Enhancement of a 2D front-tracking algorithm with a non-uniform distribution of Lagrangian markers

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_{: Febres Soria, Mijail and Legendre, Dominique}

Enhancement of a 2D front-tracking algorithm with a non-uniform distribution

of Lagrangian markers. (2018) Journal of Computational Physics, vol. 358. pp.

173-200. ISSN 0021-9991

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Enhancement

of

a

2D

front-tracking

algorithm

with

a non-uniform

distribution

of

Lagrangian

markers

Mijail Febres,

Dominique Legendre

∗

InstitutdeMécaniquedesFluidesdeToulouse(IMFT)–UniversitédeToulouse,CNRS-INPT-UPS,Toulouse,France

a

b

s

t

r

a

c

t

The 2D front tracking method is enhanced to control the development of spurious velocitiesfornon-uniformdistributionsofmarkers.ThehybridformulationofShinetal. (2005)[7]isconsidered.Anewtangentcalculationisproposedforthecalculationofthe tensionforceatmarkers.Anewreconstructionmethodisalsoproposedtomanage non-uniform distributions of markers. We show that for boththe staticand thetranslating sphericaldrop testcasethespuriouscurrentsare reducedtothemachine precision.We also show thatthe ratio ofthe Lagrangian gridsize 1s overthe Eulerian gridsize 1x hastosatisfy1s/1x>0.2 forensuringsuchlowlevelofspuriousvelocity.Themethodis foundtoprovideverygoodagreementwithbenchmarktestcasesfromtheliterature.

1. Introduction

Numerical simulations of two-phase flows requires the resolution of moving/deforming fluid interfaces. Among the methods reportedin theliterature, thefront-tracking methodpresentsitself asa promisingalternative. Oneof themain advantages of the front-tracking method for the treatment of interfaces is that the surface tension contribution can be simplified.Indeed,insteadofcalculatingalocalcurvatureasseeninvolume-trackingmethodslikeVOF[1–3] orLevel Set [4–6],the surface tension contributionin the front-tracking method is determined through the calculation of tangential forces onthe edges(or thenodes in2D)of surface elements[7].This ensuresthat the totalforce on anyclosedsurface is zero,whichisbeneficial in longsimulations [8,7].This advantagecomes withsome shortcomings.Fluidinterfacescan move anddeforminsucha waythat markersformingthefrontcantravelinthetangential directiontotheinterface, ac-cumulatinginsmallinterfaceareas,orleavinglargeinterface areaswithoutanymarkers.ThisisshowninFig. 1(a),where a risingbubbleissimulatedonauniformEuleriangridofcellsize

1

x andtheinterfaceisinitializedwithfrontelements ofsize

1

s

= 1

x.Alossofprecision is observedwhenthe densityof markerisreducedbutinstabilitiescan beobserved whenthenumberofmarkersisincreased[9,10](seeFig. 1(b)).Addingorremoving elementsfromthefrontaccordingto some thresholdofelementsize

1

s isarelatively simpleprocedure[11,12].However, itisnot clearhowtheflowisthen affected,andalgorithmiccomplexityisexpectedininterfacemergingandbreak-upproblems,aswellasin3Dcalculations. Controllinganuniformdistributionofmarkersworkswellenoughincasesinvolvinginterfaceswithmoderatedeformation [see forexample thework of[13],where an artificialtangential markervelocity isapplied], butsome perturbations can be observedon thebubblesidesasshowninFig. 2(a). Asimilarbehavior isexpectedwhen insteadofusingan artificial velocity, polynomials/functions arefittedtothe markersandareusedto keepa uniformdistribution ofthefront[14,15]. Also, markers cansuffersmall amplitudemesh-scale oscillationsthat are not physical[14]. Theapplicationofsmoothing

*

Correspondingauthor.

Fig. 1. Front-trackingmethodimplementedinJADIM(presentwork,withoutanyreconstruction/redistribution),appliedtoarisingBubblewithρ1/ρ2=10,

µ1/µ2=10,σ=24.5,g=0.98,Re=35,Eo=10 [following[17]].(a)Bubbleshapewiththeuseof63markers.(b)Zoomontheleftsideofthebubble

whenincreasing×4 thenumberofmarkers.

Fig. 2. BubblecaseshowninFig. 1(a) using63frontmarkerstreatedwith:(a)Artificialtangentialmarkervelocity.(b)Artificialtangentialmarkervelocity combinedwith: Fourthorderfilter[14]; Savitzki–Golay[16]filter.

filters[14,16], seems toalleviate this effect.The combinationofboth artificialtangential markervelocity andsmoothing filterallowstorecoverarelevantbubbleshapeasshowninFig. 2(b).Thesetreatmentsthatseemtosolvethefront-tracking methodissuesweredevelopedinthecontextofuniformdistributionsofmarkersandtheirextensionto3Dproblemsisnot obvious.

Shinetal.[7]introduceda“hybridformulation”treatmentforthesurfacetensioncontributionandreportedareduction ofspurious velocitiesdowntomachineprecision foran uniformdistributionofmarkers.Theyalsoproposed a reconstruc-tionmethodbasedonanoptimumindicatorfunctioncontourtoobtainamass-conservativemethodfornon-uniformmarker distributions.Thisapproach was reportedtobe robustenough tosolve allthe issuesdescribed above.However, their re-construction method produces a non-uniformdistribution of markers which affectsnegatively the abilityof the “Hybrid formulation”toreducespuriousvelocities.

The objectiveofthiswork is topropose a newmethod forthe calculationof thetangents used inthe capillary con-tributionthat guarantiesthereduction ofspurious velocitiestomachine precision evenfora non-uniformdistributionof markers.Wealsoproposeanewfrontreconstructionmethodthatremarkablypreservesthespuriouscurrentsreductionof thehybridformulationclosetomachineprecision. Thepresentwork isorganizedasfollows.Insection 2wedescribethe numericalimplementationof thefront-tracking methodinside thein-housecode JADIM,givingemphasis insection 3to boththenewapproachforthecalculationofthetangents andthe newmethodforthefrontreconstruction. Insection4, wevalidate thenewapproach forthecalculationofthe tangentsusingclassic spurious velocitytestsfromtheliterature: we considerthestaticandtranslating 2D droptest casesandwe analyzetheeffectofthe Laplacenumber, theCapillary

number, thegrid refinement andthetime step onthe spurious velocity development.In section 5, wevalidate the new reconstruction methodusing the same test casesand also considering the oscillating drop test case. We test the whole implementationagainst2Drisingbubblebenchmarksinsection 6.

2. Numericalmethod 2.1. Generalsolver

Forthisstudy,we usethein-housecode JADIM,developedatIMFT.The treatmentofmoving interfacesinJADIMhas been detailedinthe literature[18–22].Consideringtwo Newtonian incompressibleisothermal fluids,no masstransfer at theinterfaceandaconstantsurfacetension,thefollowing1-fluidformulationfortheNavier–Stokesequationswrites:

∇ ·

U

₌

0 (1)

∂

U

∂

t

+ (

U

· ∇)

U

= −

1

ρ

∇

P

+

1

ρ

∇

· 6 +

g

+

Fσ (2)

where U standsforthe velocityfield, P forthepressure field,

6

istheviscous stresstensor, g is theaccelerationdueto gravity, F σ isthecapillarycontribution,

ρ

and

µ

are thelocaldensityanddynamic viscosity,whichare calculatedusing theclassicalVOFfunctionC (C

₌

1 influid1):

ρ

=

C

ρ

1

+ (

1

−

C

)

ρ

2 (3)

µ

₌

C

µ

1

+ (

1

−

C

)

µ

2 (4)

Equations(1)and(2)arediscretizedusingthefinitevolumemethodinastaggeredgridandspatialderivativesarecalculated usinga second-ordercenteredscheme.Athird-orderthree-stepRunge–Kutta schemeisusedtosolvethe advectiveterms timeadvancement,asemi-implicitCrank–Nicholsonmethodisusedtotreatviscoustermsandaprojectionmethodisused to ensure the continuity condition. Besides the CFL condition, thetime step is selectedto satisfy the stability condition imposedbythecapillarycontributioninthemomentumequation[23,24]:

1

tmin

<

1

tσ

=

s

(

ρ

1

+

ρ

2

) 1

x3

8

σ

(5)

2.2. ComputingtheVoFfunctionC infront-tracking

TocalculateC fromagivenmarkerdistribution,weadopttheprocedurefoundin[15,25].Briefly,asetofmarkers iden-tifiedwithindexesk inFig. 3(a) isdistributedalongtheinterfaceformingthe“front”overtheEuleriangrididentifiedwith subscriptsi

,

j.Aroundeachfrontelemente oflength

1

se formedbythepairofmarkersk

−

1 andk,arectangle(typically

B and C inFig. 3(b))ofwidth

1

se andheight

ℓ

canbe constructedusingthenormalunitvector totheelementpointing toboth sidesoftheinterface.Thesigneddistancetothefrontiscalculatedinthecell

(

i

,

j

)

lyinginsideeachrectangleas

¯

¯

di,j

¯

¯ =

¯

¯

xe

−

xi,j

¯

¯

wherexi,j is theEuleriancell centersandxe is theelement center.Thesignofdi,j isdetermined de-pendingonwhichsideoftheinterfacethepointxi,j lies.Additionally, irregularquadrilateralscanbeconstructedforeach

markerlocated inbetweentworectangles onthe convexsideofthe interface(see thequadrilateral A in Fig. 3(b)).They areintroduced becauseregularrectangleswillnotcompletelycoverconvexareas.Insidetheseirregularquadrilaterals,di,j iscalculatedasthedistancefromthecellcenterxi,j tothecorresponding marker.Whenacell centerislocated inmore

thanonerectangletheminimumdistancetothecorrespondingelementsisselectedfordi,j.

Thedistancefunction

φ

totheinterfaceisthendeterminedineachcell

(

i

,

j

)

by:

φ

i,j

=











−

γ

if di,j

<

−

γ

,

di,j if

¯

¯

di,j

¯

¯ ≤ +

γ

,

+

γ

if di,j

>

+

γ

(6)

where

γ

isthewidthofbandformed bythe unionofallquadrilaterals,typically

γ

₌

2

√

2

1

x [see[15]].The height

ℓ

of the rectangleis chosen suchthat 0

<

γ

< ℓ

, sothe bandof width2

γ

is containedinthe unionofall the quadrilaterals constructedbefore.The calculationofthedistancefunction

φ

isrepeatedateachtime stepbutonlyforthecellslying in thevicinityoftheinterface(givenofcoursethataconsistentinitializationof

φ

isprovidedatt

₌

0).Itisworthmentioning thesimplicityofimplementationof

φ

,itsaccuracyandeconomyofcalculation[see[15,25]].

Oncethediscretefieldforthedistancefunction

φ

isknown,theVoFfunctionC isfoundthroughthemollifiedHeaviside functionineachcell

(

i

,

j

)

:

Fig. 3. Fronttrackingmethod.(a)Eulerian/Lagrangiangrid,showingEuleriancellsi,j andelementsformedbyLagrangianmarkersk;(b)Corresponding quadrilateralsconstructionusedforthecalculationofthedistancefunction(forclarity,quadrilateralsononlyonesideofthefrontareshown).

Ci,j

=











0 if

φ

i,j

<

−

ε

,

0

.

5

h

1

+

φi,j ε

+

1 πsin

³

_πφ i,j ε

´i

if

¯

¯φ

i,j

¯

¯ ≤

ε

,

1 if

φ

i,j

>

ε

(7)

where

ε

₌

√

2

1

x isapproximatelyhalfthenumericalthicknessoftheinterface.

2.3.Hybridformulationforthesurfacetensionforce

Thesurface tensionforce F σ

=

σ κ

n

δ

iscalculatedfollowingthehybrid formulationintroduced byShin etal.[7].This methodconsistsin calculatingthen

δ

contributionon theEulerian gridwhile thecurvature

κ

iscalculatedfrom the La-grangianmarkers. F σ isthencalculatedas:

Fσ

= −

σ

ρ

κ

∇

C (8)

Consideringforexampleani

+

1

/

2 faceandan j

+

1

/

2 faceofthestaggeredgrid,thediscretizationofthecapillaryforce inbothdirectionisthen:

Fσi+1/2,j

= −

σ

ρ

κ

i+1/2,j Ci₊1,j

−

Ci,j

1

x (9) Fσi,j+1/2

= −

σ

ρ

κ

i,j+1/2 Ci,j+1

−

Ci,j

1

y (10)

Thecalculationof

κ

isobtainedfromthecapillaryforce F′

σ determinedusingthemakersasfollows.Wecanwrite F′σ as

F′_σ

₌

σ κ

G (11)

wherethediscrete numericalexpressionof F′

σ andG ontotheEuleriangridare expressedintheformofasumoverthe elementse[7].Forexampleatani

+

1

/

2 cellface

F′_σi+1/2,j

₌

X

e feDi+1/2,j

(

xe

)1

se (12) Gi+1/2,j

=

X

e neDi+1/2,j

(

xe

)1

se (13)

where fe isthecapillaryforcecontributionofelemente andDi+1/2,j

(

xe

)

istheDiracdistributionfunction.Theircalculation arenowdetailed.

Thelocalforce fe atafrontelemente iscalculated(in2D)following[8]:

fe

=

Z

1se

σ κ

nds (14)

UsingtheFrenetrelationforthecurvatureofatwo-dimensionalline,

κ

n

_{= ∂}

t

/∂

s,wehave: f_e

₌

σ

¡

tk

−

tk−1

¢

Fig. 4. (a)Localforce featelemente usingtkandtk−1thetangentvectorsatmarkersk andk−1 accordingtoequation(15)(presentwork);(b)Local

force fkatmarkerk usingteandte₊1thetangentsofelementse ande+1 accordingtoequation(16).(c)Twointerfacesdefinedbytwoelementshaving

thesametangents:abovewithauniformmarkersdistributionandbelow,withanon-uniformmarkerdistribution.(Forinterpretationofthereferencesto colorinthisfigure,thereaderisreferredtothewebversionofthisarticle.)

wheretkandtk−1 arethetangentvectorsatmarkersk andk

−

1 (redcirclesinFig. 4)thatdefinethefrontelemente (see

redlineinFig. 4(a)).Thisapproachisalsousedin[11,12,26,8,9,27–29].Anewmethodforthecalculationofthetangenttk atthemarkerpositionisproposedinsection3.1.

Note that there is anotherapproach that consists in calculatingthis force atmarker position [see [30,7,31,32,25,33]]. Considering the difference between the tangents te and te₊1 of the neighbor front elements e and e

+

1 as shown in

Fig. 4(b),theforce fk atthemarkerk iscalculatedas:

f_k

₌

σ

(

te+1

−

te

)

(16)

Although bothformulationsensure that thetotal forceon anyclosedsurface isequally zero,thereisa crucialdifference betweenthetwomethods.Weconsiderthetwointerfaces( A BC )and( A′_B′_C′_{)describedbytwoelementsinFig. 4(c).The}

elements A B and A′B′(respBC andB′C′)havethesameunittangentsbutthecurvatureandsothecapillarycontributions atpoint B andB′aredifferent.However,ifthetangent fB atmarkerB iscalculatedusingequation(16),thentBC

−

tA B

=

tB′C′

−

tA′B′ andbothinterfaces( A BC )and( A′B′C′)willcontributeatmarker B withanidenticalcapillaryforce fB

=

fB′

whiletheyclearlyhavedifferentcurvatures.Theconsequenceofthisproblemwerenoticedbutnotsolvedin[7].

WenowconsiderthediscretizationoftheDiracdistributionfunction.Forexampleatan i

₊

1

/

2 cellface,Di+1/2,j

(

xe

)

is approximatedby[34] Di+1/2,j

(

xe

)

=

1

1

x

1

y

δ

µ

_x i+1/2,j

−

xe

1

x

¶

δ

µ

_y i+1/2,j

−

ye

1

y

¶

(17) where

δ(

r

)

₌











δ

∗

(

r

)

if

_|

r

_{| ≤}

1

,

1

/

2

_{− δ}

∗

(

2

_{− |}

r

_|)

if 1

<

_|

r

_{| <}

2

,

0 if

_|

r

_{| ≥}

2 (18) and

δ

∗

(

r

)

₌

3

−

2

|

r

| +

p

1

₊

4

_|

r

_{| −}

4r2 8 (19)

Cellcenterquantitiesarelinearlyinterpolatedfromthefaces.Inauniformstaggeredgrid,thisisachievedby:

F_xi′_,j

₌

1 2

(

F ′ xi+1/2,j

+

Fxi′−1/2,j

),

F′yi,j

=

1 2

(

F ′ yi,j+1/2

+

F′yi,j−1/2

)

(20) Gxi,j

=

1 2

(

Gxi+1/2,j

+

Gxi−1/2,j

),

Gyi,j

=

1 2

(

Gyi,j+1/2

+

Gyi,j−1/2

)

(21)

Thecurvatureatcellcentersisthencalculatedby:

κ

i,j

=







F′

xi,jGxi,j+F′yi,jGyi,j

σ(G2 xi,j+G2yi,j) if G2_xi,j

₊

G2_yi,j

>

0

,

0 if G2 xi,j

+

G2yi,j

=

0 (22)

178 M. Febres, D. Legendre / Journal of Computational Physics 358 (2018) 173–200 ci,j

=

(

1 if G2 xi,j

+

G2yi,j

6=

0

,

0 if G2_xi,j

₊

G2_yi,j

₌

0 (23)

andthecurvaturesatthefacecenterarerecoveredwith:

κ

i+1/2,j

=

(

κi,jci,j+κi+1,jci+1,j ci,j+ci+1,j if ci,j

+

ci+1,j

6=

0

,

0 if ci,j

+

ci+1,j

=

0 (24)

κ

i,j+1/2

=

(

κi,jci,j+κi,j+1ci,j+1

ci,j+ci,j+1 if ci,j

+

ci,j+1

6=

0

,

0 if ci,j

+

ci,j+1

=

0

(25)

2.4.Advectionofmarkers

Oncemassandmomentumequations (1)–(2)are solved andaconservativevelocity field isobtained, markersare ad-vectedintegratingintime:

dxk

dt

=

Uk (26)

usinganexplicitfirstorder(F O )scheme:

xn_k+1

=

xn_k

+

Uk

1

t (27)

wheren standsforthecurrenttimestep.ThemarkervelocityUkisinterpolatedfromtheEuleriangridthrough:

Uk

=

X

i j

1

x

1

yUi jDi j

(

xk

)

(28)

Forcomparisonpurposes,themarkeradvectionwillalsobesolved insidethemomentumthree-stepRunge–Kuttacycle andeq.(26)willbecalculatedusingthethreeorder(R K 3)scheme:

xm_k,n

₋

xm_k−1,n

1

t

= (

α

m

+ β

m

)

U

m−1,n

k (29)

where

α

m and

β

maretheRunge–KuttacoefficientsoftheNavier–Stokessolver:

α

1

= β

1

=

4

/

15

;

α

2

= β

2

=

1

/

15

;

α

3

= β

3

=

1

/

6 (30)

Thesubscriptm isthecurrentR K 3 stepandUm_k isthecorrespondingintermediatevelocity.TheR K 3 isinitializedwiththe positionandvelocityattimen:x_k0,n

₌

xn_k andU0_k,n

₌

Un_k.Thenewpositionattimen

₊

1 isx_kn+1

₌

x3_k,n.Bydefault,inallthe simulationsreportedinthiswork,themarkersadvectionwillbeperformedusingthefirstorderEulerscheme F O except

whentheRunge–KuttaR K 3 isexplicitlymentioned.

3. Improvementtothemarkermethod 3.1. Newtangentcalculationatmarkersposition

Thecalculationoftangentsatthemarkerspositionhasbeenaddressedintheliteraturebyfitting/interpolatingfunctions onthemarkerspositionandthencalculatingthetangentsanalytically[seeforexample[8,9]].Inthisworkweconsiderthat twoneighborfrontelementse ande

+

1 shareacommoncurvaturecenter,asdepictedinFig. 5.

Bysimplegeometricalconsiderations,thetangenttkatmarkerk canbecalculatedfromthetangentste andte₊1 ofthe

elementse ande

₊

1 consideringthatmarkersk

−

1,k andk

+

1 arelocatedonthesamecircle: tk

₌

q

1

se+1te

+ 1

sete+1

1

s2_e+1

_{+ 1}

s2e

+

2

1

se+1

1

sete

·

te+1

(31)

AsdetailedinAppendix B,relation(31)isan exactgeometricalrelationsonoadditionalerrorfrominterpolationis intro-ducedhere.

Fig. 5. Schematicsofafront.Inred,thediscretefrontdefinedbyelementse ande+1.Theirtangentteandte+1(notshown)areusedinthecalculation

ofthetangenttkatmarkerk.InblackthecircleofcenterC definedbymarkersk−1,k andk+1.(Forinterpretationofthereferencestocolorinthis

figure,thereaderisreferredtothewebversionofthisarticle.)

Fig. 6. Descriptionofthereconstructionprocedure.Theinterfaceshapeisapartoftheshapeofarisingbubblewithρ1/ρ2=1000,µ1/µ2=100,σ=1.96,

g=0.98, Re=35,Eo=125[17](seesection6).Circles1,2,3,4 and5 aregeneratedwiththecurvatureofthefrontelementsab,cd,de,e f and f g

respectively. Markersbeforereconstruction; MarkersafterreconstructionlocatedontheEuleriangrid.(Forinterpretationofthereferencestocolorin thisfigure,thereaderisreferredtothewebversionofthisarticle.)

3.2. Thenewfrontreconstructionmethod

It iswell knownthat markersmay driftalong theinterface duetothe fluidsmotion.Intheliterature, keepinga rea-sonable density ofmarkers on thefront can be achieved by includingand removing front elements givena certain size threshold[see[11,12]].Amoresophisticatedandautomatedprocedurecanbefoundin[9],whereaninterpolatingcurveis usedtoobtainanhomogeneousdistribution.Also,anartificialtangentialvelocitycanbecalculatedandaddedtothemarker velocity toproduce afront withauniformdistribution ofmarkers[see [13,25]].Amore robustprocedure isproposed in [7],where an optimum indicator functionis introduced topreserve the mass andisused tointersect the Eulerianfaces toproduce newmarkers.However, anincrease inspurious velocitiesisthenreportedbecauseofthenon-uniformmarker distributionresultingofthereconstruction.

In this work, we propose a new front reconstruction method with theobjective to keep the property of the marker methodforthereductionofspuriousvelocities.Themethodconsistsinalocalreconstructionoftheinterfaceusingforeach elemente acircle

C

_e ofradius Re determinedusingthepairofmarkersk

−

1 andk thatdefinee.Theintersectionbetween thiscircle

C

_e andtheEuleriangridwill bethebaseofthe reconstructionprocess. Theradius Re isdetermined usingthe FrenetrelationforthelocalcurvatureR−1

e n

=

dt

/

ds.Theradius Re isthencalculatedfortheelemente usingthetangents

tk₋1 andtk atmarkersk

−

1 andk as:

R−_e1

₌

°

°

°

°

tk

−

tk−1

1

s

°

°

°

°

(32)

Fig. 7. ZoomfromFig. 6onfrontelements(a)ab,(b)cd,(c)de and(d)e f . Beforereconstruction; Afterreconstruction.(Forinterpretationofthe referencestocolorinthisfigure,thereaderisreferredtothewebversionofthisarticle.)

Theunittangentstk₋1 andtkarecalculatedusingthenewexpressionproposedinthiswork(relation(31)insection3.1). Thecirclecenterlocationwithrespecttotheinterfaceisdefinedconsideringthesignoftheproductne

· (

tk

−

tk−1

)

.Once

thiscircle

C

_e isdetermined,itisintersectedwiththefaces oftheEuleriangrid.Inpracticetheintersectionisonlymade withtheEuleriancellsinthevicinityoftheelement.Thennewmarkersaredefinedastheintersectingpointlocatedinside the circular arc limitedby the two considered markers k

−

1 and k of element e. Since front elements do not overlap (coalescenceprocessare not addressedin thiswork),the intersectingpoints foreach elementare unique.Fig. 6 showsa typicalreconstructionprocess.Thefrontbeforereconstructionisrepresentedbytheredsquares.Foreachelementsab,cd, de, e f and f g,the circles1, 2,3, 4 and5 areconstructed.The intersectionsbetweenthesecirclesandthe Euleriangrid facesdefinethenewfrontmarkersrepresentedusingbluecirclesinthefigure.

Fig. 7 showsindetail selectedelements andtheir associated circlesused forthereconstruction. The casedepictedin Fig. 7(b) deserves a special treatment. Circle 2 intersectsthe Eulerian gridclose to a corner.In such situation,two new markers(ingreeninFig. 7(b))aregeneratedandtheyformanewfrontelementofshortlength

1

se.Thecloserthecircle maketheintersectiontothecorner,thesmalleristheresultingfrontelement (inanextremecase

1

se

=

0).The effectof havingsmallelementswillbe discussedinsection 5.Athresholdvalue

1

smin isintroducedtocontrol theminimumsize oftheelements.If

1

se

≤ 1

smin themarkersk andk

−

1 arefusedtogethertoavoidsmallelements.Theprocedureforthis fusionissimple.Followingthemethoddescribed aboveacircle

C

_e isdeterminedbasedonthetwo markerstofuse.They arereplaced byan unique markerlocatedatthecenterofthe circulararcbetweenthetwo markers.Fig. 7(b) showsthe resultofsuch fusion.Itwillbeshowninthefollowingthatthereconstruction isnotnecessaryateachtimestep.Wenote

nt thenumberofiterationsbetweentworeconstructions.Itisworthmentioningthattheradius Re inequation(32)isonly usedinthereconstructionprocessandisnotusedinthecalculationofthecapillaryforcecontribution.

4. Validationofthenewtangentcalculation

Ithasbeenreportedin [7]thatthe hybridformulationforthecapillarycontributiondescribed insection 2.3only re-ducesspurious velocities closeto machineprecision when a uniformdistribution ofmarkers isused. Inthis section,we first reproducethe test proposed in [7]for astatic dropand then we extendthe validationto the translating drop test case.Weevaluatethenewtangentformulationproposedinthisworkforthecalculationofthecapillaryforce(combination ofequations(15)and(31)).Inparticular,we testifspuriousvelocitiescanbe reducedtomachineprecision whena

non-Fig. 8. 2Dstaticdroptestcase.Thecapillaryforceiscalculatedusingtheoriginalhybridformulationof[7](equation(16)).Comparisonbetweenauniform ( )andarandomdistributionofmarkers( ).(a)Maximumnormalizedvelocityevolution.(b)Normalizedpressureat y=0.5.(Forinterpretation ofthereferencestocolorinthisfigure,thereaderisreferredtothewebversionofthisarticle.)

Fig. 9. 2Dstaticdroptestcase.Newcalculationofthecapillaryforce(equations(15)and(31)).Comparisonofauniform( )andarandomdistribution ofmarkers( ).(a)Maximumnormalizedvelocityevolution.(b)Normalizedpressureat y=0.5.(Forinterpretationofthereferencestocolorinthis figure,thereaderisreferredtothewebversionofthisarticle.)

uniformdistributionofmarkersisconsidered.Forthetestsreportedinthissection,thereconstructionproceduredescribed insection3.2isnotapplied.

4.1. Thestaticdroptestcase

A 2D drop with radius R

=

0

.

25 is initialized in a regular 50

×

50 grid of size L

×

H

₌

1

×

1. Havingall the fluid propertiessetto1,thecorrespondingLaplacenumberisLa

=

ρ

D

σ

/µ

2

₌

₀

_.

_{5.Forthistest,markersareadvectedusingthe}

firstorderFOtime integrationscheme(equation(27)).ResultsareshowninFig. 8wheretwo distributionsofmarkersare compared.Inblue,theinterfaceisinitializedwithauniformdistributionofmarkerssuchthat

1

s

= 1

x.Inred,theinterface is initialized witha random distributionof markers.The minimumand maximumelement sizes are

1

smin

=

0

.

3

1

x and

1

smax

= 1

x,respectively.The calculation ofthelocalsurface tension force f isfirst performedusingthehybrid method from[7]correspondingtoequation(16).

The maximum velocity Umax inside thedomain isreported inFig. 8(a) usingthe capillarynumber Ca

=

µ

Umax

/σ

as a functionofthenormalizedtime

τ

₌

t

/

p

ρ

D3

_/σ

_{.Whenusing anuniformdistributionofmarkers,a maximumcapillary}

numberclosetomachineprecision isobtained. Theuseofanon-uniformdistributionofmarkersincreasesthemaximum capillarynumberCa upto8ordersofmagnitude.However,thepressureonalinecrossingthecenterofthebubbleseemsto beunaffectedandcorrectlypredictedinbothuniformandnon-uniformdistributionsofmarkers.ThisisshowninFig. 8(b), wheretheradialcoordinateandthepressurearerespectivelynormalizedas

˜

r

_{= (}

x

₋

xc

)/

xc and P

˜

=

P

/(σ/

R

)

where xc is thepositionofthebubblecenter.

Thesametestisnowrepeatedusingthenewmethodproposedhereforthecalculationofthecapillaryforce(equations (15)and(31)).Ascanbe seeninFig. 9(a),inboth uniformandnon-uniformdistributions ofmakers thecapillarynumber isnowclosetomachineprecision.Pressureaty

=

0

.

5 obviouslyshowsnovariationforbothdistributions(seeFig. 9(b)).

The influenceoftheLaplacenumber La onthespurious velocitiesforthenon-uniformdistributionofmarkersis now analyzed. The density

ρ

andthe dynamic viscosity

µ

ofboth fluids are set to 1,while the surface tensionis varied to

Fig. 10. EffectofLa onthespuriouscurrentconvergenceforanon-uniformdistributionofmarkers.(a)Maximumvelocity: ,La=0.5; ,La=12; ,La=120; ,La=1200; ,La=12000.(b)MaximumandRMSvelocities: ,La=0.5; ,La=12; ,La=120; ,La=1200;

La=12000.(Forinterpretationofthereferencestocolorinthisfigure,thereaderisreferredtothewebversionofthisarticle.)

Fig. 11. Static drop test case. Laplace number effect on the pressure jump. Relative error for ,1Pmax; ,1Pav g.

considertheLaplacenumbersLa

=

0

.

5

,

12

,

120

,

1200 and12000. ThecorrespondingresultsareshowninFig. 10(a) where theevolutionofthenormalizedmaximumvelocity isreportedusingthecapillarynumberCa.TheLaplacenumberaffects theinitialevolutionofthevelocity, butinallcasesastabilizationisalwaysachievedwithsimilarvaluesclosetomachine precision. InFig. 10(b) the RMSdimensionless velocity isshown up to

τ

₌

0

.

2.The rateof convergencein time forthe RMSvelocity isquitesimilar tothat ofthemaximumvelocity, its valuebeingone orderofmagnitudesmaller. Notethat resultsclosetomachineprecisionwherealsofoundwithaVOF-CSFmethod[35]andwiththeLevelSetSharpSurfaceForce method[20]butalongertimeforconvergencewasobservedforthesemethods.

Wedefinetheaveragepressurejump

1

Pav g andthemaximumpressurejump

1

Pmax,bothnormalizedbytheLaplace pressure

σ

/

R.Theformerisfoundbytakingthedifferenceoftheareaweightedaverageofthepressureinside(0

.

5

<

C

_≤

1) andoutside(0

≤

C

<

0

.

5)thedrop,whilethelatteriscalculatedbythedifferencebetweenthemaximumandtheminimum pressureinsidethedomain.Therelativeerrorfor

1

Pmax and

1

Pav g aredefinedas

|

1

− 1

PmaxR

/σ

|

and

¯

¯

1

_{− 1}

Pav gR

/σ

¯

¯

, respectively.TheyarereportedasfunctionoftheLaplacenumberinFig. 11.Theerrorin

1

Pmaxhasatendencytodecrease with La,while

1

Pav g it is close to 0

.

01% for any La.A grid test dependence isreported in Fig. 12. The valuesof both theMaximumandRMSvelocitiesremainsclosetomachineprecisionforallthegridconsidered.However,weobservethe growthofthe maximumandRMSvelocitieswithgrid resolution(Fig. 12(a)). Thesametrend isfound forthemaximum pressurejumperror(Fig. 12(b))whiletheerrorintheaveragepressureshowsadecreasewithgridresolution.Thereason oftheobservedlackofconvergencemightbeduetoround-offerrorsthatcanbeimportantatthislevelofprecision.Taking equation(17)asanexample,itisclearthatwhen

1

x

= 1

y

= 1 →

0 then Di,j

→ ∞

.Analytically,thisdoesnotaffect

κ

i,j inequation(22),sincethedenominatorin

1

2ofDi,j vanishesthroughF′_{σ and}G .However,thesuccessiveoperationswith

1

2 mightbringround-offerrors as

1 →

0.Toavoidtheerror,the surfacetensionforce contributionshouldbe rewritten usingDi,jdefinedasDi,j

= δ(

xi,j

−

x_k′

)δ(

yi,j

−

y′_k

)

butthissimplificationonlyworksinuniformEuleriangrids.Theanalysis andcorrectionofround-offerrors(ifpossible)mustbeextendedtoallroutinesinthesolver,ataskthatwasoutsideofthe scopeofthiswork.

Nextweevaluatetheeffectofthetime step

1

t onthereduction ofspurious velocitiesandthe F O and R K 3 schemes

Fig. 12. Staticdroptestcase.EffectofR/1x (1τ=0.15).(a) ,MaximumVelocity; ,RMSVelocity.(b)Relativeerrorfor ,1Pmax; , 1Pav g.

Fig. 13. Staticdroptestcase.Effectof1t (R/1x=12.5).(a) ,MaximumVelocity(F O ); ,RMSVelocity(F O ); ,MaximumVelocity(R K 3); ,RMSVelocity(R K 3).(b)Errorfor ,1Pmax(F O ); ,1Pav g(F O ); ,1Pmax(R K 3); ,1Pav g (R K 3).(Forinterpretationofthe

referencestocolorinthisfigure,thereaderisreferredtothewebversionofthisarticle.)

wherethetime stephasbeennormalizedusingthecharacteristictime as

1τ

_{= 1}

t

/1

tσ where

1

tσ isdefinedineq.(5). Forall

1τ

tested,bothmaximumandRMSvelocitiesdecreasewith

1τ

andstabilizetoavalueclosetomachineprecision.

F O and R K 3 schemes give similar spurious velocitiesexcept for the larger time step considered (

1τ

≈

1

.

8) where the difference is about 2 orders ofmagnitude. Thisis expected giventhat R K 3 is a highorder scheme ofadvection so the difference betweenthe two schemes willbe noticeable whenincreasing the time step. Thereis no noticeabledifference between F O andR K 3 fortherelativeerrorsofthepressurejump

1

Pmax and

1

Pav g (Fig. 13(b)).Therelativeerrorforthe maximumpressure jump

1

Pmax showsaclearconvergencewiththereductionofthetime stepwhereas theerrorinthe averagepressureisalmostconstant(

_≈

0

.

01%).

4.2. Thetranslatingdroptestcase

The translatingdroptest caseisamoredifficultconfigurationforthe controlofspurious velocities[35,24,20].Indeed, thecombinedeffectofthecurvaturecalculationandtheinterfaceadvectioncontrolsthedevelopmentofspuriouscurrents [20].Weconsideratranslatingdropofradius R

=

0

.

2 insideaunitsquare domaininitializedwithauniformvelocityfield

U0 (Fig. 14).Onthehorizontalboundariesthesymmetryconditionisimposed,whileontheverticalboundariesaperiodic

condition isprescribed.ThefluidspropertiesarevariedtotestdifferentLaplaceLa andCapillaryCa numbers,keepingthe samedensityandviscosityinsideandoutsidethedrop.Markersareinitiallyrandomlydistributedonthedropsurfacewith elementsizesrangingbetween

1

smin

=

0

.

3

1

x and

1

smax

= 1

x.ThetimescaletU

=

D

/

U0 isusedtodefinethenormalized

timeas

τ

₌

t

/

tU.

Inthereferenceframeofthetranslatingdrop,thevelocityshouldremainszero.So,wedefinethenormalizedvelocities

˜

Umax

=

max

(

|

U

−

U0

|)/

U0andU

˜

R M S

= |

UR M S

−

U0

| /

U0tocharacterizethedevelopmentofthespuriouscurrents.The

evo-lutionofU with

˜

timeisshowninFig. 15.Themaximumvelocityrapidlystabilizestovaluesclosetomachineprecisionfor alltheLaplacenumbersLa considered.TheRMSvelocityhoweverdoesnotperfectlyconvergewithtime,atrendthatwas

Fig. 14. Domain definition for the translating drop test case.

Fig. 15. Translatingdroptestcase.TimeevolutionofthemaximumvelocityU˜maxfor: ,La=1.2; ,La=120; ,La=1200; ,La=12000;

andfortheRMSvelocityU˜R M Sfor: ,La=1.2; , La=120; ,La=1200; ,La=12000,whilekeepingW e=0.4 andR/1x=12.8. (Forinterpretationofthereferencestocolorinthisfigure,thereaderisreferredtothewebversionofthisarticle.)

Fig. 16. Translatingdroptestcase.EffectoftheLaplacenumberLa.(a) ,MaximumVelocityU˜max; ,RMSVelocityU˜R M S.(b)Relativeerrorfor

,1Pmax; ,1Pav g,whilekeepingW e=0.4 andR/1x=12.8.

noticedin[35] whereit isstatedthat advectioninducesperturbation onthe shapeofthe interface(hence thecurvature calculation).Fig. 16(a) showstheeffectoftheLaplacenumberonthemaximumandtheRMSvelocitykeepingaconstant WebernumberW e

=

ρ

U2

0D

/σ

=

0

.

4.ThemaximumandRMSvelocitiesbothincreasewiththeLaplacenumber,their

max-imumvaluesbeingU

˜

_≈

1

×

10−13 andU

˜

R M S

≈

1

×

10−14,closetomachineprecision. Therelativeerrorforthemaximum pressurejumpdecreaseswithLa,whiletheaveragepressurejumpisalmostconstant(Fig. 16(b)).

TheeffectofthegridrefinementisshowninFig. 17forW e

=

0

.

4 andLa

=

12000.Thesimulationsareconductedwith thetimestep

1τ

₌

0

.

15 imposedby thestabilitycriteria(5)forthefinestgrid(R

/1

x

=

50).Fig. 17reportsforboththe

Fig. 17. TranslatingdroptestcaseforW e=0.4 andLa=12000.EffectofR/1x (1τ=0.15).(a) ,MaximumU ;˜ ,RMSU .˜ (b)Percenterrorfor ,1Pmax; ,1Pav g.

Fig. 18. Translating drop test case. Normalized pressure profile for R/1x=50, La=12000 and W e=0.4 at: (a)τ=0.0804; (b)τ=1.6089.

velocity andthe pressure the same behavior as observed forthe static drop test case. The spurious velocity magnitude andtheerrorofthemaximumpressuredonotdecreasewiththegridrefinementasdiscussedinsection4.1.Howeverboth maximumandRMSvelocitiesremainclosetomachineprecisionwithvalueslessthan7

_×

10−13and4

_×

10−14,respectively. Notethat thislevelofspurious velocityissignificantlylower thanthosereportedin[35]and[20].The pressureprofileis alwayssharpandnoperturbationcouldbedetectedasshowninFig. 18.

We consider the effectof the time step

1

t in Fig. 19(a) for La

=

12000, W e

=

0

.

4 and R

/1

x

=

12

.

8. Here the F O

and R K 3 schemesforthemarkeradvectionprovidedifferentlevelofspuriousvelocities.Fortheerrorinthepressurethe behaviorissimilartotheoneobservedforthesteadydroptestcase.Thedifferenceforthevelocityismoresignificant.The magnitudeofthespuriousvelocitiesremains toavery lowlevelbutnoclearconvergenceisshownwith

1τ

.Infact,the maximumofspuriousvelocityisobservedforthesmallest

1τ

≈

0

.

01 forboththemaximumandRMSspuriousvelocities. This may be explained with the complex coupling between surface tension force calculation andinterface advection as outlinedin[20].Whenwe reducethetimestep

1

t moretimesteps arerequiredtoachieve thesametimeincreasing the numberofoperationsandtheamplificationoftheperturbations. Forthesamereason,thereisnoadvantage ofusingthe

R K 3 schemewhenmarkersaretransportedwithauniformvelocity.

We end the translating drop test case by considering the effect of the flow capillary number. In practice, numerical simulations areaffectedbythedevelopmentofspuriousvelocitieswhenthecharacteristiccapillarynumberoftheflow is notlargeenoughcomparedtothecharacteristiccapillarynumberofthespuriouscurrents.Forthedroptestcase,wedefine the drop Capillary number as Ca

=

µ

U0

/σ

and the spurious velocity capillarynumbers as

f

Camax

=

max

(

|

U

−

U0

|)

µ/σ

and

f

CaR M S

= |

UR M S

−

U0

|

µ/σ

forthemaximumandRMSspuriousvelocities,respectively.Toprovideacompleteviewof

the method performance, we modify thefluid properties to covera large range ofthe drop Capillary numbersuch that 10−10

_≤

Ca

≤

1.ResultsareshowninFigs. 20forboththespuriousvelocityandtheerrorinthepressurejump.Both

f

Camax and

f

CaR M S are linearlyincreasing withCa buttheir magnituderemainsalways about10ordersofmagnitudelowerthan

Ca.ForCa

=

O

(

1

)

we observethat theerrorinthemaximumpressure jumpissignificantly increasedwhile theerrorin theaverage pressurejumpislessaffected.Fig. 21(a) reportsthegrid testconductedforCa

=

10−10_{,thesmallercapillary}

Fig. 19. Translatingdroptestcase.Effectofthetimestep1t forLa=12000,W e=0.4 andR/1x=12.8.(a) ,MaximumVelocity(F O ); ,RMS Velocity(F O ); ,MaximumVelocity(R K 3); ,RMSVelocity(R K 3).(b)Errorfor ,1Pmax(F O ); ,1Pav g(F O ); ,1Pmax(R K 3);

,1Pav g(R K 3).(Forinterpretationofthereferencestocolorinthisfigure,thereaderisreferredtothewebversionofthisarticle.)

Fig. 20. TranslatingdroptestcasewithR/1x=12.8.EffectofthedropcapillarynumberCa.(a) ,Maximumspuriousvelocity; ,RMSspurious velocity.(b)Relativeerrorfor ,1Pmax; ,1Pav g.

forboth the spurious velocity magnitude andthe error inthe pressure are very similar to those reportedin Fig. 17.In particularthemagnitudeofthespuriousvelocitiesincreaseswiththegridrefinementbutitremainsataverysmalllevel. Wecanconcludethattheperformanceofthemethodisnotsensibleoftheflowcapillarynumber.

5. Assessmentofthereconstructionprocedure

We have demonstrated so far that our new method for the capillary calculation witha non-uniform distribution of markerscanmaintainspurious velocitiesclosetothoseobtainedwithauniformdistribution.Theobjectiveofthissection isnowtoevaluatethenewreconstructionprocessdescribedinsection3.2.Firstwechecktheabilityofthereconstruction methodto reproducea givenshape. Then thereconstruction procedure isused forthesimulation ofboth thestaticand translatingdroptestcases.

5.1. Accuracyofthereconstructionmethod

Totestthe accuracyofthereconstruction process,we consideran ellipsoidalinterfaceof equation x2

_/

₉

+

_y2

_/

₄

=

_{1 in}

a12

×

12 box[see[36] fordetailsontheinitialization].Theellipsefrontisinitializedwithmarkersuniformlydistributed. Then the markersare redistributed on thegrid cells following the reconstruction procedure. Wemeasure the maximum relativeerrorbetweenthemarkers y-coordinatesobtainedafterreconstructionandtheexactellipseequation.Wevarythe ratio

1

s

/1

x where

1

s istheaveragelengthoftheelementsand

1

x isthe Euleriangridsize.InFig. 22(a) thenumberof elements N is varied from10 to100 for afixed grid (16

×

16 cells) whilein Fig. 22(b),thegrid size

1

x is varied from

Fig. 21. Translatingdroptestcase.EffectofthegridspacingR/1x forCa=10−10_{.(a) ,Maximumspuriousvelocity; ,RMSspuriousvelocity.}

(b) Relativeerrorfor ,1Pmax; ,1Pav g.

Fig. 22. Effectof1s/1x forthereconstructionoftheellipsex2_/9 +y2_/4

=1:(a)VaryingthenumberofLagrangianmarkers1siforthegrid16×16,

(b) VaryingtheEuleriangridsize1xifor100markers.

16

×

16 to168

_×

168 cellsforafixednumberofelements(N

₌

10).Forclaritywenote

1

xi(resp.

1

si)whenthegridsize (resp.theelementlength)isvaried.

Fig. 22(a) shows that the error decreases when decreasing

1

si

/1

x on a given grid. Indeed, the shape description is improved by theinitial marker distributionwhen their numberis increased. The maximumerroris around 0

.

4% for the maximumratiotested(

1

si

/1

x

=

2

.

1).Fig. 22(b) alsoshowsthedecreaseoftheerrorwhendecreasing

1

s

/1

xi.Whenthe Euleriangridresolutionisimprovedforafixednumberofmarkers(increasein

1

s

/1

xi),theerrorontheshapeincreases. Indeed,when

1

s

/1

xi isincreasedfora fixednumberofelements,the circle

C

e generatedfromthe twomarkersofeach elemente intersectsanincreasingnumberofcellsgeneratingnewmarkerslocatedonthecircle

C

_e thatisalocal approxi-mationoftheellipseshape.Theerrorwiththeellipsoidalshapeofreferenceisthenincreased.Thus,frontelementsoflarge sizecan bringerrorsinthetransferofinformationbetweenthefrontandtheEuleriangrid.Finally,accordingtothistest, theelementlength

1

s shouldbechosensmallerthanthegridsize

1

x.

5.2. Optimumvalueforthethreshold

1

smin

It hasbeenreportedin[7]that thereconstructionprocess ofanon-uniformdistributionofmarkersaffectsthehybrid formulation capacity to reduce spurious velocities. During the reconstruction process small elements can be generated, typicallywhentheinterfacepositionisclosetoacorneroftheEuleriangrid.Thequestionaddressedisthissectionconcerns the minimumsize

1

smin allowed foran element inorderto keep thespurious currentscloseto machine precision. The staticdrop testcaseasdefinedin section 4.1isfirst considered.A dropofradius R

=

0

.

25 is introduced ina2D box of size L

×

H

₌

1

×

1 describedusingauniformgridspacing

1

x

=

L

/

25.Thedensitiesanddynamic viscositiesaresettoone. Thesurface tensionischosensuchthat La

=

12000.Thesimulationstartswithauniformfrontwith

1

s

/1

x

₌

1

.

0 and the time step

1

t

₌

10−5 isused.Thereconstructionprocedure isappliedevery nt

=

100 time steps.Thetestedvaluesforthe

Fig. 23. Staticdroptestcase.Effectof1sminonthedevelopmentofthespuriousvelocities.(a)MaximumCapillary numberbasedonthespuriousvelocities; 1smin=01x; 1smin=0.11x; 1smin=0.21x; 1smin=0.31x; 1smin=0.41x.(b)Relative errorforthepressurejump: 1Pmax; 1Pav g.(Forinterpretationofthereferencestocolorinthisfigure,thereaderisreferredtothewebversionofthisarticle.)

Fig. 24. Translatingdroptestcase.Effectof1sminonthedevelopmentofthespuriousvelocities.(a)Maximumvelocity; 1smin=0.01x; 1smin=0.11x; 1smin=0.21x; 1smin=0.31x; 1smin=0.41x.(b)Zoom.(Forinterpretationofthereferencestocolorinthisfigure,the readerisreferredtothewebversionofthisarticle.)

minimumvalue

1

smin inthereconstruction are

1

smin

=

0,0

.

1,0

.

2, 0

.

3 and0

.

4 times

1

x.

1

smin

/1

x

=

0 meansthatany elementoflength

1

s isconserved.If

1

s

≤ 1

sminthenthemarkersarefusedtogetherfollowingtheprocedureexplainedin section3.2.The maximumCapillarynumberbasedonthespurious velocitiesisreportedinFig. 23(a).There,two typesof evolutionareobserved.For

1

smin

/1

x

=

0 (blue),

1

smin

/1

x

=

0

.

1 (red)and

1

smin

/1

x

=

0

.

2 (orange),theCapillarynumber stabilizes to a value close to 10−5 while for

1

smin

/1

x

=

0

.

3 (black) and

1

smin

/1

x

=

0

.

4 (green) the Capillary number remains closetomachine precision. It isclearthat the condition

1

smin

>

0

.

2

1

x is requiredto obtainspurious velocities comparableto those obtainedwith a uniform distribution. For

1

smin

≤

0

.

2

1

x, the maximum Ca stabilizes to the value observedinFig. 8(a) (redline)whentheforce f iscalculatedatthemarkerposition(equation(16)).

Fig. 23(b) showstherelativeerrorforthemaximumandaveragepressurejumpsasafunctionof

1

smin

/1

x.The maxi-mum(resp.average)pressurejumperrordecreases(resp.increases)with

1

smin

/1

x.For

1

smin

/1

x

>

0

.

2 botherrorsmeet roughlywithavaluearound0

.

05%.

Thetranslatingdroptestcaseisnowconsideredwiththesamedomain,gridspacing,timestepandfluidproperties.The Laplacenumber is La

=

12000 andthevelocity U0 isselected to have W e

=

0

.

4. Thefront is initializedwith auniform

distributionofmarkerssatisfying

1

s

/1

x

₌

1.Thereconstructionisperformedeverynt

=

100 timesteps.Resultsareshown in Fig. 24. The normalized maximum velocity U seems

˜

to convergefor all the cases except forthe ratio

1

smin

/1

x

=

0 (any element ofsize

1

s being allowed).The zoom in Fig. 24(b) shows that only for the ratios

1

s

/1

x

>

0

.

2 (black and greenlines)thespuriouscurrentarestabilizedat

τ

₌

1.Theerrorinthepressurejumpseemstobeindependentof

1

smin (Fig. 25).Therelativeerrorintheaverageandmaximumpressurejumpsarearound0

.

001% and0

.

0002%,respectively.

Tounderstandwhyasmallvaluefor

1

smin amplifiesthespuriousvelocity,weconsiderequation(15).Substitutionoftk andtk−1 givenbyequation(31)inthelocalcapillaryforcecontribution fe

/σ

=

tk

−

tk−1 ofthefrontelemente yields:

Fig. 25. Translating drop test case: effect of1sminon the relative error for the pressure jump. 1Pmax; 1Pav g.

Fig. 26. Translatingdroptestcase.Effectofthenumberoftimestepntbetweentworeconstructions.(a) ,MaximumnormalizedspuriousvelocityU ;˜

,RMSnormalizedspuriousvelocityU .˜ (b)Relativeerrorforpressurejump; 1Pmax; 1Pav g.

fe

/

σ

=

te

1

se₊1

+

te+1

1

se

q

1

s2_e+1

_{+ 1}

s2e

+

2

1

se₊1

1

sete

·

te+1

−

q

te−1

1

se

+

te

1

se−1

1

s2e

+ 1

s2e−1

+

2

1

se

1

se−1te−1

·

te (33)

From relation (33)we seethat fe

/σ

→

0 as

1

se

→

0 asexpected. However, thisrelation showsthat when

1

se−1

→

0 (resp.

1

se+1

→

0) the contribution of the tangent te−1 (resp. te+1) in the calculation of fe does not vanish. Indeed, consideringforexamplethelimitofrelation(33)when

1

se−1

→

0 weget:

f_e

/

σ

=

q

te

1

se+1

+

te+1

1

se

1

s2

e+1

+ 1

s2e

+

2

1

se+1

1

sete

·

te+1

−

te−1

Infact,when

1

se−1

/1

se

→

0 (resp.

1

se+1

/1

se

→

0)makerk

−

1 andk

−

2 (resp.k andk

+

1)becomesveryclosecompared to neighboring markers so that tk−1

→

te−1 (resp. tk

→

te+1), giving a non-expected importance of the corresponding

elementse

₋

1 (resp.e

₊

1)inthecapillaryforcecontributionofelemente.Thisparticularbehaviorexplainswhyathreshold for

1

s isrequired.

5.3. Frequencyofreconstruction

The frequency of reconstruction is now analyzed for the caseof the translating drop for La

=

12000 and W e

=

0

.

4. Differentnumbersoftimestepsnt

=

10,50,100 and200 betweentworeconstructionsarecompared.Resultsareshownin Fig. 26.Themagnitudeofthespurious velocitiesarekeptclosetomachineprecision forallthevaluesofnt.However, the spurious velocitiesincreasewhenincreasingthefrequencyofreconstruction.Indeed,errorsinduced bythecircle-segment lineintersectionareamplifiedwhenthefrequencyofreconstructionisincreased.Theaverageerrorinboththeaverageand meanpressurejumpisconstantforallthefrequencyofreconstructionconsidered.

Fig. 27. 2Doscillatingdroptestcaseusing: Fronttracking(present); Level-Set-CSF.(a)MaximumVelocityevolution.(b)KineticEnergyevolution. (c) MaximumPressurejump.(d)AveragePressurejump.(Forinterpretationofthereferencestocolorinthisfigure,thereaderisreferredtothewebversion ofthisarticle.)

5.4.2Doscillatingdroptestcase

Weclosethisseriesoftestswithatestproposedin[36]andreproducedin[7]foralargeLaplacenumberLa.Anelliptical drop of equation x2

/

0

.

0032

₊

y2

/

0

.

0022

₌

1 is initialized inside a 0

.

01

×

0

.

01 box with periodic boundary conditions. Propertiesare setsuchthat thedensityandviscosityjumps are1000 and 10 respectively. Surfacetensionis

σ

₌

0

.

1 and thecorrespondingLaplacenumberisLa

=

5

_×

105_{.Thenumberoftimestepsbetweentworeconstructionsisn}

t

=

1000 and thegrid ismadeof64

×

64 uniform cells.Afterinitialization thedropshape oscillatesandconverges toa circularshape whilethevelocityinthedomainvanishes.

Thistestisofparticularinterest sinceaccordingto[37],forlarge densityratios andhighLaplacenumbers La,intense spuriousvelocitiesshouldbeobservedandshoulddestroytheinterface.Weexperiencedthiswhenwetriedforcomparison purposeto performthe simulationwiththe VOF-FCT-CSF(FCT forFlux-Corrected Transport)methodofJADIM. The VOF-FCT-CSFmethodproducescatastrophicresultsnotshownhere,butwecanmentionthatittook15000 timestepstodestroy theinterface.Soinsteadwe usedtheLevelSet-CSFmethodinJADIMforthecomparison[see[20] fordetails onboththe VOF-FCT-CSFandtheLevelSet-CSFmethods].

Fig. 27(a) showstheevolutionofthemaximumvelocityinsidethedomain.Itcanbeseenthatthevelocityisdecreasing withtime forthe present front-tracking methodwith reconstruction, while the Level Set-CSF method seems to reach a minimumvalue atUmax

≈

10−2.Thetendencyisconfirmedwhenlookingatthekineticenergyevolution(Ek

=

R

ρ

U2dV ).

FortheLevel Set-CSFmethodthekineticenergyreachesa minimumEk

≈

10−8 whileforourfront-trackingmethodEk is stilldecreasingattheendofthesimulation(seeFig. 27(b)).Figs. 27(c) and27(d)showthemaximumandaveragepressure jumpsnormalizedusing

σ

/

req wherereq istheequivalentradiusbasedontheinitialellipsearea.Thepresentfront-tracking implementationreproducesacorrectstabilizedpressurejumpwhileitdoesnotconvergefortheLevel-setmethod.Notethat novolumecorrection[20]was appliedhereandwefoundthat thedroparea startsincreasingwithtimeafteroscillations haveceased(t

_≈

3)fortheLevel-Set-CSFmethod.Att

_≈

5,theLevel-Set-CSFmethodshowsanincreaseof25% ofthedrop areawhiletheerrorwasfoundtobelessthan1%withthefronttrackingmethod.

Fig. 28. Initialization of the bubble rising test.

Table 1

Densityratio,viscosityratio,ReynoldsnumberandEötvösnumberforthetwoselectedcasesforthesimulationofa2Drisingbubble.

Case ρ1/ρ2 µ1/µ2 σ g Re Eo

1 10 10 24.5 0.98 35 10

2 1000 100 1.96 0.98 35 125

6. Benchmarkingthefronttrackingmethod

The newfronttracking implementationisnow comparedto simulationsofreferencefromthe literature.We have se-lected the simulationof 2D risingbubbles [17]. Thesetestsallow ustoevaluate relatively largeinterface deformation to assesstherobustnessofboththenewlocalsurface tensionforcecalculationandthereconstructionmethod.Aninitial cir-cularbubbleofradius R

=

0

.

25 isintroducedintoaboxcontainingaliquid(seeFig. 28).Noslipandsymmetryconditions areappliedonthehorizontalandtheverticalboundaries,respectively.

A constant gravity field g is imposed. We introduce the Eötvös number Eo

=

4

ρ

1g R2

/σ

and the Reynolds number

Re

=

ρ

12R3/2g1/2

/µ

basedonthecharacteristicvelocity

√

g R andthebubblediameter.Thefluidpropertiesarereportedin

Table 1forthetwocasesconsidered.

Four uniformgrids are made with

1

x

=

1

/

40

,

1

/

80

,

1

/

160 and 1

/

320 and the corresponding time steps

1

t

_{= 1}

x

/

16 areused.In[17],threedifferentcodeshavesimulatedthesecases.Thecode1(TUDortmund,Inst.ofAppliedMath.)uses a FEM-Level Set method,thecode 2(EPFL Lausanne, Inst. ofAnalysis andSci. Comp.) uses a FEM-Level Setmethod and thecode3(Univ.Magdeburg,Inst.ofAnalysis andNum.Math.)usesaFEM-ALEmethod.Fordetails wereferthereaderto [17].Forthetwocasesconsideredhere,theinterfaceisinitializedwithanon-uniformdistributionofmarkers.Themarkers aregeneratedbyintersectingtheinitialbubbleshapewiththeEuleriangrid,withtheconstraint

1

s

/1

x

>

0

.

2.Thebubble centerpositionYc,circularity

˘

c andrisevelocityVc arecalculatedasfollows:

Yc

=

P

i,jCi jyi jAi j Ab (34)

˘

c

₌

2

π

L

r

Ab

π

(35) Vc

=

P

i,jCi jvi jAi j Ab (36)

Intheserelations,theindexi j denotetheEuleriangridcells. Ai j, AbandL arethecellarea,thebubbleareaandthebubble perimeter,respectively. Ab andL arecalculatedas:

Ab

=

X

i j Ci jAi j (37) L

₌

X

k

1

sk (38)

Threemeasuresoftherelativedifferencesei(i

=

1

,

..

3)betweenthemethodsareintroduced[see[17]].Foranyquantityqt, theyaredefinedas

Fig. 29. Numericalsimulationofa2Drisingbubbleinastagnantfluid(case1:Re=35 andEo=10,ρ1/ρ2=100 andµ1/µ2=10).(a)Bubbleshape,

(b) bubblecircularity˘c,(c)bubblecenterpositionYc and(d)bubblerisevelocityVc.Thefrequencyofthefrontreconstructionis ,nt=10; ,

nt=50; ,nt=100; ,nt=200.(Forinterpretationofthereferencestocolorinthisfigure,thereaderisreferredtothewebversionofthis

article.) e1

=

P

t

|

qt,ref

−

qt

|

P

t

|

qt,ref

|

(39) e2

=

à P

t

|

qt,ref

−

qt

|

2

P

t

|

qt,ref

|

2

!

1/2 (40) e3

=

maxt

|

qt,ref

−

qt

|

maxt

|

qt,ref

|

(41)

whereqt,re f isthequantityofreference.Whenagridconvergenceisperformed,thereferencequantityqt,re f isthequantity

qt obtainedwiththefinestgrid.Thevaluesofthedifferentquantitiesprovidedbythebenchmarkaregivenathttp://www. featflow.de/en/benchmarks/cfdbenchmarking/bubble.html.

6.1. RisingBubble(case1)

Theparametersofcase1are Re

=

35, Eo

₌

10,

ρ

1

/ρ

2

=

100 and

µ

1

/µ

2

=

10.Thefrequencyofreconstructionistested

withnt

=

10,50,100 and200 usingthegridsize

1

x

=

1

/

40.ResultsareshowninFig. 29andthedifferencesinTable 2. Bubbleshape,circularity,centroidpositionandrisevelocityseemtobeindependentofnt.ZoomsreportedinFigs. 30(a) and 30(b)arenecessarytoseesomesmalldifferencesbetweenthedifferentsimulations.Table 2summarizesforeachcasethe differencewiththereferencegivenbycode1.Thereportedvaluesconfirmthatthedifferentfrequenciesofreconstruction consideredprovidesimilarresults.Forthenexttestswehavechosennt

=

100.Notethattheprocessoffrontreconstruction couldalsobetriggeredbytestingateachtimesteptheminimumandmaximumfrontelementsizeandactingaccordingly. Inthisworkhowever,weprescribeafixedfrequencyofreconstructionforsimplicity.

Table
2

Numericalsimulationofa2Drisingbubbleinastagnantfluid(case1:Re=
35
andEo=
10,ρ1/ρ2
=
100
andµ1/µ2
=
10).Effectofthereconstructionfrequencyusing1x=
1/40.Thereferenceforthedifference calculationisprovidedbycode1.

nt e1 e2 e3

Circularityc˘

10 2.80e−03 3.28e−03 6.07e−03 50 2.76e−03 3.23e−03 6.01e−03 100 2.81e−03 3.28e−03 6.18e−03 200 2.68e−03 3.11e−03 5.84e−03 Centroid position Yc

10 2.41e−03 3.37e−03 5.69e−03 50 2.53e−03 3.52e−03 5.96e−03 100 2.46e−03 3.43e−03 5.79e−03 200 2.38e−03 3.28e−03 5.46e−03 Rise velocity Vc

10 1.30e−02 1.47e−02 2.19e−02 50 1.23e−02 1.40e−02 2.17e−02 100 1.26e−02 1.43e−02 2.15e−02 200 1.34e−02 1.49e−02 2.17e−02

Fig. 30. Zoomof(a)Fig. 29(b) and(b)Fig. 29(d).Thefrequencyofthefrontreconstructionis ,nt=10; ,nt=50; ,nt=100; ,nt=200.

(Forinterpretationofthereferencestocolorinthisfigure,thereaderisreferredtothewebversionofthisarticle.)

The effectof thegrid size isnow reportedin Fig. 31.The simulations are clearly convergingto the caseofreference (code1with

1

x

=

1

/

320).Interestingly,wecannoticethatthereportedresultsarefewimpactedbythegridresolution.The coarsestgrid

1

x

₌

1

/

40 (inblue)givesverycloseresultstotheonesobtainedwiththefinestgrid

1

x

=

1

/

320.Circularity seems to be themost affectedvariable by thecoarsest grid.In Fig. 31(b),the maximumdifference is observedat t

₌

2, the grid

1

x

=

1

/

40 gives c

˘

₌

0

.

897 while the grid ofreference

1

x

=

1

/

320 gives c

˘

₌

0

.

901,a difference lessthan 0.5%. According toFig. 31(a),thedifferenceismainlylocated atthe maximumofcurvatureofthe bubble.The resultsobtained with

1

x

=

1

/

320 arenow compared inTable 3with theresults obtainedwithcode 1[17] usingthe samegrid. Avery good agreementisfound forallthe consideredquantities. Themaximumrelative differenceis 2

.

75

×

10−3 _{forthe rising}

velocity. Additionally, we show inAppendix A a verygood agreementbetweenthisnewfront trackingmethodandboth theVOF-FCT-CSFandtheLevel-Set-CSFmethodsimplementedinourcodeJADIM[18,24,20].Themassconservationisalso comparedandourimprovedFronttrackingmethodconservesmassataverygoodlevelof10−3_{% forthistestcase.}

6.2. RisingBubble(case2)

Theparameters ofcase2are Re

=

35, Eo

=

125,

ρ

1

/ρ

2

=

1000 and

µ

1

/µ

2

=

100.Thesimulationismadeusing

1

x

=

1

/

320 andthefrequencyofreconstructionisnt

=

100.Thecomparisonofthebubbleshapebetweenthedifferentcodesis showninFig. 32.Asobservedwithcode3usingaFEM-ALEmethod[17],ourfronttrackingmethodcannothandlebreak-up automatically, instead,thesimulationkeep stretchingthebubbleskirt.However, exceptintheskirt zoneofbreak-up,the bubbleshapeisfoundinverygoodagreementwiththeothercodes.

Thebubblecircularity,centroidpositionandrisevelocityarereportedinFig. 33.Thecircularityevolutionforthepresent fronttrackingmethodisnearlyidenticaltothatofcode1uptot

_≈

2

.

3.Aftert

_≈

2

.

3,thecircularitycalculationisaffected bythesmallbubblesgeneratedbytheruptureofthebubbleskirt(seeFig. 33(a)).TheCentroidpositionshowninFig. 33(b)

Fig. 31. Evolutionofa2Drisingbubbleinstagnantfluid(case1:Re=35 andEo=10,ρ1/ρ2=100 andµ1/µ2=10)for1x: ,1/40; ,1/80;

,1/160; ,1/320; ,Ref.1/320.(Forinterpretationofthereferencestocolorinthisfigure,thereaderisreferredtothewebversionofthis article.)

Table 3

Comparisonbetweenthepresentfronttrackingmethodandcode1[17]forthe2Drisingbubbletestcase 1 (Re=35,Eo₌10,ρ1/ρ2=100 andµ1/µ2=10).Thegridis1x=1/320.

e1 e2 e3

Circularityc˘ 9.75e−05 1.22e−04 2.62e−04 Centroid position Yc 8.69e−05 1.35e−04 2.58e−04

Rise velocity Vc 1.90e−03 2.10e−03 2.75e−03

isvirtuallyindependentofthebreak-upprocess.Therisevelocityisthemostsensiblequantityforallthecodesconsidered includingthepresentmarkermethod(seeFig. 33(c))andthemaximumdifference observedattheendofthesimulation israngingfrom6.1% withcode 1to0.9%withcode2.The difference betweenourfront trackingmethodandcode1are reportedinTable 4,confirming whatFig. 33shows:averygoodagreementisfoundandthelargestdifference isobtained forthecircularity.

7. Conclusions

Inthis workwe have improvedthe 2D front trackingmethod andwe haveimplemented it insideour in-housecode JADIM.Thehybridformulation[7]forthecalculationofthesurfacetensionforcehasbeenreconsideredwiththecalculation oftangentsatmarkersposition.Spuriouscurrentsarethenreducedtomachineprecisionforbothuniformandnon-uniform distributionofmarkers.The newimplementationwas testedagainstthe staticandtranslatingdroptest casesfordiverse Laplaceandcapillarynumbers.Testsonthetimestep

1

t andgridspacing

1

x revealsthatbothtimeandgridconvergence cannotbeobservedatthislevelofprecisionduetotheeffectofround-offerror.

Thisnewapproach tocalculate tangentsallowed usalso topropose a newreconstruction schemeby whichthe front canberedistributed/reconstructedautomaticallyatprescribedtimeintervals.Thisreconstructionschemeisbasedoncircles

Fig. 32. Bubbleshapefortherisingbubbleofcase2(Re=35,Eo=125,ρ1/ρ2=1000 andµ1/µ2=100)attimet=3 for1x=1/320 simulatedwith:

,code1; ,code2; ,code 3; ,presentmarkermethodinJADIM.(Forinterpretationofthereferencestocolorinthisfigure,thereader isreferredtothewebversionofthisarticle.)

Fig. 33. Evolutionof(a)thecircularity,(b)thebubble centerand (c)andthe risingvelocityofthe2D risingbubbleofcase2(Re=35, Eo=125,

ρ1/ρ2=1000 andµ1/µ2=100)for1x=1/320: ,code1; ,code2; ,code3; ,PresentmarkermethodinJADIM.(Forinterpretation