RayleighB ID: 389573
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84,N1PHYSICALREVIEWLETTERS3J Rayleigh-BénardConvectioninaVerticallyOscillatedFluidLayerJeffreyL.RogersandMichaelF.Schatz*SchoolofPhysics,GeorgiaInstituteofTechnology,Atlanta,Georgia30332-0430JonathanL.BougieandJackB.SwiftDepartmentofPhysicsandCenterforNonlinearDynamics,UniversityofTexas,Austin,Texas78712Received2July1999 84,N1PHYSICALREVIEWLETTERS3J FIG.1.Convectionpatternsarevisualizedusingshadow-graphyandcharacterizedbyfourdimensionlessquantities:PrandtlnumberPr ,drivingfrequency ,displacementamplitude andRayleighnumber ,whilethekinematic,thermaldiffusivity,thermalexpansionco-efficient,forcingfrequency(Hz),amplitudeandgravitationalacceleration .(a)Hspiraldefectchaos.(b)CoexistingHrollsandhexagons.(c)Srollsnearonset.(d)Srolls.(e)HrollswithlocalizeddomainsofSrolls.(f)SrollscontaininggrainboundariesoverlayingaweakpatternofHrollsandcellsandPrheldconstant(Fig.1)whileincreasingandatvariousfixedvaluesofHconvectionoccursforsmall[Fig.2(a)].Withoutos-cillations()spiraldefectchaosarisesforinagreementwithpreviousexperiments[6].Withoscilla-tions(atfixed),spiraldefectchaosmodulatedatpersistsforasignificantrangein).Withincreasingthenumberofspi-raldefectsdecreaseasmoreregularstateswhosemorphol-ogydependsonemerge.Foremergingpatternsaretypicallymultiarmspiralswhichre- FIG.2.Phasediagramandcomparisonoflinearstabilitypre-dictionstoexperimentsat.Thephasediagram(a)con-tainsregionsofconduction,convectionwithHandSmodulations,aswellascoexistingH-Spatterns.Marginalsta-bilitycurvescomputedfortheconductionstatesubjectedtoH(solidline)andS(dashed)perturbationsagreewiththemea-suredvaluesof(a)and(b)attheonsetofHandSconvection.ThemeasuredtransitiontocoexistingpatternsfrompureHandSstatesiscomparedtothemarginalstabilitypredictionsfor.Themaximumdisplace-correspondstoanaccelerationofduceinarmnumber,eventuallybecomingtargetsastheconductionstateisapproached.Atlarger)spiraldefectchaosbecomesapatternofnearlyparal-lelrollstendingtoterminateperpendiculartothesidewallsandpossessingseveralfociattheboundaries;thenum-beroffociandcurvatureoftheassociatedrollsdecreaseswithincreasing.Thetransitionwithincreasingspiraldefectchaostoparallelrollsisreminiscentofthewell-studiedtransitioninunmodulatedRayleigh-Bconvectionfordecreasing[16].Foruniformparallelrollsortargetslosestabilitywithincreas-asdomainsofhexagonsform[Fig.1(b)].Thesestatesofhexagonsandrollsortargetsoccuronlyforanarrowrange()ofbeforelosingstabilitytoconductionwithasmalladditionalincreasein.Withintheexperimentalresolutionin)nohystere-sisisobservedinthetransitionbetweenthehexagon-rollstatesandconduction.ThenonhysteretictransitionandmorphologyofthesepatternsareconsistentwithothermodulatedRayleigh-Bnardexperimentsinvolvingtime-periodicdrivingofthebottomplatetemperature[17].Sconvectionisobservedforsufficientlylarge[Fig.2(a)].TheonsetofSpatternsoccursasauniformpatchofrolls;nohysteresisorhexagonsareobserved,consistentwiththeStemporalsymmetrythatexcludesthreewaveinteractions[2].Withincreasingrolldomainsformwithgrainboundariesatthedomainintersections.Therolldomainsmergewithfurtherin-creasein,leadingtotheformationofdisclinationsthatmayinteract[Fig.1(c)].Forsufficientlylarge,eitherasingleconvexdisclinationor,lessfrequently,aspiralarisescenteredwithintheconvectioncell.Thesepatternsexperienceskew-varicoseinstabilitiesleadingtorepeated 84,N1PHYSICALREVIEWLETTERS3J FIG.3.HandSpatternsrotateinoppositedirections.(a)AtthemotionofasingleHroll(dashedline)isfollowedintimeatintervalsof.(b)AtthemotionofaSdisclination(brightwhiteregion)isfollowedintimeatintervalsof.(c)Thedimensionlessrotationrateversusnucleationofdislocations;additionallythepatternsmaymoveoffcenter[Fig.3(b)].Withincreasingasinglerolldomainformswithfewdislocationsandalongwave-lengthdistortion[Fig.1(d)].Patternsqualitativelysimi-lartoFig.1(d)havebeenpreviouslyobservedinrotatingnardconvection[18].Followingthemethoddescribedbypreviousinvesti-gators[10,11]weperformedalinearstabilityanalysisoftheconductivestate.TheresultingpredictionsforbothcriticalRayleighnumbersandcriticalwaveareingoodagreementwiththeexperi-mentallyobservedvaluesatonsetofbothHandSconvection(Fig.2).ForHconvection,modulationenhancesthestabilityofconduction()whilebelowitsunmodulatedvalueconsistentwithpreviousmodulatedRayleigh-Bexperiments[17].Inaddition,forSconvectiondecreasewithincreasing(Fig.2).Forpa-rametervaluesnotstudiedhereispredictedtodropbelowwForpatternsundergonearlysolid-bodyro-tationwhereHandSstatesrotateoppositedirections(Fig.3).Forfixed)andincreasingfromzero,theonsetofrotationoccursnear.Patternsdeviatesomewhatfromidealsolid-bodyrotationbecausepointdefectsandgrainboundariescon-tinuallypropagatewithintherotatingpatterns.Globalrotationrateincreaseswithexceptneartheconduc-tionboundarieswhererotationslowsaspatternsweaken[Fig.3(c)].Agivenrotationdirectionisselectedandmain-tainedbythepatternsthroughoutthedurationofanexperi-mentaltrial.Patternsdonotequallyselectclockwiseandcounterclockwisedirections;in62separateexperimentsHstatesrotatedcounterclockwiseinofthetrials.Inallcases,HandSpatternsrotateinoppositedirections.Ro-tationsarequalitativelyrobustagainstperturbationsfrom FIG.4.CoexistingHandSpatterns(a)()maybedecomposedbyfilteringinthewavenumberdomain(b)toyieldbothH(c)andS(d)components;inthiscase,bothcomponentsequalpowertothewavenumberspectrum(b)andexhibitmodelockingofthewave .(e)TherelativepowercontributedbyHandScomponentstowavenumberspectrachangesabruptlyasafunctionofforconstantVerticallinesmarkthemeasuredcoexistenceboundaries.tiltingtheapparatusoffthevertical,changingthesidewallstosquaresymmetryandasymmetriccoolingofthetopplate.Forconductionisnolongerstable;insteadHandSpatternscompeteandcoexistoverarangeoftweenthepurestates[Fig.2(a)].Asisincreased,pureHstateslosestabilitytomixedpatternswherelocalizedpatchesofSrollsformaboutHdefectsandareadvectedalongasthedefectspropagate.Atslightlylargerger inFig.4(e)],Srollsbegintoformper-pendiculartoHupflowsthroughoutthepattern[Fig.1(e)].ThewavenumberofemergingSrolls()isclosetothesecondharmonicoftheHpatternwavenumber().Asmallchangeinin inFig.4(e)]yieldsstateswhereHpatternsoflocalhexagonal,square,andrhombicsymmetriesaremixedwithrollsoftheScom-ponentperpendiculartothecellfaces[Figs.4(a),4(c),and4(d)].Forthesestates,theHandScomponentscontributeequalpowertothewavenumberspectraandhavemode-lockedwavenumbers( ).Withfurther 84,N1PHYSICALREVIEWLETTERS3J smallincreasesinin inFig.4(e)],theScomponentdominatesthepowerspectraand,concurrently,thewavenumberratiounlocks( )asabruptly.TheScomponentformsdomainsofincreas-inglylargersizeastheHcomponentgraduallyweakens[Fig.1(f)].UponcrossingthephaseboundarywithpurelySstates[Fig.2(a)]rollswithalong-wavelengthdistortionaretypicallyobserved[Fig.1(d)].Theexperimentallydeterminedphaseboundariesseparatingcoexistingstatesfromthepurepatternstrackthemarginalstabilitycurvesforthe[Fig.2(a)].For,theHmarginalstabilitycurveisinnearlyexactagreementwiththephaseboundarybetweencoexistingandpureSstates.ThissuggeststheSbasestatefromwhichHconvectionbifurcatesdifferslittlefromconductioninaspatiallyaveragedsense.SpatialFourierspectrasupportthisviewpointsincethehighermodesofSpatternscannotoverlapwiththesmallerwavenumberHfundamental.Bycontrast,theexperimentallydeterminedphaseboundarybetweencoexistingandpureHstatesliesabovetheSmarginalstabilitycurve,suggestingthatHconvectioninhibitstheonsetofSconvectionduetowavenumberinteraction.EvidenceforthisinhibitoryeffectisfurtherbolsteredbytheobservationthatSconvectionfirstappearsnearHpatterndefects.Theamplitudeofconvectiveflowisgenerallysuppressedinthecoresofpatterndefects[19]and,therefore,anyinhibitoryeffectofHconvectiononSpatternsshouldbeweakerneardefects.Moreover,apreviousstabilityanalysisoftheHbasestatesuggeststheonsetofSconvectionisdelayedbythepresenceofHconvection[10].Thesemultiplelengthscaleconvectionpatternsdifferqualitativelyfromcoexistingwavelengthstatesinspatiallyseparatedomainsobservedinopticalsystems[20]aswellasquasiperiodic[21]andsuperlattice[22]statesreportedinFaradayexperiments.Threewaveinteractions(reso-nanttriads)areresponsibleformultiscaleFaradaypat-terns;itseemsdoubtfulresonanttriadsareimportantintheconvectionpatternsdescribedhereduetotheStem-poralsymmetryandlargedifferencebetweenResonanttriadsmaybeintroducedinconvectionpatternsbynon-Boussinesqeffectsandforthecurrentexperimentwithheatingfromabovesquaresandquasiperiodicstruc-tureshavebeenpredicted[12].ThisworkissupportedbytheNASAOfficeofLifeandMicrogravitySciencesGrantsNo.NAG3-2006(attheGeorgiaInstituteofTechnology)andNo.NAG3-1839(attheUniversityofTexas-Austin). Electronicaddress:jeff@einstein.physics.gatech.edu*Electronicaddress:mike.schatz.physics.gatech.edu[1]P.Coullet,T.Frisch,andG.Sonnino,Phys.Rev.E2087(1994).[2]M.C.CrossandP.C.Hohenberg,Rev.Mod.Phys.,851[3]M.Boucif,J.E.Wesfried,andE.Guyon,Eur.J.Mech.A,,641(1991).[4]C.Szwaj,S.Bielawski,D.Derozier,andT.Erneux,Phys.Rev.Lett.,3968(1998).[5]W.vanSaarloosandJ.D.Weeks,Phys.Rev.Lett.,290[6]S.Morris,E.Bodenschatz,G.Ahlers,andD.S.Cannell,Phys.Rev.Lett.,2026(1993).[7]P.M.GrashoandR.L.Sani,J.FluidMech.,783[8]G.Z.GershuniandD.V.Lyubimov,ThermalVibrationalConvection(Wiley,Chichester,1998).[9]G.Ahlers,P.C.Hohenberg,andM.Lcke,Phys.Rev.A,3493(1985);,3519(1985).[10]R.Clever,G.Schubert,andF.H.Busse,J.FluidMech.,663(1993).[11]R.Clever,G.Schubert,andF.H.Busse,Phys.FluidsA2430(1993).[12]U.E.VolmarandH.W.Mller,Phys.Rev.E,5423[13]F.H.Busse,J.FluidMech.,625(1967).[14]B.A.Malomed,A.A.Nepomnyashchy,andM.I.Tri-belshy,Phys.Rev.A,7244(1990).[15]E.Bodenschatz,J.R.deBruyn,G.Ahlers,andD.S.Can-nell,Phys.Rev.Lett.,3078(1991).[16]S.Morris,E.Bodenschatz,D.S.Cannell,andG.Ahlers,Physica(Amsterdam),164(1996).[17]C.W.Meyer,D.S.Cannell,andG.Ahlers,Phys.Rev.A,8583(1992).[18]Y.Hu,R.E.Ecke,andG.Ahlers,Phys.Rev.E,6928[19]M.C.Cross,Phys.Rev.A,1065(1982).[20]E.Pampaloni,S.Residori,S.Soria,andF.T.Arecchi,Phys.Rev.Lett.,1042(1997),andreferencestherein.[21]W.S.EdwardsandS.Fauve,J.FluidMech.,123[22]C.Wagner,H.W.Mller,andK.Knorr,Phys.Rev.Lett.,308(1999);H.ArbellandJ.Fineberg,Phys.Rev.Lett.,4384(1998);A.Kudrolli,B.Pier,andJ.P.Gollub,Physica(Amsterdam),99(1998).