Multifractal Concentrations of Heavy Particles in Random Flows

Abstract

Small-size impurities such as dust or droplets suspended in turbulent incompressible flows typically have a finite size and a mass density larger than the carrier fluid. They cannot be described as simple passive tracers, that is point-like particles with negligible mass advected by the fluid; an accurate model for their motion must take into account inertia effects. These inertial particles generally interact with the fluid through a viscous Stokes drag and thus their motion typically lags behind that of passive tracers. The dynamics of the latter is governed by a conservative dynamical system when the carrier flow is incompressible (because volume is conserved), but inertial particles have dissipative dynamics. While an initially uniform distribution of tracers remains uniform at any later time, the spatial distribution of inertial particles develops strong inhomogeneities. Such a phenomenon of preferential concentration refers to the presence of regions with either extremely high or low concentrations. Their characterization plays an essential role in natural and industrial phenomena. Instances are optimization of combustion processes in the design of Diesel engines [1], the growth of rain drops in sub-tropical clouds [2], the formation of the planets in the Solar system [3], coexistence between several species of plankton [4], etc. For such applications it is recognized that a key problem is the prediction of the collision or reaction rates and their associated typical time scales. The time scales obtained using diffusion theory exceed by one or several orders of magnitude those observed in experiments or numerical simulations. A full understanding of particle clustering and, in particular, of the fine structures appearing in the mass distribution is crucial for identifying and quantifying the mechanisms responsible for this drastic reduction in time scales. We propose here an original approach leading to a systematic description of inertial particles clustering. This approach is in part inspired by recent breakthroughs in the study of passive scalar advection by turbulent flows, using Lagrangian techniques [5]. Preferential concentrations can be interpreted as the convergence of particle trajectories onto certain dynamically evolving sets in the position–velocity phase space called attractors. Use of dissipative dynamical systems tools and, in particular, of