The emptying of a bottle as a test case for assessing interfacial momentum exchange models for Euler–Euler simulations of multi-scale gas-liquid flows

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The emptying of a bottle as a test case for assessing interfacial momentum exchange models for Euler–Euler

simulations of multi-scale gas-liquid flows

Samuel Mer, Olivier Praud, Hervé Neau, Nicolas Merigoux, Jacques Magnaudet, Véronique Roig

To cite this version:

Samuel Mer, Olivier Praud, Hervé Neau, Nicolas Merigoux, Jacques Magnaudet, et al.. The emptying of a bottle as a test case for assessing interfacial momentum exchange models for Euler–Euler simu- lations of multi-scale gas-liquid flows. International Journal of Multiphase Flow, Elsevier, 2018, vol.

106, pp. 109-124. �10.1016/j.ijmultiphaseflow.2018.05.002�. �hal-01801495�

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To link to this article : DOI:10.1016/j.ijmultiphaseflow.2018.05.002 URL : https://doi.org/10.1016/j.ijmultiphaseflow.2018.05.002

To cite this version : Mer, Samuel and Praud, Olivier and Neau, Hervé and Merigoux, Nicolas and Magnaudet, Jacques and Roig, Véronique The emptying of a bottle as a test case for assessing interfacial momentum exchange models for Euler–Euler simulations of multi-scale gas-liquid flows. (2018) International Journal of Multiphase Flow, vol. 106. pp. 109-124. ISSN 0301-9322

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The emptying of a bottle as a test case for assessing interfacial momentum exchange models for Euler–Euler simulations of multi-scale gas-liquid flows

Samuel Mer

^a

, Olivier Praud

^a

, Herve Neau

^a

, Nicolas Merigoux

^b

, Jacques Magnaudet

^a

, Veronique Roig

^a

aInstitut de Mécanique des Fluides de Toulouse, IMFT, Université de Toulouse - CNRS, Toulouse, France

bElectricité de France R&D, 6 quai Wattier, Chatou 78401, France

Keywords:

Euler–Euler formulation Bubbly flow

Multi-scale flow

a b s t r a c t

Simulatinggas-liquidflowsinvolvingawiderangeofspatialandtemporalscalesandmultipletopological changesremainsamajorchallengenowadays,asthecomputationalcostassociatedwithdirectnumeri- calsimulationstillmakesthisapproachunaffordable.Acommonalternativeisthetwo-fluidEuler–Euler formulationthatavoidssolvingallscalesatthepriceofsemi-empiricalclosuresofmass,momentumand energyexchangesbetweenthetwofluids.Manyofsuchclosuresareavailablebuttheirperformancesin complexflowsarestillindebate.Closuresconsideringseparatelylargegasstructuresandsmallerbub- blesandmaking thesetwo populationsevolveand possiblyexchangemassaccordingtotheirinterac- tionswiththesurroundingliquidhaverecentlybeenproposed.Inordertoassessthevalidityofsome oftheseclosures,wecarryoutanoriginalexperimentinasimpleconfigurationexhibitingarichsucces- sionofhydrodynamicevents,namelytheemptyingofawaterbottle.Wesimulatethisexperimentwith theNEPTUNE_CFDcode,usingthreedifferentclosureapproachesaimedatmodellinginterfacialmomen- tumexchangeswithvariousdegreesofcomplexity.Basedonexperimentalresults,weperformadetailed analysisofglobal and localflowcharacteristics predicted byeachapproachtounveilits potentialities andshortcomings.Althoughallofthemarefoundtopredictcorrectlytheoverallfeaturesoftheemp- tyingprocess,strikingdifferencesareobservedregardingthedistributionofthedispersedphaseandits consequencesintermsofliquidentrainment.

1. Introduction

Gas-liquid flows involving a broad range of bubble sizes are ubiquitous ingeophysical andengineeringconfigurations andap- plications,suchasmagmaticchimneys,submarineexplosions,bub- blecolumnsornuclearsafety,tomentionjustafew.Insuchsitua- tions,thegasphasefrequentlyinvolvesawiderangeofspatialand temporalscales,fromlargegaspocketstosmalldispersedbubbles.

Moreover,dramaticallydifferentflowregimesmaybeencountered, characterized bydistinctinteraction mechanismsbetweenthegas phaseandthecarryingliquid.Simulatingsuchflowsremainsama- jor challenge nowadays, although massive efforts have been de- voted during the last two decades to develop modelling strate- giesaimed at computingmultiphase flows(Prosperetti andTryg- gvason,2007).

Thesestrategies differ accordingto the level ofaccuracy they target andthecomputational resources they require. A firstclass of numerical techniques based on Direct Numerical Simulation (DNS)oftheNavier–Stokesequationsshares thesamemainchal- lengeconsisting in precisely localizinginterfaces in the flow do- main and imposing the proper jump conditions across them.

Three main approaches have been proposed to track interfaces, namely the Volume Of Fluid (Hirt and Nichols, 1981), Level Set (Osher andSethian, 1988) andFront Tracking(Unverdi andTryg- gvason, 1992) methods (see also Scardovelli and Zaleski, 1999;

Sethian andSmereka, 2003; Tryggvason etal., 2001 forreviews).

Since then, these techniques have become mature and are now widely used to get insight into detailed mechanisms governing flow configurations withincreasing complexity.For bubblyflows, thismayrangefromthoseinvolvingasinglebubblerisingatlarge Reynolds number(Cano-Lozano et al., 2016) to dispersed bubbly flows with up to O(¹⁰³) ^{bubbles moving at moderate Reynolds} number(BunnerandTryggvason,2002).

DespitethepotentialitiesofferedbytheseDNSapproachesand their improvements associated with local grid refinement tech- niques(Popinet,2009),their computational coststill makesthem unable to simulate complex configurations, especially those in- volving multiplecoalescence and break-up sequences. Such com- plextwo-phaseflowsareusuallycomputedusingthemuchcruder Euler–Eulerapproachbasedontheso-calledtwo-fluidmodel(Ishii, 1975; Ishii and Hibiki, 2006). In this framework, the governing equationsare obtained aftervolume-averaging (or more formally ensemble-averaging)thelocalbudgets,sothatunknowntermsoc- curatinterfaces.Asinanyaveragingscheme,closuresarerequired to express these terms with respect to the primitive variables andtheirgradients.Theaccuracyofthesimulationsthendepends tremendouslyonthevalidityoftheseclosures.Manyofthemhave been proposed for each flow configuration, e.g. separated flows, dispersed bubbly or particulate flows, etc. (see Drew and Pass- man,1999andBalachandarandEaton,2010forreviews).Inbubbly flows for instance, assuming non-deformable and mono-disperse bubbles, momentum interfacial exchange is usually modelled by considering drag, added-mass and shear-induced lift forces act- ing onindividual bubbles,supplemented with turbulent diffusio- phoresisandlubrificationeffectswhenthecarryingflowisturbu- lentandwalls are present, respectively. Applications ofthe two- fluidapproachtosuchflows, possiblywithphase change,maybe found forinstance in Mimouni et al.(2010, 2011a, 2011b). Simi- larly,specific closureswithvariousdegreesofsophistication have been developed to simulate ‘slug’ flow configurations (Issa and Kempf, 2003; Issa et al., 2006) and separated nearly-horizontal flows(Valléeetal.,2008).

The key limitationof the above closuresis that they are spe- cificto theconfiguration forwhich they were calibrated andare unableto properlymodeltheinterfacialexchangemechanisms at workinanothertypeofflow.Thislimitationcanonlybeovercome ifthemodellingapproachismadeabletorecognizewhichconfig- urationispresentatagivenpositioninspaceandtime.Twomain streamsofapproachesweredevelopedduringthelasttwodecades toreach this goal.The first ofthem consists inswitching locally fromDNS(based on eitherthe VolumeOfFluid orthe Level Set approach)tothetwo-fluidformulationwhereverinterfacesexhibit acharacteristicsizeoftheorderofthegridcell(ˇCerneetal.,2001;

Tomiyama et al., 2006; Yan and Che, 2010). This technique was successfullyemployed to compute severalgas-liquid flows domi- nated by fragmentation, e.g. a two-phase vortex or the unstable Rayleigh-Taylorconfiguration.Thesecondapproachconsistsinex- tendingthe two-fluid modelto an arbitrarynumber of‘fields’ or

‘phases’,eachofthemcorresponding toa specificflow configura- tionorclass oftwo-phase entities (e.g.smallbubbles, large bub- bles, slugs, etc). Occurrence of each of these configurations at a giventime andposition hasto be identified inorder toevaluate thecorrespondingvolumefraction.Aseach‘phase’hasitsownve- locityfield,momentumclosureshavetobeformulatedtoproperly account for the interaction betweentwo of them. Thisapproach hasforinstancebeenappliedtotheupwardbubblypipeflowwith severalwidely distinct bubble sizes and possible mass exchange betweenthem,duetophasechange(Krepperetal.,2008).

Althoughtheabovemethodologywasinitiallydesignedtodeal withdispersedflows,itmaybeappliedtoseparatedflowsaswell, provided one is able to (i) properly define a criterion allowing theoccurrenceofthe‘separated’configurationtobedetected,and (ii) derive realistic closure laws for the various separated flow regimesaccordingtotheinterfaceroughness.Thisistheessenceof theLargeInterfaceModel(LIM)designedbyHenriques(2006)and Coste(2013),aswell asthatoftheAlgebraicInterfacialAreaDen- sity (AIAD) model promoted by Höhne and Vallée (2010) and Deendarlianto et al.(2011). Mixed configurations in which sepa- ratedanddispersed regions coexist within the flow mayalso be

tackledwithintheframeworkofthen-fieldapproach,providedthe abovecriterionallows‘LargeInterfaces’(hereinafterabbreviatedas LI)correspondingtotheseparatedconfigurationtobedisentangled fromsmall-scaleinterfaces,anddistinctclosurelawsareemployed forthedispersed andseparatedregions. Thisideayieldedseveral differentmodellingapproaches,suchastheGeneralizedTwo-Phase Flowmodel(GENTOP,Hanschetal.,2012)ortheGeneralizedLarge InterfaceModel(GLIM,Merigouxetal.,2016).Examplesofapplica- tionofthistype ofapproachto agasjetimpingingafree surface andabubblecolumnwithbubblesburstingatthefreesurfacemay befoundinthefirstreference.

Still in the context of the two-fluid and n-field formulations, severalattempts were recentlycarried out to achieve a morere- alistic and accurate treatment of LI by taking explicitly into ac- count surface tension effects (Bartosiewicz et al., 2008; Štrubelj etal.,2009; Gadaetal., 2017).Atechnicaldifficulty arisesinthis type of approach,dueto the naturaltendency fornumericaldif- fusion tospread stiff volumefraction gradients.Sharpeningtech- niqueshavebeenproposed tocounteractthiseffectandmaintain well-defined separated‘phases’, so that the LImayremain prop- erly defined over time. A cutoff length must also be defined, so that interfaces with a characteristic size smaller than this criti- cal length are no longer resolved and interactions between the corresponding dispersed phase and the continuous one are en- tirely modelled with the help of empirical closurelaws. Last, an exchange procedurecombiningnumerical requirementsandbasic physical principleshas to be designed to allow a LI to break up intosmallerbubbles,andsuch bubblestocoalesceandgeneratea LI.Suchanapproachhasbeenimplementedbothintheaforemen- tionedGENTOPformulation(Montoyaetal.,2015),andintheNEP- TUNE_CFDcodewhereitistermedtheLargeBubbleModel(LBM, Denèfleetal.,2015;Mimounietal.,2017).Preliminaryassessment of thisapproach in canonical configurations, such as the Kelvin–

HelmholtzandRayleigh–Taylorinstabilities,wasreportedbyFleau etal.(2015,2016).

The aim of the presentpaper is to assess the validity of the above LIM, GLIM and LBM approaches implemented in the NEP- TUNE_CFDsoftware, by consideringan academicbutalready sig- nificantly complex flow configuration and performing a one-to- one comparisonbetweenoriginalexperimentscarried outin that flowandcomputationsmakinguseoftheabovethreemodels.The selected two-phase configuration, namely theemptying of a wa- ter bottle, is especially relevant for checkingsuch modelling ap- proaches,asitexhibitsawiderangeoftemporalandspatialscales.

Largeairbubbleswithdiametersoftheorderofthebottleneckare periodicallygeneratedandrisewithinthebottleuntiltheyburstat thefreesurfacebelowthetopofthebottle.Whileascending,these largebubblesundergosuccessivebreak-upevents,yieldingswarms ofsmallerbubbles,partofwhichmaycoalesceagainandpartici- pateintotheregenerationandreconfigurationofthelargebubble population.

Few computational studies have been performed so far on this flow configuration. The most noticeable is that of Geigeretal.(2012) who simulateditwiththe helpof theOpen- Foam softwarein theframework ofa Volume OfFluidapproach.

They mainly focused on the influence ofgeometrical parameters and bottle inclination on the emptying time. However they as- sumed the liquid and air phases to be both isothermal and in- compressible.Asweshallseelater,thelatterassumptionishighly questionable.

Thepresentpaperisorganizedasfollows.Section2introduces themulti-fieldformulationanddetailsthevariousapproachesem- ployed to model interfacial momentum exchanges in the NEP- TUNE_CFD code. The experimental and computational configura- tionsaredescribedinSection3.Section4discussestypicalresults obtainedthroughbothapproachesonsome quantitiescharacteriz-

ingtheoveralldynamicsofthesystem.Section5focusesonlocal characteristics ofthe two-phase flow andexamines theinfluence oftheaforementionedmodelsontheevolutionandstatisticaldis- tributionofthesecharacteristics.FinallySection6summarizesthe main findings ofthe study anddraws some prospects forfuture work.

2. n-fieldApproachandinterfacialexchangemodelsin NEPTUNE_CFD

All simulations presented in this paper were performed with therelease4.0oftheNEPTUNE_CFDcodejointlydevelopedbyEDF, CEA, IRSN and Framatome for more than a decade. The govern- ing equations considered in thiscode are based onan extension to n fields of the two-fluid model (Ishii and Hibiki, 2006). This techniqueis particularlywell-suited fordispersedflows involving asinglesizeofparticles,dropsorbubbles.Inthiscontext,thede- scriptionreducestotwofields,namelyacontinuouscarryingliquid phase anda dispersedone. Ingeneral, mass,momentumanden- ergyconservationequationsaresolvedforeachfield,withtheas- sumptionthat theysharethesamepressurefield.Fromanumeri- calviewpoint,allgoverningequationsconsideredinNEPTUNE_CFD are discretized using a finite volume technique with collocated variables. The grid is unstructuredand involves arbitrary shaped cells. A second order linear upwind scheme is used to update the volume fraction of each field. The velocity field is advanced thanks to a fractional step technique while the pressure field is computed with the help of the SIMPLE algorithm (Patankar and Spalding, 1972). An iterative coupling between energy and mass balances isusedtoenforcethesimultaneousconservationofboth quantities(Mimounietal.,2008).

2.1. Primaryequationsofthen-fieldmodel

Weassumean isothermalflow,sothatonlythemassandmo- mentum conservationequationsneedto beconsidered. Velocities andvolume fractions, together withfluid properties(density and viscosity)aredefinedforeachfieldkateverypointofthedomain.

Thevolumefractions,

α

k,satisfy

k

α

k=1, (1)

Foreachfieldk,themassbalanceiswrittenas

∂ ∂

^t

⁽ α

k

ρ

k

)

⁺

∇

^·

( α

k

ρ

ku_k

)

⁼

p=k

p→k, (2)

where

ρ

kandu_karethedensityandvelocityoffieldk,andp→k

denotesthemassexchangeratebetweenfieldspandk.Foranyp andk,thismassexchangeratemustsatisfy

p→k+

k→p=0. (3)

Intwo-fluidgas-liquidconfigurations,Eq.(3)merelyexpressesthe interfacialmassbalance, sothat p→k istheinterfacialmass ex- changeratepossiblyduetophasechange.Nosuchphasechangeis consideredinthesimulationstobediscussedlater.However,when severaldistinctfieldsare employedtorepresententitiesofdiffer- entsizeswithinthesamephysicalphase(e.g.largeandsmallbub- bleswithinthegasphase),coalescenceandbreak-upeventsmake the mass of each of these fieldsvary, so that p→k is generally nonzero.

Themomentumbalanceforphasekiswrittenas

∂ ∂

^t

⁽ α

k

ρ

ku_k

)

+

∇

^·

( α

k

ρ

ku_ku_k

)

=−

α

k

∇

^P+

α

k

ρ

kg +

∇

·

α

k

μ

k

( ∇

^uk+^T

∇

^uk

)

+

p=k

I_p→k, (4)

where

μ

kis theviscosityoffield k andI_p→krepresents themo- mentumexchange ratebetweenfieldspandk.The lattermaybe splitintheform

I_p→k=I^H_p→k+

p→ku^I_pk, (5)

whereu^I_pk isthe velocity at the interface betweenphases pand k,and thefirst term inthe right-hand side, I^H_p

→k,represents the momentumexchange duetohydrodynamicforces,whilethesec- ondismerelythemomentum exchangeassociatedwiththemass exchangeratebetweenthetwophases.Theinterfacialmomentum exchangetermhastobemodelledtoclosethesetofequations.

Ifaninterfacialtension,

σ

pk,maybedefinedbetweenphasesp andk,thecorrespondinginterfacialmomentumbalanceimplies Ip→k+Ik→p= lim

V→0

1 V

σ

pk

κ

pknpkdA^I, (6) wheren_pkistheunitnormaltothe p−kinterface,

κ

pk=−

∇

·n_pk isthecorresponding meancurvature,dA^I istheelementaryinter- facialareaand^{denotesthecontrolvolume(which incomputa-} tionalpracticecorresponds tothegridcell), thevolume ofwhich isV. In what follows, capillaryeffects are neglected in the LIM andGLIMapproaches.Weshallspecifyinduecoursehowandun- derwhichconditionstheyaretakenintoaccountintheLBM.Note that,providedtheinterfacialvelocity u^I_pk iscontinuousacrossthe interface,theleft-handsideofEq.(6)reducestoI^H_p→k+I^H_k→p,ow- ingtoEq.(3).

Inthenextfoursubsections,wedetailthevariousclosuresand detection criteria used to express the interfacialmomentum ex- changeinthethreeaforementioned modelsimplementedinNEP- TUNE_CFD.Asummary ofthecharacteristicsandclosurelawsin- volvedineachofthesemodelsisprovidedinTable1.

2.2.Thedispersedbubblyflowmodel

Ina bubbly flow withmono-dispersed bubbles, the above set ofequationsreducesto theusual two-fluid formulation.The cor- respondingtwofieldsarereferred toasthecontinuousliquid(cl) andthedispersedgas(dg)phases,respectively.Insuchaflow,pro- videdwalleffectsandturbulentdispersionareabsentorhaveneg- ligibleeffects, the interfacialmomentum exchange isassumed to resultfrom the sum of three independent contributions, namely a viscous dragforce, F^D,an addedmass force, F^AM,and a shear- inducedliftforce,F^L (Mimounietal.,2011b).

Themomentumtransferresultingfromviscousdragiswritten as

F^D_cl→dg=1

8A^I

ρ

lCD

||

^udg−u_cl

|| (

^ucl−u_dg

)

, (7) where A^I=lim_V→0V¹

dS^I denotes the rate of interfacial area perunit volumewhich,fora mono-dispersedbubbledistribution, maybeexpressedasafunctionofthevolumefractionofthedis- persed phase throughthe well-known relation A^I=6

α

dg/d_dg, d_dg denotingthebubblediameter.ThedragcoefficientC_D dependson the bubble Reynolds number, Re_dg=

ρ

l

||

^udg−u_cl

||

^ddg/

μ

l, on the Bond number, Bo_dg=(

ρ

l−

ρ

^g)^gd2_dg/

σ

⁽

σ

^{being the surface ten-}

sionoftheliquid),andonthevolumefraction,

α

dg.NEPTUNE_CFD makes use of the empirical correlations established by Ishii and Zuber(1979) toexpressC_D asafunction ofRe_dg,Bo_dg and

α

dg in the various regimesencountered withrising bubbles, possibly in denseconfigurations.

ThecontributionofaddedmasseffectstoI^H_p→kiswritteninthe form(Zuber,1964)

F^AM_cl→dg=C_AM1+2

α

dg

1−

α

dg

α

dg

ρ

l

Du_cl Dt −dudg

dt , (8)

Table 1

Summary of the three closure models for the momentum interfacial exchange rate, I ^H. Model

Number of gas fields

Cutoff length cg / dg

Diameter of dispersed bubbles

Forces acting on LI

Forces acting on dispersed bubbles LIM 1 — — I ^H_cl→cg= F ^F_cl→cg( Eq. (10) ) I ^H_cl→dg= F ^P_cl→cg( Eq. (11) )

+ F ^P_cl→cg( Eq. (11) )

I ^H_cl→cg= F ^F_cl→cg( Eq. (10) ) I ^H_cl→dg= F ^D_cl→dg( Eq. (7) )

GLIM 1 l c≈ d g + F ^P_cl→cg( Eq. (11) ) + F ^AM_cl→dg( Eq. (8) )

+ F ^L_cl→dg( Eq. (9) ) I ^H_cl→cg= F ^σ_cl→cg( Eq. (10) ) I ^H_cl→dg= F ^D_cl→dg( Eq. (7) ) LBM 2 l c= 8 d g + F ^D_cl→cg( Eqs. (15) –(17) ) + F ^AM_cl→dg( Eq. (8) ) + F ^L_cl→dg( Eq. (9) )

where C_AM=0.5 is the added mass coefficient of an isolated sphere, ^D_Dt^ucl = ^∂_∂^u_t^cl +u_cl.

∇

^ucl and ^du_dt^dg= ^∂_∂^udg_t +u_dg.

∇

^udg are the localaccelerationsoftheliquidanddispersedphases,respectively.

Thesetwo distinctexpressions of theaccelerations areconsistent withthetheoreticalexpressionsknowntoholdinaninviscidfluid (Auton et al., 1988). In the limit of small-but-finite volume frac- tions,theweightingby

α

dgreducesto

α

dg(¹+3

α

dg)^whichisalso consistentwiththeoreticalresults(BiesheuvelandSpoelstra,1989).

Last, themomentum transferresulting fromshear-inducedlift effectsisassumedtotaketheform

F^L_cl→dg=CL

α

dg

ρ

l

(

^ucl−udg

)

×

ω

cl, (9) where

ω

cl=

∇

×u_clisthelocalvorticityintheliquidphase.When theBondnumberissmall,theliftcoefficientC_Lissetto0.5which is the theoretical value corresponding to a sphere translating in auniform inviscid shear flow (Auton, 1987), a resultalso known tohold for spherical bubbles moving in a viscous fluid provided Re_dg࣡10² (LegendreandMagnaudet,1998).WhentheBondnum- ber is of O(¹), deformation alters the magnitude of the shear- induced lift force and may even reverse it (Adoua et al., 2009).

Inthisregime, the empiricalcorrelation C_L=F(^Bodg,Re_dg) ^estab- lishedbyTomiyamaetal.(2002)inasimpleshearflowisapplied.

2.3.TheLargeInterfaceModel

The n-field formulation maybe adapted to the description of separated(or‘stratified’)two-phaseflows,forinstancetodealwith situationsinwhichahigh-speedgasshearsaliquidlayer.Twoof the main attempts in that directionare the Algebraic Interfacial AreaDensity (AIAD) modelimplemented in the ANSYSCFX code (HöhneandVallée,2010;Deendarliantoetal.,2011)andtheLarge InterfaceModel(LIM)developed byCoste(2013) withinthe NEP- TUNE_CFDcode.TheLIMcomprisestwomainingredients,namely a recognition algorithm aimed at detecting ‘large’ interfaces (LI) andaspecificclosurefortheinterfacialmomentumexchangerate.

2.3.1. Detectionofalargeinterface

In this model, a LI is captured thanks to a three-cell stencil (so-calledLI3Calgorithm): onecell containstheinterface, witha mixtureof thetwo phases (characterizedby a non-zero value of theproduct

α

cl

α

^{cg),whereasonecelloneachsideoftheinterface}

isonly filled withone phase. The former cell is identified based onthemagnitudeofthe liquidvolumefractiongradient, ||

∇α

cl||, whichforaseparatedconfigurationisidenticaltotherateofinter- facialareaperunit volume,A^I.The detectionisachievedbycom- paringeachcomponentof

∇α

cltoaprescribedthresholdvalue.A cellissaidtocontain aLIifatleastoneofthesecomponentsex- ceedsthethreshold.Then,theneighbouringliquidandgascellsare identifiedbymovingawayfromthatcellinthedirectionnormalto theinterface,characterizedbytheunitvectorn_cl=

∇ α

cl

||∇ α

cl

||

^−1.

Knowingthecomponentsofn_cl andthedistributionof

α

cl allows

thepositionandorientation oftheinterface withinthe interface- containingcell to be determined. With thisinformation athand, distances betweentheinterface andthe neighbouringcollocation pointsforthegasandliquidvelocitiesmaybeevaluated.Thetan- gential componentsof these neighbouringvelocities are also ob- tainedbyprojectingthemontotheinterfaceplane,makingitpos- sible to evaluate the gradient of thesetangential components in thedirectionnormaltotheinterface.Thentheshearstressonboth sidesof the interface is estimatedby applying a near-wall treat- mentqualitativelysimilartothatroutinelyemployedtodetermine theso-calledfrictionvelocityinaturbulentboundarylayerovera rigidwall.Here, continuityofshearstresses acrosstheLIimplies that the friction velocities inthe two phases, u^∗_cg andu^∗_cl, satisfy

ρ

gu^∗2_cg=

ρ

lu^∗2_cl. Coste (2013) devised a complex procedure aimed attakinginto account theinfluence ofthe possiblesubgrid-scale roughnessoftheLIonu^∗_cgandu^∗_cl.Forthis,hereferredtothestate diagramestablishedbyBrocchiniandPeregrine(2001a,b)todistin- guishbetweensmooth,wavyand‘knobbly’interfaces.Moredetail onthedetectionalgorithmanddeterminationofthefrictionveloc- itymaybefoundintheoriginalreference(Coste,2013).

2.3.2. Momentumexchangeacrossalargeinterface

OnceacellisrecognizedtocontainaLIandthefrictionveloc- ities are determined, the tangential componentof the interfacial momentumexchangeratewithinitisassumedtoresultfromthe frictionalforcedensity

F^F_cl→cg=

ρ

gu^∗2_cgA^I

(

^ucl−ucg

)

||

^ucl−ucg

||

^{. (10)}

Anadditionalcontributionto I^H_cl→cgis added inthe normal direc- tionforapurelynumericalpurpose.Theroleofthistermistoen- forcetheequalityofthenormalcomponentsofu_clanducgonaLI.

Indeed,thesetwovelocityfieldsbeingtreatedasindependentvari- ablesandnointerfacialtensionactingatcl−cginterfaces,thereis ingeneralnochancethatthisequalityisachieved‘naturally’.Con- sequentlyitisenforcedbyintroducingapenalizationforcedensity, F^P_cl→cg,intheform

F^P_cl→cg=

α

cl

α

cg

( α

cg

ρ

g+

α

cl

ρ

l

)

^Cτ_t^[

(

^ucl−ucg

)

·ncl]ncl, (11) where tisthenumericaltimestep,C_τisanempiricalcoefficient, andtheweightingfactor

α

cl

α

^{cgensuresthatFP}cl→cgisnonzero only in cellscontaininga LI. Thanks to theabove definition ofF^P_cl→cg, anydifferencebetweenthenormalcomponentsofu_cl anducgona LIresultsinalargemagnitudeofthisartificialforceif tissmall.

Hence,selectingalargevalue ofC_τ/ t guaranteesthatthediffer- encebetweenthetwonormalvelocitycomponentsremainsnegli- giblysmallincellscontainingaLI.

The LIM approach summarized above was specifically devel- oped to simulate separated two-phase flows. This model does not apply to a dispersed configuration.Indeed, although no LIis

Fig. 1. Schematic representation of the role of f cand βin the weighting procedure of the momentum closure defining the GLIM approach. The square represents the unit grid cell .

present in that case(as rightly identified by the detectionalgo- rithm), mostcells exhibit non-zero values of the product

α

cl

α

^cg.

Thus,iftheLIMapproachisselected,thepenalizationforcedensity definedinEq.(11) isappliedtothesecells,providingamisrepre- sentationofthemomentumexchangebetweenthetwophases.

2.4. TheGeneralizedLargeInterfaceModel

The idea underlying thismodel is tocombine the capabilities ofthosedescribedinSections2.2and2.3,soastosimulatemulti- scaleflowconfigurationsinwhichLIanddispersedbubblescoexist withintheflowdomain.

To this end, the interfacial momentum transfer closure has to be adapted to the local flow configuration. The first step is to detect LI throughout the domain by using the algorithm de- scribed in Section 2.3.1. Similar to the LIM approach, the mo- mentum exchange closure I^H_cl→cg=F^F_cl→cg+F^P_cl→cg is applied to cells containinga LI. Then, in cellsthat do not contain a LI but have a non-zero gas volume fraction, the dispersed bubbly flow modelofSection2.2withaninterfacialmomentumexchangerate I^H_cl→dg= F^D_cl→dg+F^AM_cl→dg+F^L_cl→dg isapplied. Toavoid superimpos- ingbothmodelsincellswhere

α

cl

α

cg=0,thetotalmomentumex- changeratewithinsuchcellsisdefinedinacompositemanneras

γ

^IHcl→dg+(¹−

γ

)^IH_cl→cg^{.The weighting}pre-factor,

γ

^{,isselectedin}

theform(Merigouxetal.,2016)

γ

=

β (

¹−Min

(

^fc,1

) )

^{with fc}=6V

Sn·

∇α

cl, (12) where S denotes the area ofthe outer surface that boundsthe controlvolume^,andnistheouterunitnormaltothatsurface.

InEq.(12),

β

^{isapre-factorthat smoothlyvariesfrom0to 1ac-}

cordingtothelocalvalue oftheliquidvolume fraction,

α

cl.More specifically,

β

^{is set to 0if}

α

cl<0.5, which is considered to cor- respond to locationsinvolving a LI. Conversely,

β

=1 if

α

cl>0.7, where the flow structure is considered to be dominated by the presenceofdispersedbubbles.Last,

β

^{followsthelinearvariation}

β

=(

α

cl−0.5)/0.2if0.5≤

α

cl≤0.7.However,afractionofaLImay still bepresentincellswith

α

cl>0.5.Insuch cells,itisobviously desirable to lower the influence of the dispersed phase and in- creasethatoftheLI.Thecorrectionfunctionfchelpssatisfyingthis requestbylowering

γ

^iftherateofinterfacialareaassociatedwith theLIpresentinthecell(whichisapproximately||

∇α

cl||)isofthe sameorder asthesurface-to-volumeratioofthe gridcell,S/V (seeFig.1).

Itisimportanttorealizethatinthismodel,the‘dispersed’(dg) and ‘continuous’ (cg) gas phasesactually refer to the same field.

It is only the weighting factor,

γ

^{, that selects whetherthe local}

flow structure is close to a separated or a dispersed configura- tion.Thisselectioncriterionallows the‘generation’ofadispersed or continuous gas phase only in a limited number ofconfigura- tions, mostlydriven bygrid resolutionandnumericallimitations.

Morespecifically, startingfroma configurationinwhichall inter- faces are properly resolved, the GLIM approach may later detect

Fig. 2. Schematic of the inter-phase couplings involved in the Large Bubble Model (adapted from Denèfle et al., 2015 ).

a dispersed phase either iftwo LI get very close to one another (which alsohappensifthe curvatureofan interface becomeslo- cally verylarge), or iftheinterfacial regioncontaininga LIgrad- ually thickens, owing to numerical diffusion. Conversely, starting fromapurelydispersedgas-liquidconfiguration,aLImayonlybe creatediftheflowconditionsorboundaryconditionsforcethegra- dientofthegasvolumefractiontoreachlocallyalargevalue(e.g.

aplume ofrising bubbleshitting ahorizontalwall).Last,it must bekeptinmindthatthedispersedphaseisassumedtobemono- dispersedandthat theingredientsbrought totheGLIM approach bythemodels describedinSections2.2and2.3donot allowthe bubblediameter,d_dg, tobe predicted. Insteadthis diametermust beprescribedasafunctionoftheaveragecellsize, a.Inthesim- ulationstobedescribedbelow,thisratioissettod_dg/ a≈0.4(see Section3.2).

2.5.TheLargeBubbleModel

The LBM (Denèfle, 2013; Denèfle etal., 2015;Mimouni etal., 2017) is the most sophisticated of the models considered in the presentstudy.Here,distinctdispersed(dg)andcontinuous(cg)gas phasesare considered, making the LBM a 3−field model involv- ingseparatecl→cgandcl→dgmomentumexchangeclosures.The cl→dgclosureisachievedthankstotheforcedensityexpressions detailedinSection 2.2,whereas thelarge gasstructures are con- sideredasacontinuousphase separatedfromthe carryingliquid byLIthatarefullyresolvedandexperienceacapillaryforce.

Comparedto LIMandGLIM, the extra modellingeffortin the LBM rests essentially in the treatment of the LI which, in addi- tion to the computationof the capillaryforce, involves an inter- face sharpening procedure anda specific closure forthe interfa- cialdragforcecouplingtheclandcgfields.Furthermore,massex- changetermsassociatedwithpossiblecoalescenceandfragmenta- tionneedtobeimplementedinthedgandcgfields.Fig.2,adapted fromDenèfle etal.(2015),summarizesthephasecouplingproce- duresinvolvedintheLBMapproach.

2.5.1. Computationofthecapillaryforceandinterfacesharpening procedure

The capillary force that takes place at these LI is evaluated thankstotheContinuumSurface Force(CSF)approachdevisedby Brackbillet al.(1992),in whichthe interfacialsurfaceforce den- sity,i.e.theright-handsideofEq.(6),istransformedintoavolume forcedensity,F^σ_k.Forphasek=clorcg,thelatteriswrittenas F^σ_k =

α

k

σκ

kp

∇α

k with p=cg,cl, (13) where the interface mean curvature,

κ

kp=−

∇

·n_kp, is com- puted using the unit normal pointingtoward phase k, i.e.n_kp=

∇ α

k

||∇ α

k

||

^−1.

Defining and computing the mean curvature, say

κ

lg, at a liquid-gas interface makes sense only if this interface may be properly identified and described on the computational grid. A well-known issue encountered when attempting to track sharp interfaces using a purely hyperbolic transport equation for the volume fraction is the smearing of the discontinuity (Rudman, 1997;1998; SatoandNiˇceno,2012).Acommonapproachtolimit thiseffectconsistsinintroducinganadditionalviscous Hamilton- Jacobiequationensuring anartificialcompressionoftheinterface (Sethian,1999).NEPTUNE_CFDfollowsthispathbysolvingthead- ditionalequation(OlssonandKreiss,2005;Olssonetal.,2007)

∂α

k

∂τ

⁺

∇

·

α

k

(

¹−

α

k

)

ⁿkp =

∇

²

α

k, k,p=cl,cg. (14) Thepseudo-time,

τ

^,andthe pseudo-viscosity,

^{,are directlycon-}

nectedwiththelocalcellsize, .FollowingŠtrubeljetal.(2009), values

τ

= ₃₂ and

= ₂ are respectively selected. With this choice,thefinal thicknessoftheinterfaciallayerwithinwhich

α

k

variesfrom0to1is

δ

α=5 whatevertheinitialspreadingofthe

α

k distribution.Eq.(14)issolved inconservativeforminorderto improvemassconservation(Fleau,2017).

2.5.2. Dragforceonlargeinterfaces

Since the capillaryforce is takenintoaccount anda sharpen- ing procedure is applied, an artificial penalization force like that ofEq.(11)isnolongerneededintheLBM approachtoguarantee negligibly small relative normal velocities on a LI. Moreover, ex- tensivetestsofthe LBM carriedout by Fleau(2017) usingafric- tional drag force model only taking into account the fluid den- sitiesyielded quite poorresults in thecase of large bubbles ris- ingin low-viscosity liquids,suggestingthat fluid viscositiesmust enterthe interfacialdraglaw.Similar findings were noticed with the AIAD model in applications involving free surfaces, and led HöhneandVallée(2010) andPorombkaandHöhne (2015) tode- velopalternativeclosurelawsfortheinterfacialdraglawtobeap- pliedonsuch interfaces.Thatfrictionaldraglawsindependent of thefluid viscositiesfail to properlymimic the momentum trans- fer at gas-liquid interfaces is no surprise since the flow in the liquid close to such an interface almost obeys a shear-free con- ditionratherthan a no-slipone (provided contaminationby sur- factantsisnegligible).Animportantconsequenceofthisdifference isthat thedrag force on uncontaminated bubbles grows linearly ratherthan quadratically withthe rise velocity (Batchelor, 1967).

In NEPTUNE_CFD, this state of affairs led Fleau (2017) (see also Mimouniet al., 2017) to assume that the interfacialmomentum transfertowardacgentityinvolvesacharacteristiclengthscalelg

andis proportionalto

μ

ll_g(^ucl−u_cg),so thatthe momentumex- changerateperunitvolumemaybewrittenintheform

F^Db_cl→cg=

α

^cg

μ

ll⁻_g²

(

^ucl−ucg

)

^{. (15)}

Thisclosureisusedaslong as

α

^cg≤0.3,which isusually consid- eredastheupperlimitofthedispersedbubblyflowregimeunder homogeneousconditions(Taiteletal.,1980).Applicationofasimi- larreasoningtothe‘opposite’caseoflargedropsmovingwithina gasinregionswhere

α

cg≥0.7yields

F^Dd_cl→cg=

α

cl

μ

^gl−l²

(

^ucl−ucg

)

, (16) wherel_l is thelengthscale characterizingtheinterfacialmomen- tum exchange in that configuration. In the intermediate range 0.3<

α

^cg<0.7, the momentum exchange rate is assumedto vary linearlyinbetweentheabovetwolaws,namely

F^D_cl→cg=f_bdF^Db_cl→cg+

(

¹−f_bd

)

^FDdcl→cg, (17) with f_bd(

α

cg)=2.5×(⁰.7−

α

cg)^{.Itisworth pointingout thatthe} closureprovidedbyEq.(16)ismuchmorequestionablethanthat ofEq.(15)since thegasflow almostobeysano-slipconditionat

aliquid-gasinterfaceif

μ

^g/

μ

l 1,suggestingthataquadraticdrag lawwouldphysicallybemoreappropriateinthatcase.

Thelengthscalesl_i(i=g,l)^{involvedinEqs.(15)–(17)arecon-} sideredto dependon the product

α

cl

α

cg (which mayrangefrom 0 to 0.25) in such a way that they both tend toward d_dg/3√

2 when

α

cl

α

^cg→0 (the factor 3√

2 ensures that Stokes’ drag law is recovered in that limit),whereas they become proportional to

α

ci

||∇α

ci

||

^{−1 for}

α

cl

α

cg࣡0.1anda linearvariation isassumedfor intermediatevaluesof

α

cl

α

^cg.Since

||∇α

ci

||

^−1isproportionalto thankstotheinterfacesharpeningprocedure,thisdefinitionmakes the characteristiclength scales ofthe momentum exchange atLI dependonthecellsize.

2.5.3. Cutoff lengthscaleandmassexchangesbetweenthe continuousanddispersedgasphases

Numericalrequirementsassociatedwiththecomputationofthe meancurvature

κ

lg ofcl−cginterfacesdeterminethecharacteris- tic size of thesmallest gas structures that can be fully resolved.

Thiscriticalsize,lc,istakenasthecutoff lengthscalecorrespond- ing tothe separationbetweenthe dgandcggas phases. InNEP- TUNE_CFD,thecomputationof

κ

lg involvesa5-cellstencilineach grid direction. It is then a simple matter to show that the sten- cilsrequiredtocomputethecurvatureatthetwooppositepoints of thediameter ofa circular interface overlap ifthis diameteris lessthan5√

2≈7.2gridcells(Denèfleetal.,2015).Forthisreason, thevaluelc/ =8isselected.Notingthatthecorrespondingcriti- calmeancurvatureis

κ

c=2/lc=(⁴ )⁻¹,onehas

κ

cV(

δ

α )⁻¹= 1/20. Hence the cutoff criterion maybe generalizedby requiring that thecapillary force isapplied onlyif

κ

lgV

||∇ α

k

||

^{−1 is less}

than 1/20 (k=cl,cg). Interfaces satisfyingthis criterion are con- sideredasLIandthecorrespondinggasstructuresbelongtothecg phase.Conversely, neithertheinterface sharpeningprocedurenor thecapillaryforce densityF^σ_k isapplied tointerfacesexhibitinga meancurvaturelargerthan

κ

c.

The corresponding smallgasentities are transferred to thedg phaseasdescribedbelow(see Eq.(19)). Massexchangesbetween thecganddgfieldsariseduetocoalescenceandbreak-upevents.

Inthepresentcontext,thedefinitionoftheseeventsreliesonnu- mericalratherthanphysicalcriteria,sincethedistinctionbetween the cganddgfields dependsentirelyon thegrid resolution.Nu- merical coalescence of dg bubbles giving rise to a cg gas struc- ture isassumed to occur when (i) the local gasfraction exceeds thecriticalvalue

α

dgc=0.3(Taiteletal.,1980), and(ii)thenorm ofthelocalgradient ofthegasvolume fraction,||

∇α

dg||,exceeds a thresholdvalue. Asthe interface sharpeningprocedure yields a typical volume fraction gradient

||∇α

cl

||

=

||∇ α

^cg

||

=

δ

α⁻¹ within interfacialregions,thisthresholdissetto(²

δ

α)^{−1.Basedonthese} requirements, therate atwhich the dispersed phase coalesces is expressedintheform

cg+=

α

^cg

α

dg

ρ

^gC+_t^H

( α

dg−

α

dgc

)

^H

(

²

δ

α

||∇ α

dg

||

⁻¹

)

, (18) where Hdenotes the Heavisidefunction andC₊ isan O(¹) ^con- stant. Similarly, the rate at which cg gas entities turn into dis- persedbubblesisassumedtobe

cg−=

α

cg

α

dg

ρ

g

C₋

tH

( δ

α

||∇ α

k

|| κ

lg

κ

c −1

)

, (19) whereC₋ isanother O(¹) ^{constant. The net mass exchange rate} fromthedispersedgasfieldtowardthecgfieldisthen

dg→cg=

cg+−

cg−. (20)

Notethat thepresence ofthe

α

^cg

α

dg pre-factorincg+ andcg−

ensures that mass exchanges may only take place at locations wherebothgasfieldsarepresent.Theinteractionofthedispersed gasphasewiththecarryingcontinuousliquidtakesplacethrough

Fig. 3. Schematic of the experimental set-up.

theinterfacialforcesdefinedinEqs.(7)–(9),assumingthedgphase tobemonodispersedandmadeofbubbleswithdiameterd_dg lc. SimilartotheGLIMapproach,d_dghastobespecifiedasafraction of a(seeSection3.2).

3. Experimentalandcomputationalconfigurations 3.1. Experimentalset-up

The experimental set-up is sketched in Fig.3. It consists ofa Plexiglas^TM cylindricalvesselwithdiameterD=114mmandheight L=800 mm. The top endof the cylinder isclosed with a blank flange.Conversely,itsbottomsectionisopenedthankstoacentral circularthin-walledholewithadiameterd=35mm.Thisholehas abevellededgemaking asharp20°anglewiththecylinderbase.

The bottle neck is closed by a gate mounted on two hinges and equippedwithfiveelectromagnetsensuringafastopening.Before a test, the gate isclosed, thevalves v_in andvout are opened and the bottle is filledup to an altitude z=z₀ with tapwater, using thecirculationpump.Oncefillingiscompleted,thepumpisturned off andthe two valves are closed.Asvout remains opened during the filling process,the initial pressure atthe top of thebottle is theatmosphericpressure,patm.Thegateisquicklyopenedattime t=0.Waterstartstoflowoutofthebottleinasuccessionofjets separatedbythegenerationoflargeairbubblesattheneck.Then thesebubblesrisewithinthewatercolumnandreachthefreesur- faceafterasequenceofcomplexreconfigurations.

Theairpressureatthecentreofthetopendofthebottle,ptop, ismonitoredwithapressuresensor(Keller,PR-23).Imagesofthe emptyingprocessarerecordedwithaCMOScamera(PhotonLines, PCO1200HS)ataspeedof400fps.Theseimagesmakeitpossible to followthedisplacementofthe upperfreesurfaceandgivein- sightintothebubbledynamics.Thecameraandthepressuresen- soraresynchronizedthroughaTTLsignalthattriggerstheelectro- magnet,thusthegateopening.

3.2. Computationalset-upandconditions

ThecomputationalgeometryissketchedinFig.4.Similartothe experimental device, the ‘numericalbottle’ consistsof a cylinder withdiameterD=114mmandheightL=800mm.Torelaxgrid

Fig. 4. Cross-sectional view of the computational domain; g = −g e z denotes the gravity vector.

constraints,thebottleneckwithdiameterd=35mmisassumed tobeathincylinderwithheight

δ

=5mm,thusslightlydiffering fromthebevelledgeometryused intheexperiment. No-slipcon- ditionsare imposed on all wallsandintersections betweenwalls andiso-

α

l surfaces take place at rightangle, corresponding to a 90°contact angle.

The outlet boundary condition plays a crucial role in the present configuration. To prevent the liquid outflow and bubble generationatthe bottleneck frombeingdisturbedby theoutlet, thelatterismovedaway fromtheneckbyaddingabuffer region withlengthL_r=0.1m anddiameterDbelowthe bottleneck,as depictedinFig.4.

Theinitialconditionsrefertotheexperimentalcasewithanini- tialwaterheightz0/L=0.75.Waterandairareinitiallyatrestand theinitialpressureinbothairregionsistheatmosphericpressure.

Waterproperties at 20° C are extracted from the CATHARE rou-

Fig. 5. Evolution of the pressure at the top of the bottle for an initial water height z 0/L = 0 . 75 . The inset shows a closer view at the oscillations in the range 10 s ≤t ≤13 s.

tines encapsulated in NEPTUNE_CFD, namely

ρ

l=997.24kg.m⁻³,

μ

l=1×10⁻³Pa.sand

σ

=7.28×10⁻²N.m. Airviscosityissetto

μ

^g=1.8×10⁻⁵ Pa.s.To assessthe possible role ofaircompress- ibility, incompressible simulations as well as compressible ones (in which the propagation of density/pressure waves is fully re- solved)arecarriedout.Inincompressiblecases,airdensityissetto

ρ

^a=1.20 kg.m⁻³. Incontrast,whencompressibilityis takeninto account,airisassumedto behaveasan isothermalidealgas, the densityofwhichobeys

ρ

^a=p(^RairT)⁻¹,wherepdenotesthelocal pressure,R_air=287.06J.kg⁻¹.K⁻¹ is theperfect gasconstant, and thetemperatureTis setto293.15K.Under suchisothermal con- ditions,the densityderivative withrespect to pressureis merely (

∂ ρ

^a/

∂

^p)T=(^RairT)^−1.

With the above physical properties, the capillary length of thefluid set-up isl_σ=(

σ

/

ρ

lg)^1/2≈2.7mm (g denoting gravity).

Henced/l_σ≈13,whichguaranteesthatcapillaryeffectsintheneck regionhaveanegligibleinfluenceontheemptyingdynamics.

The simulations are run over 5 s, with a constant time step of0.05ms. All simulations discussed beloware carried out ona three-dimensional computational domain with approximately 2.1 millionscells, whichyieldsan averagegridsize _a≈1.6mmcor- respondingtoadimensionlesslengthratio a/D≈0.014.Withthis grid,thediameterd_dgprescribedforthedispersedbubblesinboth the GLIM and LBM approaches is set to 1 mm. It yields a bub- ble volume about 15% of the averaged cell volume, ³_a, which isconsistent withthe assumptionof thepresence ofa dispersed phaseandallowsustoproperlyresolveasignificantpartofthegas content.The maincharacteristicsofthethreeclosuremodelsem- ployedinthecomputationsare summarizedinTable1.Westress that no turbulence model is used in any of these computations.

Thecalculationsareruninparallelmodeon100processorsofthe EOSsupercomputerfromtheCALMIPsupercomputingmesocentre.

Computationsmaking use ofthe LIM,GLIM andLBM approaches consumeapproximately7500,5200and11,500hintotalCPUtime, respectively.

4. Globalresults

4.1.Preliminaryexperimentalobservations

The evolution of the pressure ptop throughout the emptying process is shownin Fig. 5 in the caseof an initial waterheight z₀/L=0.75.

Fig. 6. Evolution of the pressure difference p top−p atm obtained in experiments (blue line), incompressible (black line) and compressible (yellow line) simulations with the LIM approach, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Just after the opening, the pressure immediately drops from patmtoalowervalue,patm− p^init.Thisinitialdropismerelydue to the hydrostaticpressure corresponding to the initial height of thewatercolumn,sothat p^init≈

ρ

lgz₀.Thepressurethenrisesal- mostlinearlyovertimeuntilitgoesbacktotheatmosphericpres- sure,p_atm,attheendoftheemptyingprocess.

A closer look at the pressure signal (see the inset in Fig. 5) reveals almost sinusoidal oscillations around a slowly increasing value,p^eq_top(^t),correspondingtotheinstantaneoushydrostaticequi- librium.TheseoscillationshaveatypicalperiodTos≈0.20s,much shorterthantheemptyingtime.Theyfindtheirrootinthealterna- tionofbubblegeneration andliquidejectionevents,asdescribed by Clanet and Searby (2004). The evolution of the equilibrium pressure, p^eq_top,obeystherelation p^eq_top(^t)=patm−4gMw(^t)/(

π

^D2), where Mw(t) is the massof waterin thebottle at time t (hence 4Mw(t)/(

ρ

l

π

^D2)istheinstantaneouswaterheight).Inthepresent case, thelinearevolutionof p^eq_top yieldsa constantmassflowrate dMw/dt≈ −0.19 kg.s⁻¹, corresponding to a cross-sectional aver- aged velocity Uav≈0.2 m.s⁻¹ at the bottle neck, which yields a Reynolds number based on the neck diameter of approximately 7000.

4.2. Influenceofaircompressibility

Atypicalpressuresignalrecordedbythepressuresensorisdis- playedinFig.6,stillwiththeinitialconditionz₀/L=0.75.Pressure oscillationsstart rightafterthegate openingandare presentun- tilthe endof theemptying process.Based on their experiments, ClanetandSearby(2004)attributedtheseoscillationstothecom- pressibilityofthe airbufferatthe topofthe bottle.However, an oscillatorybehaviourwasalsoobservedintheincompressiblesim- ulations carriedout by Geigeretal.(2012).Toassesswhetheror not compressibility has to be takeninto account to properly re- produce theemptying process,we carriedout both compressible andincompressiblesimulations.Thepressuresignalcorresponding tobothsetsofconditionsiscomparedwiththeexperimental sig- nalinFig.6.OnlycomputationalresultsobtainedwiththeLIMap- proacharedisplayedinthisfigurebuttheothertwomodelsreveal similartrends.

Large differences are observed with the two modelling as- sumptions.When airis considered incompressible, thecomputed pressure signal does not display the sinusoidal oscillations ob- served inexperiments.Itratherfollows anoisyevolutioncharac- terizedby small-amplitudehigh-frequency fluctuations. Themag- nitude of the initial drop and the subsequent linear increase of