INTERMITTENT DISTRIBUTION OF HEAVY INERTIAL PARTICLES IN TURBULENT FLOWS Alain Pumir , Gregory Falkovich Institut Non Linéaire de Nice, 1361, route des Lucioles, F-06560, Valbonne, France Physics of Complex Systems, The Weizmann Institute of Sciences, Rehovot 76100, Israel Summary The phenomenon of preferential concentration of inertial particles is studied by following lagrangian trajectories. Elementary properties of the coarse-grained distribution of heavy particles in simple turbulent flows are investigated by direct numerical simulations. In the small Stokes number case, we compute the coarse-grained particle distribution, , and we demonstrate that the second moment behaves as an approximate power law : . The dependence of the exponent as a function of the Reynolds and of the Stokes number is studied in the small Stokes number limit. Our results show a strong dependence of the level of fluctuation of the particle distribution as a function of the Reynolds number. INTRODUCTION Inertial particles advected by a turbulent flow are known to have very inhomogeneous distributions [1]. In clouds, the clustering of water droplets is known to have several important consequences [2],in particular in enhancing the collision rate between droplets, thus dramatically accelerating the formation of rain [3]. The origin of this effect is due to parti- cle inertia : particles are advected by an effective velocity field , which is compressible ( ) even when the surrounding fluid motion, is incompressible ( ). This effect is often refered to as preferential concentration. In the limit of weak particle concentration, the density of particles, , obeys : (1) In an incompressible turbulent flow, with a Kolmogorov scale , the divergence of the particles’ velocity field acts as a source term of density fluctuations at scales [1]. As fluid elements get squashed by turbulence, the particle distribution reaches finer and finer scales, down to a cutoff length scale, , determined by the diffusive process and/or by the size of the inertial particles. The clustering of particles is characterized here by the coarse-grained density of particles over a volume of size , . This quantity can be expressed in terms of the properties of the trajectories of particles in the flow, conditioned by the stretching occuring along the trajectory [1, 3]. Using direct numerical simulations (DNS) of turbulent flows, we compute the properties of the second moment of the fluctuating quantity : and its dependence as a function of the parameters in the problems, such as the Reynolds number, , the Stokes number, , and size, . A power law dependence is found numerically : . In the limited range ( ) studied here, a strong variation of the exponent is found. LAGRANGIAN APPROACH Inertial particles of density , of radius , evolve in a fluid of density and of viscosity subject to an incompressible turbulent motion, described by an eulerian velocity field . In the low concentration case, particles can be considered as purely passive. Introducing and the Stokes time, , the velocity of inertial particles satisfies [2] : (2) We restrict ourselves here to the case where the time is short compared to the Kolmogorov time scale, , so the Stokes number, is small. The velocity is well approximated by , where is the lagrangian derivative of the velocity field : . The velocity field is compressible : , where is the velocity derivative tensor : . The source of concentration fluctuations is related to the compressibility of the velocity field ; it acts therefore most strongly at the Kolmogorov size, [1, 3]. To estimate the coarse-grained distribution of particles over a size , , at location and time , one has to trace back in time to the previous time , and to the location , such that the trajectory starting at , and location ends up at at location . The flow maps a small volume of characteristic size at to a volume whose largest scale is at time . The intuitive picture is that particle concentration fluctuations accumulate at a scale as long as it takes to expand the volume to a size as the trajectory evolves backwards in time. The coarse-grained particle concentration is [1] : (3) The statistical weight associated with each trajectory is . As a consequence, the second moment of the distribution of is simply , where ( ) refers to eulerian average (average over a set of trajectories). In the small Stokes number limit, the difference is small. We thus simply follow the compression along lagrangian trajectories of the flow to obtain an estimate of the quantity .