34
Air Pollution and Turbulence: Modeling and Applications
is positioned) at inertial-range scales. On the other hand, the FDT is dissipative,
the mean energy dissipation per unit mass ε(ν) tends to a fi nite positive limit in the
infi nite Reynolds number limit (ν → 0, where ν is the kinematic viscosity).
In order to study turbulent dispersion, it is important to defi ne a fl uid particle.
By the notion fl uid particle, we mean a very small control volume of characteristic dimension much larger than the molecular scale but much smaller than the
Kolmogorov microscale (Saffman, 1960; Hunt, 1982). The continuum forming such
a fl uid particle remains intact at the least during a time interval suffi ciently large
compared with the interval to be considered during the transport process. Any
exchange with its direct surroundings is purely molecular in nature. The dimension
of the fl uid particle implies that it can be regarded as part of the fl uid continuum,
and its centroid responds to all scales of turbulent motion (Venkatram, 1988). The
turbulent dispersion is distinct from the molecular diffusion process according to
the kinetic theory of gases at least for the following two reasons: fi rst, because of
the intensive interactions between the fl uid particles, there may occur a continuous
exchange of a transferable property; second, as will be shown, there is a correlation in time between properties of a fl uid particle at subsequent instants. Because of
this memory behavior the turbulence diffusion process may not be considered as a
Markov process (a stochastic process that has zero memory of the past and a future
that is a function of the present and some statistical rule for the transition).
The aim of this chapter is to present and discuss some general characteristics
of a FDT. The present analysis is based on plausible hypothesis, which were
experimentally observed, that was mathematically represented by Kolmogorov.
Phenomenological characteristics (associated to a FDT), known as self-similarity,
scale-invariance within the inertial subrange, localness of interaction, and the turbulent energy spectrum, will be assumed and employed for the derivation of the different turbulent parameters. Based on these characteristics associated to a FDT and
on the Taylor statistical diffusion theory, turbulent eddy diffusivities for a convective
boundary layer (CBL) will be derived.
Furthermore, in this chapter we study two phenomena associated to the turbulence and that frequently play a fundamental role in describing contaminants dispersion in a PBL. One of these phenomena concerns the turbulence decaying in a
CBL. This dynamical process, which occurs in the characteristic sunset transition
time (1 h), is responsible for sustaining turbulence in the residual layer (RL). Some
contaminant sources (stacks) are localized in this RL and consequently a description
of the turbulent diffusion in this environment of decaying turbulence is of fundamental importance to parameterize turbulent terms that appear in analytical and
numerical mathematical models of air pollution.
The second phenomenon occurs in situations characterized by low wind speed
(LWS). The study of LWS conditions is of interest, partly because the simulation
of airborne pollutant dispersion in these conditions is rather diffi cult. In fact, most
of existing regulatory dispersion models become unreliable as the wind speed U
approaches zero, so that their application is generally limited to U > 2.0 m/s. In
these conditions, the contaminant plume is unlikely to have any defi nable travel
and dispersion is governed by meandering (low-frequency horizontal wind oscillations). As a consequence, the more the wind speed decreases the more the standard
© 2010 by Taylor and Francis Group, LLC
Air Pollution and Turbulence: Modeling and Applications
is positioned) at inertial-range scales. On the other hand, the FDT is dissipative,
the mean energy dissipation per unit mass ε(ν) tends to a fi nite positive limit in the
infi nite Reynolds number limit (ν → 0, where ν is the kinematic viscosity).
In order to study turbulent dispersion, it is important to defi ne a fl uid particle.
By the notion fl uid particle, we mean a very small control volume of characteristic dimension much larger than the molecular scale but much smaller than the
Kolmogorov microscale (Saffman, 1960; Hunt, 1982). The continuum forming such
a fl uid particle remains intact at the least during a time interval suffi ciently large
compared with the interval to be considered during the transport process. Any
exchange with its direct surroundings is purely molecular in nature. The dimension
of the fl uid particle implies that it can be regarded as part of the fl uid continuum,
and its centroid responds to all scales of turbulent motion (Venkatram, 1988). The
turbulent dispersion is distinct from the molecular diffusion process according to
the kinetic theory of gases at least for the following two reasons: fi rst, because of
the intensive interactions between the fl uid particles, there may occur a continuous
exchange of a transferable property; second, as will be shown, there is a correlation in time between properties of a fl uid particle at subsequent instants. Because of
this memory behavior the turbulence diffusion process may not be considered as a
Markov process (a stochastic process that has zero memory of the past and a future
that is a function of the present and some statistical rule for the transition).
The aim of this chapter is to present and discuss some general characteristics
of a FDT. The present analysis is based on plausible hypothesis, which were
experimentally observed, that was mathematically represented by Kolmogorov.
Phenomenological characteristics (associated to a FDT), known as self-similarity,
scale-invariance within the inertial subrange, localness of interaction, and the turbulent energy spectrum, will be assumed and employed for the derivation of the different turbulent parameters. Based on these characteristics associated to a FDT and
on the Taylor statistical diffusion theory, turbulent eddy diffusivities for a convective
boundary layer (CBL) will be derived.
Furthermore, in this chapter we study two phenomena associated to the turbulence and that frequently play a fundamental role in describing contaminants dispersion in a PBL. One of these phenomena concerns the turbulence decaying in a
CBL. This dynamical process, which occurs in the characteristic sunset transition
time (1 h), is responsible for sustaining turbulence in the residual layer (RL). Some
contaminant sources (stacks) are localized in this RL and consequently a description
of the turbulent diffusion in this environment of decaying turbulence is of fundamental importance to parameterize turbulent terms that appear in analytical and
numerical mathematical models of air pollution.
The second phenomenon occurs in situations characterized by low wind speed
(LWS). The study of LWS conditions is of interest, partly because the simulation
of airborne pollutant dispersion in these conditions is rather diffi cult. In fact, most
of existing regulatory dispersion models become unreliable as the wind speed U
approaches zero, so that their application is generally limited to U > 2.0 m/s. In
these conditions, the contaminant plume is unlikely to have any defi nable travel
and dispersion is governed by meandering (low-frequency horizontal wind oscillations). As a consequence, the more the wind speed decreases the more the standard
© 2010 by Taylor and Francis Group, LLC