A penalization method for the simulation of bubbly flows

HAL Id: hal-02077455

https://hal.archives-ouvertes.fr/hal-02077455

Submitted on 22 Mar 2019

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

A penalization method for the simulation of bubbly flows

Antoine Morente, Jérôme Laviéville, Dominique Legendre

To cite this version:

Antoine Morente, Jérôme Laviéville, Dominique Legendre. A penalization method for the sim- ulation of bubbly flows. Journal of Computational Physics, Elsevier, 2018, 374, pp.563-590.

�10.1016/j.jcp.2018.07.042�. �hal-02077455�

OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible

Any correspondence concerning this service should be sent

to the repository administrator: tech-oatao@listes-diff.inp-toulouse.fr This is an author’s version published in: http://oatao.univ-toulouse.fr/23459

To cite this version:

Morente, Antoine and Laviéville, Jérôme and Legendre, Dominique A penalization method for the simulation of bubbly flows. (2018) Journal of Computational Physics, 374. 563- 590. ISSN 0021-9991

Official URL:

https://doi.org/10.1016/j.jcp.2018.07.042

Open Archive Toulouse Archive Ouverte

A penalization method for the simulation of bubbly flows

Antoine Morente

^a

^,

^b

, Jérôme Laviéville

^a

, Dominique Legendre

^b

^,∗

aEDF(ÉlectricitédeFrance)R&D,6QuaiWatier,78400Chatou,France

bInstitutdeMécaniquedesFluidesdeToulouse,IMFT,UniversitédeToulouse,CNRS–Toulouse,France

Keywords:

Bubblyflows Penalizationmethod Shearfreecondition Bubbleinducedagitation

Thisworkisdevotedtothedevelopmentofapenalizationmethodforthesimulationof bubblyflows.Sphericalbubblesare consideredasmovingpenalizedobstacles interacting withthefluidandanumericalmethodforensuringtheshearfreeconditionattheliquid–

bubble interfaceisproposed.Three test-cases(curvedchannel,inclinedchannel and3D translatingbubble)areusedtovalidatetheaccuracyofthediscretizationensuringtheslip conditionattheinterface.Numericalsimulationsofarisingbubbleinaquiescentliquidare performedformoderateReynoldsnumbers.Consideringbubbleterminal velocities,initial accelerations and wake decay, the effect ofthe penalization viscosity used to ensure a uniformvelocityinthepenalizedobjectisdiscussed.Finally,simulationsofbubbleswarms havebeencarriedoutinafullyperiodicboxwithalargerangeofvoidfractionsfrom1% to 15%.Thestatisticsprovidedbythesimulationscharacterizingthebubble-inducedagitation arefoundinremarkableagreementwiththeexperiments.

1. Introduction

Buildingreliablesimulation toolsabletopredicta widerangeofmultiphaseflows regimesisa majorchallengeinin- dustry.Today,such simulationsofboiling flows(dispersedorseparate phases)can becarriedout usingthelocal 3DCFD code

NEPTUNE_CFD

[12] developedintheframeoftheprojectNEPTUNE(EDF,CEA,Areva,IRSN).Makingaccuratesimulations ofbubblyflowsisofimportanceforprovidingclosureslawsonmomentumtransfer,bubble–liquidinteractionandinduced turbulence.BunnerandTryggvason[5,6] andEsmaeeliandTryggvason[9] carriedoutsimulationswithnon-deformableand deformable bubblesproviding PDFsofthe bubblevelocity. Roghair etal. [28] developeda newdrag correlation forbub- blesinbubbleswarmsatintermediateandhighReynoldsnumbersthroughtheir numericalsimulations. Directnumerical simulations (DNS) were performed by Roghair etal.[29] to studythe behavior ofa swarm of rising airbubbles forthe comparisonoftheliquidenergyspectraandbubblevelocityprobabilitydensityfunctions(PDFs)withexperimentaldata.

Togetalocaldescriptionoftheflow,theDNSsimulationapproachesofliquid–bubbleflowsarewidelyused.Thenotable variationamongalltheDNSapproachesconcernstherepresentationandthenumericaltreatmentofthebubblesinterface:

thebody fittedapproach proposedby McLaughlin [19] fora single bubble;thefront trackingmethodby Tryggvason and Unverdi[32,35] requiringmarkersontheinterface.Twomajordrawbacksarisefromthiskindofapproach:aclearlimitation on both the Reynolds number andthe number of bubbles. DNS of bubbly flows are often limited to moderate bubble Reynoldsnumbers (Re

=

^{O(10–100)) forsphericalbubbles [32,41]. Recently,the simulationof 16rising bubbles at Re}

=

*

Correspondingauthor.

E-mailaddress:legendre@imft.fr(D. Legendre).

Fig. 2.UseoftheBnaetal.[4] methodfortheinterfacevolumetracking.(a)Contoursofthevolumefractionforasinglebubble;(b)volumefractioninthe correspondingcutplane.

Table 1

Singlebubbletestcaseforthevolumeconservation.Therelativeerrorsattheendofthe simulationandtheCPUtimetsofthesimulationarereportedforthefourgridsconsidered.

20³ 40³ 80³ 160³

4.5×10⁻¹⁶ 2.4×10⁻¹⁷ 3.1×10⁻¹⁸ 3.9×10⁻¹⁹

ts(ms) 35 52 84 103

The bubblesare transportedin aLagrangian way.When the velocity uⁿ_b⁺¹ of abubble attime tn+¹ is calculated(see section2.8),thebubblemasscenterxⁿ_bisthendeterminedusingthefirstorderEulerexplicitscheme:

xⁿ_b⁺¹

=

^xn_b

+

tuⁿ_b⁺¹ (2)

Once,theupdatedpositionofthebubbleisknown,thebubblesurface Fⁿ⁺¹ attimen

+

^{1 iscalculatedthroughequation} (1).WeintroducetheclassicalVoFfunctionCasthevolumefractionofthecontinuousfluid.TheVoFfunctionCⁿ⁺¹attime n

+

^{1 iscomputedfromtheinterfacepositionFn+1 usingtheVoFfunction}initializationfoundinBnaetal.[4] (Fig.2).We stressthattheVoFfunctionC isnottransportedherebutdeterminedfromthebubblesurfaceequation F.Wealsodefine

α

f

=

^{C and}

α

b

=

¹

−

^{C thevolumefractionoftheliquidandthegas,}respectively.

Wetestedthemethodonasingle bubbletestcasetoevaluateitsprecision andinparticularthemasslossduring the simulation.Forthistest, we considera sphericalbubble witharadius rb

=

⁰

.

2 m centered ina 1 m

×

^{1 m}

×

^{1 m cubic} computationaldomain.AuniformgridofN cellsisusedandthecellsizeish0.Thecomputationaldomainisperiodic.The bubbleisinitiallylocatedatthecenterofthecomputationaldomain.Att₀ inauniformvelocityu_b

= (

0

.

10

,

0

.

15

,

0

.

20

)

m

/

s is imposed tothe bubble.The errorand CPUmeasurements (

andts) are made att_f,when the bubblehas traveled a distancecorrespondingto1 m.Thevelocityofthebubbleischosentoavoidanycaseoffavorablealignmentwitharawof cells.

Thevolumeofthebubbleismeasuredas V_C

=

^h3₀

N

i=¹

C_i (3)

andweintroducetherelativeL1error definedas:

= |

^Vth

−

^VC

|

Vth

(4)

whereV_th isthetheoreticalvolume.Fourdifferentgrids20³,40³,80³and160³areconsidered,leadingtotheratioofcells perradius4,8,16and32,respectively.Therelativeerrors isalmostconstantduringthesimulationanddoesnotdepend onthepositionofthebubbleonthegrid.Thevalueof attheendofthesimulationandthetimetsofthesimulationare reportedinTable1.The resultshighlightthree characteristicsofthe selectedVoFmethod:the highprecision ofthecolor functioncomputation, evenforthesmallestratioofcells perradius,thedecrease oftheerrorwithmesh refinementand thecontroloftheCPUtime.

Fig. 12.Effectofthepenalizationviscosityμponthevelocityfieldinsidethebubble;leftfigure:μp=^{1 Pa}.s,rightfigure:μp=^{100 Pa}.s.Thecontoursof thevelocitymagnitudeinsidethebubbleareplottedinwhite,thecontourofthecolorfractioninpurple.

toa 2-Fluideulerianmultiphaseframe,inwhicha momentumequationforeachphase,liquidandbubble,isweightedby theirrespectivevolumefraction

α

f and

α

b.Theformulationofthesystemofequationisthefollowing:

α

f

ρ

f

∂

u

∂

t

+ α

f

ρ

fu

.∇

^u

= − α

f

∇

^P

+ α

f

μ

f

∇

^2u

+ α

f

ρ

fg

+ α

f

α

b

ρ

b

u_b

−

^u

τ α

_b

ρ

_b

^∂

^ub

∂

t

+ α

_b

ρ

_bu_b

.∇

u_b

= − α

_b

∇

P

+ α

_b

μ

p

∇

²u_b

+ α

_b

ρ

_bg

− α

_f

α

_b

ρ

_b^ub

⁻

^u

τ

(23)

The velocities appearing in the penalization term are considered at time n

+

^{1 for a more stablecoupling between the} velocity of the bubble and the hydrodynamic action of the liquid. The aim is now to express the penalization term

α

b

ρ

b

(

u_bⁿ⁺¹

−

^un+1

)/ τ

inequation (7), given the fluid velocity uⁿ anduⁿ⁺¹ at time n andn

+

^1, respectively. The con- tribution from thefluid on the bubblesis expressed fromthe liquidmomentum andis injected inthe bubbleequation.

Giventhattheconvectiveterm

α

b

ρ

bu_b

.∇

^ub

=

^{0 andtheviscousterm}

α

b

μ

p

∇

^2ub

=

^{0 becauseu}bisimposedtobeuniform inthepenalizedbubble,weobtainthefollowingformulationofthepenalizationterm(seeAppendix):

ρ

_b^ub

n+¹

−

^un+1

τ ⁼ ρ

_b

^t

τ ₊

t

u_bⁿ

−

^un+1

t

+ ρ

_f

^t

τ ₊

t

uⁿ⁺¹

−

^un

t

+

^un+1

. ∇

^un

− μ

_f

_∇

²uⁿ

+ ( ρ

b

− ρ

f

)

t

τ +

tg

(24)

whichisinjectedintheliquidmomentumequationtoobtainthereversecouplinginthefluid:

ρ

f

1

− α

b

t

τ +

t

uⁿ⁺¹

−

^un

t

+

^un+1

.∇

^un

− μ

f

∇

^2un

= −∇

^Pn

+

ρ

f

+ α

b

( ρ

b

− ρ

f

)

t

τ +

t

g

+ α

b

ρ

b

u_bⁿ

−

^un+1

τ +

t

(25)

The couplingmethodis definedby thesemi-implicitsystemofequations (24) and(25). Solving thissystemgivesthe valuesofu_bⁿ⁺¹anduⁿ⁺¹.Anotabledrawbackofthiscouplingmethodconcernsanon-uniform solutionforthevelocityfield insidethebubbleasshowninFig.12(left) aftersolvingthesystem.Theformulationofthepenalizedterm

(

u_bⁿ⁺¹

−

^un+1

)/ τ

impliesthatauniformvelocityshouldbefoundinsidethepenalizedregion.Toensurethatcondition,wehadtointroduce a penalization viscosity

μ

p.

μ

f is imposed to

μ

p in equation (25) inside the penalized domain.

μ

p has not a physical meaningsinceashear-free conditionisimposedfortheliquidatthebubblesurface(seesection 2.6).Theroleof

μ

p isto setthevelocityfieldinsidethebubbleasuniformaspossibleasshowninFig.12(right).Theartificialpenalizationviscosity

μ

p isusedfortheviscous stresscalculationonthefacesjoiningtwopenalizedcellsasshowninFig.13.Weobservedno influence ofthepenalization viscosity

μ

p onthe stability ofthesimulations. Howeverthe penalizationviscosity

μ

p was observedtoimpactthecomputationalcostofthesimulations.TheCPUtimeofthesimulationsisincreasedwhenincreasing

μ

p duetothepoorconditioningofthematrixinvolvedinthe resolutionofthecouplingmethodof

NEPTUNE_CFD

.The impactofthepenalizationviscosity

μ

p ontheresultsisfurtherdiscussedinthefollowingsections.

3. Validationofthenumericalmethod

Test cases are now proposed to validate the enhancement of the penalization method for the simulation of bubbly flows. Forthatpurpose,we willconsiderinthissectionthe flowina2D curvedchannel withfixed boundaries,the flow

Fig. 15.Curvedchanneltestcase.(a)Pressurefieldattheendofthesimulation.Inwhite:isocontoursofthepressurefield.(b)Velocityfieldattheendof thesimulation,colorationbymagnitude.

Fig. 16.Curved channel test case. Zoom of the velocity field near the interface; Left: Original method, Right: new penalization method.

In orderto determinethe convergenceorderofthe method,simulations havebeenrun withsixdifferentgrids made of 4,8,16, 32,64,and128cells in thegap ofthechannel. The resultsare reportedin Fig.17.The erroron the x- and y-componentsofthevelocitybothdecreasewithanorderof0.75whilethepressuredecreasesshow afirstorderconver- gence.Theconvergenceclosetofirstorderisattributedtotheimposedslipcondition basedonafirstorderextrapolation of the fluid velocityat thesurface ofthe penalized domain.The convergence maybe improvedsince other penalization methods offer better convergence rates such as 2 for the velocity and 1 for the pressure [10,37]. Additional numerical developmentsrequiredtoimprovetheconvergenceratemaybeconsideredforfuturedevelopmentsin

NEPTUNE_CFD

. 3.2. Flowinsidea2Dinclinedchannelwithmovingwalls

For this second test case, the configuration studied is a 2D inclined channel as shownin Fig. 18a. The sides of the channel (

1 and

2) aretwo parallellines definedbythevector

(

0

.

1

,

0

.

8

)

.The dimensionsare Lx

=

⁰

.

1 m, L y

=

²

×

^Lx.

The gapofthe channel isr

=

⁰

.

25 m. Outside ofthefluid domain,thepenalized domain

isset toa constantvelocity u_b

= −

^u0

= − (

u0x

,

u0y

) = ( −

⁰

.

01

, −

⁰

.

08

)

m

.

s⁻¹ parallel to thewall asshownin Fig.18b.The slipcondition isimposed on both surfaces. The fluid properties are

ρ

_f

₌

1000 kg

.

m⁻³ and

μ

_f

₌

0

.

1 Pa

.

s so that the Reynolds number is Re

= ρ

fr

|

^u0

| / μ

f

=

^{20.Theinitial conditionsareimposedto u}

(

t

=

⁰

, (

x

,

y

)) =

^{0 and P}

(

x

,

y

) =

^{0 insidethefluid region}

f.At t

=

^{0 thefluid velocity is imposed to u}

=

^u0 atthe inlet of the domain while thepressure is P

=

^{0 atthe outlet. The} fluid velocity is thusparallel to thewall andin theopposite directionof thevelocity imposed inthe penalized domain.

Due totheslipcondition,themoving wallsareexpectednotto interactwiththefluid sothatsolutionub

(

x

,

y

) =

u0 and P

(

x

,

y

) =

^{0 isexpectedinthe channel.The} simulationsare run fort

=

^{5 s until thesteadystate isreached. Thevelocity} fieldsarepresentedinFig.19fortheoriginal penalizationmethodandthenewone.Thegridismadeof20

×

^{40 cellsin} eachdirection.Thenewmethodclearlyshowauniformvelocityfieldparalleltothewallwhilesomeoscillationsattributed tothecalculationofthedivergenceinthemass conservationareobservedwiththeoriginalmethod.Simulationshavebeen

Table 2

3Dtranslatingbubbletestcase.Evolutionofthe L₂ erroronthevelocity through3timeiterations.The200³gridisused.

Method t1 t2 t3

||^u−^u0||2

||u₀||2 CR 4.19×^{10−16 4}.14×^{10−16 4}.24×¹⁰⁻¹⁶ NCR 1.76×^{10+1 2}.61×^{10+1 1}.31×¹⁰⁺¹

Fig. 23.Convergence study forP, (Vx,Vy,Vz)inL1norm for the bubble moving at zero relative velocity in the flow test case.

4. Riseofasinglebubbleinaquiescentliquid

Theaimofthissectionistosimulatetheriseasinglebubbleinaliquidatrestinordertotestthemomentumcoupling methodproposedinsection2.8.Thepresentedformulationmakespossibleadynamicinteractionbetweenthefluidandthe bubble.WeconsidertwodifferentmoderateReynoldsnumbersRe

=

^{17 andRe}

=

^{71 basedonthebubblediameterd}b

=

^2rb andubT themodulusofitsterminalvelocityubT.ForsuchReynoldsnumbers,thebubbleissphericalanditstrajectorycan beobtainedwith[18]:

m_fC_Mdu_b dt

= −

¹

2C_D

ρ

f

π

r²_bu_b²

−

^mfg (27)

where mf

= ρ

f4

3

π

r_b³, CM

=

¹

/

2 isthe added mass coefficient ofa single spherical bubbleand thehistory force can be consideredofsecondorderforsuchReynoldsnumbers.Thedragcoefficientisgiventhroughthecorrelation[20]:

C_d

=

¹⁶

Re 1

+ (

8

/

Re

+

¹

/

2

(

1

+

³

.

315

/

Re^0.5

)

⁻¹ (28)

Equation(27) canbeeasilysolved,leadingtotheterminalvelocityu_bT andthetransientevolutionofthebubblerising velocity.Asitwillbediscussed,thecouplingformulationintroducesadependencetothepenalizationviscosity.Inparticular, we willfocus on thebubble initial accelerationdu_b

/

dt

= −

^{2g (balancebetweenbubble} accelerationand gravityvolume forcing), the terminalvelocity ubT (balancebetweenpressure-viscous drag withgravity volume forcing),andthe bubble wakeshapebecauseofitsimportanceforthedevelopmentofbubbleinducedagitation[26].

We consider the configuration described inFig. 24. A1 mm spherical bubble islocated at the bottomof a tank-like computationaldomain

(

xb

,

yb

,

zb

) = (

0

.

25db

,

0

.

25db

,

0

.

2db

)

.Att0,thebubblestartsrisinguntilreachingitsterminalvelocity.

The simulationis stopped after the steadystate is reached, fora final simulation time t_f depending on theconsidered Reynolds number. Simulations have been carriedout fortwo different fluid viscosities leading to two differentterminal velocities andthe twoReynolds numbers Re

=

^{17 and Re}

=

^{71.The listof theparameters used isgivenin Table3. The} penalizationparameter

τ

issetto

τ =

^{10−14s.Wewanttoconsiderheretheinfluenceoftheinternal}penalizationviscosity

μ

p onthebubblemotion.Thetime step

t isimposed followingtheCFLcriteria

|

^u

|

^t

/

x

=

⁰

.

5.Threegridrefinements havebeentested:4,8and16cellsintheradiusofthebubble.Notethatwhenconsidering4cellsinthebubbleradius,the boundarylayerthicknessestimatedas

δ ∼

^rb

/

Re^1/2andthegridsizehavethesameorderofmagnitudeforRe

=

^71.

ThedimensionsofthecomputationaldomainshowninFig.24are5d_b

×

^5db

×

^25db.48simulationsintotalhavebeen carriedoutforawiderangeofinternalviscosities,eightforeachReynoldsnumberandgridrefinement.Thecomputational resources havebeenadaptedtotherefinementofeachgrid: 6

×

^24,8

×

^24,12

×

^{24 CPU unitswereused}respectivelyfor 4,8and16cells intheradiusofthebubble,leadingtothecorresponding CPUtimes45 min,2 hours and12 hours fora singlesimulation.

Fig. 24.Singlebubblerise.(a)Initialconfigurationforthesimulation.Thebubbleislocatedatthebottomofthetank.Thereportedverticalplaneisused forthewakedescription.(b)Locationsofthereportedwakeprofiles.

Table 3

Setofnumericalparametersforthesimulationofasinglebubblerise.

db(mm) ρb(kg.m⁻³) ρf (kg.m⁻³) μf (Pa.s) tf (s) |^ubT|(cm.s⁻¹) Re

1.0 50 1000 0.005 0.06 8.90 17

1.0 50 1000 0.0022 0.12 15.60 71

4.1. Initialaccelerationandterminalvelocity

The timeevolution ofthebubbleisreported inFig.25for Re

=

^{17 and Re}

=

^{71 anddifferent}penalization viscosities

μ

p. The evolutions clearly reveal the effectof the penalization viscosity

μ

p onthe evolution ofthe bubble velocity. An appropriatepenalizationviscosity

μ

p needstobe selectedtoreproducecorrectlyboththeinitialacceleration(addedmass effect) andtheterminal velocity ofthebubble. ForeachReynolds number,andforeach ratioof cells per radius,we are alwaysabletoreproduceasatisfactoryevolutionofthebubblerise.

AsummaryofthetendsobservedinFig.25isshowninFig.26.Theoptimumvalueofthepenalizationviscosity

μ

p for agooddescriptionoftheinitialaccelerationiscomparedtotheonegiventhebestterminalvelocity. Theyarereportedas afunctionofthegridspacing.DependingonboththegridresolutionandtheReynoldsnumberanappropriatepenalization viscosity

μ

p hastobe selected.It isnot herepossibletopropose aclear scalingof

μ

p andthisaspectofthe numerical methodneedstobeimprovedforprovidingpredictivesimulationsofsinglebubbledynamics.However,theobjectiveishere toprovideamethodabletodescribebubblyflows.Asitwillbeshowninthenextsection,theimpactofthepenalization viscosity

μ

p issignificantlyreducedwhenconsideringabubbleswarm.

4.2. Maximalvorticity

Inadditiontothevelocityfields,wealsocomputedthemaximalvorticity

max thatdevelopsatthebubblesurface for each meshrefinementandeach Reynoldsnumber.Thisquantity isofinterest sinceit controlsthebubbledrag forceand thedevelopmentofitswake.Themaximalvorticityforasphericalbubblemovingsteadilyinaviscousfluidisgivenbythe followingexpression[17]:

ω

max

= |

^ubT

|

r_b

16

+

³

.

315Re^1/2

+

^3Re

16

+

³

.

315Re^1/2

+

^{Re (29)}

Thisexpression hasbeenestablished through DNSsimulationsfor Reynoldsnumbersranging from0.1to5000. The evo- lutions are shown inFig. 27. The maximal vorticity measured inthe simulations tends to getcloser to the DNSresults (Equation (29))withthemeshrefinement.Themaximumvelocity andvorticity atabubblesurface aredirectlylinked by therelation u_max

=

^rb

ω

max

/

2 [17]. Thegood convergenceofthemaximal vorticityto theexpectedvalue impliesthat our simulationsreproduce acorrectfluidvelocity closetothebubblesurface.The streamlinesandthe velocitymagnitudefor

Fig. 29.Singlebubblerise.VelocityprofileinthewakeofarisingbubbleasindicatedinFig.24.(a)Re=^17,(b)Re=^{71.FirstlineX}=^0,secondline X=^2R/3.

is knowntofollowthestandardfar-wakebehavior thatwillbe usedhereforthecomparison.The correspondingvelocity deficitprofileparalleltotheX-directionisgivenby(Batchelor,1967[3]):

u_wake

∼

^ubTQ 4

π ν

_fZexp

−

^ubT

X² 4

ν

_fZ

(30)

where

ν

f

= μ

f

/ ρ

f isthecinematicviscosityofthefluid andQ

=

^FD

/ ρ

fu_bT isdeterminedby integrationoverthewake and is directly related to the magnitude FD of the drag force of the body.Given u_bT and FD, we obtain the far wake description.ThenumericalresultsarepresentedforthetwocasesinFigs.29for X

=

^{0 and X}

=

^2rb

/

3 andforthe3mesh refinementsconsidered:4,8and16cellsinthebubbleradius.Thevelocitydecayinthebubblewaketendstogetcloserto relation(30) withthemeshrefinementforbothReynoldsnumbersRe

=

^{17 andRe}

=

^71.

5. Mono-dispersedbubbleswarm

The final step of thevalidation process is thesimulation of bubblesswarm forboth a highReynolds numbersand a highnumberofbubbles.Theobjectiveistoverifythewholenumericalmethod,inordertodemonstratetheeffectiveness of the so-built numerical tool for bubbly flows simulations. The validation strategy is defined as follows. Experimental investigations of the flow generated by a homogeneous population of bubbles rising in water havebeen carried out by Zenitetal.(2001)[42],Garnieretal.(2002)[11],Ribouxetal.(2010)[26] andColombetetal.(2015)[8].Theexperiments providearathercompletedescriptionofboththebubblesmotionandtheinducedagitationintheliquid.Theideahereis toreproduceinrelativesimilarconditionsthoseexperimentswithournumericaltool,andmakeadirectcomparisonwith the relevantinformationcharacterizing abubbleswarm such asthevelocity PDFsofboth thebubbles andthefluid. The flow issupposed to befullyperiodic ineach direction.The collisions betweenthebubbles areassumedfullyelastic.The simulationparametersareshowninTable4.Forallthesimulationsreportedinthissection,thetimestepisthetimestep imposedbytheCFLconditiondividedby5.Thestatisticalmeasurement(PDF)isstartedafterreachingastabilizedagitated

Table 4

Setofnumericalparametersforthebubbleswarmsimulations.

db(mm) ρb(kg.m⁻³) ρf (kg.m⁻³) μf (Pa.s) μp/μf V0(cm.s⁻¹) Re0 α0

1.0 50 1000 0.0008 100 22.5 281 0.008

state ofthemixture.The simulationsare rununtil thePDFsareconverged.We note here

α

theglobalvoidfraction. The analysisofthe resultsrequiresfornormalizationpurposethe computation oftherise velocity V₀ of asingle bubble.We note Re0 and

α

0 thecorresponding Reynoldsnumberandreferencevoidfraction, respectively.Thevaluesof V0, Re0 and

α

0 arereportedinTable4.

Beforeadetailedinvestigationofthebubbleswarm,we presentsome preliminarysimulationstotestthemeshrefine- ment,thedomainsizeandthepenalizationviscosity.

5.1. Influenceofthemeshrefinement

Wefirstconsider theinfluenceofthemeshrefinementwithaspecialattentiontotheliquidandbubblevelocity fluc- tuations.Weintroducethreegridsrefinements R0

,

R1 andR2 correspondingrespectivelyto3,6and12cells inthebubble radius. The computational domain is chosen as

=

^D1

= [

⁰

;

⁰⁰⁵

] × [

⁰

;

⁰⁰⁵

] × [

⁰

;

⁰²⁵

]

^{(Fig. 31). Three void fractionare} consideredhere:

α ₌

5%

,

10% and15%,correspondingto Nb

=

⁶⁰

,

120 and180 bubbles,respectively.ThePDFsofthefluid velocity andthebubblevelocity forboththehorizontalandverticalcomponentsare reportedinFig.30.Statisticsonthe flowhavebeenmeasuredbetweent

=

⁰

.

2 s,oncethefluidishomogeneouslyagitated,andt_f

=

^{2 s.Asshown,thegridhas} nosignificantinfluenceonthebubblehorizontalvelocityfortheconsideredrefinements.Theothervelocitycomponentsare moresensitivetothegridbutaclearconvergencewiththegridisobservedandcloseresultsarefoundfortherefinements R1 andR2.Theaveragebubblerisingvelocityincreasesas

xdecreases.Thisisduetoan underestimationofthebubbles risingvelocityoncoarsegrids,mostlyduetotherelativehighReynoldsnumberconsideredhere(Re

^{300)becauseofthe} boundarylayeratthebubblesurfacethatdecreasesasRe^−1/2.Theaveragerisingvelocityisevolvinginthesamewayfor eachvoidfraction. Thevarianceofthehorizontalandverticalfluid velocitiesincreasesslightlywiththemeshrefinement.

The simulationsperformedwith themeshrefinement R0, R1 and R2 required respectively12,81and350 CPUhours of calculationfor6

×

^{28 CPUunits.Inordertosave}computationalresources,therefinementR1isusedforallthesimulations reportedinthefollowingsections.

5.2. Influenceofthedomainsize

Wewantnowtoconsidertheinfluenceofthedomainsizeontheresultsandontheconvergenceofthestatistics.For eachsimulation,thenumberofbubblesN_bisdeducedfromthevolumeofthecomputationaldomainandtheimposedvoid fraction

α

.Ifthenumberofbubblesinthe domainreachesarelativelow value, thestatisticalmeasurements willnot be abletodescribe preciselythebubble-inducedagitationofthefluid.Thecomputationaldomains D1

= [

⁰

;

^5db

] × [

⁰

;

^5db

] × [

⁰

;

^25db

] = [

⁰

;

^Lx

] × [

⁰

;

^Ly

] × [

⁰

;

^Lz

]

^{, D}2

= [

⁰

;

^Lx

] × [

⁰

;

^Ly

] × [

⁰

;

^2Lz

]

^{and D}3

= [

⁰

;

^2Lx

] × [

⁰

;

^2Ly

] × [

⁰

;

^Lz

]

^{are considered} (see Fig.31).Comparedto D1,for D2 the lengthof thedomaininthe

(

O z

)

directiondoubles,while forD3 thedomain is double in both horizontal directions

(

O x

)

and

(

O y

)

. Three void fractions have been studied:

α ₌

5%, 10% and 15%, corresponding respectivelyto Nb

=

^{60,120 and}180 bubblesfor

=

^D1, Nb

=

^{120,240 and}360 bubblesfor

=

^D2 and N_b

=

^{240,480 and} 720 bubbles for

=

^D3.Statistics onthe flowhavealso beenmeasured betweent

=

⁰

.

2 s,oncethe fluid ishomogeneously agitated, andt_f

=

^{2 s. The}simulations requiredapproximately 4,6 and8days ofcalculation for thedomains D1

,

D2

,

D3 using6

×

^{28 CPUunits.ThenormalizedPDFsplottedinFig.32showverysimilarresultsforeach} componentof the liquidand bubblevelocity when using the domains D₁, D₂ and D₃ for the voidfractions considered (

α ₌

5%,10% and15%).

Weconcludethatperforming simulationsona computationaldomainlargerthan D1 inanydirectionwillnotprovide anyadditionalinformationon thePDFs.Allthe simulationspresented inthefollowingsectionsare carriedout usingthe computationaldomainD1.

5.3. Influenceofthepenalizationviscosity

μ

p

Intheprevious section,we detailedtheinfluenceofthepenalization viscosity

μ

p onthedynamicsofa single bubble risingina quiescentliquid.Foreach bubbleReynoldsnumberandmeshrefinementconsidered, anoptimum valueof

μ

p has tobe determined for agood description ofthe bubble velocity evolution.As we deal now withbubble swarms,we wantto examine againtheinfluence of

μ

p onthe results.Forthispurpose,three simulations havebeencarried out for

α =

^10%,correspondingtothreedifferentvalues,

μ

p

=

⁰

.

1,1 and10 Pa

.

s.ThemeshrefinementisR2(6cellsinthebubble radius)andthedomainisD1.Thesimulationsarestoppedatt_f

=

^{4 s.WeextractedthePDFsofthehorizontalandvertical} components of the fluid and bubble velocities for

μ

p

=

⁰

.

1, 1 and 10 Pa

.

s (Fig. 33). Foreach component ofthe liquid andbubblevelocities, theagreement isperfect betweenthePDFs fordifferentvaluesof

μ

p. Asa conclusion, thechoice of

μ

p hasnoinfluence onthesimulations ofbubbleswarms.A cleardifferentbehaviorwas observed forthesimulation

Fig. 30.Bubble swarm. Effect of the mesh refinement on the normalized PDFs of the liquid velocity (left) and the bubbles velocity (right).

of asingle bubblewhereresultsare sensitiveto thechoice ofthepenalizationviscosity

μ

p.Wewere not abletoclearly determinetheoriginoftheobserveddifferencebetweenthesinglebubblecaseandthebubbleswarmcase.Webelievethat the maincandidateisthesolverused in

NEPTUNE_CFD

fortheinversionofthesemi-implicitsystem(24)–(25).Insingle bubbleconfigurations,wethinkthatthepresenceofthepenalizationviscositycausesasingularityinthematrixtoinverse while inthebubbleswarm configuration,thebubblesingularitiesare distributedinall thedomainimproving thematrix conditioning,explainingtheindependenceoftheresultswiththeviscosityofpenalization.Adeeperanalysisofthesolver performancesneedtobeconductedtoclearlyexplainthisnumericalbehavior.

We now discuss the influence ofthe numerical value of

μ

p on the CPUtime. We have observed that the CPUtime strongly increaseswhen

μ

p increases.Foreach simulation performed(

μ

p

=

⁰

.

1,1 and 10 Pa

.

s)we report thefinal CPU timeattf

=

^{2 s inTable5.Again,wenotetheincreaseoftheCPUtimeas}

μ

p increases.HowevertheCPUtimeincreaseis onlyaround7% forachangeofoneorderofmagnitudefor

μ

p (0.1to1 Pa.sand1to10 Pa.s),whichremainsacceptablein termofmanagementofcomputingresources.

Fig. 31.Bubble swarm. Computational domains used for the study of the domain size effect (a)D1, (b)D2, (c)D3.

Table 5

Bubbleswarm.EvolutionofthefinalCPUtimefordifferentvaluesofμp. μp=⁰.1 Pa.s μp=^{1 Pa.s} μp=^{10 Pa.s}

CPU time (hours) 72 77 83

5.4. Bubbleandfluidagitation

Simulations havebeencarried out fordifferentvoid fractionsranging from

α =

²

.

5% to 20%,leading to an increasing numberofbubblesintheflowasshowninFig.34.

Theaveragebubblerisingvelocity

<

ubz

>

isreportedinFig.35asafunctionofthevoidfractionfor0

.

002

≤ α _≤

0

.

20.

The evolution is compared to the experiments. The diameter d_b

=

^{1 mm is considered in the present work while the} diametersconsidered intheexperiments ared_b

=

¹

.

6,2

.

1 and2

.

5 mm inRiboux etal.[26],d_b

=

¹

.

4 mm inZenit etal.

[42],anddb

=

³

.

5 mm inGarnieretal.[11].Thevelocitiesarenormalizedbythevelocity V0 ofasinglerisingbubble.The resultsarefoundinverygoodagreementwithboththeexperimentsandthedecreasinglawV₀

(

1

− α

^0.49

)

.

Variances ofthe bubblerising velocity

<

u_bz²

>

arecompared in Fig.35 withthecorresponding data available inthe literature. The experimental results highlight the independence of the variance to the void fraction, tendency which is also remarkably observed forthe present simulations. This behavior show that the contribution of the path oscillations dominatesoverbubbleinteractions,effectthatisproperlyreproducedinoursimulations.Wealsonotethatthevariancein thepresentwork

(<

u_bz²

> ≈

²

×

^{10−3 m2}

.

s⁻²

)

significantlydifferwiththevaluesobtainedintheexperiments

(<

u_bz²

> ≈

1

.

2

×

^{10−2 m2}

.

s⁻²

)

forthe three diameters considered. Clearly we are not able to reproduce the same level ofbubble agitationasreportedby theseexperiments.The experimentswere performedforhigherbubblediameters,corresponding tohigherbubbleReynoldsnumbersandsignificantlydeformedbubbles.Forsuch conditions,singlebubblemovefollowing zig-zag orhelicoidal path.This may be a possible explanation of thelower level ofbubble agitation observed withour sphericalbubbles.However, morerecentlyColombetetal.(2015) [8] also obtainedintheirexperiments alower levelfor thebubbleagitationwhenthebubblemotionismeasuredusingparticletrackingvelocimetrybasedonimagestakenwitha fixedfocallens.Theyreport

<

u_bz²

> ≈

³

×

^10−2m2

.

s⁻² upto

α =

^{10%,avaluecomparabletoour}simulations.Asdiscussed in [26] the measure by dual optical probe ofthe variance of the bubble agitation maybe significantly perturbed when bubblesareoblatespheroidmovingwithoscillatingvelocityandorientation.

Wenowfocusonthedynamicsoftheliquid.Fig.36showsthenormalizedPDFoftwocomponentsofthefluidvelocity forthepresentwork

(

d_b

=

^{1 mm}

)

,the consideredvoidfractions rangingfrom

α =

²

.

5% to

α ₌

15%. ThePDFsare scaled following

( α / α

0

)/

V0 initiallyproposedby Risso[27].Wehavealsochecked(notshownhere)thatthebest superposition ofourPDF isalsoobtainedwith

α

^0.4.ThePDFsofthehorizontalvelocity(rightcolumn)are symmetric,meaningthatthe simulations are ableto restore the anisotropy property,the flow beingstatisticallyaxisymmetric around the bubble,and

Fig. 32.Bubble swarm. Effect of the domain size. Normalized PDFs of vertical and horizontal liquid (left) and bubble (right) velocity fluctuations.

thedistributionofbubblesinthehorizontaldirectionbeinguniform. Alowerliquidagitationisobservedinthehorizontal direction.Howeverthecomparisonfortheverticalfluidagitationisverygoodforboththeshapeandthelevelofagitation.

TheshapeoftheverticalbubblevelocityPDFsisclearly non-symmetricduetotheentrainmentofthefluidinthewakeof thebubbles,implyingthatupwardfluctuationsaremoreprobable.

6. Conclusions

Anewpenalizationmethodforthesimulationofbubblyflowshasbeenproposed.Theoriginalityofthepresentedwork relies ontheextension oftheclassic penalization methodsdevelopedforsolid objectsforthe simulationofbubbles.The bubbles are seen as moving objects whose motionis determined by the action of the fluid on them, through the new coupling method presented. A second notable point is the numerical methodology implemented to ensure a shear-free conditionatthesurfaceofthepenalizeddomain.

Fig. 35.Bubbleswarm.(Left)Averagebubblevelocity<ubz>and(right)variance<u_bz²>ofthebubblevelocitynormalizedbythevelocityV0ofasingle risingbubbleasafunctionofthegasvolumefractionα.Redsquares,presentnumericalwork.Experiments:• ^{Ribouxetal.(2010),}◦^Zenitetal.

(2001),^{Garnieretal.(2002).}

Fig. 36.Bubble swarm. Normalized PDFs of vertical (left) and horizontal (right) liquid velocity fluctuations.

bubble velocitywiththevoidfraction, a bubbleagitationindependentofthe voidfraction, anda liquidagitationinboth horizontalandverticaldirectionsscalingas

α

^0.4.

The results obtainedthroughthispaper demonstratetheviability ofthe numericaltool forthe simulationsof bubbly flows.Despitetheassumptionsmadeonthebubbles(sphericalshape,non-deformability,elasticcollisions)wewereableto reproducethemaincharacteristicofbubbleinducedagitationandconfirmtheoriginoftheinducedagitationresultingfrom wake interactions.Oneofthemaininterestofthemethodisitscapacitytodealeasily witha highnumberofbubblesat moderateandhighvoidfraction. Thesimulationspresentedinthispaperinvolveupto720bubblesforabubbleReynolds number of Re

^{300 and for a void fractionup to 20%. In this paperwe show that our numerical approach is able to} reproducethemaincharacteristicsofbubblyflows.

This numerical methodwas originally built to dodeal with two kindsof objectives. The short-term objectivewas to develop a numericaltool allowing us to performliquid–bubbles simulations ina simplified frame (no break-up, no coa- lescence, nomass transfer, elastic collisions).We proposed asuccessful validation strategy toqualify the abilitiesof our tool anditcan nowbe usedtoinvestigatemono-dispersedbubblyflows andtowork onclosurerelationsforEuler/Euler simulations.Thesecondobjectiveistoconsiderbubblyflowsrelativelyclosetonuclearapplications.First,themethodwill be extended toconsider poly-dispersedbubblyflows. Then deformationandcoalescence effects,aswell asmasstransfer will beintroduced inthenumericalmethodwithrelativeease,meanwhile consideringdeformation upto breakupevents willbemorechallenging.

Acknowledgements

TheauthorsthankEDFR&Dforthecomputationalresourcesprovidedfortherealizationofthiswork.Thecomputations wereperformedusingATHOSandPORTHOSsupercomputers.

Appendix

Wedetailinthisappendixthederivationofthecoupledrelations(24) and(25).Westartfromthetwo-phaseformalism describedinequation(23).Duetotheuniformityofthepenalizedvelocity u_b inthebubble,

∇

^ub

=

^{0 and}

∇

^2ub

=

^0.Each equationisfirstdividedby

α

f and

α

b fortheliquidandthebubblephase,respectively.Thepressuregradient

∇

^{P isthen} expressedfromeachequationas:

∇

^P

= − ρ

f

∂

u

∂

t

− ρ

fu

. ∇

^u

+ μ

f

∇

^2u

+ ρ

fg

+ α

b

ρ

b

u_b

−

^u

τ

⁽³¹⁾

∇

^P

= − ρ

_b

^∂

^ub

∂

t

+ ρ

_bg

− α

_f

ρ

_b^ub

⁻

^u

τ

⁽³²⁾

Theseequationsarelinkedthrough thesamepressuregradient.Combiningequations(31) and(32) leadstothefollowing relation:

( α

b

+ α

f

) ρ

b

u_b

−

^u

τ ^{= −} ρ

b

∂

u_b

∂

t

+ ρ

f

∂

u

∂

t

+ ρ

fu

. ∇

^u

− μ

f

∇

^2u

+ ( ρ

b

− ρ

f

)

g (33) with

α

b

+ α

f

=

^{1.Animplicitschemeisusedforthe}penalizationtermandtheresultingsemi-discreteformulationwrites as:

ρ

b

u_bⁿ⁺¹

−

^un+1

τ ^{= −} ρ

b

u_bⁿ⁺¹

−

^ubn

t

+ ρ

f

uⁿ⁺¹

−

^un

t

+

uⁿ⁺¹

.∇

uⁿ

− μ

f

ρ

f

∇

²uⁿ

+ ( ρ

b

− ρ

f

)

g (34) Theuⁿ_b⁺¹ termsareregroupedontheleftsideoftheequation.Equation(34) becomes:

ρ

b

ubn+1

−

uⁿ⁺¹

τ ⁼ ρ

b

t

τ +

t

ubn

−

uⁿ⁺¹

t

+ ρ

f

t

τ +

t

uⁿ⁺¹

−

uⁿ

t

+

^un+1

.∇

^un_f

− μ

f

ρ

f

∇

^2un

+ ( ρ

_b

₋ ρ

_f

)

t

τ ₊

tg

(35)

Thisequationcorrespondstotheimplicitformulationofthepenalizationvelocity.Thisexpressionistheninjectedintothe discretizedformofthefluidequationtoobtain:

ρ

_f

1

− α

_b

^t

τ ₊

t

uⁿ⁺¹

−

^un

t

+

^un+1

. ∇

^un

− μ

f

ρ

_f

^∇

2uⁿ

= −∇

^Pn

+

ρ

_f

₊ α

_b

( ρ

_b

₋ ρ

_f

)

t

τ ₊

t

g

+ α

b

ρ

b

ubn

−

uⁿ⁺¹

τ +

t

(36)

References

[1]B.J.Alder,T.E.Wainwright,Studiesinmoleculardynamics.I.Generalmethod,J.Chem.Phys.31 (2)(1959)459–466.

[2]P.Angot,C.H.Bruneau,P.Fabrie,Apenalizationmethodtotakeintoaccountobstaclesinincompressibleviscousflows,Numer.Math.81 (4)(1999) 497–520.

[3]G.K.Batchelor,AnIntroductiontoFluidDynamics,CambridgeUniversityPress,1967.

[4]S.Bnà,S.Manservisi,R.Scardovelli,P.Yecko,S.Zaleski,NumericalintegrationofimplicitfunctionsfortheinitializationoftheVOFfunction,Comput.

Fluids113(2015)42–52.

[5]B.Bunner,G.Tryggvason,Dynamicsofhomogeneousbubblyflows.Part2.Velocityfluctuations,J.FluidMech.466(2002)53–84.

[6]B.Bunner,G.Tryggvason,Effectofbubbledeformationonthepropertiesofbubblyflows,J.FluidMech.495(2003)77–118.

[7]D.K.Clarke,H.A.Hassan,M.D.Salas,EulercalculationsformultielementairfoilsusingCartesiangrids,AIAAJ.24 (3)(1986)353–358.

[8]D.Colombet,D.Legendre,F.Risso,A.Cockx,P.Guiraud,Dynamicsandmasstransferofrisingbubblesinahomogenousswarmatlargegasvolume fraction,J.FluidMech.763(2015)254–285.

[9]A.Esmaeeli,G.Tryggvason,AdirectnumericalsimulationstudyofthebuoyantriseofbubblesatO(100)Reynoldsnumber,Phys.Fluids17 (9)(2005) 093303.

[10] A.Etcheverlepo,Développementdeméthodesdedomainesfictifsausecondordre,Doctoraldissertation,UniversitéSciencesetTechnologies-Bordeaux I,2013.

[11]C.Garnier,M.Lance,J.L.Marié,Measurementoflocalflowcharacteristicsinbuoyancy-drivenbubblyflowathighvoidfraction,Exp.Therm.FluidSci.

26 (6–7)(2002)811–815.

[12]A.Guelfi,D.Bestion,M.Boucker,P.Boudier,P.Fillion,M.Grandotto,etal.,NEPTUNE:anewsoftwareplatformforadvancednuclearthermalhydraulics, Nucl.Sci.Eng.156 (3)(2007)281–324.

[13]Y.Hallez,D.Legendre,Interactionbetweentwosphericalbubblesrisinginaviscousliquid,J.FluidMech.673(2011)406–431.

[14]T.Kempe,J.Fröhlich,Animprovedimmersedboundarymethodwithdirectforcingforthesimulationofparticleladenflows,J.Comput.Phys.231 (9) (2012)3663–3684.

[15]T.Kempe,M.Lennartz,S.Schwarz,J.Fröhlich,Imposingthefree-slipconditionwithacontinuousforcingimmersedboundarymethod,J.Comput.Phys.

282(2015)183–209.

[16]J.Kim,D.Kim,H.Choi,Animmersed-boundaryfinite-volumemethodforsimulationsofflowincomplexgeometries,J.Comput.Phys.171 (1)(2001) 132–150.

[17]D.Legendre,Ontherelationbetweenthedragandthevorticityproducedonacleanbubble,Phys.Fluids19 (1)(2007)018102.

[18]J.Magnaudet,I.Eames,Themotionofhigh-Reynolds-numberbubblesininhomogeneousflows,Annu.Rev.FluidMech.32 (1)(2000)659–708.

[19]J.B.Mclaughlin,Numericalsimulationofbubblemotioninwater,J.ColloidInterfaceSci.184 (2)(1996)614–625.

[20]R.Mei,J.F.Klausner,C.J.Lawrence,AnoteonthehistoryforceonasphericalbubbleatfiniteReynoldsnumber,Phys.Fluids6 (1)(1994)418–420.

[21]R.Mittal,G.Iaccarino,Immersedboundarymethods,Annu.Rev.FluidMech.37(2005)239–261.

[22]A.Morente,APenalizationMethodfortheSimulationofBubblyFlows,Thèsededoctorat,InstitutNationalPolytechniquedeToulouse,2018.

[23]C.S.Peskin,Flowpatternsaroundheartvalves:anumericalmethod,J.Comput.Phys.10 (2)(1972)252–271.

[24]C.S.Peskin,Numericalanalysisofbloodflowintheheart,J.Comput.Phys.25 (3)(1977)220–252.

[25]G.Riboux,D.Legendre,F.Risso,Amodelofbubble-inducedturbulencebasedonlarge-scalewakeinteractions,J.FluidMech.719(2013)362–387.

[26]G.Riboux,F.Risso,D.Legendre,ExperimentalcharacterizationoftheagitationgeneratedbybubblesrisingathighReynoldsnumber,J.FluidMech.

643(2010)509–539.

[27]F.Risso,K.Ellingsen,Velocityfluctuationsinahomogeneousdilutedispersionofhigh-Reynolds-numberrisingbubbles,J.FluidMech.453(2002) 395–410.

[28]I.Roghair,Y.M.Lau,N.G.Deen,H.M. Slagter,M.W.Baltussen, M.V.S.Annaland, J.A.M.Kuipers,Onthedragforceofbubblesinbubbleswarmsat intermediateandhighReynoldsnumbers,Chem.Eng.Sci.66 (14)(2011)3204–3211.

[29]I.Roghair,J.M.Mercado,M.V.S.Annaland,H.Kuipers,C.Sun,D.Lohse,Energyspectraandbubblevelocitydistributionsinpseudo-turbulence:numerical simulationsvs.experiments,Int.J.Multiph.Flow37 (9)(2011)1093–1098.

[30]S.Schwarz,T.Kempe,J.Fröhlich,Animmersedboundarymethodfor thesimulationofbubbleswith varyingshape,J.Comput.Phys.315(2016) 124–149.

[31]H.Sigurgeirsson,A.Stuart,W.L.Wan,Algorithmsforparticle-fieldsimulationswithcollisions,J.Comput.Phys.172 (2)(2001)766–807.

[32]G.Tryggvason,B.Bunner,A.Esmaeeli,D.Juric,N.Al-Rawahi,W.Tauber,etal.,Afront-trackingmethodforthecomputationsofmultiphaseflow,J.

Comput.Phys.169 (2)(2001)708–759.

[33]M.Uhlmann,Animmersedboundarymethodwithdirectforcingforthesimulationofparticulateflows,J.Comput.Phys.209 (2)(2005)448–476.

[34]M.Uhlmann,Interface-resolveddirectnumericalsimulationofverticalparticulatechannelflowintheturbulentregime,Phys.Fluids20 (5)(2008) 053305.

[35]S.O.Unverdi,G.Tryggvason,Afront-trackingmethodforviscous,incompressible,multi-fluidflows,J.Comput.Phys.100 (1)(1992)25–37.

[36]S.P.VanderPijl,A.Segal,C.Vuik,P.Wesseling,Amass-conservinglevel–setmethodformodellingofmulti-phaseflows,Int.J.Numer.MethodsFluids 47 (4)(2005)339–361.

[37]S.Vincent,J.C.B.DeMotta,A.Sarthou,J.L.Estivalezes,O.Simonin,E.Climent,ALagrangianVOFtensorialpenaltymethodfortheDNSofresolved particle-ladenflows,J.Comput.Phys.256(2014)582–614.

[38]S.Vincent,A.Sarthou,J.P.Caltagirone,F.Sonilhac,P.Février,C.Mignot,G.Pianet,AugmentedLagrangianandpenaltymethodsforthesimulationof two-phaseflowsinteractingwithmovingsolids.Applicationtohydroplaningflowsinteractingwithrealtiretreadpatterns,J.Comput.Phys.230 (4) (2011)956–983.

[39]D.Wunsch,TheoreticalandNumericalStudyofCollisionandCoalescence–StatisticalModelingApproachesinGas-DropletTurbulentFlows,Doctoral Dissertation,Thèsededoctorat,InstitutNationalPolytechniquedeToulouse,2009.

[40]T.Ye,R.Mittal,H.S.Udaykumar,W.Shyy,AnaccurateCartesiangridmethodforviscousincompressibleflowswithcompleximmersedboundaries, J. Comput.Phys.156 (2)(1999)209–240.

[41]X.Yin,D.L.Koch,Lattice-BoltzmannsimulationoffiniteReynoldsnumberbuoyancy-drivenbubblyflowsinperiodicandwall-boundeddomains,Phys.

Fluids20 (10)(2008)103304.

[42]R.Zenit,D.L.Koch,A.S.Sangani,Measurementsoftheaveragepropertiesofasuspensionofbubblesrisinginaverticalchannel,J.FluidMech.429 (2001)307–342.