204
Air Pollution and Turbulence: Modeling and Applications
Lagrangian stochastic dispersion models (LSDM), also named Lagrangian particle models (LPM), are numerical models aimed at the simulation of these processes,
being able to account for fl ow and turbulence space-time variations. Emissions in
the atmosphere are simulated using a number of fi ctitious particles named “ computer
particles.” Each particle represents a specifi ed pollutant mass. We call particle a fl uid
portion containing the emitted substance, having dimensions appropriate to follow
the motion of the smallest turbulence eddies present in the atmosphere (of the order
of the Kolmogorov scale), but containing a number of molecules large enough to
allow disregarding the effect of the single molecule. Under the hypothesis, accurately demonstrated, that dispersion due to molecular motion is negligible compared
to turbulent dispersion, it can be thought that these particles possess a concentration of their own that is preserved during the motion. It is assumed that the particles passively follow the turbulent motion of air masses in which they are. Particles
are moved following the turbulent eddies, thus describing random trajectories. To
prescribe this behavior, particle velocities are subject to a random forcing, thus these
are stochastic type models, whereas Eulerian models are deterministic. As a consequence, the emitted mass concentration can be calculated from the space distribution
of particles at a particular time.
Particle mean motion in the computation domain, which simulates the airborne
pollutant motion in the real domain (atmosphere), is prescribed by the local mean
wind. Particle dispersion (operated by turbulent eddies) is obtained from random
speeds. These last are the solutions of stochastic differential equations, reproducing the statistical characteristics of the local atmospheric turbulence. In such a way,
different parts of the plume can be liable to different atmospheric conditions. This
approach allows producing more realistic simulations in complex conditions, which
can be reproduced by traditional models with diffi culty.
The models considered here are single-particle type; this means that the trajectory of each particle represents an individual statistical realization in a turbulent fl ow
characterized by certain initial conditions and physical constraints. Thus, the motion
of any particle is independent of the other particles, and consequently the concentration fi eld must be interpreted as an ensemble average. The basic relationship, for an
instantaneous source located in x 0 is (Csanady, 1973)
=
0 0
( , )
( , | , )
C x t Q P x t x t
(8.1)
where
Q is the total emitted mass
C(x, t) is the ensemble mean concentration (the mass of particles within a small
volume surrounding x at time t)
P(x, t | x 0 , t 0 ) is the probability that a particle that was at x 0 at time t 0 arrives at x
at time t
In LSDM, to compute P(x, t | x 0 , t 0 ) it is necessary to release a suffi ciently large
number of particles, to follow their trajectories, and to calculate how many of them
arrive in a small control volume surrounding x at time t. It is worth noting that particles move in the computational domain, without any grid, using as input the values
© 2010 by Taylor and Francis Group, LLC
Air Pollution and Turbulence: Modeling and Applications
Lagrangian stochastic dispersion models (LSDM), also named Lagrangian particle models (LPM), are numerical models aimed at the simulation of these processes,
being able to account for fl ow and turbulence space-time variations. Emissions in
the atmosphere are simulated using a number of fi ctitious particles named “ computer
particles.” Each particle represents a specifi ed pollutant mass. We call particle a fl uid
portion containing the emitted substance, having dimensions appropriate to follow
the motion of the smallest turbulence eddies present in the atmosphere (of the order
of the Kolmogorov scale), but containing a number of molecules large enough to
allow disregarding the effect of the single molecule. Under the hypothesis, accurately demonstrated, that dispersion due to molecular motion is negligible compared
to turbulent dispersion, it can be thought that these particles possess a concentration of their own that is preserved during the motion. It is assumed that the particles passively follow the turbulent motion of air masses in which they are. Particles
are moved following the turbulent eddies, thus describing random trajectories. To
prescribe this behavior, particle velocities are subject to a random forcing, thus these
are stochastic type models, whereas Eulerian models are deterministic. As a consequence, the emitted mass concentration can be calculated from the space distribution
of particles at a particular time.
Particle mean motion in the computation domain, which simulates the airborne
pollutant motion in the real domain (atmosphere), is prescribed by the local mean
wind. Particle dispersion (operated by turbulent eddies) is obtained from random
speeds. These last are the solutions of stochastic differential equations, reproducing the statistical characteristics of the local atmospheric turbulence. In such a way,
different parts of the plume can be liable to different atmospheric conditions. This
approach allows producing more realistic simulations in complex conditions, which
can be reproduced by traditional models with diffi culty.
The models considered here are single-particle type; this means that the trajectory of each particle represents an individual statistical realization in a turbulent fl ow
characterized by certain initial conditions and physical constraints. Thus, the motion
of any particle is independent of the other particles, and consequently the concentration fi eld must be interpreted as an ensemble average. The basic relationship, for an
instantaneous source located in x 0 is (Csanady, 1973)
=
0 0
( , )
( , | , )
C x t Q P x t x t
(8.1)
where
Q is the total emitted mass
C(x, t) is the ensemble mean concentration (the mass of particles within a small
volume surrounding x at time t)
P(x, t | x 0 , t 0 ) is the probability that a particle that was at x 0 at time t 0 arrives at x
at time t
In LSDM, to compute P(x, t | x 0 , t 0 ) it is necessary to release a suffi ciently large
number of particles, to follow their trajectories, and to calculate how many of them
arrive in a small control volume surrounding x at time t. It is worth noting that particles move in the computational domain, without any grid, using as input the values
© 2010 by Taylor and Francis Group, LLC