Closure of two dimensional turbulence the role of pressure gradients

Inverse energy cascade regime of two dimensional turbulence is investigated by means of high resolution numerical simulations. Numerical computations of conditional averages of transverse pressure gradient increments are found to be compatible with a recen

Closureoftwodimensionalturbulence:theroleofpressuregradients

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G.Bo etta1,M.Cencini2,3andJ.Davoudi4,5

DipartimentodiFisicaGeneraleandINFM,Universit`adiTorino,ViaPietroGiuria1,I-10125Torino,ItalyDipartimentodiFisica,andINFM,Universit`adiRoma”laSapienza”,PiazzaleAldoMoro5,00185Roma,Italy

3

ObservatoiredelaC oted’Azur-NiceBoulevarddel’Observatoire,B.P.4229F-06304NICECedex44

Max-Planck-Institutef¨urPhysikKomplexerSysteme,N¨othnitzerStr.38,01187Dresden,Germany

5

FachbereichPhysik,PhilippsUniversit¨at,Renthof6,35032Marburg,Germany

Inverseenergycascaderegimeoftwodimensionalturbulenceisinvestigatedbymeansofhigh

resolutionnumericalsimulations.Numericalcomputationsofconditionalaveragesoftransversepressuregradientincrementsarefoundtobecompatiblewitharecentlyproposedself-consistentGaussianmodel.Ananalogousloworderclosuremodelforthelongitudinalpressuregradientisproposedanditsvalidityisnumericallyexamined.Inthiscasenumericalevidenceforthepresenceofhigherordertermsintheclosureisfound.Thefundamentalroleofconditionalstatisticsbetweenlongitudinalandtransversecomponentsishighlighted.PACSnumber(s):47.27.-i,47.27.Ak,05.40.-a

arXiv:n

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1 [nlin.CD] 28 May 2002

Theexistenceoftwosimultaneousinertialrangesintwo-dimensionalturbulence,asaconsequenceofcoupleden-ergyandenstrophyconservation,isoneofthemostim-portantphenomenainstatistical uidmechanicsAtvariancewith3D-turbulence,theenergyinjectedintothesystematscale f owstowardthelargescales,whiletheenstrophycascadesdownonthesmallscales.Becauseoftheinverseenergycascade,theNavier-Stokesequations,

tui+uj jui= ip+ν 2ui+fi

(1)

whichruletheevolutionofanincompressible( iui=0)velocity eld,cannotreachasteady-stateunlessanen-ergysinkatlargescalesisadded.Alternativelyonecanconsideranensembleofsolutionsof(1)witha xeden-ergyvaluebelowthecondensationlevel[2],i.e.withanintegralscaleL(t)(growingintimeast3/2)stillmuchsmallerthanthesystemsize.Becauseofthescal-ingofthecharacteristictimes,thesmallscales(iner-tialrange)inthesystem f r Lcanbeconsid-eredinastationarystate.Oneofthemostchalleng-ingproblemsistounderstandthestatisticsofvelocity uctuations u(r)=u(x+r) u(x)Inhomo-geneousandisotropicturbulenceitamountstostudythejointprobabilitydensityfunction(PDF)P(U,V,r)oflongitudinalUandtransverseVvelocitydi erenceswhere u=Ux +Vy andx =r

whereε≡ fiui istherateofenergyinputandthe

conditionaltransversal,Py,v≡ yp|U,V,r ,andlon-gitudinal,Px,u≡ xp|U,V,r ,pressuregradientin-1

UV2P(U,V,r)=

rr

2

2

ε( U+ V)+ UPx,u+ VPy,vP(U,V,r),

1

(2)

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