Instability of Taylor-Couette Flow between Concent

1InstabilityofTaylor-CouetteFlowbetweenConcentricRotatingCylindersHua-ShuDou1,2,BooCheongKhoo2,andKhoonSengYeo21TemasekLaboratories,NationalUniversityofSingapore,Singapore1192602FluidMechanicsDivision,DepartmentofMechanicalEngineeringNationalUniversityofSingapore,Singapore119260,SINGAPOREEmail:[e-mail];[e-mail]ienttheoryisusedtostudytheinstabilityofTaylor-Couetteflowbetweenconcentricrotatingcylinders.Thistheoryhasbeenstrictlyderivedinourpreviousworks.Inourpreviousstudies,theenergygradienttheorywasdemonstratedtobeapplicableforwall-boundedparallelflows.ItwasfoundthatthecriticalvalueoftheenergygradientparameterKmaxatsubcriticaltransitionisabout370-389forwall-boundedparallelflows(whichincludeplanePoiseuilleflow,pipePoiseuilleflowandplaneCouetteflow)belowwhichnoturbulenceoccurs.Inthispaper,thedetailedderivationforthecalculationoftheenergygradientparameterintheflowbetweenconcentricrotatingcylindersisprovided.Thetheoreticalresultsforthecriticalconditionofprimaryinstabilityobtainedareinverygoodagreementwiththeexperimentsfoundinliterature.Themechanismofspiralturbulencegenerationforcounter-rotationoftwocylindersisalsoexplainedusingtheenergygradienttheory.TheenergygradienttheorycanalsoservetorelatetheconditionoftransitioninTaylor-CouetteflowtothatinplaneCouetteflow.Thelatterreasonablybecomesthelimitingcaseoftheformerwhentheradiiofcylinderstendtoinfinity.Itisourcontentionthattheenergygradienttheoryispossiblyuniversalforanalysisofflowinstabilityandturbulenttransition,andisvalidforbothpressureandsheardrivenflowsinbothparallelandrotatingflowconfigurations.Keywords:Instability;Transition;Taylor-Couetteflow;Rotatingcylinders;Energygradient;Energyloss;Criticalcondition.1.IntroductionTaylor-CouetteflowreferstotheproblemofflowbetweentwoconcentricrotatingcylindersasshowninFig.1[1-4].ThisterminologywasnamedaftertheworksofG.I.Taylor(1923)andM.Couette(1890).ThisproblemwasfirstinvestigatedexperimentallybyCouette(1890)andMallock(1896).Couetteobservedthatthetorqueneededtorotatetheoutercylinderincreasedlinearlywiththerotationspeeduntilacriticalrotationspeed,afterwhichthetorque2increasedmorerapidly.Thischangewasduetoatransitionfromstabletounstableflowatthecriticalrotationspeed.Taylorwasthefirsttosuccessfullyapplylinearstabilitytheorytoaspecificproblem,andsucceededinobtainingexcellentagreementoftheorywithexperimentsfortheflowinstabilitybetweentwoconcentricrotatingcylinders[5].Taylor’sgroundbreakingresearchforthisproblemhasbeenconsideredasaclassicalexampleofflowinstabilitystudy[6-8].Inthepastyears,theproblemofTaylor-Couetteflowhasreceivedrenewedinterestsbecauseofitsimportanceinflowstabilityandthefactthatitisparticularlyamenabletorigorousmathematicaltreatment/analysisduetoinfinitesimaldisturbances[1-3].Forthestabilityofaninviscidfluidmovinginconcentriclayers,LordRayleigh[9]usedthecirculationvariationversustheradiustoexplaintheinstabilitywhilevonKarman[10]employedtherelativerolesofcentrifugalforceandpressuregradienttointerprettheinstabilityinitiation.Theirgoalwastodeterminetheconditionforwhichaperturbationresultingfromanadversegradientofangularmomentumcanbeunstable.Inhisclassicpaper,Taylor[5]presentedamathematicalstabilityanalysisforviscousflowandcomparedtheresultstolaboratoryobservations.Taylorobservedthat,forsmallratioofthegapwidthtothecylinderradiiandforagivenrotatingspeedofoutercylinder,whentherotationspeedoftheinnercylinderislow,theflowremainslaminar;whentherotationspeedoftheinnercylinderexceedsacriticalvalue,instabilitysetsinandrowsofcellularvorticesaredeveloped.Whentherotatingspeedisincreasedtoanevenhighervalue,thecellrowsbreakdownandaturbulencepatternisproduced.Heproposedaparameter,nowcommonlyknownastheTaylornumber,()12/ReRhT=,tocharacterizethiscriticalconditionforinstability.Here,ReistheReynoldsnumberbasedonthegapwidth(h)andtherotationspeedoftheinnercylinder,andR1istheradiusoftheinnercylinder.ThecriticalvalueoftheTaylornumberforprimaryinstabilityis1708asobtainedfromlinearanalysis.Thisvalueagreeswellwithhisexperiments[1-3].However,theproblemofTaylor-Couetteflowisstillfarfromcompletelyresolveddespiteextensivestudy[11-17].Forexample,thelimitingcaseofTaylor-CouetteflowwhentheratioofthegapwidthtotheradiitendstozeroshouldagreewiththatofplaneCouetteflow.Thus,thecriterionforinstabilityshouldreflectthisphenomenon.Therearetworecentworkstryingtoaddressthisissuetosomedegreeofsuccess[18-19].OnemayobservethatTaylor’scriterionisnotappropriatewhenthislimitingcaseisstudiedbecauseplaneCouetteflowisjudgedtobealwaysstableduetoTaylornumberassuminganullvalueusingTaylor’scriterion.ThismaybeattributedtothefactthatTaylor’scriteriononlyconsideredtheeffectofcentrifugalforce,anddoesnotincludethekinematicinertiaforce.Therefore,itisreckonedtobesuitableforlowRe3numberflowswithhighcurvature.ForrotatingflowwithhigherRenumberandlowcurvature,itmaytransittoturbulenceearlierandyetdoesnotviolateTaylor’scriterion.Recently,Dou[20,21]proposedanewenergygradienttheorytoanalyzeflowinstabilityandturbulenttransitionproblems.Inthistheory