A mass-momentum consistent, Volume-of-Fluid method for incompressible flow on staggered grids

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A mass-momentum consistent, Volume-of-Fluid method for incompressible flow on staggered grids

T. Arrufat, M. Crialesi-Esposito, D. Fuster, Y. Ling, L. Malan, S. Pal, R.

Scardovelli, G. Tryggvason, S. Zaleski

To cite this version:

T. Arrufat, M. Crialesi-Esposito, D. Fuster, Y. Ling, L. Malan, et al.. A mass-momentum consistent,

Volume-of-Fluid method for incompressible flow on staggered grids. Computers and Fluids, Elsevier,

2021, 215, pp.104785. �10.1016/j.compfluid.2020.104785�. �hal-03099353�

ContentslistsavailableatScienceDirect

Computers and Fluids

journalhomepage:www.elsevier.com/locate/compfluid

A mass-momentum consistent, Volume-of-Fluid method for incompressible flow on staggered grids

T. Arrufat

^a

, M. Crialesi-Esposito

^b

, D. Fuster

^a

, Y. Ling

^f

, L. Malan

^a,c

, S. Pal

^a

, R. Scardovelli

^d

, G. Tryggvason

^e

, S. Zaleski

^a,∗

aSorbonne Université et CNRS, Institut Jean Le Rond d’Alembert, UMR, Paris 7190, France

bCMT-Motores Térmicos, Universitat Politécnica de Valéncia, Camino de Vera, s/n, Edificio 6D, Valencia, Spain

cInCFD, Dept. of Mechanical Engineering, University of Cape Town, South Africa

dDIN - Lab. di Montecuccolino, Università di Bologna, Bologna I-40136, Italy

eDept. of Mechanical Engineering, Johns Hopkins University, Baltimore, USA

fDept. of Mechanical Engineering, Baylor University, Waco, TX, USA

a rt i c l e i nf o

Article history:

Received 30 November 2018 Revised 11 August 2020 Accepted 2 November 2020 Available online 19 November 2020 Keywords:

Multiphase flows Navier-Stokes equations Volume-of- Fluid Surface tension Large density contrast

a b s t r a c t

Thecomputationofflowswithlargedensitycontrastsisnotoriouslydifficult.Toalleviatethedifficultywe consideradiscretizationoftheNavier-Stokesequationthatadvectsmassandmomentuminaconsistent manner.Incompressibleflowwithcapillaryforcesismodeledandthediscretization isperformedona staggeredgridofMarkerand Celltype.TheVolume-of-Fluidmethodisusedtotracktheinterfaceand aHeight-Functionmethodisusedtocomputesurfacetension.Theadvectionofthevolumefractionis performedusingeithertheLagrangian-Explicit/CIAM(Calculd’InterfaceAffineparMorceaux)methodor theWeymouthandYue(WY)Eulerian-Implicitmethod.TheWYmethodconservesfluidmasstomachine accuracyprovided incompressibilityissatisfied.To improvethestabilityofthesemethodsmomentum fluxesareadvectedinamanner“consistent” withthevolume-fractionfluxes,thatisadiscontinuityof the momentumis advectedatthe same speed as adiscontinuityofthe density. To findthe density onthestaggered cellsonwhichthevelocity iscentered,anauxiliaryreconstruction ofthedensity is performed. Themethodis tested foradropletwithout surfacetensioninuniform flow,foradroplet suddenly accelerated inacarryinggas atrestatvery largedensity ratio withoutviscosity orsurface tension,fortheKelvin-Helmholtzinstability,fora3mm-diameterfallingraindropand foranatomizing flowinair-waterconditions.

© 2020TheAuthors.PublishedbyElsevierLtd.

ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/)

1. Introduction

Multiphase flows aboundin nature,buttheir stableandaccu- ratecomputationremainselusiveinmanycases.Asacaseinpoint, manynumericalmethodsusedfortwo-phase incompressibleflow arestronglyunstableforlargedensitycontrastsandlargeReynolds numbers. Experience withsuch simulations showsthat the pres- enceofsurfacetensionisan aggravatingfactor.Thelarge density contraststhatareofinterestareair/waterorgas/liquid-metal,with

ρ

l/

ρ

^{g of the orderof 103 or104. The} large densitycontrasts are a difficulty whetherone dealswithanyofthe threemajor inter-

∗Corresponding author.

E-mail address: zaleski@dalembert.upmc.fr (S. Zaleski).

faceadvectionmethods,Level-Set,Volume-of-Fluid(VOF)orFront- Tracking,orwithcombinationssuchasCLSVOF.(Thetermdensity contrastis preferableto densityratiosince it encompassesratios bothmuchlargerthanoneandmuchsmallerthanone.)

Severalmethodshavebeenused toalleviate thehigh-density- contrastdifficulties. It hasbeen observed by severalauthorsthat making the momentum-advection method conservative improves the situation. For incompressible flow, “momentum-conserving”

methodshavebeeninitiallyproposed by[1],andbyseveralother authorssince [2–7].These methodshavebeen showntoimprove thestabilityofthenumericalresultsinvarioussituations.Inpar- ticular,liquid-gasflows with verycontrasted densities, asforex- ample intheprocess ofatomization,causeserious problemsthat areresolvedbyusingmomentum-conservingmethods.Inthatcase anotheroft-suggestedsolutionistoincreasethenumberofequa-

https://doi.org/10.1016/j.compfluid.2020.104785

0045-7930/© 2020 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license ( http://creativecommons.org/licenses/by/4.0/ )

tions fromthestandardfourequationstofive,sixorsevenequa- tions, by introducing new field variables in each phase. The ad- ditionofonemoredensity

ρ

i,momentum

ρ

iu_i orenergyvariable

ρ

ie_iincreasesthenumberofequations.Theauthorsofref[8].used sevenequations,thoseof[9]usedfiveequationsandsixequations wereusedin[10].Thelastthreereferencesalsouseamomentum- conserving formulation. Severalauthors, including some of those cited above, have argued that the difficulty may come fromgas velocities of order ug being mixedwith liquid densities of order

ρ

l.Boththe

ρ

l andu_g scales are largeandthe appearance ofan nonphysical

ρ

lu²_g dynamic pressure scale could create numerical pressure fluctuationsofthesame orderandnonphysicalpressure spikes, asthe one nicelyillustrated in [11] inthe front partofa suddenlyacceleratedsmalldroplet.Onewayofavoidingthisnon- physicalmixtureofliquidandgasquantitiesistoextrapolateliq- uid andgaspressure andvelocity inthe“other” phase, asinthe ghost fluidmethod.Thisextrapolationstrategy wasusedsuccess- fullyin[12].

Itmaybearguedthatawaytoavoidthisnumericaldiffusionof liquidandgasquantitiesistoadvectthevolumefractionandthe conserved quantities that depend on it (density, momentum and energy) in a consistent manner. In incompressible flow inwhich we specializeinthispaper,itmeansthat thevolumefractionand momentumorvelocitymustbeadvected inthesameway.Thisis equivalenttorequestthatthediscontinuityoftheHeavisidefunc- tion H, marking the phase transition, should be advected atthe same speed as the discontinuity in momentum. This can be ex- pressed by the following consistency requirement: ifmomentum isinitiallyexactlyproportionaltovolumefraction,itshouldremain soafteradvection.WecallsuchamethodVOF-consistent.

To satisfy thisrequirement the idea is to solve the advection equation formomentumwiththesamenumericalschemethatis used for the VOFcolor function.This consistency property mini- mizes the nonphysical transferof momentum fromone phase to anotherduetothedifferencesinthenumericalschemesused.The consistencyisespeciallyimportantwhendealingwithfluidswitha large densitycontrastwhereasmallnumericalmomentumtrans- fer fromthe dense phase to the light phase results in large nu- mericalerrorsinthevelocityfieldwhichinturncreatesnumerical instabilities.

In this work, we present a modification of the classical momentum-preserving scheme proposed by [1] for the case of a staggered grid and VOF method. The scheme is then not momentum-conservingalthoughitisbuiltbydiscretizationofthe momentum conservingformulationoftheadvectionterms,sowe rathercallitmass-momentumconsistent.

Thepaperisorganizedasfollow:thesecondsectiondealswith the continuum mechanics formulation for incompressible flow and sharp interfaces. Section 3 describes our numerical method, starting with an overview of already-known methods for spa- tial discretization, time-stepping, and VOF advection. We con- tinuewiththenewmomentumadvection-VOF-consistentmethod.

Section 4isdevotedtotestsofthemethod,followedbyaconclu- sion.

Among the authors,GretarTryggvason, Ruben Scardovelli,Yue Ling andStéphane Zaleskihavebeeninvolvedintheconstruction ofthebaseoftheParisSimulatorVOFandFront-Trackingcodethat wasusedtoimplementandtesttheideasinthispaper.ParisSim- ulator isitself basedon a Front-Trackingcode developedby Gre- tar Tryggvason,SadeghDabiriandJiacai Lu.DanielFusterwasin- volved in the developmentof the momentum advectionmethod consistentwithVOFadvection,withhelpfromTomasArrufat,Leon Malan and Yue Stanley Ling. The Kelvin-Helmholtz analysis and testing were done by Stéphane Zaleski. The fallingraindroptest- ingandthecorrespondingfiguresweredonebyTomasArrufatand SagarPal.Thelargedensitydropletandshearlayertestsweredone

bySagarPal.Theatomisationtestingwasdone byMarcoCrialesi- Esposito.

2. Navier–Stokesequationswithinterfaces

We model flows with sharp interfaces defined implicitly by a characteristic function H(^x,t) ^{defined such that fluid 1 corre-} spondstoH=1andfluid2toH=0.Theviscosity

μ

^anddensity

ρ

^{arecalculatedasanaverage}

μ

=

μ

1H+

μ

2

(

¹−H

)

,

ρ

=

ρ

1H+

ρ

2

(

¹−H

)

. (1) Thereisnophasechangesotheinterface,almostalwaysasmooth differentiable surface S, advances at the speed of the flow, that isV_S=u·n whereuis thelocal fluid velocity andna unit nor- mal vector perpendicular to the interface. Equivalently the inter- facemotioncanbeexpressedinweakform

∂

^tH+u·

∇

^H=0, (2)

which expresses the fact that the singularity ofH, located on S, moves at velocityV_S=u·n. Forincompressible flows, which we willconsiderinwhatfollows,wehave

∇

^{·u=0. (3)}

The Navier–Stokes equations for incompressible, Newtonian flow withsurfacetensionmayconvenientlybewritteninoperatorform

∂

t

( ρ

^u

)

=L

( ρ

,u

)

−

∇

^{p (4)}

whereL=Lconv+Ldiff+Lcap+Lext sothat theoperatorL isthe sumofadvective,diffusive,capillaryforceandexternalforceterms.

Thefirsttwotermsare

Lconv=−

∇

^·

( ρ

^uu

)

, Ldiff=

∇

^{·D, (5)}

where D is a stress tensor whose expression for incompressible flowis

D=

μ

∇

^u+

( ∇

^u

)

^T

, (6)

where

μ

^{iscomputedfromHusing(1).Thecapillarytermis} Lcap=

σκδ

^Sn,

κ

^{=1/R1+1/R2, (7)}

where

σ

^{is the surface tension coefficient, n is the unit normal}

perpendiculartotheinterface,

κ

^{isthesumoftheprincipalcurva-}

turesand

δ

S isaDiracdistribution concentratedonthe interface.

Weassume aconstantcoefficient

σ

^.FinallyLext representsexter- nalforcessuchasgravity.

3. Method

3.1. Spatialdiscretization

We assume a regular cubic orsquare grid.This can be easily generalizedtorectangular orcuboid grids,andwithsome efforts to quadtreeandoctree grids. We alsouse staggeredvelocity and pressuregrids,asrepresentedintwodimensionsinFig.1.

Thecontrolvolumesurroundingthepressurepisusedforother scalarquantities,suchasthedensity

ρ

^{andvolumefractionC.The}

control volumesofthevelocity components,u₁ andu₂,andmo- mentumcomponentsareshiftedrespectivelyhalfcellhorizontally and vertically, with respect to the pressure control volume. The useof staggered control volumeshasthe advantage ofsuppress- ingneutralmodesoftenobservedincollocatedmethodsbutleads tomorecomplexdiscretizations(see[13]foradetaileddiscussion.) Thistypeofstaggeredrepresentationiseasilygeneralizedtothree dimensions, and the discrete version of the continuity Eq. (3) is rathercompactonsuchagrid

u_1;i+1/2,j,k−u_{1;i−1/2,j,k}

^{x +}

u_2;i,j+1/2,k−u_{2;i,j−1/2,k}

^y

Fig. 1. Representation of the staggered spatial discretization in two dimensions. The pressure p _i,jis located at the center of its control volume (light color area); the horizontal velocity component u 1;i+1/2,jis stored in the middle of the right edge of the pressure control volume and at the center of its control volume (dash-dotted area); the vertical velocity component u 2;i,j+1/2is stored in the middle of the top edge and at the center of its control volume (dashed area).

+u_3;i,j,k+1/2−u_{3;i,j,k−1/2}

^{z =0. (8)}

In what follows,we shall use thesubscript f=m±, with the integer index m=1,2,3, tonote theface of anycontrol volume located in thepositive or negativeCartesian directionm,andn_f fortheunitnormalvectorofface f pointingoutwardsofthecon- trolvolume.Onacubicgridthespatialstepis^x=^y=^z=h sothediscretecontinuityequationbecomes

∇

^h·u=3

m=1

(

^um++um−

)

/h=0, (9)

whereu_f=um±=u·n_f isthevelocitycomponentnormaltoface f.The discretization of the interface location is performed using a VOFmethod. VOF methods typically attempt to solve approxi- matelyEq.(2)whichinvolvestheHeavisidefunctionH,whosein- tegral inthe cell ^{indexedby i},j,k definesthe volume fraction C_i,j,kfromtherelation

h³C_i,j,k=

Hdx. (10)

C_i,j,krepresentsthefractionofthecellfilledwithfluid1,takento bethereferencefluid.

3.2. Timemarching

Thevolumefractionfieldisupdatedas

Cⁿ⁺¹=Cⁿ+LVOF

(

^Cn,uⁿ

τ

/h

)

, (11) where LV OF represents the operator that updatesthe Volume of Fluid data given the velocity field. Once volume fraction is up- dated,thevelocity fieldisupdated ina coupleofsteps.Aprojec- tionmethodisfirstused,inwhichaprovisionalvelocityfieldu^∗is computed

ρ

^n+1u∗=

ρ

^nun+

τ

L^hconv

( ρ

ⁿ,uⁿ

)

+

τ

L^h_diff

( μ

ⁿ,uⁿ

)

+L^hcap

(

^Cn+1

)

+L^hext

(

^Cn+1

)

. (12)

Itgoeswithoutsayingthattheaboveoperatorsdependonthedis- cretizationtime step

τ

^{andspatialstephaswell asthefluidpa-}

rameters. The discussion of the L^hconv operator is the main point ofthispaper.Inthesecondstep,theprojectionstep,thepressure gradient forceisaddedtoyieldthevelocity atthenewtimestep uⁿ⁺¹=u^∗−

τ

ρ

ⁿ⁺¹

∇

^{hp. (13)}

Thepressureisdeterminedbytherequirementthatthevelocityat theendofthetimestepmustbedivergencefree

∇

^h·uⁿ⁺¹=0, (14)

whichleadstoaPoisson-likeequationforthepressure

∇

^h·

τ

ρ

ⁿ⁺¹

∇

^hp=

∇

^{h·u∗. (15)}

3.3. Volume-of-fluid

Inthis section we detailonly the necessarysteps to illustrate the momentum advectionmethod based onthe VOF method.To simplifythe presentation we rescale spaceand time variables so thatthecellvolumeandthetimesteparebothequalto1.Anyve- locitycomponentubecomesu=u

τ

/handtheC_i,j,kvalue isalso themeasure ofthevolume ofreferencefluid incell i,j,k. Atthe beginning of any simulation the volume fraction field is initial- izedwiththeVofilibrarydescribedin[14]and[15].Thisallowsa highlyaccuratenumericalintegrationofthemeasureoffluidvol- umes.

3.3.1. Normalvectorandplaneconstantdetermination

TheVOFmethodproceedsintwosteps,reconstructionandad- vection.InthefirststepweconsideraPLICreconstructionineach cell cutby the interface wherethe unit normalvector niscom- putedwiththe MYCmethod describedin[13].Wethen consider thecolinearnormalvector mwhosecomponentssatisfy therela- tion

|

^mx

|

+

|

^my

|

+

|

^mz

|

=1.GiventhevolumeV=C_i,j,kincelli,j,k occupiedby fluid1 andthenormalmwe consider thefamilyof planes

m·x=

α

^{. (16)}

Bychangingthevalueoftheplane constant

α

^{adifferentvolume}

offluid1iscutinthecell.Thecorrectvalueof

α

^{isdeterminedby}

theresolutionofacubicequation[16].

3.3.2. Generalsplit-directionadvection

The interface reconstruction at timetn is then used to obtain thepositionoftheinterfaceandthevolumesC_i,j,katthenextdis- cretetimet_n+1.Thefollowingdiscussionofmomentumadvection isbased on two VOFadvectionmethods, Lagrangian Explicit and Weymouth and Yue’s schemes. We first describe their common features.

Afteradditionandsubtractionofatermproportionaltotheve- locitydivergence,Eq.(2)leadsto

∂

tH+

∇

·

(

^uH

)

=H

∇

·u. (17) Thisequationisintegratedinthetimestepandcellvolume

C_iⁿ_,⁺_j,k¹−C_i,j,kⁿ =−

facesf

F_f^(c)+ _tn+1

tn

dt

H

∇

·udx, (18)

wherethe first termon the right-hand side is the sumover the cellfaces f ofthefluxesF_f^(c)of(^uH)^{.Obviouslythe}“compression”

term on the right-hand side disappears for incompressible flow, howeveritisessentialinsplit-advectionmethods.Intheprevious equationthefluxF_f^(c)is

F_f^(c)= tn+1

tn

dt

f

u_f

(

^x,t

)

^H

(

^x,t

)

^dx, (19)

where u_f=u·n_f. Once an approximation for the evolution of (^uH) ^{duringthe time step is chosen, a} four-dimensionalintegral remainstobecomputedinEq.(19).Thetwomethodsweconsider hereare directionally split andare alsodesignedto preserve the property 0≤C_i,j,k≤1 which we callC-bracketing. Itis important

Fig. 2. Formation of the areas V 1, V 2and V 3by Lagrangian advection in the horizontal direction: initial reconstruction with the horizontal velocities on the faces of the central cell (left); segments and areas V iafter Lagrangian advection (right). In three dimensions rectangles become cuboids.

Fig. 3. Eulerian flux representation for advection in the horizontal direction: same initial reconstruction and horizontal velocities of Fig. 2 (left); fluxes, or areas V 1and V 3, are calculated directly from the interface reconstruction in each cell (right).

topreserveC-bracketinginordertoavoidarbitraryadditionorre- movalofmass.Furthermoretheydonotproduceinthebulkofthe twofluidsdeviationsfromthecorrectvalues0and1.

Directional splitting results in the breakdown of equation(18)intothreeequations

C_i^n,l+1_,j,k −C_i^n,l_,j,k=−Fm^(c−⁾−F_m^(c₊⁾+cm

∂

m^hum, (20)

where the superscriptl=0,1,2 is the substep index, i.e.C_i^n,_,j⁰_,k= C_i,j,kⁿ andCⁿ_i,^,_j,k³ =C_i,j,kⁿ⁺¹.Thefacewithsubscriptm−isthe“left” face indirectionmwithF_m^(c₋⁾≥0iftheflowislocallyfromrighttoleft.

A similar reasoningapplies to the“right” facem+.We have also approximatedthecompressiontermin(18)by

tn+1 tn

dt

H

∂

mumdxcm

∂

m^hum, (21)

withnoimplicitsummationrule.IntheRHSof(20)and(21)the flux termsF_f^(c) andthepartialderivative

∂

^{mummustbeevaluated}

withthesamediscretizedvelocities.Inparticular,

∂

m^humisafinite difference or finite volume approximation of the spatial deriva- tive of the mthcomponent ofthe velocity vector indirectionm, andthe“compressioncoefficient” cm approximatesthecolorfrac- tion. Its exact expression is dependent on the advectionmethod and it preserves theC-bracketingcondition. Since the coefficient cmmaynot bethesamealong thethreeCartesian directions,the

sum

mcm

∂

m^hum is not necessarily vanishing even ifthe flow is incompressible. Aftereach advectionsubstep(20),theinterfaceis reconstructedwiththeupdatedvolumesC_i^n,l+1

,j,k ,thenthefluxesF_f^(c) arecomputedforthenextsubstep.

Ineach substep thevelocityfield indirectionmisusually de- pendent onlyonthespatialcomponentxm,um(^xm)^.Thisapprox- imation ensures thatthe fluid areasfluxingacrossa cellside are rectangular in two dimensions, as shown in Figs. 2 and 3. The multi-dimensionalityoftheflowisconsideredinunsplitmethods, where thefluxingareasaredescribedby morecomplexpolygons [17].

3.3.3. Lagrangianexplicitadvection

TheLagrangianExplicit/CIAMmethodreferstoaspecifictype ofsplitadvectionanditismostnaturallyexplainedasaLagrangian transport of the Heaviside function [18,19]. The velocity field is linearly interpolated between the face velocities um− on the left and um+ on the right, so that um(^xm)=−um−(¹−xm)+um+xm, with the origin of xm on the left face. The equation of motion dxm/dt=um(^xm) ^{is integrated witha first-order in time, explicit}

u_m(^xnm)approximation,togetinrescaledvariables

xⁿ_m⁺¹=−um−+

(

¹+um++um−

)

^xnm. (22) Thistransformationgivesthepositionofadvectedpointsasafunc- tionoftheoriginalpositionandcompressesdistancesalongdirec- tionmbyafactor(¹+um++um−)^{.Pointsoverfacesandlinearin-} terfaceareadvectedinthesameway,andinthetwo-dimensional caseofFig.2the advectionsubstep resultsinthethreecontribu- tionsV₁,V₂ andV₃ to the central cell.The intermediate value of thecolorfunction inthecentral cellwill begivenby thesumof thesethreecontributions.

Thereisacorrespondencebetweenthegeometricalinterpreta- tionoftheLagrangianExplicitadvectionandthedefinition(19)of F_f^(c).Forexample,forthecentralcellofFig.2thefluxontheleft faceisfromlefttoright,sinceu_1;i−1/2,j>0.Thenwithm=1and f=1−, we have V₁=−F₁⁽₋^c)>0 and u₁₋=u·n₁₋<0. The final expressionofthesubstepis

C_iⁿ_,^,_j^l_,⁺_k¹=C_iⁿ_,^,_j^l_,k

(

¹+um++um−

)

−F_m^(c₋⁾−F_m^(c₊⁾, (23) which showsthat the approximation ofthe derivative is

∂

m^hum= um++um−and the compression coefficient is

cm=C_i^n,l_,j,k. (24)

Inorderto removespurious asymmetriesintheflow itisimpor- tanttochangetheorderofsplitadvectionsateachtimestep.Then thecompressioncoefficientinthehorizontaladvection,m=1,can beassociatedtothefirstsubstep, l=0,intimestep n,andtothe lastsubstep,l=2,inthenexttimestep.ThesequenceofthreeLa- grangiansubsteps(20)doesnotresultinvolumeconservation

C_iⁿ_,⁺_j,k¹−C_i,j,kⁿ =−

facesf

F_f^(c)+ 3

m=1

cm

(

^um++um−

)

. (25)

Whilethefluxtermscanceluponintegrationoverthedomain,the sumofthecompressivetermsdoesnotvanishsincecmchangesat eachsubstepl.

3.3.4. WeymouthandYue’sadvection

The(WY)splitadvectionisexactlymass-conserving[20].Inthis methodthecompressioncoefficientisindependentofdirectionm, sothatcm=c,andisdefinedas

c=H

C_iⁿ_,j,k−1/2

(26) whereH isa one-dimensional Heavisidefunction,that isc=0if C_i,j,kⁿ <1/2andc=1ifC_i,j,kⁿ ≥1/2.ThefluxesF_f^(c)arealsodefined differently.Thereferencephase fluxedthroughtheleftfaceindi- rectionm=1isequaltothevolumefractioninacuboidofwidth

u_i−1/2,j,k adjacenttotheleft face f=1−.Thisfluxedvolume cor- respondsto“EulerianImplicit” (EI)advectionintheterminologyof [19]andis representedin2DbytheareaV₁ofFig.3.Usingthese definitions, Weymouth andYue were ableto show that the final resultobeysC-bracketing[20].

Thissplitadvectionschemeconservesvolumeatmachineaccu- racy.Indeedthesummationofthreesubsteps(20)resultsin

C_iⁿ_,⁺¹_j,k−C_iⁿ_,j,k=−

facesf

F_f^(c)+c 3

m=1

∂

m^hum. (27) Since3

m=1

∂

m^hum=3

m=1(^um++um−)^isthefinite-volumeexpres- sionfor

∇

·u,itdisappearsandmassisconservedattheaccuracy withwhichcondition(3)issatisfied.

3.3.5. Clipping

Thealgorithmthat hasbeencodedinvolvesanumberofaddi- tionalstepsdesignedtoavoidunwantedeffectsofarithmeticfloat- ing point round-off error. The mostimportantone isclipping: at the end of each directional advection,the values ofC_i,j,k are re- setsothatC_{i jk}issetto0ifC_{i jk}<

^corto1ifCi jk>1−

^c.When

thereisnosurfacetensionthechoice

^c=10⁻¹² workswell.Oth- erwise

^c=10⁻⁸givesmorestableresultswithsmootherinterface shapes. Thisstronger clippingis anecessityforsome simulations withWY,asweobservethatWYproducesmanymore“wisps”,i.e.

cellswithtinyvaluesof1−C_i,j,kinsidefluid1orC_i,j,kinsidefluid 2.Wehavenotyetbeenabletodeterminetheoriginofthisneed foramoreforcefulclippingwithWY,butitcouldberelatedtothe factthat theCIAMmethodhasageometricalinterpretation,while WYisintrinsicallyalgebraicinnature.

3.4. Momentum-advectionmethods

3.4.1. Advectionofagenericconservedquantity

Consider the advectionofa genericconserved quantity

φ

^{by a}

continuousvelocityfield

∂

t

φ

+

∇

·

( φ

^u

)

=0. (28) We assume that

φ

^{is smoothly varying except on the interface}

where it may be discontinuous. Indeed finding a correctscheme for the advectionof thisdiscontinuity, atthe same speed asthe advectionofthevolumefraction,isthegoalofthepresentstudy.

The smoothnessoftheadvected quantityawayfromtheinterface is verified forthe density

ρ

, the momentum

ρ

^{uor the internal}

energy

ρ

^{e.Wefirstintegrate(28)intime}

φ

iⁿ,⁺j,k¹−

φ

_i,j,kⁿ =−

facesf

F_f^(φ). (29)

Thesumontheright-handsideisthesumoverfaces fofcelli,j,k ofthefluxesF_f^(φ) of

φ

,whicharedefinedinthesamewayasthe colorfunctionfluxesF_f^(c)in(19)

F_f^(φ)= tn+1

tn

dt

f

u_f

(

^x,t

) φ (

^x,t

)

^dx. (30)

Inorderto“extract” thediscontinuityweintroducethecharacter- isticfunctionH(^x,t)

F_f^(φ)= tn+1

tn

dt

f

[ufH

φ

+uf

(

¹−H

) φ

^]dx, (31) andrewriteitas

F_f^(φ)=

φ

¯1

_tn+1

tn

dt

f

ufHdx+

φ

¯2

_tn+1

tn

dt

f

uf

(

¹−H

)

^dx, (32) wherethefaceaverages

φ

¯^s,s=1,2,are

φ

¯^s=

tn+1

tn dt _f

φ

^ufHsdx

tn+1

tn dt _fufHsdx , (33)

andH₁=H,H₂=1−H. Expression(32) can be written interms of the fluxesF_f^(c) andF_f^(1−c), thissecond one being obtained by replacingHwith1−Hin(19)

F_f^(φ)=

φ

¯^1Ff^(c)+

φ

¯^2Ff^(1−c). (34)

3.4.2. Cloningthetracers

When a cell is cut by the interface, and the field

φ

^{is not}

smooth, it becomes difficult to estimate the integrals in (32). A possibilityistodefinetwonewfields

φ

1 and

φ

2,with

φ

^s=

φ

^in-

sidephases,then

φ

=H

φ

1+

(

¹−H

) φ

2. (35) This is morecostly in memory usage but simplifies considerably thecomputationoftheaveragesin(32).Thetwoequations(2)and (28)arenow replacedby threeequations,the samevolumefrac- tionEq.(2)and

∂

^t

φ

¹⁺

∇

^·

( φ

^1u

)

=0,

∂

^t

φ

²⁺

∇

^·

( φ

^2u

)

=0. (36) ThethreeEqs.(2)and(36)nowimply(28).Theadditionofapair of“cloned” variablestodealwithlargedensitycontrastsissimilar tothemethodsusedfortheresolutionofthemomentumanden- ergyequationsforcompressibleflow. ForexampleSaurel andAb- grallused two density, momentum andenergy variables intheir seven-equationsmodel[8],whileAllaire,ClercandKokhuse two densityvariablesintheirfive-equationsmodel[9].Theadditionof aclonedtracervariableinincompressibleisothermalflowwasalso implementedbyPopinetinthe“Basilisk” code[21].

3.4.3. Advectionofthedensityfield

The density

ρ

(^x,t) ^{obeys(28)with}

φ

=

ρ

^{. Moreoverwe con-}

sider a divergence-free velocity field, with constant density in eachphase.Wecanextractthedensitytriviallyfromtheintegrals (33)toobtainexactly

ρ

¯^s=

ρ

^s.Thefluxof

ρ

^isthen

F_f^(ρ)=

ρ

1F_f^(c)+

ρ

2F_f^(1−c). (37) Usingthisfluxdefinitionfor

ρ

,andanyVOFmethodforthefluxes of the color function, one obtains a conservative method for

ρ

, since eq. (29) evolves

ρ

^{as a difference of fluxes. Thus the total}

massisconserved. Howeverthisresultisnot consistentwiththe advectionofthecolorfunctionintheCIAMcase,asCIAMdoesnot conservevolumesexactly(see(25)).Asaresulttheadvectionof

ρ

isnotconsistentwiththeadvectionofC.

Thisparadoxmaybe resolvedifone noticesthatthecompres- siontermismissingin(28).Forconsistencythecompressionterm shouldbekept,andtheadvectionequationforaconservedquan- titybecomes

∂

^t

φ

⁺

∇

^·

( φ

^u

)

=

φ ∇

^{·u. (38)}

Itisthenpossibletodefinetheevolutionof

φ

^{throughasequence}

ofdirectionally-splitoperationswhichareequivalenttotheopera- tionsperformedonthecolorfunction

φ

ⁿ_i,j,k^,l+1−

φ

_i,j,k^n,l =−Fm^(φ)− −F_m^(φ)₊ +

φ

˜1^mcm⁽¹⁾+

φ

˜2^mc⁽m²⁾

∂

m^hum (39)

whereF_m^(φ)_± aredefinedin(34),thecellaverages

φ

˜s^mare

φ

˜^ms =

tn+1

tn dt

φ

^Hs

∂

m^humdx

tn+1

tn dt Hs

∂

m^humdx , (40)

andc⁽_m¹⁾=cmisthecompressioncoefficientoftheVOFadvection, while c_m⁽²⁾=1−cm is that ofthe symmetric color fraction 1−C.

Specificallyfor

ρ

^thisgives

ρ

i^n,l+1,j,k −

ρ

i^n,l,j,k=−F_m^(ρ)₋ −F_m^(ρ)₊ +C_m^(ρ), (41)