A time splitting projection scheme for compressible two-phase flows. Application to the interaction of bubbles with ultrasound waves

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To link to this article : DOI:10.1016/j.jcp.2015.09.019

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To cite this version : Huber, Grégory and Tanguy, Sébastien and

Béra, Jean-Christophe and Gilles, Bruno A time splitting projection

scheme for compressible two-phase flows. Application to the

interaction of bubbles with ultrasound waves. (2015) Journal of

Computational Physics, vol. 302. pp. 439-468. ISSN 0021-9991

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A

time

splitting

projection

scheme

for

compressible

two-phase

flows.

Application

to

the

interaction

of

bubbles

with

ultrasound

waves

Grégory Huber

a

,

b

,

Sébastien Tanguy

a

,

b

,

∗

,

Jean-Christophe Béra

c

,

d

,

Bruno Gilles

c

,

d

a_{Université de Toulouse, France}

b_{Institut de Mécanique des Fluides de Toulouse, France}

c_{Inserm U1032, Lab. of Therapeutic Applications of Ultrasound, Lyon, France} d_{Université Claude Bernard Lyon 1, Lyon, France}

a

b

s

t

r

a

c

t

Keywords:

Two-phaseflows

LowMachcompressiblesolver Levelset

Ghostfluid Projectionmethod

Thispaperisfocusedonthenumericalsimulationoftheinteractionofanultrasoundwave withabubble.Ourinterestistodevelopafullycompressiblesolverinthetwophasesand toaccountforsurfacetensioneffects.

Asthevolumeoscillationofthe bubbleoccursinalow Machnumberregime,aspecific care must be paid to the effectiveness of the numerical method which is chosen to solvethecompressibleEulerequations.Threedifferentnumericalsolvers,anexplicitHLLC (Harten–Lax–vanLeer-Contact)solver[48],apreconditioningexplicitHLLCsolver[14]and the compressible projection method[21,53,55], are described and assessed with aone dimensional sphericalbenchmark.Fromthispreliminary test, wecan concludethatthe compressibleprojectionmethodoutclassestheothertwo,whetherthespatialaccuracyor thetimestepstabilityareconsidered.

Multidimensionalnumericalsimulationsarenextperformed.Asabasicimplementationof thesurfacetensionleadstostrongspuriouscurrentsandnumericalinstabilities,aspecific velocity/pressuretimesplittingisproposedtoovercomethisissue.Numericalevidencesof theefficiencyofthisnewnumericalschemeareprovided,sinceboththeaccuracyandthe stabilityoftheoverallalgorithmareenhancedifthisnewtimesplittingisused.Finally,the numericalsimulationoftheinteractionofamovinganddeformablebubblewithaplane waveispresentedinordertobringouttheabilityofthenewmethodinamorecomplex situation.

1. Introduction

Theinteraction ofabubble withultrasoundwaves canleadto volumeoscillations andtodeformationsofthebubble, generatingmicrostreaminginitsvicinity.Iftheamplitudeoftheultrasoundwavesissufficientlylarge,thebubblecollapses anditcanbreakupintodaughterbubbles,generatingashockwave duringthephaseofseparation.Thesephenomenacan be usefulin manyindustrial andmedicalapplications.In waste waterpurification, chemical,pharmaceutical, mechanical

*

Correspondingauthor.

and food industry, as well as for ordinary instrument cleaning, sonicators and sonochemical reactors are based on the action ofbubbles submittedto ultrasound.Inmedicine,currentclinical applicationsof high-intensityfocused ultrasound, namelyextracorporealkidneystonedestructionandprostatetumorablation,alsoinvolveultrasoundbubbles[32].Moreover, promisinginvestigationsconcernnewtherapymodalitiesspecificallybasedonthesephenomena,aimingnotablytotargeted drugdelivery[39],extracorporealsonothrombolysis[41]andcellsonoporation[33].Theseultrasoundtechniquesaresubject to active research, concerning in particularthe interaction betweena bubbleand asolid wall or a livingcell [28,52,58]. Numerical andtheoreticalinvestigations arerequiredto helpunderstanding themechanismsofbubbleaction involvedin theseapplications.

The development of numerical methods for compressible solvers in the framework of interface capturing (Level-Set method) or interface-tracking methods (Front-Tracking method) for two-phase flows is still an active topic of research. Many existing works [5,8,16,25,35,47]propose numerical strategies tocompute theinteraction of bubbles ordrops with shockwaves.Inthesestudies,fullexplicitshock-capturingsolversare usedforthesimulationsofcompressibleflows.Itis wellknownthatthesesolversareperfectlysuitedtocomputetheformationandthepropagationofshockwaves,butworks poorlyinthelowMachnumberregime[14,49].

As a result, such solvers are notefficient to compute thecollapseof abubble dueto theinteraction withultrasound waves.Indeed,thecollapseofthebubblecanbedividedindifferentstepsinvolvingveryheterogeneousvaluesofthelocal Mach number.Inthefirstone,thevolumeofthebubbleoscillatesduetotheinteractionwiththeultrasoundwaves.This initial step occursessentially ina low Mach number regime (in theliquid phase). In the second one, if the ultrasound wave amplitudeissufficientlystrong,abubblebreak-upoccursandleadstotheformationofashockwave.Therefore,the computation ofthissecond steprequirestheuseofa fullexplicitshockcapturingsolvertocompute accuratelytheshock wavepropagationwhereasthissolverisnotefficienttocomputetheinitialstepofthephysicalprocess.IntheRef.[42],the authorshaveperformednumericalsimulationswithaHLLCsolveroftheexpansionofabubbleinteractingwithultrasound wavesina one-dimensionalsphericalcoordinatesystem. Inthat paper,theypresentcomputationswithanumberofgrid points up to 10 000 in order to achieve a sufficient spatial convergence of the computation with only one bubble. This drasticconstraintonthespatialcellsizelimitstremendouslytheabilityofexplicitcompressiblesolvers,todealwithmore complex problemsinvolvingmultidimensional features,asit occursinmanypractical situations.Forexample, during the interaction ofa bubblewithan ultrasoundplane wave, thebubblecandisplace andundergo non-sphericaldeformations. A multidimensional modeling is relevantin manyothers situations ofinterest, asthe bubble collapseclose to a wall in ordertocomputethemechanicalstressesactingonawall,orifanincreasingcomplexityapproachisconsidered,tomodel thecollectiveeffectsduringthecollapseofseveralbubblesinteractingwithultrasoundwaves.

Severalotherstudies[38,57]havebeendedicatedtotheinitialstepofthebubbleoscillationbeforethecollapse.Asthe Mach numberremains lowinthisstep,thesestudiesassume thatthe liquidremainsincompressibleandthatthe bubble pressureisspatiallyconstant.ThetemporalvalueofthepressureinthebubblecanthenbeimposedasaDirichlet bound-arycondition tosolve thepressurefield intheincompressibleliquid.Whereastheseapproachesprovidevery satisfactory results to compute the volume bubble oscillation since an incompressiblesolver works perfectly well inthe Low Mach numberregime,theyarelimitedintheirrepresentationofthephysicalreality.Indeed,theyareunabletocapturetheshock formation during thebubblecollapseandits propagationinthe liquidphase.Moreover, itis not alwaysrelevant to con-siderthatthepressureremainsuniforminsidethebubbleifitisstronglydeformed,sincethevariationsofcurvaturealong the interface willinvolvevariations onthe pressurejumpcondition. Thus, novelnumericaltools,whichallow computing accurately thevarioussteps ofthebubblecollapse,arestillrequiredtoimproveouroverallunderstandingofthiscomplex phenomenon.Inparticular,afewstudies[4,13,37,56]havebeendedicatedtothedevelopmentofnumericalmethodsto per-formthedirectnumericalsimulationsoftheinteractionofaninterface,separatingacompressibleliquidandacompressible gas,withanacousticoranultrasoundwaveinthelowMachnumberregime.

In theframeworkofone-phase flow,differentnumericalstrategies[2,14,18,21,49,50,54,55] existtoovercometheissue of computingcompressiblelowMach numberflow. Ausual strategyconsistsinenhancing aHLLC (Harten–Lax–van Leer-Contact) solver witha preconditioningmethod [14,49] toimprovethe spatial accuracy inthelow Mach numberregime. However, when an explicit temporaldiscretization isused, thissimplemethod leads to atime step even morestringent thantheclassicalHLLCsolver(alreadytoosmallforthelowMachnumberregime).Animplicittemporaldiscretizationcan be usedtoincreasethetimestep,butitsimplementationismuchmorecomplex[27].Anotherkindofmethods,basedon the incompressibleprojection method,canbe considered.Thiscompressibleprojection methodconsistsina semi-implicit temporal scheme tosolve the compressibleEuler equations[21,53–55]. Itallows splitting the computation ofconvective termsinanexplicittemporalschemeandtheacoustictermsinanimplicittemporalscheme.IncomparisontoHLLCsolver, itimprovesboththespatialaccuracyinthelowMachnumberregimeandthetemporalstabilityofthenumericalscheme, sincethesoundvelocitycanberemovedfromtheconvectivetimestepconstraint.Ithaseverbeenusedintheframework oftwo-phaseflowsin[17]butonlyfortheliquidphasesincetheauthorsmadetheassumptionofauniformgaspressure. Moreover,thenumericalmethod,whichisdescribedinthatpaper,donotallowtoaccountforsurfacetensioneffect.

Thus, weproposeinthispaper,anewtime splittingprojectionschemeforcompressibletwo-phase flows,whichtakes intoaccountsurfacetensioneffectsandallowscomputingthepressure,thedensityandthevelocityfieldinthetwophases. The interfacemotionisdescribed withaLevelSetmethod[36,43].Thejumpconditionsare imposedfollowingthe frame-workoftheGhostFluidMethodtocomputeanaccurateandsharpdescriptionoftheinterface.Ourapproachmixesfeatures ofnumericalmethodsdedicatedtoincompressible[19,30]andsupersoniccompressible[8]two-phaseflows.Themain

nov-eltyofthispaperisanewtimesplittingcompressibleprojectionmethodtocomputeaccuratelythesurface tensioneffects inthecompressiblelowMachnumberregime.Inparticular,thisnewmethodpreventsfromnumericalinstabilitiesoccurring whenthejumpconditiononpressure,duetosurfacetension,isroughlyincorporatedinthepressurecomputation.

Anumericaltesthasbeendesignedinaonedimensionalsphericalconfigurationandcanbecomparedtothesolutionof theRayleigh–Plessetequation.ComparisonsbetweenanexplicitHLLCsolver,apreconditionedexplicitHLLCanda compress-ibleprojection methodarenext presented.Asthecompressibleprojectionmethodoutclassestheothertwo,theextension ofthe compressibleprojection method tomultidimensional flows accountingforsurface tension effectsis proposed. The requirementtodevelop anewtimesplittingtoincludeaccurate andstablecomputationsofsurfacetensionishighlighted, andtheconvergenceofthenewmethodisshownonamultidimensionaltest-case.Finally,asimulationofamovingdroplet, whichundergoesnon-sphericaldeformationsduetotheinteractionwithanultrasoundplanewave,ispresented.

2. Governingequationsandnumericalmethodsforaone-phaseflow

2.1. Governingequations 2.1.1. Eulerequationssystem

Inviscidcompressibleflowscanbecomputedbysolvingthefollowingsystemofpartialdifferentialequations(1)which expressrespectivelytheconservationofthe mass,momentum andenergy.Thissystem, knownasEulerequations,canbe expressedwithaconservativeformulation:

∂

ρ

∂

t

+ ∇ · (

ρ

u



)

=

0

∂ρ



u

∂

t

+ ∇ · (

ρ



u

⊗ 

u

)

+ ∇

p

=

0

∂

ρ

E

∂

t

+ ∇ · ((

ρ

E

+

p

)

u



)

=

0 (1)

Orwithaprimitiveformulation:

∂ρ

∂

t

+ 

u

· ∇

ρ

+

ρ

∇ · 

u

=

0

∂



u

∂

t

+ 

u

· ∇

u

+

∇

p

ρ

=

0

∂

p

∂

t

+ 

u

· ∇

p

+

ρ

c 2

_{∇ · }

_u

₌

₀ (2)

where

ρ

isthedensity,u the



velocity, p the pressure,E thetotalenergy(E

=

e

+

1₂



u

· 

u,wheree istheinternalenergy) andc thesoundspeed.

Asinviscidflowsareconsidered,allthedissipativeeffectsareneglected,thustheentropyisalsoaconservativevariable. Itcanbeexpressedwiththefollowingequation:

Ds

Dt

=

0 (3)

Inone-dimensionalCartesiancoordinate,System(2)reads

∂

W

∂

t

+

A

(

W

)

∂

W

∂

x

=

0 (4) withW

= (

ρ

u p

)

T,and A

(

W

)

=

⎛

⎜

⎝

u

ρ

0 0 u _ρ1 0

ρ

c2 u

⎞

⎟

⎠

(5)

Threeeigenvalues ofAmatrixare

λ

1

=

u

−

c,

λ

2

=

u and

λ

3

=

u

+

c,anditsthreelefteigenvectorsare

l1

=

⎛

⎝

₋

0

ρ

c 1

⎞

⎠ ,

l2

=

⎛

⎝

−

c 2 0 1

⎞

⎠ ,

l3

=

⎛

⎝

ρ

0c 1

⎞

⎠ .

The compressible Euler equationsSystem (2) is hyperbolic which means that the solutions of System(2) have wave properties:adisturbancepropagatesalongthecharacteristicsofthesystem.Thismathematicalpropertyisthebasisofthe acousticalwavespropagationinfluidmechanics.

Ifanincompressibleflowisconsidered,theEulerequationssimplywrites:

∇ · 

u

=

0

∂



u

∂

t

+ 

u

· ∇

u

+

∇

p

ρ

=

0 (6)

2.1.2. Systemclosure:equationofstate

ThesystemofEulerequationsisnotcomplete,andanequationofstatemustbeaddedtoclosethesystem.Anexpression ofthesoundspeedinthefluidisrequired.Letusremindthedefinitionofthesoundspeed:

c2

=

∂

p

∂

ρ



s=cste

(7)

Ifaperfectgasisconsidered, p

=

ρ

rT (r

=

R

/

M where R

=

287,058J kg−1K−1),thesoundspeedis:

c2

=

γ

p

ρ

.

(8)

If a liquid isconsidered, the Tait equation of state [8] can be used, p

=

B

((

_ρρ

0

)

γ

₋

₁₎

₊

_p

0. It leads to the following relationforthesoundspeedinaliquidphase

c2

=

γ

B

ρ

0

(

ρ

ρ

0

)

γ−1

.

(9)

withthefollowingvalueforwater: p0

=

105Pa andB

=

3,31

×

108Pa. 2.2. Numericalmethods

WepresentinthissectionsomeusualnumericalmethodsusedtosolvethecompressibleEulerequations. 2.2.1. Riemanntypesolver

This solver is commonly used to compute compressibleflows. Forthe sake ofsimplicity, we detail the method ina one dimensionalCartesian coordinatesystem,butmultidimensionalextensionscanbe easilyderived. TheEulersystemin conservativeform(1)canbeexpressedas:

∂

U

∂

t

+

∂

F

∂

x

=

0 (10)

withU

= (

ρ ρ

u

ρ

E

)

T,F

= (

ρ

u

ρ

u2

+

p

(

ρ

E

+

p

)

u

)

T. TheGodunovschemereads

Un_i+1

=

Un_i

−



t



x

(

F

n

i+1/2

−

Fin−1/2

)

(11)

FluxesarecomputedusingtheHLLCsolver[48].

FHLLC_i+1/2

=

F

(

U_iL_+1/2

,

U_iR_+1/2

)

(12)

ThestateatedgemeshUL

i+1/2andUiR+1/2iscalculatedwithaWENOscheme[15]onthecharacteristicvariablesvectors X_iL_+1/2and X_iR_+1/2 whichreads

X

=

(

p

−

ρ

cu

) (

p

+

ρ

c2

) (

p

+

ρ

cu

)

T

(13)

XL

i+1/2and XiR+1/2arerespectivelycalculatedwiththeleftstencilandtherightone. Themaximumtimestepisestimatedby



tHLLC

=

max

(

_|u+xc|

,

|u−xc|

).

OtherRiemannsolversexistsuchtheRoesolver[40],theHLLorHLLEsolvers[7],buttheHLLCsolveristhemost com-monlyusedduetoitsaccuracytocomputehighMachnumberflowssuchasshockwavespropagation.However,iftheMach number islow,asforthe simulationswhich areconsidered inthispaper,its efficiencyconsiderablydecreases.Therefore, we arenowgoingtointroducesomespecificnumericalmethodswhichareabletocomputeaccuratelycompressibleflows inthelowMachnumberregime.

2.2.2. LowMachpreconditioning

ThemisleadingbehavioroftheHLLCsolverforlowMachnumbercanbetheoreticallyjustified[14,27,49]by demonstrat-ingthatthecompressibleEulerequationsdoesnotconvergetotheincompressibleEulerequationswhentheMachnumber M tends to zero.Thisdemonstration,basedon anasymptoticexpansion depending onMach number,isbriefly reminded here.

CompressibleEulersystemlimitwhentheMachnumberMatendstozero TheprimitiveformulationoftheEulerequations(2)is

expressedwithdimensionlessvariablestocarryouttheasymptoticexpansion.Thevariablesarewrittenas

α

= ˜

α

[

α

]

,where

[

α

]

isa characteristicscale and

α

˜

isthe corresponding dimensionlessvariable.For sake ofsimplicity,a one-dimensional Cartesiancoordinatesystemisstillconsidered.Choosing

[

u

]

= [

x

]/[

t

]

,theEulerdimensionlesssystemcanbeexpressed:

∂

ρ

˜

∂ ˜

t

+ ˜

u

∂

ρ

˜

∂

x

˜

+ ˜

ρ

∂

u

˜

∂

x

˜

=

0

∂

u

˜

∂ ˜

t

+ ˜

u

∂

u

˜

∂

x

˜

+

[

p

]

[

ρ

][

u

]

2 1

˜

ρ

∂

p

˜

∂

x

˜

=

0

∂

p

˜

∂ ˜

t

+ ˜

u

∂

p

˜

∂

˜

x

+

[

ρ

][

c

]

2

[

p

]

ρ

˜

˜

c 2

∂

u

˜

∂

x

˜

=

0 (14)

Thecorrespondingentropydimensionlessequationreads:

D

˜

s

Dt

˜

=

0 (15)

InordertoshowuptheMachnumber,wechose

[

p

]

= [

ρ

][

c

]

2_{.Thedimensionlesssystembecomes}

∂

ρ

˜

∂ ˜

t

+ ˜

u

∂

ρ

˜

∂

x

˜

+ ˜

ρ

∂

u

˜

∂

x

˜

=

0

∂

u

˜

∂ ˜

t

+ ˜

u

∂

u

˜

∂

˜

x

+

1 Ma2 1

˜

ρ

∂

p

˜

∂

˜

x

=

0

∂

p

˜

∂ ˜

t

+ ˜

u

∂

p

˜

∂

x

˜

+ ˜

ρ

c

˜

2

∂

u

˜

∂

x

˜

=

0 (16)

Wenow calculatethelimit ofthisprevious systembyan asymptoticanalysis. Eachvariableis expressedas

α

=

α

0

+

α

1

+



2

α

2 where



−→

0.System (16)isexpandedby considering that theMach numbertends tozero,so Ma

=



. To alleviatethenotation,

∼

isremoved,butintherestofthedemonstrationallthevariablesaredimensionless.

Atorder



−2_,weobtain

∂

p0

∂

x

=

0 (17) attheorder



−1_,

∂

p1

∂

x

=

0 (18)

andattheorder0,

∂ρ

0

∂

t

+

u0

∂

ρ

0

∂

x

+

ρ

0

∂

u0

∂

x

=

0

∂

u0

∂

t

+

u0

∂

u0

∂

x

+

1

ρ

0

∂

p2

∂

x

=

0

∂

p0

∂

t

+

ρ

0c 2 0

∂

u0

∂

x

=

0 (19)

We can observe that the limit of the compressible Eulersystem doesnot matchto the incompressible Eulersystem whichreadsafterthesameasymptoticanalysisprocedure:

ρ

=

cste

∂

u0

∂

x

=

0

∂

u0

∂

t

+

1

ρ

0

∂

p2

∂

x

=

0 (20)

Indeed,thevelocitydivergenceisdifferentofzero.Toinsurethatthevelocityfieldwillbedivergence-free,acoefficient canbeaddedinthepressureequationoftheSystem(19)allowingtouncouplethetwotermsoftheequationforthelimit calculation.ThisideahasbeeninitiallyproposedbyTurkel[49].

∂

ρ

0

∂

t

+

u0

∂

ρ

0

∂

x

+

ρ

0

∂

u0

∂

x

=

0

∂

u0

∂

t

+

u0

∂

u0

∂

x

+

1

ρ

0

∂

p2

∂

x

=

0

∂

p0

∂

t

+

M 2

_ρ

0c20

∂

u0

∂

x

=

0 (21)

ThispreconditioningallowstheSystem(21)totendtotheincompressibleSystem(20)whentheMachnumbertendsto zero,with M

=

Ma. TheSystem(21)ishyperbolic,andthefollowing wavespeedscan bedefined:u, u

+

c∗₊ andu

−

c∗₋,

where c∗₊

=

(

1

−

M 2

₎

_u

₊



₍

_M2

₋

₁

₎

_u2

₊

_4M2_c2 2 c∗₋

=

(

M 2

₋

₁

₎

_u

₊



₍

_M2

₋

₁

₎

_u2

₊

_4M2_c2 2 (22)

Thesewavesspeedsareonlyusedforthecomputation ofthefluxes, sincetheoverallschemeremains identicaltothe HLLCsolver[14]andthemaximumtimestepisestimatedby



tLM

=

M



tHLLC [27].

2.2.3. Semi-implicitscheme:projectionmethod

This methodhasbeenoriginally proposedby Yabe[55] andhasbeenpursued byXiao[53] andKwatra etal.[21] for one phase flows. The main idea consists ina splitting of the Eulerequations in two subsystems. The first one contains the convective partof theEulerequations, andthesecond one contains the acousticpart oftheEuler equations.As the second partissolvedwithanimplicittemporaldiscretization,therestrictivetimestepconditionduetothepropagationof acoustic wavesisremoved.Thus, thetimestep restrictiononlydependsonthemaximumconvective velocity.As aresult a significant benefiton theoverall time stepcan be expectedifthismethodis usedin alow Mach numberregime. For example in [21], theauthors report a speed-upof 300on the time step incomparison to aHLLC solver. Moreover, this compressible projection method isidentical tothe incompressibleprojection methodifthe Mach numbertends to zero. Thisremarkablefeatureguaranteesthatthissolverwillbehaveaccuratelyintheincompressiblelimit.Themethodisnow brieflydescribedbyapplyingthefollowingsplittingtothesystemofEulerequations:

⎛

⎝

ρ

ρ

u



ρ

E

⎞

⎠

t

+

F1

+

F2

=

0 (23) with F1

= ∇.

⎛

⎝

ρ



u

ρ

⊗ 

u



u

ρ

Eu



⎞

⎠

(24) and F2

=

⎛

⎝

_∇

0p

∇ · (

p



u

)

⎞

⎠ =

0 (25)

F1 istheadvectionpartandF2 thenon-advectionpart.

Advectionsub-system The first sub-system contains the massconservation terms,andthe advectionpart ofvelocity and pressureequations.Itreads:

∂

ρ

∂

t

+ 

u

· ∇

ρ

+

ρ

∇ · 

u

=

0

∂



u

∂

t

+ 

u

· ∇

u

=

0

∂

p

∂

t

+ 

u

· ∇

p

=

0 (26)

Inone-dimensionalCartesiancoordinate,theadvectionsub-system reads

∂

W

∂

t

+

A

(

W

)

∂

W

∂

x

=

0 (27)

A

(

W

)

=

⎛

⎝

u0

ρ

u 00 0 0 u

⎞

⎠

(28)

ThematrixAhas3eigenvalueswhichareall

λ

=

u,andthreelefteigenvectors:

l1

=

⎛

⎝

00 0

⎞

⎠ ,

l2

=

⎛

⎝

01 0

⎞

⎠ ,

l3

=

⎛

⎝

00 1

⎞

⎠ .

Because l1 is zero, the A matrix is non-diagonalizable, consequently the advection sub-system is not hyperbolic. Nev-ertheless, the time step restriction is calculated with



t

=

_Maxx_(u). This sub-system is solved with an explicit temporal discretization.

Acousticalsub-system TheremainingtermsoftheEulersystemaregroupedinthesecondsub-system.Itcorrespondstothe acousticpartofthemainsystem.IfanexplicitHLLCsolverisused,thisacousticpartisresponsibleforthestringenttime steprestrictionduetothepropagationofacousticwaves.

∂

ρ

∂

t

=

0

∂

u



∂

t

+

∇

p

ρ

=

0

∂

p

∂

t

+

ρ

c 2

_{∇ · }

_u

₌

₀ (29)

Inone-dimensionalCartesiancoordinates,System(29)reads

∂

W

∂

t

+

A

(

W

)

∂

W

∂

x

=

0 (30) with A

(

W

)

=

⎛

⎝

00 00 0_ρ1 0

ρ

c2 0

⎞

⎠

(31)

Threeeigenvalues ofAmatrixare

λ

1

= −

c,

λ

2

=

0 and

λ

3

= +

c,anditsthreelefteigenvectorsare

l1

=

⎛

⎝

₋

0

ρ

c 1

⎞

⎠ ,

l2

=

⎛

⎝

10 0

⎞

⎠ ,

l3

=

⎛

⎝

ρ

0c 1

⎞

⎠ .

Consequently, System (29) is hyperbolic which is relevant to correctly compute the acoustical waves propagation. If an implicittemporaldiscretizationisappliedtosolvethissubsystem,the‘acoustical’timesteprestrictionwillberemoved,the restrictionwillbebasedonthemaximumoftheconvectionvelocity u,andneitheronu

+

c nor u

−

c asitwouldbewith afullyexplicitsolver.

Timediscretization Following the semi-implicit strategy introduced above, the time discretization of System (2) is now described:

ρ

n+1

₋

_ρ

n



t

+ ∇ · (

ρ

u



)

n

₌

₀



un+1

− 

un



t

+ (

u

· ∇

u

)

n

₊

(

∇

p

)

n+1

ρ

n+1

=

0 pn+1

−

pn



t

+ (

u

· ∇

p

)

n

_{+ (}

_ρ

_c2

₎

n+1

₍

_{∇ · }

_u

₎

n+1

₌

₀ (32)

Firstly,thefluiddensityisupdatedbysolvingthemassconservationequationwhichisusedinaprimitiveformulation:

ρ

n+1

=

ρ

n

− 

t

((

u



· ∇

ρ)

n

+

ρ

n

∇ · 

un

)

(33)

Tocomputethepressure,theenergyconservationequationcanbeexpressedasfollows:

Thefollowingtimesplitting,typicalofincompressibleprojectionmethods,cannextbeapplied:



u∗

= 

un

− 

tu



n

· ∇

un (35) and



un+1

= 

u∗

− 

t

∇

p n+1

ρ

n+1 (36)

Byusingtheequation(36),wecanexpressthedivergenceofthevelocityfieldasafunctionofu∗ andpn+1.

∇ · 

un+1

= ∇ · 

u∗

− 

t

∇ ·



_∇

pn+1

ρ

n+1



(37)

Thisrelationcanbeinjectedinthepressureequation(34)inordertoobtainthefollowingsystemforthepressure:

pn+1

− 

t2

(ρ

c2

)

n+1

∇ ·



∇

p

ρ

n

₊1

=

p∗

− 

t

(ρ

c2

)

n+1

∇ · 

u∗ (38)

with p∗

=

pn

− 

t

(

u



· ∇

p

)

nthepressuresolutionoftheadvectionsub-system.

The system(38) is a Helmholtz-type equation which can be solved asa linear systemprovided we can compute an approximationof

(

ρ

c2

₎

n+1_{.Oncethepressurefieldisknown,thevelocityfieldcanbecomputedwiththeEq.(36).}

Theequation(38)isadiscreteformulationresultingfromSystem(2).Wehavetoremarkthatthisformulationiscorrect ifisentropicflowsareconsidered.Thisconditionisalwaysrespectedinthispapersinceviscouseffectsandheatconduction are neglected. Thewaywe useto formulatethe conservationenergyequation wouldbe still validforentropicflows but only fora perfectgasequation ofstate, providedthat additionalterms duetoviscous friction andheat conductionwere added.Amoregeneralframeworkwhichallowsdealingwithcompressibletwo-phaseflowsinvolvingviscousfriction,heat conductionwithanyequationofstateisproposedin[4].Inthisreference,apressureevolutionequationhasbeenderived, thisequationlookssimilartoEq.(38),butitaccountsforanadditionaltermonthecouplingbetweenthepressureevolution andtheentropictemperatureevolution.

3. Governingequationsandnumericalmethodsfortwo-phasesflows

Toourknowledge,thecompressibleprojectionmethodhasneverbeenappliedtomultidimensionalsimulationsof two-phaseflowsaccountingforsurfacetensioneffect.Weproposeintherestofthispaperanumericalmethodbasedonanew time splittingin ordertoachieve thistask.Therelevance ofournewmethodwill behighlighted intheresultsection of thispaper.

3.1. Governingequations

Theequationsareidenticaltotheonephasecase(System(2)),exceptforthemomentumconservationequationwhere thesurfacetensionforcemustbeincluded.

∂

ρ

∂

t

+ 

u

· ∇

ρ

+

ρ

∇ · 

u

=

0

∂

u



∂

t

+ 

u

· ∇

u

+

∇

p

ρ

=

σ κ

n



δ



ρ

∂

p

∂

t

+ 

u

· ∇

p

+

ρ

c 2

_{∇ · }

_u

₌

₀ (39)

Where

σ

isthesurfacetensioncoefficient,

κ

isthelocalcurvatureoftheinterface,n is



thenormalvectorattheinterface, and

δ

 isaDiracdistributionlocalizedontheinterface

.

Anothersignificantdifference liesinthe descriptionofthedensityfield whichisdiscontinuousinthegridcells which are crossed bytheinterface. Someequationsofstate havebeendescribed earliertocompute thespeedsoundinthe gas phase(perfectgaslaw)andintheliquidphase(Taitequation).

3.2. Level-setmethod

TheinterfacemotioniscapturedbysolvingthefollowingconvectionequationofaLevelSetFunction[36]:

∂φ

∂

t

+ 

uint

· ∇φ =

0 (40)

where u



int isthe interface velocity. Ifthe normal componentof thevelocity field iscontinuous across theinterface the

extrapolationcouldnotbeusedinmorecomplexsituationswhereshockwavesorphasechangeoccursincethesesituations involveadiscontinuityofthenormalcomponentofthevelocityfield,seethefollowingreferencesformoredetailsonthis topic [3,11,34,45,46]. However asthe test-cases which are considered in this paper do not induce any shock waves or discontinuities ofthe normalvelocity component, we haveobserved that the standardextrapolation ofthe velocity field usedforincompressibleflows,u



int

= 

u,waswellsuitedforthesimulationspresentedinthispaper.

The non-zerolevelcurves ofthelevel-setfunction are usefultocompute interface geometricalpropertiessuch asthe curvature:

κ

= ∇ · 

n (41) where



n

=

∇φ

∇φ

(42)

AreinitializationequationmustbesolvedattheendofeachtimesteptoensurethattheLevelSetfunctionwillremain a signeddistance [43].It iswell known that thisfurtherequation can inducesome slightdisplacements ofthe interface positionduring thereinitialization.Howeverifsufficientlyrefinedgrids areusedthisspurious displacementtendstoward zero.

∂

d

∂τ

=

sign

(φ)(

1

− ∇

d

)

(43)

Inthisequation,

τ

isafictitioustime.ThesteadysolutionofEq.(43)providesafunctiond whichisthesigneddistanceto theinterface.

3.3. Projectionmethod 3.3.1. Basicprojectionmethod

We describe in this section, a first attempt to incorporate the surface tension in the framework of the compressible projection method.Themethod,which isproposed here, isdirectlyinstigated by theclassicalapproach usedto compute the surface tension termfor incompressible two-phase flows. In particular, the following splitting is carried out on the velocityfield:



u∗

= 

un

− 

tu



n

· ∇

un (44)



un+1

= 

u∗

− 

t

∇

p n+1

ρ

n+1

+ 

t



σ κ



n

δ



ρ

n

₊1 (45)

Thissplittingisusedinthefollowingworks[19,23,44]intheframeworkoftheGhostFluidMethodforincompressibleflows. This method allows imposing sharp jump conditionsacross the interface withan accurate discretization of the singular terms.Sharpapproximationsofsingulartermsas

δ

 havebeenderived in[30] inordertoavoidanartificialsmoothingof

thesetermsontheinterface.Ifacompressibleprojectionmethodisconsidered,itleadsnaturallytothefollowingHelmholtz equationforthepressure:

pn+1

− 

t2

(ρ

c2

)

n+1

∇ ·



∇

p

ρ

n

₊1

=

p∗

− 

t

(ρ

c2

)

n+1

∇ · 

u∗

+ 

t

(ρ

c2

)

n+1

∇ ·



σ κ

n



δ



ρ

n

₊1 (46)

Ina firsttime, we haveapplied thisquitesimplemethod, butitisillustrated inFig. 9 that itprovides very poorresults duetostrongparasiticcurrentswhichdevelop ontheinterface.WhereastheGhostFluidMethodisknownasan accurate methodtocomputethesurfacetensionintheframeworkofprojectionmethodsforincompressibleflowssinceitproduces parasiticcurrentswithverylowamplitudes[19,44],its generalizationtothecompressibleprojection methodseems tobe morechallenging.Formoredetails,refertoAppendix A.

3.3.2. Splittingmethodforsurfacetensionresolution

We present now a new time splitting method in order to perform accurate and stable computations accounting for surface tension effects. As stated above, the Ghost Fluid Method provides an accurate and stable representation of the surfacetensioneffectsifitisusedwithanincompressibleprojectionmethod.AccordingtotheHelmholtz–Hodgetheorem, thecompressiblevelocityfieldcanbedecomposedasthesumofthegradientofascalarfieldwithadivergence-freevector field. We propose to incorporatethe surface tension inthe part ofthe velocity field which is divergence-free.Thus, the surfacetensiontermwillbecomputedina similarwayasforan incompressibleprojection method.Forthatpurpose,we isolatethesurfacetensioncontributiononthesolutionsofsystem(39)bysplittingeachvariableasfollowing:

ρ

=

ρ

0

+

ρ

ST



u

= 

u0

+ 

uST

p

=

p0

+

pST

withthesubscript’ST’whichindicatesthecorrespondingvariabletakesintoaccountthesurfacetensioneffects.Thevector field



uST isthedivergence-freepartoftheHodgedecompositionofthevelocityfield.Usingthisdecomposition,thesystem (39)canbesplittedintwosub-system(48)and(49).Transportandacousticaltermsarecontainedinthefirstsub-system:

∂ρ

0

∂

t

+ 

u

· ∇

ρ

0

+

ρ

∇ · 

u0

=

0

∂

u



0

∂

t

+ 

u

· ∇

u

+

∇

p0

ρ

=

0

∂

p0

∂

t

+ 

u

· ∇

p0

+

ρ

c 2

_{∇ · }

_u 0

=

0 (48)

andthesurfacetensiontermsareincludedinthesecondsub-system(49):

∂ρ

ST

∂

t

+ 

u

· ∇

ρ

ST

+

ρ

∇ · 

uST

=

0

∂



uST

∂

t

+

∇

pST

ρ

=

σ κ

n



δ



ρ

∂

pST

∂

t

+ 

u

· ∇

pST

+

ρ

c 2

_{∇ · }

_u ST

=

0 (49)

Spurious currents due to surface tension come from the second sub-system. Recall that u



ST is defined to respect the

divergence-free condition. System (49) is incompressible:

∇.

uST

=

0

⇐⇒

ρ

ST

=

cste. For sake of simplicity, we choose

ρ

ST

=

0 and

ρ

=

ρ

0.TheSystem(49)becomessimply:

∂



uST

∂

t

+

∇

pST

ρ

=

σ κ

n



δ



ρ

∇ · 

uST

=

0 (50)

Thus,thepressurepST fieldcanbeclassicallycomputedbysolvingaPoissonequation:

∇ ·



∇

pST

ρ



= ∇ ·



σ κ



n

δ



ρ



(51)

Finally,thewholesystem,thatwehavetosolve,canbesummarizedasfollows:

∂ρ

∂

t

+ 

u

· ∇

ρ

+

ρ

∇ · 

u

=

0

∂

u



0

∂

t

+ 

u

· ∇

u

+

∇

p0

ρ

= 

0

∂

p0

∂

t

+ 

u

· ∇

p0

+

ρ

c 2

_{∇ · }

_u 0

=

0

∂

u



ST

∂

t

+

∇

pST

ρ

=

σ κ

n



δ



ρ

∇ · 

uST

=

0 (52) whereu



= 

u0

+ 

uST andp

=

p0

+

pST. 3.3.3. Timediscretization

WedescribenowthetemporaldiscretizationusedtosolvetheoverallSystem(52).Inafirsttime,theinterfacemotion iscomputed:

φ

n+1

= φ

n

− 

t

(

u



· ∇φ)

n (53)

Next,thedensityfieldcanbeupdatedbysolvingthefollowingequation:

ρ

n+1

=

ρ

n

− 

t

(

u



· ∇

ρ)

n

+

ρ

n

∇ · 

un

.

(54)

Inthethirdstep,thepressure pST fieldissolvedduetotheEq.(51)correspondingtoaPoissonequationwhichleads toa

symmetricpositivedefinitelinearsystem:

∇ ·



∇

pST

ρ

n

₊1

= ∇ ·



σ κ

n



δ



ρ

n

₊1 (55)



u∗₀

= 

un₀

− 

tu



n

· ∇

un (56)

p∗₀

=

pn₀

− 

tu



n

· ∇

pn₀ (57)

Inthefifthstep,thefollowingHelmholtztypeEq.(58)forthepressure p0iscomputedbysolvingalinearsystem:

pn₀+1

(ρ

c2

₎

n+1

− 

t 2

_{∇ ·}



∇

p0

ρ

n

₊1

=

p∗0

(ρ

c2

₎

n+1

− 

t

∇ · 

u ∗ 0 (58)

Letusnotice, that theresolution ofthisequation has beeneased by dividing theEq. (38)by ‘

ρ

c2_{’,inorder toobtain a} symmetricdefinitepositivelinearsystem.

(

ρ

c2

)

n+1 iscomputedknowingthedensityattime

(

n

+

1)thankstoEq.(33).As theflowisassumedtobeisentropic,thegasobeystotheLaplacelaw: _ρpγ

=

cste.ThankstoEq.(8),weobtain

(

ρ

c2

)

n+1

=

γ

p0

ρn+1

ρ0



forair.Forwater,wejustuseEq.(9):

(

ρ

c2

₎

n+1

₌

_γ

_B

ρn+1 ρ0



. Finally,thevelocityfieldiscomputedby



un+1

= 

un₀+1

+ 

u_STn+1

= 

u∗₀

+ 

un_ST

− 

t



∇

pn₀+1

ρ

n+1

+

∇

pn_ST+1

ρ

n+1

−



σ κ

n



δ

ρ

n

+1



(59)

Theredistancealgorithmisthenappliedattheendofthetemporaliteration. 3.3.4. Spatialdiscretization

Thespatialdiscretizationsaredetailedinthissectionfora2DCartesianmesh,butitcanbeeasilygeneralizedtoan ax-isymmetricmeshora3DCartesianmesh.AlloftheconvectivederivativesarecomputedusingaWENOscheme[15]which iswell-knownforitsaccuracyandstability.Asitisusualforanincompressibleprojectionmethod,velocitycomponentsare computedonstaggeredgrids relativetothe centeredgridfortheinterface,densityandpressurefields.Thus,thevelocity divergenceiscomputedasfollows:

∇ · 

ui,j

=

ui+1/2,j

−

ui−1/2,j



x

+

vi,j+1/2

−

vi,j−1/2



y (60)

where u and v are the velocity components following the x-axis and the y-axis, and the components of the pressure gradientare

∂

p

∂

x



i+1/2,j

=

pi+1,j

−

pi,j



x (61)

∂

p

∂

y



i,j+1/2

=

pi,j+1

−

pi,j



y (62)

ThisstandarddiscretizationoftheLaplaceoperatorisusedtocomputethepressure p0 andpST:

∇ ·



∇

p

ρ

n

₊1 i,j

=

pn_i++₁1_,j−pn_i,+_j1 ρ_in₊+₁1_/2,j

−

pn_i,+_j1−pn_i−+₁1_,j ρn_i−+₁1_/2,j



x2

+

pn_i,+_j+1₁−pn_i,+_j1 ρn_i,+_j+1_1/2

−

pn_i,+_j1−pn_i,+_j−1₁ ρn_i,+_j−1_1/2



y2 (63)

Forthesegmentsofthegridwhicharecrossedbytheinterface,thedensityisinterpolatedontheborderofthe computa-tionalcellswithaharmonicaverageasitisusuallydone intheframeworkoftheGhostFluidMethodforincompressible two-phase flows[19,30,44].The discretizationofthesurface tensiontermintheEq.(55)isalsoperformedfollowingthe efficientapproachdevelopedinthesepapers.Aspreviouslystated,thisapproachallowsasharpdescriptionofthesingular termsbyremarkingtheequivalencybetweenEq.(51)andEq.(64):

∇ ·



∇

pST

ρ

n

₊1

=

0 with [pST]

=

σ κ

(64)

See[23,29]formoredetailsonthisspecificpoint. 3.3.5. Ghostfluidmethod

Sincethepioneeringworksof[8,19,30],theGhostFluidMethodbecameapopularapproachtodealwithdiscontinuities in heterogeneous media, see for example the following Refs. [1,3,11,12,23,34,47,45,46,51]. However, it is noteworthy to remind thatglobaldenominationcan referto manydifferenttechniques.In particular,ifthe basicprincipleof theGhost FluidMethod isalways the same,its implementation can bevery different whetheran explicitor an implicitresolution isperformed. Indeed,ifan explicitresolution isconsidered,an extrapolation ofaknown fieldmust be computedonthe other sideofthe interfaceinorderto builda continuousfield beforethediscretization.Onthe otherhand,ifan implicit discretizationiscarriedout,thesingulartermsareinterpolatedontheinterfaceinordertoimposethecorrectvalueofthe

jump conditions.Thus,asthe development presentedabove wasmainly focused onthecomputation ofthe pressure,we haveessentiallyreferredtonumericaltoolswellsuitedtoimposejumpconditionswithanimplicittemporaldiscretization. Unlike the pressure field, an explicit algorithm is used to update the density field. Therefore, it is required to build a continuousextrapolationofthedensityfieldofeachphaseontheothersideoftheinterface.Itisperformedbytransporting thecorrespondingdensityintheghostfieldsalongthenormalsn to



theinterfacebyaniterativeresolutionofthefollowing equation[8]:

∂

ρ

∂τ

± 

n

· ∇

ρ

=

0 (65)

where

τ

is a fictitioustime. Moreover, thedensity values closeto the interface (in a layer thickness ofone cell in the real field)areupdated bythe extendedfield computedwithEq.(65).Thismethod,knownasthe“isobaric fix”,hasbeen proposed in[9]topreventfromadensityovershootingclosetothe interface.Letusremarkthatitisnot requiredtouse any extrapolationto compute theexplicitconvective derivative on pressure inEq.(48)since thissplit pressuredoesnot containthejumpconditionduetothesurfacetensioneffects.

3.3.6. Someconsiderationsonthetimestepstabilityandonthecostofatemporaliteration

Aclassical time-stepconstraintaccountingforthestability conditionson convection,andsurface tensionisimposed if thecompressibleprojectionmethodisused.



tconv

=

1 2



x max



u



(66)



tSurf _Tens.

=

1 4



ρ

liq



x3

σ

(67)

Finally,theglobaltimesteprestrictioncanbecomputedwiththefollowingrelation:

1



t

=

1



tconv

+

1



tSurf _Tens. (68)

As it hasbeen explainedpreviously, forlow Mach numberflows, the compressibleprojection methodstrongly alleviates thetime steprestrictionincomparisontoan explicitHLLCsolver.However,one couldpointoutthisimprovementonthe temporal stability leads to an important additional cost of a temporal iteration since one linear system must be solved to compute the pressurefield (twolinear systemsifthe surface tensionistaken intoaccount), unlike the classicalHLLC solver, whichisonlybasedonanexplicittemporaldiscretization.Thislastassertioncould berelevantinsomesituations, for instanceifan IncompleteCholeskyConjugate Gradient (ICCG) methodis used tosolve the linearsystem. But herein, theBlackBoxMulti-Gridmethod[6]isusedtocomputethesolutionofourlinearsystems.Thismethod,whichisperfectly suited to deal with Poisson(or Helmholtz) type-equation withstrongly inhomogeneous diffusion coefficient [31], allows savinganimportantcomputationaltime,tosuchanextentthatsolvingthelinearsystemisnolongerthetemporallimiting factor, butinstead themanycallsto theWENOschemeforcomputingthe convectivederivatives.Therefore,thecostofa temporaliterationcanbeapproximatelythesamewithacompressibleprojectionmethodoraHLLCsolver.

4. Results

4.1. Rayleigh–Plessettheory

The Rayleigh–Plessetequation describestheradius temporalevolutionofa sphericalbubblesurrounded byan incom-pressiblefluidinaninfinitemedium.Itreads

Rd 2_R dt2

+

3 2



dR dt



2

=

pB

(

t

)

−

2σR

−

p∞

ρ

L (69)

whereR isthebubbleradius,pB thepressureinthebubblesupposedtobeuniform, p∞ thepressureoftheliquidfarfrom thebubbleand

σ

thesurfacetensioncoefficient.

ItisderivedfromtheincompressibleEulerequations(20)inone-dimensionalsphericalgeometry,asshownin[26].No analyticalsolutionhasbeenfoundyet,butanumericalsolutioncanbeeasilycomputed.

YangandProsperetti[57]proposedanextensionoftheRayleigh–Plessetequationaccountingforthecontainmenteffects bysupposingthattheincompressibleliquiddomainisfinite.Thismodifiedequationisexpressedasfollows:



S

−

R S



Rd 2_R dt2

+ [

2

−

(

S2

+

R2

)(

S

+

R

)

2S3

]



dR dt



2

=

pB

(

t

)

−

2σR

−

pS

(

t

)

ρ

L (70)

where S istheradiusoftheincompressibleliquiddomain,andpS thepressureoftheliquidatthedistanceS ofthebubble

Fig. 1. Comparison between Rayleigh–Plesset and Rayleigh–Plesset–Prosperetti equation. The containment cannot be neglected.

Theinfluenceofthecontainmenteffectonthetemporalevolutionofthebubbleradiusisinvestigatedonthefollowing configuration presentedin [42].An air bubble issurrounded by water. On the domain boundary, a spherical ultrasound wavefront is generated. The surface tension is neglected. R0 is the initial radius. The wavefrontis simulated by setting thepressure: pS

(

t

)

=

p0

−

Asin

(

wt

).

Forthistestcase, R0

=

10−5 m, S

=

32

×

10−5 m, p0

=

105 Pa, A

=

4

×

105 Pa and w

=

5

×

105rad s−1.

The bubble radius evolution hasbeen plotted inthe Fig. 1 to compare the solution ofthe Rayleigh–Plesset equation with the solution of the Rayleigh–Plesset–Prosperetti equation. We observe that the amplitude of the radius oscillation is increasing and the time ofthe collapse is reducedif the containment effectis accounted for.As we target to design a relevant benchmark for the validation of our numerical approaches, we can conclude from this preliminary test that benchmarkshouldbebasedonareferencesolutionwhichincludesthecontainmenteffectinourreferencesolution,since asimulationmustbeperformedinafinitesizecomputationaldomain.

Theinfluenceoftheliquidcompressibilitycanbeconsideredby solvingthemorecomplexKeller–Miksisequation[20]. Thisequationprovidessimilar resultsastheRayleigh–Plessetequation ontheconfigurationwhichisconsideredabove.To ourknowledge,itdoesnotexist anyequationsaccountingforboththecontainment andthecompressibilityoftheliquid phase. Therefore, asit has been demonstratedabove that the containment should not be neglected, the solutionof the Rayleigh–Plesset–Prosperettiequationwillbeconsideredintherestofthispaperasourreferencesolution.

4.2. Comparisonbetweenthedifferentcompressiblenumericalmethodsona1Dsphericalbubbleoscillatingtestcase

Inthissection,weperformacomparative studybetweenthedifferentnumericalmethodswhichhavebeenpreviously presentedbyusingthetest-casedescribedinSection 4.1.Thispreliminarytestiscomputedinaonedimensionalspherical coordinatesysteminordertoassessthesuitabilityofthedifferentsolverstodealwithlowMachnumberflows.Therefore, asitisnotrequiredtoaccountforthesurfacetension,itwillbeneglected.

Inafirsttime,theHLLCsolverhasbeentested.TheresultsarepresentedinFig. 2fordifferentmeshsizes.Asithasbeen previously reportedin[42],ifthissolverisused, averyimportantnumberofgrid cellsis requiredtoobtainan accurate solution.ThistestillustratesthelowefficiencyofsuchasolverinthelowMachnumberregime,sincethemaximumMach numbermeasuredinthewaterisabout10−2.Asexpected,wecanconcludethissolverisnotsuitabletoperformnumerical simulationsinthelowMachnumberregimeoftheinteractionbetweenaliquid–gasinterfacewithacousticalorultrasound waves.

AssessmentsofthepreconditioningLowMach HLLCsolverarenowpresentedonthesametest-caseforseveralvalues oftheMachcoefficient.Thebubbleradiusisdiscretizedwith16computationalcellsattheinitialtime. M

=

1 corresponds tothe HLLCsolverwithout theLowMachpreconditioning.It isclearinFig. 3 thatthelower theMach coefficientis,the moreaccuratearetheresults.DuetotheLowMachpreconditioning,asufficientaccuracycanbeachievedwithmuchless computationalcells.However,ifthismethodisused,thetimestepwhichwasalreadysmallwiththeclassicalHLLCsolver, isevensmaller. Animplicitcomputation methodcanbeperformed[27],butitismuchmorecomplex sinceitleads toa couplednon-linearsystem.

TheresultsobtainedwiththecompressibleprojectionmethodhavebeenplottedinFig. 4.Wefirstnotethatthespatial convergenceisstronglyenhancedincomparisontotheresultsobtainedwiththeHLLCsolver.Inparticular,relevantresults canbeobtainedwithveryfewpointsintheinitialbubbleradius.Secondly,duetothesemi-implicittemporaldiscretization, thetime stepis16 timeslargerthan



tHLLC.Thus, ifalow Machnumberflow isconsidered,incomparisonto theHLLC

Fig. 2. ConvergencestudyofHLLCsolverontheRayleigh–Plesset–Prosperettitestcase.Thenumberofcellswhichisindicatedistheonecontainedinthe initialradiusofthebubble.Thisnumbermustbemultipliedby32toobtainthetotalnumberofcells.

Fig. 3. Influenceofthe

M parameter

onthespatialaccuracyoftheLowMachPreconditioning methodwith16computationalcellsontheinitialbubble radius.Thetimestepis

M times

smallerthantheacousticaltimestepfromtheHLLCsolver.

Fig. 4. Convergencestudyofthecompressibleprojectionmethod.Thenumberofcellswhichisindicatedistheonecontainedintheinitialradiusofthe bubble.Thetotalnumberofcellsis32timeshigher.

Fig. 5. Comparisonbetweenthedifferentnumericalmethodswith16computationalcellstodiscretizetheinitialradiusofthebubble.TheMachcoefficient ofpreconditioning issetto

M

=1/4.

Table 1

Convergencestudyofthedifferentnumericalmethods.

N is

thenumberofcellsintheinitialbubbleradius.

N HLLC Preconditioning HLLC (M=1

4) Projection method

Relative error Order Relative error Order Relative error Order

4 91.84% 35.69% 11.95% 8 74.76% 0.30 15.45% 1.21 6.34% 0.65 16 36.86% 1.02 6.99% 1.14 3.23% 0.92 32 18.03% 1.03 3.31% 1.08 1.58% 0.97 64 8.98% 1.01 1.65% 1.01 0.75% 1.03 128 4.71% 0.93 0.86% 0.94 0.33% 1.18 256 3.46% 0.44 0.12% 1.47

solver,thismethodallowssavingcomputationaltimebothbyusingafewernumberofgridcellsandbyperformingafewer numberoftemporaliterations.

Theresultsobtainedwiththethreedifferentsolversare plottedinFig. 5forcomputationsinvolving16computational cells to discretize the initial bubble radius. As stated before, the HLLC solver provides very poorresults in comparison to other solvers. Moreover, its time step restriction is stringent, since its value is about



tHLLC

=

2

×

10−10 s. The Low

MachpreconditioningHLLCmethodincreasessignificantlythespatial accuracyoftheHLLCsolver. However,thetimestep restrictionisevenmorestringentthanwiththesimpleHLLCsolver,sinceitisequalto:



tHLLCPrecond

=

5

×

10−

11_s.Finally,it highlightsthatthespatialaccuracyofthecompressibleprojection methodisimprovedincomparisontotheothertwo.As thismethodalsoallows usingahighertimestep(16timesthanHLLC one):



tProj

=

32

×

10−10s, wecanconcludefrom

thesepreliminarysimulationsthatthecompressibleprojectionmethodoutclassestheothertwoforcomputingthevolume oscillationsofabubbleinalowMachnumberregime.

IntheTable 1,theevolutionoftheerroronthebubbleradiusisreportedfordifferentcomputationalgridsandatagiven time.Wecanconcludethatthethreesolversarefirstorderaccurateinspacewhenthecoarsestgridsareused,whereasthe valuesoftheerrorsareverydifferent. Moreoverwecanobservethattheaccuracyofthecompressibleprojectionmethod increaseswhenthespatial resolutionincreases, itmeansthat thecomputation tendstowardan asymptoticregimewhich couldleadtoanalmostsecondorderaccuracy.Ontheotherside,theaccuracyoftheHLLCSolversseemstodecreasewhen thespatial resolutionisincreasing.It confirmsthat thesesolversare notwell suitedtocompute thiskindofapplications involvinglowMachnumberregime.

Therefore,thiscompressibleprojectionmethodwillbeusedinthefollowingtoperformmultidimensionalcomputations accountingforsurfacetension.

4.3. Multidimensionalcomputationsoftheinteractionofabubblewithanultrasoundwavefront 4.3.1. Shapeoscillation

Wepresentinthissectionsomecomputationsoftheshapeoscillationofadeformedbubble.Thisclassicaltest-casefor thenumericalsimulation ofincompressibletwo-phase flows [10,22]willallow usbringing out thesuitability ofthetime

Fig. 6. Temporal evolution of the bubble displacement in the x direction.

Table 2

Comparisonoftheshapeoscillationfrequencybetweentheoryandcomputations.

Number of cells Oscillation frequency (kHz) Relative error (%)

32×32 110.62 1.45

64×64 111.09 1.03

128×128 111.46 0.70

256×256 111.53 0.63

splittingprojectionschemeforcompressibletwo-phaseflowsinthestrictlyincompressiblelimit.Theoscillationfrequency isprovidedbythefollowingrelation[24]:

fl

=

2

π



σ

R3

(

l

−

1

)

l

(

l

+

1

)(

l

+

2

)

l

ρ

L

+ (

l

+

1

)ρ

G (71)

where R istheequivalent radiusofthedroplet,andl theoscillationmodenumber.Inthistestcase,weonlyconsiderthe model

=

2,and

ρL

=

103_{kg m}−3_,

_ρG

₌

_{1.226kg m}−3_and

_σ

₌

₇

_×

₁₀−2 _{Pa m.}

Initially,theinterfaceisdefinedbyanellipsoidequation:

x2 a2

+

y2 b2

+

z2 c2

=

1 (72)

witha

=

10−5 _m,b

₌

₁₀−5_{m andc}

₌

_1.25

_×

₁₀−5_{m.ThiscorrespondstoR}

₌

_1.077

_×

₁₀−5_m.

The numericalcomputations are performedonan axisymmetric Cartesian mesh.Moreover, an additionalsymmetry is imposedfollowingtheradialplaneinordertosavecomputationaltime.Thewholecomputationaldomainisasquarewith alengthsideequalto2

×

10−5 _m.

We observethepositionoftheinterface crossingtheradial planeovertime inFig. 6,andwe comparetheoscillations frequencyfordifferentmesheswiththetheoreticalfrequency f2

=

112,24kHz.A goodagreementbetweenthe computa-tions andthetheory isreportedintheTable 2,sincethe relativeerroronthe frequencyisalways lessthan 2%.We also remarkintheFig. 6thattheoscillationamplitudeisdecreasingwithtimeduetothenumericaldissipationofthenumerical scheme.Howeverbyusingmorerefinedgrids,thisnumericaldissipationtendstozeroandtheoscillationamplitudeiswell maintainedalongthesimulation.IntheFig. 7,theinterfacelocation,thepressurefieldandthevelocityfieldhavebeen plot-tedinordertohighlightthesuitablebehaviorofthenumericalsolveronthisspecificsituationinvolvinganincompressible flow.Inthedroplet,wenoticethatthepressurefielddirectlydependsontheinterfacecurvature.

4.3.2. Interactionofabubblewithasphericalwavefront

Wenowinvestigatetheabilityofthecompressibleprojectionmethodtocompute multidimensionalbubbleoscillations. Thesurfacetensionisnowconsidered,andittendstokeepthebubblespherical.Theinitialbubbleradiusis R0

=

10−5 m. Spherical boundary conditions are imposed on the boundaryof the computational field in orderto generate a spherical wavefront. At R1,the wavefrontiscreatedby settingthepressure onthe boundary: pS

(

t

)

=

p0

−

Asin

(

wt

+

S

/

c

),

where

S

=

4

×

10−5 _{m, p}

0

=

105 Pa, A

=

8

×

104 Pa, w

=

5

×

105 rad s−1,

σ

=

7

×

10−2 Pa m andc isthesoundspeedinwater computedwithEq.(9).

Fig. 7. Interface location, pressure (Pa) and velocity (m s−1_{) fields at different times. The grid contains 256×256 computational cells.}

Fig. 8. Comparison between the solutions of the Rayleigh–Plesset–Prosperetti equation with and without surface tension effects.

Eqs. (69) and(70) show that the surface tension decreases the amplitude of the bubble volume. This observationis verifiedinFig. 8.Theseeffectsaresignificantformicro-bubbles.Indeed,thepressurejumpacrosstheinterfaceisinversely proportionaltothebubbleradius.Forthistest,with

σ

=

7

×

10−2_{Pa m thepressurejumpis} 2σ

R0

=

14000Pa whichcannot

beneglectedcomparedtothemagnitudeofthepressure p0.

Thecomputationsare performedonan axisymmetricCartesian mesh.Asbefore,a symmetryis imposedonthe radial plane.Thesizesofthecomputationalfieldarelr

=

4

×

10−5 _{m andlz}

₌

₄

_×

₁₀−5 _{m.Inafirsttime,wehaveattemptedto} performthistestcasebyusingthebasicprojectionmethoddescribedintheSection3.3.1.Wehaveobservedthismethod

Fig. 9. Comparisonbetweenthebasicprojectionmethod(left)andthetime-splittingcompressibleprojectionmethodforsurfacetension(right).Thegrids contain256×256 computationalcells.Thankstosplittingprojectionmethod,spuriouscurrentsdisappear.

Fig. 10. Temporal evolution of the bubble radius. Comparison between the computation and the reference solution for different grids.

was subjecttonumericalinstabilitiesifthesurface tensioneffectistakenintoaccount.Seeforexamplethesnapshotsof velocity fieldintheFig. 9,attimet

=

1.55

×

10−6 _{s withameshcontaining 256}

_×

_{256 cells,whereverystrongspurious} currents haveappeared ontheinterface. Moreover, wehave observedthat withthebasicprojection method,the fineris themesh,thestrongerarethespuriouscurrents.Thisleadstoaquiteunstablemethodwhichrequiressomeimprovements. ThisobservationhasmotivatedustodesignthenewtimesplittingmethodwhichispresentedinSection3.3.2toovercome thisdifficulty.Formoredetailsonthesemisleadingbehaviors,refertoAppendix A.

InFig. 9,asnapshot ofthevelocityfieldhasalsobeenplottedwiththetime-splittingcompressibleprojection method for surface tension. If this method is used, a clean velocity field is obtained since the unstable spurious currents have vanished. Therefore, thistime-splitting willbe used inall the simulations whichare presented inthe restof thispaper. Inparticular,thespatialconvergenceonthebubbleradiustemporalevolutionisstudiedinFig. 10.Itclearly demonstrates that the numericalalgorithm converges tothe referencesolution.Whereas, the simulation,which isperformedwith the coarsestgrid(only 16gridcellsintheinitialradius), providesaccurateresultontheamplitude ofthefirstoscillation,we observethat 128computationalcells intheinitialbubbleradiusare requiredtocomputeaccurately theamplitudeofthe fifthoscillation.Itcorrespondsto512

×

512 cellsinthewholedomain.Inaddition,thetemporalconvergenceonthebubble radiustemporalevolutionisalsostudiedinFig. 11.Itclearlyshowsthetemporalconvergenceofthenumericalalgorithm, since thediscrepancybetweenthetheory andthe simulationdecreaseswhen thetime stepdecreases.Witha 128

×

128 grid, dividingthetimestep(Eq. (68)) by8allows toobtainaboutthe256

×

256 grid accuracywithoutdividingthetime step.

In the Table 3, the variation of the error on the maximum radius during the first oscillation is reported for several computational grids.Wecanobserveasimilarbehaviorasforthe1Dtest-case,sincethespatial accuracyisfirstorderfor the coarsestgrids andnext it increasesto asecond order accuracy, whichis compatiblewiththesecond order centered schemeswhichareusedinseveralstepsofthecompressibleprojectionmethod.