Clusters and coherent voids in particle-laden wake flow

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ContentslistsavailableatScienceDirect

International Journal of Multiphase Flow

journalhomepage:www.elsevier.com/locate/ijmulflow

Clusters and coherent voids in particle-laden wake ﬂow

Zhaoyu Shi

^a

, Fengjian Jiang

^b^,^∗

, Lihao Zhao

^c^,^a

, Helge I Andersson

^a

aDepartment of Energy and Process Engineering, Norwegian University of Science and Technology, 7491 Trondheim, Norway

bDepartment of Ships and Ocean Structures, SINTEF Ocean, 7052 Trondheim, Norway

cAML, Department of Engineering Mechanics, Tsinghua University, 10 0 084 Beijing, China

a rt i c l e i nf o

Article history:

Received 25 January 2021 Revised 10 April 2021 Accepted 26 April 2021 Available online 3 May 2021 Keywords:

Coherent voids Smooth edges Cylinder wake ﬂow Numerical simulation

a b s t r a c t

Inertialpointparticlessuspendedinatwo-dimensionalunsteadycircularcylinderﬂowatRe=100are studiedbyone-waycoupledthree-dimensionalnumericalsimulations.Thestrikingclusteringpatternin the near-wakeisstrongly correlatedwiththe periodicallyshed Kármán vortexcells.The particlesare expelledfromthevortexcoresduetothecentrifugalmechanismandcoherentvoidsencompassingthe localKármáncellsarethereforeobserved.Theparticleclusteringattheupstreamsideofeachvoidhole formasmoothedge,wheretheparticlevelocitymagnitudeisconsistentlylowerthanatthedownstream edgeofthevoids.Thetrajectoriesoftheseparticlesoriginatefromthesideofthecylinderwherethesign ofvorticityisoppositetothatofthevortexencompassedbythecorrespondingvoidhole.Theparticles areseentodeceleratealongasubstantialpartoftheirtrajectories.Particleinertiaisparameterizedby meansofaStokesnumberSkandsmoothedgesaroundthevoidholesstillexistwhenSkisincreased, althoughtheirformationisdelayedduetolargerinertia.Increasinginertiacontributestoadecouplingof theparticleaccelerationfromtheslipvelocity,whichalmostcoincidedatSk=1.

ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/)

1. Introduction

Particle-laden ﬂows around natural obstacles and man-made bluff bodies are frequently encountered in nature and industry.

Even though the entering flow is regular and smooth, the fluid motion in the wake of the obstacle becomes more complex and comprises a variety of characteristic length and time scales. In- sightinparticletransportationanddispersionmechanismsincom- mon bluff-body wakes maypromote mitigation ofindustrial pol- lution andabetter understandingofnaturalphenomena, such as scouring around offshore wind turbine foundations and clogging ofsteamgeneratorsinnuclearpowerplantsatlargescales,aswell as enhancing the mixing performance of microfluidicreactors at small scales. Particularly, microfluidicsystems are commonly ap- pliedinthelab-on-a-chipbiologicaldevices,whereintheflowsare almost always laminardefinedby a tinypore scale.Anumber of experimental andnumericalstudiesinvestigatedthemicroparticle suspensions,e.g.particlemigrationandtrappinginmicrochannels (Leeetal.,2010;Gijsetal.,2010;DressaireandSauret,2017).The mixingordemixingprocessescanbemanipulatedwiththefunda-

∗Corresponding author.

E-mail address: Fengjian.Jiang@sintef.no (F. Jiang).

mentalinsightsofgeneralparticle-ladenlaminarvortexﬂows.The currentunderstanding ofthebehaviorofinertialparticlesevenin thesimplestbluff-bodywakes isincompleteandnoclueexiststo howtheremarkableparticleclusteringpatternsarise.

Besidestheoccurrenceofparticle-ladenwakeflowsinavariety ofdifferentsettings,particleadditivesplayacrucialroleindyeand smokeflowvisualizations. Cimbalaetal.(1988)utilizedasmoke- wirevisualizationtechniquetoexplorethewakeformedbehinda circularcylinder atrelatively low Reynolds numbers.Theyaimed toinvestigatethefar-wakebehaviorbutrealizedthatparticlesre- leasedfromasmoke-wireretainedacellularpatternlongafterthe characteristicKármánvortices havelargely diffused.Thisobserva- tionwasinterpretedasaninertiaeffectofparticlessincesmokein airiscomposedoftinyaerosol-typeparticleswhosemassissignif- icantlygreaterthanthemassofthesurroundingairmolecules.The persistentsmokepatternshowedthatdiffusionofsmokeaerosols isextremely slowandtheeffectiveSchmidtnumber,i.e.theratio ofviscousdiffusivitytothediffusivityofsmokeparticlesinair,is ordersofmagnitudelargerthanunity.Thetinysmokeparticlesdo thereforenotactaspassivetracersandwebelieve thattheeffect ofparticleinertiacanbe understoodintermsofa particleStokes numberratherthanbeinglumped intoan effectiveSchmidtnum- ber.TheStokesnumber(Sk)isadoptedasameasureofparticlein- ertia,typicallydefinedastheratiobetweentheparticlerelaxation time

τ

^p^and^acharacteristictimescale

τ

f oftheﬂow.

https://doi.org/10.1016/j.ijmultiphaseﬂow.2021.103678

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etyofconﬁgurations.

Experimentsby Yang etal.(2000)showed snapshotsofparti- cledistributionsfortwodifferentparticlesizes,forwhichparticles withSkoforderunitywere observedtopresentthemostdistinc- tive pattern. In another perturbation study withthe Stokes drag forceparameterizedasasmalldisturbance,Burnsetal.(1999)dis- cussed the presence of a periodic attractor in a Kármán vortex street modeled aspoint vortices. The so-called dynamical attractor appeared asan indicator ofparticlefocusing. Haller andSap- sis (2008) extended the scope to finite-size particles and con- structed a slow manifold to describe the asymptotic attraction along trajectories of low-Sk particles. Burger et al., (2006) com- paredthe one-wayandtwo-waycoupledresults inan oscillating hot-air circularcylinderwakeflow atlaminarconditionbydirect numerical simulations (DNSs). Afterwards, Yao et al. (2009) and Luoetal.(2009)alsoperformedDNSsofacircularcylinderwake flow andpresented clusteringpatternsina better-resolved wake.

Theflowfieldexhibitedvorticesofdifferentscales,determinedby Reynolds number Re, which significantly affected the way of instantaneous particle clustering. A subsequent work (Zhou et al., 2011) additionallyshowed thetemporal development ofthe particle distributions at a benchmarking Re=100. Although the fo- cus of Haugen and Kragset (2010) and more recent work by Aarnes etal. (2019) wason particle-cylinderimpaction, glimpses ofparticledispersion patternsinthenearwakeatRe=100were alsopresented.

Numerous efforts havebeen made in explorations of the underlying physical mechanisms of particle clustering, for instance in the paradigmatic homogeneous isotropic turbulence (HIT) and wall-bounded ﬂows. The centrifugal effect induced by local vortices was recognized to interpret the tendency of particle clustering in strain-dominant regions (Squires andEaton, 1991). This strain-vorticity-selection mechanism also applies in mixing-layers andshearﬂows(EatonandFessler,1994).Previousnumericaland experimentalobservationsinHITshowedthatthemaximumclus- teringappearedataStokesnumber(Sk_η)basedontheKolmogorov timescale

τ

ηclosetounity(WangandMaxey,1993;Alisedaetal., 2002;Salazaretal.,2008;Petersenetal.,2019).

The sameoptimalStokes numberforthe maximumclustering wasalsoreportedforchannel flow(Kulicketal.,1994) andwake flowbehindathickplane(Tangetal.,1992;Yangetal.,2000).The explanationofvortexejectionatthedissipativescalescanalsobe extended to the inertialrange andaffected by self-similar multi- scaleeddies(YoshimotoandGoto,2007;Becetal.,2007).However, thecentrifugalmechanismastheonlyeffectfailsforSk_ηsubstan- tiallylargerthanunity.GotoandVassilicos(2008)proposedanal- ternative explanation for particles withmoderateStokes number, i.e.thesweep-stickmechanism.Thisprocesssuggeststhatparticles preferentiallyconcentrateinregionswithlowfluidaccelerationin- stead of low vorticity both in 2D and3D (Coleman and Vassili- cos, 2009). Additionally, Bragg and Collins, (2014) carefully com- paredother potential factorsaccountingforthe clusterformation atabroad-scalespectrum,inwhichastatisticalmodel(Zaichikand Alipchenkov,2007)basedonradialdistributionfunctions(RDFs)in

(TGV)ﬂow.Onemaywonderwhethertheabovementionedmech- anisms are acting also in these and other non-turbulent ﬂows.

Tang et al. (1992) proposed a stretching-folding mechanism, but only formixing-layer flows, in whichparticles tend toaggregate inthethinregionsoutliningtheboundariesoftheLSS,regardedas stretchingprocess.Thefoldingprocessisrelatedtothevortexpair- ing (Wen etal., 1992). This interpretation isstill ambiguousand notapplicabletocylinderwakeflows.RajuandMeiburg(1997)indicated the action of centrifugalejection in analytical models of a 2D solid-bodyvortex anda point vortex. The centrifugalforce pointingoutwardsfromthevortexcoreopposesotherforcesuntil an optimal ejectionrate isreached. Candelier etal. (2004)high- lighted another dissipative effect of the history force on parti- cletrajectorieswhichinfluencedtheejectionrateinsimpleflows.

DaitcheandTél(2011, 2014)improved theanalysisin a modeled Kármán vortexflowandexplicitlypointedout thesuppressionof concentration orattractiondueto thehistoryforce forfinite-size inertialparticles(e.g.bubbles,oildroplets).However,thesestudies on clusteringmechanisms are based on simplified and2D flows, the explorations on real-life 3D vortex flows are still scarce. Re- cently,Jayarametal.(2020)studiedparticleclusteringinanevolv- ingTGVflow andobservedclusteringinstrain-ratedominatedar- easaftertheLSSshadbeenbroken downtosmall-scaleeddiesso that aneffective Stokesnumber,based onthe viscousdissipation timescale,becameoforderunity.

Toobjectivelycharacterizeparticleclustering,quantitativemea- surements of clustering have been escalated from box counting (Alisedaetal.,2002),correlationdimension(Tangetal.,1992;Bec etal., 2007), RDFs (ZaichikandAlipchenkov, 2007; Irelandet al., 2016)toVoronoï diagrams(Monchauxetal.,2010).Thetwolatter onesare currently the mostwidely used approaches, of which a RDF isspatially-averaged foreach manually selectedlength-scale and thus fails to provide information of individual clusters. By contrast, information about instantaneous local clustering is ac- cessible via Voronoï diagrams forwhich the number of particles should be carefully chosen in terms of statistical analysis. A re- centexperimental HIT studybyPetersen etal.(2019) foundhigh consistency between the two approaches. Mohammadreza and Bragg (2020) extensively applied Voronoï cells to investigate the effectsofReandgravityonclusteringinHITusingDNS.Moreover, Shi et al. (2020) applied Voronoï diagrams to analyze a striking bow-shockclusteringinthevicinityofacircularcylinder.Alterna- tivedynamicaltechniques,suchasﬁnite-timeLyapunovexponents (Bec et al., 2006a;Jacobs andArmstrong,2009; Daitcheand Tél, 2014) andaccelerationanalysis(Bec etal.,2006b),which relyon particletrajectories,aremainlyeffectiveforparticleswithmodest inertia,i.e.smallSk.Anotherapproachisbasedontheapplication ofanEulerianvelocity-accelerationstructurefunction(Gibertetal., 2012), showingthe tendency of clusteringin high-strain regions.

The reviewarticle byMonchaux etal.(2012) provides acompre- hensiveoverviewof thedifferenttechniques usedto characterize particleclustering.

Themotivationofthepresentstudyistoinvestigateclustering ofheavy inertial particlesin thenearwake ofa circularcylinder

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ﬂow at Re=100. Ofparticular interest is to providean in-depth exploration of a striking phenomenon, namely the formation of coherent voids, andhow andwhy these voids are bounded by a smoothedgeconsistingofdenselyconcentratedparticles.Thisphe- nomenonisnotanumericalartifactbutarealityofpracticalinter- est, forinstancewithrespectto homogenizationofparticleaddi- tives.Thephenomenonhasbeenobservedbeforeinsomenumeri- calsimulationstudies(Yaoetal.,2009;HaugenandKragset,2010;

Aarnes et al., 2019), but beenleft almost unnoticedandwithout beingscrutinized.Thereforeacoupleofquestionsremaintobean- swered: why andhowdo particle clusteringappearin a laminar Kármán vortex street? How do the characteristic coherent voids form and evolve in spaceand time?How are the particle structures correlated withthe local vortices? These questions will be addressedinthe presentpaperwhichcanbe considered asa se- queltothedifferentlyfocusedpaperbyShietal.(2020).

The paperis organizedasfollows: themathematical problem, thesimulationmethodandnumericaldetailsareﬁrstdescribedin Section 2.ResultsforSk=1particlesarepresentedanddiscussed inthreesub-sections.The travelingfeatures ofthecoherentvoids andtheirsmoothedgesareconsideredinsub-sections3.1and3.2, respectively,whereasthekeyobservationsmadefromparticletra- jectories are addressed in 3.3, in terms of both group and individual trajectories.Finally,inordertoelucidatethecrucialrole of particle inertia, comparisons of results obtained forsome differ- entStokesnumbersare madeinSection4,beforeconclusionsare drawninSection5.

2. Problemformulationandnumericalmethods 2.1. Equationsofmotions

πμ

^a

β

^[^uf@p−up

(

^x^p^,^t

)

^]^, ⁽¹⁾

which results from the slip velocityUs=u_f@_p−up(^xp,t)^. ^Herein, u_f@_pisthelocalﬂuidvelocityatparticlepositionx_p.Theempirical correctionfactor

β

=C_DRep/24isunityinthe StokesianlimitRep= 0,whereRep=2a

||

^u^p−u_f@_p

||

/

ν

^is^the ^particle^Reynolds^number.

Themodifiedfinite-Re_pdragcoefficientC_DisafunctionofRe_pes- timatedas

CD= 24

Rep

(

¹+0.15Re⁰_p^.⁶⁸⁷

)

+ 0.42

1+4.25×10⁴Re⁻_p¹^.¹⁶ (2)

Table 1

The average particle Reynolds number Re pand peak value Re p,mfor each Sk at time t ^∗.

Sk 1 3 5 8 12 16

Average Re p 0.092 0.258 0.472 0.718 1.001 1.323 Maximum Re p,m 0.833 1.859 2.834 3.796 4.797 5.798

forRep<3×10⁵ (Cliff etal., 1978,Chapter5).The empiricaldrag lawEq.(2)iswell-justifiedunderasteadyuniformflow atmod- erateRep(from1upto10³)(BagchiandBalachandar,2003).From apractical perspective, Eq.(1)isalsowidespread inmostknown studies. The simplified Maxey-Riley (M-R) equations (Maxey and Riley,1983)giveustheLagrangianparticleinformationassuch, ap=dup

dt =C_DRep

24

τ

^p

⁽

^u^f@^p⁻^u^p

⁾

^,

dxp

dt =up. (3)

^is

the particle response time to changes in the local ﬂuid velocity and

τ

f=D/U₀ denotes the nominaltime scale of the ﬂow ﬁeld.

Analternativetimescaleofthefluidfield associatedwithKármán vortices can be defined asthe inverseof the maximumabsolute spanwisevorticity,i.e.

τ

fv=1/

| ω

z

|

^.^We^believe^that^this^vorticity-

based time scale is physically more relevantthan the time scale ofthevortexshedding.An’effectiveStokesnumber’basedonthis vorticity-timescaleisthusdeﬁnedasSke=

τ

p/

τ

fv=

| ω

z

| τ

p,where

| ω

^z

|

^decaysdownstream.Inotherwords,forsuchsingledominat- ing timescaleﬂow, it isonlya scalingmatter whethertopresent the results withan effective orthe nominalSk numberand the observed physics are not going to be affected by the choice of timescale. Table1 showsboththe averageandpeak value ofRep

obtainedfromsampleswithin [−0.5D,15.8D]alongstreamwiseX- direction.NeverthelessRe_pincreasesasSkislarger,themaximum Rep isonly5.8forall considered Sk-numberscomputedfromthe relativelysmallslipvelocityUs. Thisindicatesthe applicabilityof the ﬁnite-Re_p correction in Eq. (2). Note that we adopt a ’slid- ingmotion’collisionmodelatthesurfaceofthecylinder,wherein the particlewall-normal velocity reducesto zerowhile the wall- tangentialvelocity component preserves, details ofwhich are re- portedintherecentworkbyShietal.(2020).

2.2. Computationaldetails

Fig. 1. (color online) Schematic of the instantaneous particle ﬁeld with superimposed blue and red whirls illustrating oppositely rotating Kármán vortex cells. (For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the web version of this article.)

Table 2

Particle simulation framework for each Sk in two groups.

Case no. Injected particle (Y/D) ×(Z/D) Injection area Sample region (X/D) Sample number

number per timestep in total

1 9 (–4.096, 4.096) ×(0, 2.048) (–2.5, 15.8) 4 . 7 ×10 ⁴

2 2 (–2.048, 2.048) ×(0, 1.024) (–2.5, 16.5) 6 . 1 ×10 ³

After runningthe simulationfor400

τ

f,the unsteadyﬂow has developed into a strictly periodic vortex shedding regime. The time-averageddragcoeﬃcientC_d=2F_d/

ρ

fU₀²LDandstandarddevi- ation of thelift-coeﬃcientC_l₋_rms=2F_l₋_rms/

ρ

fU₀²LD (F_d dragforce, F_l₋_rms root-mean-square liftforce, Lcylinder length)are obtained to be1.391and0.240,respectively.TheStrouhal numberSt= f

τ

f

is calculated as 0.168 via fast Fourier transformation (FFT). The measuredparametersareallwithintherangesreportedbyothers, see Ferziger and Tseng (2003).In a post-processing step, we de- ducedthespanwisevorticity

ω

∇

^u−(

∇

^u)^T^]^and^S=

1

2[

∇

^u+(

∇

^u)^T^]^are^therotation-ratetensorandstrain-ratetensor, respectively).

Inertialparticlesareseededintotheﬂowfromtheinletatini- tial velocity U₀. The particle simulation, starting at 400

τ

f, lasts 8 vortex shedding periods (total number of timesteps 9448). At each timestep istep=1,2,...,9448,acertain numberofnewpar- ticles are released into the ﬂow and others are leaving the do- main through the outlet. Fig. 1 showsthe snapshotofthe particle distribution andthe extractedvortex coresfrom DNSdata at thelasttimestepi_step=9448.Theterminologyusedafterwardsare depictedalongwiththecorrespondingschematictopology.

Six different Stokes numbers are considered, i.e. Sk= 1,3,5,8,12,16. Unlike the tracer particles approaching the cylinder wall without colliding, inertial particles lose part of their kinetic energy due to collision with the cylinder surface.

We perform two groups of particle simulations with different seedings shown inTable 2. The first caseserves for the purpose to exhibitthe particleclusteringwiththesufficientsamplesused inSec 3.1and3.2.In thesecondsimulation,we intendtooutput the essential quantities along each particle trajectory at every timestep. The heavy dataset requires to reduce the number of injected particles,therefore,theinjectionareaisonlyaquarterof that inthefirstsimulation.Inthepost-processing,we onlyselect theusefulparticlesamplesalongthestreamwiseX-directionwhich are nearthecylinderandinvolvedwithvortices inthewake.The totalparticlenumbersofthetwocasesusedforstatisticalanalysis inSec3.3,aredifferentbyafactorof8.

3. Resultsanddiscussion 3.1. Travelingof’voidholes’

TheinstantaneousparticleconcentrationseeninFig.1exhibits a peculiarrepetitivepattern, which intuitively reﬂectsthe underlying Kármán vortex street. The characteristic concentration pat- ternoftheSk=1particleswillbethoroughlyexploredinthenext three sub-sections withfocus on the formation and propagation of thevoid holes (empty areas). Before the closing ofthe paper, Stokes number effects will be addressed in Section 4. Since the ﬂowatRe=100istwo-dimensionalandtwo-componential,i.e.the only two velocity components u and

v

are independent of Z, all particles are projected into a single XY-plane. Fig. 2 shows how void holes formin the shadowof the cylinderand travel down- streamduringone sheddingperiodT=D/(^St·U₀)≈6D/U₀,starting at time t^∗≈47D/U₀, i.e. about 8 shedding periods after the start of the particle seeding. Five leaf-shaped void holes, labeled from0to4,canbeseenintheuppermostpanelattimet^∗.Eachof thevoidholesarepartiallyoverlappingwithalocalvortexcell,vi- sualizedbymeansofthespanwisevorticity

ω

^z^.Consecutivesnap- shots,separatedT/8intime,areshowninthesubsequentplots,in whichwefocusonthemotionandshapeofholeno1.Theshape ofthevoidholeisapproximatedbyaleaf-shapedcontour(incyan) ofwhichthecentroidisindicatedbyacross.Weobservethatthe voidholetranslatesdownstreamwithnegligiblelateralexcursions.

Thevoidholebecomesmoreuprightwithtime,whichrepresents a very modest clockwise rotation, i.e. in the same sense as the rotation ofthe underlying Kármán vortex. The upper partof the peripheryofthe holebecomesmorerounded, whereasthelower cusp becomes peakier. The length of the perimeterof hole no 1 increasesby ca 35% duringthe sheddingperiod considered, thus reﬂecting the increasing area of the void.A possible void distor- tion likely caused by an insuﬃcient blockage ratio

α

=Ly/D can thus be excluded in view of the almost proportional increase of perimeterandarea.Afterahalfsheddingcyclethetopologyofthe concentration patternexhibitsa mirror-symmetry withrespect to the X-axis of that of the original pattern att^∗.After a complete sheddingcycleT,thevoidhole no1hastranslatedtotheearlier positionofvoidholeno3,whereasvoidholeno-1hastakenover theoriginalpositionofvoidholeno1.

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Fig. 2. (color online) Time-sequence of particle distributions during one vortex shedding period T . The characteristic snapshots of the clustering patterns of Sk = 1 particles are T / 8 apart in time. The approximate outline and the centroid (+) of hole no 1 are colored in cyan and the perimeter is indicated at the top. The spanwise vorticity ﬁeld ωzD/U 0is superimposed on the particle distributions at time t ^∗, t ^∗+ T / 2 and t ^∗+ T .

Let usnowtake a closerlookat howthe voidholes travel in Fig. 3, wherethe fluid velocity field now ischaracterized by the scalar Q-field. The centers ofKármán vortices are markedby or- angedotswhosepositionisdecidedbythelocalmaximaofQ,and the grey chunks represent fluid withQ>0. Unlikethe centroids ofthevoidholes(black+),whichtranslatedownstreamwithneg- ligible lateral excursions, the center of the anti-clockwise vortex associated withvoidholeno 0iswell belowthe Y-axis,whereas

thecenteroftheclockwisevortexcellcorrespondingtoholeno1 isabovethesymmetryplane. Althoughthevoidholesencompass theKármánvortexcells,thecentroidofagivenholedoesnot co- incidewiththecenterofaKármánvortex.Thisobservationcannot readilybemadefromtheFig.2.Itisworthtonotethattherelative distancebetweenthevortexcoreandthevoidcentroidpersists,i.e.

thatthevortexcoreremainsmostoffsetfromthesymmetryplane ofthewake(Y=0).AccordingtoFig.3,however,thedisplacement

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Fig. 3. (color online) Particle distributions and Qat time (a) t ^∗+ T / 2 and (b) t ^∗+ T . The Sk = 1 particles are colored by the local value of Q, i.e. Q f@p. The grey chunks inside the void holes represent parts of the ﬂow ﬁeld with Q > 0 and the orange dots show positions of local maxima of Q. The translational distance of the centroids (+) of void holes no 0 and no 1 can be measured over the time interval T / 2 . The numbering of the void holes refers to the top panel in Fig. 2 .

between the vortex coreand the void centroid decreases in the downstream direction, ca 10.2% and22.1% forvoid no 0 andno 1,respectively,despitethatthedisplacementatupstreamislarger than thatatdownstream.The reasonforthedisplacementdimin- ishingwithtimeislikelytobeattributedtothegraduallychanging shape ofthevoids,whichshiftsthecentroidsclosertothevortex coreswhereastheupstreamvoidholeexpandsataslowerrate.

Theparticles havebeencoloredbylocalQ atparticleposition, i.e.Q_f@_p.Thevastmajorityoftheparticlesisbluish,whichmeans that they are in a strain-rate dominated region characterized by negativeQ-values.OnecanthereforeinferthattheSk=1particles areexpelledfromtherotation-dominatedKármánvortexcellsand swepttothebordersofthevoidholes.However,itissurprisingto observethemanyreddishparticleslocatedinthetwoshearlayers emanatingfromthesurfaceofthecylinder.Thecentrifugalmech- anism is active when an inertialparticle triesto followa curved trajectory whereas a particlein aseparated shearlayer followsa mildlycurvedpath.Thatiswhynumerousparticlescanbeseenin the shearlayers, inspite ofQ beingpositive.Here, positive Q_f@_p impliesthattherotationrateexceedsthestrainrate.

By measuring the traveling distance of the centroids of void holeno 0andhole no1overthetime interval0.5T betweenthe two plots, the translationalspeed of the two holes can be estimated as 0.796U₀ and 0.854U₀. The higher speed of hole no 1 reflects the monotonically decreasing velocity deficit withdown- stream distanceX. This reduction ofthe velocity deficit andthe accompanyingattenuationofthestrengthofthevortexcorescon- tribute to the persistent shape of the void holes. The traveling speed of the holes seems to be consistent with the estimated speed0.895U₀ oftheKármán vortexcells. Theslightlylowertrav- elingspeedoftheholescanbeascribedtotheinertiaoftheSk=1 particles.Thewavelength

λ

^x^,^i.e.^the^distance^between^two^succes-

sivevortexcoreswiththesamesignof

ω

^z^,^is^about^5.284^D^.^These

estimates ofthe vortexspeed and separationare consistent with datareportedfromtheDNSstudybyMowlavietal.(2016),albeit

inpresenceofasomewhatinsuﬃcientblockageratiointheircom- putationalset-up.

3.2. Tracking’smoothedges’

An interesting phenomenon can be observed already from Fig. 2, namely that particles are densely clustered and form a smoothedgeattheupstreamsideofthevoidholes.Theparticles arecomparativelymoredispersedaround thedownstreamsideof the void holes, and the edge of the hole is rather blurred near thecusp.Carefulinspectionofplotsbyothers,notably(Yaoetal., 2009;HaugenandKragset,2010) and(Aarnesetal., 2019),shows the same tendencies, but this peculiar phenomenon has neither been addressed nor explained before. To this end we introduce Voronoï diagrams in Fig. 4 to investigate the local particle concentration.The inverse ofthe area A ofa Voronoï cell is amea- sureoftheparticleconcentration.Theleaf-shapedvoidholescom- priseVoronoï cells whichareorder-of-magnitudeslarger thanthe surroundingVoronoï cells. This cellpattern resemblesreticulated veinswiththethinnestcell-shapeattheupstream(left)halfofthe voidhole.Theseneedle-likeVoronoï cells resultfromthe densely populatedsmoothedgeofthevoidhole.

The particlesin Fig.4 are coloredby local Q_f@_p/A in orderto emphasize particles around the void holes better than in Fig. 2. No particles downstream of X=5D are seen to be in rotation- dominated(Q>0)areasandthevastmajorityoftheparticlesare instrain-dominated (Q<0) areas.Particlesnearthecenter-plane (Y = 0) are greyish/white (Q≈0), i.e. neither affected by strain nor by rotation.Recall from Fig.2 that the vorticity is negligible betweenthevortexcellsandwecannowinferthatalsothestrain- rateisnegligiblysmallinthisarea (betweenthewavyred-dashed lines).

Howisthesmoothedgeofavoidholeformedandmaintained?

Letusexaminethehistoryoftheparticleswhichformthesmooth edgeof holesno 3and4attime t^∗.The particlesatthe edgeof

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Fig. 4. (color online) Voronoï diagrams at time t ^∗. The Sk = 1 particles are colored by Q f@p/A , where A is the area of the corresponding Voronoï cell. Q f@p/A ≈0 in the white band between the two undulating dashed red lines. (For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the web version of this article.)

holeno 3arebacktracked intimeandplottedinFig.5(a)atfour earliertimeinstantsT/4apart.Thefast-movingparticlesthatmake upthesmoothedgeofvoidholeno3(encompassingaclockwise- rotating Kármán vortex) seem to stem from the smooth edge of hole no1, whichnow residesattheearlier positionofvoid hole no3.Theparticles,especiallythoseformingtheupperpartofthe smoothedge,seemtotranslatecollectivelyalmostasanensemble.

|

^a^p

|

^with^a^distinct ^componentⁱⁿ^the^positiveY-direction.Thisis consistentwiththerelativelyhighvelocities

|

^u^p

|

^of^particles^at^the

smoothedgeofholeno4.Thesigniﬁcantcomponentofapperpen- dicular to u_p suggeststhat theseparticles are deﬂectedupwards, i.e.towardsthesymmetryplaneofthewake.

Contrary to Fig.6, inwhich the particles were backtracked in time, Fig.7showsthe accelerationa_p ofallparticles atthe same instant oftime t^∗. Themagnitudeofthe particleaccelerationde- cays withstreamwise distance (the colorchanges gradually from blue to green). Particularly strong accelerationsare seen on both sides of the void holes andmost notably near the center ofthe

corresponding Kármán vortices. These accelerations are towards downstream at the smooth edge but upstream at the blurred downstreamside ofeachvoidhole.The reasonthat theaccelera- tionvectorsroughlypointtothecenteroftheKármánvortexcells isinordertomaketheparticlesfollowacirculartrajectoryaround thevortexcores(orangedot).Thehigh

|

^a^p

|

^exhibits^similar^undu-

lationsaboutthemid-plane(Y=0)asthewhitebandinFig.4. The slip velocity vector Us determines the drag force F in Eq.(1) and thereforealso the particle acceleration ap in Eq.(3). Thestreamwiseandcross-streamcomponentsofUsareshownin Fig.8,wherethestreamwisecomponentU_s,xinFig.8(a)alternates between positive and negative values, whereas the cross-stream componentUs,yinFig.8(b)retainsthesamesignonagivensideof thewake.Onemaynoticethatthetwocomponentsoftheslipve- locityvectorareofsurprisinglysimilarmagnitudes.Thisisthecase because the primary causeof velocity slip is the Kármán vortex cells. A positive slipvelocity component in Fig.8(a) implies that the particles are lagging behind the fluid motion, whileU_s,x<0 representsparticlesleadingthelocalfluid. Thisisfullyconsistent withthedirectionsoftheaccelerationvectorsonthetwosidesof thevoidholesinFig.7.Thecross-streamslipvelocityinFig.8(b)is consistentlynegativeintheupperpartofthewake(Y>0),i.e.in thedirectionofnegativeY.Thisreflectsthattheparticlesaremov- ing fastertowards thecenterline inareaswhere aclockwise vor- texgivesrisetoanegativefluidvelocitycomponentuy(^x^p,t)^.^See, forinstance in Fig.7, the densely populated bluishparticles just downstream of void hole no 1 which encompasses a clockwise- rotatingKármán vortex.Similarly, weobservethatU_s,y>0inthe lower part of the wake. We therefore conclude that the cross- streamslip velocity tends todrive all particles towardsthe sym- metryplaneY=0.

3.3. Particletrajectories

3.3.1. Grouptrajectories

Now,afterhavingexaminedthecoherenceofthevoidholesin the cylinder wake, we are shifting our attentionto how inertial particlesthatclusteraround avoidholearetravelingthroughthe Kármánvortexcellsinthenearwake.Fig.9showsthetrajectories of particles landing on both sides of void hole no 2. The coherent voidsappear wherethe first pairofcounter-rotating Kármán cellsarerollingupanddetachfromtheseparatedshearlayers.We observe fromthe upper trajectoriesin Fig. 9 that these particles werefirstacceleratedastheypassedthecylinderandthedetached shearlayer.Mostofthesetrajectorieswerebentdownwardsbythe first clockwise vortex that shed fromthe upper shear layer. The particlesthereafter experienceda substantialdeceleration, caused by an anti-clockwise Kármán vortex shed from the lower shear

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Fig. 5. (color online) Backtracking Sk = 1 particles that formed the smooth edge of void hole no 3 (a) and void hole no 4 (b) at t ^∗in time. The backtracked particles are colored by their velocity magnitude |u p| at time t ^∗, whereas the grey particles are all at t ^∗. The earlier time instants are T / 4 apart in (a) and separated by T / 3 in panel (b).

The arrows are particle velocity vectors.

Fig. 6. (color online) Backtracking Sk = 1 particles that formed the smooth edge of void hole no 4 at t ^∗in time (same particles as in Fig. 5 (b)). The backtracked particles are colored by their acceleration magnitude |a p| at time t ^∗, whereas the grey particles all are at t ^∗. The earlier time instants are separated by T / 3 in time. The arrows are particle acceleration vectors.

Fig. 7. (color online) Particle acceleration a pat time t ^∗. The point particles are colored according to the magnitude |a p|and a grey arrow represents the particle acceleration vector. An orange dot shows the position of local maximum of Qin the four vortex cores (embedded in void holes nos 1–4) and the whirls illustrate the sense of ﬂuid rotation.

Fig. 8. (color online) Slip velocity U s= u f@p−u pat time t ^∗. (a) Streamwise component U s,x; (b) cross-stream component U s,y; ’+’ and ’-’ denote areas with positive and negative dominance.

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Fig. 9. (color online) Left: Trajectories of Sk = 1 particles landing on the upstream (smooth) and downstream (blurred) edges of void hole no 2 at time t ^∗, colored by the instantaneous particle velocity magnitude |u p|. The age ranges of particles terminating on upstream and downstream are [18.5, 26.1] and [26.2, 27.3] measured in D/U 0, respectively. The age of a particle is deﬁned as the time elapsed since the particle was entered at the inﬂow boundary. Right: Trajectories of younger Sk = 1 particles with ages about 19.8 D/U 0(upper panel) and 17.8 D/U 0(lower panel).

Fig. 10. (color online) Trajectories of Sk = 1 particles landing on the upstream smooth edge of a coherent void in the wake at time t ^∗. (a) void hole no 1; (b) hole no 2;

(c) hole no 3; (d) hole no 4. The black dots show the particle distribution at time t ^∗and the colors represent the instantaneous particle velocity magnitude |u p| along the trajectories ranging from 0.4 (blue) to 1.3 (red). The whirls illustrate the sense of vortex rotation. (For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the web version of this article.)

layer,beforethetrajectoriesreachedthesmoothedgeofvoidhole no 2 attime t^∗.In contrast,particles passing below the cylinder maintainedtheirhighspeedevenduringthestretchwheretheup- perparticlesdecelerated.Furtherdownstream,however,thelower trajectoriesarebentupwardsduetoananti-clockwiseKármánvor- tex,buttheseparticles exhibitedamoremodest decelerationbe- fore they endedup atthe downstream side ofvoidhole no2 at timet^∗.Trajectoriesofyoungerparticles,i.e.particlesinjectedlater thanthosetrackedintheleftpanel,areshowntotheright.Asex- pected, theupperandlower trajectoriesremainsymmetric about themidplaneY=0untilvoidholeno0isapproached.Theparti- clevelocitymagnitude

|

^up

|

3.3.2. Individualtrajectories

The variations ofthe velocity of particles that cluster atboth sides ofa void holeseen in Fig. 10are now explored further by means oftrajectories of two particles released at the same time but landing atdifferent sides of voidhole no 3.The trajectories of thetwo particlesare plottedinall panelsofFig.12, toenable comparisonswiththestreamwiseandcross-streamcomponentsof theslipvelocity,particlevelocityandparticleacceleration.Theco- incidingvariationsoftheslipvelocityU_sandaccelerationa_pcom- ponents at thisStokes number(Sk=1) suggest that theﬁnite-Re correctionfactor

β

ⁱⁿÊq.⁽¹⁾îs^close^toûnity.

We observe from Fig. 12(a, c) that both particles are deceler- atedastheyareapproachingthecylinderduetotheadversepres- sure gradient in the stagnation zone. Both particles start to ac- celerate at X ≈ −0.6D andattain maximum acceleration as they passtheshouldersofthecylinder.Theparticlepassing belowthe cylinder (labeled P_low) decelerates again alreadyfrom X ≈ +1.2D (light green dot) and is accordingly slowed down severely, until anothermildaccelerationsetsin.Theother particlepassingabove the cylinder (labeled Pup) experiences a longer acceleration pe-

riod and accordingly achieves a relatively higher velocity before astretch ofdeceleration.The decelerationstage ofboth particles, causedbythelocalKármán vortices,extendsover4–5D.The par- ticlesarethereafteracceleratedonceagainsothatthestreamwise particle velocity componentup,x recovers to approximatelyU₀ as theparticleslandontherespectiveedgesofvoidholeno3attime t^∗.Theparticlesare nowfollowingtheﬂuidmorecloselyandthe slipvelocityUs,x andtheaccelerationap,x areonlymodestlyposi- tive.

Contrarytothedecreasingup,xinfrontofthecylinder,themag- nitudeofthecross-streamparticlevelocityup,yisrapidlyincreas- inguntilX≈0whereasign-changeofap,ycanbeobserved.Both particlescontinuetomoveawayfromthesymmetryplaneatY = 0fora little while,until up,y changes signatX ≈2D.Thereafter, the particle(P_low) originatingat thelower side of the cylinderis dragged bythe ﬂuidinthe positiveY-direction,whereas thepar- ticle (Pup) passing above the cylinder is dragged in the opposite direction.

4. Stokesnumbereffect

After having explored how inertial Sk=1 particles are en- trained into the wake and yet leave coherent voids, we finally take a brief look at how particles withhigher inertia behave in thesameunsteadywakeflow.ParticleswithStokesnumberhigher thanunityare believedtobemoreweakly coupledtothecarrier flowduetotheirlargerinertia.Theinstantaneousparticleconcen- trations inFig. 13 show that the more inertial particles are able topenetrate theshear layers thatseparate fromthe shouldersof the cylinderand thereafter proceed inthe downstream direction relatively unaffected by thealternatingly shed vortex cells, asvi- sualized by colorcontoursof spanwisevorticity

ω

^z^. ^This^leads^to

the formation of a void shadow in the nearwake, in contrast to the void cells formed by the Sk=1 particles in Fig. 2. The void shadow extends further downstream with increasing inertia, i.e.

higherSk,andtherebypostpones theappearanceofcoherentvoid cellsencompassingtheKármánvortexcells.Theincreasingparticle concentrationinhigh-vorticityregionsreﬂectsthatthemoreiner- tialparticlesbehaveballisticallyandessentiallydecouplefromthe local ﬂuid. The gradually increasing decoupling is caused by the reduction ofthe factorin front ofthe slipvelocity Us inEq.(3). Although coherent voids are formed further downstream in the wakewithincreasingSk,smoothedgesattheupstreamsideofthe voidholes can be observedat all Stokes numbers considered. At thedownstreamsideofthevoids,however,moreparticlesaredis-

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Fig. 12. (color online) Trajectories of two different particles landing on the upstream ( P low, circles) and downstream ( P up, squares) side of void hole no 3 at time t ^∗are plotted in all four panels. (a, b) Particle acceleration a pand slip velocity U salong the trajectories; (c, d) Particle velocity u palong the trajectories. Streamwise components are shown in panels (a, c) to the left and cross-stream components in panels (b, d) to the right. The light and dark green dots in (a, c) identify the locations where the streamwise acceleration a p,xof the slow (low) and fast (up) particles changes sign. Similarly, the green dots in (b, d) identify the locations where the cross-stream acceleration a p,y

changes sign, whereas the red dots indicate where the cross-stream velocity u p,ychanges sign. (For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the web version of this article.)

Fig. 13. (color online) Instantaneous distribution of heavier particles at time t ^∗superimposed with the spanwise vorticity ωz. (a) Sk = 3 (b) Sk = 5 (c) Sk = 8 (d) Sk = 12 .

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Fig. 14. (color online) KDEs of particle velocity at time t ^∗for six different Stokes numbers. (a)Particle velocity u p,xin the streamwise direction; (b)particle velocity u p,yin the cross-stream direction. Particles are sampled within X ∈ [ −0 . 5 D, 15 . 8 D ] .

Fig. 15. (color online) KDEs of particle acceleration (solid lines) and slip velocity (dashed lines) at time t ^∗ for three different Stokes numbers. (a) a p,xand U s,x; (b) a p,yand U s,y. Particles are sampled within X ∈ [ −0 . 5 D, 15 . 8 D ] .

persed aroundthecuspsofthevoids,ascomparedwiththeclus- teringoftheSk=1particles.

In addition to the phenomenological description of the particle concentration patterns in Fig. 13, we also make quantitative comparisons oftheinstantaneous distributionsofthe particleve- locity, particle acceleration, and slip velocity for some different Stokes numbers to further investigate the effect of particle inertia. The peaks of the six KDEs of the streamwise particle velocity in Fig.14(a) become higher withincreasing inertia whilethe peak locationsmonotonicallyapproachthefreestreamvelocityU₀. AsimilarlyincreasingtrendofthecentralpeaksoftheKDEsofthe cross-stream particle velocity are seen in Fig. 14(b),whereas the secondary peaksatthe tailsofthe KDE-distributions vanishwith increasing particleinertia. Particle inertiamakes thecross-stream velocity u_p,y moreconcentrated aroundzeroandin particularfor the Sk=16 particles for which the two secondary peaks do no longer exist.The leftofthetwo near-bycentralpeaksareconsis- tentlyhigherthantherightone.Webelievethatthisiscausedby thewaketopologyattheparticulartimeinstantt^∗,andweconjec- turethattherightoneofthetwocentralpeakswillbethehigher att^∗+1/2T.

In Fig.15,the streamwiseandcross-stream componentsofa_p orUsexhibitfairlysimilarKDEdistributions.However,weobserve a signiﬁcant difference betweenap,y andUs,yfor thetwo highest Stokes numbers, namely that the latteris much ﬂatterthan that of ap,y. With higher inertia most of the particles experience in-

signiﬁcant accelerations in spite ofa non-negligible slipvelocity.

The Sk=1 particles represent an exception, for which the KDEs of the particle acceleration and slip velocity almost coincide, as seeninFig. 8(c). Thissuggeststhat theﬁnite-Recorrection

β

≈1 for the Sk=1 particles, while

β

^deviates ^from ^unity ^for ^higher

Stokesnumbersinaccordancewiththesemi-empiricalcorrelation inEq.(2).Thisexplainsthedecouplingbetweenparticleaccelera- tionapandslipvelocityUsasaneffectofinertia.

5. Conclusions

We performed three-dimensional numerical simulations of particle-ladenfluidflowaroundacircularcylinderatRe=100.The unsteady2Dflow isladen withinertialsphericalparticlescharac- terizedby adimensionlessStokesnumber(Sk).Althoughprevious studiesdealtwithparticledispersionincylinderwakeflowand/or vortex streets from various aspects, the topology of the particle concentration hasneverbeenanalyzedindetail,similarlyto how wehavedone inthepresentmanuscript.We focusedtheinvesti- gationonSk=1particles,forwhichitisobservedthattheinertial particles are expelled fromcertain parts of the flow field which accordinglyappearascoherentvoids.ThelocalKármánvortexcells inthenearwakecontributedtotheformationofthesevoids.Each coherentvoidencompassesaKármán vortexshedfromthecylin- der.Aninterestingphenomenonisthatparticlesthatclusteratthe smooth upstream edge ofa voidholesurrounding a Kármán cell