Emptying of a bottle: How a robust pressure-driven oscillator coexists with complex two-phase flow dynamics

an author's

https://oatao.univ-toulouse.fr/23959

https://doi.org/10.1016/j.ijmultiphaseflow.2019.05.012

Mer, Samuel and Praud, Olivier and Magnaudet, Jacques and Roig, Véronique Emptying of a bottle: How a robust

pressure-driven oscillator coexists with complex two-phase flow dynamics. (2019) International Journal of

Multiphase Flow, 118. 23-36. ISSN 0301-9322

Emptying

of

a

bottle:

How

a

robust

pressure-driven

oscillator

coexists

with

complex

two-phase

flow

dynamics

Samuel

Mer

∗

,

Olivier

Praud

,

Jacques

Magnaudet

,

Veronique

Roig

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

Keywords:

Bubbly flow Multi-scale flow Taylor bubble

a

b

s

t

r

a

c

t

Theemptying ofabottleisone ofthemostcommon two-phaseflows encountered ineveryday’slife fluidmechanics.Wereportonadetailedexperimentalanalysisofthisflowconfigurationcoveringawide rangeofneck-to-bottlediameterratios,d∗_{,andinitialfillingratios,F.Thejointuseofapressuresensor} atthetopofthebottleandashadowgraphtechniquetotracktheevolutionoftheupperliquidsurface allowstheaveragegasvolumefraction inthefluidcolumntobecomputed throughoutthedischarge. Variationsofthebottleemptyingtime,gascontent,andcharacteristicsofthepressuresignalasa func-tionofd∗ and/orF areanalyzed,andscalingorevolutionlawsarederived.A specificfocusisputon theinitialtransientstagesfollowingtheneckopening,especiallyonthenatureand growthofthefirst bubblethatreachestheupperfreesurface.Overawiderangeofd∗_{,thisbubbletakestheformofalarge} sphericalcaporTaylorbubble,therisespeedofwhichiscontrolledbythebottlediameter.Sucha bub-bledirectlyemergesfromtheneckforlargeenoughd∗_{,butratherresultsfromsuccessivecoalescenceof} smallerbubblesforsmalld∗_.Whateverd∗_{,thevolumeofthisleadingbubbleincreasesovertime,since} afraction ofthesmallbubblesswarmtrailedinitswakemergeswithitsbackface. This coalescence-inducedgrowthisquantifiedthroughasimplemodel.

1. Introduction

Liquid-gas two-phase flows are ubiquitous in Nature and ev-eryday’s life fluid mechanics. Who has never been amazed by the formation of bubble trails in a glass of Champagne or by the frothy foam sitting on top of a beer? Some of the physical phenomena atstakein bubbly drinks were recentlyreviewed by

Zenit and Rodríguez-Rodríguez (2018). However bubbles are not involvedonlyinalcoholicdrinksandoneofthefirstbubbly two-phase flows encountered in a human life probably standsin the emptyingofawaterbottle.Despiteitscommonnessandapparent simplicity, thisflow exhibitsrich physics andcan be seen either as a complex nonlinearoscillator (Kohiraet al., 2012) or a com-plexmulti-scalegas-liquidflow(Meretal.,2018),orevenaliquid sandglass - asthe liquid dischargeremains almost constant dur-ingthewholeprocess(SchmidtandKubie,1995).Indeed,allalong thedischarge,largeairbubbleswithdiametersoftheorderofthe bottle neck are periodicallygenerated andrise within the bottle, untiltheyburstatthefreesurfacebeneathits top.While ascend-ing, these large bubbles undergo successive break-up sequences, yielding swarms of smaller bubbles, part ofwhich may coalesce

∗ _{Corresponding author.}

E-mail addresses: [email protected] (S. Mer), [email protected] (J. Magnaudet).

againandparticipateintotheregenerationandreconfigurationof thelargebubblepopulation.

Thefirststudiesofthisflowconfigurationreportedinthe liter-atureassessedtheinfluenceofgeometricalparameterssuchasthe bottleshapeandneckdiameter(orshape)ontheglobalemptying time(e.g.Whalley,1987;Whalley,1991;SchmidtandKubie,1995; KordestaniandKubie, 1996;TangandKubie, 1997). These contri-butions showed that: (i) the liquid discharge rate remains con-stantduringthewholeemptyingprocesswhatevertheinitial fill-ingratio;(ii)theaveragedpressuredifferencebetweenthetopair buffer andthe ambientmedium scales well withthe hydrostatic head corresponding to the height of the water column; (iii) the emptyingtime is influencedby the bottle inclination angle,neck shape,andliquidtemperature. Previousstudieswereextendedby

Héraud(2002),Clanetetal.(2004)andClanetandSearby(2004), hereinafterreferredtoasCS,tocoverawiderangeofbottleshapes anddimensions.Inparticular,CSderivedageneralscalinglawfor theemptyingtime Te ofcylindrical bottleswithlength Land

di-ameterDintheform

Te≈ 3.0 L



gDd ∗−5/2 , (1)

where d∗=d/D is the normalized neck diameter and g denotes gravity.Thisscaling lawemphasizes thefactthat thegeometrical ratiod∗playsanessentialroleintheemptyingdynamics.

A second class of studies analyzed theemptying process asa non-linearoscillator,theliquidmassandtopairbufferplayingthe role of a mass and a spring, respectively. These studies focused onsmalld∗, aconditionunderwhich theflow exhibitsan on-off oscillatorybehavior.Tehranietal.(1992)proposedamodelaimed at predicting the pressure fluctuation amplitude in the top air buffer in an idealized configuration. In their work, the bottle is reduced to a cubic tank, with a typical size of 25cm, and the neck to a 60cm long straw with a diameter up to 25mm, i.e. d∗<0.1. The straw-like neck geometryresults ina large pressure loss at the bottle neck, which triggers the occurrence of on-off oscillations. Under these conditions, four stages were identified during one oscillation period: (i) liquid downflow; (ii) bubble formationand rise inthe neck; (iii)bottle re-pressurization, and (iv) neck refill. Results provided by the model revealed good agreement with the experiments. Kohira et al. (2012) carried out an experimental andtheoretical study of singleand coupled on-off plastic bottle oscillators. They observed the emergence of in- and anti-phase synchronization, depending on the nature of the coupling. Jia et al. (2016) modeled these two distinct syn-chronization scenarios by considering time-dependent nonlinear termsresulting from the asymmetry between the liquidejection and bubble admission processes. CS predicted the evolution of theoscillationperiod duringtheemptyingprocess by integrating the Euler equation throughout the flow, assuming a schematic variationofthe velocitydistribution closeandfarfromthe neck. Their approach is not limited to the on-off oscillating behavior and can be applied to larger d∗. Its predictions reveal a fairly goodagreementwithexperimentaldataoverabroadrangeofd∗, suggesting that it captures the dominant mechanisms by which theneckdiameterinfluencestheoscillationperiod.

In a recent work, we highlighted how rich the bottle emp-tying process is from the point of view of multi-scale gas-liquid flow processes (Mer et al., 2018). In particular, we showed that thisconfiguration provides a complete and difficult test case for validatingmulti-scalemodelingapproachescurrentlyunder devel-opment(Deendarlianto etal., 2011; Hanschetal., 2012;Montoya etal., 2015;Gadaetal.,2017; Mimounietal.,2017).Therigorous validationofsuchmodelingapproaches,andofthecomputational codes in which they are implemented, requires detailed and reliable experimental databases in which the gas phase within the water column is properly characterized. Local gas volume fractionmeasurementsmakeuseofopticalprobes(Suzanneetal., 1998;Raimundo etal., 2016),image processing(Lau et al., 2013) or LDA techniques (Kulkarni et al., 2001; Gandhi et al., 2008). However, giventhe highlyintermittent nature ofthe flow inthe presentcase, extractingrelevantindicatorsfromlocalgasfraction measurements with the goal of validating multi-scale models is notstraightforward.Conversely,couplingalocalpressure determi-nationwithan optical technique,such asshadowgraph,maygive access to the average gas volume fraction in the water column. More precisely, assuming the pressure to be uniform within the airbuffer at thetop ofthe bottle, andequal to the atmospheric pressureoutsidethebottle,the couplingofadifferentialpressure sensorlocatedwithin thisbuffer withan opticaldetectionofthe uppersurfaceoftheliquidcolumnallowsthe space-averagedgas volumefractiontobedetermined.

The presentpaperreportsonan experimental investigation of the emptyingprocess in water‘bottles’ withvarious neck diam-eters, based on the latter approach. By considering contrasting flowconditions,it aimsatclarifyingtherole ofthetime-evolving two-phase flow that developswithin the bottle on the emptying dynamics.Under some ofthese flow conditions, the fluid region

is merely crossed by bubbles rising almost inline. Conversely, it rather looks as a dense andstrongly agitated gas-liquid mixture under other conditions. Last, the flow configuration may evolve alongtheemptyingprocess,fromaquiescentliquidwithlarge ris-ing bubbles in the early stage, to a dense and strongly agitated bubbly flow when the liquid content has decreased significantly. The common thread ofthis investigation isto gain some insight intothe influenceofthiscomplextwo-phase flow onthe oscilla-torybehavioroftheoverallsystemandviceversa.

The experimental facility along with the measurement tech-nique are presented in Section 2. Section 3 reports qualitative observations of the flow structure and evolution, first describ-ing the successive stages encountered during the emptying pro-cess within the top air buffer and the two-fluid column. Then, an overviewof the two-phase flow structure and gas content is provided inSection3.2.Global resultsare discussedin Section4, startingwithvariationsoftheemptyingtime withthe bottle ge-ometryin Section4.1,while some features ofthepressure signal are analyzed inSection 4.2.The coupled measurementtechnique is used to determine the evolutions of the free surface and the gas content in thewater column; theseevolutions are discussed inSections4.3and4.4,respectively.Section5focusesontheearly stagesoftheemptyingprocess,withaspecialemphasisonthe ini-tialpressure transient(Section 5.1) andthe characteristics ofthe firstrisinggasentity(Section5.2).Section6summarizesthemain findings ofthe study and draws some prospects, especially with respecttomodelingissues.

2. Experimentalset-upandmeasurementtechniques

Thedevice,measurementtechniquesandexperimentalprotocol aresimilar tothoseemployedby Meretal.(2018).Their descrip-tionissummarizedbelowforthesakeofself-consistency. 2.1. Experimentalset-up

Theexperimentalset-up(Fig.1)consistsofaPlexiglasTM

cylin-drical vesselwithdiameterD₌114mmandheight L₌800mm. The top end of thecylinder is closed witha blank flange, while its bottom section hasa central circularthin-walled hole with a diameterd.Theholewasmanufacturedwithabevelededge mak-ing asharp20◦ angle withthe cylinderbase.Thisbottompartis interchangeable and allows the neck diameterto be varied from 13 to 80 mm. The bottle neck is closed with a gate mounted ontwo hinges andequippedwithfiveelectromagnetsensuring a fast opening.Before a test, the gate is closed,the valves vin and

vout are opened and the bottle is filled up to an altitude z=z0

with tapwater, using the circulation pump. Oncefilling is com-pleted,the pumpisturned off andthe two valvesare closed.As vout remains opened during the filling process, the initial

pres-sure at the top of the bottle is the atmospheric pressure, patm.

Thegateisquicklyopenedattime t=0.Waterstartsto flowout of the bottle in a succession ofjets separated by the generation of large airbubbles at the neck.Then, these bubbles rise within the watercolumnand reachthe free surfaceafter asequence of complex topological changes. In what follows, the tank normal-izedoutletdiameter,d∗,andinitialfillingratio,_F=z0/L,are

var-iedintherange0.18≤ d∗_{≤ 0.7and0.24≤ F≤ 1,respectively.The}

densityand kinematicviscosity of the waterused in the experi-ments, performed at room temperature, are

ρ

l=997kgm−3 and

ν

l=1× 10−6 m2s−1,respectively.

2.2. Measurementtechniques

The air pressure at the center of the top end of the bot-tle, ptop,ismonitored thanksto a pressuresensor(Keller,PR-23).

Fig. 1. Schematic of the experimental set-up and measurement means.

Fig. 2. The three successive steps in the image processing procedure for d ∗_{= 0 . 22} and F = 0 . 75 . (a) Raw image; (b) image obtained after background removal; (c) ver- tical intensity profile (in red) obtained by averaging horizontally the pixel intensity between the two green lines in ( b ). The detection of the dominant peak makes it possible to follow the displacement of the upper free surface, indicated by the cyan line. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Correspondingdataareacquiredatafrequency fp=1000Hz.

Im-ages oftheemptyingprocess aresimultaneously recordedwitha PCOTM _{Dimax camera with a full resolution of 2000× 2000}

pix-elsoperatingatafrequency fim=200Hz.Thecylindricalbottleis

back-lightedbymeansofaLEDpanel(seeFig.1).Thecameraand pressuresensoraresynchronizedthrougha TTLsignalwhichalso triggerstheelectromagnet,thusthegateopening.

Recorded imagesarepost-processedto extractthevertical po-sition, hint, ofthe upperfree surface. Fig.2 displayseach step of

theimage processingalgorithm.First,areferenceimage,recorded

before the beginning of the experiment, is subtracted from the rawimage(a) toobtainthe image(b). Thelatterdisplaysablack backgroundonwhichopticaldiopters,materializinggas-liquid in-terfaces, appear asgray shades since they are associated witha higherpixelintensity.The pixelintensityisaveraged horizontally betweenthetwo green lines indicated onimage (b) to construct theverticalintensityprofile, displayedin redon image(c). Start-ingfromthe top,thefirst peakpinpoints theupperair-water in-terface.Thankstoan adaptivethresholdingtechnique, thevertical positionofthisinterface, indicatedby thecyanlineonimage (c), isdetected. Atthe beginning of the emptyingprocess, when the freesurfaceisflatandwell defined(suchasinFig 2),the uncer-tainty in the determination of h_int is approximately 2mm. Later, whenlargebubblesburstatthefreesurface, theygeneratea sig-nificantsloshingofthelatter(e.g.intherightpanelofFig.6).The uncertaintythen increases significantly and isestimated, forlate timesandlargeopenings,tobeoftheorderof20mm.

Atypicalevolutionofhint(t)isshowninFig.3ford∗=0.22and

F=0_.75.Thisevolutionrevealsthattheupperfreesurfacemoves downwithanearlyconstantspeed throughouttheemptying pro-cess, aspreviously observed by Kordestani andKubie (1996)and CS. As shownin the insetof Fig. 3,the free surfaceposition ac-tuallyexhibitssmalloscillationssuperimposedonaregular evolu-tion,owingtoperiodicejectionofliquidandbubblegenerationat theneck.The free surfacepositioncannot bereliably detected at theveryendoftheprocess,owing toopticalconstraints.Forthis reason,allevolutionsofhint(t)presentedinthepaperaretruncated

ataminimumvalue,hint/L=0.06.

3. Qualitativeobservationsoftheemptyingprocessand

two-phaseflowstructure

3.1. Successivestagesoftheemptyingprocess

The emptying-generated flow may be seen as resulting from thecouplingbetweentwo fluidsystemstriggered bythe periodic

Fig. 3. Evolution of the position of the upper free surface during the emptying pro- cess for d ∗_{= 0 . 22 and F = 0 . 75 . The inset provides a closer view at the oscillations} of the free surface position during the time interval 22 s ≤ t ≤ 25 s.

Fig. 4. Characteristic time scales of the three successive stages of the emptying pro- cess as a function of d ∗_{for F = 0 . 77 . Green squares: τ/ T e ; blue dots: T os / T e ; purple} diamonds: T fb / T e . Based on measurements, T os was taken to be 0.2 s in all cases. (For

interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

generationofairbubblesattheneck.Thetwosystems,thetopair buffer and the aerated water column, experience two successive transientsbefore their couplingleadsto a slowly-varyingregime duringwhichthepressure inthetopairbufferoscillates periodi-cally.Beforetheveryfirstbubblegeneratedattheneckreachesthe topoftheliquidcolumn,theairbufferformsaclosedgasvolume. Duringthis initial stage, a fast pressure transientwith a charac-teristicdecaytime

τ

is detectedwithin thebuffer. Then,thefirst bubblereachesthewater-airinterfaceattimeT_fb.Afterthese tran-sients,a slowlyvaryingtwo-phase flowsetsinandlastsuntil the emptyingprocess iscompleted attime Te.Throughoutthis

‘fully-developed’stage,regular oscillationswithperiod Tos are detected

inthepressuresignalrecordedatthetopofthebottle.

In the rangeofinitial filling ratiosinvestigated inthe present study (_{F≥ 0.}24), all successive stages described above were al-waysobserved. Depending on the control parameters d∗ and F, the characteristictime ratios

τ

/Te, Tfb/Te and Tos/Te mayvary by

morethan one orderof magnitude, asshownin Fig.4. Selecting Te asthereferencetimeallowstherelativedurationofeachstage

ofthe evolution tobe compared withthe duration of thewhole

emptyingsequence;anyotherchoicedidnotprovideabetter scal-ing.Consideringtheselargevariationsofthetimeratios,itislikely thatvaryingd∗andF yieldscontrastingflowdynamics.Whatever d∗,theperiodTos ismuchsmallerthantheemptyingdurationTe,

implying that an oscillatory behavior takesplace throughout the discharge, evenwiththelargestnecks.The characteristictimeTfb

correspondingto theriseofthefirstbubbleisapproximatelyone order ofmagnitudelarger than Tos whatever d∗.Hence, for large

enough F,beforethefirst bubblereachestheupperfree surface, severalbubbleshaveenteredtheliquidcolumnandmayevenhave alreadybrokenup orcoalesced.Asd∗ increases,Tfb maybe upto

40% of Te. Consequently, the transient stage lasts fornearly half

ofthetotal emptyingtimeundersuch circumstances, andcannot be neglected. During such long transients,the first rising bubble isa large Taylorbubble,asshownintheright panelin Fig.5.In contrast,shorttransientscorresponding tosmalld∗ involvemuch smaller, regularlyspaced risingbubbles (leftpanel in Fig.5). The specificities of these transients are discussed in more detail in

Section5.InFig.4,thedampingtime

τ

oftheveryfirstpressure transientis seen tovary fromone-tenth toone-half T_fb when d∗ increases.Thevariousstagesoftheemptyingprocess existforall F aslong astheinitial volumeof liquidismuch largerthan the volumeofthefirstbubbleemittedattheneck.Howevertheir rel-ativeduration maydepend on the initial filling ratio,since their respectivecharacteristictimesvarydifferentlywithF.

3.2. Overallevolutionofthetwo-phaseflowcontent

Fig.6displaysimagesofthegascontentinthewatercolumnat t=2.5s fornormalizedneckdiametersranging fromd∗=0.18to d∗=0.7.Thistimeinstant istypical oftheslowly-varyingregime that succeeds the initial transient. The cyan markeron each im-agecorrespondstothedetectedfreesurfaceposition.Asexpected, the largerthe diameterof the openingis,the faster thecylinder empties. The flow dynamicsmay be characterized by a Reynolds number,Re=Ved/

ν

l,basedonthemeanliquidvelocityattheexit,

Ve,andneckdiameter,d.Sincemassconservationimpliesthatthe

emptying time, Te, satisfies the equality d2VeTe=FLD2, the

cor-responding Reynolds number is Re=FLD2_/

₍

_ν

lTed

)

. In Fig. 6, Re

ranges from 3000 for d∗=0.18 to 22,500 for d∗=0.7. Alterna-tively,one maydefinethe ReynoldsnumberReD=d∗Rebasedon

the cylinder diameter, D, and average emptying velocity, _FL/Te.

This Reynolds number ranges from 540 for d∗=0.18 to 15,750 ford∗=0.7. The Bond number, Bo=

ρ

lgd2/

σ

, with

σ

the

water-airsurface tension,ofthe large bubbles witha size ofthe order of d that form periodically at the neck is in the range 85–970, while their Weber number,We=

ρ

lVe2d/

σ

, is in the range 380–

1200.Such bubblesarehighlydeformableandmaybreakup eas-ily,owingtotheagitationofthehigh-Reynolds-numberflowinthe fluid column.It mustbe noticed thatthe dimensionlessnumbers Re, Bo andWecharacterizing the flow dynamicsare independent ontheinitialfillingratio,F.

Dependingontheneckdiameter,abroadvarietyoftwo-phase flowconfigurationsisobserved,rangingfromisolatedbubbles ris-inginarelativelyquiescentambientfluid-forsmallopenings-to aflowwhich,qualitatively,lookslikeachaoticfoamwhend∗gets closetounity.Wedidnotobserveanyclearsharptransitionfrom oneflowregime totheother butratherasmoothandcontinuous variationwithd∗.Foragivend∗,wedidnotnoticeanysignificant effect ofF on the observed flow regime provided Te>>Tfb.For

smallopeningdiameters,typicallyd∗₌0_.18and0.22,bubbles pe-riodicallygeneratedatthebottomofthetankrisetowardthefree surface along oscillatory paths andonly exhibit weak size varia-tions.Astheneckdiameterincreases,largerbubblesaregenerated. Although some of them may maintain their integrity through-out their rise, the majority undergoes successive ‘catastrophic’

Fig. 5. Visualization of the gas content in the bottle during the initial transient ( t = 1 . 5 s) for five different normalized neck diameters and the same initial filling, F = 0 . 77 . From left to right: d ∗_{= 0 . 18 , 0 . 26 , 0 . 39 , 0 . 53 , 0 . 70 . The cyan marker in each panel indicates the detected free surface position. The computed mean gas volume fractions α} are 3, 6, 12, 17 and 41% , respectively.

Fig. 6. Visualization of the gas content in the bottle in the fully-developed regime ( t = 2 . 5 s) for the same five experiments as in Fig. 5 . The corresponding computed mean gas volume fractions α are 3, 5, 11, 16 and 35% , respectively.

Fig. 7. Variation of the normalized emptying time, T e / (FL/



gD) , with respect to the normalized neck diameter d ∗ _{for different initial filling ratios, F. Each point} gathers results obtained in several independent experiments. The color scale refers to the initial filling, with from dark to light: F = 1 . 0 , 0 . 74 , 0 . 5 and 0.26. The solid line corresponds to the fit T e / (FL/



gD) = 3 . 0 d ∗−5/2

.

reconfigurations andtheir average size decreases, owing to frag-mentation.Thisyields achaotic bubbleswarmcharacterized bya broadbubblesizedistribution,asobservedinFig.6ford∗≥ 0.39.It isexpectedthatthisdistributionbroadensasReisincreased. Nev-ertheless,how its widthand peak value vary withRe isunclear, since thesecharacteristics are controlled by a complex combina-tionofbreak-upandcoalescenceevents.Despitethelimitationsof visualobservation,Fig.6suggeststhatlargebubblesand/orstrong spatialmodulationsofthegasvolumefraction,theextentofwhich increaseswithd∗,arepresentintheflow,atleastuptod∗=0.53. This provides evidence that the two-phase flow structure keeps somememoryoftheperiodicgasinjectionattheneck.

Experimental observations at t=2.5s suggest that the aver-age gas volume fraction in the fluid column increases with the diameter of the opening. The quantitative determination of the space-averagedgas volumefraction,



α

,confirms thisvisual ob-servation.Morespecifically,wefind



α

=3,5,11,16and35%for d∗=0.18,0.26,0.39,0.53 and 0.70, respectively. Time variations of



α

areanalyzedinSection4.4.

4. Macroscopicflowcharacteristics

4.1.Variationsoftheemptyingtimewiththeneckdiameterand bottlefillingratio

The emptyingtime, Te,defined asthetime beyondwhich the

measuredpressureinthetopairbuffer,ptop,remainsconstantand

equalto patm,decreases withboththeneckdiameterandthe

ini-tialvolumeofwater,asmaybeseen inFig.7.Thisfiguregathers resultsobtainedwithvariousinitialwaterheightsandneck diam-eters(onepointrepresentstheresultofseveralexperiments).The theoreticalpredictionbyCSisfoundtobeingoodagreementwith observations.Hence,the modelinitiallydeveloped forF=1may beextendedtoarbitraryinitialfillingratiosintheform

Te

3.0FL/



gD≈ d

∗−5/2

, (2)

where3.0L/



gD correspondsto theemptyingtime under condi-tionsd∗=1andF=1(Dumitrescu,1943).Remarkably,thepower law(2)holds fromd∗≈ 0.25 tod∗=0.7,and evendown to d∗= 0.11 provided the initial filling ratio is large enough, typically F0.75. Thatthissimple predictionholds throughoutthiswide

Fig. 8. Evolution of the pressure in the top air buffer for d ∗_{= 0 . 22 and F = 0 . 75 ,} along with the TTL signal triggering the gate opening. The inset provides a closer view to the pressure oscillations during the time interval 22 s ≤ t ≤ 25 s. Green: raw signal; yellow: Gaussian smoothing of the raw signal yielding the equilibrium pres- sure p top(t) . (For interpretation of the references to colour in this figure legend, the

reader is referred to the web version of this article.)

range of relative openings and initial filling ratios indicates that the physicalmechanisms governingthe emptyingprocess remain unchanged,despite thelargedifferencesintheflowstructure ob-served in Fig. 6. For a given relative opening in that range, the emptyingtimeincreaseslinearlywiththeinitial watervolumein the bottle, itself proportional to FLD2_{. This observation suggests}

thatnomemoryeffectisinvolvedintheoverallemptyingprocess, atleastintherangeof_{F investigated}here.Thistrendisalso con-sistent withthe factthat themeanmassflux (whichmaybe de-fined by applyinga suitable slidingtime averageto the instanta-neousliquidflowrate, seeSection 4.2)remains constant through-outthedischarge,asobservedbySchmidtandKubie(1995).

Deviationsfromlinearityarenoticedford∗࣠0.25andF0.75. Forsuch smalld∗,thesmaller_{F the}shorterthenormalized emp-tying time,suggestingthat Te increasesfasterwithF in that

pa-rameter range. As pointed out by CS, this is because the model neglectstheliquidaccelerationclosetotheopeningandthe non-linearity of the pressure and velocity oscillations, both of which become increasingly important in the momentum balance as d∗ andFL (i.e. theoscillatingliquid mass)become small.As d∗ fur-therdecreases,capillaryeffectscomeintoplay.Closetothe block-ing limit (observed totake placefor d∗≈ 0.07,i.e.d=3c,where

c=

(

σ

/

(

ρ

lg

))

1/2≈ 2.7mm is the capillary length), the emptying

processexhibitsanintermittentbehaviorduetothestabilizing in-fluence of surfacetension. While liquid ejection and bubble en-trance alternate regularly most of the time, resting periods may takeplacewithoutanyregularity.Owingtothisintermittency,the emptyingtimeslightlyvariesfromoneexperimenttoanother. 4.2. Evolutionofthepressureinthetopairbuffer

The evolution of the pressure ptop in the air buffer is

dis-playedinFig.8ford∗=0.22andF=0.75.Startingwiththe ini-tial value ptop=patm, the pressure immediately drops after the

gateisopened.Thepressuredropisclosetothehydrostatic pres-sure variation corresponding to the height of the liquid column,

ρ

lgFL. Then, ptop increasesalmost linearlyover time, until it

re-coversits initial value ptop

(

0

)

=patm attheendofthe discharge.

Fig. 9. Evolution of the pressure difference, p top(t) − p atm , vs. the normalized

air volume in the bottle, for F = 0 . 74 and various neck diameters. Each color line corresponds to a different neck diameter, with from dark to light: d ∗₌ 0 . 7 , 0 . 53 , 0 . 44 , 0 . 39 , 0 . 35 , 0 . 31 , 0 . 26 , and 0.22. The dashed line represents the ref- erence linear evolution of the hydrostatic component, −ρl gFL (1 − t ∗) .

oscillatesaboutaslowlyincreasingvalue, ptop,witha

characteris-tic period Tos∼ 0.2 s, much shorter than the emptyingtime (the

orderofmagnitudeofwhichisTe∼ 75sinthiscase).

The slowly varying contribution ptop

(

t

)

is obtainedby

convo-luting the pressure signal with a Gaussian kernel with a stan-dard deviation

σ

G=10/ fp=0.01s. Thispressure component

cor-respondstotheequilibriumhydrostaticpressure,patm−

ρ

lghint

(

t

)

,

andobeystherelation

ptop

(

t

)

− patm=−

4

π

gMw

(

t

)

D2 , (3)

whereMw(t) istheremainingmassofwaterinthebottleattime

t. In the presentcase, p_top

(

t

)

is found to evolve linearly, which indicates that the mean mass flow rate −dMw/dt stays constant

throughoutthedischarge.

As Fig.6 revealed,theneckdiameter hasadeep influenceon thestructureofthetwo-phaseflowthroughouttheemptying pro-cess. However, Fig. 9 shows that all pressure evolutions collapse on a mastercurve whatever d∗,provided time isrescaled appro-priately. The suitable dimensionless time is 1_{− F}

(

1_{− t}∗

)

_, with t∗=t/Te, which corresponds to the normalized volume of airin

thebottleattimet.Indeedthisdimensionlesstimevanisheswhen the bottle isinitially full ofwater andbecomesunity atthe end oftheprocess whenitis entirelyfilled withair.Existence ofthis mastercurveindicatesthattheslowlyvaryingcontributionptop

(

t

)

growslinearlyovertimewhateverd∗,andsuggeststhatthe phys-ical mechanismsdrivingthe long-termevolutionofthe emptying processareindependentofd∗,incontrastwiththeaveragegas vol-umefraction



α

whichstronglydependsond∗.AsFig.10shows, theoverall evolutionoftheslowlyvaryingpressure componentis also independentof theinitial filling ratio.Nevertheless, the col-lapseisnotperfectjustaftertheopening.ForallF,ptop

(

0

)

=patm

butthereturntotheequilibriumpressureis essentiallygoverned byeffectsofcompressibilityanddependsontheinitialvolumeof the air buffer, 1− F. The duration of the transient (analyzed in more detail in Section 5.1) increases withthe openingdiameter, asalreadyseeninFig.4.Afterthistransient,theflowevolutionno longerdependsontheinitialconditions.

The pressureoscillations thatsubsist atlargertimes represent the rapidly-varying contribution to the pressure signal. They are associated with the repeated occurrence of bubbles at the neck

Fig. 10. Evolution of the pressure in the top air buffer, p top ( t ), vs. the normalized air

volume in the bottle, for a normalized neck diameter d ∗_{= 0 . 31 and various initial} filling ratios. Each color line corresponds to a different initial filling ratio, with from dark to light: F = 1 . 0 , 0 . 74 , 0 . 5 , and 0.26.

andcruciallydepend ofthegascompressibility(Meretal.,2018). We found the measured frequency, fos,to be closeto 5Hz in all

cases,withonlyaweakdependenceond∗.Moreover,fos isingood

agreementwiththe predictionofCS underall experimental con-ditionsweconsidered.Accordingtothisprediction,theoscillation frequencydependsonthe gasthermodynamic characteristicsand currentwaterheightas

fCS=



γ

pvpatm

ρ

gL2



hint(t) L



1−hint(t) L



, (4)

where

isafunctionthatdependsonhint(t)/L,d∗andD/L,

ρ

gand

γ

pvdenotingthegasdensityandadiabaticindex,respectively.That

fos is close to fCS in all cases is noticeable,as partof our

exper-imentsare performedwith openingratios well beyondthe max-imum value d∗=0.44 considered by CS. With such large open-ings, large Taylor bubbles dominate the initial transient, asseen inFig. 5.A dense unsteadybubbly flow resultingfromthe quick break-upofthe airvolumeentering thenecktakesplaceintheir wake. Despite this specific flow structure, which strongly differs fromthoseobserved withsmallerd∗,fos isstill correctly

approx-imatedby (4)andremains inbetween4Hz and5Hzthroughout thedischarge.

Beyondtheinitialtransient,pressurefluctuationsgraduallylose theirleft-rightsymmetryoveratimeperiod.Thisisespeciallytrue neartheendofthedischarge,inwhichthesysteminvolvesstrong nonlinearcouplings. Forthisreason, we found appropriate to in-troducearootmeansquare(rms)pressurefluctuation.This time-dependentfluctuationisdefinedas p=



(

ptop− ptop

)

2



11/2,where



·



1standsforalocaltimeaverageovera1swindow,i.e.

approx-imatelyfiveoscillationperiods.Theevolutionofpbeyondthe ini-tialtransientis showninFig.11forneckdiametersrangingfrom d∗=0.18tod∗=0.39andaninitialfillingratioF=0.74.Forlarge enoughd∗, theemptyingtime becomesofthesame orderasthe time window over whichthe average is performed. Forthis rea-son, the evolution of p becomes meaningless for d∗࣡0.39. For lowerd∗,theamplitudeofpressurefluctuationsisseentoincrease withtheneckdiameter.Thisisbecausepressureoscillationsinthe air buffer are associated with the alternation of liquid ejections andbubbleadmissionsthroughtheneck(Mer etal.,2018),which makes their amplitudedependon thesize ofthegenerated bub-bles,hence on d. Twosuccessivestages maybe identified in the

Fig. 11. Evolution of the rms pressure fluctuation p _{for an initial filling ratio F =}

0 . 74 and various neck diameters. Each color line corresponds to a different neck diameter, with from dark to light: d ∗_{= 0 . 39 , 0 . 35 , 0 . 31 , 0 . 26 and 0.22.}

evolutionofpforagivend∗.Thefirstofthem,uptot∗_{≈ 0.8,} cor-respondsto a fully-developed regime characterized by a periodic generationofbubblesattheneck.Duringthisstage,theamplitude ofpdecreasesalmostlinearlyovertimeataratewhichincreases withtheneckdiameter.Forinstance,p decreasesfrom0.2kPaat t∗=0.2to0.1kPaatt∗=0.8ford∗=0.22,whileitgoesfrom0.6 kPa to 0.35 kPa duringthe same time period for d∗=0.39. The originofthislineardecreaseisstillunclear.

A second stage takes place for t∗࣡0.8, during which a rapid decrease of p toward zero is observed. At the very end of the discharge,allpevolutionscollapseontoasinglecurve,indicating thattheamplitudeofpressurefluctuationsnolonger dependson thesize oftheopening.Inthislatestage, theseparationofscales betweenthewaterheight,h_int,andtheneckdiameter,d,nolonger holds.A strongcoupling betweenthe bubblerelease atthe neck andthedisplacement ofthe freesurfacetakes place(see movies accompanyingthepaper).

4.3.Descentofthefreesurface

The evolution of the upper free surface position is shown in

Fig.12 forvarious neckdiameters andan initial filling ratioF= 0.77.Twodistinct stagesare observedwhend∗≥ 0.26.In thefirst ofthem,thepositionofthefreesurfaceremainsroughlyconstant, i.e. the water height does not vary. Small-amplitude oscillations aboutthispositionarenonethelessobservedbuttheseoscillations arequicklydamped.Thisfirststage coincideswiththeriseofthe firstairbubblethroughthewatercolumn.Thevolumeofthis lead-ingbubble,togetherwiththatoftheswarmofsmallerbubblesin itswake(seeFig.5),compensatesforthevolumeofliquidthat ex-itsfromthebottle,leavingh_intunchanged. Whenthefirst bubble reachesthe free surface andbecomes part of the air buffer, the volume ofthe latter increases suddenly. Consequently, hint(t)

ex-hibitsa downward step rightat thisinstant, asFig. 12 confirms. Thiseventdoesnot affect theexiting liquid flowrate. The dura-tionofthis firststage is determinedby the time Tfb requiredfor

thefirstbubbletoreachthefreesurface,which,inafirst approxi-mation,increaseslinearlywithF,similartoTe.ForF=0.77,this

timeisfoundtobeconstantandoftheorderof1.9sford∗≥ 0.31, whichshowsthattherisespeedofthecorrespondingbubbleis in-dependentofd∗whenthediameteroftheopeningislargeenough. Lower d∗ yield slightly larger Tbf. The height of the hint-step is

Fig. 12. Evolution of the normalized vertical position h int / L of the upper free surface

for various normalized neck diameters d ∗_{and an initial filling ratio F = 0 . 77 . Each} color line corresponds to a different neck diameter, with from light to dark: d ∗₌ 0 . 7 , 0 . 53 , 0 . 39 , 0 . 26 and 0.18.

governedby thebubblevolume andisoftheorderof30mmfor d∗₌0_.39_,increasing to 200mm ford∗₌0_.70.The height ofthe step is expected to increase with F, since increasing the initial filling rateincreases the time available forcoalescence events to occuratthebackfaceoftheleadingbubble(seeSection5.2).This first stage exists forall neck diameters.However, ford∗࣠0.2, its durationandtheheightofthestepbecomemuchsmallerthanTe

andFL,respectively,whichmakesthemtoosmalltobeidentified inFig.12.

Inthenextstage,thefreesurfacedescent ischaracterizedbya constantvelocity.Itsevolutionfollowsthesimplelaw

hint

(

t

)

/L=F

(

1− t∗

)

. (5)

Using(2),theverticalvelocityofthefreesurfaceisthenfound toobey Uf s



gd ≈ 1 3d∗2. (6)

It increases with the relative diameter of the opening, from 5mms−1 ford₌20mmto145mms−1ford₌80mm.

4.4. Meangasvolumefractioninthefluidcolumn

Fromtheequilibriumpressureandthepositionofthefree sur-face, we can estimate the mean gas volume fractionin the fluid columnas



α

=1−patm− ptop

ρ

lghint

, (7)

wherehint istheequilibriumheight ofthecolumndeterminedby

convolutingthesignalh_int(t)withaGaussiankernel.Theaccuracy inthedeterminationof



α

isdirectlyrelatedtothatinthe detec-tionofthefreesurfaceheight,i.e.



α

∼

α



hint/hint.Wequantified

thecorresponding uncertaintyby examiningtheinterfacialregion onaselection ofrawimagesandconsidering howhint isaffected

by local fluctuations. The estimated uncertaintywas found to be negligibleatthebeginning oftheemptyingprocessbutincreases significantlywhentheinterfacegetsclosertotheneck.Fort∗≈ 0.8, considering



hint≈ 2cm,hint≈ 16cmand

α

≈ 0.3,theuncertainty

on

α

is estimated to be of the order of 0.04, i.e.



α

/

α

≈ 12%. We used themeasurement technique describedinSection 2.2 to determine the evolution of



α

for various neck diameters (the

photographs displayed in Figs. 5 and 6 provide snapshotsof the correspondingflows).ThisevolutionisshowninFig.13,usingthe normalizedtimet∗.Againweidentifythetwodistinctregimes ob-served inFig. 12.That is, the gas volume fractionfirst increases linearly, which corresponds to the liquid beingejected from the bottleandreplacedby anequivalentvolumeofgas.Oncethefirst bubble has reached the free surface, less gas remains entrapped withinthefluidcolumn,whichleadstoasuddendecreaseof



α

. For d∗=0.7, the burst ofthe first bubble occursatt∗=0.4 and yieldsa50%reductionof



α

.Thisphenomenologytakesplacefor allneckdiameters,astestifiedbythepresenceofakinkinallfour evolutions of



α

corresponding to d∗≥ 0.39inFig.12. Neverthe-less, ford∗࣠0.3the volume ofthe bubble that first reachesthe freesurfaceistoosmalltoleaveasignatureinFig.13.

Beyond this first stage, the meangas volume fraction contin-uously increases over time ata rate whichalso goeson increas-ing. This is because most of the large bubbles contained within the fluid column break up into a large number of smaller bub-bles as they rise. Small bubbles having a significantly lower rise speed thanlarge ones,the volumeoccupiedby thegasphase in-creasesduringtheemptyingprocess.Inaddition,asthedischarge goes on, the overall agitation in the fluid increases. Indeed, the first bubbles generated at the neckrise in a relatively quiet wa-tercolumninwhichthefragmentationrateislow.Incontrast,the subsequentlarge airbubblesevolvewithin amuchmoreagitated medium inwhich fragmentationismore likelyto occur.This en-hanced agitation concurs to produce smaller bubbles which stay longer withinthefluid column,making therateofchangeof



α

increasethroughouttheprocess.

Fig. 13 indicates that



α

isan increasing function ofthe rel-ative neck diameter whatever t∗. This feature is a direct conse-quence of the relative volume of bubbles generated at the neck with frequency fos, which grows like d∗3. Nevertheless, rescaling

the experimental curves by d∗3 doesnot make them collapseon asinglemastercurve.Thereasonforthismayberevealedby set-tingupasimplemodelallowingtheobservedevolutionstobe pre-dicted beyondtheinitial transient,assuming negligible compress-ibilityofthegaswithinthetwo-fluidcolumn.Theevolutionofthe gasvolumefractionwithinthecolumnisdrivenbythedifference between the air flux entering through the neck, q, and the one leavingthroughtheupperfreesurface,Q=

π

D2_U

f s/4.Thegasflux

attheneckisapproximatelyq=

ζπ

d3_/

₍

_6T

os

)

,where

ζ

isanO

(

1

)

pre-factorcharacterizingtheactualsizeofbubblesdetachingfrom the neck. For a given discharge, we keep

ζ

constant, justas Tos

(see Section 4.2). Beyondthe initial transient, thevolume ofthe two-phase column evolves as Vt p

(

t

)

=

π

D2/4



FL− Uf st



. Equat-ingthetime rate-of-changeofthegasvolumewithinthecolumn,



α

m

(

t

)

Vt p

(

t

)

,withthedifferencebetweentheenteringand

leav-ing gas fluxesand integratingover time yields the predictedgas volumefraction



α

mas



α

m=

α

0+ 2 3

ζ

d∗3D/Tos− Uf s FL− Uf st

(

t− t0

)

, (8)

where

α0

istheinitialvolumefractionwhichwe settothe mea-sured



α

at t₌t0=2s, since the model does not account for

the formationofthe bubbleswarm duringtheinitial stage t<t0.

The pre-factor

ζ

isobtainedthrough a least-squarefit ofthe ex-perimentaldata.Modelpredictionsarecomparedwith experimen-talevolutions inFig.14,revealingagoodagreementford∗<0.35. Forlarger diameters,thepredictions deteriorate, presumablyasa consequence ofthehigher levelofagitation within thefluid col-umn.Morespecifically,owingtothisintenseagitation, fragmenta-tionproducesalargernumberofsmallbubbles.Asignificant frac-tionofthemneverreachesthetopairbuffer,asthecorresponding bubblesareentraineddownwardsbytheoutgoingliquidand even-tuallyejectedthroughtheneck.

Fig. 13. Evolution of the space-averaged air volume fraction, α , in the fluid column for F = 0 . 77 and various normalized neck diameters, d ∗_{. Each} color line corresponds to a neck diameter, with from light to dark: d ∗₌ 0 . 7 , 0 . 53 , 0 . 44 , 0 . 39 , 0 . 31 , 0 . 26 , 0 . 22 and 0.18.

Fig. 14. Assesment of the space-averaged air volume prediction (8) for the four smallest d ∗ _{displayed in Fig. 13 . The solid lines and dots refer to αm and α ,} respectively.

5. Earlytimesofthedischarge

5.1. Pressuretransient

The transition fromthe initial state in which the pressure in the air buffer equals the atmospheric pressure, to the regime in whichitgradually oscillatesaround aslowly-evolvingequilibrium hydrostaticcomponent, isillustrated inFig. 15 for d∗=0.44 and F=0.75.This transientischaracterized by an exponential decay oftheamplitudeofoscillations,until theyreach aslowly-varying amplitudeoftheorderof1kPa.Thisinitialtransientmaybeseen astheimpulseresponseofthesystemtothepressuredisturbance generatedbytheinitialcondition,herethesuddengateopening.

As shown inFig. 15, the amplitude of pressure oscillations is accuratelyfittedwithanexponentialfunctionoftheformAe−t/τ+ B, where

τ

is the characteristic decay time, A the initial ampli-tudeofoscillationsandBtheiramplitudeatthebeginningofthe slowly-evolvingregime.Caremustbetakenwhenapplyingsucha

Fig. 15. Pressure oscillations during the initial transient for d ∗_{= 0 . 44 and F = 0 . 75 .} Solid line: recorded amplitudes; diamonds: extrema of oscillations; dots: abso- lute value of the extrema used in the fitting procedure; dotted line: fitting curve

Ae−t/τ_{+ B, with A = 6 . 2 kPa, τ= 0 . 34 s and B = 0 . 8 kPa.}

nonlinearfittingprocedureonascarcedataset.Forsmallneck di-ameters,belowd∗≈ 0.26,thedamping time isof thesame order asthe oscillations periodandthereforecan notbe accurately es-timated.Thisiswhyonlyneckdiametersd∗≥ 0.26areconsidered inthissection.ValuesofBagreequalitativelywiththe experimen-taldeterminationsofpreportedinSection4.2atthebeginningof theslowly-evolvingregime.

The amplitude of oscillations, normalized by the hydrostatic pressuredifference,

ρ

_lg_FL, is displayedin Fig. 16 (left). This fig-ureshowsthatAisentirelydeterminedbytheinitialconditions -thepressureptop

(

0

)

=patmintheairbufferandthefillingratioF

- anddoesnotdependond∗.Incontrast,theneckdiameterdeeply influences

τ

,whichincreasesfrom0.18sford∗=0.26to0.92sfor d∗=0.7.Considering apower law

τ

∝dn_{,nis foundto varyfrom}

n≈ 1.3 ford∗࣠0.4 to n≈ 1.7 for d∗=0.7. which suggeststhat

τ

isapproximatelyproportional tod3/2_{throughoutthewhole range}

ofneck diameters. Note that this exponent isclearly lower than n=2,rulingouttheintuitiveideathatthedecaymightberelated

toviscouseffectsintheneckregion,inwhichcase

τ

wouldscale asd2_/

_ν

l.The d3/2-scaling maybe understood by considering that

the deformability ofthe first bubbles entering the watercolumn dampstheinitialpressureoscillationbyconvertingpartofthe cor-responding energyinto interfacialenergyof gas-liquidinterfaces. Toputthisscenarioonafirmerbasis,considertheisolatedsystem madeoftheairbuffer+liquidcolumn,beforetheveryfirstbubble detachesfromtheneckandtheveryfirstliquidjetisejected out-wards. Assuming theliquid tobe incompressible, the mechanical workinthissystemresultssolelyfromtheactionofthepressure within theairbuffer withvolumeVab

(

t

)

, andmakes thebulk

in-ternalenergyvaryby−ptop

(

t

)

dVabwhenthebuffervolumevaries

bydVab.Assuming thattheupperfree surfaceremains flat,i.e.its

areadoesnotvary,thesurfaceenergy

σ

Ab

(

t

)

associatedwith

air-waterinterfacesinvolvedinthe systemvariesby

σ

dAb,owingto

the work of surfacetension resulting frominterfacial area varia-tionsintheneckregion.Ifdissipationwere negligibleinthis sys-tem,theinternalenergyofthewholesystemwouldbeconserved, implyingthatthenegativeworkcorrespondingtoanexpansionof the air buffer would be exactly balanced by a positive work re-sulting froman increase of the interfacialarea inthe neck, such that

σ

dAb

(

t

)

=ptop

(

t

)

dVab. This iswhere the neck diameter

en-tersthescalingofthedecaytime

τ

.Keepinginmindthatthe cap-illarytimeattheneckscalesas

ρ

ld3/

σ

,wenormalize

τ

by2

π

/

ω

2,

where

ω

2 stands for the radian frequency of the lowest mode

of capillaryoscillations of a nearly-spherical bubble with diame-ter d (so-called l₌2 mode), known to be

ω

2=



96

σ

_/

(

ρ

_ld3

₎



1/2

(Lamb, 1932). As Fig. 16 (right)shows,

τ

isadequately re-scaled inthisway,althoughsomescatterremains,andthedimensionless decaytime

ω

2

τ

/2

π

isofO

(

1

)

asexpected.

The initialtransientmaybe considered to endatt≈ 2.5

τ

,for whiche−t/τ≈ 0.08.AccordingtoFig.4,the timeTfbatwhichthe

first bubble reaches the upper free surface exceeds 2.5

τ

for all neck diameters such that d∗࣠0.5. For t<Tfb, the air buffer

be-haves as a closed thermodynamic system with a constant mass. Itsvolume,_Vab

(

t

)

₌ πD2

4

(

L− hint

(

t

))

,isknownfromtherecordof

thefreesurfacepositionhint(t).Takingintoaccounttheinitial

con-dition ptop

(

0

)

=patm and assuming that air behaves as an ideal

gas,wecomputedpressurevariationsintheairbufferduringthe wholetransientusingtherecordedvariationsofVab

(

t

)

,after

pos-tulatingeitheran isothermaloranisentropicevolution,thelatter withanadiabaticindex

γ

pv=1.4.

Fig. 16. Variation of the normalized amplitude A

ρlgFL of pressure oscillations (left), and decay time

ω2τ

2π of the pressure transient (right), with respect to the normalized neck diameter d ∗_.

Fig. 17. Evolution of the pressure in the air buffer during the first two seconds for

d∗_{= 0 . 44 and F = 0 . 75 . Black: recorded signal; green and blue: predicted evolution} assuming isentropic and isothermal evolutions, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

As Fig. 17reveals, the actual pressure evolution standsin be-tween thesetwo extremeassumptions, butsignificantly closerto the isothermal evolution. A similar comparison was carried out forallexitdiameters,leadingtothesameconclusion.Thisimplies that,followingthescenariodiscussedabove,onlypartofthework resultingfromthe displacementofthe upperfreesurfaceis con-vertedintosurfaceenergyattheneck,whiletherestisdissipated

inthe bulk andatthe walls. As an aside,the fact that the pres-surepredictionbasedontheexperimentaldeterminationofhint(t)

isconsistent withthe recordedsignal ptop(t) alsoprovides an

in-direct check of the consistency of the pressure and free surface heightmeasurements.

5.2.Firstrisinggasentity

The evolution of the gas volume fraction discussed in

Section 4.4 revealed the influence of the first rising gas entity (hereinafter abbreviated asFRGE) on the globalevolution of the system,mostnotablyforlarged∗.Thisentitydiffersfromall bub-bles subsequentlyemitted atthe neck,as itis the onlyone that rises in a quiescent liquid. This distinctive condition deeply af-fectsits shape, stability andrise speed. Toget more insight into thesefeatures, theinterface height detectionalgorithm described inSection2.2wasadaptedtoalsodetectthesecondpeakpresent intheverticalpixelintensitydistribution,asitcorrespondstothe apex of the FRGE. The position of the tail of the FRGE was also detected in configurations with d∗≥ 0.44.Fig. 18 shows how the geometryoftheFRGE varieswiththeneckdiameterby thetime itsapexreachesthemedianhorizontalplane ofthebottle.Inthis figure,theinitialfillingrateiskeptfixed,withF=0.77.

Whatever d∗>0.18, it turns out that the FRGE takes the form of a spherical cap bubble or a Taylor bubble, the vol-ume of which significantly increases with d∗. When d∗≥ 0.39, the length-to-diameter ratio of the Taylor bubble is of O

(

1

)

or more. The apex rise speed, Vap, was calculated for vertical

po-sitions 0.44≤ z/L≤ 0.69, then averaged over this range. Fig. 19

showsthatVap≈ 0.36ms−1 whatever d∗.Based on this estimate,

thecharacteristicReynoldsnumberoftheFRGE,Reap=VapD/

(

2

ν

l

)

,

is approximately 2× 104_{. Spherical cap bubbles rising in an}

Fig. 18. Visualization of the first rising gas entity (FRGE) by the time its apex reaches the vertical position h/L = 0 . 5 , for F = 0 . 77 and various neck diameters. The time required for the FRGE to reach h/L = 0 . 5 is specified in each image. The cyan, red and magenta lines indicate the detected free surface, FRGE apex and back face position, respectively. The configuration corresponding to d ∗_{= 0.18 was not reported because the coalescence rate is too low for bubbles periodically emitted at the neck to generate} a spherical cap bubble.

Fig. 19. Rise speed of the FRGE apex for various normalized neck diameters. Solid line: V ap = 0 . 34



gD . Inset: shape of the front part of the corresponding FRGE in

Fig. 18 with, in black, the leading-order term r = Dzap in the potential flow so-

lution (lines have been shifted vertically to improve readability, and horizontally to make the apex coincide with the tube axis in all cases). Colors specify the cor- respondence between the front shape (inset) and the rise speed (top curve), with from dark to light: d ∗_{= 0 . 22 , 0 . 31 , 0 . 39 , 0 . 53 and 0.70.}

unbounded domain or in tubes have been widely studied since

Dumitrescu(1943)andDaviesandTaylor(1950)(whoconsidered bubblesizesyieldingReynoldsnumbersuptoReap≈ 1.2× 104);see

thereviewsby Wegener and Parlange (1973)and FabreandLiné (1992). The bubblerise speed is knownto scale withthe square rootofthecurvatureradius atthe apex.ThatVap is independent

ofd∗suggeststhat allbubbleshaveasimilar curvatureinpresent experiments.This is confirmedin the insetof Fig. 19, wherewe superimposedallfrontshapescorrespondingtobubblesshownin

Fig.18,extractedbyacontourdetectionalgorithm.Thecrucialrole ofconfinementby thelateral wall isemphasized by the leading-orderpotentialflowsolution,whichpredictstheshapeintheapex regiontobeclosetor=



Dzap,withzaptheverticaldistancefrom

the apex (Davies and Taylor, 1950; Héraud, 2002; Clanet et al., 2004). Thissolution makes it clear that thebubble shape, hence Vap,doesnotdependond∗.

Aminimumgasvolumemustbepresentinthetwo-phase col-umnforasphericalcaporTaylorbubbletoform.Thiscritical vol-umeisavailable since theveryfirst bubbleformsattheneckfor large enough d∗.Conversely, it isreached only afterseveral coa-lescence events for smallneck diameters (see the accompanying movies andcompare those corresponding to d∗=0.18 and0.31). Thisiswhynosphericalcapbubblewasobservedwithd∗=0.18, owing to the insufficient number of such events allowed by the limited height of the device. That successive coalescence events are requiredto generatea spherical cap bubble hasan influence onthetime requiredforthe FRGE toreach agivenvertical posi-tion.AsFig.18indicates,thistimeincreaseswithdecreasingd∗,in linewiththesmallercoalescencerateobservedforsmalld∗.

A simple model may be set up to get some insight into the growth of the Taylor bubble due to the coalescence of small bubblesatits back.Theidea istoconsiderthefluidcolumnwith volumeVt p=

π

D2FL/4atthe bottomofwhichsmallbubblesare

injected periodically, and examine the gas flow rate exchanged betweenthe two markedlydifferent bubbles populations present in this column, namely the large spherical cap or Taylor bubble and the swarm of small bubbles. The large bubble is character-ized by the vertical position of its apex, hap(t), and back face,

hba(t). As shown in Fig. 20 (left), these positions evolve with

differentspeeds, depending ond∗.Based on thesetwo positions, the volume of the large bubble, VT, may be approximated by

consideringthatitconsistsofahalfsphereandacylinder,sothat VT=

π

D2/4



hap− hba− D/2



+

π

D3_{/12.Small bubblesare located}

inthe rearvolume, _Vr=

π

D2hba/4,andoccupyan actual volume

Vsw=



α

sw



Vr, where



α

sw



denotes the gas volume fraction in

theswarm.Knowingthe twopositions,hap(t)andhba(t),thetotal

volumeofthecolumn,_Vt p

(

t

)

,andtheglobalgasvolume fraction,



α

(t), makes it possible to determine Vsw=



α

Vt p− VT, hence



α

sw



=Vsw/Vr. The evolution of



α

sw



(t) is reported in Fig. 20,

whereitmaybecomparedwiththatof



α

(t)whichriseslinearly during this stage. As time proceeds, the swarm volume fraction exhibits a clear decrease on which oscillations resulting from fluctuations (especially along the lateral wall) in the position of theback faceofthelargebubblesuperimpose.Thesefluctuations reflectthechaotic sloshingofthisbackfaceinduced bytheburst oftrailingbubbles.Toovercomethecorrespondinglargepotential uncertaintyinthemeasurementofh_ba(t), wefittedthedatasoas toretain onlythedominanttrendineachseries.Itisnoteworthy that for t→Tfb,



α

sw



(t) consistently decreases down to values

closeto those measured for



α

(t) rightaftert₌T_{f b} (Fig. 13), as conservation of the gas volume requires. Assuming that bubbles periodicallyinjectedattheneckfeedtheswarmofsmallbubbles but not the large bubble itself because its back face stands al-readyfarfromtheneckduringthe sequenceunderconsideration (hba≥ 15cmaccordingtoFig.20),wemaywrite

dVT

dt =Csw→T, (9)

dVsw

dt =Se− Csw→T, (10)

whereSeisthesourcetermresultingfromtheperiodicadmission

ofair atthe neck,and_Csw→T is the volume flowrate exchanged

between the swarm and the large bubble, owing to coalescence and break-up events happeningin the wake of the latter. Using linear or cubic polynomial fits of hap(t),



α

(t) and hba(t),

espe-cially to remove fluctuations in the latter, (9) and (10) may be employedto determinethesource/sinktermsSe

(

t

)

andCsw→T

(

t

)

.

Corresponding predictionsare reported inFig. 20.The prediction forSeisfoundtobewithin10%oftheexpectedtheoreticalvalue,

π

d3_/(6T

os), giving credit to the post-processing procedure. The

exchange term Csw→T

(

t

)

is always positive, which ensures the

growthof theTaylor bubble.Inall cases,Csw→T isinitially larger

than Se. Presumably, the reason why the coalescence rate is so

largeduringthisveryearlystagestandsinthestrongacceleration experienced bysmallbubbles entrappedinthewakeofthe large bubble and sucked toward its back face. Due to the imbalance betweenCsw→T

(

t

)

andSe

(

t

)

atearly time, thegasvolume inthe

swarm decreases, despite the gas flow rateentering the column. As the largebubble rises, Csw→T

(

t

)

decreases, making



α

sw



vary

muchmoreslowly.AsFig.20shows, thelargerd∗,thehigherthe initial value of _Csw→T

(

t

)

is. This is whythe final volume of the

Taylor bubble increases with the normalized neck diameter, as doesthe magnitudeof thejump experienced by the freesurface andthegasvolumefractionatt=Tf b(Figs.12and13).

6. Summaryandconcludingremarks

Thisstudyprovidesnewexperimentalinsightsintotheflow re-sulting fromthe emptyingof avertical cylindrical bottle initially filled with water up to a certain level. The oscillatory nature of this flow, resulting from the coupling between the compressible top airbuffer andthe two-phase columncrossed by gas bubbles generatedatthe neck,wasexamined indetail byCS. Present ex-perimentswere performedwithsimilarneck-to-cylinderdiameter

Fig. 20. Influence of the coalescence of small bubbles on the growth of the initial Taylor bubble ( t ≤ T fb ) for the largest three neck diameters, with from light to dark: d ∗=

0 . 7 , 0 . 53 and 0.44. Left: evolution of the apex position, h ap ( t ) (dashed line), and back face position, h ba ( t ) (dots) (as h ap does not depend on d ∗, its evolution is only reported

for d ∗_{= 0 . 7 ); solid line: fit used to determine the sink and source terms involved in the model. Center: evolution of the total gas volume fraction, α ( t ) (dashed line), and} swarm volume fraction, αsw ( t ) (dots); solid line: fit of experimental data. Right: fit-based estimates of the sink and source terms involved in the model, normalized by

πd3 /(6 T

os ); solid line: source term S e(t) due to the periodic admission of bubbles at the neck; dashed line: volumetric flow rate C sw→T(t) exchanged between the Taylor

bubble and the trailing swarm.

ratiosd∗,butsmallercylinderheight-to-diameterratios,L/D, com-pared to these referenceexperiments. The initial filling ratio,F, was varied over a wide range.The operating conditions selected forL/D and_{F allowed}usto magnifythe relativeduration ofthe initialtransientstageofthedischarge.

We confirmed therobustnessof the flow oscillatory behavior. Thatis,whatevertheneckdiameter,thedominantresponseofthe flowtotheneckopeningisperiodic,evenduringtransients involv-ing the riseof largeTaylor bubbles.Although thepressure signal becomesmorecomplexwhend∗increases,theperiodTos remains

correctlypredictedbythemodelderivedbyCS.Providedd∗࣡0.25, the globalemptyingtime, Te, isalso well predictedby (2)

what-everthe initialfilling ratio,whichextendsthe originalprediction bythesameauthorstoarbitraryvaluesofF.

Theinitialstageoftheflowinvolvestwodistinctbutpartly(or, forlargeexitdiameters,almostentirely)superimposedtransients. The firstof them,alsothe fastestinmostcases,resultsfromthe initialexpansionofthetopairbuffer,thevolumeofwhichhasto adaptinstantaneously toanewpressureconditionwhenthegate opens. We showed that, duringthis first transient, the evolution ofthepressuredisturbanceintheairbufferresultsfromthe com-bined effect of thehydrostatic pressure head,

ρ

lgFL (which

gov-erns its amplitude), and the deformability of bubbles that form atthe neck(which governs itsdecay andimposes a d∗3/2-scaling tothecharacteristicdampingtime).Thusthepressuredisturbance ensures a complete coupling of the flow, from the neck to the top airbuffer, andits dynamics involvesgravity andcapillary ef-fects, together withtwowidely differentspatialscales,FL andd. As the mass enclosed in the air buffer remains unchanged dur-ing this early stage, the upper free surface andthe pressure ad-just themselves,so as to ensure an approximate isothermal evo-lution. The second transientresults fromthe riseof the firstgas entitywithinthefluidcolumn.Thisentitymaytake theformofa stringofwobblingbubblesoralargeTaylorbubble,dependingon d∗.Asthisgasentityrises,therelativegasvolumewithinthefluid columnprogressivelyincreaseswhileliquidisejectedthroughthe neck, allowing the total volume of thecolumn to remain almost unchanged.Ford∗࣡0.2,thefirstrisinggasentityexhibitsa spher-icalcapshape,eventuallyturningintoalargeTaylorbubble,aftera

sufficientnumberofcoalescenceeventshavetakenplace.Assoon asits frontradiusofcurvaturebecomesoftheorderofthe cylin-der radius, this bubblerises witha constant speed whatever d∗. During its growth, andfor large enough d∗, we could determine thegasvolumefraction



α

sw



inthebubbleswarmtrailedby the

largebubble.Weset upa simplemodeltoexplain thegrowthof theTaylorbubblevolume,usingasimplemassconservation argu-mentgivingaccesstotherateatwhichairistransferredfromthe swarm to the large leading bubble. The corresponding exchange volumeflux,_Csw→T,isalwayspositiveanditsmagnitudeincreases

withd∗, reflecting the largernumber of topologicalchanges that takeplacewhentheexit-to-cylinderdiameterratioincreases.

Beyond this initial transient, a slowly-evolving discharge pro-cesstakesplace.Themeanpressureinthetopbufferincreases lin-earlyovertime,whilethemeanliquidmassdecreaseswitha con-stantrate. High-frequencyoscillationsinduced byperiodic bubble formationatthenecksuperimposeontheselineardynamics.Their amplitude decreases linearly over time during most of the dis-charge. However,duringthe verylast stage, thisamplitudetends tozeroatad∗-independentrate, presumablybecausethe remain-ingoscillationsresultfromadirectcouplingbetweentheneckand thelocal fluctuationsof the free surfaceposition.For a givenF, contrasting gas-liquid flows typologies were observed inthe liq-uid region, depending on d∗. The flow structure was also found tostronglyevolveovertime,withmarkeddifferencesbetweenthe transientstageandthesubsequentslowly-evolvingstage.

A natural extension of this work would consist in examining in detail the bubble formation process at the neck throughout the discharge. It would in particular shed light on the sensitiv-ityofthisprocess tothe internal agitationofthe two-phase col-umn,whichgraduallyincreasesduringthedischarge,owingtothe chaotic motionwithin thebubble swarm andthesloshingof the free surface. Another aspect worthy of investigation would con-sistinexamining theinfluenceoftheneck’slength.As shownby

Tehrani et al. (1992), increasing this length may change the dy-namicsof oscillations by addinga largepressure loss inthe sys-tem.Howthesechanges arereflectedinthe dynamicsofbubbles releasedfromtheneckisunknownandcouldbeinvestigatedwith themethodsemployedhere.