m.cencini inertial particles in turbulent flows warwick, july 2006 inertial particles in turbulence...

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

InertialInertial particles in particles in turbulenceturbulence

Massimo CenciniMassimo Cencini CNR-ISC Roma

INFM-SMC Università “La Sapienza” Roma [email protected]@roma1.infn.it

In collaboration with:

J. Bec, L. Biferale, G. Boffetta, A. Celani, A. Lanotte, S. Musacchio & F. Toschi


Problem:Problem: Particles differ from fluid tracers their dynamics is dissipative due to inertia one has preferential concentration

GoalsGoals:: understanding physical mechanisms at work,characterization of dynamical & statistical properties

Main assumptionsMain 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.


Rain drops in cloudsRain drops in clouds (G. Falkovich et al. Nature 141, 151 (2002))

clustering enhanced collision rate

Formation of planetesimals in Formation of planetesimals in thethe

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 Optimization of combustion processes inprocesses in diesel enginesdiesel engines (T.Elperin et al. nlin.CD/0305017)


Equations of motion & Equations of motion & assumptionsassumptions

Dissipative range physics

Heavy particles

Particle Re <<1

Dilute suspensions: no collisions

η<<a

fpρρ >>1vRe

a<<= νaa

Stokes number

Response time Stokes Time

(Maxey & Riley Phys. Fluids (Maxey & Riley Phys. Fluids 2626, 883 (1983)), 883 (1983)) Kolmogorov ett Kolmogorov ett

u(x,t) (incompressible) fluid velocity fieldu(x,t) (incompressible) fluid velocity field


PhenomenologyPhenomenology

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 rateAs 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 islow-pass filtered


Direct numerical Direct numerical simulationssimulations

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)


DNS summaryDNS summary

Resolution 1283, 2563, 5123

Pseudo spectral code

Normal viscosity

Code parallelized MPI+FFTW

Platforms: SGI Altix 3700, IBM-SP4

Runs over 7 - 30 days

N3 5123 2563 1283

Tot #particles 120Millions

32Millions 4Millions

Fast 0.1 500.000 250.000 32.000

Slow 10 7.5Millions 2Millions 250.000

Stokes/Lyap (15+1)/(32+1)

(15+1)/(32+1)

15+1

Traject. Length

900 +2100 756 +1744 600+1200

Disk usage 1TB 400GB 70GB


Particle Clustering Particle Clustering

Important in optimization of Important in optimization of reactions, reactions,

rain drops formation….rain drops formation….

Characterization of fractal aggregatesCharacterization of fractal aggregates

Re and St dependence in turbulence?Re and St dependence in turbulence?

Some studies on clustering: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)


Two kinds of clusteringTwo kinds of clusteringParticle preferential concentration is observed both

in the dissipativedissipative and in inertialinertial range

Instantaneous p. distribution in a slice of width ≈ 2.5η. St = 0.58 R = 185


Small scales clusteringSmall 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


Correlation dimensionCorrelation dimension D2 weakly depending on Re

Maximum of clustering for

Particles preferentially concentrate in Particles preferentially concentrate in

the hyperbolic regions of the flow.the hyperbolic regions of the flow.

Maximum of clustering seems to beMaximum of clustering seems to beconnected to preferential connected to preferential concentrationconcentration

but Counterexample: inertial p. in Kraichnan flow

(Bec talk)

Hyperbolic non-hyperbolic


Multifractal distributionMultifractal distribution

Intermittenc

y in the mass

distribution


Lyapunov dimensionLyapunov dimension

d D1 provides information similar to D2

can be seen as a sort of “effective” compressibility


Inertial-range clusteringInertial-range clustering


Characterization of clustering in the inertial Characterization of clustering in the inertial rangerange (Preliminary & Naive) (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: Coarse grained mass:

St=0 ==> Poissonian

St0 ==> deviations from Poissonian. How do behave moments and PDF of the coarse grained mass?


PDF of the coarse-grained massPDF 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)


Collapse of CG-mass Collapse of CG-mass momentsmoments

Inertial range


Sketchy argument for Sketchy argument for ss/r/r5/35/3

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

The argument can be made more rigorous in terms of the dynamicsdynamics of the quasilagrangian mass distributionof the quasilagrangian mass distribution and using the rate of

volume contraction. But the crucial assumption is


Scaling of accelerationScaling of accelerationControversial 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 assumingthat the sweeping by the large scales is the leading term

We cannot exclude that the other spectra may be observed at higher Re


Single point acceleration Single point acceleration propertiesproperties

Some recent studies on fluid acceleration: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)

Probe of small scale intermittencyProbe of small scale intermittency

Develop Lagrangian stochastic modelsDevelop Lagrangian stochastic models

What are the effect of inertia?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).


Acceleration statisticsAcceleration statisticsAt 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)


Two mechanismsTwo mechanisms

Preferential concentration

Filtering


Preferential concentration & Preferential concentration & filtering filtering

Heavy particles acceleration

Fluid acc. conditioned on p. positions good at

St<<1

Filtered fluid acc. along fluid traj. good at St>1


Preferential concentrationPreferential concentration

Fluid acceleration

Fluid acc. conditioned on particle positions

Heavy particle acceleration


FilteringFiltering

Fluid acceleration

Filtered fluid acc. along fluid trajectories

Heavy particle acceleration


Dynamical featuresDynamical featuresFrom passive tracers studies we know

that wild acceleration events comefrom trapping in strong vorticestrapping in strong vortices.

(La Porta et al 2001)

(Biferale et al 2004)

Inertia expels particles from strongvortexes ==> acceleration

depletion(a different way to see the effect of

preferential concentration)


ConclusionsConclusionsTwo kinds of preferential concentrations in

turbulent flows:Dissipative range: intrinsic clustering (dynamical attractor) tools borrowed from dynamical system concentration in hyperbolic regionInertial 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!


ThanksThanks


Then assuming

With the choice

Mass conservation

One sees that pr, (t) can be Related to pr, (t-T(r,)) hence all the statisticalProperties depend on T(r,).

From which

Hence if a=a0

Where we assumed that a p.vel. Field can be defined

m.cencini inertial particles in turbulent flows warwick, july 2006 inertial particles in turbulence...

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