Continuous models of the motion of inertial particles in laminar and turbulent flows based on the Fokker-Planck equations

Abstract

A methodology of continuous description of the motion of inertial particles in hydrodynamic flows has been developed. This methodology involves the following main elements: the equations of motion for a single particle in a given velocity field of the carrier medium in the presence of a random disturbing force such as white noise; the Fokker-Planck equation with a diffusion term in velocity space, which is treated as an instantaneous equation relative to the large-scale motion; the moment (for the Fokker-Planck equation) relations which are treated as instantaneous continuous equations in physical space; the closing of the continuous equations on the basis of phenomenological relations or by means of an approximate asymptotic solution of the Fokker-Planck equation for a short particle dynamic relaxation time. The direct interaction of the particles and their reciprocal action on the motion of the carrier medium are not taken into account. In the case of laminar motion, the random disturbing force results from Brownian fluctuations (the interaction between the particles and the molecules of the carrier medium) and the Fokker-Planck equation for this situation can be written in a natural way. For turbulent motion of the carrier medium it is assumed that the fluctuating field can be broken down into small- and large-scale motions (in relation to the particle dynamic relaxation time); in the random force, besides Brownian perturbations the small-scale turbulent fluctuations are included, while the velocity field is averaged over the small scales. In this case the Fokker-Planck equation (and its moment analogs) becomes an instantaneous equation with respect to the large-scale turbulent fluctuations. The continuous equations are analyzed and in the case of the turbulent motion elementary closing methods are proposed. Estimates of the small-scale turbulent particle diffusion coefficient are given.