Applied Mathematics


Model of aerosols turbulent transport


The mathematical models of the turbulent transport of aerosols in the atmosphere are based mainly on the hypothesis that the eddy diffusivity for the small-sized particles is proportional to the eddy diffusivity of a heat and the latter value is assumed proportional to the eddy viscosity. In turn the turbulent eddy viscosity in the stratified flows is determined from the empirical formulas or from the k-e model, or from the turbulent kinetic energy model.

Turbulent boundary layer model also can be developed for the case of the atmospheric aerosols transport. This model based on the viscous heat-conducting gas transport equation and on the dynamical model of the dust cloud. The aerosol is considered as sets of identical, small-sized particles, which move chaotically under influence of thermal fluctuations and are involved in macroscopic movement together with the airflow. The mass concentration of aerosol particles is considered so small, that the influence of particles to gas dynamic can be neglected. The dynamic parameters of the aerosol particles can be averaged to exclude of the chaotic thermal motion, then its can be described by the continuous functions: the numerical concentration and the aerosol particles flow velocity. Thus at the description of aerosol dynamics we shall take into account the Brownian diffusion, weight and inertia of particles. The motion of mono-disperse small particles in a dust cloud are governed by the transport and momentum equations:



where np is the particles numerical density, up is the particles flow velocity vector, Dp is the particle diffusion coefficient.

The diffusion coefficient of the small particles is given by the Einstein's formula: , where k=1.38*10-23J/K is the Boltzmann constant, mp is the particle mass, tp is the time relaxation parameter which for the spherical particles in the Stokes' regime is given by , dp is the particle aerodynamic diameter, rs is the particle material density.

Utilized the transformation method explained in the books [1,2], one can derive the equation system for the aerosol turbulent transport in the case when the aerosol particles are formed due to the condensation from the vapor phase as follows






where  are the random function of the particles flow velocity and numerical density, accordingly;  is the vertical turbulent transport rate of aerosol, , is the particles turbulent kinetic energy in the small volume , is the Schmidt number of aerosol,  is the latent heat,  is the rate of the phase transition which depends on the air temperature and vapor pressure as  , here  is the gas constant of the vapor phase.



Trunev A. P. Theory of Turbulence and Turbulent Transport in the Atmosphere. WIT Press, 2001.

Trunev A. P. Theory of Turbulence, Russian Academy of Sciences, Sochi, 1999.

Trunev, A. P., Diffuse processed in turbulent boundary layer over rough surface, Air Pollution III, Vol.1. Theory and Simulation, eds. H. Power, N. Moussiopoulos & C.A. Brebbia, Comp. Mech. Publ., Southampton, pp. 69-76, 1995.

Trunev, A. P., Similarity theory and model of turbulent dusty gas flow over large-scale roughness, Abstr. of Int. Conf. On Urban Air Quality: Monitoring and Modelling, University of Hertfordshire, Institute of Physics, London, p. 3.8, 1996.

Trunev, A. P., Similarity theory for turbulent flow over natural rough surface in pressure and temperature gradients, Air Pollution IV. Monitoring, Simulation and Control, eds. B. Caussade, H. Power & C.A. Brebbia, Comp. Mech. Pub., Southampton, pp. 275-286, 1996.

Trunev, A. P., Similarity theory and model of diffusion in turbulent atmosphere at large scales, Air Pollution V. Modelling, Monitoring and Management, eds. H. Power, T. Tirabassi & C.A. Brebbia, CMP, Southampton-Boston,  pp. 109-118, 1997.