On two-phase flow solvers in irregular domains with contact line

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pen

A

rchive

T

OULOUSE

A

rchive

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uverte (

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DOI:10.1016/j.jcp.2016.06.013

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To cite this version :

Lepilliez, Mathieu and Popescu, Elena Roxana and Gibou, Frédéric

and Tanguy, Sébastien On two-phase flow solvers in irregular

domains with contact line. (2016) Journal of Computational Physics,

vol. 321. pp. 1217-1251. ISSN 0021-9991

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On

two-phase

flow

solvers

in

irregular

domains

with

contact

line

Mathieu Lepilliez

a,b,c

,

Elena

Roxana Popescu

a

,

Frederic Gibou

d,e

,

Sébastien Tanguy

a,∗

a_{InstitutdeMécaniquedesFluidesdeToulouse,2bisalléeduProfesseurCamilleSoula,31400Toulouse,France} b_{CentreNationald’EtudesSpatiales,18AvenueEdouardBelin,31401ToulouseCedex9,France}

c_{AirbusDéfence& Space,31AvenuedesCosmonautes,31402ToulouseCedex4,France}

d_{DepartmentofMechanicalEngineering,UniversityofCalifornia,SantaBarbara,CA93106-5070,UnitedStates} e_{DepartmentofComputerScience,UniversityofCalifornia,SantaBarbara,CA93106-5110,UnitedStates}

Keywords:

Sharpinterfacemethods Irregulardomains Implicitviscosity Contactlines Levelsetmethod Ghost-FluidMethod

Wepresentnumericalmethods thatenablethedirect numericalsimulationoftwo-phase flowsinirregulardomains.Amethodispresentedtoaccountforsurfacetensioneffectsin ameshcellcontainingatriplelinebetweentheliquid,gasandsolidphases.Ournumerical methodis based onthelevel-set method tocapture theliquid–gasinterfaceand onthe single-phaseNavier–Stokessolverinirregulardomainproposedin[35]toimposethesolid boundaryinanEulerianframework.Wealsopresentastrategyfortheimplicittreatment oftheviscoustermandhowtoimposebothaNeumannboundaryconditionandajump conditionwhensolving for thepressure field. Special careisgiven on howto takeinto accountthe contactangle, the no-slipboundary condition for thevelocity fieldand the volumeforces.Finally,we presentnumericalresults intwoandthreespatialdimensions evaluatingoursimulationswithseveralbenchmarks.

1. Introduction

Multiphaseflows areubiquitous inmostmodern processing technologyand arecritical to theunderstandingofawide

range of physical and biological phenomena. Consequently, the design of numerical methods enabling the simulation of

multiphaseflowsisthesubjectofintenseresearch.Whenconsideringsolidboundaries,explicitdescriptionsofthegeometry

have demonstratedresults ofsuperior accuracy inthecaseofone-phaseflows,whencompared toimplicitrepresentations

ofthegeometry.Indeed, body-fittedgridscanbeconstructedtoenabletheaccuratetreatmentofboundaryconditionsand

to capture the rapid variations of the solution near walls, e.g. the turbulent boundary layer in high speed flows. In the

caseofmultiphaseflows, however, body-fittedapproachesareless practicalsinceconformingmeshmustbereconstructed

several time duringthe courseofa simulation.This is,for example, thecase ofthe simulationofbubbly flows or sprays,

where the fluid interface can be strongly deformed and stretched. The automatic meshing necessary in such cases is a

difficulttaskandinturn,theresulting meshcouldlackthequalityneeded toguaranteeaccuratenumericalresults.

Since the pioneer work of Peskin [38], introducing the immersed boundary method, an important research effort has

been devotedtothedevelopment ofEulerianfluidsolversinirregulardomains.Thesemethodshavetheadvantagethatno

remeshingisnecessaryduringthecourseofasimulationinthecaseofuniformgrids.Ifadaptive gridsareemployed,then

the remeshingisstraightforward sincethe free boundaryis embeddedinthe grid. Twoclasses ofmethodshave emerged

in the Eulerian framework: ‘delta’ formulations, where the Dirac distribution representing surface forces is approximated

by a regularized function [54] (and the references therein) and ‘sharp’ approaches, where jump conditions at the free

boundaryareenforcedatthediscretelevel [15,19,22,32,40,43,52,53,64,66](andthereferencestherein).Formulationsbased

onregularizedDiracdistributionsprovideapathwaytoeasilyextendsingle-phasetomultiphasesolvers[8,49].Ontheother

hand, these methods sufferfromdrawbacks that limittheir useinsome applications due totheinherent smearingofthe

solution near interfaces. Althoughefficient adaptive meshrefinement techniques [2,20] can help reduce the extent ofthe

smearingbyimposingincreasedresolutionwhereneeded, ‘sharp’numericalmethodsarestilldesired insome applications.

Thesemethods,however,aredifficulttodesign duetothedifficultyofimposingboundaryconditionsimplicitly.

While several works have been presented in thelast decade on immersed boundarymethods for incompressible

one-phase flows [4,25,31,33–35,56,57,59,63,65], only afew [29,36,58] have been dedicatedto incompressible two-phase flows,

especially inthecase whereacontact lineoftheliquid–gasinterface isformedon theembedded solidboundary. Insuch

a situation, two different interfaces must bemanaged in a computational cell, and the numerical solver for the pressure

mustimpose botha jumpconditionbetween theliquidandthegasto accountfor surfacetensioneffects and aNeumann

boundarycondition betweenthefluidand thesolidphasetoimposetheno-slip boundarycondition.

An attractive immersed boundary method for one-phase flows, based on a finite volume discretization of the Laplace

operator, has been proposed in [35]. Itenables to impose solid boundary conditions on the pressure and on thevelocity

with a second-order spatial accuracy and to maintain the symmetric definite positiveness of the resulting linear system.

Consequently, standardmethodssuch asthepreconditionedconjugategradient can beused toinvertefficiently thelinear

system.

Inthispaper,we proposeto extendtheworkof[35]to themorecomplex situation oftwo-phaseflows.Indeed, toour

knowledge, only a few methods are dedicated to thenumerical simulation of multiphase flows in irregular domains. As

two-phase flows are involved, surface tension has to be accounted for, even in the case where solid, liquid and gas are

present in thesame computationalcell. To that extent, we have developed a newnumerical scheme to account for both

a pressure jumpcondition due tothesurface tensionand aNeumanncondition on thesoliddomain inthesamecells,as

described in section 3.2. This spatialdiscretization enables thetreatment of volume forces inside themesh cells that are

crossedbythecontactline.Insection3.3,specificdetailsaregivenonhowtoperformanimplicittemporaldiscretizationof

theviscoustermswithacoupledlinearsysteminvolvingallthevelocitycomponents,whileimposingthejumpconditionon

viscosity.Insection3.4,wedescribehowtoenforcetheDirichletboundaryconditiononthevelocityfieldincomputational

cells crossed by the solid boundary, in order to ensure the no-slip boundary condition. To our knowledge, solving such

a coupled system with an immersed boundary condition has never been addressed in the existing literature. Finally, we

discuss insection 3.5howtoimposetheappropriatecontactanglebyextrapolatingtheliquid–gaslevel-setfunctioninside

thesoliddomainwithaniterativePDE.

Severalbenchmarksareproposedinsection4.Inthefirstpartofsection4,one-phaseflow simulationsarepresentedin

order to compare twopossible approachesto compute theviscous terms intwo and threespatial dimensions.Next,

test-cases involvingtwo-phaseflows in acomplex geometry(dropletdeposited on aslanted wall, half-filled rotatingspherical

tank) are presented and compared to theoretical solutions in order to highlight the behavior of the proposed numerical

methods for thecomputation of the surface tensionand the volume forcesin the grid cellsthat areboth crossed bythe

liquid–gas interface and the solid boundary. Orders of accuracy are difficult to determine for such simulations, however

grid sensitivitystudiesarepresentedforallthebenchmarksinordertoascertainthatallthecomputationsconvergetothe

correct solution.

2. Equationsandstandardprojectionmethods

The Navier–Stokes equations describe the motion of fluids at the continuum level. However, their formulations and

approximations depend on how surface forcesare represented. Nevertheless, standard state-of-the-art numerical

approxi-mationsarebasedontheprojectionmethodforsingle-phaseflows,introducedbyChorin[11].

2.1. Single-phaseflows

Consider a domain Ä_{= Ä}l∪ Äs with boundary ∂Ä. The regions Äl and Äs represent the fluidand solid regions,

re-spectively. Theboundarybetweenthefluidand thesolidisdenotedbyŴs.Theincompressible Navier–Stokesequationsfor

one-phaseNewtonianflowsarewrittenas:

∇ ·

u

₌

0 in

Ä

l

,

ρ

µ ∂

u

∂

t

+ (

u

· ∇)

u

¶

= −∇

p

₊

µ

1

u

₊

ρ

g in

Ä

l

,

where t theistime,

ρ

thefluiddensity,u_{= (}u,v,w) thevelocity field,

µ

theviscosity assumed constant, p the pressure and g theaccelerationduetogravity.

Fig. 1. The computational domain in the case of two-phase flows in irregular domains.

Inthecaseofsingle-phaseflows,ChorinusedtheHodgedecompositionofvectorfields,todesignathree-stageprojection method to solvetheNavier–Stokes equations[11]:first, given a velocity un at timetn₌n1t,anintermediate velocity u∗

canbecomputedforatimestep1t withoutconsideringthepressurecomponent:

u∗

₌

un

_{− 1}

t

µ

¡

un

_{· ∇}

¢

un

₋

µ

1

u n

ρ

−

g

¶

.

Second,thepressurefield pn+1 _{servesasthescalarfunctionoftheHodgedecomposition, satisfying:}

∇ ·

µ ∇

p n+1

ρ

¶

=

∇ ·

u ∗

1

t

,

(1)

withhomogeneousNeumannboundaryconditionson ∂Äandnon-homogeneousNeumannboundaryconditionon Ŵs:

ns

·

∇

p

ρ

¯

¯

¯

¯

_Ŵ s

=

ns

·

¡

u∗

−

us

¢¯

¯

Ŵs

,

where us is thespecifiedvelocity fieldon thesolid’sboundary. Finally, thefluidvelocity un+1 isdefined at thenewtime

steptn+1 astheprojectionofu∗ ontothedivergence-freespace: un+1

₌

u∗

_{− 1}

t

∇

p

n₊1

ρ

.

2.2. Two-phaseflows

Consideracomputationaldomain,Ä,thatcontainssolid,liquidandgasregionsdenotedbyÄs,Äl andÄg,respectively1

(see Fig. 1).We call Ŵ theliquid’s boundary and n its outwardnormal. Likewise, we callŴs thesolid’s boundary and ns

its outward normal. Finally, ∂Ä denotes the boundary of Ä. The incompressible Navier–Stokes equations for Newtonian

two-phase flowscanbedescribed differently, whetherthejump conditionsareconsidered asfunctionsin theentire

com-putational domain,orassharpjumpconditionslocallyappliedtothedensity,viscosityand pressurefield.Inwhatfollows,

we brieflypresentthemainformulationsthat areused, keepinginmindthatanimplicitformulationoftheviscous terms

must be favored to perform computations on irregular domains because an immersed Dirichlet boundary condition has

to beimposed on thevelocity field. Indeed, asasubcell resolution isused to compute thisboundary condition, temporal

stabilityofthecomputationcannotbeachievedbyusinganexplicittemporaldiscretization,see[18]formoredetails.

2.2.1. The“delta”formulation

The“delta”formulation[44,49]writestheNavier–Stokesequationsfortwo-phaseflowsas:

∇ ·

u

₌

0

,

ρ

µ ∂

u

∂

t

+ (

u

· ∇)

u

¶

= −∇

p

₊

ρ

g

_{+ ∇ ·}

³

2

µ

D

´

₊

σ κ

n

δ

Ŵ

,

where

σ

isthesurface tension,

κ

is theinterfacelocalmeancurvature, δŴ amultidimensionalDirac distributionlocalized

attheinterface,and D therate-of-deformationtensordefinedas:

D

₌

∇

u

+ ∇

u

T

2

.

Thedensityand theviscosityarepiecewiseconstantandonlyvaryacross theinterface.Theycanthenbedefinedas:

ρ

₌

ρ

l

+

HŴ

(

ρ

g

−

ρ

l

),

µ

₌

µ

l

+

HŴ

(

µ

g

−

µ

l

),

with HŴ denotingtheHeaviside distribution,equalto 1in theliquidand 0inthegasphase, withthefollowingdefinition

fordensity andviscosityjumpconditions:

[

ρ

_{] =}

ρ

l

−

ρ

g

,

[

µ

_{] =}

µ

l

−

µ

g

.

When applying a projection method to the “delta” formulation, one obtains thefollowing explicit discretization: first,

solveforanintermediatevelocity fieldu∗₁ with

u∗₁

₌

un

_{− 1}

t





¡

un

_{· ∇}

¢

un

₋

∇ ·

³

2

µ

n+1_Dn

´

ρ

n+1

−

σ κ

n+1

ρ

n+1 n

δ

Ŵ

−

g





.

In therestof thepaper,we omitthe subscriptson

κ

and

µ

, whichare alwaysestimatedattn+1_{.Second, solveaPoisson}

equationtodetermine theHodgefield:

∇ ·

µ ∇

p n₊1

ρ

n+1

¶

=

∇ ·

u ∗ 1

1

t

.

And finallyproject u∗

1 ontothedivergence-freevelocityfield:

un+1

₌

u∗₁

_{− 1}

t

∇

p n₊1

ρ

n₊1

.

ThisnumericalschemeisusuallyusedintheframeworkoftheContinuumSurfaceForce(CSF)model,whichrequiresthe

definitionofsmoothedfunctionsδǫ and Hǫ toapproximate theDirac andHeavisidedistributions.Theease of

implementa-tionofthismethodcomes withdrawbackofartificiallythickeningtheinterface.UnliketheCSFmodel,the“sharp”interface

approach[24,39,50]buildsasharpdiscretizationofsingularsourceterms,avoidingtheintroductionofafictitiousinterface

thickness.Some detailsonthetheoreticalequivalencebetween thejumpcondition formulationand the“delta”formulation

isdetailedin[26,27].

2.2.2. TheGhost-FluidPrimitiveviscousMethod

Asplittingontheviscous-stresstensorcanbeapplied:

∇ ·

³

2

µ

D

´

₌

2

µ

_{∇ ·}

³

D

´

₊

2

_[

µ

_]

∂

un

∂

n

δ

Ŵn

=

µ

1

u

₊

2

_[

µ

_]

³

nT

_{· ∇}

u

_·

n

´

δ

Ŵn

.

(2)

Thisleadstoanewpossibledescriptionoftheintermediatevelocity,whereajumpcondition intheviscoustermhastobe

imposed.Thereadermayfindadetaileddemonstrationofequation(2)inAppendix A.

In[24],theauthorsproposedasharpinterfacemethodforincompressibletwo-phaseflowsbasedontheprinciplesofthe

Ghost-FluidMethod[15,28].Thisapproachiswidelyused intheliterature,eventhoughitisfirst-orderaccurateforsolving

the Poisson equation withjump conditions. In [21], asecond-order approachhas been developed, but notyet applied to

Navier–Stokes equations.TheGhost-FluidMethodintroducedin[24]canbedescribedbrieflyasproposedin[26]:first,the

intermediatevelocity u∗ 2 isupdatedwith u∗₂

₌

un

_{− 1}

t

µ

¡

un

_{· ∇}

¢

un

₋

µ

1

u ∗

ρ

n+1

−

g

¶

.

Next, thePoissonequationforthepressureissolvedwiththeappropriatejumpcondition:

∇ ·

µ ∇

p n+1

ρ

n+1

¶

=

∇ ·

u ∗ 2

1

t

+ ∇ ·

Ã

σ κ

₊

2

_[

µ

_]

¡

nT

_{· ∇}

u

_·

n

¢

n

ρ

n+1

δ

Ŵn

!

.

un+1

₌

u∗₂

₋

1

t

ρ

n+1

Ã

∇

pn+1

₋

µ

σ κ

₊

2

_[

µ

_]

³

nT

_{· ∇}

u

_·

n

´

n

¶

n

δ

Ŵ

!

.

Thismethodhas beendubbedtheGhost-FluidPrimitiveviscous Method(GFPM)in[26].In thisframework,the

compu-tation oftheviscousterms requiresto determineexplicitly thejump conditionon theprojected viscousstresses. As these

jumpconditionsdependonthenumericalsolution,achievingafullyimplicittemporaldiscretizationoftheviscoustermsis

notastraightforwardtaskinthecaseoftheGFPM.Dependingon whethertheCSFformulationortheGFPM formulationis

used, theintermediatevelocity inthepredictionstepisdifferent.Even thoughthesetwo methodsareformallyequivalent,

theyleadtodifferentdefinitionsofthepredictedvelocityfieldintheprojectionstep.

2.2.3. TheGhost-FluidConservativeviscousMethod

In[50],Sussmanetal.introducedanotherformulationfor theintermediatevelocityfield,which refersto asthe Ghost-FluidConservativeviscousMethod(GFCM)in[26].Inthiscase,theintermediatevelocity u∗

3 isobtainedas: u∗₃

₌

un

_{− 1}

t

µ

¡

un

_{· ∇}

¢

un

₋

1

ρ

n+1

∇ ·

³

2

µ

Dn

´

₋

g

¶

.

Next, thePoissonequationissolvedtodeterminethepressure field:

∇ ·

µ ∇

p n+1

ρ

n+1

¶

=

∇ ·

u ∗ 3

1

t

+ ∇ ·

µ

_{σ κ}

n

δ

Ŵ

ρ

n+1

¶

.

Finallyinthecorrectionstep,thephysicalvelocity fieldcanbedeterminedwiththepressurefieldpreviouslycomputed:

un+1

₌

u∗₃

₋

1

t

ρ

n₊1

(

∇

p

n₊1

−

σ κ

n

δ

Ŵ

).

In order to remove the O¡1x2¢

time step restriction incurred by the viscosity term, the following implicit temporal

discretization isused, referred toin thispaperasthe Ghost-FluidConservativeMethod withan Implicitscheme(GFCMI):

theintermediatevelocityu∗

4isupdatedas:

ρ

n+1u∗₄

_{− 1}

t

_{∇ ·}

³

2

µ

D∗

´

₌

ρ

n+1

¡

un

_{− 1}

t

¡¡

un

_{· ∇}

¢

un

₋

g

¢¢ ,

(3)

which leadstoalargelinearsystemwherethethreevelocitycomponentsarecoupled.Thesubsequentstepsaresimilarto

theoneusedinGFCM,withthecomputationofthepressurefield:

∇ ·

µ ∇

p n₊1

ρ

n₊1

¶

=

∇ ·

u ∗ 4

1

t

+ ∇ ·

µ

_{σ κ}

n

δ

Ŵ

ρ

n₊1

¶

,

followed bythecorrectionstep:

un+1

₌

u∗₄

₋

1

t

ρ

n+1

(

∇

p

n+1

−

σ κ

n

δ

Ŵ

).

2.2.4. TheGhost-FluidSemi-ConservativeMethod

Finallyin[26]theauthorsproposedasemi-conservativeformulationfordiscretizingtheviscousterm,referredtointhis

paperastheGhost-FluidSemi-Conservative MethodwithanImplicitscheme(GFSCMI).Startingwiththesplitting:

∇ · (

2

µ

D

)

= ∇ · (

µ

∇

u

₊

µ

∇

Tu

)

=∇ · (

µ

_∇

u

)

_{+ ∇ · (}

µ

_∇

Tu

)

=∇ · (

µ

_∇

u

)

_{+ ∇}

Tu

_{· ∇}

µ

₊

µ

_{∇ · (∇}

Tu

)

=∇ · (

µ

∇

u

)

+ [

µ

]∇

Tu

_·

n

δ

Ŵ

,

where _{∇ · (}

µ

_∇u) isaformalequivalentoftheone-phaseflow formulationofviscouseffects. Thereforeeachcomponent is

solved independently fromthe others,asthe cross-termsofthe viscousterm in _[

µ

_]∇T_u

·nδŴ are enforcedexplicitly asa

jumpcondition.Thepredictionstepis:

ρ

n+1u∗₅

_{− 1}

t

_{∇ ·}

³

µ

_∇

u∗₅

´

₌

ρ

n+1

Ã

un

_{− 1}

t

µ

¡

un

_{· ∇}

¢

un

₋

g

¶

!

.

∇ ·

µ ∇

p n+1

ρ

n+1

¶

=

∇ ·

u ∗ 5

1

t

+ ∇ ·

à ¡

σ κ

_{+ [}

µ

_]

¡

nT

_{· ∇}

u

_·

n

¢

n

¢

ρ

n+1 n

δ

Ŵ

!

.

(4)

Finally, thecorrectionsteptocomputethedivergencefree velocityis:

un+1

₌

u∗₅

₋

1

t

ρ

n₊1

Ã

∇

pn+1

₋

µ

σ κ

_{+ [}

µ

_]

¡

nT

_{· ∇}

u

_·

n

¢

n

¶

n

δ

Ŵ

!

.

(5)

Remark.Theimplementationofanimplicittemporaldiscretizationoftheviscoustermsiseasierwiththismethod(GFSCMI)

than GFCMI.Indeed if GFSCMIisused,eachvelocity componentcan becomputedbysolvingasimple linearsystem

(sym-metricdefinitepositive)thatdoesnottreatcross-derivativesimplicitly.However,eventhoughGFCMItreatscross-derivatives

implicitly, afew iterations ofGauss–Seidel suffice forconvergence and theresults are superior toGFSCMI. Thesemethods

are compared in section 3.4 to offer some insight on how velocity boundary conditions should be enforced on irregular

domains.

2.2.5. Anoteonthetimestepconstraint

Letusremarkthat,unlikeGFCMI,GFSCMIdoesnotfullyremovethestabilityconstraintonthetimestepduetoviscosity,

sinceanexplicitpartdependingontheviscosityjumpconditionremainsinequation(5).Therefore,asithasbeendiscussed

in[26],astabilityconstraintdependingontheviscosityjumpconditionmustbeimposedtoensurestabilityofthemethod.

Specifically,thefollowingstandardtimestepconstraintsfortheconvectionandthesurfacetensioneffects,mustbeimposed

toensurethetemporalstabilityofthecomputation:

1

tconv

=

1

x max

_|

u

_|

,

1

tsurf_tens

=

1 2

r

ρ

l

σ

1

x 3/2

_,

foraglobaltimesteprestriction of: 1

1

t

=

1

1

tconv

+

1

1

tsurf_tens

.

(6)

IfGFSCMIisused,thefollowingconstraintmustbeaddedtoaccountforthejumpconditiononviscosityinequation(5):

1

t_visc

₌

ρ

[

µ

_]

1

x 2

_,

where

ρ

istheaveragedensity ofthetwophases.Thisleadstoanewglobaltimesteprestriction givenby:

1

1

t

=

1

1

tconv

+

1

1

tsurf_tens

+

1

1

tvisc

.

2.3. Anoteonsolid’sboundaryconditionsformultiphaseflows

As stated earlier, when irregular domains are handled, Neumann boundary conditions on the domain boundaries are

enforcedonŴs thefluid–solidinterfaceforthepressurefield.ItleadstothefollowingrelationforGFCMI:

ns

·

∇

p

ρ

¯

¯

¯

¯

Ŵs

=

ns

·

¡

u∗

−

us

+

σ κ

n

δ

Ŵ

¢¯

¯

Ŵs

,

and forGFSCMIas:

ns

·

∇

p

ρ

¯

¯

¯

¯

Ŵs

=

ns

·

Ã

u∗

₋

us

+

³

σ κ

_{+ [}

µ

_]

¡

nT

_{· ∇}

u

_·

n

¢

n

´

n

δ

Ŵ

!¯

¯

¯

¯

¯

_Ŵ s

.

Onecannoticethatjumpconditionshavetobetakenintoaccountontheirregulardomainaswell.Section3willdiscuss

thenumericaldiscretizationofthese boundaryconditions.

3. Numericalmethods

Inthissection,weprovidethedetailsofthenumericalmethodsusedinourtwo-phaseflowsolverinirregulardomains.

First, we present the level-set method and the spatial discretizations of the two different projection methods that we

consider forirregular domains.Wethenprovidethedetailsoftheimplicitdiscretizationoftheviscoustermsand howthe

Fig. 2. StandardMACgridconfiguration:thescalarvariablesaresampledatthecells’centers(circles),thex-componentofthevelocityfieldissampled ontheverticalfaces(bluetriangles),andthey-componentofthevelocityfieldissampledonthehorizontalfaces(redtriangles).Theirregulardomain isrepresentedbytheshadedareaÄs.(Forinterpretationofthereferencestocolorinthisfigurelegend,thereaderisreferredtothewebversionofthis

article.)

3.1. Capturingthemovinginterface

Inthiswork, weusethelevel-setmethod ofOsherand Sethian [37]torepresent theinterfacebetween thetwofluids.

A signedand continuouslevel-set functionφ isdefined in adomain Ä suchasÄ−_{= {}x_{: φ(}x)<0_}, Ä+_{x_{: φ(}x)>0_} and

Ŵ_{= {}x_{: φ(}x)₌0_} whereŴ representstheinterfacecapturedbythefollowingtransportequation:

∂φ

∂

t

+

u

· ∇φ =

0

.

(7)

Oneofthebenefitsforusingthelevel-setfunctionisitsregularpropertyinthewholedomain.Sussmanetal.[49]

devel-opedanalgorithmallowingtomaintainthispropertybyre-initializingthelevel-set functionwiththefollowingequation:

∂

d

∂

τ

=

Sign

(φ) (

1

− |∇

d

|),

(8)

where Sign(φ) is a regularized signed functiondefined by Sussman et al. in [49] and d the update of the function φ.

Equation (8) iterates for a few fictitious time step

τ

to finally converge to a continuous signed distance function in the

whole domain.Thus,geometricpropertiessuchasn theoutwardunitnormalvectortotheinterfaceand

κ

thelocalmean

curvaturecanbeaccuratelycomputed:

n

₌

∇φ

|∇φ|

,

and

κ

(φ)

= ∇ ·

n

.

Spatialderivativesinequations(7)and(8)arecomputedwiththeWENO-Zscheme[5],andthetemporalderivativewith

athirdorderTVDRunge–Kuttascheme.Letusnoticethat thetemporalintegrationisfully coupledwiththeNavier–Stokes

solver[24].

3.2. Discretizationoftheprojectionmethodinirregulardomains

The method developed in this study to solve thePoisson equation (4) on irregular domains is based on Ng, Min and

Gibou’salgorithm [35]. Thisnumericalmethodpresentsseveral attractivefeatures.Indeedthespatialdiscretization forthe

pressure Poisson equation is second order, for all computationalnodes, includingthe irregular domain nodes crossed by

thesolid–fluid interfacewhere theembeddedNeumannBoundary conditionis enforced.Moreover,theresulting matrix of

thelinearsystemisstillsymmetricdefinitepositivewhichallowsusingaclassicalPreconditioned ConjugateGradient.More

efficientsolvers,astheBlackBoxMultiGrid[13],canalsobeusedtospeed-upsimulations.

Consideravectorfieldu∗ _{inthedomainÄ,separatedbytheinterfaceŴs intotwodistinctsubdomainsÄf andÄs,such}

as Ä_{= Ä}f ∪ Äs. As Äs would correspond to a solid media, weonly solvethe Navier–Stokes equations in Äf. Let φs be

another level-setfunctionsuch asÄs= {x: φs(x)>0},Äf{x: φs(x)<0} and Ŵs= {x: φs(x)=0}. Weconsider aMAC grid

configuration and a cell Ci,j= [i−1/2,i+1/2]× [j−1/2,j+1/2], partially covered with Äf. For moreclarity we only

consider a2Dexampletoillustratethemethodwhichcanbeextendedto3D.TheconfigurationisillustratedinFig. 2.

Consideringafinitevolumeapproach,weintegratethelefthandside ofequation(4)over Ci,j and applythedivergence

theorem:

Z

Ci,j∩Äf

∇ ·

µ ∇

_ρ

p

¶

d A

₌

Z

∂(Ci,j∩Äf) n

_·

µ ∇

p

ρ

¶

dl

,

withd A anddl differentialareaandlength.Similarlyfortherighthandside:

Z

Ci,j∩Äf

∇ ·

u∗d A

₊

Z

Ci,j∩Äf

∇ ·

µ

FS T

ρ

¶

d A

₌

Z

∂(Ci,j∩Äf) n

_·

u∗dl

₊

Z

∂(Ci,j∩Äf) n

_·

µ

FS T

ρ

¶

dl

,

where FS T canrepresent thesurface tensioneffects and thejump conditionsdue toviscosity depending onthe choiceof

theintermediatevelocity:

• ForGFCMI,FS T =

σ κ

nfδf. • ForGFSCMI,FS T= ³

σ κ

_{+ [}

µ

_]Ŵ ¡ nT_{· ∇}u_·n¢n´ nδf.

Wenow onlyconsider thecontribution ofallthecomponents of∂(Ci,j∩ Äf), anddefinethelengthfractionLi,j ofthe

facecovered bytheirregular domain_{x_|φs(x)≤0} whichcanbelinearlyapproximatedherefor[i−1₂]× [j−1₂,j+1₂]by:

L_i−1 2,j

=



















































1

y

φ

_i−1 2,j−12

φ

_i−1 2,j−12

− φ

i−12,j+12 for

φ

_i−1 2,j−12

<

0 and

φ

i−12,j12

>

0

,

1

y

φ

i− 1 2,j+12

φ

_i−1 2,j+12

− φ

i−12,j−12 for

φ

_i−1 2,j−12

>

0 and

φ

i−12,j12

<

0

,

1

y for

φ

_i−1 2,j−12

<

0 and

φ

i−12,j12

<

0

,

0 for

φ

_i−1 2,j−12

>

0 and

φ

i−12,j12

>

0

.

(9)

Formoreclaritythesubscript“s”isomitted,butitisthefunctionφs thatisusedtoestimatethelengthfractions. Using

thelengthfractionestablishedinequation(9),andsampledvaluesatthecenterofthecell,weobtain:

−

Z

∂(Ci,j∩Äf) n

_·

µ ∇

p

ρ

¶

dl

_≃

L_i −12,j

ρ

_i−1 2,j

µ

_p i,j

−

pi₋1,j

1

x

¶

+

L_i +12,j

ρ

_i+1 2,j

µ

_p i,j

−

pi₊1,j

1

x

¶

+

L_i,j−1 2

ρ

_i,j−1 2

µ

_p i,j

−

pi,j−1

1

y

¶

+

L_i,j+1 2

ρ

_i,j+1 2

µ

_p i,j

−

pi,j+1

1

y

¶

−

Z

Ci,j∩Ŵ n

_·

µ ∇

p

ρ

¶

dl

,

where R

Ci,j∩Ŵ istheintegralover theinterfacewiththeirregular externalboundary.Similarly,weobtain:

−

Z

∂(Ci,j∩Äf) n

_·

u∗dl

_≃

L_i −12,ju ∗ i₋1₂,j

−

Li₊1₂,ju∗i₊1₂,j

+

Li,j₋1₂v∗i,j₋1₂

−

Li,j₊1₂v∗i,j₊1₂

−

Z

Ci,j∩Ŵ n

_·

u∗dl

.

ForcellswherebothŴ andŴs arepresent,respectivelythefluid–fluidinterfaceandthefluid–solidinterface,the

Ghost-Fluid Method [15,28] is applied on eachpart of Ŵ:let βi,j= _ρ1_i

,j bea diffusion coefficient in the cell, computed with a

harmonic averageofthevalues β+ in theregion whereφ ispositive and β− for thenegativeregion, and a(xŴ)=

σ κ

the

correspondingjumpfunctionforGFCMI.Fortheinterfacecrossingacellborderbetween xi,j and xi₊1,j:

β

i+1/2,j

=

β

+

β

−

β

+

_θ

_{+ β}

−

₍

1

_{− θ)}

,

and a

(

xŴ

)

i₊1/2,j

=

σ κ

i,j

θ

+

σ κ

i₊1,j

(

1

− θ),

with

θ

₌

|φ

i+1,j

|

|φ

i,j

| + |φ

i₊1,j

|

.

Thenonecandefineg∗ i,j asin[26]: g_iL_,j

_{= ±β}

_i−1 2,ja

(

xŴ

)

i−12,j g R i,j

= ±β

i+12,ja

(

xŴ

)

i+12,j g B i,j

= ±β

i,j−12a

(

xŴ

)

i,j−12 g T i,j

= ±β

i,j+12a

(

xŴ

)

i,j+12 and finallyobtain:

−

Z

∂(Ci,j∩Ä−) n

_·

µ

_{σ κ}

_n f

δ

f

ρ

¶

dl

_≃

L_i −12,jg L i,j

+

Li₊1₂,jg R i,j

+

Li,j₋1₂g B i,j

+

Li,j₊1₂g T i,j

−

Z

Ci,j∩Ŵ n

_·

µ

_{σ κ}

_n f

δ

f

ρ

¶

dl

.

If GFSCMI is used, one can notice that the formulation is almost the same, except that the function a(xŴ) will be

completedwiththejumpduetotheviscousterm:

a

(

xŴ

)

i+1/2,j

= θ

³

σ κ

+ [

µ

]

Ŵ

¡

nT

· ∇

u

·

n

¢

n

´

i,j

+ (

1

− θ)

³

σ κ

+ [

µ

]

Ŵ

¡

nT

· ∇

u

·

n

¢

n

´

i₊1,j

,

and

−

Z

∂(Ci,j∩Ä−) n

_·





³

σ κ

_{+ [}

µ

_]

Ŵ

¡

nT

_{· ∇}

u

_·

n

¢

n

´

n_f

δ

f

ρ





dl

≃

L_i−1 2,jg L i,j

+

Li₊1₂,jgiR,j

+

Li,j₋1₂giB,j

+

Li,j₊1₂gTi,j

−

Z

Ci,j∩Ŵ n

_·





³

σ κ

_{+ [}

µ

_]

Ŵ

¡

nT

· ∇

u

·

n

¢

n

´

nf

δ

f

ρ





dl

.

Finally,oneobtainsthefollowingproblemwithNeumannboundaryconditionsontheembeddedsolidboundary:

L_i −12,j

ρ

_i−1 2,j

µ

_p i,j

−

pi₋1,j

1

x

¶

+

L_i +12,j

ρ

_i+1 2,j

µ

_p i,j

−

pi₊1,j

1

x

¶

+

L_i,j −12

ρ

_i,j−1 2

µ

_p i,j

−

pi,j₋1

1

y

¶

+

L_i,j +12

ρ

_i,j+1 2

µ

_p i,j

−

pi,j₊1

1

y

¶

=

L_i −12,ju ∗ i−12,j

−

L_i +12,ju ∗ i+12,j

+

L_i,j −12v ∗ i,j−12

−

L_i,j +12v ∗ i,j+12

+

L_i−1 2,jg L i,j

+

Li+12,jg R i,j

+

Li,j−12g B i,j

+

Li,j+12g T i,j

.

As Ng,Minand Gibou demonstratedin[35],theabove discretizationformsa symmetricpositive definite linearsystem

forthepressurefield.

3.3. Spatialdiscretizationoftheviscousterms

As illustrated inSection 2, the implementation ofthe viscosity terms dependson the formulation ofthe intermediate

velocity,thusgeneratingdifferentdiscretizationsthatareformallyequivalents.TheimplicitformulationsforbothGFCMand

GFSCMarethoroughlydetailedinthefollowingsection.

3.3.1. GFCMIdiscretization

Wehaveseeninsection2that theequationoftheintermediatevelocityforGFCMIis:

ρ

n+1u∗

_{− 1}

t

_{¡∇ ·}

¡

2

µ

D∗

_{¢¢ =}

Frhs

,

(10)

withFrhs theright-handsideofequation(10):

Frhs

=

ρ

n+1

¡

un

− 1

t

¡¡

un

· ∇

¢

un

−

g

¢¢.

(11)

Inorderto computetheintermediatevelocity field,onemustcomputeallthetermsoftheright-handside ofequation

(11). Theadvection termissolved withafifth orderWENO-Z scheme[5].Let usremind thestress tensorformulationfor

thediscretizationoftheviscousterm:

D

₌

1 2

³

∇

u

_{+ ∇}

Tu

´

with

_∇

u

₌









∂

u

∂

x

∂

u

∂

y

∂

v

∂

x

∂

v

∂

y









.

The discretizationis presentedasa 2Ddiscretization, but canbeeasily extended toa 3Dformulation. Byapplying the divergence operator:

∇ ·

³

2

µ

D

´

₌









∂

∂

x

µ

2

µ

∂

u

∂

x

¶

+

∂

∂

y

µ

µ

µ ∂

u

∂

y

+

∂

v

∂

x

¶¶

∂

∂

y

µ

2

µ

∂

v

∂

y

¶

+

∂

∂

x

µ

µ

µ ∂

u

∂

y

+

∂

v

∂

x

¶¶









.

Byprojectingontheex direction:

∇ ·

³

2

µ

D

´

_·

ex

¯

¯

¯

_i +12,j

=

∂

∂

x

µ

2

µ

∂

u

∂

x

¶¯

¯

¯

¯

_i +12,j

+

∂

∂

y

µ

µ

µ ∂

u

∂

y

+

∂

v

∂

x

¶¶¯

¯

¯

¯

_i +12,j

,

(12) with:

∂

∂

x

µ

2

µ

∂

u

∂

x

¶¯

¯

¯

¯

_i +12,j

≃

2

µ

i₊1,j

Ã

_u i₊3₂j

−

ui₊1₂,j

1

x

!

−

2

µ

i,j

Ã

_u i₊1₂,j

−

ui₋1₂,j

1

x

!

1

x

,

∂

∂

y

µ

µ

∂

u

∂

y

¶¯

¯

¯

¯

_i +12,j

≃

µ

_i+1 2,j+12

Ã

_u i+12,j+1

−

ui+12,j

1

y

!

−

µ

_i+1 2,j−12

Ã

_u i+12,j

−

ui+12,j−1

1

y

!

1

y

,

and

∂

∂

y

µ

µ

∂

v

∂

x

¶¯

¯

¯

¯

_i +12,j

≃

µ

_i+1 2,j+12

Ã

_v i+1,j+12

−

vi,j+12

1

x

!

−

µ

_i+1 2,j−12

Ã

_v i+1,j−12

−

vi,j−12

1

x

!

1

y

.

Likewise,byprojectingontheey|_i,j+1

2 direction:

∇ ·

³

2

µ

D

´

_·

ey

¯

¯

¯

_i ,j₊1₂

=

∂

∂

x

µ

µ

µ ∂

u

∂

y

+

∂

v

∂

x

¶¶¯

¯

¯

¯

_i,j +12

+

∂

∂

y

µ

2

µ

∂

v

∂

y

¶¯

¯

¯

¯

_i,j +12

,

(13) with:

∂

∂

x

µ

µ

∂

u

∂

y

¶¯

¯

¯

¯

_i ,j₊1₂

≃

µ

_i+1 2,j+12

Ã

_u i₊1₂,j₊1

−

ui₊1₂,j

1

y

!

−

µ

_i−1 2,j+12

Ã

_u i₋1₂,j₊1

−

ui₋1₂,j

1

y

!

1

x

,

∂

∂

x

µ

µ

∂

v

∂

x

¶¯

¯

¯

¯

_i,j +12

≃

µ

_i+1 2,j+12

Ã

_v i₊1,j₊1 2

−

vi,j+12

1

x

!

−

µ

_i−1 2,j+12

Ã

_v i,j₊1 2

−

vi−1 j+12

1

x

!

1

x

,

and

∂

∂

y

µ

2

µ

∂

v

∂

y

¶¯

¯

¯

¯

_i,j +12

≃

2

µ

i,j₊1

Ã

_v i,j₊3 2

−

vi,j+12

1

y

!

−

2

µ

i,j

Ã

_v i,j₊1 2

−

vi,j−12

1

y

!

1

y

.

Inordertocomputerightfullytheviscoustermattheintermediatetimestep,onemustsolveacoupledlinearsystemof

two matriceswith9diagonalspervelocitycomponentsas:

ai+1,ju∗_i+3 2,j

+

ai,ju∗_i−1 2,j

+

a_i+1 2,j+12u ∗ i+12,j+1

+

a_i+1 2,j−12u ∗ i+12,j−1

+

α

u∗ i+12,j

−

b_i+1 2,j+12v ∗ i+1,j+12

+

b_i+1 2,j+12v ∗ i,j+12

−

b_i+1 2,j−12v ∗ i+1,j−12

+

b_i−1 2,j−12v ∗ i,j−12

=

Fx_rhs and c_i +12,j+12v ∗ i₊1,j₊1₂

+

ci₋1₂,j₊1₂v∗i₋1 j₊1₂

+

ci,j+1v∗i,j₊3₂

+

ci,jv∗i,j₋1₂

+ β

v∗i,j₊1₂

−

b_i +12,j+12u ∗ i₊1₂,j₊1

−

bi₋1₂,j₊1₂u∗i₊1₂,j

+

bi₊1₂,j₊1₂u∗i₋1₂,j₊1

+

bi₋₂1,j₊1₂ui∗₋1₂,j

=

Fy_rhs

with Fx_rhs and Fy_rhs the right-hand side terms from GFCMI projection method in equation (11), respectively projected

onto thex- and y-directions,respectively. Thematrixcoefficientsarededucedfromequations(10),(12)and (13)as:

ai₊1,j

= −

2

µ

i₊1,j

1

x2

1

t

,

bi₊1₂,j₊1₂

=

µ

_i+1 2,j+12

1

x

1

y

1

t

,

ci+12,j+12

= −

µ

_i+1 2,j+12

1

x2

1

t

,

ai,j

= −

2

µ

i,j

1

x2

1

t

,

bi₊1₂,j₋1₂

=

µ

_i+1 2,j−12

1

x

1

y

1

t

,

ci−12,j+12

= −

µ

_i−1 2,j+12

1

x2

1

t

,

a_i +12,j+12

= −

µ

_i+1 2,j+12

1

y2

1

t

,

bi₋1₂,j₊1₂

=

µ

_i−1 2,j+12

1

x

1

y

1

t

,

ci,j+1

= −

2

µ

i,j₊1

1

y2

1

t

,

a_i +12,j−12

= −

µ_i +1₂,j−1₂ 1y2

1

t

,

bi₋1₂,j₋1₂

=

µ

_i−1 2,j−12

1

x

1

y

1

t

,

ci,j

= −

2

µ

i,j

1

y2

1

t

,

(14)

α

₌

ρ

n+1 i₊1₂,j

−

ai+1,j

−

ai,j

−

ai+12,j+12

−

ai+12,j−12

,

(15) and

β

₌

ρ

n+1 i,j₊1₂

−

ci+12,j+12

−

ci−12,j+12

−

ci,j+1

−

ci,j

.

(16) Since the linear system is diagonally dominant,2 it can be solved efficiently with a few Gauss–Seidel iterations (<20 iterationsfortypicalmultiphaseflowconfigurations,suchasthebenchmarksofsection4).Fora3Dsystem,athirdcoupled

matrix hastobeaccountedfor, resultingina15-diagonalmatrixpervelocitycomponents.

3.3.2. GFSCMIdiscretization

GFSCMI projection proposed in [26] enables an easier implementation for an implicit temporal discretization of the

viscousterms,albeitsemi-implicit:

ρ

n+1_u∗ 5

− 1

t

∇ ·

¡

µ

_∇

u∗₅

_{¢ =}

Frhs

=

ρ

n+1

µ

un

_{− 1}

t

¡ ¡

un

.

_∇

¢

un

₋

g

¢

¶

.

(17)

Theviscoustermcanbecomputedas:

∇. (

µ

_∇

u

)

₌









∂

∂

x

µ

µ

∂

u

∂

x

¶

+

∂

∂

y

µ

µ

∂

u

∂

y

¶

∂

∂

x

µ

µ

∂

v

∂

x

¶

+

∂

∂

y

µ

µ

∂

v

∂

y

¶









.

Byprojectinginthex-direction:

∇ · (

µ

_∇

u

)

_·

ex

|

_i+1 2,j

=

∂

∂

x

µ

µ

∂

u

∂

x

¶¯

¯

¯

¯

_i +12,j

+

∂

∂

y

µ

µ

∂

u

∂

y

¶¯

¯

¯

¯

_i +12,j

,

(18) with:

∂

∂

x

µ

µ

∂

u

∂

x

¶¯

¯

¯

¯

_i +12,j

≃

µ

i+1,j

Ã

_u i₊3₂j

−

ui₊1₂,j

1

x

!

−

µ

i,j

Ã

_u i₊1₂,j

−

ui₋1₂,j

1

x

!

1

x

,

and

∂

∂

y

µ

µ

∂

u

∂

y

¶¯

¯

¯

¯

_i +12,j

≃

µ

_i+1 2,j+12

Ã

_u i+12,j+1

−

ui+12,j

1

y

!

−

µ

_i+1 2,j−12

Ã

_u i+12,j

−

ui+12,j−1

1

y

!

1

y

.

Likewise,byprojectinginthe y-direction:

∇ · (

µ

_∇

u

)

_·

ey

¯

¯

_i ,j₊1₂

=

∂

∂

x

µ

µ

∂

v

∂

x

¶¯

¯

¯

¯

_i ,j₊1₂

+

∂

∂

y

µ

µ

∂

v

∂

y

¶¯

¯

¯

¯

_i ,j₊1₂

,

(19)

2 _{Depending onthetime stepconstraint,ifρ>> 6a}

i j, thenthesystemisdiagonallydominant,whichisalwaysthecase in ourconfigurations, cf.

with:

∂

∂

x

µ

µ

∂

v

∂

x

¶¯

¯

¯

¯

_i ,j₊1₂

≃

µ

_i+1 2,j+12

Ã

_v i₊1,j₊1₂

−

vi,j₊1₂

1

x

!

−

µ

_i−1 2,j+12

Ã

_v i,j₊1₂

−

vi₋1 j₊1₂

1

x

!

1

x

,

and

∂

∂

y

µ

µ

∂

v

∂

y

¶¯

¯

¯

¯

_i,j +12

≃

µ

i,j₊1

Ã

_v i,j₊3₂

−

vi,j₊1₂

1

y

!

−

µ

i,j

Ã

_v i,j₊1₂

−

vi,j₋1₂

1

y

!

1

y

.

Inordertosolvefortheviscoustermintheintermediatetimestep,onemustsolvealinearsystemineachdirection:

ai₊1,ju∗_i +32j

+

ai,ju∗_i −12,j

+

a_i +12,j+12u ∗ i₊1₂,j₊1

+

ai₊1₂,j₋1₂ui∗₊1₂,j₋1

+

α

u∗i₊1₂,j

=

Fx_rhs

,

and c_i +12,j+12v ∗ i+1,j+12

+

c_i −12,j+12v ∗ i−1 j+12

+

c_i,j+1v∗_i,j+3 2

+

c_i,jv∗_i,j−1 2

+ β

v∗ i,j+12

=

F_{y_rhs}

,

with Fx_rhsand Fy_rhstheright-handsidetermsfromGFSCMIprojectionmethodinEq.(17),respectivelyprojectedontothe x- and y-direction.Thecoefficients abovearegivenbyequations(14),(15)and(16).

As explained before, this formulation is quite straightforward as each component is decoupled from the others. This

section is concluded by some comments on how to define the density

ρ

n+1

i,j₊1 2

and

ρ

n+1

i₊1 2,j

, as well as all the viscosity constants

µ

_i±1

2j±12.The definitionsmustbecompatible withthecomputationofthecorrectionstep,which itselfrelieson

the way the density is computed when the Poisson equation for the pressure is solved. We use the computation of the

harmonicaveragewhengridcellsarecrossedbythefluid–fluidinterface,asproposedin[24,28]:

1

ρ

n₊1

=

1

ρ

+

_ρ

− 1

ρ

+

θ

+

1

ρ

−

(

1

− θ)

=

_ρ

₋ 1

θ

₊

ρ

+

₍

1

_{− θ)}

.

3.4. Velocityboundaryconditions

On thesolid–fluidinterface,we imposethefollowingDirichlet boundarycondition toenforcethesolid’svelocity us on

thefluidtoensuretheno-slipcondition: u

_|

_Ŵs

₌

us

.

In[18],theauthorsintroducedasecond-orderaccuratenumericalschemetoenforceaDirichletboundaryconditionforthe

Poissonequationinirregulardomains.Wefollowthisapproachtoimplicitlyimposetheno-slipconditionwhenconsidering

theviscousterm,i.e.inthecaseofthelinearsystemsforbothGFCMIandGFSCMIgiveninsection3.3.2. Theprojectiononthex-directionofthestresstensorinthecaseofGFCMIgives:

∇ ·

³

2

µ

D

´

_·

ex

¯

¯

¯

_i +12,j

=

∂

∂

x

µ

2

µ

∂

u

∂

x

¶¯

¯

¯

¯

_i +12,j

+

∂

∂

y

µ

µ

µ ∂

u

∂

y

+

∂

v

∂

x

¶¶¯

¯

¯

¯

_i +12,j

.

(20)

Thediscretizationofequation(20)gives:

∇ ·

³

2

µ

D

´

_·

ex

|

_i+1 2,j

≃

2

µ

i₊1 j

µ ∂

u

∂

x

¶

i₊1,j

−

2

µ

i,j

µ ∂

u

∂

x

¶

i,j

1

x

+

µ

_i+1 2j+12

µ ∂

u

∂

y

¶

i₊1₂,j₊1₂

−

µ

_i+1 2j−12

µ ∂

u

∂

y

¶

i₊1₂,j₋1₂

1

y

+

µ

_i+1 2j+12

µ ∂

v

∂

x

¶

i+12,j+12

−

µ

_i+1 2j−12

µ ∂

u

∂

x

¶

i+12,j−12

1

y

.

Eachtermisdiscretizedasfollows: • Discretizationof ∂u ∂x ¯ ¯ ¯ ¯ i₊1,j : Ifφ_i+1 2,jφi+32j>0,thecell h

i₊1₂,i₊3₂i_×[ j] isentirelycoveredwithfluid.Inthiscase,theapproximationis:

∂

u

∂

x

¯

¯

¯

¯

_i +1,j

=

u_i +32j

−

ui+12,j

1

x

.

Ifφ_i +12,jφi+ 3 2j≤

0,therearetwopossiblesituations:eitherthecellpoint³i₊1₂,j´isinthesoliddomain,whichmeans that u_i

+12,j=0,orthecellpoint ³

i₊ 3₂,j´isinthesoliddomain.Inthisscenario, themethodof[18]isapplied,giving:

∂

u

∂

x

¯

¯

¯

¯

_i +1,j

=

uŴ

−

u_i+1 2,j

θ 1

x

,

withuŴ thevelocityboundaryconditionandθ 1x thelengthfractionofthecellcoveredbythefluid,hence:

θ

₌

|φ

i+

1 2,j

|

|φ

i₊1₂,j

| + |φ

i₊3₂j

|

.

Theinterpolatedlengthfractioniscomputedonastaggeredcellpointinordertokeepsecond-orderaccuracy with:

φ

_i+1 2,j

=

φ

i+1,j

+ φ

i,j 2 and

φ

i+32,j

=

φ

i+2,j

+ φ

i+1,j 2

.

• Discretizationof ∂u ∂x ¯ ¯ ¯ ¯ i,j : Ifφ_i+1 2,jφi−12,j>0 thecell h

i₋1₂,i₊1₂i_×[ j] isentirelycoveredwithfluid.Inthiscase,theapproximationis:

∂

u

∂

x

¯

¯

¯

¯

_i ,j

=

u_i+1 2,j

−

ui−12,j

1

x

.

Ifφ_i+1

2,jφi−12,j≤0,therearetwopossiblesituations:eitherthecellpoint ³

i₊1₂,j´isinthesoliddomain,whichmeans that u_i

+12,j=0,orthecellpoint ³

i₋ 1₂,j´isinthesoliddomain.Inthiscase,themethodof[18]isapplied,giving:

∂

u

∂

x

¯

¯

¯

¯

_i,j

=

u_i+1 2,j

−

uŴ

θ 1

x

,

with

θ

=

|φ

i+12,j

|

|φ

i+12,j

| + |φ

i−12,j

|

.

• Discretizationof ∂u ∂y ¯ ¯ ¯ ¯ i₊1 2,j+12 : Ifφ_i+1 2,j+1φi+12,j>0,thecell h

i₊1₂i_×[ j,j₊1] isentirelycoveredwithfluid.Inthiscase,theapproximationis:

∂

u

∂

y

¯

¯

¯

¯

_i +12,j+12

=

u_i +12,j+1

−

ui+12,j

1

y

.

If φ_i+1

2,j+1φi+12,j≤0, there are two possible situations: either the cell point ³

i₊1₂,j´ is in the solid domain, which means that u_i

+12,j=0, orthe cellpoint ³

i₊1₂,j₊1´ isin thesolid domain.In thiscase, themethodof [18]is applied, giving:

∂

u

∂

y

¯

¯

¯

¯

_i +12,j+12

=

uŴ

−

u_i+1 2,j

θ 1

y

,

with

θ

=

|φ

i+12,j

|

|φ

i₊1 2,j

| + |φ

i+12,j+1

|

.