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Leiden, August Leiden, August 2006 2006 Massimo Cencini Massimo Cencini Clustering of Inertial particles in turbulent flows Clustering of inertial Clustering of inertial particles in turbulence particles in turbulence Massimo Cencini Massimo Cencini CNR-INFM Statistical Mechanics and Complexity Università “La Sapienza” Rome CNR- Istituto dei Sistemi Complessi, Via dei Taurini 19, Rome [email protected] [email protected] with: J. Bec, L. Biferale, A. Lanotte, S. Musacchio & F. Toschi (nlin.CD/0608045)

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

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Page 1: Massimo Cencini Massimo Cencini Clustering of Inertial particles in turbulent flowsLeiden, August 2006 Clustering of inertial particles in turbulence Massimo

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

[email protected]@roma1.infn.it

with:

J. Bec, L. Biferale, A. Lanotte, S. Musacchio & F. Toschi

(nlin.CD/0608045)

Page 2: Massimo Cencini Massimo Cencini Clustering of Inertial particles in turbulent flowsLeiden, August 2006 Clustering of inertial particles in turbulence Massimo

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?

Page 3: Massimo Cencini Massimo Cencini Clustering of Inertial particles in turbulent flowsLeiden, August 2006 Clustering of inertial particles in turbulence Massimo

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

Page 4: Massimo Cencini Massimo Cencini Clustering of Inertial particles in turbulent flowsLeiden, August 2006 Clustering of inertial particles in turbulence Massimo

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

Page 5: Massimo Cencini Massimo Cencini Clustering of Inertial particles in turbulent flowsLeiden, August 2006 Clustering of inertial particles in turbulence Massimo

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))

Page 6: Massimo Cencini Massimo Cencini Clustering of Inertial particles in turbulent flowsLeiden, August 2006 Clustering of inertial particles in turbulence Massimo

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

Page 7: Massimo Cencini Massimo Cencini Clustering of Inertial particles in turbulent flowsLeiden, August 2006 Clustering of inertial particles in turbulence Massimo

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

Page 8: Massimo Cencini Massimo Cencini Clustering of Inertial particles in turbulent flowsLeiden, August 2006 Clustering of inertial particles in turbulence Massimo

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

Page 9: Massimo Cencini Massimo Cencini Clustering of Inertial particles in turbulent flowsLeiden, August 2006 Clustering of inertial particles in turbulence Massimo

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

Page 10: Massimo Cencini Massimo Cencini Clustering of Inertial particles in turbulent flowsLeiden, August 2006 Clustering of inertial particles in turbulence Massimo

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))

Page 11: Massimo Cencini Massimo Cencini Clustering of Inertial particles in turbulent flowsLeiden, August 2006 Clustering of inertial particles in turbulence Massimo

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?

Page 12: Massimo Cencini Massimo Cencini Clustering of Inertial particles in turbulent flowsLeiden, August 2006 Clustering of inertial particles in turbulence Massimo

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

Page 13: Massimo Cencini Massimo Cencini Clustering of Inertial particles in turbulent flowsLeiden, August 2006 Clustering of inertial particles in turbulence Massimo

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))

Page 14: Massimo Cencini Massimo Cencini Clustering of Inertial particles in turbulent flowsLeiden, August 2006 Clustering of inertial particles in turbulence Massimo

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)

Page 15: Massimo Cencini Massimo Cencini Clustering of Inertial particles in turbulent flowsLeiden, August 2006 Clustering of inertial particles in turbulence Massimo

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

Page 16: Massimo Cencini Massimo Cencini Clustering of Inertial particles in turbulent flowsLeiden, August 2006 Clustering of inertial particles in turbulence Massimo

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

Page 17: Massimo Cencini Massimo Cencini Clustering of Inertial particles in turbulent flowsLeiden, August 2006 Clustering of inertial particles in turbulence Massimo

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

Page 18: Massimo Cencini Massimo Cencini Clustering of Inertial particles in turbulent flowsLeiden, August 2006 Clustering of inertial particles in turbulence Massimo

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

Page 19: Massimo Cencini Massimo Cencini Clustering of Inertial particles in turbulent flowsLeiden, August 2006 Clustering of inertial particles in turbulence Massimo

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?