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Leiden, AugustLeiden, August 20062006Massimo CenciniMassimo Cencini Clustering of Inertial particles in turbulent flows
Clustering of inertialClustering of inertial particles in turbulenceparticles in turbulence
Massimo CenciniMassimo 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.itMassimo.Cencini@roma1.infn.it
with:
J. Bec, L. Biferale, A. Lanotte, S. Musacchio & F. Toschi
(nlin.CD/0608045)
Leiden, AugustLeiden, August 20062006Massimo CenciniMassimo Cencini Clustering of Inertial particles in turbulent flows
What we know and what we want to What we know and what we want to knowknow
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?
Leiden, AugustLeiden, August 20062006Massimo CenciniMassimo Cencini Clustering of Inertial particles in turbulent flows
MotivationsMotivations
Rain Drops Rain Drops formation formation
In warm cloudsIn warm clouds1.1. CCN activationCCN activation2.2. CondensationCondensation3.3. CoalescenceCoalescence
(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
Leiden, AugustLeiden, August 20062006Massimo CenciniMassimo Cencini Clustering of Inertial particles in turbulent flows
MotivationMotivation
Sprays & Sprays & optimizationoptimizationof combustion of combustion processes inprocesses in diesel enginesdiesel engines (T.Elperin et al. nlin.CD/0305017)
From Bracco et al. (Phys. Fluids 1999)
Protoplanetary disk1. Migration of dust to the
equatorial plane
2. Accretion of planetesimals from 100m to few Km
3. Gravitation & collisions coalescence -> planetary embryos
Main issue: time scalesAerosols
Leiden, AugustLeiden, August 20062006Massimo CenciniMassimo Cencini Clustering of Inertial particles in turbulent flows
Heavy particle dynamicsHeavy particle dynamics
Particles with (Kolmogorov scale)
Heavy particles
Particle Re <<1Very dilute suspensions: no collisions passive particlesno gravity
η<<a
fpρρ >>
1vRea
<<= νaa
Stokes number
Drag: Stokes Time
(Maxey & Riley Phys. Fluids (Maxey & Riley Phys. Fluids 2626, 883 (1983)), 883 (1983))
Leiden, AugustLeiden, August 20062006Massimo CenciniMassimo Cencini Clustering of Inertial particles in turbulent flows
PhenomenologyPhenomenology
Mechanisms at work: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 Stη , i.e. particles may arrive very close with very different velocities
Leiden, AugustLeiden, August 20062006Massimo CenciniMassimo Cencini Clustering of Inertial particles in turbulent flows
DNS summaryDNS summary
1TB
900 +2100
(15+1)/(32+1)
7.5Millions
500.000
120Millions
5123
15+1(15+1)/(32+1)Stokes/Lyap
70GB400GBDisk usage
600+1200756 +1744Traject. Length
250.0002MillionsSlow 10 η
32.000250.000Fast 0.1 η
4Millions32MillionsTot #particles
12832563N3
NS-equation +
Particles with &
Tracers
STATISTICSTRANSIENT (1-2 T)+BULK ( 3-4 T)
SETTINGSSETTINGSmillions of particles and tracers millions of particles and tracers injected randomly & injected randomly & homogeneously homogeneously with initial vel. = to that of the with initial vel. = to that of the fluidfluid
NOTESNOTES
Pseudo spectral code with Pseudo spectral code with
resolutionresolution12812833
,, 25625633
, 512, 5123 3 - - ReRe=65, =65,
105, 185105, 185
Normal Normal viscosityviscosity
Leiden, AugustLeiden, August 20062006Massimo CenciniMassimo Cencini Clustering of Inertial particles in turbulent flows
Two kinds of clusteringTwo kinds of clusteringParticle clustering is observed both
in the dissipativedissipative and in inertialinertial range
Instantaneous p. distribution in a slice of width ≈ 2.5η. Stη = 0.58 R = 185
Leiden, AugustLeiden, August 20062006Massimo CenciniMassimo Cencini Clustering of Inertial particles in turbulent flows
Clustering at r<Clustering at r<ηη• Velocity is smooth we expect fractal distribution
• At these scales the only relevant time scale is η thus everything must be a function of StStηη & Re& Re only
correlation dimension
Leiden, AugustLeiden, August 20062006Massimo CenciniMassimo Cencini Clustering of Inertial particles in turbulent flows
Correlation dimensionCorrelation dimension
Stη is the only relevant parameterMaximum of clustering for Stη1D2 almost independent of Re, (Keswani & Collins (2004) ) high order statistics?
Maximum of clustering seems to Maximum of clustering seems to bebeconnected to preferential connected to preferential concentrationconcentrationconfirming the traditional confirming the traditional scenarioscenario
Though is non-generic: counter Though is non-generic: counter example example Kraichnan flowsKraichnan flows (Bec, MC, Hillenbrand (Bec, MC, Hillenbrand 2006)2006)
Hyperbolic non-hyperbolic
Particles preferentially Particles preferentially concentrate concentrate in hyperbolic regionsin hyperbolic regions
Prob. to be in non-hyperbolic pointsProb. to be in non-hyperbolic points
The preferential concentrationThe preferential concentrationis also evidenced by lookingis also evidenced by lookingat the fluid acceleration at the fluid acceleration conditioned on particle conditioned on particle positions positions aa((XX,t),t)
(Bec,Biferale, Boffetta, Celani, MC, Lanotte,(Bec,Biferale, Boffetta, Celani, MC, Lanotte,
Musacchio & Toschi (2006))Musacchio & Toschi (2006))
Leiden, AugustLeiden, August 20062006Massimo CenciniMassimo Cencini Clustering of Inertial particles in turbulent flows
Inertial-range clusteringInertial-range clustering
•Voids & structures Voids & structures from from ηη to L to L
•Distribution of Distribution of particles over particles over scales?scales?
•What is the What is the dependence on Stdependence on Stηη? ? Or what is the Or what is the proper parameter?proper parameter?
Leiden, AugustLeiden, August 20062006Massimo CenciniMassimo Cencini Clustering of Inertial particles in turbulent flows
Preliminary considerationsPreliminary considerationsParticles should not distribute self-similarlyCorrelation functions of the density are not power law(Balkovsky, Falkovich & Fouxon 2001)
Natural expectationNatural expectationIn analogy with the dissipative clustering since
at scale r the typical time scale is r=-1/3r2/3
the only relevant parameter should be Stthe only relevant parameter should be Strr
Leiden, AugustLeiden, August 20062006Massimo CenciniMassimo Cencini Clustering of Inertial particles in turbulent flows
It works in Kraichnan flowsIt works in Kraichnan flows
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
Gaussian random flow with no-time correlationGaussian random flow with no-time correlationIncompressible, homogeneous and isotropicIncompressible, homogeneous and isotropic
(Bec, MC & Hillenbrand 2006; nlin.CD/0606038)(Bec, MC & Hillenbrand 2006; nlin.CD/0606038)
h=1 dissipative rangeh=1 dissipative range
h<1 inertial rangeh<1 inertial range
Local corr
ela
tion
Local corr
ela
tion
dim
en
sio
nd
imen
sio
n
Note that tracers limitNote that tracers limitIs recovered for StIs recovered for Strr ->0 ->0
(i.e. for (i.e. for 0 or r0 or r))
Leiden, AugustLeiden, August 20062006Massimo CenciniMassimo Cencini Clustering of Inertial particles in turbulent flows
In turbulence?In turbulence?*PDF of the coarse-grained mass: number density of*PDF of the coarse-grained mass: number density of particles ( N in total ) at scale r, weighting each cellparticles ( N in total ) at scale r, weighting each cell with the mass it contains, with the mass it contains, natural (Quasi-Lagrangian)natural (Quasi-Lagrangian) measure to reduce finite N effects at measure to reduce finite N effects at ρρ<<1<<1
*Poisson for tracers (*Poisson for tracers (=0) deviations already for =0) deviations already for <<1<<1
Result on Kraichnan suggestsResult on Kraichnan suggests
PPr,r,((ρρ)= )= PPSt(r)St(r)((ρρ))
But is not!But is not!
r=L/16r=L/16*For *For ρρ<<1 <<1 algebraic tails (voids)algebraic tails (voids)
Leiden, AugustLeiden, August 20062006Massimo CenciniMassimo Cencini Clustering of Inertial particles in turbulent flows
Why does not work?Why does not work?Kraichnan model: Kraichnan model:
•no-time correlationsno-time correlations•no-sweepingno-sweeping•no-structuresno-structures
In Turbulence we have allIn Turbulence we have all
2d-NS Inverse cascade:2d-NS Inverse cascade: strong correlation strong correlation betweenbetweenparticle positions and zero acceleration pointsparticle positions and zero acceleration points
In 2d Kinematic flowsIn 2d Kinematic flows: : (no-sweeping)(no-sweeping) still still clusteringclusteringbut no correlations with zero acceleration pointsbut no correlations with zero acceleration points
(Chen, Goto & Vassilicos 2006)(Chen, Goto & Vassilicos 2006)Working hypothesisWorking hypothesis
May be sweeping is playing some May be sweeping is playing some rolerole
Leiden, AugustLeiden, August 20062006Massimo CenciniMassimo Cencini Clustering of Inertial particles in turbulent flows
The contraction rateThe contraction rate
[Maxey (1987)][Maxey (1987)]
Effective compressibilityEffective compressibilitygood for r<<good for r<<ηη for St for Stηη<<1<<1
[Balkovsky, Falkovich & Fouxon (2001)][Balkovsky, Falkovich & Fouxon (2001)]
No - sweeping No - sweeping
Yes - sweepingYes - sweeping
Assume that the argument remains valid also for StAssume that the argument remains valid also for Strr-->0 >0
(reasonable for r enough large & (reasonable for r enough large & not too large) not too large)ThenThen
ThoughThough
we cannot exclude we cannot exclude
finite Re effectsfinite Re effects
Leiden, AugustLeiden, August 20062006Massimo CenciniMassimo Cencini Clustering of Inertial particles in turbulent flows
NumericsNumerics
Non-dimensional contraction rateNon-dimensional contraction rate
The collapse confirms The collapse confirms that the contraction ratethat the contraction rateis indeed the proper time is indeed the proper time scalescale
Uniformity is recoveredUniformity is recoveredvery slowly going to thevery slowly going to thelarge scales, e.g. muchlarge scales, e.g. muchslower than for Poissonslower than for Poissondistributiondistribution 9/59/5
Leiden, AugustLeiden, August 20062006Massimo CenciniMassimo Cencini Clustering of Inertial particles in turbulent flows
Summary & Conclusions Summary & Conclusions Description of particle clustering for moderate St number
and moderate Re number in the dissipative and inertial range
r<<η strong clustering, everything depends on Stη & very weakly on Reη<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 propertiesLarger Re studies necessary to confirm the picture
Leiden, AugustLeiden, August 20062006Massimo CenciniMassimo Cencini Clustering of Inertial particles in turbulent flows
Role of Sweeping on Role of Sweeping on accelerationacceleration
A short historyA short history
Tennekes 1975Tennekes 1975 points out the importance of sweeping for multitime points out the importance of sweeping for multitime statistics and pressure/accelerationstatistics and pressure/acceleration
Van Atta & Van Atta & Wyngaard 1975 1975 experimental evidence of k experimental evidence of k-5/3-5/3 for pressure for pressureYakhot, Orzag & She 1989Yakhot, Orzag & She 1989 RG--> k RG--> k-7/3-7/3 for pressure for pressureChen & Kraichnan 1989Chen & Kraichnan 1989 importance of sweeping for multitime statistics importance of sweeping for multitime statistics
RG does not consider sweeping from the outset RG does not consider sweeping from the outset Nelkin & Tabor 1990Nelkin & Tabor 1990 importance of sweeping for acceleration & pressure importance of sweeping for acceleration & pressureSanada & Shanmugasundaram 1992 numerics on multitime and pressure
confirming the important role of sweeping
More recently• Vedula & Yeung 1999Vedula & Yeung 1999 doubts on k doubts on k-5/3-5/3 for pressure but observed for pressure but observed• Gotoh & Fukayama 2001Gotoh & Fukayama 2001 both k both k-5/3-5/3 and k and k-7/3 -7/3 are observed, is kare observed, is k-5/3 -5/3
spurious or a finite Re effect?spurious or a finite Re effect?