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To cite this version:

Dalmon, Alexis and Kentheswaran, Kalyani and Mialhe, Guillaume and Lalanne, Benjamin and Tanguy, Sébastien Fluids-membrane interaction with a full Eulerian approach based on the level set method. (2020) Journal of Computational Physics, 406. ISSN 0021-9991

Official URL: https://doi.org/10.1016/j.jcp.2019.109171

Fluids-membrane interaction with a full Eulerian approach based on the level set method

Alexis Dalmon

^a

,

^b

, Kalyani Kentheswaran c,

^d

. Guillaume Mialhe

^a

. Benjamin Lalanne

^c

, Sébastien Tanguy

^a

,*

• lnsrîrut de Mécanique des Fluides de Toulouse, Université de Toulouse, CNRS, INPf, UPS, Toulouse, France b Centre National d'Etudes Sparîales, 18 Avenue Edouard Belin, 31401 Toulouse Cedex 9, France c Laboratoire de Génie Otimique, Université de Toulouse, CNRS, /NPf, UPS, Toulouse, France d TB/, Université de Toulouse, CNRS, INRA, /NSA, Toulouse, France

ARTICLE INFO Keywords:

Membrane Level set Ghost fluid

1. Introduction

ABSTRACT

A fully Eulerian approach to predict fluids-membrane behaviours is presented in this paper. Based on the numerical mode( proposed by li et al. (2012), we present a sharp methodology to account for the jump conditions due to hyperelastic membranes. The membrane is considered infinitely thin and is represented by the level set method. lts deformations are obtained from the transport of the components of the left Cauchy­

Green tensor throughout time. Considering the linear or a hyperelastic material law, the surface stress tensor is computed and gives the force exerted by the membrane on the surrounding fluids. The membrane force is taken into account in the Navier-Stokes equations as jump conditions on the pressure and on the velocity derivatives by imposing suitable singular source terrns in cells crossed by the interface. To prevent stability issues, an extension algorithm has been developed to remove the normal derivatives of the scalar fields specific to the membrane. ln particular, a subcell resolution at the interface of the extrapolated variable is proposed for increasing the accuracy of the extension algorithm.

These improvements are validated by comparing our numerical results with benchmarks from the literature. Moreover, a new benchmark is proposed for fluids with both different viscosities and different densities to target applications where a gas and a liquid phase are separated by a membrane.

The fluids-membrane interaction study of this paper is part of a global project on propellant sloshing in satellite tanks.

The latter phenomenon happens during a satellite manoeuvre and can be a major disturbance of the stability. The tanks contain Jiquid propellant and gas to maintain a sufficient pressure within the tank. During a manoeuvre, inertial forces lead to a motion of the fluids and thus of the centre of mass. This generates disturbing forces and torques on the whole structure which may deteriorate the quality of satellite imaging. Considering simple tanks, numerical methods have been developed in our home-made code DIVA (Dynamics of Interface for Vaporisation and Atomisation) to model propellant sloshing in

* Corresponding author.

E-mail address: tanguy@imft.fr (S. Tanguy).

hnps:/ /doi.org / 10.1016/j .jcp.2019.109171

micro-gravity conditions[1,2]. Aparametric studyhas been done on typical rotationalmanoeuvres exerted by satellites in space and has been validated by comparisons with data from the FLUIDICS (FLUId DynamICs in Space) experiment, performed in the InternationalSpace Station (ISS) [1,3,4]. The work presented in this paperis a first step towards the extension of the sloshing study to diaphragm tanks for which a hyperelastic membrane separates the liquid propellant andthegas.The modellingoftheinteractionbetweenthemembraneandthefluidswithinthetank iscrucial.Itmustbe predictedaccuratelyandthusnumericaldevelopmentsarerequiredtodoso.

The fluid-membrane interaction is a challenging problem to solve numerically. Peskin [5] developed the Immersed Boundary Method(IBM)topredictincompressibleflowswithmoving elasticboundaries.Thesefirstapplicationswere the modelling of blood flows in the heart [6] and has been extended to many biological problems cited in [7]. With this method,theNavier-StokesequationsaresolvedonaCartesianmeshandthemembraneisdescribedbyLagrangianmarkers.

The force exerted by themembrane isdeduced fromthe position ofthe markers andisinterpolated onto theCartesian meshusingDiracdeltafunctions.TheNavier-Stokesequationsarethensolved withtheforcing termcorresponding tothe elastic contributionfromthemembrane.Finally,thepredictedfluid velocityisused toupdate thelocationofthe marker points definingthe membrane.This methodologyis repeatedforeach time step.The smoothedDirac functionsinducea numericalsmearingofthemembraneinthefluidgridwhichmayaffecttheaccuracyofthemethod.

The Immersed InterfaceMethod (IIM) of Levequeand Li[8] replaces theinterpolated forcing term dueto the elastic membrane with sharp jump conditions.This method has been introduced for elliptic equationsand extended to Stokes flows [9] and incompressible viscous flows [10]. The artificial smearing of the elastic force in the fluid grid generates spurious velocitiesatthemembraneandasmoothingofthepressurefieldwhereadiscontinuousjumpmustappear.With the IIM, the normal componentof theelastic force is enforced through a jump condition on thepressure when solving the Poisson equation. The tangential componentof the force induces jumps in the derivativesof the velocity across the membrane.

TheMaterialPointMethod(MPM)describesboththesolidandthefluidphasesusingLagrangianmarkersinthewhole computational domain [11]. Thismethod hasbeen usedin thecaseof fluid-membraneinteraction [12] and allows large deformationsofthelinearelasticmembrane.

Full Eulerian approachesdo not useLagrangian particles to follow themembrane motionand tocompute theelastic forces.CottetandMaitre[13] usethelevelsetmethodtofollowthemotionofan elasticmembraneimmersedinafluid.

Without thereinitialisation algorithmofthe levelsetfunction[14],thevariation ofthelevelset gradientcanbe directly relatedtothemembranestretching.Theauthorsusethispropertytoenforcetheelasticmateriallawonthezerolevelset andtointegratetheresultingforceintheCartesianmeshusingdeltafunctions.However,thismethodonlygivesthemem- brane stretchingandlimitsthecomplexity ofthe membranemodel.Moreover, themembraneisstillartificiallythickened bythedeltafunctions.

Another full Eulerian method consists in defining the components of the deformation tensor of the membrane as scalar variables in the whole computational domain [15]. This approach is based on the work of Sugiyama [16]

which developed a full Eulerian fluid-structure interaction model working with hyperelastic solid bodies. The position of each phase, fluid and solid, is described by the volume-of-fluid function (VOF) [17] and only one set of govern- ing equations is solved in the whole domain. In the fluid region, the stress tensor contains the pressure term and the viscous stress tensor, in the solid region, the stress tensor is deduced from the solid deformation and the hy- perelastic material law. The left Cauchy-Green deformation tensor is updated throughout time thanks to a transport equation in the whole computational domain. This allows to follow the deformation of the hyperelastic solid in a Eu- lerian manner. Then, the solid stress tensor can be computed following the hyperelastic material law and enforced in the Cartesian mesh. Ii et al. [15] extend this method to fluid-membrane interactions, which means that the solid re- gion is reduced to a codimension-one subspace. Following the methodology of Barthes-Biesel and Rallison [18], the deformation tensor of the membrane is defined asa solid deformation tensor projected onto the tangent plane of the membrane. The normal projection of the membrane deformation tensor is not considered because the membrane ma- terial is supposed to be incompressible. The membrane stresses are computed thanks to the deformation tensor and the hyperelastic material law and the resulting force is integrated in the Navier-Stokes equations with delta functions.

Nevertheless, this method presents losses of accuracy due to the numerical dissipation of the smoothed Dirac func- tion. Moreover, instability issues may appear over long time periods because the values of the deformation tensor far from the membrane may evolve chaotically with the fluid velocity and may influence the computation at the mem- brane.

In this paper,we propose to improvethe full Eulerian method ofIi et al.[15] byadding some aspects of the sharp methodologyoftheIIM.Moreover,we extendthismodeltohandledifferentfluidsoneachsideofthemembrane.Insec- tion 2, themodellingof two-phase flows andthe membranemodel aredescribed. The numerical methodsimplemented in the codeare explainedin section 3.More particularly,the discretisation ofthe jumpconditions fromthe GhostFluid Method [19,20] isdetailed.Furthermore,an extension algorithmbased onasubcellresolution hasbeendevelopedto re- duce the numerical instability by removing the spurious normalderivatives. The validation of the methodology is done in section 4 withseveral benchmarksof growing complexity from the literature:from the simple caseof a bubble ris- ing dueto a surface tension gradient untilthe much more complexcases of a stretched membraneseparating different fluidsanda capsule immersedinashear flow.Finally, thepaperisconcludedby some remarksandperspectivesinsec- tion5.

2. Eulerianfluidmembranemodel 2.1. Two-phaseflowmodel

We consider a domain with a boundary ∂ which contains two differentfluids defined by ⁺ and ⁻ such as =⁺∪⁻.Theinterfacebetweenthetwofluidsregionsisdenotedanditsoutwardnormalvectorisn.Eachfluidis incompressibleandNewtonianandfollowstheNavier-Stokesequations.

∇ ·

^u

=

⁰

,

(1)

ρ ∂

u

∂

t

+ (

u

· ∇)

^u

= −∇

^p

+ ∇ ·

2

μ

D

^¯¯

,

(2)

withu=(u,v,w)thevelocityfield,t thetime,

ρ

thefluiddensity,

μ

thefluidviscosity, pthepressureand D¯¯ therateof deformationtensordefinedas

D

¯¯ = ∇

^u

+ ∇

^uT

2

.

(3)

Considering theentire domain , specialcare must be takenat the interface between the two fluidsregions. The followingjumpconditionsmustbeaccountedfor,

[ ρ ]

= ρ

⁺

− ρ

⁻

,

(4)

[ μ ]

= μ

⁺

− μ

⁻

,

(5)

[

ⁿ

· ¯¯ σ _]

₌

f

,

(6)

with

σ

¯¯ thestresstensorandfthelocalforcedensityattheinterface.Inthecaseoftwo-phaseflowswithsurfacetension, thelocalforce densitybecomes f=

γ κ

nwith

γ

thesurface tensionand

κ

= ∇ ·^{nthe meancurvatureoftheinterface.}

Whenamembraneseparatesthetwofluids,thelocalforceisdefinedinsection 2.2.

Consideringtheentiredomain,wedefinethedensityandviscosityfieldas

ρ (

x

) = ρ

⁻

+ ( ρ

⁺

− ρ

⁻

)

H

(

x

),

(7)

μ (

x

) = μ

⁻

+ ( μ

⁺

− μ

⁻

)

H

(

x

),

(8)

withHtheHeavisidefunctiondefinedasH(x)=^{0 in−}andH(x)=^{1 in+}. TheNavier-Stokesequationsintheentiredomaincanthereforebewrittenas

∇ ·

^u

=

⁰

,

(9)

ρ ∂

u

∂

t

+ (

u

· ∇)

^u

= −∇

^p

+ ∇ ·

2

μ

D

^¯¯

+ δ

f

,

(10)

withδ theDiracfunctionlocatedattheinterface.

2.2. Membranemodel

Inthissection,the largedeformablemembrane modelofBarthes-BieselandRallison[18] isdescribed.The localforce densityofthe membraneon thefluidsis computedwitha fullEulerian formulation. Thisformulationisinspired by the workofSugiyamaetal.[16] for3DsolidproblemsandadaptedtothemembranesbyIietal.[15].

Inastress-freestate,eachparticleofasolidisdefinedbyitscoordinatevectorX.Wedenotebyx(X,t)theposition,at timet,ofaparticlelocatedinX inthestress-freestate.Thelinkbetweenthecurrentstateandthestress-freestateisthe deformationgradienttensor F¯¯ definedas

F

¯¯ = ∂

x

∂

X

.

(11)

Thematerialtimederivativeofthedeformationgradienttensor F¯¯ follows

dF

¯¯

dt

= (∇

^us

)

F

¯¯ ,

(12)

with u_s the velocity of the membrane particles. This velocity vector is obtained by extending u from its values atthe membranetowardthenormaldirection.Theextensionmethodisdefinedinsection3.6.

Now, we consider the special case ofa membrane which thicknessis very small compared to its other dimensions.

Therefore, we will neglect the thicknessand representthe membrane as a surface in a 3D space. Eachfibre dX of the membraneinthereferencestatebelongstothetangentplaneofthemembrane.Similarly,eachdeformedfibre dxbelongs to thetangent plane of themembrane in the currentstate. Thismeans that only thecomponents perpendicular to the normaldirectionofthemembranemustbe consideredbythedeformationgradienttensor.LetnR betheoutwardnormal of the membrane in the reference state and n the outward normal in the current state, the previous condition can be expressedrespectivelyas F¯¯s.nR=^0andn.F¯¯s=^0with F¯¯s thesurface deformationgradienttensorofthemembrane.This tensorcanthenbewritten

F

¯¯

s

= ¯¯

PF

¯¯

P

¯¯

R

,

(13)

with P¯¯ = ¯¯^I−ⁿ⊗^{nthesurfaceprojectiontensorinthecurrentstateand}P¯¯_R= ¯¯^I−ⁿR⊗ⁿRinthereferencestate.Thelocal deformationofthemembranecanbeobtainedwiththesurfaceleftCauchy-Greentensor,

B

¯¯

s

= ¯¯

^FsF

¯¯

_s^T

.

(14)

The Cauchy-Greentensor B¯¯s issymmetricanditseigenvaluesλ²₁ andλ²₂ correspondtothesquareofthetwoprincipal strainsofthemembraneinitstangentplane andthethirdeigenvaluezerocorresponds tothedeformationinthenormal directionwhichhasnoexistencebecauseoftheprojectionsperformedpreviously.Thescalarinvariantsofthesurfaceleft Cauchy-Greentensorare

I1

=

tr

(

B

¯¯

s

) = λ

²₁

+ λ

²₂

,

(15)

I2

=

¹ 2

tr

(

B

¯¯

s

)

²

−

tr

(

_B

¯¯

²

s

)

= λ

²₁

λ

²₂

,

(16)

I3

=

det

(

B

¯¯

s

) =

0

.

(17)

Thisstraintensor,considering(13) and(14),canbewrittenthankstoanintermediatetensorG¯¯_s,

B

¯¯

_s

= ¯¯

^PG

¯¯

_sP

¯¯

with G

¯¯

_s

= ¯¯

^FP

¯¯

_RF

¯¯

^T

.

(18) Considering(12) and(18),thematerialtimederivativeofG¯¯sisgivenby

dG

¯¯

s

dt

= (∇

us

)

G

¯¯

s

+ ¯¯

Gs

(∇

us

)

^T

.

(19)

In the caseof hyperelasticmaterials, the stress tensor

σ

¯¯ can be written accordingto the surface strain energy func- tion W(I1,I2,I3) which depends on the left Cauchy-Green tensor and its scalar invariants. Inthe case of a membrane, Barthes-BieselandRallison[18] showthatthesurfacestresstensor

σ

¯¯s is,

¯¯

σ

s

= √

² I₂

∂

W

∂

I₁B

¯¯

_s

+

^I2

∂

W

∂

I₂ P

¯¯

.

(20)

Severalstrainenergyfunctions[21–24] existtodescribedifferenthyperelasticbehaviours.Allfunctionsdependingonthe scalarinvariantsoftheCauchy-Greentensorcan beconsideredwiththismethodology.Wewillconsider inthisstudythe strain energyfunctionoftheneo-Hookeansolid [25],adapted tothemembranecaseby consideringtheincompressibility ofthemembraneas

W

=

^Es 6

(

I1

+

¹

I2

−

³

),

(21)

withEsthesurfaceelasticmodulusdefinedasEs=^EhRwithEtheYoung’smodulusofthemembraneandhR thethickness ofthemembraneinthereferencestate.Thesurfacestresstensorbecomesforthismaterialmodel,

¯¯

σ

s

=

^Es 3

√

I₂

B

¯¯

_s

−

P

¯¯

I₂

.

(22)

Finally, the local force densityexerted by the membrane is given by the surface divergence ofthe membranestress tensor,

f

= ∇

s

· ¯¯ σ

s

,

(23)

with ∇s the surface gradient operator definedby ∇s= ¯¯^P.∇^.Considering the distanceproperty of thelevel set function, whichimplies(n· ∇)n=^{0,andthegeometricpropertiesofthemembrane,some}simplificationscanbemade,

P

¯¯ σ ¯¯

_s

_{= ¯¯} σ

_s

,

n

· ¯¯ σ

_s

₌

0

,

(24) andthelocalforcebecomes

f

= ∇ · ¯¯ σ

s

.

(25)

Thelocalforcedensitycanbedividedintotwocomponents,respectivelytowardthenormalandthetangentdirections ofthemembranesuchas

fn

=

^f

·

ⁿ

= − ¯¯ σ

s

: ∇

ⁿ

= − σ ¯¯

s

∇ ·

ⁿ

,

(26)

becausethestresstensorissymmetric,and

f_τ

= ¯¯

Pf

= ¯¯

P

(∇ · ¯¯ σ

s

).

(27)

3. Numericalmethods

Thissection describesspecificnumerical methodsimplementedto modelthemembrane behaviourandits interaction withthesurroundingfluids.Thesedevelopmentshavebeenintegratedinthehome-madecodeDIVA.Thissolverisbasedon severalnumericalmethodsdedicatedtothecomputationoftwo-phaseflows.TheDIVAcodecanalsoconsiderliquid-vapour phasechange[26–30] andcompressibleflows[31].Complexgeometrycanbeaccountedforbyusingtheirregulardomain methodproposedin[32] forsinglephaseflowsandextendedtotwo-phaseflowsinirregulardomainsin[2].Inthisstudy, the interface between thetwo fluidscorresponds to the membrane andis represented by the level set method[33,14].

The Ghost-Fluid method[19,34–36,20,37] is used to consider the sharpjump conditions atthe membrane. It should be pointedoutherethatGhostFluidMethodisfirstorderaccurateforimposingjumpconditions,butsomerecentworkshave proposedextensionstosecondorderaccuracyasin[38–40].

3.1. Interfacetrackingmethod

Thelevelsetmethod[33,14] isusedtotracktheinfinitelythinmembranethroughouttime.Wedefinethescalarfieldφ whichcorrespondstothesigneddistancefromthemembrane.Eachfluidregioncorrespondstothesignofthelevelset functionas⁺= {^x:φ (x)>0}^,−= {^x:φ (x)<0}^{andthemembrane}correspondstothezerolevelset= {^x:φ (x)=⁰}^. Themotionofthemembraneisupdatedbysolvingthefollowingtransportequation

∂φ

∂

t

+

^u

· ∇ φ =

⁰

.

(28)

Tokeepthesigneddistancepropertyofthelevelsetfunctionthroughouttime,weconsiderthereinitialisationalgorithm [14].Thefollowingequationissolvediterativelytocorrectthedistancebetweeneachlevelset

⎧ ⎨

⎩

∂

d

∂ τ ⁺

^sign

^(φ)(

¹

^{− ||∇}

^d

^{|| ) =}

⁰

^,

d

( τ ₌

0

) = φ,

(29)

with

τ

a fictitioustime along which the reinitialiseddistance function d is corrected tomaintain the distanceproperty

||∇^d||=^{1.Thesmoothedsignfunctionsign}(φ)isdefinedin[19,14].

Thelevelsetmethodallowsustocomputethegeometricpropertiesofthemembranesuchasitsoutwardnormalusing simpledifferencing,

n

= ∇ φ

||∇φ|| ,

(30)

andthus,theprojectiontensor P¯¯ = ¯¯^I−ⁿ⊗^{ncanbecomputedinthewholedomain.}

The spatial derivatives of the transport equation and the reinitialisation algorithm are computed using the WENO-Z scheme[41].ThetemporalschemeofthetransportequationisthesecondorderTVDRunge-Kuttascheme.Thereinitialisa- tionalgorithmissolvedattheendofeachtimestep.

3.2. Two-phaseflowsolvers

Inthissection,we describethemethodologyto solvetheNavier-Stokesequationswithtwodifferentwaystoconsider the membrane contribution. The first one is the “delta” formulation forwhich the elastic force is a source termin the right-handside.Thesecondmethodintegratestheelasticforcethroughsharpjumpconditionsonthepressureandonthe velocityderivatives.

3.2.1. The“delta”formulation

The“delta”formulation[42,14] solvestheNavier-Stokesequationsfortwo-phaseflowsbysmoothingtheinterfaceonan artificialthickness.ThemethodisbasedonaprojectionmethodinspiredbyChorin[43] inthecaseofsinglephaseflows.For two-phaseflows,themethodweuseisderivedfromtheGhost-FluidviscousConservativeMethodwithanImplicitscheme ofLepilliez etal.[2], inspiredby theworkofSussmanetal.[37].First,givenavelocity fielduⁿ atthetime tⁿ=ⁿt,an intermediatevelocityu^∗ iscomputedwithoutconsideringthepressureterm,

ρ

ⁿ⁺¹u^∗

−

t

∇ ·

2

μ

ⁿ⁺¹D

^¯¯

^∗

= ρ

ⁿ⁺¹

uⁿ

−

t

uⁿ

· ∇

^un

+

t

δ

εfⁿ

.

(31)

The viscoustermisconsideredimplicitlytoavoiditstime steprestriction.Thisleadstotheresolutionofalargelinear systemwherethe threecomponentsofthevelocity arecoupled.The advectiontermiscomputedwithWENO-Z schemes [41] andthe viscoustermwithcentral differencing schemes[2].The localforce fⁿ ismultiplied bythe smoothedDirac distributionδ_ε definedbelowandaddedtotheright-handside.Astheresultingmatrixisdiagonallydominant,thesystem issolvedusingafewstepsoftheGauss-Seidelalgorithm.

Then, the pressure field pⁿ⁺¹ serves asthe scalar potential function of the Hodge decomposition which satisfies the followingPoissonequation

∇ ·

∇

^pn+1

ρ

ⁿ⁺¹

= ∇ ·

^u∗

t

,

(32)

with homogeneous Neumann boundary conditions on ∂. The resolution of this equation is done with the Black Box Multigridmethod[44,45] toreducethecomputationtime.

Finally,thevelocityfielduⁿ⁺¹isdefinedastheprojectionoftheintermediatevelocityu^∗ontothedivergence-freespace

uⁿ⁺¹

=

^u∗

−

t

ρ

ⁿ⁺¹

^∇

^p

n+¹

.

(33)

In the framework ofthe Whole Domain Formulation, the densityand viscosity fieldsare updated with the level set functionusingasmoothedHeavisidedistribution

ρ (

x

) = ρ

⁻

+ ( ρ

⁺

− ρ

⁻

)

H_ε

(

x

),

(34)

μ (

x

) = μ

⁻

₊ ( μ

⁺

₋ μ

⁻

)

H_ε

(

x

),

(35)

withHε(x)thesmoothedHeavisidedistributiondefinedas

H_ε

(

x

) =

⎧ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎩

0 if

φ (

x

) < − ε ,

1

2

1

+ φ (

x

) ε ⁺

sin

( π φ (

x

)/ ε ) π

if

| φ (

x

) | < ε ,

1 if

φ (

x

) > ε ,

(36)

with

ε

thefictitiousthicknessofthemembrane,whichisequaltotwoorthreetimesthesizeofacell[46].Similarly,the smoothedDiracdistributionδε(x)isdefinedasthederivativeofthesmoothedHeavisidefunction,

δ

ε

(

x

) =

⎧ ⎪

⎨

⎪ ⎩

0 if

|φ (

x

)| > ε ,

1

2

ε

1

+

^cos

π φ (

x

) ε

if

|φ (

^x

)| < ε .

⁽³⁷⁾

3.2.2. Thesharpformulation

Letusconsiderthatnisthenormalvectortothemembraneand

eτ1,eτ2 aretwoorthogonalvectorsbelongingtothe tangentplaneofthemembrane.Theno-slipconditiononbothsidesofthemembraneenforces[^u]=^{0 andgives}

∂

u

∂

e_τ1

=

^{0 and}

∂

u

∂

e_τ2

=

⁰

.

(38)

Theincompressibilitycondition∇ ·^u=^{0 inbothfluidsensuresthatthereisnojumpatthemembrane[}∇ ·^u]=^{0 and} givesthefollowingrelationfrom[47]

∂

u

∂

n

·

ⁿ

=

⁰

.

(39)

The densityandviscosityfieldsarepiecewiseconstantandtheirjumpsatthemembranecorrespondtoequations(4) and (5).Withthesharpformulation,thelocaldensityforcecorresponding tothemembranecontributionisexpressedasthree primaryjumpconditions:

[

^p

]

=

²

μ ^∂

^u

∂

n

·

ⁿ

+

^f

·

ⁿ

=

²

[ μ _]

^∂

^u

∂

n

·

ⁿ

+

^fn

,

(40)

μ ^∂

^u

∂

n

·

^eτ1

+

μ ^∂

^u

∂

e_τ1

·

ⁿ

= −

P

¯¯

f

·

^eτ1

= −

^fτ

·

^eτ1

,

(41)

μ ^∂

^u

∂

n

·

e_τ2

+

μ ^∂

^u

∂

e_τ2

·

n

= −

P

¯¯

f

·

e_τ2

= −

f_τ

·

e_τ2

.

(42)

ThesejumpconditionshavebeenprovenbyXuandWang[48] insinglephaseflowsandextendedtopiecewiseconstant viscosity by Tan et al.[49,47]. In thisstudy, we only consider theseprimary jump conditions which account forall the physicalphenomenaofinterestforthisstudy.Itisnoticeablethathigheraccuracynumericalmethodshavebeenachieved in[49,47,48] inthesimplercaseoflinearelasticity, byimposing furthersecondary jumpconditions onthesecond order velocity derivativesandfirst order pressurederivatives.However, these numericalmethods havenot beengeneralised to morecomplexconfigurationssuchashyperelasticmembranes.

FollowingtheGhost-FluidConservativeviscousMethod(GFCM),Lalanneetal.[46] showthatonlythenormalcomponent oftheforcemustbeconsideredintheprojectionandthecorrectionstepsbecausetheviscouscomponentisalreadytaken intoaccountinthepredictorstepoftheprojectionmethod.Theremainingjumpconditiononthepressureisthen

[

p

]

=

f

·

n

=

fn

.

(43)

Fromequations(38),thejumpconditions(41) and(42) canbewrittenas

μ ^∂

^u

∂

n

·

^eτ1

+

[

μ

]

∂

u

∂

e_τ1

·

ⁿ

= −

^fτ

·

^eτ1

,

(44)

μ ^∂

^u

∂

n

·

e_τ2

+

[

μ

]

∂

u

∂

e_τ2

·

n

= −

f_τ

·

e_τ2

.

(45)

Followinga similar approachto that ofLalanneetal.[46], thesource termsdueto thetangential componentsofthe elastic forceare considered asjumps thatare onlyenforced on thefirstterm:

μ

^∂u

∂n

·^eτk withk= [¹,2]^{.The second} terms ofequations(44) and(45) will be zeroif thereis no viscosityjump, evenin thecase wherethe tangential com- ponent ofthe local force isnot zero. Thisapproach still stands with a viscosityjump andhasbeen used in[50] where severalsimulationswithsurfactantswereperformed.Consequently,thesplittingofthejumpconditionsallowsasimplenu- mericalimplementationofthesourcetermsinthecellscutbythemembrane.Indeed,thejumpconditionsonthetangent componentsoftheviscosity-scaledvelocitygradientinthefirststepoftheGFCMcanbetakenintoaccountbyimposing

μ ^∂

^u

∂

n

·

^eτk

= −

^fτ

·

^eτk

,

(46)

withk= [¹,2]^{.Becausethejumpofthenormalcomponentofthe}viscosity-scaledvelocitygradientisonlyduetothejump ontheviscosity,thefollowingsourcetermisconsideredinthefirststepofthealgorithm

μ ^∂

^u

∂

n

= − ¯¯

Pf

= −

f_τ

.

(47)

Insection4.1,avalidationofthismethodologytodealwithjumpsofbothtangentialandnormalstressesispresented bycomparingsimulationsandthetheoreticalsolutionofadropletrisinginasurfacetensiongradientintheStokesregime.

Thesecondtermsofequations(44) and(45) arezerointhiscase,withorwithoutviscosityjump.Asthesimulationsmatch tothetheoreticalsolution,thiscomparisondemonstratestherelevanceofthepresentedapproach.Forageneralcasewith nonzeroterms[

μ

] ∂u

∂eτ₁ ·^nand[

μ

] ∂u

∂eτ₂ ·^{nandviscosityjump,weassumethatallthetermsofequations(44) and(45)} arewelltakenintoaccount.Toourknowledge,ademonstrationforthisgeneralcasehasneverbeenprovidedandfurther investigationwouldbeappreciated.Itappearsfromthisdiscussionthatthekey-pointistointroduceaconsistentnumerical approximation of the source term resulting fromthe jump condition. Finally, the jump condition will be automatically satisfiedifaconsistent approximationofthesourcetermisaddedattherightplace.Other generalconsiderationsonthis specificpoint canbefound inAppendixB.From this,some analogiescanbefound betweenjumpcondition formulations andδformulationsfortwo-phaseflowsNavier-Stokesequations.

TosolvetheNavier-Stokes equationsfortwo-phaseflows withthe GhostFluidviscousConservativeMethod,thesame projectionmethodinspiredby[43] isconsideredwithjumpconditionstodescribethemembraneforce.First,theinterme- diatevelocityiscomputedusingthesameimplicitschemefortheviscousterm,

⎧ ⎪

⎨

⎪ ⎩

ρ

ⁿ⁺¹u^∗

−

t

∇ ·

2

μ

ⁿ⁺¹D

¯¯

^∗

= ρ

ⁿ⁺¹

uⁿ

−

t

uⁿ

· ∇

^un

,

μ ^∂

^u

∂

n

= −

^fτ

,

⁽⁴⁸⁾

with the source termdue to the tangent componentof the singular force imposed as a jump on the viscous stress. It is noteworthythat, unlike thedelta formulation, thepredictionstep ofthe projection methodonlycontains this tangent component,sincethenormalcomponentisaccountedforintheprojectionandcorrectionsteps.Theanalogybetweensharp methodsusingjumpconditionsandthe“delta”formulationisthoroughlyexplainedin[46].Thelinearsystemresultingfrom (48) isasinglelinearsystemwhereallthevelocitycomponentsarecoupled.Thisallowsafullyimplicitdiscretizationofthe viscoustermwithoutadditionaltimestepconstraintsduetoviscosityjumpanditcanbesolvedeasilywithaGauss-Seidel algorithm aspointed out by Lepilliez etal.in [2]. The numericalimplementation ofthe jumpconditions on theviscous stresstensorisdetailedinsection3.5.

ThenthePoissonequationissolvedwith

∇ ·

∇

^pn+1

ρ

ⁿ⁺¹

= ∇ ·

^u∗

t

,

(49)

withthefollowingpressurejumpconditionaccountingforthenormalcomponentofthelocalforce

[

^p

]

=

^fn

,

(50)

andthefollowingjumpconditiononthepressurenormalgradient

n

· ∇

^pn+1

ρ

ⁿ⁺¹

=

μ

n

·

uⁿ

ρ

ⁿ⁺¹

,

(51)

details ofwhichcan befound inAppendixB.The numericaldiscretizationofthissingularsource termis identicaltothe onedescribedbyLiuetal.in[20] andisdetailedinsection3.4.

These equations based on a jump condition formulation can alsobe rewrittenin a similar way withsingular source termsinsteadofjumpcondition.Itwillgiveamathematicallyidenticalsystemthatwillexpressas

ρ

ⁿ⁺¹u^∗

−

t

∇ ·

2

μ

ⁿ⁺¹D

^¯¯

^∗

= ρ

ⁿ⁺¹

uⁿ

−

t

uⁿ

· ∇

^un

+

tf_τ

δ

,

(52)

to compute the velocity fieldin thepredictor step.Next,the pressurefield can be computedwiththe appropriate jump conditionwiththefollowingequation

∇ ·

∇

^pn+1

ρ

ⁿ⁺¹

= ∇ ·

^u∗

t

+ ∇ ·

f_nn

δ

ρ

ⁿ⁺¹

,

(53)

which is identical to equations (49) and (50). Indeed, the term 2[

μ

]

∂u

∂n·^{nis included in} ∇ ·^{u∗,for more details, see} AppendixB.Finally,thevelocityiscorrected,

uⁿ⁺¹

=

u^∗

−

t

ρ

ⁿ⁺¹

^(∇

^p

n+¹

−

fnn

δ

).

(54)

Byconsideringthatmodificationsofthenumericalschemesaccountingforjumpconditionsactassharpapproximations ofsingular sourceterms,itcanbeunderstoodthat thetwopresentedformulationsareidentical.Seeforinstance[46] for moredetailsonthisspecificpoint.

3.3. Discretisationofthemembranemodel

Considering the Cartesian mesh, the standard MACgrid is used: the scalar variables are located in the centreof the mesh cells andthe components ofthe velocity field are staggered at the cell faces in each direction. Unlikein [15], all thecomponentsofthedifferentsolidtensorsarelocatedatthecentreofthemeshcells inthisstudy.Thelocationofthe differentvariablesina2DmeshcellisillustratedinFig.1.

At each time step, theposition ofthe interface is updated using(28). Thesix componentsof theintermediate strain tensorG¯¯sin3Darethencomputedusingequation(19)

G

¯¯

ⁿ_s⁺¹

= ¯¯

Gⁿ_s

−

t

(

usn

· ∇)

G

¯¯

ⁿ_s

− (∇

usn

)

G

¯¯

ⁿ_s

− ¯¯

Gⁿ_s

(∇

usn

)

^T

.

(55)

The advectiontermiscomputedwiththe WENO-Zscheme[41] and thevelocityderivativesare obtainedwithcentral differencing schemes. Forexample, in 2D, the advectionterm corresponding to the first component ofthe intermediate straintensoriscomputedas

(

u_sⁿ

· ∇)

^Gn₁₁

=

^us

i,j

∂

G₁₁

∂

x

i,j

+

^vs

i,j

∂

G₁₁

∂

y

i,j (56)

with

Fig. 1.Location of the variables on a 2D mesh cell.

u_s

i,j

=

¹ 2

u_s

i−¹/2,j

+

^us

i+¹/2,j

andv_s

i,j

=

¹ 2

v_s

i,j−¹/2

+

^vs

i,j+¹/2

.

(57)

Consideringnowthederivativesofthex-componentofthevelocity,

∂

us

∂

x

i,j

=

^us

|

i+¹/2,j

−

^us

|

i−¹/2,j

x and

∂

us

∂

y

i,j

=

^us

|

i,j+¹/2

−

^us

|

i,j−¹/2

y

,

(58)

with

u_s

i,j±¹/2

=

¹ 4

u_s

i−¹/2,j

+

^us

i+¹/2,j

+

^us

i−¹/2,j±¹

+

^us

i+¹/2,j±¹

.

(59)

The computation ofthe components ofthe surface left Cauchy-Green tensor andthe stress tensoratthe cell centres followsequations(14) and(20).

Thecomponentsofthesurfacedensityforcefarecomputedatthecentreofthefacesofthecellsfollowing(25).Con- sideringthesharpmethodologyandtheforcedecomposition(26,27),thenormalcomponentoftheforceiscomputedatthe centreofthecellandthetangentcomponentsarestaggeredatthecellfaces.Consideringa2D example,thediscretisation ofthenormalcomponentoftheforce,following(26),is

f_n

i,j

= σ

11

i,j

∂

n_x

∂

x

i,j

+ σ

12

i,j

∂

n_x

∂

y

i,j

+ σ

21

i,j

∂

n_y

∂

x

i,j

+ σ

22

i,j

∂

n_y

∂

y

i,j

= σ

11

i,j

n_x

|

i+1,j

−

ⁿx

|

i−1,j

2

x

+ σ

12

i,j

n_x

|

i,j+1

−

ⁿx

|

i,j−1

2

y

+ σ

21

i,j

n_y

|

i+1,j

−

ⁿy

|

i−1,j

2

x

+ σ

22

i,j

n_y

|

i,j+¹

−

ⁿy

|

i,j−¹

2

y

.

(60)

Thetangentcomponentoftheforceinthex-directionfrom(27) gives

f_τ

·

^ex

i+1/2,j

=

^P11

i+1/2,j

∂ σ

11

∂

x

i+¹/2,j

+ ∂ σ

12

∂

y

i+¹/2,j

+

^P12

i+1/2,j

∂ σ

12

∂

x

i+¹/2,j

+ ∂ σ

22

∂

y

i+¹/2,j

=

P11

i+1/2,j

σ

11

|

i+¹,j

− σ

11

|

i,j

x

+ σ

12

|

i+¹/2,j+¹/2

− σ

12

|

i+¹/2,j−¹/2

y

+

^P12

i+¹/2,j

σ

12

|

i+¹,j

− σ

12

|

i,j

x

+ σ

22

|

i+1/2,j+1/2

− σ

22

|

i+1/2,j−1/2

y

,

(61)

with

P_kl

i+1/2,j

=

¹ 2

P_kl

i+1,j

+

^Pkl

i,j

and

σ

_kl

i+1/2,j+1/2

=

¹ 4

σ

_kl

i,j

+ σ

_kl

i+1,j

+ σ

_kl

i,j+1

+ σ

_kl

i+1,j+1

.

(62)

Withthe“delta”formulation,thediscretisationoftheelasticforce correspondstoequation(61) with ¯¯I insteadof P¯¯ to considerall the componentsofthe force. Thismethodis identicalinthe other directionsandcan easily be extrapolated in 3D.

3.4. Jumpconditiononthepressure

ThepressurejumpistakenintoaccountfollowingtheGhostFluidMethod[19,20].ThePoissonequationonthepressure (53) canbewrittenas

∇ · (β ∇

^p

) =

^{R H S}

,

(63)

withβ=¹/

ρ

thediffusioncoefficient.Theequation containsjumpconditionsonthepressuredenoteda= ^fn andonthe diffusioncoefficient[β].

Thediscretisationofthe2DPoissonequationatpoint(i,j)gives

β

i+¹/2,j

_p

i+1,j−pi,j

x

− β

i−¹/2,j

_p

i,j−pi−1,j

x

x

+ β

i,j+¹/2

_p

i,j+1−pi,j

y

− β

i,j−¹/2

_p

i,j−pi,j−1

y

y

=

^{R H S}i,j

+

^gi,j

,

(64) with(βi+¹/2,j,βi−¹/2,j,βi,j+¹/2,βi,j−¹/2)theharmonicaveragesofthediffusioncoefficientatthecentreofthecellborders.

Iftheinterfacecrossestherightmeshsegment[^xi,j,xi+¹,j]^,

β

i+¹/2,j

=

⎧ ⎪

⎪ ⎨

⎪ ⎪

⎩

β

⁺

β

⁻

β

⁻

θ

^R

+ β

⁺

(

1

− θ

^R

)

^if

φ

i,j

<

0 and

φ

i+¹,j

>

0

, β

⁺

β

⁻

β

⁺

θ

^R

+ β

⁻

(

1

− θ

^R

)

^if

φ

i,j

>

0 and

φ

i+¹,j

<

0

,

(65)

with

θ

^R

= |φ

i+1,j

|

|φ

i+¹,j

| + |φ

i,j

| .

(66)

Similarly,iftheinterfacecrossestheleftmeshsegment[^xi−¹,j,x_i,j]^,

β

i−¹/2,j

=

⎧ ⎪

⎪ ⎨

⎪ ⎪

⎩

β

⁺

β

⁻

β

⁻

θ

^L

+ β

⁺

(

1

− θ

^L

)

^if

φ

i,j

<

0 and

φ

i−¹,j

>

0

, β

⁺

β

⁻

β

⁺

θ

^L

+ β

⁻

(

1

− θ

^L

)

^if

φ

i,j

>

0 and

φ

i−¹,j

<

0

,

(67)

with

θ

^L

= | φ

_i−1,j

|

|φ

i−1,j

| + |φ

i,j

| .

(68)

The g_i,jtermin(64) correspondstothejumpsenforcedwhentheinterfacecrossesatleastoneofthefourneighbouring meshsegments.ThelatteraredenotedbythesuperscriptsR,L,TandBwhichcorrespondrespectivelytotheright,left,top andbottomborders.Asageneralrule,

g_i,j

=

^g_i^R_,j

+

^g_i^L_,j

+

^g_i^T_,j

+

^g_i^B_,j

.

(69)

Eachoneofthesevaluesexistsonlyifthemembranecrossesthemeshsegmentsandthenequals

g_i^R_,j

= ± β

i+¹/2,ja^R

x²

,

g_i^L_,j

= ± β

i−¹/2,ja^L

x²

,

g^T_i,j

= ± β

i,j+¹/2a^T

y²

,

g_i^B_,j

= ± β

i,j−¹/2a^B

y²

,

(70)

with±correspondingtotheoppositesignofφi,jand

a^R

=

^ai,j

θ

^R

+

^ai+¹,j

(

1

− θ

^R

),

a^L

=

^ai,j

θ

^L

+

^ai−¹,j

(

1

− θ

^L

),

(71) a^T

=

^ai,j

θ

^T

+

^ai,j+1

(

1

− θ

^T

),

a^B

=

^ai,j

θ

^B

+

^ai,j−1

(

1

− θ

^B

).

(72) Theextensionoftheseschemesto3Dproblemsisstraightforward.

3.5. Jumpconditiononthevelocityderivatives

The discretisationofeach componentoftheviscous-stresstensorisdescribed inthissection withaspecificemphasis onhowtoenforcesuitablejumpconditions.Consideringa2Dexample,thedivergenceoftheviscous-stresstensorgives

∇ · (

2

μ

D

¯¯ ) =

⎛

⎝

∂

∂x

2

μ

^∂_∂^u_x

₊

_∂^∂_y

μ

_∂^∂u_y

₊

^∂_∂^v_x

∂

∂x

μ

∂u

∂y

+

^∂_∂^v_x

+

_∂^∂_y

2

μ

^∂_∂^v_y

⎞

⎠ .

(73)