A three-phase VOF solver for the simulation of in-nozzle cavitation effects on liquid atomization

an author's

https://oatao.univ-toulouse.fr/26916

https://doi.org/10.1016/j.jcp.2019.109068

Giussani, Filippo and Piscaglia, Federico and Sáez Mischlich, Gonzalo and Hèlie, Jérôme A three-phase VOF solver

for the simulation of in-nozzle cavitation effects on liquid atomization. (2020) Journal of Computational Physics,

406. 109068-109106. ISSN 0021-9991

A

three-phase

VOF

solver

for

the

simulation

of

in-nozzle

cavitation

effects

on

liquid

atomization

F. Giussani

a,b,

∗

,

F. Piscaglia

a

,

G. Saez-Mischlich

c

,

J. Hèlie

b

a_{Dept.ofAerospaceScienceandTechnology(DAER),PolitecnicodiMilano,Italy} b_{ContinentalAutomotiveSAS,France}

c_{ISAE-SUPAERO,UniversitédeToulouse,France}

a

b

s

t

r

a

c

t

Keywords:

Three-phaseLES-VOF Internalnozzleflows Cavitation Primaryatomization Injection

OpenFOAM

Thedevelopmentofasingle-fluidsolversupportingphase-changeandabletocapturethe evolution ofthree fluids, two ofwhich are miscible, intothe sharp interfacecapturing VolumeofFluid(VOF)approximation,ispresented.Thetransportofeachphase-fractionis solvedindependentlybyaflux-correctedtransportmethodtoensuretheboundednessof thevoidfractionoverthedomain.Theclosureofthesystemofequationsisachievedby acavitationmodel,thathandlesthephasechangebetweentheliquidandthefuelvapor and it also accounts for the interaction with the non-condensable gases. Boundedness and conservativenessof the solver inthe transport of the volume fraction are verified on two numerical benchmarks: a two-dimensional bubble rising in a liquid column and a cavitating/condensing liquid column. Finally, numerical predictions from large-eddy simulations are compared against experimental results available from literature; in particular, validation against high-speed camera visualizations and Laser Doppler Velocimetry (LDV) measurements of cavitating microscopic in-nozzle flows in a fuel injectorisreported.

1. Introduction

Cavitationplays apivotalrole inachievingfineratomizationofsprayto favoran improvedfuel economyandreduced emission levelsduring combustion[1,2];however,itmayalso causea significant reductioninthenozzle volumetric effi-ciencyandinthestabilityofthespray[3] andtopotentialdamageoftheinjectorcomponents,leadingtoreducedreliability oftheinjector.Ininjectornozzles,afterasurfacespotisinitiallysurroundedbyacavitatingflowregion,ittendstoerode atan acceleratedpace:cavitation pits increasethe turbulenceoftheflow andcreatecrevices thatact asnucleationsites fornewcavitationbubbles,thusleadingtoanavalancheeffect.Fromhighspeedcameravisualizationsontransparentglass nozzles,two differentforms ofcavitationshavebeendistinguished [4,5] andthey are knownas“geometry-induced” and “vortex”(orstring)cavitationrespectively.Geometry-inducedcavitationisinitiatedatsharpcornerswherethepressurefalls belowthesaturationvalue [6–8] becauseofasuddenflow detachmentandtheaccompanyingrecirculationregion.String (orvortex) cavitation,conversely,developsbytheevolutionofthevorticity whichallowstheformationofgeometry-scale vortices andissignificantly influencedby thewallsandtheinteractionwithother vortices[9];additionally,low pressure regionsinthecentersofthevortices inthenozzlecangenerateaphase-changeorentrapandstabilizebubblesthat were

*

Correspondingauthorat:Dept.ofAerospaceScienceandTechnology(DAER),PolitecnicodiMilano,Italy. E-mailaddress:filippo.giussani@polimi.it(F. Giussani).

entrained intheirproximity, similarlytowhatisobservedinhydromachines[10,11].Vortexcavitationintheinjector noz-zles wasfirst observedbyKim [12] andsince thenithasbeendescribed infurtherstudiesperformedinenlargednozzle replicas andwas termed as“string cavitation” [13,14]. The main differencesbetweenvortex cavitation inpropellers and turbinesandthoseinfuelinjectorsarisefromthegeometricconfigurationofthenozzleandtheoperatingconditionsofthe flow. Stringcavitationinnozzleflows developsinveryconfinedvolumes, thatmayallow formationoflarge-scalevortices relativetothenozzlegeometry,whereeachcavitatingvortexmayinterferewithothervorticesandwheretheinfluenceof the wallscanbe significant. Also,huge pressuredrops infuelinjectors areencountered within veryshortdistances (few hundreds ofmicrometers) whilethelifetimeoftheformedvorticalstructureisusuallyonlyafractionoftheinjection pe-riod. Cavitationstrings areusuallyformedduring fuelinjectioninareaswherelarge-scalevorticalstructuresdevelop:this happens when localpressure level islower than the vapor pressure ofthe fuel. Ina typical nozzle geometry,cavitation vortices arelocatedbetweentheseparationpointontheneedlesurface andtheseparationpointattheholeinletcorner, and wherethere issharpflow turning inside thesac volume oftheinjector. Unlikegeometricalcavitation, string cavita-tionispresentinanynozzlegeometry:withsac-typeandValveCoveredOrifice-type(VCO)nozzles,witheithercylindrical ortapered holes,whoseinletcan beeithersharporrounded. Cavitationmodelinginhigh-pressureinjectioninvolvesthe simulation ofmultiphase(in thecontextofthispaperrefers toliquidandgas)andmulticomponent (severalinstances of the same phase)flows and poses great challenges. Thesechallenges are dueto the presence of the interfaces between phasesandlarge ordiscontinuous propertiesvariations acrossinterfacesbetweenphasesand/or components;additionally,

the modeling ofsmall-scale interactions betweenphases andcomponents has a significant impact on macroscopic flow

properties,thisiswhyLESturbulencemodelingisoftenrequired.Twoapproachesarecommonlyusedforthesimulationof multiphase andmulticomponent flows.In thefirstapproach,eachphase and/orcomponentisconsidered tofilladistinct volumeandtheinterfacesbetweenthephasesand/orcomponentsarecapturedexplicitly.Thisapproachisageneralization of two-fluidapproach [15];typicalapplications[16–18] include thepredictionofthe motionoflargebubblesin aliquid, the motion of liquid after a dam break, the prediction of jet breakup, andthe captureof any liquid-gas interface. The mutualinteraction attheinterface canbe describedasan interfacialmomentum transferand, wheninterfacial massand energy transferare involved, they alsoneed to be includedin theequation sets. In orderto modelthe transferof mass forcavitation witha minimumsetofequationsforclosure, an equation ofstate tocorrelatedensityandspeed ofsound with pressure andtemperature is required:no additional transport equations are used forthe vapor phase, whose void fraction is determined by the mixture composition. Similarly to barotropicmodels, the densityis a function of pressure only. In the second approach, the phasesand/or components are spatially averaged to lead to a homogeneous mixture; relativevelocity amongthephasesisneglected,whichimpliestheabsenceofclosureforthetransferofmass,momentum andenergyattheinterfaceswhilethermalequilibriumamongthedifferentphasesisusuallyassumed. Thesemethodsare generallyidentified bythe VolumeofFluid(VOF) [19],the LevelSet(LS) [20] andthe CoupledLevel Set-VolumeofFluid (CLSVOF)[21–23].Thenumberofphasefractionequationssolvedcanvarydepending onthenumberoffluidinterfacesto capture.Fluidproperties,suchasdensityandviscosity,sharplyvaryacrosstheinterfaceofthedifferentphases;finally,the rateofthetransferofmassiscontrolled bysourcetermbuiltonRayleigh-Plessetequation [24–27].Severalattemptshave beendonetocombinethepotentialityoftheVOFwiththesimplicityofmixturemodel.In[28] thecavitatingfluidmixture (liquidandvapor)isconsideredasprimaryphasewhilethenon-condensablegasisthesecondaryphase;inthiscase,only one interfaceiscapturedbysolvingthephase-fractionequationforthenon-condensable(NC)gasandthevoidfractionof the cavitatingfluid mixtureis equalto

(

1

−

α

nc

)

;the volumefractionofthe fuelvapor inthecavitating fluid mixtureis

estimatedfromthemixturecomposition,wherethedensitiesofthedifferentcomponentsarecomputedthroughnon-linear equationsofstate.In[29],amasstransfermodelpublishedin[30] wasextendedtoaneight-equationtwo fluid-modelto includenon-condensablegases.Othermethodstodescribeathree-phaseflowwhileconsideringnon-condensablegasesare the useofthehomogeneous mixturemodelcombined withabarotropictwo-fluid cavitation model[31], orthecoupling ofatwo-fluidapproachwithVOF[32].Inthelattercase, atwo-fluid approachisusedtodescribetheinteractionbetween liquidandvaporinthenozzle,whileVOFisusedtomodelthejetformation.Allthementionedmodelshaveincommon theaspectthattheycaptureasingleinterfacebetweenthenon-condensablegasesandamulti-componentmixture[33,34]. Barotropic modelsare widely usedfor complex simulations becausethey are simpleto implement andnumerically sta-ble.On theother hand,oneofthe mainlimitationsusinga barotropicequation ofstate isinthe underestimationofthe vorticity change,because itdoesnot account forthemisalignmentbetweenthe gradientof pressureandthegradient of density

(

∇

ρ

× ∇

p

)/

ρ

2 _{[35],unlessa non-linearcorrelationbetweenpressureanddensityisused[33]. Thiscontribution,}

called baroclinity,isimportanteither incompressiblefluids andalsoinincompressible andinhomogeneousfluidsandit is identifiedby theinterface ina VOFmethod.Anotherchallenge whencavitation ismodeled usingbarotropicmodels is in the definitionof an appropriate equation ofstate forthe mixture,which includesair inaddition to liquidandvapor. Capture theinterface betweencoexistingmiscible phases(fuel vapor andnon-condensablegases inthiswork)ininjector nozzles maybeimportantinpresenceofswirlcavitation andhydraulic flipregime [36,37],whena severedetachmentof flowpockets[37,38] transportedawayfromtheholeallowsnon-condensablegasestoflowbackintothenozzle.This hap-pensbothinsimplified,straight,centralholeinjectors[39] andinnon-axialnozzles,inwhichlargepressurefluctuationsare observedinthenozzle.Recently,attemptstoextendVOF,inordertoincludeairintransportandincavitationmodelhave been done [40],[41] where eachphase isconsidered asa compressiblefluid butthecavitation modelused[42,43] have been previouslydevelopedunderincompressibleformulation.Conversely, in[44] eachphase isconsideredincompressible andisothermalbutthechangeofdensityisaddressedattheinterface andkeepsinconsiderationthe presenceoftheair

insidecavitationmodel.Amulti-fluidquasi-VOFmodelwiththetransportofthreephaseshasbeenproposedalsoin[45], consideringdifferentvelocitiesamongphasesandthusmomentumtransferrateamonginterfaces.Althoughseveralauthors havedevelopeddifferentmethodologiestodescribecavitation andthe jetformation,anysimplebenchmarkhasnotbeen found inliteraturein ordertoprovide anyfurther informationaboutthemassconservationguaranteed bythe cavitation model,eitherusingtheRayleigh-Plessetequation[24,27] orusingabarotropicequationofstatecoupledwithvaporquality. SometestshavebeenperformedusingashocktubeorRayleighbubblecollapsetestcase[46–49],whichdealwithsurface captureregardlessofmassconservationwhenthephasechangeoccurs.

The objectiveofthisstudyis topresent thedevelopment ofa three phaseVOFsolver forthree incompressiblefluids (liquidfuel,fuelvaporandnon-condensablegases),twoofwhicharemiscible,wherefuelcavitation/condensationis mod-eledthroughtheRayleigh-Plessetequation.Threedifferenttestcasesareproposed forthesolvervalidation:a)amodified versionofthetestcasepresentedin[50,51] toexploretheaccuracyofthesolverincapturingthreeinterfaceswithoutphase change.DifferentformulationstocomputethesurfacetensionintheVOFmethodhavebeeninvestigatedandacomparison amongthem hasbeenshown; b)a novel one-dimensionaltest case, that consistsofa 1Dcolumnhalf filledwith liquid, wherenon-condensable gas(air)stays above theliquid.The hydrostaticpressurefield resulting fromtheweight force of eachfluidinquiescentstateproducesapressuregradient



p attheinterface,lettingtheonsetofcavitationintheliquid. Lateron,condensationofthevaporisenforcedtoreachtheinitialstate.Therelativemassconservationerrorismonitored, togetherwithotherrelevantquantities,toprovetheconservativenessofthesolver;c)thethree-dimensionalinternal-nozzle flowofaninjector,whoseexperimentalresultsareavailableintheliteratureatoneoperatingpoint[52].Thediscretization ofthegoverningequationsusedinthisstudyisbasedonthefinite-volumeapproachasimplemented inOpenFOAM [53]. Mass andmomentum are solved usingthe pressure-implicit split-operator (PISO)algorithm [54]. The cavitation andthe condensation termhave beenincluded in a semi-implicitformulation ofthe phase-fraction equations,where a flux cor-rectedtransporttechnique[55] isusedtopreserveboundednessofthesolution;cavitationmodelingfollowsthetheoryby SchnerrandSauer[43] withtheextensionsproposedbyYuan[56].Theimplementedthree-phasesolverisabletocapture theinterfaceofthreephases,namelytheliquid,thecondensablegas(vapor)andthenon-condensablegas(air).Turbulence is modeled using LES: large turbulent scales are resolved, while smaller scales are modeled [57–59]. This separation of scalesisexplicitlyorimplicitly[60,61] obtainedbyfilteringoutthesmallflowscales thatcannotbeproperlyrepresented bythemesh[57];theireffectmustbemodeledonthefilteredfield bymeansofthesocalledsubgrid-scale (SGS)model. Althoughthemultiphasenatureoftheproblem,theuseofLESmodelsisalsoverypopularinmulti-phasesingle-fluidVOF simulations[62–67].ItisworthmentioningthatseveralnumericalstudieshavebeenledwithUnsteadyReynoldsAveraged NaviersStokes (URANS)equationsbutthisapproximationcan significantlyunderestimatetheformationandtheextentof cavitationduetoanoverestimatedturbulentviscosityinthecavitatingzones[68–72].AcomparativestudybetweenURANS andLESmodels[46] showsthatURANSmodelsfailtopredict theincipientcavitation whentheinletflowpressureisnot farfromthepressureatthenozzleoutlet,whileLESisabletobettercapturethecavitationonsetthankstoabetter charac-terizationofthedifferentflowscales.ApossiblesolutiontoovercomethelimitationsofURANSwhenappliedtocavitation modeling consistsinreducing theeddy-viscositypredictedby the turbulencemodel[68];thisapproach looks promising, butitsvaliditydoesnotseemgeneral;LESlooksthereforetobethebestapproachfortheproblemdiscussed,despiteofits highcomputationalcost.

The paper is organized as follows: the theory of the three phase solver withphase changeis discussed from Sec. 2

to Sec. 5; the discretized solution of the phase fraction equations inpresence ofphase-change, and its coupling to the segregatedsolutionofthegoverningequationsispresentedinSects.6,7andSec.8.Then,therisingbubbletest case[50] andanewone-dimensionalliquid-columnbenchmarktestcasehavebeenusedforvalidationinSects.10and11.Finally, thesimulationofathree-dimensionalinternalnozzleflowanditsvalidationispresentedinSec.12.Mainconclusionsare summarizedinSec.13.

2. Phase-fractionequationsintheVOFsolver

The cavitating fluid,the vapor andthe non-condensablegas inthe three-phase flow arerepresented ina single-fluid approximation as a mixture of phases, in which the phase-fraction distribution includes a sharp yet resolved transition betweenthephases.

Analgebraic-type VoFmethodbelongingto thefamilyoftheinterface-capturing methods[73],isused tocapturethe interface;morespecifically,theinterfaceisvisualizedbythecontourofascalarfunction,thatisassumedtobetheiso-value (setto0.5inthiswork)ofthevoidfractionofthephaseconsidered.Eachphasei hasapartialvolume Vi,thatisafraction

ofthevolumeV ofthecellelement(Vi

⊆

V )anditisdefinedbyitslocalvolumefraction

α

i

∈

[0;1]:

α

i

=

Vi V (1) with: 3



i=1

α

i

=

1 (2)

a“mixture”density:

ρ

=



i

α

i

ρ

i (3)

anda“mixture”viscosity:

μ

=



i

α

i

μ

i (4)

It is importanttonote that densityin thesolver varieswithpressure,through the phasetransport equations.The effect oftheheattransferonthetemperature,thatshouldbeaccountedbysolvingtheenergyequation,isnotconsideredinthe presentwork.Thecompletesystemofequationsforthree-phaseflowwithphasechangearethephase-fractionequations, thatarewrittenas:

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎩

∂

α

l

∂

t

+ ∇ · (

U

α

l

)

= −

Sα

ρ

l

∂

α

v

∂

t

+ ∇ · (

U

α

v

)

=

Sα

ρ

v

∂

α

nc

∂

t

+ ∇ · (

U

α

nc

)

=

0 (5)

In thesystemofequations(5), Sα isa sourcetermtomodelthe phase-change(cavitationorcondensation) attheliquid interfacethroughthecavitationmodelandcouplestheeffectsofthecavitationwiththeevolutionoftheinterfacedirectly:

Sα

=

ρ

v

ρ

l

ρ

+

α

nc

(

ρ

l

−

ρ

nc

)

D

α

v

Dt (6)

In Eq.(6) thesubscriptsl and v areadoptedforliquidandvapor (thatareinvolvedinthe phasechange) respectively, while the subscript nc isadopted for non-condensable gases. It is important to note that the closure of the system of equations(5) inpresenceofacavitation/condensationsourceterm Sα requirestoexplicitlyresolvethetransportofathird phase fraction(non-condensable phase),toincludeacavitation modelandtocoupletheequationswiththecompatibility condition (2);inthisway,thesystemisclosedandimplicitlybounded,thanksto(2).Inabsenceofsourceterms,3-phase VOFsolversusually calculatethevoidfractionofnon-condensablegases directlyfromEq.(2),that issufficientforclosure only in that case. Theseaspects are discussed in detail inAppendix A andAppendix B,where the derivationof the full systemofequationsisshownandtheformulationofthesourcetermsforthephasechangeisalsodescribed.

2.0.1. Counter-gradienttransportinVOF

In theFVframework, numericaldiffusion,which isveryhighinthe transporttermin second-orderspatial discretiza-tion,“smears” thesharpliquid-gasinterface.InOpenFOAM, thestrategycommonlyfollowedinmultiphaseVOFsolversto modelthetransportofthevoidfractionconsistsinaddingaconvectiontermwhichcompressestheinterfaceandpreserves boundedness: this is similar to what is done forthe treatment of the scalar-flux second-moment closure, used for the “counter-gradient” transportinsomecomplexcombustionmodels describingthedynamic ofturbulent flames[74].Inthe VOFtreatment,acommonclosureusedforcounter-gradienttransporthastheform:

∇ ·

[Uc

α

(

1

−

α

)

] (7)

whereUc isthecompressionvelocityattheinterfacebetweenthephases,whichisaconsequenceofthedifferentdensities

andtheterm

α

(

1

−

α

)

ensuresboundedness[55].IntheVOFsolverused,thecompressionvelocityismodeledas:

Uc

=

cα

|

U

|ˆ

ni j (8)

Theemployedformulationofthecompressionvelocityis: Uc

=

min



cα

|

U

|,

max

(

|

U

|)



ˆ

ni j (9)

ThediscretizedformofEq.(7) isaflux(counter-gradientterm,Eq.(38))computedatthecellfacesusingaTotalVariation Diminishing(TVD)scheme.Inthiswork,theTVDschemeusediscalled

interfaceCompression

scheme[55],inwhich thelimiter

ψ

tocomputethefluxisdefinedas:

ψ (φ

P

, φ

N

)

=

min

max

1

−

max



1

−

4

· φ

P

· (

1

− φ

p

),



1

−

4

· φ

N

· (

1

− φ

N

)

,

0



,

1



(10)

ψ

isboundedbetween0and1;

φ

P isthevalueofthevariable,definedatthecellcenter,wherethelimiterisapplied;

φ

N

Thecompressionrateshouldbesetinordertoensureinterfacesharpness.Higher valuesofthecompressionratemight introducenumericalinstabilityorslowconvergence.ThetermCα inEq.(9) is thecompressioncoefficientanditissetto unityinthiswork.Thecompressioncoefficient Cα isa binarycoefficientwhichswitches interfacesharpeningon(Cα

≥

1) or off (Cα

=

0). With Cα set to 0 fora givenphase pair, there is no imposed interface compression resulting in phase dispersion accordingto themulti-fluid model.IfCα is setto 1,sharp interface capturingisapplied andVOF-style phase fractioncapturingoccurs, forcinginterfaceresolutiononthemesh.Ifacompressiontermisnotapplied,theinterfacewill be very diffuse. In most applications, it is suggesteda cα of the order of unity [55]. A complete discussion aboutthe interfacecompressionmethodandthecompressioncoefficientcanbefoundin[75].

To ensure that the compression term does not bias the solution, it should only introduce flow of

α

, normal to the interface,inthedirectionofthevolumeaverageinterfacenormaln

ˆ

i j.Forathree-phasesolverithasbeencomputedasnet

gradientofthephasei-thattheinterface[53]:

ˆ

ni j

=

α

j

∇

α

i

−

α

i

∇

α

j



α

j

∇

α

i

−

α

i

∇

α

j



(11)

In the convention adopted (see Fig. 1), n

ˆ

i j always points towards the lighter fluid. A common practice isto use the

compressiontermonlywheresurfacesharpnesswantstobepreserved:intheproposedformulation,theconvection-based termisusedonlytocompresstheinterface betweentheimmiscible(liquid fuel)andthemisciblephases(fuelvaporand non-condensablegases).Thephase-fractionequationsforthethreephaseVOFtaketheform:

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

∂

α

l

∂

t

+ ∇ · (

α

lU

)

+

_

∇ · (

α

l

α

vUclv

)

+ ∇ · (

_

α

l

α

ncUclnc

_

)

compression term, liquid-vapor + liquid-non condensable gases

= −

ρ

v

ρ

+

α

nc

(

ρ

l

−

ρ

nc

)

D

α

v Dt

∂

α

v

∂

t

+ ∇ · (

α

vU

)

+

_

∇ · (

α

_

l

α

vUclv

_

)

compression term, liquid-vapor

=

ρ

l

ρ

+

α

nc

(

ρ

l

−

ρ

nc

)

D

α

v Dt

∂

α

nc

∂

t

+ ∇ · (

α

ncU

)

+

_

∇ · (

α

l

_

α

ncUclnc

_

)

compression term, liquid-non condensable gases

=

0

(12)

Fig. 1. Schematicoftheinterfacebetweentwofluids.fσ isthesurfaceforceperunitinter-facialarea;n andˆ κarenamelytheinterfacenormalandthe

interfacecurvature.Accordingtothesignconventionadopted,fσ isalwaysorientedtowardstheconcaveinterfaceandn alwaysˆ pointstowardslighter

coupleoffluids[76].

InEqs.(12),noadditionaltermstomodelinterfacecompressionbetweenmisciblephasesareused;numericaldiffusion isassumedto be sufficientto modelthesmall diffusionofmassatthe interface whenconvection isdominant(i.e.with largevaluesofReynoldsandlowSchmidt numbers).Differentmodelingapproachesforsubgrid-scale computationofthe masstransfer areonlyproposed forlow-ReandhighSchmidt numberflows [77–79].The term Dαv

Dt inEqs.(12) includes

theeffectsofthephasechange(cavitation/condensation)anditisthereforelinkedtothecavitation/condensationmodel. 3. MassconservationintheVOFsolver

Duringcavitationandcondensation,liquidandvaporphasearebothaffectedbyastrongvariationofdensity.Thelatter influences significantly the numerics of the segregated solver. Therefore, to ensure the stability of the solver under the aforementionedcondition, thecontinuityequation isusedinits non-conservative formassuggestedin[80] and whichit hasalreadybeenused[43,81,44]:

∇ ·

U

= −

1

ρ

D

ρ

Theadvantageofusingvolumefluxesratherthanmassfluxes(conservativeform)consistsofhavingcontinuousvolume fluxesattheinterface,thusfavoringthesolutionofpressurecorrectionequation.Withphasechangeandthreephases,Eq. (13) canberewrittenas:

∇ ·

U

=

ρ

l

−

ρ

v

ρ

+

α

nc

(

ρ

l

−

ρ

nc

)

D

α

v

Dt (14)

ThederivationoftheRHSofEq.(14) isreportedinAppendixA. 4. MomentumconservationintheVOFsolver

Themomentumequationreads:

∂ (

ρ

U

)

∂

t

+ ∇ · (

ρ

U

⊗

U

)

= −∇ ˆ

p

+ ∇ ·

τ

+

fσ

+

SU

−

g

·

x

∇

ρ

(15)

where p is

ˆ

a modified pressure, that iscalculated by removing thehydrostatic partfromthe staticpressure p,

τ

is the deviatoric stress tensor, SU includes the sourceterms, fσ isthe surface force per unit inter-facialarea calculatedat the

fluid interface in the control volume, g isthe gravitational acceleration.The term

−

g

·

x

∇

ρ

in the RHSof Eq.(15) is a consequenceoftheremovalofthemodifiedpressure p from

ˆ

thestaticpressure:

ˆ

p

=

p

−

ρ

g

·

x (16) whichyields:

∇ ˆ

p

= ∇

p

− ∇(

ρ

g

·

x

)

= ∇

p

−

ρ

g

−

g

·

x

∇

ρ

(17) so:

−∇

p

+

ρ

g

= −∇ ˆ

p

−

g

·

x

∇

ρ

(18)

The useof p in

ˆ

themomentumequationfavorsamorestablesolutionofthedensityjumps atthesharpinterfaceand simplifiestheimplementationandsetupoftheboundaryconditionsonpressure.InEq.(15),fσ isdefinedas:

fσ

=

σ κ

n

ˆ

δ (

x

−

xs

)

(19)

where

σ

isthefluidsurface tensioncoefficient in[N/m],n is

ˆ

theunitvector normaltotheliquidinterface,whosecenter is locatedin xs,

δ

isthe Diracfunctionto ensurethat the force isapplied onlyattheliquid interface,

κ

istheinterface

curvature

[

m−1

]

,whichisdefinedas:

κ

≡ −∇ · (ˆ

n

·

Sf

)

(20)

where Sf is the cell faces surface area vector defined asthe scalar product betweenthe cell faces normal andthe cell

facearea.InEq.(19),fσ isalwaysorientedtowardstheconcaveinterface(Fig.1).Itisimportanttonote thattheinterface curvatureinEq.(19) and(20) usedistheoneoftheinterfaces ofthephasewithhighestdensity(liquidinthiscase):

ˆ

n

=

∇

α

l

∇

α

l



(21)

Thesurfacetensioncoefficient

σ

appearinginEq.(19),hasbeenwrittenasanaverageofthesurfacetensionsweighted withthephase-fractionscomputedinthecontrolvolume:

σ

=

α

v

σ

lv

+

α

nc

σ

lnc

α

v

+

α

nc

(22)

where

σ

lv isthesurfacetensionbetweentheliquidfuelandfuelvapor,while

σ

lncisthesurfacetensionbetweenliquidfuel

andnon-condensablegases.Similarlytoviscosityanddensity,thesurfacetensioncoefficient

σ

forthemixtureiscomputed asaweighted-average,wheretheweightingfactorsarethevoidfractions;theconceptofmisciblephasesimpliesthattheir surfacetensioncoefficientiszero,so

σ

doesnotincludethesurfacetensionbetweenfuelvaporandnon-condensablegases. Finally,tocomputethesurfacetensionforce,thetermnδ (x

ˆ

−

xs

)

inEq.(23) mustbealsomodeled.TheContinuousSurface

Force(CSF)approximation[82] isthereforeused,yieldingto:

fσi

=

σ κ

∇

α

i (23)

It mustberemarkedthatCSFisthesimplestmodelcommonlyusedinVOFsolvers.Despiteitspopularity,ithasbeen proven (insurfacetensiondominated/drivenflow)thatitdoesnotguaranteemomentumconservation[83],leading tothe onset ofphysically unrealistic velocities at the interface. This parasitic current can be relevant when the flow is highly

dominatedbysurface tensionforces.Forthatreason, severaldifferentformulationsastheContinuum-Surface-Stress(CSS) approximation [84] andtheGhost fluidMethod(GFM)[85] have beenproposed.The formerisbasedon theintegral for-mulation ratherthan on thevolumetric one, andgivesmore advantages such asinclusion ofthe tangential stresses due to a variablesurface tension (i.e. Marangoni Effect). However, since it was onlyapplied within high-order (spline-based) front-trackinginterfacedescriptionframework[86,87],CSFisratherpreferredforitssimplicity,especiallyifsurfacetension is not dominant. On the contrary, the GFMis a technique used to handlesharp transitions andin the caseof capillary forces,it explicitly introduces the singular pressure jumpcondition intothe discretization equations.Eachphase is then artificiallyextendedacrosstheinterface,producingghostcellswhichcontainpropertiesoftheextrapolatedphaseusedfor thediscretization scheme,removingthe tendencyfortheadjacentinter-facialcells todiffuseduetothe sharptransition. Thismethodhasbeenalsoextendedbyseveralauthors[88–90] butalwaysbasedonLSandCLSVOFsincetheycancompute inamoreaccuratewaytheinterfacecurvature[91].

IfLES-specific filtering operationsare applied tothe momentum andto thephase-fraction equationsin a single-fluid multiphase solver, additional sub-grid terms appear. Theseterms can be identified underthe so calledinterface-specific sub-grid contributions. The term coming from the filtering applied to the term representing the surface tension forces (fσi

=

σ κ

∇

α

i) inthemomentumequationisa Sub-gridCurvature Tensor(SCT); whiletermscomingfromthefilteringof

the phaseadvectionterms (

∇ · (

U

α

i

)

) areknown asSub-grid MassTransfer terms(SMT). Publishedworkson two-phase

flowinterfacial flows withoutphase-change[92–94] provedthatinspecific flowconfigurations[92], thosesub-gridterms cannotbeneglected.AtleastthreepossibleclosuretermshavebeenpublishedtomodeltheSub-gridCurvatureTensor:the Sub-GridSurface Dynamics(SGSD)model[95],theSub-grid curvaturemodel[96] andtheADM-

τ

[97,98].No workshave beenfoundbytheauthorswheretheSub-grid MassTransfertermhasbeenmodeled,evenifin[92,99] itisreportedthat insome cases(likethe“separation ofphase”test case)itmayhavethesameorderofmagnitudeofthefilteredresolved advectionterm. Ontheother hand,thevapor phase-fraction

α

v ishardlysubjectto high-frequencyfluctuation according

to the definition of Eq. (B.5), because the pressure fluctuations only acts directly on the time-derivative of the bubble radius R [65]. Since cavitation is modeled by several micro-bubbles, their status is little sensitive to pressure pulsation ratherthan tothemeanpressure distribution;forthisreason, thesub-grid scalerelatedtotheSMT shouldbe negligible with respect to the filtered vapor volume fraction field in the cavitation regions where the mean pressure is relatively uniform. Moreover, in the flow regions with sharp pressure gradient, the vapor volume fraction is itself small enough to allow SMT to be neglected. It must be underlined that, the proposed closure termsfor SCT havebeen developed for explicit filteringapproach; when deconvolutionis performed, the velocity obtainedisnot strictly equal to U

=

U

+

Usgs,

beingthedeconvolutionoperatorapproximated.Finally,the“separationofphase”test[92],usedforthesub-grid interface analysis, is dominatedby the surface tensionforces effects. According to authors’research, no published works studied the contribution of the sub-grid interface with highspeed flows including phase-change: such a study wouldtherefore represent a big step towards an improved mathematical description of the formulation of LES single-fluid VOF solvers withphase-change. It isworth to remind that inthe FiniteVolume (FV)approach, both the computationalgrid andthe discretizationoftheoperatorsimplicitlyactasatop-hatfiltertotheequations[100].SincemostoftheCFDsolversinthe FVframeworkare usuallylimitedtosecond orderaccuracy[101],theSCTandtheSMTtermcomingfromfilteringofthe equationswouldbeprobablybiasedbythediscretizationschemesandbythegrid[102,103],evenwhenexplicitfilteringis applied.

Thefilteringprocedure wouldproduceanotheradditionalterminthephase-fractionequations(Eqs.(12))with phase-change.However,noinformationaboutthesetermswasfoundinliterature.

5. Cavitationmodel

Cavitation mayconsist either of smallbubbles (bubbly-flow cavitation)or maycontain large pocketsof vapor (cloud cavitation)[104];withasharpinterface-capturemethod,thebubblemustbelargerthanthecelltobeaccuratelyresolved, otherwise a sub-model is needed. In the approach followed in this work, a sub-model for cavitation is always used to provide an expressionforthe term Dαv

Dt . Thisisrequired toclose thesystemof governingequations(12), (14) and (15).

Theratesoffuelvaporizationandcondensationaredeterminedby asimplificationoftheRayleigh-Plessetequationwhich assumessphericalbubblesofradius R subjecttouniformpressurevariations.Sphericalbubblesarethenrepresentedbya fractionof thevapor phase inthe computationalcell; from[81] andconsidering thatliquid, vaporandnon-condensable gasesmaycoexistinacontrolvolume,itfollows:

Vv

=

Nb 4 3

π

R 3

₌

_n 0Vl 4 3

π

R 3 ₍₂₄₎

where Vv and Vl are respectivelythe volume ofthe vapor andthe liquid inthe computational cell of volume V, Nb is

thenumberofspherical bubblesofradius R in thecomputational cellandn0 isdefinedasthebubbleconcentration per

unit volumeofpure liquid.TheuseofrelationsofEq.(24) requiresan a-prioriknowledge ofthenucleiconcentrationn0

andan estimation oftheir initial radius R. Some measurements ofcavitation nucleiwere carried out a fewdecades ago on waterusing CavitationSusceptibility Meter(CSM) andHolographic measurement [105]; even though the holographic

measurements haveproved tobe moreaccuratethan CSM,both techniquescannot detectbubblesatthe sub-micrometer scale andthey mayomitmany additionalnuclei. In addition, thegrowth ofthe smallestbubbles is affectedby the sur-face tension,that is not considered [25] to havea simple correlation describingthe bubblegrowth. Aproper estimation of the surface tensionwould requirea numericalapproach to determinethe bubble growthrate, butit wouldrequire a significant increase ofthe computational cost [25]. Nucleationcan be originated eitherby homogeneous and by hetero-geneous nucleiaswell(airdissolved intheliquid,particles,etc.) [104,106].Forthesakeofsimplicity,onlyhomogeneous nuclei have been considered in the present model; as a consequence, it is not straightforward to set the value of this parameter forthe caseof high-pressure injection, since measurements of nuclei are not available in literature. For fuel injection, itisusually acceptedthat thenumberofnuclei,duetoimpurities,arelarge enoughthatthey shouldnot influ-encetheresultsofthemodel.Dissolvedgascould alsocontribute[107],butasafirststep,inthepresentstudytheyare not considered asnucleon precursors. The complete expression forthe rate offuel vaporization, derived in Appendix B, is: D

α

v Dt

=

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎩

3αvD R_Dt R+R4 4 3πn0  ρ+αnc(ρv−ρnc) ρ+αnc(ρ_l−ρnc) 

= −

3αvmax(p−psat,0)  2 3ρ_l|p−1psat| R+R4 4 3πn0  ρ+αnc(ρv−ρnc) ρ+αnc(ρ_l−ρnc) 

=

α

v



Dαv Dt

+ i f p

>

psat αl4πn0R2 D RDt 1+R3 4 3πn0  ρ+αnc(ρv−ρnc) ρ+αnc(ρ_l−ρnc) 

= −

αl4πn0R 2_min(p−p sat,0)  2 3ρ_l|p−1psat| 1+R3 4 3πn0  ρ+αnc(ρv−ρnc) ρ+αnc(ρ_l−ρnc) 

=

α

l



Dαv Dt

− i f p

<

psat (25)

whichcanberewrittenasanetcontributionbetweencavitationandthecondensation:

D

α

v Dt

=



D

α

v Dt

+

α

v

+



D

α

v Dt

−

α

l (26)

Inthetransportequationfortheliquidphase

α

l inthesystem(12) D_Dtαv isreplacedbyrewritingEq.(26) intheform:

D

α

v Dt

=



D

α

v Dt

+

α

v

+



D

α

v Dt

−

α

l

=



D

α

v Dt

+

(

1

−

α

nc

−

α

l

)

+



D

α

v Dt

−

α

l

=



_D

α

v Dt

−

−



D

α

v Dt

+



α

l

+



D

α

v Dt

+

(

1

−

α

nc

)

(27)

whileinthetransportequationofthevaporphase

α

v,stillinthesystem(12),Eq.(26) ismanipulatedtowrite D_Dtαv inthe

form: D

α

v Dt

=



D

α

v Dt

+

α

v

+



D

α

v Dt

−

α

l

=



D

α

v Dt

+

α

v

+



D

α

v Dt

−

(

1

−

α

v

−

α

nc

)

=



_D

α

v Dt

+

−



D

α

v Dt

−



α

v

+



D

α

v Dt

−

(

1

−

α

nc

)

(28)

Both inEq.(27) and(28),thefirstterminthesquare bracketsisthecoefficient ofapartoftheequation thatwillbe implicitlysolved, whiletheremaining partisexplicitlysolved. Asit willbeexplained inthefurthersections,solving the source terminasemi-implicitfashionfavorsimprovednumericalstabilityandboundedness.Thefinal formofthesystem describingthetransportofthethreephasefractionsreads:

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

∂

α

l

∂

t

+ ∇ · (

α

lU

)

+

_

∇ · (

α

l

α

vUclv

)

+ ∇ · (

_

α

l

α

ncUclnc

_

)

compression term, liquid-vapor + liquid-non condensable gases

= −

ρ

v

ρ

+

α

nc

(

ρ

l

−

ρ

nc

)



D

α

v Dt

−

−



D

α

v Dt

+



α

l

+



D

α

v Dt

+

(

1

−

α

nc

)



∂

α

v

∂

t

+ ∇ · (

α

vU

)

+

_

∇ · (

α

_

l

α

vUclv

_

)

compression term, liquid-vapor

=

ρ

l

ρ

+

α

nc

(

ρ

l

−

ρ

nc

)



_D

α

v Dt

+

−



D

α

v Dt

−



α

v

+



D

α

v Dt

−

(

1

−

α

nc

)



∂

α

nc

∂

t

+ ∇ · (

α

ncU

)

+

_

∇ · (

α

l

_

α

ncUclnc

_

)

compression term, liquid-non condensable gases

=

0

(29)

6. Solutionalgorithm

ThecoderesolvesthegoverningequationsbytheFiniteVolume(FV)SolutionMethod;acell-centeredformulationwith co-locatedarrangementisusedforthesequentialsolutionofthegoverningequationsonapolyhedralmesh.Thesegregated solution of the governing equations (mass and momentum) is achieved by a pressure-velocity coupling algorithm. The turbulentviscosity

μ

t ismodeledusingtheWall-AdaptingLocalEddy-viscositymodel(WALE)[108],whichhasbeenproved

tobesuitableforwall-boundedflowsandsingle-fluidapproach[9]. 7. Discretizedformofthephase-fractionequations

Thesolverhasbeendevelopedinordertomodelhigh-speedinjectionandprimaryatomization.Thisaverychallenging taskforinterfacetrackingandcapturingmethods.Inthisregard,themostimportantfactorthatmustbeconsideredisthe mutualeffectofturbulenceandcavitation.Breakup andcavitationprocessesaredominatedby surfaceinstabilities,which areaffectedbyturbulence,bytheboundaryconditionsandbythenumerics.Turbulenceintheliquid,andtoalesserextent inthegasphase,stronglyinfluencesthepredictionsintheinjectionbreakup;thefactthatsurfacestructuresbeingresolved areofasimilarspaceandtime-scaletosmall,butnotthesmallest,turbulentstructuressuggeststhatthisinteractioncannot be realisticallyrepresented by traditionalRANS modeling andthat LESturbulenceis themostappropriate approach. The useofLESimposestightconstraintsonthenumericsinthecasesetup:highresolutionschemesandaccuratenumericsare requiredfordifferentiation, inordertopreservetheenergyassociatedwiththeresolved turbulentstructuresandtoavoid a numerical error working as artificialdissipation [102]. On the other hand, high-order methods applied to high-speed flows incomplexgeometries mayleadto instabilities;forthisreason, specialcaremust betakenin thediscretizationof theconvectionofmomentum andofthetemporal derivatives.Inparticular, thenumericalfluctuationcreatedby theVOF approach,possiblycomingfromthecompressionandmostly fromthecavitation/condensationsourceterms,arepreserved by second-order time-differencingschemes; asaresult, a wrong accumulationof energymaybe found and, inturn,the simulationisdestabilized.The stencilofthediscretizationforthephase-fractionequationsofsystem(29) makesuseofa first-ordertimedifferencingscheme,asithappenswhenfirstorderhyperbolicPDEareused[109]:

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

(

α

n_l+1

−

α

n_l

)

p



t Vp

+



f F_fn

=

Bn_l p



_D

α

v Dt

−

−



D

α

v Dt

+



n p

α

n_l+1 p Vp

+

B n lp



_D

α

v Dt

+



n p

(

1

−

α

nc

)

npVp

(

α

n+1 v

−

α

nv

)

p



t Vp

+



f Fn_f

=

Bn_v p



_D

α

v Dt

+

−



D

α

v Dt

−



n p

α

nv+p1Vp

+

B n vp



D

α

v Dt

−



n p

(

1

−

α

nc

)

npVp

(

α

n+1 nc

−

α

nnc

)

p



t Vp

+



f Fn_f

=

0 (30)

where the subscript P in the equationsindicates that quantities are defined at the centerof the Control Volume (CV);



fFi,f isthesumoftheconvectivefluxesforthei-thphase(seeSec.8)and:

Bn_l p

=



−

ρ

v

ρ

+

α

nc

(

ρ

l

−

ρ

nc

)



n p (31)

Bn_vp

=



ρ

l

ρ

+

α

nc

(

ρ

l

−

ρ

nc

)



n p (32)

Systemofequations(30) isthensolvedexplicitlyasfollows:

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

α

_ln+1 p

=

α

_ln p

−



fFni,f t Vp

+

B n lp



Dαv Dt

+



n p

(

1

−

α

nc

)

np



t 1

−

Bn lp



t



Dαv Dt

−

−



Dαv Dt

+



n p

α

nv+p1

=

α

n vp

−



fFin,f t Vp

+

B n vp



Dαv Dt

−



n p

(

1

−

α

nc

)

np



t 1

−

Bnvp



t



Dαv Dt

+

−



Dαv Dt

−



n p

α

n+1 ncp

=

α

n ncp

−



t Vp



f F_in_,f (33)

TotalVariationDiminishing(TVD)[110] isapplied inthecalculationof

α

n+1 lp and

α

n+1

vp ,toensureastablesolutionand

boundednessofthephasefractionwithlargedensityratios.Inthisway,theoscillationsinthesolutionneardiscontinuities inthephase-fractionequations(representedbytheinterfaceoftheVOF)aresmoothedandmonotonicityispreserved.Phase transitionleads to largevalues ofthefluxes:thisis particularlyapparent inthecalculation ofthe transportequation for theliquidphase.Ithasadirecteffectonthestabilityofthesolverandmayputconstraintsinthetime-stepadvancement. Similar considerations maybe drawn forthetransport equation ofthe vapor phase,butin thiscasethecontribution on thefluxesderivingfromthephasetransition(condensation)isnotsolargeasfortheliquidandaboundedsolutioncanbe achieved. Tostabilizethesolution ofthetransportequation fortheliquidfractionin(29), theterm

α

l

∇ ·

U isaddedand

subtractedonitsRHS.Theequationisthenwrittenas:

∂

α

l

∂

t

+ ∇ · (

U

α

l

)

+ ∇ · (

Uclv

α

l

α

v

)

+ ∇ · (

Uclnc

α

l

α

nc

)

= −

ρ

v

ρ

+

α

nc

(

ρ

l

−

ρ

nc

)

D

α

v Dt

−

α

l

∇ ·

U

+

α

l

∇ ·

U

=

= −

ρ

v

ρ

+

α

nc

(

ρ

l

−

ρ

nc

)

D

α

v Dt

−

α

l Dαv Dt

(

ρ

l

−

ρ

v

)

ρ

+

α

nc

(

ρ

l

−

ρ

nc

)

+

α

l

∇ ·

U

=

=

D

α

v Dt

ρ

l

ρ

v

ρ

+

α

nc

(

ρ

l

−

ρ

nc

)



−

1

α

l

+

α

l

(

1

ρ

l

−

1

ρ

v

)



+

α

l

∇ ·

U

=

=

D

α

v Dt

ρ

l

ρ

v

ρ

+

α

nc

(

ρ

l

−

ρ

nc

)

Al

+

α

l

∇ ·

U (34)

ThediscretizedformofEq.(34) reads:

(

α

_ln+1

−

α

_ln

)

p



t Vp

+



f F_fn

=

Bn_l p



_D

α

v Dt

−

−



D

α

v Dt

+



n p

α

_ln+1 p Vp

+

Bn_l p



_D

α

v Dt

+



n p

(

1

−

α

nc

)

npVp

+

α

nlp



f Un_f

·

Sf (35) where Bn_l p

=



ρ

l

ρ

v

ρ

+

α

nc

(

ρ

l

−

ρ

nc

)

Al



n p (36)

Thefinalexpressiontocalculatetheliquidfraction

α

n+1 l in(29) is:

α

_ln+1 p

=

α

n lp

(

1

+



fUnf

·

Sf t Vp

)

−



fFfn t Vp

+

B n lp



Dαv Dt

+



n p

(

1

−

α

nc

)

np



t 1

−

Bn_l p



t



Dαv Dt

−

−



Dαv Dt

+



n p (37)

Multiple calculations of the system of equations (33), with updated values of the phase fractions, are performed to favoraboundedandmoreaccuratesolution.Thisprocedureisrepeatediterativelyuntiltheglobalconservationofthevoid fractionsis reached. Itis importanttonote that, similarlyto whatis done inthecalculation ofthe specietransport, the last phase (non-condensable gases in the present work)is usually solved asthe complement to reach the unity. In the caseofcavitatingflows,asdiscussedinthepreviousparagraph,thesolutionofthephasefractionequationsisrequiredfor closure;thismeansthat,aftertheiterativeprocedurejustdescribedtocalculateindependently

α

l,

α

vand

α

nc iscompleted,

the (small)residual errormust be added tothe non-condensable phase fractionbefore the pressure-velocity couplingis calculated.

8. Discretizationoftheconvectivefluxesinthephasefractionequations(MULES)

The implemented systemofphase-fraction equationsfor thethree-phase VOF,Eq. (12), have beendiscretized follow-ing the MultidimensionalUniversal Limiter withExplicit Solution (MULES), to ensure boundednessandconsistency even in presenceof flow cavitation andcondensation.The method to solvethe phase fractionequations isfundamentally ex-plicitandintroduces astrictCourantnumberlimitwithadirectimpactontime stepadvancement;timestepsub-cycling, commonlyusedtoenlargetime-stepsinVOFsolvers,isappliedheretoensureconsistency andboundednessofthe solu-tionwithstrongcavitation/condensation. Oneofthecriticalissues withthe VOFmethodused isthediscretizationofthe advectiveterminEq.(37),thatincludeseithertheconvectivefluxesandthecounter-gradientterm(compressivefluxes):



f F_in_,f

=



f

α

n_i fUf

·

Sf

  

convective fluxes

+



f

α

n_i f



j=i

α

n_j fUCi j f

·

Sf







counter-gradient transport

=



f F_αn f

+



f F_Cn f (38)

Numerical diffusivityof first order schemes mightsmear the interface;on the other hand, higherorder schemes are unstableandmaycausenumericaloscillations.Itisthereforeneededtoderiveadvectionschemesabletokeeptheinterface sharp andto produce monotonic profiles ofthe colorfunction. In the modified systemof phase fractionequations (33), theFlux Correct Transport(FCT)technique hasbeen applied:flux limitersare computedbyan iterativeprocedure which allowstheuseofhigh-orderschemespreservingboundedness,massconservationandsharpinterfacecapturing.Thetheory originally formulatedin[111] was furtherextendedtomulti-dimensionalproblemsin[112]. Themethodinvolvesseveral stages of calculation: first, the discretization of the advective term Fn_αf is provided by a higher FH_f,i and a lower order FL_f,i (obtained applying a monotonic and a diffusive advective scheme) flux approximation; Then, an anti-diffusive flux (FA) isdefinedtoattemptandreduce thenumericaldiffusionresultingfromthelowerorderscheme.Anestimate ofthe

anti-diffusivefluxesFA_{forthei-thphaseequationisgivenby:}

FA_f,i

=

FH_f,i

−

FL_f,i (39)

Anti-diffusivefluxesFA

f,iarelimitedto FCf,ibyaflux-limitingtechnique[112] basedonthecalculationofaTVDlimiter

λ

topreventundershootsandovershootsinthephasefractioninthecontrolvolume:

FC_f,i

= λ

FA_f,i with

λ

∈ [

0

,

1

]

(40)

being

λ

=

f

(

Fn

αf

)

afunctionofthevoidfractions.

9. Definitionoftestcasesandsetup

Simulationshavebeenperformedonthreedifferentcases,inordertotestthenumericalpropertiesofthenewly imple-mentedsolver(inthefollowingreferredas interPhaseChangeMixingFoam)intermsofabilityto:a)capturetheinterfaces, whilemaintainingthemsharp;b)preservetheconservativenessandtheboundednessofthesolutionofthephase-fraction equationswithphase-change.Finally,therobustnessofthesolveranditsapplicationtothedescriptionofrealflowphysics istestedona largeparallelsimulation ofaninjector geometry.Validationtestcasesthat willbe discussedinthefurther sectionsare:

1) the evolutionofa two-dimensionalbubblerisingin aliquidcolumn[50],to testthe abilityofthesolverto properly capturetheinterfacebetweenfluidsofdifferentdensities;

2) the studyoftheevolution ofafree-surfaceinapartially cavitating/condensingliquidcolumn, toverifythe conserva-tivenessandboundednesswhilephase-fractionequationsaresolvedwithphase-change;

3) acavitatingflowevolvinginsideaninjectornozzle[52,113]. 10. Bubblerisinginaliquidcolumn

The two-dimensional bubblerising ina liquid column [50] has been proposed as a validation test-caseto study the ability of the multiphasesolvers to capturean interface. The first test case studiedconsists ofa two-dimensional rising bubble problem,where agas bubbleimmersedina chamber filledwithliquid movesuntilit breaksup. The casesetup, the boundary conditionsandthephysical propertiesofthe fluid aredescribed in Fig.2. Forcesactingon thebubble are surfacetensionandgravity.Thedomainhasanaspectratiowidth/height

=

0.5;no-slipboundaryconditionsonthevelocity are setat the upperandlower boundaries, whilefree-slip is applied atthe rightand left bounds;gravity g isoriented towards thenegative y direction.At timet

=

0 s,thebubblecenterislocatedat

(

x

,

y

)

= (

0

.

5

,

0

.

5

)

andthebubbleradius is rb,0

=

0

.

25 m. Thistest-casehasbeen alreadyusedby theauthors in[114] to validate a2-phase VOFsolverandit is

nowusedagainwithsomenecessarymodificationsinthecasesetup:asshowninFig.2,theevolutionofthebubbleinthe surroundingliquid(identifiedbythevoidfraction

α

1)isnowcapturedbythetransportoftwoidenticalphases(

α

2

=

α

3).

Inthistest,phasechangeisdisabled.Itisexpectedthatthesolutionofthethree-phasesolvertendstothesolutionofthe two-phasesolverof[114].ThisisnottrivialintheVOFframework,sinceglobalconservationofthephasefractionsismore difficult asthe numberofphasesincreases andina single-fluidsolver it isstrictly linked to the calculationof thefluid properties(seeEqs. (3) and(4)).

Parameter Value Unit

ρ1 1000 kg/m3 ρ2=ρ3 1 kg/m3 μ1 10 kg m−1s−1 μ2=μ3 0.1 kg m−1s−1 g −0.98 m/s2 σ 1.96 N/m Re 35 – Eo 125 –

Fig. 2. Bubblerisinginaliquidcolumn:casesetup,boundaryandinitialconditions.Fourdifferentgridresolutionsaretested:a)40×80cells;b)80× 160cells;c)160×320cells;d)320×640cells(referencesolution).

Thephysicalpropertiesofinterestofthefluid,listedinFig.2,are:

– theEötvösnumberEo,definedastheratiobetweenthebuoyancyforceandsurfacetension:

Eo

=

ρ

1U

2 gL

σ

(41)

– theReynoldsnumberReoftheliquid,definedas:

Re

=

ρ

1UgL

μ

1

(42)

whereL

=

2rb,0 isthecharacteristiclengthscaleandUg

=



2grb,0 isthecharacteristicrisingvelocity.AthighvaluesofEo,

thebubbleshapewillbesomethinginbetweentheshapeobservedfortheskirtedandthedimpledellipsoidal-capregimes, implying that a breakup islikely to occur [115].Simulations at highvaluesof Eoare challenging forinterface capturing

algorithmsandcanyieldtodifferentpredictionsoftheevolutionandoftheformationofnewlycreateddroplets.Following theworkof[50], theevolutionofthebubblehasbeeninvestigatedforatotaltime T

=

L

/

Ug anda fixedtime stepwith



t

=

1

/(

2Nx

)

shas beenusedfortime marching. Inthesurrounding regionofthe bubbleinitial conditionsforthevoid

fractionsare

α

2

=

α

3

=

0 and

α

1

=

1;in thebubbleregion,

α

1

=

0,

α

2

=

1 and

α

3

=

0 (left half)and

α

2

=

0 and

α

3

=

1

(righthalf)areset.Thisleadstoastair-casedshapedinterface,soapreliminarysimulationwithoutgravity(g

=

0)isneeded toobtainasmoothinitialbubbleshape.Theresultsfromthezero-gravityprecursorsimulationarethenusedastheinitial conditionfortheactualsimulation.

Quantitative validation ofthe described code extensions,based on geometricalmetrics proposed in[50], is now pre-sented. Fora faircomparisonwith[50],thegrid usedisCartesianwitharesolution Nx

×

2Nx cells,being Nx 40,80,160

and320respectively(fourgridsweretestedintotal).Themonitoredquantitiesfromthesimulationswere: a) bubblecenterofmass:

xc

=

˜

A

(

α

2

+

α

3

)

xcdxd y

˜

A

(

α

2

+

α

3

)

dxd y (43)

b) degreeofcircularity foratwo-dimensionaldomain[116],beingbubbleAreadefinedas Ab

=

π

r2eq:

C

=

perimeter of equivalent circle

actual perimeter of the bubble

=

2

π

req

˜

A

(

∇

α

2

+ ∇

α

3

)

dxdy

=

2

π



Ab π

˜

A

(

∇

α

2

+ ∇

α

3

)

dxdy

=

2

π



Ab π

˜

A

(

∇

α

2

+ ∇

α

3

)

dxdy (44)

wherereq istheequivalentradius,definedas:

req

=



Ab

π

=

˜

A

(

α

2

+

α

3

)

dxdy

π

(45)

TheC parameterisequaltounityforaperfectlycircularbubbleandlowerthanunityforothercases; c) meanrisingvelocity:

uc

=

˜

A

(

α

2

+

α

3

)

U dxd y

˜

A

(

α

2

+

α

3

)

dxd y (46)

Results inthissection are organized as follows:using theprocedure of[50], simulations on thefour grids (40

×

80, 80

×

160,160

×

320and320

×

640cellsrespectively)arepresented,tomonitorthegrid-dependencyoftheresults.Ina secondstep,thesolutionfromthefinestgridistakenasreferencesolutionanditiscomparedwithCFDsimulationsfrom threeincompressibleinterfacialflow codes,namely:a)TP2D [117,118] andFreeLIFE[119],that arebasedonthe level-set approachappliedonastaticgrid;b)MooNMD[120] wheretheinterfaceistrackedinaLagrangianmannerandinnermesh points arethenprojected ontotheinterfaceby solvingalinearelasticityproblem.The evolutionintimeofthequantities described above is studied on four grids of different resolution: results are reported in Fig. 3 and in Table 1, while in Figs. 4 and5 the graphicalevolution of the bubblewith differentgrids is reported. Bubblecircularity (Fig. 3b) showsa monotonic diminishing tendency,that doesnot seem very influencedby the meshresolution, untilthe beginning ofthe bubblebreak-up.Conversely,meanbubblerisingvelocity (Fig.3c)doesnotpresentamonotonictendencyandthiscanbe justifiedbythelocaldeformation ofthebubble,thatisincreasingintime.Twolocalmaximaareclearly visible:whilethe bubble is rising, its velocity reaches a first maximum ucmax,1,whose position looks independent by the meshresolution

Fig. 3. Two-dimensionalbubblerisinginaliquidcolumn,validationtestcase.Evolutionintimeof:(a)bubblecentroidlocationC ;(b)bubblecircularity C∈ [0;1]and(c)bubblerisingvelocityuc;circularityisequaltounityifthebubbleshapeisaperfectcircle.Testshavebeenperformedby inter-PhaseChangeMixingFoamusingfourdifferentgrids:....40×80cells;–––80×160cells;—160×320cells; — 320×640cells.