No abstract

The preferential concentration of inertial particles in a turbulent velocity field occurs when the particle and fluid time constants are commensurate. We propose a straightforward mathematical model for this phenomenon and use the model to study various scaling limits of interest and to study numerically the effect of interparticle collisions. The model comprises Stokes' law for the particle motions, and a Gaussian random field for the velocity. The primary advantages of the model are its amenability to mathematical analysis in various interesting scaling limits and the speed at which numerical simulations can be performed. The scaling limits corroborate experimental evidence about the lack of preferential concentration for a large and small Stokes number and make new predictions about the possibility of preferential concentration at large times and lead to stochastic differential equations governing this phenomenon. The effect of collisions is found to be negligible for the most part, although in some cases they have an interesting antidiffusive effect.

We develop an efficient algorithm for detecting collisions among a large number of particles moving in a velocity field, when the field itself is possibly coupled to the particle motions. We build on ideas from molecular dynamics simulations and, as a byproduct, give a literature survey of methods for hard sphere molecular dynamics. We analyze the complexity of the algorithm in detail and present several experimental results on performance which corroborate the analysis. An optimal algorithm for collision detection has cost scaling at least like the total number of collisions detected. We argue, both theoretically and experimentally, that with the appropriate parameter choice and when the number of collisions grows with the number of particles at least as fast as for billiards, the algorithm we recommend is optimal.

The motion of an inertial particle in a Gaussian random field is studied. This is a model for the phenomenon of preferential concentration, whereby inertial particles in a turbulent flow can correlate significantly. Mathematically the motion is described by Newton's second law for a particle on a 2-D torus, with force proportional to the difference between a background fluid velocity and the particle velocity itself. The fluid velocity is defined through a linear stochastic PDE of Ornstein–Uhlenbeck type. The properties of the model are studied in terms of the covariance of the noise which drives the stochastic PDE. Sufficient conditions are found for almost sure existence and uniqueness of particle paths, and for a random dynamical system with a global random attractor. The random attractor is illustrated by means of a numerical experiment, and the relevance of the random attractor for the understanding of particle distributions is highlighted.