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Heavy particles in turbulent flows Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte with: J. Bec, L. Biferale, G. Boffetta, A. Celani, M. Cencini, S. Musacchio, F. Toschi Alessandra Lanotte CNR ISAC Lecce (Italy)

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Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte Outline Introduction Physical systems Observations Model Details of numerical simulations What we measured Short summary of some results Core of the talk Small scales clustering Inertial scales clustering

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Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte Where do we find heavy particles? Formation of planetesimals in the solar system ( A. Bracco et al. Phys. Fluids 2002 ) Control of combustion processes in diesel engines (see T.Elperin et al. nlin.CD/ ) In clouds, dust storms, fires volcano eruption.. (see e.g. K. Sassen, Nature 2005)

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Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte What can we observe/measure in a lab? Lagrangian turbulence has always suffered the lack of accurate space & time measurements now particles can be accurately tracked ! From Cornell group: frame rate : 1000fps; 4x4 cm area. State-of-the-art Lagrangian experiments (tracers) Ott & Mann exp. at Risø, 3D PTV - Re 300 Pinton exp. at ENS, Doppler track. Re =740 Bodenshatz exp. at Cornell, fast CCD Re =1000

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Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte Heavy particles in wind tunnel turbulence Z. Warhaft experiment at Cornell Re 250 water droplets = 20 micron High-speed camera: 2D frames Sampling time 1/100 then also other experiments in complex geometries: e.g. channel flows,..

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Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte The Model finite size impurities of size much smaller than the flow dissipative scale much heavier than the fluid particle Reynolds number low very dilute suspension : no role of collisions no back reaction on the flow

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Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte Simplified equations Under previous assumption we can simplify original eqs: (M. Maxey & J. Riley, Phys Fluids 1983) Parameters: Stokes time --> Stokes number Density ratio X only Stokes drag (water in air =0.001)

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Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte Something we know about inertia… Try to understand physical mechanisms and identify relevant parameters for statistical description… 1. Ejection of heavy particles from vortices --> experience smaller acceleration 3.Very strong concentration fluctuation --> particle distribute on clusters 2. Particle have finite response time to fluid fluctuations --> smoothing and filtering of fast time scales (since Maxey, Eaton, Fessler, Squires, …)

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Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte A large numerical “experiment” simplest To start with the simplest situation To have good statistics To build up a database for common use The lab particles in the flow box

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Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte Details about the DNS N3N Tot #particles120Millions32Millions4Millions Fast 0.1 Slow 10 7.5Millions2Millions Stoke/Lyap(15+1)/(32+1) 15+1 Traject. Length Disk usage1TB400GB70GB Lagrangian Particles with 15 Lagrangian Tracers Initial conditions particles and tracers injected randomly & homogeneously with initial veloc. = fluid veloc. STATISTICS TRANSIENT ( 1-2 T)+BULK ( 3-4 T) Re = 65, 105, 185 Pseudo Spectral Code, MPI Normal viscosity

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Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte How long do we wait for the stationary mass distribution? Coarse-grained mass in the j-th cell of side l= 2 x St=0 St=0.48 St=0.27 St=0.9 St=1.6 St=3.3 St=0.16

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Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte Just a quick overview about few things: Acceleration Conditioned analysis

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Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte Why study acceleration ? Urban reshape, Old Shangai Steel factory, Taranto Acceleration is relevant for Lagrangian Stochastic Models for relative dispersion (see e.g. Sawford, Ann. Rev. Fluid Mech. 2001)

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Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte Acceleration for tracers a P(a) K41 prediction Multifractal (Biferale, Boffetta, Celani, Devenish, AL, Toschi 2004) Tracers acceleration can be very well described in terms of the multifractal model Phenomenological model for small scale fluctuations 1024^3 DNS

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Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte Acceleration for heavy particles Increase St Increase Re Two coexisting effects preferential concentration at low St filtered dynamics at higher St No simple phenomenological model for particles at varying Stokes and Reynolds numbers !

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Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte Comparison with experiments A. Gylfason, S. Ayyalasomayajula, E. Bodenschatz, Z. Warhaft, PRL submitted 2006 St=0.09 St=0.15 Water drops in air: clearly polydisperse flow !

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Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte Particles and 3d flow structures White: non-hyper regions Black: hyperbolic regions St =0.16St = 0.8 St = 3.3 Particles preferentially concentrate in hyperbolic regions Such effect is clearly evident by looking at the fluid acceleration conditioned on particle positions a(X,t) (Bec, Biferale, Boffetta, Celani, Cencini, AL, (Bec, Biferale, Boffetta, Celani, Cencini, AL, Musacchio & Toschi 2006) Musacchio & Toschi 2006) hyperbolic non-hyperbolic P nonhyper

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Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte Summarising Acceleration statistics depends on two mechanisms: 1. Preferential concentrat. of particles effective at small St 2. Filtering due to particles response time effective at large St A very small amount of inertia expels particles out of intense structures: strong correlation with flow at small St; at larger St, because of filtering, particles can not follow the flow: no correlation with flow at larger St Can we better understand clustering ?

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Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte One motivation Strong particle concentration fluctuations have an impact on climate in different ways Reflective power of the atmosphere due to aerosols scattering and absorption is crucial for climatological models Desert dusts are particularly active ice-forming agents. They can affect clouds formation.

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Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte Rain droplets formation due to clustering Enhanced collision rates may explain rapid rain formation Rain drop size 2mm coalescence Droplet size 0.02mm condensation CCN size 0.2-2micron nucleation preferential concentration + gravity (warm) cloud large scale L=100m; dissipative scale = 1mm; Re=10 7

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Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte Only a small scale feature? Particle clusters & voids are observed both dissipativeinertial in the dissipative and in inertial range Slice of width ≈ 2.5 . Particles with St = 0.58; R = 185

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Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte Observables at small scales r < Space density of particles pairs (useful for collisions, pair dynamics) Probability to find 2 particles at a distance smaller than r Another common observable is the radial distribution function g(r) It is O(1) for tracers, it diverges as r--> 0 for inertial particles (or in compressible flows). is the correlation dimension (Grassberger 1983 ; Hentschel Procaccia, 1983) r

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Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte Probability and D 2 Velocity is smooth: we expect fractal distribution (with power law tails) At these scales, the only relevant time scale is thus everything should depend on St & Re only

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Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte Shape of correlation dimension D 2 Optimal Stokes number for maximal clusterization No Reynolds dependence (as in Collins & Keswani 2004) Similar behaviour at higher order D q Particles positions correlate with low values of acceleration (for 2d flows Chen, Goto, Vassilicos 2006) Maximum of clustering seems to be connected to preferential concentration, confirming classical scenario preferential concentration, confirming classical scenario

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Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte What happens at larger scales < r < L? Can particles of Stokes time feel effects of time scales t r >> ? How do particles distribute out of vortical regions? What are the proper parameters to describe the mass distribution?

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Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte Inertial range observables Probability Distribution Function of the coarse-grained particle density: Given N particles, we compute number density of particles within a cell of scale r, weighting each cell with the mass it contains: Quasi-Lagrangian measure a natural measure to reduce finite N effects at <<1 due to voids r

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Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte Quasi-Lagrangian mass density Tracers behave according to uniform Poisson distribution Particle show deviations, already there for very small such deviations become stronger with r=L/16

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Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte Algebraic tails at low density <<1 we have (tracers limit, uniform) (non zero prob. to have empty areas) St These empty regions can play a relevant role in many physical issues

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Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte How do we understand this PDFs? Particles should not distribute self-similarly i.e. Deviations from a uniform distribution are not scale-invariant (Balkovsky, Falkovich & Fouxon 2001) No simple rescaling of the mass distributions We note however that for the mass PDF these two limits are equivalent: fixed and r ∞ (large observation scale) fixed r and small inertia) Both limits give a uniform particle distribution. So…

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Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte So there could be a parameter, rescaling the mass distribution, which relates Stokes times and observations scales r At scale r, the eddy-turn-over time scale is r = -1/3 r 2/3, in analogy with dissipative scales, we could define: Is this time scale relevant particle clustering in the inertial range?

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Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte Unfortunately not so simple! This simple analogy works in synthetic flows: e.g. Kraichnan flows no time correlation no spatial structures no large scale-sweeping (Bec, Cencini & Hillerbrand 2006) (Bec, Cencini & Hillerbrand 2006) But it does not work in real turbulence where all these features are present… X

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A different observation [Maxey (1987)] Effective compressibility good for r<< for St <<1 [Balkovsky, Falkovich & Fouxon (2001)] Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte [Maxey (1987)] Suppose the argument remains valid also for finite r & This is the contraction- rate of a particle volume of of size r and Stokes time particleflow

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Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte Numerical Results Non-dimensional contraction rate Collapse of the coarse-grained mass PDF for different values of Uniformity is recovered going to the large scales But very slowly

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Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte Deviation from uniformity: 2nd moment can give some better information

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Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte Conclusions 1. clustering at small scales r < The only relevant number for particle dynamics is St = / Particles concentrate onto a multi-fractal set, whose dimension depends on the Stokes number only (or just very weakly depends on Reynolds) Optimal finite Stokes number for clusterization: St ~ 0.6 (unpredictable..) This global picture is the same as in smooth random flow (see Bec 2005; Bec, Celani, Cencini,Musacchio 2005) We gave a description of particle clustering for moderate St and moderate Re 200 numbers 2. clustering at inertial range scales < r < L concentration fluctuations are relevant also for the inertial range scales uniformity of mass distribution is recovered very slowly at large scale if the contraction rate , and not St r, is the proper number to rescale mass statistics - ---> sweeping is important (Bec, Biferale, Cencini, AL, Musacchio, Toschi PRL submitted 2006)

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Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte Perspectives A better understanding of the statistics of fluid acceleration (rather than vorticity) seems crucial to understand clustering Conversely inertial particles can be used as probes for acceleration properties Larger Re studies are necessary to confirm the picture Currently performing DNS to study rain drops growth

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A common database :

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Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte END

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Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte St=0 St=3.31 Particles with different inertia inside a vortex

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Istituto di Scienze dell’Atmosfera e del Clima Alessandra Lanotte Clusters & voids 2d slice (512x512x4) at Stokes 0.16 (blue) 0.8 (red) 1.33 (green)

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