Two-fluid model of the truncated Euler Equations

A phenomenological two-fluid model of the (time-reversible) spectrally-truncated 3D Euler equation is proposed. The thermalized small scales are first shown to be quasi-normal. The effective viscosity and thermal diffusion are then determined, using EDQNM

Two- uidmodeloftheTruncatedEuler’s

Equations

´GiorgioKrstulovica,1Marc-EtienneBracheta,2

arXiv:0710.2462v1 [physics.flu-dyn] 12 Oct 2007dePhysiqueStatistiquedel’EcoleNormaleSup´erieure,associ´eauCNRSetauxUniversit´esParisVIetVII,24RueLhomond,75231Paris,FranceaLaboratoireAbstractAphenomenologicaltwo- uidmodelofthe(time-reversible)spectrally-truncated3DEulerequationisproposed.Thethermalizedsmallscalesare rstshowntobequasi-normal.Thee ectiveviscosityandthermaldi usionarethendetermined,usingEDQNMclosureandMonte-Carlonumericalcomputations.Finally,themodelisvalidatedbycomparingitsdynamicswiththatoftheoriginaltruncatedEulerequation.1IntroductionItiswell-knownthatthe(inviscidandconservative)truncatedEulerequationadmitsabsoluteequilibriumsolutionswithGaussianstatistics,equipartitionofkineticenergyamongallFouriermodesandthusanenergyspectrumE(k)～k2[1].Recently,Cichowlasetal.[2,3]observedthattheEulerequation,withaverylarge(severalhundreds)spectraltruncationwavenumberkmax,haslong-lastingtransientswhichbehavejustasthoseofhigh-Reynoldsnumberviscous ow;inparticulartheyfoundanapproximatelyk 5/3inertialrangefollowedbyadissipativerange.Howissuchabehaviorpossible?Itwasfoundthatthe

highest-kmodesthermalizeat rst,displayingak2spectrum.Progressivelythethermalizedregionextendstolowerandlowerwavenumbers,eventuallycoveringthewholerangeofavailablemodes.Atintermediatetimes,whenthethermalizedregimeonlyextendsoverthehighestwavenumbers,itactsasathermostatthatpumpsouttheenergyoflarger-scalemodes.Notethatsimilark 5/3/k2spectrahavealreadybeendiscussedinthewaveturbulenceliterature1

2krstulov@lps.ens.frbrachet@lps.ens.fr

A phenomenological two-fluid model of the (time-reversible) spectrally-truncated 3D Euler equation is proposed. The thermalized small scales are first shown to be quasi-normal. The effective viscosity and thermal diffusion are then determined, using EDQNM

(e.g.,[4])andweremorerecentlyobtained,withinasimpledi erentialclosure,inconnectionwiththeLeithmodelofhydrodynamicturbulence[5].

Thepurposeofthepresentworkistobuildaquantitativetwo- uidmodelfortherelaxationofthe3DEulerequation.Insection2,afterabriefrecallofbasicde nitions,thestatisticsofthethermalizedsmallscalesarestud-iedduringrelaxation.Theyareshowntobequasi-normal.Ournewtwo- uidmodel,involvingbothane ectiveviscosityandathermaldi usion,isintro-ducedinsection3.Thee ectivedi usionlawsarethendetermined,usinganEDQNMclosurepredictionanddirectMonte-Carlocomputations.ThemodelisthenvalidatedbycomparingitspredictionswiththebehavioroftheoriginaltruncatedEulerequation.Finallysection4isourconclusion.

2RelaxationdynamicsoftruncatedEulerequations

2.1Basicde nitions

ThetruncatedEulerequations(1)areclassicallyobtained[1]byperformingaGalerkintruncation( v(k)=0forsupα|kα|>kmax)ontheFouriertransform v(x,t)=v (k,t)eik·xofaspatiallyperiodicvelocity eldobeyingthe(unitdensity)three-dimensionalincompressibleEulerequations, tv+(v· )v= p, ·v=0.Thisprocedureyieldsthefollowing nitesystemofordinarydi erentialsequationsforthecomplexvariablesv (k)(kisa3Dvectorofrelativeintegers(k1,k2,k3)satisfyingsupα|kα|≤kmax)

iv β(p,t) vγ(k p,t) tv α(k,t)= Pαβγ(k)2p(1)

wherePαβγ=kβPαγ+kγPαβwithPαβ=δαβ kαkβ/k2andtheconvolutionin(1)istruncatedtosupα|kα|≤kmax,supα|pα|≤kmaxandsupα|kα pα|≤kmax.

Thistime-reversiblesystemexactlyconservesthekineticenergyE=kE(k,t),wheretheenergyspectrumE(k,t)isde nedbyaveragingv (k ,t)onsphericalshellsofwidth k=1,

1E(k,t)=|v (k ,t)|2.2k k/2<|k |<k+ k/2 (2)

A phenomenological two-fluid model of the (time-reversible) spectrally-truncated 3D Euler equation is proposed. The thermalized small scales are first shown to be quasi-normal. The effective viscosity and thermal diffusion are then determined, using EDQNM

2.2SmallScalesStatistics

PerhapsthemoststrikingresultofCichowlasetal.[2]wasthespontaneousgenerationofa(timedependent)minimumofthespectrumE(k,t)atwavenum-berkth(t)wherethescalinglawE(k,t)=c(t)k2starts.Thus,theenergydissi-patedfromlargescalesintothetimedependentstatisticalequilibriumisgivenby

Eth(t)=

kth(t)<kE(k,t).(3)

Inthissectionweusetheso-calledTaylor-Green[6]initialconditionto(1):thesingle–modeFouriertransformofuTG=sinxcosycosz,vTG= uTG(y, x,z),wTG=0.

Inordertoseparatethedynamicsoflarge-scale(k<kth)andthestatisticsofsmall-scales(k>kth)wede nethelowandhigh-pass ltered elds

f(r)=<

kik·r F(k)fke(4)

(5)f>(k)=1 f<(r)

wheref( r)isanarbitrary eldandfkitsFouriertransform;wehavechosen kth11+tanh|k| ,with k=1/2.F(k=This lterallowsustode nethelarge-scalevelocity eldv<andthespatiallydependentthermalizedenergy(orheat)http://doc.guandang.neting

1>><(vivj),wede nethelocalheatthetraceoftheReynold’stensor[7],Rij=as

1 >2 <Q(r)=(v)(r).2(6)

Byconstructionofthe lters,(4-5)theheatspatialaverageisequaltothedissipatedenergy(3)<Q(r)>=Eth.Fig.1ashowsa2DcutoftheheatQonthesurfacez=π,whereacoldzoneisseentobepresentatthecenteroftheimpermeablebox(x=[0,π],y=[0,π],z=[0,π]).AnisosurfaceofthehottestzonesisdisplayedonFig.1b.Isisapparentonboth guresthatQ(r)isnotspatiallyhomogeneous.

2.3Heatdi usion

ThesimplestquantitiestostudyinordertoquantifytheevolutionofQ,arethespatialaverageQ(t)= Q(r,t) andtherootmeansquarevariation

A phenomenological two-fluid model of the (time-reversible) spectrally-truncated 3D Euler equation is proposed. The thermalized small scales are first shown to be quasi-normal. The effective viscosity and thermal diffusion are then determined, usi …… 此处隐藏：16270字，全部文档内容请下载后查看。喜欢就下载吧 ……