etc.),
the time, and space scales considered.
Thanks to the present computer power, the use of LSDM is growing even as
regulatory models (see, for instance, the offi cial German Federal Environmental
Agency air dispersion model—AUSTAL2000: Luft TA 2002).
8.2 GENERALITIES ON LAGRANGIAN STOCHASTIC MODELS
Thomson (1987) showed that the criterion for selecting the correct model for the
diffusion of scalars in a turbulent fl ow is the “well-mixed condition”: particles that
are initially uniformly distributed must remain so. The LSDM that are based on
the generalized Langevin equation satisfy the well-mixed condition. It is worth
noting that these models have a unique solution in one dimension only, while they
do not have a unique solution in two- or three-dimensional fl ows (Sawford and
Guest, 1988).
The position of each particle, at each time step, is obtained by numerically integrating the following 3-D equations (Thomson, 1987; Anfossi and Physick, 2005;
Ferrero, 2005):
=
+
d
( , , )
( , , )d ( )
i
i
i j
j
u a x u t
b x u t W t
(8.2)
=
⋅
d ( )
( ) d
i
i
x t
u t t
(8.3)
where
x i is the position vector of each particle
u i its corresponding Lagrangian velocity vector
dW j is the incremental Wiener process that is Gaussian with zero mean and a variance of dt (random velocity fl uctuation)
The term a i of Equation 8.2 is a deterministic term, representing the friction force
exerted by the fl ow on the particle, and the term b i,j is a stochastic term, representing
the random accelerations caused by pressure fl uctuations. b i,j is obtained from the
Kolmogorov theory of local isotropy in the inertial subrange (Monin and Yaglom,
1965; Thomson, 1987) and has the following expression:
= δ
⋅ ε
0
ij
ij
b
C
(8.4)
where
ε is the dissipation rate of turbulent kinetic energy
C 0 is a numerical constant
© 2010 by Taylor and Francis Group, LLC