Time-resolved measurement of concentration fluctuations in a confined bubbly flow by LIF

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Time-resolved measurement of concentration

fluctuations in a confined bubbly flow by LIF

Elise Alméras, Sébastien Cazin, Véronique Roig, Frédéric Risso, Frédéric

Augier, Cécile Plais

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Eprints ID : 15871

To link to this article : DOI:10.1016/j.ijmultiphaseflow.2016.03.011

URL :

http://dx.doi.org/10.1016/j.ijmultiphaseflow.2016.03.011

To cite this version : Alméras, Elise and Cazin, Sébastien and Roig,

Véronique and Risso, Frédéric and Augier, Frédéric and Plais, Cécile

Time-resolved measurement of concentration fluctuations in a

confined bubbly flow by LIF. (2016) International Journal of

Multiphase Flow, vol. 83. pp. 153-161. ISSN 0301-9322

Time-resolved

measurement

of

concentration

fluctuations

in

a

confined

bubbly

flow

by

LIF

Elise

Alméras

a,b,∗

_,

_Sébastien

_Cazin

a

_,

_Véronique

_Roig

a

_,

_Frédéric

_Risso

a

_,

_Frédéric

_Augier

b

_,

Cécile

Plais

b

a Institut de Mécanique des Fluides de Toulouse (IMFT), Université de Toulouse, CNRS-INPT-UPS, Toulouse, France b IFP Energies nouvelles, Rond-point de l’échangeur de Solaize, BP 3, Solaize 69360, France

Keywords:

Confined bubbly flow Mixing

LIF and spectrometry

a b s t r a c t

The present work investigates the mixing of a low-diffusivity dye in a swarm of bubbles at high Reynolds numberconfinedinaHele-Shawcellforgasvolumefractionsrangingfrom1.4to5.4%.Apatchofa fluo-rescentdyeisinjectedwithintheswarmand,duringitsmixing,itsconcentrationismeasuredatagiven locationinanobservation volumeof4.5mm2 _{bymeansofLaserInducedFluorescenceatafrequency}

of250Hz.Aspectrometerisusedtoanalysethelightissuedfromtheobservationvolumeand to dis-tinguishthefluorescedlightfromotherlightsources.Simultaneously,thebubbledistributionaroundthe observationvolumeisimagedwithahighspeedcamerasynchronisedwiththespectrometerinorderto assesstheLIFtechniqueinbubblyflow.Thankstothegoodtimeresolution,rapidand intense concen-trationfluctuationscorrespondingtodyepatchespassingthroughtheobservationvolumearerecorded andaresuperimposed toaslowglobalevolution.Thisslowglobal evolutionshowsfirstanincreaseof theconcentrationandthenanexponentialdecreaseduetothemixingbybubble-inducedagitation.This exponentialdecay,whichisincompatible withadiffusionprocess,isconsistentwiththe transportby dyecaptureinbubbleswakesthatarequicklydampenedbytheshear-stressatthewalls.Theone-point statisticsoftheconcentrationfluctuations(probabilitydensityfunctionandspectrum)alsopointoutthat mixinginaconfinedbubblyflowisintermittentandconvective.

1. Introduction

Bubbles columnsare commonlyused forchemicaland biolog-ical processes to enhance the mixingof chemical species by the bubble rising motions without using mechanical devices. Macro-scopic mixing times are usually estimated by correlating tracer concentrationsmeasuredontheglobalsystemoronseveral loca-tions (Pandit andJoshi, 1983). Macroscopicdescription of mixing haslimitations. Forinstance,recently, McClure etal.(2015),have pointed out the difficulty to identifythe impact ofthe measure-mentlocationonthemacroscopicmixingtime.Alocaldescription of mixing in bubbly flows is thus preferable butis still missing. Eventhemainmechanismsremainunclear.Thislackofknowledge mainly comes from thedifficulties to perform precise concentra-tionmeasurements inbubblyflows. Asolutiontogetaround this problemconsists instudying mixinginan array of fixed spheres mimicking a bubble swarm (Besnaci, 2012; Besnaci et al., 2010).

∗ _{Corresponding author at: Physics of Fluids Group, Faculty of Science and Tech-}

nology, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherland.

E-mail address: e.o.almeras@utwente.nl (E. Alméras).

WhiteandNepf(2003)and TaninoandNepf(2008)havealso stud-ied mixingin randomarrays ofobstacles inthe context of envi-ronmentalfluid mechanics. These studies underline the presence oftwomainmechanismsinvolvedinthemixingofaweakly diffu-sivedyeinflowsatrelativehighReynoldsnumbers.Thefirstone isdueto thedirectinteractions ofthe dyewiththeobstacles. In particular thedye can be caught inthe wakes and also strongly twisted.The second mixingmechanismis thespecific turbulence inducedby interactions betweenthe wakes(Amoura,2008;Risso etal.,2010;Ribouxetal.,2013).Inabubblyflow,thetwomixing mechanismsdescribedpreviouslyarestillpresentandthemobility ofthebubbleshastobetakenintoaccounttoo.Therelative con-tributions ofthese various mechanisms maydepend on theflow configuration, on the structure of the wakes and of the random agitationinthe liquidphase. Ina bubblyflow confinedin athin gap,where turbulencecannotdevelop, mixingturnsout toresult fromdyecaptureinthewakesandtobe incompatiblewitha dif-fusionprocess (Bouche etal., 2013). Onthe contrarymixingin a three-dimensionalhomogeneousbubblyflowisduetoturbulence induced by bubbles and can be modelled by an equivalent Fick law (Alméras et al., 2015). Up to now the description of mixing

ina bubblyflow wasrestricted tolow acquisition frequencyand nomeasurementofconcentrationfluctuationshasbeenperformed asfarasweknow.Theissueishoweverimportanttohavedeeper knowledgeofmixingmechanisms.

The purposeof thisstudy is twofold.From the point of view ofthemetrology,weintroduceanoriginalexperimentaltechnique devotedtolocal concentrationmeasurements ina bubblyflow at high acquisition frequency. This technique is based on Laser In-ducedFluorescence(LIF)andusesaspectrometer inorderto dis-tinguish the fluoresced light from other light sources. Previous techniquesusedinourteamto investigateconfinedbubblyflows are based on a global lighting of the cell, which prevents them frombeingapplied tothree-dimensionalbubblecolumnsbecause ofthe optical light occultationcaused by the bubbles. Since the presenttechniqueonlyconsidersasmallmeasurementvolumeby means ofthe use ofoptical fibres,its extension to three dimen-sionalcasesispossible.Somelightdistortionsoftheconcentration field are neverthelessstill present. In the presentwork, we have focusedonaconfinedbubblyflowinordertobeableto simulta-neouslyusethe newlocal measurement technique together with aflowvisualizationbymeans ofa camera,whichisonlypossible inatwo-dimensionalcolumn.Bythisway,ithasbeenpossibleto developasignalprocessingmethodthatallowedustoremove un-ambiguouslythespurious partsoftheconcentration signalwhich wereperturbed by bubblespassing closeorthroughoutthe mea-surementvolume. Thanks to thismethod, reliable measurements ofconcentrationfluctuationswithinaconfinedbubbleswarmhave beenobtained,andforthefirsttime,PDFsandtimespectraof con-centrationfluctuationshavebeenmeasuredina bubblyflow.The applicationofthepresenttechniquetoathree-dimensional geom-etryisleftforafuturework.

Fromthepointofviewofthephysics,theresultshaverevealed theroleoftheintermittenttransportbythebubblesandexplained whythemixingina two-dimensional bubblyflow cannotbe de-scribedbyapurediffusionprocess.

The paper is organisedas follows.The Section 2 presentsthe experimentalset-upwhilethe Section 3describesthe instrumen-tationusedtomeasuretheconcentration.In Section4the valida-tionprocedure ofthe concentration measurementis described. A physicalanalysisofmixinginaconfinedbubblyflowisperformed in Section5andconclusionsaregivenin Section6.

2. Experimentalset-up

The mixing experiments take place in a thin cell which is presented in Fig. 1. The cell is composed of two glass plates of 400 mm wide, 800 mm high and separated by a gap of width

w=1mm.Itisfilledwithanaqueoussolutionofmagnesium sul-fate(MgSO4).Saltofmagnesiumisfirstdissolvedindistilledwater

ataconcentration of5× 10−2 _{mol/Landmixedup usinga}

mag-neticstirrer. Thesolution is then filteredin order toremove the residualsolid. Adding a smallamount of saltavoids the bubbles coalescencewithout affectingsignificantly thephysical properties ofthedistilledwater(Boucheetal.,2013).Theliquidisinitiallyat restandbubblesaregeneratedbycapillarytubesofinnerdiameter 0.6mm whichareregularly distributedatthebottomofthe cell. Thegasinjectorsareconnectedtoalargeenoughpressurizedtank inorderto providea stationarygas flow rate, whichis variedto adjustthe gasvolumefractionfrom1to 5.4%.Theaveraged two-dimensionalequivalentdiameterofthebubblesd=√4A/

π

(where

Aistheprojectedareaofthebubbleonthecellplan)variesfrom 3.9 ± 0.4 mm to 4.2 ± 0.4 mm asthe gas volume fraction in-creasesinthisrange(Boucheetal., 2012).Thebubblediameteris thereforelargecomparedtothegapsize(w_d ≪ 1)andbubblesare flattened.Byvarying thegas volumefraction, the Reynolds num-berdefinedbyRe=ρVνd (whereVistherisingvelocityofasingle

Laser Op!cal fiber Highpass filter (570 nm) Backlight Camera w=1 mm Dye injector Gas injectors 800 mm x z y

Fig. 1. Experimental set-up and instrumentation. The lighted volume is denoted by the small cylinder (in green), the observation volume by large cylinder (in orange), the field of view of the camera by the parallelepiped (in red). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

bubble,

ρ

isthe densityofthe liquidand

ν

its kinematic viscos-ity)evolvesfrom450 to500 andtheReynoldsnumber basedon thegapRe

(

w_d

)

2_{goesfrom25to30.SinceRe≫ 1andRe}

₍

w

d

)

2≫ 1, theflowis dominatedby inertiaandthusdiffers fromthe classic Hele-Shaw regime. The liquid perturbations induced in the bub-ble wakes are localised justbehindthe bubbles andare strongly attenuated by theshear stress at sidewalls (Roig etal., 2012). In thatcase,collectiveinstability andturbulencecannotdevelopdue totheconfinement(Boucheetal.,2014).

Mixing experiments are carried out 400 mm above the gas injectors where the gas volume fraction is homogeneously dis-tributedandthebubbleshavereachedtheirterminalvelocity.They consist in injecting a passive fluorescent dye within the swarm through a long capillarytube of inner diameter0.6 mm (Fig. 1). A volume of 1 mL of rhodamine WT at an initial concentration

c0=10−4 mol/Lisinjectedduring2sbyavolumetricpump.

Dur-ing theinjection,a jet ofdyeis generatedanddampens approxi-matelyinaroundoneviscoustime,Tv= w2

ν =1s.Afterattenuation

ofthejet,mixinginducedbybubbles canbestudied(Fig.2).The time evolution of the concentration in an observation volume of 4.5mm3 _{ismeasured usingLaser InducedFluorescence(LIF)and}

spectrometry. For3gas volume fractions (

α

= 1.4, 2.8and 5.4%), concentrationsignalsarerecordedat3differentdistancesHfrom thedyeinjectortip(H=5,10and15cm).Foreachcase,5 experi-mentshavebeenperformedtoimprovethestatisticalconvergence oftheresults.

It is worth mentioning that the advective transport of dyein thecellismainlyachievedbyavelocityfieldparalleltotheplane of thecell dependingon the coordinatein the direction perpen-diculartothatplane.Inparticular,thefrictionatthewallsandthe velocitygradientthroughoutthegapplayamajorroleinthe struc-tureofthe bubblewakes asshownby Roig etal.(2012), Bouche etal.(2014) and Filellaet al.(2015).Withina viscous time scale

w2_/(4

_ν

_{),thein-gapvelocityprofilegeneratedby abubblepassage}

Fig. 2. Image of the mixing process. α= 5 . 3% .

Moreover, the measurement technique described below doesn’t giveanyinformationaboutthedistributionofthedyeinthedepth direction.Itonlyprovidesameasurementoftheconcentration av-eragedoverthegap.

3. ConcentrationmeasurementbyLIFandspectrometry coupledwithimaging

Measurementofconcentrationinbubblyflows byoptical tech-niques mustbe carefullycheckedto avoidoptical biasduetothe presenceofbubbles.Concentrationmeasurementshave,thus,been carried out by means of two independentand synchronized sys-tems that provide complementary information about the precise meaningofthemeasuredsignals.

Thefirstone isbasedonLIFmeasured byspectrometry.A flu-orescentdye, therhodamine WT,isexcitedby acontinuouslaser ofwavelength532nm(YAG532,10M,Lasiris)placedononeside of the cell (Fig. 1). The laser beam illuminates a cylindrical vol-umeof0.5mmdiameterand1mmdepththroughthecellwhich iscalledhereafter thelightedvolume.The absorptionspectrumof the rhodamine WTranges from480 to590 nm andits emission spectrumrangesfrom560to680nm(Fig.3).Thefluorescedlight isthen collectedinacylindricalvolumeof2.4mm diameterand 1mmdepthnamedobservationvolumebyanopticalfibredisposed ontheother sideofthecell(Fig.1).Asthelaserbeamisdirectly pointedinthedirectionoftheopticalfibre,ahigh-passfilterwith acut-off wavelengthof570nm(OG570,MellesGriot)isplaced be-tweenthecellwallandtheopticalfibretoprotectthe spectrome-tersensorfromthelaserbeam.Theopticalfibreisconnectedtoa spectometerUSB2000+developedbyOceanOptics.The spectrom-eterisusedtomeasurethespectrumofthecollectedlightE(

λ

,t) ateachtimetintherangeofwavelengths

λ

from340to1030nm witha resolution of0.3 nm/pixel anda maximal acquisition fre-quencyof 1kHz. Practically,our measurementswere recorded at anacquisitionfrequencyequalto250Hzwithanintegrationtime equal to 4 ms in order to get enough photons to ensure a reli-ablemeasurement.Theconcentrationofthedyeisthencalculated by integrating, foreach time step, the spectrum of the collected lightinarangeofwavelengthsincludedintheemissionspectrum of rhodamine WT(Fig. 3). Byusing a spectrometer, the range of

Fig. 3. Absorption and emission spectra of rhodamine WT, and emission spectrum of the led panel used for the imaging technique. The marked spectral band corre- sponds to the part of the spectrum of the collected light used to determine the concentration measured by spectrometry.

Fig. 4. Comparison of the signals measured by the spectrometer (black) and by imaging (gray) . α= 3 . 7% . H = 5 cm.

wavelengthsfor theintegration ofthe emission spectrum canbe selectedverypreciselyandcanevenbechosenafterwards.In prac-tice,theconsidered rangeisrestrictedto570–610nm becauseof thepresenceoftheredLEDpannelusedtovisualizetheflowfield aroundtheobservationvolume(Fig.3).Providedthatthedye con-centration is sufficiently low (see next section for more details), the instantaneous concentration is calculated by the following expression:

c

(

t

)

=K

! λ=610nm

λ=570nm

Fig. 5. Impact of the presence of bubbles in the vicinity of the measurement volume on the measurement. Bubble outside the measurement volume (A), Reflection of fluoresced light on the bubble interface (B), Occultation of the lighted volume (C). The observation volume is denoted by the orange circle on the image, the lighted volume may be seen at its centre on the image in cases A or B.

wherecistheconcentrationofthedyeintheobservationvolume attimet,Kisacoefficientdeterminedbycalibration(seenext sec-tion) and E is the spectrum of the light collected by the optical fibre.

The second system combines LIF and shadowgraphy and is based on imaging. It allows to visualize the bubbles around the

observationvolumeandsimultaneously provide informationabout theconcentration of thefluorescent dye. Ared LED panel is dis-posed on the same side as the laser (Fig. 1). It illuminates the field around the observation volume and reveals the bubbles as shadowsontheimages.Thespectrumofthelightemittedby the LEDpanelshowsapeakat633nmandinvolvesonlywavelengths largerthan610nmwhichdonot exciterhodamineWT(Fig.3).It isalsowellseparatedfromthemaximumofintensityofthe emis-sionspectrumlocalisedat587nm.Integratingtheemission spec-trumonlyintherangeofwavelength570–610nm(Eq.1)ensures thattheLEDpaneldoesn’tdisturbtheconcentrationmeasurement by the first system using spectrometry. On the opposite side of thecell,acamera(PhotronAPXRS3000)witha105-mmoptic im-agesafieldofview of2× 2 cm2 _{around theobservationvolume}

(Fig.1).Ahigh-passfilterwithacut-off wavelengthof540nm is placed infront ofthe camera to protect it from the laser beam. Thecamerathus registerslight emittedby LIFandlight fromthe LEDpanel, eventually occultedby bubblepassages. The

measure-ment by imaging givestwo different pieces of information.First, the concentration inside the observation volume is measured by summingthegreylevelsofeachpixelsresultingfromtheLIFlight intensityand fromthe constant light intensityof the LED panel. Second,the bubbledistributionaroundthe observationvolumeis visualizedbyshadowimaginginaregionof2× 2cm2_illuminated

byaconstantlightintensityoftheLEDpanel.

4. Validationoftheconcentrationmeasurementsprovidedby spectrometry

Fig. 6. Concentration measurements by spectrometry: raw data (gray), after signal processing (black). α= 2 . 4% . H = 10 cm.

Fig. 7. C oncentration measurement by LIF and spectrometry with an homogeneous concentration of rhodamine WT at c = 10 −4 mol/L: raw data (gray), after signal pro-

cessing (black). α= 5 . 4% .

t=4s ort=5 s forexample)butare interrupted by strong up-wardordownwardsuddenpeakswhichhaveacharacteristictime of approximately 0.05 s. This time is very close to the duration of a bubble passage (T_b= d

V ≈ 0.04 s),which suggeststhat these strongpeaksareduetothepassagesofthebubblesinthe obser-vationvolume.Thisisconfirmedbythesimultaneousexamination of both concentration signalsandthe distribution ofthe bubbles around the observationvolume, which is denoted by the orange circle in Fig. 5. The distribution of the bubbles is given atthree differentinstantsdenotedAtoC.ThecaseAcorrespondstoan in-stantbetweentwo strong maximaandminima,the caseB(resp.

Fig. 8. Calibration of LIF measurements towards dye concentration. ( I cref is the in-

tensity of the fluoresced light obtained at the concentration c re f = 2 × 10 −4 mol/L).

Fig. 9. Illustration of the two timescales present in the concentration signal. Sig- nal of concentration (black). Fitting of the mean concentration by an exponential relation ( Eq. 3 ) (blue). α= 5 . 4% . H = 5 cm.

Fig. 10. Exponential decrease time as a function of the gas volume fraction for vari- ous elevations of the measurement volume. Dash lines are fits by k / α, with k = 13 . 7 s for H = 5 cm, k = 21 . 5 s for H = 10 cm and k = 41 . 9 s for H = 15 cm. (Error bars correspond to the standard deviation of individual measurements).

thebubbleinterface (caseB). Thesestrong reflectionsand refrac-tionseffectsarelocalizedonfewpixelsonthebubblesurface(see greenarrowintheinsertof Fig.5B).Theconcentration measure-ment by imagingis thus not able to see them unlike the spec-trometer. A strong downwardpeak is caused on both signals by theoccultationofthelightedvolumebyabubble(caseC).Wecan thereforeconcludethatthesestrong peaksare notrepresentative ofrealfluctuationsofthedyeconcentrationandthatan appropri-atesignalprocessinghastobedevelopedtoremovethem.

The signal processing aimsat localizingthe temporalposition of the spurious peaks. This operation is performed by calculat-ing the temporal derivative of the raw signal. As the temporal derivative generated by the strong peaks are much larger than those from real concentration fluctuations, they are detected by applyingathresholdtothesignal derivative.Athresholdequalto ±2× 10−4 _molL−1_s−1 _{wassuitableforallconsideredvaluesof}

_α

andH.A part of the signal extending ±0.012 s around each de-tected extremum hasalso been removedin order to localisethe entire part of the signal corresponding to spurious peaks. Both thethreshold andthe time extension of ±0.012s have been de-terminedfromthe comparisonof thespectrometer rawsignal to theimagesobtainedbyshadowimaging.Byinspectingtheimages, itis possible to detect precisely thestart and the endof a bub-blepassingthrough themeasurementvolume.The corresponding partofthe signal isremovedand replacedby a linear interpola-tion.Thelinearinterpolation isofcourse questionnablesincethe concentration is not defined in the region occupied by the bub-bles.Similarmethodshavebeenusedforprocessingliquid veloc-itymeasurements by hot-film anemometry in bubbly flows. Dis-cussionoftheirconsequencesonthesignalstatisticscanbefound in Serizawa etal.(1983)and Suzanne etal.(1998).Herewe have checked that the present interpolation has not significant influ-ence upon PDFs of concentration fluctuations. Concerning spec-tra, differentother techniques have beenused for their determi-nationfor velocity fluctuationsfrom signals that are interrupted by the passages of the bubbles, such as smoothing the disconti-nuitiesby a Gauss function (Lance andBataille, 1991) or consid-eringonlyintervalsbetweenbubbleswherethesignalis

continu-Fig. 11. Temporal evolution of the concentration fluctuations (up). After normaliza- tion by the mean concentration (down). α= 5 . 4% . H = 5 cm.

ous(Martínez-Mercadoetal.,2010;Boucheetal.,2014).However, all techniqueseventually leadto asimilar spectruminpower −3 of the wavenumber. We are therefore confident that the present signal processing preservesthe characteristic of the spectrum of theconcentrationfluctuations. Fig.6presentsatypicalsignal mea-suredbythespectrometerbefore(greycurve)andaftersignal pro-cessing (black curve). We can see that most ofthe strong peaks havebeenremoved.Theperturbations inducedbyresidual reflec-tions orpartial occultationshavebeenquantifiedby applyingthe signal processingtoa situationwheretheHele-Shawcellisfilled withahomogeneousconcentrationofrhodamineWTatc0=10−4

mol/Land

α

=5.4% (Fig.8). Theprocessed signal(black curve)is obviously moreregular than the rawsignal (grey curve)but still shows some fluctuations due to residual perturbations. A minor amountofthemcomesfromthefactthattheobservationvolume is muchlarger than thelighted volume.The signal-to-noiseratio couldthereforebeslightlyenhanced byhavingasmaller observa-tionvolume,whichmustremaininanycaselargerthanthelighted volume.Theresidualperturbations canbequantified bythe stan-darddeviationofthesignalafterprocessing(Fig.8).Forthismost critical case, the standard deviationis equal to 2.1× 10−7 _mol/L,

whichcorrespondsto2× 10−3_{c0.Residualperturbationsare}

there-fore negligible andreliable concentration measurements are per-formed ina confined bubblyflow forgas volumefractions up to 5.4%.

Fig. 12. Left: Probability density function of the concentration fluctuation calculated on the first six seconds (black) and on the six last seconds (grey). Right: PDF of the fluctuations normalised by the mean concentration. Results obtained from 4 independent runs at H = 5 cm.

mol/L. Note that the intensity of the fluoresced light Ihas been normalized by the intensity of the fluoresced light measured at concentrationcref=2× 10−4mol/L.Theintensityofthefluoresced

light evolves linearly for concentrations lower than c=1× 10−4

mol/L. Eq.1isthusvalidforthewholerangeofconcentration con-sideredhere.

Ingeneral,theintensityofthefluorescedlightdependsalsoon laserintensity,pHandtemperature.Concerningthelaserintensity, ithasbeencheckedbyusingneutraldensityfilterthat the inten-sity of the fluoresced light evolves linearly withthe laser inten-sity.Moreover,the Fig.8showsthatthepotentialvariationsofthe measuredconcentrationinducedbyfluctuationsofthelaser inten-sitycan beneglected andthat nophotobleachingoccurs. Possible pHvariationscanalsobeendisregarded inthepresentcasesince all theexperiments havebeenconductedwiththesameaqueous solution. Finally, the temperaturehas been recordedduring each mixingexperiments andonlyvarieswithin 20.0to 20.8°C. Tem-peraturevariationswerethereforetoosmalltosignificantly influ-encethemeasurements.

5. Results

The maximum of concentration is reached at an instant that depends on the gas volume fraction

α

and the elevation H of themeasuring point. Thismaximum is detected foreach consid-eredcaseandonlythedecreasingpartoftheconcentrationsignal, whichisnotinfluencedbythedetailsoftheinitialinjection condi-tions,isconsideredhereafter. Fig.9showsatypicaltimeevolution ofthe concentration measured by the spectrometer for

α

=5.4% andH=5 cm.Two main features, involving two different times scales,areclearlyvisible.Thefirstone isaglobaldecreaseofthe concentrationonatimeTm oftheorderoftenseconds.Thisslow decayhasbeencheckedtobe reproduciblefordifferenttests per-formed atthesame

α

andH.The second one is thepresence of intenseandrandomfluctuationscharacterizedbyatimescaleTfof theorderofafewtenthsofasecond.Theconcentrationsignalcis thusthesumofaslowlytime-varyingmeanconcentration

_⟨

c

_⟩

and randomfluctuationsc′:

Foreachindependentrun,thetimeevolutionofthemean con-centrationiswell describedby a decreasingexponential function (Fig.9),

⟨

c

_⟩

(

t

)

=c

(

t0

)

exp

"

−t_T− t0 mi

#

, (3)

wheret0 is the instant where the maximum of concentration is

reachedandTmi isthecharacteristictime ofdecreaseforasingle runi.Foreach

α

andH,themeancharacteristictime ofdecrease

TmiscalculatedbyaveragingthetimesTmi overthefive indepen-dentruns anderrorbars are estimatedfromthe standard devia-tion ofindividual measurements. Fig. 10 shows that Tm depends onboth

α

andH.AtagivenH,Tmdecreases withthegasvolume fraction,whichisnotsurprisingsincethe mixingefficiencyis ex-pectedtobeanincreasingfunctionof

α

.Itiswelldescribedbythe functionk/

α

(dash linesin Fig.10),withk=13.7sforH=5cm,

k=21.5sforH=10cmandk=41.9sforH=15cm.Foragiven

α

, Tm increases withH and is multipliedby a factor 2 between

H=5cmandH=15cm,indicatingthatthemixingprocessslows downwhenthedistancefromtheinjectionincreases.The depen-denceofTmwiththedistanceHhastobelinkedtothediminution ofthe concentration gradients. The timescaleof mixing depends both onthe structure of thebubbly flow andon the gradient of concentration.Foranormaldiffusiveprocess,thetimescalewould begivenby

&

2_{/D.ThediffusioncoefficientDdependsontheflow}

structureandisthereforeindependentoftheelevationH.However, thesquare length,

&

2₌

₍

1

Cd

2_C

dx2

)

−1,increaseswithHas

concentra-tiongradientsdecrease.EvenifthemixinginaHele-Shawcell can-notbedescribedbyanormaldiffusionprocess,itisexpectedthat thetimescaleTmincreasesasthegradientofconcentration dimin-ishes.Inthesamecellbutbymeansofaplanarlaserinduced flu-orescentmethod with two dyes, Bouche etal. (2013) have mea-suredthetimeevolutionofthetotalmassofdyecontainedwithin awindowof115× 70mm2 _{located atH=4cm.Theyhavealso}

foundan exponential decrease ofthis globalconcentration. Their resultobtained in a large measurement window (also shown in

Fig.10) isvery close to the presentlocal measurement obtained atH=5cm,indicatingthatthesizeofthemeasurementwindow hasaweak influenceonthemean decreaseoftheconcentration.

Boucheetal.(2013)haveshownthattheobservedexponential de-cayisincompatiblewithapurely diffusiveprocessandhave con-cludedthat the captureand theadvection ofthe dyewithin the bubblewakesisprobablythemainmechanismresponsibleforthe mixinginaconfined cell.Thisremains to beconfirmedby time-resolvedmeasurementsofthelocalconcentration.

The following analysis of the fluctuations of concentration is performed by considering measurements recorded at H= 5 cm where the dynamics of the signals remains good over a long enoughperiod oftime. Statistics are carriedout over 4 indepen-dentrunsof12s,whichallowstogetareasonablestatistical con-vergence of PDFs and spectra. Instantaneous fluctuations of the concentrationareobtainedbysubtractingthemeanconcentration of each independent test determined by the exponential fitting

(Eq.3)totheinstantaneoussignalofconcentrationfort≥ t0:

c′

(

t

)

=c

(

t

)

− c

(

t0

)

exp

"

−t− t_T 0 mi

#

. (4)

Theupper Fig.11showsthetemporalevolutionofthe concentra-tion fluctuations for a typical case at

α

=5.4%. Similarly to the meanconcentration, theamplitude of the fluctuationsdecreases. However,weseeinthelower Fig.11that,oncenormalizedbythe mean concentration, the amplitude of the fluctuations seems no longerevolvingwithtime.Thiscan becheckedbycomparingthe probabilitydensityfunctions(PDFs)ofthefluctuationsdetermined fromthefirsthalfofthesignal(0≤ t− t0≤ 6s)tothatdetermined

fromthesecond part(6 s≤ t− t0≤ 12s). Fig.12showsthePDFs

Fig. 13. Spectra of the normalized fluctuations of concentration for various αat

H = 5 cm. The vertical dash lines show the spectral range [5–25 Hz] where a –3 slope is observed.

averagedover4independenttestsofboththeraw(ontheleft)and the normalized (on the right) concentration fluctuations for two differentgasvolumefractions(

α

=1.2% and

α

=5.4%).Whilethe widthoftherawPDFsareverydifferent,itturnsoutthatthe nor-malizedPDFsofthetwopartsofthesignalnicelymatch.Provided theyarerescaledbytheslowlytime-evolvingmeanconcentration, we can thus conclude that the fluctuations of concentration are statistically stationary in time. Moreover, the PDF of concentra-tionfluctuationspresentthesamepropertiesastheoneofvertical velocity fluctuations of the liquid (Bouche et al., 2014). Both are non-Gaussianwithstrongpositivefluctuationsmoreprobablethan strongnegativefluctuations.Inparticular,thisimpliesthatthe up-ward transport ofthe dyeisdifferent fromthe downward trans-port.Itisthereforenotcompatiblewithanormaldiffusionprocess whichwouldbecharacterizedbyasymmetrictransportinthe ver-ticaldirection. Forliquid fluctuations,thelong tailonthe sideof upward fluctuationsis known to be the signature of the bubble wakes. Here, the tailobserved forpositive concentration fluctua-tions is thus probablyrelatedto patches ofdye that are trapped andentrainedinthebubblewakes.

A spectral analysis of the concentration fluctuations has also beencarriedout fordifferent

α

andH. Fig.13 showsthespectra ofthenormalizedconcentrationfluctuationsaveragedover4 inde-pendentrunsof12s each forthe3studiedgas volumefractions

α

atH=5cm.Itisobservedtobeweaklydependentof

α

andH

andtoshow,inanycase, anevolutionasthepower-3ofthe fre-quencyinbetween5and25Hz.Thespatialspectrumoftheliquid velocity fluctuations,measured by Bouche etal.(2014),showeda similarsubrangeinpower−3ofthewavenumberforwavelengths in betweendand 5d, which correspondto the same frequencies as observed here: V_d ≈ 5 Hz and ₅V_d ≈ 25 Hz. The timescales of concentrationfluctuationsarethereforecloselyrelatedtothoseof theliquidvelocityfluctuations.Concerningtheliquidfluctuations,

Finally, we can concludefrom the analysisof the fluctuations thatthemixinginaconfinedbubblyflowiscontrolledbythe agi-tationintheliquidphaseandisduetorandompassagesof finite-sizepatchesofdye.

6. Conclusion

Anewtechniquebasedonlaserinducedfluorescenceand spec-trometry has beendeveloped to measure thelocal instantaneous concentrationofasolutewithinabubblyflow.Itinvolvesaspecific signal processingtogetridofspuriouspeakscausedbythe pres-enceofbubblesinthevicinityofthemeasurementvolume.Thanks tocomparisonswithatechniquebasedonhigh-speedimaging,the method has been validated and proved to provide reliable mea-surementswithatimeresolutionof4ms.Time-resolved measure-ments of the dye concentration have thus been obtained, giving accessforthefirsttimetoanaccuratedescriptionoflocal fluctua-tionsofconcentrationinabubblyflow.

Thetechniquehasbeenappliedtotheinvestigationofthe mix-ing of a passive dye initially injected at a given position within a homogeneous bubbly flow confined in a thin gap cell. Time records of the concentration have been obtained at various dis-tances above the dye injectionpoint and forvarious gas volume fractions. The concentration signal involves a mean component thatslowlyevolvesintimeandrapidrandomfluctuationsthatare thesignatureofthepassagesofpatchesofdyeinthemeasurement volume.

As alreadyobservedforthe globalevolutionofthe totalmass of dyecontainedwithin a large measurement volume by Bouche etal.(2013),themeanconcentration decaysintime byfollowing anexponentiallaw.Thetimescaleofdecayisobservedtoincrease with the distance fromthe injection point and to decrease with thegasvolumefraction.Italsoseemstobeweaklydependenton thesizeofthemeasurementvolume.

The fluctuationsof concentration have been analysed in both time andfrequencydomains. Oncenormalizedby themean con-centration,they turnout tobe statisticallystationaryandinvolve timescales similar to those of the velocity fluctuationsgenerated inthevicinityofthebubbles.TheirPDFhasanasymmetricshape similar to that of the vertical liquid fluctuations, with a tail for large positivefluctuations. Thissuggeststhat thetransport ofdye isdifferentintheupwardanddownwarddirectionandisnot com-patiblewithapurelydiffusiveprocessofmixing. Thespectrumof thefluctuationsshowsanevolutioninpower−3ofthefrequency inthesamerangeasthespectrumofthevelocityfluctuations pre-viouslymeasuredby Boucheetal.(2014).Asfortheliquidvelocity fluctuations (Risso2011),theconcentrationcanthusbeinterpreted asacollectionofrandompatchesofdyesofvarioussizes.This re-vealsa peculiarmechanismofmixingwhichis strongly intermit-tentandconvective.

Futureworkshouldbedevotedtoderivationofaphysical mix-ingmodelbasedonthecapture,transportandreleaseofthedyein

thebubblewakes.Theapplicationofthepresenttechnique to in-vestigatethe fluctuationsofconcentration ina three-dimensional densebubblecolumnisalsoplaned.

Acknowledgements

Theauthorswouldliketodedicatethisworktothememoryof thelateEmmanuellaBouche.

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