1. Inertial particles in self- similar random flows Jérémie Bec CNRS, Observatoire de la Côte d’Azur, Nice Massimo Cencini Rafaela Hillerbrand

2. Rain initiation Warm clouds 1 raindrop = 10 9 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

3. Formation of the solar system Protoplanetary disk after the collapse of a nebula (I) Migration of dust toward the equatorial plane of the star (II) Accretion  10 9 planetesimals from 100m to few km (III) Merger and growth  planetary embryos  planets Problem = time scales ? From Bracco et al. (Phys. Fluids 1999)

4. Very heavy particles Impurities with size (Kolmogorov scale) and with mass density viscous drag Passive suspensions: no feedback of the particles onto the fluid flow (e.g. very dilute suspensions) Stokes number: ratio between response time and typical timescale of the flow (turbulence: ) with

5. 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.

6. 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)

7. 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 and Large-scale Stokes number: No dependence for smooth velocity fields ( )

8. 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

9. Phenomenology of the dynamics The loops play a fundamental role: Flux of probability from to, so that Events during which (and hence ) becomes very small Prevent from vanishing Probable mechanism ensuring mixing of the dynamics

10. Smooth case Single dimensionless parameter: Stokes number Exponential separation of the particles

11. 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 inter- particle separation is given by

12. 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

13. Numerics different colours = different Same qualitative picture reproduced for different values of Roughness weakens the maximum of clustering Local Stokes number Local correlation dimension

14. 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 size” exponent

15. Numerics Local Hölder exponent of the particle velocity Local Stokes number Free particles Fluid tracers

16. 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: 1 st Lyapunov exponent in the smooth case

17. Large Stokes - smooth flows Same argument applies to the large deviations of the stretching rate

18. Statistics of velocity differences PDFs of velocity differences also rescale at large Stokes numbers: Power- law tails

19. Power law tails

20. Tails related to large loops Cumulative probability Simplification of the dynamics: noise + loops 1 st contribution: should be sufficiently small to initiate a large loop Radius estimated by Prob to enter a sufficiently large loop Fraction of time spent at

21. Prediction for the exponent 2 nd 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 Clustering weakens when roughnessincreases

22. 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 “2 nd order” Add compressibility: what are the different regimes present? Are the regimes observed for tracers also present? Do they appear only in the singular limit ? Are there other regimes? Open questions

23. 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 Particles do not sample uniformly the flow (their position correlates with fluid velocity statistics) Open questions