Low-wavenumber forcing and turbulent energy dissipation

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1、Low-wavenumberforcingandturbulentenergydissipationCharlesR.DoeringandNikolaP.PetrovDepartmentofMathematicsandMichiganCenterforTheoreticalPhysicsUniversityofMichigan,AnnArbor,MI48109,USAE-mailaddresses:doering@umich.eduandnpetrov@umich.edu1IntroductionInmanyDirectNumeri

2、calSimulations(DNS)ofturbulenceresearchersinjectpowerintothefluidatlargescalesandthenobservehowit“propagates”tothesmallscales[1,2,3,4,5,6,7,8,9,10,11,12].Onesuchtypeofstirringistotaketheforcef(x,t)tobeproportionaltotheprojectionofthevelocityu(x,t)oftheflowontoitslowestFo

3、uriermodes,whilekeepingtherateofinjectedexternalpowerconstant.Inthispaperweperformasimplebutrigorousanalysistoestablishboundsontherelationshipbetweentheenergydissipationrate(whichisthesameastheinjectedpower)andtheresultingReynoldsnumber.Whilethisanalysiscannotgivedetai

4、ledinformationoftheenergyspectrum,itdoesprovidesomeindicationofthebalanceofenergybetweenthelower,directlyforced,modes,andthoseexcitedbythecascade.Thisworkisanextensionoftheanalysisin[13,14,15],wheretheforceisfixed(notafunctionalofthevelocity).Considerfluidinaperiodicd-di

5、mensionalboxofsidelengthℓ.Theallowedwavevectorskareoftheformk=2πa,wherea∈Zdisad-dimensionalℓvectorwithintegercomponents.LetLbethesubsetofwavevectorsthathavethesmallestpossiblewavenumber(namely,2π);Lconsistsof2delements:ℓL={±2πe,...,±2πe}.TheoperatorPprojectsthevectorfie

6、ldℓ1ℓdXu(x,t)=uˆ(k,t)eik·xkontothesubspacespannedbytheFouriercomponentswithwavevectorsinL:XPu(x,t)=uˆ(k,t)eik·x.(1)k∈LarXiv:physics/0404049v1[physics.flu-dyn]9Apr2004Obviously,PmapsL2intoL2vectorfields;infact,PuisC∞inthespatialvariables.Theprojectionalsopreservestheinco

7、mpressibilityproperty.Thatis,if∇·u(x,t)=0,then∇·Pu(x,t)=0.2CharlesR.DoeringandNikolaP.PetrovTheNavier-Stokesequationis1u˙+(u·∇)u+∇p=ν∆u+f,(2)ρwithf(x,t)takenintheformPu(x,t)f(x,t)=ǫ.(3)1kPu(·,t)k2ℓd2R1wherek·kstandsfortheL2-norm,kPu(·,t)k:=

8、Pu(x,t)

9、2ddx2.22Thischoice

10、offorcingensuresthattheinputpowerisconstant:Zu(x,t)·f(x,t)ddx=ℓdǫ.(4)Inthisapproachǫ,νandℓarethe(only)controlparamete