Dynamics of a deformable body in a fast flowing soap film

SunghwanJung1,KathleenMareck1,MichaelShelley1,andJunZhang2,1

AppliedMathematicsLaboratory,CourantInstituteofMathematicalSciences,NewYorkUniversity,251MercerStreet,NewYork,NewYork10012,USA

2

DepartmentofPhysics,NewYorkUniversity,4WashingtonPlace,NewYork,NewYork10003,USA

(Dated:February2,2008)

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arXiv:physics/0606090v2 [physics.flu-dyn] 5 Sep 2006Westudythebehaviorofanelasticloopembeddedinaflowingsoapfilm.Thisdeformableloopiswettedintothefilmandisheldfixedatasinglepointagainsttheoncomingflow.Weinterpretthissystemasatwo-dimensionalflexiblebodyinteractinginatwo-dimensionalflow.Thiscoupledfluid-structuresystemshowsbistability,withbothstationaryandoscillatorystates.Initsstationarystate,theloopremainsessentiallymotionlessanditswakeisavonK´arm´anvortexstreet.Initsoscillatorystate,theloopshedstwovortexdipoles,ormorecomplicatedvorticalstructures,withineachoscillationperiod.Wefindthattheoscillationfrequencyoftheloopislinearlyproportionaltotheflowvelocity,andthatthemeasuredStrouhalnumberscanbeseparatedbasedonwakestructure.

PACSnumbers:47.32.ck,47.54.De,47.20.-k

Thewakeflowbehindarigidobstacleisacentralob-jectofstudyinfluidmechanics.Whentheoncomingflowvelocityexceedsathreshold,vorticesareshedbehindtheobstacle[1].Atypicalwakeiscomposedofsuccessiveed-diesofalternatingsign–the“vonK´arm´anvortexstreet”–andisobservedoverawiderangeofflowvelocitiesandbodyshapes[2,3].Thefrequencyofvortexshedding(f)isdeterminedbytheflowvelocity(V)andtheobjectsize(d),whoserelationiscapturedbythenearconstancyoftheStrouhalnumber,St=df/V[3].

Thedynamicsofarigidobjectwhichmovesfreelyinthedirectionperpendiculartotheflowisofinterestinmanyindustrialandbiologicalapplications[4,5,6].Lat-eralmotionofanobjectcanbeinducedbyinteractionwiththeflowandisoftencalledthevortex-inducedvi-bration(VIV).Atlowflowvelocities,thebodystartstooscillatesidewayswithsmallamplitude(lessthan0.4timesbodydiameter).ItsassociatedwakestructureisagainavonK´arm´anvortexstreet.However,furtherin-creaseofflowvelocitycausestheobstacletooscillateinphasewiththevortexshedding,andasaresult,aseriesofdipolesareshedinstead[7,8].

Settlingbodiesorrisingbubbles,wherethebalanceofgravitationalanddragforcessetthevelocity,alsoexhibittransitionsastheyinteractwiththeirwakes.Forex-ample,aslowlysettlingsedimentingspherefallsstraightdownwards[9]butaboveacertainsedimentationvelocity,thesphere’smotionbecomesperiodicanditstrajectoryaspiralorzigzag[10].Adeformableobject,suchasadropletorbubble,canbehavesimilarlyevenasitsshapenowchanges[11,12].Finally,studieshaveshowntheinstability(andbista-bility)ofslenderdeformablebodiestolateraloscilla-tionsinquasi-2Dsoap-filmflows[13],andofheavyde-formablesheetstolateraloscillationsinfast3Dflows[14,15,16,17].Inthesecases,thesystemcorrespondstotheflappingofaflaginastiffbreeze.

Flowingsoapfilmprovidesapracticaltemplateuponwhichtostudythedynamicsofanearly2Dflow[18,19].Theexperimentalsetuphasbeenintroducedearlier[13,18,19,20].Inthiswork,weintroduceadeformableclosedbodyintoafastflowingsoap-film.Twothinny-lonwires(0.3mmindiameter)separateatanozzle(0.5mminnerdiameter)attachedtothebottomofareser-voir.Thereservoircontainssoapywatermaintainedatafixedpressurehead,thusfixingtheflux.Astopcockregulatestheflowratethroughthenozzle.Thenylonwiresextenddownwardstoacollectionbox2.4mbe-low.Drivenbygravity,thesoapfilmflowsdownwards.Owingtoairdrag,aterminalvelocityisreachedapprox-imately60cmbelowthenozzlewithavelocityprofilenearthecenterclosetobeinguniform(velocitydiffer-encesarewithin20%ofthemean,over60%ofthespanaboutthemidline).Fromopticalinterferencepatterns,thefilmthicknessisfoundtovarysmoothlyacrossthefilmbyabout15%ofitsaveragethickness.

Weuseathinrubberloop(0.2mmthick)asthede-formablestructure.Theloopwetsintothesoap-filmandissupportedfromitsinnersideagainsttheflow.Theloopismuchthickerwhencomparedtothefilmthickness(0.003mm)thatthefluidpresumablydoesnotpenetrateovertheloop.Sixloopsofdifferentcircumferences(5–7.5cm)areused.Wefindthatforregimesstudiedhere,theloopappearstoundergoonlybendingdeformations,andnotstretchingorcompression,asitslengthshowsnomeasurableincreaseordecrease.Currently,wedonotunderstandwhatbalanceofeffectssetstheenclosedareaoftheloop,whichisanimportantconstraintonthepossibledynamics.However,wedofindthat,onceex-perimentalconditionsarefixed,andtheloopisinafixedstateofdynamics,theenclosedareachangesverylittleintime(e.g.∼5%fora5cmloop).However,betweendifferentstatesorconditions,theenclosedareacanvarybyfactorsoftwoorthree.

AlaserDopplervelocimeter(LDV;ModelLDP-100,TSIInc.)isusedtorecordtheupstreamvelocityV.

FIG.1:Flowstructuresbehinda5cmloopat2.2m/sflowve-locity.Thecoupledfluid-structuresystemshowsbistability:(a)thestationarystate;theloopremainsessentiallymotion-lessanditswakeisavonK´arm´anvortexstreet.Theloopisdeformedbytheflowintoateardropshape.(b)theos-cillatorystate;theloopshedstwovortexdipoleswithineachoscillationperiod.

Micron-sizedparticles(TiO2)areseededintotheflowforLDVmeasurements.Flowstructuresarevisualizedusinginterferencepatternsfrommonochromaticillumination(low-pressuresodiumlampsoperatingatwavelength585nm).Themoviesofthewakeflowtogetherwiththelooparerecordedusingahighspeedcameraat1000framespersecond.

Theinteractionbetweentheloopandtheflowisquitecomplicated.Inourexperiments,weobservebistablestates,onestationaryandanotheroscillatory(seeFig.1aandb),thatco-existoverarangeofflowvelocities.Atleastintheconditionsconsideredhere,wedonotobservespontaneoustransitionsbetweenthesetwostates.How-ever,atransitionfromthestationarytotheoscillatorystatecanbeinducedbyexternallyperturbingtheloop,orbyabruptlychangingtheflowvelocity.

Inthestationarystate,theloopbehavesasarigidhoopandhasateardropshapewithhighercurvatureonthetopthanonthesides(Fig.1a).Acharacteristic

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FIG.2:(a)At2.2m/sflowvelocity,eightsnapshotsofa5cmloopanditscentroidareshown.Theoscillationperiodoftheoscillatingloopis52ms.Thesnapshotsoftheflexiblelooparepresentedassolidloopsat6msintervalsandthetrajectoryofthecentroidoftheloopwith2msintervals.Theloopsaresequentiallynumbered.(b)Wakestructurewhentheloopcentroidisatthefarleft.Theloopstartstoshedacounter-clockwisevortex.Thewakestructureisschematicallyshownin(c).

lengthscale(D)oftheloop,itswidthinthefilm,is1cm.Theflowvelocity(V)variesfrom1.5to2.5m/sandthekinematicviscosity(ν)ofsoapfilmisapproximately0.04cm2/s.Thefrequencyofvortexshedding(fs)variesfrom20to50Hz.Basedonthesecharacteristicnumbers,weestimatetheReynoldsnumber(Re)andStrouhalnumber(Sts)forthesystemtobe

Re=

VD

V

∼0.2,

(1)

wherethesubscriptsstandsforvortexshedding,sincetheStrouhalnumberiscalculatedbasedonthevorticalstructureofthewake.Figure1ashowsthedeformingbodyanditsvorticalwakeusinga5cmloopandflowvelocityof2.2m/s.Ascanbeseen,vorticesofalternatingsignaresuccessivelyproduced.

Intheco-existingoscillatorystate,showninFig.1batthesameparametersasabove,theloopnowoscillatesperiodicallyinthehorizontaldirection,andthevorticalwakebehinditisquitedifferent.Forlowflowvelocity,twodipolepairsareshedduringeachoscillationperiod.Suchawakestructureisalsoobservedbehindoscillatingcylindersandisreferredtoasthe2Pmode[8].

Figure2ashowsboththepositionoftheloopatseveraltime-pointsduringoneperiodofoscillation,andthepathtakenbyitscentroid.Duringtheoscillation,theloopcontinuouslychangesitsshape,anditscentroidmovesalongafigure-eighttrajectory(Fig.2a).Thisfigure-eightshapeisduetothefactthatthefrequencyofoscillationinthestream-wisedirectionistwicethatinthetransversedirection.Thishasbeenobservedinthemotionsofa

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FIG.3:Frequencyofanoscillatingloop(floop)versusrescaled√velocity(VL).Wetestover6differentlooplengths(5cm(△),5.5cm(▽),6cm(£),6.5cm(Q),7cm(󰀁),and7.5cm(󰀄)).Theopensymbolsareintheoscillatorystateandtheclosedonesareinthestationarystate.Thefrequencyoftheoscillatingloopislinearlyproportionaltotherescaledvelocity.

FIG.4:Strouhalnumberoftheloop(StL)versustheflowvelocity(V)fordifferentlooplengths(5cm(△),5.5cm(▽),6cm(£),6.5cm(Q),7cm(󰀁),and7.5cm(󰀄)).Opensymbolsare2Pmodeandclosedonesareflag-likemodes.TheStrouhalnumbersofthetwomodesarewellseparated,andtheStrouhalnumbersof2Pmodesarecloseto0.2whereasthoseofflag-likemodesareabove0.25.

flappingflag[13]andVIVsystems[5,21].Also,thelooposcillatesinphasewiththatvortexshedding(Fig.2bandc).Whentheloopisatfarright(orleft),aclockwiseturning(orcounter-clockwise)vortexisshed.

Byusingloopsofseveraldifferentlengths,wefindalin-earrelationbetweentheoscillation√frequencyoftheloop(floop)andarescaledvelocityVL(seeFig.3),whereδisthefilmthickness,athethicknessoftheloop,andLthelooplength.Ourresultsfromloopsofdiffer-entlengthsanddifferingflowvelocitiesallcollapseontoasinglelinewithslopeofabout0.27.Thisoffsetofthisaffinerelationsuggestsabifurcationtooscillationatafiniteflowvelocity;linearextrapolationtofloop=0sug-gestsacriticalrescaledvelocityofabout20,whichisunfortunatelybeyondthereachofthisexperiment.

Tobetterunderstandtherelationbetweenoscillationfrequencyandflowvelocity,weproposeasimplemodelfortheoscillationsofanelongatedloopwithlongitudi-nallengthLldrivenbya“liftforce”inthedirectionperpendiculartothestream.Theliftforce(F)istakenasproportionaltoρV2Llδwhereρisthedensityoffluid,Vthefluidvelocity.Hence,F=(1/2)CLρV2Llδ,whereCLisaliftcoefficient.Typically,Llisproportionaltotheloopcircumference,L.Atanangleθinclinedtotheflowstream,CLisproportionaltosinθ[22].Forsmallθ,sinθ∼xcm/ycmwhere(xcm,ycmisthecenterofmass(centroid)location.Therefore,weapproximatetheliftforceasF=mx¨cm≈(1/2)ρV2Llδxcm/ycm,wherex¨cmistheaccelerationinthetransversedirectionandmis

thetotalbodymass.Inthisexperiment,themassofthe(wetted)loopismuchgreaterthanthatoftheenclosedfluid.Hence,weassumethatthetotalbodymass(m)isproportionaltoρLLa2whereρListhedensityoftheloop.Also,they-componentofthecentroid,ycm,andthelengthLlareassumedtobeproportionaltothelengthoftheloopifthebodyiselongatedduetotheflow.Withthetrivialsolutionforthex-componentofcentroidasxcm=Ceiωt,weobtainanexpressionfortheoscillation√frequency:ω=2πfloop∝VL.Thisisconsis-tentwithourobservationsandunderliesourrescalingofthedatainFig.3.Putdifferently,thisissimplytheoscillationfrequencyofahangingpendulumwherethegravitationalforceisreplacedbyadragforce.

Astheflowvelocityincreases,amorecomplicatedmodeintheoscillatorystatecanbeobserved(leftpanelinFig.4).Inthiscase,theloopshedsmorethanfourvor-ticesoverasingleperiodofoscillation.Werefertothiswakestructureasaflag-likemode.Weusethisterminol-ogybecausethewakenowresemblesmorethatbehindaflappingflag(see[13]),andbecausethebodyitselflookselongatedand”flag-like”.ThisisbecausetheenclosedareaisnowsmallerinrelationtoL2thanforthebodyinthe2Pmode.Tocharacterizethelooposcillationandthewakestructure,weagaindefineaStrouhalnumber,nowusingtheoscillationfrequencyoftheloopitself,orStL=Afloop/VwhereAistheouteramplitudeofoscil-

0.80S0.6/S0.40.2L0.6/A0.411.21.41.6V1.8(m/s)22.22.42.6FIG.5:Enclosedarea(normalizedbyL2/4π)andamplitude(normalizedbyL)oftheoscillatingloopfor6cmloop.Areaandamplitudeabruptlychangeasthewakestructuretran-sitsfrom2Pmodes(opensymbols)toflag-likemodes(closedsymbols).Asthevelocityincreasesfurther,theamplitudedecreasesduetothelargedragforce.

lationasindicatedinFig.4.ThecorrespondingStrouhalnumbersfor2Pmodesandforflag-likemodesareshowninFig.4.The2PmodeyieldsStLapproximately0.2andtheflag-likemodesyieldvaluesabove0.25.

ThisseparationintherespectiveStrouhalnumbersiscausedbyadiscontinuouschangeoftheamplitudeAfromthe2Pmodetotheflag-likemode.Figure5showsthesimultaneoustransitionsofenclosedarea(S)andam-plitude(A)oftheoscillatingloop.Athighflowvelocity,thestreamwiseextensionoftheloopincreasesduetothehigherdrag,andthispresumablycausestheobservedrelativedecreaseinenclosedareaintheflag-likemode.Duetoitsnowhigheraspectratio,theloop’samplitudeincreases(compareleftandrightpanelsinFig.4).Thecausesofthisabruptchangeinenclosedloopareaandoscillationamplituderemainsanopenquestion.Follow-ingthisabruptchange,thefrequencyofvortexsheddingalsoincreasespresumablyduetothehigheraspectra-tio.Unlikethe2Pmode,theoscillationoftheloopandvortexsheddingarenotin-phase,andthewakebecomesmorecomplicated.

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