Pattern formation in inclined layer convection

KarenE.Daniels,BrendanB.Plapp,andEberhardBodenschatz

LaboratoryofAtomicandSolidStatePhysics,CornellUniversity,Ithaca,NY14853-2501

(December21,1999)

Abstract

WereportexperimentsonthermallydrivenconvectioninaninclinedlayeroflargeaspectratioinafluidofPrandtlnumber

.Weobservedanumberofnew

nonlinear,mostlyspatio-temporallychaotic,states.Atsmallanglesofinclinationwefoundlongitudinalrolls,subharmonicoscillations,Busseoscillations,undulationchaos,andcrawlingrolls.Atlargerangles,inthevicinityofthetransitionfrombuoyancy-toshear-driveninstability,weobserveddriftingtransverserolls,localizedbursts,anddriftingbimodals.Foranglespastvertical,whenheatedfromabove,wefounddriftingtransverserollsandswitchingdiamondpanes.47.54.+r,47.27.Te,47.20.Bp,47.20.Ft,47.20.-k,05.45.Jn

TypesetusingREVTEX

1

Rayleigh-B´enardconvection(RBC)ofaverticalfluidlayerheatedfrombelowhaslongservedasaparadigmforpatternformingsystems[1,2].Variationsthatalterthesymmetries,suchasro-tationaroundaverticalaxis[3,4]andverticalvibrations[5]continuetoleadtoimportantinsights.Anothercase,ofparticularmeteorologicalandoceanographicinterest,isRBCofafluidlayerin-clinedwithrespecttogravity.Thissystemisnotonlywellsuitedforthestudyofbuoyancyandshearflowdriveninstabilities,butmayalsoserve,alongwithliquidcrystalconvection[6],asaparadigmforanisotropicpatternformingsystems.

AswithRBC,theonsetofinclinedlayerconvection(ILC)occurswhenthetemperaturedif-ference,

,acrossthelayerissufficientforconvectionrollstoform.Themaindifferencefrom

RBCisthat,inILC,thepatternlessbasestateischaracterizednotonlybyalineartemperaturegradientbutalsobyasymmetry-breakingshearflow.AsshowninFig.1,thecomponentofgrav-itytangentialtothefluidlayer,

,causesbuoyantfluidtoflowupalongthewarmplateanddown

alongthecoldplate.Forsmallanglesofinclination,buoyancydominatesovershearflow(large,

small),andtheprimaryinstabilityistolongitudinalrolls(LR)whoseaxesarealigned

withtheshearflowdirection[7].Withincreasing,buoyancyeffectsdecreaseandforbuoyancyisstabilizing.Aboveacriticalangle

(

)theshearflowcausesaprimaryinsta-

bilitytotransverserolls(TR)withroll-axesperpendiculartotheshearflow[7].Thefewpriorexperiments[8–10]onILCshowedreasonableagreementwiththelineartheory[7,11–13].TheseexperimentsalsodemonstratedthatLRareunstabletosomeformofundulations[8–10],inquali-tativeagreementwiththeory[12,13],butthequantitativedetailsofthestatewereinaccessibleduetoexperimentallimitations.

HerewereportthefirstexperimentalresultsonpatternformationinILCforlargeaspect-ratiosystemsinarangeofinclinationangles

,i.e.,fromhorizontal(heatedfrom

below)topastvertical(heatedfromabove).Wefoundmanyunpredictedstateswhenincreasing

abovethecriticaltemperaturedifference.For

weobservedlongitudinal

rolls,subharmonicoscillations,Busseoscillations,undulationchaos,andcrawlingrolls.Intheneighborhoodofthecodimensiontwopoint[11]forthermalandsheardriveninstability(

),weobserveddriftingbimodals,driftingtransverserolls,andlocalizedlongitudinaland

2

transversebursts.Forinclinations

wefounddriftingtransverserolls,switchingdiamond-

panes,andlongitudinalbursts.Mostofthesenovelstateswerespatio-temporallychaoticandwerefoundveryclosetoonset,wheretheoreticalprogressshouldbepossible.

Experiment:Ourexperimentalapparatusconsistedofawater-cooledpressurechambercon-tainingaconvectioncellofdiameter10cm,subdividedintotwolargeaspectratiorectangularcells.Theexperimentaldesignwassimilartotheonedescribedin[14].Theopticallyflatupperandlowerplateoftheconvectioncellconsistedof

cmthicksinglecrystalsapphireandsingle

crystalsilicon,respectively.Thesapphireplatewascooledbyawaterbath,whilethesiliconplatewasheatedbyanelectricfilmheater.Theconvectionpatternswerevisualizedbytheusualshad-owgraphtechnique[14].Thesidewallswereconstructedof9layersofnotebookpaper,providingthebestpossiblethermalmatchingbetweencellboundariesandthefluid.Asmeasuredinterfer-ometrically,theplateswereparallelto

m.Theconvectioncellwashousedinapressure

chamber,whichheldboththecoolingwaterandtheconvectinggasto(to

)bar,regulatedC.Throughout

bar.Thetemperaturesofthetwoplateswereregulatedto

theexperimentthemeantemperaturewaskeptconstantat

C.Wedeterminedthe

cellheightbymeasuringthepatternwavenumberatonsetfortheoreticalvalueof

andcomparingitwiththe

.Wefound

mand

=

mfortwosets

ofexperiments.Thetwoconvectioncellshadasizealldata,thePrandtlnumberwasviscosity

and

.For

asdeterminedfrom[14],withthekinematic

andthermaldiffusivity.Theverticalthermaldiffusiontimewas

s.

Inclinesfrom(horizontal)to

(

pastvertical)werepossible,withanaccuracyof

.

Following[2]wecalculatedtheBoussinesqnumbertoestimatenon-Boussinesqeffects.At

forthecorrespondinghorizontallayer

for

wefound

,puttingthe

flowintotheBoussinesqregime.Forlargerangles

increasedlinearlyto3.0forthelargesttem-

peraturedifferencesinvestigated.Weobservedthesameconvectivepatternsinbothconvectioncells.

Onsetofconvection:InILC,theforwardbifurcationtoLRispredictedtooccuratthecriticalRayleighnumber

where

(isthethermal

3

expansioncoefficientand

isthecriticaltemperaturedifference).Thethresholdfortheforward

bifurcationtosheardrivenTRatlargeinclinationangleismorecomplicated,andcanonlybedeterminednumerically[12,13,15].Wedetermined

forconvectionbyquasi-staticallyincreasing

instepsof1mKevery

minutespastthepointwhereconvectionwasobservableandthendecreasingthetemperature

differencesimilarly.Forallanglesweobservedforwardbifurcations.Figure2showsthemeasured

,aswellasthetheoreticallypredictedonsetsforboththebuoyancy-driven(longitudinal)and

theshear-driven(transverse)instabilities[15].WefoundagreementwiththeoryfortheonsetofLR:theexperimentallyobservedvaluewas

with

.

Wedidnot,however,observethetheoreticallypredictedstationaryTR,butinsteaddriftingTR(DTR)ataslightlylargercriticalRayleighnumber.Thedriftdowntheinclinemaybeattributedtothebrokensymmetryacrossthelayerwhichiscausedbythetemperaturedependenceofthefluidparameters(non-Boussinesqeffects).Veryinterestingisthevicinityofthetheoreticallypredictedcodimensiontwopointat

[15],whereLRandTRhavethesameonsetvalue.

Experimentally,wefoundaforwardbifurcationtoDTRaboverange

,andinthe.Asshown

DTRloststabilitytodriftingbimodals(DB)above

inFig.3,DBconsistofasuperpositionofLRandDTR.Here

isthe

reducedcontrolparameter.Theoretically,FujimuraandKelly[11]predictedaforwardbifurcationtotransverserolls,whichlosestabilitytobimodalsat

inanarrowangularregion.

Wefindgoodagreementwiththesepredictions,butwiththedifferencethattheexperimentallyobservedpatternsaredrifting.

Nonlinearstates:Figure4showsthemeasuredphaseboundariesforthetenobservednonlin-earconvectivestates.Atlowangles(

),LRarestableupto

abovewhichthenovel

stateofsubharmonicoscillations(SO)setsin.Theseoscillationsarecharacterizedbyapearl-necklace-likepatternofbright(cold)spotsthattravelalongastandingwavepatternofwavyrolls.AsshowninFig.5,theseoscillationsappearinpatcheswhoselocationchangesintime.Typi-calfrequenciesoftheoscillationsweremeasuredtobe1to3cyclesper

.Arecenttheoretical

analysishasshownagreementwiththisvalue[18].Withfurtherincreasein,localizedpatches

4

oftravelingoscillationsburstintermittently.Within

theamplitudeoftherolls’waviness

increases,thepatterntearstransversetotherollsasshownintheupperleftcornerofFig.5,andfadesawayleavinganalmostparallelrollstate.For

and

,weobservedpatchesofthewell-knownBusseoscillations(BO)

coexistingwithpatchesoftheSO.AsshownbythedottedlineinFig.4,ourdatafortheonsetoftheBOagreeswellwiththetheoreticalpredictioncalculatedfor

[12].Itissurprising,

however,thatbothoscillations(SOandBO)coexistaslocalizedpatchesinthesamecell.Atintermediateangles(

),wheretheinitialinstabilityistoLRwefoundwith

increasingthatLRwereunstabletoundulations.Althoughtheexperimentallydeterminedvaluefortheinstability

agreeswellwiththetheoreticalprediction(seeFig.4)[12,13,15],

wedidnotobserveastationarypatternofundulations,butadefect-turbulentstateofundulationchaos(UC).AsnapshotofUCisshowninFig.6a.Atinthedirectiontransversetotherollsontimescales

,theUCbeginsto“twitch”

.Withincreasing,theamplitude

ofthetwitchingincreasesandtherollseventuallytear,withtheends“crawling”inthedirectiontransversetotheoriginalrolls.Asnapshotofcrawlingrolls(CR)isshowninFig.6b.Inthevicinityofthecodimension-twopoint,at

,weobserveddriftingbimodalsquiteclose

toonset.AsshowninFig.4,forsmallanglestheexistenceregionofthepureDBislimitedbylocalizedtransversebursts(TB),whileforlargeanglesbyDTR.AsnapshotoftransverseburstsandtheevolutionofasingleburstisshowninFig.7.InthisregionofphasespacetheLRoccurinpatchesthatgrowanddecayintermittentlywhileTBnucleateinhighamplitudeLR-regions.AsshowninthetimeseriesinFig.7,TBgrowovertheperiodofafewAbove

andthendecayrapidly.

theDBareunstabletolocalizedlongitudinalbursts(LB)asshowninFig.8a.As

showninFig.8b–j,afewlongitudinalrollsgrowlocallytolargeamplitudeandthenquicklyfade.Withbothtypesofbursts,theburstsincreasebothindensityandfrequencywheneventuallydevelopingintoaturbulentstateatPast

isincreased,

.

,wecontinuedtoobservesheardrivenconvectionpatterns.DTRaretheprimary

instability;however,theyareunstabletoswitchingdiamondpanes(SDP)atcharacterizedbyspatio-temporallychaoticswitchingontime-scalesof

5

.Thestateis

from

to

oflargeamplituderegionsofDTR,asseeninFig.9a.At

SDPareunstabletoLB,which

inthisregionofphasespacearedenserbuttravellessdistancethaninTR,asshowninFig.9b.Conclusion:Inclinedlayerconvectionintheweaklynonlinearregimedisplaysarichphasediagram,withtendifferentstatesaccessibleovertherangeofparametersinvestigated.Thephasespacenaturallydividesintoseveralregionsofcharacteristicbehaviorwhichhavesofarbeenchar-acterizedsemi-quantitatively.Allstatesbutlongitudinalandtransverserollsarespatio-temporallychaotic.Mostinstabilitiesoccurredveryclosetoonsetandfurthertheoreticaldescriptionshouldbepossible.Especiallyinterestingistheburstingbehavior,whichmayberelatedtoturbulentburstsinothershearflows[19].

WethankF.H.BusseandW.Peschforimportantdiscussionsonthestabilitycurvesandtheoreticaldescriptionsofvariousstates.E.B.acknowledgesthekindhospitalityofProfH.LevineattheUniversityofCaliforniaatSanDiegowherepartofthismanuscriptwasprepared.WegratefullyacknowledgesupportfromtheNSFundergrantDMR-9705410.

6

REFERENCES

PresentlyatCenterforNonlinearDynamics,UniversityofTexasatAustinEmail:eb22@cornell.edu

[1]M.C.CrossandP.C.Hohenberg,Rev.Mod.Phys.65,851(1993).

[2]E.Bodenschatz,W.Pesch,andG.Ahlers,Annu.Rev.FluidMech.32,709(2000).[3]K.M.S.Bajaj,J.Liu,B.Naberhuis,andG.Ahlers,Phys.Rev.Lett.81,806(1998).[4]Y.Hu,W.Pesch,G.Ahlers,andR.E.Ecke,Phys.Rev.E58,5821(1998).

[5]J.L.Rogers,M.F.Schatz,J.L.Bougie,andJ.B.Swift,Phys.Rev.Lett.(tobepublished).[6]L.KramerandW.Pesch,Annu.Rev.FluidMech.27,515(1995).

[7]Y.ChenandA.J.Pearlstein,J.FluidMech.198,513(1989)andreferencestherein.[8]J.E.Hart,J.FluidMech.47,547(1971).

[9]D.W.Ruth,G.D.Raithby,andK.G.T.Hollands,J.FluidMech.96,481(1980).[10]J.N.ShadidandR.J.Goldstein,J.FluidMech.215,61(1990)andreferencestherein.[11]K.FujimuraandR.E.Kelly,J.FluidMech.246,545(1993);inBifurcationPhenomena

andChaosinThermalConvection,editedbyH.H.Bau,L.A.Bertram,andS.A.Korpela,ASME:HTD,214,73,(1992).

[12]R.M.CleverandF.H.Busse,J.FluidMech.81,107(1977).[13]F.H.BusseandR.M.Clever,J.Eng.Math.26,1(1992).[14]J.R.deBruyn,etal.,Rev.Sci.Instrum.67,2043(1996).[15]W.Pesch(privatecommunication).[16]MPEG

movies

of

spatio-temporally

chaotic

states

are

available

at

http://milou.msc.cornell.edu/ILCmovies.

7

[17]K.E.DanielsandE.Bodenschatz(unpublished).[18]F.H.BusseandR.M.Clever(privatecommunication).

[19]E.KnoblochandJ.Moehlis,inPatternformationincontinuousandcoupledsystems,edited

byM.Golubitzki,D.Luss,andS.H.Strogatz(Springer-Verlag,Berlin,1999).

8

y

ggg vd(a)

θ

T2

(b)

T1

zy

T1

z

T2

gg gθ

FIG.1.Schematicofthebaseflow.(a)Heatedfrombelowand(b)heatedfromabove;cellheight,

gravitationalacceleration,shearflow,andtemperaturedifference

with

.

8Rc / 17086420

0

20

40

60θ (o)

80

100

120

Longitudinal~ 1/cosθ

Transverse

FIG.2.Onsetoflongitudinalrolls()anddriftingtransverserolls().Alsoplottedarethepredicted

onsetsforlongitudinal(dashed)andtransverserolls(solid)[15].

9

FIG.3.DigitallyenhancedshadowgraphimageofbimodalsdriftingfromlefttorightinCell1for

,

.Theleftsideofthecellishigherthantheright,withwarmup-flowtotheleftand

colddown-flowtotheright.Therollsattheedgesofthecellarecausedbysidewallimperfections.MPEGmoviesofallstatesareavailableonline[16].

10

10.00BOSO0.10LBTBDBDTRSDP1.00CRε0.01758085900.10LRUCLBDBSDPTB0.010LR3060θ (o)

DTR90120FIG.4.(,)phase-spaceshowingtheboundariesbetweenthedifferentnonlinearstates.LR(longitu-dinalrolls),BO(Busseoscillations),SO(subharmonicoscillations),UC(undulationchaos),CR(crawlingrolls),DTR(driftingtransverserolls),DB(driftingbimodals),LB(longitudinalbursts),TB(transversebursts),andSDP(switchingdiamondpanes).ThedottedlineisthepredictedonsetofBusseoscillationsfor

[12],thedashedlineisthepredictedonsetofundulations[15],andthesolidlinesareguides

totheeye.Opencircles(UC)weremeasuredviadefectdensity[17],opendiamonds(SDP)weremeasuredviacorrelationlength[17],andtheremainderweremeasuredvisually.Theinsetshowsamagnificationofthecodimension-tworegion.

11

FIG.5.ContrastenhancedshadowgraphimageofsubharmonicoscillationsinCell1,withaturbulentburstintheupperleftcorner.

,

.Warmup-flowtotheleftandcolddown-flowtotheright.

12

FIG.6.Digitallyenhancedshadowgraphimagesofconvectionstatesattionchaosatright.

inCell1.(a)Undula-

.(b)Crawlingrollsat

.Warmup-flowtotheleftandcolddown-flowtothe

13

FIG.7.Digitallyenhancedimagesof(a)transverseburstsinspatio-temporallychaoticlongitudinalrolls,at

and

inCell1.(b–u)time-evolutionofasingleburstattime-intervals

.

Warmup-flowtotheleftandcolddown-flowtotheright.

14

FIG.8.Digitallyenhancedimagesof(a)longitudinalburstsatj)time-evolutionofasingleburstattime-intervalstheright.

and

inCell1.(b–

.Warmup-flowtotheleftandcolddown-flowto

15

FIG.9.Digitallyenhancedshadowgraphimagesof(a)switchingdiamondpanes(and(b)longitudinalburstswithindiamondpanes(andcolddown-flowtotheright.

,

)

,

)inCell1.Warmup-flowtotheleft

16