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Editor-in-Chief

TheodoreE.Simos

AcademicianLaboratoryofAppliedMathematicsforSolvingInterdisci- plinaryProblemsofEnergyProduction,UlyanovskStateTechnicalUniver- sity,Ulyanovsk,RussianFederation,

SouthUralStateUniversity,Chelyabinsk,Russia,

Sectionof Mathematics,Departmentof CivilEngineering, Democritus UniversityofThrace,Xanthi,Greece

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PUBLISHER Elsevier Inc.

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Applied Mathematics and Computation 411 (2021) 126490

ContentslistsavailableatScienceDirect

Applied Mathematics and Computation

journalhomepage:www.elsevier.com/locate/amc

Investigating the effects of viscosity and density ratio on the numerical analysis of Rayleigh-Taylor instability in two-phase ﬂow using Lattice Boltzmann method: From early stage to equilibrium state

Bahrul Jalaali

^a

, Muhammad Ridlo Erdata Nasution

^b

, Kumara Ari Yuana

^c

, Deendarlianto

^d^,^e

, Okto Dinaryanto

^a^,^∗

aDepartment of Mechanical Engineering, Faculty of Aerospace Technology, Institut Teknologi Dirgantara Adisutjipto, Blok R Lanud Adisutjipto, Yogyakarta 55198, Indonesia

bDepartment of Aerospace Engineering, Faculty of Aerospace Technology, Institut Teknologi Dirgantara Adisutjipto, Blok R Lanud Adisutjipto, Yogyakarta 55198, Indonesia

cDepartment of Informatics, Faculty of Computer Science, Universitas Amikom Yogyakarta Jl Ringroad Utara, Yogyakarta 55281, Indonesia

dDepartment of Mechanical and Industrial Engineering, Faculty of Engineering, Universitas Gadjah Mada, Jalan Graﬁka 2, Yogyakarta 55281, Indonesia

eCenter for Energy Studies, Universitas Gadjah Mada, Sekip K-1A Kampus UGM, Yogyakarta 55281, Indonesia

a rt i c l e i nf o

Article history:

Received 25 June 2020 Revised 23 June 2021 Accepted 28 June 2021

Keywords:

Liquid-liquid two-phase ﬂow Rayleigh-Taylor instability Lattice-Boltzmann method Viscosity

Density ratio

a b s t ra c t

Thegravitationalliquid-liquidtwo-phaseflowwasnumericallyinvestigatedbyusinglat- tice Boltzmannmethod(LBM). The methodwas implementedforanalyzing amodelof Rayleigh-TaylorInstability(RTI).Thefeasibilityofthispresentnumericalapproachwasin- vestigatedbyperformingconvergencetest,andvalidatingtheobtainedresultswiththose obtained from experiments as well as other preceding numerical methods. Qualitative andquantitativecomparisonswereexamined,wherebygoodagreementsarenotedforall cases. Parametricstudies werealsoconductedby varyingbothofReynoldsand Atwood numbers toinvestigatetheeffects ofviscosityand density ratioonthebehavior offlu- ids interaction.Basedontheobtainedoutcomesofthisnumericalapproach, thepresent LBMwasabletosuccessfullysimulatethecompletephenomenaduringRTI,i.e.:thelinear growth,secondaryinstability,bubblerisingandcoalescence,and liquidbreak-up,includ- ingturbulentmixingconditionsaswellastheequilibriumstate.Thefindingobtainedfrom thisworkmightbebeneficialintheinvestigationofparametricbehaviorindesignofpro- cessesequipmentsuchasforseparatordesign.

1. Introduction

Theliquid-liquidtwo-phaseﬂowisoftenencountered inmanyapplicationswithinpetroleumindustries[1–3],e.g.:oil- waterseparationprocess [4,5].Thegravitationalliquid-liquidtwo-phaseﬂowhasbeenbecomingtheobjectofmultiphase studies andisrelatedtothedemandofoilprocess operation[6–8].Flow instabilitieswerefrequentlyfoundintheliquid-

∗Corresponding author at: Department of Mechanical Engineering, Faculty of Aerospace Technology, Institut Teknologi Dirgantara Adisutjipto.

E-mail address: [email protected] (O. Dinaryanto).

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B. Jalaali, M.R.E. Nasution, K.A. Yuana et al. Applied Mathematics and Computation 411 (2021) 126490

Fig. 1. The growth of RTI phenomena: (a) ﬁrst stage, (b) second stage, (c) third stage, and (d) fourth stage.

liquidtwo-phaseflow[2],forinstance,fluidmixingandseparationingravityseparatordevices[8].Theoperationofgravity separatorplaysanimportantroleforseparatingfluidphasesbeforetransportingintofacilities[9].Inthedesigndevelopment offluidseparationequipment,costreductionbecomesoneofmajorconcerns.Thiscanbeachievedbyminimizingthespace andweightofequipment,aswell asacceleratingtheseparationprocess[3].Severalcomplexitiesare particularlyfound in the design ofseparator, suchas theconsideration ofresidenceandsurge timeswhich relatesto fluid’s equilibriumstate anditsperformance [7,10], whereasthoseparameters areinfluencedby physicalpropertiessuch asviscosityanddensity [10,11]andtherateofachievingtheequilibriumstateispresentedastheinstabilityrate[12–14].Hence,itisimportantto improvetheunderstandingofinstabilitybehavioroffluidinteractioninthedesignconsiderationofseparator.

The gravityseparator involvesa phenomenonwhereby thedenser fluid flowsdownwardthrough thelighter fluid due to the body force of gravity [12, 13, 15,16]. The presence of body force instratified multiphase flow withzero velocity gradientemergencesthebuoyancy-driveninstabilitycalledRayleigh-TaylorInstability(RTI).TheRTIoccurswhentwofluids withdifferentdensityhaveanopposingtrendbetweenpressureanddensitygradients[15].Itleadstoaregimeofturbulent orchaoticmixingoftwofluids. Themixingisattainedafterseveralstagesasfollows:(i)initialperturbationwave growth (Fig.1(a));(ii)themovementofheavierfluidintothelighterfluidintheformofround-toppedbubblesduetotheirdiffer- entdensities(Fig.1(b)); (iii)thedevelopmentofspikesandHelmholtzinstabilityphenomenon(Fig.1(c));(iv)theforming ofturbulent regime (Fig.1(d)) [17–20].The presenceofturbulentmixinggenerates vibrationeffects[17,21],such asres- onance, whichmayleadto significantdamage tothedevice [1].Consequently,the aforementionedphenomenashouldbe consideredinthedesignofgravityseparator.Theflowinstabilitybehaviorisimportanttobecomprehensivelyinvestigated fromitsinitialstatetothecompleteequilibriumstate(i.e.thecompleteseparationbetweentwofluids).Whereasthedepth understandingofinstabilitybehaviorindesignconsiderationofoil-gasdevicesisindispensablesinceinstabilityisanunpre- dictedphenomenon.However,tothebestofauthors’knowledge,thereexistonlyafewresearchesinvolvingtheequilibrium stateofRTI.

Severalpreviousworkshadbeenanalyticallycarriedoutbyusinglinearstabilitytheory[15].However, someemployed assumptions yielded inaccurate results. Waddell et al.[22] experimentally investigated RTI by varying the density ratio using aheptane-water mixture.Intheearly stages,spikesandbubbleswere formed andtheinstabilityratewasgrowing exponentially. Meanwhile, the instability rateinthelate stages canbe considered to increase linearly.In thisregard,the similar trendswereparticularlyfoundatlow ReynoldnumberintheresearchofLewis[23]andWilkinsonetal.[24].The experimental resultsclarifiedthattheanalyticalsolutionwasonlyvalidfortheearlystagesofRTI.Theanalytical solution started to deviate at the late stages of RTI since the forming of the turbulent mixing regime. The flow behavior in the turbulentmixingdependedonthefluidproperties,i.e.:viscosityanddensityratio,numericallyrepresentedbyReynoldsand Atwoodnumbers,respectively.Bothpropertieswerealsofoundtoaffecttheinstabilityrate[22,24–26].Thus,RTIbehavior needstobecarefullyinvestigatedbymeansofvaryingthevalueofReynoldsandAtwoodnumbers.However,thiswouldbe quitecumbersometodoinexperiments.

The investigation of material properties and behavior including instability phenomenon can be efficiently performed through the useofnumerical analysis[1, 27]. InRTI,numerical analysisisoften performedby eitherconventional mesh ormesh-lessmethods[16,17,28,48,49].Numerousnumericalmethodshadbeenabletosuccessfullysimulatetheinitial stagesofRTI,forinstance,thephase-fieldmethod[21],finiteelementapproach[29],radialbasisfunction [30]andvortex incell[31].Nevertheless,thosestudieswere stillnot abletosimulateRTIintheequilibriumstate,whilenumericalinsta- bilitywasstill foundwhensolvingacomplexgoverningequation.Leeetal.[21]usedphase-fieldmethodandsuccessfully generatedtheRTItoanequilibriumstate.Ontheotherhand,itlackedaclearphysicalexplanation.

The particle-based method that has been developed into a powerful and eﬃcient method for an immense range of ﬂuid casesis Lattice BoltzmannMethod (LBM) [32–34]. LBM has advantagesin completely linear ofconvective operator

2

(9)

(streamingprocess) thannon-linearNavier-Stokes equations,whichmakes itlessdifficulttosolve. Furthermore,LBMuti- lizestheequationofstates,andthisyieldsthatPoissonequationisnotnecessarilytobesolvedinordertoobtainpressure terms. LBMis ableto formulatethe simplemechanicalrules fordeclaringthe complexboundarycondition[34–36].Sev- eral LBM methods hadbeen developedto approximate multiphase flow cases.Gunstensenet al.proposed color-gradient LBM [37].In thestudy,inaccurateresults ofmultiphaseflow wereobtained. The free-energyLBM wasproposed by Swift et al.[38] byutilizingmore complexequation andlack ofGalileaninvariance whichwasnumericallyinefficient. Further, both color-gradientandfree-energymethodsneed tobenumericallyimproved [34,39].Thepseudopotential methodwas developedby Shan-Chen(SC).The methodpossessedseveraladvantagesinbothnumericalefficiencyandsimplergovern- ingequation.Yet,thermodynamicsconsistency,numericalaccuracy,andlimitedReynoldsnumberbecometheweakness of thismethod[34,39–40]. Heetal.[17]introduced thenovelLBM (HCZ)to overcomethelimitationoftheSC method.The method utilizedan indexfunctionto define afluid interface. Furthermore,HCZisableto facilitatea higherdensityratio andReynoldsnumber[41].CitedfromHuangetal.[42],HCZmodelcan beimprovedbymodifying theforce schemeinto that of Chao etal.[43]. TheLBM for RTIcaseshadbeen previously employed by Fakhariet al.[44].The studywasable tosimulateRTIathighReynoldsandAtwoodnumbers.However, theinvestigationofequilibriumstatewasnotperformed.

TheimplementationoftheforcingschemeisusedinthisstudytoanalyzethebehaviorRTIandbecamethedistinguishand improvisationaspectcomparedwiththerelated-previousstudies.

The aim of thispaperis to conduct the physicalphenomena analysisof RTIfrom early stage to equilibriumstate by using unconventional numericalmethod of fluid. The analysis employs a novel perspective in comparisonwith previous studiesforamodelofimmiscibleliquid-liquidtwo-phaseflowinaccordancewithpracticalcase.Assessmentsoftheimpact ofviscosityanddensityratioto thefluidinstability isperformedwherebyviscosityisrepresentedasReynoldswhilethat ofdensityratioisrelatedtoAtwoodnumbers.Inthispaper,RTIsimulationiscarriedout untiltheRTIlate-stageorafter theemerged ofsecondaryinstabilityusingmodified-LBM.TheRTIintheturbulentmixingregimeconsistsofseveralfluid phenomenasuch asbubbleandspike coalescence,emerging ofsecondary instabilityandliquid break-up.Thispaper also aims to enrich the study of late-stage instability behavior in liquid-liquid two-phase numerical model, which is, to the bestofauthors’knowledge,noteasilyfound inthecurrentliterature.Inordertoexaminetheplausibilityoftheproposed approach, theresultsof presentanalysis arecompared tothose ofother conventional meshmethodaswell asmess-less method.

2. Methods

TheobservationdomainofLBMisamesoscopicscalethatconsidersasetofparticlesasaunitwithoutconsideringthe behaviorofeachparticle[32].Inordertodeterminethepropertyofthesetofparticles,adistributionfunctionisintroduced.

Thefunctionisdeﬁnedasastatisticaldescriptionofasystemthatcharacterizesthepropertiesofparticles.Byapplyingthe law of similarity, dimensionless parameters can be used to link between the propertiesin mesoscopic and macroscopic domains[32,33].

In the presentanalysis, a discrete distribution function is employed for simulating liquid-liquidflow. The simulation adopts the HCZ model,which islinked to theauthors’ previous work of Kumaraet al.[48]. It hasa limitation that the densityratiobetweenbothfluidsshouldbe morethan0.1[42].Inthe caseoftheinvestigationoftwo phaseswithhigh- densityratio, suchasliquid-gasflow, extension analyses employingLBM can be foundin theresearches ofFakhari etal.

[44]andGeier etal.[50].Itisalsoworthwhileto considerseveralnumericalanalyses intheframeworkofﬁniteelement methodinRefs.[51,52].

In thisproposedmethod,the analysisofliquid-liquidtwophase ﬂowutilizes twodistribution functionsnamelyindex distribution functionandpressuredistributionfunction asexpressedinEqs.(1)and2,respectively. Theindexdistribution functionperformsasatrackinginterface,whilepressuredistributionfunctionisbeneﬁcialtoanalyzethephysicalproperties.

Twosourceterms(i.e.Sf_i(x,t)andSg_i(x,t))areincludedinthedistributionfunctions.BothtermsaredeﬁnedinEqs.(3)and4. Inthoseequations,thetermi(u)isdescribedas ^fⁱ_φ^eq,whereby

ϕ

^denotesîndex^function⁽Êq.⁽⁵⁾^).^Theîndex^functions^have

φ

h<

φ

^<

φ

l

f luid2,

φ

l

Thefunctionof

ψ

⁽

ϕ

^),încludedⁱⁿÊq.⁽³⁾^,îs^necessaryⁱⁿ^two-phase^flowsimulationstoimitatethephysicalintermolec- ularinteractionintermsofparticlecollisions.Influidsproblem,theparticlecollisionsareinfluencedbysurfacetensionFs

andeffectiveforce

ψ

⁽

ρ

^).

fi

(

^x+ei

δ

^t,t+

δ

^t

)

−fi

(

^x,t

)

=−

f_i

(

^x^,^t

^x^,^t

) )

f

(

^ei−u

)

.

∇ψ ( φ )

RT (3)

Sg

(

^x^,^t

)

⁼⁻²

τ

^g⁻¹

2

τ

^g

⁽

^eⁱ⁻^u

⁾

^.^[

(7)

p=

g_i−u

2.

∇ψ ( ρ ) δ

RT.ThetermsofR,Tandw_iarethegasconstant,temperature, andweightingfactors,respectively.

f_i^eq

(

^x^,^t

)

⁼^wi

φ

1+3e_i.u

c² +9

(

^ei.u

)

² 2c⁴ −3u²

2c² (10)

g^eq_i

(

^x^,^t

)

⁼^wi

p+

ρ

^RT

3e_i.u

RTc² +9

(

^ei.u

)

² 2RTc⁴ − 3u²

2RTc² (11)

F=Fs+G=

κρ∇ ∇

²

φ

⁺ ^g.

ρ

⁽¹²⁾

Inthisstudy,theexternalandbody forcesare consideredthroughtheuseofgravityforce(G)andsurfacetension(Fs).

The sumoftotalforcesisgiveninEq.(12).Theterm

κ

^relates^the^magnitudeôf^surface^tension.^Numerical^modelîsôb-

served in two-dimensionalspace withninedirections ofmotion(D2Q9).The weighting factorof theconﬁgurationis ex- pressedas:

wi=

₍

₄_/₉

₎

_, _i₌₀

(

¹^/⁹

)

^, ⁱ⁼¹^,²^,³^,⁴

(

¹/36

)

, i=5,6,7,8

Thecorrespondingdiscretevelocities(e_i)foreachdirectioncanbedeﬁnedasfollows:

⁻^π4ⁱ

√2

(

^sin

α

i,cos

α

i

)

^, ⁱ⁼³^, ⁵^, ⁷^, ⁹

⎫ ⎬

⎭

The numericalmodel utilizedinthisnumericalworks isshowninFig. 2.Inthisﬁgure, v_x andv_y denotethevelocity components,LandHrespectivelyindicatethedomainlengthandheight,gisthegravitationalforce,whilefa,f_b,ga andg_b arethedistributionfunctions.

Aperturbationwaveissubjectedtotheinterfacebetweentwoﬂuids.ThewaveisexpressedinEq.(13),wherebyAisthe wave amplitudeandndenotes thewavelength.Asmallinitialperturbation wave(y)isalsogivenby Eq.(13)asaninitial conditionwherexandnisthelengthandnumberofwave,respectively.Inthemodel,bounce-backboundaryconditionsare appliedatthetopandbottomwalls.Therightandleftsidesofthemodelareunderperiodicboundaryconditions(Eq.(14))

4

(11)

Fig. 2. Numerical model of RTI simulation.

illustratedinFig.2.Furthermore,gravityforceisappliedintheﬂuidinterfaceaswell.

y= H

2 +0.1Ancos

₂

π

^x

n

(13)

fa= f_b;ga=g_b (14)

Thesimulationresultsarepresentedindimensionlessquantities.Theutilizeddimensionlessparametersaretimestep(ts), Reynoldsnumber(Re),andAtwoodnumber(At).TheReynoldsnumberistheratiobetweeninertiaandviscousforces.Itis expressed byRe= ^L√

Lg

ν . Meanwhile,the Atwoodnumberwhichrelatedto thedensityratiobetweenboth ﬂuidscanbe deﬁnedasAt=^ρ_ρ¹₁⁻₊^ρ_ρ²₂.Thetimedomainischaracterizedusingtimesteps(ts)expressedasT=

L g . 3. Resultsanddiscussion

In thepresent study,the observationdomainis shownin Fig.3.In thisﬁgure, schematicrepresentationof spikeand bubblewithrespecttothepositionisgiven.Theanalysiswillevaluatetheheightofthebubble(h_b)andspike(h_b)amplitude atcertaintimesteps.

Agridindependencetestwasﬁrstlyconducted.Thetestwascarriedoutbyvaryingthenumberoflatticeswhilekeeping the physicalparametersatconstant values.Thistest aimsto providethevalue ofindependencelatticeswherethecalcu- lations start to obtain convergent results. The test utilizestwo caseswith differentReynolds number(i.e. Re= 256 and Re=1024).Meanwhile,thetotalnumberoflatticesarevariedasfollows:25,600;40,000;65,536;90,000;and122,500.The simulationswere carriedoutbyusinggeometrywithaspectratioof1:4forLandHvalues.TheresultsaregiveninFig.4, wherebyonlyasmallchangeofthespikeamplitudeexistsintheresultsofbothRe=256andRe=1024forlatticenumber of65,536 andabove.Meanwhile, thenumericaltime tendstoincrease exponentiallyastheincrease oflatticesnumberas clearlyshown inFig.5. Theaforementioned factsconcludethat the numberoflatticesof 65,536 canbe regarded asthe optimumvalueinthisanalysis,andwillbeusedfortherestcalculation.

Inthispaper,thevalidationsare performedby comparingtheresultsobtainedfromthecurrentanalysiswiththoseof other numerical methods [21, 31, 44–46] as well as available experiment data[22,24] from open literature. The spikes and bubblepositions are considered asquantitative parameters.Numerical validations were conductedby simulatingthe RTI model underthe flow conditionof Re= 3000 andAt = 0.5. Both LBM-based andnon-LBMmethods are utilized as comparingmethods.Inthisstudy,thenon-LBMmethodsusedforcomparisonpurposeareVortexincellmethod[31]and finitedifference-phasefieldmeshlessmethod[20].ThevalidationresultsareshowninFig.6.

In thecaseofvalidationwiththe experimentalstudy[22],anumerical modelwithphysicalparametersof At = 0.156 andRe= 6000isemployed. Besides,theexperimental studyconductedbyWilkinson covers theearly stagesofRTIwith A_t=0.15[24].Thenumericalandexperimental resultsarequantitatively andqualitativelycompared.Here, themagnitude ofbubbleamplitudeisquantitativelystudiedintermsofspikeandbubbleposition.Fig.7ashowsthecomparisonofbubble andspikepositionbetweennumericalandexperimentalresults.Fromthisﬁgure,atTofabout0.2s,thedeviationbetween both resultstends to increase. Thisphenomenon occursdue to the ﬂuidsturbulence mixingcondition. However, within

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Fig. 3. Schematic deﬁnition of bubble ( h b) and spike ( h s) amplitude.

Fig. 4. The value of spike amplitude for varied number of lattice at particular Reynolds number.

Fig. 5. The duration of simulation at certain number of lattice.

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Fig. 6. Simulation results for bubble and spike position of current model compared with others numerical study.

Fig. 7. a) The spike and bubble position of simulation and experimental of Ref. [22] result comparison. b) The amplitude of bubble comparison in early stages RTI.

theearly stage(T=0to 0.2),both simulationandexperimental valuesareingoodagreements.Fig.7bshowsthebubble amplitudeofpresentstudycomparedwiththoseofRefs.[22]and[24].Goodagreementsarealsoobservedduringtheearly stage of RTI.Furthermore,thequalitative comparisonresults areshownin Fig.8.In thisﬁgure,the visual representation ofearly stageRTIissuccessfullysimulated.Thisfact canbeexhibitedby asimilarpatternofspikesandbubblesbetween simulationandexperimentalresultsforseveralcertaintime.

Forsakeofaccuracyassessmentofthesimulation,meanabsolutepercentageerror(MAPE)iscalculatedbythefollowing equation:

MAPE=1 n

n

i=1

^X^s^,ⁱX⁻c,i^X^c^,ⁱ

^x^100% ^(14a)

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Fig. 8. Comparison of numerical (below) and experimental (above) result of Waddell et al. [22] at certain time.

Table 1

MAPE value of numerical and experiment benchmarking.

MAPE of position comparison with present result

Bubble Spike Total MAPE value Numerical Studies

Tryggvason [31] 7.17% 8.66% 7.85%

Lee et al. [20] 8.10% 5.28% 6.49%

Fakhari et al. [44] 4.12% 8.21% 5.72%

Zu et al. [45] 5.39% 9.43% 7.34%

Ren et al. [46] 3.72% 8.93% 6.32%

Experimental Study

Waddel et al. [22] 12.72% 25.62% 19.03%

InEq.(14),X_s,i istheresultofpresentanalysiswhileX_c,idenotestheresultofcomparativestudy.ThecalculatedMAPE valuesare showninTable1.The resultsreveal thattheobtainedMAPEvalue forbubblepositionbetweensimulationand experimentis12.72%whilethosebetweenthepresentsimulationandothernumericalanalysesarelessthan10%.Interms of spikeposition,MAPEvalue betweensimulationandexperimentis 25.62%whichis stillconsidered asa reasonableac- curate. Meanwhile,MAPEvalue forspikepositionisnot morethan10% ascomparedtoother numericalstudies.Thisfact yieldsthatthecurrentLBManalysisisabletoaccuratelysimulatetheliquid-liquidtwo-phaseofRTIproblem.

InordertoinvestigatetheeffectsofviscosityanddensityratioonRTIbehavior,parametricstudies varyingseveralsets of Reynolds andAtwood numbers are performed. In the parametric studyof Re, three differentReynolds numbers were varied (i.e.Re=256;512;1024)underaconstantAt =0.5. Meanwhile,variationsofAtwoodnumbersof0.1, 0.3,and0.5 arealsoutilizedintheparametricstudyofA_t underaconstantofRe=1024.Here,theinstabilityratewillbeinvestigated byconsideringthespikeandbubbleposition.Intermsofseparatordesign,instabilityrateiscloselyrelatedtothesurgeand residencetimeinﬂuidseparationprocess.TheRTIstagesareshowninFig.9.

TheresultsofparametricstudyoftheeffectofReareshowninFigs.10and11forthebubbleandspikepositions,respectively.Ingeneral,spikesandbubblepositionstendtobequicklyshiftedforRe=1024thanRe=512and256.ThelowerRe means thehigherviscous force,consequently,thebubbleandspiketraveled-distancewillbeslightlydelayedcomparedto ahigherReduetotheabilitytoholdthegravitationalforce up.Inthefigures,thediscrepanciesofbothspikesandbubble positionsbetweentheresultsofRe=1024and512withrespecttoRe=256aregiven.AtT<1,thespikesandthebubbles are shiftedconstantlyexperiencing thelineargrowthinstability.Meanwhile, thefluidroll-upemerges asaresultofshear force asshowninT= 3forRe= 1024.Althoughthe spikepositionisslightlythesame,the fluidsroll-upforRe= 1024 is firstly formed than others. The higher Releads the fluid to become less viscous, hence the fluids frictional force de- creases.Therolling-upfluidexhibitsthesecondaryinstabilityofKelvin-Helmholtzinstability(KHI).Thespikedisplacement isslightlydelayedduringtherollingupprocess.Thisphenomenonoccursduetotheinteractionbetweengravitationaland viscousforcesonthefluidinterface.Furthermore,theemergenceofKHIcausesspiketobeacceleratedatT=4.Meanwhile, for smallerRe,theinstability ratewaslow andthe effectofKHI wassuppresseddue tothe lessdeformablebehavior of fluid.AdeeperinvestigationofspikepositionresultsinatendencyofspiketofallslightlyfasterinhigherRe.AtRe=1024, thestagesofinstabilityareclearlyobserved(i.e.fluidroll-up)incomparisontothecaseswithlowerReandthistrendisin linewithRef.[28].

Fromtheaboveanalysis,byneglectingtheeffectofthedensityratio,viscosity(i.e.intermsofRe)wasfoundtoslightly influence the instability rate. However, viscosityhas a significant influencein the fluid roll-upprocess. The vortices that

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Fig. 9. RTI phase to equilibrium state.

Fig. 10. Bubble position and value of discrepancy w.r.t Re = 256.

Fig. 11. Spike position and value of discrepancy w.r.t Re = 256.

occurredduringtheinstabilitystagesareclearlycapturedinthecasewithhigherRe.Theoutcomesareingoodagreement with the previous analytical studyof [15, 46]. This aforementioned phenomena lead ﬂuidsto be more deformable (less viscous)andincrease thedurationofturbulentmixingandsurfaceoscillationphaseasshowninFigs.12.Thisassertsthat increasingRewilldelaytheequilibriumstate,albeititincreasestheinstabilityrate.

Investigation oftheeffectsofdensityratioisperformedbyvarying Atwoodnumber.The resultsareshowninFigs.13 and14forbubbleandspikes,respectively.Thedensityratiocorrespondstotheeffectofbuoyancyforce,whichaffectsthe

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Fig. 12. The duration of RTI equilibrium state process for Re variation w.r.t. A t= 0.5.

Fig. 13. Bubble position and value of discrepancy w.r.t. A t= 0.1.

Fig. 14. Spike position and value of discrepancy w.r.t. A t= 0.1.

lighter ﬂuid to arise through theheavier ﬂuid.The spike andbubblepositions in thehigher At havea tendencyto shift fasterthanlowerA_t,thereforetheyareattainingtoturbulentmixingphasequickly.

WhenTislessthan1,thebubblepositionofAt=0.5isdelayedduetotheinteractionbetweengravitationalandviscous force attheinterface.When Tismorethan2,thegravitationalforce isdominatingtheviscous force,andthiscauses the bubbletolinearlyincrease.ThebubblepositionsinbothA_t=0.5and0.3showrelativelythesametrend.Ontheotherhand, atAt=0.1,thebubblepositionislowerbecausethedensitydifferenceissmall.BothﬂuidsatlowAtarehardtobeseparated so that affectsto the bubble andposition displacement. The formation of ﬂuid roll-up occursat T = 1 forAt= 0.5and acceleratesthespikeposition.Consequently,thespikepositionofAt=0.5hasaquickerdecreasethanthatofAt=0.1and

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Fig. 15. The duration of RTI equilibrium state process for A tvariation w.r.t Re = 1024.

Fig. 16. Plot of spike and bubble velocity and the region of early stages instability.

0.3.Meanwhile,forAt=0.1,spikefallsslowerandsmootherasthefluidroll-upisholdingup,hencethespikeaccelerationis delayed.Itmaybeconcludedthatduetothelowerbuoyancyforce,A_t=0.1,itwillgraduallyrisethaninthecasewithhigher At. Thebuoyancyforce causesa lighter fluidto moveupwardsfasterthan theheavierone. Thismayrelatestothe surge andresidence timein separatordesign considerations.As showninFig. 15,the equilibriumstate forhigherAt isquickly attainedthanthatoflowerA_t.Thehigherdensityratiomeansthatthefluidsarerapidlyenteringtheturbulentmixingand equilibrium state, andconsequently yields a lower surgeand residencetime. Contrarily, fluids withsimilar densities are difficult tobe separatedandwillhavehighersurge andresidencetime. Intermsof separatordesign,such fluidbehavior shouldbeconsideredinthedesignofseparatorlength.Theaforementionedfactsshowthattheinstabilityratewasstrongly influencedbythedensityratio.Thisagreeswithsomefindingsinexistinganalyticalstudies[14,15]aswellasexperimental workinRef.[26].

IncomparisontotheeffectofRe,thediscrepancyofbothbubblesandspikepositionswithrespecttotheresultsobtained by thelowestvalue ofAt(0.1)tendtobe moreﬂuctuated.FromFigs.13and14,itisfoundthattheincreaseofAtofthree times (i.e. from 0.1 to 0.3) results in the maximumvalue of discrepancyof about 180% and 210% forbubble and spike positions,respectively.Meanwhile,inFigs.9and10,thediscrepanciesobtainedduetotheincreaseofRefrom256to1024 (fourtimes)arelessthan15%.ThesefactsindicatethatAthasmoredominanteffectsonthetraveled-distanceofspikeand bubblesandinstabilityrate,ascomparedtoRe.Thisalsoagreeswiththeﬁndingsinother availableliteraturesinRefs.[17, 20, 46]. It canbe concludedthat densityratiohas amore dominantrole, ascomparedto theviscosity, intheinstability rate.

Inthisstudy,RTIsimulationwasperformedinacompletemanner.Thesimulationcoverstheearlystagesofinstability, turbulent mixingphase,aswell asthe equilibriumstateasthelast stage. Investigationofthe earlystagesisdivided into fourregion,ascanbeseeninFig.16.Thefirstregionexhibitsthatthegravitationalforcestarttopenetrateagainstthefluids viscousforce.Thegravitationalforcewillforcethedenserfluidtopenetrateintothelighterfluid.Thiscausesbothbubbles andspiketoacceleratefastasthebuoyancyforceofthelighterfluidtendstoagainstthegravitationalforce.Inthesecond region,bubblevelocityisrelativelyconstantascomparedtothatofspike,whichstillacceleratesdownwards.Onthecenter ofgeometry,thevelocityofhigherfluidwillmovedownwardduetothegravitationalforce.Becauseofexistenceofvelocity difference,liketheBernoulliprinciple,thehigher-pressurefluidwillpenetrateintothelower-pressurefluidtoformaroll- up mushroom-shaped. The early roll-up ofspike isformed atT = 2.This roll-upformation delaysthe bubbleand spike

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Fig. 17. Velocity ﬁeld at the emerging of secondary ﬂuid roll up.

movements.Theformationoffluidrollupinspikeisfoundinthirdregion.Thespikeroll-upmechanismisinitiallyformed as a resultof fluidsshear force interaction. The formof spikes vortex exhibitssecondary instability ofKelvin-Helmholtz instability (KHI) emerged. The KHI causes fluidsto roll-up,andforms other vortexes, which initiatesfluid mixingphase.

The secondfluid roll-upappearsatT= 3.2andthevelocity fieldisshowninFig.17.Theformationofthesecond vortex yieldsthespiketore-accelerate,andthenincreasesthebubblevelocity.There-accelerationofthespikepushesheavyfluid downwardandgivesadditionalforcetothebuoyancyforce.Meanwhile,fluidroll-upformationpullsfluidwithinthefluid vortex,hencethebubblevelocityisdelayed.Asthefluidroll-upmechanismiscompleted,bubblevelocity willincreaseas shownatT>3.2. ThefourthregioninvolvesspikeinstabilitywhichisshownT>3.2. Thespikeinstabilityregionisright beforethefluidmixingstage.Bothspikesandbubbleswillbere-acceleratedwithinthespikeinstabilityregion.Inthisstudy, the bubblepositionwillexponentiallyrise toobeythelinearstabilitytheoryasreportedinRef. [2].Sharp[15]identified that, when thefluid roll-up occurredasan exhibition ofKHI, fluid vortexleadsthe fluid tobe separated inthe formof a bubblewhichstatedasliquidbreakup phenomenon.In thisstudy,theliquidbreak-up canbe successfullysimulatedas showninFig.18.

The results are shownin Figs. 19and 20for earlyand late-stageinstability,respectively. As mentioned above, in the early stage, the fluidroll-up asa resultofKHI is formed,and thebubble rises.When the spikereaches thebottom, the turbulentmixingoccursasshowninFig.20(a)atT>8.Thedenserfluidwilldirectlymove downwardasthelighterfluid arises. The remaining lighter fluid is trapped inside the heavierfluid region and the bubble will be formed dueto the

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Fig. 18. Appearance of liquid break-up during KHI formation.

Fig. 19. Early stage of RTI simuation.

Fig. 20. Late stage of RTI simulation result.

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Fig. 21. The formation of bubble amalgamation.

Fig. 22. Bubble rising at certain time.

emergenceofsurfacetensionforce.ThebuoyancyforceinducesbubblerisingasshowninFig.20(b).Theinterfaceoffluids experiencesoscillationaftertheturbulentmixingconditioncompleted,whenthefluidshavefullyexchangedbetweeneach other (Fig. 20(c)). Furthermore,the gravitational force is not strongenough to make the lighter fluid to go through the densefluid.Thefluidinterfaceoscillationwillgraduallydecreaseastimegoesby.AsshowninFig.20(d),thefluidreaches theequilibriumstate,wherebythelighterfluidissituatedabovetheheavierfluid.

TheinvestigationofbubbleamalgamationandbubbleriseisreportedinFig.21.AtFig.21(a),twobubblesareformedand mergedatT=16.8(Fig.21(b)).Duetothesimilarvalueofbothﬂuidssurfacetension,thebubblewillmergebetweeneach other.Severalsmallbubblesarethenmergedintoabiggerbubbleasaresultofbuoyancyforce.Thisphenomenonwasalso reportedbyRef. [47].Thisresultprovesthat thecurrentLBM modelisabletoaccuratelysimulatethebubblecoalescence phenomenon. InFigs.22(a) and(b),thebubble riseisobservedatT =17.2 -18.1rightafterthebubblecoalescence.The subsequentbubbleriseappearsatT=20.3asshowninFig.22(c).TheaforementionedfactsindicatethatthecurrentLBM modelcanpreciselysimulatethebubblerisephenomenon.

4. Conclusion

A numericalstudyof thephenomenaduringRTI wasconductedthrough theuse ofa Lattice-Boltzmann method.The method isproposed toanalyzethe turbulentmixingphenomenainliquid-liquidtwo-phase ﬂow. Theanalyses performed inthepresentpaperareabletoaccuratelyinvestigateacompletebehaviorofﬂuidsinteractionincludingconditionswhen instabilityoccursaswellasthephenomenaduringtheequilibriumstate.Theremarkableresultsaresummarizedasfollows:

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