Multifractal concentrations of inertial particles in smooth random flows

Abstract

Collisionless suspensions of inertial particles (finite-size impurities) are studied in two- and three-dimensional spatially smooth flows. Tools borrowed from the study of random dynamical systems are used to identify and to characterize in full generality the mechanisms leading to the formation of strong inhomogeneities in the particle concentration.

Phenomenological arguments are used to show that in two dimensions, the positions of heavy particles form dynamical fractal clusters when their Stokes number (non-dimensional viscous friction time) is below some critical value. Numerical simulations provide strong evidence for the presence of this threshold in both two and three dimensions and for particles not only heavier but also lighter than the carrier fluid. In two dimensions, light particles are found to cluster at discrete (time-dependent) positions and velocities in some range of the dynamical parameters (the Stokes number and the mass density ratio between fluid and particles). This regime is absent in the three-dimensional case for which evidence is that the (Hausdorff) fractal dimension of clusters in phase space (position–velocity) always remains above two.

After relaxation of transients, the phase-space density of particles becomes a singular random measure with non-trivial multiscaling properties, whose exponents cannot be predicted by dimensional analysis. Theoretical results about the projection of fractal sets are used to relate the distribution in phase space to the distribution of the particle positions. Multifractality in phase space implies also multiscaling of the spatial distribution of the mass of particles. Two-dimensional simulations, using simple random flows and heavy particles, allow the accurate determination of the scaling exponents: anomalous deviations from self-similar scaling are already observed for Stokes numbers as small as $10^{-4}$.