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Eprints ID : 20029
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
aa Institut de Mécanique des Fluides de Toulouse, IMFT, Université de Toulouse - CNRS, Toulouse, France b Electricité 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,asthecomputationalcostassociatedwithdirect numeri-calsimulationstillmakesthisapproachunaffordable.Acommonalternativeisthetwo-fluidEuler–Euler formulationthatavoidssolvingallscalesatthepriceofsemi-empiricalclosuresofmass,momentumand energyexchangesbetweenthetwofluids.Manyofsuchclosuresareavailablebuttheirperformancesin complexflowsarestillindebate.Closuresconsideringseparatelylargegasstructuresandsmaller bub-blesandmaking thesetwo populationsevolveand possiblyexchangemassaccordingtotheir interac-tionswiththesurroundingliquidhaverecentlybeenproposed.Inordertoassessthevalidityofsome oftheseclosures,wecarryoutanoriginalexperimentinasimpleconfigurationexhibitingarich succes-sionofhydrodynamicevents,namelytheemptyingofawaterbottle.Wesimulatethisexperimentwith theNEPTUNE_CFDcode,usingthreedifferentclosureapproachesaimedatmodellinginterfacial momen-tumexchangeswithvariousdegreesofcomplexity.Basedonexperimentalresults,weperformadetailed analysisofglobal and localflowcharacteristics predicted byeachapproachtounveilits potentialities andshortcomings.Althoughallofthemarefoundtopredictcorrectlytheoverallfeaturesofthe emp-tyingprocess,strikingdifferencesareobservedregardingthedistributionofthedispersedphaseandits consequencesintermsofliquidentrainment.
1. Introduction
Gas-liquid flows involving a broad range of bubble sizes are ubiquitous ingeophysical andengineeringconfigurations and ap-plications,suchasmagmaticchimneys,submarineexplosions, bub-blecolumnsornuclearsafety,tomentionjustafew.Insuch situa-tions,thegasphasefrequentlyinvolvesawiderangeofspatialand temporalscales,fromlargegaspocketstosmalldispersedbubbles. Moreover,dramaticallydifferentflowregimesmaybeencountered, characterized bydistinctinteraction mechanismsbetweenthegas phaseandthecarryingliquid.Simulatingsuchflowsremainsa ma-jor challenge nowadays, although massive efforts have been de-voted during the last two decades to develop modelling strate-giesaimed at computingmultiphase flows(Prosperetti and Tryg-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 thesamemain chal-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 and Tryg-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 ReynoldsDespitethepotentialitiesofferedbytheseDNSapproachesand 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,sothatunknownterms oc-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-phoresisandlubrificationeffectswhenthecarryingflowis turbu-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 ifthemodellingapproachismadeabletorecognizewhich config-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.Thesecondapproachconsistsin ex-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’hasitsown ve-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 asthatoftheAlgebraicInterfacialArea Den-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).Examplesof applica-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 more re-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 fornumerical dif-fusion tospread stiff volumefraction gradients.Sharpening tech-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.Suchanapproachhasbeenimplementedbothinthe aforemen-tionedGENTOPformulation(Montoyaetal.,2015),andinthe NEP-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,partofwhichmaycoalesceagainand partici-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 the Open-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-fieldformulationanddetailsthevariousapproaches em-ployed to model interfacial momentum exchanges in the NEP-TUNE_CFD code. The experimental and computational configura-tionsaredescribedinSection3.Section4discussestypicalresults obtainedthroughbothapproachesonsome quantities
characteriz-ingtheoveralldynamicsofthesystem.Section5focusesonlocal characteristics ofthe two-phase flow andexamines theinfluence oftheaforementionedmodelsontheevolutionandstatistical dis-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,the de-scriptionreducestotwofields,namelyacontinuouscarryingliquid phase anda dispersedone. Ingeneral, mass,momentumand en-ergyconservationequationsaresolvedforeachfield,withthe as-sumptionthat theysharethesamepressurefield.Froma numeri-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,sothatonlythemassand mo-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
kρ
k)
+∇
·(
α
kρ
kuk)
=
p=k
p→k, (2)
where
ρ
kandukarethedensityandvelocityoffieldk,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 employedtorepresententitiesof differ-entsizeswithinthesamephysicalphase(e.g.largeandsmall bub-bleswithinthegasphase),coalescenceandbreak-upeventsmake the mass of each of these fieldsvary, so that
p→k is generally nonzero.
Themomentumbalanceforphasekiswrittenas
∂
∂
t(
α
kρ
kuk)
+∇
·(
α
kρ
kukuk)
=−α
k∇
P+α
kρ
kg +∇
·α
kμ
k(
∇
uk+T∇
uk)
+ p=k Ip→k, (4)where
μ
kis theviscosityoffield k andIp→krepresents the mo-mentumexchange ratebetweenfieldspandk.The lattermaybe splitintheformIp→k=IHp→k+
p→kuIpk, (5) whereuI
pk isthe velocity at the interface betweenphases pand k,and thefirst term inthe right-hand side, IH
p→k,represents the momentumexchange duetohydrodynamicforces,whilethe sec-ondismerelythemomentum exchangeassociatedwiththemass exchangeratebetweenthetwophases.Theinterfacialmomentum exchangetermhastobemodelledtoclosethesetofequations.
Ifaninterfacialtension,
σ
pk,maybedefinedbetweenphasesp andk,thecorrespondinginterfacialmomentumbalanceimpliesIp→k+Ik→p= lim V→0 1 V
σ
pkκ
pknpkdA I, (6)wherenpkistheunitnormaltothe p− kinterface,
κ
pk=−∇
· npk isthecorresponding meancurvature,dAI istheelementary inter-facialareaanddenotesthecontrolvolume(which in computa-tionalpracticecorresponds tothegridcell), thevolume ofwhich isV. In what follows, capillaryeffects are neglected in the LIM andGLIMapproaches.Weshallspecifyinduecoursehowand un-derwhichconditionstheyaretakenintoaccountintheLBM.Note that,providedtheinterfacialvelocity uI
pk iscontinuousacrossthe interface,theleft-handsideofEq.(6)reducestoIH
p→k+I H k→p, ow-ingtoEq.(3).
Inthenextfoursubsections,wedetailthevariousclosuresand detection criteria used to express the interfacialmomentum ex-changeinthethreeaforementioned modelsimplementedin NEP-TUNE_CFD.Asummary ofthecharacteristicsandclosurelaws in-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-videdwalleffectsandturbulentdispersionareabsentorhave neg-ligibleeffects, the interfacialmomentum exchange isassumed to resultfrom the sum of three independent contributions, namely a viscous dragforce, FD,an addedmass force, FAM,and a shear-inducedliftforce,FL (Mimounietal.,2011b).
Themomentumtransferresultingfromviscousdragiswritten as FD cl→dg= 1 8A I
ρ
lCD||
udg− ucl||
(
ucl− udg)
, (7) where AI=lim V→0V1dSI denotes the rate of interfacial area
perunit volumewhich,fora mono-dispersedbubbledistribution, maybeexpressedasafunctionofthevolumefractionofthe dis-persed phase throughthe well-known relation AI=6
α
dg/ddg, ddg denotingthebubblediameter.ThedragcoefficientCD dependson the bubble Reynolds number, Redg=
ρ
l||
udg− ucl||
ddg/μ
l, on the Bond number, Bodg=(
ρ
l−ρ
g)
gd2dg/σ
(σ
being the surface ten-sionoftheliquid),andonthevolumefraction,α
dg.NEPTUNE_CFD makes use of the empirical correlations established by Ishii and Zuber(1979) toexpressCD asafunction ofRedg,Bodg andα
dg in the various regimesencountered withrising bubbles, possibly in denseconfigurations.ThecontributionofaddedmasseffectstoIH
p→kiswritteninthe form(Zuber,1964) FAM cl→dg=CAM 1+2
α
dg 1−α
dgα
dgρ
l Ducl 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 — — IH cl→cg = F Fcl→cg ( Eq. (10) ) IHcl→dg = F Pcl→cg ( Eq. (11) ) + F P cl→cg ( Eq. (11) ) IH cl→cg = F Fcl→cg ( Eq. (10) ) IHcl→dg = F Dcl→dg ( Eq. (7) )
GLIM 1 lc ≈ dg + F Pcl→cg ( Eq. (11) ) + F AMcl→dg ( Eq. (8) )
+ F L cl→dg ( Eq. (9) ) IH cl→cg = F σcl→cg ( Eq. (10) ) IHcl→dg = F Dcl→dg ( Eq. (7) ) LBM 2 lc = 8 dg + F Dcl→cg ( Eqs. (15) –(17) ) + F AMcl→dg ( Eq. (8) ) + F L cl→dg ( Eq. (9) )
where CAM=0.5 is the added mass coefficient of an isolated sphere, Ducl Dt = ∂ ucl ∂t +ucl.
∇
ucl and dudg dt = ∂udg ∂t +udg.∇
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,theweightingbyα
dgreducestoα
dg(
1+3α
dg)
whichisalso consistentwiththeoreticalresults(BiesheuvelandSpoelstra,1989). Last, themomentum transferresulting fromshear-inducedlift effectsisassumedtotaketheformFLcl→dg=CL
α
dgρ
l(
ucl− udg)
×ω
cl, (9) whereω
cl=∇
× uclisthelocalvorticityintheliquidphase.When theBondnumberissmall,theliftcoefficientCLissetto0.5which 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 providedRedg࣡102 (LegendreandMagnaudet,1998).WhentheBond num-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 CL=F(
Bodg,Redg)
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
clα
cg),whereasonecelloneachsideoftheinterface isonly filled withone phase. The former cell is identified based onthemagnitudeofthe liquidvolumefractiongradient, ||∇α
cl||, whichforaseparatedconfigurationisidenticaltotherateof inter-facialareaperunit volume,AI.The detectionisachievedby com-paringeachcomponentof∇α
cltoaprescribedthresholdvalue.A cellissaidtocontain aLIifatleastoneofthesecomponents ex-ceedsthethreshold.Then,theneighbouringliquidandgascellsare identifiedbymovingawayfromthatcellinthedirectionnormalto theinterface,characterizedbytheunitvectorncl=∇
α
cl||∇
α
cl||
−1. Knowingthecomponentsofncl andthedistributionofα
cl allowsthepositionandorientation oftheinterface withinthe interface-containingcell to be determined. With thisinformation athand, distances betweentheinterface andthe neighbouringcollocation pointsforthegasandliquidvelocitiesmaybeevaluated.The tan-gential componentsof these neighbouringvelocities are also ob-tainedbyprojectingthemontotheinterfaceplane,makingit pos-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∗2cg=ρ
lu∗2cl. Coste (2013) devised a complex procedure aimed attakinginto account theinfluence ofthe possiblesubgrid-scale roughnessoftheLIonu∗cgandu∗cl.Forthis,hereferredtothestate diagramestablishedbyBrocchiniandPeregrine(2001a,b)to distin-guishbetweensmooth,wavyand‘knobbly’interfaces.Moredetail onthedetectionalgorithmanddeterminationofthefriction veloc-itymaybefoundintheoriginalreference(Coste,2013).2.3.2. Momentumexchangeacrossalargeinterface
OnceacellisrecognizedtocontainaLIandthefriction veloc-ities are determined, the tangential componentof the interfacial momentumexchangeratewithinitisassumedtoresultfromthe frictionalforcedensity
FF cl→cg=
ρ
gu∗2cgAI(
ucl− ucg)
||
ucl− ucg||
. (10) Anadditionalcontributionto IHcl→cgis added inthe normal direc-tionforapurelynumericalpurpose.Theroleofthistermisto en-forcetheequalityofthenormalcomponentsofuclanducgonaLI. Indeed,thesetwovelocityfieldsbeingtreatedasindependent vari-ablesandnointerfacialtensionactingatcl− cg interfaces,thereis ingeneralnochancethatthisequalityisachieved‘naturally’. Con-sequentlyitisenforcedbyintroducingapenalizationforcedensity, FP
cl→cg,intheform
FPcl→cg=
α
clα
cg(
α
cgρ
g+α
clρ
l)
Cτ
t[
(
ucl− ucg)
· ncl]ncl, (11)where
tisthenumericaltimestep,Cτisanempiricalcoefficient, andtheweightingfactor
α
clα
cgensuresthatFPcl→cgisnonzero only in cellscontaininga LI. Thanks to theabove definition ofFPcl→cg, anydifferencebetweenthenormalcomponentsofucl anducgona LIresultsinalargemagnitudeofthisartificialforceif
tissmall. Hence,selectingalargevalue ofCτ/
t guaranteesthatthe differ-encebetweenthetwonormalvelocitycomponentsremains negli-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 c and 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 detection algo-rithm), mostcells exhibit non-zero values of the product
α
clα
cg. Thus,iftheLIMapproachisselected,thepenalizationforcedensity definedinEq.(11) isappliedtothesecells,providinga misrepre-sentationofthemomentumexchangebetweenthetwophases.2.4. TheGeneralizedLargeInterfaceModel
The idea underlying thismodel is tocombine the capabilities ofthosedescribedinSections2.2and2.3,soastosimulate multi-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 IH
cl→cg=FFcl→cg+FPcl→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 IH
cl→dg= FDcl→dg+FAMcl→dg+FLcl→dg isapplied. Toavoid superimpos-ingbothmodelsincellswhere
α
clα
cg=0,thetotalmomentum ex-changeratewithinsuchcellsisdefinedinacompositemannerasγ
IHcl→dg+
(
1−γ
)
IclH→cg.The weightingpre-factor,γ
,isselectedin theform(Merigouxetal.,2016)γ
=β
(
1− Min(
fc,1)
)
with fc=6VSn·
∇α
cl, (12)where S denotes the area ofthe outer surface that boundsthe controlvolume
,andnistheouterunitnormaltothatsurface. InEq.(12),
β
isapre-factorthat smoothlyvariesfrom0to 1 ac-cordingtothelocalvalue oftheliquidvolume fraction,α
cl.More specifically,β
is set to 0ifα
cl<0.5, which is considered to cor-respond to locationsinvolving a LI. Conversely,β
=1 ifα
cl>0.7, where the flow structure is considered to be dominated by the presenceofdispersedbubbles.Last,β
followsthelinearvariationβ
=(
α
cl− 0.5)
/0.2if0.5≤α
cl≤ 0.7.However,afractionofaLImay still bepresentincellswithα
cl>0.5.Insuch cells,itisobviously desirable to lower the influence of the dispersed phase and in-creasethatoftheLI.Thecorrectionfunctionfchelpssatisfyingthis requestbyloweringγ
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 of configura-tions, mostlydriven bygrid resolutionandnumericallimitations. Morespecifically, startingfroma configurationinwhichall inter-faces are properly resolved, the GLIM approach may later detectFig. 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 becomes lo-cally verylarge), or iftheinterfacial regioncontaininga LI grad-ually thickens, owing to numerical diffusion. Conversely, starting fromapurelydispersedgas-liquidconfiguration,aLImayonlybe creatediftheflowconditionsorboundaryconditionsforcethe gra-dientofthegasvolumefractiontoreachlocallyalargevalue(e.g. aplume ofrising bubbleshitting ahorizontalwall).Last,it must bekeptinmindthatthedispersedphaseisassumedtobe mono-dispersedandthat theingredientsbrought totheGLIM approach bythemodels describedinSections2.2and2.3donot allowthe bubblediameter,ddg, tobe predicted. Insteadthis diametermust beprescribedasafunctionoftheaveragecellsize,
a.Inthe sim-ulationstobedescribedbelow,thisratioissettoddg/
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,mass ex-changetermsassociatedwithpossiblecoalescenceand fragmenta-tionneedtobeimplementedinthedgandcgfields.Fig.2,adapted fromDenèfle etal.(2015),summarizesthephasecoupling proce-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,Fσk.Forphasek=cl orcg,thelatteriswrittenas
Fσk =
α
kσκ
kp∇α
k with p=cg,cl, (13) where the interface mean curvature,κ
kp=−∇
· nkp, is com-puted using the unit normal pointingtoward phase k, i.e.nkp=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_CFDfollowsthispathbysolvingthe ad-ditionalequation(OlssonandKreiss,2005;Olssonetal.,2007)∂α
k∂τ
+∇
·α
k(
1−α
k)
nkp =∇
2α
k, k,p=cl,cg. (14) Thepseudo-time,τ
,andthe pseudo-viscosity,,are directly con-nectedwiththelocalcellsize,
.FollowingŠtrubeljetal.(2009), values
τ
= 32 and= 2 are respectively selected. With this
choice,thefinal thicknessoftheinterfaciallayerwithinwhich
α
k variesfrom0to1isδ
α=5whatevertheinitialspreadingofthe
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) usinga fric-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) to de-velopalternativeclosurelawsfortheinterfacialdraglawtobe ap-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
llg(
ucl− ucg)
,so thatthe momentum ex-changerateperunitvolumemaybewrittenintheformFDb
cl→cg=
α
cgμ
ll−2g(
ucl− ucg)
. (15) Thisclosureisusedaslong asα
cg≤ 0.3,which isusually consid-eredastheupperlimitofthedispersedbubblyflowregimeunder homogeneousconditions(Taiteletal.,1980).Applicationofa simi-larreasoningtothe‘opposite’caseoflargedropsmovingwithina gasinregionswhereα
cg≥ 0.7yieldsFDd
cl→cg=
α
clμ
gl−2l(
ucl− ucg)
, (16) wherell is thelengthscale characterizingtheinterfacial momen-tum exchange in that configuration. In the intermediate range 0.3<α
cg<0.7, the momentum exchange rate is assumedto vary linearlyinbetweentheabovetwolaws,namelyFD
cl→cg=fbdFDbcl→cg+
(
1− fbd)
FDdcl→cg, (17) with fbd(
α
cg)
=2.5×(
0.7−α
cg)
.Itisworth pointingout thatthe closureprovidedbyEq.(16)ismuchmorequestionablethanthat ofEq.(15)since thegasflow almostobeysano-slipconditionataliquid-gasinterfaceif
μ
g/μ
l 1,suggestingthataquadraticdrag lawwouldphysicallybemoreappropriateinthatcase.Thelengthscalesli(i=g,l
)
involvedinEqs.(15)–(17)are con-sideredto dependon the productα
clα
cg (which mayrangefrom 0 to 0.25) in such a way that they both tend toward ddg/3√ 2 when
α
clα
cg→0 (the factor 3√2 ensures that Stokes’ drag law is recovered in that limit),whereas they become proportional toα
ci||∇α
ci||
−1 forα
clα
cg࣡0.1anda linearvariation isassumedfor intermediatevaluesofα
clα
cg.Since||∇α
ci||
−1isproportionaltothankstotheinterfacesharpeningprocedure,thisdefinitionmakes the characteristiclength scales ofthe momentum exchange atLI dependonthecellsize.
2.5.3. Cutoff lengthscaleandmassexchangesbetweenthe continuousanddispersedgasphases
Numericalrequirementsassociatedwiththecomputationofthe meancurvature
κ
lg ofcl− cg interfacesdeterminethe characteris-tic size of thesmallest gas structures that can be fully resolved. Thiscriticalsize,lc,istakenasthecutoff lengthscale correspond-ing tothe separationbetweenthe dgandcggas phases. In NEP-TUNE_CFD,thecomputationofκ
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.Notingthatthecorresponding criti-calmeancurvatureis
κ
c=2/lc=(
4)
−1,onehasκ
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 densityFσk isapplied tointerfacesexhibitinga meancurvaturelargerthan
κ
c.The corresponding smallgasentities are transferred to thedg
phaseasdescribedbelow(see Eq.(19)). Massexchangesbetween thecganddgfieldsariseduetocoalescenceandbreak-upevents. Inthepresentcontext,thedefinitionoftheseeventsrelieson nu-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(
2δ
α)
−1.Basedonthese requirements, therate atwhich the dispersed phase coalesces is expressedintheformcg+=
α
cgα
dgρ
gC+
tH
(
α
dg−α
dgc)
H(
2δ
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-persedbubblesisassumedtobecg−=
α
cgα
dgρ
g C−tH
(
δ
α||∇α
k||
κ
lgκ
c − 1)
, (19)whereC− isanother O
(
1)
constant. The net mass exchange rate fromthedispersedgasfieldtowardthecgfieldisthendg→cg=
cg+−
cg−. (20)
Notethat thepresence ofthe
α
cgα
dg pre-factorincg+ and
cg−
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 tobemonodispersedandmadeofbubbleswithdiameterddg lc. SimilartotheGLIMapproach,ddghastobespecifiedasafraction of
a(seeSection3.2).
3. Experimentalandcomputationalconfigurations 3.1. Experimentalset-up
The experimental set-up is sketched in Fig.3. It consists ofa PlexiglasTM 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 vin 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.As vout remains opened during the filling process,the initial pressure atthe top of thebottle is theatmosphericpressure,patm.Thegateisquicklyopenedattime t=0.Waterstartstoflowoutofthebottleinasuccessionofjets separatedbythegenerationoflargeairbubblesattheneck.Then thesebubblesrisewithinthewatercolumnandreachthefree sur-faceafterasequenceofcomplexreconfigurations.
Theairpressureatthecentreofthetopendofthebottle,ptop, ismonitoredwithapressuresensor(Keller,PR-23).Imagesofthe emptyingprocessarerecordedwithaCMOScamera(PhotonLines, PCO1200HS)ataspeedof400fps.Theseimagesmakeitpossible to followthedisplacementofthe upperfreesurfaceandgive in-sightintothebubbledynamics.Thecameraandthepressure sen-soraresynchronizedthroughaTTLsignalthattriggersthe electro-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-slip con-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 withlengthLr=0.1m anddiameterDbelowthe bottleneck,as depictedinFig.4.
Theinitialconditionsrefertotheexperimentalcasewithan ini-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.sandσ
=7.28× 10−2N.m. Airviscosityissettoμ
g=1.8× 10−5 Pa.s.To assessthe possible role ofair compress-ibility, incompressible simulations as well as compressible ones (in which the propagation of density/pressure waves is fully re-solved)arecarriedout.Inincompressiblecases,airdensityissettoρ
a=1.20 kg.m−3. Incontrast,whencompressibilityis takeninto account,airisassumedto behaveasan isothermalidealgas, the densityofwhichobeysρ
a=p(
RairT)
−1,wherepdenotesthelocal pressure,Rair=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 islσ=
(
σ
/ρ
lg)
1/2≈ 2.7mm (g denoting gravity). Henced/lσ≈ 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≈ 1.6mm cor-respondingtoadimensionlesslengthratio
a/D≈ 0.014.Withthis grid,thediameterddgprescribedforthedispersedbubblesinboth 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 maincharacteristicsofthethreeclosuremodels em-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−
pinit.Thisinitialdropismerelydue to the hydrostaticpressure corresponding to the initial height of thewatercolumn,sothat
pinit≈
ρ
lgz0.Thepressurethenrises
al-mostlinearlyovertimeuntilitgoesbacktotheatmospheric pres-sure,patm,attheendoftheemptyingprocess.
A closer look at the pressure signal (see the inset in Fig. 5) reveals almost sinusoidal oscillations around a slowly increasing value,peqtop
(
t)
,correspondingtotheinstantaneoushydrostatic equi-librium.TheseoscillationshaveatypicalperiodTos≈ 0.20s,much shorterthantheemptyingtime.Theyfindtheirrootinthe alterna-tionofbubblegeneration andliquidejectionevents,asdescribed by Clanet and Searby (2004). The evolution of the equilibrium pressure, peqtop,obeystherelation peqtop(
t)
=patm− 4gMw(
t)
/(
π
D2)
, where Mw(t) is the massof waterin thebottle at time t (hence 4Mw(t)/(ρ
lπ
D2)istheinstantaneouswaterheight).Inthepresent case, thelinearevolutionof peqtop yieldsa constantmassflowratedMw/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
Atypicalpressuresignalrecordedbythepressuresensoris dis-playedinFig.6,stillwiththeinitialconditionz0/L=0.75.Pressure
oscillationsstart rightafterthegate openingandare present un-tilthe endof theemptying process.Based on their experiments,
ClanetandSearby(2004)attributedtheseoscillationstothe com-pressibilityofthe airbufferatthe topofthe bottle.However, an oscillatorybehaviourwasalsoobservedintheincompressible sim-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.OnlycomputationalresultsobtainedwiththeLIM ap-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 anoisyevolution charac-terizedby small-amplitudehigh-frequency fluctuations. The mag-nitude of the initial drop and the subsequent linear increase of
Fig. 7. Correlation between the pressure signal at the top of the bottle and the formation cycle of large bubbles at the neck. Top: pressure signal, p top ( t ). Bottom: a large
bubble rises in the water column after having been released from the neck (left panel in ( a ) and ( b )); somewhat later, a new bubble forms at the neck (right panel in ( a ) and ( b )); the corresponding two instants of time are identified with stars on the pressure signal. ( a ) numerical simulation with z 0 /L = 0 . 75 using the LIM approach (interfaces
are identified using the iso-contour αcl = 0 . 5 ); ( b ) experiment with z 0 /L = 1 .
the equilibrium pressure are in qualitative agreement with the characteristics of the peqtop
(
t)
componentof the experimental sig-nal.HoweverweshallseeinSection4.4thattherearesignificant quantitative differences with a direct consequence regarding the predicted time-averaged flow rate. In contrast, when air com-pressibility istakenintoaccount, thepressure signaldisplaysthe expected oscillations,with an amplitudeand a frequencyin rea-sonable agreement with the experimental findings, except dur-ing the initial transient when the magnitude of the oscillations is clearly underestimated. From thiscomparison, it may be con-cluded that air compressibility is essential to correctly repro-duce pressureoscillations.Consequentlyall numericalresults dis-cussed below were obtained through compressible simulations. The underestimate of the oscillation amplitudeduring the initial transient may have several origins. One of them could be that the evolution is not strictly isothermal, given the magnitude of the sudden pressure drop. There may also be some influence of the bevelled shape of the neck which is not considered in the computations.Torevealtheimportanceofaircompressibilityontheflow dy-namics, theconnectionbetweenthepressure signal atthetopof thebottleandthelifecycleoflargebubblesgeneratedatits neck isdisplayedinFig.7.Thisfigureshowsthattheobservedpressure oscillationsarethedirectsignatureofthewaterjetsandlarge air bubblessuccessivelygeneratedatthebottleneck.Whentheliquid flowsout(leftpanelsin(a)and(b)),thefreesurfacemoves down-ward,inducinganexpansionoftheairbufferatthetopofthe bot-tle, henceadropinthelocalpressure.As ptop(t)decreases below the hydrostatic equilibriumvalue peqtop
(
t)
,an upward force starts actingonthewatercolumn,makingthegenerationofanew bub-bleattheneckanditsreleaseatthebottomofthewatercolumn possible. The free surfacethen movesupward, making the pres-sureincrease,whichcorrespondstothestartofanewcycle(right panelsin(a)and(b)).4.3.Influenceoftheinterfacialmomentumexchangemodelonthe pressureinthetopairbuffer
Fig. 8 reveals the influence of the interfacial momentum ex-changemodel(LIM,GLIMorLBM)ontheevolutionofthepressure recordedat the top ofthe bottle. As alreadynoticed, a transient takes place during the first 1s following the gate opening. This firststage ischaracterized by pressureoscillations, theamplitude ofwhichdecreasesexponentially untilitreachesan almost time-independent value closeto 1 KPa. All three models are found to severelyunderestimatetheoscillationamplitudeduringthatstage. Whiletheinitial pressuredroprecordedintheexperimentis ap-proximately8kPa,its predictedamplitudebarelyexceeds6kPa. It isnoteworthythat allthreecomputationalcurvesperfectly super-impose during the first three oscillations. This suggests that the initialtransientismainlygovernedbycompressibilityeffects,with little influence of the detailed flow dynamics. Beyond this tran-sient, say fort>1.5 s, the pressure oscillates around an equilib-riumvalue.Allmodelspredictfairlywelltheoscillationamplitude aswellastheequilibriumpressureduringthatsecondstage.
Experimentalandcomputationaldeterminationsofthetime pe-riod, Tos, of these oscillations are compared in Fig. 9. The the-oretical prediction derived by Clanet and Searby (2004) is also shownasreference.Accordingtothisprediction,theoscillation pe-rioddependsonthegasthermodynamiccharacteristicsandwater height in the form Tos=L
(
γ
pvptop/ρ
g)
−1/2(
z(
t)
/L)
with(
z)
≈{
zL
(
1−zL)
}
1/2,whereγ
pvisthe adiabatic index,i.e.the ratio of the twospecificheats ofthegas,andz(t)isthecurrentheightofthe waterinthebottle.The experimental and computational periods are obtained throughaslidingFouriertransformofthecorresponding pressure signals,usingasuccessionof1stimewindowswitha50%overlap betweentwoconsecutivewindows.Inordertolimittheinfluence ofthefinite-sizewindow effectontheresults,time windowsare projectedonahalfcosine.ThetheoreticalpredictionofClanetand Searby(2004) isfoundto be ingeneralagreementwith
observa-Fig. 8. Evolution of the pressure at the top of the bottle. The experimental sig- nal (blue line) is compared with computational predictions obtained with the three different interfacial momentum exchange models (yellow: LIM, green: GLIM, pur- ple: LBM). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 9. Evolution of the pressure oscillation period throughout the emptying pro- cess. Predictions of the various interfacial momentum exchange models are com- pared with experimental data and theoretical predictions provided by the model of
Clanet and Searby (2004) (black solid line). Symbols obey the colour code defined in the caption of Fig. 8 .
tions.Itslightlyunder-predictstheactualperiodbutcorrectly cap-turesits slow increase. Allthree interfacialmomentum exchange modelsgivesimilarpredictions,althoughLBMisseentobe gener-allymarginallymoreaccurate.Allmodelsslightlyoverestimatethe actualoscillationperiod,especiallyduringthefirsthalfofthe pro-cess.Thediscrepancymaybeupto10%andreducesastimegoes on.
The above observations indicate that the pressure signal is barely sensitive to the details of the interfacial momentum ex-change.Allthree modelsare found toprovide reasonable predic-tionsoftheamplitudeandtimeperiodofpressureoscillations, ex-ceptduringtheinitialtransient.
4.4.Influenceoftheclosuremodelontheliquidflowrate
The computed and experimentally determined average liquid flow rates are reported in Table 2. When air compressibility is notconsidered,theemptyingprocessissignificantlysloweddown, withamassflowrateunderestimatedbymorethan35%.This con-firmsthat air compressibilityplays an essential role inthe over-all flow dynamics and cannot be ignored in computations. Once compressibilityistakenintoaccount,themeanflowrateobtained with the LIM approach is seen to be in very good agreement with the experimental value, whereas the GLIM and LBM
com-putations inwhichthe dispersedbubble diameterissetto 1mm under-predict it by 13%.This difference maybe anticipatedfrom
Fig. 8,where a slightdownshift ofthe equilibriumpressure pre-dicted by the GLIM and LBM formulations is noticed. Neverthe-less,weregard themuchcloseragreementobtainedwiththeLIM approachasmostlyfortuitous, sincetheprecise value ofthe pre-dicted flow rate depends on the closures used in the interfacial momentum exchangemodel.Thisis easilyunderstood by consid-eringthebudgetofthetotal (i.e.kineticplus potential) mechani-calenergy,eM=
α
clρ
l(
12u2cl+gz)
,oftheliquidenclosedwithinthe cylinder.FromEq.(4)itisstraightforwardtoshowthatd dt Vc eMdV+ Sn eMucl· nndS = Sn
α
cl(
−Pnn+2μ
lScl· nn)
dS −2μ
l Vcα
clScl:ScldV+ Vc p=cl Ip→cl· ucldV, (21)where Scl= 12
(
∇
ucl+T∇
ucl)
, nn is the outer unit normalto the neck surface, Sn, and Vc is the volume of the cylinder. The last termintheright-handside ofEq.(21) istherateofwork result-ingfromthemomentumexchangebetweentheliquidandthegas phasep(p=cl,cg)withinthecylinder.Duetothepresenceofthis term,itisclearthattherateatwhichtheinitialpotentialenergy oftheliquidis convertedintokinetic energydependsdirectlyon Ip→cl, henceonthe closuresdiscussed inSection 2.Forinstance, anychangeintheevaluationoftheinterfacialfrictionvelocity,u∗cg, involvedonlargeinterfacesintheLIMandGLIMapproaches(seeEq.(10)), orin that ofthe length scales lg andll involved inthe LBMformulation(seeEqs.(15)and(16),respectively)modifiesthis balance,which directlyimpactsthe liquidflowrate. Asimilar ef-fectisexpectedtotakeplacewhentheclosurelawsusedtomodel themomentum exchange withthedispersed phase are modified. To illustrate this influence, we carried out two additional simu-lations based on the GLIM andLBM approaches, with ddg set to 2mm.Thecorrespondingresultsareprovidedinthelasttworows ofTable2.Comparedtothoseobtainedwithddg=1mm,theflow rateisfoundtobeincreasedby1%and1.2%,respectively.
This influence of the closure models on the flow rate makes usconsiderthat thetwo mostrobustindicationsconveyedbythe results displayed in Table 2 are that (i) the LIM, GLIM and LBM formulationsinwhichaircompressibilityistakenintoaccount all provideareasonableestimateofthemeanflowrate,andthat(ii) underpresentconditions,thispredictionislowerbytypically10% with the lattertwo approaches, compared to that obtainedwith the LIM formulation. Thislowering originates in the much more significant presence of a dispersed phase inthe simulations per-formed with the former two models, as shown in Figs. 10 and
11below.In bothcases,the riseofdispersed bubbles induces an average upward motionin the liquidthrough the interfacial mo-mentum exchange model. Thus, these small bubbles somewhat hamperthe liquidoutflow,which yields a slightlyslower empty-ing dynamics.In otherterms,a largerpartoftheinitial potential energyoftheliquidistransferredtointerfaceswhentheGLIMor LBMformulationisused,reducingthepartthat canbeconverted intokinetic energy,hencethe liquidflowrate. InTable 2, the ef-fect ofan increase ofddg to 2mmis seento be slightlystronger intheLBMsimulation,whichisnosurprisesincealarger partof thegascontentis treatedasdispersedinthisformulation,owing tothelargercutoff lengthinvolved(seeTable1).
Havingdoubledddg dividesthe numberofbubblesin the dis-persedphase bya factorofeight,aschangingddgleavesthe vol-ume fraction
α
dg unchanged and only modifies the momentum transferred fromthe liquidto thedispersed bubbles.This iswhy thecorresponding effectis smallwithboth models.Moresignifi-Table 2
Influence of the selected model on the prediction of the time-averaged water flow rate. In the experiment, this quantity is determined by dividing the initial mass of liquid by the total emptying time, while in computations it is obtained by time-averaging the instantaneous liquid flow rate throughout the simulation.
Nature of the simulation Momentum exchange model
Emptying mass flow rate
dMliq / dt [g.s −1 ]
Qnum Qexp (Qexp − Q num) /Q exp [%]
incomp. LIM −123 . 0 −191 . 2 −35 . 5 comp. LIM −189 . 4 −191 . 2 −1 . 0 comp. GLIM - d dg = 1 mm −166 . 4 −191 . 2 −13 . 0 comp. LBM - d dg = 1 mm −166 . 3 −191 . 2 −13 . 1 comp. GLIM - d dg = 2 mm −168 . 0 −191 . 2 −12 . 2 comp. LBM - d dg = 2 mm −168 . 3 −191 . 2 −12 . 0
Fig. 10. Visualization of the large bubbles in the water column at t = 4 . 2 s . Inter- faces are identified with the iso-contour αcl = 0 . 5 .
cantvariationsoftheflowraterelatedtothepresenceofthe dis-persedgasphasewouldpresumablybeobservedifthecutoff crite-rionweretuned,eitherbymodifyingtheway
β
andfcaredefined inEq.(12)intheGLIMapproach,orbychangingtheratiolc/in theLBMrepresentation.Wedidnotattemptsuchtestsuptonow butplantoperformtheminthefuture.
5. Influenceoftheinterfacialmomentumexchangemodelon thelocalflowcharacteristics
5.1. Qualitativeobservations
Thethreemodels mostlydifferinthewaylargegasstructures involvingLIcoexistwithsmallerbubblesandinteractwiththe sur-roundingwaterflow.Toobtainmoreinsightintothepotential dif-ferencesbetweenthesemodels,examinationofsuitablelocal char-acteristicsofthegasandliquiddynamicsisrequired.
Asnapshotof thedistribution ofthelarge gasbubbles within thebottle,aspredictedbyeachmodel,isdisplayedinFig.10.With theLIMapproach,thelargebubblesformedattheneckrisetoward thefree surfacewithlittlesizevariation.Thisisnosurprisesince
nomomentumtransfermechanismtowardthedispersedphase ex-istsinthat model.Actually,inthepresentconfigurationinwhich theinitialinterface reducestoa singleLI, smallbubblescan only becreateddueto thenumericalsmearingofthesuccessive inter-faces,which,astimeproceeds,preventssomeofthemfrombeing consideredasLIbythedetectionalgorithm(seeSection2.3.1).The observed behaviour is significantly different with the other two models. In both cases, the large bubbles generated at the bottle neckundergosubstantialsuccessivereconfigurationsandtheir av-eragesizedecreasesastheyrise,owingtofragmentation.Withthe LBM,thefragmentationrateissuchthatthecontinuousgasphase isvirtually absent intheupperpartof thewatercolumn, having beentransferredearliertothedispersedphase.
Fig. 11 is similar to Fig. 10 but the computational snapshots havebeencoloured accordingto thelocal volumefractionof the dispersed phase,so as tobetter reveal the complete structure of themulti-scalebubbleswarm(avideobasedontheGLIM simula-tionislinkedtothepaper).Thecorrespondingexperimental snap-shot(withz0/L=1asinitialcondition)isalsoshown,toserveas
a reference.This experimental snapshothighlights thebroad dis-tribution of air bubble sizeswithin the bottle. Close to the bot-tom, air is mostly contained within a few large bubbles which are subjectto fragmentationand gradually evolve in a swarm of much smaller bubbles as they rise, although some of them suc-ceedinmaintainingtheirintegrity andarestillpresentinthe up-per part of thewater column. Fig.11 confirmsthat the LIM ap-proachseverely underestimates the amount oftopological recon-figurations experienced by large bubbles, asalmost no dispersed gasphaseiscreated.Theresultsprovidedbytheothertwomodels appearqualitatively morerealistic,witha gradual increase ofthe volume fractionof thedispersed phase as one gets closerto the upperfree surface. This volume fractionseems to be larger with theGLIMapproach, especiallyintheupperhalf ofthewater col-umn.However,itmustbekeptinmindthatthecriteriabywhich largegasstructuresturnintodispersedbubblesdifferbetweenthe GLIMandLBMapproaches;thismaybethereasonwhymore en-titiesbelongtothedispersedphaseintheformerattheinstantof timeselectedinFig.11.
5.2.Gasvolumefraction
Toobtainamorequantitativeinsightintotheabilityofthe var-iousmodelstoreproducetheformationofadispersed gasphase,
Fig. 12 displays the horizontal cross-sectional average of the gas volumefraction,
α
g,attwo differentpositionswithin thewater column.Atthelowerposition(10cmabovetheneck),thethreemodels yieldsimilarevolutionsoftheaveragegasfraction(Fig.12(a)).This evolutionis characterized bya periodic successionoflarge sharp peaks, each of which corresponds to the crossing of the control volumebyalargebubble.Thisdynamicsjustreflectstheperiodic detachmentof large bubbles fromthe neck. The peak amplitude