MULTISCALE ANALYSIS AND COMPUTATION FOR THE 3D INC

MULTISCALEANALYSISANDCOMPUTATIONFORTHE3DINCOMPRESSIBLENAVIER-STOKESEQUATIONSTHOMASY.HOU,DANPINGYANGy,ANDHONGYURANzAbstract.Inthispaper,weperformasystematicmultiscaleanalysisforthe3DincompressibleNavier-Stokesequationswithmultiscaleinitialdata.Therearetwomainingredientsinourmultiscalemethod.TherstoneisthatwereparameterizetheinitialdataintheFourierspaceintoaformaltwo-scalestructure.Thesecondoneistheuseofanestedmultiscaleexpansiontogetherwithamultiscalephasefunctiontocharacterizethepropagationofthesmallscalesolutiondynamically.Byusingthesetwotechniquesandperformingasystematicmultiscaleanalysis,wederiveamultiscalemodelwhichcouplesthedynamicsofthesmallscalesubgridproblemtothelargescalesolutionwithoutaclosureassumptionorunknownparameters.Furthermore,weproposeanadaptivemultiscalecomputationalmethodwhichhasacomplexitycomparabletoadynamicSmagorinskymodel.Wedemonstratetheaccuracyofthemultiscalemodelbycomparingwithdirectnumericalsimulationsforbothtwoandthreedimensionalproblems.Intwodimensionalcase,weconsiderdecayingturbulencewhileinthethreedimensionalcaseweconsiderforcedturbulence.Ournumericalresultsshowthatourmultiscalemodelnotonlycapturetheenergyspectrumveryaccurately,itcanalsoreproducesomeoftheimportantstatisticalpropertiesthathavebeenobservedinexperimentalstudiesforfullydevelopedturbulentows.Keywords.Multiscaleanalysis,turbulencemodeling,3DNavier-Stokesequations.AMSsubjectclassications.Primary:76M50,76F05,Secondary:76F65,76M221.Introduction.Wedevelopasystematicmultiscaleanalysisforthe3DincompressibleNavier-Stokesequationswithmultiscaleinitialdata.Theunderstandingofscaleinteractionsforincompressibleowshasbeenamajorchallenge.ForhighReynoldsnumberows,thedegreesoffreedomaresolargethatitisalmostimpossibletoresolveallsmallscalesbydirectnumericalsimulations.Derivinganeectiveequationforthelargescalesolutionisveryusefulforengineeringapplications.Therehavebeensomewell-knownLargeEddySimulation(LES)modelsavailableintheliteratureseee.g.[31,24,12,11,27,18,28].However,manyoftheLESmodelsarebasedonsomeclosureassumptionswhichcannotbeveried,andtheycontainunknownparameters.ItwouldbedesirabletoderiveamoresystematicLESmodelwhichdoesnotcontainunknownparametersandcanbejustiedbymultiscaleanalysis.Ontheotherhand,thenonlinearandnonlocalnatureoftheNavier-Stokesequationsmakesitdiculttoperformmultiscaleanalysis.Oneoftheimportantquestionsistounderstandhowsmallscalesaregeneratedandpropagateintimeandwhetherthemultiscalestructureofthesolutionispreserveddynamically.OurmultiscaleanalysisismotivatedbythepreviousworkofMcLaughlin-Papanicolaou-Pironneau(MPPforshort)onthe3DEulerequationsusinghomogenizationtechniques[26].ToconstructamultiscaleexpansionforthesolutionoftheEulerequations,theymadeanimportantassumptionthattheoscillationisconvectedbythemeanow.Usingmultiscaleexpansiontechniques,MPPobtainedaperiodiccellproblemforthevelocityeldandthepressure.However,itisnotclearwhethertheresultingcellproblemhasasolutionthatisperiodicinboththefastspacevariableyandthefasttimevariable.Additionalassumptionswereimposedonthesolutionofthecellprobleminordertoderiveavariantofthekmodel.InspiredbythepioneeringworkofMcLaughlin-Papanicolaou-Pironneau[26],therehavebeenmanysubsequentcontributionsinthisarea,seee.g.[3,8,30,2,6,7,5].Inthispaper,wegeneralizethemultiscaleanalysisofMPPtoproblemswithinnitelymanynon-separatedscalesanddevelopanovelmultiscaleanalysisfortheincompressibleEulerandNavier-Stokesequations.Therearetwokeyingredientsinourmultiscaleanalysis.TherstoneistoreformulatethesolutionintheFourierspaceintoaformaltwo-scalestructure.Thesecondoneistointroduceamultiscalephasefunctiontocharacterizethepropagationofthesmallscales.Usingthisnewmultiscalephasefunction,thetwo-scalestructureoftheinitialconditionispreserveddynamically.Attheend,wederiveamultiscaleAppliedandComput.Math,217-50,Caltech,Pasadena,CA91125.Email:[e-mail],EastChinaNormalUniversity,Shanghai,200062,China.Email:[e-mail],217-50,Caltech,Pasadena,CA91125.Email:[e-mail]ynamicsofthesmallscalesubgridproblemtothelargescalesolution.Further,wedevelopasimpliedmultiscalemodelinwhichonlytheaverageofthemultiscalephasefunctionisusedinthemultiscaleexpansion.Thissignicantlysimpliesthecomputationofthecellproblems.BasedonthemultiscaleanalysiswedevelopfortheincompressibleEulerequations,wehavedesignedaneectivemultiscalecomputationalmethodtosolvethe3DincompressibleNavier-Stokesequations.Inordertoreducethecomputationalcost,wedesignaneectiveadaptivemethodwhichupdatesonlyasmallnumberofcellproblemsateachtimestep.Thisoersconsiderablecomputationalsavings.Theadaptivemultiscalemethodismuchmoreecientthanadirectnumericalsimulation,andbutisslightlymoreexpensivetotheSmagorinskyLESmodel[31,12].Ontheotherhand,withonlyamodestextracomputationalcostcomparedwiththeLESmodel,ourmultiscalemodeloersbetteraccuracyforthelargescalesolutionthantheLESmodelandhasthecapabilityofcomputingsomesubgridstatist