Local dissipation properties and collision dynamics in a sustained homogeneous turbulent suspension composed of finite size particles

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pen

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rchive

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OULOUSE

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rchive

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uverte (

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To cite this version : Brändle de Motta, Jorge César and Estivalezes,

Jean-Luc and Climent, Eric and Vincent, Stéphane Local dissipation

properties and collision dynamics in a sustained homogeneous

turbulent suspension composed of finite size particles. (2016)

International Journal of Multiphase Flow, vol. 85. pp. 369-379. ISSN

0301-9322

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Local

dissipation

properties

and

collision

dynamics

in

a

sustained

homogeneous

turbulent

suspension

composed

of

finite

size

particles

J.C.

Brändle

de

Motta

a,b,c,∗

_,

_J.L.

_Estivalezes

b,c

_,

_E.

_Climent

c

_,

_S.

_Vincent

d a CORIA, Normandie Université, UMR CNRS 6614, Site Universitaire du Madrillet, 76801 Saint-Etienne du Rouvray, France b ONERA, The French Aerospace Lab, 2, avenue Edouard Belin, 31055 Toulouse, France

c Institut de Mécanique des Fluides de Toulouse (IMFT), Université de Toulouse, CNRS-INPT-UPS, Toulouse, France

d Université Paris-Est Marne-la-Vallée, UMR CNRS 8208, Laboratoire MSNE, 5 boulevard Descartes, 77454 Marne-la-Vallée, France

Keywords: Particulate-flows Turbulence Finite-size particles Local dissipation Collision statistics

a

b

s

t

r

a

c

t

Particulateflowsarepresentinmanyapplicationsandtheeffectofparticlesizeisstillnotwell under-stood.Thepresent paperdescribes threecases ofsustained homogeneousturbulenceinteractingwith particles.Simulations correspondtothree particle-fluiddensity ratios and 3%volume fractionin zero gravityfield.Fullyresolvedparticlesimulationsarebasedonfictitiousdomainandpenaltymethod.

Thelocaldissipationaroundparticlesisstudiedaccordingtothedensityratio.Spatialdescriptionof theaverageddissipationisprovided.Collisionstatisticsarealsoinvestigated.Theintercollisiontimeand theangleofcollisionarecomparedtothekinetictheory.Theeffectoftheinterparticlefilmdrainageis highlightedbysimulatingthesameconfigurationswithandwithoutlubricationmodel.

1. Introduction

Particle-laden flows can be found innatural environment and many industrial applications. For example, sediments in rivers, solid particlesin fluidizedbeds,droplets incloudsorin combus-tionchambers(Yang,2003)couldbetreatedasparticles. Depend-ing ontheregime,theturbulencecancompletelygovernthe par-ticles’behaviour(spatialdistribution,collisionrate,…).Inthecase ofmoderatelyconcentrated todenseflows, thepresenceof parti-cles can also influence the turbulence,Squires and Eaton (1990),

Boivinetal.(1998).

Dimensionless parameters characterizing particle laden-flows are:

• the density ratio

ρ

=ρp

ρf, with

ρ

p and

ρ

f the density of the

solidandfluidrespectively, • particlevolumefraction,

• the Stokes number, defined as the ratio betweenthe particle response time

τ

p and a characteristic flow time scale

τ

f. For

theturbulent case, theKolmogorov time scaleis oftenchosen for

τ

f.InthiscasetheStokesnumberisStk.Nevertheless,this

∗ _{Corresponding author at: CORIA, Normandie Université, UMR CNRS 6614, Site} Universitaire du Madrillet, 76801 Saint-Etienne du Rouvray, France.

E-mail addresses: jorge.brandle@coria.fr (J.C. Brändle de Motta), estivalezes@ onera.fr (J.L. Estivalezes), climent@imft.fr (E. Climent), stephane.vincent@ u-pem.fr (S. Vincent).

scalelosesitsrelevancetopredictfinitesizeparticledynamics,

i.e.,whentheparticlesizeislargerthantheKolmogorovlength scale,Luccietal.(2011).

• forfinite-sizeparticles, anewdimensionlessparameterhas to be considered, i.e.theratiobetweenthe particleradius Rand theKolmogorovlengthscale

η

.

The motion of finite size particles in turbulent flow has re-ceived much less attentionthan smallparticles. Indeed,the size of the particles induces newcoupling phenomena betweenfluid and particles, for example with flow scales larger than the Kol-mogorov scales. In this case, classical Lagrangian pointwise ap-proachesarenotabletomodelthesecouplingsasscaleseparation isnotsatisfied.Duetoavailable CPU resourcesthecomputational costoffinite-sizeparticlesimulationswasnot affordableuntil re-centyears.Morerecently,withincreasingpowerofmassively par-allelcomputers,thefinite-sizeeffectonparticledynamicshas mo-tivatedmanystudies.

In Fig. 1, an extensive collection of relevant studies is pre-sented. Experimental data, pointwise simulations and finite-size simulationsareplottedtogetheraccordingtoSt_k,

ρ

and R

η.Studies

aredepicted withfilledcircles forsimulations andopen symbols forexperiments.Inaddition,pointwisesimulations havebeen re-portedassquare symbols.It canbeobservedthat pointwise sim-ulations are focused on large density ratios, i.e. mostly gas/solid motions, whereas finite-size particle studies are concerned with Stokes numbers and density ratios ranging from 1 to 100. In http://dx.doi.org/10.1016j.ijmultiphaseflow.2016.07.003

Fig. 1. Representation of turbulent particulate flows according to the density ra- tio, Stokes number based on Kolmogorov scale and R

η for finite size configura-

tions. The square symbols represent the pointwise simulations (red: Ferrante and Elghobashi, 2003 , green: Ahmed and Elghobashi, 20 0 0 , blue: Sundaram and Collins, 1997 , brown: Elghobashi and Truesdell, 1993 ), the open symbols hold for exper- imental data (red: Xu and Bodenschatz, 2008 , green: Elhimer et al., 2011 , blue:

Qureshi et al., 2008 , brown: Tanaka and Eaton, 2010 , orange: Bellani and Vari- ano, 2012 , yellow: Wood et al., 2005 , dark blue: Poelma et al., 2007 , black: Fessler et al., 1994 ) and the filled circles correspond to the finite size particle simulations (red: Lucci et al., 2010 , green: Corre et al., 2008 , blue: Zhang and Prosperetti, 2005 , brown: Cate et al., 2004 , orange: Homann and Bec, 2010 , yellow: Cisse et al., 2013 , dark blue: Gao et al., 2011 and Shao et al., 2012 , purple: present paper). The size of the circle is proportional to the ratio between the particles radii and the Kol- mogorov length scale. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

order to contribute to the understanding of finite-size particle ladenflows,fully-resolvedturbulentsimulationsareconsidered in thepresentworkformoderatedensityratioscorrespondingto liq-uid/solidsuspensionsandStkrangingfrom10to100.These

simu-lationsarealsoreportedinFig.1 (purplecircles).

The main results obtained in the literature are now summarized:

• Turbulencemodulation:Thedispersedphasecanmodifythe tur-bulent characteristics.This modulation can be split intothree contributions. Thefirst one isdueto theforce exerted by the particles onthe fluid.Forpointwisemodels, itcan bewritten astheensembleaveraging



u· f



obtainedfromthescalar prod-uct of the fluid velocity atthe position of the particleu and theforce f(see forexampleFerranteandElghobashi,2003).It wasconcludedthatmodulationdependsontheStokesnumber basedontheKolmogorovtimescale.FordifferentStokes num-bers,itwasobservedthatpointparticlescouldenhance turbu-lenceordissipateenergy.Thesecondcontributionisatransfer of energyfrom large to small scales, Ferrante andElghobashi (2003),Boivinetal.(1998).Finally,athirdcontributioncomes fromthemodificationoflocaldissipationaroundtheparticles. Thisistypicallya finite-sizeeffectwhichcannotbe accounted forbyclassicalpointwisesimulations.

The effect of finite size particles on turbulence modulation wasstudiednumericallyinZhang andProsperetti (2005).It is shownthattherateofdecayoftheturbulenceismore impor-tant forthefinitesize casethanforsingle-phaseand particu-late pointwisesimulations.Actually,it isstill necessaryto un-derstandandexplain thiseffect.From theinstantaneous dissi-pationfieldsobtainedinthesimulationsofCateetal.(2004),it can beobservedthat a localdissipationappears around parti-cles.Later,thestudyofLuccietal.(2010) verifiesthatthe aver-ageddissipationincreasesaroundparticles.Inordertoquantify thiseffect,theauthorscomputedtheaveraged dissipation

up-streamanddownstream ofparticles. Thelocal increase ofthe dissipationismoreimportantinthefrontanditsspatial exten-sionisaboutonediameterlong.Asimilarconclusionisdrawn inCisseetal.(2013) inwhichatentativeexplanationisgiven: thewake generated behindthe particles decreases the turbu-lentdissipationlevel.Theseresultsareconfirmedbythe exper-imentsofTanakaandEaton(2010) concerningthestudyof dis-sipationaroundparticlessettlinginaturbulentflow.The dissi-pationismultipliedby3inthefrontoftheparticles.Thelocal dissipationwillbeanalysedinthepresentpaper.

Thelocaldissipationisnot theonlyfinitesizeeffectin turbu-lencemodulation.Infact,asshownbyPoelmaetal.(2007), fi-nitesizeparticlescanmodifytheturbulencestructureswithout modifyingtheenergycontent.Finally,Luccietal.(2010) shows thatthecouplingterm



u· f



ispositive(energyenhancement) forfinitesizeparticles.Forpointwiseparticles,thisterm gener-allydissipatesenergy.

• Collisionregime:Thestudyofthecollisionregimeisimportant formanyapplicationssuchasthepredictionofcoalescencerate ofdroplets. The collision rate represents thenumber of colli-sionsperunittime.Itcanbeobtainedfromtherelative veloc-ityoftheparticles|wr|andtheradialdistributionatcontactg0,

asfollows, fc= 1

τ

c = 1 2g0n 2 p

π

(

2R

)

2

|

wr

|

(1)

where



.



istheensembleaverage,

τ

c isthecollisiontime and

np is the particle number density. The particle collision

fre-quency in turbulent flow is relatively well described for two cases.In thefirst case, forparticles withsmall response time (tracers),thecollisionsaredrivenbythelocalshearintheflow. Thecollisionratecanbecorrelatedtothedissipationrateofthe flow,SaffmanandTurner(1956).Inthesecondcase,wherethe particlesareinertial,theTchen–Hinzecorrelation(Hinze,1975; Tchen,1947) linkstheparticle kinetic energytothe turbulent statistics.Thus,thecollisionratecanbecomputed,Abrahamson (1975). For intermediate cases, many models have been pro-posed(KruisandKusters, 1997;Lavieville, 1997;Williamsand Crane,1983).

Forfinitesizeparticles,onlythestudyofCateetal.(2004) has addressedcollisionstatistics.Thisstudyshowedthatsecondary collisionsappear.The secondarycollisionsareconsecutive col-lisions between pairs that stay correlated during a relatively longtime.Thesecollisionsarenotpredictedbyprevious mod-els.Inthepresentpaper,thesecondarycollisionfrequencywill beanalysed.

Thisarticlepresentsastudyofasustainedhomogeneous turbu-lentflowseededwithfinite-sizeparticles.First ofall,the numer-icalmethodissummarizedinSection 2.Thechoiceofthe turbu-lentparticleladenflowparametersisdiscussedinSection3.From this turbulent case, two major phenomena are studied. The first one is the averaged flow around particles which is described in

Section 4. The second phenomenon is the collisional regime de-tailed inSection 5 where we focuson lubrication effectson col-lisionalregime. Conclusionsandperspectivesare finally drawnin thelastsection.

2. Simulationoffinitesizeparticles

The modellingandsimulationoffully resolvedfinite size par-ticles inturbulent incompressible andisothermal fluid is investi-gatedbymeansofasinglefluidmodel(Kataoka,1986)generalized forparticulateflows(Vincentetal.,2014).Thekeyideaistousea fictitiousdomainapproachinwhichanEuleriandescriptionofthe two-phase flow is introduced by a characteristic function C, also calledvolumeoffluid(VOF). Thisfunctionis0inthefluid phase

and1 inthesolid phase. Itallows to locateanypoint ina given phase bydefining theinterface asC=0.5.Thesinglefluid model isimplementedonfixedstructured Cartesianmeshesnotadapted totheshapeofthefluid/particleinterfaces,sothattheparticlesare fictitiousdomainsembeddedinthecalculation grid.The formula-tion ofthesinglefluidmodel isthefollowing(see Vincentetal., 2014 formoredetails)

∇

· u=0 (2)

ρ



∂

_∂

u_t +

(

u·

∇

)

u



=−

∇

p+

∇

·



μ



∇

u+

∇

t_u



_+f si (3) dC dt =

∂

C

∂

t +u·

∇

C=0 (4)

whereuandparethevelocityfieldandthepressurerespectively inbothphases, tthetime,

ρ

and

μ

thedensityandtheviscosity oftheequivalentfluid(fluidandsolidphases).Thedensityandthe viscosity of the equivalentfluid are formulated according to the real physical properties of the two different media andthe VOF function byusingarithmetic orharmonicweighting basedonthe localvalueofC.Inordertoenforcethesolidbodyresponseof par-ticles,the viscosityofthe solidphase is300 timesthefluid one. Due to theEulerianrepresentation ofthe two-phase flow, a spe-cificfourwaycouplingtermfsi isaddedtothemomentum

equa-tionsinordertomodelcollisionsandlubrication.Thisforce distri-butionresultsfromadampandspringcollisionrepresentationand alubricationmodeldependingonthedistancebetweeninteracting particles(BrändledeMottaetal.,2013).

From a numericalpointof view,thedivergencefree andsolid constraints of the fictitious domain approach are approximated by finite volumes, an implicit viscous tensorial penalty method (Randrianarivelo et al., 2005; Ritz andCaltagirone, 1999) and an augmented Lagrangian minimization algorithm adapted for two-phase flows (Vincent et al., 2011). By using penalty methods and augmentedLagrangian techniques, the velocity, pressure and fluid/solidconstraintaresatisfiedinstantaneouslyandinacoupled manner ateach time step.This way,the CFDcode is robust and physically relevant. Concerning the advection Eq. (4) of the VOF function C, it is approximated in a Lagrangian manner in order toavoiddeformationoftheparticleshape commonlyobservedin standard VOFmethods (Rider andKothe,1998). In ourapproach, thecentreofmassoftheparticleisadvectedinaLagrangianway withthevelocityobtainedthankstomassandmomentumbalance equationi.e.,Eqs.(2)and(3).Afterthisadvectionstep,the analyt-icalsphericalshapeoftheparticleisprojectedontotheCartesian gridtoupdatetheCfunction.Thefinalapproximationoftheglobal one fluidmodel forresolved scale particlesconvergesspatially to secondorder.

Themodelandnumericalmethodshavebeenwidelyvalidated on particle settling, Poiseuille and rotating flows, collisions and also on liquid/solid fluidized bed simulations in Ritz and Calta-girone(1999),Randrianariveloetal.(2005),BrändledeMottaetal. (2013) andVincentetal.(2014).

Acollisionmodelisusedtoreproduceparticle–particle interac-tionandrebound.Duringcollisions,aforceisexertedonthe parti-clevolume,fsi.Twodifferentforcesaretakenintoaccountinorder

to model theeffect of lubrication(fluid drainage) andthe solid– solidcollisionforce.Theseforcesdependonthedimensionlessgap betweenparticlesurfacesandonthenormalrelativevelocitygiven by:



=

(

X a p− Xbp

)

· n − 2R R , un=



Va p− Vbp



· n 2 , (5)

wherenistheunitvectorconnectingthecentresofthetwo par-ticles.Adetailedpresentationofthismodelandvalidationagainst experimentsaregiveninBrändledeMottaetal.(2013).

Inordertocomparetheeffectoflubricationmodel,each sim-ulation presented in the following section has been carried out twice.A firstsimulation hasbeendone withonly thesolid–solid collision model.A second one with both forces,solid–solid colli-sionandlubrication.

Thesolid–solidcollision ismodelledby a springanddash-pot force Fs=− me



π

2_+[ln

₍

_e d

)

]2



[Nc



t]2 R



−2meln

(

ed

)

[Nc



t] un (6) where me=



m−1a +m−1b



−1

is the reduced mass, ed is the dry

restitution coefficient, defined as the ratio between the velocity after and before the collision, and Nc



t is the numerical

colli-siontime.Inallsimulationsed=0.97,thusthesolidcollisionsare

quasi-elastic.Thenumberoffluid timesteps forthesolid numer-icalcollisiontimeNcisfixedto 8.Thisforceisactivatedata

dis-tance



ss.Thisthresholdisfixedto0forthesimulationsinwhich

thelubricationforceisactivatedandto0.1whenthisforce isnot takeninto account.In order tocompare similar collision withor without lubrication,



ss without lubrication has to be equal the

thresholddistance



_al forlubrication.

Thesecond force is amodelof hydrodynamicinteraction dur-ingthedrainageoftheliquidfilmbetweentheparticles.This hy-drodynamic interactionis not captured directlyby the numerical solution of the Navier–Stokes equations because the grid size is largerthan the filmthicknessclose tocontact. The force is com-putedfromthe analytical expressions(Brenner,1961; Cooleyand O’Neil,1969) Fl

(



,un

)

=−6

πμ

fRun[

λ

(



)

−

λ

(



al

)

] (7)

λ

= 1 2



− 9 20ln

(



)

− 3 56



ln

(



)

+1.346+O

(



)

. (8)

The lubrication force is activated for



₌



_al=0_.1 which is equivalentto0.6



x.Whentheparticlesarenearoverlappingthe forceiskeptconstant.Ithasbeenshownthattheadditionofthese two forces ensures the expectedlubricated rebound in a viscous fluid(BrändledeMottaetal.,2013).

3. Macroscopicproperties 3.1. Energydistribution

Basedonfullyresolvedsimulationsweaimatgettingmore in-sight onthe collisional regime offinite size particles ina turbu-lentflow.Inordertoobtainconvergedcollisionstatisticsoverlong times,it isnecessary to simulatesustainedturbulence. Todo so, turbulence is sustained by using the linear forcing proposed by

Rosalesand Meneveau (2005). In our implementation the veloc-ityfielduisrescaledtou+ateachtimestepastokeepaconstant kineticenergyfixed atthebeginning ofthesimulationu2

0 within

thesimulationdomain:

u+=



_ u20d

v

 u2d

v

u (9)

The turbulence forcing is unsuitable for particle flow energy balanceanalysis(Luccietal.,2010).Inthepresentpapersuch de-tailedanalysis isnot provided.In additionthe caseconsidered is dilute,forthisreasonthe two-waycouplingeffectremains negli-gible.Weexpectinthepresentpaperthatphysicsofthelocal dis-sipationandcollisionregimearenotaffectedbythelinearforcing. A2563_{meshisusedfordirectnumericalsimulationsenforcing}

adequateresolutionoftheKolmogorovscale

η

(with

η

/



x=0.56). Simulatingone largeeddy turn-overtime takes10h CPU on512 coresofanSGIAltixIcesupercomputer.

Fig. 2. Energy spectra with and without particles. Solid line for single phase flow DNS. Dotted, dash-dotted and light dashed lines stand for density ratio equal to 1, 2 and 4, respectively. The vertical dash line represents the wavenumber correspond- ing to the particle diameter. Inset is a close-up for high wavenumbers content of spectra.

Inordertoobtaintheinitialturbulentflow, asingle-phase tur-bulence is initialized with a predefined homogeneous isotropic turbulence spectrum (Mansour and Wray, 1994; Trontin et al., 2010). Then, single-phase simulations with sustained turbulence are achieved with the previous forcing scheme until reaching steadystatespectrum.TheTaylormicro-scaleReynoldsnumberof theturbulent flowcomputedfromthisspectrumisRe_λ=73.This moderateReynoldsnumbercorrespondstoamoderatelengthscale separation:theratioofTaylormicro-scaleoverKolmogorovscaleis

λ

/

η

=17.First,theflowisforcedfromitsinitialcondition(analytic spectrum)to reachstatisticalsteadystate. Then,512 particlesare seededandatransienttimeisconsideredbeforestartingaverages fortwo-phaseflows.Theparticulateflowisconsideredtobefully resolved when the particleradius containsat least6 grid points correspondingto R=0.59

λ

=10

η

. Theparticlevolume fractionis 3%.Toobtainconvergedparticlestatistics, simulationsare carried outover12largeeddyturn-overtimes.Inthepresentpaperthree differentsimulationsare presentedforsolid tofluid densityratio

ρ

equalto1,2and4.Nogravityeffectisconsidered.

A study of the spatial spectrum has been carried out to ver-ify that the presence of particles has only a minorinfluence on the turbulence statistics. In Fig. 2, the single phase flow spec-trum of energy is compared to two-phase flow simulations for thethreedensityratios.Onlyslightmodificationsare observedat wavenumbershigher than theinverse of the particleradius. The energyspectrumisformedovertheentiredomain(accountingfor thesolid body motion ofthe particles). As shownby Lucci etal. (2010),theinclusionofthesolid domaincreatessomeoscillations on the wavenumbers larger than the inverse of the particle ra-dius. As shown in Fig. 2, our simulations stand in a very dilute regimeyieldingtheseoscillationstobenegligible.Linearforcingis madewithouttakingintoaccountthedensityratiooftheparticles,

Eq.(9).Forthisreasontheenergyontheentiredomainincreases when

ρ

increases.Becausethecaseconsideredisverydiluteonly aweakincreaseofenergyathighwavenumbersisobserved.

Inordertoconfirmtheweakeffectoftheparticlesonthe car-ryingfluidturbulence(thelocaleffectwillbediscussedlater)and overtakethe problemofthecomputationofthespatialspectrum includinggridpointsinside theparticles,thetemporalLagrangian spectrumhasbeencomputed.

This spectrum is computed from200000 Lagrangian fluid el-ements (point particles following the flow) which are randomly seededinthefluiddomain.Theirtrajectoriesareintegratedwhile the turbulent suspension is simulated. Then, the Lagrangian ve-locity auto-correlation function, R_Lf= Vi(t0)·Vi(t0−t)

V_i(t)2_ , is computed

Fig. 3. Temporal spectrum of the single phase simulation and three configurations of two-phase flows. These spectra were computed from tracer particle trajectories. Solid line for single phase flow DNS. Dotted, dash-dotted and light dashed lines stand for density ratio equal to 1, 2 and 4 respectively.

fromthose trajectoriestoevaluate theenergytemporal spectrum

E

(

ω

)

= 1 2F

(

R

f L

)

F∗

(

R

f

L

)

,where F and F∗ are the Fourier transform

andits conjugate.TheLagrangianspectrumisgiveninFig.3.This statistical quantity is really a measure of the turbulence of the carryingflowandisnotflawedby accountingsimultaneouslythe fluidandsolidinstatistics.Based onFig.3,we canconcludethat the carryingflow statistics remain constant forthe three consid-ereddensityratios.

Tocharacterizetheparticleresponseto fluidflow fluctuations, two different Stokes numbers are determined, based onthe Kol-mogorovtimescale andthelarge eddyturnovertime.The Stokes number based on the Kolmogorov time scale is relatively large

St_k=

{

26,52,104

}

,asaconsequence,theparticlesareinertial rel-ative to the Kolmogorov scales. The preferential concentration of inertialparticles inlow vorticity regions hasbeencommonly ob-served and quantified for small particles (pointwise or particles smallerthantheKolmogorovlengthscale).ParticleswhoseStokes numberbasedontheKolmogorovtimescalearenearunityare ex-periencingstrongpreferentialconcentration.Indeedtoour knowl-edge, nothing on preferential concentration has been published withparticleresolvedsimulations. Foroursimulations,theStokes number based on the large eddy turnover time is less than 10 (StE=

{

1.5,3.0,6.0

}

)meaningthat thetimeassociatedtothe

par-ticlesis equivalentto thisflowtime scale. Evenforthis interme-diate scale no clusters formation was observed. Statistics on the radialpairdistributionfunctiondidnotshow evidencesof prefer-ential concentration. Different explanations can be proposed. The firstone isthat the simulationtime (12eddyturn-over times)is notlongenoughfortheparticlestomigrateinaccumulationzones. The second explanation is that the physical life-time of the low vorticity regions atlarge scales is too shortto give enough time fortheparticlestocluster.Thesegregationandclusteringoffinite sizeparticlesisstillanopensubjecttostudy.

3.2. Particlestatistics

The dynamics of particles smaller than the Kolmogorov scale hasbeen studiedextensively.Characterizing therelationbetween theparticlestatistics(kineticenergyandthevelocity autocorrela-tion time)and theturbulence propertieshas beenestablished in

Tchen (1947), Hinze(1975),Deutsch (1992),Lavieville(1997) and

Fede (2004).In Table 1 statistics are reportedfor finite-size par-ticles and scaled by the fluid tracer statistics. As expected the

Table 1 Particle statistics. ρ q2 p q2 f _q2 p q2 f T.−H. TL Tf L 1 0.85 1 1.31 2 0.79 0.5 1.45 4 0.74 0.2 1.65

particle kinetic energy, averaged over the Np particles during Nt

timestepsq2

p= 12

jiV2i(tj)

NtNp ,isreducedwhenthedensityratio

in-creaseswhile thevelocity autocorrelationtime increases.This re-duction is predicted by Tchen–Hinze theory (Hinze,1975; Tchen, 1947):



q2 p q2 f



T.–H. = TLf τp+b 2 TLf τp+1 (10) where,b= 3ρf 2ρp+ρf = 3 2ρ+1.

However,theparticlekinetic energyislessimportantthanthe prediction of the Tchen–Hinze theory,

(

q2p

q2

f

)

T.–H.

. Indeed, Tchen– Hinzetheoryisbasedonanestimateofthefluidkineticenergyat theposition ofparticlesassuming that theirsize ismuchsmaller thantheKolmogorovlengthscale.Inourcase,theparticlesare sig-nificantly larger than theKolmogorov length scale therefore they are notexperiencingthefull fluidfluctuations(largeparticles be-havelikeaspatiallowpassfilteroftheflowkineticenergy).

The Lagrangianvelocity autocorrelationtime TL=0∞RL

(

t

)

dt=

_∞

0 

V_i(t₀)·Vi(t0−t)

V_i(t)2_ dt is presented in Table 1. This autocorrelation

timeTL isnormalizedbytheautocorrelationtimeofthefluid

par-ticles T_Lf obtainedthanks to the autocorrelation function R_Lf (see

Section 3.1). It characterizes the typical time a particle takes to changeitsvelocitydirection.ThepresentsimulationsshowthatTL

increasesasdensityratiorises.Similartrendisobservedfor iner-tial point-particles simulations. As can be seenin Table 1, what-everthe densityratiois,even forneutrallybuoyantparticles,the normalizedautocorrelationtime isgreater than1,clearlyshowing finitesizeeffect.

Analysingtheaccelerationstatisticsprovidesinsightsonthe dy-namicresponse ofparticlestothe turbulenceforcing. Ithasbeen showninliterature (Calzavarinietal.,2009; Qureshietal., 2008; Voth et al., 2002) that the probability density function of finite size solid particles acceleration P(a) can be approximated by an exponential lawexp−|a|β _where

_β

_{≤ 1. As showed by Bec et al.}

(2006) the constant

β

approaches unity forinertial particles (here forhighdensityratio

ρ

).InFig.4(a)and(b)two p.d.f.ofparticle accelerationsnormalizedbytheirr.m.s.valueshavebeenreported for the different simulations. In Fig. 4(a) the p.d.f. has been ob-tained withoutconsidering the accelerationoccurringduring col-lisions. Forthiscase, thep.d.f.closelyfollowsan exponential law with

β

=1whateverthedensityratio.Therefore,thisisameasure ofaccelerationstatisticsstrictlyrelatedtotheinteractionwith tur-bulenceintheverydiluteregime(nocollision).

Concerning, the case where acceleration during collision is takeninto accountforthe p.d.f.acceleration(Fig. 4(b)),thesame exponentialbehaviourisobservedforlowaccelerationvalues.This isnolongerthecaseforlargervalues.Indeedthetailsofthep.d.f. in Fig. 4(b)) are shifted towards highervalues. This is relatedto rarebutstrongcollisioneventsontheacceleration.

In Qureshi etal.(2008) an experimental studyhas been con-ducted on the interaction between finite size soap bubbles and turbulent air channel flow. It was shown that the r.m.s. accel-eration scaled by the fluid dissipation rate(A0=a2r.m.s./

ε

t−3/2

ν

1/2)

a

b

Fig. 4. Probability density function of particle acceleration. Exponential p.d.f. is given as a reference, exp −|a|β

, β= 1 . (a) Without collision events. (b) With colli- sion events.

Fig. 5. Dimensionless acceleration r.m.s. compared to density ratio. Present simula- tions (cross) and experiments of Qureshi ( Qureshi et al., 2008 ) (squares) and their respective exponential law.

decreases as

ρ

increases. In order to compare to this result the dissipation of the turbulent flow

ε

_f is estimated in the present simulationsfromthespectrum:2

ν

₀∞

κ

2_E

₍

_κ

₎

_d

_κ

_{.InQureshietal.} (2008) an exponentialdecrease hasbeenobserved from

ρ

₌1 to

ρ

=10asA0=2.8

ρ

−0.6.From oursimulationsthesametrendcan

beobservedas15

ρ

−0.94_(Fig.5).

4. Dissipationdistributionaroundparticles 4.1. Movingframearoundparticles

To evaluate the average flow around moving particles a spe-cificpost-processinghas beenapplied toinstantaneous DNSflow fields.A local framework is attached to each particle centre and itistranslatedwiththe particlevelocity.Aunit vector isaligned

withtheinstantaneousparticlevelocityeVp= Vp

Vp.Astheaverage

flow isaxi-symmetric, this flow field is describedin the moving particleframeworkina{r,

θ

}plane,withrthedistancetothe par-ticlecentreand

θ

theanglewiththeunitvector. Astaggered po-largridisattachtotheparticleframework.Theradialdistribution {ri}isrefinedneartheparticlesurfacetoprovideabetter

descrip-tionofboundary layers. Theangle incrementisalso not uniform tokeepratherconstantthevolumeofeachcellforagivenradiiri.

Thischoiceismadeinordertoconvergesmoothlythestatisticsfor eachcell.The surfaceofthecell between

θ

j and

θ

j+1 is

approxi-matedby2

π

r_isin

(

θj+θj+1

2

)

d

θ

j.Inourcase180anglesare usedin

the[0..180]interval.

Thedifferentstepsofthepost-processingarethefollowing: (1)The DNSflow field isknown on thestaggered fixed

Carte-siangrid.

(2)For each radius ri a spherical shell around the particle is

defined. The valuesofthephysicalquantities(velocity, dis-sipation...) are linearly interpolated at the intersection be-tween thespherical shell and the Cartesian grid usingthe surroundingCartesiangridvalues.

(3)For eachinterpolation location, theangle betweeneVp and

the radiusconnectingthislocation withtheparticle centre iscomputed.Then, thevalue isaddedtotheaverageofthe cellwhichnodeisreferencedby

θ

∈[

θ

j,

θ

j+1].

This algorithm isapplied for each particleandfor manytime stepschosenarbitrarilytoobtainconvergedstatistics.

In Fig. 6 the streamlines resulting from the average velocity flow around particles are presented for different density ratios

ρ

=1,2and4.

As shown by Cisse et al. (2013) the absence of fore-and-aft symmetryisrelatedto theconditional averagingofthe flowina frameworkmoving withtheparticle.Thishasbeenobserved also forflowaroundatracerparticleinCisseetal.(2013).

4.2.Dissipationrateinamovingframearoundparticles

Animportantfeatureoftwo-phaseturbulenceisthatthe pres-enceof particles generates modulation of turbulence.Depending ontheparticlepropertiesandtheflowconfiguration,particlescan enhanceordampfluidflowturbulence(thisfundamentalquestion isstillopen).Fullyresolvedsimulationsoffinitesizeparticles pro-videinformationwhich cannot beobtainedwithpoint-wise two-waycouplingsimulations.Thedissipationofenergyisalocaleffect occurringneartheparticlesurfaceandinitswake.

The dissipation

ε

=2

ν

(

∂ui ∂x_j +

∂u_j

∂x_i

)

2 is computed on the

Carte-sian grid and averaged as presented in the previous paragraph.

Fig. 7 showsthe spatial distribution of dissipation scaled by

ε

_∞ whichis the overall turbulent dissipation of the flow. Apeak of dissipation is located in the front of the particle. In the present simulationsthis peak reach2.5the average fluiddissipation. The magnitudeofthisdissipationincreaseswithdensityratio.This is theresultoftheprogressivedecorrelationof theparticlevelocity andthelocal flowwhich increasesfor largeparticle density.The dissipationincreasesaroundparticlesoveradistancesmallerthan oneradiusinagreementwithdifferentauthors.

In theexperimentsofTanakaandEaton(2010) thedissipation aroundasedimentingparticleinaturbulentflowisgiven.The par-ticlesfall in a turbulencechamber.The particles are much heav-ier than the surrounding gas and their radii are similar to the Kolmogorovlength scale. Theturbulencearound theparticles are measuredbya PIVsystem. Inthisexperimentthedissipation ob-servedaround theparticleshasincreasedto 3.5timesthe turbu-lencedissipation.

Fig. 6. Intensity of the average relative velocity and streamlines around a reference particle for each two-phase flow configuration.

Numerically,severalstudieshavebeenconducted.Inthe simu-lations ofaturbulent particleladen flowpresentedby Cateetal. (2004) instantaneous dissipation fieldsare presented. Itcould be seen that the local dissipation increases around the particles. In thispaper, no average dissipation is provided. InNaso and Pros-peretti(2010) afixedparticlewithradius4timestheKolmogorov scaleis simulatedinafullyturbulent flow. Theaveraged dissipa-tionfora givendistanceiscomputedshowingan increase ofthe dissipation around 4timesthe averageddissipation ata distance ofoneradius.Theaverageddissipationforagivendistanceisalso provided by Wang et al. (2014) for simulations with free parti-clesinaturbulentflowforthesameparticlesize(R₌4

η

)andfor a density ratio

ρ

=4. The method used is based on the Lattice-Boltzmann method.In this casethe dissipation is 6 times larger aroundtheparticlesthantheaverageddissipationinthefluid.

InLuccietal.(2010) simulationsoffinitesizeparticlesare car-riedoutwiththeIBMmethodfordifferentradiiR∈[8.2..17.7]and

Fig. 7. Spatial distribution of average dissipation for each two-phase flow configu- ration (left column). Close-up around the particle surface (right column). Left: com- plete average domain. Right: detail around the particles.

densityratios

ρ

∈[2.53..10].Theycomputedtheaveraged dissipa-tionfora givendistancefortwo regionsofthe flow:thefront of theparticles(equivalentto

θ

∈[0..90]inthepresentnotation)and thebackoftheparticle(

θ

∈[90..180]).Asinthepresentpaper,the dissipationinthefrontoftheparticleislargerandtheinfluenceis observedoveroneradiusdistance.

Finally,Cisseetal.(2013) providesadetailedstudyofslipping velocity and local dissipation around buoyant particles in a tur-bulent flow. This numerical study is based on a pseudo-spectral Navier–Stokes solution and an IBM method to account for fully-resolved particles.Different radii areconsidered (R _∈[8.5..33.5]

η

) that permit to study the influence of the particles size. In this case the average of the dissipation is split in three regions: up-stream (

θ

∈ [0..45]), transverse (

θ

∈ [45..135]) and downstream (

θ

∈[135..180]). Again, the dissipation in the front of the parti-clesislargerthanattherear.Onthethreeregionstheincreaseof dissipationislimitedtooneradius.

The magnitude ofdissipation (front andrear of particles)has beencomparedinFig.8 withtheresultsofLuccietal.(2010) and

Cisse etal.(2013).The profiles ofdissipation atthe frontand at therearoftheparticlesarereported.

The orders ofmagnitudeofdissipation found inthe literature fromthesurfaceoftheparticleto3timestheparticlesradiusare similar toour computations.Inaddition,forall the caseswe ob-servethatthedissipationincreaseislessimportantattheparticle rear. As inLucci etal.(2010) the dissipation peakincreases with thedensityratio.

Eveniftherearemajordifferencesbetweenthewaythe com-putationsarecarriedout,thenumericalapproachandthe

simula-Fig. 8. Dissipation in the front and at the rear of the particle. Present simulations compared to Lucci et al. (2010) and Cisse et al. (2013) .

tionparameters,themagnitudeofthedissipationincreaseagrees. Nevertheless,dissipationseemstobegreaterforthebuoyantcase ofCisseetal.(2013) thanforoursimulations.Weexplainthis dif-ference because the particles are larger than in our simulations,

R=17

η

insteadofR=10

η

, andthus, they aremore inertial. An-otherexplanationcouldbethattheturbulenceisstrongerthanin thepresentsimulations,Re_λ=160.

Asaconclusion,thepresentsimulationsconfirmtheresults ob-tainedbyseveralauthors(Cisseetal.,2013;Luccietal.,2011;Naso andProsperetti,2010;TanakaandEaton,2010):theparticles mod-ifylocallythe turbulence.Thismodificationis alwaysan increase ofthelocaldissipationoftheturbulence.Thisdissipationislarger inthe frontof theparticles andits spatialextension is lessthan oneradius.Themagnitudeofthisdissipationincreaseswith parti-cleinertia.

5. Collisionstatistics

5.1. Collisionswithsolidcontactonly

Understanding thecollision dynamics is a keypoint for mod-elling suspension flows and predicting coalescence events for drops. In this section, only the solid–solid model has been acti-vatedwithoutlubrication(Eq.(6)).

We have followed the present procedure to form statistics. If we call



t_coll the time between two collisionsfor a given parti-cle, and



tcoll,i the averaged of all the



tcoll concerning the ith

particle,the inter-collision time

τ

c for Np particles isdefined as

τ

c=





tcoll,i



Np=

1

Np

Np

i=1



tcoll,i.Anotherwaytocompute

τ

cisto

countthenumberofcollisionNcollduringthetotaltimeofanalysis



T,andfortheNp particlestoobtain

τ

c= _2NNp

coll



T.

Thischaracteristictimecanbeestimatedtheoretically(Eq.(11)) fromtheaveragerelativevelocityofparticles

|

wr

|

asproposedin

Table 2

Comparison of inter collision times for various den- sity ratios. The reference Kolmogorov time is τk =

0 . 0635 s . ρ [ τn,1 c τk .. τn,2 c τk] τcth τk

Based on computations Theory Eq. (12)

1 [22.6 .. 24.5] 21.5 2 [19.3 .. 20.8] 22.4 4 [16.2 .. 17.1] 23.1

SaffmanandTurner(1956).

1

τ

c = 1 2g0n 2 p

π

(

2R

)

2

|

wr

|

(11)

Whenthemotionofparticlesisfullyuncorrelated,therelative ve-locityofparticles

|

wr

|

canbeevaluatedfromtheparticlekinetic

energyq2 p,Abrahamson(1975): 1

τ

th c = 1 2g0n 2 p

π

(

2R

)

2



16

π

23q 2 p (12)

InTable2 theintercollisiontime isreportedfordifferent den-sity ratios. Twodifferent inter collision times have been consid-ered.The first one

τ

th

c is obtainedthanks toEq.(12), by

extract-ing the particle kinetic energyq2

p from simulations with the

as-sumption that g0 is equal to 1 (particle dilute flow) and using

theeffectivecollisionradius(1.1Rbecausethecollisionisactivated for



₌0_.1)andnp= ₍₂512_{π )}3 theparticlesnumberdensity.The

sec-ondinter collision time

τ

n

c isdirectlyobtained byestimatingthe

timebetweentwocollisionsinsimulations.Thisestimatehasbeen achievedwithtwomethods:

τ

n,1

c Weaverageallthetimesbetweentheendofacollisionand

the beginning of the next collision for all the particles. In this estimate the solid contact, which is not instantaneous withoursoftspheremodel,isnottakenintoaccount.

τ

n,2

c Istheratiobetweenthetotalnumberofcollisionsoccurring

duringthesimulationandthesimulationtime.

The first method givesa lower bound of

τ

c whilethe second

oneprovidesanupperbound.

Theordersofmagnitudeofintercollisiontimesarecomparable whatevermethodused.Thegapbetween

τ

th

c ,

τ

cn,1,

τ

cn,2 isalways

lessthan 15%. As expected from Eq.(12),

τ

th

c is increasing with

thedensityratioasqpisdecreasing.Incontrast,thecollisiontime

measuredisdecreasingwithincreasing densityratio.Onecan ex-plainthis behaviour by considering that the evaluationof

τ

th

c is

doneundertheassumptionofg0=1.

Inordertounderstandthosediscrepanciestheprobability den-sityfunctionofinter-collisiontimesisnowconsidered.In simula-tions,thep.d.f.isobtainedbycountingthenumberofoccurrences ofeach intercollisiontime



t_coll forall theparticles.Thep.d.f.is dividedbythetotalnumberofoccurrences.Fromthekinetic the-oryof granularflows (uncorrelatedmotion ofparticles)the p.d.f. follows: P

(



tcoll

)

= 1

τ

th c exp



−



tcoll

τ

th c



(13) In Fig.9,theprobability densityfunctionis reportedforthree densityratios.Acomparisonisprovidedbetweensimulationsand kinetictheory.

The p.d.f.obtainedwithsimulationsfollowsthekinetic theory for



tcoll<4

τ

c.Forlargervaluesof



tcoll,themeasuredp.d.f.has

alargedispersionduetothesmallnumberofoccurrencesthatare counted.Forthis reason, thecomparison to thekinetic theory is notrepresentativefor



tcoll>4

τ

c.

Fig. 9. Inter-collision time probability density function. Solid line corresponds to

Eq. (13) , ( τc = τcth for kinetic theory and τc = τcn, for the measured p.d.f.). 1

Fig. 10. Probability density function of the angle between particle velocity at con- tact. Theory for uncorrelated motion is given by Eq. (14) (solid line). A sketch of particle velocity at contact is provided for θcoll = { 0 , 90 , 180 } degrees.

Ifwe focus onthe smallinter collision times,thecomparison between kinetic theory and measured p.d.f. shows a larger dif-ference for

ρ

=1than for

ρ

=4.This is the consequenceof the particlemotionthat followsmoreeasily theflow andcomesinto contactwithslightlymorecorrelatedtrajectorieswhen

ρ

₌1.This discrepanciesfor

ρ

=1couldbeinferredtotheeffectofsecondary collisionsthat willbe affectedbythelubricationmodel.A discus-siononthesecondarycollisionsisgiveninSection5.2.

Inordertodescribemorepreciselythecollisionregime, statis-ticsofthevelocityorientationswhencollisionoccursareanalysed. For uncorrelated motion of particles, the p.d.f. of the angle be-tween two particlevelocities atcontactispredictedby Lavieville (1997): f(θcoll) = ∞ 0 ∞ 0 8πsin(θcoll)Va2Vb2  V2 a+Vb2− 2VaVbcos(θcoll)fv(Va)fv(Vb)dVadVb π 0 ∞ 0 ∞ 0 8πsin(θ )Va2Vb2  V2 a +Vb2− 2VaVbcos(θ )fv(Va)fv(Vb)dVadVbdθ (14) wherefvisthevelocityp.d.f.

In Fig.10 thekinetic theory corresponding to Eq.(14) forthe collisionanglep.d.f.iscomparedtosimulationresults.AGaussian profilehasbeenused forfv correspondingto thebehaviour

mea-suredinoursimulations.

Whatever thedensityratio is,the maximumof the measured p.d.f. is obtainedfor 20°<

θ

coll < 40°,whereas the kinetic

the-oryprovidesapeakat108°.Thisischaracterizedbyashiftofthe p.d.f.fromlargeto low anglesindicating thatparticles comeinto contactwithnearlyparallelvelocities,evidencingaparticleto par-ticletrajectory correlation. Thisshiftdecreases forhigherdensity ratio showingthat particleinertia yields larger decorrelation be-tween the particles andthe carrying fluid flow. From this result,

Fig. 11. Probability density function of the relative velocity at contact (theory and simulations).

it isobviousthat therelative velocity atcontactshouldbe lower thantheoneobtainedwiththekinetictheorydiscussedbelow.

We nowcompute the relativevelocity atcontactwr=2un,coll,

whereun,collistherelativevelocitygivenbyEq.(5) atthe

begin-ningofcontact.ForuncorrelatedvelocityparticleswithaGaussian velocitydistribution,thep.d.f.ofwrisgivenby

fcoll

(

|

wr

|

)

= 2



8/3q2 p



2

|

wr

|

exp



−

|

wr

|

8/3q2 p



(15) InFig.11,we comparethep.d.f.oftherelativevelocityat con-tact obtained with the kinetic theory and simulation results. In

Eq.(15),thekineticenergyq2

pisobtainedfromthesimulations.

As alreadyobservedforp.d.f. oftheangle,a clearshiftofthe p.d.f. ofvelocity is also reportedin simulations compared to the kinetictheory.Asexpected,therelativevelocityatcontactislower inthesimulationsthaninanuncorrelatedmotion.Particleinertia increasesdecorrelationofparticlemotionandfluidflowyieldinga betteragreementwithkinetictheoryforthelargestdensityratio.

The relative velocity at contact characterizes the energy in-volved incollisions.When collisionsoccurina viscousfluid,part of this energy is dissipated by viscous effect during the fluid drainage. As a result, the velocity after the collision is reduced. It has been shown that the impact Stokes number defined as

Stcoll= 29

Ru_n,collρp

μ is a dimensionless parameter which can

deter-mine the collision regime (fromdry contactto viscous rebound). Inordertoestimatetheenergydissipatedduringthecollisionthe restitutioncoefficient eisdefinedastheratiobetweenthe veloc-ity after the collision and before the collision. Based on experi-ments, the correlation proposed by Legendre et al. (2005), _ee

d =

exp

(

− 35

St_coll

)

, givesthe relationbetweenthe restitution coefficient

andtheStokes number(the restitutioncoefficient isscaledby its value fora dry contacted). InFig. 12 the p.d.f. ofimpact Stokes

numbers is presented aswell asthe distribution of theeffective restitution coefficientsbased onthe p.d.f.ofthe relative velocity,

Fig.11.

Itcanbe seenthatforthecase

ρ

=1mostofcollisionsare in the Stokes range corresponding to no rebound (e ࣃ0). Rebound will notoccur andparticles willstay stucktogether.Forthe case

ρ

=4 thevalues of theStokes numberscover a wide range cor-responding to restitution coefficients lower than 0.5. This means thatfortherangeofphysicalparameterswesimulated,lubrication effects duringparticleencounters are importantandmustbe ac-countedfor.

5.2. Theeffectoflubricationoncollisionstatistics

The effect of lubrication on collisions will be studied for cases

ρ

=1 and

ρ

=4. The lubrication model corresponding to

Fig. 12. Probability density function of the impact Stokes number obtained from the relative velocity. Inset plot represents the p.d.f. of the restitution coefficient ob- tained with the correlation given by Legendre et al. (2005) .

Eq. (8) is now activated at distance 1.1R similarly to the solid– solid model in the previous section. The solid–solid collision modelisnot modifiedandisactivatedonlyforthe particlesthat overlap.

Several physical quantities and statistics will be compared in ordertohighlighttheeffectoffilmdrainageoncollisions. The ki-neticenergyofparticleagitation,theautocorrelationtime,the ve-locityandaccelerationdistributionsarenotdrasticallymodifiedby includingtheeffectoflubrication.Afirstdifferencebetween sim-ulationswithandwithoutlubricationmodelappearsonthe aver-ageddistancebetweenparticles.Forexample,themeandistanceto thenearest neighbourisdecreasedbyabout5%,BrändledeMotta (2013).Thesameresultcanbeobservedintheradial distribution function: for small distances the probability to observe a parti-cleincreases,meaning that particles tends to stay closer toeach other.Theactivationoflubricationbetweentwoparticlesdecreases therelative velocitybetweenthem duringthe encounteryielding alongertimeofnearbyinteraction.

Herewefocusonthemodificationofthecollisionalregime.In ordertoanalysethiscollisionalregimeitisnecessarytomakethe difference betweentwo different interactions:the encounter, de-fined as the activation of lubrication model, and the solid–solid collision,definedastheactivationofsolid–solidmodel.

Theinter-encountertimeobtainedastheaveragetimebetween twoactivationsofthelubricationmodelis{21.6,16.1}

τ

k for

ρ

=

{

1,4

}

respectively.Theelapsedtimebetweentwosuccessive inter-actionsisclosetotheresultsobtainedforthesimulationswithout lubricationmodel:

τ

_cn,1=

{

22.6,16.2

}

τ

_k.Itshowsthatthe encoun-tersbetweenparticlesdonotdependonthelubricationmodelbut onlyontheflowdynamicswhichcarriestheparticlestowards con-tact.InFig.13 thep.d.f.oftheinter-encounterandinter-collision timesarepresentedforthesimulationswithandwithout lubrica-tionmodel.

Inallcasesanexponentialdecayisobservedforlargeinter col-lisionorinterencountertimes.Thisisresultingfromthe uncorre-latedmotioninagreementwiththekinetictheory.

When thelubrication is considered, the ratioof collisions oc-curringatsmalltimesincreases.Asaconsequence,thelevelofthe exponentiallaw decreases.Forthe smallinter collisiontimes the exponentiallawisnomorevalid.Thiseffectisenhancedfor

ρ

₌1_, butalsopresentfor

ρ

=4.Because oftheincreaseofsmall



tcoll

occurrences,theintercollisiontime

τ

cn,1isreducedfrom22.6

τ

kto

2.5

τ

_kforneutrallybuoyantparticles.Thisisexplainedby the oc-currenceofmanybody solid–solidcollisionsduringtheactivation ofthelubricationmodel.Thelubricationslowsdowntheparticles duringseparationafter a collision,increasing solid–solidcollision events.These phenomena have been also reportedin Cate et al. (2004),so-calledsecondarycollisions.

Fig. 13. Inter-collision time probability density function for simulations with and without lubrication model. Fitted exponential law is given for solid–solid collisions for simulations with lubrication model. For comparison purpose the tcoll time is

scaled by the inter collision time obtained without lubrication model (see Table 2 ). The kinetic theory ( Eq. (13) ) corresponds to solid line.

As proposed by Cate et al.(2004) the p.d.f. can be split into twocontributions.Thefirst one,relativetoprimary collisionshas anexponentialtailcorrespondingtolongtimecollisions:P

(

t

)

=

α τcexp

(

−

t_coll

τc

)

.The secondone, relative tosecondary collisionsis

obtainedasthe differencebetweenthe overallp.d.f.andthefirst contribution.

α

is a parameter representingthe gapbetweenthe realp.d.f.andthekinetictheory.Itisobtainedby fittingthep.d.f. forlong inter-collisiontimes. The exponential law of the kinetic theoryisrecoveredwhen

α

=1.

From oursimulationsweobtain

α

forthesolid–solidcollisions withthelubricationmodel:

• For

ρ

=1,

α

=7%:only7pathsover100arereproducedbythe kinetic theory,theother 93are free pathsbetweensecondary collision.InthesimulationsofCateetal.(2004),

α

wasranging between8%and11%whichagreeswithourresults.

• For

ρ

=4:theoccurrenceofconsecutivecollisionsisless pro-nounced.Theinter-collisiontimeisreducedonlyfrom16.2

τ

k

to14.6

τ

k.Theparameter

α

isequalto77%. Here,particle

in-ertiadrivestheparticlestowardsmoreenergeticcollisionsand rebounds. The effectoflubricationis weakerandcannot keep theparticlestogetheraftercollision.

6. Conclusion

The present paperprovides an analysis of a sustained turbu-lentflow seededwithfinite-sizeparticlesbased onfullyresolved simulations. Based on a numerical method previously developed forthiskindofsimulation(BrändledeMottaetal.,2013;Vincent etal.,2014),specificcharacteristicsoftheparticleflowhavebeen investigatedsuch asthelocaldissipationaroundparticlesandthe collisiondynamics.Thesephenomenawereunderinterestasonly fewworksweredevoted tothemintheliterature.Three configu-rationsweresimulatedwithdifferentdensityratios.

From a macroscopic point of view simulations have provided the following results: First of all, Eulerian as well as Lagrangian turbulent energy spectra are reported. It has to be noticed that thelaterhasneverbeenachievedwithresolvedscalesimulations. Theconclusionisthatwhateverthedensityratiotheenergy spec-traarealmostthesame,withonlysmalldifferencesatthelargest andsmallestscales.Otherwise,theaccelerationoftheparticles is modifiedbytheinertiaandthecollisions.Thissecondeffectis ob-servedontheaccelerationp.d.f.andwasnotreportedinprevious works.

A specific point ofinterest inthe presentwork wasthe aver-ageddissipation around particles. Simulations confirm that dissi-pationincreasesneartheparticle.Ononehand,dissipationis

con-centratedinthefrontoftheparticleandisextendedoverone ra-dius. On the other hand, dissipation increases with the particles inertia,almost2timesthefluidturbulentdissipation.

Thecollisionregimewasalsoanalysedinthispaper.Firstofall, itisshownthat whenparticleinertia increasesthecollisionscan bedescribedbykinetictheory.Thisisverifiedforthep.d.f.ofthe intercollision time,theangleofthecollisionandtherelative ve-locityat contact.These two last statistics were not oftenstudied and provide interesting physical information. The divergence be-tweenmeasuredp.d.f.andthekinetictheoreticalp.d.f.fortheless inertialparticlesarehereexplained:theparticlepairvelocitiesare correlatedwhentheyareneareachother.Thisisaneffectoftheir correlationwiththesurroundingfluidvelocity.Theparticleto par-ticlecorrelationshaveastronginfluenceontheenergyofthe col-lisions whichdecreases,especially forneutrallybuoyantparticles. Forthislastconfiguration,themeasuredimpactStokesnumberis nearthestickingparticleconfiguration.Thestickingeffectcan be interpretedasalackofreboundthatisproducedby energy dissi-pationduetolubrication.Thisisconfirmedbyacomparisonofthe simulationwithandwithoutlubrication model.This studyshows that forneutrallybuoyantparticles the lubricationforce prevents particles to separate after the solid collision. Thus, several solid collisionappears afterthefinal separation.Thatcouldhavemany effectsonphysicalapplications,forexampleonthecreationof ag-gregates.

Acknowledgement

We aregrateful foraccesstocomputational facilities ofCINES andCCRT underProject numberx20162b6115.Thisworkwasalso grantedforHPCresourcesatCalMipundertheProject2013-P0633. TheauthorsthankP.Fedeforfruitfuldiscussionsaboutcollisional regimes.

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