fluid method in axisymmetric coordinates Interface reconstruction and advection schemes for volume of Journal of Computational Physics

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Journal of Computational Physics

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Interface reconstruction and advection schemes for volume of fluid method in axisymmetric coordinates

Ananthan Mohan, Gaurav Tomar

^∗

DepartmentofMechanicalEngineering,IndianInstituteofScience,Bangalore,India

a rt i c l e i nf o a b s t ra c t

Articlehistory:

Availableonline31August2021 Keywords:

Volumeoffluid Axisymmetriccoordinates Multiphasesimulation

Volume of fluid (VOF) method is a sharp interface method employed for simulations of two phase flows.Interface inVOF is usually represented usingpiecewise linearline segments ineachcomputationalgridbasedonthe volumefractionfield.WhileVOF for cartesian coordinatesconserve mass exactly, existing algorithms do not show machine- precisionmassconservationforaxisymmetriccoordinatesystems.Inthiswork,wepropose analyticformulaeforinterfacereconstructioninaxisymmetriccoordinates,similartothose proposedbyScardovelliandZaleski(2000)[12] forcartesiancoordinates.Wealsopropose modificationsto theexisting advectionschemes inVOF foraxisymmetriccoordinatesto obtainhigheraccuracyinmassconservation.

©^{2021ElsevierInc.Allrightsreserved.}

1. Introduction

Multiphaseflows areubiquitiousinseveralindustrial applications.Inthe lastthreedecades,therehasbeenasurgein thenumericalmethodsandalgorithmsforsimulationsofcomplexmultiphaseflows.Therehavebeenseveraldifferenttypes ofinterfacecapturingstrategiesthathavebeenproposedfortwo-phaseflows.ThemostpopularofthesearetheLevelset method,Volume ofFluid(VOF) method,andFronttrackingscheme [1,2]. VOFmethodswithgeometric advectionstrictly conservethevolumeofthetwophases.

Several improvementshave been madesince the inception ofthe method (see Hirt andNichols [3–8]).The simplest and earliest representation for interface reconstruction is simple line interface calculation (SLIC) in which the interface is approximatedby horizontal orvertical lines.Subsequently, piecewise lineinterface construction (PLIC) was introduced wheretheinterfaceisapproximatedasalinearlineatanangleinthecell[9].Higherorderinterfaceconstructionhasbeen proposed (suchasParabolic reconstruction by [4]),butconsidering theassociatedcomputational cost andcomplexity for geometricadvection,PLICisusuallypreferred.

In PLICbased method theinterface isapproximated asa line segment givenby m.x=^{a,where m}=(mx,my) is the unit normal to the interface anda is the line constant. The reconstruction step in VOF methods consists of two steps, the computation ofm andthen the computation ofline constant a such that the volume cut by the interface matches thevolumefractionofthecell.Theinterfacenormaliscomputedusingthevolumefractiongradientbyusingan accurate gradient computing algorithm like Youngs method [9], a least-squares method (e.g.: (E)LVIRA [7]) or a height function method [10].Anotherpopularmethodforinterface reconstructionistheMoment-of-Fluid(MoF)method [11], whichuses the volume fractionvalue ofthe cell along withthe geometric centroids of the individual fluid volumeswithin the cell

*

Correspondingauthor.

E-mailaddress:gtom@iisc.ac.in(G. Tomar).

https://doi.org/10.1016/j.jcp.2021.110663

0021-9991/©^{2021ElsevierInc.Allrightsreserved.}

to reconstructthe interface.The MoFmethods require one to tracktheadvectionof thegeometric centroids ofthefluid volumesalong withthevolumefraction. Inthe present,work we areprimarily interestedinthecomputation ofthe line constantgivenaninterfacenormalintheframeworkofVOFmethods.Neverthelessthereconstructionalgorithmpresented inthestudyisquite genericforstructuredgrids.

ScradovilliandZaleski [12] proposedanalyticalformulaeforthepiecewiselinearreconstructionoftheinterfaceincarte- sian coordinates that ledto a significantspeedupover the earlieriterativeschemes.However, forcurvilinear coordinates (suchasaxisymmetric coordinatesystem), theproposed analytical formulaecannot be employeddirectly. Diot etal.[13]

proposedasemi-analyticformulaeforarbitraryconvexcellswhichrequiredtheuseofNewtonsiterationforaxisymmetric coordinatesystem. Inthepresentwork,we derivesimilar analyticformulaeforaxisymmetriccoordinates,whichresultin aspeedupof∼^{28 overtheiterative}counterparts (Brent’srootfinding method).Further,we demonstratethattheexisting interfaceadvectionschemesinVOFforaxisymmetriccoordinatesarenotstrictlymassconserving.Inthisstudy,wepropose modifications tothecurrentoperatorsplitalgorithmsthat resultinmachine-precisionmassconservationinaxisymmetric coordinates.Weshowtheefficacyoftheproposedalgorithmsusingseveraltestcases.

The paperisorganizedasfollows.Wefirst presentanalytical formulaefortheinterface reconstructionschemesin ax- isymmetriccoordinatesinsection2.Insection3,weproposemodificationsintheexistinginterfaceadvectionalgorithmfor axisymmetricVOFandpresenttestcasestoshowtheefficacyofthescheme.Finally,insection4,wediscusstheimportant conclusions.

2. Interfacereconstructionscheme

Interfacereconstructioninthevolumeoffluid(VOF)methodrequiresthevolumefractionfield.Usingthevolumefraction field,apiece-wiselinearorahigherorderinterfaceisconstructedinagivengrid-cell.Interfacereconstructionisanintegral partofthegeometricadvectionschemestoensuremassconservationpropertyoftheVOFmethod[2].Initialconditionfor a multiphaseflow simulationrequires theinitial distributionofthevolume fractionfield, usually providedasan implicit functionofthespatialcoordinates.VOFIlibrary [14] isanopensourcelibrarytoinitialize theliquidvolumefractionfieldin cartesiancoordinatesystemsaccurately.InVOFI,forcellscutbytheinterface(seeFig.1),PLICreconstructionmethod[9,6]

isemployedtoapproximatetheinterfaceasalinearlinesegment,

m

.

x

=

^a

,

(1)

where mis thelocalnormal attheinterface, x is apoint on theplane, anda isthe normaldistanceofthe originfrom the plane.Analyticalrelation,givenby ScardovelliandZaleski [12],betweenthevolume fractionandthelineconstant is employedtodeterminethelineconstanta.Thus,fortwo-dimensionalandthree-dimensionalCartesiancoordinatesystems, VOFI library canbe directly employed foraccurate assignment of the initialvolume fraction field on a givendiscretized domain usingan implicitequation oftheinterface. However, incurvilinearcoordinatesystems,fora givenimplicitfunc- tion,thepiece-wiselinearinterface constructedfromVOFIwouldrequirecomputationofthevolumefractionfieldusinga formulaspecific tothecurvilinearcoordinates.Forinstance,foraxisymmetriccoordinatesystem, themodified Gaussarea (shoelace)formulaforcomputationoftheareaofaconvexpolygonisgivenby,

V

= π

3

n

i=1

(

xi

+

xi+¹

) (

xiyi+¹

−

xi+¹yi

)

⁽²⁾

where(xi,yi)fori=¹,...,n(withxn+¹=^x1andyn+¹=^y1)arethecoordinatesoftheverticesofaconvexpolygonordered counterclockwiseasshownintheFig.1.

Thus,forinitializationofthevolumefractionfield,C,onceweobtainthelinearinterfaceineachgridcellusingtheVOFI library,weusetheaboveformulatocomputethevolume,V,andassignvolumefractionineachgridcellas,

C

=

^V

2

π

r_c

x

y

.

(3)

Here rc isthedistanceofthecenterofthe cellfromtheaxisofsymmetry,andxandy arethe grid-cellsizes inthe radial (r) andaxial(y) directions,respectively.Wenote that theabove procedureis followedessentiallyto minimize the errorinthevolume-fractionduringinitialization.Toillustratethis,weinitialize atorusofminorradiusr_t=⁰.25 andmajor radiusr=⁰.50 inthecenterofacomputationaldomainofsize1×^{1 asshownintheFig.2.Thevolumeofthetoruscanbe} analyticallycomputedasVt=²

π

²rr_t²,wherethemajorradius,r,isthedistancetothecenterofthetorusfromtheaxisof symmetry.Wecomparetheresultsforinitialized volumeforvariousgridsizesbycomputingtherelativeerrorwithrespect tovolumeobtainedusinganalyticalformula,giveninTable1.

Thus, we haveshownthat forcurvilinearcoordinatesvolume fractionfield can beinitialized up tomachine accuracy.

Now, we derive an analytic relation forPLICreconstruction for axisymmetric coordinateson the lines ofScardovelli and Zaleski[12].WeuseYoungsmethod [9] togettheinterfacenormal(minequation (1))fromfluid-1(C=^{1)tofluid-2C}=⁰ andisgivenbym= −∇^C/|∇^C|^{.TocompletethePLICinterface}reconstruction foragivenC,inadditiontothenormalm,

Fig. 1.Coordinatesoftheverticesofasimplepolygoncutbytheinterfaceorderedcounterclockwise.Herex3,y3andx4,y4representstheendpointsof thePLIClinesegmentwithinterfacenormalmandlineconstanta.

Table 1

Resultsforrelativeerrorinvolumeduringinitializa- tionofatorusofradius,rt=0.25,fordifferentgrid sizes.

Relative error in volume:E=^|V−Vt^Vt|

Grid Current Solver

16×16 5.4×10⁻¹⁶

32×32 7.1×10⁻¹⁶

64×64 1.8×10⁻¹⁶

128×^{128 5}.4×¹⁰⁻¹⁶

Fig. 2.Reconstructedinterfacefor16meshpointsofatorusofradiusrt=0.25 andmajoraxisr=0.50 initialized atthecenterof1×1 domainwhere reconstructedPLIClinesegmentsforeachmixedcellisdepicted.

we alsoneedtoobtain thelineconstant,a,whichis thenormaldistanceofthe interfacefromoneofthe verticesofthe computationalcell.Inwhatfollows,wepresentamethodologytogetthelineconstant(a)analyticallyforagiveninterface normalvectorandthevolumefractionofamixedcell.

As discussed in[12],using an analytical relationbetween thevolume fraction(C),interface normal(m) andthe line constant (a), we can implement an i f −^else−^{i f} −^end−^{i f constructto determine the line constant, a. Thisapproach} iscomputationally muchmoreefficientcompared tothealternative iterativeapproach togetthelineconstant.Giventhe equationoftheinterface,m1x+^m2y=^a,allcombinationsofm1,m2(suchthatm²₁+^m22=^{1)canbereducedtooneofthe} cases showninFig.3, eitherby changingthe originorby changingthe referencefluidfromfluid-1 to fluid-2,such that bothm₁ andm₂arepositiveandtheleftbottomcornerofthemixedcellunderconsiderationiscontainedinfluid-1.Fig.3 showsallthepossibleconfigurationsforinterfacearrangementwithm1≥^{0 andm}2≥^0.

Fig. 3.Variouscaseswhichcanariseinthestandardconfigurationoftheinterfacewithm1,m2≥0 andfluid1 isoccupyingthebottomleftcornerofthe cell.

WefirstdiscussCaseAshownintheFig.3andsimilarprocedurecanbefollowedtoobtainrelationsforothercases.For theaxisymmetriccoordinatesystem,theshadedareashownintheFig.3forCaseAisgivenby,

V

= π

3

(

x₀

+

^x2

)

^3m1 m₂

− π

3

(

x₀

+

^x1

)

^3m1

m₂

− π

x²₀y₁

.

(4)

Using theequation oflinem₁x+^m2y=^{a we have x}1=^a/m₁−(m₂y)/m₁ and x₂=^a/m₁.In thepresentstudy,we assumex=y,butthesameanalysiscanbeeasilyextendedforx=y.Substitutingx1andx2intheequation(4) and collectingthetermsinpowersofa,weobtain,

π

y m²₁

a²

+

− π

m₂

y²

m²₁

+

²

π

yx₀ m1

a

+

π

m²₂

y³

3m²₁

− π

y²m₂x₀ m1

=

^V

.

(5)

Thus,wehaveananalyticalrelationbetweenthevolumeandthelineconstanta.Wenotethattheaboverelationholdstrue onlywhentheinterfacecutsthroughthetopandthebottomedgesofthecellshownforCaseAinFig.3:(x2−^x0)≤x, y1=y and y2=^{0.These conditions yield the boundson the values ofa: m}2y≤^a≤^m1x.Substituting the above boundsforaintheequation(5),yieldtheboundsonthelimitingvolumes:

V₁

= π

y

3

x²m²₁

+

³

xm₁

(

2m₁x₀

−

ym₂

) +

ym₂

(

ym₂

−

^3m1x₀

)

3m²₁

,

(6)

and

V₂

= π

y²m₂

(

ym₂

+

^3m1x₀

)

3m²₁ (7)

ForagivenvolumefractionC andinterfacenormal(m1,m2),thevolume occupiedby fluid-1 intheconfigurationsshown intheFig.3isgivenby:V =²

π

rxyCwherer=^x0+x/2 isthedistancefromtheaxisofsymmetrytothecellcenter.

If V1≤^V≤^V2 then theanalytical relation givenby equation (5) can be used to determinethe lineconstant a.Forthe quadratic equation ina givenby equation (5), we note that onlyone of the rootswill satisfy the requiredboundson a forcaseA.ForcasesBandC,we obtaincubicequationsthatcanbe solvedfora usingCardano’sformulaorusingBrent’s methodtofindtheappropriaterootwithnecessaryboundsforthelineconstant.Followingthesameapproachwe canget boundsonvolumeforcaseDas:

V₃

= π

y²

(−

²

ym₁

+

³

ym₂

−

^3m1x₀

+

^6m2x₀

)

3m₂ (8)

and

V4

= π

y²m1

(

y

+

^3x0

)

3m2

.

(9)

Fig. 4shows all the possibleconfigurations of theinterface and thevolume bounds whichseparate each caseasthe interface normalinradial directionvariesfromminimumtoamaximum.Fig.4clearlyshowsthatthevariousboundsfor thecasesshowninFig.3donotoverlapandprovideauniquecriterionforcomputingthelineconstanta.Thus,wecanuse thefollowingalgorithmtoclassifyeachcase.

Welistbelowtheanalyticalrelationbetweenthelineconstant(a)andthevolume(V)foreachcase:

CaseA

π

y m²₁

a²

+

− π

m2

y²

m²₁

+

²

π

yx0

m1

a

+

π

m²₂

y³

3m²₁

− π

y²m2x0

m1

=

^V

.

(10)

Fig. 4.BoundsfordifferentcasesshowninFig.3asafunctionofthex-componentoftheinterfacenormal,m1.Theinterfacialcellisplacedatadistance of1 fromtheaxisofsymmetrywithx=y=^{1.ThetotalvolumeofthecellisV}cell=²πgivenbythetopboundaryintheplot.

Data:V,m1

Result:Identificationofthecaseinwhichthestandardinterfacebelongsto.

ifm1≥^√1₂ ^then if V≥V1 then

CaseB elseif V≤^V2 then

CaseC else

CaseA // V1>V>V2

end else

if V≥^{V3 then} CaseB elseif V≤V4 then

CaseC else

CaseD // V3>V>V4

end end

Algorithm 1:Classification of the standard case of the reconstructed interface wherem1,m2≥0.

CaseB

− π

3m²₁m2

a³

+

π

y m²₁

− π

x0

m₁m₂

a²

+

π (

y

+

x0

)

²

m₂

− π

m2

y²

m²₁

+

²

π

yx0

m₁

− π

x²₀ m₂

a

+

− π

y

(

y

+

^x0

)

²m1

m2

+ π

m²₂

y³

3m²₁

− π

y²m2x0

m1

− π

m₁x³₀

3m2

+ π

m1

(

y

+

^x0

)

³ 3m2

=

V

(11)

CaseC

π

3m²₁m₂

a³

+ π

x₀

m₁m₂ a²

=

^{V (12)}

CaseD

a

=

²

π

y³m1

+

³

π

y²m1x0

+

^3m2V

3

π

y

(

y

+

2x0

)

⁽¹³⁾

WenoteherethattheothercasescanbereadilytransformedintooneofthecaseslistedintheFig.3byeitherchanging thefluid(byusing(1−^{C)insteadofC tocomputethevolumeandinvertingtheinterfacenormalm) orby changingthe} origin(keepingthelocationoftheaxis-of-symmetrysamebutinvertingitsdirection).

Wenowcomparetheabovedescribedanalyticalmethodwiththeiterativemethodforfindingthelineconstantforthe caseA giveninFig. 3.The relativeerrorin thelineconstant is giveninTable 2forthe analyticalanditerativemethods withdifferenttolerances.Wenotethattheiterativemethodtoreachanerrorwithatoleranceof10⁻⁸ isabout28 slower comparedtotheanalyticalmethod.

Table 2

RelativeerrorinlineconstantaandtheratioofCPUtimerequiredbyiterativemethod(Brent’salgorithm)to thatrequiredbyanalyticalmethodfor10000 repetitionsforcaseAshowninFig.3fordifferenttolerancesused intheiterativemethod.

Comparision between analytical and iterative reconstruction methods

Method Tolerance Relative Error titerative/tanalytical

Analytical − ^{0 1}

Iterative 1.0×10⁻⁴ 1.8×10⁻⁴ 14.2

Iterative 1.0×10⁻⁶ 1.1×10⁻⁶ 21.2

Iterative 1.0×10⁻⁸ 6.7×10⁻¹¹ 28.3

Oncethelineconstant,a,isobtained,thepositionoftheendpointsofthelinearapproximation oftheinterfacecanbe computed thuscompleting the construction ofa planar interface ina givencomputational cell. As discussedearlier, this moreprecise descriptionoftheinterface withinthegrid cellallows geometricadvectionwhichgivestheVOFmethodits strictmassconservationpropertywhilemaintainingasharpinterface.

In what follows,we discuss an operator split algorithm forthe geometric advection ofthe interface in axisymmetric coordinates. Wenote that the straightforwardextension ofthe 2D cartesianalgorithmdoesnot yield accurate results,as alsoindicatedbytheresultsobtainedfromtheexistingopensourcecodes.

3. Advectionoftheinterface

We present herea scheme for accurate geometric advection ofthe volume fraction in theaxisymmetric coordinates.

We have used a uniformgrid to describe thevariables with volume fractionbeing storedat the cell centers (Ci,j). The incompressible fluid flow is determined by the velocity field which is defined atthe cell faces (ui+¹/2,j,vi,j+¹/2). Here, u denotes the radial directionvelocityand v is theaxial velocity. Thevelocity field satisfies thediscrete divergencefree conditiongivenby:

(

ru

)

_i+1

2,j

− (

ru

)

_i−1 2,j

ri

x

+

^vi,j+12

−

^v_i,j−¹

2

y

=

0

.

(14)

Motionoftheinterfaceisgovernedbytheadvectionequationforthevolumefractionfield,

∂

C

∂

t

+

^u

· ∇

^C

=

^{0 (15)}

Forincompressiblefluids,conservationoftheindividualvolumesofthetwofluidsresultsintheconservationofmass.

In the volume offluid method,geometric advection ofthe volume fractionfield is employed instead of using a higher orderscalaradvectionschemetosolveequation(15) since itlimitsthediffusionofvolumefractionfield thusmaintaining a sharpinterface andatthe sametime ensuresthe conservationofmasstohighdegree ofaccuracy [2].Givenavolume fractionfield,reconstructedinterfaceandsolenoidalvelocityfield,wecansolvetheequation(15) usinganoperatorsplitting algorithm consistingofan r-sweep anda y-sweepfollowing[15].Inordertoemploy anoperator splittingalgorithm,the advectionoftheinterface(equation (15)),using∇ ·^u=^{0,canbewrittenas:}

∂

C

∂

t

+ ∇ · (

uC

) =

C

(∇ ·

u

).

(16)

The RHS in equation (16) is the dilatation term or the divergence correction term and is needed in an operator split approachsinceineachstepofanoperatorsplitalgorithmthevolumefractionC isadvectedfirstinonedirectionandthen inthe otherdirection(s). Thediscrete divergencetermorthedilatationtermisincludedinthe formulationsince ineach step ofan operatorsplitalgorithm thevolume fractionC isadvected ina one dimensionalflowwhich isnot divergence free.Givenavolumefraction(Cⁿ_i,j)andvelocityfield(uⁿ_i+1/2,j,vⁿ_i,j+1/2) atthenth timestep,thediscretized equation(16) isgivenby,

Cⁿ_i,⁺_j¹

=

^Cn_i,j

+

t r_i,j

x

δ

V_i−1/2,j

− δ

V_i+1/2,j

+

t

y

δ

V_i,j−1/2

− δ

V_i,j+1/2

+

Cⁿ_i,j

t ri,j

x

ri+¹/2,juⁿ_i+1/2,j

−

ri−¹/2,juⁿ_i−1/2,j

+

t

y

vⁿ_i,j+1/2

−

vⁿ_i,j−1/2

(17)

where δVi+¹/2,j=(ruC)ⁿ_i+1/2,j is the amount of volume fraction fluxed through the right cell face. Similarly, fluxes δVi−¹/2,j,δVi,j+¹/2 andδVi,j−¹/2canbecomputedforothercellfaces.

Usingoperatorsplitting,wecansplittheaboveequationasfollows:

C^∗_i,j

=

^Cn_i,j

+

t r_i,j

x

δ

V_i−1/2,j

− δ

V_i+1/2,j

+

^Cc_i,j

t r_i,j

x

r_i+1/2,juⁿ_i+1/2,j

−

^ri−1/2,juⁿ_i−1/2,j (18)

Cⁿ_i,⁺_j¹

=

^Ci^∗,j

+

t

y

δ

V_i,j−1/2

− δ

V_i,j+1/2

+

^C_i^c_,j

t

y

vⁿ_i,j+1/2

−

^vn_i,j−1/2 ⁽¹⁹⁾

where C_i^∗_,j istheintermediate valueofthe volumefraction. Here, C_i^c_,j isafunction basedonwhich wecan getdifferent operator splitting approaches. If C^c_i,j=^C∗_i,j ^{then we get Sussmanand Puckett’s operator split} approach [15]. An implicit schemeisusedinthefirstdirectionandanexplicitschemeintheseconddirectiontomaintaintheconservationofvolume fraction [16].The orderofsweepofdirectionisalternatedevery timestep[17] (“Strangspliting”)to achievesecond order accuracyintime.

Ifweuse, C^c_i,j

=

_{1 C}n i,j

≥

⁰

.

5

0 Cⁿ_i,j

<

0

.

5 (20)

whereC_i^c_,jdependsonlyontheprevioustimestepvalueofvolumefractionwegettheaxisymmetricformofWeymouth andYue’soperatorsplit approach [18].Thismethodisexplicitinboththesweep directions.Foranoperatorsplit method tobestrictlyconservativeithastofulfill,followingWeymouthandYue [18],asetofconcurrentrequirements,

1. Thevolumefluxtermsareconservative,and 2. thedilatationtermsumstozero,and

3. noclippingorfillingofacellisneededtoimpose0≤^C≤^1.

Usinggeometricadvection(describedbelow)forvolumefluxingandadivergencefreevelocity field,bothSussmanand Puckett’sandWeymouthandYue’smethodssatisfythefirsttworequirements.SussmanandPuckett’smethoddoesrequire clipping orfillingofacell toimpose 0≤^C≤^{1 andthereforeisnotstrictly} conservative.Howeverthe massconservation errorsintroducedinthismethodaresmallandthusithasbeenwidelyadoptedpreviously.WeymouthandYue [18] proved that,byimposingagridCourantnumberrestrictionof,

t

N

d=¹

^ux^d_d

<

¹

2 (21)

in their method one can satisfy thethird requirement forthe operator split approachto be conservative. Here N is the number ofsweep directions and u_d and x_d are the maximum velocity andgrid size in each of those directions.Thus by usingWeymouth andYue’soperator splittingapproachwe canpreserve themass exactlywhenadvecting thevolume fractionfield.

The volumefluxthrough cellfaces,δV_{cell−f ace},iscomputedgeometrically.Consider theschematic inFig.5,wherethe shadedregionshowsthevolumeoffluid-1 inthecelltobefluxed throughtherightface(δVi+¹/2,j).Consideringtheface velocity(ui+¹/2,j)tobepositive,thefluxcanbecomputedas,

δ

V_i+1

2,j

= (

ru

)

_i+1/2,jV

2

π

r

r

y (22)

where V isthevolume offluid 1 fluxedthroughtherightface(shownastheshaded regioninFig.5),r isthedistance in the radial directionwhich contains the volume advected in this timestep and r is the distance to the center of this volumefromtheaxisofsymmetry.HerethevolumeV dependsontheorientationoftheinterface.Wecancalculaterby considering theconservationofvolume fluxedthrough therightface andsolving theresultingquadratic equation,which yields,r=^r_i+¹

2,j− r²

i+¹2,j−^2r_i+¹

2,juet.Usingthesectionofthepiece-wisereconstructedinterfacelyinginthevolume tobefluxedthroughthecell-faceovert time-stepandemployingtheGaussareaformula,givenbyequation(2),wecan calculatethevolumecutbythisregion.

TheabovesmallcorrectionincomputingralongwiththeaccurateGaussformulaforaxisymmetricsimulationsallows us to improve upon the existing volume fraction advection schemes. Existing schemes [15] modify the velocity in 2D algorithms by using ru for velocity field and use 2D geometric advection scheme which results in a third order error (O(t²h)),wherehisthegridsizeandt isthetimestep)inmassconservation.Toillustratethisgeometricfluxingerror, consider the radial advection of the volume fraction field for a full cell (C=^{1) as shown in Fig. 6. The shaded region} representsthevolumeadvectedthroughtherightfaceofthecellwitharadialcellfacevelocityue.

The existing schemes compute the volume as V_c=²

π

(r_e−^ue₂^t)(u_et)y,where r_e is the distanceof theeast cell facefromtheaxisofsymmetry,tisthetimestep,andy istheheightofthecell.Whereas,theproposedschemeyields

Fig. 5.The fluxed volume through the right face of the cell whenui+1/2,jis positive.

Fig. 6.The shaded region is the volume advected through the east cell face in a single timestep.

theexactvolume,V =²

π

(r_e−₂^r)ry withr=^re−

r²_e−^2reu_et.Thus,theerrorinvolumecalculationisgivenby, E=

π

(u_et)²y.

Wevalidate theproposedmodificationswiththefollowingtestcasesandcomparewiththeresultsobtainedusingthe opensourcemultiphaseflowsolverGerris [19] whichusesasimplerextensionofthecartesianoperatorsplitapproachfor geometricfluxing.

3.1. Advectionofatorus

Inthistestcase,atorusofradius0.25 isinitialized at(0.35,0.5)inacomputationaldomainofsize(2.0,1.0).Thetorus is advectedunderthe steadystate velocityof u=⁰.1/r forr>0.05 and v=^{0,where r isthedistancefromthe axisof} symmetry. The fluid is advected forward in time t=⁷.8125 and then the velocity is reversed so asto advect the fluid backwardsintime tilltime t=¹⁵.625.Inordertoillustratethe effectofaccurate geometric fluxing,we advectthetorus using SussmanandPuckett’s method fora grid Courant numberof 1 andcompare theresults withthat obtained using Gerris flowsolver. For a grid sizeof x=y=¹/128, thisresultsin a time step oft=⁰.0078125. Thus the fluid is advected a 1000 timestepsforward intime andthen backwards.As seen fromtheFig. 7,after1000 timesteps thetorus ishighlycompressedduringtheadvectionaslessarea(duetoaxisymmetry)occupiesthesamevolumeaswemoveaway fromtheaxisofsymmetry.Wenotethatthefinalinterface shapematchesverywellwiththeinitialpositionofthetorus, thusvalidatingouralgorithm.

Therelativeerrorinthevolumebetweentheinitialandfinaldistributionoffluid-1 forvariousnumberofforwardand backwardadvectiontimestepsisgiveninTable3.Thecorrespondingrelativechangeinthevolumeobtainedforthesame testcasesimulatedusingGerrisflowsolverisalso givenforcomparison.Wenotethattheerrorobtainedfromthepresent schemesishighly accurateincomparisontothoseobtainedfromGerris(Table3).

WerepeatthistestfordifferentCourantnumbersusingbothSussmanandPuckett’sandWeymouthandYue’sadvection schemes andcomparethe resultswiththoseobtainedusingGerrisflow solver.Table 4givestherelativeerrorinvolume

Fig. 7.Interfaceshapeafter1000 timestepsforwardandbackwardafteradvectingthetorusofradius0.25 withagridCourant(CFL)numberof1 advected usingSussmanandPuckett’smethodforgridCourantnumberof1.

Table 3

Resultsforrelativeerrorinthevolumeforatorusofradius0.25 advectedinradial directionforwardandbackwardintimefordifferentnumberoftimesteps.

Relative error in volume

Number of timesteps CurrentSolver (SussmanandPuckett)

Gerris Solver

1 5.2×^{10−15 1}.1×¹⁰⁻⁵

10 2.9×^{10−14 7}.2×¹⁰⁻⁵

100 4.8×^{10−13 2}.1×¹⁰⁻⁴

1000 3.4×^{10−10 3}.8×¹⁰⁻⁴

Table 4

Resultsforrelativeerrorinthevolumeforatorusofradius0.25 advectedinradialdirectionforwardandback- wardintimefordifferentgridCourantnumbers.

Effect of Courant number on relative error in volume

Grid Courant number Gerris Solver CurrentSolver (SussmanandPuckett)

CurrentSolver (WeymouthandYue)

0.1 9.5×10⁻⁵ 1.4×10⁻⁹ 1.9×10⁻¹²

0.2 3.9×10⁻⁶ 1.9×10⁻¹⁰ 2.4×10⁻¹²

0.3 1.3×^{10−4 7}.1×^{10−11 2}.6×¹⁰⁻¹³

0.4 2.4×^{10−4 1}.0×^{10−10 2}.5×¹⁰⁻¹⁵

forvariousgridCourantnumbers.Weobservethat usingWeymouthandYue’soperatorsplittingapproachalongwiththe interface reconstruction algorithm presented we can obtain machine accurate volume conservation.We observe a small increaseinerrorasCourantnumberdecreaseswhichcanbeattributedtothe largenumberoftimestepswhichresultina largernumberofinterfacereconstructioneventsthusleadingtolargererroraccumulation.

As suggestedby Kotheetal. [20],simplelinearadvectiontest casesdonotreveal theefficacyofadvectionalgorithms appropriately.Thus, we furthertest theefficacyofthealgorithm by subjectingitto amore severetest caseofadvection ofatorus inaHill’svortex [21] withasuperimposedradial velocityfield whichresultsina divergencefree velocityfield uptomachine-precision.Thisisaxisymmetricequivalentofthecircleinavortextestcasefor2Dcases[20].Forthisvelocity field,theinterfaceundergoesstrongtopologicalchangesincludingfragmentationandmergingduetostrongsheareffects.A torusofradius0.1 isinitialized at(0.2,0.8)inacomputationaldomainofsize(1.0,1.0)withL=⁰.5.Thefluidisadvected underhighlystrainedsteadystatevelocityfieldgivenby(forr≥⁰.05)

u

=

0

.

1

r

L (y−L)

L

+

^0._r⁰⁵ (23) v

=

⁰

.

1

1

−

y−^L L

₂

−

²

_r

L

2

.

(24)

The fluidis advectedforwardintime t=⁴.0 and thenthevelocity isreversed soastoadvect thefluid backwardsin time tilltime t=⁸.0.Fora gridsizeofx=y=¹/128,andgridCourantnumberof0.1 thisresultsina time stepof t=¹.0×^{10−3. Thefluid is advected} 4000 timestepsforwardin time andthenthe velocityis reversed toadvect 4000 timestepsbackwardsintime.

Fig. 8.Interfaceshapeafter4000 timestepsforwardandbackwardafteradvectingthetorusplacedinavortexusingWeymouthandYue’soperatorsplit approachforagridCourantnumberof0.1.

Table 5

Resultsforrelativeerrorinvolumeforatorusofradius0.1 placedinacomplexvelocityfieldadvectedforwardandbackwardintimefordifferentgrid Courantnumbers.

Effect of Courant number on relative error in volume

Grid Courant number Gerris Solver CurrentSolver

(SussmanandPuckett)

CurrentSolver (WeymouthandYue)

0.1 2.7×10⁻⁴ 3.5×10⁻⁸ 2.7×10⁻¹²

0.2 1.6×10⁻⁴ 3.4×10⁻⁷ 2.2×10⁻¹⁵

0.3 8.1×10⁻⁵ 7.6×10⁻⁷ 4.1×10⁻¹⁵

0.4 5.4×10⁻⁴ 4.2×10⁻⁶ 1.9×10⁻¹⁵

Fig. 9.Interfaceshapeafterevery1000 timestepscomparedagainstthetrajectoriesofLagrangianmarkerparticlesforsimulationusingagridCourant numberof0.1 usingWeymouthandYue’soperatorsplittingapproach.

As seenfromFig. 8after4000 timesteps theshape oftheinterface ishighly distorted duetohighly strainedvelocity field.Thefinalinterfaceshapematchesverywellwiththeinitialpositionofthetoroidthusvalidatingouralgorithm.

WerepeatthistestfordifferentCourantnumbersusingbothSussmanandPuckett’sandWeymouthandYue’sadvection schemes andcomparethe resultswiththoseobtainedusingGerrisflow solver.Table 5givestherelativeerrorinvolume forvariousgridCourantnumbers.Weobservethat relativeerrorusingSussmanandPuckett’soperatorsplittingapproach isanordersmallercomparedtotheresultsfromGerrisflowsolverevenforlargernumberoftimesteps.UsingWeymouth andYue’sschemeweobservemachineaccuratevolumeconservation.

Fig. 10.Interfaceshapeaftertimet=^{1.Theinterfaceisflattened againstthetopwallandstretcheddue totheunderlyingvelocity.Evenwhenthe} cross-sectionalareachangesthevolumeofthetoroidalbubbleismaintainedveryaccurately.

Table 6

Resultsforrelativeerrorinthevolumeforabubbleinstagnationpointflowfortf inal=1.0 fordifferenttoleranceofPressure-Poissonsolverforboth SussmanandPuckett’sandWeymouthandYue’sadvectionmethods.

Toleranceof PoissonSolver

Divergenceof velocityfield

Relativeerror (SussmanandPuckett)

Relativeerror (WeymouthandYue)

1.0×10⁻⁴ 1.1×10⁻⁴ 1.1×10⁻⁴ 2.3×10⁻⁵

1.0×10⁻⁶ 5.1×10⁻⁵ 1.1×10⁻⁶ 1.5×10⁻⁶

1.0×10⁻⁸ 2.6×10⁻⁷ 2.5×10⁻⁶ 2.3×10⁻⁹

1.0×10⁻¹⁰ 2.67×10⁻⁹ 2.6×10⁻⁶ 4.8×10⁻¹⁰

1.0×^{10−12 1}.5×^{10−11 2}.6×^{10−6 1}.7×¹⁰⁻¹¹

1.0×^{10−14 7}.5×^{10−13 2}.6×^{10−6 1}.4×¹⁰⁻¹¹

InordertoillustratetheaccuracyofthereconstructedinterfacewepopulatetheinitialinterfacepositionwithLagrangian marker particles andadvect them withthe underlying velocity witha fifth order accurate time integrator [22] fortime t=⁴.0. Wecomparethe final positionsofthe markerparticleswiththat oftheadvected interface usingWeymouth and Yue’sadvectionschemeandfora gridCourantnumberof0.1.The Lagrangianmarkerparticles matchthe positionofthe reconstructedinterfaceverywellasshowninFig.9thusdemonstratingtheefficacyofthealgorithm.

3.2. Bubbleinastagnationpointflow

Inthistest caseweimplementtheVOFalgorithmpresented inthispaperfora morecomplexflow. Wesolve Navier- Stokesequationsinonefluidformgivenby:

ρ (

C

) ∂

u

∂

t

+

^u

.∇

^u

= −∇

^p

+ ∇ · μ (

C

)

∇

^u

+ (∇

^u

)

^T

+ ρ (

C

)

g

+

^fγv (25)

whereuandparethevelocityvectorandpressure,respectively,

ρ

(C)and

μ

(C)arethefluiddensityandviscositywhich areafunctionofvolumefractionfield,C.WeuseChorin’sprojectionmethod [23] tosolvetheaboveequation(25) where wediscretized theadvectiontermusingasecondorderENOscheme [24] andthediffusiontermsusingcentraldifferencing.

Surface tension forces, fγ

v are acting only at the interface and have been modeled as volumetric body force using the continuumsurfaceforcemodelofBrackbill,Kothe,andZemach [25].TheinterfaceiscapturedusingCLSVOFalgorithmgiven bySussmanandPuckett [15].Thisalgorithmismassconservingandcalculatesthecurvatureandsurfacenormalwithhigh accuracywhichisusedforsurfacetensionforcecalculation.Theinterfaceisadvectedbysolvingtheadvectionequationsfor level-setfunction, φ,andvolumefraction, C.We initializeatoroidalbubbleofradius 0.1 at (0.2,0.5)inacomputational domainofunitsize,1×^{1.Thebottomboundaryhasaninletvelocityofunityintheupwardaxialdirectionandtheright} boundary hasoutflowboundaryconditions.The topboundary actsasa rigidwall withno-slipandimpermeable surface.

The densityratioandviscosityratiois10 withthe Laplacenumberofthebubble,La= ^ρ_μ^D2^σ =⁰.048.Theincomingaxial velocitydragsthebubbleandflattens itagainstthetopwall,stretchingitintheaxialdirectionconsiderably(see Fig.10).

Even though the fluid interface undergoes drastic change in its shape, the volume is conserved with a high degree of accuracy.

AnaccuratelycomputeddivergencefreevelocityfieldisessentialforaccuratevolumeconservationinVOFadvection.In numericalsimulations ofincompressiblemultiphaseflows usingprojectionmethod,thisisachievedbysolving apressure Poissonequation toahighdegreeofaccuracyby settinga smalltolerancevalueforsolvingthe matrixsystem.Theorder

oferrorinthedivergencefree conditionforvelocityisoftheorderofthesettolerancevalue.Solving Poissonequation is the mostexpensivestep inthe entiresimulation [2], thus studyingthe effectoflimiting toleranceforPoisson solveron theconservationofvolumeinVOFmethodsisrelevanttodeterminetheaccuracy ofthe simulation.Using theabovetest caseforagridsizeof ₁₂₈¹ andtimestepof5.0×^{10−4,westudytheeffectofaccuracyof}discretized velocityfieldinmass conservationofthetwoVOFadvectionschemespresentedinthispaper.Thedivergenceofthevelocityfieldisthesumof eachdiscretized valueofdivergenceofvelocityatthecellcentergivenby(14) foreachcomputationalcell.

Table 6gives theeffect ofdivergenceof discretized velocityfield in volume conservationforVOFadvectionschemes.

SussmanandPuckett’soperator splitapproachresults infairlyuniformerrorinvolume conservationforlowertolerances sincetheerrorduetoclippingandfillingofthecellstomaintaintheboundednessofvolumefractionfieldcontributesthe mosttothe error.Asthe divergencefreenature ofthevelocity field deterioratestheerrorinvolume conservationscales downwiththetolerance.DuetotheverysmallgridCourantnumberusedinthesimulationeven forahighly divergence freevelocity fieldthereissmallerrorinthevolumeconservationwhenWeymouthandYue’soperatorsplittingmethodis employed.TherelativeerrorscalesuniformlywiththetoleranceandisbetterthanSussmanandPuckett’smethod.

4. Conclusions

In the presentwork, we havepresented severalimprovements for the implementationof volume of fluid methodin axisymmetriccoordinates.Wehavepresentedanalyticalrelationsforthereconstructionofpiecewiselinearinterfaceinax- isymmetriccoordinatessimilartothosegivenbyScardovelliandZaleski [12] forcartesiancoordinates.Theproposedscheme substantially reducesthecomputationalcostin comparisonto theiterativeschemes usuallyemployed forthereconstruc- tion. Further, we showed that for axisymmetric coordinate system, machine-precision advectionof volume fraction field canbeachieved.Weillustratedtheimprovementsbycomparingtheresultswiththepopularopensourcemultiphaseflow solverGerris.Finally,wewouldliketonotethatsimilarmodificationsintheadvectionschemeforvolumeoffluidmethod inother curvillinearcoordinatesystems(suchasellipticcoordinates)canbederived usingtheapproachpresentedinthis work.

CRediTauthorshipcontributionstatement

AnanthanMohan: