A penalization method for the simulation of bubbly flows

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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

A

penalization

method

for

the

simulation

of

bubbly

flows

Antoine Morente

a

,

b

,

Jérôme Laviéville

a

,

Dominique Legendre

b

,

∗

a_{EDF(ÉlectricitédeFrance)R&D,6QuaiWatier,78400Chatou,France}

b_{InstitutdeMécaniquedesFluidesdeToulouse,IMFT,UniversitédeToulouse,CNRS–Toulouse,France}

Keywords: Bubbly flows Penalization method Shear free condition Bubble induced agitation

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 majorchallengein in-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 for bub-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

=

*

Corresponding author.

Fig. 2. Use

of the Bna et al. [

4] method for the interface volume tracking. (a) Contours of the volume fraction for a single bubble; (b) volume fraction in the

corresponding cut plane.

Table 1

Single bubble test case for the volume conservation. The relative errors at the end of the simulation and the CPU time tsof the simulation are reported for the four grids considered.

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 u_bn+1 of abubble attime tn+1 is calculated(see

section2.8),thebubblemasscenterxn_bisthendeterminedusingthefirstorderEulerexplicitscheme:

xn_b+1

=

xn_b

+ 

t un_b+1 (2)

Once,theupdatedpositionofthebubbleisknown,thebubblesurface Fn+1 attimen

+

1 iscalculatedthroughequation (1).WeintroducetheclassicalVoFfunctionC asthevolumefractionofthecontinuousfluid.TheVoFfunctionCn+1_attime n

+

1 iscomputedfromtheinterfacepositionFn+1 usingtheVoFfunctioninitializationfoundinBnaetal.[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 inauniformvelocityub

= (

0

.

10

,

0

.

15

,

0

.

20

)

m

/

s is imposed tothe bubble.The errorand CPUmeasurements (



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

Thevolumeofthebubbleismeasuredas

VC

=

h30 N



i=1

Ci (3)

andweintroducetherelativeL1error



definedas:



=

|

Vth

−

VC

|

Vth

(4)

whereVth 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. Effect

of the penalization viscosity

μpon the velocity field inside the bubble; left figure: μp=1 Pa.s, right figure: μp=100 Pa.s. The contours of

the velocity magnitude inside the bubble are plotted in white, the contour of the color fraction in purple.

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

∇

2_u

₊

_α

f

ρ

fg

+

α

f

α

b

ρ

b ub

−

u

τ

α

b

ρ

b

∂

ub

∂

t

+

α

b

ρ

bub

.

∇

ub

= −

α

b

∇

P

+

α

b

μ

p

∇

2_u 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

(

ubn+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

bub

.

∇

ub

=

0 andtheviscousterm

α

b

μ

p

∇

2ub

=

0 becauseubisimposedtobeuniform inthepenalizedbubble,weobtainthefollowingformulationofthepenalizationterm(seeAppendix):

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

+

u n+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

+

u n+1

_.

_∇

_un

₋

_μ

f

∇

2un



= −∇

Pn

+



ρ

f

+

α

b

(

ρ

b

−

ρ

f

)



t

τ

+ 

t



g

+

α

b

ρ

b ubn

−

un+1

τ

+ 

t (25)

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

(

ubn+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. Curved

channel test case. (a) Pressure field at the end of the simulation. In white: isocontours of the pressure field. (b) Velocity field at the end of

the simulation, coloration by magnitude.

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 afirstorder

conver-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

ub

= −

u0

= −(

u0x

,

u0 y

)

= (−

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

3D translating bubble test case. Evolution of the L2 error on the velocity through 3 time iterations. The 2003_{grid is used.}

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 for P, (Vx,Vy,Vz)in L1norm for the bubble moving at zero relative velocity in the flow test case.

4. Riseofasinglebubbleinaquiescentliquid

Theaimofthissectionistosimulatetheriseasinglebubbleinaliquidatrestinordertotestthemomentumcoupling methodproposedinsection2.8.Thepresentedformulationmakespossibleadynamicinteractionbetweenthefluidandthe bubble.WeconsidertwodifferentmoderateReynoldsnumbersRe

=

17 andRe

=

71 basedonthebubblediameterdb

=

2rb andubT themodulusofitsterminalvelocityub T.ForsuchReynoldsnumbers,thebubbleissphericalanditstrajectorycan beobtainedwith[18]: mfCM dub dt

= −

1 2CD

ρ

f

π

r 2 bub2

−

mfg (27)

where mf

=

ρ

f4₃

π

r_b3, CM

=

1

/

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

Cd

=

16

Re 1

+ (

8/Re

+

1/2(1

+

3.315/Re

0.5

₎

−1 ₍₂₈₎

Equation(27) canbeeasilysolved,leadingtotheterminalvelocityu_{b T} andthetransientevolutionofthebubblerising velocity.Asitwillbediscussed,thecouplingformulationintroducesadependencetothepenalizationviscosity.Inparticular, we willfocus on thebubble initial accelerationdub

/

dt

= −

2g (balance betweenbubble accelerationand gravityvolume forcing), the terminalvelocity ub T (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 tf 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

=

10−14_{s.Wewanttoconsiderheretheinfluenceoftheinternalpenalizationviscosity}

μ

p onthebubblemotion.Thetime step



t isimposed followingtheCFLcriteria

|

u

|

t

/

x

=

0

.

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

δ

∼

rb/Re1/2andthegridsizehavethesameorderofmagnitudeforRe

=

71.

ThedimensionsofthecomputationaldomainshowninFig.24are5db

×

5db

×

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

×

24,8

×

24,12

×

24 CPU unitswereusedrespectivelyfor 4,8and16cells intheradiusofthebubble,leadingtothecorresponding CPUtimes45 min,2 hours and12 hours fora singlesimulation.

Fig. 24. Single

bubble rise. (a) Initial configuration for the simulation. The bubble is located at the bottom of the tank. The reported vertical plane is used

for the wake description. (b) Locations of the reported wake profiles.

Table 3

Set of numerical parameters for the simulation of a single bubble rise.

db(mm) ρb(kg.m−3) ρf (kg.m−3) μf (Pa.s) tf (s) |ub T| (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 anddifferentpenalization 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

=

|

ub T

|

rb 16

+

3.315Re1/2

₊

_3Re 16

+

3.315Re1/2

₊

_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 umax

=

rb

ω

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

Fig. 29. Single

bubble rise. Velocity profile in the wake of a rising bubble as indicated in Fig.

24. (a) Re=17, (b) Re=71. First line X=0, second line X=2R/3.

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

uwake

∼

ubTQ 4

π ν

fZ exp



−

ubT X2 4

ν

fZ



(30)

where

ν

f

=

μ

f

/

ρ

f isthecinematicviscosityofthefluid andQ

=

FD/

ρ

fubT isdeterminedby integrationoverthewake and is directly related to the magnitude FD of the drag force of the body.Given ub T 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

Set of numerical parameters for the bubble swarm simulations.

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 preliminarysimulationstotestthemesh refine-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,andtf

=

2 s.Asshown,thegridhas nosignificantinfluenceonthebubblehorizontalvelocityfortheconsideredrefinements.Theothervelocitycomponentsare moresensitivetothegridbutaclearconvergencewiththegridisobservedandcloseresultsarefoundfortherefinements

R1 andR2.Theaveragebubblerisingvelocityincreasesas



x decreases.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.Inordertosavecomputationalresources,therefinementR1isusedforallthesimulations

reportedinthefollowingsections.

5.2. Influenceofthedomainsize

Wewantnowtoconsidertheinfluenceofthedomainsizeontheresultsandontheconvergenceofthestatistics.For eachsimulation,thenumberofbubblesNbisdeducedfromthevolumeofthecomputationaldomainandtheimposedvoid 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

]

, D2

= [

0

;

Lx

]

× [

0

;

Ly

]

× [

0

;

2Lz

]

and D3

= [

0

;

2Lx

]

× [

0

;

2Ly

]

× [

0

;

Lz

]

are considered (see Fig.31).Comparedto D1,for D2 thelengthof 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 and180 bubblesfor



=

D1, Nb

=

120,240 and360 bubblesfor



=

D2 and Nb

=

240,480 and 720 bubbles for



=

D3.Statistics onthe flowhavealso beenmeasured betweent

=

0

.

2 s,oncethe

fluid ishomogeneously agitated, andtf

=

2 s. Thesimulations requiredapproximately 4,6 and8days of calculation 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.Thesimulationsarestoppedattf

=

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

Bubble swarm. Evolution of the final CPU time for different values of μp.

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 db

=

1 mm is considered in the present work while the diametersconsidered intheexperiments aredb

=

1

.

6,2

.

1 and2

.

5 mm inRiboux etal.[26],db

=

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

<

u2

bz

>

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

<

u2

bz

>

≈

3

×

10−2m2

.

s−2 upto

α

=

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

Wenowfocusonthedynamicsoftheliquid.Fig.36showsthenormalizedPDFoftwocomponentsofthefluidvelocity forthepresentwork

(

db

=

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}

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. Bubble

swarm. (Left) Average bubble velocity

<ubz>and (right) variance <ubz2>of the bubble velocity normalized by the velocity V0of a single rising bubble as a function of the gas volume fraction α. Red squares, present numerical work. Experiments: •  Riboux et al. (2010), ◦Zenit et al. (2001), Garnier et al. (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 ub 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

∇

2_u

₊

_ρ

fg

+

α

b

ρ

b ub

−

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 ub

−

u

τ

= −

ρ

b

∂

ub

∂

t

+

ρ

f

∂

u

∂

t

+

ρ

fu

.

∇

u

−

μ

f

∇

2_u

_{+ (}

_ρ

b

−

ρ

f

)

g (33)

with

α

b

+

α

f

=

1.Animplicitschemeisusedforthepenalizationtermandtheresultingsemi-discreteformulationwrites as:

ρ

b ubn+1

−

un+1

τ

= −

ρ

b ubn+1

−

ubn



t

+

ρ

f



un+1

−

un



t

+

u n+1

_.

_∇

_un

₋

μ

f

ρ

f

∇

2_un



+ (

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

+

u n+1

_.

_∇

_un f

−

μ

f

ρ

f

∇

2_un



+ (

ρ

b

−

ρ

f

)



t

τ

+ 

tg (35)

Thisequationcorrespondstotheimplicitformulationofthepenalizationvelocity.Thisexpressionistheninjectedintothe discretizedformofthefluidequationtoobtain:

ρ

f



1

−

α

b



t

τ

+ 

t

 

un+1

−

un



t

+

u n+1

_.

_∇

_un

₋

μ

f

ρ

f

∇

2_un



= −∇

Pn

+



ρ

f

+

α

b

(

ρ

b

−

ρ

f

)



t

τ

+ 

t



g

+

α

b

ρ

b ubn

−

un+1

τ

+ 

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