Clustering of inertial particles in turbulence

1
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
• Massimo.Cencini_at_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 -gt 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 ltlt1
• Very dilute suspensions no collisions
• passive particles
• no gravity

(Maxey Riley Phys. Fluids 26, 883 (1983))
Stokes number
Drag Stokes Time
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Phenomenology

• Mechanisms at work
• Ejection of heavy particles from vortices
  preferential 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
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 Pseudo spectral code with
resolution 1283, 2563, 5123 - Re?65, 105,
185 Normal viscosity
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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
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Clustering at rlt?

• 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
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Correlation dimension

• 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)
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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
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It works in Kraichnan flows
Gaussian random flow with no-time
correlation Incompressible, homogeneous and
isotropic
h1 dissipative range hlt1 inertial range
Local correlation dimension
Note that tracers limit Is recovered for Str
-gt0 (i.e. for ??0 or r??)
(Bec, MC Hillenbrand 2006 nlin.CD/0606038)
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In turbulence?
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 ?ltlt1 Poisson for tracers
(?0) deviations already for ?ltlt1
For ?ltlt1 algebraic tails (voids)
Result on Kraichnan suggests Pr,?(?) PSt(r)(?)

• But is not!

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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
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
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Summary Conclusions

• Description of particle clustering for moderate
  St number and moderate Re number in the
  dissipative and inertial range
• rltlt? strong clustering, everything depends on
  St? very weakly on Re
• ?ltrltL very slow recovery of uniformity, and the
  statistics depends on the contraction rate.
  Dominance of voids --gt 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

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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--gt 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?