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Inertial particles in selfsimilar random flows Jérémie Bec CNRS, Observatoire de la Côte d’Azur, Nice Massimo Cencini Rafaela Hillerbrand Rain initiation • Warm clouds 1 raindrop = 109 droplets Growth by continued condensation way = too slow • Collision/Coalescence: Polydisperse suspensions with a wide range of droplet sizes with different velocities Larger, faster droplets overtake smaller ones and collide Droplet growth by coalescence Formation of the solar system • Protoplanetary disk after the collapse of a nebula Migration of dust toward the equatorial plane of the star (II) Accretion 109 planetesimals from 100m to few km (III) Merger and growth planetary embryos planets (I) From Bracco et al. (Phys. Fluids • Problem = time scales ? Very heavy particles • Impurities with size (Kolmogorov scale) and with mass density viscous drag with • Passive suspensions: no feedback of the particles onto the fluid flow (e.g. very dilute suspensions) • Stokes number: ratio between response time Clustering of inertial particles • Different mechanisms involved in clustering: Delay on the flow dynamics (smoothing) Ejection from eddies by centrifugal forces Dissipative dynamics due to Stokes drag • Idea: find models to disentangle these effects in order to understand their signature on the spatial distribution and dynamical properties of particles. Fluid flow = Kraichnan • Gaussian carrier flow with no time correlation Incompressible, homogeneous, isotropic = Hölder exponent of the flow • -correlation in time no structure, no sweeping • Relevant when (Fouxon-Horvai) Reduced dynamics • Two-point motion can be written as a system of SDE with additive noise (smooth case: Piterbarg 2D, Wilkinson-Mehlig 3D) + Time 2D: + Boundary conditions on Large-scale Stokes number: and Phenomenology of the dynamics • stable fixed line • Close to this line, noise dominates and behave as two independent Ornstein– Uhlenbeck processes • Far away, the quadratic terms dominate and trajectories perform loops from to QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. Phenomenology of the dynamics QuickTime™ et un décompresseur sont requis pour visionner cette image. The loops play a fundamental role: • Flux of probability from to that • Events during which (and hence becomes very small , so ) Smooth case • Single dimensionless parameter: Stokes number • Exponential separation of the particles Rough case • For , the dynamics can be rescaled and depends only on a local Stokes number [Falkovich et al.] • If we drop the boundary condition, the only lengthscale is the initial value of . The interparticle separation is given by Correlation dimension • Behaviour of when • Fractal mass distribution: • Smooth case: both when and when • Rough case: scale-dependent Stokes number when and thus Information on clustering is given by the local correlation dimension: expected to depend only upon and Numerics Local correlation dimension different colours = different QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. Local Stokes number • Same qualitative picture reproduced for different values of Velocity differences • Typical velocity difference between particles separated by Important for applications (approaching rate + multiphasic models) small-scale behaviour: Hölder exponent for the “particle velocity field” • Smooth case: function of the Stokes number • Rough case: (infinite inertia at small scales) Relevant information contained in the “finite Numerics Local Hölder exponent of the particle velocity Fluid tracers QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. Local Stokes number Free particles Large Stokes number behaviour • Relevant asymptotics for smooth flows + gives the small-scale behaviour in the rough case • Idea: [Horvai] with fixed Any statistical quantity should depend only on in this limit but depends also only on for the original system Example: 1st Lyapunov exponent in the smooth Large Stokes - smooth flows QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. • Same argument applies to the large deviations of the stretching rate QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. Statistics of velocity differences • PDFs of velocity differences also rescale at large Stokes numbers: Powerlaw tails QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. Power law tails QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. Tails related to large loops • Cumulative probability • Simplification of the dynamics: noise + loops • Prob to enter a sufficiently large loop Fraction of time spent at • 1st contribution: should be sufficiently small to initiate a large loop Radius estimated by QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. Prediction for the exponent • 2nd contribution: Approximation of the dynamics by the deterministic drift Fraction of time spent at is – confirmed by • numerics • Power law with same exponent at large positive and Smooth case QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. • Clustering weakens when Open questions More on Kraichnan flows: Move to mass dynamics instead of two-point motion. Does this model catch the formation of voids in the particle distribution? Understanding of the dynamical flow singularity at and Questions related to the uniqueness of trajectories Different from tracers: breaking of Lipschitz continuity is “2nd order” Add compressibility: what are the different Open questions Toward realistic flows: Does large-Stokes rescaling apply in turbulent flows? Important for planet formation (density ratio ) Measure of relative velocity PDFs in real flows: are the algebraic tails also present? Effect of time correlation? Problems = • Rescaling with the turnover time is wrong