Anomalousvelocity fluctuations in particulateturb ulent channelflow 1

Koji Fukagataa� c, SaidZahraib � c, ShunsukeKondoa, Fritz H. Barkc

aDepartmentof QuantumEngineeringandSystemsScience, University of Tokyo7-3-1Hongo,Bunkyo-ku,Tokyo113,Japan

bABBCorporateResearch, 72178 Vasteras,SwedencFaxenLaboratoriet,Kungl.TekniskaHogskolan,10044 Stockholm,Sweden

Abstract

Gas-particleturbulent channelflow at Reτ� 644, loadedwith copperparticlesat a mass

flow ratio of 2%, is studiednumericallyby largeeddysimulation(LES) coupledwith La-grangianparticletracking(LPT). Inter-particlecollisionsandcorrectionof dragforcein thevicinity of walls areaccountedfor. Focusis madeon theinfluenceof particlewall bound-ary conditionsandtheir influenceon the statisticalstructureof the flow. It is shown thataccordancewith experimentaldataby Kulick et al. (1994)canbe improved if a mecha-nismwhich cansuppressthedirect re-entrainmentof particlesafter the impactat thewallis present.Presentresultshows that inter-particlecollisionsmayplay animportantrole inthe re-distribution of particlemomentumamongdifferentcomponentseven at low massloadingconditions.

1 Intr oduction

Gas-particleturbulentflowsoftenappearin industrialprocesses.Examplesof sucharepollutioncontrolprocesses,foodprocessingandmaterialprocessing.Althoughmany theoreticalstudieshavebeendoneonsuchflow, they arelimited in rathersim-plegeometrieswith homogeneousturbulenceor simpleshearturbulence.However,in the real world applications,the flow is alwaysconfinedby solid walls. More-over, in many casesdepositionandentrainmentat walls areplaying an importantrole in theprocess.Therefore,for many industrialsystems,detailedunderstandingof wall-boundedgas-particleflows is of primaryinterest.

1 Dedicatedto Dr. GadHetsronion his65thbirthday.

Published in Int. J. Multiphase Flow 27 (2001), 701-719. c�

Elsevier Science

Whenthe particle loading is high, the structureof turbulenceis modifiedby thepresenceof particles,as hasbeenobserved in variousexperimentssincethe pi-oneeringwork by Hetsroniand Sokolov (1971) who studiedthe suppressionofturbulenceby addingdropletsin aturbulentjet.Suchturbulencemodulationoccursin wall-boundedflows,too,asstudiedby ZisselmarandMolerus(1979)andMaedaetal. (1980)to nameafew. Resultsof theseexperimentsconcerningtheturbulencemodulationarereviewedin Hetsroni(1989)andGoreandCrowe(1989).

Experimentalinvestigations,however, have not beenableto provide completesetof datafor understandingthedynamicsof theprocess.Turbulencemodulationis anextremelycomplex phenomenafar beyond the limit of applicability of analyticalmethods.The studiesof turbulencemodulationphenomenonis also importantinindustrialapplicationsin additionto its academicvalue,asit maylargely increaseor decreasethe transportof heatand mass.Thereforefurther and more detailedinvestigationshouldbedonebasedon a solid basis.Thefirst stepof suchinvesti-gationof a complex wall-boundedgas-particleflow maybeanaccuratepredictionof motion andstatisticsof particlesin simplercasessuchasa dilute flow. In thatcase,the numberdensityof particlesis sufficiently low suchthat the structureofturbulenceis not largely modified.Theknowledgeon sucha dilute casewill bealarge building block to achieve the final goal,namelyto understandthe influenceof particleson thestructureof fluid turbulence.

Due to the explosive developmentof computercapacities,simulationsof wall-boundeddilute gas-particleturbulent flows have becomepossibleusingratherdi-rectmethods.An exampleis theLagrangianparticletracking(LPT), in which thetrajectoriesof individualparticlesarecomputedby integratingtheequationof mo-tion for a particle.The turbulent fluid flow is usuallysimulatedusinga directnu-mericalsimulation(DNS) or a largeeddysimulation(LES) in anEulerianframe.

Wang and Squires(1996) performedLPT simulationscoupledwith LES (LPT-LES) of gas-particleturbulent channelflows at Reτ � 180 andReτ � 644.A dy-namicsubgridscalemodelwasusedfor LES. Lycopodium,glassandcopperpar-ticlesof differentdiameterswereused.While theinfluenceof fluid flow to particlemotion wasaccountedfor, the inverseinfluenceof the motion of particleto fluidflow wasneglectedby anassumptionof having verydilute flow, i.e. one-way cou-pling. Thecomputedstatisticsfor thecasesof Reτ � 180werein excellentagree-mentwith thesimulationdataby RousonandEaton(1994),whousedDNSinsteadof LES.Fukagataetal. (1998b)alsoperformedone-waycouplingLPT-LESfor thesamecases.Excellentagreementwasfoundwith theLPT-LES databy WangandSquires(1996).For thecaseswith Reτ � 644,WangandSquires(1996)comparedtheir resultswith the experimentaldatafor the casesfor massloading,Z, of 2%by Kulick et al. (1994).Largediscrepancieswereobserved.For thecaseof 70 µmparticles,for instance,thesimulationpredictedmuchhighermeanparticleveloc-ity andmuchweaker wall-normalvelocity fluctuationsin thewholechannel.Thebimodality in thevelocity distribution nearthewall observedin theexperimentbyKulick etal. (1994)wasnot reproducedin their simulations,either.

2

Tanakaet al. (1997)performedLPT-LES taking into accountthe influenceof theparticlemotionto fluid flow andtheinter-particlecollisions,i.e.four-waycoupling.They focusedon the casesof highermassloading,Z � 20%,which correspondsto the casesreportedin experimentalinvestigationsby Kulick et. al (1994).Theyfoundthattheinter-particlecollisionshaveastronginfluenceon thestatistics.Thepredictedwall-normalroot-mean-square(RMS)particlevelocityfluctuationsin thebulk region of thechannelfor thecaseof 70 µm copperparticlesat Z � 20%wasabout75%of theexperimentaldata.This is twiceaslargeastheresultof asimula-tion neglectingtheinter-particlecollisions.

Fukagataet al. (1999)revisitedthecasewith 70 µm copperparticlesat the lowestmassloading,Z � 2%, in theexperimentof Kulick et al. (1994).They performedfour-way couplingLPT-LES consideringtheincreaseof dragin thevicinity of thewall (Faxen, 1923; Brenner, 1961) and found that inter-particle collisions havestronginfluenceon the statistics.It hasbeenassumedbeforethat the massflowratio, Z � 2%, which is even lower thanthoseexaminedby Tanakaet al. (1997),is so dilute as to be able to neglect the inter-particle collisions. The considera-tion of increaseof dragcoefficients in the vicinity of the wall was found to fur-therincreasethewall-normalRMSvelocity fluctuations.It wasalsofoundthatthewall-normalRMSvelocityfluctuationsincreasesastheincreaseof particlenumberdensitynearthewall. Althoughtheagreementbetweensimulationandexperimentwasimprovedascomparedto a simpleone-way couplingsimulationwithout dragincreasenearthewall, discrepancieswerestill significant.However, theseresultsobviouslysuggestthattheaccordancebetweentheLPT simulationandexperimen-tal datawould possiblybe improvedby furtheraccumulationof particlesnearthewall.

Themainobjectiveof thepresentpaperis to studytheinfluenceof accumulationofparticlesnearthewall on thestructureof thedilute particulateturbulentflow nearthe wall andto understandthe reasonof discrepanciesobservedbetweensimula-tionsandexperiments.For this purpose,differentboundaryconditionswhich willenhancetheaccumulationof particlesnearthe wall areexamined.The fluid flowis modeledusingLES andthemotionof particlesaresimulatedby LPT. Two-wayparticle-fluidcoupling,particle-particlecollisionsandwall-particleinteractionsareaccountedfor.

The paperis organizedin the following way: In section2, the theoreticalmodelis introduced.The studiedcasesanddifferentboundaryconditionsusedin thosecasesaredescribedin section3. Thecomputationalprocedureis outlinedin section4. The resultsarepresentedanddiscussedin section5. Finally, in section6, theresultsaresummarizedandconcluded.

3

2 Theoretical model

Similarly as the previous work introducedin the previous section,the turbulentchannelflow wassimulatedusingLESwith amodifiedSmagorinsky subgridscale(SGS)model(Zahraiet al., 1995),andthe particlesweretracked individually byintegratingtheparticleequationof motion.

Thegoverningequationfor theLES,i.e.filteredcontinuityandNavier-Stokesequa-tions,for an incompressiblefluid with density, ρ f , andkinematicviscosity, ν, canbeexpressedas

∂ufi

∂xi� 0 (1)

and

∂ufi

∂t � ∂∂xj

����uf

i ufj

pρ f δi j � � ν � νS���� ∂uf

i

∂xj� ∂uf

j

∂xi

������� � ψi , (2)

whereufi is thefilteredfluid velocity, p is thefilteredpressure.TheSGSviscosity,

νS, wasmodeledby ananisotropicversionof theSmagorinsky model.For details,theinterestedreaderis referredto Zahraietal. (1995).Thelastterm,ψi , representstheforcefrom particles.Similarly asin thepreviouswork (Fukagataetal., 1998a),ψi was modeledas a body force with the samemagnitude,but in the oppositedirection,astheaveragedsurfaceforceson particles,i.e.

ψi � 1ρ f V ∑m� dup

i

dt

gi � , (3)

whereV denotesthevolumeof thefilter, i.e.thecomputationalcell,m is theparticlemass.Thesummationin eqn(3) is takenfor all theparticlesin thatcell. Influencesof thepresenceof particleson theSGSmodelwasneglected.

Theparticleequationof motionusedin thepresentstudycanbewrittenas

dupi

dt � Ci1τp� 1 � 0 � 15Re0 � 687

p ��� ufi

up

i ��� gi , (4)

whereupi anduf

i arethe i-componentof theparticlevelocity andthefictiousfluidvelocity at particlecenter, respectively. The first term of the particleequationofmotionis thedragforceandthesecondtermis thegravitationalforceterm.Effectsof inter-particlecollisionswill bequantifiedin thenumericalschemein section4.

4

TheStokesrelaxationtime,τp, is definedas

τp � d2 ρp

18νρ f , (5)

whered is theparticlediameterandρp is theparticledensity. TheparticleReynoldsnumber, Rep, is definedby

Rep �����up �uf � dν

. (6)

TheReynoldsnumbercorrectionfactor � 1 � 0 � 15Re0 � 687p � in eqn(4) wastakenfrom

the empirical formula by Schiller andNaumann(1933).This formula is, strictlyspeaking,valid for steadyconditionsonly. However, effectsdueto particleaccel-erationwereignoredsimilarly to thework by otherresearcherssuchasWangandSquires(1996)andTanakaet al. (1997).

The correctionfactorfor the dragforce term,Ci , representsthe correctionduetothepresenceof thewall. For thedirectionsparallelto thewall, theexpressionbyFaxen(1923),

C1 � C3 � 1 916 ! d

2y " � 18 ! d

2y " 3

45256 ! d

2y " 4 116 ! d

2y " 5 #%$ 1

,

(7)

wasused.For the directionnormalto the wall, a modelformula (Fukagataet al.,1999),

C2 � '& 1 9

8 ! d2y " � 1

2 ! d2y " 3 (*)

1

exp ! a � 2yd

b� ",+ #%$ 1

, (8)

wasapplied.This modelformula is basedon theapproximatesolutionby Wakiya(1960)andtheexactsolution,in a form of infinite series,by Brenner(1961).Theconstant,b, in themodelformulais 1 in orderto obtainsimilarexponentialbehaviornearthe wall asthe solutionby Brenner. Due to numericalreasons,it wassettob � 0 � 9999in the simulations.The constanta wasset to 2 � 686 (Fukagataet al.,1999).

The wall correctionson drag introducedabove are basedon the assumptionofsmall,or possiblymoderatelysmall,valuesof Rep. Accordingto theexperimentby

5

Fig. 1. Geometryof thechannel.

Hallouinetal. (1998),thecorrectionfactorby Brenner(1961)significantlyoveres-timatesthedragcoefficient for particlesatRep - 2 � 5. Unfortunately, noexpressionis availableatpresentfor adragcorrectionfactornearawall atmoderateReynoldsnumbersandin ashearor in turbulence.

3 Problem description

Thechannelhasahalfwidth,δ, of 2cmin y directionandis assumedto beinfinitelylarge in thestreamwise,x, andthespanwise,z, directions.The dimensionsof thecomputationaldomainis πδ . 2δ . 0 � 5πδ in x, y andz directions,asshown in Fig.1, with periodicboundaryconditionsappliedin x andz directions.Hereafter, forconveniencein notation,x, y andz directionsaresometimesalsocalledas1, 2 and3 directions,especiallywhenthedirectionappearsasasubscriptto variables.

For the fluid, air with a kinematicviscosity, ν, of 1 � 52 . 10$ 5m2/s wasassumed.Thewall-shearReynoldsnumberof theundisturbedfluid flow, Reτ, definedby

Reτ � uτ δν

(9)

whereuτ is theshearvelocity, wassetto 644.

6

Table1The restitutioncoefficient, r, Hamaker constant,A, the numberof particlesusedin thesimulation,N0, andcomputedmassflow ratio,Z, for eachcase.

r A /10 102 20 J3 N0 Z /%3Case1 1 0 1500 1.99

Case2 0.5 0 1600 2.04

Case3 0 0 1800 1.95

Case4 0 18 1900 1.98

Case5 0 180 2500 1.97

For particles,copperparticleswith adiameterof 70µm,correspondingto avalueofd 4 � 2 � 3 in wall unit,wereassumed.ThecorrespondingStokesrelaxationtime,τ 4p ,is 2000wall unit.Gravitationalforcewassetto g41 � 1 � 27 . 10$ 3 andg42 � g43 � 0.Thesevaluescorrespondto thedownwardgravity of 9.8m/s2 in thephysicalspace.

Fivedifferentcases,Case1 - Case5, wereexaminedaccordingto differentmodelsfor particle-wall impact in order to investigatethe influenceof accumulationofparticlesnearthewall of eachcaseasgivenin Table 1.

Theparticle-wall impactin Case1 - Case3 is modeledasbouncewith a restitutioncoefficient, r. If a particleimpactthewall with velocity up

2, its velocity incrementafterimpactis givenby

∆up2 � � 1 � r � up

2 . (10)

Thestreamwiseandspanwisevelocities,up1 andup

3, wereassumedto beunchangedthroughtheimpact.Therestitutioncoefficient, r wassetto 1, 0.5and0 in Case1,Case2 andCase3, respectively.

For Case4 andCase5, a modelwith a wall surfacepotentialwasusedin orderto further enhancethe accumulationof particlenearthe wall. In this model,it isassumedthata particlewith a diameter, d, andat a separationdistanceto a planewall of ys � y

� d 5 2� , experiencesanattractive force,potentialof which, Φ, canbeapproximated(seee.g.Friedlander, 1977)as

Φ � y�6� A d12ys

, (11)

whereA is known asHamakerconstant(Hamaker, 1937).

Usingthiswall potential,thefollowingboundaryconditionswereintroduced.When

7

a particlehits thewall, thewall-normalcomponentof themomentumis lost, sim-ilarly as in Case3. In thesecases,again,the streamwiseandspanwisevelocitieswereassumedto beunchangedthroughthe impact.If thewall-normalvelocity ofthe particlebecomesgreaterthana critical velocity, Vc, dueto re-accelerationbyfluid dynamicalforcesor collisionswith otherparticles,theparticleis re-entrained

with a velocity, 7 � up2 � 2 V2

c . Otherwisetheparticlestaysat thewall. Thecriticalvelocity, Vc, which is theminimumvelocity necessaryfor escapingfrom thewallpotential,is givenby

Vc �98 2Φ � y0 �m

, (12)

wherem is the massof particle.The minimum separationlength,y0, was set aty0 � 4 A, which is similar to thatusedby Li andAhmadi (1994)who studiedthedepositionproblemin a turbulentparticulateflow.

TheHamakerconstantbetweenacopperparticleandaglasswall, materialsusedintheexperimentby Kulick et al., canbecomputedasA � 18� 0 . 10$ 20 J usingtheHamakerconstantsof copperandglass(Dahneke,1972),Acopper � 28� 3 . 10$ 20 JandAglass � 11� 5 . 10$ 20 J, in therelationby Boehmeet al. (1969),

A �;: Acopper Aglass . (13)

In Case4, theHamaker constantcalculatedabove, A � 18� 0 . 10$ 20 J, wasused.In Case5, a valueof ten timeslarger thanthat in Case4, A � 180 . 10$ 20 J, wasusedin orderto seetheeffectof wall potentialmoreclearly.

4 Computational procedure

Thenumberof cellsusedin LESare32 . 42 . 96in x, y andzdirections.Themeshis stretchedin y directionsothatthecellsarefiner nearthewall andcoarserin thecenterof thechannel.Thecorrespondingmeshspacingin eachdirectionis shownin Table2. Thesuperscript,� , denotesthevariablesin wall unit.

At the beginningof the simulation,the particleswerehomogeneouslydistributedin the whole channelwherea fully developedvelocity field for fluid hadalready

Table2Meshspecifications.

Nx Ny Nz ∆x< ∆y<min ∆y<max ∆z<32 42 96 63.2 2.9 109.7 10.5

8

Particle 2 at time t

Particle 1at time t

Particle 2at time (t+dt)if collision were not considered

Particle 1at time (t+dt)if collision werenot considered

x (t+dt)R*i

x (t)Ri

x (t) + k (x (t+dt)−x (t))Ri

R* Ri i

Fig. 2. Computationof a inter-particlecollision.

beencomputed.The initial particle velocitieswere set to the local fluid veloci-ties.For thecomputationof theparticleequationof motion,eqn(4), a fourth orderLagrangianinterpolationwasusedfor obtainingthe fluid velocity at the particlelocations.BoththefilteredNavier-Stokesequationandtheparticleequationof mo-tion wereintegratedusinga third orderAdams-Bashforthschemewith a time stepof ∆t 4 � 0 � 322. The statisticswereaccumulatedfor about4000wall time unitsafter that theflow hadbeenfully developed,typically afterabout10000wall timeunits from the initial state.The convergenceof the flow statisticswasassuredbymonitoringtheglobal balanceof momentumandenergy of particles,similarly asin Fukagataet al. (1998b).

Thenumberof particlesusedin thesimulationsvariedfor differentcases,asshownin Table1, in orderto keepthemassflow ratio,Z, definedby

Z � ∑mp up=ρ f uf dV

, (14)

wheremp is themassof a particle,constant.Resultsshow thatZ � 2 > 0 � 05%inthefully developedflow for all casesdespiteall changesin thestructureof theflowdueto thechangesin particleboundaryconditions,asshown in Table1.

9

The computationalprocedurefor the inter-particle collisionsusedin the presentstudy is as the following. At first, provisional positionsof the particlesat time� t � ∆t � are computedwithout taking into accountcollisions,as representedbydottedcirclesin Fig. 2. For the judgementwhethercollisionsoccurbetweentwoneighbouringparticles,the methoddescribedby Yamamotoet al. (1998) canbeused.Namely, two particles,labeledasParticle1 andParticle2 asshown in Fig 2,canbejudgedto collideif theequationaboutthedistancebetweentheirtrajectories,

���xR � t ��� k � �xR? � t � ∆t � �xR � t �@� � 2 � d2 , (15)

hastwo real roots,k1 andk2 (k1 A k2), and the valueof smallerreal root, k1, isbetween0 and1, i.e.0 A k1 A 1. Here, �xR � t � is thepositionof Particle1 relative toParticle2 at time t, i.e.

xRi � t �B� x C 1Di � t � x C 2Di � t � , (16)

and �xR? � t � ∆t � is theprovisionalpositionof Particle1 relative to Particle2 at time� t � ∆t � . If theseparticlesare judgedto collide, the normalunit vector from thecenterof Particle2 to thecenterof Particle1 at thecontact,�nR, canbecomputedas

nRi � 1

d E xRi � t �F� k1 � xR?

i � t � ∆t � xRi � t �@�HG . (17)

By assummingherea perfectlyelasticcollision, thevelocity incrementof Particle

1 andParticle2 dueto thecollision,∆u C 1Di and∆u C 2Di , canbecomputedby�III� III� ∆up C 1Di � uR

j � t � nRj n

Ri ,

∆up C 2Di �J� uRj � t � nR

j nRi ,

(18)

whereuRi � t � is therelativevelocitybeforecollision,

uRi � t �K� u C 1Di � t � u C 2Di � t � . (19)

A perfectlyelasticcollision wasassumedin the presentstudy. For moregeneralexpressionincluding an inelasticcollision, particle rotationand friction betweencolliding particles,readeris referredto Yamamotoet al. (1998).

10

0

5

10

15

20

25

1 10 100

UL +f

y+

Fig. 3. Meanvelocity of the undisturbedflow at Reτ� 644. MONPM , LES in this study;

———, experimentby Kulick etal. (1994).

0

0.5

1

1.5

2

2.5

3

1 10 100

u+f

rms,

v+f

rms,

w+

frm

s

y+

Fig. 4. RMS levels of the undisturbedflow at Reτ� 644. MON*M , u< f

rms; - - 0 - -, v< frms;QRQRQ@S6QRQRQ , w< f

rms; LES in this study. ———, u< frms; - - - - -, v< f

rms; experimentby Kulick et al.(1994).

5 Results

In thissection,reportedareresultsfrom simulationsof particulateturbulentchannelflow whereinter-particlecollisionsareaccountedfor. Thefocusof thedescriptionis placedon the investigationof effects of particle wall boundaryconditionsonstatisticalbehavior of theflow.

Fig. 3 andFig. 4 show themeanvelocity andtheRMS levelsof velocity fluctua-tions of the fluid in a flow without any particles.Although the meshspacingin ydirectionin thecenterof channel,∆y4max � 109� 7, is likely to becoarsefor a largeeddysimulation,comparisonof thesestatisticswith experimentaldataof Kulick

11

et al. (1994)indicatedthatthefluid velocity field wascomputedwith a reasonableaccuracy.

It wasfoundthatthestatisticsof thefluid werenot differentfrom thosein unladenflow for all the casesstudiedhere.The observation that the statisticsof fluid isunaffectedby presenceof 70 µm copperparticlesfor the caseof Z � 2% is inaccordancewith the experimentalobservation by Kulick et al. (1994),exceptforthewall-normalvelocity fluctuationsof which a slight reductionof RMS valueinbulk of the channelascomparedto that in the unladenflow wasobserved in theexperiment.

Figs.5.a-cshow themeanandRMSvaluesof particlevelocities.Only Case1,Case3 andCase5 arepresented,becauseno significantdifferencescould be observedbetweenCase1 andCase2, andbetweenCase3 andCase4. The meanparticlevelocity , U 4 p, in Case5 is in agoodagreementwith theexperimentaldata,exceptfor thenear-wall region,y4PT 10.Thevaluesof U 4 p areslightly over-predictedinCase1 andunder-predictedin Case3 in wholechannel.An abruptincreaseof U 4 p

in y4UT 10 region in theexperimentaldatawasnot reproducedin any caseof thepresentsimulations.

In all cases,theprofile of U 4 p show betteragreementwith theexperimentaldatain the presentsimulationsthanthat of the one-way couplingsimulationby WangandSquires(1996).Onemayconsiderthatthis improvementof thesimulationdatain thepresentstudyover thosein WangandSquiresis causedby the introductionof extra effects, i.e. inter-particlecollisionsanddragcorrectionnearthe wall, inthepresentstudy. However, it wasobservedin thepreviousstudy(Fukagataet al.,1999)thatthedifferencesin meanparticlevelocitiescomputedusingfour-waycou-pling andone-way couplingsimulationswerenot significant,andthattheincreaseof dragcoefficient nearthewall hadonly a little effect on themeanvelocity. Fur-thermore,thediscrepancy observedhereis similar to thatappearswhentheflow ofparticlesis not fully developed,seeFukagataetal. (1998b).Therefore,wecancon-cludethatthediscrepancy in themeanvelocity is morelikely dueto aninsufficientdevelopmenttimeusedin thesimulationby WangandSquires(1996).

TheRMS level of streamwiseparticlevelocity, u4 prms, in Case1 shows similar val-

uesasthoseof theexperimentaldatain y4 - 30 region.Theprofile in Case3 andthat in Case5 aresimilar andhave a local maximumvalueas in the caseof theexperimentaldata.Thepeakof theprofile in Case5 almostcoincideswith theex-perimentaldata.However, Case3 andCase5 predictedsignificantlyhighervaluesin theouterregion,y4 - 30 thanthosefrom experiments.

The RMS level of wall-normalvelocity, v4 prms, in Cases1 is almosthalf of that in

the experiment,while in Case3 andCase5 it shows a goodagreementwith theexperimentaldata.

The probability distribution function (PDF) of the streamwiseparticle velocityaroundy4 � 12 planeis shown in Fig. 6. A clearbimodaldistribution similar to

12

a)

0

5

10

15

20

25

1 10 100

UL +p

y+

b)

0

1

2

3

4

5

6

1 10 100

u rm

s+

p

y+

c)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1 10 100

v rm

s+

p

y+

Fig. 5. Particle statistics. a) mean velocity; b) RMS streamwisevelocity; c) RMSwall-normalvelocity. MVNWM , Case1; - - 0 - -, Case3; - - S - -, Case5. — - —, one-waycoupling LES by Wang & Squires(1996); X , experimentby Kulick et al. (1994),Z � 2%; ( ——–, undisturbedfluid. )

13

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0 5 10 15 20

P(u

Y +p )u+p

Fig. 6. PDF of streamwiseparticlevelocity at y< � 12 plane. M,NPM , Case1; - - 0 - -,Case3; - - S - -, Case5; X , experimentby Kulick etal. (1994),Z � 2%.

theexperimentaldatais only observedin Case5. Theprofile for Case3 alsoshowsa weakbimodality. Assumingthat thesetwo modescanbeseparatedat u4 p � 12,themeanvalueof lower mode,U 4 p

L ,

U 4 pL � 1

PL

12=$ ∞

u4 pP � u4 p � du4 p , (20)

themeanvalueof highermode,U 4 pH ,

U 4 pH � 1

PH

4 ∞=12

u4 pP � u4 p � du4 p , (21)

Table3Meanvelocitiesandprobabilitiesof lower andhighermodes.

U < pL U < p

H PL PH PH Z PL

Case3 7.21 15.29 0.82 0.18 0.22

Case4 6.75 15.33 0.79 0.21 0.27

Case5 6.08 15.82 0.65 0.35 0.54

Experimentaldata 6.26 16.52 0.57 0.43 0.75

14

a) b)

u+p

y+

0

5

10

15

20

25

1 10 100

u+p

y+

0

5

10

15

20

25

1 10 100

c)

u+p

y+

0

5

10

15

20

25

1 10 100

Fig. 7. PDFcontourof streamwiseparticlevelocity. a)Case1; b) Case3; c) Case5. Differ-encebetweenlevel linesis 0.02.

andtheprobabilityof eachmode,PL andPH ,

PL � 12=$ ∞

P � u4 p � du4 p (22)

and

PH � 4 ∞=12

P � u4 p � du4 p , (23)

canbe calculatedusingthe datain Fig. 6, asshown in Table3. It wasfound thatthemeanvaluesandtheprobabilitiesof modescalculatedfrom thesimulationdatabecamecloserto thevaluesof theexperimentaldataasthewall potentialwasin-creased.

The global behavior of the PDF for all y4 positionsin half of thechannelis pre-

15

a) b)

0

5

10

15

20

-0.4 -0.2 0 0.2 0.4 0.6 0.8

u+p

v+p

0

5

10

15

20

-0.4 -0.2 0 0.2 0.4 0.6 0.8

u+p

v+p

c)

0

5

10

15

20

-0.4 -0.2 0 0.2 0.4 0.6 0.8

u+p

v+p

Fig.8.Phasespacedistributionof particlevelocityaty< � 12.a)Case1; b) Case3; c) Case5.

sentedin Fig. 7. The valuesof P � u4 p � arenormalizedsuchthat the integral overu4 p, at a fixed y4 , is unity. According to Fig. 7, bimodality is not observed atany position in the channelin Case1, while it is pronouncedin the region of20 T y4[T 50 in Case3 and 12 T y4 T 25 in Case5. In the near-wall region,y4\T 10, high velocity particlescomingtoward thewall aredeceleratedbasicallyby low velocity fluid in Case1. In Case3 andCase5, however, the velocity ofparticlesseemsto drop fasterandno or few particleswith high velocitiescanbeobserved nearthe wall. In the bulk region, y4 - 40, high velocity particlesarepresentin all cases,while theexistenceof low velocity particleswhich causehighvaluesof u4 p

rms is evidentonly in Case3 andCase5.

16

1e-13

1e-12

1e-11

1e-10

1e-09

1e-08

1e-07

1e-06

1e-05

0.0001

1 10 100

N+, f

+ c

y+

N+

f+

c

Fig. 9. Particlenumberdensity, N < , andcollision frequency, f <c . –N –, N < for Case1; - -0 - -, N < for Case3; QRQRQ^]OQRQRQ , f <c for Case1; - Q`_WQ -, f <c for Case3.

It canbeconcludedfrom theseobservationson PDF that the profilesof U 4 p andu4 p

rms in theregionwherebimodalityexistswill largelydependon theratiobetweenthe total numberof particlesin the highermodeandthat in the lower mode.Al-thoughthemeanvalueof eachmodeat y4 � 12 in Case3 seemsto besimilar tothatin theexperiment,thevaluesof U 4 p andu4 p

rmsaty4 � 12becamedifferentfromtheexperimentdueto thedifferencein theratio betweenhigherandlower modes,PH 5 PL. In Case5, thevaluesof U 4 p andu4 p

rms arecloserto the experimentaldatabecausePH 5 PL becomescloserto theexperimentalvalue.

The causeof the occurrenceof this bimodality in Case3 andCase5 canalsobeinvestigatedby studyingasnapshotof correlationsof (u4 p, v4 p) takenin theregion10 T y4aT 20,which is shown in Fig. 8. Differentgroupsof particlescanbeidenti-fiedhere.Thefirst groupconsistsof particleswith highvaluesof u4 p andnegativevaluesof v4 p. Theseparticlesarethosecomingfrom thecenterof thechannelandmoving toward thewall. Thesecondgroupconsistsof theparticleswith interme-diatevelocity andalmostzerowall-normalvelocity. In Case1, anotherfamily ofhigh velocity particlescan be identified, this time with a positive value of v4 p,which correspondsto particlesleaving the near-wall region afteran impactat thewall. In Case3 andCase5, this lastgroupof particlesmoving away from thewall,have a low streamwisevelocity, typically lessthan6, i.e. high velocity particlesmoving away from thewall aremissing.

From the resultspresentedabove, it canbe concludedthat the suppressionof theentrainmentprocessreducesthe maximumvalue of the lower modewhile a re-ducednumberof particleswith low streamwisevelocity resultsin an increaseofthemaximumvalueof thehighermode.Thestatisticalbehavior in Case3 andthatin Case5 wasfound to be qualitatively the same.The quantitative differencebe-tweenthesewasonly dueto the differentrateof entrainmentfrom the near-wall

17

-7

-6

-5

-4

-3

-2

-1

0

1

1 10 100

∆uY +p

y+

Fig. 10. Streamwisevelocity changeat collision of particle approachingto the wall.MbMVNcMOM , Case1; - - 0 - -, Case3.

region.Thereforein thefollowing discussionsaboutinter-particlecollisions,com-parisonsaremadeonly betweenCase1 andCase3. ThesamediscussionsarevalidbetweenCase1 andCase5.

Fig. 9 shows thenumberof particlesin a wall unit volume,N 4 , andthecollisionfrequency, f 4c , definedasthenumberof collisionsin awall unit volumeperawallunit time.In bothCase1 andCase3, collisionsin thevicinity of thewall occursfarmorefrequentlythanin the bulk flow. The particlenumberdensityat the samplepoint nearestto thewall in Case3 is about10 timeslarger thantha in Case1 andthe collision frequency is about100 timeslarger. In thebulk region, the collisionfrequenciesandnumberdensityhavesimilar valuesin Case1 andCase3.

Fig. 10 shows the meanchangeof streamwisevelocity of particles,herelabeledasParticle 1, dueto collisions.The labelwasdefinedsuchthat Particle 1, oneofthe colliding pair, hasalwaysa larger approachingvelocity toward the wall thanthe otherone,Particle 2. In both cases,the streamwisevelocity of theseparticlesin averagedecreasedafter thecollisions.The tendency is mostclearin thebufferregion,say20 T y4OT 60,in Case3.Thisphenomenacanbeexplainedbyascenarioconsistingof thefollowing steps:1) aparticlewhichhasobtainedhighstreamwisevelocity in the centerof the channelapproachestoward the wall; 2) the particlecollideswith a particlein thecloudof low velocity particlesexisting in thebufferregionsand3) the particle largely losesits streamwisemomentum.Suchdrop ofstreamwisevelocity is alsocontributing to thedecreaseof PDFof highervelocitymodein thebuffer region.

Thecorrelationsbetweendifferentcomponentsof velocity of Particle1 relative toParticle2, u4 R andv4 R, beforeandaftercollisions,areplottedin Fig. 11. In Case3, thedistribution is highly anisotropicbeforecollisionsandis isotropicafter thecollisions.Thismeansthatthestreamwisecomponentof thevelocity is partly con-

18

a) b)

-20

-15

-10

-5

0

5

10

15

20

-15 -10 -5 0 5 10 15

u+R

v+R

-20

-15

-10

-5

0

5

10

15

20

-15 -10 -5 0 5 10 15

u+R

v+R

Fig. 11. Correlationbetweenu< R andv< R. a) Case1; b) Case3. d , beforecollisions; 0 ,aftercollisions.

vertedto wall-normalvelocity componentdueto collisions.Althoughthenumberof samplepointsis few in Case1, asimilar behavior is observed.

The high RMS wall-normal velocity observed in Case3 can be relatedto morefrequentcollisions in the near-wall region thanthosein Case1. Due to large re-laxationtime of particles,suchhigh wall-normalvelocitiesin thenear-wall regionalsoresultin highv4 p

rms in thebulk of theflow. Valuesof v4 prmsbecomeevenhigherin

thebulk flow, asshown in Fig. 5.c.,becausetherelative importanceof theparticleswith highwall-normalvelocity increasesasthetotalnumberof particlesdecreases.

Makinguseof all theinformationabove,thepresenceof bimodalprobabilitydistri-bution functioncanbeexplainedasfollows.With purelyelasticcollisionsbetweenparticleswith long relaxationtime and the wall, particlesenteringthe near-wallregion leave it after an impact,and therebya reflectionappearsat the wall. Theparticlelosespartof its energy only dueto viscouseffects.How largeit will bede-pendsontherelaxationtimeof theparticleandthedirectionof its motionrelativetothewall. In suchcase,particlesof differentstreamwisevelocitieswouldbepresentin thenear-wall regionwith amono-modalprobabilitydistribution function.

On the otherhand,if thereis any mechanismthat canpreventall or partsof par-ticles to move freely away from thewall, a cloudof low velocity particleswill beformedin thevicinity of thewall. Partsof theseparticleswill be injectedinto theflow. A particlewhich entersthenear-wall region meetsthe low velocity cloudofparticlesandlosesits directionalvelocitydueto particle-particlecollisionsin anef-ficientway. Thiscreatesagroupof particleswith aprobabilitydistributionfunctionstronglydependenton theconditionsat thewall.

For thelattercaseof flow, with certaincombinationof parameters,thetwo family

19

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1 10 100

v rm

s+

p

y+

Fig. 12.RMS levelsof wall-normalparticlevelocity of Case3 with andwithout near-walldragcorrection.- - 0 - -, with correction;- - e - -, without correction. X , experimentby Kulick et al. (1994),Z � 2%; ( ——–, undisturbedfluid. )

of particles,i.e. the onewith low streamwisevelocity creatednearthe wall andtheonewith high streamwisevelocity comingfrom theouterregion, canco-existat a certaindistanceto the wall. If their meandirectionalvelocity is sufficientlydifferent,a bimodalprobabilitydistribution functionwill beobserved.

It shouldbementionedthattheaboveexplanationof thecauseof bimodalityof theprobabilitydistribution function is a detailedversionof thatgivenby Kulick et al.(1994)with a slight modification.Here,it is claimedandshown thatanadditionalmechanismthanhydrodynamicaleffectsis neededto traptheparticlenearthewallto createbimodalprobabilitydistribution function.

Finally, thecreationof thelow velocitycloudof particlesnearthewall andaneffi-cientscatteringof highvelocityparticlesgiveaclearcontributionto non-directionalvelocitieswhich can be observed in the whole channel.As a result, sufficientlyhigherRMS levels in thewall-normalcomponentof theparticlevelocity, in goodagreementwith experimentaldata,were predictedwhen an attractive force waspresentnearthewall.

As mentionedin section2, theformulausedfor near-wall dragcorrectionis validfor low particleReynoldsnumbersonly. Useof that formula might have overes-timatedthe drag force sincethe simulationdatashow, seee.g. Fig. 7b, that theprobability densityof particleswith high Reynoldsnumbers,sayaround10, i.e.u4 p f 4, in thenear-wall regionis notnegligible. In orderto ensurethediscussionsmadeabove,Case3 wasre-computedwithoutthenear-wall dragcorrection.Fig.12andFig. 13 show the RMS wall-normalparticlevelocity andPDF of streamwiseparticlevelocity in Case3 with andwithout near-wall dragcorrection.However,the RMS levels of wall-normalparticlevelocity andthe PDF of streamwisepar-ticle velocity seemnot to be sensitive to the choiceof formula of near-wall dragcorrection.

20

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0 5 10 15 20

P(u

Y +p )u+p

Fig. 13. PDF of streamwiseparticlevelocity of Case3 with andwithout near-wall dragcorrection.- - 0 - -, with correction;- - e - -, without correction. X , experimentbyKulick etal. (1994),Z � 2%.

6 Conclusions

Analysesweremadeto clarify thegeneraldiscrepancy foundbetweenresultsfromnumericalsimulationsand data from physical experimentsof dilute particulatechannelflow. The flow of 70 µm copperparticlesin a turbulent channelat lowmassloadingof 2% investigatedby Kulick et al. (1994)waschosenasa referencecase.Fluid flow wasmodeledusingLES.Motion of particleswasfollowedby LPTtakingintoaccountbinaryparticle-particlecollisions.Theinfluenceof wall-particleinteractiononthestatisticalbehavior of theflow wasinvestigatedby usingdifferentparticleboundaryconditionsat walls.

Thepresentanalysesshowed that the impactmodelfor particlesnearthewall, orin a moregeneralterm,themodelusedfor theparticle-wall interactionhasa verystrongeffectonthestructureof theflow in thewholechannel.Thisobservationcanbeattributedto thefact that in a dilute flow of particleswith long relaxationtime,thehistorythateachparticleexperienceshasa directinfluenceon all thestatistics.Secondly, particle-particlecollisionsareunexpectedlyof crucial importanceevenat low massloading.Thereasonsfor this couldbeclearlyillustrated.

In thepresentstudy, thebimodalityof theprobabilitydistributionfunctionthatwasobserved in experimentalinvestigationsbut not in thepastnumericalsimulations,couldbeobservedclearly. It wasshown thatthis bimodalityis causedby a mecha-nismthatcouldsuppressa directre-entrainmentof particlesafteranimpactat thewall. Suchaneffect will increasethenumberdensityof particlesin thevicinity ofthewall andcreatesa sourceof low velocityparticlesto beinjectedinto theflow.

Thelow velocity particlesin thenear-wall region interactswith high velocity par-ticlescomingfrom thebulk of theflow andresultsin scatteringof thoseparticles.This scatteringnearthewall increasestheRMS levelsof thewall-normalcompo-

21

nentof theparticlevelocity in thewholechannel.

The bimodality of the probability function andthe increaseof RMS levels of thewall-normalcomponentsmentionedabovewereenhancedwhentheparticlesweretrappedin a potentialnearthewall. The useof suchpotentialchangestheproba-bility distribution functionof particlesnearthewall, increasesthenumberdensitynearthe wall andcontributesto scatteringof high velocity particlesenteringthenear-wall region.

The lack of bimodality of the probability distribution function and sufficientlylower RMS levels in the wall-normal componentof the particle velocity in pre-vious simulationsof the casestudiedhereare thereforeattributed to the useofpurelyelasticcollisionsat thewall andtheneglectof inter-particlecollisions.

Although the presentsimulationscould reproducethe anomalousbehavior of ve-locity fluctuationssimilar to thoseobserved in thephysicalexperiment,thecauseof the phenomenacanbe different.As examples,small electrostaticchargesandsurfaceroughnessmay have similar amplifying effects on the statisticsof parti-cleflows.Accordingto thepresentresults,wall-particleinteractionmustbetreatedwith specialcarein bothexperimentalandnumericalinvestigationsof diluteflow ofheavy particles.Thesituationis certainlydifferentfor flowsathighparticlenumberdensityandparticleswith shortrelaxationtime,but not likely to belesscomplex.

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