Massimo Cencini Massimo Cencini Clustering of Inertial particles in turbulent flowsLeiden, August 2006 Clustering of inertial particles in turbulence Massimo Cencini CNR-INFM Statistical Mechanics and Complexity Università “La Sapienza” Rome CNR - Istituto dei Sistemi Complessi, Via dei Taurini 19, Rome with: J. Bec, L. Biferale, A. Lanotte, S. Musacchio & F. Toschi ( nlin.CD/ )

Massimo Cencini Massimo Cencini Clustering of Inertial particles in turbulent flowsLeiden, August 2006 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?

Massimo Cencini Massimo Cencini Clustering of Inertial particles in turbulent flowsLeiden, August 2006Motivations Rain Drops formation In warm clouds 1. CCN activation 2. Condensation 3. Coalescence (Pruppacher and Klett, 1998) (Falkovich, Fouxon and Stepanov, Nature 2002) Enhanced collision rate of water droplets by clustering may explain the fast rate of rain drop formation, which cannot be explained by condensation only

Massimo Cencini Massimo Cencini Clustering of Inertial particles in turbulent flowsLeiden, August 2006Motivation  Sprays & optimization of combustion processes in diesel engines ( T.Elperin et al. nlin.CD/ ) From Bracco et al. (Phys. Fluids 1999) 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

Massimo Cencini Massimo Cencini Clustering of Inertial particles in turbulent flowsLeiden, August 2006 Heavy particle dynamics Heavy particle dynamics Particles with (Kolmogorov scale) Heavy particles Particle Re <<1 Very dilute suspensions: no collisions passive particles no gravity Stokes number Drag: Stokes Time (Maxey & Riley Phys. Fluids 26, 883 (1983))

Massimo Cencini Massimo Cencini Clustering of Inertial particles in turbulent flowsLeiden, August 2006Phenomenology Mechanisms at work: 1. Ejection of heavy particles from vortices preferential concentration 2. Finite response time to fluid fluctuations (smoothing and filter of fast time scales) 3. 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

Massimo Cencini Massimo Cencini Clustering of Inertial particles in turbulent flowsLeiden, August 2006 DNS summary 1TB (15+1)/(32+1) 7.5Millions Millions (15+1)/(32+1)Stokes/Lyap 70GB400GBDisk usage Traject. Length Millions Slow 10   Fast 0.1   4Millions32MillionsTot #particles N3N3 NS-equation + Particles with & Tracers STATISTICS TRANSIENT (  1-2 T)+BULK (  3-4 T)SETTINGS millions of particles and tracers injected randomly & homogeneously with initial vel. = to that of the fluid NOTES NOTES Pseudo spectral code with resolution 128 3, 256 3, Re =65, 105, 185 Normal viscosity

Massimo Cencini Massimo Cencini Clustering of Inertial particles in turbulent flowsLeiden, August 2006 Two kinds of clustering Particle clustering is observed both dissipativeinertial in the dissipative and in inertial range Instantaneous p. distribution in a slice of width ≈ 2.5 . St  = 0.58 R = 185

Massimo Cencini Massimo Cencini Clustering of Inertial particles in turbulent flowsLeiden, August 2006 Clustering at r<  Velocity is smooth we expect fractal distribution St  & Re At these scales the only relevant time scale is   thus everything must be a function of St  & Re only correlation dimension

Massimo Cencini Massimo Cencini Clustering of Inertial particles in turbulent flowsLeiden, August 2006 Correlation dimension   St  is the only relevant parameter   Maximum of clustering for St   1   D 2 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) Hyperbolic non-hyperbolic Particles preferentially concentrate in hyperbolic regions Prob. to be in non-hyperbolic points 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, (Bec,Biferale, Boffetta, Celani, MC, Lanotte, Musacchio & Toschi (2006)) Musacchio & Toschi (2006))

Massimo Cencini Massimo Cencini Clustering of Inertial particles in turbulent flowsLeiden, August 2006 Inertial-range clustering Voids & structures from  to LVoids & structures from  to L Distribution of particles over scales?Distribution of particles over scales? What is the dependence on St  ? Or what is the proper parameter?What is the dependence on St  ? Or what is the proper parameter?

Massimo Cencini Massimo Cencini Clustering of Inertial particles in turbulent flowsLeiden, August 2006 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 =  -1/3 r 2/3 the only relevant parameter should be St r

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

Massimo Cencini Massimo Cencini Clustering of Inertial particles in turbulent flowsLeiden, August 2006 In turbulence? *PDF of the coarse-grained mass: number density of particles ( N in total ) at scale r, weighting each cell particles ( N in total ) at scale r, weighting each cell with the mass it contains, natural (Quasi-Lagrangian) with the mass it contains, natural (Quasi-Lagrangian) measure to reduce finite N effects at  <<1 measure to reduce finite N effects at  <<1 *Poisson for tracers (  =0) deviations already for  <<1 Result on Kraichnan suggests P r,  (  )= P St(r) (  ) But is not! r=L/16 *For  <<1 algebraic tails (voids)

Massimo Cencini Massimo Cencini Clustering of Inertial particles in turbulent flowsLeiden, August 2006 Why does not work? Kraichnan model: no-time correlationsno-time correlations no-sweepingno-sweeping no-structuresno-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

Massimo Cencini Massimo Cencini Clustering of Inertial particles in turbulent flowsLeiden, August 2006 The contraction rate The contraction rate [Maxey (1987)] Effective compressibility good for r<<  for St  <<1 [Balkovsky, Falkovich & Fouxon (2001)] No - sweeping Yes - sweeping Assume that the argument remains valid also for St r ->0 (reasonable for r enough large &  not too large) Then Though Though we cannot exclude we cannot exclude finite Re effects finite Re effects

Massimo Cencini Massimo Cencini Clustering of Inertial particles in turbulent flowsLeiden, August 2006Numerics Non-dimensional contraction rate 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

Massimo Cencini Massimo Cencini Clustering of Inertial particles in turbulent flowsLeiden, August 2006 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  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

Massimo Cencini Massimo Cencini Clustering of Inertial particles in turbulent flowsLeiden, August 2006 Role of Sweeping on acceleration A short history  Tennekes 1975 points out the importance of sweeping for multitime statistics and pressure/acceleration  Van Atta & 1975 experimental evidence of k -5/3 for pressure  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 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 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 Gotoh & Fukayama 2001 both k -5/3 and k -7/3 are observed, is k -5/3 spurious or a finite Re effect? spurious or a finite Re effect?