On probability density function equations for particle dispersion in a uniform shear flow

Abstract

The paper examines a fundamental discrepancy between two probability density function (PDF) models, the kinetic model (KM) and generalized Langevin model (GLM), currently used to model the dispersion of particles in turbulent flows. This discrepancy is manifest in particle dispersion in an unbounded simple shear flow where model predictions for the values of the streamwise fluid-particle diffusion coefficients are not only different but are of opposite sign. It is shown that this discrepancy arises through a neglect of the inertial convection term in the GLM equation for the mean carrier flow velocity local to a particle which eventually leads to algebraic forms for the particle-fluid diffusion coefficients. Evaluating this term for a Gaussian process leads to identical results for both PDF formulations. This also resolves a fundamental long-standing discrepancy in previous forms reported for the passive scalar diffusion coefficients in a simple shear flow where similar assumptions were made. Avoiding this assumption, the exact solutions are given for the dispersion of particles in this simple shear flow case derived from the solution of the GLM PDF equation which show explicitly the dependence on the particle response time and the strain rate, both normalized on the integral timescale of the turbulence. The analysis shows that the particle diffusion coefficient in the streamwise direction is negative when the strain rate $\,{\geq}\,$ a certain value. The origin of negative diffusion coefficients is explained and their influence is shown in the way in which the mean concentration and mean velocity flow fields of the particle and carrier flow (seen by the particle) evolve with time for particles released from the centre of the shear.