M.Cencini Inertial particles in turbulent flows Warwick, July 2006 Inertial particles in turbulence Massimo Cencini CNR-ISC Roma INFM-SMC Università “La Sapienza” Roma In collaboration with: J. Bec, L. Biferale, G. Boffetta, A. Celani, A. Lanotte, S. Musacchio & F. Toschi

M.Cencini Inertial particles in turbulent flows Warwick, July 2006 Problem: Problem: Particles differ from fluid tracers their dynamics is dissipative due to inertia one has preferential concentration Goals : Goals : understanding physical mechanisms at work, characterization of dynamical & statistical properties Main assumptions Main assumptions : collisionless heavy & passive particles in the absence of gravity In many situations it is important to consider finite-size (inertial) particles transported by incompressible flows.

M.Cencini Inertial particles in turbulent flows Warwick, July 2006 Rain drops in clouds  Rain drops in clouds ( G. Falkovich et al. Nature 141, 151 (2002))  clustering  enhanced collision rate  Formation of planetesimals in the solar system solar system ( J. Cuzzi et al. Astroph. J. 546, 496 (2001); A. Bracco et al. Phys. Fluids 11, 2280 (2002)) Optimization of combustion processes in diesel engines Optimization of combustion processes in diesel engines ( T.Elperin et al. nlin.CD/ ) 

M.Cencini Inertial particles in turbulent flows Warwick, July 2006 Equations of motion & assumptions Dissipative range physics Heavy particles Particle Re <<1 Dilute suspensions: no collisions Stokes number Response time Stokes Time (Maxey & Riley Phys. Fluids 26, 883 (1983)) Kolmogorov ett u(x,t) (incompressible) fluid velocity field

M.Cencini Inertial particles in turbulent flows Warwick, July 2006 Phenomenology Preferential concentration: particle trajectories detach from those of tracers due to their inertia inducing preferential concentration in peculiar flow regions. Used in flow visualizations in experiments Dissipative dynamics: The dynamics is uniformly contracting in phase-space with rate As St increases spreading in velocity direction --> caustics This is the only effect present in Kraichnan models Note that as an effect of dissipation the fluid velocity is low-pass filtered

M.Cencini Inertial particles in turbulent flows Warwick, July 2006 Direct numerical simulations After the fluid is stabilized simulation box seeded with millions of particles and tracers injected randomly & homogeneously with For a subset the initial positions of different Stokes particles coincide at t=0 ~2000 particles at each St tangent dynamics integrated for measuring LE Statistics is divided in transient(1-2ett) + Bulk (3ett)

M.Cencini Inertial particles in turbulent flows Warwick, July 2006 DNS summary Resolution 128 3, 256 3, Pseudo spectral code Normal viscosity Code parallelized MPI+FFTW Platforms: SGI Altix 3700, IBM-SP4 Runs over days N3N Tot #particles120Millions32Millions4Millions Fast Slow 107.5Millions2Millions Stokes/Lyap(15+1)/(32+1) 15+1 Traject. Length Disk usage1TB400GB70GB

M.Cencini Inertial particles in turbulent flows Warwick, July 2006 Particle Clustering  Important in optimization of reactions, rain drops formation…. rain drops formation….  Characterization of fractal aggregates  Re and St dependence in turbulence? Some studies on clustering:  Squires & Eaton Phys. Fluids 3, 1169 (1991)  Balkovsky, Falkovich & Fouxon Phys. Rev. Lett. 86, 2790 (2001)  Sigurgeirsson & Stuart Phys. Fluids 14, 1011 (2002)  Bec. Phys. Fluids 15, L81 (2003)  Keswani & Collins New J. Phys. 6, 119 (2004)

M.Cencini Inertial particles in turbulent flows Warwick, July 2006 Two kinds of clustering Particle preferential concentration 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

M.Cencini Inertial particles in turbulent flows Warwick, July 2006 Small scales clustering Velocity is smooth we expect fractal distribution Probability that 2 particles are at a distance correlation dimension D 2 Use of a tree algorithm to measure dimensions at scales

M.Cencini Inertial particles in turbulent flows Warwick, July 2006 Correlation dimension  D 2 weakly depending on Re Maximum of clustering for Particles preferentially concentrate in the hyperbolic regions of the flow.  Maximum of clustering seems to be connected to preferential concentration but Counterexample: inertial p. in Kraichnan flow (Bec talk) Hyperbolic non-hyperbolic

M.Cencini Inertial particles in turbulent flows Warwick, July 2006 Multifractal distribution  Intermittency in the mass distribution

M.Cencini Inertial particles in turbulent flows Warwick, July 2006 Lyapunov dimension d  D 1 provides information similar to D 2  can be seen as a sort of “effective” compressibility

M.Cencini Inertial particles in turbulent flows Warwick, July 2006 Inertial-range clustering

M.Cencini Inertial particles in turbulent flows Warwick, July 2006 Characterization of clustering in the inertial range (Preliminary & Naive) From Kraichnan model ===> we do not expect fractal distribution (Bec talk and Balkovsky, Falkovich, Fouxon 2001) Range too short to use local correlation dimension or similar characterization Coarse grained mass: St=0 ==> Poissonian St  0 ==> deviations from Poissonian. How do behave moments and PDF of the coarse grained mass?

M.Cencini Inertial particles in turbulent flows Warwick, July 2006 PDF of the coarse-grained mass r ss Deviations from Poissonian are strong & depends on  s, r Is inertial range scaling inducing a scaling for Kraichnan results suggest invariance for (bec talk)

M.Cencini Inertial particles in turbulent flows Warwick, July 2006

M.Cencini Inertial particles in turbulent flows Warwick, July 2006 Collapse of CG-mass moments Inertial range

M.Cencini Inertial particles in turbulent flows Warwick, July 2006 Sketchy argument for  s /r 5/3 True for St<<1 (Maxey (1987) & Balkovsky, Falkovich & Fouxon (2001)) Reasonable also for St(r)<<1 (i.e. in the inertial range) <-- Rate of volume contraction <-- from the equation of motion The relevant time scale for the distribution of particles is that which distinguishes their dynamics from that of tracers can be estimated as dynamics The argument can be made more rigorous in terms of the dynamics of the quasilagrangian mass distribution of the quasilagrangian mass distribution and using the rate of volume contraction. But the crucial assumption is

M.Cencini Inertial particles in turbulent flows Warwick, July 2006 Scaling of acceleration Controversial result about pressure and pressure gradients (see e.g. Gotoh & Fukayama Phys. Rev. Lett. 86, 3775 (2001) and references therein) Our data are compatible with the latter Note that this scaling comes from assuming that the sweeping by the large scales is the leading term We cannot exclude that the other spectra may be observed at higher Re

M.Cencini Inertial particles in turbulent flows Warwick, July 2006 Single point acceleration properties Some recent studies on fluid acceleration:  Vedula & Yeung Phys. Fluids 11, 1208 (1999)  La Porta et al. Nature 409, 1011 (2001) ; J. Fluid Mech 469, 121 (2002)  Biferale et al. Phys. Rev. Lett. 93, (2004)  Mordant et al. New J. Phys. 6, 116 (2004)  Probe of small scale intermittency  Develop Lagrangian stochastic models  What are the effect of inertia? Bec, Biferale, Boffetta, Celani, MC, Lanotte, Musacchio & Toschi J. Fluid. Mech. 550, 349 (2006); J. Turb. 7, 36 (2006).

M.Cencini Inertial particles in turbulent flows Warwick, July 2006 Acceleration statistics  At increasing St: strong depletion of both rms acc. and pdf tails.  Residual dependence on Re very similar to that observed for tracers. ( Sawford et al. Phys. Fluids 15, 3478 (2003); Borgas Phyl. Trans. R. Soc. Lond A342, 379 (1993)) DNS data are in agreement with experiments by Cornell group (Ayyalasomayajula et al. Phys. Rev. Lett. Submitted)

M.Cencini Inertial particles in turbulent flows Warwick, July 2006 Two mechanisms Preferential concentration Filtering

M.Cencini Inertial particles in turbulent flows Warwick, July 2006 Preferential concentration & filtering  Heavy particles acceleration  Fluid acc. conditioned on p. positions good at St<<1  Filtered fluid acc. along fluid traj. good at St>1

M.Cencini Inertial particles in turbulent flows Warwick, July 2006 Preferential concentration  Fluid acceleration  Fluid acc. conditioned on particle positions  Heavy particle acceleration

M.Cencini Inertial particles in turbulent flows Warwick, July 2006 Filtering  Fluid acceleration  Filtered fluid acc. along fluid trajectories  Heavy particle acceleration

M.Cencini Inertial particles in turbulent flows Warwick, July 2006 Dynamical features From passive tracers studies we know that wild acceleration events come trapping in strong vortices from trapping in strong vortices. (La Porta et al 2001) (Biferale et al 2004) Inertia expels particles from strong vortexes ==> acceleration depletion (a different way to see the effect of preferential concentration)

M.Cencini Inertial particles in turbulent flows Warwick, July 2006 Conclusions Two kinds of preferential concentrations in turbulent flows : Dissipative range: intrinsic clustering (dynamical attractor)  tools borrowed from dynamical system  concentration in hyperbolic region Inertial range:  voids due to ejection from eddies  Mass distribution recovers uniformity in a self-similar manner ( DNS at higher resolution required, experiments? )  open characterization of clusters ( minimum spanning tree….?? )  Preferential concentration together with the dissipative nature of the dynamics affects small scales as evidenced by the behavior of acceleration  New experiments are now available for a comparative study with DNS, preliminary comparison very promising!

M.Cencini Inertial particles in turbulent flows Warwick, July 2006 Thanks

M.Cencini Inertial particles in turbulent flows Warwick, July 2006 Then assuming With the choice Mass conservation One sees that p r,  (t) can be Related to p r,  (t-T(r,  )) hence all the statistical Properties depend on T(r,  ). From which Hence if a  =a 0 Where we assumed that a p.vel. Field can be defined