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Particles in turbulence

Federico Toschi

NCAR, Boulder CO, 14 August 2012

ICTR International Collaboration for Turbulence Research

/ TN & WI Federico Toschi

• Introduction

• Motivation to study particles in turbulence

• Short review of results on point particles

• A “point particle” model for finite size particles

• Fully deformable droplets...

• Conclusions

Dictionary...

• Tracer: a tiny particle moving following the streamlines of the flow (e.g. a fluid molecule)

• Heavy particle: • density >> fluid density • size << dissipative scale

• Light particle: • density << fluid density• size << dissipative scale

• Large particle:• D <= 10 η

• Deformable “particle”• deforming droplets with surface tension

Where is Lagrangian turbulence ?

Plankton

Pollu)onDust,stoms

Combus)on

Eulerian vs. Lagrangian turbulence

Leonardo da Vinci; circa 1500 (translated by Ugo Piomelli): “Observe the motion of the surface of the water, which resembles that of hair, which has two motions, of which one is caused by the weight of the hair, the other by the direction of the curls; thus the water has eddying motions, one part of which is due to the principal current, the other to the random and reverse motion.”

Eulerian turbulence

Inertial range

Richardson energy cascade

The “standard model”

Multi-"actal modelParisi-Frisch 1995

Tracers acceleration

Acceleration multi-fractal view

with probability

and for the large scale:

The small scale fluctuates !!!

The large scale fluctuates !!!

Acceleration multi-fractal view

From standard multifractal arguments:

Supposing without intermittency (K41 case)

Also able to make other predictions, i.e. acceleration variance conditioned to velocity value:

…with h=1/3

L. Biferale, G. Boffetta, A. Celani, B. Devenish, A. Lanotte, and F. Toschi. Multi#actal statistics of lagrangian velocity and acceleration in turbulence. Physical Review Letters, 93:064502, 2004.

Acceleration p.d.f. result for tracers

Multifractal predictionK41prediction

Lagrangian velocity statistics

Does it exist and how to estimate ?

We do not have much fantasy! So we “copy” from Eulerian turbulence: structure functions

In Eulerian turbulence we have

Magnifying glass: Local Scaling Exponents

The local exponents ζp(τ) act as magnifying glass, probing locally the value of the scaling exponents. By comparing with S2 we also take advantage of the Extended Self Similarity (ESS)

Power law scaling plateaux for local scaling exponents

Biferale, L., Bodenschatz, E., Cencini, M., Lanotte, A. S., Ouellette, N. T., Toschi, F., & Xu, H. (2008). Lagrangian structure functions in turbulence: A quantitative comparison between experiment and direct numerical simulation. Physics Of Fluids, 20(6), 065103. doi:10.1063/1.2930672

DNS & Experiments: no discrepancies !!

S2 vs. τ : resolution far from satisfactory!!

Local scaling exponents

DNS and experiment show a very good agreement

MF: Lagrangian velocity statistics

Bridge between eulerian and lagrangian description:

We assume that and are linked by the typicaleddy turn over time at the given spatial scale

MF: Lagrangian structure functions

Multifractal prediction for the Lagrangian structure functions

whereSame D(h) that forthe Eulerian field !!

This allow us to actually predict the following value:

Velocity structure functions (tracers)

Eulerian (space) -> Lagrangian (time)

Start from Eulerian

Lagrangian turbulence IS universal

ICTR A. Arneodo, R. Benzi, J. Berg, L. Biferale, E. Bodenschatz, A. Busse, E. Calzavarini, B. Castaing, M. Cencini, L. Chevillard, R. Fisher, R. Grauer, H. Homann, D. Lamb, A. S. Lanotte, E. Leveque, B. Luthi, J. Mann, N. Mordant, W.-C. Muller, S. Ott, N.T. Oullette, J.-F. Pinton, S.B. Pope, S.G. Roux, F. Toschi, X. Xu, P.K. Yeung

How to model (point) particles in turbulence ?

Neutral HeavyLight

Faxen pointwise particle model

Numerical study of particles in turbulence

Bubbles β=3, St=0.6 and Reλ=78

Year N Rλ Reference

1972 32 35 Orszag, Patterson, PRL 28 76

1981 128 84 Rogallo, NASA rep. 1981PRL 58 547 (1987)

1991 256 150 Vincent, Meneguzzi, JFM 25 1Sanada, PRA 44 6480

1993 512 200 She et al., PRL 70 3251

2001 1024 460 Gotoh, Fukuyama, PRL 86 3775

2003 2048 730 Kaneda et al. Phys.Fluids 15 L21

2006 4096 1200 Kaneda et al. J. Turb. 7 20

Floating Point Operation for eddy turnover time: FLOP ∝ N3 log(N) x N (at high N flop=660 N3 log(N) for time step)Memory: ram ∝ N3 (at high N, RAM=182 N3 byte)

best fit: Reλ=2.26 N0.75

K41: Reλ≈N2/3

Moore law for DNS of HI turbulence

k-5/3Spectral flux

N Reλ η L TL τη T δx Np

512 183 0.01 3.14 2.1 0.048 5 0.012 0.96 106

1024 284 0.005 3.14 1.8 0.033 4.4 0.006 1.92 106

Lagrangian database(x(t),v(t),a(t)=-∇p+ν∆u)with high temporal resolution

Energy spectrum

CINECA keyproject 10243 DNS+tracers

Pseudo spectral code - dealiased 2/3 rule - normal viscosity - 2 millions of passive tracers- code fully parallelized with MPI+FFTW - Platform IBM SP4 (sust. Performance 150Mflops/proc) - 50000 cpu hours - duration of the run: 40 days

2048 400 0.0025 3.14 1.8 0.02 5.9 0.003 2 109

Lagrangian database (x(t),v(t),u(t),∂iuj (t)) at high resolution

Energy spectrum

DEISA 20483 DNS with tracers & heavy

Pseudo spectral code - dealiased 2/3 rule - normal viscosity - 2 billions of passive tracers & heavy particles- code fully parallelized with MPI+FFTW - Platform SGI Altix 4700 - 400000 cpu hours - duration of the run: 40 days over 3 months.

512 183 0.01 3.14 2.1 0.048 5 0.012 1.0 108

Lagrangian database (x(t),v(t),u(t),∂iuj (t)) at high resolution

CASPUR 5123 DNS tracers & heavy & light

Pseudo spectral code - dealiased 2/3 rule - normal viscosity - 100 millions of passive tracers & heavy/light particles- code fully parallelized with MPI+FFTW - Platform IBM SP5 1.9 GHz - 30000 cpu hours - duration of the run: 30 days.

tracerbubble heavy

64 different particles classes (ß,St)

Heavy particles - parametersL3 2563 5123 20483

Total particles 32 Mparticles 120 Mparticles 2,1 Gparticles

Stokes/ LyapStokes 16/32 16/32 21

Slow dumps 10 τη 2.000.000 7.500.000 101.888.000

Fast dumps 0.1 τη 250.000 500.000 203.776

dt 8 10-4 4 10-4 1.1 10-4

Time step ch0+ch1 756 + 1744 900 + 2100 11000+39000

τη 0.0746 0.0466 0.02

τ 0.0, 0.0120, 0.0200, 0.0280, 0.0360, 0.0440, 0.0520, 0.0600, 0.0680, 0.0760, 0.0840, 0.1000, 0.1200, 0.152, 0.200, 0.248

0.0, 0.00753454, 0.0125576, 0.0175806, 0.0226036, 0.0276266, 0.0326497, 0.0376727, 0.0426957, 0.0477187, 0.0527418, 0.0627878, 0.0753454, 0.0954375, 0.125576, 0.155714

0, 0, 0, 0.0032, 0.0032, 0.0032, 0.012, 0.012, 0.012, 0.02, 0.02, 0.02, 0.04, 0.06, 0.1, 0.2, 0.4, 0.6, 0.8, 1, 1.4

Disk space used 400 GByte 1 TByte 6.3 Tbytes

http://mp0806.cineca.it/icfd.php

iCFDdatabase2

Heavyparticle

Filtered tracer

Tracers at particle position

All effects of inertia, in a slide...

Bubbles Tracers Heavy

Preferential concentration Filtering of turbulent fluctuations

Toschi and Bodenschatz. Lagrangian properties of particles in Turbulence. Ann. Rev. Fluid Mech. (2009) vol. 41 pp. 375-404

Heavy particle’s acceleration

Filteredacceleration

Fluid acceleration at particle’s position -20%

Acceleration at varying Stokes no model

More general picture of forces

• Behavior of acceleration vs. (St, β)

Toschi and Bodenschatz. Lagrangian properties of particles in Turbulence. Ann. Rev. Fluid Mech. (2009) vol. 41 pp. 375-404

/ TN & WI Federico Toschi 33

Preferential concentration

β = 0, heavyβ = 1, tracerβ = 3, bubble

Poisson distribution

St=0.6

tracerbubble heavy

105 particles

Kaplan-Yorke dimension: DKY

As in Bec Phys. Fluids (2003), Bec JFM (2005), Bec et al. Phys. Fluids (2006)

Particle equations of motion defines a dissipative dynamical systemAttractor’s dimension in the (x,v) space: Kaplan-Yorke dimension DKY

6 Lyapunov exponents computed by tracking

stretchingrates

Standard orthonormalization Gram-Schmidt procedure adopted

Federico Toschi - ETC12 Marburg 2009

Kaplan-Yorke dimension Balance between contraction and expansion

Heavy min. at St≈0.5, DKY ≈2.6 Light min at

St≈1, DKY≈1.4

Close to fractal dimension of vortex filaments in turbulence? Dω≈1.1(Moisy & Jimenez JFM04)

DKY =3 ± 0.01

P2(r) Probability to find a couple of particle whose distance is below r.

At r << η P2(r) = A rD2

Correlation dimension D2

Similar features as DKY

fractal dimension hierarchy: D2 ≤ D1 = DKY

Finite size particles

Motivation: Marine snow

Particle aggregation processes and how they affect particles in the marine environment. Biological aggregation (e.g., fecal pellet production) and physical aggregation by (a) shear and (b) differential sedimentation form large, heterogeneous, rapidly settling particles in the surface waters. In deeper waters, fragmentation and repackaging of this material by zooplankton are the dominant processes that affect aggregate sizes and properties. Microbes decompose material throughout the water column.

Burd, A. B., & Jackson, G. A. (2011). Particle Aggregation. Annu. Rev. Marine. Sci., Annual Review of Marine Science, 1(1), 65-90. Annual Reviews.

Bacteria: small scale view

Bacteria: large scale view

http://earthobservatory.nasa.gov/IOTD/view.php?id=41385

Plankton: small scale view

Plankton: large scale view

"Van Gogh" AlgaeImage courtesy EROS/USGS/NASA

In the style of Van Gogh's "Starry Night," massive congregations of greenish phytoplankton swirl in dark water around Sweden's Gotland (see map) island in a satellite picture released this week by the U.S. Geological Survey (USGS).

The image of the Baltic Sea island is 1 of 40 in the new Earth as Art 3 collection, the latest compilation of Landsat pictures chosen for their artistic quality.

"The collected images are authentic and original in the truest sense," Matt Larsen, the USGS's associate director for Climate and Land Use Change, said in a statement. "These magnificently engaging portraits of Earth encourage us all to learn more about our complex world."

Population explosions, or blooms, of phytoplankton, like the one shown here, occur when deep currents bring nutrients up to sunlit surface waters, fueling the growth and reproduction of these tiny plants, according to the USGS.

(Related: "The Best Pictures of Earth: Reader Picks of NASA Shots.")

Published November 19, 2010

Experiments

Recent experiments on acceleration statistics with D ≥ η particles:

- G. Voth et al. J.Fluid Mech. (2002)

- N. Qureshi et al. Phys. Rev. Lett. (2007)

- R. Volk et al. Physica D (2008)

- H. Xu et al. Physica D (2008)

- N.Qureshi et. al EPJ B (2008)

- R. D. Brown et. al PRL (2009)

- R. Volk et al. JFM (2011)

- Experimentally most studied case: neutrally buoyant particles.

Von Karman Flow, Corne! USA, Goettingen, Lyon

Wind tunnelgrid turbulenceGrenoble

Instrumented particle

• Instrumented tracer for Lagrangian measurementsin Rayleigh-Bénard convectionInstrumented tracer for Lagrangian measurementsin Rayleigh-Bénard convection

Shew, W. L., Gasteuil, Y., Gibert, M., Metz, P., & Pinton, J. (2007). Instrumented tracer for Lagrangian measurements in Rayleigh-Benard convection. Review of Scientific Instruments, 78(6), 065105. doi:10.1063/1.2745717

Instrumented tracer for Lagrangian measurements in Rayleigh-Bénard convection

Tracking the dynamics of translation and absolute orientation of a sphere in a turbulent flowRobert Zimmermann, Yoann Gasteuil, Mickael Bourgoin,Romain Volk, Alain Pumir, and Jean-Francois Pinton

Finite size particles

These clear, colorless spheres made of sodium polyacrylate - the superabsorbent polymer found in diapers - grow from 4 to 20 mm in diameter. Hydrated spheres have the refractive index of water, so they are invisible in water. Students can monitor growth rate or calculate before and after volumes as spheres hydrate over 12 hr.

Each pack contains 50 g (about 1,100 spheres). Re-use objects multiple times. Allow 12 hr to air dry.

Finite size particles experiments: GoettingenAbstract: CA.00007 : Flow around finite-size neutrally buoyant Lagrangian particles in fully developed turbulence DFD 2010

Mathieu Gibert (Max Planck Institute for Dynamics and Self-Organization)

Simon Klein (Max Planck Institute for Dynamics and Self-Organization)

Antoine B\'erut (Max Planck Institute for Dynamics and Self-Organization)

Eberhard Bodenschatz (Max Planck Institute for Dynamics and Self-Organization)

By using an innovating technique based on Lagrangian Particle Tracking (LPT), we have been able to follow the motion of finite-size neutrally buoyant particles together with the trajectories of tracer particles in the surrounding fluid. The particles we study have diameters of about 200 times the dissipative scale of the flow, and their density is almost that of the fluid. The experiments are conduced in a von Karman swirling water flow at Taylor microscale Reynolds numbers up to 500. By measuring the full motion of the big particles (translation and rotation), we are able to ``sit'' in their frame of reference and measure the flow properties around them. We will report experimental results on the flow properties and its correlations with the big particle trajectories in this Lagrangian frame.

Finite size particle: Lyon

Figure 1. (Colour online) Experimental set-up. (a) Geometry of the turbulence generator. (b)Schematics of the von Karman flow in water. (c) Principle of the LDV using wide beams

(eLDV) – top view of the experiment. PM denotes location of the photomultiplier which

detects scattering light modulation as a particle crosses the interference pattern created at

theintersection of the laser beams. The eLDV measures, for one particle at a time, the

evolution of its velocity component ux (t ) along the particle trajectory.

Volk, R., Calzavarini, E., Leveque, E., & Pinton, J. F. (2011). Dynamics of inertial particles in a turbulent von Kármán flow. Journal Of Fluid Mechanics, 668, 223-235. doi:10.1017/S0022112010005690

Phenomenology (neutrally buoyant particle)

Voth et al. (2002) Qureshi et al. (2007)

For D>η, acceleration variance decreases when D is increased

Phenomenology (neutrally buoyant particle)

Qureshi et al. (2007)

Early experiments were showing acceleration PDF very weakly dependent on D and non-Gaussian, recently things changed.

Volk et al. (2011)

From the Point Particle model (PP)...

dilute suspension (no collisions), no particle feedback on the flow

Point-like heavy/neutral/light inertial particles: Stokes drag, fluid acceleration & added mass

0 ≤ β ≤3

A minimal model for finite-sized particles: the Faxén Corrected model (FC)

Acceleration statistics of finite-sized particles in turbulent flow: the role of Faxen forcesE. Calzavarini , R. Volk, M. Bourgoin, E. Leveque, J-F. Pinton and F. Toschi, J. Fluid Mech. (2009)

Accounts for the nonuniformity of the flow at the particle scale

volume average

surface average

Sphere Average

Approximation:

Gaussian Average

with first order Faxén corrections

Efficient numerical implementation

PP model (neutrally buoyant particles)

• Acceleration variance is D independent• Acceleration PDF is D independent• Correlation time of Acceleration does not

grow with D, but reduces.

arms / <Dtu>rms

=1same acc.as the fluid

St=4 D/η ≈ 7

See:Volk et al. Physica D (2008)

Acceleration Variance from:Point-Particle and Faxén corrected models

DNS (1283 ) Reλ =75

FCheavy

neutral

/<a f2 >

Acceleration Flatness

DNS (1283 ) Reλ =75

F( af ) PP

FCheavy

neutral

F(a) → F(af)

PDF of acceleration: exp vs. FC simulations

DNS (5123 ) Reλ =180EXP (Qureshi et al. 2007) Reλ =160 EXP (Volk et al. 2011)

Acceleration Variance: comparison

More refined FC model for finite size particles

Van Dyke, M. (1982). An Album of Fluid Motion. (M. Van Dyke, Ed.) (12nd ed. p. 176). Parabolic Press, Inc.

Impact of trailing wake drag on the statistical properties of finite-sized particles in turbulence

More refined particle model

Non-Stokesian drag force(model for the wake drag)

History force(unsteady Stokes drag)

Faxen corrected modelwith Stokes drag

Schiller-Naumann (1933)

Particle Reynolds number

Focus on the case Reλ ~ 31 and neutrally buoyant particles to compare with fully resoled DNS by

Velocity variance

particle

Acceleration variance

a = [Du/Dt]v (sphere)

a = [Du/Dt]v (Gauss)

(with improved volume averages)

Acceleration variance

Large Eddy Simulation

Filtered velocity field Filtered energy spectra

Resolved scales

Unresolved scales

/ TN & WI PAGE 10/01/2010

Effects of droplet deformability

/ TN & WI PAGE 10/01/2010

Droplets deformability: key issues

• Physics: finite size particles plus surface tension• Transfer of energy from fluid to elastic modes (and

viceversa)• How is turbulence affected by the presence of droplets?• How do properties of (deformable) droplets differ from

rigid droplets ?

Experimental investigations

Gopalan and Katz. Turbulent Shearing of Crude Oil Mixed with Dispersants Generates Long Microthreads and Microdroplets. PRL, 104, 054501 (2010)

Prakash, Tagawa, Martinez-Mercado, Sun, LohseExperiment in the Twente water tunnel.

Bubbles deform in presence of flows.

Experimental investigations

Some recent numerical work

Qian et al., Simulation of bubble breakup dynamics in homogeneous turbulence. Chem. En#. Commun. 193, 1038 (2006)

Some recent numerical work

J.J. Dersksen and H.E.A. Van Den Akker, Chem. En#. Res. (2006)

• Turbulence

• Inertial force

• Surface tension force

• Weber number

/ TN & WI PAGE 10/01/2010

Dimensionless numbers

J.O. Hinze, A.I.Ch.E, (1955)

/ TN & WI PAGE 10/01/2010

Elongation in a stationary flow

H. Stone, “Dynamics of drop deformation and breakup in viscous fluids” Annu Rev Fluid Mech (1994) vol. 26 pp. 65-102

/ TN & WI PAGE 10/01/2010

d > dmax: Droplet breaksd < dmax: Droplet does not break

Hinze 1955

J.O. Hinze, A.I.Ch.E, (1955)

/ TN & WI PAGE 10/01/2010

Numerical approach

Biferale, L., Perlekar, P., Sbragaglia, M., & Toschi, F. (2012). Convection in Multiphase Fluid Flows Using Lattice Boltzmann Methods.Physical Review Letters, 108(10).

D3Q19 BGK LB model

with multicomponent Shan-Chen

Technique inspired to the continuum Boltzmann equation

Lattice Boltzmann Method (LBM)

Shan and Chen. Lattice Boltzmann Model for Simulating Flows with Multiple Phases and Components. Phys. Rev. E 47, 1815 (1993).

LBM: multicomponent SC

Random phases generated from Ornstein-Uhlenbeck process

Convincing LBM to go turbulent

Forcing: Large scale forcing in first two Fourier modes

LBM: Energy and enstrophy

Acceleration of a fluid parcel

SC: acceleration for single component flow

LBM: Energy and acceleration

• JUGENE (FZJ-JSC IBM Blue Gene/P)• 23.5RM (about 15Mhours)• 32-64 kprocs• I/O HDF5• Fully parallel code

Simulations

/ TN & WI PAGE 10/01/2010

Droplet breakup in turbulence

/ TN & WI PAGE 10/01/2010

Towards a stationary state…

/ TN & WI PAGE 10/01/2010

Droplet radius vs. time

0 5 t/!eddy 15 20 25

"=0.5%"=5%

/ TN & WI PAGE 10/01/2010

Re N density ratio

(liquid /saturated

vapor)

viscosity

G dissipative

scale(LU)

D Hinze

D(LBE)

We Volume fraction

Eddy turnove

RUN0 80 512 0.5/0.5 0.005 - 6 - - -

RUN1 80 512 2.038/0.362

0.005 0.03 6 24.2 24+/-1 0.075 0.3%RUN2 80 512 1.757/0.08

80.005 0.08 6 39.5 36+/-1 0.033 0.3%

RUN0 40 128 0.5/0.5 0.005 - 3

RUN1RUN2RUN3RUN4

40 128 2.038/0.362

0.005 0.03 3 8.65 11,13,15,18

0.12 0.07,0.5,5,10

50,50,50,560

Simulation parameters

/ TN & WI PAGE 10/01/2010

pdf of radius and volumes

/ TN & WI PAGE 10/01/2010

d32 (LBM)

40 14.5 3.5 10-8

80 24 2.85 10-9

Droplet PDF: Re dependence

/ TN & WI PAGE 10/01/2010

Droplet size distribution

0.6 1 5 10 20

!=0.07%

!=0.5%

/ TN & WI PAGE 10/01/2010

Verification of Hinze criterion

1!10-2

1!10-1

3!10-1

1!101 1!102$/#"

4 1!104 5!104

#=1.6e-3,%=0.3%, N512A#=1.7e-3,%=0.3%, N512B

#=1.6e-3,%=0.07%, N128A #=1.6e-3,%=0.5%, N128B

#=1.6e-3,%=5%, N128C #=1.6e-3,%=10%, N128D

/ TN & WI PAGE 10/01/2010

Trajectories look smooth but acceleration is very noisy

Droplet trajectory vs. acceleration

/ TN & WI PAGE 10/01/2010

PDF of acceleration

/ TN & WI PAGE 10/01/2010

Droplet deformation

Volume VSurface S

Deformation parameter is the dimensionless surface i.e. normalized wrt the surface of the equivalent sphere with same volume V

/ TN & WI PAGE 10/01/2010

Deformation, dissipation and breakups

7 14 t/!eddy 21 28 0

! breakup

/ TN & WI PAGE 10/01/2010

Deformation vs. local Weber

-1.0 -0.5 log10 Wes 0.5 1.0

-0.2lo

S*=We0.066

/ TN & WI PAGE 10/01/2010

The boring (?) life of non-breaking drop

/ TN & WI PAGE 10/01/2010

Minimal bibliography

Droplets in turbulenceDroplet size distribution in homogeneous, isotropic turbulence, P. Perlekar, L. Biferale, M. Sbragaglia, and F. Toschi, arXiv:1112.6041(submitted to Phys. Fluids).Convection in Multiphase Fluid Flows Using Lattice Boltzmann Methods, L. Biferale, P. Perlekar, M. Sbragaglia and F. Toschi, Physical Review Letters, 108(10) (2012).

Numerical TechniqueA Lattice Boltzmann method for turbulent emulsions, L. Biferale, M. Sbragaglia, S. Srivastava, and F. Toschi, J. Phys.: Conf. Ser., 318, 052017 (2011) .Unified %amework for a side-by-side comparison of different multicomponent algorithms: lattice Boltzmann vs phase field model, L. Scarbolo, D. Molin, P. Perlekar, A. Soldati, and F. Toschi, arXiv:1112.5547 (submitted to J. Comp. Phys.)

The end.

PublicationsL. Biferale and G. Boffetta and A. Celani and A. Lanotte and F. Toschi,Lagrangian statistics in fu!y developed turbulenceJournal of Turbulence, 7 (2006) 1-12 10.1080/14685240500460832.

T. H. Van Den Berg and S. Luther and I. M. Mazzite!i and J. M. Rensen and F. Toschi and D. Lohse,Turbulent bubbly flowJournal of Turbulence, 7 (2006) 1-12 10.1080/14685240500460782,

L. Biferale and F. Toschi,Joint statistics of acceleration and vorticity in fu!y developed turbulenceJournal of Turbulence, 6 (2006) 1-8 10.1080/14685240500209874.

J. Bec and L. Biferale and G. Boffetta and A. Celani and M. Cencini and A. Lanotte and S. Musacchio and F. Toschi,Acceleration statistics of heavy particles in turbulenceJournal of Fluid Mechanics, 550 (2006) 349-358 10.1017/S002211200500844X M. Cencini and J. Bec and L. Biferale and G. Boffetta and A. Celani and A. Lanotte and S. Musacchio and F. Toschi,Dynamics and statistics of heavy particles in turbulent flows

L. Biferale and J. Bec and G. Boffetta and A. Celani and M. Cencini and A. Lanotte and S. Musacchio and F. Toschi,Inertial Particles in Turbulence

Luca Biferale and Massimo Cencini and Alessandra S. Lanotte and Federico Toschi,Velocity Statistics of Tracers and Inertial Particle in Turbulent Flows: the effects of vortex trapping

J. Bec and L. Biferale and G. Boffetta and M. Cencini and A. Lanotte and S. Musacchio and F. Toschi,Lyapunov exponents

Publications

L. Biferale and G. Boffetta and A. Celani and A. Lanotte and F. Toschi,Particle trapping in three-dimensional fu!y developed turbulencePhysics of Fluids, 17 (2005) 021701 10.1063/1.1846771 L. Biferale and G. Boffetta and A. Celani and B.J. Devenish and A. Lanotte and F. Toschi,Lagrangian statistics of particle pairs in homogeneous isotropic turbulencePhysics of Fluids, 17 (2005) 115101 10.1063/1.2130742

F. Toschi and L. Biferale and G. Boffetta and A. Celani and B.J. Devenish and A. Lanotte,Acceleration and vortex filaments in turbulenceJournal of Turbulence, 6 (2005) 1-10 10.1080/14685240500103150

L. Biferale and G. Boffetta and A. Celani and B.J. Devenish and A. Lanotte and F. Toschi,Multiparticle dispersion in fu!y developed turbulencePhysics of Fluids, 17 (2005) 111701 10.1063/1.2130751 L. Biferale and G. Boffetta and A. Celani and B.J. Devenish and A. Lanotte and F. Toschi,Multi&actal statistics of Lagrangian velocity and acceleration in turbulencePhysical Review Letters, 93 (2004) 064502 10.1103/PhysRevLett.93.064502

Irene M. Mazzite!i and Detlef Lohse and Federico Toschi,Effect of microbubbles on developed turbulencePhysics of Fluids, 15 (2003) L5-L8, 10.1063/1.1528619

particles in turbulence - university corporation for ... · / tn & wi federico toschi toc •...

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