An efficient mass-preserving interface-correction level set/ghost fluid method for droplet suspensions under depletion forces

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Zhouyang GE, Jean-Christophe LOISEAU, Outi TAMMISOLA, Luca BRANDT - An efficient

mass-preserving interface-correction level set/ghost fluid method for droplet suspensions under

depletion forces - Journal of Computational Physics - Vol. 353, p.425-459 - 2018

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An

efficient

mass-preserving

interface-correction

level

set/ghost

fluid

method

for

droplet

suspensions

under

depletion

forces

Zhouyang Ge

∗

,

Jean-Christophe Loiseau

1

,

Outi Tammisola,

Luca Brandt

LinnéFlowCentreandSeRC(Swedishe-ScienceResearchCentre),KTHMechanics,S-10044Stockholm,Sweden

a

b

s

t

r

a

c

t

Keywords: Multiphaseflow Levelsetmethod Ghostfluidmethod Colloidaldroplet Depletionforce

Aimingforthesimulationofcolloidaldropletsinmicrofluidicdevices,wepresentherea numerical methodfortwo-fluid systemssubjecttosurfacetension anddepletion forces among the suspended droplets. The algorithm is based on an efficient solver for the incompressibletwo-phaseNavier–Stokesequations,and usesamass-conservinglevelset methodtocapturethefluidinterface.Thefournovelingredientsproposedhereare,firstly, an interface-correctionlevelset (ICLS)method; global massconservationisachievedby performinganadditionaladvectionneartheinterface,withacorrectionvelocityobtained bylocallysolvinganalgebraicequation,whichiseasytoimplementinboth2Dand3D. Secondly,wereportasecond-orderaccurategeometricestimationofthecurvatureatthe interfaceand,thirdly,thecombinationoftheghost fluidmethodwiththefast pressure-correction approach enabling an accurate and fast computation even for large density contrasts.Finally,wederiveahydrodynamicmodelfortheinteractionforces inducedby depletionofsurfactantmicellesandcombineitwithamultiplelevelsetapproachtostudy short-rangeinteractionsamongdropletsinthepresenceofattractingforces.

1. Introduction

Inthe fieldofcolloidalscience, muchprogresshasbeen madeonthesynthesis ofelementarybuildingblocks(Fig. 1) mimickingmolecularstructurestoelaborateinnovativematerials,

e.g. materials

withcompletethreedimensionalbandgaps [1–4]. Thebasicelements ofsuch colloidalmolecules areparticles ordroplets lessthan one millimeterinsize, andtheir self-assembly reliesoneither lengthybrownian motionorcarefulmicrofludicdesigns, ontop oftypical colloidal interac-tions,

e.g. depletion

attractionandelectrostaticrepulsion[5–7].Regardlessoftheapproach,however,questionsremainwhy thecolloidalparticles/dropletsundergocertainpathtoorganizethemselvesandhowsuchprocesscanbecontrolledand op-timized.Sincefulldataarenotyetaccuratelyaccessiblefromexperimentsinsuchminiaturesystems,computersimulations willbeusefultoprovidesupplementalinformation.

Scalingdowntomicroscaleappearsfirsttobeaconvenienceforthenumericalsimulationsofmulticomponentand mul-tiphasesystemsasthenon-linearNavier–Stokes(NS)equationscanbereducedtothelinearStokesequations.Thisallows

*

Correspondingauthor.

E-mailaddresses:zhoge@mech.kth.se(Z. Ge),jean-christophe.loiseau@ensam.eu(J.-C. Loiseau),outi@mech.kth.se(O. Tammisola),luca@mech.kth.se

(L. Brandt).

Fig. 1. Self-assembled colloidalclusters.a)Electronmicrographofasuspensionoftripletclusters.Scalebar,30μm.b–e)Closeupofdoublet,triplet, quadruplet,andquintupletclusters.Scalebars,10

μ

m.Furtherdetailsareavailablein[7],photographcourtesyofDr.JoshuaRicouvier.

theuseofboundaryintegralmethods(BIM)[8],

e.g. most

recentlytheGGEM-basedBIM[9,10]solvingtheStokesequations in generalgeometries. However,it isalsopossibletouse theconventionalunsteady, fractional-step/projection-methodNS solveratlowReynoldsnumber,combinedwithaninterfacedescriptionmethod[11,12].Thelatterapproachismore versa-tile,probablylessdifficultto implement,andenjoysa richliteratureofstandard numericaltechniques.Here,inview ofa richrangeofpossibleapplicationsandconsideringalsotherapiddevelopmentofinertialmicrofluidics(whereinertialeffects are usedtobettercontroltheflowbehavior)wetaketheapproachofsimulatingtheincompressible, two-fluidNSas out-linedin[13].Thesplittingprocedureproposedin[13]enablestheuseoffastsolversforthepressurePoissonequationalso forlargedensityandviscositycontrasts.Theremainingchoicethenistobemadeamongtheavailableinterface-description methods.

Generally, there are two categories ofmethods to resolve an interface in a NS solver, i.e. front-tracking methods and front-capturingmethods.Anexampleofthefront-trackingmethodistheimmersedboundarymethod(IBM)[14,15].Using Lagrangian pointsinamoving frame,IBMcanofferahighinterfaceresolutionwithouttheneedtodeformtheunderlying meshinthefixedframe.However,thecouplingofthetwomeshesreliesonaregularizeddeltafunction,whichintroduces certain degreesof smearing. Moreover, large interface deformation requires frequent mesh rearrangement; andtopology changes may haveto be handledmanually. Theseconstraints make IBMtypically more expensiveandless appealingfor dropletsimulations.

Front-capturingmethods,ontheotherhand,areEulerianandhandletopologychangesautomatically;theyaretherefore easier to parallelize to achieve higherefficiency. One of such methods is the volume-of-fluid (VOF) method[16], which definesdifferentfluidswithadiscontinuous colorfunction.Themain advantageofVOFis itsintrinsicmassconservation. It suffers however from inaccurate computations of the interface properties, e.g. normals and curvatures. This makes it less favorable forsimulations ofmicrofluidicsystemswhere surface tensionis thedominanteffect andrequiresaccurate modelling.

Anotherpopularfront-capturingmethodisthelevelset(LS)method[17,18].ContrarytoVOF,LSprescribestheinterface through a(Lipschitz-)continuousfunctionwhichusually takestheformofthesigned distanceto theinterface.Underthis definition, normalsandcurvatures ofthe interface can bereadily andaccurately computed. However, theproblemwhen simulating incompressible flows is that mass loss/gain may occur and accumulate because the LS function embeds no volume information.In addition, errors can also arise from solving the LSadvection equation and/or the reinitialization equation, aprocedure commonlyrequiredto reshapetheLSintoadistancefunction.Therefore,additionalmeasures have tobetakentoensuremassconservation.

Manydifferentapproacheshavebeenproposed tomakeLSmass-conserving,whichcanbeclassifiedintothefollowing four methodologies. The first approach is to improve the LS discretization and reinitialization so that numerical errors are reduced. In practice,one can increase theorder of LSfluxes [19], minimize the displacement ofthe zero LSduring reinitialization [20,19],oremploylocalmeshrefinement[21–23].Bydoingso,masslosscanbe greatlyreduced,although theLSfunctionisstillinherentlynon-conservative.ThesecondremedycouplestheLSwithaconservativedescription(e.g. VOF) orLagrangian particles.Forexample,thehybridparticlelevelsetmethod[24],thecoupledlevelsetvolume-of-fluid (CLSVOF)method[25],themass-conservinglevelset(MCLS)method[26],ortherecentcurvature-basedmass-redistribution method[27].Withvaryinglevelofcoupling,thesemethodscanusuallypreservemassreallywell;thedrawbackisthatthe complexityandsomeinaccuracy(duetointerpolation,reconstruction,

etc.)

oftheothermethodwillbeimported.Thethird approach improves massconservationby addinga volume-constraintin theLSorNS formulation. Examples ofthiskind includetheinterface-preservingLSredistancingalgorithm[28]andthemass-preservingNSprojectionmethod[29].Finally, one can alsosmartly modifythe definitionofthe LS, such asthehyperbolic-tangent levelset [30], toreduce the overall massloss.

Withthephysicalapplicationofcolloidaldropletsinmind,andusingideasfromsomeoftheabove-mentionedmethods, we heuristicallyproposeaninterface-correctionlevelset(ICLS)method.TheessentialideaofICLSistoconstructanormal

velocitysupportedonthedropletinterfaceanduseitinanadditionalLSadvectiontocompensateformassloss,inaway similar to inflating a balloon. Becauseno coupling withVOFor Lagrangian particles is required,the simplicityand high accuracyoftheoriginalLSmethodispreserved,yettheextracomputationalcostofthisprocedureisnegligible.

Providedamass-preservinglevelsetmethod,thecoupledflowsolvermustalsoaccuratelycomputethesurfacetension, a singular effectof thenormal stress on theinterface. This isparticularly importantformicrofluidicsystems; assurface tension scales linearly with the dimension, it decays slower than volumetric forces (e.g. gravity) when the size of the systemreduces.Tohandlesuchdiscontinuities,oneapproachisthecontinuumsurfaceforce(CSF)[31],originallydeveloped for the VOF method,later extended to the LS[18]. Although easy to implement, CSF effectively introduces an artificial spreading ofthe interface by regularizing thepressure difference,and itcan become erroneouswhen two interfacesare withinitssmoothingwidth.Asecond,non-smearingapproachistheghostfluidmethod(GFM).Proposedinitiallyforsolving compressibleEulerequations[32],GFMprovidesafinite-differencediscretizationofthegradientoperatorevenifthestencil includesshocks. Ithasbeen provento converge[33] andwas soonapplied fortreating thepressure jumpinmultiphase flows [34]. We note that although the GFM can be reformulated in a similar wayto the CSF[35,36], its treatment for discontinuousquantitiesissharpinthefinitedifferencelimit.

SeveralimplementationoptionsoftheGFMweresuggestedin[34,35,37].Here,wefollowthemethodologyof[37],

i.e.

usingtheGFMforthepressurejumpduetosurfacetensionwhileneglectingtheviscouscontribution.Aswillbediscussed later, thischoice isespeciallysuitableformicrofluidicapplicationswherethecapillaryeffectisstrong.Toefficientlysolve forthe pressure, we furthercombine theGFM witha fastpressure-correction method (FastP*) [13]. Such a combination enables a directsolve of thepressure Poisson equation using theGauss elimination inthe Fourierspace; it is themost efficient when the computational domain is periodic, but it also applies to a range of homogeneous Dirichlet/Neumann boundary conditionsvia fastsine/cosinetransforms [38], see

e.g. a

recent open-source distribution[39].Using a second-order accurate,grid-converging interface curvature estimation,we will show that the coupledICLS/NS solver canhandle largedensity/viscositycontrastsandconvergesbetweenfirstandsecondorderinbothspaceandtime.

Finally,auniquechallengetothesimulationofcolloidaldropletsisthemodelingofnear-fieldinteractions.Itisknown thattwoormorecolloidscaninteractviadispersion,surface,depletion,andhydrodynamicforces[5].Apartfromthe hydro-dynamicforceswhichisdetermineddirectlyfromtheNS,andthedispersionforceswhicharisefromquantummechanical effects,thedepletionandsurface forcesmustbemodelled.Theseforcescan beeitherattractionorrepulsionandare typ-icallycalculated fromthegradientofa potential.Based oncolloidal theory,we propose anovel hydrodynamicmodel for thedepletion force intheframework oftheICLS/NSsolver. Ourmethodreliesontwo extensions:

i) extending

thesingle levelset(SLS)functiontomultiplelevelset(MLS)functions;and

ii) extending

theGFMforcomputationofthegradientof depletionpotential.MLShasthebenefitsthateachdropletwithinacolloidalclustercanbetreatedindividually,isallowed to interactwiththe other droplets, andis guarded fromits own mass loss.MLS also preventsnumerical coalescenceof dropletswhenthey gettooclose.Thecomputational complexity,proportional tothenumberofMLSfunctions(l)andthe numberofcellsineachdimension(N),ishigherthanSLS.However,wenotethatmanytechniquesexisttoreducetheCPU costand/or memoryconsumption if

lN

d _(d

₌

_{2or 3) islarge. Fordetailedimplementations ofsuchoptimized algorithms} wereferto[40–42].Inthepresentpaper,wewilldemonstratetheself-assemblyofcolloidaldropletsusingonedropletper MLSfunction.

Thepaperisorganizedasfollows.InSec.2,thegoverningequationsfortheincompressible,two-phase flowarebriefly presented.InSec.3,theclassical signed-distanceLSmethodologytogetherwithsomecommonlyusednumericalschemes isdiscussed.Wethen introducetheICLSmethodinSec. 4,starting fromthederivationendingwithademonstration.We furtherprovideageometricestimationoftheinterfacecurvaturetailoredtotheGFMinSec.5.ThecompleteICLS/NSsolver is outlined inSec. 6,including a detaileddescription ofthe implementationandthree examples ofvalidation. InSec. 7, weproposeaMLS/GFM-based methodforthemodelingofnear-fielddepletionpotential.Finally,wesummarizetheoverall methodologyinSec.8.

2. Governingequationsforinterfacialtwo-phaseflow

ThedynamicsoftheincompressibleflowoftwoimmisciblefluidsisgovernedbytheNavier–Stokesequations,written inthenon-dimensionalform

∇ ·

u

=

0, (1a)

∂

u

∂

t

+

u

· ∇

u

=

1

ρ

i



− ∇

p

+

1 Re

∇ ·



μ

i

(

∇

u

+ ∇

uT

)



+

1 F rg

,

(1b)

where u

=

u

(

x

,

t

)

is the velocity field, p

=

p

(

x

,

t

)

is the pressure field, and g is a unit vector alignedwith gravity or buoyancy.

ρ

i and

μ

i are the densityand dynamic viscosity ratios of fluid i (i

=

1 or 2) and the referencefluid. These propertiesareconstant ineachphase andsubjectto ajumpacrossthe interface,whichwe denoteas

[

ρ

]



=

ρ

2

−

ρ

1 for densityand

[

μ

]



=

μ

2

−

μ

1 forviscosity.Forviscousflows,thevelocityanditstangentialderivativesarecontinuousonthe interface[43].However,thepressureisdiscontinuousduetothesurfacetensionandtheviscosityjump,

i.e.

[

p

]



=

1 W e

κ

+

2 Re

[

μ

]

n T

_{· ∇}

_u

_·

_n

_,

₍₂₎

where

κ

istheinterfacecurvature,andn isthenormaltotheinterface.Ifthesurfacetensioncoefficient,

σ

˜

,variesonthe interfacethetangentialstressisalsodiscontinuous.Inthispaper,weassumeconstantanduniform

σ

˜

.InEqs.(1b)and(2), Re,We,andFrare,respectively,theReynolds,Weber,andFroudenumbers,definedas

Re

=

ρ

˜

1U

˜

˜

L

˜

μ

1

,

W e

=

ρ

˜

1U

˜

2

˜

_L

˜

σ

,

F r

=

˜

U2

˜

g

˜

L

,

(3)

where U ,

˜

L,

˜

ρ

˜

1,

μ

˜

1,and g denote

˜

thereferencedimensionalvelocity,length, density,dynamicviscosity,andgravitational acceleration.Notethat

ρ

1

=

1 and

μ

1

=

1 (i.e.wedefinefluid1asthereferencefluid).

3. Classicallevelsetmethodology

In thelevel setframework, theinterface



isdefinedimplicitlyasthe zerovalue ofa scalarfunction

φ (

x

,

t

)

,

i.e.



=

{

x

| φ(

x

,

t

)

=

0

}

. Mathematically,

φ (

x

,

t

)

can be any smooth or non-smooth function; but it isclassically shaped asthe signedEuclideandistancetotheinterface[44,18],

viz.

φ (

x

,

t

)

=

sgn

(

x

)

|

x

−

x

|,

(4)

where x denotes the closest pointon the interfacefrom nodalpoint x,and sgn

(

x

)

isa signfunction equal to1 or

−

1

depending on whichside ofthe interface itlies.Fortwo-phase problems withsingle levelset, sgn

(

x

)

provides anatural “colorfunction”forphaseindication.Furthermore,withthisdefinition,geometricpropertiessuchastheunitnormalvector,

n,andthelocalmeancurvature,

κ

,canbeconvenientlycomputedas n

=

∇φ

|∇φ|

,

(5)

κ

= −∇ ·

n

.

(6)

3.1. Advection

ThemotionofafluidinterfaceisgovernedbythefollowingPDE

∂φ

∂

t

+

u

· ∇φ =

0, (7)

where u is the flow velocity field. Despite of its simple form, obtaining an accurate and robust solution to Eq. (7) is challenging. Fortwo-fluid problems, state-of-the-art level set transport schemes includethe high-order upstream-central (HOUC) scheme [19],the weighted essentially non-oscillatory(WENO) scheme[43],the semi-Lagrangian scheme[45],or the semi-jet scheme [46]. Quantitative comparisons of theseschemes in various test cases can be found in[19,46]. We note that thechoice of the schemeis case-dependent, i.e. depending onthe smoothnessof theoverall level set field or thestiffness ofEq.(7).Forflowsinvolvingmoderatedeformations, HOUCisusuallysufficientandmostefficient.Formore complexflows,WENOorsemi-Lagragian/jetschemescombinedwithgridrefinementmightbepursed.Inthepresentstudy, weuseeitherHOUC5orWENO5(5denotesfifth-orderaccuracy)toevaluate

∇φ

.

ForthetemporaldiscretizationofEq.(7),weuseathree-stagetotal-variation-diminishing(TVD)third-orderRunge–Kutta scheme[47].Denoting f

(φ)

= −

u

· ∇φ

,itupdates

φ

fromtimelevel

n to n

+

1 inthreesub-steps

⎧

⎪

⎨

⎪

⎩

φ

1

= φ

n

+ 

t

·

f

(φ

n

)

φ

2

=

3₄

φ

n

+

1₄

φ

1

+

1₄



t

·

f

(φ

1

)

φ

n+1

=

1₃

φ

n

+

2₃

φ

2

+

2₃



t

·

f

(φ

2

).

(8)

Finally, we note that Eq. (7) does not need to be solved in the entire computational domain, as only the near-zero values are used to identifythe interface andcompute its curvature.This motivatedthe so-callednarrow band approach [48,40],whichlocalizesthelevelsettotheinterfaceusingindexarrays.Combinedwithoptimaldatastructures[41,42],fast computation andlowmemoryfootprint maybe achievedatthe sametime.Inourimplementation,westore allthelevel setvalueswhileonlyupdatethoseinanarrowband,

i.e. solving

φ

t

+

c

(φ)

u

· ∇φ =

0 withthecut-offfunctiongivenas

c

(φ)

=

1 if

|φ| <

γ

0 otherwise, (9)

where

γ

=

6



x as additionaldistanceinformationisrequiredto modeldropletinteractions (Sec.7). Thisisequivalentto [40]withasimplified

c

(φ)

.

Zalesak’s disk. The Zalesak’sdisk[49],

i.e. a

slotteddiscundergoingsolidbodyrotation,isastandardbenchmarktovalidate levelsetsolvers. Thedifficultyofthistest liesinthetransport ofthesharpcornersandthethinslot,especiallyin under-resolvedcases.Theinitialshapeshouldnotdeformundersolidbodyrotation.Hence,bycomparingtheinitiallevelsetfield

Fig. 2. Comparisonoftheinitialinterfaceanditsshapeafteronefullrotationfordifferentmeshresolutions.Solidlinesdepicttheinitialinterface.Two differentschemeshavebeenusedtoevaluatethegradients,namelyHOUC5(dashedlines)andWENO5(dash-dottedline).(Forinterpretationofthecolors inthisfigure,thereaderisreferredtothewebversionofthisarticle.)

andthatafteronefullrotationonecancharacterisethedegreeofaccuracyofanumericalsolver.Here,theparametersare chosen sothata diskofradius 0

.

15,slotwidthof0

.

05 iscenteredat

(

x

,

y

)

= (

0

,

0

.

25

)

ofa

[−

0

.

5

,

0

.

5

]

× [−

0

.

5

,

0

.

5

]

box. Theconstantvelocityfieldisgivenas

u

= −

2

π

y

,

v

=

2

π

x

.

(10)

Threedifferentmeshresolutionshavebeenconsidered,namely50

×

50,100

×

100 and200

×

200.Fig. 2depictstheshape oftheinterface afterone fullrotationofthe disk,solvingEq.(7)only.Along withtheresultsoftheHOUC5 scheme(red dashed line), the shape of the interface obtainedusing the WENO5 scheme (green dash-dottedline) is also reportedin thisfigure. Both schemes yieldgood resultson finegrids, butHOUC5 clearly outperforms WENO5 onthe coarsestmesh consideredhere.

3.2. Reinitialization

Althoughthelevelsetfunctionisinitializedtobe asigned-distance,itmaylosethispropertyastimeevolves, causing numericalissuesparticularlyintheevaluationofthenormalandthecurvature[18].Inordertocircumventtheseproblems, anadditionaltreatmentisrequiredtoconstantlyreshape

φ

intoa distancefunction,

i.e.

|∇φ|

=

1.Thiscanbedone either withadirect,fastmarchingmethod(FMM)[17],orbyconvertingitintoatime-dependentHamilton–Jacobiequation[18]

∂φ

∂

τ

+

S

(φ

0

)(

|∇φ| −

1)

=

0, (11)

where

τ

isapseudo-time,andS

(φ

0

)

isamollifiedsignfunctionoftheoriginallevelset,usuallydefinedas

S

(φ

0

)

=

⎧

⎪

⎪

⎨

⎪

⎪

⎩

−

1 if

φ

0

<

−

x 1 if

φ

0

>



x φ0  φ2₀+x2 otherwise. (12)

ComparingwithFMM,thesecondapproachallows theuseofhigherorderschemes (e.g. WENO5)andiseasy to paral-lelize;hence,ithasbeenamuchmorepopularchoice.However,aspointedoutbyRussoandSmereka[20],usingregular upwindingschemesfor

∇φ

neartheinterfacedoesnotpreservetheoriginallocationofthezerolevelset.Thiscanleadto massloss,especiallyifthelevelsetisfarfromadistancefunctionandEq.(11)needstobeevolvedforlongtime.A simple solutionistointroducea“subcellfix”[20],whichpinstheinterfaceinthereinitializationbymodifyingthestencil. Beauti-fullyasitworksinredistancingthelevelset,thismethodishoweveronlysecondorderaccurate andthusnotwell-suited forevaluatingcurvature.Itsfourthorderextension[50]suffersfromstability issuesandmayrequireaverysmall pseudo-timestep[22].Basedontheseobservations,inthispaperwesolveEq.(11)usingtheclassicalWENO5 [43]andthesame SSP-RK3[47].The reinitializationisnot performedatevery physicaltime step,butdependsontheadvectionvelocity. In ourapplications,ittypicallyrequiresonetotwoiterationsofEq.(11)pertentoahundredtimesteps.

Distorted elliptic field. In ordertoillustratetheredistancing procedure,atest casesimilar totheone in[20] isconsidered. Definetheinitiallevelsetas

φ (

x

,

y

,

0)

=

f

(

x

,

y

)

⎛

⎝



x2 4

+

y2 16



−

1

⎞

⎠ ,

Fig. 3. Illustrationofthereinitializationprocedure.Theshapeoftheellipsoidisdepictedasthethicksolidline.Thedashedlinesthendepictiso-contours ofφ(x,y)rangingfrom−1 to1 byincrementsof0.25.(Forinterpretationofthecolorsinthisfigure,thereaderisreferredtothewebversionofthis article.)

Fig. 4. 2D illustrationofthemasscorrection.Thesolidlinerepresentstheinterface.Thearrowsindicatethenormalcorrection-velocitylocatedatcell centersofthegrid.

with f

(

x

,

y

)

adistortionfunctionthatleavesonlythelocationoftheinterface(anellipse)unchanged.Theinitialcondition is displayed in Fig. 3(a), where the shape of the ellipse is depictedas the thick blue line; the red dashed lines depict iso-contoursof

φ

rangingfrom

−

1to1.Clearly,thisinitialconditionisfarfrombeingequidistant.However,as

φ (

x

,

y

,

τ

)

is evolvedunderEq.(11),iteventuallyconvergestowardsasigned-distancefunctionasseeninFig. 3(b)and(c).

4. Interface-correctionlevelset(ICLS)method

Itisknownthatclassicallevelsetmethodsleadtomasslosswhenappliedtomultiphaseflows,partiallybecausethere isnounderlyingmassconservationinthelevelsetformalism,partiallybecauseofthereinitializationprocedure.Suchmass losscansometimes bereducedorevenremovedby usingthevariousapproacheslistedinSec.1,

e.g. the

CLSVOFmethod [25] or the hybrid particle level set method [24]. However, doing so oftenmakes the level set schemes complicated to implementandlessefficient.Tomaintainthesimplicityoftheoriginallevelsetmethod,weproposeanalternativeapproach toconservemassby performingsmallcorrectionsneartheinterface.Becausesuchcorrectionsaredonebydirectlysolving aPDE(sameasEq.(7)),theproposedmethodisstraightforwardtoimplementinboth2Dand3D.Meanwhile,becausethe correction doesnotneedtobeperformedateverytime step,theadditionalcostisalsonegligible.Below,wefirstpresent thederivationofthecorrection-velocity,thenwedemonstratethemassconservationwithanexample.

Let



divide adomaininto twodisjointsubsets



1 (e.g. a droplet) and



2 (e.g. theambientfluid), and V denote the volume of



1 (Fig. 4). Withoutloss ofgenerality, we let

φ <

0 in



1,and

φ >

0 in



2.The rateofchange of V can be writtenastheintegralofanormalvelocity ucdefinedon



[29],

i.e.





n

·

ucd



=

δ

V

δ

t

,

(13)

where n isthe outward-pointingnormalfromthe interface



.If

−δ

V

/δ

t corresponds tothe massloss overan arbitrary periodoftime(itdoesnothavetobethetimestepofthelevelsetadvection),thenuccanbethoughtasasurfacevelocity

thatcorrectsthevolumebyanamount

δ

V

/δ

t, hencecompensatingthemassloss.Inotherwords,ifuc isknown,thenthe followingPDEcanbesolved,

∂φ

∂

t

+

uc

· ∇φ =

0, (14)

afterwhichthemasslossaccumulatedover

δ

t is removed.

Toobtainsuchasurface correction-velocityuc,weintroduceaspeedfunction fs,an auxiliarypressure pc,andexpress therateofchangeofuc as

duc

dt

= −

fs

∇

pc

.

(15)

Here, pc canbeimaginedasanon-dimensionalcorrection-pressurein



1.If fs

=

1,thephysicalinterpretationofEq.(15) isanalogoustotheinflationofaballoonby

δ

V under pressure

p

c overtime



t. Itismoreevidentrewritingucintheform oftheimpulse-momentumtheorem(perunit“mass”oftheinterface)

uc

= −

t



0

∇

pcdt

,

(16)

inwhichthecorrection-velocityiszeroat

t

=

0,andwerequireaunitspeedfunction.Ingeneral,substitutingEq.(16)into Eq.(13)resultsin t



0 dt



 n

· (−

fs

∇

pc

)

d



=

δ

V

δ

t

.

(17)

Inorderfor

∇

pc tobecompatiblewithuc,pc hastobedifferentiatedattheinterface.Usinga1DregularizedHeaviside functionof

φ

,suchas H

(φ)

=

⎧

⎪

⎨

⎪

⎩

1 if

φ >



1 2



1

+

φ_

+

_π1sin(π_φ

)



if

|φ| 



0 otherwise, (18)

with



=

1

.

5



x the halfsmoothingwidth, thecorrection-pressure anditsgradientinEq.(17)canbe convenientlywritten as pc

=



1

−

H

(φ)



p0

,

(19) and





∇

pc

= −





δ



(φ)

∇φ

p0

,

(20)

where

δ



(φ)

isthederivativeof

H

(φ)

,and

p

0isaconstant.Notethatn

·∇φ = |∇φ|

,wecandenote



 fs

δ



(φ)

|∇φ|

d



=

Af

andexpresstheconstantpressurealgebraically p0

=

δ

V

δ

t

1

Af



t

,

(21)

by substituting Eq. (20) into (17), and approximating the time integration to first order, i.e.



₀tAfdt

=

Af



t. Finally, Eqs.(15) (20)and(21)canbecombinedtogive

uc

(φ)

=

δ

V

δ

t fs

δ



(φ)

Af

∇φ,

(22) or uc

(φ)

=

δ

V

δ

t fs Af

∇

H

(φ).

(23)

Onceuc isfound,Eq.(14)canbesolvedforonetimesteptocorrectthemassloss.Here,wehaverequiredabounded supportforuc,

i.e. u

c

=

0 for

|φ|





(see Fig. 4).Thereare twobenefits ofspreadingthesurface velocity.First, itallows an easy handling ofthe interface location, asuc only dependson a 1D Dirac deltafunction ofthe level set.The choice of

δ



(φ)

canalsobe differentfromthetrigonometricformimplied fromEq.(18); however,weprove inAppendix Athat

the discretization errorof



_n

·

ucd



is always zero,independent of

δ



(φ)

. The important point here is we spread the correction-velocity rather than the interface. The interface remains sharp, as it is implicitly represented by the level set function.The secondbenefitofspreading uc isthatitgreatly reducestherisk ofnumericalinstability.Asuc issupported on a2



bandaround theinterface,themaximal nodalvalue ofuc scaleswith1

/



.Inourtests,we haveneverfoundits non-dimensionalvaluetoexceed1.Therefore,theCFLconditionsimposedbyEq.(14)issatisfiedaslongasweusethesame temporalscheme(e.g. RK3)forsolvingEq.(7)andEq.(14).Lastly,weremindthereaderthatourcorrection-velocitydiffers conceptuallyfromtheextension-velocityproposedforsolvingStefanproblems[51,52].Theextension-velocitybydesignwill keepthelevelsetadistancefunction;whilethedesignprinciplehereistopreservetheglobalmass.Thisdistinctionisclear comparingtheconstructionproceduresofthetwovelocities.

Afinalquestionisthechoiceofthespeedfunction fs,actingasapre-factorforucinEq.(22)or(23).Tothebestofthe authorsknowledge,thereisnosimple,universally-validcriteriaforsuchcorrections.Twopossiblewaysare

fs

≡

1 uniform speed

κ

(φ)

curvature-dependent speed. (24)

The uniformspeed willobviouslyresultinafixed strength

δ

V

/δ

t

/

Af forthe velocitydistribution.Inthe caseofastatic spherical droplet,thisis theidealchoicefor fs,sincethedroplet shouldremaina sphere.Inmoregeneralcases,when a fluid interface is subjectto deformationsortopological changes,a curvature-dependent speed maybe moreappropriate. Thisis basedontheassumptionthat localstructuresofhighercurvatureorregions wheretheflowcharacteristics merge tend to be under-resolved [24]; hence, they are more prone to mass losses. Indeed, a linearcurvature weight has been adoptedbymanyanddemonstratedtoproduceaccurateresultsindifferentcontexts[27,53].Furthermore,

κ

/

Af reducesto 1

/

Af whenthecurvatureisuniform.Therefore,wecanrewriteEq.(23)usingacurvature-dependentspeed

uc

(φ)

=

δ

V

δ

t

κ

(φ)

Af

∇

H

(φ).

(25)

Clearly,thiscorrection-velocityislargerinhighlycurvedparts,andsmallerinflatterparts.Itthusincludes“local” informa-tionwhilemaintaining“global”massconservation.Standardcentral-differencediscretizationapplies,wherethecomponents of uc canbe obtainedateitherthecellfaces orcellcenters.Thecomputation of

κ

(φ)

iscrucialandwill bepresentedin thenext section.Westressthatsuchacurvature-dependenceisnot unique.Inprinciple,onecanchoosedifferent weight-functions, andvalidatethechoice basedonthespecificapplications.Practically,thedifferenceisexpectedtobe negligible sincethemasslossremainssmall(typicallyaround10−5)ateachcorrectionstep.

Aftercorrectingthelevelsetona2



bandaroundtheinterface,areinitializationstepisrequiredtoredistancethevalues withintheentirenarrowband(2

γ

).Thetwoprocedurescanbereadilycombined,sinceitisnotnecessarytoperformmass correction atevery time step. Also, because the formalism iscast in a level set frame, generalization from 2D to 3D is trivial. Comparingwithother mass-preservingmethods, theadditionalcomputational costofICLS issmall.Thisis dueto thesimplealgebraicexpressionofuc (Eq.(25)),andonlyonesolveofEq.(14)isrequired;whereasatypicalVOF-coupling method involvessolving anothersetoftransport equations[25], orreconstructingtheinterface by an iterativeprocedure [27].

Insummary,theICLSmethodproceedsbyperformingthefollowingsteps: 1. Advect

φ

nfromtime

t

n _to

_t

n+1 _{withEq.(7),usingtheflowvelocityu}n_. 2. Ifreinitializationwillbeexecuted(otherwise,gotostep3):

(a) PerformmasscorrectionwithEq.(14),usinguc fromEq.(25). (b) Reinitialize

φ

n+1withEq.(11).

3. Exitthelevelsetsolver.

Deforming circle. To assessthe performanceofICLS onmass conservation,we test thestandardbenchmark ofacircle de-formed byasingle vortex.Here, thecircleofradius0

.

15 isinitiallycenteredat

(

x

,

y

)

= (

0

.

5

,

0

.

75

)

ofa[0

,

1]

×

[0

,

1] box. Thevelocityisimposeddirectlyandcanbeobtainedfromthestreamfunction

ψ (

x

,

y

,

t

)

=

1

π

sin 2

₍

_π

_x

₎

_sin2

₍

_π

_y

₎

_cos



π

t T



,

where T is traditionallysetto8.Underthisflow,thecirclewillbe stretchedtomaximumat

t

=

T

/

2 andrewoundtoits initial conditionat

t

=

T . Althoughformulatedsimply,accuratelytransportingtheinterfacewithoutmasslossisadifficult task.

We perform this test on three differentmeshesusing the complete levelset solver: HOUC5 is used forthe level set advection,WENO5isusedforreinitializationevery5to20timesteps,themasscorrectionisperformedevery5to10time steps; andthe time step is chosen such that



t

/

x

=

0

.

32.Fig. 5 showsthe shapesof the filament/circle att

=

4 and t

=

8 at various resolutions.From the upperpanel,it isclearly seen that thefilament hasa longer tailandheaddueto masscorrection; aswe increasetheresolution,thedifferencebecomes smaller.Thelowerpanel ofFig. 5depictsthefinal

Fig. 5. Interfaceatt=4 andt=8 fordifferentmeshes.Thesolidblacklinesindicatesimulationswithoutmasscorrection,thesolidbluelinesindicate simulationswiththecurrentmasscorrectionmethod,thegreendashedlinesin(b)(d)(f)indicatetheoriginalcircle.(Forinterpretationofthereferencesto colorinthisfigurelegend,thereaderisreferredtothewebversionofthisarticle.)

Fig. 6. Relativevolumelossforthreedifferentmeshes.Dashedlinesindicatesimulationswithoutmasscorrection;solidlinesindicatesimulationswith masscorrection.

shapes,ideallytheinitialcircleifthemotionistotallypassive.Someartifactsarevisibleduetothefactthatthefilamentis alwaysunder-resolvedatthemaximumstretchingandthelevelsetwillautomaticallymergethecharacteristicstoyieldan entropysolution[17].Wenotethatthefinaloutcomecanbetunedbymodifyingthefrequencyofthereinitialization/mass correction,atrade-offbetweentheappearanceandthemassloss.However,theobjectivehereistodemonstratethemass conservationenforcedbyICLS, whichisclearlyillustrated inFig. 6.Forpassivetransportinvolvinglargedeformations, we recommendparticle-basedmethods[24].Examplesofdroplets/bubblesinphysicalconditionsusingICLSwillbe shownin thevalidations(Sec.6.5)andapplications(Sec.7)below.

5. Curvaturecomputation

Curvature computation is crucial to interfacial flows in the presence of surface tension, as inaccurate curvature can resultinunphysicalspuriouscurrents[23,37],andevenmoresoinourcasewhenweapplycurvature-dependentinterface corrections.Inthissection,wefirstbrieflydescribethecalculationofcell-centercurvatures;

i.e.,

thecurvatureevaluatedat thesamenodalpositionasthelevelsetfunction.Then,weintroduceageometricapproachfortheestimationofinterface curvaturescorresponding tothezerolevelset.Thesecondstepisspeciallytailoredtotheghost fluidmethodthatwillbe presentedinSec.6.2.

Fig. 7. Estimation of the interface’s curvature from neighboring cells.

5.1. Cell-center curvature

FromEq.(6),thecurvature

κ

canbeevaluatedas

κ

= −

φ

y y

φ

2 x

+ φ

xx

φ

2y

−

2φx

φ

y

φ

xy

(φ

2x

+ φ

2y

)

3/2 (26) andas

κ

M

= −



(φ

y y

+ φ

zz

)φ

2x

+ (φ

xx

+ φ

zz

)φ

2y

+ (φ

xx

+ φ

y y

)φ

z2

−

2φx

φ

y

φ

xy

−

2φx

φ

z

φ

xz

−

2φy

φ

z

φ

yz



(φ

x2

+ φ

2y

+ φ

2z

)

3/2 (27)

in 2D and3DCartesian coordinates,respectively, wherethesubscript M denotes themeancurvature [17].The curvature can be determined from theseexpressions usingsimple central finite-differences. It hasto be noted, however,that such evaluation of

κ

involves second derivativesof the level set field

φ (

x

)

.As a consequence,if the calculation of

φ

is only second-order accurate, the resulting

κ

will be oforder zero.Tononetheless retain a gridconverging

κ

, one canuse the compactleast-squaresschemeproposedbyMarchandiseetal.[54].Theirapproachprovidesasecond-order,gridconverging evaluationofthecell-centercurvature.Itmoreoversmearsoutundesiredhighfrequencyoscillationspossiblyintroducedby thevelocityfield.Asimilarprocedurehasalsobeenadoptedinotherworks[37,27].

The principleoftheleastsquaresapproachistosolvean over-determined linearsystem, Ax

=

b,where A isamatrix builtfromthelocalcoordinates,x isaunknown arraycontaining thereconstructedlevelsetvaluesanditsspatial deriva-tives,andb istheoriginallevelsetfield.Thedetaileddescriptionscanbe foundin[54].Here,weonlynote thatthelevel set functionremains unmodifiedafterthisstep.From apractical pointofview,provided themeshconsidered isuniform in all directions,thepseudo-inverse ofthe matrix A onlyneeds to beevaluated once andappliedcloseto theinterface. Therefore,thecomputationalcostofthisleast-squarescalculationisnegligible.

5.2. Interface curvature

The least-squaresapproachdescribed intheprevious sectiononlyallows onetocompute thenodalcurvature

κ

ofthe level set field

φ

. For computations using the GFM(Sec. 6.2), one might however require an accurate evaluation of the curvature at theexact location of the interface.Provided a grid-convergingcell-center curvature, theactual curvature at the interface canbe interpolated fromitsneighboring cells weighted bythe levelset [55,56].Here we presentaslightly differentbutrobustalgorithmtoestimatetheinterfacecurvature,withastraight-forwardgeometricalinterpretation. 2D estimation. Suppose theinterface



cutsthroughtwoadjacentcells,

(

i

,

j

)

and

(

i

+

1

,

j

)

,wherethecell-centercurvatures

κ

i,jand

κ

i+1,jareknown.In2D,wecandeterminetheradiusofcurvatureateachcelldirectlyfrom

κ

i,j

= −

1 ri,j

,

κ

i+1,j

= −

1 ri₊1,j

,

(28)

asillustratedinFig. 7.Sincethelevelsetisdefinedasthesigneddistancetotheinterface,



mustbe tangenttoacircle of radius

|φ

i,j

|

centered at

(

i

,

j

)

, and parallel to the contour line of



i

= {

x

|φ = φ

i,j

}

(otherwise they will not remain equidistant). We alsoknow



lies between

(

i

,

j

)

and

(

i

+

1

,

j

)

,thenit mustpass through P (see Fig. 7). Since



and



i areparallelandthereisonlyonelinenormaltobothcurvespassingthroughP,

r

i,jand O P must originatefromthesame point,

O .

Thenweget

Table 1

Gridconvergenceofthecurrentinterfacecurvaturecalculationinboth2Dand3D.

Grid points per diameter 16 32 48 64

L∞ 2D 1.144×10−2 _2.904×10−3 _1.285×10−3 _7.227×10−4

L∞ 3D 1.527×10−2 _3.888×10−3 _1.732×10−3 _9.753×10−4

|

O P

| =

ri,j

−

s

φ

i,j

.

(29)

where

s

isasignfunctionequalto1 iftheinterfacewrappingthenegativelevelsetisconvex,andequalto

−

1 ifconcave.

Thesameargumentholdsforcell

(

i

+

1

,

j

)

,whichyields

|

O Q

|

=

ri+1,j

−

s

φ

i+1,j.Wecanthereforewritetheradiusof theinterfacecurvaturebetween

(

i

,

j

)

and

(

i

+

1

,

j

)

as

r

=

|

O P

| + |

O Q

|

2

,

(30)

sothattheinterfacecurvaturebecomes

κ



=

2

κ

_i−_,j1

+

κ

_i−₊1_1,j

+

s

(φ

i,j

+ φ

i+1,j

)

.

(31)

Theabove derivationprovidesa relationbetweentheinterface curvatureandthatattheadjacentcell-centersinthe x direction.Similarresultscanbeobtainedinthe y direction (e.g. between

φ

i,j and

φ

i,j−1).Theassumptionswehavemade hereare1) thecell-center curvaturesare accurateand2) theinterface curvaturesat P and Q are thesame,so that O P and

O Q are

co-centered(or,

|

O P

|

≈ |

O Q

|

≈ |

O R

|

).Thesecondassumptionisessentiallyasub-cellapproximation,andwe expectittobevalidaslongastheinterfaceiswell-resolved.Oneexceptionwehavefoundiswhentwointerfacesarecloser thanabout2



x, thelocallevelset fieldwilldevelop “corners”.In thatcase, thecell-centercurvaturesare erroneousand theunderlyingassumptions werequireherearenot fulfilled.Wedonot discussthat caseinthepresentpaper.However, wedemonstrateinthenextsectionthatasecond-orderconvergenceisachievedwhentheinterfaceisresolved.

3D estimation. In threedimensions,themeancurvatureofasurfacecanbewrittenas

κ



= −(

1 r1

+

1 r2

),

(32)

where

r

1 and

r

2 arethetwoprincipalradiicorresponding tothemaximalandminimalplanarradiusofcurvature.Note that we do not needto approximate theinterface as asphere since there isalways a plane wherethe previous picture (Fig. 7)holds.Underthesameassumptionasforthe2Dcase,that theinterface atP and Q have thesameprincipalradii (hencethesamecurvature),onecanagainrelatethenodalcurvaturestotheirnearbyinterfaceas

κ

i,j,k

= −(

1 r1

+

s

φ

i,j,k

+

1 r2

+

s

φ

i,j,k

),

κ

i+1,j,k

= −(

1 r1

+

s

φ

i+1,j,k

+

1 r2

+

s

φ

i+1,j,k

),

(33)

where s is thesame signfunction definedforthe 2D case. Comparingequations (32)and (33), it isnaturalto expand

Eq.(33)intoaTaylorseriesandtoapproximatetheinterfacecurvaturedirectlyas

κ



=



i+1

κ

i

−



i

κ

i+1



i+1

−



i

+

O

(



2_i

,



_i2₊₁

),

(34) where



i

=

s

φ

i,j,k

.

(35)

Sincethelevelsetmustchangesignacrosstheinterface,Eq.(34)isalwaysdefinedanditreducestotheexactvalueifthe cellcenterhappenstobeontheinterface.Similarly,thewholeprocedureisrepeatedinthe y and z directions.

Finally,inordertoensurearobust estimation,weperforman additionalquadraticleastsquaresapproximation onthe curvature field near the interface, similar to [54]. This procedure takes place before the 3D estimation (Eq. (34)), and essentially improves theaccuracy of cell-centercurvatures by removing possiblehigh-frequency noise. We note that the secondaveragingisoptional,anddifferentmethodscanbefoundinliteraturetoevaluatethecell-centercurvatures[50].In thepresentpaper,theleastsquaresapproachmentionedinSec.5.1isusedforallthecases.

Toassesstheaccuracyofourinterfacecurvatureestimation,wecalculatethe

L

_∞ normofacircle/sphereofradius0

.

25 centered in a unit square/cube.Table 1 summarizes theerror afterone stepof the calculationson differentresolutions, whicharealsoplottedinFig. 8.Clearly,second-orderconvergenceisachievedinboth2Dand3Dcases.

Fig. 8. Second order convergence of the interface curvature computation in both 2D and 3D.

6. SolutionoftheNavier–Stokesequations

Inthissection,weoutlinetheflowsolverdevelopedfromthatofBreugem[57] forparticle-ladenflows.Afteradvancing thelevelsetfrom

φ

n to

φ

n+1,thedensityandviscosityfieldsareupdatedby

ρ

n+1

=

ρ

1Hs

(φ

n+1

)

+

ρ

2

(1

−

Hs

(φ

n+1

)),

(36a)

μ

n+1

=

μ

1Hs

(φ

n+1

)

+

μ

2

(1

−

Hs

(φ

n+1

)),

(36b) where Hs

(φ)

=

1 if

φ >

0 0 otherwise, (37)

isasimplestepfunction.

Next,apredictionvelocity u∗iscomputedbydefiningRUnas

RUn

= −∇ · (

unun

)

+

1 Re



1

ρ

n+1

∇ ·



μ

n+1

₍

_∇

_un

_{+ (∇}

_un

₎

T

₎



₊

1 F rg

,

(38)

whichistheright-handsideofthemomentumequation(1b)excludingthepressuregradientterm.Integratingintimewith thesecond-orderAdams–Bashforthscheme(AB2)yields

u∗

=

un

+ 

t



3 2RU n

₋

1 2RU n−1



.

(39)

Toenforceadivergence-freevelocityfield(Eq.(1a)),we proceedbysolvingthePoissonequationforthepressureasin thestandardprojectionmethod[58],

i.e.

∇ ·



1

ρ

n+1

∇

p n+1



=

1



t

∇ ·

u ∗

_.

₍₄₀₎

Thesurfacetensionbetweentwofluidsisalsocomputedduringthisstep,usingtheghostfluidmethod[32] (Sec.6.2).This allows foran accurateandsharpevaluationofthepressurejumpevenatlarge densitycontrasts[37].Finally,thevelocity atthenexttimelevelisupdatedas

un+1

=

u∗

−



t

ρ

n+1

∇

p

n+1

_.

₍₄₁₎

6.1. Fast pressure-correction method

In the above outline,a Poissonequation for thepressure (Eq. (40)) mustbe solved ateach time step.Thisoperation takesmostofthecomputationaltimeintheprojectionmethod,asitisusuallysolvediteratively.Inaddition,theoperation count ofiterativemethodsdependsontheproblemparameters(e.g. densityratio) andtheconvergencetolerance[13].On

theotherhand,DongandShen[59]recentlydevelopedavelocity-correctionmethodthattransformsthevariable-coefficient Poissonequationintoaconstant-coefficientone.TheessentialideaistosplitthepressuregradientterminEq.(40)intwo parts,onewithconstantcoefficients,theotherwithvariablecoefficients,

i.e.

1

ρ

n+1

∇

p n+1

_→

1

ρ

0

∇

pn+1

+ (

1

ρ

n+1

−

1

ρ

0

)

∇ ˆ

p

,

(42)

where

ρ

0

=

min

(

ρ

1

,

ρ

2

)

and p is

ˆ

theapproximatepressureattime level

n

+

1.Thissplittingreducestotheexactformof Eq.(40)withinthelower-densityphase,whileitsvalidityinthehigher-densityphaseandattheinterfacedependsonthe choiceofp.

ˆ

Later,DoddandFerrante[13]showedthatbyexplicitlyestimatingp from

ˆ

twoprevioustimelevelsas

ˆ

p

=

2pn

−

pn−1

,

(43)

theresultingvelocity fieldinEq.(41)willbesecond-order accurateinboth spaceandtime,independent oftheinterface advectionmethod.Furthermore,ifthecomputationaldomainincludesperiodicboundariesorcanberepresentedbycertain combination ofhomogeneous Dirichlet/Neumann conditions[38], the constant-coefficient part of Eq. (42)can be solved directlyusingGausseliminationintheFourierspace.SuchaFFT-basedsolvercanleadtoaspeed-upof10

−

40 times,thus thenamefastpressure-correctionmethod(FastP*).Followingthisapproach,Eqs.(40)and(41)aremodifiedas

∇

2_pn+1

_{= ∇ ·}



₁

₋

ρ

0

ρ

n+1

)

∇ ˆ

p



+

ρ

0



t

∇ ·

u ∗ ₍₄₄₎ and un+1

=

u∗

− 

t



1

ρ

0

∇

pn+1

+



1

ρ

n+1

−

1

ρ

0



∇ ˆ

p



.

(45)

6.2. Ghost fluid method

As discussed before, surface tension is commonly computed using the continuum surface force (CSF) model [31], in whichthepressurejumpacrossaninterfaceisrepresentedasaforcingtermontheright-handsideofEq.(1b).Despiteits simplicity,CSFintroducesan unfavorablesmearing inthedensityandpressureprofiles,resultinginanartificialspreading of the interface (typically over a thicknessof 3



x). An alternative approach is the so-calledghost fluid method (GFM), originallydevelopedbyFedkiwetal.[32]tocapturetheboundaryconditionsintheinviscidcompressibleEulerequations. UnlikeCSF,GFMenablesanumericaldiscretizationofthegradientoperatorwhilepreservingthediscontinuityofthe differ-entiatedquantity.ItwasextendedtoviscousflowsbyKangetal.[34]andhasbeensuccessfullyutilizedinmultiphaseflow simulations,see

e.g.

[37,60,61].

RecallfromEq.(2)thatthepressurejumphastwocomponents,onearisingfromthesurfacetension,theotherfromthe viscositydifferenceofthetwofluids. In[34],acompletealgorithmisprovidedtocompute thetwocontributions,making thedensity,viscosity,andpressureallsharp.However,havingasharpviscosityprofilerequiresanextrasteptoevaluatethe divergenceofthedeformationtensor(seeEq.(38)).Thatis,forcellsadjacenttotheinterface,thesecondderivativesofthe velocitymustbeevaluatedusingthetechniquesdevelopedin[62,34].However,rewritingEq.(2)as

[

p

]



=

1 Re



κ

Ca

+

2

[

μ

]

n T

_{· ∇}

_u

_·

_n



,

(46)

revealsthatsurface tensionisthedominanttermwhen theCapillarynumber,

Ca

=

W e

/

Re, issmall.Fortheapplications we areinterested in,

e.g. colloidal

droplets inmicrofluidicchannels,

Ca is

oftheorder of10−5.Therefore,in thepresent implementation,weregularizetheviscosityprofile(i.e. replacing

H

s

(φ)

inEq.(36b)withH

(φ)

inEq.(18)) anduseGFM onlyforthepressurejump.

6.2.1. Spatial discretization

Eqs.(38), (44), and(45)are discretized on astandard staggered grid usinga second-order conservative finitevolume method.Itis equivalentto centraldifferencesinall threedirectionsifthemesh isuniform. Adetaileddescriptionofthe discretizationoftheindividualtermscanbe foundin[13],Sec. 2.2.1.Forbrevity,weshowhereonlythe2Devaluationsof

∇

p and

∇

2_{p due toGFM.}

As sketched in Fig. 9, computing

∇

2p at node

(

i

,

j

)

requires three entries of p in each direction. If CSFis used, all gradienttermscanbeevaluatedwiththestraightforwardcentral-difference,

i.e.

(

∇

2p

)

i,j

=

ps_i−1,j

−

2p_is_,j

+

ps_i+1,j



x2

+

ps_i,j−1

−

2p_is_,j

+

ps_i,j+1



y2

.

(47)

However,thepressureatthecellsadjacenttotheinterfacewillhavetobesmearedout;hencewedenotethemwithps_.In orderforthepressuretobesharp,GFMcreatesanartificialfluid(the“ghost”fluid)andassumesthatthediscontinuitycan

Fig. 9. Schematicofthe2Dstaggeredgridwherepressurelocatesatcellcentersandvelocitycomponentslocateatcellfaces.Thecurvedlinespecifiesthe interface;filledandemptycirclesindicatediscontinuouspressure(ordensity)valuesinphase1and2,respectively.

beextendedbeyondthephysicalinterface.Thatis,ifweknowthecorrespondingjumpsofpressure,thenitsderivativescan beevaluatedwithoutsmearingbyremovingsuchjumps.FortheparticularcasedepictedinFig. 9,Eq.(47)canbere-written as(see[62]fortheintermediatesteps)

(

∇

2p

)

i,j

=

pi−1,j

−

2pi,j

+

pi+1,j



x2

−

[

p

]

i,j



x2

−

1



x



∂

p

∂

x



i+1/2,j

+

pi,j−1

−

2pi,j

+

pi,j+1



y2

−

[

p

]

i,j−1



y2

,

(48)

wherewerecall

[·]

i,jdenotesthediscontinuityfromfluid1tofluid2atcell

(

i

,

j

)

(samefor

[·]

i,j−1,

etc.).

TodeterminethejumptermsinEq.(48),wefirstnotethatthevelocityanditsmaterialderivativesacrosstheinterface ofviscousflowsarecontinuous[34,37],resultingin



1

ρ

n+1

∇

p n+1





=

0

.

(49)

Furthermore,owingtothesplittingthat allowsustosolve onlyforaconstant-coefficient Poissonequation(Eq.(44)),Eqs. (42)and(49)leadto



1

ρ

0

∇

pn+1





+



(

1

ρ

n+1

−

1

ρ

0

)

∇ ˆ

p





=

0

,

(50)

whichalsoimpliesthatthepressuregradienttermsarecontinuouseverywhere(e.g. thesubscriptcanbe

(

i

+

1

/

2

,

j

)

),along anydirection.

Denotingtheright-handsideofEq.(44)asR P , itisdiscretizedas R Pi,j

=



1

−

ρ

0

ρ

_in₊+₁1_/2,j

)

∂

p

ˆ

∂

xi+1/2,j

−



1

−

ρ

0

ρ

_in₋+₁1_/2,j

)

∂

p

ˆ

∂

xi−1/2,j





x

+



1

−

ρ

0

ρ

n_i,+_j+1_1/2

)

∂

p

ˆ

∂

y_i,j+1/2

−



1

−

ρ

0

ρ

n_i,+_j−1_1/2

)

∂

p

ˆ

∂

y_i,j−1/2





y

−

1



x



(1

−

ρ

0

ρ

n+1

)

∂

p

ˆ

∂

x



i+1/2,j

+

ρ

0



t



_u∗ i,j

−

u∗i−1,j



x

+

v∗_i,j

−

v∗_i,j−1



y



,

(51)

againusingGFM[62].ComparingEqs.(48)and(51),wenotethatthejumpofthefirstderivativescancelsout recognizing Eq.(50).Withamodifiedright-handside, R P∗,definedas

R P∗_i,j

=

R Pi,j

+

1



x



(1

−

ρ

0

ρ

n+1

)

∂

p

ˆ

∂

x



i+1/2,j

,

(52)

thediscreteformofEq.(44)reducesto pn_i−+₁1_,j

−

2pn_i,+_j1

+

pn_i++₁1_,j



x2

+

p_in_,+_j−1₁

−

2pn_i,+_j1

+

pn_i,+_j+1₁



y2

=

[

p

]

n_i,+_j1



x2

+

[

p

]

n_i,+_j−1₁



y2

+

R P∗i,j

.

(53)

Eq.(53)isstillnotreadytosolve,sincethepressurejumpsforthefirstpointawayfromtheinterface (e.g.

[

p

]

n_i,+_j1)are notknown.Following[37],weperformaTaylorseriesexpansionaround



,

[

p

]

n_i,+_j1

= [

p

]

n_+1

+ (

xi

−

x

)



∂

p

∂

x



n+1 

+

O

((

xi

−

x

)

2

),

(54)

where

[

p

]

n_+1

=

κ

,x

/

W e, and

κ

,x isestimatedfromEq.(31)in2D andfromEq.(34)in3D,along the

x direction

using

φ

_in_,+_j1 and

φ

_in₊+₁1_,j.Thejumpofthepressuregradientattheinterfacecanbesimilarlyexpandedat

(

i

,

j

)



∂

p

∂

x



n+1 

=



∂

p

∂

x



n+1 i,j

+

O

(

x

−

xi

),

(55) resultingin

[

p

]

n_i,+_j1

=

κ

,x W e

+ (

xi

−

x

)



∂

p

∂

x



n+1 i,j

+

O

((

xi

−

x

)

2

).

(56)

UsingEq.(50),wecanre-writeEq.(56)as

[

p

]

n_i,+_j1

=

κ

,x W e

+ (

xi

−

x

)



(1

−

ρ

0

ρ

n+1

)

∂

p

ˆ

∂

x



i,j

+

O

((

xi

−

x

)

2

),

(57)

wherethejumptermontheright-handsidecanbeexplicitlycalculatedusingthefamilyofidentitiesoftheform[34]

[

A B

] = [

A

] ˜

B

+ ˜

A

[

B

], ˜

A

=

a A1

+

b A2

,

a

+

b

=

1. (58)

AlthoughEqs. (57)and(58)lead to a second-orderpressure jump,it ismuch simplerto keep onlythe leading-order term,

i.e.

[

p

]

n_i,+_j1

=

κ

,x

W e

+

O

(

xi

−

x

).

(59)

Thisway,thepressurejumpvariesonlywiththelocalcurvature,remainsinvariantacrosstheinterface,andissecond-order accuratewhenthedensityisuniform.Forthetestcasesshownbelow,Eq.(59)isused.Thus,thecompletediscretizationof Eq.(44)reads p_in₋+₁1_,j

−

2pn_i,+_j1

+

pn_i++₁1_,j



x2

+

p_in_,+_j−1₁

−

2pn_i,+_j1

+

pn_i,+_j+1₁



y2

=

1 W e



κ

,x



x2

+

κ

,y



y2



+

R P∗_i,j

,

(60)

with

R P

∗_i,jdefinedinEq.(52)correspondingtoFig. 9.

Clearly,theresultinglinearsystem(Eq.(60))hasastandardpositivedefinite,symmetriccoefficientmatrix,anditcanbe solveddirectlyusingtheFFT-basedfastPoissonsolver(Sec.6.1).Careshould beexercisedwhena nodalpointcrossesthe interfaceinmorethanonedirection.Inthosecases,theinterfacecurvatureofeachcrossingdirectionmaybedifferentand itshallnotbeaveraged.Otherwise,theprojection(Eq.(44))andcorrection(Eq.(45))stepscanbecomeinconsistent,making thevelocity notdivergence-free.Additionally, whentakingthegradient ofthepressure-correction term;e.g. its derivative alongthe

x direction,

thecorrectdiscretizationshouldbe

∂

p

ˆ

∂

xi,j

=



ˆ

pi+1,j

− (

2

[

p

]

ni+1,j

− [

p

]

n−1 i+1,j

)



− ˆ

pi,j



x

.

(61)

Afterremovingthejump,thedivergenceofthebracketterminEq.(44)isevaluatedinthesamewayasin[13]. Finally,wecanre-writeEqs.(44)and(45)compactlyas

∇

2_pn+1

_{= ∇}

2 g

[

p

]



+ ∇ ·



1

−

ρ

0

ρ

n+1

)

∇

gp

ˆ



+

ρ

0



t

∇ ·

u ∗

_,

₍₆₂₎ un+1

=

u∗

− 

t



1

ρ

0

∇

g pn+1

+



1

ρ

n+1

−

1

ρ

0



∇

gp

ˆ



,

(63)

where

∇

gand

∇

2g

[

p

]

 denote,respectively,thegradientoperatorconsideringthejumpandtheextrajumptermsfromthe