Inertial particles in turbulence

1. Inertial particles in turbulence Massimo Cencini CNR-ISC Roma INFM-SMCUniversità “La Sapienza” Roma Massimo.Cencini@roma1.infn.it In collaboration with: J. Bec, L. Biferale, G. Boffetta, A. Celani, A. Lanotte, S. Musacchio & F. Toschi

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

3. Rain drops in clouds(G. Falkovich et al. Nature 141, 151 (2002)) clustering enhanced collision rate  Formation of planetesimals in the solar system (J. Cuzzi et al. Astroph. J. 546, 496 (2001); A. Bracco et al. Phys. Fluids 11, 2280 (2002)) Optimization of combustion processes indiesel engines(T.Elperin et al. nlin.CD/0305017)

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

5. 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 Phenomenology

6. 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)

7. DNS summary Resolution1283,2563, 5123 Pseudo spectral code Normal viscosity Code parallelized MPI+FFTW Platforms: SGI Altix 3700, IBM-SP4 Runs over 7 - 30 days

8. Particle Clustering Important in optimization of reactions, 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)

9. Two kinds of clustering Particle preferential concentration 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

10. Small scales clustering • Velocity is smooth we expect fractal distribution • Probability that 2 particles are at a distance • correlation dimension D2 Use of a tree algorithm to measure dimensions at scales

11. Correlation dimension D2 weakly depending on Re Maximum of clustering for Hyperbolic non-hyperbolic 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)

12. Multifractal distribution  Intermittency in the mass distribution

13. Lyapunov dimension d D1 provides information similar to D2 can be seen as a sort of “effective” compressibility

14. Inertial-range clustering

15. 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?

16. PDF of the coarse-grained mass r 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)

18. Collapse of CG-mass moments Inertial range

19. Sketchy argument for s/r5/3 <-- from the equation of motion <-- Rate of volume contraction True for St<<1(Maxey (1987) & Balkovsky, Falkovich & Fouxon (2001)) Reasonable also for St(r)<<1 (i.e. in the inertial range) The relevant time scale for the distribution of particles is that which distinguishes their dynamics from that of tracers can be estimated as The argument can be made more rigorous in terms of the dynamics of the quasilagrangian mass distribution and using the rate of volume contraction. But the crucial assumption is

20. 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

21. Single point acceleration properties 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). 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, 064502 (2004) Mordant et al. New J. Phys. 6, 116 (2004)

22. 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)

23. Two mechanisms Preferential concentration Filtering

24. 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

25. Preferential concentration Fluid acceleration Fluid acc. conditioned on particle positions Heavy particle acceleration

26. Filtering Fluid acceleration Filtered fluid acc. along fluid trajectories Heavy particle acceleration

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

28. 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!

29. Thanks

30. Mass conservation Where we assumed that a p.vel. Field can be defined With the choice One sees that pr, (t) can be Related to pr, (t-T(r,)) hence all the statistical Properties depend on T(r,). Then assuming From which Hence if a=a0