Clustering of inertial particles in turbulence

1. Clustering of inertialparticles in turbulence Massimo Cencini CNR-INFM Statistical Mechanics and Complexity Università “La Sapienza” Rome CNR- Istituto dei Sistemi Complessi, Via dei Taurini 19, Rome Massimo.Cencini@roma1.infn.it with: J. Bec, L. Biferale, A. Lanotte, S. Musacchio & F. Toschi (nlin.CD/0608045)

2. What we know and what we want to know Statistical characterization of clustering in turbulence (no-gravity, passive suspensions) • Very small scales: particle concentration fluctuations are very strong and their statistics depend on the Stokes number and correlate with the small scale structures of the flow [‘80s--now: Maxey, Eaton, Fessler, Squires, Zaichik, Wilkinson, Collins, Falkovich, ….] • Inertial range scales: evidence for strong fluctuations also a these scales (2d-NS[Boffetta, de Lillo & Gamba 2004; Chen, Goto & Vassilicos 2006] ) statistical characterization, what are the relevant parameters?

3. Motivations Rain Drops formation In warm clouds CCN activation Condensation Coalescence Enhanced collision rate of water droplets by clustering may explain the fast rate of rain drop formation, which cannot be explained by condensation only (Pruppacher and Klett, 1998) (Falkovich, Fouxon and Stepanov, Nature 2002)

4. Motivation Protoplanetary disk Migration of dust to the equatorial plane Accretion of planetesimals from 100m to few Km Gravitation & collisions coalescence -> planetary embryos Main issue: time scales Aerosols Sprays & optimization of combustion processes in diesel engines (T.Elperin et al. nlin.CD/0305017) From Bracco et al. (Phys. Fluids 1999)

5. Heavy particle dynamics Particles with (Kolmogorov scale) Heavy particles Particle Re <<1 Very dilute suspensions: no collisions passive particles no gravity (Maxey & Riley Phys. Fluids 26, 883 (1983)) Stokes number Drag: Stokes Time

6. Phenomenology Mechanisms at work: Ejection of heavy particles from vorticespreferential concentration Finite response time to fluid fluctuations (smoothing and filter of fast time scales) Dissipative dynamics in phase-space: volumes are contracted & caustics for high values of St , i.e. particles may arrive very close with very different velocities

7. DNS summary N3 5123 2563 1283 Tot #particles 120Millions 32Millions 4Millions Fast 0.1  500.000 250.000 32.000 Slow 10  7.5Millions 2Millions 250.000 Stokes/Lyap (15+1)/(32+1) (15+1)/(32+1) 15+1 Traject. Length 900 +2100 756 +1744 600+1200 Disk usage 1TB 400GB 70GB NS-equation + & Particles with Tracers SETTINGS millions of particles and tracers injected randomly & homogeneously with initial vel. = to that of the fluid STATISTICS TRANSIENT (1-2 T)+BULK ( 3-4 T) NOTES Pseudo spectral code with resolution 1283,2563, 5123 - Re=65, 105, 185 Normal viscosity

8. Two kinds of clustering Particle clustering is observed both in the dissipative and in inertial range Instantaneous p. distribution in a slice of width ≈ 2.5. St = 0.58 R= 185

9. Clustering at r< • Velocity is smooth we expect fractal distribution • At these scales the only relevant time scale is thus everything must be a function of St & Reonly correlation dimension

10. Correlation dimension Hyperbolic non-hyperbolic The preferential concentration is also evidenced by looking at the fluid acceleration conditioned on particle positions a(X,t) (Bec,Biferale, Boffetta, Celani, MC, Lanotte, Musacchio & Toschi (2006)) Particles preferentially concentrate in hyperbolic regions Prob. to be in non-hyperbolic points • St is the only relevant parameter • Maximum of clustering for St1 • D2 almost independent of Re, (Keswani & Collins (2004) ) high order statistics? Maximum of clustering seems to be connected to preferential concentration confirming the traditional scenario Though is non-generic: counter example Kraichnan flows(Bec, MC, Hillenbrand 2006)

11. Inertial-range clustering • Voids & structures from  to L • Distribution of particles over scales? • What is the dependence on St? Or what is the proper parameter?

12. Preliminary considerations Particles should not distribute self-similarly Correlation functions of the density are not power law (Balkovsky, Falkovich & Fouxon 2001) Natural expectation In analogy with the dissipative clustering since at scale r the typical time scale is r=-1/3r2/3 the only relevant parameter should be Str

13. It works in Kraichnan flows Gaussian random flow with no-time correlation Incompressible, homogeneous and isotropic h=1 dissipative range h<1 inertial range Local correlation dimension Note that tracers limit Is recovered for Str ->0 (i.e. for 0 or r) (Bec, MC & Hillenbrand 2006; nlin.CD/0606038)

14. In turbulence? r=L/16 *PDF of the coarse-grained mass: number density of particles ( N in total ) at scale r, weighting each cell with the mass it contains, natural (Quasi-Lagrangian) measure to reduce finite N effects at <<1 *Poisson for tracers (=0) deviations already for <<1 *For <<1 algebraic tails (voids) Result on Kraichnan suggests Pr,()= PSt(r)() But is not!

15. Why does not work? • Kraichnan model: • no-time correlations • no-sweeping • no-structures • In Turbulence we have all 2d-NS Inverse cascade: strong correlation between particle positions and zero acceleration points In 2d Kinematic flows: (no-sweeping) still clustering but no correlations with zero acceleration points (Chen, Goto & Vassilicos 2006) Working hypothesis May be sweeping is playing some role

16. The contraction rate Effective compressibility good for r<< for St<<1 [Balkovsky, Falkovich & Fouxon (2001)] [Maxey (1987)] Assume that the argument remains valid also for Str->0 (reasonable for r enough large &  not too large) Then No - sweeping Yes - sweeping Though we cannot exclude finite Re effects

17. Numerics The collapse confirms that the contraction rate is indeed the proper time scale Uniformity is recovered very slowly going to the large scales, e.g. much slower than for Poisson distribution9/5 Non-dimensional contraction rate

18. Summary & Conclusions Description of particle clustering for moderate St number and moderate Re number in the dissipative and inertial range • r<< strong clustering, everything depends on St & very weakly on Re • <r<L very slow recovery of uniformity, and the statistics depends on the contraction rate. Dominance of voids --> algebraic tails for the pdf of the coarse- grained mass • A better understanding of the statistics of fluid acceleration (in the inertial range) may be crucial to understand clustering and conversely inertial particles may be probes for acceleration properties • Larger Re studies necessary to confirm the picture

19. Role of Sweeping on acceleration • A short history • Tennekes 1975 points out the importance of sweeping for multitime statistics and pressure/acceleration • Van Atta & Wyngaard 1975 experimental evidence of k-5/3 for pressure • Yakhot, Orzag & She 1989 RG--> k-7/3 for pressure • Chen & Kraichnan 1989 importance of sweeping for multitime statistics RG does not consider sweeping from the outset • Nelkin & Tabor 1990 importance of sweeping for acceleration & pressure • Sanada & Shanmugasundaram 1992 numerics on multitime and pressure confirming the important role of sweeping • More recently • Vedula & Yeung 1999 doubts on k-5/3 for pressure but observed • Gotoh & Fukayama 2001 both k-5/3 and k-7/3 are observed, is k-5/3 • spurious or a finite Re effect?