On the combined effects of surface tension force calculation and interface advection on spurious currents within Volume of Fluid and Level Set frameworks

O

pen

A

rchive

T

OULOUSE

A

rchive

O

uverte (

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DOI:10.1016/j.jcp.2015.04.054

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To cite this version :

Abadie, Thomas and Aubin, Joelle and Legendre, Dominique On the

combined effects of surface tension force calculation and interface

advection on spurious currents within Volume of Fluid and Level

Set frameworks. (2015) Journal of Computational Physics, vol. 297.

pp. 611-636. ISSN 0021-9991

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On

the

combined

effects

of

surface

tension

force

calculation

and

interface

advection

on

spurious

currents

within

Volume

of

Fluid

and

Level

Set

frameworks

T. Abadie

a

,

b

,

J. Aubin

b

,

D. Legendre

a

,

∗

a_{UniversityofToulouse,InstitutdeMécaniquedesFluidesdeToulouseCNRS/INPT/UPS,1AlléeduProfesseurCamilleSoula,31400Toulouse,} France

b_{UniversityofToulouse,LaboratoiredeGénieChimiqueCNRS/INPT/UPS,4AlléeEmileMonso,BP-84234,31030Toulouse,France}

a

b

s

t

r

a

c

t

Keywords:

Volumeoffluid Levelset Spuriouscurrents Continuumsurfaceforce Heightfunction Taylorflow

This paper deals with the comparison of Eulerian methods to take into account the capillarycontributioninthevicinityofafluid–fluidinterface.Eulerianmethodsare well-known to produce additional vorticity close to the interfacethat leads to non-physical spuriouscurrents.Numericalequilibriumbetweenpressuregradientandcapillaryforcefor the staticbubble testcase withinaVOF framework hasbeen reached in[35] withthe height-function technique[14,35].However, once the bubble is translatedin auniform flow,spuriouscurrentsaremaintainedbyslighterrorsinducedbytranslationschemes.In thiswork,twomainpointsareinvestigated:theabilityofVolumeofFluidandLevelSet methodstoaccuratelycalculatethecurvature,andthemagnitudeofspuriouscurrentsdue toerrorsinthecalculationofthecurvatureafteradvectioninbothtranslatingandrotating flows.Thespuriouscurrentssourcetermisexpressedfromthevorticityequationandused todiscussandcomparethemethods.Simulationsofgas–liquidTaylorflowatlowcapillary numbershowthattheflowstructureandthebubblevelocitycanbesignificantlyaffected byspuriouscurrents.

1. Introduction

Numericalsimulationsofindustrialprocesses,aswellasacademicsituationsofteninvolvetwoimmisciblefluids.A num-berofcomputationalmethodshavebeendevelopedoverthepastdecadetoimprovethecomputationofmultiphaseflows. The numericalmethods to simulatemultiphase flows can be classified intotwo main groups: the “Lagrangian” methods andthe“Eulerian”methods.Inthefirstclass ofmethods,theinterfaceis generallytrackedusingLagrangian markers,e.g.

Front-Tracking [49,36]or thePoint-Setmethod [48].The interface is locatedexplicitly andthecalculation ofgeometrical properties(normaltotheinterfaceandcurvature)ishighlyaccurate.However,theimplementationofsuchmethodsin3D isnot straightforward andspecificalgorithms are neededto dealwiththe distributionofmarkers, aswell aschangesin topology.Thesecond groupconsistsofanimplicitrepresentationofthephasesineachcellwithanadditionalscalarfield. The mostcommon approachesare the Volume ofFluid(VOF) [23,28,7,17] andthe Level Set (LS) methods [45,43,22,46]. VOFmethods are generallywell suitedto conservethe massofthe phasesandappearto be a naturalchoicein afinite

volumeframework,whileLSmethodsareknowntoallowbettercomputationofthegeometricalpropertiesoftheinterface. Eulerianmethodshavebeenshowntobewellsuitedtodealwithvariousconfigurations,includingasinglebubblerisingin aliquid[7],ajetordropbreaking[35,39],coalescence[46],aswellasatomizationwithanumberofinclusionsofdifferent sizes[19].

Within this Eulerian representation of two-phase flow, great effort has been dedicated to two main features: the transport of the interface andthe consideration ofcapillary forces.When dealing with flows where capillaryforces are preponderant, such asthesimulation ofTaylorflow (or slugflow)inmicrochannels, careneeds to betaken inthe com-putation ofsurfacetensionforces.Thissuggeststhat thenormaltotheinterface andthecurvatureneedto beaccurately estimated.Thiscapillaryforcelocatedontheinterfacecanberepresentedinaspatiallyfilteredwayusinganinterface thick-nesswiththeContinuumSurface Forcemethod[8]orinasharpwayusingthelocationoftheinterfaceatasub-celllevel,

e.g. GhostFluidMethod(GFM)[27]andtheSharpSurfaceForce(SSF)[18].Withinimplicitrepresentationsoftheinterface, manymethodsconsiderthesuccessivederivativesofthescalarfieldrepresentingtheinterface.Morerecently,[14]and[35]

haveshownthattheconstructionofheightfunctionsallowsabetterapproximationoftheinterfacecurvature.Indeed,this method consistsoffindingthe positionoftheinterface witha goodaccuracyby addingsuccessivevolume fractionsina columnoffluid(seeSection 3.2).Usingthisheightfunctiontechnique,[35] achievedanexactnumericalbalancebetween surface tensionforcesandpressurejumpwiththeeliminationofspuriouscurrentsinthecaseofastaticinterface. Inthe caseof inviscidfluidswherethere isno viscousdissipation tobalance thekinetic energyproduced byspurious currents,

[35] still observeda decreaseintheintensityofspurious currentsovertime duetonumericaldissipation.The numerical dampingofspurious currentswhendealing withinviscidfluidswas alsoobservedin[12] withaheight-function method and a finite-volume compressible flow solver where the intensityof spurious currentsdecreased to zeroafter a certain time. This numericaldissipation was furtherstudied in[19] and[12] withthe dampingofthe oscillationsofan inviscid droplet. Anumericalviscositywasestimatedby fittingthedampingoftheamplitude oftheoscillationsandasexpected, thenumericaldissipationdecreasesasthemeshisrefined.

However, it was also shownin[35] that thecoupling betweentransport schemes,surface tension force withNavier– Stokes equationsand curvature estimation still needs improvements. [15] showedthat the advectionstep has no direct influence onthe spurious currentswitha balanced-forcesurface tensionalgorithm when theexact curvatureisimposed on bothcartesianandtetrahedralmeshes. However,asitwillbecomeclearinSection4,imposingtheexactcurvature al-lowsthe development ofspurious currentsto be avoidedsince thespurious currentsource termiscanceled. Incases of practical interest, thecurvature cannot be imposedandthe advectionstep introduces errorsin thevolume fractionfield therebyleadingtocurvaturegradients.Asaconsequence,thezerovelocityfieldexpectedintheframeofreferencemoving withthebubble(like forthestaticcase[35])isnot recoveredwhen theinterfaceistranslatedinauniformflow [35,12]. Theseobservationshavemotivatedourwork.Differentnumericalmethodsimplementedinthesameflowsolverhavebeen comparedintermsofthemagnitudeofspuriouscurrentsandpressurejumpevaluationonthebasisoffourtestcases:the staticbubblecaseforwhichanumberofresultsareavailableintheliterature;thetranslating bubble,whichseems more related tophysicalflows;a bubbleina rotatingflow;andthe dynamicsofTaylorbubbleina circularmicrochannel. The objectivehereistopayattentiontothecouplingbetweentheinterfacetransportequationandtheNavier–Stokesequations. Finally,thedynamicsofTaylorbubblesinmicrochannelsisconsideredbecauseitappearstoberepresentativeoftheability of a particularmethod to deal withspurious currentssince these flows are dominatedby surface tension (lowcapillary numberandlowWeber number).Aswillbeshown, thedevelopmentofspurious currentsinsuch flowscanpromotethe development ofnon-physicalrecirculationareasandconsequently,erroneousslipvelocitybetweenthebubblevelocityand meanvelocityintheliquidslug.

2. Numericalschemes

2.1. Spatialandtemporaldiscretizations

The numericalcodeusedforthisstudyistheJADIM code, whichhasbeendevelopedto simulatedispersedtwo-phase flows andused to simulate various multiphase flows systems [7,17,2,29,41,42]. The interface iscaptured by an Eulerian descriptionofeachphaseonafixedgridwithvariabledensityandviscosity.Undertheassumptionsthat(i)thefluidsare Newtonianandincompressible,(ii)thereisnomasstransferattheinterface,(iii)theflowisisothermaland(iv)thesurface tensionisconstant,thefluidflowcanbedescribedbytheclassicalonefluidformulationoftheNavier–Stokesequations:

∇ ·

U

=

0

,

(1)

∂

U

∂

t

+ (

U

· ∇)

U

= −

1

ρ

∇

P

+

1

ρ

∇ ·  +

g

+

Fσ,s

,

(2)

where



istheviscousstresstensor,g istheaccelerationduetogravity,Fσ,s isthecapillarycontributionwhosecalculation

isdescribed inSection 3,and

ρ

and

μ

arethelocaldensityanddynamic viscosity,respectively.Thedensityandviscosity arededucedbylinearinterpolationfromthevolumefractionC ofonephaseineachcomputationalcell:

ρ

=

C

ρ

1

+ (

1

−

C

)

ρ

2

,

(3)

wherethevolumefractionisC

=

1 incellsfilledwithfluid 1,C

=

0 incellsfilledwithfluid 2and0

<

C

<

1 incellsthat are cutby theinterface. Thisvolume fractioniseithercalculateddirectlyby solving anadvectionequation withtheVOF method,ordeducedfromadistancefunctionandtheapproximationoftheHeaviside functionasdefinedbyEq.(5)with theLSmethod: C

= ˜

H

(φ)

=

⎧

⎪

⎪

⎨

⎪

⎪

⎩

0 if

φ <

−

ε

,

0

.

5



1

+

φ

ε

+

1

π

sin



π

φ

ε



if

|φ| ≤

ε

,

1 if

φ >

ε

,

(5)

where

ε

=

√

2



x ishalfthenumericalthicknessoftheinterface.

Inbothcases,VOFandLS,an advectionequationisneededtocalculatethevolumefractionandthedistancefunction respectively.Theadvectionschemesemployedinthisstudyaredescribedinthenextsection.

Theequationsarediscretized onastaggeredgridusingafinitevolume methodandall spatialderivativesare approxi-matedusingsecond-ordercentered schemes.Thetimeschemeusedtocompute theadvectivetermsintheNavier–Stokes equations is a third-order Runge–Kuttatype scheme,while the viscous stresses are solved usinga semi-implicitCrank– Nicolsonmethod.Theincompressibility isensuredusingaprojectionmethod,whichconsistsinsplittingthevelocityfield intotwo contributions: a rotationalone, whichgives apredictedvelocity field calculated semi-implicitly,anda potential one,obtainedfromapressure correctionsolutionofa pseudo-Poissonequation, whosedivergenceiszero.Furtherdetails onthenumericalalgorithmsconcerningthespatialdiscretization,aswellasthetimeadvancementprocedure,canbefound in[32,10,30]whiledetailsontheinterfacetrackingandrelatedreferencesaregiveninthefollowing.

2.2. Advectionschemes

In this section, we present the advection schemes that havebeen used to characterize the spurious currents in the different simulations carried out inthis work. Whatever the method used in this work, the Eulerian description of the two-phaseflowprobleminvolvestheadvectionequationofascalar

ψ

:

∂ψ

∂

t

+

U

· ∇ψ =

0 with



ψ

=

C (volume fraction) in VOF methods,

ψ

= φ

(signed distance function) in LS methods. (6)

TwodifferentVOFschemeshavebeenused:ageometricaloneusingaPiecewiseLinearInterfaceCalculation(PLIC)and a second one basedon Flux CorrectedTransport (FCT)schemes without interface reconstruction. Inaddition, aLevel Set schemebasedonthesameadvectionequation butwithanadditionalredistancingequationtokeepthelevelsetfunction asadistancefunctionhasbeenused.Thesethreemethodsaredescribedbelow.

2.2.1. VOF-PLIC

In many VOFmethods employed to capture a fluid–fluid interface, a reconstruction technique is used to control the thicknessoftheinterface.Inthe2-dimensionalVOF-PLICschemeusedinthisstudy,thegeometricalreconstructionofthe interfaceisbasedontheVOFTools librariesdevelopedbyLópez andHernández[31].Thenormalsusedtodeterminethe lin-earreconstructionarecalculatedinthesamewayasthosecalculatedforthesurfacetensionforce(seeSection3).Fromthe calculatednormalsandthegivenvolumefractionatthebeginningofthetimestep,theinterfaceisthenrepresentedbya segmentineachinterfacialcellusingtheVOFTool libraries[31].Withstaggeredgrids,thevolumefractionenclosedinthe volumeadvectedbyeachvelocitycomponentcanbefluxedthroughthecorrespondingcellface.Thesevolumesadvectedby thevelocitycomponentsui,jandvi,j inthex andy directionsarerespectivelyui,j



t



y andvi,j



t



x.Theinterfacebeing approximatedby a segmentineach interfacial cell,the volumefractionenclosed intheseadvected volumesisestimated with the VOFTool libraries [31]. The advection scheme is split directionby direction[40].This implies that despite the volumefractionisfluxedfromonecelltoanother,asmallgainorlossofmasscanoccur. Thisgeometricalreconstruction oftheinterfacewithsegments(orplanesin3-dimensions)allowsasharprepresentationoftheinterface.

2.2.2. VOF-FCT

IntheVOF-FCT methodimplementedintheJADIM code,thelocation andthicknessoftheinterfacearebothcontrolled byan accuratealgorithmbasedonFlux-CorrectedTransport schemes[51,7].Theadvectionequation(6)canbe writtenas (with

ψ

=

C ):

∂

C

∂

t

+ ∇ · (

C U

)

=

C

∇ ·

U

.

(7)

Despiteadivergencefree velocity fieldused forthe advectionofC (

∇ ·

U

=

0),the schemeissplitdirectionby direction andthevelocityfieldisnot divergencefree ineach direction.Thus,asthevolumefractionisnotequalineverysub-step, theschemeisnotexplicitly conservative.Inaddition,duetotheshearnormaltotheinterface,adrawbackofthemethods withoutinterface reconstruction isthenon-physicalspreading oftheinterface overa non-negligible thicknessofcells. To correctthisexcessivespreadingoftheinterface,thevelocity fieldismodifiedsothat thevolumefractionofall interfacial

cells inthedirectionnormaltotheinterface isadvected withthevelocityontheinterfacerepresentedbytheiso-contour

C

=

0

.

5.However,the modifiedvelocity fieldisgenerallynotdivergencefree andcontributestoa gainorlossofmassof one ofthephases.Tosolve thisproblem, analgorithm thatredistributesmassina globalmanneris used[7]similarlyto whatcanbedoneinLSmethodsbychangingthecontourrepresentingtheinterface[11].

WhilethesemodificationsintheFluxCorrectedTransportschemesignificantlyimprovetheinterface advection[7],the interfacestillspreadsovermorethanthreecellsincertaincases,andnotablyinsimulationswheretheinterfaceisstretched. Inaddition,forsimulationsofflows atReynoldsnumbersclosetoone,severalhundredsofthousandsiterationsare often requiredduetotheslowestablishmentoftheflowandtheconstraintofstabilityinducedbythecapillaryforce[8].Although thespreadingoftheinterfaceisnotsignificantduringonetimestep,itcanbecomenon-negligible whenalargenumberof iterations are performed.Toavoidthisunphysicalspreading oftheinterfacethat leadsto thediffusevariation ofphysical properties, aswell asthe pressure jumpat theinterface, a rough cut-off algorithm hasbeen employed when necessary. ThisconsistsinimposingthevolumefractionC

=

0 (orC

=

1)ifthevolumefractiontakesanintermediatevaluewhilethe interface doesnot crossthecell,thereby neglectingstructuresthinnerthanthedimensionofthecell.Thisresultsinlocal gains/lossesofmassinproximityoftheinterfaceandisthereforeappliedbeforetheglobalredistributionofmass.

Thealgorithmisgivenin2-dimensionsandtheextensionto3-dimensionsisstraightforward: if

0

<

Ci,j

<

1

if



C−_i−nc,j

>

0 & C−_i+nc,j

>

0 & C_i−_,j−nc

>

0 & C_i−_,j+nc

>

0



if

(

Ci,j

>

0

.

5

)

Ci,j

=

1

if

(

Ci,j

<

0

.

5

)

Ci,j

=

0

where C−_i+k,j+l

= (

Ci,j

−

0

.

5

)

× (

Ci+k,j+l

−

0

.

5

)

andnc isapproximatelyhalfthe permittednumber ofcells fornumerical thickness.Inourcases,nc

=

1 and 2 havebeentestedwiththetestpresentedinSection6.Ithasbeenfoundthatthecut-off algorithmcombinedwiththevelocityextensionandglobalmassredistributiondonotsignificantlychangetheintensityof the spuriouscurrents(see Appendix Aforthecomparison).Inaddition,ithasbeenobservedinpure advectionteststhat there is nospreading ofthe interface whenusing finemeshesand asaresult thecut-off algorithm isnot calledduring the advectionstep [1].Thus, the VOF-FCT advectionscheme withinterface velocity extension, globalmass redistribution andnc

=

2 seemsto bethe bestcompromise betweeninterface thicknessandspuriouscurrents andwillbe usedinthe followingwiththenotationVOF-FCT.Furtherdetailsontheeffectsofeachfeatureofthescheme(velocityextension,mass redistributionandcut-off)onpureadvectiontestscanbefoundin[7,1].

2.2.3. LS

The Level Setmethod[45,11]involves asigned distancefunctioninsteadof avolume fractionin eachcell thatallows theprecisecomputationofgeometricalproperties(normalandcurvature)oftheinterface.Thisprecisionisensuredbythe smoothtransitionfromonephasetotheotherandisenhancedwhentheLevelSetfunctiongradientmagnitudeisconstant inthedomain.Thedistancefunctionobeystheadvectionequation(6),whichcanbewrittenas(with

ψ

= φ

):

∂φ

∂

t

+ ∇ · (φ

U

)

=

0

.

(8)

The advectionofthesigneddistancefunction

φ

inEq.(8)isadvancedintimewithathirdorderRunge–Kuttascheme andtheadvectivetermsarediscretizedviaaconservativeWENO5scheme[13].

In orderto keep theLevel Set functionasclose aspossibleto asigned distancefunction, aredistancing procedure is appliedusingEq.(9) [43].

∂φ

∂

τ

+

sgn

(φ) (

|∇φ| −

1

)

=

0

.

(9)

Thispartialdifferential equationissolvedusingaWENO5scheme[13,26]fortheadvectivetermsandathirdorderRunge– Kuttaschemeforthetemporaldiscretization.TheHamiltonianfluxesarecalculatedwithaGodunovoperator.

3. Surfacetensionforce

Incapillaryflows,careneedstobetakeninthecalculationofthecapillaryforceFσ,s

= −

σ

/

ρ

(

∇ ·

n

)

n

δ

I,where

σ

isthe surfacetension,n thenormaltotheinterfaceand

δ

IistheDirac distributionlocalizingtheinterface.Thenumericalmethod used tosolvethe interfacialforce istheContinuum SurfaceForce (CSF) proposedby Brackbilletal.[8].The surfaceforce

Fσ,s istransformedintoavolumeforce Fσ,v bydistributingitseffectsovergridpointsinthevicinityoftheinterface ina

regionthatisafewcellsthick: Fσ,v

= −

σ

ρ

κ

∇

C

.

(10)

The localizationoftheinterface isavailable throughanon-zero gradientofthevolume fractionandthe calculationof thecurvature

κ

isdetailedinthefollowing.

Fig. 1. (a)SchematicrepresentationofthecapillaryforcewithaContinuumSurfaceForcemodelonstaggeredgrids.Adaptivestencilfortheconstruction ofheightfunctionsin(b)aVolumeofFluidformulation;(c)aLevelSetformulation.Thedashedlinerepresentstheinterfacewhilethedash-dottedline representstheiso-contourwherethecurvatureiscalculatedintheclassiccontinuumsurfaceforcemodel.

TheapproachesconsideredinthisworkaremostlyvariantsoftheCSFmethodproposedin[8]thatdifferinthewaythe curvatureisdiscretized.Themannerinwhichthecapillarysurfaceforceisconvertedintoavolumeforceandisspreadover thefinitevolumemeshisnotaffectedexceptintheSharpSurfaceForce(SSF)method[18].TheCSFmethodsbasedonthe calculationofthecurvatureby meansofthedivergenceofnormalvectorsorheight functionsaredifferentiatedby calling themClassic ContinuumSurface Force (CCSF)andHeightFunction ContinuumSurface Force (HFCSF).However, theSharp Surface Force[18],whichroughlysets thepressurejumpbetweencells thatare cutby theinterface,slightlydiffersfrom theContinuumSurface Forceinthat thereisnodiscretizationofaDeltaFunctiontoestimate thesurface oftheinterface thatcuttheinterface.TheSSFformulation,resultsinasharppressurejumplikethatobtainedwiththeGhostFluidMethod

[27]orthepressurecorrectionproposedin[36].

3.1. VOFClassicContinuumSurfaceForce(VOF-CCSF)

FromanEulerianrepresentationinvolvingascalarfunction,thegeometricalpropertiesoftheinterfacecanbecalculated fromthesuccessivederivativesofthescalarfunction:

n

δ

I

= −∇

C

,

(11)

κ

= −∇ ·



∇

C

||∇

C

||



.

(12)

Sincethevolumefractionvariesfromzerotooneoverathicknessoftwotothreecells,ithasbeenshownthata smooth-ing procedure allows adecrease inthe errorsdue tothe discretizationof thegradient anddivergenceoperators [14,17]. However, [15] alsoshowedthat oversmoothingthevolume fractionfieldleads to erroneousresultsanddoesnotallow a bubblerising inliquidatresttobe simulatedcorrectly.The filteringprocedureusedin thisstudyhasbeen showntobe ableto deal withdifferenttwo-phase flowproblems[7,17,2] anditconsistsin successivelyapplyinga 3

×

3 convolution matrix(Kconv

= [

0 1

/

16 0

;

1

/

16 3

/

4 1

/

16

;

0 1

/

16 0

]

) in2-dimensionstothevolumefractionfield.Inthepresent simulations,thesmoothingstepinvolves12 and6 iterationstocalculatethecurvatureandthenormal(localizationand ori-entationoftheforce),respectively,asrecommendedin[17]inwhichthesameflowsolverandthesameconvolutionmatrix wereused.InJADIM,thecurvatureiscalculatedatthecenterofthestaggeredcontrolvolumeasillustratedinFig. 1(a)and thedivergenceoftheunitnormaltotheinterfaceiscalculatedinaconservativeway[6]:

F_σ,v

=

−

σ

ρ

V

V

∇ ·



∇

C

∇

C





∇

CdV (13)

≈

−

σ

ρ

V

∇

C

∂S



∇

C

∇

C





·

ncelldS

,

(14)

where

ρ

= (

ρ1

+

ρ2

)/

2 isthemeandensity.Althoughtheuseofthemeandensityisnotconsistentwiththepressureforce inEq.(2),it hasbeenshowntostabilizesimulations anddecreasethe intensityofspuriouscurrents [8].The gradientof volumefractioninEq.(14)isdiscretizedinthesamewayasthepressuregradientinEq.(2).

3.2. VOFHeightFunctionContinuumSurfaceForce(VOF-HFCSF)

Theheightfunctiontechnique[14,35]allowsthegeometricalpropertiesoftheinterfacetobecalculatedaccuratelyfrom itsposition,whichisobtainedbysummingthevolumefractionsoffluidcolumns.Thestencilsareorientedalongthemain componentofthenormaltotheinterface andthey areadapted ineverycolumnuntilacellofvolumefractionofzerois reachedinthedirectionofnegativegradientofvolumefractionandavolumefractionofoneinthepositivedirection(see

Fig. 1(b)). Oncethe position ofthe interface is evaluated, the curvatureis calculated fromthe plane curve y

=

f

(

x

)

via Eq.(15):

κ

=

y

1

+

y2

3/2

.

(15)

Inaddition,thelocaldensityinthecell

ρ

isusedinVOF-HFCSFsincetheuseof

ρ

isnotnecessaryforstability purposes withthismethod.

3.3. LSClassicContinuumSurfaceForce(LS-CCSF)

IntheLS-CCSFmethod,thesamediscretizationasthatusedintheVOF-CCSFmethod(Eq.(14))isemployed. Fσ,v

=

−

σ

ρ

V

∇

C

∂S



∇φ

∇φ



·

ncelldS

.

(16)

Note that, likein VOF-HFCSF,the localdensityin thecell

ρ

isused inLS-CCSF. In CSFmethods coupledto Level Set techniques [45], the deltaDirac function is generally approximated by the derivative of the smooth Heaviside function definedinEq.(5)(

˜δ

I

=

dH

˜

(φ)/

d

φ

)andisapplieddirectlywhilethegradientoftheLevelSetfunctionisusedforcalculating the orientation of theforce (n

δ

I

= ∇φ/|∇φ| ˜δ

I). Here, the curvatureis derived fromthe divergence ofthe normalto the interface representedbythedistancefunction(

κ

= ∇ · (∇φ/|∇φ|)

)asitisusuallydone.However,thedifferenceresidesin the estimationofthe localizationandtheorientation, whichare both deducedfromthe volumefractioncalculated from the LevelSet function(Eq.(5)), i.e. n

δ

I

= −∇

C , sincenaturalequilibriumisreachedwiththisformulationassoonasthe curvatureisuniformalongtheinterfaceandboththepressureandvolumefractiongradientsinEq.(18)arediscretizedin thesameway.Weobservedinpreliminaryteststhatthisformulation(Eq.(16))providesbetterresultsintermsofspurious currents,aswellasbetterconvergencerates.

3.4. LSHeightFunctionContinuumSurfaceForce(LS-HFCSF)

SimilarlytotheVOF-HFCSFmethod,inan LSframeworkthecurvaturecanbeestimatedfromthepositionoftheinterface (Eq. (15)) instead ofcalculating the divergence of the normal to the interface asin the previous section. The Level Set method forthe advectionequation isthesameasintheLS-CCSF methodbutthecurvatureiscalculatedvia theposition oftheinterface,whichcanbefoundbylinearinterpolationormoreaccuratelyusingquadraticinterpolationswiththefour cellssurroundingtheinterface(seeFig. 1(c)).Theproceduretolocatetheinterfaceisthesameasthatusedin[34].Inorder to improvethestabilityofthequadraticinterpolationsneardiscontinuities,aminmodoperatorwasintroduced in[34] on the second derivativesof theLevel Setfunction calculated oneach side ofthe interface.The curvature isalsocalculated withEq.(15).

3.5. LSSharpSurfaceForce(LS-SSF)

IntheSharpSurfaceForcemodel[18],thesurfacetensionforceisnon-zero onlyinthecellscrossedbytheinterface;this is differenttotheCSFmethod,whichimposes theLaplace pressurejumpcontinuously along theinterface.The curvature is calculatedfollowing thesame procedureasinthe LS-CCSF method.Forthehorizontaldirection, ifthe control volume centeredonthex-componentofthevelocityiscrossedbytheinterface,i.e.

φ

i−1

× φ

i

<

0,thesurfacetensionforceis:

Fσ,v

= −(

σ κI

)/(

ρ



x

) ,

(17)

where

κ

I isthecurvatureinterpolatedontheinterface

κ

I

=

|φ

i

|

κ

i−1

+ |φ

i−1

|

κ

i

|φ

i

| + |φ

i−1

|

.

4. Originofspuriouscurrents,vorticitysource

Insimplecases,suchasastaticoratranslatingbubbleordrop,themomentumconservationequation(2)reducesto:

0

= −∇

P

+

σ κ

∇

C

,

(18)

suchthat,whentakingthecurlofEq.(18),thecurvatureshouldsatisfy:

Spurious currents are generated when thiscondition isnot satisfied. In the particularcase ofa surface witha constant curvature(e.g. asphericalbubbleordrop),theflowwillbecurlfreeifthecomputedcurvatureisconstant.Ifthecurvature is maintained constant during the simulation [18,15], the source term is zero and no spurious currents develop. As a consequence,theeliminationofspuriouscurrentsrequirestheuseofbalanced-forcealgorithms [18,22,35]withconsistent discretizationofpressure andcapillaryforcestosatisfy Eq.(18).Curvaturegradients assources ofvorticityare identified whenwritingthevorticityequationfromEq.(2):

∂

ω

∂

t

+ (

U

· ∇)

ω

+ (

ω

· ∇)

U

=

μ

ρ

∇

2

_ω

₋

σ

ρ

∇

κ

× ∇

C

.

(20)

From Eq.(20),itis seenthat the surface tensionforce can contribute asasource termforvorticity productionwhen condition (19)isnotsatisfied.Whenlookingattheinitialstageofthesimulations,thissourceterm(

σ

∇

κ

× ∇

C

∼

σ

/

D3₎

induces theamplificationof vorticityduring the firsttime steps(

ρ

∂

ω

/∂

t

∼

ρ

Uσ

/(

D



t

)

) andthecorresponding spurious currentsareofintensityUσ givenby:

Uσ,trans

∼

σ



t

ρ

D2

.

(21)

Thespuriouscurrentsthereforeinitiallygrowlinearlywith

σ

aspreviouslyobservedin[18,15],aswellaswiththetimestep

[18].Thislinearevolutionwiththesurfacetensionis alsoinagreementwiththedimensionalanalysisofthemomentum equationperformedin[15].

Whenconsideringlongtimesimulations,Eq.(20)showsthatasteadystatecanbereachedforthespuriouscurrentsof magnitudeUσ .Ifviscous effectsaredominant,thespuriouscurrentsresultingfromthevorticitysourcetermarebalanced bytheviscousterm(

μ

∇

2

_ω

_∼

_μ

_Uσ

_/

_D3_):

Uσ,visc

∼

σ

μ

.

(22)

Thus,theintensityofspuriouscurrentscanbe writtenusingacharacteristiccapillarynumber(Ca

=

μ

Uσ,visc

/

σ

∼

c where

c isaconstant). Thisisconsistentwithprevious works[28,39,22,17,50,2,15] inwhichthemagnitudeofspuriouscurrents wasrelatedempiricallyto

σ

/

μ

.

Wheninertiaisdominating,thespuriouscurrentsresultingfromthevorticitysourcetermarecontrolledbytheinertia term(

ρ

(

U

· ∇)

ω

∼

ρ

U2_σ

/

D2)leadingtothecharacteristicvelocity:

Uσ,in

=



σ

ρ

D

.

(23)

Thisis thecharacteristicvelocity usedinthe inviscidproblem[35].Thesetwo characteristicvelocitiesare closelyrelated since

U_σ,visc

=

√

La U_σ,in

,

(24)

where La

=

ρ

D

σ

μ

2 isthe Laplace number.In thetestsreported inthis work,the Laplace numberhasbeen varied inthe

range1

.

2 to 12000 and theReynoldsnumbervarieswithintherange

[

0

.

69

;

600

]

.Inthisstudy,Uσ,visc hasbeenchosen tomakethevelocitiesdimensionless.

5. Staticbubble

The first test casethat we consider is the 2-dimensional staticbubble [35]. Acylindrical interface is initialized in a continuous phase without gravity and both fluids haveequal density and viscosity. The Laplace number is La

=

12000. Onlyaquarterofthebubbleofradius R0

=

0

.

4,placedinthebottomlefthandsideofasquarecomputationaldomainof

length1 issimulated.Symmetryboundaryconditionsareappliedontheleftandbottomboundarieswhileno-slipboundary conditionsareapplied onthetopandrightboundaries. Theexactsolutionofthevelocity fieldshouldremain zerointhe wholedomainandthepressureshouldobeytheLaplacepressurejumpattheinterface.

Theanalysisoftheintensityofspuriouscurrentsandpressurejumpsisbasedonthefollowingerrornorms:

– Umax isthemaximumabsolutevelocityinthewholedomainandCamax

=

μ

Umax

/

σ

isitsdimensionlessformusedin thefollowing,

–

|∇

κ

× ∇

C

|

is the normof the source termin Eq. (20)and isused to characterize the vorticity production andthe associatedspuriouscurrents,

–



Ptotal is the pressure jump betweenthe average pressurein the bubble(C

≥

0

.

5) andthe average pressure inthe continuousphase(C

≤

0

.

5),

Fig. 2. Intensity ofthe sourceterm|∇κ× ∇C|inthe vorticityequation(20)andvelocityfield(bydecreasingspuriouscurrentsfrom(a)to(f))for

R0/x=12.8 andLa=12000.

The numericalparameters considered correspondto thoseusedby Popinet [35] whereaftera transient evolution,the bubble shape reached a numericalequilibrium witha uniform curvature tangential to the interface, leading to an exact balancebetweensurfacetensionandpressureforces,andtheeliminationofspuriouscurrents.Oncethisnumericalbalance betweenpressureandcapillaryforcesisreached,thevelocityfieldiszero,theuniformcurvaturetangentialtotheinterface keepsthesamevalueovertimeandthepressureobeystheLaplacepressurejumpwithanaccuracythatismesh-dependent

[35].Thetimeismadedimensionlesswiththecapillarytimescalet∗_σ

=



ρ

D3

σ

.

Therelevance ofthespacediscretizationbetweenpressuregradientandsurfacetensionforcehasfirstlybeentestedby imposingtheexactcurvatureinthewholedomainforthecalculationofFσ,v.Themaximumdimensionlessvelocityinthe

wholedomainreachedCamax



5

.

08

×

10−18showingthatEq.(19)issatisfiedtomachineaccuracysothatthesourceterm inEq.(20)isnegligibleandnospuriouscurrentsdevelop.Thus,theexactbalancebetweenthevolumefractionandpressure gradients isverifiedandthespurious currentsobservedinthefollowingcanbeattributedtotheerrorsinthecalculation of the curvature andmore precisely, inthe gradients of curvature along the interface dueto the resolution ofa curved interfaceonacartesiangrid.

Fig. 2showsthevelocityfieldandtheintensityofthesourceterminthevorticityequationforthedifferentmethods.It isclearlyseenthatthemaximumintensitiesofvelocityandvorticitysourcearecollocated.

Fig. 3showstheevolutionofthemaximumintensityofthespuriouscurrentswiththedifferentmethods.WithinaVOF framework, itis clearlyseen that afterapproximately30t∗_{σ ,}the heightfunction curvaturecalculationallows thespurious currentstobedecreasedbyapproximatelysixordersofmagnitudewiththeVOF-FCT-HFCSFmethodwhencomparedwith thestandardcurvaturecalculation(VOF-FCT-CCSF)andthespuriousvelocitiesareclosetomachineaccuracywith VOF-PLIC-HFCSF,asin[35].ThecurvaturecalculatedwiththeHFCSFmethodsisexactlylocatedattheinterfacewhileitisthemean value ofthe curvatureinthecontrol volumecrossed bythe interfacein CCSFmethods(see Fig. 1).The errorinduced by the locationofthecurvaturecalculation promotescurvaturegradients,whichare sourceofvorticityinthe vicinityofthe interface (Eq. (20)). As expected, the Classic ContinuumSurface Force method givesbetter results when coupled witha distancefunction thanavolume fraction.Withinan LSframework, Fig. 3showsthat bothcontinuous andsharpmethods (LS-CCSFandLS-SSF)are closeintermsoftheintensityofspuriouscurrents.Thesimilaritiesintheresultsobtainedwith the LS-CCSF andLS-SSFmethods areconsistent withtheobservations of[18] who foundsimilar spurious velocitieswith continuousandsharpsurfacetensionmodels.

Inan LS context,theheightfunctioncurvaturecalculation improvestheresultsbymorethanone orderofmagnitude. However, the exactbalance betweenpressure andcapillaryforcesthat is achievedina VOFframework, withorwithout

Fig. 3. Evolutionofthemaximumintensityofthespuriouscurrentsinthecomputationaldomainovertime.Legend:( )VOF-FCT-CCSF;( ) VOF-FCT-HFCSF;( )VOF-PLIC-HFCSF;( )LS-CCSF;( )LS-HFCSF;( )LS-SSF;( )LS-CCSFwithoutredistancing;( )LS-HFCSFwithout redistancing;( )LS-SSFwithoutredistancing.

Fig. 4. Evolutionofthemaximumintensityofthesourceterm|∇κ× ∇C|inthevorticityequation(20)overtime.Legend:( )VOF-FCT-CCSF;( )

VOF-FCT-HFCSF;( )VOF-PLIC-HFCSF;( )LS-CCSF;( )LS-HFCSF;( )LS-SSF.

reconstruction, is not obtained withthe LS-HFCSF method.Interestingly, when the redistancing step inthe LS transport schemeisskipped,the spuriouscurrentsare significantlydecreasedanditispossibleto obtaintheexactequilibrium be-tweenpressure andcapillaryforceswithbothSharpSurfaceForce (LS-SSF)andHeightFunctionContinuumSurface Force (LS-HFCSF).Indeed,bothmethodsestimatethecurvatureattheinterpolatedpositionoftheinterfaceandtheinterface os-cillatesaroundthepositionofnumericalequilibriumwhilethevelocitiestendtowardszero.Thisbalancecannotbereached when the redistancing step is activated because the position of the interface (represented by the iso-contour

φ

=

0) is slightlymovedduring theredistancingstep [34],thereby maintainingthespuriouscurrents.Spurious currentsaremainly duetotheredistancingstepinLSmethodswhenthecurvatureiscalculatedontheinterface(LS-SSFandLS-HFCSF). Never-theless,theredistancingstepwillbeusedinthefollowingasitisnecessarytomaintainthelevelsetfunctionasadistance function.

ThetemporalevolutionoftheintensityofvorticitysourceisreportedinFig. 4.Agoodcorrelationbetweenthissource termandthespuriousvelocitiesisobserved.Indeed,thecurvaturegradientsandthereforethevorticitysourcetermalmost vanishwiththeVOF-FCT-HFCSF methodandarereducedto10−12 withtheVOF-PLIC-HFCSF.Ontheotherhand,withthe othermethods,thisvorticitysourcedoesnotvanishandisbalancedbyviscousdissipation(seeSection4).Thesetrendsare similartothoseobservedwiththemaximumintensityofspuriouscurrents.

The balancebetweenpressurejump andsurface tensionforcesobserved withboththe VOF-FCT-HFCSFand VOF-PLIC-HFCSFmethodshasbeenverifiedfordifferentLaplacenumbersrangingfrom120 to12000 anddifferentmeshesinvolving between 10 and 50 cells per bubble radius. The intensity of spurious currents slightlyincreases with mesh refinement withtheVOF-PLIC-HFCSFmethodbutthemaximumspuriouscurrentscapillarynumberremainsbelow10−11_.Thespatial

Fig. 5. Convergencewithspatialresolutionofmaximalspuriouscurrentsvelocities.Legend:( )VOF-FCT-CCSF;( )VOF-FCT-HFCSF;( ) VOF-PLIC-HFCSF;( )LS-CCSF;( )LS-HFCSF;( )LS-SSF;( )ConservativeLevelSet–GhostFluid[16];( )RefinedLevelSetGridMethod[22]; ( )VOF-CELESTE[15];( )ConservativeLevelSet-CSF(finiteelementandLa=100)[50];( )front-tracking[36];(· · ·)Umax/Uσ∼ (R0/x)−1;(—) Umax/Uσ∼ (R0/x)−2.

obtainedwiththeVOF-FCT-CCSF methoddonotdecreasewithgridrefinement.Sincethenumberofsmoothingstepsthat spreads the interface ona given numberofcells has beenkept constant, thephysical length ofsmoothingis smallerso that the filterisless efficientwhen thegrid spacingisdecreased. [14] drew a similar conclusionaboutthe convergence withmesh refinementofcurvatureerrors computedwitha convolution methodatthefirsttime step.Witha smoothing length proportional to the mesh size (4



x intheir study), the rateof convergence ofcurvature errors is

−

1 while the rateofconvergenceis2whenthesmoothinglengthisnotmeshdependent.However, theyalsomentioned thattheratio ofthisconstant physicallength ofsmoothingoverthegridsize canbecomelarge withfinemeshesinordertoreach the same accuracysotheconvolution wouldinvolvea largeconvolution stencil,leadingto non-negligiblecomputationalcost. Anotherpossibilitywouldbetouseadaptiveconvolutionstencilsdependingontheratioofthemeshsizeoveranestimated curvature.ResultsfromtheliteraturearealsoshowninFig. 5anditisseenthattheVOFmethodproposedin[15]inwhich the curvatureiscalculatedwitha least-squarefitof second-orderTaylorseriesexpansion oftheconvolutedvolume frac-tion allowsthespurious currentstobedecreasedby severalorderofmagnitudeswhen comparedwithstandard VOF-CSF method.Interestingly, althoughthecurvature iscalculatedfroma convolutedvolume fractionfield, thespurious currents showaconvergenceratebetweenoneandtwowithmeshrefinement.SpuriouscurrentswithLS-CCSF,LS-SSFandLS-HFCSF decreasewitharateofconvergencecloseto2.Inaddition,bothLS-CCSFandLS-SSFagreewellwiththeConservativeLevel Set methods oftheliterature. The presentLS-CCSF methodandthe CLS-CCSFmethodof[50] show similar spurious cur-rentintensities, aswell assimilartrends.AlthoughtheLevel Setfunctionin[50] isnot adistancefunctionbutasmooth regularized indicatorfunction,thecurvature,thelocalizationandthenormalofthesurface tensionforce arecalculatedin a similarwayasthatpresentedinSection3.3.AsimilarConservativeLevel Setmethodispresentedin[16],theLevel Set functionvariessmoothlyasanhyperbolictangentacrosstheinterfaceandaGhostFluidmethodisusedtocalculatesurface tensionforce.ItisseenthatthismethodallowsthespuriouscurrentstobeslightlydecreasedwhencomparedwithLS-CCSF andLS-SSFmethodsbutitalsopresentsalower rateofconvergence.However,itisseenthattheheightfunctionmethods (withLS,VOF-FCTandVOF-PLIC)givebetterresultsintermsofparasiticcurrents.Finally,itisnotedthattheRefinedLevel Set Gridmethodproposed in [22](where thegrid isrefined attheinterface)leads to verylow spurious currents,which are ofthesame orderasthoseobtainedwitha front-trackingmethodin[36] andsimilarrates ofconvergencewithgrid refinementarefound.

6. Translatingbubble

Thesecond testcaseconsistsofauniformflowfieldthattranslatesthebubbleasproposed in[35].The previousstatic case allows tostate whether asurface tension schemeis well-balancedor not,andwhether thecurvature calculation is accurate ornot. [15] presented a test casewhere a bubbleis advected andthe exact curvature isimposed andshowed that no spurious currents develop in agreement withEq. (19).Here we consider the combined effects of the curvature calculationandtheadvectionoftheinterfaceinordertoevaluatetheprecisionandtherobustnessofthecouplingbetween advection schemes, Navier–Stokes solver and capillary termcalculation. A uniform horizontal velocity U0 is imposed in

the wholedomainwithperiodicboundary conditionsonlateralsidesandsymmetryboundaryconditionson thetopand bottom.Both fluidshaveequaldensityandviscosity.Thecylindricalbubble/dropletshouldmove withtheexternalflowat the samevelocity U0,thevelocity fieldinthe frameofreferencemovingwiththebubbleshould ideallybezeroandthe

Fig. 6. Intensityofthesourceterm|∇κ× ∇C|inthevorticityequation(20)andvelocityfieldafterthetranslationofadistance1.25D intheframeof referencemovingwiththebubble(bydecreasingspuriouscurrentsfrom(a)to(f))forR0/x=12.8,La=12000 andWe=0.4.

(

u

,

v

) (

x

,

y

)

= (

U0

,

0

) ,

(25) p

(

x

,

y

)

=



_σ

R0 in the bubble

,

0 outside

.

(26)

In this test case, the time is made dimensionless withthe characteristic advectiontime t0

=

D

/

U0. Note that a similar

test casewithtypicalgas–liquiddensityandviscosityratiosispresentedinAppendix B.Theresults obtainedintermsof spuriouscurrentsareverysimilartothetestspresentedherewithdensityandviscosityratiosofunity.

ThespuriousvelocityandthevorticityproductionfieldsobtainedwiththedifferentmethodsareshowninFig. 6. Simi-larlytothestaticcase,themaximumintensitiesofvorticityproductionandspuriousvelocityarecollocated.Itisobserved that the maximumintensityof spurious currentsin VOF-FCT-HFCSF simulationdoes not tendtowards machine accuracy inthistest caseandisofthe sameorderof magnitudeasintheVOF-FCT-CCSF simulation.However, whilethe spurious velocities are uniformlyspread around theinterface inthe VOF-FCT-CCSF method,the spurious currents are localizedin spacewhenusingtheheightfunctiontechniqueforthecurvaturecalculation.Thisdifferentbehavior betweenthedifferent curvaturecalculation techniquesisalsoobserved withthe VOF-PLICandLStransportschemes. Therefore,theaverage in-tensityofspuriouscurrentsisdecreasedwiththeheightfunctioncurvaturecalculationwhereasthemaximumintensityof spuriouscurrentsisofthesame orderwithboththeheightfunction andcurvaturecalculationfromthedivergenceofthe unitnormaltotheinterfaceforagiventransportscheme.

6.1. Timeevolutionofspuriouscurrents

ThetemporalevolutionofthemaximumvelocityofspuriouscurrentsisreportedinFig. 7.Foragiventransportscheme (VOF-FCT,VOF-PLICorLS),theestimationofthecurvaturefromtheheightfunction(HFCSF) leadstosimilarintensitiesof spurious currents tothose obtainedwiththe calculation ofthe curvaturefromthe divergenceof theunit normal tothe interface (CCSF andSSF). Thespurious currentsobtainedwithCCSF andSSFmethods are ingood agreement withthose obtainedinthestaticcase. Ontheother hand,withtheheight functioncurvaturecalculation (VOF-FCT-HFCSF, VOF-PLIC-HFCSF,LS-HFCSF), spurious currentsare enhanced inthis testcasewhen compared withthestaticcase. The accuracyof theheightfunctioncurvaturecalculationisthereforehighlightedsincespuriouscurrentsareverysensitivetotheerrorsin theshapethat areintroduced intheadvectionstep,especiallywiththeVOF-FCT andVOF-PLICschemes.Inthistest,itis

Fig. 7. Temporalevolutionofthemaximumspuriouscurrentsvelocityintheframeofreferencemovingwiththebubblefor R0/x=12.8,La=12000, We=0.4.Legend:( )VOF-FCT-CCSF;( )VOF-FCT-HFCSF;( )VOF-PLIC-CCSF;( )VOF-PLIC-HFCSF;( )LS-CCSF;( )LS-HFCSF; ( )LS-SSF.

Fig. 8. Maximumspuriouscurrentscapillarynumber(a)andvorticityproduction(b)versustheLaplacenumberforR0/x=12.8 andWe=0.4.Legend:

( )VOF-FCT-CCSF;( )VOF-FCT-HFCSF;( )VOF-PLIC-CCSF;( )VOF-PLIC-HFCSF;( )LS-CCSF;( )LS-HFCSF;( )LS-SSF;( ) VOF-PLIC-HFCSF@Gerris[35];(—)Camax∼La−1/3.

seen that spurious velocitiesareinduced by themethodused tocaptureandadvectthe interface.Indeed,thisflow con-figurationshowsthatVOF-FCTandVOF-PLICtransportschemesgeneratealmostsimilarspuriouscurrents,witha stronger intensitythanthatobtainedwithan LSmethod,whateverthemethodusedtocalculatethecurvatureandthesurfaceforce (continuousorsharp).

6.2. EffectsoftheLaplacenumber

TheeffectsoffluidpropertiesontheintensityofthespuriouscurrentshavefirstbeenconsideredbyvaryingtheWeber number.OurresultsobtainedwiththeVOF-PLIC-HFCSFmethodareclosetothoseobtainedwiththeGerris collocated finite-volumecodein[35]thatusesageometricalVOFtransportschemeoftheinterface(PLIC)andtheCSFmethodiscoupledto theheightfunctioncurvaturecalculation.Withmostofthemethods,thecapillarynumberbasedonthemaximumintensity ofspuriouscurrentsdoesnot dependstronglyontheWebernumber,asmentionedin[35].Sinceapproximatelythesame evolutionisfoundwithallthemethods,wehavefocusedontheeffectoftheLaplacenumberonthespuriouscurrents.The evolutionofthespuriousvelocities,intermsofcapillarynumber,asafunctionoftheLaplacenumberisreportedinFig. 8(a), whichalsoincludesresultsfrom[35].Notethat in[35],thevelocitiesare madedimensionlesswiththeadvectivevelocity

U0.To comparetheseresultswiththe presentstudyinterms ofcharacteristiccapillarynumber,the velocitiesfrom[35]

Fig. 9. MaximumspuriouscurrentscapillarynumberasafunctionofspatialresolutionforLa=12000.Legend:( )VOF-FCT-CCSF;( ) VOF-FCT-HFCSF;( )VOF-PLIC-CCSF;( )VOF-PLIC-HFCSF;( )LS-CCSF;( )LS-HFCSF;( )LS-SSF;( )VOF-PLIC-HFCSF@Gerris[35];(· · ·)

Umax/Usigma∼ (R0/x)−1;(—)Umax/Usigma∼ (R0/x)−2.

fora givenadvectiveWeber numberWe0.Ourresultsobtainedwiththe VOF-PLIC-HFCSFmethodarein goodagreement

withtheresultsfrom[35],despiteaslightlylowerinfluenceofthefluidproperties(Camax

∼

La− 1

4).Whateverthetransport

scheme,VOF-PLIC,VOF-FCTorLS,thespuriouscurrentsgeneratedwiththeCCSFmethodshowalmostnodependencywith thefluidpropertiesandareclosetothoseobtainedinthestaticcase.TheLS-SSFmethodalsogivessimilarspuriouscurrent intensities andtrendsto theLS-CCSF method.However, theHFCSFmethod coupledwitheitherVOF-PLIC, VOF-FCTor LS showsapproximatelythe sametrend,witha decreasein thespurious capillarynumberasthe Laplacenumberincreases (Camax

∼

La−

1 4_).

From ageneralpoint ofview,inthe rangeofLaplaceandWeber numberssimulatedin thepresentwork(1

.

2

<

La

<

12000 and0

.

4

<

We

<

30)with12

.

8 cellsperbubbleradius,VOF-FCT-CCSFandVOF-PLIC-CCSFarealmostequivalentsince theerrorsintheadvectionaresmoothed.ThespuriouscurrentsgeneratedwiththeVOF-PLIC-HFCSFareofthesameorder ofmagnitude.Theheightfunction coupledwiththeVOF-FCTtransportschemeleadsto enhancedspuriouscurrentswhen compared withthesamecurvature calculationcoupledwiththe VOF-PLICtransport schemeandthisis attributedtothe slightinterfacediffusion.Finally,thespuriouscurrentsobtainedinan LSframeworkarereducedbyafactorbetween 2 (La

=

1

.

2 andWe

=

0

.

4)andapproximately100 (La

=

12000 andWe

=

30)whencomparedwiththeminimumonesobtainedin aVOFframework.Itisalsointerestingtopointoutthattheintensityofthespuriouscurrentsinthistestcaseisessentially driven bythetransport scheme,i.e. LS schemesgive better resultsthanVOF-PLIC andfinally VOF-FCT transportschemes. However, the trends observed when increasing the Laplace numberappear to depend on the method forthe curvature calculation, i.e. almostno dependencyon theLaplace numberforthe curvaturederived fromthe interface normalanda slightdecreasewiththeheightfunctioncurvaturecalculation.

Fig. 8(b)showsthemaximumintensityofthevorticity sourcetermandconfirms thatthevorticity productionmainly dependson the transport scheme andthe discretization.Indeed, the vorticity production inLS methods issmaller than in VOFmethods, except with the LS-HFCSF method where strong curvaturegradients, which are localized both in time andspace, appear.As expected, thespurious currentsproductionfora givenmethodisconstant ina firstapproximation throughouttherangeofLaplacenumbersconsidered despiteslightvariationsduetothecouplingbetweenspurious veloc-ities,advectionerrors,curvaturegradients andvorticityproduction.Theseobservationsshowthatthedecreaseinspurious currentsastheLaplacenumberincreasesisnotrelatedtoadecreaseinvorticityproduction.Thus,thedevelopmentofthe vorticitysourcetermintospuriouscurrentsandthebalancebetweenthemdependsontheLaplacenumber.

6.3. Convergencewithspatialresolution

6.3.1. Intensityofspuriouscurrents

Theconvergencewithspatialresolutionofthemaximumvelocity isshowninFig. 9 fora givenLaplacenumber(La

=

12000)andtwodifferentWeber numbers(We

=

0

.

4 andWe

=

30). Asobserved in[17] forthestaticcase, VOF-FCT-CCSF doesnotconvergewithspatialresolutionandthisisalsotruefortheVOF-PLIC-CCSFmethodsinceboth methodsbehave similarlyasexplainedbefore.Itisnotsurprisingthataslightincreaseinspuriouscurrentswithincreasingthenumberof nodesisobservedsincethe numberofsmoothingiterations iskeptconstant andthus, thefilteringprocedureactsovera thinnerregion.VOF-FCT-HFCSFdoesnotconvergeeitherwhenthenumberofnodesisincreasedandthetrendisevenworse withtheLS-HFCSFmethod.VOF-PLIC-HFCSFpresentsalmostafirstorderconvergencerate.Thesedifferentbehaviors witha

Fig. 10. Shape errors(a);curvatureerrors(b)asafunctionofspatialresolutionforLa=12000 andWe=30.Legend:( )VOF-FCT-CCSF;( ) VOF-FCT-HFCSF;( )VOF-PLIC-CCSF;( )VOF-PLIC-HFCSF;( )LS-CCSF;( )LS-HFCSF;( )LS-SSF;LS-SSF;(· · ·)E∼ (R0/x)−1;(—) E∼ (R0/x)−2.

givencurvaturecalculationanddifferenttransportschemesshowtheimportanceoftheerrorsintroducedintheadvection step that are captured withan accurate curvature calculation,such asthe height function method. Finally, LS-CCSF and LS-SSFpresentnearsecondorderconvergencerates.ThemethodsshowsimilartrendswithintherangeofWebernumbers considered. The differencesreside inslightlylower ratesofconvergence atWe

=

30 andthe spurious velocitiesobtained withtheVOF-FCT schemearegreater thanthoseatWe

=

0

.

4.Sincethereisno majordifferenceandthecalculationtime isfoundtodecreasewhenthebubblevelocityincreases(i.e. forhighWebernumbers),allotherspatialconvergencestudies havebeencarriedoutatWe

=

30.

6.3.2. Shapeerrors

Itisinterestingtonotetheconsistencybetweenthepreviousobservationsaboutspuriouscurrentsandtheerrorsonthe shapeofthebubble,Emaxanditscurvature,E

(

κ

)

max aftertranslation.Indeed,itisshowninFig. 10(a)thattheerroronthe shape isminimizedwiththeLSmethodandismaximumwiththeVOF-FCTtransportscheme.Despitethedifferentshape errors obtainedwithVOF-FCT-CCSFandVOF-PLIC-CCSFmethods,theerrors onthecalculatedcurvatureare similardueto thesmoothingprocedureused.However,withtheheightfunctioncurvaturecalculation,spuriouscurrentsarise duetothe advectionofthe interfaceanditis notsurprisingtoobserve anincrease inthecurvatureerrors withthe VOF-FCT-HFCSF method when compared withthe VOF-PLIC-HFCSF method, which minimizesthe advectionerrors in a VOFframework. While theerrorontheshapedecreaseswithgridrefinementwiththeLS-HFCSFmethod,themaximumerroron the cur-vature andthe vorticity source termincrease, asitis thecase withthemaximum spurious velocity. It isobserved with the LS-CCSF that the maximumcurvature errorsare enhanced when compared withLS-SSF orVOF-PLIC-HFCSF methods whereasthespuriouscurrentsgeneratedarelowertothoseobtainedwiththeVOF-PLIC-HFCSFmethodandsimilartothose obtainedwiththeLS-SSFmethod.Thiscanbeduetothefactthatthesameaccuracyisachievedinthecurvature calcula-tionforboththesharpandcontinuousformulationsbutthedifferenceresidesintheinterpolationofthecurvatureatthe interface inthesharpformulation.Thus, theLS-CCSF introducesagradientofcurvaturenormaltotheinterface,whilethe maincontributioninthegenerationofspurious currentscomesfromthetangentialgradient(seeEq.(19)).[38]conducted asimilarmeshconvergencestudyofcurvatureerrorswiththetranslationofacircle.Itisinterestingtonotethatalthough theNavier–Stokesequationswerenotsolvedintheiradvectiontest,thecurvatureerrorsdidnotconvergeeitherwithmesh refinement in a VOF framework with the curvature calculated from the divergence of the unit normal to the interface, whichisinagreementwiththeresultspresentedfortheVOF-FCT-CCSFandVOF-PLIC-CCSFschemesinFig. 10(b).However, theydidnotobserveconvergenceofthecurvatureerrorswithaLevelSetfunctionwhenrefiningthemesh,whilefirstand second orderconvergencerates are observedwith LS-CCSFand LS-SSFmethods respectively. Thisis probablydueto the fact that their LevelSet functionisused intheframework ofa CLSVOFmethod andit hasbeenobserved thatthe order ofconvergenceofcurvatureandspuriouscurrentsisaroundoneorderlowerwithCLSVOFmethodsthanwithLSmethods in thecaseof astaticbubble [33].In addition,[38] presented anewmethod consistingofadvecting the normalsto the interfaceinordertocomputenormalsandcurvaturesaccurately.Closetosecondorderconvergenceratewasobtainedwith translationandrotationadvectiontests.

Inaddition,theratesofconvergenceofthemaximumcurvatureerrorforthedifferentmethods areclosetothose ob-tainedforthemaximumspuriouscurrentintensity.Indeed,goingbackoverthedimensionalanalysispresentedinSection4

reso-Fig. 11. Ratioofvorticitysourcetermtocurvatureerror(a)andratioofspuriouscurrentscapillarynumbertovorticitysourceterm(b)asafunctionof thespatialresolutionLa=12000 andWe=30.Legend:( )VOF-FCT-CCSF;( )VOF-FCT-HFCSF;( )VOF-PLIC-CCSF;( )VOF-PLIC-HFCSF; ( )LS-CCSF;( )LS-HFCSF;( )LS-SSF.(a)(· · ·)E∼ (R0/x);(—)E∼ (R0/x)2.(b)(· · ·)E∼ (R0/x)−1;(—)E∼ (R0/x)−2;(– – –) E∼ (R0/x)−3.

lutionofthevelocity,vorticityproductionandcurvaturecanberelated.Theuseofthischaracteristiclengthisjustifiedsince thepresentanalysisconcerns spuriousflows wherethe derivativesinthevorticity equation (20)quantify errorsoverthe mesh(discretizationerrors,aswell asspuriousvelocityandvorticityfieldsgradients)insteadofvariationsduetophysical phenomena,whichwouldberelatedtophysicalcharacteristiclengths(e.g. thebubblediameter).Thus,thevorticitysource termiswrittenasafunctionoftheerrorinthecurvaturecalculationasfollows:

∇

κ

× ∇

C

∼

E

(

κ

)



x2 (27)

Inaddition,thebalancebetweenvorticityproductionandtheviscoustermcanthenbewrittenas:

μ

ωmax



x2

∼

σ

∇

κ

× ∇

C (28)

i.e. Camax



x3

∼ ∇

κ

× ∇

C (29)

Fig. 11(a)showstheratioofvorticityproductiontocurvatureerrorasafunctionofthemeshsize.Goodagreementwith Eq.(27)isobserved(divergenceoforder2)forallthemethods.Itisinterestingtopointoutthatforagivencurvatureerror, thevorticitysourcetermisminimizedwithstandardcurvaturecalculationswhencomparedwithheightfunctionsincethe curvatureerrorsare localizedinspaceandthecurvaturegradientisenhanced whileitisspreadalong theinterface with smoothedmethods (CCSF).Fig. 11(b)showstheratioofmaximumspurious currentintensityto thevorticitysource term asafunction ofthespatial resolution.Itis seenthat thisratiodecreases asthemeshsize decreaseswhereas therateof convergenceisapproximatelyoneorderlowerthantheoneexpectedfromEq.(29),i.e. closetosecond orderconvergence insteadofthird order.Thisdecreaseintherateofconvergenceisattributedtonon-linear effectsinthevorticityequation. Asaconsequence,curvatureerrorsandmaximumspuriouscurrentintensityshowsimilarratesofconvergencewithspatial resolution(

∼ |∇

κ

× ∇

C

|

x2).Finally,note that foragivenvorticity production,theVOFmethods minimizethe spurious capillarynumberwhencomparedwithLSmethods(seeFig. 11(b))exceptfortheLS-HFCSFmethodwherestrongoscillations of the curvature and the vorticity source term occur instantaneously and lead to a decreased ratio of spurious current intensitytovorticitysourceterm.Theseobservationsallowustoconcludethatspurious currents,vorticityproductionand curvaturegradientsarecloselyrelated.Thesensitivityofthevorticityproductiontothetransportschemeandtheadvection errorsis enhancedwiththe heightfunction curvaturecalculation whencomparedwiththe smoothedstandardcurvature calculation.

6.3.3. Laplacepressurejump

Anotherfeatureofthecouplingbetweenthesurfacetensionscheme,thecurvaturecalculationandthetransportscheme isthe accuracyinthepressure jumpestimation.The decreaseinspurious velocitiesdoesnotnecessarilylead toa better pressure jumpcalculation.Indeed, thepressure jumpdependson theerroron thecurvature whilespurious currentsare relatedtoitsderivative sincethesourceofvorticityresidesincurvaturegradientstangentialtotheinterface.Forinstance,

[17] showedthat increasingthenumberofiterations inthefilteringprocedureofthevolumefractionleadstoadecrease inspuriousvelocitiesbutthenumericalthicknessofthepressurejumpincreaseswithfilteringandacompromisetherefore

Fig. 12. Normalizedpressurejump.Closeupof(a)thepressureinthebubble;(b)thepressurejumpatthebubblerearcap.Legend:( )VOF-FCT-CCSF; ( )VOF-FCT-HFCSF;( )VOF-PLIC-CCSF;( )VOF-PLIC-HFCSF;( )LS-CCSF;( )LS-HFCSF;( )LS-SSF.

Fig. 13. Pressureerrorsasafunctionofspatialresolutionfor La=12000 andWe=30.Legend:( )VOF-FCT-CCSF;( )VOF-FCT-HFCSF;( )

VOF-PLIC-CCSF;( )VOF-PLIC-HFCSF;( )LS-CCSF;( )LS-HFCSF;( )LS-SSF;(· · ·)E∼ (R0/x)−1;(———)E∼ (R0/x)−2.

needs tobefoundtoaccuratelycalculatetheLaplacepressurejump.Inthepresentsimulations,theerrorsinthepressure jump were generally insensitive to the changes in the Weber number andno explicit tendency was observed withthe Laplacenumber.Fig. 12showsthepressureprofilethroughthebubblealongthehorizontalplaneofsymmetry.Itisseenin

Fig. 12(a)that thepressureatthe centerofthebubbleisbettercalculatedwithin an LSframework(lessthan 0

.

3% error) thaninaVOFframework.AlthoughthepressureatthecenterofthebubblewiththeVOF-PLIC-HFCSFmethodisnotsofar fromthe pressureestimatedinthe LSsimulations,thepressure fieldhas peaksaroundthe interface,whereas itismuch moreuniforminan LSframework.

Fig. 13(a) reports the maximum pressure jump errors asa function of spatial resolution. No convergence with grid spacingis observedwithanymethod.The importanceofthe transportschemewhen usingtheheightfunction curvature calculation is againhighlighted. The VOF-PLIC transport schemeallows the maximum pressurejump errors to be signif-icantly decreased when compared with the VOF-FCT advection scheme. However, the errors remain greater than those obtained witheitherVOF-FCT-CCSF or VOF-PLIC-CCSFmethods, whichare againsimilar. The LSmethods andmainly the LS-CCSF method allow themaximumpressure jump errorsin thedomain tobe minimized. Finally, thetransitionregion ofthepressurejumpacrosstheinterface atthebubblerearcapisillustratedinFig. 12(b)andquantified withthespatial convergenceoftheerroronthetotalpressurejump



Ptotal

/(

σ

/

R0

)

inFig. 13(b).Itisseenthatcomparedwiththe

contin-uousformulationofthesurfacetensionforce,theheightfunctionreducesthetransitionregionandespeciallywithinaVOF framework sincethe smoothingprocedureapplied toreduce spuriouscurrentsin theCCSFmethodsspreads thepressure jump. However, itis clearthat the sharpsurface force is themostaccurate since there isno numericalthicknessof the

Fig. 14. Intensityofthesourceterm|∇κ× ∇C|inthevorticityequation(20)andvelocityfieldafterthetranslationofadistance1.25D intheframeof referencemovingwiththebubble(bydecreasingspuriouscurrentsfrom(a)to(f))forR0/x=12.8,La=12000 andWe=0.4.

interface. The LS-SSF method also showsa better rateof convergence (slightly lessthan 2) than all the other methods, whichallconvergewithfirstorder.

7. Rotatingbubble

A bubble placed in a rotating flow has been used previously to characterize the efficiency of the transport scheme

[6] butnot to study the efficiency ofthe coupling between interface advectionand the surface tension force. This test is interesting since the displacement is not uni-directional like forthe translating bubble,which movesalong the mesh direction.Theoretically,thevelocityintheframeofreferencemoving withthebubbleshouldbe zero,providingthereare nospuriouscurrents.AlthoughthepressureineachphaseisnotconstantduetotheadvectivetermsoftheNavier–Stokes equations,whichare not nullinthiscase, thepressure jumpatthe interface still obey theLaplacelaw sinceno viscous shearstress is inducedby the rotatingflowfield givenby Eq.(30).The theoretical velocityandpressure fieldsare given byEqs.(30)–(31).Imagesoftheconfigurationandthespuriouscurrentsgeneratedinsuch aflowafteronerevolutionare illustratedinFig. 14.

(

u

,

v

) (

x

,

y

)

= (

U0

×

y

,

−

U0

×

x

) ,

(30) p

(

x

,

y

)

=

ρ

U 2 0 2



(

x

−

x0

)

2

+ (

y

−

y0

)

2



+



_σ

R0 in the bubble

,

0 outside

.

(31)

TheevolutionofthemaximumintensityofspuriouscurrentswithtimeisreportedinFig. 15.Inthistestcase,whichis slightlymorecomplex,thepredominanteffectsofthetransportschemeonthespuriouscurrentsthathavebeenobserved withthe translating caseare enhanced. Fig. 15 showsthat the spurious currentsare clearly controlled by the transport scheme. Indeed,with both curvature calculations, the use ofthe VOF-PLIC scheme leads to decreased spurious currents when compared with those obtained withthe VOF-FCT scheme. This decrease is accentuatedwhen compared withthe translating case. Despitethe smoothingprocedure,spurious currentsobtainedwiththe VOF-PLIC-CCSFarereducedwhen comparedwiththeVOF-FCT-CCSFmethod.Thismeansthattheshapeerrorsintroducedbytheadvectionsteparecaptured in the curvaturecalculation and overcomethe errors associated withsmoothed curvaturecalculations. Nevertheless,the spurious currents obtained withthe VOF-PLIC transport schemes remain approximately3 to 4 times greater than those

Fig. 15. TemporalevolutionofthemaximumvelocityofspuriouscurrentsintheframeofreferencemovingwiththebubbleforR0/x=12.8,La=12000

andWe=30.Legend:( )VOF-FCT-CCSF;( )VOF-FCT-HFCSF;( )VOF-PLIC-CCSF;( )VOF-PLIC-HFCSF;( )LS-CCSF;( )LS-HFCSF; ( )LS-SSF.

Fig. 16. DimensionlessmaximumvelocityofspuriouscurrentsasafunctionofspatialresolutionforLa=12000 andWe=30.Legend:( )

VOF-FCT-CCSF;( )VOF-FCT-HFCSF;( )VOF-PLIC-CCSF;( )VOF-PLIC-HFCSF;( )LS-CCSF;( )LS-HFCSF;( )LS-SSF;(· · ·)Umax/Usigma∼

(R0/x)−1;(———)Umax/Usigma∼ (R0/x)−2.

obtainedwithinan LSframeworkwherebothcontinuousmethods(LS-HFCSFandLS-CCSF)andthesharpmethod(LS-SSF) giveidenticalspuriousvelocities.

Similarly towhat happensinthe translatingcase, the maximumandmeanvelocitiesgenerallydo notdepend onthe Webernumber.ThetrendsobservedfortheevolutionoftheintensityofspuriouscurrentswiththeLaplacenumberarealso qualitativelysimilartothoseinthetranslatingcase.However,theyaremorehomogeneousthaninthetranslatingcase,and thevelocityscalesapproximatelyasCamax

∼

La−1/4 formostofthemethods,exceptLS-CCSFandLS-SSF(Camax

∼

La−1/6).

7.1. Convergencewithspatialresolution

The convergence of the different methods with spatial resolution is presented in Fig. 16, which shows the capillary numberbased onthemaximumvelocity asa functionofthegrid resolution.Aspreviously observedwiththetranslating case, withtheVOF-FCT-HFCSFscheme,theintensityofspuriouscurrentsismaximumandincreaseswithmeshresolution. Within a VOF-FCT framework, i.e. without interface reconstruction, the intensity of spurious currents is lower with the CCSFmodelthanwiththeHFCSFschemebutthesametrendwithmeshrefinementisobserved.Similarly,theintensityof spuriouscurrentsincreasesasthemeshisrefinedwiththeVOF-PLIC-CCSF.Atlowresolution,thesameintensityofspurious currentsisfoundwiththeVOF-PLIC-HFCSFmethodbutitdecreaseswithmeshresolution.However,therateofconvergence isapproximatelyonethird,whichissmallerthaninthetranslatingcase(closetoone).ConcerningtheLStransportscheme that minimizesspurious currents,while theLS-CCSF andLS-SSFmethods showedarateofconvergence about1

.

5 in the translating case,themagnitudeofthespuriouscurrentsdecreasesinthisflowconfigurationaslongasthenumberofcells

Fig. 17. Rangeofmaximumspuriouscapillarynumbergeneratedwiththedifferentmethodsfor1.2≤La≤12000;0.4≤We≤30 and6.4≤R0/x≤51.2.

Foreachmethod,thestatic,translatingandrotatingcasesarereportedfromlefttoright.Thesymbolsrepresenttheaveragevaluesofspuriousvelocities obtainedwithintherangeoffluidpropertiesand spatialresolutioncovered andtheerrorbar correspondstotheminimumandmaximum valuesof spuriousvelocitiesobtained.

Table 1

Summaryofthemaincharacteristicsofthetransportschemesandcalculationsofthesurfacetensionforceusedforthestatic,translatingandrotatingtest cases.Theratesofconvergencewithspatialresolution,aswellasthedependenciesontheLaplacenumber,havebeensimplifiedtomakethecomparison easierandstressonlythemajordifferences.Inordertoallowaneasycomparisonbetweenthemethodsatacommonresolutionof12.8 cellsperbubble radius,thespatialresolutionisexpressedwithamodifieddimensionlessbubbleradiusR∗0/x=R0/(12.8x).Thecoefficientofthescalinglawstherefore

corresponds to the spurious current intensity for R0/x=12.8 and La=1.

Method – spurious currents magnifier Camaxstatic case Camaxtranslating case Camaxrotating case

VOF-FCT-CCSF – curvature calculation 10−3 1×10−3

_R∗ 0 x 0.4 2×10−2La−14 _R∗ 0 x 0.4

VOF-FCT-HFCSF – advection errors 1.4×10−8

 R∗0 x −1.6 2×10−2La−14  R∗0 x 0.4 6×10−2La−14  R∗0 x 0.4

VOF-PLIC-CCSF – curvature calculation see VOF-FCT-CCSF 7×10−4

 R∗0 x 0.2 7×10−3La−14  R∗0 x 0.2

VOF-PLIC-HFCSF – advection errors 10−12 5×10−3La−14

 R∗0 x −0.7 5×10−3La−14  R∗0 x −0.3

LS-CCSF – curvature calculation and redistancing step 6.1×10−5

 R∗0 x −1.8 4×10−5  R∗0 x −1.4 5×10−4La−16

LS-HFCSF – redistancing step and advection errors 3.6×10−6

 R∗0 x −1.3 3×10−4La−14  R∗0 x 1.4 10−3La−14  R∗0 x 0.8 LS-SSF – redistancing step 5.7×10−5  R∗0 x −2.1 5×10−5  R∗0 x −1.7 5×10−4La−16

per bubbleradius islower than 13 butthen stabilizeswhen refiningthe meshfurther. TheLS-HFCSF methodshowsthe sametrendasinthetranslatingcaseandexhibitsnoconvergencewithspatialresolution.

Concerning the pressure errors and their convergence with grid refinement, the observations do not differ from the translating case, exceptthat thedifferencesbetweenVOF-FCT andVOF-PLIC transportschemes are againenhanced.As it hasbeenobserved withthe spuriousvelocities,the rateofconvergence forthe pressurejumpdecreases inthistest case whencomparedwiththetranslatingone.TheLS-CCSFmethodallowsthemaximumpressurejumperrortobeminimized but none of the methods converge with spatial resolution. The errors obtained withthe VOF-FCT-HFCSF and LS-HFCSF methodsincreasewithdecreasingthecellsize.Itisinterestingtonoteadifferencewiththetranslatingcase;althoughthe SharpSurface Force formulation still imposes asharp pressurejump, thismethoddoesnot appear tobe superior tothe VOF-PLIC-HFCSF,LS-CCSForLS-HFCSFmethods,whichgivethebestresultsinthisrotatingflowregardingthetotalaverage pressurejump.Finally,theLS-CCSFmethodseemstobeagoodcompromisebetweenspuriouscurrentintensityandpressure jumpcalculation. TheVOF-PLIC-HFCSFmethodalsoshowsgoodresults concerningthepressure jumpestimationthat are closetothoseobtainedwiththeLS-CCSFmethod.

8. Summaryofstatic,translatingandrotatingcases

Asummaryoftherangeofspuriouscurrentsobservedinthisstudyforthedifferentmethodsandtestcasesisreported inFig. 17andinTable 1throughapproximatecorrelationswiththeLaplacenumberandspatialresolution.Thesuperiorityof