Kolmogorov’s-Theory

1Lecture9-Kolmogorov’sTheoryAppliedComputationalFluidDynamicsInstructor:AndréBakker©AndréBakker(2002-2005)©FluentInc.(2002)2Eddysize•Kolmogorov’stheorydescribeshowenergyistransferredfromlargertosmallereddies;howmuchenergyiscontainedbyeddiesofagivensize;andhowmuchenergyisdissipatedbyeddiesofeachsize.•Wewillderivethreemainturbulentlengthscales:theintegralscale,theTaylorscale,andtheKolmogorovscale;andcorrespondingReynoldsnumbers.•Wewillalsodiscusstheconceptofenergyanddissipationspectra.3JetsattwodifferentReynoldsnumbersRelativelylowReynoldsnumberRelativelyhighReynoldsnumberSource:Tennekes&Lumley.Page22.4Turbulenteddies•ConsiderfullyturbulentflowathighReynoldsnumberRe=UL/.•Turbulencecanbeconsideredtoconsistofeddiesofdifferentsizes.•An‘eddy’preludesprecisedefinition,butitisconceivedtobeaturbulentmotion,localizedoveraregionofsizel,thatisatleastmoderatelycoherentoverthisregion.•Theregionoccupiedbyalargereddycanalsocontainsmallereddies.•Eddiesofsizelhaveacharacteristicvelocityu(l)andtimescalet(l)l/u(l).•Eddiesinthelargestsizerangearecharacterizedbythelengthscalel0whichiscomparabletotheflowlengthscaleL.•Theircharacteristicvelocityu0u(l0)isontheorderofther.m.s.turbulenceintensityu’(2k/3)1/2whichiscomparabletoU.•Heretheturbulentkineticenergyisdefinedas:•TheReynoldsnumberoftheseeddiesRe0u0l0/isthereforelarge(comparabletoRe)andthedirecteffectsofviscosityontheseeddiesarenegligiblysmall.)'''(2222121wvuuukii5Integralscale•Wecanderiveanestimateofthelengthscalel0ofthelargereddiesbasedonthefollowing:–Eddiesofsizel0haveacharacteristicvelocityu0andtimescalet0l0/u0–Theircharacteristicvelocityu0u(l0)isontheorderofther.m.s.turbulenceintensityu’(2k/3)1/2–Assumethatenergyofeddywithvelocityscaleu0isdissipatedintimet0•Wecanthenderivethefollowingequationforthislengthscale:•Here,(m2/s3)istheenergydissipationrate.Theproportionalityconstantisoftheorderone.Thislengthscaleisusuallyreferredtoastheintegralscaleofturbulence.•TheReynoldsnumberassociatedwiththeselargeeddiesisreferredtoastheturbulenceReynoldsnumberReL,whichisdefinedas:3/20kl1/220ReLklk6Energytransfer•Thelargeeddiesareunstableandbreakup,transferringtheirenergytosomewhatsmallereddies.•Thesesmallereddiesundergoasimilarbreak-upprocessandtransfertheirenergytoyetsmallereddies.•Thisenergycascade–inwhichenergyistransferredtosuccessivelysmallerandsmallereddies–continuesuntiltheReynoldsnumberRe(l)u(l)l/issufficientlysmallthattheeddymotionisstable,andmolecularviscosityiseffectiveindissipatingthekineticenergy.•Atthesesmallscales,thekineticenergyofturbulenceisconvertedintoheat.7Richardson•L.F.Richardson(“WeatherPredictionbyNumericalProcess.”CambridgeUniversityPress,1922)summarizedthisinthefollowingoftencitedverse:BigwhirlshavelittlewhirlsWhichfeedontheirvelocity;Andlittlewhirlshavelesserwhirls,Andsoontoviscosityinthemolecularsense.8Dissipation•Notethatdissipationtakesplaceattheendofthesequenceofprocesses.•Therateofdissipationisdetermined,thereforebythefirstprocessinthesequence,whichisthetransferofenergyfromthelargesteddies.•Theseeddieshaveenergyoforderu02andtimescalet0=l0/u0sotherateoftransferofenergycanbesupposedtoscaleasu02/t0=u03/l0•Consequently,consistentwithexperimentalobservationsinfreeshearflows,thispictureoftheenergycascadeindicatesthatisproportionaltou03/l0independentof(athighReynoldsnumbers).9Kolmogorov’stheory•Manyquestionsremainunanswered.–Whatisthesizeofthesmallesteddiesthatareresponsiblefordissipatingtheenergy?–Asldecreases,dothecharacteristicvelocityandtimescalesu(l)andt(l)increase,decrease,orstaythesame?TheassumeddecreaseoftheReynoldsnumberu0l0/byitselfisnotsufficienttodeterminethesetrends.•TheseandothersareansweredbyKolmogorov’stheoryofturbulence(1941,seePope(2000)).•Kolmogorov’stheoryisbasedonthreeimportanthypothesescombinedwithdimensionalargumentsandexperimentalobservations.10Kolmogorov’shypothesisoflocalisotropy•Forhomogenousturbulence,theturbulentkineticenergykisthesameeverywhere.Forisotropicturbulencetheeddiesalsobehavethesameinalldirections:•Kolmogorovarguedthatthedirectionalbiasesofthelargescalesarelostinthechaoticscale-reductionprocessasenergyistransferredtosuccessivelysmallereddies.•HenceKolmogorov’shypothesisoflocalisotropystatesthatatsufficientlyhighReynoldsnumbers,thesmall-scaleturbulentmotions(ll0)arestatisticallyisotropic.•Here,thetermlocalisotropymeansisotropyatsmallscales.Largescaleturbulencemaystillbeanisotropic.•lEIisthelengthscalethatformsthedemarcationbetweenthelargescaleanisotropiceddies(llEI)andthesmallscaleisotropiceddies(llEI).FormanyhighReynoldsnumberflowslEIcanbeestimatedaslEIl0/6.222'''uvw11Kolmogorov’sfirstsimilarityhypothesis•Kolmogorovalsoarguedthatnotonlydoesthedirectionalinformationgetlostastheenergypassesdownthecascade,butthatallinformationaboutthegeometryoftheeddiesgetslostalso.•Asaresult,thestatisticsofthesmall-scalemotionsareuniversal:theyaresimilarineveryhighReynoldsnumberturbulentflow,independentofthemeanflowfieldandtheboundaryconditions.•ThesesmallscaleeddiesdependontherateTEIatwhichtheyreceiveenergyfr