Multifractal concentrations of inertial particles in smooth random flows

Abstract

Collisionless suspensions of inertial particles (finite-size impurities) are studied in twoand three-dimensional spatially smooth flows. Tools borrowed from the study of random dynamical systems are used to identify and to characterise 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 (nondimensional 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) remains always 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 .

Droplet growth in warm turbulent clouds

“…For values of St not too large, the RDF for a monodisperse system typically takes the form of a power law which reflects the multi-scale self-similar nature of droplet clustering (e.g. Figures 7 and 8 of Goto and Vassilicos, 2006;Bec, 2005;Bec et al, 2007). This multi-scale clustering has been accounted for in terms of the sweep-stick mechanism, which explains why droplet clustering mimics the multi-scale clustering of vanishing fluid acceleration points in a turbulent flow (Coleman and Vassilicos, 2009).…”

Acceleration statistics of heavy particles in turbulence

“…In particular, we look at the sign of the discriminant (see e.g. Chong, Perry &Cantwell 1990 andBec 2005):…”

“…In parallel with experimental effort, theoretical analysis (Balkovsky, Falkovich & Fouxon 2001, Falkovich & Pumir 2004, Bec, Gawedzki & Horvai 2004, Zaichik, Simonin & Alipchenkov 2003 and numerical simulations (Boivin, Simonin & Squires 1998, Reade & Collins 2000, Zhou, Wexler & Wang 2001, Chun et al 2005 are paving the way to a thorough understanding of inertial particle dynamics in turbulent flows. Recently, the presence of strong inhomogeneities characterised by fractal and multifractal properties have been predicted, and found in theoretical and numerical studies of stochastic laminar flows (Balkovsky, Falkovich & Fouxon 2001, Bec, Gawedzki & Horvai 2004, Bec 2005, in two dimensional turbulent flows (Boffetta, De Lillo & Gamba 2004) and in three dimensional turbulent flows at moderate Reynolds numbers in the limit of vanishing inertia (Falkovich & Pumir 2004).…”

Dynamics and statistics of heavy particles in turbulent flows

“…Progresses in the characterization of the statistical features of particle clusters have been achieved by studying inertial particles evolving in laminar stochastic flows [12,22,23,24,25] and two dimensional turbulent flows [26]. Experimental results are reviewed in Ref.…”