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Transport of finite-size particles in a turbulent Couette flow: The effect of particle shape and inertia

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This is an author’s version published in: http://oatao.univ-toulouse.fr/21072

To cite this version:

Wang, Guiquan

and Abbas, Micheline

and Yu, Zhaosheng and Pedrono,

Annaig

and Climent, Éric

Transport of finite-size particles in a turbulent

Couette flow: The effect of particle shape and inertia. (2018) International

Journal of Multiphase Flow, 107. 168-181. ISSN 0301-9322

Official URL:

https://doi.org/10.1016/j.ijmultiphaseflow.2018.05.014

Open Archive Toulouse Archive Ouverte

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Transport

of

finite-size

particles

in

a

turbulent

Couette

flow:

The

effect

of

particle

shape

and

inertia

Guiquan

Wang

a,b,c

,

Micheline

Abbas

b,c,∗

,

Zhaosheng

Yu

d

,

Annaïg

Pedrono

a

,

Eric

Climent

a,c

a Institut de Mécanique des Fluides de Toulouse, IMFT, Université de Toulouse, CNRS, Toulouse, France b Laboratoire de Génie Chimique, Université de Toulouse, CNRS, Toulouse France

c FERMaT, Université de Toulouse,CNRS, INPT, INSA, UPS, Toulouse, France d Department of Mechanics, Zhejiang University, Hangzhou 310027, China

a

b

s

t

r

a

c

t

The transport offinite-size particlesina turbulent plane Couette flowhas been studiedby particle-resolvednumericalsimulationsbasedontheForceCoupling Method.Theinfluenceofthedeviation of particleshapefromsphericitywasaddressed,usingneutrallybuoyantspheroidswithaspectratio rang-ingfrom0.5to2.Theparticletransportwas comparedtothecasewheretheinertiaofspherical par-ticleswasvariedbyconsideringdifferentparticledensities(whilekeepingcomparableStokesnumber). Thiswork hasshownthatclosetothewall, thesymmetryaxisofoblateparticlesisalmostparallelto the wall-normaldirectionand themajor axisofprolateparticlestends toalign intheflowdirection. Bothtypesofparticleshaveakayakingtypeofrotationinthecoreregionwhichyieldshomogeneous collisions.Thespatialparticledistributionisstronglycorrelatedtocoherentstructures.However,strong deviations occur for the mostinertial particleswhichaccumulate inthe near wallregion. Successive stagesofaccumulationandreleaseinthestreaksareobservedwhiletheregenerationcycleofturbulence proceeds.Thecaseofmasslessbubblesischaracterizedbyaverystrongcorrelationwiththelargescale vorticesthatspanoverthedepthoftheCouettegap.Eventhoughtheparticlesdonotmodifydrastically theflow,theyhavesomeeffectonthefluctuatingenergy,assuggestedfromthepdf.Thiseffectismore clearfornon-sphericalparticlescomparedtothesphericaloneswithvariousdensityratios.

1. Introduction

Wall-bounded turbulent flows arepopulated by coherentflow structuresthatarelargelyresponsibleforenhancingheatandmass transfer(Robinson,1991).Understanding theparticledynamics in thesecoherentflow structures is fundamentaltounderstand and predictparticletransport,entrainmentanddepositionin environ-mentalsystemsorindustrialprocesses.Forexample particlescan formparticle streaksnearthewall wherethevorticalflow struc-turescreatesuitableconditionsforparticleentrainment,and par-ticipatetoparticledepositionbyconveyingthemfromthecore re-giontothewallregionKaftorietal.(1995).

The dynamics of wall-bounded turbulent flows is very rich, andthe literature on this subject is abundant. We will focus in thisintroductiononsomeaspectsrelatedtoourinvestigation.The keystructures inwall-boundedturbulencearethespatially coher-ent,temporally evolving, large-scale streaks (wall-normal sweeps

and ejections) and large-scale streamwise vortices. The turbu-lenceissustainedthrough asequenceoflinear(lift-up)and non-linear(stretchingandbreak-up)structureinteractions,formingthe so-called regeneration cycle (Hamilton et al., 1995; Jiménez and Pinelli,1999).Weareinterestedinthetransportoffinite-size par-ticlesinwall-boundedturbulentflows.Byfinite-sizewemeanthat the particle diameter is comparable to or larger than the small scalesoftheflow.

Particlescanmodulatetheflowregime iftheysufficiently per-turb the turbulenceregeneration cycle. Forexample,in turbulent plane Couetteflow, Wangetal.(2017)haveshownindetails that neutrallybuoyantfinite-sizeparticleswithmoderatevolume con-centration hardly alter the coherent structures in plane Couette flow.ThiswasinagreementwiththeobservationofBrandt(2014). However particleswiththe samesize andconcentration enhance theturbulenceinpressuredrivenflow,asobservedexperimentally by Matas et al. (2003)and confirmed later by many simulations (Yu etal., 2013;Loiseletal., 2013;Picano etal.,2015).Inamore recentpaper Wanget al.(2018),we have shownthat turbulence enhancementinpressure-drivenflowisduetotheaccumulationof

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particles inejectionregions, leadingtothemodulation ofthe re-generationcyclestages(mainlythelift-upandthevortex stretch-ing mechanisms). All theseafore-mentioned studies were carried outwithsphericalparticles.

Iftheparticlesaremuchsmallerthatthesmalleddies,theyare consideredaspointwise.MarchioliandSoldati(2002)haveshown that the sweep and ejection events are effective in transferring these particles toward the wall and the core respectively. Point-wiseinertialparticlesareknowntoaccumulateinhighstrainrate regions while bubbles tend to move towards the coreof vortical regions(BalachandarandEaton,2010).

As for non-spherical particles, they experience forces and torques that dependon particleorientation. Inwall-bounded tur-bulentflow,their axiswasshowntopreferentially orienttowards the fluid vorticity direction(Parsa et al., 2012; Voth andSoldati, 2017).Theorientationdynamicsofnon-sphericalparticlesdepends on their aspect ratioAr. Evenin the simplest case, a pure shear

laminarflow,theparticlerotationratedependsonits aspectratio (followingJeffery’sorbit(Jeffery,1922)).Whencoupledto transla-tion,particlerotationleadstocross-streamlinemotioneveninthe absenceofparticleinertia.Inaddition,non-sphericalparticles ex-perience torque from the local flow deformation, which possibly leadstopreferentialalignment.Becauseoftherichorientation dy-namics,theflowstatisticsoftwo-phaseflowsmightdependonthe deviation of particles from sphericity, in a non-trivial way(Voth andSoldati,2017;Einarssonetal.,2015b;Dabadeetal.,2016).

Most studies performed with anisotropic particles considered particle transport by fluid, without accounting for the parti-cle two-way coupling with the fluid flow structures. For non-spherical particles which size is small compared to the small-est turbulent flow structures, many properties of particle mo-tion are similar betweendifferentturbulent flows since the fluid strain isdominated by smallscales, which share universal prop-erties. Zhang et al. (2001) have shown that prolate ellipsoidal inertial particles accumulate in the viscous sublayer of a di-lute turbulent channel flow (in the ejection regions like spher-ical particles), and the aspect ratio influences their deposition rate. Inthe sameflowconfiguration,Mortensenetal.(2008) and

Marchioli et al. (2010) observed in addition that prolate parti-cles exhibit preferential orientation in the streamwise direction, especially near thechannel walls, the aspect ratiohaving a neg-ligible effect on particle distribution in the flow. Recently, the particle rotation dynamics was a subject of interest in addi-tion to particle orientation, both being dependent on the parti-cle position: whether nearthe channel center (similarto homo-geneous turbulence) ornearthe wallswherethe shearisstrong.

Zhaoetal.(2015b)foundthatnearthechannelcenter,inertia-free spheroidswere evenly distributedacross thechannel, the flattest disks(Ar=0.01) mainlytumblingintheplane normaltothe

vor-ticity, whereas the longest (Ar=50) rods spinning in the

vortic-itydirection,asinhomogeneousisotropicturbulence(Byronetal., 2015). Inertia was found to reduce the preferential spinning or tumbling leading to more isotropic rotation. Near the channel walls, Zhao et al. (2015b) observed that preferential orientation ofspheroidsinthestreamwisedirectionisinducedby the coher-entflow structures.Whenthefeedback forcingfromtheparticles onto the flow is taken into account, turbulence intensity reduc-tionwasobservedforachannel flowladen withprolateparticles (Zhaoetal.,2015a).

Studieson turbulentflowsladen withfinite-sizenon-spherical particlesarescarce.ExperimentsofParsaetal.(2012) inisotropic turbulencewithnearlyisolatedneutrallybuoyantrod-likeparticles with the axis being larger than the Kolmogorov scale confirmed thepreferential alignmentoftheparticleaxiswiththelocalfluid vorticity. Based on numerical simulations, they have shown that theparticlermsrotationratedependsstronglyontheaspectratio

withanabruptreductionof80%occurringforArbetween0.5and

2.Do-Quanget al.(2014) showedthat finite-sizefibers in turbu-lentchannelflowbehavedifferentlyfrompointwiseparticles.They accumulate in high-speed streaks,staying there dueto collisions withthewall.Inthechannelcore,theyconfirmedthatfibersalign withthemeanflowvorticitydirection.Closertothewallthefibers tumblein the shear plane. Veryclose to the walls they become alignedintheflowdirection.ForoblateparticleswithAr=1/3in

turbulentchannelflow,NiaziArdekanietal.(2017)observedadrag reductionduetotheabsenceofanear-wallparticlelayerwhichis otherwisefound when particlesare spherical. Theymainlyfound thatthesymmetryaxisoftheoblateparticlestendtobe preferen-tiallyorientednormaltothechannelwallsinthenearwallregion. Totheauthors’knowledge,thetransportofnon-sphericalparticles inplaneturbulentCouetteflowconfigurationhasnotbeen consid-eredsofar.

We investigate in this paper a plane Couette flow, as it is a genericandsimple configurationtopoint outinteresting features ofthecomplexdynamicsofnon-spherical(prolate andoblate) fi-nitesizeparticles.Wefocusedparticularlyonparticlerotation dy-namics,orientationandtransportwithinlarge-scalestructures.We consider spheroidalparticles of minor radiusrp=Ly/40 with

as-pectratiofrom0.5to2(whererpandLyaretheparticleand

Cou-ette gapsize respectively). Theirbehavior iscompared to that of inertialspherical particles ofradius rp=Ly/40 withdensityratio

from0 to5. The Reynoldsnumber isslightly above theonset of turbulence.Thepaperisorganizedasfollows.InSection2,the nu-mericalmethodusedforthisstudyisdetailed,withaspecial em-phasisonitsextensiontosimulateellipsoidalshapeparticles. Val-idationstestsarealsopresented.Section3providesthedetails of thesuspension flow simulations carriedout inturbulent Couette configuration,withthecharacteristicdimensionlessnumbers.The resultsaredescribedinSection 4,wherewe mainlyfocuson the particle spatial distribution, their rotational dynamics, their resi-dencetimeinflowrotationalstructuresandcorrelationofparticle distributionwith main turbulentflow structures. The paper ends withaconclusiononthemajorfindings.

2. Simulationmethodandvalidation

Direct numerical simulations of single-phase flows are per-formedbyusingthecodeJADIMforanincompressibleNewtonian fluid (Calmet and Magnaudet, 1997). The unsteady 3-D Navier– Stokesequationsdiscretized ona staggeredgridare integratedin spaceusing the finite volume method. All terms involved in the balanceequationsarewrittenina conservativeformandare dis-cretizedusingsecond ordercenteredschemes inspace.The solu-tionis advancedin time by asecond-order semi-implicit Runge– Kutta/Cranck Nicholsontime stepping procedureand incompress-ibilityis achievedby apressure correctionwhich issolutionof a Poissonequation.

Numerical simulations of particle trajectories and suspension flow dynamics are based on multipole expansion of momen-tumsource terms addedto the Navier–Stokes equations(namely Force-Coupling Method as described in Maxey and Patel (2001),

Lomholt and Maxey (2003) and Climent and Maxey (2009)). The comparison of FCM with other methods that belong to the class ofFictitious Domain methods can be found in a review by

Maxey(2017)forparticulateflows.FCMisbasedonalow-order, fi-niteforcemultipolerepresentationoftheeffectoftheparticleson thesurroundingfluidflow.Althoughno-slipboundaryconditionis notstrictlyenforcedattheparticlesurface,theflowfieldperfectly agreeswiththeexactsolutionwithin atypical distanceof10% of theradiusawayfromtheparticlesurface.Optimal resultsare ob-tainedwithFCM onauniformgridwithonly6meshgridsalong thediameter.Comparedtootherfictitiousdomainmethodssuchas

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directforcing methods(immersedboundarymethodorimmersed bodymethod)weobtainthesamelevelofaccuracybutwithless computingefforts. Flowdynamics iscoupledto Lagrangian track-ingof particles.The fluid isassumedto filltheentiresimulation domain,includingtheparticlevolume.Thefluidvelocityand pres-surefieldsaresolutionsofcontinuityEq.(1)andmomentum bal-anceEqs.(2)and(3).

· u=0 (1)

ρ

DDut =−

p+

μ∇

2u+f

(

x,t

)

(2) fi

(

x,t

)

= Np  n=1 Fn i 

(

x− Y n

(

t

))

+Gn i j

x j 

(

x− Yn

(

t

))

(3)

The spatialforce distributionf(x,t) inthemomentumbalance

Eq.(3)accountsforthepresenceofparticlesintheflow.Itis writ-tenasamultipoleexpansiontruncatedafterthesecondterm.The first term of the expansion called the monopole represents the forceFn thattheparticleexertsonthefluidduetobuoyancy,

par-ticleinertia andparticle-to-particlecontactforces (Eq.(23)). This monopole term balances the drag, added-mass, lift and history forces(ClimentandMaxey,2003).

Fn=

(

m p− mf

)



gdVdtn



+Fn ext (4)

Thesecond term,calleddipole,isbasedonatensorGnsumof

twocontributions: ananti-symmetric partAn i j=

1

2



i jkTn isrelated

tothetorqueTn appliedontheparticle,whichisacombinationof

theexternaltorqueandinertialterms(ClimentandMaxey,2009), andasymmetricpartwhichaccountsfortherigidityoffinite-size particle(LomholtandMaxey,2003).

Tn=Tn ext

(

Ip− If

)



d



n dt



(5)

whereIp(resp.If)istheparticle(resp.appropriatefluid)rotational

inertia.ThesymmetricpartSn

i j thataccountsfortheresistanceofa

rigidparticletolocalstrain byensuringzeroaveragedeformation insidetheparticlevolume,Eq.(6).

En i j

(

t

)

= 1 2 



u i

xj+

uj

xi





(

x− Yn

(

t

))

d3x=0 (6)

The particle finite-size is accountedfor by spreading the mo-mentumsourcetermsaroundtheparticlecenterYnusinga

Gaus-siansphericalenvelope.Foranellipsoidalparticlehavingits prin-cipalaxesalignedwiththeaxesanditscenterlocatedattheorigin ofthereferenceframework,theimplicitequationofthesurfaceis writtenintheformofEq.(7).

x2 a2 + y2 b2+ z2 c2 =1 (7)

wherea,bandcarethelengthsofitssemi-axes.

In the frame of the Force Coupling Method, the Gaussian en-velopes are adapted in order to take into account the particle shape.Therefore,thegeneralizedGaussianenvelopescanbe writ-tenforthemonopoleinEq.(8)anddipoletermsinEq.(9), follow-ingLiuetal.(2009):



(

x

)

=

(

2

π

)

−3/2

(

σ

1

σ

2

σ

3

)

−1 exp



−1 2



x2

σ

12 + y2

σ

22 + z2

σ

32



(8) 

(

x

)

=

(

2

π

)

−3/2

(

σ

 1

σ

2

σ

3

)

−1 exp



−1 2



x2

σ

 1 2 + y2

σ

 2 2 + z2

σ

 3 2



(9)

Itwasshownbythesameauthorsthatthewidthsofthe Gaus-sian envelopes

σ

i (Eq. (10)) and

σ

i (Eq. (11)) are related to the

semi-axisa,b,csimilarlytoasphericalshape.

σ

1=a/

π

;

σ

2=b/

π

;

σ

3=c/

π

(10)

σ

 1=a/

(

6 √

π

)

1/3;

σ

 2=b/

(

6 √

π

)

1/3;

σ

 3=c/

(

6 √

π

)

1/3 (11)

Inadditiontothe updateoftheparticlepositions,itis impor-tanttoupdatetheparticleorientationintime.Thegeneral orienta-tionofanellipsoidisdeterminedfromtheorthogonalunitvectors

p1, p2 andp3 followingthe ellipsoid semi-axes.They rotateasa

rigidbody.Therefore,foraparticlen,theirevolutionintimeis ob-tainedfromtheparticlerotationvelocity



nas:

dpn i

dt =



n× pn

i (12)

Thetransformationbetweenthefixedcoordinateaxesof refer-enceandthe instantaneous semi-axes of an ellipsoidis specified bytheorthogonalmatrixQ:

Q=[p1T,p2T,p3T] (13)

ThegeneralformoftheGaussianenvelopisthenwrittenas:



(

x

)

=

(

2

π

)

−3/2

(

σ

1

σ

2

σ

3

)

−1 exp

−1 2x TQT

σ

−2 1 0 0 0

σ

2−2 0 0 0

σ

3−2

Qx

(14) 

(

x

)

=

(

2

π

)

−3/2

(

σ

1

σ

2

σ

3

)

−1 exp

−1 2x TQT

σ

−2 1 0 0 0

σ

2−2 0 0 0

σ

3−2

Qx

(15)

Theseparatetemporalintegrationofthethreeparticleunit vec-torsinEq.(11)canleadto someinconsistencyintheparticle an-gulardynamics.Therefore,followingNikraveshetal.(1985)we re-placethetime integrationofseparate unitvectorspi bythe time

integrationofthequaternionqofaunit vectorofanorientational axisofrotation.Thederivativeintimeofthequaternionisrelated totheparticlerigidmotionasfollows:

dqn dt = 1 2

(

A n

)

T ×



n (16)

whereATisthematrixdetailedinEq.(17).

AT=

−q1 −q2 −q3 q0 q3 −q2 −q3 q0 q1 q2 −q1 q0

(17)

TherotationmatrixR(Eq.(18))

R=2



q2 0+q21− 1/2 q1− q0q3 q1q3+q0q2 q1q2+q0q3 q20+q22− 1/2 q2q3− q0q1 q1q3− q0q2 q2q3+q0q1 q20+q23− 1/2



(18)

allows to obtain the projection of a vector M from the particle coordinatesystemontotheCartesianframeofreferenceMusing:

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Fig. 1. Schematic representation of the repulsive force and torque at the center of a pair of ellipsoids if their surface is closer than a distance of 0 . 1( O αAα + 

OβAβ) .

With two perpendicular axes of the same length and a third oneshorter(resp.longer),theellipsoidiscalledoblate(resp. pro-late) spheroid. Only spheroids will be considered in this work, withindifferentflowconfigurations.

Furthermore,theparticletranslationandrotationvelocitiesare obtainedfromalocalweightedaverageofthevolumetricfluid ve-locity(resp. rotationalvelocity)field over theregion occupiedby theparticle(Eqs.(20)and(21)).

Vn

(

t

)

=  u

(

x,t

)



(

x− Yn

(

t

))

d3x (20)



n

(

t

)

=1 2 

(

× u

(

x,t

))



(

x− Yn

(

t

))

d3x (21)

Particle trajectoriesare then obtainedfromnumerical integra-tionoftheequationofmotionasinEq.(22).

dYn

dt =V

n (22)

This modeling approach allows calculating the hydrodynamic interactions witha moderatecomputationalcost.Inorderto cap-ture correctly the dynamics of dilute suspension flow, four grid points per particle radius (for spherical particles) are usually re-quiredwhenthemonopoleforceisnotzero,andinthecasewhere onlydipoleforcingisrelevant,threegridpointsperparticleradius aresufficient.Forthesimulationofellipsoids,weensure3–4grid pointsforwithinthesemi-minoraxisdirection.

The repulsivemodel relatedtothe distancebetweentwo par-ticles (expressed as Eqs. (23) and (24)) is important to prevent the overlapping of non-spherical moving particles because it se-riously influences the particle orientation andconsequently their interaction with the flow. For a pair of ellipsoids (

α

) and (

β

), thecollisionbarrierisactivatedwhentheparticlesareveryclose. The minimal distance between the surfaces of two ellipsoids, or the surface of one ellipsoid and the wall, is calculated follow-ing Pope(2008)by successiveapproximations.Theiterationsstop when the difference between two successive iterations, TOL=



Aα,iAβ,i



min



Aβ,i−1Aα,i−1



min, is less than 0.001a (a beingthe

length of the major semi-axis). Note that for the aspect ratios used forthispaper, i = 8–10iterationsare needed ifwe require

TOL≤ 0.001aandi=4–6iterationsifTOL≤ 0.01a.

The contactforce is added at the points of “contact” Aα and

Aβ (which are the closest points between both particle surface) iftheirdistanceislessthan0.1

(



OαAα



+



OβAβ



)

.Asshown inFig. 1,therepulsive forceatthe contactpoint Aα isnormalto the plane tangent to the surfaceatthat point. This force is acti-vated assoon as the two dashed ellipsoidal lines intersect with each other, the expandingfactor being1.1. Since the directionof

the force does not necessarily pass through the ellipsoid center, therepulsiveforce(Fαβc )issupplementedby atorque(Tαβc ), both

ofthembeingappliedatthecenteroftheellipsoid(Oα).Theforce andtorqueappliedonparticle(

α

)areformulatedinEqs.(23)and

(24). Fαβc =Fre f



1−



AαAβ



2

(

0.1



OαAα



+0.1



OβAβ



)

2



AαAβ



AαAβ



(23) Tαβc =OβAβ× Fαβc (24)

However, ifthe particles actually overlap witheach other,we impose a constant magnitude equal to Fref. Fref is simply scaled withtheStokesdragforceFd=6

πμγ

a2appliedona

correspond-ingspherebasedoncharacteristicparticlerelativevelocityinshear flow

γ

a,where ais thesemi-major axisand

γ

istheshearrate. Inturbulent flow simulations throughthispaper, Fref waschosen

such that the number of overlapping particles was found to be lessthan1%ofthetotalparticlenumber.Simultaneously,theforce (Fβαc =−Fαβc )andtorque(Tβαc =OαAα× Fβαc )areappliedat of

theotherellipsoid(

β

).Theforcesandtorquesonbothparticlesare addedasexternalforcesandtorquestotheFCMinEqs.(4)and(5). Whenparticlesareveryclose,lubricationduetofluiddrainage withinthegapbetweenparticlesurfacesisnot capturedbyFCM. This limits our present study to dilute or moderately concen-trated suspensions. The repulsive model we used reproduces a softcontactbetweenparticles assumingnorebound.Asshownin

de Motta etal. (2013), when particles approach each other with a low impactStokes numbers (St<10), kinetic energyis damped byviscousdissipationinthefluidgapandno-reboundoccurs.For all the simulations under investigation in this paper, the impact Stokesnumberisbelowthecriticalvalueforactualrebound.

Manytestswereperformedtovalidatethenumericalapproach, especiallyparticlerotationundershearflowandtheinteractionof apairofparticles.Someofthemwillbediscussedbelow.

2.1. Interactionoftwospheroidsinshearflow

ThistestisdedicatedtovalidatethecalculationoftheStresslet ofanellipsoidinalinearflow,andtoverifyifthecollisionbarrier issuitable to recover theinteraction of two ellipsoids.The refer-encecasethatweuseforcomparisonisobtainedwiththe Bound-aryElementmethodofPozrikidis(2006)forStokesflow.

We consider a pair of prolate spheroids in a linear flow. The initial orientation of the particles is

ϕ

0=

π/

2 and

θ0

=0 (p is

orientedinthe flow directioninitially). Particlecenters are sepa-ratedinthestreamwisedirectionbyadistance

xbefore interac-tion.Thetwoparticleaxesarerotatingintheshear-gradient(x–y) plane duringtheinterception. We selectparticle sizea=0.1,b= c=0.05anddomainsizeLx× Ly× Lz=1.5× 1× 0.5witha

mesh-grid of 95× 62× 32 points. The particle Reynolds number in the FCMsimulationisRep=

γ

a2=0.1,where

γ

istheshearrate.

Fig.2 showsa sequence ofparticle positions andorientations that compare very well with the result of Pozrikidis (2006) ob-tained with the same initial condition. The initial separation is

x=−10a,

y=1.5a,

z=0. Snapshots of pair particle posi-tionsandorientationsare selectedatdimensionlessinstants

γ

t= 0, 2.5, 5, 7.5, 10, 12.5, 15.

Fig.3showstheevolutionoftheparticleangularvelocity



z=

d

θ/d

t andshearStressletG12togetherwiththecorresponding

evo-lutionofan isolatedparticle. TheresultsfromFCM agree quanti-tatively well withPozrikidis (2006) forStokes flow. As statedby thisauthor,theparticleinterceptionhasonlyaweakeffectonthe effectiveviscosity andthiseffect is expectedto become stronger when theparticles interactunder lubrication.The lubrication ef-fectisnot includedneitherinFCM norinthespectral boundary-elementmethodforStokesflow.

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(7)

Fig. 6. Instantaneous angular velocity along the vorticity vector in a shear flow for spheroid (oblate Ar = 0 . 5 and prolate Ar = 2 − 5 ) in a suspension with volume fraction ϕ = 0 . 058 . FCM at Re p = 0 . 1 compared with Stokes solu- tion of Jeffery. Colors distinguish four aspect ratios. Black: Ar = 0 . 5 , blue: Ar = 2 , red: Ar = 3 and green: Ar = 5 . Symbols are results from Daghooghi and Boraz- jani (2015) at Re p = 0 . 01 where ◦: Ar = 2 , : Ar = 3 and  : Ar = 5 . (For inter- pretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

here. Furthermore, we performed numerical simulations using the direct-forcing fictitious domain(DF/FD) methodfrom Yu and Shao (2007). The method is a non-Lagrange-multiplier version of the distributed-Lagrange-multiplier/fictitious-domain (DLM/FD) methodproposed by Glowinski etal.(1999). Itsaccuracy and ro-bustnessare fullydemonstratedinYu andShao(2007),in partic-ularatlowReynoldsnumbersandforneutrally-buoyantparticles. Both DF/FDand FCM are used insimilar configurations, andthe resultsaredirectlycompared.

2.2.1. Effectofaspectratio

The first test is realized at relatively low Rep (based on the

semi-major axis and the shear rate) and aspect ratio Ar that ranges between 0.5 and 5. p lays in the shear plane (x–y) and will stay there (since

ϕ

˙=0). Fig. 6 displays the angular ve-locity (



z=d

θ

/dt) obtained by the FCM with Rep=0.1

com-pared to the Jefferyorbit (Stokes flow), andRep=0.01 fromthe

workofDaghooghiandBorazjani(2015)whousedthecurvilinear immersed-boundarymethod(CURVIB).Theangularvelocity result-ingfromFCMsimulationsareclosetothetheoreticalsolutionover mostof theperiod. Fortheprolate particle,the largest deviation takes place near the maximumvelocity, when the prolate major axisisalmostalignedwiththesheardirection.Atthesame angu-larorientation,thesimulationfromCURVIBshowsevenasmaller rotation rate at that orientation, especially when the symmetric axispisperpendiculartotheflowdirection(e.g.

θ

isanodd mul-tiplesof

π

/2).

2.2.2. EffectofparticleReynoldsnumber

ThesecondtestconsiderstheeffectoftheReynoldsnumberon theellipsoidrotation.InthecaseofaprolatespheroidwithAr=2. The Reynolds number Rep ranges between 0.1 and 10. Fig. 7(a)

showsthat increasing the flowinertia tends to decreasethe par-ticleangularvelocity,especiallynearthepeaks,inagreementwith thesimulationsofDaghooghiandBorazjani(2015).

Simulations usingtheFCM andDF/FDwere runinthe caseof anoblate(Ar=0.5) andprolate(Ar=2)spheroidswiththe semi-minoraxislength equalto0.05inaunitcubiccomputational do-mainattwoReynoldsnumbers(Rep=1.0andRep=4.0).Thegrid

spatialresolutionisof64× 64× 64inFCMand128× 128× 128in DF/FD.Therotationalaxisoftheoblatespheroidisinitiallyaligned with streamwise direction whereas it is aligned with the wall-normaldirectionfortheprolatespheroid.Theparticleangular tra-jectoryshowninFig.7(b),iscalculatedfrombothFCM andDF/FD methods.Theresultsobtainedfrombothmethodsagreeverywell

Fig. 7. Effect of the Reynolds number on the instantaneous angular velocity ( ωz = dθ/d t). ( a ): a prolate spheroid ( Ar = 2 ) about the vorticity vector in a shear

flow. The volume fraction is ϕ = 0 . 058 . Jeffery orbit. FCM: Rep = 0 . 1 , Rep = 1 and Rep = 10 . Symbols are results from Daghooghi and Boraz- jani (2015) where + : Re p = 0 . 1 , ∗: Re p = 1 and ◦: Re p = 10 ; ( b ) is the comparison between FCM (line) and DF/FD (dash line): the top panel shows the symmetry axis of an oblate spheroid ( Ar = 0 . 5 ) rotating about the vorticity vector at two Reynolds numbers, black is Re p = 1 and red is Re p = 4 ( Re p is based on its semi-major axis). The bottom panel shows the symmetry axis of a prolate spheroid ( Ar = 2 ) rotating about the vorticity vector. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

forthe oblate spheroid whereas there is a smalldiscrepancy for theprolatespheroid.Ingeneral,theFCMslightlyoverestimatesthe maximumprolate particlevelocity (correspondingto the orienta-tion along the sheardirection), dueto lower resolution near the particleendpoints.

2.2.3. Particledrifttostableorbit

Thethirdtestisrelatedtotheorientationaldynamicsofan el-lipsoidwhichaxis ofsymmetryisinitially not intheshearplane neitherparallelto thevorticity axis.Jeffery(1922)suggestedthat

the particles will tend to adopt that motion which, of all the mo-tionspossibleunder theapproximated equations,corresponds to the leastdissipationofenergy.Thereforethesteadystateorbitofa pro-latespheroidalparticleinashearflow tendstowardthespinning motion (particle major axis alignedwith the vorticity direction), whereastheoblatespheroidal particlestendtowardthetumbling motion (the axis of symmetry rotate in the shear plane). After 3 decades,Saffman (1956) showedthat when theflow inertia is finite but small and the deviation from sphericity is small, the orbitat equilibrium is unchangedwith respect to the inertialess regime,forboth typesofspheroids, afinding that waslater con-firmedby SubramanianandKoch(2006). Thiswasrevisitedafter recent simulations of (Qi andLuo, 2003; Yu et al., 2007; Huang etal., 2012;Rosénetal., 2014) whofoundtheopposite:tumbling isthestableorbitforaprolate spheroid,whereasspinning isthe stable one for oblate spheroid using a wide range of aspect ra-tio 1/3<Ar<3 and particle Reynolds number Rep<15. This was

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Fig. 8. Comparison of FCM and DF/FD: orientation vector p of a prolate and an oblate spheroid in simple shear flow at finite-size Reynolds number ( Re p = 1 . 0 ). ( a ) cos θx ( p x ), cos θz ( p z ), red is for DF/FD and black is for FCM, the top panel corresponds to oblate spheroid and bottom panel for prolate particle; ( b ) projection of p on the unit sphere corresponding to ( a ) and the blue solid circle is the initial position of particle center, red cross is for DF/FD and black dot is for FCM, the top panel corresponds to oblate spheroid and bottom panel for prolate particle. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

later demonstrated theoretically by Einarsson et al. (2015a). To-gether with Dabade etal. (2016) they have shownthat for thin oblates(closetodisks)withaspectratiosAr<1/7.3,bothtumbling andspinningarestableorbits,whichmakesthedilutesuspension rheologynotuniquelydefinedintheory.

Inthistest,neutrallybuoyantprolate(Ar=2) andoblate(Ar= 0.5)particlesareconsideredandRep=1(basedonthesemi-major

axis and the shear rate). The initial orientation is

ϕ

0=

π/

4 and

θ0

=

π/

2.ThedomainsizeissetequaltoLx=Ly=Lz=5max

(

a,b

)

forbothsimulations.TheorientationorbitsobtainedwithFCMand DF/FDare compared in Fig. 8.Directional cosines px andpz of a

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prolate and an oblate spheroid are plottedin Fig. 8(a).The con-vergenceofaprolate(resp.oblate)particletowardstabletumbling (resp.log-rolling)orbitsisaveryslowprocessatlowRep=0.125.

TheconvergenceisfasterathigherRep(figureisnotshownhere).

The angulartrajectory ofthe oblate spheroidobtainedby FCM is ingoodagreementwithDF/FD.Asfortheprolatespheroid,inthe first several orbits, FCM is in good agreement with DF/FD. Then the FCM tends to be slower than DF/FD to reach a stable state. We plot theorientation vector p on theunit sphere inFig. 8(b). A nearlyclosed orbitisobserved fortheprolatespheroid afterit drifts toastabletumblingstatewhereasorientationvector pgets to aconvergence pointforthe oblatespheroid afterit drifts toa stablelog-rollingstate.

3. Suspensionflowconfiguration

The turbulent plane Couette flow is generated by two walls moving in opposite directions with equal velocities, the dimen-sions of the domain being the same as the minimal flow unit

Wang et al.(2017). Table 1 reports on all numerical parameters. The length andvelocityare scaled by wall unitsy+≡ yuτ/ν, and

u+≡ u/uτ where uτ=



τ

w/ρf,

τ

w being the wall shear stress.

Although the Reynolds number in this paper is only slightly above the onsetof turbulence(far fromthe regime ofdeveloped turbulence), we can give an estimate of the Kolmogorov scale

η

/

δ

ν≈ 1.5atthewalland

η/δ

ν=

(

κ

y+

)

0.25≈ 2.0inthecentral

re-gion(Pope,2000).AllthecasesofTable1fall intherangewhere theForce CouplingMethod captureswellthe particleresponseto flow fluctuations (Wang et al., 2017). The results with neutrally buoyantspheroidalparticlesarecomparedwithdifferent particle-to-fluiddensityratios.ThesizeratiobetweentheCouettegapand the radius of smallest spherical particles rp used for this study

isLy/rp=40.TheparticleReynoldsnumberRep

(

Vr1/3rp

)

2is

basedonlocalshearrate



=

|

du/dy

|

andtheeffectiveradius eval-uated from the particle volume reff≡ (Vr)1/3rp. Vr is the ratio of

thespheroidvolumetothereferencespherevolume(seeTable1). The value of Fref in Eq. (23) has been set to 5Fd for all simula-tions.TheStokesnumberSt≡ (2

ρ

p/9

ρ

f)Reptakesintoconsideration

the increase ofinertia due to particlevolume orto density ratio

ρ

r

ρ

p/

ρ

f. The Stokesnumber definedheregivesvalues closeto

the definitionbased on therelaxationtime ofnon-spherical par-ticles (summarized in Voth and Soldati (2017)). Forexample for

Ar=2(respAr=0.5),theStokesnumberis1.52(resp2.42)larger

than the reference particle Stokes number when calculated from

Voth andSoldati (2017) whereas the increase is 1.59 (resp 2.51) timesinthepresentstudy.

4. ResultsonsuspensionflowinpCf

4.1. Particlespatialdistribution

Fig. 9(a and b) show the spheroidal particle distribution and orientationinthe(y–z)plane.Thespheroids,likespheresasshown in Wang et al. (2017), tend to rather accumulate in the center ofthevortices,whereas thestrongejectionregions arequasi-free fromparticles.It isobservedthat atmoderateinertiaspatial par-ticle distribution is not influenced by inertia when the particle shape is spheroidal (see bottom profiles in Fig. 9(c)).This obser-vation holds forspherical particles withdensity ratiolower than 2. However, whenthe densityratio isequal to5 (see concentra-tion profilesinFig. 9(c)for10%concentration),the probabilityof findingparticlesaccumulatedclosetoCouettewallsissignificantly increased (peaks in the red dashed line). Thisis consistent with what has beenobserved forinertial pointwisespherical particles inturbulence(BalachandarandEaton,2010). However,thisis dif-ferent with inertial finite-size particles in pressure-driven

turbu-Fig. 9. ( a ) and ( b ) show the velocity magnitude and particle distribution with its orientation of C500-5-05-1 (oblate) and C500-5-2-1 (prolate particles), respectively. The figures are chosen when the large scale streaks are the strongest. The isocon- tours represent the instantaneous velocity magnitude in y –z slice. ( c ) Particle con- centration profiles for different particle shapes and density ratios. (For interpreta- tion of the references to color in the text, the reader is referred to the web version of this article.)

lence (Fornari etal., 2016), who found the local volume fraction increasesdrastically atthe centerline.Particle spatial distribution ishardly affectedbytheshape(atleasttotheaspectratiowe in-vestigated),asitwasalreadyobservedforpointwiseinertial non-sphericalparticlesinachannelflow(Mortensenetal.,2008; Mar-chiolietal.,2010).Indeed,foralower aspectratio(Ar=1/3)and volumeconcentrationlowerthan10%,theconcentrationprofileof oblateparticles(NiaziArdekanietal., 2017)issimilartospherical particles(Picanoetal.,2015)inchannelflow.

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Fig. 10. ( a ) Wall-normal profiles of the particle orientation angles θand ϕ(projections of the p vector). The angle between the symmetry axis ( p or −p ) with the positive axis ( + x or + z) is used. ∗and ◦ stand for oblate and prolate particles respectively. ( b –d ): Profiles of the particle absolute angular velocity in streamwise ( |ωx| ), wall-normal ( |ωy| ) and spanwise ( |ωz| ) directions normalized by the average shear rate γ. The shape effect is shown for y / h < 1 and density ratio effect for y / h > 1 on the same graph.

4.2.Particlerotationdynamics

The rotation dynamics of spheroids in laminar plane Cou-ette flow was already discussed in Section 2.2. As shown by

Rosénetal.(2014),tumblingisthestablerotationorbitofa pro-latespheroid,whereasspinningisthestableregimeforoblate par-ticles. Also in Section 2.2, we have shown that oblate and pro-latespheroidsbothtendtomigratetowardsthecoreofalaminar pCf,no matter iftheinitial orientation oftheir symmetry axis is alignedwiththe vorticity (along z, the spanwisedirection) or in theshearplane(x–yplane).

Consider a spheroid withtheunit vector p along the symme-tryaxis. The projection ofp onthe Cartesian frame of reference is given in Eq. (25), where the angles

ϕ

and

θ

are defined in

Fig.10(b).Theparticleismainly tumblingwhen

ϕ

iscloseto90° orspinningif

ϕ

iscloseto0.Theangle

θ

indicatesifthe symme-tryaxisisratherorientedinthestreamwisedirectionoralongthe sheardirection.

In Fig. 10(a), we show the inclination angleof the symmetry axis instead of using cosine to avoid confusion in the averages sincethecosinefunctionisnotlinear.Foroblatespheroids,both

θ

and

ϕ

arerelativelyhigh,whichmeansthatoblatespheroidstend tomovewiththesymmetryaxisalmostparalleltothewall-normal directionespecially inthenearwallregion,indicating thatoblate particleshavemoretumblingactivitythan spinning.Thisis simi-lartooblate spheroidswithAr=1/3inturbulentpressure-driven

flow(NiaziArdekanietal.,2017).However,prolatespheroidstend rather to align their major axis in the flow direction especially closetothewallandtotumble(

ϕ

islargeand

θ

issmallin aver-age).Thisisconsistentwiththeobservationsinturbulent pressure-drivenflowbyDo-Quangetal.(2014).

The threecomponentsofparticleabsoluteangularvelocityare shown in Fig. 10(b–d). Near the walls, the particles rotate pre-dominantlyalong the spanwise directiondueto the meanshear. Inthecoreregion thedominantcomponentisthe rotationalong thewall-normaldirection(

ω

y).Thisis dueto thegradient ofthe

streamwisevelocityinspanwisedirection(

u/

z)whichisformed by the gradient of streamwise velocity between low-speed (neg-ativeu) and high-speed streaks (positive u) in spanwise direc-tion.Forboth typesofspheroids, theparticlerotation rateis

de-creased in the spanwise direction whereas it is increased in the othertwo directions,whencompared tothesphericalparticle ro-tation rate. In the core region, the rotation ratein the three di-rections arenon-zero,for bothprolate andoblatespheroids. This indicatesa kayakingtypeofmotionsimilartowhat hasbeen ob-served forprolate spheroidsinturbulent pressure-driven flow by

Do-Quang et al. (2014). For spherical particles, the density ratio andconcentration(from5to10%)havebothanegligibleeffecton therotationalratesinthethreedirections.

Thekayaking typeofrotation inthecoreregioncan yield ho-mogeneous collisions.The threecomponents oftheparticle colli-sionforcesare plottedinFig.11(a),wherethey were averagedin thehomogeneousstreamwiseandspanwisedirections.Inthecore region,thedominantcollisionforcebetweensphericalparticlesis in streamwise direction which is both due to particle collisions in (x–z) plane dueto high- and low-speed streaks in that plane, and to the meanshear in the (x–y) plane (which is non-zero at the Couette center).Thiseffect isenhanced by particle inertiaas shown in Fig. 11(b). Near the Couette walls, the dominant com-ponent isin the wall-normaldirectionwhich isdue tocollisions occurring when particles are swept towards the walls. Fig. 11(c andd) show the collision forcesscaled by Fref.These forces give

an indicationon themomentum transferby particle interactions. It is clear to see that at equal volume fraction, all the collision force componentare much stronger when the particles are not spherical.

4.3. Probabilitydensityfunctionofvelocityfluctuations

Tocharacterize the particle transport by fluid flow structures, we focus on the buffer layer region, 0.15<y/Ly<0.5 (10<y+<

40), where the regeneration cycle governs the flow behavior (Jiménez, 2013). The probability density functions(PDF) of parti-clevelocityfluctuationshelpstodescribeifstrongandweak fluc-tuations of particles are similar to that of the fluid. In Fig. 12(a and b), the PDFs of streamwise and wall-normal velocity fluctu-ations are shown, for simulations realized withdifferent particle shapesanddensities.EverypanelcomparesthePDFoftheparticle velocityfluctuationswith,on onehand,thefluidsurroundingthe particles in thetwo-phase simulations (which isnot expected to

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Fig. 11. Profiles of the repulsive force components ( F i ) in directions x black, y blue and z red. In ( a ) simulation results with spherical particles C500-5-1-1(s) are com- pared to the simulations with prolate particles C500-5-2-1. In ( b ) the effect of inertia is shown with cases C500-10-1-0 and C500-10-1-5. Corresponding to ( a ) and ( b ), ( c ) and ( d ) are the ratio between the repulsive force components with F d . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Table 1

Parameters of the numerical simulations. The Reynolds number of the singe-phase flow is Re b ≡ U w h/ν= 500 where U w = 0 . 5 is half of the relative wall velocity and h = L y / 2 is half of the Couette gap. r p = L y / 40 is the radius of the reference sphere and A r is the aspect ratio between symmetry axis with rotation axis. The Stokes number is low near the Couette center (where the average shear rate is low) and maximum near the walls. Simulations start with a random initial seeding of particles which reach a steady statistical distribution after half period of the regeneration cycle. Then, statistics are formed over ∼ 500 time units ( h / U w ).

Domain size: L x × L y × L z = 0 . 88 π× 1 . 0 × 0 . 6 π

Case (%) Ar ρr Vr L+ y Reτ Rep ( max ) St ( max ) Line type Nx × N y × N z = 30 × 86 × 32 Single-phase – – – – 81 40.2 – – + Shape effect Nx × N y × N z = 182 × 66 × 128 C500-5-1-1(l) 5 1 1 8 80.6 40.3 12.5 2.78 Nx × N y × N z = 280 × 100 × 256 C500-5-1-1(s) 5 1 1 1 80.6 40.3 3.75 0.83 C500-5-05-1 5 0.5 1 4 80.1 40.0 6.9 1.53 C500-5-2-1 5 2 1 2 81.4 40.2 3.5 0.78 Inertia effect Nx × N y × N z = 382 × 134 × 256 C500-10-1-0 10 1 1 . 25 10 −3 1 84.7 42.4 3.75 1 . 1 10 −3 C500-10-1-1 10 1 1 1 84.1 42.1 3.75 0.83 C500-10-1-2 10 1 2 1 85.1 42.5 3.75 1.67 C500-10-1-5 10 1 5 1 87.6 43.9 3.75 4.15

besignificantlydifferentfromtheparticlevelocity)andthesingle phasefluidflowfluctuationsontheotherhand.

The PDF of the wall-normal velocity fluctuations are almost Gaussian withzeromeanandsymmetricforallcasesreportedin

Fig.12.ThePDFoftheparticlevelocityfluctuations,andthatofthe surroundingfluidinthetwo-phasesimulationsareverysimilarto thesinglephasecase,withaslightreductionofthepeakatzero. Theskewnessofthisdistributionwiththewall-normalvelocityin the range−0.1<

v

/Uw<0.1 isalmostzeroforsingle-phaseflow

(theskewnessisequalto0.034).Thisindicatesthattheintensities ofwall-normalvelocity fluctuationsininward andoutward direc-tionsarestatisticallyequalinthebufferlayerofturbulentpCf.

The PDF of the streamwise velocity fluctuations are bimodal, withone velocity peak positive velocity andanotherone at neg-ativevelocityrelatedtotheejectionevents.Theintensityof nega-tiveuinlow-speedstreaksisstronger(butwithalower probabil-ity) thanthepositive u inhigh-speedstreaksinthebuffer layer,

whichissimilartowhathasbeenobservedinturbulent pressure-drivenflow(Kimetal.,1987).Particlesincreaseslightlythe proba-bilityofnegativefluctuations,andshiftthepeakofpositive veloc-ityfluctuationstowardsmallervalues.

4.4.ResidencetimeofparticlesinLSVs

In homogeneous and isotropic turbulence, the time scale of thecoherent motionofparticles is comparableto thelarge-eddy turnovertime (Bhatnagar etal., 2016). The most energetic struc-tures of a turbulent plane Couette flow in the regime of weak turbulence,consist inpairsofcontra-rotatingLargeScale Vortices whichsizeiscomparabletotheCouettegapandlargescalestreaks (Komminahoetal.,1996).TheLargeScaleVortices(LSVs)carry sig-nificantfractionofturbulentkineticenergy(Pirozzolietal.,2014). InWangetal.(2017),we foundthatthelightestparticlestend tobe trapped in the LSVswhereas heavy particles tend tomove outward. The outward motionofparticles by centrifugation from

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Fig. 12. Probability density functions of velocity fluctuations in the buffer layer ( 10 < y + < 40 ). Symbols represent the particle velocity (circles) and the single- phase flow fluctuations (crosses). In addition, the PDF of fluid fluctuation in a shell of thickness 0.5 a around the particle surface is represented by lines which color leg- end according to Table 1 ). Both streamwise and wall-normal velocity components are shown, and the PDFs are averaged over ∼ 400 time units. ( a ) shows the effect of particle shape and ( b ) the effect of particle-to-fluid density ratio. (For interpre- tation of the references to color in this figure legend, the reader is referred to the web version of this article.)

the flow coherent vortices influences particle dispersion, leading ornottopreferentialaccumulation(Marshall,1998).Wehave cal-culatedthe particle residencetime in a vortex (mainlythe large rolls) using the temporal evolution of the wall-normal position. InFig.13(a),thewall-normal positionof oblateandprolate neu-trallybuoyantspheroidsare plottedovertime, inaddition tothe spherical particle trajectories. The spheroids behave qualitatively likeneutrally-buoyantsphericalparticles,withaclearperiodic os-cillatorymotionbetweenbothwalls. Twodistinct motionscanbe observedintheparticletrajectories:arotationinasingleLSVfor instancefromitoiiandsoon(dashedlineofFig.13a),anda ro-tationofa particlein aLSV followed by itstransfer to theother counter-rotatingLSVfromiiitoiv.

In orderto havea quantitative measureofthe residencetime inonevortex,wecalculatedEq.(29)thetemporalauto-correlation oftheparticlewall-normalposition.

Ryy

(

t

)

=

yp

(

t

)

yp

(

t+

t

)

y2prms

(29)

where yp is the fluctuation of the wall-normal particle position withrespect to theaverage value, which wasverified to be hin the simulations (on average,the particles scan all the simulation domainequally).

Fig.13(b andc) show thetemporal auto-correlationfunctions. Theauto-correlationfunctioncharacterizestheparticlelargescale oscillatory motion. It becomesnegative when theparticle passes from one half ofthe Couette gapto the other, then almost zero when the particle leaves the large scale vortex. In Fig. 13(b and c) thereare twosetsofstatistics.The particlesthatwere trapped inaunique LSVwere computedinoneset,whereas theparticles transferred fromoneLSV totheother were computedinanother set.

The meancharacteristicresidencetime ofparticles inasingle LSV(setI)is ∼ 100Uw/hand∼ 150Uw/hforparticlesthatare

trans-ferredfromoneLSVtoanotherLSV(setII).Thesmallerandlighter spherical particles have shorter periods in a single LSV whereas they have longer periods when they move from one LSV to the other.TheresidencetimeofparticlesinasingleLSVapproximately coincides withthe period of the regeneration cycle observed by

Hamiltonetal.(1995),whichindicatesthestrongrelationbetween transportprocessoffinite-sizeparticlesandthethreesub-stepsof theregenerationcycleinturbulentplaneCouetteflow.

4.5. Correlationofparticledistributionwithflowstructures

The Reynolds shear stress contributions are classically di-vided into four quadrants: Q1(+u’, +v’), Q2(−u’, +v’), Q3(−u’, −v’), Q4(+u’, −v’) where + means positive and m̧eans negative values of the velocity fluctuations. Beside large scale vortices, the x-independent streaks contribute the most to turbulent ki-netic energy.Thex-independentstreaks predominantlyconsist in Q2

(

−u,+

v



)

and Q4

(

+u,

v



)

regions (ejection and sweep re-spectively) which makethe largest contributions tothe Reynolds shear stress. They are offset from the interaction quadrants, Q1

(

+u,+

v



)

and Q3

(

−u,

v



)

, which are counter-gradient type motions (Wallace, 2016). The energyof thismode decreases dur-ingitsbreakdowntox-dependentstreaks(wavystreaks).Weshow here that the accumulation of particles in the sweep and ejec-tionregions iscorrelated, intime,to theevolution ofthestreaks and therefore to the regeneration cycle. For the temporal evolu-tionofthestreakymotion,itisrepresentedbythemodalanalysis of the flow fluctuating energy. The Fourierdecomposition of the energyoverstreamwiseandspanwisedirections,asintroducedby

Hamiltonetal.(1995),iswrittenasfollows:

M

(

kx=m

α

,kz=n

β

)



Y2 Y1 [uiui

(

m

α

,y,n

β

)

]dy



1/2 (30)

whereY1 andY2 standfortheintegrationboundsinwall-normal

direction,(

α

,

β

)arethefundamentalwavenumbersinstreamwise and spanwise directions definedas (2

π

/Lx, 2

π

/Lz), andm andn

are integers. Any turbulent structure can be represented by one mode (m

α

,n

β

). Forinstance, the mode (0, n

β

) with n =0 is an x-independent structure and the mode (m

α

, n

β

) with m =0 is thex-dependentstructure(e.g.streaksconfinedinthestreamwise direction). The temporal evolution of the x-independent streaks (mode (0,

β

)) is displayed in blue in Fig. 14(a–f), for suspen-sionflows withdifferent particleshapesanddensities. Moreover,

Fig14presentsthetemporalevolutionofthepercentageof parti-clescontainedwithin thestreakyregionsQ2andQ4.The calcula-tion wasrealizedonly in thebuffer layers where the sweepand ejection events are strong, i.e. nearthe wallsat 10<y+<40. It canbe noticedthat morethan halfoftheparticles arecontained withinthesweepandejectionregions.Thefluctuationsintimeof particleconcentrationintheseregions arein-phasewiththe fluc-tuationof energycontainedin thex-independentstreaks. Thisis

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Fig. 13. ( a ) Temporal evolution of the wall-normal position of a particle in turbulent pCf at Reynolds number 500 for different particle shapes and densities: ρr = 1 . 25 10 −3 ; ρr = 5 and Ar = 0 . 5 ; Ar = 2 . ( b ) and ( c ) show the temporal auto-correlation functions of the wall-normal particle position fluctuation. The line style of ( b ) and ( c ) is shown in Table 1 . Set I noted in these figures contains the statistics of particles trapped in one large scale vortex and set II contains particles transferred from one LSV to the other. The criteria used to attribute each particle to set I, set II or neither of the two sets, is based on tmin at which the minimum of R yy corresponding to each particle occurs: if tmin U w / h < 60, particle belongs to set I; if 60 < tmin U w /h < 100 , particle belongs to set II. Overall, 10–20 percents of the total number particles belong to each set.

Fig. 14. Temporal evolution of mode (0, n β) representing turbulent kinetic energy contained in the x-independent streaks (blue lines), and of the local particle percentage (ratio of particles in Q2 and Q4 to the total number of particles) in both ejection and sweep event regions (red dot lines). ( a, c, e ) show the particle shape effect with the cases from top to bottom: C50 0-5-1-1, C50 0-5-05-1 and C500-5-2-1. ( b, d, f ) show the effect of particle density, using from top to bottom: C50 0-10-1-0, C50 0-10-1-2 and C500-10-1-5. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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particularlyevidentatlowStokesnumbers(C500-5-2-1and C500-10-1-0).Inthesamecontext,wefoundthatthepercentageof par-ticlescontainedintheQ1andQ3regions(notshownhere)is cor-relatedwiththe flow circulation,which isout-of-phase withthe timeevolutionofx-independentstreakenergy.

5. Conclusion

TheForceCouplingMethodhasbeenusedforthesimulationof spheroidalparticles.Themethodhasbeenvalidatedforthespecific casesofprolate andoblate ellipsoids. The configurationof a sin-glespheroidalparticlein shearflowhasproven thatFCM is suit-abletohandlethedynamicsoffinite-sizeparticles.Theagreement withtheoretical predictionson thestableorbitisgood andfinite Reynoldseffectshavebeencomparedtoreferencedatasimulation fromliterature.Then,resultsonapairofspheroidsinteractingina shearflowhavebeensuccessfullycomparedtoreferencenumerical datafromboundaryelementmethodforStokesflows.

We have investigated the spatial distribution and orientation statisticsoffinite-sizeparticlesinturbulentplaneCouetteflowfor differentinertiaandshapes.Whenparticlesareneutrallybuoyant, theiroveralldynamicsisdrivenbythelargescaleturbulent struc-turesofthefluidflowuptoStokesnumbers(St≈ 5).However,we haveobservedthatinertial sphericalparticlestend toaccumulate moreinthenearwallregion(for10% volumeconcentration)than spheroids.Thiseffectcanbeattributedtoenhancedcentrifugation ofdense particles by longitudinal vortices and more diffusion of non-sphericalparticlesduetotheirrotationaldynamicsand hydro-dynamicinteractionswiththewallsandamongthesuspension.

Because of the rich orientation dynamics, the flow statistics of two-phase flows might depend on the deviation of particles fromsphericity,ina non-trivialway.Regardingrotational dynam-ics,manyfeatures ofourstudyareconfirmingwhathasbeen ob-servedina turbulentchannel flow.Oblateparticles tendtomove withthe symmetryaxisalmost parallelto thewall-normal direc-tionespeciallyinthenearwallregionwheretheyhavemore tum-bling activity than spinning. However, prolate spheroids tend to aligntheirmajor axisintheflow directionespecially closeto the wall.Nearthewalls, theparticles rotatepredominantlyalongthe spanwisedirectionduetothemeanshear.Inthecoreregion,the dominantcomponentistherotationalong thewallnormal direc-tionwithakayakingtypeofmotionyieldingtodirectinteractions amongspheroidsenhancingthespreadingofthesuspension.

The typical residencetime ofa singleparticle ina largescale vortex is equal to the characteristic time scale of the regenera-tioncycleofturbulence.Atequivalentvolume fraction,the parti-cledistributionofspheroidsintheflowisnotsignificantlyaltered by their shape (prolate oroblate). Particlesare on average more presentinsidethelargescalestreamwisevortices,comparedtothe x-independentstreaks.However,instantaneousparticlespatial dis-tributiondependsonthesuccessivesteps ofturbulence regenera-tioncycle.Theejection regionsare seededby moreparticles dur-ingstreak formation(when the x-independentstructures are en-ergetic)andtheyreleaseparticlesduringstreakbreakdown(when theenergyofx-independentstructures isreduced).Duringstreak formation(resp.breakdown),theflow circulationdecreases (resp. increases),andtheQ1regionmainlylocatedinsidelargescale vor-tices loses (resp.gains) particles, migratingtoward (resp. coming from)largescalestreaks.

Thenextstepstowardsthenumericalmodelingofreal suspen-sionswouldbetoincludetheeffectsofpolydispersityinsizeand shapesforgivenmaterialpropertiesofparticlesandthenevaluate theoverallresponseofthesuspensioninordertoproposeeffective rheologicalpropertiesforengineeringapplications.

Acknowledgments

Thisworkwasgranted accessto theHPCresourcesofCALMIP undertheallocation2016 and2017-P1002.G.Wangwouldliketo thankDr.WenchaoYuforhelpfuldiscussionsonthedataanalysis. Z. YuacknowledgesthesupportfromtheNationalNaturalScience FoundationofChina(GrantNo.91752117).

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Figure

Fig.  1. Schematic representation of the repulsive force and torque at the center of a pair of ellipsoids if their surface is closer than a distance of 0
Fig.  6. Instantaneous angular velocity along the vorticity vector in a shear flow for spheroid (oblate Ar = 0
Fig.  8. Comparison of FCM and DF/FD: orientation vector p of a prolate and an oblate spheroid in simple shear flow at finite-size Reynolds number ( Re  p  = 1
Fig. 9 (a and b) show the spheroidal particle distribution and orientation in the ( y –z ) plane
+5

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